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4,700 | 5,256 | Parallel Direction Method of Multipliers
Huahua Wang , Arindam Banerjee , Zhi-Quan Luo
University of Minnesota, Twin Cities
{huwang,banerjee}@cs.umn.edu, [email protected]
Abstract
We consider the problem of minimizing block-separable (non-smooth) convex
functions subject to linear constraints. While the Alternating Direction Method of
Multipliers (ADMM) for two-block linear constraints has been intensively studied
both theoretically and empirically, in spite of some preliminary work, effective
generalizations of ADMM to multiple blocks is still unclear. In this paper, we
propose a parallel randomized block coordinate method named Parallel Direction
Method of Multipliers (PDMM) to solve optimization problems with multi-block
linear constraints. At each iteration, PDMM randomly updates some blocks in
parallel, behaving like parallel randomized block coordinate descent. We establish
the global convergence and the iteration complexity for PDMM with constant step
size. We also show that PDMM can do randomized block coordinate descent on
overlapping blocks. Experimental results show that PDMM performs better than
state-of-the-arts methods in two applications, robust principal component analysis
and overlapping group lasso.
1
Introduction
In this paper, we consider the minimization of block-seperable convex functions subject to linear
constraints, with a canonical form:
J
J
X
X
min f (x) =
fj (xj ) , s.t. Ax =
Acj xj = a ,
(1)
{xj ?Xj }
j=1
j=1
where the objective function f (x) is a sum of J block separable P
(nonsmooth) convex functions,
Acj ? Rm?nj is the j-th column block of A ? Rm?n where n = j nj , xj ? Rnj ?1 is the j-th
block coordinate of x, Xj is a local convex constraint of xj and a ? Rm?1 . The canonical form
can be extended to handle linear inequalities by introducing slack variables, i.e., writing Ax ? a as
Ax + z = a, z ? 0.
A variety of machine learning problems can be cast into the linearly-constrained optimization problem (1) [8, 4, 24, 5, 6, 21, 11]. For example, in robust Principal Component Analysis (RPCA) [5],
one attempts to recover a low rank matrix L and a sparse matrix S from an observation matrix M,
i.e., the linear constraint is M = L + S. Further, in the stable version of RPCA [29], an noisy matrix
Z is taken into consideration, and the linear constraint has three blocks, i.e., M = L + S + Z. Problem (1) can also include composite minimization problems which solve a sum of a loss function and
a set of nonsmooth regularization functions. Due to the increasing interest in structural sparsity [1],
composite regularizers have become widely used, e.g., overlapping group lasso [28]. As the blocks
are overlapping in this class of problems, it is difficult to apply block coordinate descent methods
for large scale problems [16, 18] which assume block-separable. By simply splitting blocks and introducing equality constraints, the composite minimization problem can also formulated as (1) [2].
A classical approach to solving (1) is to relax the linear constraints using the (augmented) Lagrangian, i.e.,
?
(2)
L? (x, y) = f (x) + hy, Ax ? ai + kAx ? ak22 ,
2
1
where ? ? 0 is called the penalty parameter. We call x the primal variable and y the dual variable. (2)
usually leads to primal-dual algorithms which update the primal and dual variables alternatively.
While the dual update is simply dual gradient descent, the primal update is to solve a minimization
problem of (2) given y. If ? = 0, the primal update can be solved in a parallel block coordinate
fashion [3, 19], leading to the dual ascent method. While the dual ascent method can achieve massive parallelism, a careful choice of stepsize and some strict conditions are required for convergence,
particularly when f is nonsmooth. To achieve better numerical efficiency and convergence behavior
compared to the dual ascent method, it is favorable to set ? > 0 in the augmented Lagrangian (2)
which we call the method of multipliers. However, (2) is no longer separable and solving entire
augmented Lagrangian (2) exactly is computationally expensive. In [20], randomized block coordinate descent (RBCD) [16, 18] is used to solve (2) exactly, but leading to a double-loop algorithm
along with the dual step. More recent results show (2) can be solved inexactly by just sweeping the
coordinates once using the alternating direction method of multipliers (ADMM) [12, 2]. This paper
attempts to develop a parallel randomized block coordinate variant of ADMM.
When J = 2, ADMM has been widely used to solve the augmented Lagragian (2) in many applications [2]. Encouraged by the success of ADMM with two blocks, ADMM has also been extended to solve the problem with multiple blocks [15, 14, 10, 17, 13, 7]. The variants of ADMM
can be mainly divided into two categories. The first category considers Gauss-Seidel ADMM
(GSADMM) [15, 14], which solves (2) in a cyclic block coordinate manner. In [13], a back substitution step was added so that the convergence of ADMM for multiple blocks can be proved. In
some cases, it has been shown that ADMM might not converge for multiple blocks [7]. In [14], a
block successive upper bound minimization method of multipliers (BSUMM) is proposed to solve
the problem (1). The convergence of BSUMM is established under some fairly strict conditions: (i)
certain local error bounds hold; (ii) the step size is either sufficiently small or decreasing. However,
in general, Gauss-Seidel ADMM with multiple blocks is not well understood and its iteration complexity is largely open. The second category considers Jacobian variants of ADMM [26, 10, 17],
which solves (2) in a parallel block coordinate fashion. In [26, 17], (1) is solved by using two-block
ADMM with splitting variables (sADMM). [10] considers a proximal Jacobian ADMM (PJADMM)
by adding proximal terms. A randomized block coordinate variant of ADMM named RBSUMM
was proposed in [14]. However, RBSUMM can only randomly update one block. Moreover, the
convergence of RBSUMM is established under the same conditions as BSUMM and its iteration
complexity is unknown.
In this paper, we propose a parallel randomized block coordinate method named parallel direction
method of multipliers (PDMM) which randomly picks up any number of blocks to update in parallel,
behaving like randomized block coordinate descent [16, 18]. Like the dual ascent method, PDMM
solves the primal update in a parallel block coordinate fashion even with the augmentation term.
Moreover, PDMM inherits the merits of the method of multipliers and can solve a fairly large class
of problems, including nonsmooth functions. Technically, PDMM has three aspects which make it
distinct from such state-of-the-art methods. First, if block coordinates of the primal x is solved exactly, PDMM uses a backward step on the dual update so that the dual variable makes conservative
progress. Second, the sparsity of A and the number of randomized blocks are taken into consideration to determine the step size of the dual update. Third, PDMM can randomly update arbitrary
number of primal blocks in parallel. Moreover, we show that sADMM and PJADMM are the two extreme cases of PDMM. The connection between sADMM and PJADMM through PDMM provides
better understanding of dual backward step. PDMM can also be used to solve overlapping groups in
a randomized block coordinate fashion. Interestingly, the corresponding problem for RBCD [16, 18]
with overlapping blocks is still an open problem. We establish the global convergence and O(1/T )
iteration complexity of PDMM with constant step size. We evaluate the performance of PDMM in
two applications: robust principal component analysis and overlapping group lasso.
The rest of the paper is organized as follows: We introduce PDMM in Section 2, and establish
convergence results in Section 3. We evaluate the performance of PDMM in Section 4 and conclude
in Section 5. The technical analysis and detailed proofs are provided in the supplement.
Notations: Assume that A ? Rm?n is divided into I ? J blocks. Let Ari ? Rmi ?n be the i-th row
block of A, Acj ? Rm?nj be the j-th column block of A, and Aij ? Rmi ?nj be the ij-th block of
A. Let yi ? Rmi ?1 be the i-th block of y ? Rm?1 . Let N (i) be a set of nonzero blocks Aij in the
2
? i = min{di , K} where
i-th row block Ari and di = |N (i)| be the number of nonzero blocks. Let K
K is the number of blocks randomly chosen by PDMM and T be the number of iterations.
2
Parallel Direction Method of Multipliers
Consider a direct Jacobi version of ADMM which updates all blocks in parallel:
xt+1
= argminxj ?Xj L? (xj , xtk6=j , yt ) ,
j
y
t+1
t
= y + ? ?(Ax
t+1
? a) .
(3)
(4)
where ? is a shrinkage factor for the step size of the dual gradient ascent update. However, empirical
results show that it is almost impossible to make the direct Jacobi updates (3)-(4) to converge even
when ? is extremely small. [15, 10] also noticed that the direct Jacobi updates may not converge.
To address the problem in (3) and (4), we propose a backward step on the dual update. Moreover,
instead of updating all blocks, the blocks xj will be updated in a parallel randomized block coordinate fashion. We call the algorithm Parallel Direction Method of Multipliers (PDMM). PDMM first
randomly select K blocks denoted by set Jt at time t, then executes the following iterates:
t
? t ) + ?jt B?jt (xjt , xtjt ) , jt ? Jt ,
xt+1
jt = argmin L? (xjt , xk6=jt , y
(5)
yit+1 = yit + ?i ?(Ai xt+1 ? ai ) ,
(6)
? it+1
y
(7)
xjt ?Xjt
=
yit+1
? ?i ?(Ai x
t+1
? ai ) ,
where ?i > 0, 0 ? ?i < 1, ?jt ? 0, and B?jt (xjt , xtjt ) is a Bregman divergence. Note xt+1 =
t
(xt+1
Jt , xk?J
/ t ) in (6) and (7). (6) and (7) update all dual blocks. We show that PDMM can also do
? i = min{di , K}. ?i and
randomized dual block coordinate ascent in an extended work [25]. Let K
?i can take the following values:
K
1
, ?i = 1 ?
?i =
.
(8)
?
?
Ki (2J ? K)
Ki
In the xjt -update (5), a Bregman divergence is addded so that exact PDMM and its inexact variants
can be analyzed in an unified framework [23, 11]. In particular, if ?jt = 0, (5) is an exact update. If
?jt > 0, by choosing a suitable Bregman divergence, (5) can be solved by various inexact updates,
often yielding a closed-form for the xjt update (see Section 2.1).
To better understand PDMM, we discuss the following three aspects which play roles in choosing ?i
and ?i : the dual backward step (7), the sparsity of A, and the choice of randomized blocks.
Dual Backward Step: We attribute the failure of the Jacobi updates (3)-(4) to the following observation in (3), which can be rewritten as:
?
xt+1
= argminxj ?Xj fj (xj ) + hyt + ?(Axt ? a), Acj xj i + kAcj (xj ? xtj )k22 .
(9)
j
2
In the primal xj update, the quadratic penalty term implicitly adds full gradient ascent step to the
dual variable, i.e., yt +?(Axt ?a), which we call implicit dual ascent. The implicit dual ascent along
with the explicit dual ascent (4) may lead to too aggressive progress on the dual variable, particularly
when the number of blocks is large. Based on this observation, we introduce an intermediate variable
? t to replace yt in (9) so that the implicit dual ascent in (9) makes conservative progress, e.g.,
y
? t + ?(Axt ? a) = yt + (1 ? ?)?(Axt ? a) , where 0 < ? < 1. y
? t is the result of a ?backward
y
? t = yt ? ??(Axt ? a).
step? on the dual variable, i.e., y
Moreover, one can show that ? and ? have also been implicitly used when using two-block ADMM
with splitting variables (sADMM) to solve (1) [17, 26]. Section 2.2 shows sADMM is a special case
of PDMM. The connection helps in understanding the role of the two parameters ?i , ?i in PDMM.
Interestingly, the step sizes ?i and ?i can be improved by considering the block sparsity of A and
the number of random blocks K to be updated.
Sparsity of A: Assume A is divided into I ? J blocks. While xj can be updated in parallel,
the matrix multiplication Ax in the dual update (4) requires synchronization to gather messages
from
coordinates
jt ? Jt . For updating the i-th block of the dual yi , we need Ai xt+1 =
P all blockt+1
P
t
jt ?Jt Aijt xjt +
k?J
/ t Aik xk which aggregates ?messages? from all xjt . If Aijt is a block of
3
P
zeros, there is no ?message? from xjt to yi . More precisely, Ai xt+1 = jt ?Jt ?N (i) Aijt xt+1
jt +
P
t
k?J
/ t Aik xk where N (i) denotes a set of nonzero blocks in the i-th row block Ai . N (i) can be
considered as the set of neighbors of the i-th dual block yi and di = |N (i)| is the degree of the i-th
dual block yi . If A is sparse, di could be far smaller than J. According to (8), a low di will lead to
bigger step sizes ?i for the dual update and smaller step sizes for the dual backward step (7). Further,
as shown in Section 2.3, when using PDMM with all blocks to solve composite minimization with
overlapping blocks, PDMM can use ?i = 0.5 which is much larger than 1/J in sADMM.
Randomized Blocks: The number of blocks to be randomly chosen also has the effect on ?i , ?i .
1
If randomly choosing one block (K = 1), then ?i = 0, ?i = 2J?1
. The dual backward step (7)
1
1
vanishes. As K increases, ?i increases from 0 to 1 ? di and ?i increases from 2J?1
to d1i . If
updating all blocks (K = J), ?i = d1i , ?i = 1 ? d1i .
PDMM does not necessarily choose any K combination of J blocks. The J blocks can be randomly
partitioned into J/K groups where each group has K blocks. Then PDMM randomly picks some
groups. A simple way is to permutate the J blocks and choose K blocks cyclically.
2.1
Inexact PDMM
If ?jt > 0, there is an extra Bregman divergence term in (5), which can serve two purposes. First,
choosing a suitable Bregman divergence can lead to an efficient solution for (5). Second, if ?jt is
sufficiently large, the dual update can use a large step size (?i = 1) and the backward step (7) can be
removed (?i = 0), leading to the same updates as PJADMM [10] (see Section 2.2).
Given a continuously differentiable and strictly convex function ?jt , its Bregman divergence is
defiend as
B?jt (xjt , xtjt ) = ?jt (xjt ) ? ?jt (xtjt ) ? h??jt (xtjt ), xjt ? xtjt i,
(10)
where ??jt denotes the gradient of ?jt . Rearranging the terms yields
?jt (xjt ) ? B?jt (xjt , xtjt ) = ?jt (xtjt ) + h??jt (xtjt ), xjt ? xtjt i,
(11)
which is exactly the linearization of ?jt (xjt ) at xtjt . Therefore, if solving (5) exactly becomes
difficult due to some problematic terms, we can use the Bregman divergence to linearize these
problematic terms so that (5) can be solved efficiently. More specifically, in (5), we can choose
?jt = ?jt ? ?1j ?jt assuming ?jt is the problematic term. Using the linearity of Bregman divert
gence,
1
B?jt (xjt , xtjt ) = B?jt (xjt , xtjt ) ?
B?jt (xjt , xtjt ) .
(12)
?jt
For instance, if fjt is a logistic function, solving (5) exactly requires an iterative algorithm. Setting
?jt = fjt , ?jt = 21 k? k22 in (12) and plugging into (5) yield
X
?
t
xt+1
yt , Ajt xjt i+ kAjt xjt +
Ak xtk ?ak22 +?jt kxjt ?xtjt k22 ,
jt = argmin h?fjt (xjt ), xjt i+h?
2
xjt ?Xjt
k6=jt
which has a closed-form solution. Similarly, if the quadratic penalty term ?2 kAcjt xjt +
P
?
c
2
c t
2
k6=jt Ak xk ? ak2 is a problematic term, we can set ?jt (xjt ) =
2 kAjt xjt k2 , then
?
c
t
2
t
B?jt (xjt , xjt ) = 2 kAjt (xjt ? xjt )k2 can be used to linearize the quadratic penalty term.
In (12), the nonnegativeness of B?jt implies that B?jt ? ?1j B?jt . This condition can be satisfied
t
as long as ?jt is more convex than ?jt . Technically, we assume that ?jt is ?/?jt -strongly convex
and ?jt has Lipschitz continuous gradient with constant ?, which has been shown in [23].
2.2
Connections to Related Work
Consider the case when all blocks are used in PDMM. There are also two other methods which
update all blocks in parallel. If solving the primal updates exactly, two-block ADMM with splitting
variables (sADMM) is considered in [17, 26]. We show that sADMM is a special case of PDMM
when setting ?i = J1 and ?i = 1 ? J1 (Appendix B in [25]). If the primal updates are solved
inexactly, [10] considers a proximal Jacobian ADMM (PJADMM) by adding proximal terms where
4
the converge rate is improved to o(1/T ) given the sufficiently large proximal terms. We show that
PJADMM [10] is also a special case of PDMM (Appendix C in [25]). sADMM and PJADMM are
two extreme cases of PDMM. The connection between sADMM and PJADMM through PDMM can
provide better understanding of the three methods and the role of dual backward step. If the primal
update is solved exactly which makes sufficient progress, the dual update should take small step, e.g.,
sADMM. On the other hand, if the primal update takes small progress by adding proximal terms,
the dual update can take full gradient step, e.g. PJADMM. While sADMM is a direct derivation of
ADMM, PJADMM introduces more terms and parameters.
In addition to PDMM, RBUSMM [14] can also randomly update one block. The convergence
of RBSUMM requires certain local error bounds to be hold and decreasing step size. Moreover,
the iteration complexity of RBSUMM is still unknown. In contast, PDMM converges at a rate of
O(1/T ) with the constant step size.
2.3
Randomized Overlapping Block Coordinate Descent
Consider the composite minimization problem of a sum of a loss function `(w) and composite
regularizers gj (wj ):
L
X
min `(w) +
gj (wj ) ,
(13)
w
j=1
which considers L overlapping groups wj ? Rb?1 . Let J = L + 1, xJ = w. For 1 ? j ? L,
denote xj = wj , then xj = UTj xJ , where Uj ? Rb?L is the columns of an identity matrix and
extracts the coordinates of xJ . Denote U = [U1 , ? ? ? , UL ] ? Rn?(bL) and A = [IbL , ?UT ] where
bL denotes b ? L. By letting fj (xj ) = gj (wj ) and fJ (xJ ) = `(w), (13) can be written as:
J
X
fj (xj ) s.t. Ax = 0.
(14)
min
x
j=1
b?J
where x = [x1 ; ? ? ? ; xL ; xL+1 ] ? R
. (14) can be solved by PDMM in a randomized block
coordinate fashion. In A, for b rows block, there are only two nonzero blocks, i.e., di = 2. ThereK
fore, ?i = 2(2J?K)
, ?i = 0.5. In particular, if K = J, ?i = ?i = 0.5. In contrast, sADMM uses
?i = 1/J 0.5, ?i = 1 ? 1/J > 0.5 if J is larger.
Remark 1 (a) ADMM [2] can solve (14) where the equality constraint is xj = UTj xJ .
(b) In this setting, Gauss-Seidel ADMM (GSADMM) and BSUMM [14] are the same as ADMM.
BSUMM should converge with constant stepsize ? (not necessarily sufficiently small), although the
theory of BSUMM does not include this special case.
3
Theoretical Results
We establish the convergence results for PDMM under fairly simple assumptions:
Assumption 1
(1) fj : Rnj 7? R ? {+?} are closed, proper, and convex.
(2) A KKT point of the Lagrangian (? = 0 in (2)) of Problem (1) exists.
Assumption 1 is the same as that required by ADMM [2, 22]. Assume that {x?j ? Xj , yi? } satisfies
the KKT conditions of the Lagrangian (? = 0 in (2)), i.e.,
? ATj y? ? ?fj (x?j ) ,
(15)
?
Ax ? a = 0.
(16)
fj0 (xt+1
j )
t+1
?fj (xt+1
j )
= a. Let
During iterations, (16) is satisfied if Ax
?
where ?fj be the
subdifferential of fj . For x?j ? Xj , the optimality conditions for the xj update (5) is
t+1
t+1
t
t
?
hfj0 (xjt+1 )+Acj [yt +(1??)?(Axt?a)+Acj (xt+1
j ?xj )]+?j (??j (xj )???j (xj )), xj ?xj i ? 0 .
When Axt+1 = a, yt+1 = yt . If Acj (xt+1
? xtj ) = 0, then Axt ? a = 0. When ?j ? 0, further
j
t+1
t
assuming B?j (xj , xj ) = 0, (15) will be satisfied. Note x?j ? Xj is always satisfied in (5) in
5
PDMM. Overall, the KKT conditions (15)-(16) are satisfied if the following optimality conditions
are satisfied by the iterates:
Axt+1 = a , Acj (xt+1
? xtj ) = 0 ,
(17)
j
t
B?j (xt+1
(18)
j , xj ) = 0 .
The above optimality conditions are sufficient for the KKT conditions. (17) are the optimality conditions for the exact PDMM. (18) is needed only when ?j > 0.
Let zij = Aij xj ? Rmi ?1 , zri = [zTi1 , ? ? ? , zTiJ ]T ? Rmi J?1 and z = [(zr1 )T , ? ? ? , (zrI )T ]T ?
RJm?1 . Define the residual of optimality conditions (17)-(18) as
I
J
X
?X
? t+1
t 2
r t+1
2
t+1
t
? z kPt +
?i kAi x
? ai k2 +
R(x ) = kz
?j B?j (xt+1
(19)
j , xj ) .
2
2 i=1
j=1
t+1
) ? 0, (17)-(18) will be
where Pt is some positive semi-definite matrix and ?i = JK
? i . If R(x
K
?
?
satisfied and thus PDMM converges to the KKT point {x , y }. Define the current iterate vt =
(xtj , yit ) and h(v? , vt ) as a distance from vt to a KKT point v? = (x?j ? Xj , yi? ):
h(v? , vt ) =
I
J
X
KX 1
?
kyi? ? yit?1 k22 + L?? (xt , yt ) + kz? ? zt k2Q +
?j B?j (x?j , xtj ) , (20)
J i=1 2?i ?
2
j=1
where Q is a positive semi-definite matrix and L?? (xt , yt ) with ?i = K?2(J?K)
+ d1i ? JK
? i is
K
i (2J?K)
I
X
(?i ? ?i )? r t
L?? (xt , yt ) = f (xt ) ? f (x? ) +
hyit , Ari xt ? ai i +
kAi x ? ai k22 .
(21)
2
i=1
The following Lemma shows that h(v? , vt ) ? 0.
Lemma 1 Let vt = (xtj , yit ) be generated by PDMM (5)-(7) and h(v? , vt ) be defined in (20).
K
Setting ?i = 1 ? K?1 and ?i = K? (2J?K)
, we have
i
h(v? , vt ) ?
where ?i =
J?K
? i (2J?K)
K
i
I
?X
J
X
?
?i kAri xt ? ai k22 + kz? ? zt k2Q +
?j B?j (x?j , xtj ) ? 0 .
2 i=1
2
j=1
+
1
di
?
K
?i
JK
(22)
? 0. Moreover, if h(v? , vt ) = 0, then Ari xt = ai , zt = z? and
B?j (x?j , xtj ) = 0. Thus, (15)-(16) are satisfied.
In PDMM, yt+1 depends on xt+1 , which in turn depends on Jt . xt and yt are independent of Jt . xt
depends on the observed realizations of the random variable ?t?1 = {J1 , ? ? ? , Jt?1 } .The following
theorem shows that h(v? , vt ) decreases monotonically and thus establishes the global convergence
of PDMM.
Theorem 1 (Global Convergence) Let vt = (xtj , yit ) be generated by PDMM (5)-(7) and v? =
K
(x?j ? Xj , yi? ) be a KKT point satisfying (15)-(16). Setting ?i = 1 ? K?1 and ?i = K? (2J?K)
, we
i
i
have
0 ? E?t h(v? , vt+1 ) ? E?t?1 h(v? , vt ) , E?t R(xt+1 ) ? 0 .
(23)
The following theorem establishes the iteration complexity of PDMM in an ergodic sense.
?T =
Theorem 2 (Iteration Complexity) Let (xtj , yit ) be generated by PDMM (5)-(7). Let x
PT
1
K
t
? and ?i = K
? i (2J?K) , we have
t=1 x . Setting ?i = 1 ? K
nP i
o
I
1
J
? 2
?? (x1 , y1 ) + ? kz? ? z1 k2 + PJ ?j B? (x? , x1 )
ky
k
+
L
2
j
i
j
j
Q
i=1
j=1
K
2?i ?
2
Ef (?
xT ) ? f (x? ) ?
,
T
2
I
?
0
X
? h(v , v )
r T
2
? ? ai k2 ?
?i kAi x
E
.
T
i=1
where ?i =
K
?i ,
JK
Q is a positive semi-definite matrix, and the expectation is over Jt .
6
4
residual (log)
2
1
0
?1
?2
?3
2
1
0
?1
?2
200
300
400
500
600
700
800
time (s)
?5
0
8.1
8.05
8
7.95
PDMM1
PDMM2
PDMM3
GSADMM
RBSUMM
sADMM
7.85
?4
100
8.15
7.9
?3
?4
?5
0
PDMM1
PDMM2
PDMM3
GSADMM
RBSUMM
sADMM
3
residual (log)
PDMM1
PDMM2
PDMM3
GSADMM
RBSUMM
sADMM
3
objective (log)
4
50
100
150
200
250
iterations
7.8
50
100
150
200
250
300
time (s)
Figure 1: Comparison of the convergence of PDMM with ADMM methods in RPCA.
Table 1: The best results of PDMM with tuning parameters ?i , ?i in RPCA.
time (s) iteration residual(?10?5 ) objective (log)
PDMM1
118.83
40
3.60
8.07
PDMM2
137.46
34
5.51
8.07
PDMM3
147.82
31
6.54
8.07
GSADMM 163.09
28
6.84
8.07
RBSUMM 206.96
141
8.55
8.07
sADMM1
731.51
139
9.73
8.07
Remark 2 PDMM converges at the same rate as ADMM and its variants. In Theorem 2, PDMM
can achieve the fastest convergence by setting J = K = 1, ?i = 1, ?i = 0, i.e., the entire matrix A
is considered as a single block, indicating PDMM reduces to the method of multipliers. In this case,
however, the resulting subproblem may be difficult to solve, as discussed in Section 1. Therefore,
the number of blocks in PDMM depends on the trade-off between the number of subproblems and
how efficiently each subproblem can be solved.
4
Experimental Results
In this section, we evaluate the performance of PDMM in solving robust principal component
analysis (RPCA) and overlapping group lasso [28]. We compared PDMM with ADMM [2] or
GSADMM (no theory guarantee), sADMM [17, 26], and RBSUMM [14]. Note GSADMM includes BSUMM [14]. All experiments are implemented in Matlab and run sequentially. We run
the experiments 10 times and report the average results. The stopping criterion is either when the
residual is smaller than 10?4 or when the number of iterations exceeds 2000.
RPCA: RPCA is used to obtain a low rank and sparse decomposition of a given matrix A corrupted
by noise [5, 17]:
1
(24)
min kX1 k2F + ?2 kX2 k1 + ?3 kX3 k? s.t. A = X1 + X2 + X3 .
2
where A ? Rm?n , X1 is a noise matrix, X2 is a sparse matrix and X3 is a low rank matrix.
A = L + S + V is generated in the same way as [17]1 . In this experiment, m = 1000, n = 5000
and the rank is 100. The number appended to PDMM denotes the number of blocks (K) to be chosen
in PDMM, e.g., PDMM1 randomly updates one block.
Figure 1 compares the convegence results of PDMM with ADMM methods. In PDMM, ? = 1
and ?i , ?i are chosen according to (8), i.e., (?i , ?i ) = {( 51 , 0), ( 14 , 12 ), ( 13 , 13 )} for PDMM1, PDMM2
and PDMM3 respectively. We choose the ?best?results for GSADMM (? = 1) and RBSUMM
11
(? = 1, ? = ? ?t+10
) and sADMM (? = 1). PDMMs perform better than RBSUMM and sADMM.
Note the public available code of sADMM1 does not have dual update, i.e., ?i = 0. sADMM should
be the same as PDMM3 if ?i = 13 . Since ?i = 0, sADMM is the slowest algorithm. Without
tuning the parameters of PDMM, GSADMM converges faster than PDMM. Note PDMM can run
in parallel but GSADMM only runs sequentially. PDMM3 is faster than two randomized version
of PDMM since the costs of extra iterations in PDMM1 and PDMM2 have surpassed the savings
at each iteration. For the two randomized one block coordinate methods, PDMM1 converges faster
than RBSUMM in terms of both the number of iterations and runtime.
The effect of ?i , ?i : We tuned the parameter ?i , ?i in PDMMs. Three randomized methods (RBSUMM, PDMM1 and PDMM2) choose the blocks cyclically instead of randomly. Table 1 compares the ?best?results of PDMM with other ADMM methods. In PDMM, (?i , ?i ) =
1
http://www.stanford.edu/ boyd/papers/prox algs/matrix decomp.html
7
0.5
PA?APG
S?APG
PDMM
ADMM
sADMM
0.3
0.2
0.1
0
0
0
PA?APG
S?APG
PDMM
ADMM
sADMM
0.4
objective
objective
0.4
0.3
0.2
0.1
50
100
time (s)
150
200
0
0
1
21
41
61
81
101
?1
residual (log)
0.5
?2
?3
?4
200
400
600
iteration
800
1000
?5
20
30
40
50
60
70
time (s)
Figure 2: Comparison of convergence of PDMM and other methods in overlapping group Lasso.
{( 12 , 0), ( 13 , 12 ), ( 12 , 21 )}. GSADMM converges with the smallest number of iterations, but PDMMs
can converge faster than GSADMM in terms of runtime. The computation per iteration in
GSADMM is slightly higher than PDMM3 because GSADMM updates the sum X1 + X2 + X3 but
PDMM3 can reuse the sum. Therefore, if the numbers of iterations of the two methods are close,
PDMM3 can be faster than GSADMM. PDMM1 and PDMM2 can be faster than PDMM3. By
simply updating one block, PDMM1 is the fastest algorithm and achieves the lowest residual.
Overlapping Group Lasso: We consider solving the overlapping group lasso problem [28]:
X
1
min
kAw ? bk22 +
dg kwg k2 .
w
g?G
2L?
(25)
where A ? Rm?n , w ? Rn?1 and wg ? Rb?1 is the vector of overlapping group indexed by
g. dg is some positive weight of group g ? G. As shown in Section 2.3, (25) can be rewritten
as the form (14). The data is generated in a same way as [27, 9]: the elements of A are sampled
from normal distribution, b = Ax + with noise sampled from normal distribution, and xj =
(?1)j exp(?(j ? 1)/100). In this experiment, m = 5000, the number of groups is L = 100, and
dg = L1 , ? = L5 in (25). The size of each group is 100 and the overlap is 10. The total number of
blocks in PDMM and sADMM is J = 101. ?i , ?i in PDMM are computed according to (8).
In Figure 2, the first two figures plot the convergence of objective in terms of the number of iterations
and time. PDMM uses all 101 blocks and is the fastest algorithm. ADMM is the same as GSADMM
in this problem, but is slower than PDMM. Since sADMM does not consider the sparsity, it uses
1
1
?i = J+1
, ?i = 1 ? J+1
, leading to slow convergence. The two accelerated methods, PA-APG [27]
and S-APG [9], are slower than PDMM and ADMM.
The effect of K: The third figure shows PDMM with different number of blocks K. Although the
complexity of each iteration is the lowest when K = 1, PDMM takes much more iterations than
other cases and thus takes the longest time. As K increases, PDMM converges faster and faster.
When K = 20, the runtime is already same as using all blocks. When K > 21, PDMM takes less
time to converge than using all blocks. The runtime of PDMM decreases as K increases from 21
to 61. However, the speedup from 61 to 81 is negligable. We tried different set of parameters for
2
+1
(0 ? i ? 5, ? = 0.01, 0.1, 1) or sufficiently small step size, but could not see the
RBSUMM ? ii+t
convergence of the objective within 5000 iterations. Therefore, the results are not included here.
5
Conclusions
We proposed a randomized block coordinate variant of ADMM named Parallel Direction Method of
Multipliers (PDMM) to solve the class of problem of minimizing block-separable convex functions
subject to linear constraints. PDMM considers the sparsity and the number of blocks to be updated
when setting the step size. We show two existing Jacobian ADMM methods are special cases of
PDMM. We also use PDMM to solve overlapping block problems. The global convergence and
the iteration complexity are established with constant step size. Experiments on robust PCA and
overlapping group lasso show that PDMM is faster than existing methods.
Acknowledgment
H. W. and A. B. acknowledge the support of NSF via IIS-1447566, IIS-1422557, CCF-1451986, CNS-1314560,
IIS-0953274, IIS-1029711, IIS-0916750, and NASA grant NNX12AQ39A. H. W. acknowledges the support
of DDF (2013-2014) from the University of Minnesota. A.B. acknowledges support from IBM and Yahoo.
Z.Q. Luo is supported in part by the US AFOSR via grant number FA9550-12-1-0340 and the National Science
Foundation via grant number DMS-1015346.
8
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9
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4,701 | 5,257 | Constant Nullspace Strong Convexity and
Fast Convergence of Proximal Methods under
High-Dimensional Settings
Ian E.H. Yen
Cho-Jui Hsieh
Pradeep Ravikumar
Inderjit Dhillon
Department of Computer Science
University of Texas at Austin
{ianyen,cjhsieh,pradeepr,inderjit}@cs.utexas.edu
Abstract
State of the art statistical estimators for high-dimensional problems take the form
of regularized, and hence non-smooth, convex programs. A key facet of these
statistical estimation problems is that these are typically not strongly convex under a high-dimensional sampling regime when the Hessian matrix becomes rankdeficient. Under vanilla convexity however, proximal optimization methods attain
only a sublinear rate. In this paper, we investigate a novel variant of strong convexity, which we call Constant Nullspace Strong Convexity (CNSC), where we require that the objective function be strongly convex only over a constant subspace.
As we show, the CNSC condition is naturally satisfied by high-dimensional statistical estimators. We then analyze the behavior of proximal methods under this
CNSC condition: we show global linear convergence of Proximal Gradient and local quadratic convergence of Proximal Newton Method, when the regularization
function comprising the statistical estimator is decomposable. We corroborate our
theory via numerical experiments, and show a qualitative difference in the convergence rates of the proximal algorithms when the loss function does satisfy the
CNSC condition.
1 Introduction
There has been a growing interest in high-dimensional statistical problems, where the number of
parameters d is comparable to or even larger than the sample size n, spurred in part by many modern
science and engineering applications. It is now well understood that in order to guarantee statistical
consistency it is key to impose low-dimensional structure, such as sparsity, or low-rank structure,
on the high-dimensional statistical model parameters. A strong line of research has thus developed
classes of regularized M -estimators that leverage such structural constraints, and come with strong
statistical guarantees even under high-dimensional settings [13]. These state of the art regularized
M -estimators typically take the form of convex non-smooth programs.
A facet of computational consequence with these high-dimensional sampling regimes is that these
M -estimation problems, even when convex, are typically not strongly convex. For instance, for
the ?1 -regularized least squares estimator (LASSO), the Hessian is rank deficient when n < d. In
the absence of additional assumptions however, optimization methods to solve general non-smooth
non-strongly convex programs can only achieve a sublinear convergence rate [19, 21]; faster rates
typically require strong convexity [1, 20]. In the past few years, an effort has thus been made to
impose additional assumptions that are stronger than mere convexity, and yet weaker than strong
convexity; and proving faster rates of convergence of optimization methods under these assumptions. Typically these assumptions take the form of a restricted variant of strong convexity, which
incidentally mirror those assumed for statistical guarantees as well, such as the Restricted Isometry
1
Property or Restricted Eigenvalue property. A caveat with these results however is that these statistically motivated assumptions need not hold in general, or require sufficiently large number of samples
to hold with high probability. Moreover, the standard optimization methods have to be modified in
some manner to leverage these assumptions [5, 7, 17]. Another line of research exploits a local error
bound to establish asymptotic linear rate of convergence for a special form of non-strongly convex
functions [16, 8, 6]. However, these do not provide finite-iteration convergence bounds, due to the
potentially large number of iterations spent on early stage.
In this paper, we consider a novel simple condition, which we term Constant Nullspace Strong
Convexity (CNSC). This assumption is motivated not from statistical considerations, but from the
algebraic form of standard M -estimators; indeed as we show, standard M -estimation problems even
under high-dimensional settings naturally satisfy the CNSC condition. Under this CNSC condition,
we then investigate the convergence rates of the class of proximal optimization methods; specifically
the Proximal Gradient method (Prox-GD) [14, 15, 18] and the Proximal Newton method (ProxNewton) [1, 2, 9]. These proximal methods are very amenable to regularized M -estimation problems: they do not treat the M -estimation problem as a black-box convex non-smooth problem, but
instead leverage the composite nature of the objective of the form F (x) = h(x)+f (x), where h(x)
is a possibly non-smooth convex function while f (x) is a convex smooth function with Lipschitzcontinuous gradient. We show that under our CNSC condition, Proximal Gradient achieves global
linear convergence when the non-smooth component is a decomposable norm. We also show that
Proximal Newton, under the CNSC condition, achieves local quadratic convergence as long as the
non-smooth component is Lipschitz-continuous. Note that in the absence of strong convexity, but
under no additional assumptions beyond convexity, the proximal methods can only achieve sublinear convergence as noted earlier. We have thus identified an algebraic facet of the M -estimators
that explains the strong computational performance of standard proximal optimization methods in
practical settings in solving high-dimensional statistical estimation problems.
The paper is organized as follows. In Section 2, we define the CNSC condition and introduce
the Proximal Gradient and Proximal Newton methods. Then we prove global linear convergence
of Prox-GD and local quadratic convergence of Prox-Newton in Section 3 and 4 respectively. In
Section 5, we corroborate our theory via experiments on real high-dimensional data set. We will
leave all the proof of lemmas to the appendix.
2
Preliminaries
We are interested in composite optimization problems of the form
min
x?Rd
F (x) = h(x) + f (x),
(1)
where h(x) is a possibly non-smooth convex function and f (x) is twice differentiable convex function with its Hessian matrix H(x) = ?2 f (x) satisfying
mI ? H(x) ? M I,
?x ? Rd ,
(2)
where for strongly convex f (x) we have m > 0; otherwise, for convex but not strongly convex f (x)
we have m = 0.
2.1
Constant Nullspace Strong Convexity (CNSC)
Before defining our strong convexity variant of Constant Nullspace Strong Convexity (CNSC), we
first provide some intuition by considering the following large class of statistical estimation problems
in high-dimensional machine learning, where f (x) takes the form
f (x) =
n
?
L(aTi x, yi ),
(3)
i=1
where L(u, y) is a non-negative loss function that is convex in its first argument, ai is the observed
feature vector and yi is the observed response of the i-th sample. The Hessian matrix of (3) takes
the form
H(x) = AT D(Ax)A,
(4)
2
where A is a n by d design (data) matrix with Ai,: = aTi and D(Ax) is a diagonal matrix with
??
Dii (x) = L (aTi x, yi ), where the double-derivative in L?? (u, y) is with respect to the first argument. It is easy to see that in high-dimensional problems with d > n, (4) is not positive definite so
that strong convexity would not hold. However, for strictly convex loss function L(?, y), we have
??
L (u, y) > 0 and
(5)
v T H(x)v = 0 iff Av = 0.
As a consequence v T H(x)v > 0 as long as v does not lie in the Nullspace of A; that is, the Hessian
H(x) might satisfy the strong convexity bound in the above restricted sense. We generalize this
concept as follows. We first define the following notation: given a subspace T , we let ?T (?) denote
the orthogonal projection onto T , and let T ? denote the orthogonal subspace to T .
Assumption 1 ( Constant Nullspace Strong Convexity ). A twice-differentiable f (x) satisfies Constant Nullspace Strong Convexity (CNSC) with respect to T (CNSC-T ) iff there is a constant vector
space T s.t. f (x) depends only on z = ?T (x) and its Hessian matrix satisfies
for some m > 0, and ?z ? T ,
v T H(z)v ? m?v?2 , ?v ? T
(6)
H(z)v = 0, ?v ? T ? .
(7)
From the motivating section above, the above condition can be seen to hold for a wide range of loss
functions, such as those arising from linear regression models, as well as generalized linear models
??
(e.g. logistic regression, poisson regression, multinomial regression etc.) 1 . For L (u, y) ? mL >
0, we have m = mL ?min (AT A) > 0 as the constant in (6), where ?min (AT A) is the minimum
positive eigenvalue of AT A.
Then by the assumption, any point x can be decomposed as x = z + y, where z = ?T (x),
y = ?T ? (x), so that the difference between gradient of two points can be written as
? 1
? 1
? 1 , z 2 )?z, (8)
g(x1 ) ? g(x2 ) =
H(s?x + x2 )?xds =
H(s?z + z 2 )?zds = H(z
0
0
?
? 1 , z 2 ) = 1 H(s?z + z 2 )ds is the average Hessian
where ?x = x1 ? x2 , ?z = z 1 ? z 2 , and H(z
0
? 1 , z 2 ) satisfies inequalities (2),
matrix along the path from z 2 to z 1 . It is easy to verify that H(z
(6) and equality (7) for all z 1 , z 2 ? T by just applying inequalities (equality) to each individual
?
Hessian matrix being integrated. Then we have following theorem that shows the uniqueness of z
at optimal.
Theorem 1 (Optimality Condition). For f (x) satisfying CNSC-T ,
? is an optimal solution of (1) iff ?g(?
? for some ?
? ? ?h(?
1. x
x) = ?
x).
? and z
? = ?T (?
2. The optimal ?
x) are unique.
? is an optimal solution iff 0 ? ?h(?
Proof. The first statement is true since x
x) + ?f (?
x). To prove
?1 = z
?1 + y
? 1 and x
?2 = z
?2 + y
? 2 are both optimal. Let ?x = x
?1 ?x
?2
the second statement, suppose x
?1 ? z
? 2 . Since h(x) is convex, ?g(?
and ?z = z
x1 ) ? ?h(?
x1 ) and ?g(?
x2 ) ? ?h(?
x2 ) should satisfy
??g(?
x1 ) + g(?
x2 ), ?x? ? 0.
However, since f (x) satisfies CNSC-T , by (8),
? z1, z
? z1, z
? 2 )?z, ?x? = ??z H(?
? 2 )?z ? ?m??z?22
??g(?
x1 ) + g(?
x2 ), ?x? = ??H(?
? is
for some m > 0. The two inequalities can simultaneously hold only if ??
z = 0. Therefore, z
? z , 0)?
? = ?g(?
unique at optimum, and thus g(?
x) = g(0) + H(?
z and ?
x) are also unique.
In next two sections, we review the Proximal Gradient Method (Prox-GD) and Proximal Newton
Method (Prox-Newton), and introduce some tools that will be used in our analysis.
??
1
Note for many generalized linear models, the second derivative L (u, y) of loss function approaches 0 if |u| ? ?. However, this
could not happen as long as there is a penalty term h(x) which goes to infinity if x diverges, which then serves as a finite constraint bound on
x.
3
2.2
Proximal Gradient Method
The Prox-GD algorithm comprises a gradient descent step
xt+ 12 = xt ?
1
g(xt )
M
followed by a proximal step
xt+1 = proxhM (xt+ 12 ) = arg min h(x) +
x
M
?x ? xt+ 12 ?22 ,
2
(9)
where ? ? ?2 means the Frobinius norm if x is a matrix. For simplicity, we will denote proxhM (.) as
prox(.) in the following discussion when it is clear from the context. In Prox-GD algorithm, it is
assumed that (9) can be computed efficiently, which is true for most of decomposable regularizers.
Here we introduce some properties of proximal operator that can facilitate our analysis.
Lemma 1. Define ?P x = x ? prox(x), the following properties hold for proximal operation (9).
1. M ?P x ? ?h(prox(x)).
2. ?prox(x1 ) ? prox(x2 )?22 ? ?x1 ? x2 ?22 ? ??P x1 ? ?P x2 ?22 .
2.3
Proximal Newton Method
In this section, we introduce the Proximal Newton method, which has been shown to be considerably more efficient than first-order methods in many applications [1], including Sparse Inverse Covariance Estimation [2] and ?1 -regularized Logistic-Regression [9, 10]. Each step of Prox-Newton
solves a local quadratic approximation
1
T
T
x+
t = arg min h(x) + (x ? xt ) Ht (x ? xt ) + g t (x ? xt )
2
x
(10)
to find a search direction x+ ? xt , and then conduct a line search procedure to find t such that
f (xt+1 ) = f (xt + t(x+
t ? xt ))
meets a sufficient decrease condition. Note unlike Prox-GD update (9), in most of cases (10) requires
an iterative procedure to solve. For example if h(x) is ?1 -norm, then a coordinate descent algorithm
is usually employed to solve (10) as an LASSO subproblem [1, 2, 9, 10].
The convergence of Newton-type method comprises two phases [1, 3]. In the first phase, it is possible
that step size t < 1 is chosen, while in the second phase, which occurs when xt is close enough
to optimum, step size t = 1 is always chosen and each step leads to quadratic convergence. In this
paper, we focus on the quadratic convergence phase, while refer readers to [21] for a global analysis
of Prox-Newton without strong convexity assumption. In the quadratic convergence phase, we have
xt+1 = x+
t and the update can be written as
(
)
xt+1 = proxHt xt + ?xnt
, Ht ?xnt
(11)
t
t = ?g t ,
where ?xnt
t is the Newton step when h(x) is absent, and the proximal operator proxH (.) is defined
for any PSD matrix H as
1
proxH (x) = arg min h(v) + ?v ? x?2H .
2
v
(12)
Note while we use ?x?2H to denote xT Hx, we only require H to be PSD instead of PD. Therefore,
?x?H is not a true norm, and (12) might have multiple solutions, where proxH (x) refers to any
one of them. In the following, we show proxH (.) has similar properties as that of prox(.) in
previous section.
Lemma 2. Define ?P x = x ? proxH (x), the following properties hold for the proximal operator:
1. H?P x ? ?h(proxH (x)).
2. ?proxH (x1 ) ? proxH (x2 )?2H ? ?x1 ? x2 ?2H .
4
3
Linear Convergence of Proximal Gradient Method
In this section, we analyze convergence of Proximal Gradient Method for h(x) = ??x?, where ? ? ?
is a decomposable norm defined as follows.
Definition 1 (Decomposable Norm). ? ? ? is a decomposable norm if there are orthogonal subd
J
d
J
spaces
?{Mi }i=1 with R = ?i=1 Mi such that for any point x ? R that can be written as
x = j?E cj aj , where cj > 0 and aj ? Mj , ?aj ?? = 1, we have
?
?x? =
cj , and ??x? = {? | ?Mj (?) = aj , ?j ? E; ??Mj (?)?? ? 1, ?j ?
/ E}, (13)
j?E
where ? ? ?? is the dual norm of ? ? ?.
The above definition includes several well-known examples such as ?1 -norm ?x?1 and group-?1
norm ?X?1,2 . For ?1 -norm, Mj corresponds to vectors with only j-th coordinate not equal to 0,
and E is the set of non-zero coordinates of x. For group-?1 norm, Mj corresponds to vectors with
only j-th group not equal to 0T and E are the set of non-zero groups of X. Under the definition, we
can profile the set of optimal solutions as follows.
Lemma 3 (Optimal Set). Let E? be the active set at optimal and E?+ = {j| ? ?Mj (?
?)?? = ?} be its
? is unique) such that ?Mj (?
augmented set (which is unique since ?
?) = ??
aj , j ? E?+ . The optimal
solutions of (1) form a polyhedral set
{
}
? ,
? and x ? O
X? = x | ?T (x) = z
(14)
}
{
?
? = x|x=
? j , cj ? 0, j ? E?+ is the set of x with ?
? ? ?h(x).
where O
j?E?+ cj a
Given the optimal set is a polyhedron, we can then employ the following lemma to bound the distance of an iterate xt to the optimal set X? .
Lemma 4 (Hoffman?s bound). Consider a polyhedral set S = {x|Ax ? b, Ex = c}. For any point
? ? S such that
x ? Rd , there is a x
[Ax ? b]+
,
? ?2 ? ?(S)
?x ? x
(15)
Ex ? c
2
where ?(S) is a positive constant that depends only on A and E.
The above bound first appears in [11], and was employed in [4] to prove linear convergence of
Feasible Descent method for a class of convex smooth function. A proof of the ?2 -norm version (15)
can be found in [4, lemma 4.3]. By applying (15) to the set X? , the distance of a point x to X? can be
? where the latter can
bounded by infeasible amounts to the two constraints ?T (x) = z and x ? O,
? j ? ? 0, ?j ? E?+ .
be bounded according the following lemma when cj = ?x, a
?2 . . . , a
? |E?+ | ). Suppose ?x? ? R and ?Mj (x) = 0 for j ?
Lemma 5. Let A? = span(?
a1 , a
/ E?+ .
Then
? ?22 ,
?2 ?x ? ?A?(x)?22 ? R2 ?? ? ?
? is as defined in Theorem 1.
where ? ? ?h(x) and ?
Now we are ready to prove the main theorem of this section.
Theorem 2 (Linear Convergence of Prox-GD). Let X? be the set of optimal
( solutions for problem
)
? = ?X? (x) be the solution closest to x. Denote d? = minj ?/ E?+ ? ? ??Mj (?
(1), and x
?)?? > 0.
For the sequence {xt }?
t=0 produced by Proximal Gradient Method, we have:
(a) If xt+1 satisfies the condition that
? j ? < 0,
?j ?
/ E?+ : ?Mj (xt+1 ) ?= 0 or ?j ? E?+ : ?xt+1 , a
we then have:
d2?
? t+1 ?22 ? (1 ? ?)?xt ? x
? t ?22 , ? = 2
?xt+1 ? x
? 0 ?22
M ?x0 ? x
5
(16)
(17)
(b) If xt+1 does not satisfy the condition in (16) but xt does, then
? t+1 ?22 ? (1 ? ?)?xt?1 ? x
? t?1 ?22 , ? =
?xt+1 ? x
d2?
? 0 ?22
M 2 ?x0 ? x
(18)
m
,
M ?(X? )2
(19)
(c) If neither xt+1 , xt satisfy the condition in (16), then
? t+2 ?22 ?
?xt+2 ? x
1
? t ?22 ,
?xt ? x
1+?
?=
where we recall that ?(X? ) is the constant determined by polyhedron X? from Hoffman?s
Bound (15).
? t is an optimal solution, we have x
? t = prox(?
? t,
Proof. Since x
xt ? g(?
xt )/M ). Let ?xt = xt ? x
? = H(z
? t, z
? t ). by Lemma 1, each iterate of Prox-GD
?t = M (xt+ 21 ? xt+1 ) ? ?h(xt+1 ) and H
has
? t ?22 ? ?xt+1 ? x
? t+1 ?22 ? ?xt ? x
? t ?22 ? ?xt+1 ? x
? t ?22
?xt ? x
= ??xt ?22 ? ?prox(xt ? g(xt )/M ) ? prox(?
xt ? g(?
xt )/M )?22
?
??xt ?22
? ?(xt ? g(xt )/M ) ? (?
xt ?
g(?
xt )/M )?22
+ ??t ?
(20)
? ?22 /M 2 .
?
?
Since g(xt ) ? g(?
xt ) = H?x
from (8), we have
2
?
? ?22 /M 2
? t+1 ?22 ? ??xt ?22 ? ??xt ? H?x
? t ?22 ? ?xt+1 ? x
?xt ? x
t /M ?2 + ??t ? ?
(
)
?
? ?22 /M 2
? ?xTt H/M
?xt + ??t ? ?
(21)
? ?22 /M 2 .
? m??z t ?22 /M + ??t ? ?
?
? 2 /M 2 = (H/M
?
?
?
The second inequality holds since 2H/M
?H
)(2I ? H/M
) ? H/M
. The
2
2
? t ? ? ?xt+1 ? x
? t+1 ? ? 0, that is, the distance to the optimal set
inequality tells us ?xt ? x
? t ? is monotonically non-increasing. To get a tighter bound, we consider two cases.
?xt ? x
? j ? < 0 for some j ? E?+ .
Case 1: ?M (xt ) ?= 0 for some j ?
/ E?+ or ?xt , a
j
In this case, suppose there is j ?
/ Et+ with ?Mj (xt ) ?= 0, then 2
? ?22 ? ??Mj (?t ) ? ?Mj (?
?)?? )2 ? d2? .
?)?2? ? (??Mj (?t )?? ? ??Mj (?
??t ? ?
(22)
? j ? < 0 for some j ? E?+ , then we have ?aj , a
? j ? < 0 for ?Mj (?t ) =
On the other hand, if ?xt , a
?aj . Therefore
? j ?) > 2?2 .
? j ?22 = ?2 (2 ? 2?aj , a
? ?22 ? ??Mj (?t ) ? ?Mj (?
?)?22 ? ?2 ?aj ? a
??t ? ?
Either cases we have
? t+1 ?22 ?
? t ?22 ? ?xt+1 ? x
?xt ? x
? ?22
??t ? ?
?
2
M
(
d2?
2
? 0 ?22
M ?x0 ? x
)
? t ?22 .
?xt ? x
(23)
Case 2: Both xt , xt+1 do not fall in Case 1
? defined in
? j ? ? 0, ?j ? E?+ and ?Mj (xt ) = 0, ?j ?
Given ?xt , a
/ E?+ , then x belongs to the set O
2
?
2
Lemma 3 iff ?x ? ?A?(x)?22 = 0. The condition can be also scaled as mM
?(x)?2 = 0,
R2 ?x ? ?A
where R is a bound on ?xt ? holds for ?t, which must exist as long as the regularization parameter
? > 0 in h(x) = ??x?.
By Lemma 4, the distance of point xt to the polyhedral set X? is bounded by its infeasible amount
(
)
?2
2
? t ?22 ? ?(X? )2 ?z t ? z
? ?22 +
?xt ? x
?x
?
?
(x
)?
(24)
?
t
t 2 ,
A
mM R2
2
From our definition of decomposable norm, if a vector v belongs to single subspace Mj , then ?v? = ?v?? = ?v?2 . The reason
is: By the definition, if v ? Mj , then v = cj aj for some cj > 0, aj ? Mj , ?aj ?? = 1, and it has decomposable norm ?v? = cj .
However, we also have ?v?? = ?cj aj ?? = cj ?aj ?? = cj = ?v?. The norm equals to its dual norm only if it is ?2 -norm.
6
where z t = ?T (xt ). Applying (24) to (21) for iteration t + 1, we have
? t+1 ?2 ? ?xt+2 ? x
? t+2 ?2
?xt+1 ? x
? ?2
??
??
?2
m
.
??xt+1 ?2 ? 2 2 ?xt+1 ? ?A?(xt+1 )?22 + t+1 2
2
?
M R
M
M ?(X )
For iteration t, we have
? ?2
m
?? ? ?
? t ?2 ? ?xt+1 ? x
? t+1 ?2 ?
?xt ? x
??z t ?22 + t 2
M
M
. By Lemma 5, adding the two inequalities gives
?
? ?2
??t+1 ? ?
m
m
2
2
??x
?
??z
?
+
+
t+1
t
2
M
M2
M ?(X? )2
m
m
?
??xt+1 ?2 ?
??xt+2 ?2 ,
M ?(X? )2
M ?(X? )2
which yields desired result (18) after arrangement.
? t ?2 ? ?xt+2 ? x
? t+2 ?2 ?
?xt ? x
We note that the descent in the first two cases is actually even stronger than stated above: from the
proofs, that the distance can be seen to reduce by a fixed constant. This is faster than superlinear
convergence since the final solution could then be obtained in a finite number of steps.
4
Quadratic Convergence of Proximal Newton Method
The key idea of the proof is to re-formulate Prox-Newton update (10) as
1
? (z)) + g Tt (z ? z t ) + ?z ? z t ?2Ht
z t+1 = arg min h(z + y
2
z?T
where
? (z) = arg min
y
h(z + y),
y?T ?
(25)
(26)
so that we can focus our convergence analysis on z = ?T (x) as follows.
Lemma 6 (Optimality Condition). For any matrix H satisfying CNSC-T , the update
1
?x = arg min h(x + d) + g(x)T d + ?d?2H
2
d
has
F (x + t?x) ? F (x) ? ?t??z?2H + O(t2 ),
where ?z = ?T (?x). Furthermore, if x is an optimal solution, ?x = 0 satisfies (27).
(27)
(28)
The following lemma then states that, for Prox-Newton, the function suboptimality is bounded by
only distance in the T space.
Lemma 7. Suppose h(x) and f (x) are Lipschitz-continuous with Lipschitz constants Lh and Lf .
In quadratic convergence phase (defined in Theorem 3), Proximal Newton Method has
? ?,
F (xt ) ? F (?
x) ? L?z t ? z
(29)
? = ?T (?
where L = max{Lh , Lf } and z t = ?T (xt ), z
x).
? ? ? ?. Therefore, it suffices
By the above lemma, we have F (xt ) ? F (?
x) ? L? as long as ?z t ? z
? ? to guarantee F (xt ) ? F (?
to show quadratic convergence of ?z t ? z
x) double its precision after
each iteration.
Theorem 3 (Quadratic Convergence of Prox-Newton). For f (x) satisfying CNSC-T with Lipschitzcontinuous second derivative ?2 f (x), the Proximal Newton update (10) has
LH
? ?2 ,
?? ?
?z t ? z
?z t+1 ? z
2m
? = ?T (?
where z
x), z t = ?T (xt ), and LH is the Lipschitz constant for ?2 f (x).
7
? be an optimal solution of (1). By Lemma 6, for any PSD matrix H the update ??
Proof. Let x
x=0
satisfies (27), which means
? = proxHt (?
x
x + ??
xnt ), Ht ??
xnt = ?g(?
x).
(30)
Then by non-expansiveness of proximal operation (Lemma 2), we have
? ?Ht = ?proxHt (xt + ?xnt
?xt+1 ? x
x + ??
xnt )?Ht
t ) ? proxHt (?
? ) + (?xnt
? ?(xt + ?xnt
x + ??
xnt )?Ht = ?(xt ? x
xnt )?Ht
t ) ? (?
t ? ??
(31)
?) +
= ?(z t ? z
?
?
Since for z ? T , ?Ht z?2 ? m?z?Ht , (31) leads to
1
? ?Ht ? ? ?Ht (z t ? z
? ) ? Ht (?z nt
?xt+1 ? x
z nt )?2
t ? ??
m
(32)
LH
1
? ) ? (g t ? g
? )?2 ? ? ?z t ? z
? ?22 ,
= ? ?Ht (z t ? z
m
2 m
? ? T , we have
where last inequality follows from Lipschitz-continuity of ?2 f (x). Since z t+1 , z
?
? ?Ht = ?z t+1 ? z
? ?Ht ? m?z t+1 ? z
? ?2 .
?xt+1 ? x
(33)
Finally, combining (33) with (32),
LH
? ?22 ,
? ?2 ?
?z t ? z
?z t+1 ? z
2m
?
2m
?? < L
where quadratic convergence phase occurs when ?z t ? z
.
H
(?z nt
t
5
??
z nt
t )?Ht .
Numerical Experiments
In this section, we study the convergence behavior of Proximal Gradient method and Proximal Newton method on high-dimensional real data set with and without the CNSC condition.
In particular, two loss functions ? logistic loss L(u, y)=log(1 + exp(?yu)) and ?2 -hinge loss
L(u, y)=max(1 ? yu, 0)2 ? are used in (3) with ?1 -regularization h(x) = ??x?1 , where both
losses are smooth but only logistic loss has strict convexity that implies the CNSC condition. For
Proximal Newton method we employ an randomized coordinate descent algorithm to solve subproblem (10) as in [9]. Figure 5 shows their convergence results of objective value relative to the
optimum on rcv1.1k, subset of a document classification data set with dimension d = 10, 192 and
number of samples n = 1000. From the figure one can clearly observe the linear convergence of
Prox-GD and quadratic convergence of Prox-Newton on problem satisfying CNSC, contrasted to
the qualitatively different behavior on problem without CNSC.
Prox?GD
Prox?Newton
logistic
L2hinge
?2
10
0
logistic
L2hinge
10
?2
10
?4
obj
obj
10
?4
10
?6
10
?6
10
?8
10
?8
10
0.5
1
1.5
iter
2
2.5
3
5
6
x 10
10
15
iter
20
25
30
Figure 1: objective value (relative to optimum) of Proximal Gradient method (left) and Proximal
Newton method (right) with logistic loss and ?2 -hinge loss.
Acknowledgement
This research was supported by NSF grants CCF-1320746 and CCF-1117055. C.-J.H acknowledges
support from an IBM PhD fellowship. P.R. acknowledges the support of ARO via W911NF-12-10390 and NSF via IIS-1149803, IIS-1320894, IIS-1447574, and DMS-1264033.
8
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4,702 | 5,258 | SAGA: A Fast Incremental Gradient Method With
Support for Non-Strongly Convex Composite
Objectives
Francis Bach
INRIA - Sierra Project-Team
?
Ecole
Normale Sup?erieure, Paris, France
Aaron Defazio
Ambiata ?
Australian National University, Canberra
Simon Lacoste-Julien
INRIA - Sierra Project-Team
?
Ecole
Normale Sup?erieure, Paris, France
Abstract
In this work we introduce a new optimisation method called SAGA in the spirit of
SAG, SDCA, MISO and SVRG, a set of recently proposed incremental gradient
algorithms with fast linear convergence rates. SAGA improves on the theory behind SAG and SVRG, with better theoretical convergence rates, and has support
for composite objectives where a proximal operator is used on the regulariser. Unlike SDCA, SAGA supports non-strongly convex problems directly, and is adaptive to any inherent strong convexity of the problem. We give experimental results
showing the effectiveness of our method.
1
Introduction
Remarkably, recent advances [1, 2] have shown that it is possible to minimise strongly convex
finite sums provably faster in expectation than is possible without the finite sum structure. This is
significant for machine learning problems as a finite sum structure is common in the empirical risk
minimisation setting. The requirement of strong convexity is likewise satisfied in machine learning
problems in the typical case where a quadratic regulariser is used.
In particular, we are interested in minimising functions of the form
n
1X
f (x) =
fi (x),
n i=1
where x ? Rd , each fi is convex and has Lipschitz continuous derivatives with constant L. We will
also consider the case where each fi is strongly convex with constant ?, and the ?composite? (or
proximal) case where an additional regularisation function is added:
F (x) = f (x) + h(x),
where h : R ? R is convex but potentially non-differentiable, and where the proximal operation
of h is easy to compute ? few incremental gradient methods are applicable in this setting [3][4].
d
d
Our contributions are as follows. In Section 2 we describe the SAGA algorithm, a novel incremental
gradient method. In Section 5 we prove theoretical convergence rates for SAGA in the strongly
convex case better than those for SAG [1] and SVRG [5], and a factor of 2 from the SDCA [2]
convergence rates. These rates also hold in the composite setting. Additionally, we show that
?
The first author completed this work while under funding from NICTA. This work was partially supported
by the MSR-Inria Joint Centre and a grant by the European Research Council (SIERRA project 239993).
1
like SAG but unlike SDCA, our method is applicable to non-strongly convex problems without
modification. We establish theoretical convergence rates for this case also. In Section 3 we discuss
the relation between each of the fast incremental gradient methods, showing that each stems from a
very small modification of another.
2
SAGA Algorithm
We start with some known initial vector x0 ? Rd and known derivatives fi0 (?0i ) ? Rd with ?0i = x0
for each i. These derivatives are stored in a table data-structure of length n, or alternatively a n ? d
matrix. For many problems of interest, such as binary classification and least-squares, only a single
floating point value instead of a full gradient vector needs to be stored (see Section 4). SAGA is
inspired both from SAG [1] and SVRG [5] (as we will discuss in Section 3). SAGA uses a step size
of ? and makes the following updates, starting with k = 0:
SAGA Algorithm: Given the value of xk and of each fi0 (?ki ) at the end of iteration k, the updates
for iteration k + 1 is as follows:
1. Pick a j uniformly at random.
2. Take ?k+1
= xk , and store fj0 (?k+1
) in the table. All other entries in the table remain
j
j
k+1
unchanged. The quantity ?j is not explicitly stored.
3. Update x using fj0 (?k+1
), fj0 (?kj ) and the table average:
j
"
w
k+1
k
=x ??
fj0 (?k+1
)
j
?
fj0 (?kj )
#
n
1X 0 k
+
f (? ) ,
n i=1 i i
xk+1 = proxh? wk+1 .
(1)
(2)
The proximal operator we use above is defined as
1
2
h
kx ? yk .
(3)
prox? (y) := argmin h(x) +
2?
x?Rd
In the strongly convex case, when a step size of ? = 1/(2(?n+L)) is chosen, we have the following
convergence rate in the composite and hence also the non-composite case:
2
E
xk ? x?
? 1 ?
?
2(?n + L)
k
0
x ? x?
2 +
n
f (x0 ) ? f 0 (x? ), x0 ? x? ? f (x? ) .
?n + L
We prove this result in Section 5. The requirement of strong convexity can be relaxed from needing
to hold for each fi to just holding on average, but at the expense of a worse geometric rate (1 ?
?
6(?n+L) ), requiring a step size of ? = 1/(3(?n + L)).
In the non-strongly convex case, we have established the convergence rate in terms of the average
Pk
iterate, excluding step 0: x
?k = k1 t=1 xt . Using a step size of ? = 1/(3L) we have
4n 2L
x0 ? x?
2 + f (x0 ) ? f 0 (x? ), x0 ? x? ? f (x? ) .
E F (?
xk ) ? F (x? ) ?
k
n
This result is proved in the supplementary material. Importantly, when this step size ? = 1/(3L) is
used, our algorithm automatically adapts to the level of strong convexity ? > 0 naturally present,
giving a convergence rate of (see the comment at the end of the proof of Theorem 1):
k
2
0
?
1
x ? x?
2 + 2n f (x0 ) ? f 0 (x? ), x0 ? x? ? f (x? ) .
,
E
xk ? x?
? 1 ? min
4n 3L
3L
Although any incremental gradient method can be applied to non-strongly convex problems via the
addition of a small quadratic regularisation, the amount of regularisation is an additional tunable
parameter which our method avoids.
3
Related Work
We explore the relationship between SAGA and the other fast incremental gradient methods in this
section. By using SAGA as a midpoint, we are able to provide a more unified view than is available
in the existing literature. A brief summary of the properties of each method considered in this section
is given in Figure 1. The method from [3], which handles the non-composite setting, is not listed as
its rate is of the slow type and can be up to n times smaller than the one for SAGA or SVRG [5].
2
Strongly Convex (SC)
Convex, Non-SC*
Prox Reg.
Non-smooth
Low Storage Cost
Simple(-ish) Proof
Adaptive to SC
SAGA
3
3
3
7
7
3
3
SAG
3
3
?
7
7
7
3
SDCA
3
7
3[6]
3
7
3
7
SVRG
3
?
3
7
3
3
?
FINITO
3
?
7
7
7
3
?
Figure 1: Basic summary of method properties. Question marks denote unproven, but not experimentally
ruled out cases. (*) Note that any method can be applied to non-strongly convex problems by adding a small
amount of L2 regularisation, this row describes methods that do not require this trick.
SAGA: midpoint between SAG and SVRG/S2GD
In [5], the authors make the observation that the variance of the standard stochastic gradient (SGD)
update direction can only go to zero if decreasing step sizes are used, thus preventing a linear convergence rate unlike for batch gradient descent. They thus propose to use a variance reduction approach
(see [7] and references therein for example) on the SGD update in order to be able to use constant
step sizes and get a linear convergence rate. We present the updates of their method called SVRG
(Stochastic Variance Reduced Gradient) in (6) below, comparing it with the non-composite form
of SAGA rewritten in (5). They also mention that SAG (Stochastic Average Gradient) [1] can be
interpreted as reducing the variance, though they do not provide the specifics. Here, we make this
connection clearer and relate it to SAGA.
We first review a slightly more generalized version of the variance reduction approach (we allow the
updates to be biased). Suppose that we want to use Monte Carlo samples to estimate EX and that
we can compute efficiently EY for another random variable Y that is highly correlated with X. One
variance reduction approach is to use the following estimator ?? as an approximation to EX: ?? :=
?(X ?Y )+EY , for a step size ? ? [0, 1]. We have that E?? is a convex combination of EX and EY :
E?? = ?EX + (1 ? ?)EY . The standard variance reduction approach uses ? = 1 and the estimate
is unbiased E?1 = EX. The variance of ?? is: Var(?? ) = ?2 [Var(X) + Var(Y ) ? 2 Cov(X, Y )],
and so if Cov(X, Y ) is big enough, the variance of ?? is reduced compared to X, giving the method
its name. By varying ? from 0 to 1, we increase the variance of ?? towards its maximum value
(which usually is still smaller than the one for X) while decreasing its bias towards zero.
Both SAGA and SAG can be derived from such a variance reduction viewpoint: here X is the SGD
direction sample fj0 (xk ), whereas Y is a past stored gradient fj0 (?kj ). SAG is obtained by using
? = 1/n (update rewritten in our notation in (4)), whereas SAGA is the unbiased version with ? = 1
(see (5) below). For the same ??s, the variance of the SAG update is 1/n2 times the one of SAGA,
but at the expense of having a non-zero bias. This non-zero bias might explain the complexity of
the convergence proof of SAG and why the theory has not yet been extended to proximal operators.
By using an unbiased update in SAGA, we are able to obtain a simple and tight theory, with better
constants than SAG, as well as theoretical rates for the use of proximal operators.
"
#
n
fj0 (xk ) ? fj0 (?kj )
1X 0 k
k+1
k
(SAG)
x
=x ??
+
f (? ) ,
(4)
n
n i=1 i i
"
#
n
X
1
(SAGA)
xk+1 = xk ? ? fj0 (xk ) ? fj0 (?kj ) +
f 0 (?k ) ,
(5)
n i=1 i i
"
#
n
1X 0
k+1
k
0 k
0
f (?
x) .
(6)
(SVRG)
x
= x ? ? fj (x ) ? fj (?
x) +
n i=1 i
The SVRG update (6) is obtained by using Y = fj0 (?
x) with ? = 1 (and is thus unbiased ? we note
that SAG is the only method that we present in the related work that has a biased update direction).
The vector x
? is not updated every step, but rather the loop over k appears inside an outer loop, where
x
? is updated at the start of each outer iteration. Essentially SAGA is at the midpoint between SVRG
and SAG; it updates the ?j value each time index j is picked, whereas SVRG updates all of ??s as
a batch. The S2GD method [8] has the same update as SVRG, just differing in how the number of
inner loop iterations is chosen. We use SVRG henceforth to refer to both methods.
3
SVRG makes a trade-off between time and space. For the equivalent practical convergence rate it
makes 2x-3x more gradient evaluations, but in doing so it does not need to store a table of gradients,
but a single average gradient. The usage of SAG vs. SVRG is problem dependent. For example for
linear predictors where gradients can be stored as a reduced vector of dimension p ? 1 for p classes,
SAGA is preferred over SVRG both theoretically and in practice. For neural networks, where no
theory is available for either method, the storage of gradients is generally more expensive than the
additional backpropagations, but this is computer architecture dependent.
SVRG also has an additional parameter besides step size that needs to be set, namely the number of
iterations per inner loop (m). This parameter can be set via the theory, or conservatively as m = n,
however doing so does not give anywhere near the best practical performance. Having to tune one
parameter instead of two is a practical advantage for SAGA.
Finito/MISO?
To make the relationship with other prior methods more apparent, we can rewrite the SAGA
algorithm (in P
the non-composite case) in term of an additional intermediate quantity uk , with
n
u0 := x0 + ? i=1 fi0 (x0 ), in addition to the usual xk iterate as described previously:
SAGA: Equivalent reformulation for non-composite case: Given the value of uk and of each
fi0 (?ki ) at the end of iteration k, the updates for iteration k + 1, is as follows:
n
X
1. Calculate xk :
xk = uk ? ?
fi0 (?ki ).
(7)
i=1
2. Update u with uk+1 = uk +
1
k
n (x
3. Pick a j uniformly at random.
? uk ).
4. Take ?k+1
= xk , and store fj0 (?k+1
) in the table replacing fj0 (?kj ). All other entries in
j
j
the table remain unchanged. The quantity ?k+1
is not explicitly stored.
j
Eliminating uk recovers the update (5) for xk . We now describe how the Finito [9] and MISO? [10]
methods are closely related to SAGA. Both Finito and MISO? use updates of the following form,
for a step length ?:
n
X
1X k
xk+1 =
?i ? ?
fi0 (?ki ).
(8)
n i
i=1
The step size used is of the order
P of 1/?n. To simplify the discussion of this algorithm we will
introduce the notation ?? = n1 i ?ki .
SAGA can be interpreted as Finito, but with the quantity ?? replaced with u, which is updated in the
? but in expectation. To see this, consider how ?? changes in expectation:
same way as ?,
1 k
1 k ?k
E ??k+1 = E ??k +
x ? ?kj = ??k +
x ?? .
n
n
The update is identical in expectation to the update for u, uk+1 = uk + n1 (xk ? uk ). There are
three advantages of SAGA over Finito/MISO?. SAGA does not require strong convexity to work,
it has support for proximal operators, and it does not require storing the ?i values. MISO has
proven support for proximal operators only in the case where impractically small step sizes are
used [10]. The big advantage of Finito/MISO? is that when using a per-pass re-permuted access
ordering, empirical speed-ups of up-to a factor of 2x has been observed. This access order can also
be used with the other methods discussed, but with smaller empirical speed-ups. Finito/MISO? is
particularly useful when fi is computationally expensive to compute compared to the extra storage
costs required over the other methods.
SDCA
The Stochastic Dual Coordinate Descent (SDCA) [2] method on the surface appears quite different
from the other methods considered. It works with the convex conjugates of the fi functions. However, in this section we show a novel transformation of SDCA into an equivalent method that only
works with primal quantities, and is closely related to the MISO? method.
4
Consider the following algorithm:
SDCA algorithm in the primal
Step k + 1:
1. Pick an index j uniformly at random.
f
2. Compute ?k+1
= prox?j (z), where ? =
j
Pn
3.
at location j. For i 6= j, the
1
?n and z = ??
Store the gradient fj0 (?k+1
) = ?1 z ? ?k+1
in the table
j
j
0 k+1
0 k
table entries are unchanged (fi (?i ) = fi (?i )).
At completion, return xk = ??
Pn
i
i6=j
fi0 (?ki ).
fi0 (?ki ) .
We claim that this algorithm is equivalent to the version of SDCA where exact block-coordinate
maximisation is used on the dual.1 Firstly, note that while SDCA was originally described for onedimensional outputs (binary classification or regression), it has been expanded to cover the multiclass predictor case [11] (called Prox-SDCA there). In this case, the primal objective has a separate
strongly convex regulariser, and the functions fi are restricted to the form fi (x) := ?i (XiT x), where
Xi is a d?p feature matrix, and ?i is the loss function that takes a p dimensional input, for p classes.
To stay in the same general setting as the other incremental gradient methods, we work directly with
the fi (x) functions rather than the more structured ?i (XiT x). The dual objective to maximise then
becomes
?
?
2
n
n
1 X
X
?
1
D(?) = ??
?i
?
f ? (??i )? ,
2
?n i=1
n i=1 i
where ?i ?s are d-dimensional dual variables. Generalising the exact block-coordinate maximisation
update that SDCA performs to this form, we get the dual update for block j (with xk the current
primal iterate):
(
2 )
?n
k
1
k+1
?
k
k
x +
?j = ?j + argmax ?fj ??j ? ??j ?
??j
(9)
.
2
?n
?aj ?Rd
In the special case where fi (x) = ?i (XiT x), we can see that (9) gives exactly the same update as
Option I of Prox-SDCA in [11, Figure 1], which operates instead on the equivalent p-dimensional
dual variables ?
? i with the relationship that ?i = Xi ?
? i .2 As noted by Shalev-Shwartz & Zhang [11],
the update (9) is actually an instance of the proximal operator of the convex conjugate of fj . Our
primal formulation exploits this fact by using a relation between the proximal operator of a function
and its convex conjugate known as the Moreau decomposition:
?
proxf (v) = v ? proxf (v).
This decomposition allows us to compute the proximal operator of the conjugate via the primal
proximal operator. As this is the only use in the basic SDCA method of the conjugate function,
applying this decomposition allows us to completely eliminate the ?dual? aspect of the algorithm,
yielding the above primal form of SDCA. The dual variables are related to the primal representa0
tives ?i ?s through
P ?i = ?fi (?i ). The KKT conditions ensure that if the ?i values are dual optimal
k
then x = ? i ?i as defined above is primal optimal. The same trick is commonly used to interpret Dijkstra?s set intersection as a primal algorithm instead of a dual block coordinate descent
algorithm [12].
The primal form of SDCA differs from the other incremental gradient methods described in this
section in that it assumes strong convexity is induced by a separate strongly convex regulariser,
rather than each fi being strongly convex. In fact, SDCA can be modified to work without a separate
regulariser, giving a method that is at the midpoint between Finito and SDCA. We detail such a
method in the supplementary material.
1
More precisely, to Option I of Prox-SDCA as described in [11, Figure 1]. We will simply refer to this
method as ?SDCA? in this paper for brevity.
2
This is because fi? (?i ) =
inf
?i? (?
? i ).
?
? i s.t. ?i =Xi ?
?i
5
SDCA variants
The SDCA theory has been expanded to cover a number of other methods of performing the coordinate step [11]. These variants replace the proximal operation in our primal interpretation in the
previous section with an update where ?k+1
is chosen so that: fj0 (?k+1
) = (1??)fj0 (?kj )+?fj0 (xk ),
j
j
P
k+1
1
0 k
does
where xk = ? ?n
i fi (?i ). The variants differ in how ? ? [0, 1] is chosen. Note that ?j
k+1
not actually have to be explicitly known, just the gradient fj0 (?j ), which is the result of the above
interpolation. Variant 5 by Shalev-Shwartz & Zhang [11] does not require operations on the conju?n
gate function, it simply uses ? = L+?n
. The most practical variant performs a line search involving
the convex conjugate to determine ?. As far as we are aware, there is no simple primal equivalent
of this line search. So in cases where we can not compute the proximal operator from the standard
SDCA variant, we can either introduce a tuneable parameter into the algorithm (?), or use a dual
line search, which requires an efficient way to evaluate the convex conjugates of each fi .
4
Implementation
We briefly discuss some implementation concerns:
? For many problems each derivative fi0 is just a simple weighting of the ith data vector.
Logistic regression and least squares have this property. In such cases, instead of storing
the full derivative fi0 for each i, we need only to store the weighting constants. This reduces
the storage requirements to be the same as the SDCA method in practice. A similar trick
can be applied to multi-class classifiers with p classes by storing p ? 1 values for each i.
? Our algorithm assumes that initial gradients are known for each fi at the starting point x0 .
Instead, a heuristic may be used where during the first pass, data-points are introduced oneby-one, in a non-randomized order, with averages computed in terms of those data-points
processed so far. This procedure has been successfully used with SAG [1].
? The SAGA update as stated is slower than necessary when derivatives are sparse. A just-intime updating of u or x may be performed just as is suggested for SAG [1], which ensures
that only sparse updates are done at each iteration.
? We give the form of SAGA for the case where each fi is strongly convex. However in
practice we usually have only convex fi , with strong convexity in f induced by the addition
of a quadratic regulariser. This quadratic regulariser may be split amongst the fi functions
evenly, to satisfy our assumptions. It is perhaps easier to use a variant of SAGA where the
regulariser ?2 ||x||2 is explicit, such as the following modification of Equation (5):
#
"
1X 0 k
k+1
k
0 k
0
k
f (? ) .
x
= (1 ? ??) x ? ? fj (x ) ? fj (?j ) +
n i i i
For sparse implementations instead of scaling xk at each step, a separate scaling constant
? k may be scaled instead, with ? k xk being used in place of xk . This is a standard trick
used with stochastic gradient methods.
For sparse problems with a quadratic regulariser the just-in-time updating can be a little intricate. In
the supplementary material we provide example python code showing a correct implementation that
uses each of the above tricks.
5
Theory
In this section, all expectations are taken with respect to the choice of j at iteration k + 1 and
conditioned on xk and each fi0 (?ki ) unless stated otherwise.
We start with two basic lemmas that just state properties of convex functions, followed by Lemma 1,
which is specific to our algorithm. The proofs of each of these lemmas is in the supplementary
material.
Pn
Lemma 1. Let f (x) = n1 i=1 fi (x). Suppose each fi is ?-strongly convex and has Lipschitz
continuous gradients with constant L. Then for all x and x? :
L??
?
2
hf 0 (x), x? ? xi ?
[f (x? ) ? f (x)] ? kx? ? xk
L
2
6
?
1 X 0 ?
?
2
kfi (x ) ? fi0 (x)k ? hf 0 (x? ), x ? x? i .
2Ln i
L
Lemma 2. We have that for all ?i and x? :
"
#
1X 0
1X
1X 0 ?
0 ? 2
?
?
kfi (?i ) ? fi (x )k ? 2L
fi (?i ) ? f (x ) ?
hfi (x ), ?i ? x i .
n i
n i
n i
Lemma 3. It holds that for any ?ki , x? , xk and ? > 0, with wk+1 as defined in Equation 1:
2
2
2
E
wk+1 ? xk ? ?f 0 (x? )
? ? 2 (1 + ? ?1 )E
fj0 (?kj ) ? fj0 (x? )
+ ? 2 (1 + ?)E
fj0 (xk ) ? fj0 (x? )
2
? ? 2 ?
f 0 (xk ) ? f 0 (x? )
.
Theorem 1. With x? the optimal solution, define the Lyapunov function T as:
2
1X
1 X
0 ? k
T k := T (xk , {?ki }ni=1 ) :=
fi (?ki ) ? f (x? ) ?
fi (x ), ?i ? x? + c
xk ? x?
.
n i
n i
1
1
1
Then with ? = 2(?n+L)
, c = 2?(1???)n
, and ? = ??
, we have the following expected change in the
Lyapunov function between steps of the SAGA algorithm (conditional on T k ):
1
E[T k+1 ] ? (1 ? )T k .
?
Proof. The first three terms in T k+1 are straight-forward to simplify:
#
"
1
1 1X
1X
k+1
k
fi (?i ) = f (x ) + 1 ?
fi (?ki ).
E
n i
n
n n i
"
#
1 X
0 ? k+1
1
1 1 X
0 ? k
?
E ?
fi (x ), ?i ? x
= ? f 0 (x? ), xk ? x? ? 1?
fi (x ), ?i ? x? .
n i
n
n n i
For the change in the last term of T k+1 , we apply the non-expansiveness of the proximal operator3 :
2
2
c
xk+1 ? x?
= c
prox? (wk+1 ) ? prox? (x? ? ?f 0 (x? ))
2
? c
wk+1 ? x? + ?f 0 (x? )
.
We expand the quadratic and apply E[wk+1 ] = xk ? ?f 0 (xk ) to simplify the inner product term:
2
2
cE
wk+1 ? x? + ?f 0 (x? )
= cE
xk ? x? + wk+1 ? xk + ?f 0 (x? )
2
2
= c
xk ? x?
+ 2cE wk+1 ? xk + ?f 0 (x? ), xk ? x? + cE
wk+1 ? xk + ?f 0 (x? )
2
2
= c
xk ? x?
? 2c? f 0 (xk ) ? f 0 (x? ), xk ? x? + cE
wk+1 ? xk + ?f 0 (x? )
2
2
? c
xk ? x?
? 2c? f 0 (xk ), xk ? x? + 2c? f 0 (x? ), xk ? x? ? c? 2 ?
f 0 (xk ) ? f 0 (x? )
2
2
+ 1 + ? ?1 c? 2 E
fj0 (?kj ) ? fj0 (x? )
+ (1 + ?) c? 2 E
fj0 (xk ) ? fj0 (x? )
. (Lemma 3)
The value of ? shall be fixed later. Now we apply Lemma 1 to bound ?2c? f 0 (xk ), xk ? x? and
2
Lemma 2 to bound E
fj0 (?kj ) ? fj0 (x? )
:
2
2
2
c?
E
fj0 (xk ) ? fj0 (x? )
cE
xk+1 ? x?
? (c ? c??)
xk ? x?
+ (1 + ?)c? 2 ?
L
2
0 ? k
2c?(L ? ?)
k
?
?
f (x ) ? f (x ) ? f (x ), x ? x? ? c? 2 ?
f 0 (xk ) ? f 0 (x? )
L
"
#
X
X
1
1
fi (?ki ) ? f (x? ) ?
+ 2 1 + ? ?1 c? 2 L
fi0 (x? ), ?ki ? x? .
n i
n i
3
Note that the first equality below is the only place in the proof where we use the fact that x? is an optimality
point.
7
Function sub-optimality
100
10?4
10?4
10?8
10?8
10?8
10?8
10?12
10?12
10?12
10?12
10?4
5
10
15
20
5
10
15
20
3 ? 10?2
10?1
10?4
5
10
15
20
102
101
5
10
5
10
15
20
15
20
100
100
10?1
10?2
2 ? 10?2
10?2
5
10
15
20
Finito perm
5
10
15
20
10?1
5
Gradient evaluations / n
Finito
SAGA
SVRG
10
15
SAG
20
SDCA
LBFGS
0
4
128
101016
10
Figure 2: From left to right we have the MNIST, COVTYPE, IJCNN1 and MILLIONSONG datasets. Top
row is the L2 regularised case, bottom row the L1 regularised case.
We can now combine the bounds that we have derived for each term in T , and pull out a frac
2
tion ?1 of T k (for any ?
at this point). Together with the inequality ?
f 0 (xk ) ? f 0 (x? )
?
?2? f (xk ) ? f (x? ) ? f 0 (x? ), xk ? x? [13, Thm. 2.1.10], that yields:
h
D
Ei
2c?(L ? ?)
1
1
E[T k+1 ] ? T k ? ? T k +
?
? 2c? 2 ?? f (xk ) ? f (x? ) ? f 0 (x? ), xk ? x?
?
n
L
#
" X
E
1
1
1 XD 0 ?
1
?1
2
k
?
k
?
+ 2(1 + ? )c? L ?
fi (?i ) ? f (x ) ?
fi (x ), ?i ? x
+
?
n
n i
n i
2
2
1
1
+
? ?? c
xk ? x?
+ (1 + ?)? ?
c?E
fj0 (xk ) ? fj0 (x? )
.
(10)
?
L
Note that each of the terms in square brackets are positive, and it can be readily verified that our
1
1
1
, c = 2?(1???)n
, and ? = ??
), together with
assumed values for the constants (? = 2(?n+L)
? = 2?n+L
ensure that each of the quantities in round brackets are non-positive (the constants were
L
determined by setting all the round brackets to zero except the second one ? see [14] for the details).
1
Adaptivity to strong convexity result: Note
when
that
using the ? = 3L step size, the same c as
?
1
1
above can be used with ? = 2 and ? = min 4n , 3L to ensure non-positive terms.
2
Corollary 1. Note that c
xk ? x?
? T k , and therefore by chaining the expectations, plugging
in the constants explicitly and using ?(n ? 0.5) ? ?n to simplify the expression, we get:
2
E
xk ? x?
? 1 ?
?
2(?n + L)
k
0
x ? x?
2 +
n
f (x0 ) ? f 0 (x? ), x0 ? x? ? f (x? ) .
?n + L
Here the expectation is over all choices of index j k up to step k.
6
Experiments
We performed a series of experiments to validate the effectiveness of SAGA. We tested a binary
classifier on MNIST, COVTYPE, IJCNN1 and a least squares predictor on MILLIONSONG. Details
of these datasets can be found in [9]. We used the same code base for each method, just changing the
main update rule. SVRG was tested with the recalibration pass used every n iterations, as suggested
in [8]. Each method had its step size parameter chosen so as to give the fastest convergence.
We tested with a L2 regulariser, which all methods support, and with a L1 regulariser on a subset
of the methods. The results are shown in Figure 2. We can see that Finito (perm) performs the
best on a per epoch equivalent basis, but it can be the most expensive method per step. SVRG is
similarly fast on a per epoch basis, but when considering the number of gradient evaluations per
epoch is double that of the other methods for this problem, it is middle of the pack. SAGA can be
seen to perform similar to the non-permuted Finito case, and to SDCA. Note that SAG is slower
than the other methods at the beginning. To get the optimal results for SAG, an adaptive step size
rule needs to be used rather than the constant step size we used. In general, these tests confirm that
the choice of methods should be done based on their properties as discussed in Section 3, rather than
their convergence rate.
8
1515
20
0
References
[1] Mark Schmidt, Nicolas Le Roux, and Francis Bach. Minimizing finite sums with the stochastic
average gradient. Technical report, INRIA, hal-0086005, 2013.
[2] Shai Shalev-Shwartz and Tong Zhang. Stochastic dual coordinate ascent methods for regularized loss minimization. JMLR, 14:567?599, 2013.
[3] Paul Tseng and Sangwoon Yun. Incrementally updated gradient methods for constrained and
regularized optimization. Journal of Optimization Theory and Applications, 160:832:853,
2014.
[4] Lin Xiao and Tong Zhang. A proximal stochastic gradient method with progressive variance reduction. Technical report, Microsoft Research, Redmond and Rutgers University, Piscataway,
NJ, 2014.
[5] Rie Johnson and Tong Zhang. Accelerating stochastic gradient descent using predictive variance reduction. NIPS, 2013.
[6] Taiji Suzuki. Stochastic dual coordinate ascent with alternating direction method of multipliers.
Proceedings of The 31st International Conference on Machine Learning, 2014.
[7] Evan Greensmith, Peter L. Bartlett, and Jonathan Baxter. Variance reduction techniques for
gradient estimates in reinforcement learning. JMLR, 5:1471?1530, 2004.
[8] Jakub Kone?cn?y and Peter Richt?arik. Semi-stochastic gradient descent methods. ArXiv e-prints,
arXiv:1312.1666, December 2013.
[9] Aaron Defazio, Tiberio Caetano, and Justin Domke. Finito: A faster, permutable incremental
gradient method for big data problems. Proceedings of the 31st International Conference on
Machine Learning, 2014.
[10] Julien Mairal. Incremental majorization-minimization optimization with application to largescale machine learning. Technical report, INRIA Grenoble Rh?one-Alpes / LJK Laboratoire
Jean Kuntzmann, 2014.
[11] Shai Shalev-Shwartz and Tong Zhang. Accelerated proximal stochastic dual coordinate ascent
for regularized loss minimization. Technical report, The Hebrew University, Jerusalem and
Rutgers University, NJ, USA, 2013.
[12] Patrick Combettes and Jean-Christophe Pesquet. Proximal Splitting Methods in Signal Processing. In Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer,
2011.
[13] Yu. Nesterov. Introductory Lectures On Convex Programming. Springer, 1998.
[14] Aaron Defazio. New Optimization Methods for Machine Learning. PhD thesis, (draft under
examination) Australian National University, 2014. http://www.aarondefazio.com/pubs.html.
9
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4,703 | 5,259 | Time?Data Tradeoffs by Aggressive Smoothing
John J. Bruer1,*
Joel A. Tropp1
Volkan Cevher2
Stephen R. Becker3
1 Dept. of Computing + Mathematical Sciences, California Institute of Technology
2 Laboratory for Information and Inference Systems, EPFL
3 Dept. of Applied Mathematics, University of Colorado at Boulder
* [email protected]
Abstract
This paper proposes a tradeoff between sample complexity and computation time
that applies to statistical estimators based on convex optimization. As the amount of
data increases, we can smooth optimization problems more and more aggressively
to achieve accurate estimates more quickly. This work provides theoretical and
experimental evidence of this tradeoff for a class of regularized linear inverse
problems.
1
Introduction
It once seemed obvious that the running time of an algorithm should increase with the size of the input.
But recent work in machine learning has led us to question this dogma. In particular, Shalev-Shwartz
and Srebro [1] showed that their algorithm for learning a support vector classifier actually becomes
faster when they increase the amount of training data. Other researchers have identified related
tradeoffs [2, 3, 4, 5, 6, 7, 8, 9]. Together, these works support an emerging perspective in statistical
computation that treats data as a computational resource that we can exploit to improve algorithms
for estimation and learning.
In this paper, we consider statistical algorithms based on convex optimization. Our primary contribution is the following proposal:
As the amount of available data increases, we can smooth statistical optimization
problems more and more aggressively. We can solve the smoothed problems
significantly faster without any increase in statistical risk.
Indeed, many statistical estimation procedures balance the modeling error with the complexity of the
model. When we have very little data, complexity regularization is essential to fit an accurate model.
When we have a large amount of data, we can relax the regularization without compromising the
quality of the model. In other words, excess data offers us an opportunity to accelerate the statistical
optimization. We propose to use smoothing methods [10, 11, 12] to implement this tradeoff.
We develop this idea in the context of the regularized linear inverse problem (RLIP) with random
data. Nevertheless, our ideas apply to a wide range of problems. We pursue a more sophisticated
example in a longer version of this work [13].
JJB?s and JAT?s work was supported under ONR award N00014-11-1002, AFOSR award FA9550-09-10643, and a Sloan Research Fellowship. VC?s work was supported in part by the European Commission under
Grant MIRG-268398, ERC Future Proof, SNF 200021-132548, SNF 200021-146750 and SNF CRSII2-147633.
SRB was previously with IBM Research, Yorktown Heights, NY 10598 during the completion of this work.
1
1.1
The regularized linear inverse problem
Let x\ ? Rd be an unknown signal, and let A ? Rm?d be a known measurement matrix. Assume
that we have access to a vector b ? Rm of m linear samples of that signal given by
b := Ax\ .
Given the pair (A, b), we wish to recover the original signal x\ .
We consider the case where A is fat (m < d), so we cannot recover x\ without additional information
about its structure. Let us introduce a proper convex function f : Rd ? R ? {+?} that assigns small
values to highly structured signals. Using the regularizer f , we construct the estimator
D := arg min f (x)
x
subject to Ax = b.
(1)
x
D = x\ , and we refer to this outcome as exact recovery.
We declare the estimator successful when x
While others have studied (1) in the statistical setting, our result is different in character from previous
work. Agarwal, Negahban, and Wainwright [14] showed that gradient methods applied to problems
like (1) converge in fewer iterations due to increasing restricted strong convexity and restricted
smoothness as sample size increases. They did not, however, discuss a time?data tradeoff explicitly,
nor did they recognize that the overall computational cost may rise as the problem sizes grow.
Lai and Yin [15], meanwhile, proposed relaxing the regularizer in (1) based solely on some norm
of the underlying signal. Our relaxation, however, is based on the sample size as well. Our method
results in better performance as sample size increases: a time?data tradeoff.
The RLIP (1) provides a good candidate for studying time?data tradeoffs because recent work in
convex geometry [16] gives a precise characterization of the number of samples needed for exact
recovery. Excess samples allow us to replace the optimization problem (1) with one that we can solve
faster. We do this for sparse vector and low-rank matrix recovery problems in Sections 4 and 5.
2
The geometry of the time?data tradeoff
In this section, we summarize the relevant results that describe the minimum sample size required to
solve the regularized linear inverse problem (1) exactly in a statistical setting.
2.1
The exact recovery condition and statistical dimension
We can state the optimality condition for (1) in a geometric form; cf. [17, Prop. 2.1].
Fact 2.1 (Exact recovery condition). The descent cone of a proper convex function f : Rd ? R?{+?}
at the point x is the convex cone
[(
)
D( f ; x) :=
y ? Rd : f (x + ?y) ? f (x) .
? >0
The regularized linear inverse problem (1) exactly recovers the unknown signal x\ if and only if
D( f ; x\ ) ? null(A) = {0}.
(2)
We illustrate this condition in Figure 1(a).
To determine the number of samples we need to ensure that the exact recovery condition (2) holds,
we must quantify the ?size? of the descent cones of the regularizer f .
Definition 2.2 (Statistical dimension [16, Def. 2.1]). Let C ? Rd be a convex cone. Its statistical
dimension ?(C) is defined as
f
g
?(C) := E k? C (g)k 2 ,
where g ? Rd has independent standard Gaussian entries, and ? C is the projection operator onto C.
When the measurement matrix A is sufficiently random, Amelunxen et al. [16] obtain a precise
characterization of the number m of samples required to achieve exact recovery.
2
x?
x?
nullHAL + x?
nullHAL + x?
?
?
9x : f HxL ? f Ix? M=
9x : f HxL ? f Ix? M=
DI f , x? M + x?
(a)
(b)
Figure 1: The geometric opportunity. Panel (a) illustrates the exact recovery condition (2). Panel (b)
shows a relaxed regularizer f? with larger sublevel sets. The shaded area indicates the difference
between the descent cones of f? and f at x\ . When we have excess samples, Fact 2.3 tells us that
the exact recovery condition holds with high probability, as in panel (a). A suitable relaxtion will
maintain exact recovery, as in panel (b), while allowing us to solve the problem faster.
Fact 2.3 (Exact recovery condition for the random RLIP [16, Thm. II]). Assume that the null space
of the measurement matrix A ? Rm?d in the RLIP (1) is oriented uniformly at random. (In particular,
a matrix with independent standard Gaussian entries has this property.) Then
?
m ? ? D( f ; x\ ) + C? d =? exact recovery holds with probability ? 1 ? ?;
?
m ? ? D( f ; x\ ) ? C? d =? exact recovery holds with probability ? ?,
p
where C? := 8 log(4/?).
In words, the RLIP undergoes a phase transition when the number m of samples equals ?(D( f ; x\ )).
Any additional samples are redundant, so we can try to exploit them to identify x\ more quickly.
2.2
A geometric opportunity
Chandrasekaran and Jordan [6] have identified a time?data tradeoff in the setting of denoising
problems based on Euclidean projection onto a constraint set. They argue that, when they have a large
number of samples, it is possible to enlarge the constraint set without increasing the statistical risk of
the estimator. They propose to use a discrete sequence of relaxations based on algebraic hierarchies.
We have identified a related opportunity for a time?data tradeoff in the RLIP (1). When we have
excess samples, we may replace the regularizer f with a relaxed regularizer f? that is easier to optimize.
In contrast to [6], we propose to use a continuous sequence of relaxations based on smoothing.
Figure 1 illustrates the geometry of our time?data tradeoff. When the number of samples exceeds
?(D( f ; x\ )), Fact 2.3 tells us that the situation shown in Figure 1(a) holds with high probability.
This allows us to enlarge the sublevel sets of the regularizer while still satisfying the exact recovery
condition, as shown in Figure 1(b). A suitable relaxation allows us to solve the problem faster. Our
geometric motivation is similar with [6] although our relaxation method is totally unrelated.
3
A time?data tradeoff via dual-smoothing
This section presents an algorithm that can exploit excess samples to solve the RLIP (1) faster.
3.1
The dual-smoothing procedure
The procedure we use applies Nesterov?s primal-smoothing method from [11] to the dual problem;
see [12]. Given a regularizer f , we introduce a family { f ? : ? > 0} of strongly convex majorants:
?
f ? (x) := f (x) + kxk 2 .
2
3
Algorithm 3.1 Auslender?Teboulle applied to the dual-smoothed RLIP
Input: measurement matrix A, observed vector b
1: z0 ? 0, z?0 ? z0 , ? 0 ? 1
2: for k = 0, 1, 2, . . . do
3:
yk ? (1 ? ? k )zk + ? k z? k
4:
xk ? arg minx f (x) + ?2 kxk 2 ? hyk , Ax ? bi
5:
z? k+1 ? z? k + k A?k 2 ? (b ? Axk )
6:
zk+1 ? (1 ? ? k )zk + ? k z? k+1
7:
? k+1 ? 2/(1 + (1 + 4/? k2 ) 1/2 )
8: end for
In particular, the sublevel sets of f ? grow as ? increases. We then replace f with f ? in the original
RLIP (1) to obtain new estimators of the form
D? := arg min f ? (x) subject to Ax = b.
x
(3)
x
The Lagrangian of the convex optimization problem (3) becomes
?
L ? (x, z) = f (x) + kxk 2 ? hz, Ax ? bi ,
2
where the Lagrange multiplier z is a vector in Rm . This gives a family of dual problems:
maximize g? (z) := min L ? (x, z) subject to z ? Rm .
x
(4)
Since f ? is strongly convex, the Lagrangian L has a unique minimizer xz for each dual point z:
xz := arg min L ? (x, z).
(5)
x
Strong duality holds for (3) and (4) by Slater?s condition [18, Sec. 5.2.3]. Therefore, if we solve the
dual problem (4) to obtain an optimal dual point, (5) returns the unique optimal primal point.
The dual function is differentiable with ?g? (z) = b ? Axz , and the gradient is Lipschitz-continuous
with Lipschitz constant L ? no larger than ??1 kAk 2 ; see [12, 11]. Note that L ? is decreasing in ?,
and so we call ? the smoothing parameter.
3.2
Solving the smoothed dual problem
In order to solve the smoothed dual problem (4), we apply the fast gradient method from Auslender
and Teboulle [19]. We present the pseudocode in Algorithm 3.1.
The computational cost of the algorithm depends on two things: the number of iterations necessary
for convergence and the cost of each iteration. The following result bounds the error of the primal
iterates xk with respect to the true signal x\ . The proof is in the supplemental material.
Proposition 3.1 (Primal convergence of Algorithm 3.1). Assume that the exact recovery condition
holds for the primal problem (3). Algorithm 3.1 applied to the smoothed dual problem (4) converges
?
to an optimal dual point z?
? . Let x ? be the corresponding optimal primal point given by (5). Then
the sequence of primal iterates {xk } satisfies
2 kAk kz?
?k
.
kx\ ? xk k ?
??k
The chosen regularizer affects the cost of Algorithm 3.1, line 4. Fortunately, this step is inexpensive
for many regularizers of interest. Since the matrix?vector product Axk in line 5 dominates the other
vector arithmetic, each iteration requires O(md) arithmetic operations.
3.3
The time?data tradeoff
Proposition 3.1 suggests that increasing the smoothing parameter ? leads to faster convergence of
the primal iterates of the Auslender?Teboulle algorithm. The discussion in Section 2.2 suggests that,
when we have excess samples, we can increase the smoothing parameter while maintaining exact
recovery. Our main technical proposal combines these two observations:
4
Maximal dual-smoothing of the ` 1 norm
Stat. dim. of the dual-smoothed ` 1 descent cones
102
Maximal smoothing parameter (?(m))
Normalized statistical dimension (?/d)
1
0.8
0.6
0.4
?=0
? = 0.1
?=1
? = 10
0.2
0
0
0.2
0.4
0.6
Normalized sparsity (?)
0.8
101
100
10?1
? = 0.01
? = 0.05
? = 0.1
? = 0.2
10?2
0
1
(a)
0.2
0.4
0.6
0.8
Normalized sample size (m/d)
1
(b)
Figure 2: Statistical dimension and maximal smoothing for the dual-smoothed ` 1 norm.
Panel (a) shows upper bounds for the normalized statistical dimension d ?1 D( f ? ; x\ ) of the dualsmoothed sparse vector recovery problem for several choices of ?. Panel (b) shows lower bounds for
the maximal smoothing parameter ?(m) for several choices of the normalized sparsity ? := s/d.
As the number m of measurements in the RLIP (1) increases, we smooth the dual
problem (4) more and more aggressively while maintaining exact recovery. The
Auslender?Teboulle algorithm can solve these increasingly smoothed problems
faster.
In order to balance the inherent tradeoff between smoothing and accuracy, we introduce the maximal
smoothing parameter ?(m). For a sample size m, ?(m) is the largest number satisfying
? D( f ?(m) ; x\ ) ? m.
(6)
Choosing a smoothing parameter ? ? ?(m) ensures that we do not cross the phase transition of
our RLIP. In practice, we need to be less aggressive in order to avoid the ?transition region?. The
following two sections provide examples that use our proposal to achieve a clear time?data tradeoff.
4
Example: Sparse vector recovery
In this section, we apply the method outlined in Section 3 to the sparse vector recovery problem.
4.1
The optimization problem
Assume that x\ is a sparse vector. The ` 1 norm serves as a convex proxy for sparsity, so we choose it
as the regularizer in the RLIP (1). This problem is known as basis pursuit, and it was proposed by
Chen et al. [20]. It has roots in geophysics [21, 22].
We apply the dual-smoothing procedure from Section 3 to obtain the relaxed primal problem, which
is equivalent to the elastic net of Zou and Hastie [23]. The smoothed dual is given by (4).
To determine the exact recovery condition, Fact 2.3, for the dual-smoothed RLIP (3), we must
compute the statistical dimension of the descent cones of f ? . We provide an accurate upper bound.
Proposition 4.1 (Statistical dimension bound for the dual-smoothed ` 1 norm). Let x ? Rd with s
nonzero entries, and define the normalized sparsity ? := s/d. Then
r Z
?
?
?
f
g
?
?
2
1
2
2
2
? D( f ? ; x) ? inf ?
? 1 + ? (1 + ? kxk ` ? ) + (1 ? ?)
(u ? ?) 2 e?u /2 du ?
?
?.
? ?0
d
? ?
?
?
5
150
Cost vs. sample size (` 1 norm)
?1011
? = 0.1
? = ?(m)/4
1
Average cost
Average number of iterations
Iterations vs. sample size (` 1 norm)
100
0.8
? = 0.1
? = ?(m)/4
0.6
50
1
1.5
2
2.5
3
Sample size (m)
(a)
3.5
0.4
4
?104
1
1.5
2
2.5
3
Sample size (m)
3.5
4
?104
(b)
Figure 3: Sparse vector recovery experiment. The average number of iterations (a) and the average
computational cost (b) of 10 random trials of the dual-smoothed sparse vector recovery problem with
ambient dimension d = 40 000 and normalized sparsity ? = 5% for various sample sizes m. The red
curve represents a fixed smoothing parameter ? = 0.1, while the blue curve uses ? = ?(m)/4. The
error bars indicate the minimum and maximum observed values.
The proof is provided in the supplemental material. Figure 2 shows the statistical dimension and
maximal smoothing curves for sparse vectors with ?1 entries. In order to apply this result we only
need estimates of the magnitude and sparsity of the signal.
To apply Algorithm 3.1 to this problem, we must calculate an approximate primal solution xz from a
dual point z (Algorithm 3.1, line 4). This step can be written as
xz ? ?(m) ?1 ? SoftThreshold(AT z, 1),
where [SoftThreshold(x,t)]i = sgn (x i ) ? max {|x i | ? t, 0}. Algorithm 3.1, line 5 dominates the total
cost of each iteration.
4.2
The time?data tradeoff
We can obtain theoretical support for the existence of a time?data tradeoff in the sparse recovery
problem by adapting Proposition 3.1. See the supplemental material for the proof.
Proposition 4.2 (Error bound for dual-smoothed sparse vector recovery). Let x\ ? Rd with s
nonzero entries, m be the sample size, and ?(m) be the maximal smoothing parameter (6). Given a
measurement matrix A ? Rm?d , assume the exact recovery condition (2) holds for the dual-smoothed
sparse vector recovery problem. Then the sequence of primal iterates from Algorithm 3.1 satisfies
f
g1
1
2d 2 ?(A) ? ? (1 + ?(m) kx\ k ` ? ) 2 + (1 ? ?) 2
,
kx\ ? xk k ?
?(m) ? k
where ? := s/d is the normalized sparsity of x\ , and ?(A) is the condition number of the matrix A.
For a fixed number k of iterations, as the number m of samples increases, Proposition 4.2 suggests
that the error decreases like 1/?(m). This observation suggests that we can achieve a time?data
tradeoff by smoothing.
4.3
Numerical experiment
Figure 3 shows the results of a numerical experiment that compares the performance difference
between current numerical practice and our aggressive smoothing approach.
Most practitioners use a fixed smoothing parameter ? that depends on the ambient dimension or
sparsity but not on the sample size. For the constant smoothing case, we choose ? = 0.1 based on the
recommendation in [15]. It is common, however, to see much smaller choices of ? [24, 25].
6
In contrast, our method exploits excess samples by smoothing the dual problem more aggressively.
We set the smoothing parameter ? = ?(m)/4. This heuristic choice is small enough to avoid the phase
transition of the RLIP while large enough to reap performance benefits. Our forthcoming work [13]
addressing the case of noisy samples provides a more principled way to select this parameter.
In the experiment, we fix both the ambient dimension d = 40 000 and the normalized sparsity ? = 5%.
To test each smoothing approach, we generate and solve 10 random sparse vector recovery models for
each value of the sample size m = 12 000, 14 000, 16 000, . . . , 38 000. Each random model comprises
a Gaussian measurement matrix A and a random sparse vector x\ whose nonzero entires are ?1 with
equal probability. We stop Algorithm 3.1 when the relative error kx\ ? xk k / kx\ k is less than 10?3 .
This condition guarantees that both methods maintain the same level of accuracy.
In Figure 3(a), we see that for both choices of ?, the average number of iterations decreases as sample
size increases. When we plot the total computational cost1 in Figure 3(b), we see that the constant
smoothing method cannot overcome the increase in cost per iteration. In fact, in this example, it would
be better to throw away excess data when using constant smoothing. Meanwhile, our aggressive
smoothing method manages to decrease total cost as sample size increases. The maximal speedup
achieved is roughly 2.5?.
We note that if the matrix A were orthonormal, the cost of both smoothing methods would decrease
as sample sizes increase. In particular, the uptick seen at m = 38 000 in Figure 3 would disappear
(but our method would maintain roughly the same relative advantage over constant smoothing).
This suggests that the condition number ?(A) indeed plays an important role in determining the
computational cost. We believe that using a Gaussian matrix A is warranted here as statistical models
often use independent subjects.
Let us emphasize that we use the same algorithm to test both smoothing approaches, so the relative
comparison between them is meaningful. The observed improvement shows that we have indeed
achieved a time?data tradeoff by aggressive smoothing.
5
Example: Low-rank matrix recovery
In this section, we apply the method outlined in Section 3 to the low-rank matrix recovery problem.
5.1
The optimization problem
Assume that X \ ? Rd1 ?d2 is low-rank. Consider a known measurement matrix A ? Rm?d , where
d := d 1 d 2 . We are given linear measurements of the form b = A ? vec(X \ ), where vec returns the
(column) vector obtained by stacking the columns of the input matrix. Fazel [26] proposed using the
Schatten 1-norm k?k S1 , the sum of the matrix?s singular values, as a convex proxy for rank. Therefore,
we follow Recht et al. [27] and select f = k?k S1 as the regularizer in the RLIP (1). The low-rank
matrix recovery problem has roots in control theory [28].
We apply the dual-smoothing procedure to obtain the approximate primal problem and the smoothed
dual problem, replacing the squared Euclidean norm in (3) with the squared Frobenius norm.
As in the sparse vector case, we must compute the statistical dimension of the descent cones of the
strongly convex regularizer f ? . In the case where the matrix X is square, the following is an accurate
upper bound for this quantity. (The non-square case is addressed in the supplemental material.)
Proposition 5.1 (Statistical dimension bound for the dual-smoothed Schatten 1-norm). Let X ?
Rd1 ?d1 have rank r, and define the normalized rank ? := r/d 1 . Then
(
"
1
? D( f ? ; X) ? inf ? + (1 ? ?) ? 1 + ? 2 (1 + ? kX k) 2
2
0?? ?2
d1
#)
p
(1 ? ?)
2
?1
2
2
+
24(1 + ? ) cos (?/2) ? ?(26 + ? ) 4 ? ?
+ o (1) ,
12?
as d 1 ? ? while keeping the normalized rank ? constant.
1We compute total cost as k ? md, where k is the number of iterations taken, and md is the dominant cost of
each iteration.
7
?1011
? = 0.1
? = ?(m)/4
600
Cost vs. sample size (Schatten 1-norm)
3
Average cost
Average number of iterations
Iterations vs. sample size (Schatten 1-norm)
400
? = 0.1
? = ?(m)/4
2
200
1
0
1
1.5
2
2.5
3
Sample size (m)
(a)
3.5
4
1
1.5
?104
2
2.5
3
Sample size (m)
3.5
(b)
4
?104
Figure 4: Low-rank matrix recovery experiment. The average number of iterations (a) and the
average cost (b) of 10 random trials of the dual-smoothed low-rank matrix recovery problem with
ambient dimension d = 200 ? 200 and normalized rank ? = 5% for various sample sizes m. The red
curve represents a fixed smoothing parameter ? = 0.1, while the blue curve uses ? = ?(m)/4. The
error bars indicate the minimum and maximum observed values.
The proof is provided in the supplemental material. The plots of the statistical dimension and maximal
smoothing curves closely resemble those of the ` 1 norm and are in the supplemental material as well.
In this case, Algorithm 3.1, line 4 becomes [12, Sec. 4.3]
Xz ? ?(m) ?1 ? SoftThresholdSingVal(mat(AT z), 1),
where mat is the inverse of the vec operator. Given a matrix X with SVD U ? diag(?) ? V T ,
SoftThresholdSingVal(X,t) = U ? diag (SoftThreshold(?,t)) ? V T .
Algorithm 3.1, line 5 dominates the total cost of each iteration.
5.2
The time?data tradeoff
When we adapt the error bound in Proposition 3.1 to this specific problem, the result is nearly same
as in the ` 1 case (Proposition 4.2). For completeness, we include the full statement of the result in
the supplementary material, along with its proof. Our experience with the sparse vector recovery
problem suggests that a tradeoff should exist for the low-rank matrix recovery problem as well.
5.3
Numerical experiment
Figure 4 shows the results of a substantially similar numerical experiment to the one performed for
sparse vectors. Again, current practice dictates using a smoothing parameter that has no dependence
on the sample size m [29]. In our tests, we choose the constant parameter ? = 0.1 recommended
by [15]. As before, we compare this with our aggressive smoothing method that selects ? = ?(m)/4.
In this case, we use the ambient dimension d = 200 ? 200 and set the normalized rank ? = 5%. We
test each method with 10 random trials of the low-rank matrix recovery problem for each value of the
sample size m = 11 250, 13 750, 16 250, . . . , 38 750. The measurement matrices are again Gaussian,
and the nonzero singular values of the random low-rank matrices X \ are 1. We solve each problem
with Algorithm 3.1, stopping when the relative error in the Frobenius norm is smaller than 10?3 .
In Figure 4, we see that both methods require fewer iterations for convergence as sample size increases.
Our aggressive smoothing method additionally achieves a reduction in total computational cost, while
the constant method does not. The observed speedup from exploiting the additional samples is 5.4?.
The numerical results show that we have indeed identified a time?data tradeoff via smoothing. While
this paper considers only the regularized linear inverse problem, our technique extends to other
settings. Our forthcoming work [13] addresses the case of noisy measurements, provides a connection
to statistical learning problems, and presents additional examples.
8
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4,704 | 526 | Oscillatory Model of Short Term Memory
David Horn
School of Physics and Astronomy
Raymond and Beverly Sackler
Faculty of Exact Sciences
Tel-Aviv University
Tel Aviv 69978, Israel
Marius U sher*
Dept. of Applied Mathematics
and Computer Science
Weizmann Institute of Science
Rehovot 76100, Israel
Abstract
We investigate a model in which excitatory neurons have dynamical thresholds which display both fatigue and potentiation. The fatigue property
leads to oscillatory behavior. It is responsible for the ability of the model
to perform segmentation, i.e., decompose a mixed input into staggered
oscillations of the activities of the cell-assemblies (memories) affected by
it. Potentiation is responsible for sustaining these staggered oscillations
after the input is turned off, i.e. the system serves as a model for short
term memory. It has a limited STM capacity, reminiscent of the magical
number 7 ? 2.
1
Introduction
The limited capacity (7 ? 2) of the short term memory (STM) has been a subject
of major interest in the psychological and physiological literature. It seems quite
natural to assume that the limited capacity is due to the special dynamical nature
of STM. Recently, Crick and Koch (1990) suggested that the working memory
is functionally related to the binding process, and is obtained via synchronized
oscillations of neural populations. The capacity limitation of STM may then result
from the competition between oscillations representing items in STM. In the model
which we investigate this is indeed the case.
?Present address: Division of Biology, 216-76, Caltech, Pasadena CA 91125.
125
126
Horn and Usher
Models of oscillating neural networks can perform various tasks:
1. Phase-locking and synchronization in response to global coherence in the stimuli, such as similarity of orientation and continuity (Kamen et al. 1989; Sompolinsyet al. 1990; Konig & Schillen 1991).
2. Segmentation of incoherent stimuli in low level vision via desynchronization,
using oscillator networks with delayed connections (Schillen & Konig 1991).
3. Segmentation according to semantic content, i.e., separate an input of mixed information into its components which are known memories of the system (Wang
et al. 1990, Horn and Usher 1991). In these models the memories are represented by competing cell a.'3semblies. The input, which affects a subset of these
assemblies, induces staggered oscillations of their activities. This works as long
as the number of memories in the input is small, of the order of 5.
4. Binding, i.e., connecting correctly different attributes of the same object which
appear in the mixed input (Horn et al. 1991). Binding can be interpreted as
matching the phases of oscillations representing attributes of the same object
in two different networks which are coupled in a way which does not assume
any relation between the attributes.
To these we add here the important task of
5. STM, i.e., keeping information about segmentation or binding after the input
is turned off.
In order to qualify as models for STM, the staggered oscillations have to prevail
after the input stimuli disappear. Unfortunately, this does not hold for the models
quot.ed above. Once the input disappears, either the network's activity dies out,
or oscillations of assemblies not included in the original input are turned on. In
other words, the oscillations have no inertia, and thus they do not persist after the
disappearance of the sensory input. Our purpose is to present a model of competing neural assemblies which, upon receiving a mixed input develops oscillations
which prevail after the st.imulus disappears. In order to achieve this, the biological
mechanism of post tetanic potentiation will be used.
2
Dynaillics of Short Ternl Potentiation
It was shown that following a t.etanus of electrophysiological stimulation temporary
modifications in the synaptic strengths, mostly non Hebbian, are observed (Crick
and Koch, 1990; Zucker, 1989). The time scale of these synaptic modifications
ranges between 50 111S to several minutes. A detailed description of the processes
responsible for this mechanism was given by Zucker (1989), exhibiting a rather complex behavior. In the following we will use a simplified version of these mechanisms
involving two processes with different time scales. We assume that following a prolonged activation of a synapse, the synaptic strength exhibits depression on a short
time scale, but recovers and becomes slightly enhanced on a longer time scale. As
illustrated in Fig 1 of Zucker (1989), this captures most of the dynamics of Short
Term Potentiation. The fact that these mechanisms are non Hebbian implies that
all synapses associated with a presynaptic cell are affected, and thus the unit of
change is the presynaptic cell (Crick & Koch 1990).
Oscillatory Model of Shorr Term Memory
Our previous oscillatory neural networks were based on the assumption that, in
addition to the customary properties of the formal neuron, its threshold increases
when the neuron keeps firing, thus exhibiting adaptation or fatigue (Horn & Usher
1989). Motivated by the STP findings we add a new component offacilitaion, which
takes place on a longer time scale than fatigue. We denote the dynamical threshold
by the continuous variable r which is chosen as a sum of two components, I and p,
representing fatigue and potentiation,
(1)
= all - a2p?
r
Their dynamics is governed by the equations
,dl/dt =
m
+ (l/CI
,dp/dt =
1)1
-
m
+ (1/c2 -
l)p
(2)
where m describes the average neuron activity (firing rate) on a time scale which
is large compared to the refractory period. The time constants of the fatigue and
potentiation components, Tj = c,c:' l are chosen so that TI < T2. As a result the
neuron displays fatigue on a short time scale, but recovers and becomes slightly
enhanced (potentiated) on a longer time scale. This is clearly seen in Fig. 1, which
shows the behavior when the activity m of the corresponding neuron is clamped at
1 for some time (due to sensory input) and quenched to zero afterwards.
3
_. -.
,,-
2
\
/
\f
I
\
r
\
1
,
"
.....
"-
.-.- . -. -.-
0
\
'\
'\
'\
-1
-2
""
0
.........
p. . . .
.......
....- ..-
./
.......
.......
./
40
20
60
100
80
time
Figure 1: Behavior of the dynamic threshold r and its fatigue
I and potentiation
p components, when the neuron activity m is clamped as shown. Time scale is
arbitrary. The parameters are
CI
= 1.2 C2 =
1.05
al
= 4 a2 = 1 .
We observe here that the threshold increases during the cell's activation, being
driven to its asymptotic value al c1-I.
After the release of the stimulus the dyCl
namic threshold decreases (i.e . the neuron recovers) and turns negative (signifying
127
128
Horn and Usher
potentiation). The parameters were chosen so that asymptotically the threshold
reaches zero, i.e. no permanent effect is left. In our model we will assume a similar
behavior for the excitatory cell-assemblies which carry the memories in our system.
3
The Model
Our basic model (Horn & Usher 1990) is composed of two kinds of neurons which
are assumed to have excitatory and inhibitory synapses exclusively. Memory patterns are carried by excitatory neurons only. Furthermore, we make the simplifying
assumption that the patterns do not overlap with one another, i.e. the model is
composed of disjoint Hebbian cell-assemblies of excitatory neurons which affect one
another through their interaction with a single assembly of inhibitory neurons.
Let us denote by mS'(t) the fraction of cell-assembly number Il which fires at time t,
and by m I (t) the fraction of active inhibitory neurons. We will refer to mS' as the
activity of the Ilth memory pattern. There are P different memories in the model,
and their activities obey the following differential equations
= -mS' + FT(AmS' -
Bm I - f}S' + is')
dmI/dt == -m I + FT(CM - Dm I - f}I)
(3)
M= LmS'
S'
(4)
dmS' /dt
where
f}S' and f}I are the thresholds of all excitatory and inhibitory neurons correspondingly
and is' represents the input into cell assembly Il. The four parameters ABC and
D are all positive and represent the different couplings between the neurons. This
system is an attractor neural network. In the absence of input and dynamical
thresholds it is a dissipative system which flows into fixed points determined by the
memOrIes.
This system is a generalization of the E-I model of Wilson and Cowan (1972) in
which we have introduced competing memory patterns. The latter make it into an
attractor neural network. Wilson and Cowan have shown that a pair of excitatory
and inhibitory assemblies, when properly connected, will form an oscillator. We
induce oscillations in a different way, keeping the option of having the network
behave either as an attractor neural network or as an oscillating one: we turn the
thresholds of the excitatory neurons into dynamic variables, which are defined by
f}S' = f}t;
+ brS' .
The dynamics of the new variables rS' are chosen to follow equations (1) and (2)
where all elements, r f p and m refer to the same cell-assembly 1-'. To understand
the effects of this change let us first limit ourselves to the fatigue component only,
1 and a2 = 0 in Eq. 1. Imagine a situation in which the system would flow
i.e. a1
into a fixed point mS'
1. rS' will then increase until it reaches the value cI/( C1 -1).
This means that the argument of the FT function in the equation for mS' decreases
by 9 = bCI/(Cl - 1) . If this overcomes the effect of the other terms the amplitude
mS' decreases and the system moves out of the attractor and falls into the basin
of a different center of attraction. This process can continue indefinitely creating
=
=
Oscillatory Model of Short Term Memory
an oscillatory network which moves from one memory to another. Envisage now
turning on a p/lo component leading to an r/lo behavior of the type depicted in Fig.
1. Its effect will evidently be the same as the input i/lo in Eq. (3) during the time
in which it is active. In other words, it will help to reactivate the cell-assembly /-l,
thus carrying the information that this memory was active before. Therefore, its
role in our system is to serve as the inertia component necessary for creating the
effect of STM.
4
Seglnentation and Short Term Memory
In this section we will present. results of numerical investigations of our model. The
parameters used in the following are A = C = D = 1 B = 1.10t; = 0.075 OJ =
-0.55 T = 0.05 b = 0.2 I = 2.5 and the values of ai and Ci of Fig. 1. We let n of the
P memories have a constant input of the form
i/lo
=i
/-l
= 1"
.. , n
=0
i/lo
/-l = n
+ 1,?,?, P.
(5)
An example of the result. of a system with P = 10 and n = 4 is shown in Fig. 2.
1.0 ,...
0.8
-
U1
\l.)
.....
o+J
.....
:>
.....
o+J
CJ
CO
I
,
,.,,11/
" 11? ,\ ,I ""
I
'\'
:
I
I
"
I
. ? '.
I ? .f ??
..
,
'I::
II,?::
0.6 - ,',."
. ",
,. . ., ,I,?:.
0.4 I - / '"t ? I l" r'::
.
.,
tr1
0.2
~
0.0
,
"
;
?
I
t ~
I.
oj
I,!.
I
"
::
"
.'
..
.'
,
1,1 ?
I ~r \
':
"
'
I ~
,\ ,I~I.\.
'.\1J\ ."\.1\t:.).,~~.: \ '.. 1.\,
::
.:
,
A :.
.,11::
I,
-
I
I
I
\<
., I ,'.
"1 1<
. I:
~
1 I
?
I, ::..
,,::
II
"
,.1,::
"~"
,.
'I, "
"
"
,.,,::
I,
'.
I.
\ ..
, ,I I' ,
.
" ' , \,1 I: :
I"
"
,
,,' I:: " :: '\/ I: .
',~~: I~I~"~'
Il'l
~.
"\.;~.'" ?
11,\.t
I, ~
~'" _'
',J\..J ~ '_'l
' ,
?
.A.
---------C>
4
3
2
1
a
a
25
75
50
100
125
time
Figure 2: Result.s of our model for P = 10 memories and n = 4 inputs. The first
frame displays the activities m of the four relevant cell-assemblies, and the second
frame represents their l' values. The arrow indicates the duration of the mixed
input.
129
130
Horn and Usher
Here we display the activities of the cell-assemblies that receive the constant input
and their corresponding average thresholds. While the signal of the mixed input
is on (denoted by an arrow along the time scale) we see how the phenomenon of
segmentation develops. The same staggered oscillation of the four cell-assemblies
which received an input is sustained after the signal is turned off. This indicates that
the system functions as a STM. Note that no synaptic connections were changed
and, once the system will receive a new input its behavior will be revised. However,
as long as it is left alone, it will continue to activate the cell-assemblies affected by
the original input.
We were able to obtain good results only for low n values, n ::; 4. As n is increased
we have difficulties wit.h both segmentation and STM. By modifying slightly the
paradigm we were able to feed 5 different inputs in a STM, as shown in Fig. 3.
This required presenting them at different times, as indicated by the 5 arrows on this
figure. In other words, t.his system does not perform segmentation but it continues
to work as a STM. Note, however, that the order of the different activities is no
longer maintained after the stimuli are turned off.
1.0
.-.-
I
0.8
tZl
tl)
~
.-
>
~
C)
ttl
0.6
.
....
II
, I ? l\
I ,? ?
?
?? ???
,:? ???.
I I
I
0.4
:
\:
0.2
.
:"':
? ;i
,
?:
-;
. /1 .
'1. ,I
I I ,I
., . ,
II
?? ..
.'
1
.'.
I' ::
;; :~
,?
0.0
o
o
25
75
50
100
125
time
Figure 3: Result.s for 5 inputs which are fed in consecutively at the times indicated
by the short arrows. The model functions as STM without segmentation.
Oscillatory Model of Short Term Memory
5
Discussion.
Psychological experiments show that subjects can repeat a sequence of verbal items
in perfect order a.'l long as their number is small (7 ? 2). The items may be numbers
or let.ters but can also be combinations of the latter such as words or recognizable
dates or acronyms. This proves that STM makes use of the encoded material in the
long term memory (Miller 1956). This relation between the two different kinds of
memOt'y lies in the basis of our model. Long term memory is represented by excitatory cell assemblies . Incorporating threshold fatigue into the model, it acquires the
capability of performing temporal segmentation of external input. Adding to the
threshold post tetanic potentiation, the model becomes capable of maintaining the
segmented information in the form of staggered oscillations. This is the property
which we view as responsible for STM.
Both segmentation and STM have very limited capacities. This seems to follow
from t.he oscillatory nature of the system which we use to model these functions.
In contrast with long term memory, whose capacity can be increased endlessly by
adding neurons and synaptic connections, we find here that only a few items can
be st.ored in t.he dynamic fashion of staggered oscillations, irrespective of the size
of the system. Vve regard this result as very significant, in view of the fact that
the same holds for the limited psychological ability of attention and STM. It may
indicate t.hat the oscillatory model contains the key to the understanding of these
psychological findings.
In order to validate the hypothesis that STM is based on oscillatory correlations
between firing rates of neurons, some more experimental neurobiological and psychophysical research is required. While no conclusive results were yet obtained from
recordings of t.he cortical activity in the monkey, some positive support has been obtained in psychophysical experiments. Preliminary results show that an oscillatory
component can be found in the percentage of correct responses in STM matching
experiments (Usher & Sagi 1991).
Our mathematical model is based on many specific assumptions. We believe that
our main results are characteristic of a class of such models which can be obtained
by changing various elements in our system. The main point is that dynamical
storage of information can be achieved through staggered oscillations of memory
activit.ies. Moreover, to sustain them in the absence of an external input, a potentiation capability ha.'l to be present. A model which contains both should be able
to accomodate STM in t.he fashion which we have demonstrated.
A cknowledgenlent s
M. Usher is the recipient of a Dov Biegun post-doctoral fellowship. We wish to
thank S. Popescu for helpful discussions.
References
Crick,F. & Koch,C. 1990. Towards a neurobiological theory of consciousness. Seminars in the Neurosciences 2, 263-275.
131
132
Horn and Usher
Horn,D., Sagi,D. & Usher,M. 1991. Segmentation, binding and illusory conjunctions. Neural Compo 3, 509-524.
Horn,D. & Usher ,M. 1989. Neural networks with dynamical thresholds, Phys. Rev.
A 40, 1036-1044.
Horn,D. & Usher,M. 1990. Excitatory-inhibitory networks with dynamical thresholds, Int. 1. NeuralSyst. 1, 249-257.
Horn,D. & Usher ,M. 1991. Parallel Activation of Memories is an Oscillatory Neural
Network. Neural Compo 3, 31-43.
Kammen,D.M., Holmes,P.J. & Koch C. 1990. Origin of oscillations in visual cortex:
Feedback versus local coupling. In Models of Brain Function, M.Cotterill ed., pp
273-284. Cambridge University Press.
Konig,P. & Schillen,T.B. 1991. Stimulus-dependent assembly formation of oscillatory responses: I. Synchronization, Neural Compo 3, 155-166.
Miller,G. 1956. The magical number seven plus minus two. Psych. Rev., 63,81-97.
Sompolinsky,H., Golomb,D. & Kleinfeld,D. 1990. Global processing of visual stimuli
in a neural network of coupled oscillators. Proc. Natl. Acad. of Sci. USA, 87,
7200-7204.
Schillen,T.B. & Konig,P. 1991. Stimulus-dependent assembly formation of oscillatory responses: I. Synchronization, Neural Compo 3, 155-166.
Wang,D., Buhmann,J. & von der Malsburg,C. 1990. Pattern segmentation in associative memory. Neural Compo 2, 94-106.
Wilson,H .R. & Cowan,J .D. 1972. Excitatory and inhibitory interactions in localized
populations of model neurons. Biophys. 1. 12, 1-24.
Usher,M. & Sa.gi D. 1991, in preparation.
Zucker ,R.S. 1989. Short-term synaptic plasticity. Ann. Rev. Neurosci. 12, 13-31.
PART III
SPEECH
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4,705 | 5,260 | Local Linear Convergence of Forward?Backward
under Partial Smoothness
Jingwei Liang and Jalal M. Fadili
GREYC, CNRS-ENSICAEN-Univ. Caen
{Jingwei.Liang,Jalal.Fadili}@greyc.ensicaen.fr
Gabriel Peyr?
CEREMADE, CNRS-Univ. Paris-Dauphine
[email protected]
Abstract
In this paper, we consider the Forward?Backward proximal splitting algorithm to
minimize the sum of two proper closed convex functions, one of which having
a Lipschitz continuous gradient and the other being partly smooth relative to an
active manifold M. We propose a generic framework under which we show that
the Forward?Backward (i) correctly identifies the active manifold M in a finite
number of iterations, and then (ii) enters a local linear convergence regime that we
characterize precisely. This gives a grounded and unified explanation to the typical
behaviour that has been observed numerically for many problems encompassed in
our framework, including the Lasso, the group Lasso, the fused Lasso and the
nuclear norm regularization to name a few. These results may have numerous
applications including in signal/image processing processing, sparse recovery and
machine learning.
1
Introduction
1.1
Problem statement
Convex optimization has become ubiquitous in most quantitative disciplines of science. A common
trend in modern science is the increase in size of datasets, which drives the need for more efficient
optimization methods. Our goal is the generic minimization of composite functions of the form
minn ?(x) = F (x) + J(x) ,
(1.1)
x?R
where
(A.1) J : Rn ? R ? {+?} is a proper, closed and convex function;
(A.2) F is a convex and C 1,1 (Rn ) function whose gradient is ??Lipschitz continuous;
(A.3) Argmin ? 6= ?.
The class of problems (1.1) covers many popular non-smooth convex optimization problems encountered in various fields throughout science and engineering, including signal/image processing,
2
1
machine learning and classification. For instance, taking F = 2?
||y ? A ? || for some A ? Rm?n
and ? > 0, we recover the Lasso problem when J = || ? ||1 , the group Lasso for J = || ? ||1,2 , the
fused Lasso for J = ||D? ? ||1 with D = [DDIF , Id] and DDIF is the finite difference operator,
anti-sparsity regularization when J = || ? ||? , and nuclear norm regularization when J = || ? ||? .
1
The standard (non relaxed) version of Forward?Backward (FB) splitting algorithm [3] for solving
(1.1) updates to a new iterate xk+1 according to
xk+1 = prox?k J xk ? ?k ?F (xk ) ,
(1.2)
starting from any point x0 ? Rn , where 0 < ? ? ?k ? ? < 2/?. Recall that the proximity operator
is defined, for ? > 0, as
prox?J (x) = argminz?Rn
1.2
1
2? ||z
2
? x|| + J(z).
Contributions
In this paper, we present a unified local linear convergence analysis for the FB algorithm to solve
(1.1) when J is in addition partly smooth relative to a manifold M (see Definition 2.1 for details).
The class of partly smooth functions is very large and encompasses all previously discussed examples as special cases. More precisely, we first show that FB has a finite identification property,
meaning that after a finite number of iterations, say K, all iterates obey xk ? M for k ? K. Exploiting this property, we then show that after such a large enough number of iterations, xk converges
locally linearly. We characterize this regime and the rates precisely depending on the structure of the
active manifold M. In general, xk converges locally Q-linearly, and when M is an linear subspace,
the convergence becomes R-linear. Several experimental results on some of the problems discussed
above are provided to support our theoretical findings.
1.3
Related work
Finite support identification and local R-linear convergence of FB to solve the Lasso problem,
though in infinite-dimensional setting, is established in [4] under either a very restrictive injectivity
assumption, or a non-degeneracy assumption which is a specialization of ours (see (3.1)) to the `1
norm. A similar result is proved in [13], for F being a smooth convex and locally C 2 function
and J the `1 norm, under restricted injectivity and non-degeneracy assumptions. The `1 norm is a
partly smooth function and hence covered by our results. [1] proved Q-linear convergence of FB to
solve (1.1) for F satisfying restricted smoothness and strong convexity assumptions, and J being
a so-called convex decomposable regularizer. Again, the latter is a small subclass of partly smooth
functions, and their result is then covered by ours. For example, our framework covers the total
variation (TV) semi-norm and `? -norm regularizers which are not decomposable.
In [15, 16], the authors have shown finite identitification of active manifolds associated to partly
smooth functions for various algorithms, including the (sub)gradient projection method, Newtonlike methods, the proximal point algorithm. Their work extends that of e.g. [28] on identifiable
surfaces from the convex case to a general non-smooth setting. Using these results, [14] considered
the algorithm [25] to solve (1.1) where J is partly smooth, but not necessarily convex and F is
C 2 (Rn ), and proved finite identitification of the active manifold. However, the convergence rate
remains an open problem in all these works.
1.4
Notations
Suppose M ? Rn is a C 2 -manifold around x ? Rn , denote TM (x) the tangent space of M at
x ? Rn . The tangent model subspace is defined as
?
Tx = Lin ?J(x) ,
where Lin(C) is the linear hull of the convex set C ? Rn . For a linear subspace V , we denote PV
the orthogonal projector onto V and for a matrix A ? Rm?n , AV = A ? PV . Define the generalized
sign vector
ex = PTx ?J(x) .
For a convex set C ? Rn , ri(C) denotes its relative interior, i.e. the interior relative to its affine hull.
2
2
Partial smoothness
In addition to (A.1), our central assumption is that J is a partly smooth function. Partial smoothness
of functions is originally defined in [17]. Our definition hereafter specializes it to the case of proper
closed convex functions.
Definition 2.1. Let J be a proper closed convex function such that ?J(x) 6= ?. J is partly smooth
at x relative to a set M containing x if
(1) (Smoothness) M is a C 2 -manifold around x and J restricted to M is C 2 around x.
(2) (Sharpness) The tangent space TM (x) is Tx .
(3) (Continuity) The set?valued mapping ?J is continuous at x relative to M.
In the following, the class of partly smooth functions at x relative to M is denoted as PSx (M).
When M is an affine manifold, then M = x + Tx , and we denote this subclass as PSAx (x + Tx ).
When M is a linear manifold, then M = Tx , and we denote this subclass as PSLx (Tx ).
Capitalizing on the results of [17], it can be shown that under mild transversality assumptions, the
set of continuous convex partly smooth functions is closed under addition and pre-composition by a
linear operator. Moreover, absolutely permutation-invariant convex and partly smooth functions of
the singular values of a real matrix, i.e. spectral functions, are convex and partly smooth spectral
functions of the matrix [10].
It then follows that all the examples discussed in Section 1, including `1 , `1 ?`2 , `? , TV and nuclear
norm regularizers, are partly
smooth. In fact, the nuclear
norm is partly smooth at a matrix x relative
to the manifold M = x0 : rank(x0 ) = rank(x) . The first three regularizers are all part of the
class PSLx (Tx ), see Section 4 and [27] for details.
We now define a subclass of partly smooth functions where the active manifold is actually a subspace
and the generalized sign vector ex is locally constant.
Definition 2.2. J belongs to the class PSSx (Tx ) if and only if J ? PSAx (x+Tx ) or J ? PSLx (Tx )
and ex is constant near x, i.e. there exists a neighbourhood U of x such that ?x0 ? Tx ? U
ex0 = ex .
A typical family of functions that comply with this definition is that of partly polyhedral functions [26, Section 6.5], which includes the `1 and `? norms, and the TV semi-norm.
3
Local linear convergence of the FB method
In this section, we state our main result on finite identification and local linear convergence of FB.
Theorem 3.1. Assume that (A.1)-(A.3) hold. Suppose that the FB scheme is used to create a sequence xk which converges to x? ? Argmin ? such that J ? PSx? (Mx? ), F is C 2 near x? and
??F (x? ) ? ri ?J(x? ) .
(3.1)
Then we have the following holds,
(1) The FB scheme (1.2) has the finite identification property, i.e. there exists K ? 0, such that
for all k ? K, xk ? Mx? .
(2) Suppose moreover that ?? > 0 such that
PT ?2 F (x? )PT ?Id,
where T := Tx? . Then for all k ? K, the following holds.
(3.2)
(i) Q-linear convergence: if 0 < ? ? ?k ? ?? < min 2?? ?2 , 2? ?1 , then given any
1 > ? > ?e,
||xk+1 ? x? || ? ?||xk ? x? ||,
2
where ?e = max q(?), q(?
? ) ? [0, 1[ and q(?) = 1 ? 2?? + ? 2 ? 2 .
3
(ii) R-linear convergence: if J ? PSAx? (x? + T ) or J ? PSLx? (T ), then for 0 < ? ?
?k ? ?? < min 2?? ?2 , 2? ?1 , where ? ? ? is the Lipzchitz constant of PT ?F PT ,
then
||xk+1 ? x? || ? ?k ||xk ? x? ||,
where ?2k = 1 ? 2??k + ? 2 ?k2 ? [0, 1[. Moreover, if
the optimal linear rate can be achieved is
p
?? = 1 ? ?2 /? 2 .
?
?2
? ?? and set ?k ?
?
?2 ,
then
Remark 3.2.
? The non-degeneracy assumption in (3.1) can be viewed as a geometric generalization of strict complementarity of non-linear programming. Building on the arguments
of [16], it turns out that it is almost a necessary condition for finite identification of Mx? .
? Under the non-degeneracy and local strong convexity assumptions (3.1)-(3.2), one can actually show that x? is unique by extending the reasoning in [26].
? For F = G ? A, where G satisfies (A.2), assumption (3.2) and the constant ? can be
restated in terms of local strong convexity of G and restricted injectivity of A on T , i.e.
Ker(A) ? T = {0}.
2
? When F = 21 ||y ? A ? || , not only the minimizer x? is unique, but also the rates in Theorem 3.1 can be refined further as the gradient operator ?F becomes linear.
? Partial smoothness guarantees that xk arrives the active manifold in finite time, hence raising the hope of acceleration using second-order information. For instance, one can think
of turning to geometric methods along the manifold Mx? , where faster convergence rates
can be achieved. This is also the motivation behind the work of e.g. [19].
When J ? PSSx? (T ), it turns out that the restricted convexity assumption (3.2) of Theorem 3.1 can
be removed in some cases, but at the price of less sharp rates.
Theorem 3.3. Assume that (A.1)-(A.3) hold. For x? ? Argmin ?, suppose that
J ? PSSx? (Tx? ),
(3.1) is fulfilled, and there exists a subspace V such that Ker PT ?2 F (x)PT = V for any x ?
B (x? ), > 0. Let the FB scheme be used to create a sequence xk that converges to x? with
0 < ? ? ?k ? ?? < min 2?? ?2 , 2? ?1 , where ? > 0 (see the proof). Then there exists a constant
C > 0 and ? ? [0, 1[ such that for all k large enough
||xk ? x? || ? C?k .
A typical example where this result applies is for F = G ? A with G locally strongly convex, in
which case V = Ker(AT ).
4
Numerical experiments
In this section, we describe some examples to demonstrate the applicability of our results. More
precisely, we consider solving
2
minn 12 ||y ? Ax|| + ?J(x)
(4.1)
x?R
where y ? Rm is the observation, A : Rn ? Rm , ? is the tradeoff parameter, and J is either the
`1 -norm, the `? -norm, the `1 ? `2 -norm, the TV semi-norm or the nuclear norm.
Example 4.1 (`1 -norm). For x ? Rn , the sparsity promoting `1 -norm [8, 23] is
J(x) =
Pn
i=1 |xi |.
It can verified that J is a polyhedral norm, and thus J ? PSSx (Tx ) for the model subspace
M = Tx = u ? Rn : supp(u) ? supp(x) , and ex = sign(x).
The proximity operator of the `1 -norm is given by a simple soft-thresholding.
4
Example 4.2 (`1 ?`2 -norm). The `1 ?`2 -norm is usually used to promote group-structured
S sparsity
[29]. Let the support of x ? Rn be divided into non-overlapping blocks B such that b?B b =
{1, . . . , n}. The `1 ? `2 -norm is given by
J(x) = ||x||B =
P
b?B ||xb ||,
where xb = (xi )i?b ? R|b| . || ? ||B in general is not polyhedral, yet partly smooth relative to the
linear manifold
M = Tx = u ? Rn : suppB (u) ? suppB (x) , and ex = N (xb ) b?B ,
S
where suppB (x) =
b : xb 6= 0 , N (x) = x/||x|| and N (0) = 0. The proximity operator of the
`1 ? `2 norm is given by a simple block soft-thresholding.
Example 4.3 (Total Variation). As stated in the introduction, partial smoothness is preserved under
pre-composition by a linear operator. Let J0 be a closed convex function and D is a linear operator.
Popular examples are the TV semi-norm in which case J0 = || ? ||1 and D? = DDIF is a finite
difference approximation of the derivative [22], or the fused Lasso for D = [DDIF , Id] [24].
If J0 ? PSD? x (M0 ), then it is shown in [17, Theorem 4.2] that under an appropriate transversality
condition, J ? PSx (M) where
M = u ? Rn : D? u ? M0 .
In particular, for the case of the TV semi-norm, we have J ? PSSx (Tx ) with
M = Tx = u ? Rn : supp(D? u) ? I and ex = PTx Dsign(D? x)
where I = supp(D? x). The proximity operator for the 1D TV, though not available in closed form,
can be obtained efficiently using either the taut string algorithm [11] or the graph cuts [7].
Example 4.4 (Nuclear norm). Low-rank is the spectral extension of vector sparsity to matrixvalued data x ? Rn1 ?n2 , i.e. imposing the sparsity on the singular values of x. Let x = U ?x V ? a
reduced singular value decomposition (SVD) of x. The nuclear norm of a x is defined as
J(x) = ||x||? =
Pr
i=1 (?x )i ,
where rank(x) = r. It has been used for instance as SDP convex relaxation for many problems
including in machine learning [2, 12], matrix completion [21, 5] and phase retrieval [6].
It can be shown that the nuclear norm is partly smooth relative to the manifold [18, Example 2]
M = z ? Rn1 ?n2 : rank(z) = r .
The tangent space to M at x and ex are given by
TM (x) = z ? Rn1 ?n2 : z = U L? + M V ? , ?L ? Rn2 ?r , M ? Rn1 ?r , and ex = U V ? .
The proximity operator of the nuclear norm is just soft?thresholding applied to the singular values.
Recovery from random measurements In these examples, the forward observation model is
? ? N (0, ? 2 ),
y = Ax0 + ?,
(4.2)
where A ? Rm?n is generated uniformly at random from the Gaussian ensemble with i.i.d. zeromean and unit variance entries. The tested experimental settings are
(a) `1 -norm m = 48 and n = 128, x0 is 8-sparse;
(b) Total Variation m = 48 and n = 128, (DDIF x0 ) is 8-sparse;
(c) `? -norm m = 123 and n = 128, x0 has 10 saturating entries;
(d) `1 ? `2 -norm m = 48 and n = 128, x0 has 2 non-zero blocks of size 4;
(e) Nuclear norm m = 1425 and n = 2500, x0 ? R50?50 and rank(x0 ) = 5.
5
The number of measurements is chosen sufficiently large, ? small enough and ? of the order of
? so that [27, Theorem 1] applies, yielding that the minimizer of (4.1) is unique and verifies the
non-degeneracy and restricted strong convexity assumptions (3.1)-(3.2).
The convergence profile of ||xk ?x? || are depicted in Figure 1(a)-(e). Only local curves after activity
identification are shown. For `1 , TV and `? , the predicted rate coincides exactly with the observed
one. This is because these regularizers are all partly polyhedral gauges, and the data fidelity is
quadratic, hence making the predictions of Theorem 3.1(ii) exact. For the `1 ? `2 -norm, although
its active manifold is still a subspace, the generalized sign vector ek is not locally constant, which
entails that the the predicted rate of Theorem 3.1(ii) slightly overestimates the observed one. For the
nuclear norm, whose active manifold is not linear. Thus Theorem 3.1(i) applies, and the observed
and predicted rates are again close.
TV deconvolution In this image processing example, y is a degraded image generated according to the same forward model as (4.1), but now A is a convolution with a Gaussian kernel. The
anisotropic TV regularizer is used. The convergence profile is shown in Figure 1(f). Assumptions
(3.1)-(3.2) are checked a posteriori. This together with the fact that the anisotropic TV is polyhedral
justifies that the predicted rate is again exact.
?2
10
theoretical
practical
theoretical
practical
?2
10
0
10
?2
10
?6
10
10
k x k ? x ?k
k x k ? x ?k
?4
k x k ? x ?k
theoretical
practical
10
?4
?6
10
?4
10
?6
10
?8
?8
10
10
?8
10
?10
?10
10
?10
10
380
400
420
440
460
480
500
450
500
550
600
k
650
700
750
10
800
1000
(a) `1 (Lasso)
3000
4000
theoretical
practical
?2
6000
7000
8000
(c) `? -norm
(b) TV semi-norm
2
0
10
5000
k
10
10
theoretical
practical
?2
theoretical
practical
0
10
10
?2
k x k ? x ?k
10
?6
10
k x k ? x ?k
?4
k x k ? x ?k
2000
k
?4
10
?6
10
10
?4
10
?6
10
?8
?8
10
10
?10
?8
10
?10
?10
10
10
10
350
400
450
500
250
300
350
k
(d) `1 ? `2 -norm
400
450
k
(e) Nuclear norm
500
50
100
150
200
250
300
k
(f) TV deconvolution
Figure 1: Observed and predicted local convergence profiles of the FB method (1.2) in terms of
||xk ? x? || for different types of partly smooth functions. (a) `1 -norm; (b) TV semi-norm; (c) `? norm; (d) `1 ? `2 -norm; (e) Nuclear norm; (f) TV deconvolution.
5
Proofs
Lemma 5.1. Suppose that J ? PSx (M). Then for any x0 ? M ? U , where U is a neighbourhood
of x, the projector PM (x0 ) is uniquely valued and C 1 around x, and thus
x0 ? x = PTx (x0 ? x) + o ||x0 ? x|| .
If J ? PSAx (x + Tx ) or J ? PSLx (Tx ), then
x0 ? x = PTx (x0 ? x).
Proof. Partial smoothness implies that M is a C 2 ?manifold around x, then PM (x0 ) is uniquely
valued [20] and moreover C 1 near x [18, Lemma 4]. Thus, continuous differentiability shows
x0 ? x = PM (x0 ) ? PM (x) = DPM (x)(x ? x0 ) + o(||x ? x0 ||).
6
where DPM (x) is the derivative of PM at x. By virtue of [18, Lemma 4] and the sharpness propoerty of J, this derivative is given by
DPM (x) = PTM (x) = PTx ,
The case where M is affine or linear is immediate. This conlcudes the proof.
Proof of Theorem 3.1.
1. Classical convergence results of the FB scheme, e.g. [9], show that xk converges to some x? ?
Argmin ? 6= ? by assumption (A.3). Assumptions (A.1)-(A.2) entail that (3.1) is equivalent
to 0 ? ri ? ?(x? ) . Since F ? C 2 around x? , the smooth perturbation rule of partly smooth
functions [17, Corollary 4.7] ensures that ? ? PSx? (M). By definition of xk+1 , we have
1
?k Gk (xk ) ? Gk (xk+1 ) ? ??(xk+1 ).
where Gk = Id ? ?k ?F . By Baillon-Haddad theorem, Gk is non-expansive, hence
dist 0, ??(xk+1 ) ? ?1k ||Gk (xk ) ? Gk (xk+1 )|| ? ?1k ||xk ? xk+1 ||.
Since lim inf ?k = ? > 0, we obtain dist 0, ??(xk+1 ) ? 0. Owing to assumptions (A.1)(A.2), ? is subdifferentially continuous and thus ?(xk ) ? ?(x? ). Altogether, this shows that
the conditions of [15, Theorem 5.3] are fulfilled, whence the claim follows.
2. Take K > 0 sufficiently large such that for all k ? K, xk ? Mx? and xk ? B (x? ).
(i) Since prox?k J is firmly non-expansive, hence non-expansive, we have
||xk+1 ? x? || = ||prox?k J Gk xk ? prox?k J Gk x? || ? ||Gk xk ? Gk x? ||.
(5.1)
By virtue of Lemma 5.1, we have xk ? x? = PT (xk ? x? ) + o(||xk ? x? ||). This, together
with local C 2 smoothness of F and Lipschitz continuity of ?F entails
hxk ? x? , ?F (xk ) ? ?F (x? )i =
R1
hxk ? x? , ?2 F (x? + t(xk ? x? ))(xk ? x? )idt
R1
2
= 0 hPT (xk ? x? ), ?2 F (x? + t(xk ? x? ))PT (xk ? x? )idt + o ||xk ? x? ||
2
2
? ?||xk ? x? || + o ||xk ? x? || .
(5.2)
0
Since (3.2) holds and ?2 F (x) depends continuously on x, there exists > 0 such that
PT ?2 F (x)PT ?Id, ?x ? B (x? ). Thus, classical development of the right hand side of
(5.1) yields
2
2
||xk+1 ? x? || ? ||Gk xk ? Gk x? || = ||(xk ? x? ) ? ?k (?F (xk ) ? ?F (x? ))||
2
2
2
= ||xk ? x? || ? 2?k hxk ? x? , ?F (xk ) ? ?F (x? )i + ?k2 ||?F (xk ) ? ?F (x? )||
2
2
2
2
? ||xk ? x? || ? 2?k ?||xk ? x? || + ?k2 ? 2 ||xk ? x? || + o ||xk ? x? ||
2
2
= 1 ? 2??k + ? 2 ?k2 ||xk ? x? || + o ||xk ? x? || .
(5.3)
Taking the lim sup in this inequality gives
2
2
lim sup ||xk+1 ? x? || /||xk ? x? || ? q(?k ) = 1 ? 2??k + ? 2 ?k2 .
(5.4)
k?+?
It is clear that for 0 < ? ? ? ? ?? < min 2?? ?2 , 2? ?1 , q(?) ? [0, 1[, and q(?) ? ?e2 =
max q(?), q(?
? ) . Inserting this in (5.4), and using classical arguments yields the result.
(ii) We give the proof for M = T , that for M = x? + T is similar. Since xk and x? belong to
T , from xk+1 = prox?k J (Gk xk ) we have
Gk xk ? xk+1 ? ?k ?J(xk+1 ) ? xk+1 = PT Gk xk ? ?k ?J(xk+1 ) = PT Gk xk ? ?k ek+1 .
Similarily, we have x? = PT Gk x? ? ?k e? . We then arrive at
(xk+1 ? x? ) + ?k (ek+1 ? e? ) = (xk ? x? ) ? ?k PT ?F (PT xk ) ? PT ?F (PT x? ) . (5.5)
7
Moreover, maximal monotonicity of ?k ?J gives
||(xk+1 ? x? ) + ?k (ek+1 ? e? )||
2
2
2
2
= ||xk+1 ? x? || + 2hxk+1 ? x? , ?k (ek+1 ? e? )i + ?k ||ek+1 ? e? || ? ||xk+1 ? x? || .
It is straightforward to see that now, (5.2) becomes
2
hxk ? x? , PT ?F (PT xk ) ? PT ?F (PT x? )i ? ?||xk ? x? || .
Let ? be the Lipschitz constant of PT ?F PT . Obviously ? ? ?. Developing ||PT (Gk xk ?
2
Gk x? )|| similarly to (5.3) we obtain
2
2
2
||xk+1 ? x? || ? 1 ? 2??k + ? 2 ?k2 ||xk ? x? || = ?2k ||xk ? x? || ,
where ?k ? [0, 1[ for 0 < ? ? ?k ? ?? < min 2?/? 2 , 2/? . ?k is minimized at ??2 with the
proposed optimal rate whenever it obeys the given upper-bound.
Proof of Theorem 3.3. Arguing similarly to the proof of Theorem 3.1(ii), and using in addition that
e? = ex? is locally constant, we get
xk+1 ? x? = (xk ? x? ) ? ?k PT ?F (PT xk ) ? PT ?F (PT x? )
= (xk ? x? ) ? ?k
R1
0
PT ?2 F (x? + t(xk ? x? ))PT (xk ? x? )dt,
Denote Ht = PT ?2 F (x? + t(xk ? x? ))PT 0. Using that Ht is self-adjoint, we have
PV xk+1 = PV xk .
Since xk ? x? , it follows that PV xk = PV x? for all k sufficiently large. Observing that xk ? x? =
PV ? (xk ? x? ) for all large k, we get
xk+1 ? x? = xk ? x? ? ?k
R1
0
PV ? Ht PV ? (xk ? x? )dt.
1/2
Observe that V ? ? T . By definition, Bt = Ht PV ? is injective, and therefore, ?? > 0 such that
2
2
||Bt x|| > ?||x|| for all x 6= 0 and t ? [0, 1]. We then have
||xk+1 ? x? ||
2
2
R1
2
0
1
= ||xk ? x? || ? 2?k
= ||xk ? x? || ? 2?k
R
0
2
hxk ? x? , BtT Bt (xk ? x? )idt + ?k2 ||PV ? PT ?F (xk ) ? ?F (x? ) ||
2
2
||Bt (xk ? x? )|| dt + ?k2 ||PV ? PT || ||?F (xk ) ? ?F (x? )||
2
2
2
2
2
2
2
= ||xk ? x? || ? 2?k ?||xk ? x? || + ?k2 ||PT PV ? || ||?F (xk ) ? ?F (x? )||
2
? ||xk ? x? || ? 2?k ?||xk ? x? || + ?k2 ? 2 ||PV ? || ||PV ? (xk ? x? )||
2
2
2
2
2
? ||xk ? x? || ? 2?k ?||xk ? x? || + ?k2 ? 2 ||xk ? x? || = ?2k ||xk ? x? || .
It is easy to see again that ?k ? [0, 1[ whenever 0 < ? ? ?k ? ?? < min 2? ?1 , 2?? ?2 .
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9
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absolutely:1 tested:1 phenomenon:1 correlated:1 |
4,706 | 5,261 | Simple MAP Inference via Low-Rank Relaxations
Roy Frostig?, Sida I. Wang,? Percy Liang, Christopher D. Manning
Computer Science Department, Stanford University, Stanford, CA, 94305
{rf,sidaw,pliang}@cs.stanford.edu, [email protected]
Abstract
We focus on the problem of maximum a posteriori (MAP) inference in Markov
random fields with binary variables and pairwise interactions. For this common
subclass of inference tasks, we consider low-rank relaxations that interpolate between the discrete problem and its full-rank semidefinite relaxation. We develop
new theoretical bounds studying the effect of rank, showing that as the rank grows,
the relaxed objective increases but saturates, and that the fraction in objective value
retained by the rounded discrete solution decreases. In practice, we show two algorithms for optimizing the low-rank objectives which are simple to implement, enjoy
ties to the underlying theory, and outperform existing approaches on benchmark
MAP inference tasks.
1
Introduction
Maximum a posteriori (MAP) inference in Markov random fields (MRFs) is an important problem
with abundant applications in computer vision [1], computational biology [2], natural language
processing [3], and others. To find MAP solutions, stochastic hill-climbing and mean-field inference
are widely used in practice due to their speed and simplicity, but they do not admit any formal
guarantees of optimality. Message passing algorithms based on relaxations of the marginal polytope
[4] can offer guarantees (with respect to the relaxed objective), but require more complex bookkeeping.
In this paper, we study algorithms based on low-rank SDP relaxations which are both remarkably
simple and capable of guaranteeing solution quality.
Our focus is on MAP in a restricted but common class of models, namely those over binary variables
coupled by pairwise interactions. Here, MAP can be cast as optimizing a quadratic function over
the vertices of the n-dimensional hypercube: maxx2{ 1,1}n xT Ax. A standard optimization strategy
is to relax this integer quadratic program (IQP) to a semidefinite program (SDP), and then round
the relaxed solution to a discrete one achieving a constant factor approximation to the IQP optimum
[5, 6, 7]. In practice, the SDP can be solved efficiently using low-rank relaxations [8] of the form
maxX2Rn?k tr(X > AX).
The first part of this paper is a theoretical study of the effect of the rank k on low-rank relaxations of
the IQP. Previous work focused on either using SDPs to solve IQPs [5] or using low-rank relaxations
to solve SDPs [8]. We instead consider the direct link between the low-rank problem and the IQP. We
show that as k increases, the gap between the relaxed low-rank objective and the SDP shrinks, but
vanishes as soon as k rank(A); our bound adapts to the
p problem A and can thereby be considerably
better than the typical data-independent bound of O( n) [9, 10]. We also show that the rounded
objective shrinks in ratio relative to the low-rank objective, but at a steady rate of ?(1/k) on average.
This result relies on the connection we establish between IQP and low-rank relaxations. In the end,
our analysis motivates the use of relatively small values of k, which is advantageous from both a
solution quality and algorithmic efficiency standpoint.
?
Authors contributed equally.
1
The second part of this paper explores the use of very low-rank relaxation and randomized rounding
(R3 ) in practice. We use projected gradient and coordinate-wise ascent for solving the R3 relaxed
problem (Section 4). We note that R3 interfaces with the underlying problem in an extremely simple
way, much like Gibbs sampling and mean-field: only a black box implementation of x 7! Ax is
required. This decoupling permits users to customize their implementation based on the structure
of the weight matrix A: using GPUs for dense A, lists for sparse A, or much faster specialized
algorithms for A that are Gaussian filters [11]. In contrast, belief propagation and marginal polytope
relaxations [2] need to track messages for each edge or higher-order clique, thereby requiring more
memory and a finer-grained interface to the MRF that inhibits flexibility and performance.
Finally, we introduce a comparison framework for algorithms via the x 7! Ax interface, and use it to
compare R3 with annealed Gibbs sampling and mean-field on a range of different MAP inference
tasks (Section 5). We found that R3 often achieves the best-scoring results, and we provide some
intuition for our advantage in Section 4.1.
2
Setup and background
Notation We write Sn for the set of symmetric n ? n real matrices and S k for the unit sphere
{x 2 Rk : kxk2 = 1}. All vectors are columns unless stated otherwise. If X is a matrix, then
Xi 2 R1?k is its i?th row.
This section reviews how MAP inference on binary graphical models with pairwise interactions can
be cast as integer quadratic programs (IQPs) and approximately solved via semidefinite relaxations
and randomized rounding. Let us begin with the definition of an IQP:
Definition 2.1. Let A 2 Sn be a symmetric n ? n matrix. An (indefinite) integer quadratic program
(IQP) is the following optimization problem:
max
def
x2{ 1,1}n
IQP(x) = xT Ax
(1)
Solving (1) is NP-complete in general: the MAX-CUT problem immediately reduces to it [5]. With
an eye towards tractability, consider a first candidate relaxation: maxx2[ 1,1]n xT Ax. This relaxation
is always tight in that the maxima of the relaxed objective and original objective (1) are equal.1
Therefore it is just as hard to solve. Let us then replace each scalar xi 2 [ 1, 1] with a unit vector
Xi 2 Rk and define the following low-rank problem (LRP):
Definition 2.2. Let k 2 {1, . . . , n} and A 2 Sn . Define the low-rank problem LRPk by:
max
X2Rn?k
def
LRPk (X) = tr(X T AX)
subject to kXi k2 = 1, i = 1, . . . , n.
(2)
Note that setting Xi = [xi , 0, . . . , 0] 2 Rk recovers (1). More generally, we have a sequence of
successively looser relaxations as k increases. What we get in return is tractability. The LRPk
objective generally yields a non-convex problem, but if we take k = n, the objective can be rewritten
as tr(X > AX) = tr(AXX > ) = tr(AS), where S is a positive semidefinite matrix with ones on the
diagonal. The result is the classic SDP relaxation, which is convex:
max
S2Sn
def
SDP(S) = tr(AS)
subject to S ? 0, diag(S) = 1
(3)
Although convexity begets easy optimization in a theoretical sense, the number of variables in the
SDP is quadratic in n. Thus for large SDPs, we actually return to the low-rank parameterization (2).
Solving LRPk via simple gradient methods works extremely well in practice and is partially justified
by theoretical analyses in [8, 12].
1
Proof. WLOG, A ? 0 because adding to its diagonal merely adds a constant term to the IQP objective.
The objective is a convex function, as we can factor A = LLT and write xT LLT x = kLT xk22 , so it must be
maximized over its convex polytope domain at a vertex point.
2
To complete the picture, we need to convert the relaxed solutions X 2 Rn?k into integral solutions
x 2 { 1, 1}n of the original IQP (1). This can be done as follows: draw a vector g 2 Rk on the
unit sphere uniformly at random, project each Xi onto g, and take the sign. Formally, we write
x = rrd(X) to mean xi = sign(Xi ? g) for i = 1, . . . , n. This randomized rounding procedure was
pioneered by [5] to give the celebrated 0.878-approximation of MAX-CUT.
3
Understanding the relaxation-rounding tradeoff
The overall IQP strategy is to first relax the integer problem domain, then round back in to it. The
optimal objective increases in relaxation, but decreases in randomized rounding. How do these effects
compound? To guide our choice of relaxation, we analyze the effect that the rank k in (2) has on the
approximation ratio of rounded versus optimal IQP solutions.
More formally, let x? , X ? , and S ? denote global optima of IQP, of LRPk , and of SDP, respectively.
We can decompose the approximation ratio as follows:
1
E[IQP(rrd(X ? ))]
SDP(S ? )
LRPk (X ? )
E[IQP(rrd(X ? ))]
=
?
?
IQP(x? )
IQP(x? )
SDP(S ? )
LRPk (X ? )
|
{z
} | {z }
| {z }
|
{z
}
approximation ratio
constant
1
tightening ratio T (k)
(4)
rounding ratio R(k)
As k increases from 1, the tightening ratio T (k) increases towards 1 and the rounding ratio R(k)
decreases from 1. In this section, we lower bound T and R each in turn, thus lower-bounding the
approximation ratio as a function of k. Specifically, we show that T (k) reaches 1 at small k and that
R(k) falls as ?2 + ?( k1 ).
In practice, one cannot find X ? for general k with guaranteed efficiency (if we could, we would
simply use LRP1 to directly solve the original IQP). However, Section 5 shows empirically that
simple procedures solve LRPk well for even small k.
3.1
The tightening ratio T (k) increases
We now show that, under the assumption of A ? 0, the tightening ratio T (k) plateaus early and
that it approaches this plateau steadily. Hence, provided k is beyond this saturation point, and large
enough so that an LRPk solver is practically capable of providing near-optimal solutions, there is no
advantage in taking k larger.
First, T (k) is steadily bounded below. The following is a result of [13] (that also gives insight into
the theoretical worst-case hardness of optimizing LRPk ):
Theorem 3.1 ([13]). Fix A ? 0 and let S ? be an optimal SDP solution. There is a randomized
? feasible for LRPk such that E ? [LRPk (X)]
?
algorithm that, given S ? , outputs X
(k) ? SDP(S ? ),
X
where
?
?2
? ?
((k + 1)/2)
1
1
def 2
(k) =
=1
+o
(5)
k
(k/2)
2k
k
For example, (1) =
2
?
= 0.6366, (2) = 0.7854, (3) = 0.8488, (4) = 0.8836, (5) = 0.9054.2
? under any probability distribution, so the exisBy optimality of X ? , LRPk (X ? ) EX? [LRPk (X)]
tence of the algorithm in Theorem 3.1 implies that T (k)
(k).
Moreover, T (k) achieves its maximum of 1 at small k, and hence must strictly exceed the (k) lower
bound early on. We can arrive at this fact by bounding the rank of the SDP-optimal solution S ? .
?
This is because S ? factors into S ? = XX T , where X is in Rn?rank S and must be optimal since
?
LRPrank S ? (X) = SDP(S ). Without consideration of A, the following theorem uniformly bounds
this rank at well below n. The theorem was established independently by [9] and [10]:
?
Theorem 3.2 ([9,
p10]). Fix a weight matrix A. There exists an optimal solution S to SDP (3) such
?
that rank S ? 2n.
2
The function (k) generalizes the constant approximation factor 2/? = (1) with regards to the implications of the unique games conjecture: the authors show that no polynomial time algorithm can, in general,
approximate LRPk to a factor greater than (k) assuming P 6= NP and the UGC.
3
1
1.05
R(k)
lower bound
0.9
0.9
0.85
0.85
0.75
0.75
0.7
0.7
0.65
1
2
3
4
5
6
1600
0.8
0.8
R(k) (blue) is close to it 2/(? (k))
lower bound (red) across the small k.
1500
1400
1300
1200
1
2
3
4
5
6
1100
1
2
3
k
k
(a)
SDP
Max
Mean
Mean+Std
Mean?Std
1700
objective
0.95
0.65
1800
?(k)
T(k)=LRPk/SDP
0.95
objective
rounding ratio
1
4
5
6
k
(b) T?(k) (blue), the empirical tightening ra- (c) Rounded objective values vs. k: optimal
tio, clears its lower bound (k) (red) and hits SDP (cyan), best IQP rounding (green), and
its ceiling of 1 at k = 4.
mean IQP rounding ? (black).
100?100
Figure 1: Plots of quantities analyzed in Section 3, under A 2 R
whose entries are sampled
independently from a unit Gaussian. For this instance, the empirical post-rounding objectives are
shown at the right for completeness.
Hence we know already that the tightening ratio T (k) equals 1 by the time k reaches
p
2n.
Taking A into consideration, we can identify a class of problem instances for which T (k) actually
saturates at even smaller k. This result is especially useful when the rank of the weight matrix A is
known, or even under one?s control, while modeling the underlying optimization task:
Theorem 3.3. If A is symmetric, there is an optimal SDP solution S ? such that rank S ? ? rank A.
A complete proof is in Appendix A.1. Because adding to the diagonal of A is equivalent to merely
adding a constant to the objective of all problems considered, Theorem 3.3 can be strengthened:
Corollary 3.4. For any symmetric weight matrix A, there exists an optimal SDP solution S ? such
that rank S ? ? minu2Rn rank(A + diag(u)).
That is, changes to the diagonal of A that reduce its rank may be applied to improve the bound.
In summary,
p T (k) grows at least as fast as (k), from T (k) = 0.6366 at k = 1 to T (k) = 1 at
k = min{ 2n, minu2Rn rank(A + diag(u))}. This is validated empirically in Figure 1b.
3.2
The rounding ratio R(k) decreases
As the dimension k grows for row vectors Xi in the LRPk problem, the rounding procedure incurs a
larger expected drop in objective value. Fortunately, we can bound this drop. Even more fortunately,
the bound grows no faster than (k), exactly the steady lower bound for T (k). We obtain this result
with an argument based on the analysis of [13]:
Theorem 3.5. Fix a weight matrix A ? 0 and any LRPk -feasible X 2 Rn?k . The rounding ratio for
X is bounded below as
?
? ??
E[IQP(rrd(X))]
2
2
1
1
=
1+
+o
(6)
LRPk (X)
? (k)
?
2k
k
Note that X in the theorem need not be optimal ? the bound applies to whatever solution an LRPk
solver might provide. The proof, given in Appendix section A.1, uses Lemma 1 from [13], which is
based on the theory of positive definite functions on spheres [14]. A decrease in R(k) that tracks the
lower bound is observed empirically in Figure 1a.
In summary, considering only the steady bounds (Theorems 3.1 and 3.5), T will always rise opposite
to R at least at the same rate. Then, the added fact that T plateaus early (Theorem 3.2 and Corollary
3.4) means that T in fact rises even faster.
In practice, we would like to take k beyond 1 as we find that the first few relaxations give the optimizer
an increasing advantage in arriving at a good LRPk solution, close to X ? in objective. The rapid rise
of T relative to R just shown then justifies not taking k much larger if at all.
4
4
Pairwise MRFs, optimization, and inference alternatives
Having understood theoretically how IQP relates to low-rank relaxations, we now turn to MAP
inference and empirical evaluation. We will show that the LRPk objective can be optimized via
a simple interface to the underlying MRF. This interface then becomes the basis for (a) a MAP
inference algorithm based on very low-rank relaxations, and (b) a comparison to two other basic
algorithms for MAP: Gibbs sampling and mean-field variational inference.
A binaryPpairwise Markov
random field (MRF) models a function h over x 2 {0, 1}n given by
P
h(x) = i i (xi ) + i<j ?i,j (xi , xj ), where the i and ?i,j are real-valued functions. The MAP
inference problem asks for the variable assignment x? that maximizes the function h. An MRF
being binary-valued and pairwise allows the arbitrary factor tables i and ?i,j to be transformed
with straightforward algebra into weights A 2 Sn for the IQP. For the complete reduction, see
Appendix A.2.
We make Section 3 actionable by defining the randomized relaxation and rounding (R3 ) algorithm for
MAP via low-rank relaxations. The first step of this algorithm involves optimizing LRPk (2) whose
weight matrix encodes the MRF. In practice, MRFs usually have special structure, e.g., edge sparsity,
factor templates, and Gaussian filters [11]. To develop R3 as a general tool, we provide two interfaces
between the solver and MRF representation, both of which allow users to exploit special structure.
Left-multiplication (x 7! Ax) Assume a function F that implements left matrix multiplication by
the MRF matrix A. This suffices to compute the gradient of the relaxed objective: rX LRPk (X) =
2AX. We can optimize the relaxation using projected gradient ascent (PGA): alternate between
taking gradient steps and projecting back onto the feasible set (unit-normalizing the rows Xi if the
norm exceeds 1); see Algorithm 1. A user supplying a left-multiplication routine can parallelize its
implementation on a GPU, use sparse linear algebra, or efficiently implement a dense filter.
Row-product ((i, x) 7! Ai x) If the function F further provides left multiplication by any row of
A, we can optimize LRPk with coordinate-wise ascent (BCA). Fixing all but the i?th row of X gives
a function linear in Xi whose optimum is Ai X normalized to have unit norm.
Left-multiplication is suitable when one expects to parallelize multiplication, or exploit common
dense structure as with filters. Row product is suitable when one already expects to compute Ax
serially. BCA also eliminates the need for the step size scheme in PGA, thus reducing the number of
calls to the left-multiplication interface if this step size is chosen by line search.
X
random initialization in Rk?n
for t
1 to T do
if parallel then
X
?S k (X + 2?t AX) // Parallel update
else
for i
1 to n do
Xi
?S k (hAi , Xi) // Sweep update
for j
1 to M do
x(j)
sign(Xg), where g is a random vector from unit sphere S k (normalized Gaussian)
Output the x(j) for which the objective (x(j) )T Ax(j) is largest.
Algorithm 1: The full randomized relax-and-round (R3 ) procedure, given a weight matrix A;
?S k (?) is row normalization and ?t is the step size in the t?th iteration.
4.1
Comparison to Gibbs sampling and mean-field
The R3 algorithm affords a tidy comparison to two other basic MAP algorithms. First, it is iterative
and maintains a constant amount of state per MRF variable (a length k row vector). Using the
row-product interface, R3 under BCA sequentially sweeps through and updates each variable?s state
(row Xi ) while holding all others fixed. This interface bears a striking resemblance to (annealed)
Gibbs sampling and mean-field iterative updates [4, 15], which are popular due to their simplicity.
Table 1 shows how both can be implemented via the row-product interface.
5
Algorithm
Domain
Gibbs
Mean-field
R3
n
x 2 { 1, 1}
x 2 [ 1, 1]n
X 2 (S k )n
Sweep update
Parallel update
xi ? ?Z (exp(Ai x))
xi
tanh(Ai x)
Xi
?S k (Ai X)
x ? ?Z (exp(Ax))
x
tanh(Ax)
X
?S k (X + 2?t AX)
Table 1: Iterative updates for MAP algorithms that use constant state per MRF variable. ?S k denotes
`2 unit-normalization of rows and ?Z denotes scaling rows so that they sum to 1. The R3 sweep
update is not a gradient step, but rather the analytic maximum for the i?th row fixing the rest.
x1
1 (x1 )
10x1 x2
= x1
x2
2 (x2 )
= x2
"
1 0
1
A=
2 1
1
0
10
1
10
0
#
Figure 2: Consider the two variable MRF on the left (with x1 , x2 2 { 1, 1} for the factor expressions)
and its corresponding matrix A. Note x0 is clamped to 1 as per the reduction (A.2). The optimum is
x = [1, 1, 1]T with a value of xT Ax = 12. If Gibbs or LRP1 is initialized at x = [1, 1, 1]T , then
either one will be unlikely to transition away from its suboptimal objective value of 8 (as flipping
only one of x1 or x2 decreases the objective to 10). Meanwhile, LRP2 succeeds with probability 1
over random initializations. Suppose X = [1, 0; X1 ; X2 ] with X1 = X2 . Then the gradient update
is X1 = ?S 2 (A1 X) = ?S 2 (([1, 0] + 10X1 )/2), which always points towards X1? = X2? = [1, 0]
except in the 0-probability event that X1 = X2 = [ 1, 0] (corresponding to the poor initialization
of [1, 1, 1]T above). The gradient with respect to X1 at points along the unit circle is shown on
the right. The thick arrow represents an X1 ? [ 0.95, 0.3], and the gradient field shows that it will
iteratively update towards the optimum.
Using left-multiplication, R3 updates the state of all variables in parallel. Superficially, both Gibbs
and the iterative mean-field update can be parallelized in this way as well (Table 1), but doing
so incorrectly alters the their convergence properties. Nonetheless, [11] showed that a simple
modification works well in practice for mean-field, so we consider these algorithms for a complete
comparison.3
While Gibbs, mean-field, and R3 are similar in form, they differ in their per-variable state: Gibbs
maintains a number in { 1, 1} whereas R3 stores an entire vector in Rk . We can see by example
that the extra state can help R3 avoid local optima that ensnarls Gibbs. A single coupling edge in a
two-node MRF, described in Figure 2, gives intuition for the advantage of optimizing relaxations
over stochastic hill-climbing.
Another widely-studied family of MAP inference techniques are based on belief propagation or
relaxations of the marginal polytope [4]. For belief propagation, and even for the most basic of the
LP relaxations (relaxing to the local consistency polytope), one needs to store state for every edge in
addition to every variable. This demands a more complex interface to the MRF, introduces substantial
added bookkeeping for dense graphs, and is not amenable to techniques such as the filter of [11].
5
Experiments
We compare the algorithms from Table 1 on three benchmark MRFs and an additional artificial MRF.
We also show the effect of the relaxation k on the benchmarks in Figure 3.
Rounding in practice The theory of Section 3 provides safeguard guarantees by considering the
average-case rounding. In practice, we do far better than average since we take several roundings and
output the best. Similarly, Gibbs? output is taken as the best along its chain.
Budgets Our goal is to see how efficiently each method utilizes the same fixed budget of queries to
the function, so we fix the number queries to the left-multiplication function F of Section 4. A budget
jointly limits the relaxation updates and the number of random roundings taken in R3 . We charge
3
Later, in [16], the authors derive the parallel mean-field update as being that of a concave approximation to
the cross-entropy term in the true mean-field objective.
6
algo.
Gibbs
MF
R3
Gibbs
MF
R3
Gibbs
MF
R3
Gibbs
MF
R3
parallel sweep parallel sweep
low budget
high budget
seg (50)
8.35 (23)
8.36 (23)
8.39 (15)
7.4 (19)
7.4 (26)
7.4 (17)
7.07 (33)
7.03
(9)
7.09 (23)
6.78 (31)
6.75 (12)
6.8 (25)
dataset [name (# of instances)]
dbn (108)
grid40 (8) chain (300)
1.39
(30) 14.5 (7) .473
(37)
1.3
(7) 13.6 (1) .463
(39)
1.42
(71) 13.7 (0) .538 (296)
.826
(3) .843 (0) .124
(3)
1.16
(6) 11.3 (3)
.35
(50)
1.29
(99) 11.3 (5) .418 (282)
1.26
(42) 12.5 (7) .367
(85)
1.16
(4) 11.7 (1)
.33
(39)
1.28
(62) 11.9 (0) .398 (300)
.814
(2) 1.85 (0) .132
(11)
1.1
(2) 10.9 (2) .259
(47)
1.25 (104)
11 (6) .321 (296)
Table 2: Benchmark performance of algorithms in each comparison regime, in which the benchmarks
are held to different computational budgets that cap their access to the left-multiplication routine. The
score shown is an average relative gain in objective over the uniform-random baseline. Parenthesized
is the win count (including ties), and bold text highlights qualitatively notable successes.
seg
4
580
1.3
LRP
Max
Mean
Mean+Std
Mean?Std
1.25
1.2
objective
objective
540
1.4
LRP
Max
Mean
Mean+Std
Mean?Std
520
1.1
500
1
0.95
1
2
3
4
5
6
1.25
1.2
1.05
460
LRP
Max
Mean
Mean+Std
Mean?Std
1.3
1.15
480
grid
x 10
1.35
objective
560
4
dbn
x 10
1.15
1
2
3
4
k
k
5
6
1.1
1
2
3
4
5
6
k
Figure 3: Relaxed and rounded objectives vs. the rank k in an instance of seg, dbn, and grid40. Blue:
max of roundings. Red: value of LRPk . Black: mean of roundings (? ). The relaxation objective
increases with k, suggesting that increasingly good solutions are obtained by increasing k, in spite of
non-convexity (here we are using parallel updates, i.e. using R3 with PGA). The maximum rounding
also improves considerably with k, especially at first when increasing beyond k = 1.
R3 k-fold per use of F when updating, as it queries F with a k-row argument.4 Sweep methods are
charged once per pass through all variables.
We experiment with separate budgets for the sweep and parallel setup, as sweeps typically converge
more quickly. The benchmark is run under separate low and high budget regimes ? the latter more
than double the former to allow for longer-run effects to set in. In Table 2, the sweep algorithms? low
budget is 84 queries; the high budget is 200. The parallel low budget is 180; the high budget is 400.
We set R3 to take 20 roundings under low budgets and 80 under high ones, and the remaining budget
goes towards LRPk updates.
Datasets Each dataset comprises a family of binary pairwise MRFs. The sets seg, dbn, and grid40
are from the PASCAL 2011 Probabilistic Inference Challenge5 ? seg are small segmentation models
(50 instances, average 230 variables, 622 edges), dbn are deep belief networks (108 instances, average
920 variables, 54160 edges), and grid40 are 40x40 grids (8 instances, 1600 variables, 6240 or 6400
edges) whose edge weights outweigh their unaries by an order of magnitude. The chain set comprises
300 randomly generated 20-node chain MRFs with no unary potentials and random unit-Gaussian
edge weights ? it is principally an extension of the coupling two-node example (Figure 2), and serves
as a structural obverse to grid40 in that it lacks cycles entirely. Among these, the dbn set comprises
the largest and most edge-dense instances.
4
5
This conservatively disfavors R3 , as it ignores the possible speedups of treating length-k vectors as a unit.
http://www.cs.huji.ac.il/project/PASCAL/
7
Evaluation To aggregate across instances of a dataset, we measure the average improvement over
a simple baseline that, subject to the budget constraint, draws uniformly random vectors in { 1, 1}n
and selects the highest-scoring among them. Improvement over the baseline is relative: if z is the
solution objective and z 0 is that of the baseline, (z z 0 )/z 0 is recorded for the average. We also
count wins (including ties), the number of times a method obtains the best objective among the
competition. Baseline performance varies with budget so scores are incomparable across sweep and
parallel experiments.
In all experiments,
we use LRP4 , i.e. the width-4 relaxation. The R3 gradient step size scheme is
p
?t = 1/ t. In the parallel setting, mean-field updates are prone to large oscillations, so we smooth
the update with the current point: x
(1 ?)x + ? tanh(Ax). Our experiments set ? = 0.5. Gibbs
is annealed from an initial temperature of 10 down to 0.1. These settings were tuned towards the
benchmarks using a few arbitrary instances from each dataset.
Results are summarized in Table 2. All methods fare well on the seg dataset and find solutions very
near the apparent global optimum. This shows that the rounding scheme of R3 , though elementary,
is nonetheless capable of recovering an actual MAP point. On grid40, R3 is competitive but not
outstanding, and on chain it is a clear winner. Both datasets have edge potentials that dominate
the unaries, but the cycles in the grid help break local frustrations that occur in chain where they
prevents Gibbs from transitioning. On dbn ? the more difficult task grounded in a real model ? R3
outperforms the others by a large margin.
Figure 3 demonstrates that relaxation beyond the quadratic program maxx2[ 1,1]xT Ax (i.e. k = 1) is
crucial, both for optimizing LRPk and for obtaining a good maximum among roundings. Figure 4 in
the appendix visualizes the distribution of rounded objective values across different instances and
relaxations, illustrating that the difficulty of the problem can be apparent in the rounding distribution.
6
Related work and concluding remarks
In this paper, we studied MAP inference problems that can be cast as an integer quadratic program
over hypercube vertices (IQP). Relaxing the IQP to an SDP (3) and rounding back with rrd(?) was
introduced by Goemans and Williamson in the 1990s for MAX-CUT. It was generalized to positive
semidefinite weights shortly thereafter by Nesterov [6].
Separately, in the early 2000s, there was interest in scalably solving SDPs, though not with the
specific goal of solving the IQP. The low-rank reparameterization of an SDP, as in (2), was developed
by [8] and [12]. Recent work has taken this approach to large-scale SDP formulations of clustering,
embedding, matrix completion, and matrix norm optimization for regularization [17, 18]. Upper
bounds on SDP solutions in terms of problem size n, which help justify using a low rank relaxation,
have been known since the 1990s [9, 10].
The natural joint use of these ideas (IQP relaxed to SDP and SDP solved by low-rank relaxation) is
somewhat known. It was applied in a clustering experiment in [19], but no theoretical analysis was
given and no attention paid to rounding directly from a low-rank solution. The benefit of rounding
from low-rank was noticed in coarse MAP experiments in [20], but no theoretical backing was given
and no attention paid to coordinate-wise ascent or budgeted queries to the underlying model.
Other relaxation hierarchies have been studied in the MRF MAP context, namely linear program
(LP) relaxations given by hierarchies of outer bounds on the marginal polytope [21, 2]. They differ
from this paper?s setting in that they maintain state for every MRF clique configuration ? an approach
that extends beyond pairwise MRFs but that scales with the number of factors (unwieldy versus a
large, dense binary pairwise MRF) and requires fine-grained access to the MRF. Sequences of LP and
SDP relaxations form the Sherali-Adams and Lasserre hierarchies, respectively, whose relationship is
discussed in [4] (Section 9). The LRPk hierarchy sits at a lower level: between the IQP (1) and the
first step of the Lasserre hierarchy (the SDP (3)).
From a practical point of view, we have presented an algorithm very similar in form to Gibbs sampling
and mean-field. This provides a down-to-earth perspective on relaxations within the realm of scalable
and simple inference routines. It would be interesting to see if the low-rank relaxation ideas from this
paper can be adapted to other settings (e.g., for marginal inference). Conversely, the rich literature
on the Lasserre hierarchy may offer guidance in extending the low-rank semidefinite approach (e.g.,
beyond the binary pairwise setting).
8
References
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9
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4,707 | 5,262 | Inferring synaptic conductances from spike trains
under a biophysically inspired point process model
E. J. Chichilnisky
Department of Neurosurgery
Hansen Experimental Physics Laboratory
Stanford University
[email protected]
Kenneth W. Latimer
The Institute for Neuroscience
The University of Texas at Austin
[email protected]
Fred Rieke
Department of Physiology and Biophysics
Howard Hughes Medical Institute
University of Washington
[email protected]
Jonathan W. Pillow
Princeton Neuroscience Institute
Department of Psychology
Princeton University
[email protected]
Abstract
A popular approach to neural characterization describes neural responses in terms
of a cascade of linear and nonlinear stages: a linear filter to describe stimulus
integration, followed by a nonlinear function to convert the filter output to spike
rate. However, real neurons respond to stimuli in a manner that depends on the
nonlinear integration of excitatory and inhibitory synaptic inputs. Here we introduce a biophysically inspired point process model that explicitly incorporates
stimulus-induced changes in synaptic conductance in a dynamical model of neuronal membrane potential. Our work makes two important contributions. First, on
a theoretical level, it offers a novel interpretation of the popular generalized linear
model (GLM) for neural spike trains. We show that the classic GLM is a special
case of our conductance-based model in which the stimulus linearly modulates excitatory and inhibitory conductances in an equal and opposite ?push-pull? fashion.
Our model can therefore be viewed as a direct extension of the GLM in which we
relax these constraints; the resulting model can exhibit shunting as well as hyperpolarizing inhibition, and time-varying changes in both gain and membrane time
constant. Second, on a practical level, we show that our model provides a tractable
model of spike responses in early sensory neurons that is both more accurate and
more interpretable than the GLM. Most importantly, we show that we can accurately infer intracellular synaptic conductances from extracellularly recorded
spike trains. We validate these estimates using direct intracellular measurements
of excitatory and inhibitory conductances in parasol retinal ganglion cells. The
stimulus-dependence of both excitatory and inhibitory conductances can be well
described by a linear-nonlinear cascade, with the filter driving inhibition exhibiting opposite sign and a slight delay relative to the filter driving excitation. We
show that the model fit to extracellular spike trains can predict excitatory and inhibitory conductances elicited by novel stimuli with nearly the same accuracy as
a model trained directly with intracellular conductances.
1
Introduction
The point process generalized linear model (GLM) has provided a useful and highly tractable tool
for characterizing neural encoding in a variety of sensory, cognitive, and motor brain areas [1?5].
1
inhibitory filter
nonlinearity Poisson
excitatory filter
post-spike filter
stimulus
spikes
Figure 1: Schematic of conductance-based spiking model.
However, there is a substantial gap between descriptive statistical models like the GLM and more
realistic, biophysically interpretable neural models. Cascade-type statistical models describe input
to a neuron in terms of a set of linear (and sometimes nonlinear) filtering steps [6?11]. Real neurons,
on the other hand, receive distinct excitatory and inhibitory synaptic inputs, which drive conductance
changes that alter the nonlinear dynamics governing membrane potential. Previous work has shown
that excitatory and inhibitory conductances in retina and other sensory areas can exhibit substantially
different tuning. [12, 13].
Here we introduce a quasi-biophysical interpretation of the generalized linear model. The resulting
interpretation reveals that the GLM can be viewed in terms of a highly constrained conductancebased model. We expand on this interpretation to construct a more flexible and more plausible
conductance-based spiking model (CBSM), which allows for independent excitatory and inhibitory
synaptic inputs. We show that the CBSM captures neural responses more accurately than the standard GLM, and allows us to accurately infer excitatory and inhibitory synaptic conductances from
stimuli and extracellularly recorded spike trains.
2
A biophysical interpretation of the GLM
The generalized linear model (GLM) describes neural encoding in terms of a cascade of linear,
nonlinear, and probabilistic spiking stages. A quasi-biological interpretation of GLM is known as
?soft threshold? integrate-and-fire [14?17]. This interpretation regards the linear filter output as a
membrane potential, and the nonlinear stage as a ?soft threshold? function that governs how the
probability of spiking increases with membrane potential, specifically:
Vt
rt
yt |rt
= k> xt
= f (Vt )
? Poiss(rt ?t ),
(1)
(2)
(3)
where k is a linear filter mapping the stimulus xt to the membrane potential Vt at time t, a fixed
nonlinear function f maps Vt to the conditional intensity (or spike rate) rt , and spike count yt is a
Poisson random variable in a time bin of infinitesimal width ?t . The log likelihood is
log p(y1:T |x1:T , k) =
T
X
?rt ?t + yt log(rt ?t ) ? log(yt !).
(4)
t=1
The stimulus vector xt can be augmented to include arbitrary covariates of the response such as the
neuron?s own spike history or spikes from other neurons [2, 3]. In such cases, the output does not
form a Poisson process because spiking is history-dependent.
The nonlinearity f is fixed a priori. Therefore, the only parameters are the coefficients of the filter
k. The most common choice is exponential, f (z) = exp(z), corresponding to the canonical ?log?
link function for Poisson GLMs. Prior work [6] has shown that if f grows at least linearly and at
most exponentially, then the log-likelihood is jointly concave in model parameters ?. This ensures
that the log-likelihood has no non-global maxima, and gradient ascent methods are guaranteed to
find the maximum likelihood estimate.
2
3
Interpreting the GLM as a conductance-based model
A more biophysical interpretation of the GLM can be obtained by considering a single-compartment
neuron with linear membrane dynamics and conductance-based input:
dV
dt
= ?gl V + ge (t)(V ? Ee ) ? gi (t)(V ? Ei )
= ?(gl + ge (t) + gi (t))V + ge (t)Ee + gi (t)Ei
= ?gtot (t)V + Is (t),
(5)
where (for simplicity) we have set the leak current reversal potential to zero. The ?total conductance?
at time t is gtot (t) = gl +ge (t)+gi (t) and the ?effective input current? is Is (t) = ge (t)Ee +gi (t)Ei .
Suppose that the stimulus affects the neuron via the synaptic conductances ge and gi . It is then
natural to ask under which conditions, if any, the above model can correspond to a GLM. The
definition of a GLM requires the solution V (t) to be a linear (or affine) function of the stimulus.
This arises if the two following conditions are met:
1. Total conductance gtot is constant. Thus, for some constant c:
ge (t) + gi (t) = c.
(6)
2. The input Is is linear in x. This holds if we set:
ge (xt )
=
ke > xt + be
gi (xt )
=
k i > x t + bi .
(7)
We can satisfy these two conditions by setting ke = ?ki , so that the excitatory and inhibitory
conductances are driven by equal and opposite linear projections of the stimulus. This allows us to
rewrite the membrane equation (eq. 5):
dV
dt
= ?gtot V + (ke > xt + be )Ee + (ki > xt + bi )Ei
= ?gtot V + ktot > xt + btot ,
(8)
where gtot = gl + be + bi is the (constant) total conductance, ktot = ke Ee + ki Ei , and btot =
be Ee + bi Ei . If we take the initial voltage V0 to be btot , the equilibrium voltage in the absence of a
stimulus, then the solution to this differential equation is
Z t
Vt =
e?gtot (t?s) ktot > xs ds + btot
0
=
kleak ? (ktot > xt ) + btot
=
kglm > xt + btot ,
(9)
where kleak ? (ktot > xt ) denotes linear convolution of the exponential decay ?leak? filter kleak (t) =
e?gtot t with the linearly projected stimulus train, and kglm = ktot ? kleak is the ?true? GLM filter
(from eq. 1) that results from temporally convolving the conductance filter with the leak filter. Since
the membrane potential is a linear (affine) function of the stimulus (as in eq. 1), the model is clearly
a GLM.
Thus, to summarize, the GLM can be equated with a synaptic conductance-based dynamical model
in which the GLM filter k results from a common linear filter driving excitatory and inhibitory
synaptic conductances, blurred by convolution with an exponential leak filter determined by the
total conductance.
4
Extending GLM to a nonlinear conductance-based model
From the above, it is easy to see how to create a more realistic conductance-based model of neural
responses. Such a model would allow the stimulus tuning of excitation and inhibition to differ (i.e.,
allow ke 6= ?ki ), and would include a nonlinear relationship between x and the conductances to
3
preclude negative values (e.g., using a rectifying nonlinearity). As with the GLM, we assume that
the only source of stochasticity on the model is in the spiking mechanism: we place no additional
noise on the conductances or the voltage. This simplifying assumption allows us to perform efficient
maximum likelihood inference using standard gradient ascent methods.
We specify the membrane potential of the conductance-based point process model as follows:
dV
= ge (t)(Ee ? V ) + gi (t)(Ei ? V ) + gl (El ? V ),
dt
ge (t) = fe (ke > xt ),
gi (t) = fi (ki > xt ),
(10)
(11)
where fe and fi are nonlinear functions ensuring positivity of the synaptic conductances. In practice,
we evaluate V along a discrete lattice of points (t = 1, 2, 3, . . . T ) of width ?t . Assuming ge and gi
remain constant within each bin, the voltage equation becomes a simple linear differential equation
with the solution
Is (t)
Is (t)
?gtot (t)?t
+
(12)
V (t + 1) = e
V (t) ?
gtot (t)
gtot (t)
V (1) = El
(13)
gtot (t) = ge (t) + gi (t) + gl
(14)
Is (t) = ge (t)Ee + gi (t)Ei + gl El
(15)
The mapping from membrane potential to spiking is similar to that in the standard GLM (eq. 3):
rt
= f (V (t))
(V ? VT )
f (V ) = exp
VS
yt |rt ? Poiss(rt ?t ).
(16)
(17)
(18)
The voltage-to-spike rate nonlinearity f follows the form proposed by Mensi et al. [17], where VT
is a soft spiking threshold and VS determines the steepness of the nonlinearity. To account for
refractory periods or other spike-dependent behaviors, we simply augment the function to include a
GLM-like spike history term:
(V ? VT )
> hist
f (V ) = exp
+h y
(19)
VS
Spiking activity in real neurons influences both the membrane potential and the output nonlinearity.
We could include additional conductance terms that depend on either stimuli or spike history, such as
an after hyper-polarization current; this provides one direction for future work. For spatial stimuli,
the model can include a set of spatially distinct rectified inputs (e.g., as employed in [9]).
To complete the model, we must select a form for the conductance nonlinearities fe and fi . Although
we could attempt to fit these functions (e.g., as in [9, 18]), we fixed them to be the soft-rectifying
function:
fe (?), fi (?) = log(1 + exp(?)).
(20)
Fixing these nonlinearities improved the speed and robustness of maximum likelihood parameter
fitting. Moreover, we examined intracellularly recorded conductances and found that the nonlinear
mapping from linearly projected stimuli to conductance was well described by this function (see
Fig. 4).
The model parameters we estimate are {ke , ki , be , bi , h, gl , El }. We set the remaining model parameters to biologically plausible values: VT = ?70mV, VS = 4mV, Ee = 0mV, and Ei = ?80mV .
To limit the total number of parameters, we fit the linear filters ke and ki using a basis consisting of
12 raised cosine functions, and we used 10 raised cosine functions for the spike history filter [3].
The log-likelihood function for this model is not concave in the model parameters, which increases
the importance to selecting a good initialization point. We initialized the parameters by fitting a
simplified model which had only one conductance. We initialized the leak terms as El = ?70mV
and gl = 200. We assumed a single synaptic conductance with a linear stimulus dependence,
glin (t) = klin > xt (note that this allows for negative conductance values). We initialized this filter
4
filter fits
A
B
?5.27
x 104 fit to
Actual
test data
?0.4
log likelihood
0
?0.8
C
30
L2 error
weight
0.4
estimated filter errors
?5.29
20
?5.31
10
50
100 150 200
time (ms)
0
0
5
10
minutes of training data
?5.33
0
5
10
minutes of training data
Figure 2: Simulation results. (A) Estimates (solid traces) of excitatory (blue) and inhibitory (red) stimulus
filters from 10 minutes of simulated data. (Dashed lines indicate true filters). (B) The L2 norm between
the estimated input filters and the true filters (calculated in the low-dimensional basis) as a function of the
amount of training data. (C) The log-likelihood of the fit CBSM on withheld test data converges to the log
likelihood of the true model.
the GLM fit, and then numerically maximized the likelihood for klin . We then initialized the parameters for the complete model using ke = cklin and ki = ?cklin , where 0 < c ? 1, thereby
exploiting the mapping between the GLM and the CBSM. Although this initialization presumes that
excitation and inhibition have nearly opposite tuning, we found that standard optimization methods successfully converged to the true model parameters even when ke and ki had similar tuning
(simulation results not shown).
5
Results: simulations
To examine the estimation performance, we fit spike train data simulated from a CBSM with known
parameters (see Fig. 2). The simulated data qualitatively mimicked experimental datasets, with input
filters selected to reproduce the stimulus tuning of macaque ON parasol RGCs. The stimulus consisted of a one dimensional white noise signal, binned at a 0.1ms resolution, and filtered with a low
pass filter with a 60Hz cutoff frequency. The simulated cell produced a firing rate of approximately
32spikes/s. We validated our maximum likelihood fitting procedure by examining error in the fitted
parameters, and evaluating the log-likelihood on a held out five-minute test set. With increasing
amounts of training data, the parameter estimates converged to the true parameters, despite the fact
that the model does not have the concavity guarantees of the standard GLM.
To explore the CBSM?s qualitative response properties, we performed simulated experiments using
stimuli with varying statistics (see Fig. 3). We simulated spike responses from a CBSM with
fixed parameters to stimuli with different standard deviations. We then separately fit responses from
each simulation with a standard GLM. The fitted GLM filters exhibit shifts in both peak height
and position for stimuli with different variance. This suggests that the CBSM can exhibit gain
control effects that cannot be captured by a classic GLM with a spike history filter and exponential
nonlinearity.
6
Results: neural data
We fit the CBSM to spike trains recorded from 7 macaque ON parasol RGCs [12]. The spike trains
were obtained by cell attached recordings in response to full-field, white noise stimuli (identical to
the simulations above). Either 30 or 40 trials were recorded from each cell, using 10 unique 6 second
stimuli. After the spike trains were recorded, voltage clamp recordings were used to measure the
excitatory and inhibitory conductances to the same stimuli. We fit the model using the spike trains
for 9 of the stimuli, and the remaining trials were used to test model fit. Thus, the models were
effectively trained using 3 or 4 repeats of 54 seconds of full-field noise stimulus. We compared the
intracellular recordings to the ge and gi estimated from the CBSM (Fig. 5). Additionally, we fit the
measured conductances with the linear-nonlinear cascade model from the CBSM (the terms ge and
5
A
?lters at different contrasts
B
experimental data
0.25x contrast
0.5x
1x
2x
0.03
weight
0.02
0.01
0
?0.01
(Chander & Chichilnisky, 2001)
0
50
100
150
200
time (ms)
Figure 3: Qualitative illustration of model?s capacity to exhibit contrast adaptation (or gain control). (A)
The GLM filters fit to a fixed CBSM simulated at various levels of stimulus variance. (B) Filters fit to two
real retinal ganglion cells at two different levels of contrast (from [19]).
excitatory
measured conductance
50
40
40
data mean
30
30
20
20
10
10
0
0
?10
?30
-15
0
inhibitory
50
15
?10
30
?40
?20
0
20
40
filter output
Figure 4: Measured conductance vs. output of a fitted linear stimulus filter (gray points), for both the
excitatory (left) and inhibitory (right) conductances. The green diamonds correspond to a non-parametric
estimate of the conductance nonlinearity, given by the mean conductance for each bin of filter output. For
both conductances, the function is is well described by a soft-rectifying function (black trace).
gi in eq. 11) with a least-squares fit as an upper bound measure for the best possible conductance
estimate given our model. The CBSM correctly determined the stimulus tuning for excitation and
inhibition for these cells: inhibition is oppositely tuned and slightly delayed from excitation.
For the side-by-side comparison shown in Fig. 5, we introduced a scaling factor in the estimated
conductances in order to compare the conductances estimated from spike trains against recorded
conductances. Real membrane voltage dynamics depend on the capacitance of the membrane, which
we do not include because it introduces an arbitrary scaling factor that cannot be estimated by spike
alone. Therefore, for comparisons we chose a scaling factor for each cell independently. However,
we used a single scaling for the inhibitory and excitatory conductances. Additionally, we often had
2 or 3 repeated trials of the withheld stimulus, and we compared the model prediction to the average
conductance recorded for the stimulus. The CBSM predicted the synaptic conductances with an
average r2 = 0.54 for the excitatory and an r2 = 0.39 for the inhibitory input from spike trains,
compared to an average r2 = 0.72 and r2 = 0.59 for the excitatory and inhibitory conductances respectively from the least-squares fit directly to the conductances (Fig. 6). To summarize, using only
a few minutes of spiking data, the CBSM could account for 71% of the variance of the excitatory
input and 62% of the inhibitory input that can possibly be explained using the LN cascade model of
the conductances (eq. 11).
One challenge we discovered when fitting the model to real spike trains was that one filter, typically
ki , would often become much larger than the other filter. This resulted in one conductance becoming
dominant, which the intracellular recordings indicated was not the case. This was likely due to the
fact that we are data-limited when dealing with intracellular recordings: the spike train recordings
include only 1 minute of unique stimulus. To alleviate this problem, we added a penalty term, ?, to
6
Example Cell 1
estimated filters
0.2
(conductances)
(spikes)
fit to spikes:
ge
0.1
weight
estimated conductances fit to conductance:
0
?0.1
(conductances)
(spikes)
gi
10nS
?0.2
?0.3
fit to conductance:
fit to spikes:
0
50
100 150 200
time (ms)
250ms
Example Cell 2
estimated filters
0.2
ge
0.1
weight
fit to conductance:
fit to spikes:
0
10nS
?0.1
?0.2
fit to conductance:
fit to spikes:
gi
0
50
100 150 200
time (ms)
250ms
Figure 5: Two example ON parasol RGC responses to a full-field noise stimulus fit with the CBSM. The
model parameters were fit to spike train data, and then used to predict excitatory and inhibitory synaptic
currents recorded separately in response to novel stimuli. For comparison, we show predictions of an LN
model fit directly to the conductance data. Left: Linear kernels for the excitatory (blue) and inhibitory
(red) inputs estimated from the conductance-based model (light red, light blue) and estimated by fitting a
linear-nonlinear model directly to the measured conductances (dark red, dark blue). The filters represent a
combination of events that occur in the retinal circuitry in response to a visual stimulus, and are primarily
shaped by the cone transduction process. Right: Conductances predicted by our model on a withheld test
stimulus. Measured conductances (black) are compared to the predictions from the CBSM filters (fit to
spiking data) and an LN model (fit to conductance data).
the log likelihood on the difference of the L2 norms of ke and ki :
2
?(ke , ki ) = ? ||ke ||2 ? ||ki ||2
(21)
This differentiable penalty ensures that the model will not rely too strongly on one filter over the
other, without imposing any prior on the shape of the filters (with ? = 0.05). We note that unlike
the a typical situation with statistical models that contain more abstract parameters, the terms we
wish to regularize can be measured with intracellular recordings. Future work with this model could
include more informative, data-driven priors on ke and ki .
Finally, we fit the CBSM and GLM to a population of nine extracellularly recorded macaque RGCs
in response to a full-field binary noise stimulus [20]. We used a five minute segment for model
fitting, and compared predicted spike rate using a 6s test stimulus for which we had repeated trials.
7
Excitation prediction
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0
Inhibition prediction
1
fit?to?conductance r2
fit?to?conductance r2
1
0.2
0
0.2
0.4
0.6
spike fit r2
0.8
1
0
0
0.2
0.4
0.6
0.8
spike fit r2
1
Figure 6: Summary of the CBSM fits to 7 ON parasol RGCs for which we had both spike train and
conductance recordings. The axes show model?s ability to predict the excitatory (left) and inhibitory (right)
inputs to a new stimulus in terms of r2 . The CBSM fit is compared against predictions of an LN model fit
directly to measured conductances.
A
B
GLM:
CBSM:
50 spks/s
Conductance Model
1
Rate prediction performance
0.9
0.8
on cell
0.7
0.6
0.5
0.4
0.4
250ms
off cell
0.6
GLM
0.8
1
Figure 7: (A) Performance on spike rate (PSTH) prediction. The true rate (black) was estimated using 167
repeat trials. The GLM prediction is in blue and the CBSM is in red. The PSTHs were smoothed with a
Gaussian kernel with a 1ms standard deviation. (B) Spike rate prediction performance for the population
of 9 cells. The red circle indicates cell used in left plot.
The CBSM achieved a 0.08 higher average r2 in PSTH prediction performance compared to the
GLM. All nine cells showed an improved fit with the CBSM.
7
Discussion
The classic GLM is a valuable tool for describing the relationship between stimuli and spike responses. However, the GLM describes this map as a mathematically convenient linear-nonlinear
cascade, which does not take account of the biophysical properties of neural processing. Here we
have shown that the GLM may be interpreted as a biophysically inspired, but highly constrained,
synaptic conductance-based model. We proposed a more realistic model of the conductance, removing the artificial constraints present in the GLM interpretation, which results in a new, more accurate
and more flexible conductance-based point process model for neural responses. Even without the
benefit of a concave log-likelihood, numerical optimization methods provide accurate estimates of
model parameters.
Qualitatively, the CBSM has a stimulus-dependent time constant, which allows it change gain as a
function of stimulus statistics (e.g., contrast), an effect that cannot be captured by a classic GLM. The
model also allows the excitatory and inhibitory conductances to be distinct functions of the sensory
stimulus, as is expected in real neurons. We demonstrate that the CBSM not only achieves improved
performance as a phenomenological model of neural encoding compared to the GLM, the model
accurately estimates the tuning of the excitatory and inhibitory synaptic inputs to RGCs purely from
measured spike times. As we move towards more naturalistic stimulus conditions, we believe that
the conductance-based approach will become a valuable tool for understanding the neural code in
sensory systems.
8
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9
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4,708 | 5,263 | Low-dimensional models of neural population activity
in sensory cortical circuits
Evan Archer1,2 , Urs K?oster3 , Jonathan Pillow4 , Jakob H. Macke1,2
1
Max Planck Institute for Biological Cybernetics, T?ubingen
2
Bernstein Center for Computational Neuroscience, T?ubingen
3
Redwood Center for Theoretical Neuroscience, University of California at Berkeley
4
Princeton Neuroscience Institute, Department of Psychology, Princeton University
[email protected], [email protected]
[email protected], [email protected]
Abstract
Neural responses in visual cortex are influenced by visual stimuli and by ongoing spiking activity in local circuits. An important challenge in computational
neuroscience is to develop models that can account for both of these features in
large multi-neuron recordings and to reveal how stimulus representations interact
with and depend on cortical dynamics. Here we introduce a statistical model of
neural population activity that integrates a nonlinear receptive field model with a
latent dynamical model of ongoing cortical activity. This model captures temporal
dynamics and correlations due to shared stimulus drive as well as common noise.
Moreover, because the nonlinear stimulus inputs are mixed by the ongoing dynamics, the model can account for a multiple idiosyncratic receptive field shapes with
a small number of nonlinear inputs to a low-dimensional dynamical model. We
introduce a fast estimation method using online expectation maximization with
Laplace approximations, for which inference scales linearly in both population
size and recording duration. We test this model to multi-channel recordings from
primary visual cortex and show that it accounts for neural tuning properties as
well as cross-neural correlations.
1
Introduction
Neurons in sensory cortices organize into highly-interconnected circuits that share common input,
dynamics, and function. For example, across a cortical column, neurons may share stimulus dependence as a result of sampling the same location of visual space, having similar orientation
preference [1] or receptive fields with shared sub-units [2]. As a result, a substantial fraction of
stimulus-information can be redundant across neurons [3]. Recent advances in electrophysiology
and functional imaging allow us to simultaneously probe the responses of the neurons in a column.
However, the high dimensionality and (relatively) short duration of the resulting data renders analysis a difficult statistical problem.
Recent approaches to modeling neural activity in visual cortex have focused on characterizing the responses of individual neurons by linearly projecting the stimulus on a small feature subspace that optimally drives the cell [4, 5]. Such ?systems-identification? approaches seek to describe the stimulusselectivity of single neurons separately, treating each neuron as an independent computational unit.
Other studies have focused on providing probabilistic models of the dynamics of neural populations,
seeking to elucidate the internal dynamics underlying neural responses [6, 7, 8, 9, 10, 11]. These
approaches, however, typically do not model the effect of the stimulus (or do so using only a linear
stimulus drive). To realize the potential of modern recording technologies and to progress our un1
derstanding of neural population coding, we need methods for extracting both the features that drive
a neural population and the resulting population dynamics [12].
We propose the Quadratic Input Latent Dynamical System (QLDS) model, a statistical model that
combines a low-dimensional representation of population dynamics [9] with a low-dimensional description of stimulus selectivity [13]. A low-dimensional dynamical system governs the population
response, and receives a nonlinear (quadratic) stimulus-dependent input. We model neural spike
responses as Poisson (conditional on the latent state), with exponential firing rate-nonlinearities. As
a result, population dynamics and stimulus drive interact multiplicatively to modulate neural firing. By modeling dynamics and stimulus dependence, our method captures correlations in response
variability while also uncovering stimulus selectivity shared across a population.
linear
filters
linear
dynamics
quadratic
+
population
nonlinear
function noise
spike
response
A
stimulus
...
intrinsic linear
noise update
Figure 1: Schematic illustrating the Quadratic input latent dynamical system model (QLDS).
The sensory stimulus is filtered by multiple units with quadratic stimulus selectivity (only one of
which is shown) which model the feed-forward input into the population. This stimulus-drive provides input into a multi-dimensional linear dynamical system model which models recurrent dynamics and shared noise within the population. Finally, each neuron yi in the population is influenced
by the dynamical system via a linear readout. QLDS therefore models both the stimulus selectivity
as well as the spatio-temporal correlations of the population.
2
2.1
The Quadratic Input Latent Dynamical System (QLDS) model
Model
We summarize the collective dynamics of a population using a linear, low-dimensional dynamical
system with an n-dimensional latent state xt . The evolution of xt is given by
xt = Axt?1 + f? (ht ) + t ,
(1)
where A is the n ? n dynamics matrix and is Gaussian innovation noise with covariance matrix
Q, t ? N (0, Q). Each stimulus ht drives some dimensions of xt via a nonlinear function of the
stimulus, f? , with parameters ?, where the exact form of f (?) will be discussed below. The log
firing rates zt of the population couple to the latent state xt via a loading matrix C,
zt = Cxt + D ? st + d.
(2)
Here, we also include a second external input st , which is used to model the dependence of the
firing rate of each neuron on its own spiking history [14]. We define D ? st to be that vector
PNs
whose k-th element is given by (D ? st )k ? i=1
Dk,i sk,t?i . D therefore models single-neuron
properties that are not explained by shared population dynamics, and captures neural properties such
as burstiness or refractory periods. The vector d represents a constant, private spike rate for each
neuron. The vector xt represents the n-dimensional state of m neurons. Typically n < m, so the
model parameterizes a low-dimensional dynamics for the population.
We assume that, conditional on zt , the observed activity yt of m neurons is Poisson-distributed,
yk,t ? Poisson(exp(zk,t )).
(3)
While the Poisson likelihood provides a realistic probabilistic model for the discrete nature of spiking responses, it makes learning and inference more challenging than it would be for a Gaussian
model. As we discuss in the subsequent section, we rely on computationally-efficient approximations to perform inference under the Poisson observation model for QLDS.
2
2.2
Nonlinear stimulus dependence
Individual neurons in visual cortex respond selectively to only a small subset of stimulus features
[4, 15]. Certain subpopulations of neurons, such as in a cortical column, share substantial receptive
field overlap. We model such a neural subpopulation as sensitive to stimulus variation in a linear
subspace of stimulus space, and seek to characterize this subspace by learning a set of basis vectors,
or receptive fields, wi . In QLDS, a subset of latent states receives a nonlinear stimulus drive, each
of which reflects modulation by a particular receptive field wi . We consider three different forms
of stimulus model: a fully linear model, and two distinct quadratic models. Although it is possible to incorporate more complicated stimulus models within the QLDS framework, the quadratic
models? compact parameterization and analytic elegance make them both flexible and computationally tractable. What?s more, quadratic stimulus models appear in many classical models of neural
computation, e.g. the Adelson-Bergen model for motion-selectivity [16]; quadratic models are also
sometimes used in the classification of simple and complex cells in area V1 [4].
We express our stimulus model by the function f? (ht ), where ? represents the set of parameters describing the stimulus filters wi and mixing parameters ai , bi and ci (in the case of the quadratic models). When fB (ht ) is identically 0 (no stimulus input), the QLDS with Poisson observations reduces
to what has been previously studied as the Poisson Latent Dynamical System (PLDS) [17, 18, 9].
We briefly review three stimulus models we consider, and discuss their computational properties.
Linear: The simplest stimulus model we consider is a linear function of the stimulus,
f (ht ) = Bht ,
(4)
where the rows of B as linear filters, and ? = {B}. This baseline model is identical to [18, 9] and
captures simple cell-like receptive fields since the input to latent states is linear and the observation
process is generalized linear.
Quadratic: Under the linear model, latent dynamics receive linear input from the stimulus along
a single filter dimension, wi . In the quadratic model, we permit the input to each state to be a
quadratic function of wi . We describe the quadratic by including three additional parameters per
latent dimension, so that the stimulus drive takes the form
2
fB,i (ht ) = ai wiT ht + bi wiT ht + ci .
(5)
Here, the parameters ? = {wi , ai , bi , ci : i = 1, . . . , m} include multiple stimulus filters wi and
quadratic parameters (ai , bi , ci ). Equation 5 might result in a stimulus input that has non-zero mean
with respect to the distribution of the stimulus ht , which may be undesirable. Given the covariance
of ht , it is straightforward to constrain the input to be zero-mean by setting ci = ?ai wiT ?wi , where
? is the covariance of ht and we assume the stimulus to have zero mean as well. The quadratic model
enables QLDS to capture phase-invariant responses, like those of complex cells in area V1.
Quadratic with multiplicative interactions: In the above model, there are no interactions between different stimulus filters, which makes it difficult to model suppressive or facilitating interactions between features [4]. Although contributions from different filters combine in the dynamics
of x, any interactions are linear. Our third stimulus model allows for multiplicative interactions
between r < m stimulus filters, with the i-th dimension of the input given by
r
X
f?,i (ht ) =
ai,j wi T ht wjT ht + bi wi T ht + ci .
j=1
Again, we constrain this function to have zero mean by setting ci = ?
2.3
Pr
j=1
ai,j wiT ?wj .
Learning & Inference
We learn all parameters via the expectation-maximization (EM) algorithm. EM proceeds by alternating between expectation (E) and maximization (M) steps, iteratively maximizing a lower-bound
to the log likelihood [19]. In the E-step, one infers the distribution over trajectories xt , given data
and the parameter estimates from the previous iteration. In the M-step, one updates the current parameter estimates by maximizing the expectation of the log likelihood, a lower bound on the log
likelihood. EM is a standard method for fitting latent dynamical models; however, the Poisson
observation model complicates computation and requires the use of approximations.
3
E-step: With Gaussian latent states xt , posterior inference amounts to computing the posterior
means ?t and covariances Qt of the latent states, given data and current parameters. With Poisson observations exact inference becomes intractable, so that approximate inference has to be used
[18, 20, 21, 22]. Here, we apply a global Laplace approximation [20, 9] to efficiently (linearly
in experiment duration T ) approximate the posterior distribution by a Gaussian. We note that each
fB (ht ) in the E-step is deterministic, making posterior inference identical to standard PLDS models.
We found a small number of iterations of Newton?s method sufficient to perform the E-step.
M-step: In the M-step, each parameter is updated using the means ?t and covariances Qt inferred
in the E-step. Given ?t and Qt , the parameters A and Q have closed-form update rules that are
derived in standard texts [23]. For the Poisson likelihood, the M-step requires nonlinear optimization
to update the parameters C, D and d [18, 9]. While for linear stimulus functions f? (ht ) the Mstep has a closed-form solution, for nonlinear stimulus functions we optimize ? numerically. The
objective function for ? given by
T
g(?) = ?
1 X
(?t ? A?t?1 ? f? (ht ))T Q?1 (?t ? A?t?1 ? f? (ht )) + const.,
2 t=2
where ?t = E[xt |yt?1 , ht ]. If ? is represented as a vector concatenating all of its parameters, the
gradient of g(?) takes the form
T
X
?f (ht )
?g(?)
?1
= ?Q
(?t ? A?t?1 ? f? (ht ))
.
??
??
t=2
For the quadratic nonlinearity, the gradients with respect to f (ht ) take the form
h
i
?f (ht )
?f (ht ) T 2
= 2 ai ht T wi + bi ht T ,
= ht wi ,
?wi
?ai
?f (ht )
?f (ht )
= ht T wi ,
= 1.
?bi
?ci
(6)
(7)
(8)
Gradients for the quadratic model with multiplicative interactions take a similar form. When constrained to be 0-mean, the gradient for ci disappears, and is replaced by an additional term in the
gradients for a and wi (arising from the constraint on c).
We found both computation time and quality of fit for QLDS to depend strongly upon the optimization procedure used. For long time series, we split the data into small minibatches. The QLDS E-step
and M-step each naturally parallelize across minibatches. Neurophysiological experiments are often
naturally segmented into separate trials across different stimuli and experimental conditions, making
it possible to select minibatches without boundary effects.
3
Application to simulated data
We illustrate the properties of QLDS using a simulated population recording of 100 neurons, each
responding to a visual stimulus of binary, white spatio-temporal noise of dimensionality 240. We
simulated a recording with T = 50000 samples and a 10-dimensional latent dynamical state. Five of
the latent states received stimulus input from a bank of 5 stimulus filters (see Fig. 2A, top row), and
the remaining latent dimensions only had recurrent dynamics and noise. We aimed to approximate
the properties of real neural populations in early sensory cortex. In particular, we set the dynamics
matrix A by fitting the model to a single neuron recording from V1 [4]. When fitting the model,
we assumed the same dimensionalities (10 latent states, 5 stimulus inputs) as those used to generate
the data. We ran 100 iterations of EM, which?-for the recording length and dimensionality of this
system?took about an hour on a 12?core intel Xeon CPU at 3.5GHz.
The model recovered by EM matched the statistics of the true model well. Linear dynamical system
and quadratic models of stimulus selectivity both commonly have invariances that render a particular
parameterization unidentifiable [4, 15], and QLDS is no exception: the latent state (and its parameters) can be rotated without changing the model?s properties. Hence it is possible only to compare
the subspace recovered by the model, and not the individual filters. In order to visualize subspace
recovery, we computed the best `2 approximation of the 5 ?true? filters in the subspace spanned by
4
Stimulus correlations
?0.2
B
60
80
100
0
20 40 60 80 100
20
40
60
?0.1
80
?0.2
100
true
20 40 60 80 100
60
?0.1
80
?0.2
100
0.2 0.4 0.6 0.8
real
noise vs stimulus
correlations
0.2
0.15
0.1
0.05
0
?0.05
0
0.1
true
0.2
stimulus correlation
0.4
0.3
40
fit
Noise correlations
0
20
?0.5
E
F
0.2
0.1
0
true
0.2
0.1
imaginary
?0.1
40
noise correlation
0
D 0.5 eigenvalues of A
20
probability
0.1
fit
0.2
fit
C
Total correlations
A
true
fit
0.2
0.1
20 40 60 80 100
0
20
40
60
synchronous spikes
Figure 2: Results on simulated data. Low-dimensional subspace recovery from a population of
100 simulated neurons in response to a white noise stimulus. (A) Simulated neurons receive shared
input from 5 spatio-temporal receptive fields (top row). QLDS recovers a subspace capable of
representing the original 5 filters (bottom row). (B) QLDS permits a more compact representation
than the conventional approach of mapping receptive fields for each neuron. For comparison with
the representation in panel A, we here show the spike-triggered averages of the first 60 neurons in the
population. (C) QLDS also models shared variability across neurons, as visualised here by the three
different measures of correlation. Top: Total correlation coefficients between each pair of neurons.
Values below the diagonal are from the simulated data, above the diagonal correspond to correlations
recovered by the model. Center: Stimulus correlations Bottom: Noise correlations. (D) Eigenvalues
of dynamics matrix A (black is ground truth, red is estimated). (E) In this model, stimulus and noise
correlations are dependent on each other, for the parameters chosen in this stimulation, there is a
linear relationship between them. (F) Distribution of population spike counts, i.e. total number of
spikes in each time bin across the population.
MSE (log scale)
1
0
?1
?2
?3
B
reconstruction performance
vs population size
linear
quadratic
quadratic cross
2
MSE (log scale)
A
reconstruction performance
vs experiment length
1
0
?1
?2
?3
?4
?4
200 400 600 800 1000
Population Size (# Cells)
?5
5000
10000
15000
Experiment length (# samples)
Figure 3: Recovery of stimulus subspace as a function of population size (A) and experiment duration (B). Each point represents the best filter reconstruction performance of QLDS over 20 distinct
simulations from the same ?true? model, each initialized randomly and fit using the same number
of EM iterations. Models were fit with each of three distinct stimulus nonlinearities, linear s (blue),
quadratic (green), and quadratic with multiplicative interactions (red). Stimulus input of the ?true?
was a quadratic with multiplicative interactions, and therefore we expect only the multiplicative
model (red) to each low error rates.
? i (see Fig. 2 A bottom row). In QLDS, different neurons share different filters, and
the estimated w
therefore these 5 filters provide a compact description of the stimulus selectivity of the population
[24]. In contrast, for traditional single-neuron analyses [4] ?fully-connected? models such as GLMs
[14] one would estimate the receptive fields of each of the 100 filters in the population, resulting in a
much less compact representation with an order of magnitude more parameters for the stimulus-part
alone (see Fig. 2B).
5
QLDS captures both the stimulus-selectivity of a population and correlations across neurons. In
studies of neural coding, correlations between neurons (Fig. 2C, top) are often divided into stimuluscorrelations and noise-correlations. Stimulus correlations capture correlations explainable by similarity in stimulus dependence (and are calculated by shuffling trials), whereas noise-correlations
capture correlations not explainable by shared stimulus drive (which are calculated by correlating
residuals after subtracting the mean firing rate across multiple presentations of the same stimulus).
The QLDS-model was able to recover both the total, stimulus and noise correlations in our simulation (Fig. 2C), although it was fit only to a single recording without stimulus repeats. Finally, the
model also recovered the eigenvalues of the dynamics (Fig. 2D), the relationship between noise and
stimulus correlations (Fig. 2E) and the distribution of population spike counts (Fig. 2F).
We assume that all stimulus dependence is captured by the subspace parameterized by the filters
of the stimulus model. If this assumption holds, increasing the size of the population increases
statistical power and makes identification of the stimulus selectivity easier rather than harder, in
a manner similar to that of increasing the duration of the experiment. To illustrate this point, we
generated multiple data-sets with larger population sizes, or with longer recording times, and show
that both scenarios lead to improvements in subspace-recovery (see Fig. 3).
4
Applications to Neural Data
Cat V1 with white noise stimulus We evaluate the performance of the QLDS on multi-electrode
recordings from cat primary visual cortex. Data were recorded from anaesthetized cats in response to
a single repeat of a 20 minute long, full-field binary noise movie, presented at 30 frames per second,
and 60 repeats of a 30s long natural movie presented at 150 frames per second. Spiking activity
was binned at the frame rate (33 ms for noise, 6.6 ms for natural movies). For noise, we used the
first 18000 samples for training, and 5000 samples for model validation. For the natural movie, 40
repeats were used for training and 20 for validation. Silicon polytrodes (Neuronexus) were employed
to record multi-unit activity (MUA) from a single cortical column, spanning all cortical layers with
32 channels. Details of the recording procedure are described elsewhere [25]. For our analyses, we
used MUA without further spike-sorting from 22 channels for noise data and 25 channels for natural
movies. We fit a QLDS with 3 stimulus filters, and in each case a 10-dimensional latent state, i.e. 7
of the latent dimensions received no stimulus drive.
Spike trains in this data-set exhibited ?burst-like? events in which multiple units were simultaneously
active (Fig. 4A). The model captured these events by using a dimension of the latent state with
substantial innovation noise, leading substantial variability in population activity across repeated
stimulus presentations. We also calculated pairwise (time-lagged) cross-correlations for each unit
pair, as well as the auto-correlation function for each unit in the data (Fig. 4B, 7 out of 22 neurons
shown, results for other units are qualitatively similar.). We found that samples from the model
(Fig. 4B, red) closely matched the correlations of the data for most units and unit-pairs, indicating
the QLDS provided an accurate representation of the spatio-temporal correlation structure of the
population recording. The instantaneous correlation matrix across all 22 cells was very similar
between the physiological and sampled data (Fig. 4C). We note that receptive fields (Fig. 4F) in this
data did not have spatio-temporal profiles typical of neurons in cat V1 (this was also found when
using conventional analyses such as spike-triggered covariance). Upon inspection, this was likely a
consequence of an LGN afferent also being included in the raw MUA. In our analysis, a 3-feature
model captured stimulus correlations (in held out data) more accurately than 1- and 2- filter models.
However, 10-fold cross validation revealed that 2- and 3- filter models do not improve upon a 1-filter
model in terms of one-step-ahead prediction performance (i.e. trying to predict neural activity on
the next time-step using past observations of population activity and the stimulus).
Macaque V1 with drifting grating stimulus: We wanted to evaluate the ability of the model to
capture the correlation structure (i.e. noise and signal correlations) of a data-set containing multiple
repetitions of each stimulus. To this end, we fit QLDS with a Poisson observation model to the
population activity of 113 V1 neurons from an anaesthetized macaque, as described in [26]. Drifting grating stimuli were presented for 1280ms, followed by a 1280ms blank period, with each of
72 grating orientations repeated 50 times. We fit a QLDS with a 20-dimensional latent state and 15
stimulus filters, where the stimulus was paramterized as a set of phase-shifted sinusoids at the appropriate spatial and temporal frequency (making ht 112-dimensional). We fit the QLDS to 35 repeats,
6
B
data
5
10
?20
15
0
20
0.4
0.4
0.2
0.2
0
0
?20
10 20 30 40 50 60 70 80 90
0
20
0.4
10
15
0.8
noise vs stimulus
correlation
0.6
0.4
0.2
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0.6
0.8
stimulus correlation
0
?20
0
0.4
0.2
0.2
0
0
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0
0
20
?20
0
20
?20
0
20
?20
0.4
0.4
0.4
0.4
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0.2
0.2
0
0
0
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20
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0
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?20
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?1
0
?20
0.4
0.2
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0.2
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0.2
0
0
0
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?20
0
20
?20
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20
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0
0
20
?20
0.4
0.4
0.4
0.4
0.4
0.4
0.2
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0
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0
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20
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0
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15
0.2
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10
0.4
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true
5
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0
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?20
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F
eigenvalues of A
feature 1
0.5
feature 2
0
?0.5
20
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?20
E
imaginary
noise correlation
D
10 20 30 40 50 60 70 80 90
time (s)
?0.5
0.2
0
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?20
20
?20
0.2
?20
5
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15
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5
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Total correlations
1
0.2
0
20
Simulated repeats to identical noise stimulus
C
0.4
fit
A
0
0.5
real
1
feature 3
?165ms ?132ms ?99ms ?66ms ?33ms
0ms
Figure 4: QLDS fit to V1 cells with noise stimuli. We fit QLDS to T = 18000 samples of 22
neurons responding to a white noise stimulus, data binned at 33 ms. We used the quadratic with
multiplicative interactions as the stimulus nonlinearity. The QLDS has a 10-dimensional latent state
with 3 stimulus inputs. All results shown here are compared against T = 5000 samples of test-data,
not used to train the model. (A) Top row: Rasters from recordings from 22 cells in cat visual cortex,
where cell index appears on the y axis, and time in seconds on the x. Second and third row: Two independent samples from the QLDS model responding to the same noise stimuli. Note that responses
are highly variable across trials. (B) Auto- and cross-correlations for data (black) and model (red)
cells. For the model, we average across 60 independent samples, thickness of red curves reflects 1
standard deviation from the mean. Panel (i, j) corresponds to cross-correlation between units with
indices i and j, panels along the diagonal show auto-correlations. (C) Total correlations for the true
(lower diagonal) and model (upper diagonal) populations. (D) Noise correlations scattered against
stimulus correlations for the model. As we did not have repeat data for this population, we were not
able to reliably estimate noise correlations, and thereby evaluate the accuracy of this model-based
prediction. (E) Eigenvalues of the dynamics matrix A. (F) Three stimulus filters recovered by
QLDS. We selected the 3-filter QLDS by inspection, having observed that fitting with larger number
of stimulus filters did not improve the fit. We note that although two of the filters appear similar,
that they drive separate latent dimensions with distinct mixing weights ai , bi and ci .
and held out 15 for validation. The QLDS accurately captured the stimulus and noise correlations of
the full population (Fig. 5A). Further, a QLDS with 15 shared receptive fields captured simple and
complex cell behavior of all 113 cells, as well as response variation across orientation (Fig. 5B).
5
Discussion
We presented QLDS, a statistical model for neural population recordings from sensory cortex that
combines low-dimensional, quadratic stimulus dependence with a linear dynamical system model.
The stimulus model can capture simple and complex cell responses, while the linear dynamics capture temporal dynamics of the population and shared variability between neurons. We applied QLDS
to population recordings from primary visual cortex (V1). The cortical microcircuit in V1 consists of
highly-interconnected cells that share receptive field properties such as orientation preference [27],
with a well-studied laminar organization [1]. Layer IV cells have simple cell receptive field properties, sending excitatory connections to complex cells in the deep and superficial layers. Quadratic
7
A
Stimulus correlations
Noise correlations
0.1
60
0.05
0
?0.5
?1
B
spike rate
Cell 49
0.6
0 degrees
80
100
data
20 40 60 80 100
cell index
45 degrees
90 degrees
0
?0.05
?0.1
data
20 40 60 80 100
cell index
135 degrees
180 degrees
225 degrees
500 1000 1500
time (ms)
500 1000 1500
time (ms)
500 1000 1500
time (ms)
0.4
0.2
0
stimulus off
Cell 50 0.6
spike rate
model
40
0.5
model
20
1
0.4
0.2
0
500 1000 1500
time (ms)
500 1000 1500
time (ms)
500 1000 1500
time (ms)
Figure 5: QLDS fit to 113 V1 cells across 35 repeats of each of 72 grating orientations. (A)
Comparison of total correlations in the data and generated from the model, (B) For two cells (cells
49 and 50, using the index scheme from A) and 6 orientations (0, 45, 90, 135, 180, and 225 degrees),
we show the posterior mean prediction performance (red traces) in in comparison to the average
across 15 held-out trials (black traces). In each block, we show predicted and actual spike rate
(y-axis) over time binned at 10 ms (x-axis). Stimulus offset is denoted by a vertical blue line.
stimulus models such as the classical ?energy model? [16] of complex cells reflect this structure.
The motivation of QLDS is to provide a statistical description of receptive fields in the different
cortical layers, and to parsimoniously capture both stimulus dependence and correlations across an
entire population.
Another prominent neural population model is the GLM (Generalized Linear Model, e.g. [14]; or
the ?common input model?, [28]), which includes a separate receptive field for each neuron, as
well as spike coupling terms between neurons. While the GLM is a successful model of a population?s statistical response properties, its fully?connected parameterization scales quadratically with
population size. Furthermore, the GLM supposes direct couplings between pairs of neurons, while
monosynaptic couplings are statistically unlikely for recordings from a small number of neurons
embedded in a large network.
In QLDS, latent dynamics mediate both stimulus and noise correlations. This reflects the structure
of the cortex, where recurrent connectivity gives rise to both stimulus-dependent and independent
correlations. Without modeling a separate receptive field for each neuron, the model complexity of
QLDS grows only linearly in population size, rather than quadratically as in fully-connected models
such as the GLM [14]. Conceptually, our modeling approach treats the entire recorded population
as a single ?computational unit?, and aims to characterize its joint feature-selectivity and dynamics.
Neurophysiology and neural coding are progressing toward recording and analyzing datasets of ever
larger scale. Population-level parameterizations, such as QLDS, provide a scalable strategy for
representing and analyzing the collective computational properties of neural populations.
Acknowledgements
We are thankful to Arnulf Graf and the co-authors of [26] for sharing the data used in Fig. 5, and to
Memming Park for comments on the manuscript. JHM and EA were funded by the German Federal
Ministry of Education and Research (BMBF; FKZ: 01GQ1002, Bernstein Center T?ubingen) and the
Max Planck Society, and UK by National Eye Institute grant #EY019965. Collaboration between
EA, JP and JHM initiated at the ?MCN? Course at the Marine Biological Laboratory, Woods Hole.
8
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9
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4,709 | 5,264 | Inferring sparse representations of continuous signals
with continuous orthogonal matching pursuit
Jacob L. Yates
Department of Neuroscience
The University of Texas at Austin
[email protected]
Karin C. Knudson
Department of Mathematics
The University of Texas at Austin
[email protected]
Alexander C. Huk
Center for Perceptual Systems
Departments of Psychology & Neuroscience
The University of Texas at Austin
[email protected]
Jonathan W. Pillow
Princeton Neuroscience Institute and
Department of Psychology
Princeton University
[email protected]
Abstract
Many signals, such as spike trains recorded in multi-channel electrophysiological
recordings, may be represented as the sparse sum of translated and scaled copies
of waveforms whose timing and amplitudes are of interest. From the aggregate
signal, one may seek to estimate the identities, amplitudes, and translations of the
waveforms that compose the signal. Here we present a fast method for recovering these identities, amplitudes, and translations. The method involves greedily
selecting component waveforms and then refining estimates of their amplitudes
and translations, moving iteratively between these steps in a process analogous
to the well-known Orthogonal Matching Pursuit (OMP) algorithm [11]. Our approach for modeling translations borrows from Continuous Basis Pursuit (CBP)
[4], which we extend in several ways: by selecting a subspace that optimally captures translated copies of the waveforms, replacing the convex optimization problem with a greedy approach, and moving to the Fourier domain to more precisely
estimate time shifts. We test the resulting method, which we call Continuous Orthogonal Matching Pursuit (COMP), on simulated and neural data, where it shows
gains over CBP in both speed and accuracy.
1
Introduction
It is often the case that an observed signal is a linear combination of some other target signals that
one wishes to resolve from each other and from background noise. For example, the voltage trace
from an electrode (or array of electrodes) used to measure neural activity in vivo may be recording
from a population of neurons, each of which produces many instances of its own stereotyped action
potential waveform. One would like to decompose an analog voltage trace into a list of the timings
and amplitudes of action potentials (spikes) for each neuron.
Motivated in part by the spike-sorting problem, we consider the case where we are given a signal
that is the sum of known waveforms whose timing and amplitude we seek to recover. Specifically,
we suppose our signal can be modeled as:
y(t) =
Nf J
X
X
an,j fn (t ? ?n,j ),
n=1 j=1
1
(1)
where the waveforms fn are known, and we seek to estimate positive amplitudes an,j and event
times ?n,j . Signals of this form have been studied extensively [12, 9, 4, 3].
This a difficult problem in part because of the nonlinear dependence of y on ? . Moreover, in most
applications we do not have access to y(t) for arbitrary t, but rather have a vector of sampled (noisy)
measurements on a grid of discrete time points. One way to simplify the problem is to discretize ? ,
considering only a finite set of possible time shift ?n,j ? {?, 2?..., N? ?} and approximating the
signal as
y?
Nf J
X
X
an,j fn (t ? in,j ?), in,j ? 1, ..., N?
(2)
n=1 j=1
Once discretized in this way, the problem is one of sparse recovery: we seek to represent the
observed signal with a sparse linear combination of elements of a finite dictionary {fn,j (t) :=
fn (t ? j?), n ? 1, ..., Nf , j ? 1, ..., N? }. Framing the problem as sparse recovery, one can
bring tools from compressed sensing to bear. However, the discretization introduces several new
difficulties. First, we can only approximate the translation ? by values on a discrete grid. Secondly,
choosing small ? allows us to more closely approximate ? , but demands more computation, and
such finely spaced dictionary elements yield a highly coherent dictionary, while sparse recovery
algorithms generally have guarantees for low-coherence dictionaries.
A previously introduced algorithm that uses techniques of sparse recovery and returns accurate and
continuous valued estimates of a and ? is Continuous Basis Pursuit (CBP) [4], which we describe
below. CBP proceeds (roughly speaking) by augmenting the discrete dictionary fn,j (t) with other
carefully chosen basis elements, and then solving a convex optimization problem inspired by basis
pursuit denoising. We extend ideas introduced in CBP to present a new method for recovering
the desired time shifts ? and amplitudes a that leverage the speed and tractability of solving the
discretized problem while still ultimately producing continuous valued estimates of ? , and partially
circumventing the problem of too much coherence.
Basis pursuit denoising and other convex optimization or `1 -minimization based methods have been
effective in the realm of sparse recovery and compressed sensing. However, greedy methods have
also been used with great success. Our approach begins with the augmented bases used in CBP,
but adds basis vectors greedily, drawing on the well known Orthogonal Matching Pursuit algorithm
[11]. In the regimes considered, our greedy approach is faster and more accurate than CBP.
Broadly speaking, our approach has three parts. First, we augment the discretized basis in one of
several ways. We draw on [4] for two of these choices, but also present another choice of basis that
is in some sense optimal. Second, we greedily select candidate time bins of size ? in which we
suspect an event has occurred. Finally, we move from this rough, discrete-valued estimate of timing
? to continuous-valued estimates of ? and a. We iterate the second and third steps, greedily adding
candidate time bins and updating our estimates of ? and a until a stopping criterion is reached.
The structure of the paper is as follows. In Section 2 we describe the method of Continuous Basis
Pursuit (CBP), which our method builds upon. In Section 3 we develop our method, which we call
Continuous Orthogonal Matching Pursuit (COMP). In Section 4 we present the performance of our
method on simulated and neural data.
2
Continuous basis pursuit
Continuous Basis Pursuit (CBP) [4, 3, 5] is a method for recovering the time shifts and amplitudes
of waveforms present in a signal of the form (1). A key element of CBP is augmenting or replacing
the set {fn,j (t)} with certain additional dictionary elements that are chosen to smoothly interpolate
the one dimensional manifold traced out by fn,j (t ? ? ) as ? varies in (??/2, ?/2).
The benefit of a dictionary that is expanded in this way is twofold. First, it increases the ability
of the dictionary to represent shifted copies of the waveform fn (t ? ? ) without introducing as
much correlation as would be introduced by simply using a finer discretization (decreasing ?),
which is an advantage because dictionaries with smaller coherence are generally better suited for
sparse recovery techniques. Second, one can move from recovered coefficients in this augmented
dictionary to estimates an,j and continuous-valued estimates of ?n,j .
2
In general, there are three ingredients for CBP: basis elements, an interpolator with corresponding
mapping function ?, and a convex constraint set, C. There are K basis elements {gn,j,k (t) =
gn,k (t ? j?)}k=K
k=1 , for each waveform and width-? time bin, which together can be used to linearly
interpolate fn,j (t ? ? ), |? | < ?/2. The function ? maps from amplitude a and time shift ? to KPK (k)
(1)
(K)
tuples of coefficients ?(a, ? ) = (cn,j , ..., cn,j ), so afn,j (t ? ? ) ? k=1 cn,j gn,j,k (t). The convex
constraint set C is for K-tuples of coefficients of {gn,j,k }k=K
k=1 and corresponds to the requirement
that a > 0 and |? | < ?/2. If the constraint region corresponding to these requirements is not convex
(e.g. in the polar basis discussed below), its convex relaxation is used.
As a concrete example, let us first consider (as discussed in [4]) the dictionary augmented with
0
shifted copies of each waveform?s derivative : {fn,j
(t) := fn0 (t ? j?)}. Assuming fn is sufficiently
0
smooth, we have from the Taylor expansion that for small ? , afn,j (t ? ? ) ? afn,j (t) ? a? fn,j
(t). If
0
we recover a representation of y as c1 fn,j (t)+c2 fn,j (t), then we can estimate the amplitude a of the
waveform present in y as c1 , the time shift ? as ?c2 /c1 . Hence, we estimate y ? c1 fn,j (t+c2 /c1 ) =
c1 fn (t ? j? + c2 /c1 ). Note that the estimate of the time shift ? varies continuously with c1 , c2 .
In contrast, using shifted copies of the waveforms only as a basis would not allow for a time shift
j=N?
.
estimate off of the grid {j?}j=1
Once a suitable dictionary is chosen, one must still recover coefficients (i.e. c1 , c2 above). Motivated
by the assumed sparsity of the signal (i.e. y is the sum of relatively few shifted copies of waveforms,
so the coefficients of most dictionary elements will be zero), CBP draws on the basis pursuit denoising, which has been effective in the compressive sensing setting and elsewhere [10],[1]. Specifically,
CBP (with a Taylor basis) recovers coefficients using:
2
Nf
Nf
X
X
? (1)
(1)
(1)
(2)
(1)
0 (2)
argminc
(Fn cn + Fn cn ) ? y
+
?
c
? n, i (3)
c
n
s.t. cn,i ? 0 , |cn,i | ?
2 i,n
1
n=1
n=1
2
0
Here we denote by F the matrix with columns {fn,j (t)} and F0 the matrix with columns {fn,j
(t)}.
The `1 penalty encourages sparsity, pushing most of the estimated amplitudes to zero, with higher
(1)
? encouraging greater sparsity. Then, for each (n, j) such that cn,j 6= 0, one estimates that there is
(1)
(2)
(1)
a waveform in the shape of fn with amplitude a
? = cn,j and time shift j? ? ?? = j? ? cn,j /cn,j
present in the signal. The inequality constraints in the optimization problem ensure first that we only
recover positive amplitudes a
?, and second that estimates ?? satisfy |?
? | < ?/2. Requiring ?? to fall
in this range keeps the estimated ? in the time bin represented by fn,j and also in the regime where
they Taylor approximation to fn,j (t?? ) is accurate. Note that (3) is a convex optimization problem.
Better results in [4] are obtained for a second order Taylor interpolation and the best results come
from a polar interpolator, which represents each manifold of time-shifted waveforms fn,j (t ?
? ), |? | ? ?/2 as an arc of the circle that is uniquely defined to pass through fn,j (t), fn,j (t ? ?/2),
and fn,j (t+?/2). Letting the radius of the arc be r, and its angle be 2? one represents points on this
2?
arc by linear combinations of functions w, u, v: f (t ? ? ) ? w(t) + r cos( 2?
? ?)u(t) + r sin( ? ?)v(t).
The Taylor and polar bases consist of shifted copies of elements chosen in order to linearly interpolate the curve in function space defined by fn (t ? ? ) as ? varies from ??/2 to ?/2. Let Gn,k be
the matrix whose columns are gn,j,k (t) for j ? 1, ..., N? . With choices of basis elements, interpolator, and corresponding convex constraint set C in place, one proceeds to estimate coefficients in
the chosen basis by solving:
2
Nf K
Nf
X
X
X
(1)
(K)
(k)
argminc
y ?
Gn,k cn
+ ?k
c(1)
n k1 subject to (cn,j , ..., cn,j ) ? C ?(n, j)
n=1 k=1
n=1
(4)
2
(1)
(K)
One then maps back from each nonzero K-tuple of recovered coefficients cn,j , ..., cn,j to corresponding a
?n,j , ??n,j that represent the amplitude and timing of the nth waveform present in
the jth time bin. This can be done by inverting ?, if possible, or estimating (?
an,j , ??n,j ) =
(1)
(K) 2
argmina,? k?(a, ? ) ? (cn,j , ..., cn,j )k2 .
3
Table 1: Basis choices (see also [4], Table 1.)
Interpolator
Basis Vectors
?(a, ? )
2
C
Taylor
(K=3)
0
(t)},
{fn,j (t)}, {fn,j
00
{fn,j (t)}
(a, ?a?, a ?2 )
c(1) , c(3) > 0, |c(2) | < c(1) ?
2,
2
|c(3) | < c(1) ?8
Polar
{wn,j }, {un,j },
{vn,j }
(a, ar cos( 2?
? ?),
ar sin( 2?
?))
?
p
c(1) ? 0, (c(2) )2 + (c(3) )2 ? rc(1)
rc(1) cos(?) ? c(2) ? rc(1)
SVD
{u1n,j }...{uK
n,j }.
(See Section 3.1)
(See Section 3.1)
3
Continuous Orthogonal Matching Pursuit
We now present our method for recovery, which makes use of the idea of augmented bases presented
above, but differs from CBP in several important ways. First, we introduce a different choice of basis
that we find enables more accurate estimates. Second, we make use of a greedy method that iterates
between choosing basis vectors and estimating time shifts and amplitudes, rather than proceeding
via a single convex optimization problem as CBP does. Lastly, we introduce an alternative to the
step of mapping back from recovered coefficients via ? that notably improves the accuracy of the
recovered time estimates.
Greedy methods such as Orthogonal Matching Pursuit (OMP) [11], Subspace Pursuit [2], and Compressive Sampling Matching Pursuit (CoSaMP) [8] have proven to be fast and effective in the realm
of compressed sensing. Since the number of iterations of these greedy methods tend to go as the
sparsity (when the algorithms succeed), they tend to be extremely fast when for very sparse signals. Moreover, our the greedy method eliminates the need to choose a regularization constant ?,
a choice that can vastly alter the effectiveness of CBP. (We still need to choose K and ?.) Our
method is most closely analogous to OMP, but recovers continuous time estimates, so we call it
Continuous Orthogonal Matching Pursuit (COMP). However, the steps below could be adapted in a
straightforward way to create analogs of other greedy methods.
3.1
Choice of finite basis
We build upon [4], choosing as our basis N? shifted copies of a set of K basis vectors for each
waveform in such away that these K basis vectors can effectively linearly interpolate fn (t ? ? )
for |? | < ?/2. In our method, as in Continuous Basis Pursuit, these basis vectors allow us to
represent continuous time shifts instead of discrete time shifts, and expand the descriptive power of
our dictionary without introducing undue amounts of coherence. While previous work introduced
Taylor and polar bases, we obtain the best recovery from a different basis, which we describe now.
The basis comes from a singular value decomposition of a matrix whose columns correspond to
discrete points on the curve in function space traced out by fn,j (t ? ? ) as we vary ? for |? | < ?/2.
Within one time bin of size ?, consider discretizing further into N? = ?/? time bins of size ? ?.
Let F? be the matrix with columns that are these (slightly) shifted copies of the waveform, so that
the ith column of F? is fn,j (t ? i? + ?/2) for a discrete vector of time points t. Each column of
this matrix is a discrete point on the curve traced out by fn,j (t ? ? ) as ? varies.
In choosing a basis, we seek the best choice of K vectors to use to linearly interpolate this curve. We
might instead seek to solve the related problem of finding the best K vectors to represent these finely
spaced points on the curve, in which case a clear choice for these K vectors is the first K left singular
vectors of F? . This choice is optimal in the sense that the singular value decomposition yields the
best rank-K approximation to a matrix. If F? = U?VT is the singular value decomposition, and
PK
uk , vk are the columns of U and V respectively, then kF? ? k=1 uk ?k,k (vk )T k ? kF ? Ak for
any rank-K matrix A and any unitarily invariant norm k ? k.
4
In order to use this SVD basis with CBP or COMP, one must specify a convex constraint set for the
PK
coefficients of this basis. Since afn,j (t ? i?) = k=1 auk ?k,k vik a reasonable and simply enforced
constraint set would be to assume that the recovered coefficients c(k) corresponding to each basis
vector uk , when divided by c(1) to account for scaling, be between mini ?k,k vik and maxi ?k,k vik . A
PK
simple way to recover a and ? would to choose ? = i? and a, i to minimize k=1 (c(k) ?a?k,k vik )2 .
In figure 3.1, we compare the error between shifted copies of a sample waveform f (t ? ? ) for
|? | < 0.5 and the best (least-squares) approximation of that waveform as a linear combination of
K = 3 vectors from the Taylor, polar, and SVD bases. The structure of the error as a function of the
time shift ? reflects the structure of these bases. The Taylor approximation is chosen to be exactly
accurate at ? = 0 while the polar basis is chosen to be precisely accurate at ? = 0, ?/2, ??/2. The
SVD basis gives the lowest mean error across time shifts.
Original
Waveform
0
0.5
Taylor
2
5
0
t
5
1
Polar
0
0
0
2
1
0.2
5
0
t
5
5
0
t
SVD
0.2
5
0.08
l2 error
f(t)
0.5
Approximation Error
Basis Vectors
0.06
Taylor
Polar
SVD
0.04
0.02
5
0
t
5
0.5
0
time shift
0.5
Figure 1: Using sample waveform f (t) ? t exp(?t2 ) (left panel), we compare the error introduced
by approximating f (t ? ? ) for varying ? with a linear combination of K = 3 basis vectors, from the
Taylor, polar or SVD bases. Basis vectors are shown in the middle three panels, and error in the far
right panel. The SVD basis introduces the least error on average over the shift ? . The average errors
for the Taylor, polar, and SVD bases are 0.026, 0.027, and 0.014 respectively.
3.2
Greedy recovery
Taylor: 0.027
Polar: 0.027
recoverSVD:
the time
bins
0.014
Having chosen our basis, we then greedily
in which an occurrence of each
waveform appears to be present. We would like to build up a set of pairs (n, j) corresponding to
an instance of the nth waveform in the j th time bin. (In our third step, we will refine the estimate
within the chosen bins.)
Our greedy method is motivated by Orthogonal Matching Pursuit (OMP), which is used to recover a
sparse solution x from measurements y = Ax. In OMP [11], one greedily adds a single dictionary
element to an estimated support set S at each iteration, and then projects orthogonally to adjust the
coefficients of all chosen dictionary elements. After initializing with S = ?, x = 0, one iterates the
following until a stopping criterion is met:
r = y ? Ax
j = argmaxj {|haj , ri| s.t. j ? {1, ...J}\S}
S = S ? {j}
x = argminz {||y ? Az||2 s.t. zi = 0 ? i ?
/ S}
If we knew the sparsity of the signal, we could use that as our stopping condition. Normally we do
not know the sparsity a priori; we stop when changes in the residual become sufficiently small.
We adjust this method to choose at each step not a single additional element but rather a set of
K associated basis vectors. S is again initialized to be empty, but at each step we add a timebin/waveform pair (n, j), which is associated with K basis vectors. In this way, we are adding K
vectors at each step, instead of one as in OMP. We greedily add the next index (n, j) according to:
(
)
k
X
(k) (k)
2
c
(n, j) = argminm,i min{k
cm,i gm,i ? rk2 s.t. cm,i ? C} , (m, i) ? S
(5)
cm,i
i=1
5
(k)
Here {gm,i } are the chosen basis vectors (Taylor, polar, or SVD), and C is the corresponding constraint set, as in Section 2.
In comparison with the greedy step in OMP, choosing j as in (5) is more costly, because we need
to perform a constrained optimization over a K dimensional space for each n, j. Fortunately, it is
not necessary to repeat the optimization for each of the Nf ? N? possible indices each time we add
an index. Assuming waves are localized in time, we need only update the results of the constrained
optimization locally. When we update the residual r by subtracting the newly identified waveform
n in the j th bin, the residual only changes in the bins at or near the j th bin, so we need only update
Pk
(k) (k)
the quantity mincn,j0 {k i=1 cn,j 0 gn,j 0 ? rk22 s.t. cn,j 0 ? C } for j 0 neighboring j.
3.3
Estimating time shifts
Having greedily added a new waveform/timebin index pair (n, j), we next define our update step,
which will correspond to the orthogonal projection in OMP. We present two alternatives, one of
which most closely mirrors the corresponding step in OMP, the other of which works within the
Fourier domain to obtain more accurate recovery.
To most closely follow the steps of OMP, at each iteration after updating S we update coefficients c
according to:
2
X X
K
(k) (k)
cn,j gn,j ? y
(6)
argminc
subject to cn,j ? C ? (n, j) ? S
(n,j)?S k=1
2
We alternate between the greedily updating S via (5), and updating c as in (6), at each iteration
P
PK (k) (k)
finding the new residual r = (n,j)?S k=1 cn,j gn,j ?y ) until the `2 stopping criterion is reached.
Then, one maps back from {cn,j }(n,j)?S to {a(n,j) , ?(n,j) }(n,j)?S as described in Section 2.
Alternatively we may replace the orthogonal projection step with a more accurate recovery of spike
timings that involves working in the Fourier domain. We use the property of the Fourier transform
with respect to translation that: (f (t ? ? ))? = e2?i? f?. This allows us to estimate a, ? directly via:
X
argmina,? k(
an,j e2?i??n,j f?n,j (?)) ? y?(?)k2 subject to |?n,j | < ?/2 ? (n, j) ? S
(7)
n,j?S
This is a nonlinear and non-convex constrained optimization problem. However, it can be solved reasonably quickly using, for example, trust region methods. The search space is dramatically reduced
because ? has only |S| entries, each constrained to be small in absolute value. By searching directly
for a, ? as in (7) we sacrifice convexity, but with the benefit of eliminating from this step error of
interpolation introduced as we map back from c to a, ? using ??1 or a least squares estimation.
It is easy and often helpful to add inequality constraints to a as well, for example requiring a to be
in some interval around 1, and we do impose this in our spike-sorting simulations and analysis in
Section 4. Such a requirement effectively imposes a uniform prior on a over the chosen interval. It
would be an interesting future project to explore imposing other priors on a.
4
Results
We test COMP and CBP for each choice of basis on simulated and neural data. Here, COMP denotes
the greedy method that includes direct estimation of a and ? during the update set as in (7). The
convex optimization for CBP is implemented using the cvx package for MATLAB [7], [6].
4.1
Simulated data
We simulate a signal y as the sum of time-shifted copies of two sample waveforms f1 (t) ?
4
2
t exp(?t2 ) and f2 (t) ? e?t /16 ? e?t (Figure 2a). There are s1 = s2 = 5 shifted copies of
f1 and f2 , respectively. The time shifts are independently generated for each of the two waveforms
using a Poisson process (truncated after 5 spikes), and independent Gaussian noise of variance ? 2 is
6
0
0.5
0
5
(b)
0
0.5
5
t
5
0
t
5
1
0.5
1
0
20
40 t 60
80
100
waveform 1
0.5
1
0.5
0
waveform 2
0
0.5
1
0
20
40
t 60
100
20
40
60
80
100
True
CBP SVD
1
2.5
CBP Taylor
CBP Polar
CBP SVD
COMP Taylor
COMP Polar
COMP SVD
2
1.5
1
0.5
0.5
0
0
0
20
40
60
80
100
COMP-SVD
1.5
(f )
0
.05
.1
Noise ( )
.2
.4
0.5
0.4
1
0.5
0
20
40
t
60
80
100
1.5
True
COMP SVD
1
0.3
0.2
0.1
0.5
0
80
0
1.5
0
0.5
(e)
1.5
(d)
0
1
CBP-SVD
(c)
(Misses + False Positives)/s
waveform 2
0.5
Average Hit Error
waveform 1
waveform 1
0.5
waveform 2
(a)
0
20
40
60
80
100
0
0
.05
.1
.2
.4
Noise ( )
Figure 2: (a) Waveforms present in the signal. (b) A noiseless (top) and noisy (bottom) signal with
? = .2. (c) Recovery using CBP. (d) Recovery using COMP (with a, ? updated as in (7)). (e) For
each recovery method over different values of the standard deviation of the noise ?, misses plus false
positives, divided by the total number of events present, s = s1 + s2 . (f) Average distance between
the true and estimated spike for each hit.
added at each time point. Figures 2b,c show an example noise-free signal (? = 0), and noisy signal
(? = .2) on which each recovery method will be run.
We run CBP with the Taylor and polar bases, but also with our SVD basis, and COMP with all three
bases. Since COMP here imposes a lower bound on a, we also impose a thresholding step after
recovery with CBP, discarding any recovered waveforms with amplitude less than .3. We find the
thresholding generally improved the performance of the CBP algorithm by pruning false positives.
Throughout, we use K = 3, since the polar basis requires 3 basis vectors per bin.
We categorize hits, false positive and misses based on whether a time shift estimate is within a
threshold of = 1 of the true value. The ?average hit error? of Figure 2h, 3b is the average distance
between the true and estimated event time for each estimate that is categorized as a hit. Results are
averaged over 20 trials.
We compare CBP and COMP over different parameter regimes, varying the noise (?) and the bin
size (?). Figures 2g and 3a show misses plus false positives for each method, normalized by the total
number of events present. Figures 2f and 3b show average distance between the true and estimated
spike for each estimate categorized as a hit. The best performance by both measures across nearly
all parameter regimes considered is achieved by COMP using the SVD basis. COMP is more robust
to noise (Figure 2g), and also to increases in bin width ?. Since both algorithms are faster for
higher ?, robustness with respect to ? is an advantage. We also note a significant increase in CBP?s
robustness to noise when we implement it with our SVD basis rather than with the Taylor or polar
basis (Figure 2e).
A significant advantage of COMP over CBP is its speed. In Figure 3c we compare the speed of
COMP (solid) and CBP (dashed) algorithms for each basis. COMP yields vast gains in speed. The
comparison is especially dramatic for small ?, where results are most accurate across methods.
4.2
Neural data
We now present recovery of spike times and identities from neural data. Recordings were made
using glass-coated tungsten electrodes in the lateral intraparietal sulcus (LIP) of a macaque monkey
performing a motion discrimination task. In addition to demonstrating the applicability of COMP
to sorting spikes in neural data, this section also shows the resistance of COMP to a certain kind of
error that recovery via CBP can systematically commit, and which is relevant to neural data.
7
0.8
0.7
1.5
1
0.5
(b)
500
0.5
0.4
0.3
0.2
1
1.5
Bin Width ( )
2
0
0.5
2.5
CBP Taylor
CBP Polar
CBP SVD
COMP Taylor
COMP Polar
COMP SVD
300
200
100
0.1
0
0.5
(c)
400
0.6
Computing Time
(a)
Average Hit Error
(Misses + False Positives)/s
2
1
1.5
Bin Width ( )
2
2.5
0
0.5
1
1.5
Bin Width ( )
2
2.5
Figure 3: (a) Misses plus false positives, divided by the total number of events present, s = s1 + s2
over different values of bin width ?. (b) Average distance between the true and estimated spike for
each hit for each recovery method. (c) Run time for COMP (solid) and CBP (dashed) for each basis.
0.3
0.4
0.5
0
(c)
COMP-SVD
CBP-SVD
0
1
0.5
0
0
10
20
30
40
50
60
70
80
90
100
1.5
1
0.5
0.5
1
1.5
2
time (ms)
0
0.5
0
20
1
0
0.5
0.1
0
40
60
time (ms)
80
100
0.1
0
10
20
30
40
50
60
time (ms)
1
Voltage Trace
0.5
0.5
1.5
Neuron 2
0.2
(Misses + False Positives)/s
0
Recovered Spikes
1.5
Neuron 1
Neuron 2
2
0.1
0.1
(b)
Waveforms
Neuron 1
(a)
1.5
Bin Width ( )
2
70
80
90
100
0.2
70
70.5
71
71.5
time (ms)
72
72.5
73
2.5
Figure 4: (a) Two neural waveforms; each is close to as scaled copy of the other (b) Recovery of
spikes via COMP (magenta) and CBP (cyan) using the SVD basis. CBP tends to recover smallamplitude instances of waveform one where COMP recovers large amplitude instances of waveform
two (c) Top: recovered traces. Lower panel: zooming in on an area of disagreement between COMP
and CBP. The large-ampltude copy of waveform two more closely matches the trace
In the data, the waveform of one neuron resembles a scaled copy of another (Figure 4a).The similarity causes problems for CBP or any other `1 minimization based method that penalizes large
amplitudes. When the second waveform is present with an amplitude of one, CBP is likely to incorrectly add a low-amplitude copy of the first waveform (to reduce the amplitude penalty), instead of
correctly choosing the larger copy of the second waveform; the amplitude penalty for choosing the
correct waveform can outweigh the higher `2 error caused by including the incorrect waveform.
This misassignment is exactly what we observe (Figure 4b). We see that CBP tends to report smallamplitude copies of waveform one where COMP reports large-amplitude copies of waveform two.
Although we lack ground truth, the closer match of the recovered signal to data (Figure 4c) indicates
that the waveform identities and amplitudes identified via COMP better explain the observed signal.
5
Discussion
We have presented a new greedy method called Continuous Orthogonal Matching Pursuit (COMP)
for identifying the timings and amplitudes for waveforms from a signal that has the form of a (noisy)
sum of shifted and scaled copies of several known waveforms. We draw upon the method of Continuous Basis Pursuit, and extend it in several ways. We leverage the success of Orthogonal Matching
Pursuit in the realm of sparse recovery, use a different basis derived from a singular value decomposition, and also introduce a move to the Fourier domain to fine-tune the recovered time shifts.
Our SVD basis can also be used with CBP and in our simulations it increased performance of CBP
as compared to previously used bases. In our simulations COMP obtains increased accuracy as
well as greatly increased speed over CBP across nearly all regimes tested. Our results suggest that
greedy methods of the type introduced here may be quite promising for, among other applications,
spike-sorting during the processing of neural data.
Acknowledgments
This work was supported by the McKnight Foundation (JP), NSF CAREER Award IIS-1150186
(JP), and grants from the NIH (NEI grant EY017366 and NIMH grant MH099611 to AH & JP).
8
References
[1] Scott Shaobing Chen, David L Donoho, and Michael A Saunders. Atomic decomposition by
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[3] Chaitanya Ekanadham, Daniel Tranchina, and Eero P Simoncelli. A blind deconvolution
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9
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4,710 | 5,265 | Fast Sampling-Based Inference in Balanced Neuronal
Networks
Guillaume Hennequin1
[email protected]
Laurence Aitchison2
[email protected]
M?at?e Lengyel1
[email protected]
1
2
Computational & Biological Learning Lab, Dept. of Engineering, University of Cambridge, UK
Gatsby Computational Neuroscience Unit, University College London, UK
Abstract
Multiple lines of evidence support the notion that the brain performs probabilistic
inference in multiple cognitive domains, including perception and decision making. There is also evidence that probabilistic inference may be implemented in the
brain through the (quasi-)stochastic activity of neural circuits, producing samples
from the appropriate posterior distributions, effectively implementing a Markov
chain Monte Carlo algorithm. However, time becomes a fundamental bottleneck
in such sampling-based probabilistic representations: the quality of inferences depends on how fast the neural circuit generates new, uncorrelated samples from
its stationary distribution (the posterior). We explore this bottleneck in a simple, linear-Gaussian latent variable model, in which posterior sampling can be
achieved by stochastic neural networks with linear dynamics. The well-known
Langevin sampling (LS) recipe, so far the only sampling algorithm for continuous variables of which a neural implementation has been suggested, naturally fits
into this dynamical framework. However, we first show analytically and through
simulations that the symmetry of the synaptic weight matrix implied by LS yields
critically slow mixing when the posterior is high-dimensional. Next, using methods from control theory, we construct and inspect networks that are optimally fast,
and hence orders of magnitude faster than LS, while being far more biologically
plausible. In these networks, strong ? but transient ? selective amplification of
external noise generates the spatially correlated activity fluctuations prescribed by
the posterior. Intriguingly, although a detailed balance of excitation and inhibition
is dynamically maintained, detailed balance of Markov chain steps in the resulting
sampler is violated, consistent with recent findings on how statistical irreversibility can overcome the speed limitation of random walks in other domains.
1
Introduction
The high speed of human sensory perception [1] is puzzling given its inherent computational complexity: sensory inputs are noisy and ambiguous, and therefore do not uniquely determine the state
of the environment for the observer, which makes perception akin to a statistical inference problem.
Thus, the brain must represent and compute with complex and often high-dimensional probability
distributions over relevant environmental variables. Most state-of-the-art machine learning techniques for large scale inference trade inference accuracy for computing speed (e.g. [2]). The brain,
on the contrary, seems to enjoy both simultaneously [3].
Some probabilistic computations can be made easier through an appropriate choice of representation for the probability distributions of interest. Sampling-based representations used in Monte Carlo
1
techniques, for example, make computing moments of the distribution or its marginals straightforward. Indeed, recent behavioural and neurophysiological evidence suggests that the brain uses such
sampling-based representations by neural circuit dynamics implementing a Markov chain Monte
Carlo (MCMC) algorithm such that their trajectories in state space produce sequential samples from
the appropriate posterior distribution [4, 5, 6].
However, for sampling-based representations, speed becomes a key bottleneck: computations involving the posterior distribution become accurate only after enough samples have been collected,
and one has no choice but to wait for those samples to be delivered by the circuit dynamics. For
sampling to be of any practical use, the interval that separates the generation of two independent
samples must be short relative to the desired behavioral timescale. Single neurons can integrate
their inputs on a timescale ?m ? 10 ? 50 ms, whereas we must often make decisions in less than
a second: this leaves just enough time to use (i.e. read out) a few tens of samples. What kinds of
neural circuit dynamics are capable of producing uncorrelated samples at ?100 Hz remains unclear.
Here, we introduce a simple yet non-trivial generative model and seek plausible neuronal network
dynamics for fast sampling from the corresponding posterior distribution. While some standard
machine learning techniques such as Langevin or Gibbs sampling do suggest ?neural network?type solutions to sampling, not only are the corresponding architectures implausible in fundamental
ways (e.g. they violate Dale?s law), but we show here that they lead to unacceptably slow mixing
in high dimensions. Although the issue of sampling speed in general is well appreciated in the
context of machine learning, there have been no systematic approaches to tackle it owing to a large
part to the fact that sampling speed can only be evaluated empirically in most cases. In contrast,
the simplicity of our generative model allowed us to draw an analytical picture of the problem
which in turn suggested a systematic approach for solving it. Specifically, we used methods from
robust control to discover the fastest neural-like sampler for our generative model, and to study its
structure. We find that it corresponds to greatly non-symmetric synaptic interactions (leading to
statistical irreversibility), and mathematically nonnormal1 circuit dynamics [7, 8] in close analogy
with the dynamical regime in which the cortex has been suggested to operate [9].
2
Linear networks perform sampling under a linear Gaussian model
We focus on a linear Gaussian latent variable model which generates observations h ? RM as
weighted sums of N features A ? (a1 ; . . . ; aN ) ? RM ?N with jointly Gaussian coefficients r ?
RN , plus independent additive noise terms (Fig. 1, left). More formally:
p(r) = N (r; 0, C)
and
p(h|r) = N h; Ar, ?h2 I
(1)
where I denotes the identity matrix. The posterior distribution is multivariate Gaussian, p(r|h) =
N (r; ?(h), ?), with
?1
? = C?1 + A> A/?h2
and
?(h) = ?A> h/?h2 .
(2)
where we made explicit the fact that under this simple model, only the mean, ?(h), but not the
covariance of the posterior, ?, depends on the input, h.
We are interested in neural circuit dynamics for sampling from p(r|h), whereby the data (observation) h is given as a constant feedforward input to a population of recurrently connected neurons,
each of which encodes one of the latent variables and also receives inputs from an external, private
source of noise ? (Fig. 1, right). Our goal is to devise a network such that the activity fluctuations
r(t) in the recurrent layer have a stationary distribution that matches the posterior, for any h.
Specifically, we consider linear recurrent stochastic dynamics of the form:
r
dt
2
dr =
[?r(t) + Wr(t) + Fh] + ??
d?(t)
?m
?m
(3)
where ?m = 20 ms is the single-unit ?membrane? time constant, and d? is a Wiener process of unit
variance, which is scaled by a scalar noise intensity ?? . The activity ri (t) could represent either the
1
?Nonnormal? should not be confused with ?non-Gaussian?: a matrix M is nonnormal iff MM> 6=
M> M.
2
Posterior sampling:
noise ?
Linear Gaussian latent variable model:
latent
variables
r
P(r) = N (r; 0, C)
observations
h
P(h|r) = N h; Ar, ?h2 I
network r(t)
input h(t)
Figure 1: Sampling under a
linear Gaussian latent variable model using neuronal
Left:
W network dynamics.
schematics of the generative
F
model. Right: schematics of
the recognition model. See text
for details.
membrane potential of neuron i, or the deviation of its momentary firing rate from a baseline. The
matrices F and W contain the feedforward and recurrent connection weights, respectively.
r
?1
The stationary distribution
of r is rindeed Gaussian
with a mean ? (h) = (I ? W) Fh and a cor
r >
variance matrix ? ? (r(t) ? ? )(r(t) ? ? ) t . For the following, we will use the dependence
of ?r on W (and ?? ) given implicitly by the following Lyapunov equation [10]:
(W ? I)?r + ?r (W ? I)> = ?2??2 I
(4)
Note that in the absence of recurrent connectivity (W = 0), the variance of every ri (t) would be
exactly ??2 . Note also that, just as required (see above), only the mean, ?r (h), but not the covariance,
?r , depends on the input, h.
In order for the dynamics of Eq. 3 to sample from the correct posteriors, we must choose F, W and
?? such that ?r (h) = ?(h) for any h, and ?r = ?. One possible solution (which, importantly, is
not unique, as we show later) is
2
F = (?? /?h ) A>
W = WL ? I ? ??2 ??1
and
(5)
with arbitrary ?? > 0.
In the following, we will be interested in the likelihood matrix A only insofar as it affects the
posterior covariance matrix ?, which turns out to be the main determinant of sampling speed. We
will therefore directly choose some covariance matrix ?, and set h = 0 without loss of generality.
3
Langevin sampling is very slow
Langevin sampling (LS) is a common sampling technique [2, 11, 12], and in fact the only one that
has been proposed to be neurally implemented for continuous variables [6, 13]. According to LS, a
stochastic dynamical system performs ?noisy gradient ascent of the log posterior?:
dr =
1 ?
log p(r|h) dt + d?
2 ?r
(6)
where d? is a unitary Wiener process. When r|h is Gaussian, Eq. 6 reduces to Eq. 3 for ?? = 1 and
the choice of F and W given in Eq. 5 ? hence the notation WL above. Note that WL is symmetric.
As we show now, this choice of weight matrix leads to critically slow mixing (i.e. very long autocorrelation time scales in r(t)) when N is large. In a linear network, the average autocorrelation
length is dominated by the decay time constant ?max of the slowest eigenmode, i.e. the eigenvector
of (W ? I) associated with the eigenvalue ?W?I
max which, of all the eigenvalues of (W ? I), has the
largest real part (which must still be negative, to ensure stability). The contribution
of the slowest
W?I
eigenmode to the sample autocorrelation
time
is
?
=
??
/Re
?
,
so
sampling
becomes
max
m
max
very slow when Re ?W?I
approaches
0.
This
is,
in
fact,
what
happens
with
LS
as
N
?
?. Inmax
deed, we could derive the following generic lower bound (details can be found in our Supplementary
Information, SI):
?(?? /?0 )2
WL ?I
(7)
?max
? p
1 + N ?r2
which is shown as dashed lines in Fig. 2. Thus, LS becomes infinitely slow in the large N limit
1
when pairwise correlations do not vanish in that limit (or at least not as fast as N ? 2 in their std.).
Slowing becomes even worse when ? is drawn from the inverse Wishart distribution with ? degrees
of freedom and scale matrix ? ?2 I (Fig. 2). We choose ? = N ?1+b?r?2 c and ? ?2 = ?02 (? ?N ?1)
3
1000
?max /?m
simulation (inverse Wishart)
theory (inverse Wishart)
lower bound (general)
0
L ?I
?W
max
slowing
100
factor
-0.2
-0.4
-0.6
10
?r = 0.10
?r = 0.20
-0.8
(? N (0, ?r ))
-1
1
1
10 100 1000
network size N
1
10 100 1000
network size N
-1 -0.5 0 0.5 1
pairwise corr.
Figure 2: Langevin sampling (LS) is slow in high-dimension. Random covariance matrices ? of
size N are drawn from an inverse Wishart distribution with parameters chosen such that the average
diagonal element (variance) is ?02 = 1 and the distribution of pairwise correlations has zero mean
and variance ?r2 (right). Sampling from N (0, ?) using a stochastic neural network (cf. Fig. 1) with
W = WL (LS, symmetric solution) becomes increasingly slow as N grows, as indicated by the
relative decay time constant ?max /?m of the slowest eigenmode of (WL ? I) (left), which is also
the negative inverse of its largest eigenvalue (middle). Dots indicate the numerical evaluation of the
corresponding quantities, and errorbars (barely noticeable) denote standard deviation across several
random realizations of ?. Dashed lines correspond to the generic bound in Eq. 7. Solid lines are
obtained from random matrix theory under the asssumption that ? is drawn from an inverse Wishart
distribution (Eq. 8). Parameters: ?? = ?0 = 1.
such that the expected value of a diagonal element (variance) in ? is ?02 , and the distribution of
pairwise correlations is centered on zero with variance ?r2 . The asymptotic behavior of the largest
eigenvalue of ??1 (the square of the smallest singular value of a random ? ? N rectangular matrix)
is known from random matrix theory (e.g. [14]), and we have for large N :
q
2
?
(?? /?0 )2
1
?2
L ?I
?W
?
?
(8)
N
?
1
+
b?
c
?
N
?
?O
r
max
N
b?r?2 c ? 2
This scaling behavior is shown in Fig. 2 (solid lines). In fact, we can also show (cf. SI) that LS is
(locally) the slowest possible choice (see Sec. 4 below for a precise definition of ?slowest?, and SI
for details).
Note that both Eqs. 7-8 are inversely proportional to the ratio (?0 /?? ), which tells us how much
the recurrent interactions must amplify the external noise in order to produce samples from the
right stationary activity distribution. The more amplification is required (?0 ?? ), the slower the
dynamics of LS. Conversely, one
? could potentially make Langevin sampling faster by increasing ?? ,
but ?? would need to scale as N to annihilate the critical slowing problem. This ? in itself ? is
unrealistic; moreover, it would also require the resulting connectivity matrix to have a large negative
diagonal (O(?N )) ? ie. the intrinsic neuronal time constant ?m to scale as O(1/N ) ?, which is
perhaps even more unrealistic.2
Note also that LS can be sped up by appropriate ?preconditioning? (e.g. [15, 16]), for example using
the inverse Hessian of the log-posterior. In our case, a simple calculation shows that this corresponds
to removing all recurrent connections, and pushing the posterior covariance matrix to the external
noise sources, which is only postponing the problem to some other brain network.
Finally, LS is fundamentally implausible as a neuronal implementation: it imposes symmetric synaptic interactions, which is simply not possible in the brain due to the existence of distinct classes of
excitatory and inhibitory neurons (Dale?s principle). In the following section, we show that networks
can be constructed that overcome all the above limitations of LS in a principled way.
4
General solution and quantification of sampling speed
While Langevin dynamics (Eq. 6) provide a general recipe for sampling from any given posterior
density, they unduly constrain the recurrent interactions to be symmetric ? at least in the Gaussian
2
From a pure machine learning perspective, increasing ?? is not an option either: the increasing stiffness of
Eq. 6 would either require the use of a very small integration step, or would lead to arbitrarily small acceptance
ratios in the context of Metropolis-Hastings proposals.
4
Langevin
optimal
optimal E/I
random S (? = 0.2; 0.4; 0.8; 1.6)
Newton (unconnected net)
Gibbs (update time ?m )
0 2 4 6 8 10
time lag k (units of ?m )
B
1
0.1
S ? N (0, ? 2 )
C
weight RMS
kK(k ?m )kF
kK(0)kF
1
0.8
0.6
0.4
0.2
0
?slow
A
0.01
0.01 0.1
1
?
10
S ? N (0, ? 2 )
10
1
0.1
0.01
0.01 0.1
1
10
?
Figure 3: How fast is the fastest sampler? (A) Scalar measure of the statistical dependency between any two samples collected k?m seconds apart (cf. main text), for Langevin sampling (black),
Gibbs sampling (blue, assuming a full update sweep is done every ?m ), a series of networks (brown
to red) with connectivities given by Eq. 9 where the elements of the skew-symmetric matrix S were
drawn iid. from N (0, ? 2 ) for different values of ? (see also panel B), the unconstrained optimized
network (yellow), and the optimized E/I network (green). For reference, the dashed gray line shows
the behavior of a network in which there are no recurrent interactions, and the posterior covariance
is encoded in the covariance of the input noise, which in fact corresponds to Langevin sampling
with inverse Hessian (?Newton?-like) preconditioning [16]. (B) Total slowing cost ?slow (S) when
Si<j ? N (0, ? 2 ), for increasing values of ?. The Langevin and the two optimized networks are
shown as horizontal lines for comparison. (C) Same as in (B), showing the root mean square (RMS)
value of the synaptic weights. Parameter values: N = 200, NI = 100, ?? = 1, ?m = 20 ms.
case. To see why this is a drastic restriction, let us observe that any connectivity matrix of the form
W(S) = I + ???2 I + S ??1
(9)
where S is an arbitrary skew-symmetric matrix (S> = ?S), solves Eq. 4, and therefore induces
the correct stationary distribution N (?, ?) under the linear stochastic dynamics of Eq. 3. Note that
Langevin sampling corresponds to S = 0 (cf. Eq. 5). In general, though, there are O(N 2 ) degrees of
freedom in the skew-symmetric matrix S, which could perhaps be exploited to increase the mixing
rate. In Sec. 5, we will show that indeed a large gain in sampling speed can be obtained through an
appropriate choice of S. For now, let us quantify slowness.
Let
matrix that contains all the posterior variances, and K(S, ? ) ?
? ? diag (?) be the diagonal
(r(t + ? ) ? ?)(r(t) ? ?)> t be the matrix of lagged covariances among neurons under the sta1
1
tionary distribution of the dynamics (so that ?? 2 K(S, ? )?? 2 is the autocorrelation matrix of the
network). Note that K(S, 0) = ? is the posterior covariance matrix, and that for fixed ?, ??2 and
?m , K(S, ? ) depends only on the lag ? and on the matrix of recurrent weights W, which itself
depends only on the skew-symmetric matrix S of free parameters. We then define a ?total slowing
cost?
Z ?
2
1
? 21
? 21
?slow (S) =
K(S,
?
)?
(10)
?
d?
2?m N 2 0
F
which penalizes the magnitude of the temporal (normalized) autocorrelations and pairwise cross2
correlations in the sequence of samples generated by the circuit dynamics. Here kMkF ?
P
2
trace(MM> ) = ij Mij
is the squared Frobenius norm of M.
Using the above measure of slowness, we revisit the mixing behavior of LS on a toy covariance
matrix ? drawn from the same inverse Wishart distribution mentioned above with parameters N =
200, ?02 = 2 and ?r = 0.2. We further regularize ? by adding the identity matrix to it, which
does not change anything in terms of the scaling law of Eq. 8 but ensures that the diagonal of WL
remains
bounded as N grows
large. We will use the same ? in the rest of the paper. Figure 3A
shows
??1/2 K(S, ? )??1/2
F as a function of the time lag ? : as predicted in Sec. 3, mixing is
indeed an order of magnitude slower for LS (S = 0, solid black line) than the single-neuron time
constant ?m (grey dashed line). Note that ?slow (Eq. 10, Fig. 3B) is proportional to the area under
the squared curve shown in Fig. 3A. Sample activity traces for this network, implementing LS, can
be found in Fig. 4B (top).
Using the same measure of slowness, we also inspected the speed of Gibbs sampling, another widely
used sampling technique (e.g. [17]) inspiring neural network dynamics for sampling from distributions over binary variables [18, 19, 20]. Gibbs sampling defines a Markov chain that operates in
5
discrete time, and also uses a symmetric weight matrix. In order to compare its mixing speed with
that of our continuous stochastic dynamics, we assume that a full update step (in which all neurons
have been updated once) takes time ?m . We estimated the integrand of the slowing cost (Eq. 10)
numerically using 30?000 samples generated by the Gibbs chain (Fig. 3A, blue). Gibbs sampling is
comparable to LS here: samples are still correlated on a timescale of order ? 50 ?m .
Finally, one may wonder how a random choice of S would perform in terms of decorrelation speed.
We drew random skew-symmetric S matrices from the Gaussian ensemble, Si<j ? N (0, ? 2 ), and
computed the slowing cost (Fig. 3, red). As the magnitude ? of S increases, sampling becomes
faster and faster until the dynamics is about as fast as the single-neuron time constant ?m . However,
the synaptic weights also grow with ? (Fig. 3C), and we show in Sec. 5 that an even faster sampler
exists that has comparatively weaker synapses. It is also interesting to note that the slope of ?slow at
? = 0 is zero, suggesting that LS is in fact maximally slow (we prove this formally in the SI).
5
What is the fastest sampler?
We now show that the skew-symmetric matrix S can be optimized for sampling speed, by directly
minimizing the slowing cost ?slow (S) (Eq. 10), subject to an L2 -norm penalty. We thus seek to
minimize:
?L2
2
L(S) ? ?slow (S) +
kW(S)kF .
(11)
2N 2
The key to performing this minimization is to use classical Ornstein-Uhlenbeck theory (e.g. [10]) to
bring our slowness cost under a form mathematically analogous to a different optimization problem
that has arisen recently in the field of robust control [21]. We can then use analytical results obtained
there concerning the gradient of ?slow , and obtain the overall gradient:
?L(S)
?S
=
?L
1 ?1
(? PQ)> ? (??1 PQ) + 22 S??2 + ??2 S
2
N
N
(12)
where matrices P and Q are obtained by solving two dual Lyapunov equations. All details can be
found in our SI.
We initialized S with random, weak and uncorrelated elements (cf. the end of Sec. 4, with ? = 0.01),
and ran the L-BFGS optimization algorithm using the gradient of Eq. 12 to minimize L(S) (with
?L2 = 0.1). The resulting, optimal sampler is an order of magnitude faster than either Langevin or
Gibbs sampling: samples are decorrelated on a timescale that is even faster than the single-neuron
time constant ?m (Fig. 3A, orange). We also found that fast solutions (with correlation length ? ?m )
can be found irrespective of the size N of the state space (not shown), meaning that the relative
speed-up between the optimal solution and LS grows with N (cf. Fig. 2).
The optimal Sopt induces a weight matrix Wopt given by Eq. 9 and shown in Fig. 4A (middle).
Notably, Wopt is no longer symmetric, and its elements are much larger than in the Langevin
symmetric solution WL with the same stationary covariance, albeit orders of magnitude smaller
than in random networks of comparable decorrelation speed (Fig. 3C).
It is illuminating to visualize activity trajectories in the plane defined by the topmost and bottommost
eigenvectors of ?, i.e. the first and last principal components (PCs) of the network activity (Fig. 4C).
The distribution of interest is broad along some dimensions, and narrow along others. In order to
sample efficiently, large steps ought to be taken along directions in which the distribution is broad,
and small steps along directions in which the distribution is narrow. This is exactly what our optimal
sampler does, whereas LS takes small steps along both broad and narrow directions (Fig. 4C).
6
Balanced E/I networks for fast sampling
We can further constrain our network to obey Dale?s law, i.e. the separation of neurons into separate
excitatory (E) and inhibitory (I) groups. The main difficulty in building such networks is that picking
an arbitrary skew-symmetric matrix S in Eq. 9 will not yield the column sign structure of an E/I
network in general. Therefore, we no longer have a parametric form for the solution matrix manifold
on which to find the fastest network. However, by extending the methods of Sec. 5, described in
6
optimized net.
neuron #
postsynaptic
-1
-1
presynaptic
neuron #
postsynaptic
-0.5
0
-4
ri (t)
-20
0
20
-3
0
3
-1
0
1
-20
0
20
-3
0
3
-1
0
1
8
4
20
0
-4
1
4
0
-4
500 ms
-8
100ms
40
0.5
4
0
-4
D
8
4
optimized E/I net.
0
ri (t)
20
1
0
4
0
-4
dist. of
increments
(1 ms steps)
-8
100ms
40
1
0
0
-4
1
-0.1
8
4
20
trajectories in
state space
(1 ms steps)
last PC
neuron #
postsynaptic
0
ri (t)
40
0.1
C
sample activity traces
Langevin
last PC
B
weight matrices
last PC
A
-8
100ms
-20 0 20
first PC
-3
0
3 -1 0 1
step along
E/I corr.
{first|last} PC
Figure 4: Fast sampling with optimized networks. (A) Synaptic weight matrices for the Langevin
network (top), the fastest sampler (middle) and the fastest sampler that obeys Dale?s law (bottom).
Note that the synaptic weights in both optimized networks are an order of magnitude larger than in
the symmetric Langevin solution. The first two networks are of size N = 200, while the optimized
E/I network has size N + NI = 300. (B) 500 ms of spontaneous network activity (h = 0) in each of
the three networks, for all of which the stationary distribution of r (restricted here to the first 40 neurons) is the same multivariate Gaussian. (C) Left: activity trajectories (the same 500 ms as shown
in (B)) in the plane defined by the topmost and bottommost eigenvectors of the posterior covariance matrix ? (corresponding to the first and last principal components of the activity fluctuations
r(t)). For the E/I network, the projection is restricted to the excitatory neurons. Right: distribution of increments along both axes, measured in 1 ms time steps. Langevin sampling takes steps of
comparable size along all directions, while the optimized networks take much larger steps along the
directions of large variance prescribed by the posterior. (D) Distributions of correlations between
the time courses of total excitatory and inhibitory input in individual neurons.
detail in our SI, we can still formulate the problem as one of unconstrained optimization, and obtain
the fastest, balanced E/I sampler.
We consider the posterior to be encoded in the activity of the N = 200 excitatory neurons, and add
NI = 100 inhibitory neurons which we regard as auxiliary variables, in the spirit of Hamiltonian
Monte Carlo methods [11]. Consequently, the E-I and I-I covariances are free parameters, while
the E-E covariance is given by the target posterior. For additional biological realism, we also forbid
self-connections as they can be interpreted as a modification of the intrinsic membrane time constant
of the single neurons, which in principle cannot be arbitrarily learned.
The speed optimization yields the connectivity matrix shown in Fig. 4A (bottom). Results for this
network are presented in a similar format as before, in the same figures. Sampling is almost as fast
as in the best (regularized) unconstrained network (compare yellow and green in Fig. 3), indicating
that Dale?s law ? unlike the symmetry constraint implicitly present in Langevin sampling ? is not
fundamentally detrimental to mixing speed. Moreover, the network operates in a regime of excitation/inhibition balance, whereby the total E and I input time courses are correlated in single cells
(Fig. 4D, bottom). This is true also in the unconstrained optimal sampler. In contrast, E and I inputs
are strongly anti-correlated in LS.
7
7
Discussion
We have studied sampling for Bayesian inference in neural circuits, and observed that a linear
stochastic network is able to sample from the posterior under a linear Gaussian latent variable model.
Hidden variables are directly encoded in the activity of single neurons, and their joint activity undergoes moment-to-moment fluctuations that visit each portion of the state space at a frequency
given by the target posterior density. To achieve this, external noise sources fed into the network are
amplified by the recurrent circuitry, but preferentially amplified along the state-space directions of
large posterior variance. Although, for the very simple linear Gaussian model we considered here,
a purely feed-forward architecture would also trivially be able to provide independent samples (ie.
provide samples that are decorrelated at the time scale of ?m ), the network required to achieve this
is deeply biologically implausible (see SI).
We have shown that the choice of a symmetric weight matrix ? equivalent to LS, a popular machine learning technique [2, 11, 12] that has been suggested to underlie neuronal network dynamics
sampling continuous variables [6, 13] ? is most unfortunate. We presented an analytical argument
predicting dramatic slowing in high-dimensional latent spaces, supported by numerical simulations.
Even in moderately large networks, samples were correlated on timescales much longer than the
single-neuron decay time constant.
We have also shown that when the above symmetry constraint is relaxed, a family of other solutions
opens up that can potentially lead to much faster sampling. We chose to explore this possibility
from a normative viewpoint, optimizing the network connectivity directly for sampling speed. The
fastest sampler turned out to be highly asymmetric and typically an order of magnitude faster than
Langevin sampling. Notably, we also found that constraining each neuron to be either excitatory
or inhibitory does not impair performance while giving a far more biologically plausible sampler.
Dale?s law could even provide a natural safeguard against reaching slow symmetric solutions such
as Langevin sampling, which we saw was the worst-case scenario (cf. also SI).
It is worth noting that Wopt is strongly nonnormal.3 Deviation from normality has important consequences for the dynamics of our networks: it makes the network sensitive to perturbations along
some directions in state space. Such perturbations are rapidly amplified into large, transient excursions along other, relevant directions. This phenomenon has been shown to explain some key
features of spontaneous activity in primary visual cortex [9] and primary motor cortex [22].
Several aspects would need to be addressed before our proposal can crystalize into a more thorough
understanding of the neural implementation of the sampling hypothesis. First, can local synaptic
plasticity rules perform the optimization that we have approached from an algorithmic viewpoint?
Second, what is the origin of the noise that we have hypothesized to come from external sources?
Third, what kind of nonlinearity must be added in order to allow sampling from non-Gaussian distributions, whose shapes may have non-trivial dependencies on the observations? Also, does the main
insight reached here ? namely that fast samplers are to be found among nonsymmetric, nonnormal
networks ? carry over to the nonlinear case? As a proof of principle, in preliminary simulations, we
have shown that speed optimization in a linearized version of a nonlinear network (with a tanh gain
function) does yield fast sampling in the nonlinear regime, even when fluctuations are strong enough
to trigger the nonlinearity and make the resulting sampled distribution non-Gaussian (details in SI).
Finally, we have also shown (see SI) that the Langevin solution is the only network that satisfies the
detailed balance condition [23] in our model class; reversibility is violated in all other stochastic networks we have presented here (random, optimal, optimal E/I). The fact that these networks are faster
samplers is in line with recent machine learning studies on how non-reversible Markov chains can
mix faster than their reversible counterparts [24]. The construction of such Monte-Carlo algorithms
has proven challenging [25, 26, 27], suggesting that the brain ? if it does indeed use sampling-based
representations ? might have something yet to teach us about machine learning.
Acknowledgements This work was supported by the Wellcome Trust (GH, ML), the Swiss National Science Foundation (GH) and the Gatsby Charitable Foundation (LA). Our code will be made
freely available from GH?s personal webpage.
Indeed, the sum of the squared moduli of its eigenvalues accounts for only 25% of kWopt k2F [7]. For a
P
normal matrix W (such as the Langevin solution, WL ), i |?i |2 = kWk2F , i.e. this ratio is 100%.
3
8
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4,711 | 5,266 | Information-based learning by agents in unbounded
state spaces
Shariq A. Mobin, James A. Arnemann, Friedrich T. Sommer
Redwood Center for Theoretical Neuroscience
University of California, Berkeley
Berkeley, CA 94720
[email protected], [email protected], [email protected]
Abstract
The idea that animals might use information-driven planning to explore an unknown environment and build an internal model of it has been proposed for quite
some time. Recent work has demonstrated that agents using this principle can efficiently learn models of probabilistic environments with discrete, bounded state
spaces. However, animals and robots are commonly confronted with unbounded
environments. To address this more challenging situation, we study informationbased learning strategies of agents in unbounded state spaces using non-parametric
Bayesian models. Specifically, we demonstrate that the Chinese Restaurant Process (CRP) model is able to solve this problem and that an Empirical Bayes version is able to efficiently explore bounded and unbounded worlds by relying on
little prior information.
1
Introduction
Learning in animals involves the active gathering of sensor data, presumably selecting those sensor
inputs that are most useful for learning a model of the world. Thus, a theoretical framework for
the learning in agents, where learning itself is the primary objective, would be essential for making
testable predictions for neuroscience and psychology [9, 7], and it would also impact applications
such as optimal experimental design and building autonomous robots [3].
It has been proposed that information theory-based objective functions, such as those based on the
comparison of learned probability distributions, could guide exploratory behavior in animals and artificial agents [13, 18]. Although reinforcement learning theory has largely advanced in describing
action planning in fully or partially observable worlds with a fixed reward function, e.g., [17], the
study of planning with internally defined and gradually decreasing reward functions has been rather
slow. A few recent studies [20, 11, 12] developed remarkably efficient action policies for learning
an internal model of an unknown fully observable world that are driven by maximizing an objective of predicted information gain. Although using somewhat different definitions of information
gain, the key insights of these studies are that optimization has to be non-greedy, with a longer time
horizon, and that gain in information also translates to efficient reward gathering. However, these
models are still quite limited and cannot be applied to agents in more realistic environments. They
only work in observable, discrete and bounded state spaces. Here, we relax one of these restrictions and present a model for unbounded, observable discrete state spaces. Using methods from
non-parametric Bayesian statistics, specifically the Chinese Restaurant Process (CRP), the resulting
agent can efficiently learn the structure of an unknown, unbounded state space. To our knowledge
this is the first use of CRPs to address this problem, however, CRPs have been introduced earlier to
reinforcement learning for other purposes, such as state clustering [2].
1
2
2.1
Model
Mathematical framework for embodied active learning
In this study we follow [12] and use Controlled Markov Chains (CMC) to describe how an agent
can interact with its environment in closed, embodied, action-perception loops. A CMC is a Markov
Chain with an additional control variable to allow for switching between different transition distributions in each state, e.g. [6]. Put differently, it is a Markov Decision Process (MDP) without the
reward function. A CMC is described by a 3-tuple (S , A , ?) where S denotes a finite set of states,
A is a finite set of actions the agent can take, and ? is a 3-dimensional CMC kernel describing the
transition probabilities between states for each action
?sas0 = ps0 |s,a = P (st+1 = s0 |st = s, at = a)
(1)
Like in [12] we consider the exploration task of the agent to be the formation of an accurate estimate,
b of the true CMC kernel, ?, that describes its world.
or internal model ?,
2.2
Modeling the transition in unbounded state spaces
Let t be the current number of observations of states S and Kt be the number of different states
discovered so far. The observed counts are denoted by Ct := {#1 , ..., #Kt }.
Species sampling models have been proposed as generalizations of the Dirichlet process [14], which
are interesting for non-parametric Bayesian inference in unbounded state spaces. A species sampling
sequence (SSS) describes the distribution of the next observation St+1 . It is defined by
St+1 |S1 , , St ?
Kt
X
pi (Ct )?S? + pKt+1 (Ct )
(2)
i=1
with ?S? a degenerate probability measure, see [10] for details. In order to define a valid SSS,
the sequence (p1 , p2 , ...) must sum to one and be an Exchangeable Partition Probability Function
(EPPF). The exchangeability condition requires that the probabilities depend only on the counts Ct ,
not on the order of how the agent sampled the transitions.
Here we consider one of most common EPPF models in the literature, the Chinese Restaurant Process (CRP) or Polya urn process [1]. According to the CRP model, the probability of observing a
state is
#i
pi (Ct ) =
for i = 1, ..., Kt
(3)
t+?
?
(4)
p? (Ct ) ? pKt+1 (Ct ) =
t+?
where (3) describes revisiting a state and (4) describes the undiscovered probability mass (UPM),
i.e., the probability of discovering a new state, which is then labeled Kt+1 . In the following, the set
of undiscovered states will be denoted by ?. Using this formalism, the agent must define a separate
CRP for each state action pair s, a. The internal model is then described by
b sas0 = ps0 |s,a (Ct ),
?
(5)
b sas0 is suppressed for the sake of notational ease.
updated according to (3, 4). The t index in ?
Our simplest agent uses a CRP (3, 4) with fixed ?. Further, we will investigate an Empirical Bayes
CRP, referred to as EB-CRP, in which the parameter ? is learned and adjusted from observations
online using a maximum likelihood estimate (MLE). This is similar to the approach of [22] but we
follow a more straightforward path and derive a MLE of ? using the EPPF of the CRP and employing
an approximation of the harmonic series.
The likelihood of observing a given number of state counts is described by the EPPF of the CRP [8]
Kt
Y
?Kt
?(Ct ; ?) = Qt?1
(#i ? 1)!
i=0 (? + i) i=1
2
(6)
Maximizing the log likelihood
t?1
d
Kt X 1
ln(?(Ct ; ?)) =
?
=0
d?
?
?+i
i=0
(7)
yields
Kt
(8)
1 ,
1
? 12t
ln(t) + ? + 2t
2
where (8) uses a closed form approximation of the harmonic series in (7) with Euler?s Mascheroni
constant ?. In our EB-CRP agent, the parameter ? is updated after each observation according to
(8).
?(t) ?
2.3
Information-theoretic assessment of learning
Assessing or guiding the progress of the agent in the exploration process can be done by comparing
probability distributions. For example, the learning progress should increase the similarity between
b of the agent and the true model, ?. A popular measure for comparing disthe internal model, ?,
tributions of the same dimensions is the KL Divergence, DKL . However, in our case, with the size
b models of
of the underlying state space unknown and states being discovered successively in ?,
different sizes have to be compared.
To address this, we apply the following padding procedure to the smaller model with fewer discovb has n undiscovered state transitions
ered states and transitions (Figure 1). If the smaller model, ?,
from a known origin state, one splits the UPM uniformly into n equal probabilities (Figure 1a). The
resulting padded model is given by
?
b sa?
?
b sas0 = 0
?
[Figure 1a]
? (|S?sa |?|S?b sa |) , ?
P
b
?sas0 = 1/|S? |,
(9)
s?
/ S?
[Figure 1b]
b
sa
?
?b
b
?sas0 ,
?sas0 > 0
where |S?sa | is the number of known states reachable from state s by taking action a in ?. Further,
b one adds such states and a uniform transition kernel to
if there are undiscovered origin states in ?,
potential target states (Figure 1b).
Figure 1: Illustration of the padding procedure for adding unknown states and state transitions
b of an unbounded environment in order to compare it with a
in a smaller, less informed model, ?,
larger, better informed model, ?. (a) If transitions to target states are missing, we uniformly split the
UPM into equal transition probabilities to the missing target states, which are in fact the unknown
elements of the set ?. (b) If a state is not discovered yet, we paste this state in with a uniform
transition distribution to all target states reachable in the larger model, ?.
With this type of padding procedure we can define a distance between two unequally sized models,
!
X
0
?
sas
P
b sa? ) := DKL (?sa? ||?
b sa? ) :=
?sas0 log2
DKLP (?sa? ||?
,
(10)
bP 0
?
sas
s0 ?S?
sa
and use it to extend previous information measures for assessing and guiding explorative learning
[12] to unbounded state spaces. First, we define Missing Information,
X
b :=
b sa? ),
IM (?||?)
DKLP (?sa? ||?
(11)
s?S ,a?A
3
a quantity an external observer can use for assessing the deficiency of the internal model of the agent
with respect to the true model. Second, we define Information Gain,
0
b ? IM (?||?
b s,a?s ),
IG (s, a, s0 ) := IM (?||?)
(12)
a quantity measuring the improvement between two models, in this case, between the current internal
b and an improved one, ?
b s,a?s0 , which represents an updated model after
model of the agent, ?,
0
observing a new state transition from s to s under action a.
2.4
Predicted information gain
Predicted information gain (PIG) as used in [12] is the expected information gain for a given state
action pair. To extend the previous formula in [12] to compute this expectation in the non-parametric
setting, we again make use of the padding procedure described in the last section
P IG(s, a)
:= Es0 ,?|Ct [IG (s, a, s0 )]
=
b sa? DKLP (?
b s,a??
b sa? ) +
?
||?
sa?
s,a?s0 b
b sas0 DKL (?
b sa?
?
||?sa? )
X
(13)
s0 ?S?
b sa
Here, DKLP handles the case where the agent, during its planning, hypothetically discovers a new
target state, ? ? ?, from the state action pair, s, a. There is one small difference in calculating the
s,a?? . Thus
DKLP from the previous section, which is that in equation (9) S?sa is replaced by S?
b sa
the RHS of (13) can be computed internally by the agent for action planning as it does not contain
the true model, ?.
2.5
Value Iteration
When states of low information gain separate the agent from states of high information gain in the
environment, greedy maximization of PIG performs poorly. Thus, like in [12], we employ value
iteration using the Bellman equations [4]. We begin at a distant time point (? = 0) assigning initial
values to PIG. Then, we propogate backward in time calculating the expected reward.
Q0 (s, a)
Q? ?1 (s, a)
:= P IG(s, a)
(14)
h
b sa? V? (?) +
:= P IG(s, a) + ? ?
X
i
b sas0 V? (s0 )
?
(15)
s0 ?S?
b sa
V? (s)
:= max Q? (s, a)
(16)
a
With the discount factor, ?, set to 0.95, one can define how actions are chosen by all our PIG agents
aP IG := argmax Q?10 (s, a)
(17)
a
3
Experimental Results
Here we describe simulation experiments with our two models, CRP-PIG and EB-CRP-PIG, and
compare them with published approaches. The models are tested in environments defined in the
literature and also in an unbounded world.
First the agents were tested in a bounded maze environment taken from [12] (Figure 2). The state
space in the maze consists of the |S | = 36 rooms. There are |A | = 4 actions that correspond to noisy
translations in the four cardinal directions, drawn from a Dirichlet distribution. To make the task of
learning harder, 30 transporters are distributed amongst the walls which lead to an absorbing state
(state 29 marked by concentric rings in Figure 2). Absorbing states, such as at the bottom of gravity
wells, are common in real world environments and pose serious challenges for many exploration
algorithms [12].
We compare the learning strategies proposed here, CRP-PIG and EB-CRP-PIG, with the following
strategies:
4
Random action: A negative control, representing the minimally directed action policy that
any directed action policy should beat.
Least Taken Action (LTA): A well known explorative strategy that simply takes the action it
has taken least often in the current state [16].
Counter-Based Exploration (CB): Another explorative strategy from the literature that attempts to induce a uniform sampling across states [21].
DP-PIG: The strategy of [12] which applies the same objective function as described here,
but is given the size of the state space and is therefore at an advantage. This agent uses a
Dirichlet process (DP) with ? set to 0.20, which was found empirically to be optimal for the
maze environment.
Unembodied: An agent which can choose any action from any state at each time step (hence
unembodied) and can therefore attain the highest PIG possible at every sampling step. This
strategy represents a positive control.
Figure 2: Bounded Maze environment. Two transition
distributions, ?sa? , are depicted, one for (s=13, a=?left?)
and one for (s=9, a=?up?). Dark versus light gray arrows represent high versus low probabilities. For (s=13,
a=?left?), the agent moves with highest probability left
into a transporter (blue line), leading it to the absorbing
state 29 (blue concentric rings). With smaller probabilities the agent moves up, down or is reflected back to its
current state by the wall to the right. The second transition distribution is displayed similarly.
Figure 3 depicts the missing information (11) in the bounded maze for the various learning strategies over 3000 sampling steps averaged over 200 runs. All PIG-based embodied strategies exhibit
a faster decrease of missing information with sampling, however, still significantly slower than the
unembodied control. In this finite environment the DP-PIG agent with the correct Dirichlet prior
(experimentally optimized ?-parameter) has an advantage over the CRP based agents and reduced
the missing information more quickly. However, the new strategies for unbounded state space still
outperform the competitor agents from the literature by far. Interestingly, EB-CRP-PIG with continuously adjusted ? can reduce missing information significantly faster than CRP-PIG with fixed,
experimentally optimized ? = 0.25.
Figure 3: Missing Information vs. Time for EB-CRP-PIG and several other strategies in the bounded
maze environment.
To directly assess how efficient learning translates to the ability to harvest reward, we consider the 5state ?Chain? problem [19], shown in Figure 4, a popular benchmark problem. In this environment,
agents have two actions available, a and b, which cause transitions between the five states. At each
time step the agent ?slips? and performs the opposite action with probability pslip = 0.2. The agent
receives a reward of 2 for taking action b in any state and a reward of 0 for taking action a in
5
Figure 4: Chain Environment.
every state but the last, in which it receives a reward of 10. The optimal policy is to always choose
action a to reach the highest reward at the end of the chain, it is used as a positive control for this
experiment. We follow the protocol in previous publications and report the cumulative reward in
1000 steps, averaged over 500 runs. Our agent EB-CRP-PIG-R executes the EB-CRP-PIG strategy
for S steps, then computes the best reward policy given its internal model and executes it for the
remaining 1000-S steps. We found S=120 to be roughly optimal for our agent and display the
results of the experiment in Table 1, taking the results of the competitor algorithms directly from
the corresponding papers. The competitor algorithms define their own balance between exploitation
and exploration, leading to different results.
Method
RAM-RMAX [5]
BOSS [2]
exploit [15]
Bayesian DP [19]
EB-CRP-PIG-R
Optimal
Reward
2810
3003
3078
3158 ? 31
3182 ? 25
3658 ? 14
Table 1: Cumulative reward for 1000 steps in the chain environment.
The EB-CRP-PIG-R agent is able to perform the best and significantly outperforms many of the
other strategies. This result is remarkable because the EB-CRP-PIG-R agent has no prior knowledge
of the state space size, unlike all the competitor models. We also note that our algorithm is extremely
efficient computationally, it must approximate the optimal policy only once and then simply execute
it. In comparison, the exploit strategy [15] must compute the approximation at each time step.
Further, we interpret our competitive edge over BOSS to reflect a more efficient exploration strategy.
Specifically, BOSS uses LTA for exploration and Figure 3 indicates that the learning performance
of LTA is far worse than the performance of the PIG-based models.
Figure 5: Missing Information vs. Time for EB-CRP-PIG and CRP-PIG in the unbounded maze
environment.
Finally, we consider an unbounded maze environment with |S | being infinite and with multiple
absorbing states. Figure 5 shows the decrease of missing information (11) for the two CRP based
strategies. Interestingly, like in the bounded maze the Empirical Bayes version reduces the missing
information more rapidly than a CRP which has a fixed, but experimentally optimized, parameter
value. What is important about this result is that EB-CRP-PIG is not only better but it requires no
prior parameter tuning since ? is adjusted intrinsicially. Figure 6 shows how an EB-CRP-PIG and
an LTA agent explore the environment over 6000 steps. The missing information for each state is
6
Figure 6: Unbounded Maze environment. Exploration is depicted for two different agents (a) EBCRP-PIG and (b) LTA, after 2000, 4000, and 6000 exploration steps respectively. Initially all states
are white (not depicted), which represent unexplored states. Transporters (blue lines) move the agent
to the closest gravity well (small blue concentric rings). The current position of the agent is indicated
by the purple arrow.
color coded, light yellow representing high missing information, and red representing low missing
information, less than 1 bit. Note that the EB-CRP-PIG agent explores a much bigger area than the
LTA agent.
The two agents are also tested in a reward task in the unbounded environment for assessing whether
the exploration of EB-CRP-PIG leads to efficient reward acquisition. Specifically, we assign a reward to each state equal to the Euclidian distances from the starting state. Like for the Chain problem
before, we create two agents EB-CRP-PIG-R and LTA-R which each run for 1000 total steps, exploring for S=750 steps (defined previously) and then calculating their best reward policy and executing
it for the remaining 250 steps. The agents are repositioned to the start state after S steps and the
best reward policy is calculated. The simulation results are shown in Table 2. Clearly, the increased
coverage of the EB-CRP-PIG agent also results in higher reward acquisition.
Method
EB-CRP-PIG-R
LTA-R
Reward
1053
812
Table 2: Cumulative reward after 1000 steps in the unbounded maze environment.
7
4
Discussion
To be able to learn environments whose number of states is unknown or even unbounded is crucial
for applications in biology, as well as in robotics. Here we presented a principled information-based
strategy for an agent to learn a model of an unknown, unbounded environment. Specifically, the
proposed model uses the Chinese Restaurant Process (CRP) and a version of predicted information
gain (PIG) [12], adjusted for being able to accommodate comparisons of models with different
numbers of states.
We evaluated our model in three different environments in order to assess its performance. In the
bounded maze environment the new algorithm performed quite similarly to DP-PIG despite being at
a disadvantage in terms of prior knowledge. This result suggests that agents exploring environments
of unknown size can still develop accurate models of it quite rapidly. Since the new model is based
on the CRP, calculating the posterior and sampling from it is easily tractable.
The experiments in a simple bounded reward task, the Chain environment, were equally encouraging. Although the agent was unaware of the size of its environment, it was able to learn the states
and their transition probabilities quickly and retrieved a cumulative reward that was competitive with
published results. Some of the competitor strategies (exploit [15]) required to recompute the best
reward policy for each step. In contrast, EB-CRP-PIG computed the best policy only once, yet, was
able to outperform the exploit [15] strategy.
In the unbounded maze environment, EB-CRP-PIG was able to outperform CRP-PIG even though
it required no prior parameter tuning. In addition, it covered much more ground during exploration
than LTA, one of the few existing competitor models able to function in unbounded environments.
Specifically, the EB-CRP-PIG model evenly explored a large number of environmental states. In
contrast, LTA, exhaustively explored a much smaller area limited by two nearby absorbing states.
Two caveats need to be mentioned. First, although the computational complexity of the CRP is low,
the complexity of the value iteration algorithm scales linearly with the number of states discovered.
Thus, tractability of value iteration is an issue in EB-CRP-PIG. A possible remedy to this problem
would be to only calculate value iteration for states that are reachable from the current state in the
calculated time horizon. Second, the described padding procedure implicitly sets a balance between
seeking to discover new state transitions versus sampling from known ones. For different goals or
environments this balance may not be optimal, a future investigation of alternatives for comparing
models of different sizes would be very interesting.
All told, the proposed novel models overcome a major limitation of information-based learning
methods, the assumption of a bounded state space of known size. Since the new models are based
on the CRP, sampling is quite tractable. Interestingly, by applying Empirical Bayes for continuously
updating the parameter of the CRP, we are able to build agents that can explore bounded or unbounded environments with very little prior information. For describing learning in animals, models
that easily adapt to diverse environments could be crucial. Of course, other restrictictions in these
models still need to be addressed, in particular, the limitation to discrete and fully observable state
spaces. For example, the need to act in continuous state spaces is obviously crucial for animals and
robots. Further, recent literature [7] supports that information-based learning in partially observable
state spaces, like POMDPs [17], will be important to address applications in neuroscience.
5
Acknowledgements
JAA was funded by NSF grant IIS-1111765. FTS was supported by the Director, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program of the
U.S. Department of Energy under Contract No. DE-AC02-05CH11231. The authors thank Bruno
Olshausen, Tamara Broderick, and the members of the Redwood Center for Theoretical Neuroscience for their valuable input.
8
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[19] Malcolm Strens. A bayesian framework for reinforcement learning. In ICML, pages 943?950, 2000.
[20] Yi Sun, Faustino Gomez, and J?urgen Schmidhuber. Planning to be surprised: Optimal bayesian exploration in dynamic environments. In Artificial General Intelligence, pages 41?51. Springer, 2011.
[21] Sebastian B Thrun. Efficient exploration in reinforcement learning. 1992.
[22] Jian Zhang, Zoubin Ghahramani, and Yiming Yang. A probabilistic model for online document clustering
with application to novelty detection. In NIPS, volume 4, pages 1617?1624, 2004.
9
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4,712 | 5,267 | On the Computational Efficiency of Training Neural
Networks
Roi Livni
The Hebrew University
[email protected]
Shai Shalev-Shwartz
The Hebrew University
[email protected]
Ohad Shamir
Weizmann Institute of Science
[email protected]
Abstract
It is well-known that neural networks are computationally hard to train. On the
other hand, in practice, modern day neural networks are trained efficiently using SGD and a variety of tricks that include different activation functions (e.g.
ReLU), over-specification (i.e., train networks which are larger than needed), and
regularization. In this paper we revisit the computational complexity of training
neural networks from a modern perspective. We provide both positive and negative results, some of them yield new provably efficient and practical algorithms
for training certain types of neural networks.
1
Introduction
One of the most significant recent developments in machine learning has been the resurgence of
?deep learning?, usually in the form of artificial neural networks. A combination of algorithmic
advancements, as well as increasing computational power and data size, has led to a breakthrough
in the effectiveness of neural networks, and they have been used to obtain very impressive practical
performance on a variety of domains (a few recent examples include [17, 16, 24, 10, 7]).
A neural network can be described by a (directed acyclic) graph, where each vertex in the graph corresponds to a neuron and each edge is associated with a weight. Each neuron calculates a weighted
sum of the outputs of neurons which are connected to it (and possibly adds a bias term). It then
passes the resulting number through an activation function ? : R ? R and outputs the resulting
number. We focus on feed-forward neural networks, where the neurons are arranged in layers, in
which the output of each layer forms the input of the next layer. Intuitively, the input goes through
several transformations, with higher-level concepts derived from lower-level ones. The depth of the
network is the number of layers and the size of the network is the total number of neurons.
From the perspective of statistical learning theory, by specifying a neural network architecture (i.e.
the underlying graph and the activation function) we obtain a hypothesis class, namely, the set of all
prediction rules obtained by using the same network architecture while changing the weights of the
network. Learning the class involves finding a specific set of weights, based on training examples,
which yields a predictor that has good performance on future examples. When studying a hypothesis
class we are usually concerned with three questions:
1. Sample complexity: how many examples are required to learn the class.
2. Expressiveness: what type of functions can be expressed by predictors in the class.
3. Training time: how much computation time is required to learn the class.
For simplicity, let us first consider neural networks with a threshold activation function (i.e. ?(z) =
1 if z > 0 and 0 otherwise), over the boolean input space, {0, 1}d , and with a single output in
{0, 1}. The sample complexity of such neural networks is well understood [3]. It is known that the
VC dimension grows linearly with the number of edges (up to log factors). It is also easy to see that
no matter what the activation function is, as long as we represent each weight of the network using
1
a constant number of bits, the VC dimension is bounded by a constant times the number of edges.
This implies that empirical risk minimization - or finding weights with small average loss over the
training data - can be an effective learning strategy from a statistical point of view.
As to the expressiveness of such networks, it is easy to see that neural networks of depth 2 and
sufficient size can express all functions from {0, 1}d to {0, 1}. However, it is also possible to show
that for this to happen, the size of the network must be exponential in d (e.g. [19, Chapter 20]).
Which functions can we express using a network of polynomial size? The theorem below shows that
all boolean functions that can be calculated in time O(T (d)), can also be expressed by a network of
depth O(T (d)) and size O(T (d)2 ).
Theorem 1. Let T : N ? N and for every d, let Fd be the set of functions that can be implemented
by a Turing machine using at most T (d) operations. Then there exist constants b, c ? R+ such that
for every d, there is a network architecture of depth c T (d) + b, size of (c T (d) + b)2 , and threshold
activation function, such that the resulting hypotesis class contains Fd .
The proof of the theorem follows directly from the relation between the time complexity of programs
and their circuit complexity (see, e.g., [22]), and the fact that we can simulate the standard boolean
gates using a fixed number of neurons.
We see that from the statistical perspective, neural networks form an excellent hypothesis class; On
one hand, for every runtime T (d), by using depth of O(T (d)) we contain all predictors that can be
run in time at most T (d). On the other hand, the sample complexity of the resulting class depends
polynomially on T (d).
The main caveat of neural networks is the training time. Existing theoretical results are mostly
negative, showing that successfully learning with these networks is computationally hard in the worst
case. For example, neural networks of depth 2 contain the class of intersection of halfspaces (where
the number of halfspaces is the number of neurons in the hidden layer). By reduction to k-coloring,
it has been shown that finding the weights that best fit the training set is NP-hard ([9]). [6] has
shown that even finding weights that result in close-to-minimal empirical error is computationally
infeasible. These hardness results focus on proper learning, where the goal is to find a nearly-optimal
predictor with a fixed network architecture A. However, if our goal is to find a good predictor, there
is no reason to limit ourselves to predictors with one particular architecture. Instead, we can try,
for example, to find a network with a different architecture A0 , which is almost as good as the
best network with architecture A. This is an example of the powerful concept of improper learning,
which has often proved useful in circumventing computational hardness results. Unfortunately, there
are hardness results showing that even with improper learning, and even if the data is generated
exactly from a small, depth-2 neural network, there are no efficient algorithms which can find a
predictor that performs well on test data. In particular, [15] and [12] have shown this in the case of
learning intersections of halfspaces, using cryptographic and average case complexity assumptions.
On a related note, [4] recently showed positive results on learning from data generated by a neural
network of a certain architecture and randomly connected weights. However, the assumptions used
are strong and unlikely to hold in practice.
Despite this theoretical pessimism, in practice, modern-day neural networks are trained successfully
in many learning problems. There are several tricks that enable successful training:
? Changing the activation function: The threshold activation function, ?(a) = 1a>0 , has zero
derivative almost everywhere. Therefore, we cannot apply gradient-based methods with this activation function. To circumvent this problem, we can consider other activation functions. Most
1
widely known is a sigmoidal activation, e.g. ?(a) = 1+e
a , which forms a smooth approximation of the threshold function. Another recent popular activation function is the rectified linear
unit (ReLU) function, ?(a) = max{0, a}. Note that subtracting a shifted ReLU from a ReLU
yields an approximation of the threshold function, so by doubling the number of neurons we can
approximate a network with threshold activation by a network with ReLU activation.
? Over-specification: It was empirically observed that it is easier to train networks which are larger
than needed. Indeed, we empirically demonstrate this phenomenon in Sec. 5.
? Regularization: It was empirically observed that regularizing the weights of the network speeds
up the convergence (e.g. [16]).
2
The goal of this paper is to revisit and re-raise the question of neural network?s computational efficiency, from a modern perspective. This is a challenging topic, and we do not pretend to give any
definite answers. However, we provide several results, both positive and negative. Most of them are
new, although a few appeared in the literature in other contexts. Our contributions are as follows:
? We make a simple observation that for sufficiently over-specified networks, global optima are
ubiquitous and in general computationally easy to find. Although this holds only for extremely
large networks which will overfit, it can be seen as an indication that the computational hardness of learning does decrease with the amount of over-specification. This is also demonstrated
empirically in Sec. 5.
? Motivated by the idea of changing the activation function, we consider the quadratic activation
function, ?(a) = a2 . Networks with the quadratic activation compute polynomial functions of
the input in Rd , hence we call them polynomial networks. Our main findings for such networks
are as follows:
? Networks with quadratic activation are as expressive as networks with threshold activation.
? Constant depth networks with quadratic activation can be learned in polynomial time.
? Sigmoidal networks of depth 2, and with `1 regularization, can be approximated by polynomial
networks of depth O(log log(1/)). It follows that sigmoidal networks with `1 regularization
can be learned in polynomial time as well.
? The aforementioned positive results are interesting theoretically, but lead to impractical algorithms. We provide a practical, provably correct, algorithm for training depth-2 polynomial
networks. While such networks can also be learned using a linearization trick, our algorithm is
more efficient and returns networks whose size does not depend on the data dimension. Our algorithm follows a forward greedy selection procedure, where each step of the greedy selection
procedure builds a new neuron by solving an eigenvalue problem.
? We generalize the above algorithm to depth-3, in which each forward greedy step involves an
efficient approximate solution to a tensor approximation problem. The algorithm can learn a
rich sub-class of depth-3 polynomial networks.
? We describe some experimental evidence, showing that our practical algorithm is competitive
with state-of-the-art neural network training methods for depth-2 networks.
2
Sufficiently Over-Specified Networks Are Easy to Train
We begin by considering the idea of over-specification, and make an observation that for sufficiently
over-specified networks, the optimization problem associated with training them is generally quite
easy to solve, and that global optima are in a sense ubiquitous. As an interesting contrast, note that
for very small networks (such as a single neuron with a non-convex activation function), the associated optimization problem is generally hard, and can exhibit exponentially many local (non-global)
minima [5]. We emphasize that our observation only holds for extremely large networks, which will
overfit in any reasonable scenario, but it does point to a possible spectrum where computational cost
decreases with the amount of over-specification.
To present the result, let X ? Rd,m be a matrix of m training examples in Rd . We can think of the
network as composed of two mappings. The first maps X into a matrix Z ? Rn,m , where n is the
number of neurons whose outputs are connected to the output layer. The second mapping is a linear
mapping Z 7? W Z, where W ? Ro,n , that maps Z to the o neurons in the output layer. Finally,
there is a loss function ` : Ro,m ? R, which we?ll assume to be convex, that assesses the quality of
the prediction on the entire data (and will of course depend on the m labels). Let V denote all the
weights that affect the mapping from X to Z, and denote by f (V ) the function that maps V to Z.
The optimization problem associated with learning the network is therefore minW,V `(W f (V )).
The function `(W f (V )) is generally non-convex, and may have local minima. However, if n ? m,
then it is reasonable to assume that Rank(f (V )) = m with large probability (under some random
choice of V ), due to the non-linear nature of the function computed by neural networks1 . In that
case, we can simply fix V and solve minW `(W f (V )), which is computationally tractable as ` is
1
For example, consider the function computed by the first layer, X 7? ?(Vd X), where ? is a sigmoid
function. Since ? is non-linear, the columns of ?(Vd X) will not be linearly dependent in general.
3
assumed to be convex. Since f (V ) has full rank, the solution of this problem corresponds to a global
optima of `, and hence to a global optima of the original optimization problem. Thus, for sufficiently
large networks, finding global optima is generally easy, and they are in a sense ubiquitous.
3
The Hardness of Learning Neural Networks
We now review several known hardness results and apply them to our learning setting. For simplicity, throughout most of this section we focus on the PAC model in the binary classification case, over
the Boolean cube, in the realizable case, and with a fixed target accuracy.2
Fix some , ? ? (0, 1). For every dimension d, let the input space be Xd = {0, 1}d and let H be a
hypothesis class of functions from Xd to {?1}. We often omit the subscript d when it is clear from
context. A learning algorithm A has access to an oracle that samples x according to an unknown
distribution D over X and returns (x, f ? (x)), where f ? is some unknown target hypothesis in H.
The objective of the algorithm is to return a classifier f : X ? {?1}, such that with probability of
at least 1 ? ?,
Px?D [f (x) 6= f ? (x)] ? .
We say that A is efficient if it runs in time poly(d) and the function it returns can also be evaluated
on a new instance in time poly(d). If there is such A, we say that H is efficiently learnable.
In the context of neural networks, every network architecture defines a hypothesis class, Nt,n,? ,
that contains all target functions f that can be implemented using a neural network with t layers, n
neurons (excluding input neurons), and an activation function ?. The immediate question is which
Nt,n,? are efficiently learnable. We will first address this question for the threshold activation function, ?0,1 (z) = 1 if z > 0 and 0 otherwise.
Observing that depth-2 networks with the threshold activation function can implement intersections
of halfspaces, we will rely on the following hardness results, due to [15].
Theorem 2 (Theorem 1.2 in [15]). Let X = {?1}d , let
H a = x ? ?0,1 w> x ? b ? 1/2 : b ? N, w ? Nd , |b| + kwk1 ? poly(d) ,
and let Hka = {x ? h1 (x) ? h2 (x) ? . . . ? hk (x) : ?i, hi ? H a }, where k = d? for some constant
? > 0. Then under a certain cryptographic assumption, Hka is not efficiently learnable.
Under a different complexity assumption, [12] showed a similar result even for k = ?(1).
As mentioned before, neural networks of depth ? 2 and with the ?0,1 activation function can
express intersections of halfspaces: For example, the first layer consists of k neurons computing the
P k halfspaces, and the second layer computes their conjunction by the mapping x 7?
?0,1 ( i xi ? k + 1/2). Trivially, if some class H is not efficiently learnable, then any class containing it is also not efficiently learnable. We thus obtain the following corollary:
Corollary 1. For every t ? 2, n = ?(1), the class Nt,n,?0,1 is not efficiently learnable (under the
complexity assumption given in [12]).
What happens when we change the activation function? In particular, two widely used activation
functions for neural networks are the sigmoidal activation function, ?sig (z) = 1/(1 + exp(?z)),
and the rectified linear unit (ReLU) activation function, ?relu (z) = max{z, 0}.
As a first observation, note that for |z| 1 we have that ?sig (z) ? ?0,1 (z). Our data domain is
the discrete Boolean cube, hence if we allow the weights of the network to be arbitrarily large, then
Nt,n,?0,1 ? Nt,n,?sig . Similarly, the function ?relu (z)??relu (z?1) equals ?0,1 (z) for every |z| ? 1.
As a result, without restricting the weights, we can simulate each threshold activated neuron by two
ReLU activated neurons, which implies that Nt,n,?0,1 ? Nt,2n,?relu . Hence, Corollary 1 applies to
both sigmoidal networks and ReLU networks as well, as long as we do not regularize the weights of
the network.
2
While we focus on the realizable case (i.e., there exists f ? ? H that provides perfect predictions), with a
fixed accuracy () and confidence (?), since we are dealing with hardness results, the results trivially apply to
the agnostic case and to learning with arbitrarily small accuracy and confidence parameters.
4
What happens when we do regularize the weights? Let Nt,n,?,L be all target functions that can be
implemented using a neural network of depth t, size n, activation function ?, and when we restrict
the input weights of each neuron to be kwk1 + |b| ? L.
One may argue that in many real world distributions, the difference between the two classes, Nt,n,?,L
and Nt,n,?0,1 is small. Roughly speaking, when the distribution density is low around the decision
boundary of neurons (similarly to separation with margin assumptions), then sigmoidal neurons will
be able to effectively simulate threshold activated neurons.
In practice, the sigmoid and ReLU activation functions are advantageous over the threshold activation function, since they can be trained using gradient based methods. Can these empirical successes
be turned into formal guarantees? Unfortunately, a closer examination of Thm. 2 demonstrates that
if L = ?(d) then learning N2,n,?sig ,L and N2,n,?relu ,L is still hard. Formally, to apply these networks to binary classification, we follow a standard definition of learning with a margin assumption:
We assume that the learner receives examples of the form (x, sign(f ? (x))) where f ? is a real-valued
function that comes from the hypothesis class, and we further assume that |f ? (x)| ? 1. Even under
this margin assumption, we have the following:
Corollary 2. For every t ? 2, n = ?(1), L = ?(d), the classes Nt,n,?sig ,L and Nt,n,?relu ,L are not
efficiently learnable (under the complexity assumption given in [12]).
A proof is provided in the appendix. What happens when L is much smaller? Later on in the paper
we will show positive results for L being a constant and the depth being fixed. These results will be
obtained using polynomial networks, which we study in the next section.
4
Polynomial Networks
In the previous section we have shown several strong negative results for learning neural networks
with the threshold, sigmoidal, and ReLU activation functions. One way to circumvent these hardness
results is by considering another activation function. Maybe the simplest non-linear function is
the squared function, ?2 (x) = x2 . We call networks that use this activation function polynomial
networks, since they compute polynomial functions of their inputs. As in the previous section, we
denote by Nt,n,?2 ,L the class of functions that can be implemented using a neural network of depth
t, size n, squared activation function, and a bound L on the `1 norm of the input weights of each
neuron. Whenever we do not specify L we refer to polynomial networks with unbounded weights.
Below we study the expressiveness and computational complexity of polynomial networks. We
note that algorithms for efficiently learning (real-valued) sparse or low-degree polynomials has been
studied in several previous works (e.g. [13, 14, 8, 2, 1]). However, these rely on strong distributional
assumptions, such as the data instances having a uniform or log-concave distribution, while we are
interested in a distribution-free setting.
4.1
Expressiveness
We first show that, similarly to networks with threshold activation, polynomial networks of polynomial size can express all functions that can be implemented efficiently using a Turing machine.
Theorem 3 (Polynomial networks can express Turing Machines). Let Fd and T be as in Thm. 1.
Then there exist constants b, c ? R+ such that for every d, the class Nt,n,?2 ,L , with t =
c T (d) log(T (d)) + b, n = t2 , and L = b, contains Fd .
The proof of the theorem relies on the result of [18] and is given in the appendix.
Another relevant expressiveness result, which we will use later, shows that polynomial networks can
approximate networks with sigmoidal activation functions:
?
Theorem 4. Fix 0 < < 1, L ? 3 and t ? N. There are Bt ? O(log(tL
+ L log 1 )) and
1
?
Bn ? O(tL + L log ) such that for every f ? Nt,n,?sig ,L there is a function g ? NtBt ,nBn ,?2 , such
that supkxk? <1 kf (x) ? g(x)k? ? .
The proof relies on an approximation of the sigmoid function based on Chebyshev polynomials, as
was done in [21], and is given in the appendix.
5
4.2
Training Time
We now turn to the computational complexity of learning polynomial networks. We first show that
it is hard to learn polynomial networks of depth ?(log(d)). Indeed, by combining Thm. 4 and
Corollary 2 we obtain the following:
Corollary 3. The class Nt,n,?2 , where t = ?(log(d)) and n = ?(d), is not efficiently learnable.
On the flip side, constant-depth polynomial networks can be learned in polynomial time, using a
simple linearization trick. Specifically, the class of polynomial networks of constant depth t is
contained in the class of multivariate polynomials of total degree at most s = 2t . This class can
be represented as a ds -dimensional linear space, where each vector is the coefficient vector of some
such polynomial. Therefore, the class of polynomial networks of depth t can be learned in time
t
poly(d2 ), by mapping each instance vector x ? Rd to all of its monomials, and learning a linear
predictor on top of this representation (which can be done efficiently in the realizable case, or when
a convex loss function is used). In particular, if t is a constant then so is 2t and therefore polynomial
networks of constant depth are efficiently learnable. Another way to learn this class is using support
vector machines with polynomial kernels.
An interesting application of this observation is that depth-2 sigmoidal networks are efficiently learnable with sufficient regularization, as formalized in the result below. This contrasts with corollary 2,
which provides a hardness result without regularization.
Theorem 5. The class N2,n,?sig ,L can be learned, to accuracy , in time poly(T ) where T =
2
(1/) ? O(d4L ln(11L +1) ).
The idea of the proof is as follows. Suppose that we obtain data from some f ? N2,n,?sig ,L . Based
on Thm. 4, there is g ? N2Bt ,nBn ,?2 that approximates f to some fixed accuracy 0 = 0.5, where Bt
and Bn are as defined in Thm. 4 for t = 2. Now we can learn N2Bt ,nBn ,?2 by considering the class
of all polynomials of total degree 22Bt , and applying the linearization technique discussed above.
Since f is assumed to separate the data with margin 1 (i.e. y = sign(f ? (x)),|f ? (x)| ? 1|), then g
separates the data with margin 0.5, which is enough for establishing accuracy in sample and time
that depends polynomially on 1/.
4.3
Learning 2-layer and 3-layer Polynomial Networks
While interesting theoretically, the above results are not very practical, since the time and sample
complexity grow very fast with the depth of the network.3 In this section we describe practical,
provably correct, algorithms for the special case of depth-2 and depth-3 polynomial networks, with
some additional constraints. Although such networks can be learned in polynomial time via explicit
linearization (as described in section 4.2), the runtime and resulting network size scales quadratically
(for depth-2) or cubically (for depth-3) with the data dimension d. In contrast, our algorithms and
guarantees have a much milder dependence on d.
We first consider 2 layer polynomial networks, of the following form:
(
)
k
X
>
> 2
P2,k = x 7? b + w0 x +
?i (wi x) : ?i ? 1, |?i | ? 1, kwi k2 = 1 .
i=1
This networks corresponds to one hidden layer containing r neurons with the squared activation
function, where we restrict the input weights of all neurons in the network to have bounded `2 norm,
and where we also allow a direct linear dependency between the input layer and the output layer.
We?ll describe an efficient algorithm for learning this class, which is based on the GECO algorithm
for convex optimization with low-rank constraints [20].
3
If one uses SVM with polynomial kernels, the time and sample complexity may be small under margin
assumptions in a feature space corresponding to a given kernel. Note, however, that large margin in that space
is very different than the assumption we make here, namely, that there is a network with a small number of
hidden neurons that works well on the data.
6
The goal of the algorithm is to find f that minimizes the objective
m
R(f ) =
1 X
`(f (xi ), yi ),
m i=1
(1)
where ` : R ? R ? R is a loss function. We?ll assume that ` is ?-smooth and convex.
The basic idea of the algorithm is to gradually add hidden neurons to the hidden layer, in a greedy
manner, so as to decrease the loss function over the data. To do so, define V = {x 7? (w> x)2 :
kwk2 = 1} the set of functions that can be implemented by hidden neurons. Then every f ? P2,r
is an affine function plus a weighted sum of functions from V. The algorithm starts with f being
the minimizer of R over all affine functions. Then at each greedy step, we search for g ? V that
minimizes a first order approximation of R(f + ?g):
m
R(f + ?g) ? R(f ) + ?
1 X 0
` (f (xi ), yi )g(xi ) ,
m i=1
(2)
where `0 is the derivative of ` w.r.t. its first argument. Observe that for every g ? V there is some w
> 2
>
>
with kwk2 = 1 for which g(x) = (w
the right-hand side of Eq. (2) can
Pm x) 0 = w xx w.>Hence,
> 1
be rewritten as R(f ) + ? w m i=1 ` (f (xi ), yi )xi xi w . The vector w that minimizes this
Pm 0
1
>
expression (for positive ?) is the leading eigenvector of the matrix m
i=1 ` (f (xi ), yi )xi xi .
We add this vector as a hidden neuron to the network.4 Finally, we minimize R w.r.t. the weights
from the hidden layer to the output layer (namely, w.r.t. the weights ?i ).
The following theorem, which follows directly from Theorem 1 of [20], provides convergence guarantee for GECO. Observe that the theorem gives guarantee for learning P2,k if we allow to output
an over-specified network.
Theorem 6. Fix some > 0. Assume that the loss function is convex and ?-smooth. Then if
2
the GECO Algorithm is run for r > 2?k
iterations, it outputs a network f ? N2,r,?2 for which
?
R(f ) ? minf ? ?P2,k R(f ) + .
We next consider a hypothesis class consisting of third degree polynomials, which is a subset of
3-layer polynomial networks (see Lemma 1nin the appendix) . The hidden neurons
o will be functions
Qi
from the class: V = ?3i=1 Vi where Vi = x 7? j=1 (wj> x) : ?j, kwj k2 = 1 . The hypothesis
n
o
Pk
class we consider is P3,k = x 7? i=1 ?i gi (x) : ?i, |?i | ? 1, gi ? V .
The basic idea of the algorithm is the same as for 2-layer networks. However, while in the 2-layer
case we could implement efficiently each greedy step by solving an eigenvalue problem, we now
face the following tensor approximation problem at each greedy step:
m
max
g?V3
m
1 X 0
1 X 0
` (f (xi ), yi )g(xi ) =
max
` (f (xi ), yi )(w> xi )(u> xi )(v> xi ) .
m i=1
kwk=1,kuk=1,kvk=1 m
i=1
While this is in general a hard optimization problem, we can approximate it ? and luckily, an approximate greedy step suffices for success of the greedy procedure. This procedure is given in Figure 1,
and is again based on an approximate eigenvector computation. A guarantee for the quality of approximation is given in the appendix, and this leads to the following theorem, whose proof is given
in the appendix.
Theorem 7. Fix some ?, > 0. Assume that the loss function is convex and ?-smooth. Then if the
4d?k2
GECO Algorithm is run for r > (1??
)2 iterations, where each iteration relies on the approximation
procedure given in Fig. 1, then with probability (1??)r , it outputs a network f ? N3,5r,?2 for which
R(f ) ? minf ? ?P3,k R(f ? ) + .
4
It is also possible to find an approximate solution to the eigenvalue problem and still retain the performance
guarantees (see [20]). Since an approximate eigenvalue can be found in time O(d) using the power method, we
obtain the runtime of GECO depends linearly on d.
7
d
m
Input: {xi }m
i=1 ? R ? ? R , ? ,?
?
approximate
solution to
Output: A 1??
d
X
max
F (w, u, v) =
?i (w> xi )(u> xi )(v> xi )
kwk,kuk,kvk=1
i
Pick randomly w1 , . . . , ws iid according to N (0, Id ).
For t = 1, . . . , 2d log 1?
wt
wt ? kw
tk
P
Let A = i ?i (wt> xi )xi x>
i and set ut , vt s.t:
>
T r(u>
Av
)
?
(1
?
?
)
max
t
kuk,kvk=1 T r(u Av).
t
Return w, u, v the maximizers of maxi?s F (wi , ui , ui ).
Figure 1: Approximate tensor maximization.
5
Experiments
To demonstrate the practicality of GECO to train neural networks for real world problems, we considered a pedestrian detection problem as follows. We collected 200k training examples of image
patches of size 88x40 pixels containing either pedestrians (positive examples) or hard negative examples (containing images that were classified as pedestrians by applying a simple linear classifier in
a sliding window manner). See a few examples of images above. We used half of the examples as a
training set and the other half as a test set. We calculated HoG
features ([11]) from the images5 . We then trained, using GECO,
0.1
SGD ReLU
a depth-2 polynomial network on the resulting features. We
SGD Squared
GECO
9 ? 10
used 40 neurons in the hidden layer. For comparison we trained
the same network architecture (i.e. 40 hidden neurons with a
8 ? 10
squared activation function) by SGD. We also trained a similar
7 ? 10
network (40 hidden neurons again) with the ReLU activation
6 ? 10
function. For the SGD implementation we tried the following
tricks to speed up the convergence: heuristics for initialization
5 ? 10
of the weights, learning rate rules, mini-batches, Nesterov?s mo0
0.2
0.4
0.6
0.8
1
iterations
mentum (as explained in [23]), and dropout. The test errors of
?10
SGD as a function of the number of iterations are depicted on
4
1
2
the top plot of the Figure on the side. We also mark the perfor4
mance of GECO as a straight line (since it doesn?t involve SGD
3
8
iterations). As can be seen, the error of GECO is slightly bet2
ter than SGD. It should be also noted that we had to perform a
very large number of SGD iterations to obtain a good solution,
1
while the runtime of GECO was much faster. This indicates that
GECO may be a valid alternative approach to SGD for training
0
0
0.2
0.4
0.6
0.8
1
depth-2 networks. It is also apparent that the squared activation
#iterations
?10
function is slightly better than the ReLU function for this task.
?2
Error
?2
?2
?2
?2
MSE
5
5
The second plot of the side figure demonstrates the benefit of
over-specification for SGD. We generated random examples in R150 and passed them through a
random depth-2 network that contains 60 hidden neurons with the ReLU activation function. We
then tried to fit a new network to this data with over-specification factors of 1, 2, 4, 8 (e.g., overspecification factor of 4 means that we used 60 ? 4 = 240 hidden neurons). As can be clearly seen,
SGD converges much faster when we over-specify the network.
Acknowledgements: This research is supported by Intel (ICRI-CI). OS was also supported by
an ISF grant (No. 425/13), and a Marie-Curie Career Integration Grant. SSS and RL were also
supported by the MOS center of Knowledge for AI and ML (No. 3-9243). RL is a recipient of the
Google Europe Fellowship in Learning Theory, and this research is supported in part by this Google
Fellowship. We thank Itay Safran for spotting a mistake in a previous version of Sec. 2 and to James
Martens for helpful discussions.
5
Using the Matlab implementation provided in http://www.mathworks.com/matlabcentral/
fileexchange/33863-histograms-of-oriented-gradients.
8
References
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9
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4,713 | 5,268 | Attentional Neural Network: Feature Selection Using
Cognitive Feedback
Qian Wang
Department of Biomedical Engineering
Tsinghua University
Beijing, China 100084
[email protected]
Jiaxing Zhang
Microsoft Research Asia
5 Danning Road, Haidian District
Beijing, China 100080
[email protected]
Sen Song ?
Department of Biomedical Engineering
Tsinghua University
Beijing, China 100084
[email protected]
Zheng Zhang * ?
Department of Computer Science
NYU Shanghai
1555 Century Ave, Pudong
Shanghai, China 200122
[email protected]
Abstract
Attentional Neural Network is a new framework that integrates top-down cognitive bias and bottom-up feature extraction in one coherent architecture. The
top-down influence is especially effective when dealing with high noise or difficult segmentation problems. Our system is modular and extensible. It is also
easy to train and cheap to run, and yet can accommodate complex behaviors. We
obtain classification accuracy better than or competitive with state of art results
on the MNIST variation dataset, and successfully disentangle overlaid digits with
high success rates. We view such a general purpose framework as an essential
foundation for a larger system emulating the cognitive abilities of the whole brain.
1
Introduction
How our visual system achieves robust performance against corruptions is a mystery. Although its
performance may degrade, it is capable of performing denoising and segmentation tasks with different levels of difficulties using the same underlying architecture. Consider the first two examples
in Figure 1. Digits overlaid over random images are harder to recognize than those over random
noise, since pixels in the background images are structured and highly correlated. It is even more
challenging if two digits are overlaid altogether, in a benchmark that we call MNIST-2. Yet, with
different levels of efforts (and error rates), we are able to recognize these digits for all three cases.
Figure 1: Handwriting digits with different corruptions. From left to right: random background
noise, random background images, and MNIST-2
?
?
These authors supervised the project jointly and are co-corresponding authors.
Work partially done while at Microsoft Resarch Asia
1
Another interesting property of the human visual system is that recognition is fast for low noise
level but takes longer for cluttered scenes. Testers perform well on recognition tasks even when
the exposure duration is short enough to allow only one feed-forward pass [18], while finding the
target in cluttered scenes requires more time[4]. These evidences suggest that our visual system is
simultaneously optimized for the common, and over-engineered for the worst. One hypothesis is
that, when challenged with high noise, top-down ?explanations? propagate downwards via feedback
connections, and modulate lower level features in an iterative refinement process[19].
Inspired by these intuitions, we propose a framework called attentional neural network (aNN). aNN
is composed of a collection of simple modules. The denoising module performs multiplicative
feature selection controlled by a top-down cognitive bias, and returns a modified input. The classification module receives inputs from the denoising module and generates assignments. If necessary,
multiple proposals can be evaluated and compared to pick the final winner. Although the modules
are simple, their combined behaviors can be complex, and new algorithms can be plugged in to
rewire the behavior, e.g., a fast pathway for low noise, and an iterative mode for complex problems
such as MNIST-2. We have validated the performance of aNN on the MNIST variation dataset.
We obtained accuracy better than or competitive to state of art. In the challenging benchmark of
MNIST-2, we are able to predict one digit or both digits correctly more than 95% and 44% of the
time, respectively. aNN is easy to train and cheap to run. All the modules are trained with known
techniques (e.g. sparse RBM and back propagation), and inference takes much fewer rounds of
iterations than existing proposals based on generative models.
2
Model
aNN deals with two related issues: 1) constructing a segmentation module under the influence of
cognitive bias and 2) its application to the challenging task of classifying highly corrupted data. We
describe them in turn, and will conclude with a brief description of training methodologies.
2.1
Segmentation with cognitive bias
?
M
?
?
feedback
?
?
?
M
?>?
?
?
C
? = ?(? ? ?)
?
(b)
?? = ? ? ?
? = ?(? ? ?)
?
??
?
?
M
? = ?(?? ? ?? )
?
?
(a)
?>?
?
?
C
(c)
Figure 2: Segmentation module with cognitive bias (a) and classification based on that (b,c).
As illustrated in Figure 2(a), the objective of the segmentation module M is to segment out an object
y belonging to one of N classes in the noisy input image x. Unlike in the traditional deonising
models such as autoencoders, M is given a cognitive bias vector b ? {0, 1}N , whose ith element
indicates a prior belief on the existence of objects belonging to the i-th class in the noisy image.
During the bottom up pass, input image x is mapped into a feature vector h = ?(W ? x), where
W is the feature weight matrix and ? represents element-wise nonlinear Sigmoid function. During
the top-down pass, b generates a gating vector g = ?(U ? b) with the feedback weights U . g selects
and de-selects the features by modifying hidden activation hg = h g, where means pair-wised
multiplication. Reconstruction occurs from hg by z = ?(W 0 ? hg ). In general, bias b can be a
probability distribution indicating a mixture of several guesses, but in this paper we only use two
simpler scenarios: a binary vector to indicate whether there is a particular object with its associated
weights UG , or a group bias bG with equal values for all objects, which indicates the presence of
some object in general.
2
2.2
Classification
A simple strategy would be to feed the segmented input y into a classifier C. However, this suffers
from the loss of details during M ?s reconstruction and is prone to hallucinations, i.e. y transforming
to a wrong digit when given a wrong bias. We opted to use the reconstruction y to gate the raw
image x with a threshold to produce gated image z = (y > ) x for classification (Figure 2b). To
segment complex images, we explored an iterative design that is reminiscent of a recurrent network
(Figure 2c). At time step t, the input to the segmentation module M is zt = (yt?1 > ) x, and the
result yt is used for the next iteration. Consulting the raw input x each time prevents hallucination.
Alternatively, we could feed the intermediate representation hg to the classifier and such a strategy
gives reasonable performance (see section 3.2 group bias subsection), but in general this suffers
from loss of modularity.
For iterative classification, we can give the system an initial cognitive bias, and the system produces
a series of guesses b and classification results given by C. If the guess b is confirmed by the output
of C, then we consider b as a candidate for the final classification result. A wrong bias b will lead the
network to transform x to a different class, but the segmented images with the correct bias is often
still better than transformed images under the wrong bias. In the simplest version, we can give initial
bs over all classes and compare the fitness of the candidates. Such fitness metrics can be the number
of iterations it takes C to confirm the guess, the confidence of the confirmation , or a combination
of many related factors. For simplicity, we use the entropy of outputs of C, but more sophisticated
extensions are possible (see section 3.2 making it scalable subsection).
2.3
Training the model
We used a shallow network of RBM for the generative model, and autoencoders gave qualitatively
similar results. The parameters to be learned include the feature weights W and the feedback
weights U . The multiplicative nature of feature selection makes learning both W and U simultaneously problematic, and we overcame this problem with a two-step procedure: firstly, W is trained
with noisy data in a standalone RBM (i.e. with the feedback disabled); next, we fix W and learn
U with the noisy data as input but with clean data as target, using the standard back propagation
procedure. This forces U to learn to select relevant features and de-select distractors. We find it
helpful to use different noise levels in these two stages. In the results presented below, training W
and U uses half and full noise intensity, respectively. In practice, this simple strategy is surprisingly
effective (see Section 3). We found it important to use sparsity constraint when learning W to
produce local features. Global features (e.g. templates) tend to be activated by noises and data
alike, and tend to be de-selected by the feedback weights. We speculate that feature locality might
be especially important when compositionality and segmentation is considered. Jointly training the
features and the classifier is a tantalizing idea but proves to be difficult in practice as the procedure is
iterative and the feedback weights need to be handled. But attempts could be made in this direction
in the future to fine-tune performance for a particular task. Another hyper-parameter is the threshold
. We assume that there is a global minimum, and used binary search on a small validation set. 1
3
Results and Analysis
We used the MNIST variation dataset and MNIST-2 to evaluate the effectiveness of our framework.
MNIST-2 is composed by overlaying two randomly chosen clean MNIST digits. Unless otherwise
stated, we used an off-the-shelf classifier: a 3-layer perceptron with 256 hidden nodes, trained on
clean MNIST data with a 1.6% error rate. In the following sections, we will discuss bias-induced
feature selection, its application in denosing, segmentation and finally classification.
3.1
Effectiveness of feedback
If feature selection is sensitive to the cognitive bias b, then a given b should leads to the activation
of the corresponding relevant features. In Figure 3(a), we sorted the hidden units by the associated
1
The training and testing code can be found in https://github.com/qianwangthu/feedback-nips2014-wq.git
3
input
sum
no bias
b=0
group bias
b=1
b=2
correct bias
b=8
wrong bias
(a) Top features
(b) Reconstruction
sum
activated
group bias
b=1
b=2
(c) feature selection
Figure 3: The effectiveness of bias-controlled feature selection. (a) top features selected by different
cognitive bias (0, 1, 2, 8) and their accumulation; (b) denoising without bias, with group bias, correct
bias and wrong bias (b = 1); (c) how bias selects and de-selects features, the second and the third
rows correspond to the correct and wrong bias, respectively.
weights in U for a given bias from the set {0, 1, 2, 8}, and inspected their associated feature weights
in W. The top features, when superimposed, successfully compose a crude version of the target digit.
Since b controls feature selection, it can lead to effective segmentation (shown in Figure 3(b)))
By comparing the reconstruction results in the second row without bias, with those in the third
and fouth rows (with group bias and correct bias respectively), it is clear that segmentation quality
progressively improves. On the other hand, a wrong bias (fifth row) will try to select features to
its favor in two ways: selecting features shared with the correct bias, and hallucinating incorrect
features by segmenting from the background noises. Figure 3(c) goes further to reveal how feature
selection works. The first row shows features for one noisy input, sorted by their activity levels
without the bias. Next three rows show their deactivtion by the cognitive biases. The last column
shows a reconstructed image using the selected features in this figure. It is clear how a wrong bias
fails to produce a reasonable reconstructed image.
guess
guess
1?
1?
5
5
5
5
7
5
2?
1
1
1
1
1
1
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9
9
2?
2
2
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2
2
2
3?
3?
2
2
2
2
2
2
4?
4?
2
7
7
7
7
7
7?
5?
7
7
7
7
7
7
9?
9?
2
7
7
7
7
7
(a)
(b)
Figure 4: Recurrent segmentation examples in six iterations. In each iteration, the classification
result is shown under the reconstructed image, along with the confidence (red bar, the longer the
higher confidence).
As described in Section 2, segmentation might take multiple iterations, and each iteration produces a
reconstruction that can be processed by an off-the-shelf classifier. Figure 4 shows two cases, with as4
sociated predictions generated by the 3-layer MLP. In the first example (Figure 4(a)), two cognitive
biase guesses 2 and 7 are confirmed by the network, and the correct guess 2 has a greater confidence.
The second example (Figure 4(b)) illustratess that, under high intensity background, transformations
can happen and a hallucinated digit can be ?built? from a patch of high intensity region since they can
indiscriminately activate features. Such transformations constitute false-positives (i.e. confirming a
wrong guess) and pose challenges to classification. More complicated strategies such as local contrast normalization can be used in the future to deal with such cases. This phenomenon is not at all
uncommon in everyday life experiences: when truth is flooded with high noises, all interpretations
are possible, and each one picks evidence in its favor while ignoring others.
As described in Section 2, we used an entropy confidence metric to select the winner from candidates. The MLP classifier C produces a predicted score for the likelihood of each class, and we take
the total confidence as the entropy of the prediction distribution, normalized by its class average
obtained under clean data. This confidence metric, as well as the associated classification result, are
displayed under each reconstruction. The first example shows that confidence under the right guess
(i.e. 2) is higher. On the other hand, the second example shows that, with high noise, confidences
of many guesses are equally poor. Furthermore, more iterations often lead to higher confidence,
regardless of whether the guess is correct or not. This self-fulfilling process locks predictions to
their given biases, instead of differentiating them, which is also a familiar scenario.
3.2
Classification
0.2
Table 1: Classification performance
mnist-background-noise
mnist-background-image
err rate
0.15
back-rand
back-image
RBM
11.39
15.42
0.05
imRBM
10.46
16.35
0
discRBM
10.29
15.56
DBN-3
6.73
16.31
CAE-2
10.90
15.50
PGBM
6.08
12.25
sDBN
4.48
14.34
aNN - ?rand
3.22
22.30
aNN - ?image
6.09
15.33
0.1
(a)
0
0.2
0.4
0.6
background level
0.8
1
0.25
err rate
0.2
false negative
false positive
0.15
(b)
0.1
0.05
0
1
2
3
iteration
4
5
Figure 5: (a) error vs. background level. (b) error vs. iteration number.
To compare with previous results, we used the standard training/testing split (12K/50K) of the
MNIST variation set, and results are shown in the Table 1. We ran one-iteration denoising, and
then picked the winner by comparing normalized entropies among the candidates, i.e. those with
biases matching the prediction of the 3-layer MLP classifier. We trained two parameter sets separately in random-noise background (?rand ) and image background dataset(?image ). To test transfer
abilities, we also applied ?image to random-noise background data and ?rand to image background
data. On MNIST-back-rand and MNIST-back-image dataset, ?noise achieves 3.22% and 22.3% err
rate respectively, while ?image achieves 6.09% and 15.33%.
Figure 5(a) shows how the performance deteriorates with increasing noise level. In these experiments, random noise and random images are modulated by scaling down their pixel intensity linearly. Intuitively, at low noise the performance should approach the default accuracy of the classifier
C and is indeed the case.
The effect of iterations: We have chosen to run only one iteration because under high noise, each
guess will insist on picking features to its favor and some hallucination can still occur. With more
iterations, false positive rates will rise and false negative rates will decrease, as confidence scores for
5
both the right and the wrong guesses will keep on improving. This is shown in Figure-5(b). As such,
more iterations do not necessarily lead to better performance. In the current model, the predicted
class from the previous step is not feed into the next step, and more sophisticated strategies with
such an extension might produce better results in the future.
The power of group bias: For this benchmark, good performance mostly depends on the quality of
segmentation. Therefore, a simpler approach is to denoise with coarse-grained group bias, followed
by classification. For ?image , we attached a SVM to the hidden units with bG turned on, and obtained
a 16.2% error rate. However, if we trained a SVM with 60K samples, the error rate drops to 12.1%.
This confirms that supervised learning can achieve better performance with more training data.
Making it scalable. So far, we enumerate over all the guesses. This is clearly not scalable if number
of classes is large. One sensible solution is to first denoise with a group bias bG , and pick top-K
candidates from the prediction distribution, and then iterate among them.
Finally, we emphasize that the above results are obtained with only one up-down pass. This is in
stark contrast to other generative model based systems. For example, in PGBM [15], each inference
takes 25 rounds.
3.3
MNIST-2 problem
Compared to corruption by background noises, MNIST-2 is a much more challenging task, even for
a human observer. It is a problem of segmentation, not denoising. In fact, such segmentation requires
semantic understanding of the object. Knowing which features are task-irrelevant is not sufficient,
we need to discover and utilize per-class features. Any denoising architectures only removing taskirrelevant features will fail on such a task without additional mechanisms. In aNN, each bias has its
own associated features and explicitly call these features out in the reconstruction phase (modulated
by input activations). Meanwhile, its framework permits multiple predictions so it can accommodate
such problems.
ground truth
ground truth
guess
ground truth
guess
2?
guess
2?
2
2
2
2
2
2
6?
1?
2
2
2
2
2
2
7?
6
6
6
6
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1?
7
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0?
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6
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8
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4
3?
1
1
1
1
1
1
4?
2
1
2?
1?
8?
2
5?
4?
7
7
4
(a)
(b)
4
4
4
(c)
Figure 6: Sample results on MNIST-2. In each example, each column is one iteration. The first two
rows are runs with two ground truth digits, others are with wrong biases.
For the MNIST-2 task, we used the same off-the-shelf 3-layer classifier to validate a guess. In the
first two examples in Figure 6, the pair of digits in the ground truth is correctly identified. Supplying
either digit as the bias successfully segments its features, resulting in imperfect reconstructions that
are nonetheless confident enough to win over competing proposals. One would expect that the
random nature of MNIST-2 would create much more challenging (and interesting) cases that either
defy or confuse any segmentation attempts. This is indeed true. The last example is an overlay of
the digit 1 and 5 that look like a perfect 8. Each of the 5 biases successfully segment out their target
?digit?, and sometimes creatively. It is satisfying to see that a human observor would make similar
misjudgements in those cases.
6
ground truth image
result
ground truth image
result
ground truth image
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
(a)
(b)
result
(c)
Figure 7: Sample results on MNIST-2 when adding background noises. (a) (b) (c) are examples
three groups of results, when both digits, one digit, or none are predicted, respectively.
Out of the 5000 MNIST-2 pairs, there are 95.46% and 44.62% cases where at least one digit or
both digits get correctly predicted, respectively. Given the challenging nature of the benchmark, we
are surprised by this performance. Contrary to random background dataset, in this problem, more
iterations conclusively lead to better performance. The above accuracy is obtained with 5 iterations,
and the accuracy for matching both digits will drop to 36.28% if only 1 iteration is used. Even more
interestingly, this performance is resilient against background noise (Figure 7), the accuracy only
drops slightly (93.72% and 41.66%). The top-down biases allowed us to achieve segmentaion and
denoising at the same time.
4
4.1
Discussion and Related Work
Architecture
Feedforward multilayer neural networks have achieved good performance in many classification
tasks in the past few years, notably achieving the best performance in the ImageNet competition
in vision([21] [7]). However, they typically give a fixed outcome for each input image, therefore
cannot naturally model the influence of cognitive biases and are difficult to incorporate into a larger
cognitive framework. The current frontier of vision research is to go beyond object recognition
towards image understanding [16]. Inspired by neuroscience research, we believe that an unified
module which integrates feedback predictions and interpretations with information from the world
is an important step towards this goal.
Generative models have been a popular approach([5, 13]). They are typically based on a probabilistic framework such as Boltzmann Machines and can be stacked into a deep architecture. They have
advantages over discriminative models in dealing with object occlusion. In addition, prior knowledge can be easily incorporated in generative models in the forms of latent variables. However,
despite the mathematical beauty of a probabilistic framework, this class of models currently suffer
from the difficulty of generative learning and have been mostly successful in learning small patches
of natural images and objects [17, 22, 13]. In addition, inferring the hidden variables from images
is a difficult process and many iterations are typically needed for the model to converge[13, 15]. A
recent trend is to first train a DBN or DBM model then turn the model into a discriminative network
for classification. This allows for fast recognition but the discriminative network loses the generative
ability and cannot combine top-down and bottom-up information.
We sought a simple architecture that can flexibly navigate between discriminative and generative
frameworks. This should ideally allow for one-pass quick recognition for images with easy and
well-segmented objects, but naturally allow for iteration and influence by cognitive-bias when the
need for segmentation arises in corrupted or occluded image settings.
4.2
Models of Attention
In the field of computational modeling of attention, many models have been proposed to model the
saliency map and used to predict where attention will be deployed and provide fits to eye-tracking
data[1]. We are instead more interested in how attentional signals propagating back from higher lev7
els in the visual hierarchy can be merged with bottom up information. Volitional top-down control
could update, bias or disambiguate the bottom-up information based on high-level tasks, contextual cues or behavior goals. Computational models incorporating this principle has so far mostly
focused on spatial attention [12, 1]. For example, in a pedestrian detection task, it was shown that
visual search can be sped up if the search is limited to spatial locations of high prior or posterior
probabilities [3]. However, human attention abilities go beyond simple highlighting based on location. For example, the ability to segment and disentangle object based on high level expectations
as in the MNIST-2 dataset represents an interesting case. Here, we demonstrate that top-down attention can also be used to segment out relevant parts in a cluttered and entangled scene guided
by top-down interpretation, demonstrating that attentional bias can be successfully deployed on a
far-more fine-grained level than previous realized.
We have chosen the image-denoising and image-segmentation tasks as our test cases. In the context
of image-denoising, feedforward neural networks have been shown to have good performance [6,
20, 11]. However, their work has not included a feedback component and has no generative ability.
Several Boltzmann machine based architectures have been proposed[9, 8]. In PGBM, gates on input
images are trained to partition such pixel as belonging to objects or backgrounds, which are modeled
by two RBMs separately [15]. The gates and the RBM components make up a high-order RBM.
However, such a high-order RBM is difficult to train and needs costly iterations during inference.
sDBN [17] used a RBM to model the distribution of the hidden layer, and then denoises the hidden
layer by Gibbs sampling over the hidden units affected by noise. Besides the complexity of Gibbs
sampling, the process of iteratively finding which units are affected by noise is also complicated and
costly, as there is a process of Gibbs sampling for each unit. When there are multiple digits appearing
in the image as in the case of MNSIT-2, the hidden layer denoising step leads to uncertain results,
and the best outcome is an arbitrary choice of one of the mixed digits. a DBM based architecture has
also been proposed for modeling attention, but the complexity of learning and inference also makes
it difficult to apply in practice [10]. All those works also lack the ability of controlled generation
and input reconstruction under the direction of a top-down bias.
In our work, top-down biases influence the processing of feedforward information at two levels. The
inputs are gated at the raw image stage by top-down reconstructions. We propose that this might be
equivalent to the powerful gating influence of the thalamus in the brain [1, 15]. If the influence of
input image is shut off at this stage, then the system can engage in hallucination and might get into a
state akin to dreams, as when the thalamic gates are closed. Top-down biases also affect processing
at a higher stage of high-level features. We think this might be equivalent to the processing level of
V4 in the visual hierarchy. At this level, top-down biases mostly suppresses task-irrelevant features
and we have modeled the interactions as multiplicative in accordance with results from neuroscience
research [1, 2].
4.3
Philosophical Points
The issue of whether top-down connections and iterative processing are useful for object recognition has been a point of hot contention. Early work inspired by Hopfield network and the tradition
of probabilistic models based on Gibbs sampling argue for the usefulness of feedback and iteration
[14],[13], but results from neuroscience research and recent success by purely feedforward networks
argue against it [18],[7]. In our work, we find that feedforward processing is sufficient for good performance on clean digits. Feedback connections play an essential role for digit denoising. However,
one pass with a simple cognitive bias towards digits seems to suffice and iteration seems only to confirm the initial bias and does not improve performance. We hypothesize that this ?see what you want
to see? is a side-effect of our ability to denoise a cluttered scene, as the deep hierarchy possesses the
ability to decompose objects into many shareable parts. In the more complex case of MNIST-2, performance does increase with iteration. This suggests that top-down connections and iteration might
be particularly important for good performance in the case of cluttered scenes. The architecture we
proposed can naturally accommodate all these task requirements simultaneously with essentially no
further fine-tuning. We view such a general purpose framework as an essential foundation for a
larger system emulating the cognitive abilities of the whole brain.
8
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4,714 | 5,269 | Expectation Backpropagation: Parameter-Free
Training of Multilayer Neural Networks with
Continuous or Discrete Weights
Daniel Soudry1 , Itay Hubara2 , Ron Meir2
(1) Department of Statistics, Columbia University
(2) Department of Electrical Engineering, Technion, Israel Institute of Technology
[email protected],[email protected],[email protected]
Abstract
Multilayer Neural Networks (MNNs) are commonly trained using gradient
descent-based methods, such as BackPropagation (BP). Inference in probabilistic
graphical models is often done using variational Bayes methods, such as Expectation Propagation (EP). We show how an EP based approach can also be used
to train deterministic MNNs. Specifically, we approximate the posterior of the
weights given the data using a ?mean-field? factorized distribution, in an online
setting. Using online EP and the central limit theorem we find an analytical approximation to the Bayes update of this posterior, as well as the resulting Bayes
estimates of the weights and outputs.
Despite a different origin, the resulting algorithm, Expectation BackPropagation
(EBP), is very similar to BP in form and efficiency. However, it has several additional advantages: (1) Training is parameter-free, given initial conditions (prior)
and the MNN architecture. This is useful for large-scale problems, where parameter tuning is a major challenge. (2) The weights can be restricted to have discrete
values. This is especially useful for implementing trained MNNs in precision limited hardware chips, thus improving their speed and energy efficiency by several
orders of magnitude.
We test the EBP algorithm numerically in eight binary text classification tasks.
In all tasks, EBP outperforms: (1) standard BP with the optimal constant learning
rate (2) previously reported state of the art. Interestingly, EBP-trained MNNs with
binary weights usually perform better than MNNs with continuous (real) weights
- if we average the MNN output using the inferred posterior.
1
Introduction
Recently, Multilayer1 Neural Networks (MNNs) with deep architecture have achieved state-of-theart performance in various supervised learning tasks [11, 14, 8]. Such networks are often massive
and require large computational and energetic resources. A dense, fast and energetically efficient
hardware implementation of trained MNNs could be built if the weights were restricted to discrete
values. For example, with binary weights, the chip in [13] can perform 1012 operations per second
with 1mW power efficiency. Such performances will enable the integration of massive MNNs into
small and low-power electronic devices.
Traditionally, MNNs are trained by minimizing some error function using BackPropagation (BP) or
related gradient descent methods [15]. However, such an approach cannot be directly applied if the
weights are restricted to binary values. Moreover, crude discretization of the weights is usually quite
1
i.e., having more than a single layer of adjustable weights.
1
destructive [20]. Other methods have been suggested in the 90?s (e.g., [23, 3, 18]), but it is not clear
whether these approaches are scalable.
The most efficient methods developed for training Single-layer2 Neural Networks (SNN) with binary
weights use approximate Bayesian inference, either implicitly [6, 1] or explicitly [24, 22]. In theory,
given a prior, the Bayes estimate of the weights can be found from their posterior given the data.
However, storing or updating the full posterior is usually intractable. To circumvent this problem,
these previous works used a factorized ?mean-field? form the posterior of the weights given the data.
As explained in [22], this was done using a special case of the widely applicable Expectation Propagation (EP) algorithm [19] - with an additional approximation that the fan-in of all neurons is large,
so their inputs are approximately Gaussian. Thus, given an error function, one can analytically
obtain the Bayes estimate of the weights or the outputs, using the factorized approximation of the
posterior. However, to the best of our knowledge, it is still unknown whether such an approach could
be generalized to MNNs, which are more relevant for practical applications.
In this work we derive such generalization, using similar approximations (section 3). The end result
is the Expectation BackPropagation (EBP, section 4) algorithm for online training of MNNs where
the weight values can be either continuous (i.e., real numbers) or discrete (e.g., ?1 binary). Notably,
the training is parameter-free (with no learning rate), and insensitive to the magnitude of the input.
This algorithm is very similar to BP. Like BP, it is very efficient in each update, having a linear
computational complexity in the number of weights.
We test the EBP algorithm (section 5) on various supervised learning tasks: eight high dimensional
tasks of classifying text into one of two semantic classes, and one low dimensional medical discrimination task. Using MNNs with two or three weight layers, EBP outperforms both standard BP, as
well as the previously reported state of the art for these tasks [7]. Interestingly, the best performance
of EBP is usually achieved using the Bayes estimate of the output of MNNs with binary weights.
This estimate can be calculated analytically, or by averaging the output of several such MNNs, with
weights sampled from the inferred posterior.
2
Preliminaries
General Notation A non-capital boldfaced letter x denotes a column vector with components xi ,
a boldfaced capital letter X denotes a matrix with components Xij . Also, if indexed, the components of xl are denoted xi,l and those of Xl are denoted Xij,l . We denote by P (x) the probability distribution (in the discrete case)
or density (in the? continuous case) of a random variable X,
?
P (x|y) = P (x, y) /P (y),hxi = xP (x) dx, hx|yi = xP (x|y) dx, Cov (x, y) = hxyi?hxi hyi
and Var (x) = Cov (x, x). Integration is exchanged with summation in the discrete case. For any
condition A, we make use of I {A}, the indicator function (i.e., I {A} = 1 if A holds, and zero
otherwise), and ?ij = I {i = j}, Kronecker?s delta function. If x ? N (?, ?) then it is Gaussian
with mean ? and covariance matrix ?, and we denote
? x its density by N (x|?, ?). Furthermore, we
use the cumulative distribution function ? (x) = ?? N (u|0, 1) du.
Model We consider a general feedforward Multilayer Neural Network (MNN) with connections
between adjacent layers (Fig. 2.1). For analytical simplicity, we focus here on deterministic binary
(?1) neurons. However, the framework can be straightforwardly extended to other types of neurons
(deterministic or stochastic). The MNN has L layers, where Vl is the width of the l-th layer, and
L
W = {Wl }l=1 is the collection of Vl ? Vl?1 synaptic weight matrices which connect neuronal
L
L?1
layers sequentially. The outputs of the layers are {vl }l=0 , where v0 is the input layer, {vl }l=1 are
the hidden layers and vL is the output layer. In each layer,
vl = sign (Wl vl?1 )
(2.1)
where each sign ?activation function? (a neuronal layer) operates component-wise (i.e., ?i :
(sign (x))i = sign (xi )). The output of the network is therefore
vL = g (v0 , W) = sign (WL sign (WL?1 sign (? ? ? W1 v0 ))) .
2
i.e., having only a single layer of adjustable weights.
2
(2.2)
We assume that the weights are constrained to some set
S, with the specific restrictions on each weight denoted
by Sij,l , so Wij,l ? Sij,l and W ? S. If Sij,l = {0},
then we say that Wij,l is ?disconnected?. For simplicity, we assume that in each layer the ?fan-in? Kl =
|{j|Sij,l 6= {0}}| is constant for all i. Biases can be optionally included in the standard way, by adding a constant output v0,l = 1 to each layer.
Task We examine a supervised classification learning Figure 2.1: Our MNN model (Eq. 2.2).
task, in Bayesian framework. We are given a fixed set of
N
sequentially labeled data pairs DN = x(n) , y(n) n=1
(so D0 = ?), where each x(n) ? RV0 is a data point, and
V
each y(n) is a label taken from a binary set Y ? {?1, +1} L . For brevity, we will sometimes
suppress the sample index n, where it is clear from the context. As common for supervised learning
with MNNs, we assume that for all n the relation x(n) ? y(n) can be represented by a MNN with
known architecture (the ?hypothesis class?), and unknown weights W ? S. This is a reasonable
assumption since a MNN can approximate any deterministic function, given that it has sufficient
number of neurons
[12] (if L ? 2). Specifically, there exists some W ? ? S, so that y(n) =
(n)
?
f x , W (see Eq. 2.2). Our goals are: (1) estimate the most probable W ? for this MNN, (2)
estimate the most probable y given some (possibly unseen) x.
3
Theory
In this section we explain how a specific learning algorithm for MNNs (described in section 4) arises
from approximate (mean-field) Bayesian inference, used in this context (described in section 2).
3.1
Online Bayesian learning in MNNs
We approach this task within a Bayesian framework, where we assume some prior distribution on the
weights - P (W|D0 ). Our aim is to find P (W|DN ), the posterior probability for the configuration
of the weights W, given the data. With this posterior, one can select the most probable weight
configuration - the Maximum A Posteriori (MAP) weight estimate
W ? = argmaxW?S P (W|DN ) ,
(3.1)
?
minimizing the expected zero-one loss over the weights (I {W 6= W}). This weight estimate can
be implemented in a single MNN, which can provide an estimate of the label y for (possibly unseen)
data points x through y =g (x, W ? ). Alternatively, one can aim to minimize the expected loss over
the output - as more commonly done in the MNN literature. For example, if the aim is to reduce
classification error then one should use the MAP output estimate
X
y? = argmaxy?Y
I {g (x, W) = y} P (W|DN ) ,
(3.2)
W
?
which minimizes the zero-one loss (I {y 6= g (x, W)}) over the outputs. The resulting estimator
does not generally have the form of a MNN (i.e., y =g (x, W) with W ? S), but can be approximated by averaging the output over many such MNNs with W values sampled from the posterior.
Note that averaging the output of several MNNs is a common method to improve performance.
We aim to find the posterior P (W|DN ) in an online setting, where samples arrive sequentially.
After the n-th sample is received, the posterior is updated according to Bayes rule:
P (W|Dn ) ? P y(n) |x(n) , W P (W|Dn?1 ) ,
(3.3)
for n = 1, . . . , N . Note that the MNN is deterministic, so the likelihood (per data point) has the
following simple form3
n
o
P y(n) |x(n) , W = I g x(n) , W = y(n) .
(3.4)
3
MNN with stochastic activation functions will have a ?smoothed out? version of this.
3
Therefore, the Bayes update in Eq. 3.3 simply makes sure that P (W|Dn ) = 0 in any ?illegal? configuration (i.e., any W 0 such that g x(k) , W 0 6= y(k) ) for some 1 ? k ? n. In other words, the
posterior is equal to the prior, restricted to the ?legal? weight domain, and re-normalized appropriately. Unfortunately, this update is generally intractable for large networks, mainly because we need
to store and update an exponential number of values for P (W|Dn ). Therefore, some approximation
is required.
3.2
Approximation 1: mean-field
In order to reduce computational complexity, instead of storing P (W|Dn ), we will store its factorized (?mean-field?) approximation P? (W|Dn ), for which
Y
P? (W|Dn ) =
P? (Wij,l |Dn ) ,
(3.5)
i,j,l
where each factor must be normalized. Notably, it is easy to find the MAP estimate of the weights
(Eq. 3.1) under this factorized approximation ?i, j, l
?
Wij,l
= argmaxWij,l ?Sij,l P? (Wij,l |DN ) .
(3.6)
The factors P? (Wij,l |Dn ) can be found using a standard variational approach [5, 24]. For each n,
we first perform the Bayes update in Eq. 3.3 with P? (W|Dn?1 ) instead of P (W|Dn?1 ). Then, we
project the resulting posterior onto the family of distributions factorized as in Eq. 3.5, by minimizing the reverse Kullback-Leibler divergence (similarly to EP [19, 22]). A straightforward calculation
shows that the optimal factor is just a marginal of the posterior (appendix A, available in the supplementary material). Performing this marginalization on the Bayes update and re-arranging terms, we
obtain a Bayes-like update to the marginals ?i, j, l
P? (Wij,l |Dn ) ? P? y(n) |x(n) , Wij,l , Dn?1 P? (Wij,l |Dn?1 ) ,
(3.7)
where
P? y(n) |x(n) , Wij,l , Dn?1 =
X
P y(n) |x(n) , W 0
0 =W
W 0 :Wij,l
ij,l
Y
0
P? Wkr,m
|Dn?1 (3.8)
{k,r,m}6={i,j,l}
is the marginal likelihood. Thus we can directly update the factor P? (Wij,l |Dn ) in a single step.
However, the last equation is still problematic, since it contains a generally intractable summation
over an exponential number of values, and therefore requires simplification. For simplicity, from
now on we replace any P? with P , in a slight abuse of notation (keeping in mind that the distributions
are approximated).
3.3
Simplifying the marginal likelihood
In order to be able to use the
update rule in Eq. 3.7, we must first calculate the marginal likelihood
P y(n) |x(n) , Wij,l , Dn?1 using Eq. 3.8. For brevity, we suppress the index n and the dependence
on Dn?1 and x, obtaining
Y
X
0
P (y|W 0 )
P Wkr,m
,
(3.9)
P (y|Wij,l ) =
0 =W
W 0 :Wij,l
ij,l
{k,r,m}6={i,j,l}
where we recall that P (y|W 0 ) is simply an indicator function (Eq. 3.4). Since, by assumption,
P (y|W 0 ) arises from a feed-forward MNN with input v0 = x and output vL = y, we can perform
the summations in Eq. 3.9 in a more convenient way - layer by layer. To do this, we define
?
? ?
?
Vm?1
Vm
m?1
?
?VY
XY
X
0
0
?I vk,m
? (3.10)
P (vm |vm?1 ) =
vr,m?1 Wkr,m
>0
P Wkr,m
?
?
0
Wm k=1
r=1
r=1
and P (vl |vl?1 , Wij,l ), which is defined identically to P (vl |vl?1 ), except that the summation is
0
performed over all configurations in which Wij,l is fixed (i.e., Wl0 : Wij,l
= Wij,l ) and we set
4
P (Wij,l ) = 1. Now we can write recursively P (v1 ) = P (v1 |v0 = x)
X
?m ? {2, .., l ? 1} : P (vm ) =
P (vm |vm?1 ) P (vm?1 )
(3.11)
vm?1
P (vl |Wij,l ) =
X
P (vl |vl?1 , Wij,l ) P (vl?1 )
(3.12)
vl?1
?m ? {l + 1, l + 2, .., L} : P (vm |Wij,l ) =
X
P (vm |vm?1 ) P (vm?1 |Wij,l )
(3.13)
vm?1
Thus we obtain the result of Eq. 3.9, through P (y|Wij,l ) = P (vL = y|Wij,l ). However, this
computation is still generally intractable, since all of the above summations (Eqs. 3.10-3.13) are still
over an exponential number of values. Therefore, we need to make one additional approximation.
3.4
Approximation 2: large fan-in
Next we simplify the above summations (Eqs. 3.10-3.13) assuming that the neuronal fan-in is
?large?. We keep in mind that i, j and l are the specific indices of the fixed weight Wij,l . All
the other weights beside Wij,l can be treated as independent random variables, due to the mean field
approximation (Eq. 3.5). Therefore, in the limit of a infinite neuronal fan-in (?m : Km ? ?) we
can use the Central Limit Theorem (CLT) and say that the normalized input to each neuronal layer,
is distributed according to a Gaussian distribution
p
?m : um = Wm vm?1 / Km ? N (?m , ?m ) .
(3.14)
Since Km is actually finite, this would be only an approximation - though a quite common and
effective one (e.g., [22]). Using the approximation in Eq. 3.14 together with vm = sign (um ) (Eq.
2.1) we can calculate (appendix B) the distribution of um and vm sequentially for all the layers
m ? {1, . . . , L}, for any given value of v0 and Wij,l . These effectively simplify the summations in
3.10-3.13 using Gaussian integrals (appendix B).
At the end of this ?forward pass? we will be able to find P (y|Wij,l ) = P (vL = y|Wij,l ) , ?i, j, l.
This takes a polynomial number of steps (appendix B.3), instead of a direct calculation through
Eqs. 3.11-3.13, which is exponentially hard. Using P (y|Wij,l ) and Eq. 3.7 we can now update the
distribution of P (Wij,l ). This immediately gives the Bayes estimate of the weights (Eq. 3.6) and
outputs (Eq. 3.2).
As we note in appendix B.3, the computational complexity of the forward pass is significantly lower
if ?m is diagonal. This is true exactly only in special cases. For example, this is true if all hidden
neurons have a fan-out of one - such as in a 2-layer network with a single output. However, in order
to reduce the computational complexity in cases that ?m is not diagonal, we will approximate the
distribution of um with its factorized (?mean-field?) version. Recall that the optimal factor is the
marginal of the distribution (appendix A). Therefore, we can now find P (y|Wij,l ) easily (appendix
2
B.1), as all the off-diagonal components in ?m are zero, so ?kk0 ,m = ?k,m
?kk0 .
A direct calculation of P (vL = y|Wij,l ) for every i, j, l would be computationally wasteful, since
we will repeat similar calculations many times. In order to improve the algorithm?s efficiency,
we again exploit the fact that Kl is large. We approximate ln P (vL = y|Wij,l ) using a Taylor
?1/2
expansion of Wij,l around its mean, hWij,l i, to first order in Kl
. The first order terms in this
expansion can be calculated using backward propagation of derivative terms
?k,m = ? ln P (vL = y) /??k,m ,
(3.15)
similarly to the BP algorithm (appendix C). Thus, after a forward pass for m = 1, . . . , L, and a
backward pass for l = L, . . . , 1, we obtain P (vL = y|Wij,l ) for all Wij,l and update P (Wij,l ).
4
The Expectation Backpropagation Algorithm
Using our results we can efficiently update the posterior distribution P (Wij,l |Dn ) for all the weights
with O (|W|) operations, according to Eqs. 3.7. Next, we summarize the resulting general algorithm
- the Expectation BackPropgation (EBP) algorithm. In appendix D, we exemplify how to apply the
5
algorithm in the special cases of MNNs with binary, ternary or real (continuous) weights. Similarly
to the original BP algorithm (see review in [16]), given input x and desired output y, first we perform
a forward pass to calculate the mean output hvl i for each layer. Then we perform a backward pass
to update P (Wij,l |Dn ) for all the weights.
Forward pass In this pass we perform the forward calculation of probabilities, as in Eq. 3.11.
Recall that hWkr,m i is the mean of the posterior distribution P (Wkr,m |Dn ). We first initialize the
MNN input hvk,0 i = xk for all k and calculate recursively the following quantities for m = 1, . . . , L
and all k
?k,m = ?
2
?k,m
=
Vm?1
1 X
hWkr,m i hvr,m?1 i ; hvk,m i = 2? (?k,m /?k,m ) ? 1 .
Km r=1
Vm?1
1 X
2
2
2
2
Wkr,m ?m,1 hvr,m?1 i ? 1 + 1 ? hWkr,m i hvr,m?1 i ,
Km r=1
(4.1)
(4.2)
where ?m and ? 2m are, respectively, the mean and variance of um , the input of layer m (Eq. 3.14),
and hvm i is the resulting mean of the output of layer m.
Backward pass In this pass we perform the Bayes update of the posterior (Eq. 3.7) using a Taylor
expansion. Recall Eq. 3.15. We first initialize4
2
N 0|?i,L , ?i,L
?i,L = yi
.
(4.3)
? (yi ?i,L /?i,L )
for all i. Then, for l = L, . . . , 1 and ?i, j we calculate
V
?i,l?1
=
m
X
2
2
? N 0|?i,l?1 , ?i,l?1
hWji,l i ?j,l .
Kl
j=1
(4.4)
ln P (Wij,l |Dn )
=
1
ln P (Wij,l |Dn?1 ) + ? Wij,l ?i,l hvj,l?1 i + C ,
Kl
(4.5)
where C is some unimportant constant (which does not depend on Wij,l ).
Output Using the posterior distribution, the optimal configuration can be immediately found
through the MAP weights estimate (Eq. 3.6) ?i, j, l
?
Wij,l
= argmaxWij,l ?Sij,l ln P (Wij,l |Dn ) .
(4.6)
The output of a MNN implementing these weights would be g (x, W ? ) (see Eq. 2.2). We define this
to be the ?deterministic? EBP output (EBP-D).
Additionally, the MAP output (Eq. 3.2) can be calculated directly
"
#
X 1 + hvk,L i yk
?
y = argmaxy?Y ln P (vL = y) = argmaxy?Y
ln
1 ? hvk,L i
(4.7)
k
using hvk,L i from Eq. 4.1, or as an ensemble average over the outputs of all possible MNN with the
weights Wij,l being sampled from the estimated posterior P (Wij,l |Dn ). We define the output in Eq.
4.7 to be the Probabilistic EBP output (EBP-P). Note that in the case of a single output Y = {?1, 1},
so this output simplifies to y = sign (hvk,L i).
4
Due to numerical inaccuracy, calculating ?i,L using Eq. 4.3 can generate nonsensical values (??, NaN)
if |?i,L /?i,L | becomes to large. If this happens, we use instead the asymptotic form in that limit
?i,L = ?
?i,L
?
I {yi ?i,L < 0}
2
?i,L
KL
6
5
Numerical Experiments
We use several high dimensional text datasets to assess the performance of the EBP algorithm in
a supervised binary classification task. The datasets (taken from [7]) contain eight binary tasks
from four datasets: ?Amazon (sentiment)?, ?20 Newsgroups?, ?Reuters? and ?Spam or Ham?. Data
specification (N =#examples and M =#features) and results (for each algorithm) are described in
Table 1. More details on the data including data extraction and labeling can be found in [7].
We test the performance of EBP on MNNs with a 2-layer architecture of M ? 120 ? 1, and
bias weights. We examine two special cases: (1) MNNs with real weights (2) MNNs with binary
weights (and real bias). Recall the motivation for the latter (section 1) is that they can be efficiently
implemented in hardware (real bias has negligible costs). Recall also that for each type of MNN, the
algorithm gives two outputs - EBP-D (deterministic) and EBP-P (probabilistic), as explained near
Eqs. 4.6-4.7.
To evaluate our results we compare EBP to: (1) the AROW algorithm, which reports state-of-the-art
results on the tested datasets [7] (2) the traditional Backpropagation (BP) algorithm, used to train an
M ? 120 ? 1 MNN with real weights. In the latter case, we used both Cross Entropy (CE) and
Mean Square Error (MSE) as loss functions. On each dataset we report the results of BP with the
loss function which achieved the minimal error. We use a simple parameter scan for both AROW
(regularization parameter) and the traditional BP (learning rate parameter). Only the results with
the optimal parameters (i.e., achieving best results) are reported in Table 1. The optimal parameters
found were never at the edges of the scanned field. Lastly, to demonstrate the destructive effect of
naive quantization, we also report the performance of the BP-trained MNNs, after all the weights
(except the bias) were clipped using a sign function.
During training the datasets were repeatedly presented in three epochs (in all algorithms, additional
epochs did not reduce test error). On each epoch the examples were shuffled at random order for BP
and EBP (AROW determines its own order). The test results are calculated after each epoch using
8-fold cross-validation, similarly to [7]. Empirically, EBP running time is similar to BP with real
weights, and twice slower with binary weights. For additional implementation details, see appendix
E.1. The code is available on the author?s website.
The minimal values achieved over all three epochs are summarized in Table 1. As can be seen, in all
datasets EBP-P performs better then AROW, which performs better then BP. Also, EBP-P usually
perfroms better with binary weights. In appendix E.2 we show that this ranking remains true even if
the fan-in is small (in contrast to our assumptions), or if a deeper 3-layer architecture is used.
Dataset
#Examples
#Features
Real EBP-D
Real EBP-P
Binary EBP-D
Binary EBP-P
AROW
BP
Clipped BP
Reuters news I6
Reuters news I8
Spam or ham d0
Spam or ham d1
20News group comp vs HW
20News group elec vs med
Amazon Book reviews
Amazon DVD reviews
2000
2000
2500
2500
1943
1971
3880
3880
11463
12167
26580
27523
29409
38699
221972
238739
14.5%
15.65%
1.28%
1.0%
5.06%
3.36%
2.14%
2.06%
11.35%
15.25%
1.11%
0.96%
4.96%
3.15%
2.09%
2.14%
21.7%
23.15%
7.93%
3.85%
7.54%
6.0%
2.45%
5.72%
9.95%
16.4%
0.76%
0.96%
4.44%
2.08%
2.01%
2.27%
11.72%
15.27%
1.12%
1.4%
5.79%
2.74%
2.24%
2.63%
13.3%
18.2%
1.32%
1.36%
7.02%
3.96%
2.96%
2.94%
26.15%
26.4%
7.97%
7.33%
13.07%
14.23%
3.81%
5.15%
Table 1: Data specification, and test errors (with 8-fold cross-validation). Best results are boldfaced.
6
Discussion
Motivated by the recent success of MNNs, we developed the Expectation BackPropagation algorithm (EBP - see section 4) for approximate Bayesian inference of the synaptic weights of a MNN.
Given a supervised classification task with labeled training data and a prior over the weights, this
deterministic online algorithm can be used to train deterministic MNNs (Eq. 2.2) without the need
to tune learning parameters (e.g., learning rate). Furthermore, each synaptic weight can be restricted
to some set - which can be either finite (e.g., binary numbers) or infinite (e.g., real numbers). This
opens the possibility of implementing trained MNNs in power-efficient hardware devices requiring
limited parameter precision.
7
This algorithm is essentially an analytic approximation to the intractable Bayes calculation of the
posterior distribution of the weights after the arrival of a new data point. To simplify the intractable
Bayes update rule we use several approximations. First, we approximate the posterior using a product of its marginals - a ?mean field? approximation. Second, we assume the neuronal layers have a
large fan-in, so we can approximate them as Gaussian. After these two approximations each Bayes
update can be tractably calculated in polynomial time in the size of the MNN. However, in order to
further improve computational complexity (to O (|W|) in each step, like BP), we make two additional approximations. First, we use the large fan-in to perform a first order expansion. Second, we
optionally5 perform a second ?mean field? approximation - to the distribution of the neuronal inputs.
Finally, after we obtain the approximated posterior using the algorithm, the Bayes estimates of the
most probable weights and the outputs are found analytically.
Previous approaches to obtain these Bayes estimates were too limited for our purposes. The Monte
Carlo approach [21] achieves state-of-the-art performance for small MNNs [26], but does not scale
well [25]. The Laplace approximation [17] and variational Bayes [10, 2, 9] based methods require
real-value weights, tuning of the learning rate parameter, and stochastic neurons (to ?smooth? the
likelihood). Previous EP [24, 22] and message passing [6, 1] (a special case of EP[5]) based methods
were derived only for SNNs.
In contrast, the EBP allows parameter free and scalable training of various types of MNNs (deterministic or stochastic) with discrete (e.g., binary) or continuous weights. In appendix F, we see that
for continuous weights EBP is almost identical to standard BP with a specific choice of activation
function s (x) = 2? (x) ? 1, CE loss and learning rate ? = 1. The only difference is that the input
is normalized by its standard deviation (Eq. 4.1, right), which depends on the weights and inputs
(Eq. 4.2). This re-scaling makes the learning algorithm invariant to the amplitude changes in the
neuronal input. This results from the same invariance of the sign activation functions. Note that in
standard BP algorithm the performance is directly affected by the amplitude of the input, so it is a
recommended practice to re-scale it in pre-processing [16].
We numerically evaluated the algorithm on binary classification tasks using MNNs with two or three
synaptic layers. In all data sets and MNNs EBP performs better than standard BP with the optimal
constant learning rate, and even achieves state-of-the-art results in comparison to [7]. Surprisingly,
EBP usually performs best when it is used to train binary MNNs. As suggested by a reviewer, this
could be related to the type of problems examined here. In text classification tasks have large sparse
input spaces (bag of words), and presence/absence of features (words) is more important than their
real values (frequencies). Therefore, (distributions over) binary weights and a threshold activation
function may work well.
In order to get such a good performance in binary MNNs, one must average over the output the
inferred (approximate) posterior of the weights. The EBP-P output of the algorithm calculates this
average analytically. In hardware this output could be realizable by averaging the output of several
binary MNNs, by sampling weights from P (Wij,l |Dn ). This can be done efficiently (appendix G).
Our numerical testing mainly focused on high-dimensional text classification tasks, where shallow architectures seem to work quite well. In other domains, such as vision [14] and speech [8],
deep architectures achieve state-of-the-art performance. Such deep MNNs usually require considerable fine-tuning and additional ?tricks? such as unsupervised pre-training [8], weight sharing [14]
or momentum6 . Integrating such methods into EBP and using it to train deep MNNs is a promising direction for future work. Another important generalization of the algorithm, which is rather
straightforward, is to use activation functions other than sign (?). This is particularly important for
the last layer - where a linear activation function would be useful for regression tasks, and joint
activation functions7 would be useful for multi-class tasks[4].
Acknowledgments The authors are grateful to C. Baldassi, A. Braunstein and R. Zecchina for
helpful discussions and to A. Hallak, T. Knafo and U. S?mb?l for reviewing parts of this manuscript.
The research was partially funded by the Technion V.P.R. fund, by the Intel Collaborative Research
Institute for Computational Intelligence (ICRI-CI), and by the Gruss Lipper Charitable Foundation.
5
This approximation is not required if all neurons in the MNN have a fan-out of one.
Which departs from the online framework considered here, since it requires two samples in each update.
7
i.e., activation functions for which (f (x))i 6= f (xi ), such as softmax or argmax.
6
8
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9
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4,715 | 527 | Improved Hidden Markov Model
Speech Recognition Using
Radial Basis Function Networks
Elliot Singer and Richard P. Lippmann
Lincoln Laboratory, MIT
Lexington, MA 02173-9108, USA
Abstract
A high performance speaker-independent isolated-word hybrid speech recognizer was developed which combines Hidden Markov Models (HMMs)
and Radial Basis Function (RBF) neural networks. In recognition experiments using a speaker-independent E-set database, the hybrid recognizer had an error rate of 11.5% compared to 15.7% for the robust
unimodal Gaussian HMM recognizer upon which the hybrid system was
based. These results and additional experiments demonstrate that RBF
networks can be successfully incorporated in hybrid recognizers and suggest that they may be capable of good performance with fewer parameters
than required by Gaussian mixture classifiers. A global parameter optimization method designed to minimize the overall word error rather than
the frame recognition error failed to reduce the error rate.
1
HMM/RBF HYBRID RECOGNIZER
A hybrid isolated-word speech recognizer was developed which combines neural
network and Hidden Markov Model (HMM) approaches. The hybrid approach is
an attempt to capitalize on the superior static pattern classification performance of
neural network classifiers [6] while preserving the temporal alignment properties of
HMM Viterbi decoding. Our approach is unique when compared to other studies
[2, 5] in that we use Radial Basis Function (RBF) rather than multilayer sigmoidal
networks. RBF networks were chosen because their static pattern classification
performance is comparable to that of other networks and they can be trained rapidly
using a one-pass matrix inversion technique [8] .
The hybrid HMM/RBF isolated-word recognizer is shown in Figure 1. For each
159
160
Singer and Lippmann
BEST WORD MATCH
WORD
MODELS
UNKNOWN
WORD
BACKGROUND
NOISE MODEL
_
Figure 1: Block diagram of the hybrid recognizer for a two word vocabulary.
pattern presented at the input layer, the RBF network produces nodal outputs
which are estimates of Bayesian probabilities [9]. The RBF network consists of an
input layer, a hidden layer composed of Gaussian basis functions, and an output
layer. Connections from the input layer to the hidden layer are fixed at unity
while those from the hidden layer to the output layer are trained by minimizing
the overall mean-square error between actual and desired output values. Each
RBF output node has a corresponding state in a set of HMM word models which
represent the words in the vocabulary. HMM word models are left-to-right with
no skip states and have a one-state background noise model at either end. The
background noise models are identical for all words. In the simplified diagram of
Figure 1, the vocabulary consists of 2 E-set words and the HMMs contain 3 states
per word model. The number of RBF output nodes (classes) is thus equal to the
total number of HMM non-background states plus one to account for background
noise. In recognition, Viterbi decoders use the nodal outputs of the RBF network
as observation probabilities to produce word likelihood scores. Since the outputs of
the RBF network can take on any value, they were initially hard limited to 0.0 and
1.0. The transition probabilities estimated as part of HMM training are retained.
The final response of the recognizer corresponds to that word model which produces
the highest Viterbi likelihood. Note that the structure of the HMM/RBF hybrid
recognizer is identical to that of a tied-mixture HMM recognizer. For a discussion
and comparison of the two recognizers, see [10].
Training of the hybrid recognizer begins with the preliminary step of training an
HMM isolated-word recognizer. The robust HMM recognizer used provides good
recognition performance on many standard difficult isolated-word speech databases
[7]. It uses continuous density, unimodal diagonal-covariance Gaussian classifiers
for each word state. Variances of all states are equal to the grand variance averaged
over all words and states. The trained HMM recognizer is used to force an alignment
of every training token and assign a label to each frame. Labels correspond to both
states of HMM word models and output nodes of the RBF network.
The Gaussian centers in the RBF hidden layer are obtained by performing k-means
Improved Hidden Markov Model Speech Recognition Using Radial Basis Function Networks
clustering on speech frames and separate clustering on noise frames, where speech
and noise frames are distinguished on the basis of the initial Viterbi alignment. The
RBF weights from the hidden layer to the output layer are computed by presenting
input frames to the RBF network and setting the desired network outputs to 1.0
for the output node corresponding to the frame label and 0.0 for all other nodes.
The RBF hidden node outputs and their correlations are accumulated across all
training tokens and are used to estimate weights to the RBF output nodes using a
fast one-pass algorithm [8]. Unlike the performance of the system reported in [5],
additional training iterations using the hybrid recognizer to label frames did not
improve performance.
2
DATABASE
All experiments were performed using a large, speaker-independent E-set (9 word)
database derived from the ISOLET Spoken Letter Database [4]. The training set
consisted of 1,080 tokens (120 tokens per word) spoken by 60 female and 60 male
speakers for a total of 61,466 frames. The test set consisted of 540 tokens (60
tokens per word) spoken by a different set of 30 female and 30 male speakers for
a total of 30,406 frames . Speech was sampled at 16 kHz and had an average SNR
of 31.5 dB. Input vectors were based on a mel-cepstrum analysis of the speech
waveform as described in [7]. The input analysis window was 20ms wide and was
advanced at 10ms intervals. Input vectors were created by adjoining the first 12
non-energy cepstral coefficients, the first 13 first-difference cepstral coefficients, and
the first 13 second-difference cepstral coefficients. Since the hybrid was based on
an 8 state-per-word robust HMM recognizer, the RBF network contained a total of
73 output nodes (72 speech nodes and 1 background node). The error rate of the 8
state-per-word robust HMM recognizer on the speaker-independent E-set task was
15.7%.
3
MODIFICATIONS TO THE HYBRID RECOGNIZER
The performance of the baseline HMM/RBF hybrid recognizer described in Section 1 is quite poor. We found it necessary to select the recognizer structure carefully
and utilize intermediate outputs properly to achieve a higher level of performance.
A full description of these modifications is presented in [10]. Briefly, they include
normalizing the hidden node outputs to sum to 1.0, normalizing the RBF outputs
by the corresponding a priori class probabilities as estimated from the initial Viterbi
alignment, expanding the RBF network into three individually trained subnetworks
corresponding to the ceptrum, first difference cepstrum, and second difference cepstrum data streams, setting a lower limit of 10- 5 on the values produced at the RBF
output nodes, adjusting a global scaling factor applied to the variances of the RBF
centers, and setting the number of centers to 33,33, and 65 for the first, second, and
third subnets, respectively. The structure of the final hybrid recognizer is shown in
Figure 2. This recognizer has an error rate of 11.5% (binomial standard deviation
= ?1.4) on the E-set test data compared to 15.7% (?1.6) for the 8 state-per-word
unimodal Gaussian HMM recognizer, and 9.6% (?1.3) for a considerably more complex tied-mixture HMM recognizer [10]. The final hybrid system contained a total
of 131 Gaussians and 9,563 weights. On a SUN SPARCstation 2, training time for
161
162
Singer and Lippmann
the final hybrid recognizer was about 1 hour and testing time was about 10 minutes.
BEST WORD MATCH
Figure 2: Block diagram of multiple sub net hybrid recognizer.
4
GLOBAL OPTIMIZATION
In the hybrid recognizer described above, discriminative training is performed at
the frame level. A preliminary segmentation by the HMM recognizer assigns each
speech frame to a specific RBF output node or, equivalently, an HMM word state.
The RBF network weights are then computed to minimize the squared error between the network output and the desired output over all input frames. The goal of
the recognizer, however, is to classify words. To meet this goal, discriminant training should be performed on word-level rather than frame-level outputs. Recently,
several investigators have described techniques that optimize parameters based on
word-level discriminant criteria [1, 3]. These techniques seek to maximize a mutual
information type of criterion:
Lc
C logy,
=
where Lc. is the likelihood score of the word model corresponding to the correct
Lw Lw is the sum of the word likelihood scores for all models. By
result and L
computing oC/oO, the gradient of C with respect to parameter 0, we can optimize
any parameter in the hybrid recognizer using the update equation
=
where 0 is the new value of parameter 0, () is the previous value, and TJ is a gain
term proportional to the learning rate. Following [1], we refer to the word-level
optimization technique as "global optimization."
Improved Hidden Markov Model Speech Recognition Using Radial Basis Function Networks
To apply global optimization to the HMM/RBF hybrid recognizer, we derived the
formulas for the gradient of C with respect to
the weight connecting RBF center
i to RBF output node j in subnet k; Pj, the RBF output normalization factor for
RBF output node j in subnet k; and mfl' the Ith element of the mean of center i of
subnet k. For each token of length T frames, these are given by
wt '
8C
J:lwk
U
ij
= (be; L- Pw )
w
T
<I>~
"'"' frjt{3jt
kIt'
t=1
L..J
St
and
likelihood score for word model w,
Lw / Lw Lw is the normalized word likelihood,
{ I if RBF output node j is a member of the correct word model
o otherwise,
forward partial probability of HMM state j at time t,
backward partial probability of HMM state j at time t,
unnormalized output of RBF node j of subnet k at time t,
normalized output of ith Gaussian center of sub net k at time t,
~
~,. <I>~t
I
=1
,
Ith element of the input vector for subnet k at time t,
global scaling factor for the variances of sub net k,
[th component of the standard deviation of the ith Gaussian center
of subnet k,
number of RBF output nodes in sub net k.
In implementing global optimization, the frame-level training procedure described
earlier serves to initialize system parameters and hill climbing methods are used to
reestimate parameters iteratively. Thus, weights are initialized to the values derived
using the one-pass matrix inversion procedure, RBF output normalization factors
are initialized to the class priors, and Gaussian means are initialized to the k-means
clustering values. Note that while the priors sum to one, no such constraint was
placed on the RBF output normalization factors during global optimization.
It is worth noting that since the RBF network outputs in the hybrid recognizer
are a posteriori probabilities normalized by a priori class probabilities, their values
may exceed 1. The accumulation of these quantities in the Viterbi decoders often
leads to values of (Xjt{3jt and Lw in the range of 10 80 or greater. Numerical problems
with the implementation of the global optimization equations were avoided by using
log arithmetic for intermediate operations and working with the quantity {3jt! Lw
throughout. Values of 7J which produced reasonable results were generally in the
range of 10- 10 to 10- 6
163
164
Singer and Lippmann
The results of using the global optimization technique to estimate the RBF weights
are shown in Figure 3. Figure 3( a) shows the recognition performance on the training and test sets versus the number of training iterations and Figure 3(b) tracks
the value of the criterion C = Lei L on the training and test set under the same
conditions. It is apparent that the method succeeds in iteratively increasing the
value of the criterion and in significantly lowering the error rate on the training
data. Unfortunately, this behavior does not extend to improved performance on
the test data. This suggests that global optimization is overfitting the hybrid word
models to the training data. Results using global optimization to estimate RBF
output normalization factors and the Gaussian means produced similar results.
20
%ERROR
,.---~----r-----,r-----"
TEST
C = log (Lc I L)
0
-0.2
TRAIN
-0.4
10
-0.6
TEST
-0.8
o o~--~---~--~~--~
5
10
15
20
NUMBER OF ITERATIONS
-1
0
5
10
15
20
NUMBER OF ITERATIONS
Figure 3: (a) Error rates for training and test data. (b) Criterion C for training
and test data.
5
ACCURACY OF BAYES PROBABILITY
ESTIMATION
Three methods were used to determine how well RBF outputs estimate Bayes probabilities. First, since network outputs must sum to one if they are probabilities, we
computed the RMS error between the sum of the RBF outputs and unity for all
frames of the test data. The average RMS error was low (10- 4 or less for each
subnet). Second, the average output of each RBF node was computed because this
should equal the a priori probability of the class associated with the node [9]. This
condition was true for each subnet with an average RMS error on the order of 10- 5 .
For the final method, we partitioned the outputs into 100 equal size bins between
0.0 and 1.0. For each input pattern, we used the output values to select the appropriate bins and incremented the corresponding bin counts by one. In addition, we
incremented the correct-class bin count for the one bin which corresponded to the
class of the input pattern. For example, data indicated that for the 61,466 frames
of training tokens, nodal outputs of the cepstra subnet in the range 0.095-0.105 occurred 29,698 times and were correct classifications (regardless of class) 3,067 times.
If the outputs of the network were true Bayesian probabilities, we would expect the
Improved Hidden Markov Model Speech Recognition Using Radial Basis Function Networks
relative frequency of correct labeling to be close to 0.1. Similarly, relative frequencies measured in other intervals would also be expected to be close to the value of
the corresponding center of the interval. Thus, a plot of the relative frequencies for
each bin versus the bin centers should show the measured values lying close to the
diagonal.
The measured relative frequency data for the cepstra subnet and ?2u bounds for
the binomial standard deviations of the relative frequencies are shown in Figure 4.
Outputs below 0.0 and above 1.0 are fixed at 0.0 and 1.0, respectively. Although
the relative frequencies tend to be clustered around the diagonal, many values lie
test
outside the bounds. Furthermore, goodness-of-fit measurements using the
indicate that fits fail at significance levels well below .01. We conclude that although
the system provides good recognition accuracy, better performance may be obtained
with improved estimation of Bayesian probabilities.
x:
1r-------------------------------------------------------------~?
.J
~0.9
:3 0.8
t;
wO.7
IX
IX
0 0 .6
o
~0.5
gO.4
IX
~0.3
1= 0.2
:5
~O.1
0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
RBF NETWORK OUTPUT (All Nodes)
1
Figure 4: Relative frequency of correct class labeling and ?2u bounds for the binomial standard deviation, cepstra subnet.
6
SUMMARY AND CONCLUSIONS
This paper describes a hybrid isolated-word speech recognizer which successfully
integrates Radial Basis Function neural networks and Hidden Markov Models. The
hybrid's performance is better than that of a tied-mixture recognizer of comparable
complexity and near that of a tied-mi..xture recognizer of considerably greater complexity. The structure of the RBF networks and the processing of network outputs
had to be carefully selected to provide this level of performance. A global optimization technique designed to maximize a word discrimination criterion did not
succeed in improving performance further. Statistical tests indicated that the accuracy of the Bayesian probability estimation performed by the RBF networks could
165
166
Singer and Lippmann
be improved. We conclude that RBF networks can be used to provide good performance and short training times in hybrid recognizers and that these systems may
require fewer parameters than Gaussian-mixture-based recognizers at comparable
performance levels.
Acknowledgements
This work was sponsored by the Defense Advanced Research Projects Agency. The
views expressed are those of the authors and do not reflect the official policy or
position of the U.S. Government.
References
[1] Yoshua Bengio, Renato De Mori, Giovanni Flammia, and Ralf Kompe. Global
optimization of a neural network - Hidden Markov model hybrid. Technical
Report TR-SOCS-90.22, MgGill University School of Computer Science, Montreal, Qc., Canada, December 1990.
[2] Herve Bourlard and Nelson Morgan. A continuous speech recognition system
embedding MLP into HMM. In D. Touretzky, editor, Advances in Neural
Information Processing 2, pages 186-193. Morgan Kaufmann, San Mateo, CA,
1990.
[3] John S. Bridle. Alpha-nets: A recurrent neural network architecture with a
hidden Markov model interpretation. Speech Communication, 9:83-92, 1990.
[4] Ron Cole, Yeshwant Muthusamy, and Mark Fanty. The Isolet spoken letter
database. Technical Report CSE 90-004, Oregon Graduate Institute of Science
and Technology, Beverton, OR, March 1990.
[5] Michael Franzini, Kai-Fu Lee, and Alex Waibel. Connectionist viterbi training: A new hybrid method for continuous speech recognition. In Proceedings
of IEEE International Conference on Acoustics Speech and Signal Processing.
IEEE, April 1990.
[6] Richard P. Lippmann. Pattern classification using neural networks. IEEE
Communications Magazine, 27(11):47-54, November 1989.
[7] Richard P. Lippmann and Ed A. Martin. Mqlti-style training for robust
isolated-word speech recognition. In Proceedings International Conference on
Acoustics Speech and Signal Processing, pages 705-708. IEEE, April 1987.
[8] Kenney Ng and Richard P. Lippmann. A comparative study of the practical characteristics of neural network and conventional pattern classifiers. In
R. P. Lippmann, J. Moody, and D. S. Touretzky, editors, Advances in Neural
Information Processing 3. Morgan Kaufmann, San Mateo, CA, 1991.
[9] Mike D. Richard and Richard P. Lippmann. Neural network classifiers estimate
Bayesian a posteriori probabilities. Neural Computation, In Press.
[10] Elliot Singer and Richard P. Lippmann. A speech recognizer using radial basis function neural networks in an HMM framework. In Proceedings of the
International Conference on Acoustics, Speech, and Signal Processing. IEEE,
1992.
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4,716 | 5,270 | An Autoencoder Approach to Learning
Bilingual Word Representations
Sarath Chandar A P1 ? , Stanislas Lauly2 ? , Hugo Larochelle2 , Mitesh M Khapra3 ,
Balaraman Ravindran1 , Vikas Raykar3 , Amrita Saha3
1
Indian Institute of Technology Madras, 2 Universit?e de Sherbrooke, 3 IBM Research India
[email protected], {stanislas.lauly,hugo.larochelle}@usherbrooke.ca,
{mikhapra,viraykar,amrsaha4}@in.ibm.com, [email protected]
?
Both authors contributed equally
Abstract
Cross-language learning allows one to use training data from one language to
build models for a different language. Many approaches to bilingual learning require that we have word-level alignment of sentences from parallel corpora. In this
work we explore the use of autoencoder-based methods for cross-language learning of vectorial word representations that are coherent between two languages,
while not relying on word-level alignments. We show that by simply learning to
reconstruct the bag-of-words representations of aligned sentences, within and between languages, we can in fact learn high-quality representations and do without
word alignments. We empirically investigate the success of our approach on the
problem of cross-language text classification, where a classifier trained on a given
language (e.g., English) must learn to generalize to a different language (e.g., German). In experiments on 3 language pairs, we show that our approach achieves
state-of-the-art performance, outperforming a method exploiting word alignments
and a strong machine translation baseline.
1
Introduction
The accuracy of Natural Language Processing (NLP) tools for a given language depend heavily on
the availability of annotated resources in that language. For example, high quality POS taggers
[1], parsers [2], sentiment analyzers [3] are readily available for English. However, this is not the
case for many other languages such as Hindi, Marathi, Bodo, Farsi, and Urdu, for which annotated
data is scarce. This situation was acceptable in the past when only a few languages dominated the
digital content available online and elsewhere. However, the ever increasing number of languages
on the web today has made it important to accurately process natural language data in such resourcedeprived languages also. An obvious solution to this problem is to improve the annotated inventory
of these languages, but the cost, time and effort required act as a natural deterrent to this.
Another option is to exploit the unlabeled data available in a language. In this context, vectorial text
representations have proven useful for multiple NLP tasks [4, 5]. It has been shown that meaningful representations, capturing syntactic and semantic similarity, can be learned from unlabeled data.
While the majority of previous work on vectorial text representations has concentrated on the monolingual case, there has also been considerable interest in learning word and document representations
that are aligned across languages [6, 7, 8, 9, 10, 11, 12]. Such aligned representations allow the use
of resources from a resource-fortunate language to develop NLP capabilities in a resource-deprived
language.
One approach to cross-lingual exploitation of resources is to project parameters learned from the
annotated data of one language to another language [13, 14, 15, 16, 17]. These approaches rely on a
1
bilingual resource such as a Machine Translation (MT) system. Recent attempts at learning common
bilingual representations [9, 10, 11] aim to eliminate the need of such an MT system. A common
property of these approaches is that a word-level alignment of translated sentences is leveraged to
derive a regularization term relating word embeddings across languages. Such methods not only
eliminate the need for an MT system but also outperform MT based projection approaches.
In this paper, we experiment with methods that learn bilingual word representations without word-toword alignments of bilingual corpora during training. Unlike previous approaches, we only require
aligned sentences and do not rely on word-level alignments (e.g., extracted using GIZA++, as is
usual), simplifying the learning procedure. To do so, we propose and investigate bilingual autoencoder models, that learn hidden encoder representations of paired bag-of-words sentences that are
not only informative of the original bag-of-words but also predictive of the other language. Word
representations can then easily be extracted from the encoder and used in the context of a supervised NLP task. Specifically, we demonstrate the quality of these representations for the task of
cross-language document classification, where a labeled data set can be available in one language,
but not in another one. As we?ll see, our approach is able to reach state-of-the-art performance,
outperforming a method exploiting word alignments and a strong machine translation baseline.
2
Autoencoder for Bags-of-Words
Let x be the bag-of-words representation of a sentence. Specifically, each xi is a word index from
a fixed vocabulary of V words. As this is a bag-of-words, the order of the words within x does not
correspond to the word order in the original sentence. We wish to learn a D-dimensional vectorial
representation of our words from a training set of sentence bags-of-words {x(t) }Tt=1 .
We propose to achieve this by using an autoencoder model that encodes an input bag-of-words x with
a sum of the representations (embeddings) of the words present in x, followed by a non-linearity.
Specifically, let matrix W be the D ? V matrix whose columns are the vector representations for
each word. The encoder?s computation will involve summing over the columns of W for each
word in the bag-of-word. We will denote this encoder function ?(x). Then, using a decoder, the
autoencoder will be trained to optimize a loss function that measures how predictive of the original
bag-of-words is the encoder representation ?(x) .
There are different variations we can consider in the design of the encoder/decoder and the choice of
loss function. One must be careful however, as certain choices can be inappropriate for training on
word observations, which are intrinsically sparse and high-dimensional. In this paper, we explore
and compare two different approaches, described in the next two sub-sections.
2.1
Binary bag-of-words reconstruction training with merged bags-of-words
In the first approach, we start from the conventional autoencoder architecture, which minimizes a
cross-entropy loss that compares a binary vector observation with a decoder reconstruction. We thus
convert the bag-of-words x into a fixed-size but sparse binary vector v(x), which is such that v(x)xi
is 1 if word xi is present in x and otherwise 0.
From this representation, we obtain an encoder representation by multiplying v(x) with the word
representation matrix W
a(x) = c + Wv(x), ?(x) = h(a(x))
(1)
where h(?) is an element-wise non-linearity such as the sigmoid or hyperbolic tangent, and c is a
D-dimensional bias vector. Encoding thus involves summing the word representations of the words
present at least once in the bag-of-words.
To produce a reconstruction, we parametrize the decoder using the following non-linear form:
b (x) = sigm(V?(x) + b)
v
(2)
where V = WT , b is the bias vector of the reconstruction layer and sigm(a) = 1/(1 + exp(?a)) is
the sigmoid non-linearity.
2
Then, the reconstruction is compared to the original binary bag-of-words as follows:
`(v(x)) = ?
V
X
v(x)i log(b
v (x)i ) + (1 ? v(x)i ) log(1 ? vb(x)i ) .
(3)
i=1
Training proceeds by optimizing the sum of reconstruction cross-entropies across the training set,
e.g., using stochastic or mini-batch gradient descent.
Note that, since the binary bags-of-words are very high-dimensional (the dimensionality corresponds
to the size of the vocabulary, which is typically large), the above training procedure which aims at
reconstructing the complete binary bag-of-word, will be slow. Since we will later be training on
millions of sentences, training on each individual sentence bag-of-words will be expensive.
Thus, we propose a simple trick, which exploits the bag-of-words structure of the input. Assuming
we are performing mini-batch training (where a mini-batch contains a list of the bags-of-words of
adjacent sentences), we simply propose to merge the bags-of-words of the mini-batch into a single
bag-of-words and perform an update based on that merged bag-of-words. The resulting effect is that
each update is as efficient as in stochastic gradient descent, but the number of updates per training
epoch is divided by the mini-batch size . As we?ll see in the experimental section, this trick produces
good word representations, while sufficiently reducing training time. We note that, additionally, we
could have used the stochastic approach proposed by Dauphin et al. [18] for reconstructing binary
bag-of-words representations of documents, to further improve the efficiency of training. They use
importance sampling to avoid reconstructing the whole V -dimensional input vector.
2.2
Tree-based decoder training
The previous autoencoder architecture worked with a binary vectorial representation of the input
bag-of-words. In the second autoencoder architecture we investigate, we consider an architecture
that instead works with the bag (unordered list) representation more directly.
First, the encoder representation will now involve a sum of the representation of all words, reflecting
the relative frequency of each word:
a(x) = c +
|x|
X
W?,xi , ?(x) = h (a(x)) .
(4)
i=1
Moreover, decoder training will assume that, from the decoder?s output, we can obtain a probability
distribution p(b
x|?(x)) over any word x
b observed at the reconstruction output layer. Then, we can
treat the input bag-of-words as a |x|-trials multinomial sample from that distribution and use as the
reconstruction loss its negative log-likelihood:
`(x) =
V
X
? log p(b
x = xi |?(x)) .
(5)
i=1
We now must ensure that the decoder can compute p(b
x = xi |?(x)) efficiently from ?(x). Specifically, we?d like to avoid a procedure scaling linearly with the vocabulary size V , since V will be very
large in practice. This precludes any procedure that would compute the numerator of p(b
x = w|?(x))
for each possible word w separately and normalize it so it sums to one.
We instead opt for an approach borrowed from the work on neural network language models [19, 20].
Specifically, we use a probabilistic tree decomposition of p(b
x = xi |?(x)). Let?s assume each word
has been placed at the leaf of a binary tree. We can then treat the sampling of a word as a stochastic
path from the root of the tree to one of the leaves.
We denote as l(x) the sequence of internal nodes in the path from the root to a given word x, with
l(x)1 always corresponding to the root. We will denote as ?(x) the vector of associated left/right
branching choices on that path, where ?(x)k = 0 means the path branches left at internal node l(x)k
and otherwise branches right if ?(x)k = 1. Then, the probability p(b
x = x|?(x)) of reconstructing
a certain word x observed in the bag-of-words is computed as
|?(?
x)|
p(b
x|?(x)) =
Y
k=1
3
p(?(b
x)k |?(x))
(6)
where p(?(b
x)k |?(x)) is output by the decoder. By using a full binary tree of words, the number of
different decoder outputs required to compute p(b
x|?(x)) will be logarithmic in the vocabulary size
V . Since there are |x| words in the bag-of-words, at most O(|x| log V ) outputs are required from
the decoder. This is of course a worst case scenario, since words will share internal nodes between
their paths, for which the decoder output can be computed just once. As for organizing words into a
tree, as in Larochelle and Lauly [21] we used a random assignment of words to the leaves of the full
binary tree, which we have found to work well in practice.
Finally, we need to choose a parametrized form for the decoder. We choose the following form:
p(?(b
x)k = 1|?(x)) = sigm(bl(?xi )k + Vl(?xi )k ,? ?(x))
(7)
where b is a (V -1)-dimensional bias vector and V is a (V ?1)?D matrix. Each left/right branching
probability is thus modeled with a logistic regression model applied on the encoder representation
of the input bag-of-words ?(x).
3
Bilingual autoencoders
Let?s now assume that for each sentence bag-of-words x in some source language X , we have an
associated bag-of-words y for this sentence translated in some target language Y by a human expert.
Assuming we have a training set of such (x, y) pairs, we?d like to use it to learn representations in
both languages that are aligned, such that pairs of translated words have similar representations.
To achieve this, we propose to augment the regular autoencoder proposed in Section 2 so that, from
the sentence representation in a given language, a reconstruction can be attempted of the original
sentence in the other language. Specifically, we now define language specific word representation
matrices Wx and Wy , corresponding to the languages of the words in x and y respectively. Let
V X and V Y also be the number of words in the vocabulary of both languages, which can be different. The word representations however are of the same size D in both languages. For the binary
reconstruction autoencoder, the bag-of-words representations extracted by the encoder become
?(x) = h c + WX v(x) , ?(y) = h c + WY v(y)
and are similarly extended for the tree-based autoencoder. Notice that we share the bias c before the
non-linearity across encoders, to encourage the encoders in both languages to produce representations on the same scale.
From the sentence in either languages, we want to be able to perform a reconstruction of the original
sentence in both the languages. In particular, given a representation in any language, we?d like a
decoder that can perform a reconstruction in language X and another decoder that can reconstruct in
language Y. Again, we use decoders of the form proposed in either Section 2.1 or 2.2 (see Figure 1),
but let the decoders of each language have their own parameters (bX , VX ) and (bY , VY ).
This encoder/decoder decomposition structure allows us to learn a mapping within each language
and across the languages. Specifically, for a given pair (x, y), we can train the model to (1) construct
y from x (loss `(x, y)), (2) construct x from y (loss `(y, x)), (3) reconstruct x from itself (loss
`(x)) and (4) reconstruct y from itself (loss `(y)). We follow this approach in our experiments and
optimize the sum of the corresponding 4 losses during training.
3.1
Joint reconstruction and cross-lingual correlation
We also considered incorporating two additional terms to the loss function, in an attempt to favour
even more meaningful bilingual representations:
`(x, y) + `(y, x) + `(x) + `(y) + ?`([x, y], [x, y]) ? ? ? cor(a(x), a(y))
(8)
The term `([x, y], [x, y]) is simply a joint reconstruction term, where both languages are simultanouesly presented as input and reconstructed. The second term cor(a(x), a(y)) encourages correlation between the representation of each language. It is the sum of the scalar correlations between
each pair a(x)k , a(y)k , across all dimensions k of the vectors a(x), a(y)1 . To obtain a stochastic
estimate of the correlation, during training, small mini-batches are used.
1
While we could have applied the correlation term on ?(x), ?(y) directly, applying it to the pre-activation
function vectors was found to be more numerically stable.
4
Figure 1: Left: Bilingual autoencoder based on the binary reconstruction error. Right: Tree-based
bilingual autoencoder. In this example, they both reconstruct the bag-of-words for the English sentence ?the dog barked? from its French translation ?le chien a japp?e?.
3.2
Document representations
Once we learn the language specific word representation matrices Wx and Wy as described above,
we can use them to construct document representations, by using their columns as word vector
representations. Given a document d written in language Z ? {X , Y} and containing m words,
zP
1 , z2 , . . . , zm , we represent it as the tf-idf weighted sum of its words? representations ?(d) =
m
Z
i=1 tf-idf(zi ) ? W.,zi . We use the document representations thus obtained to train our document
classifiers, in the cross-lingual document classification task described in Section 5.
4
Related Work
Recent work that has considered the problem of learning bilingual representations of words usually
has relied on word-level alignments. Klementiev et al. [9] propose to train simultaneously two neural
network languages models, along with a regularization term that encourages pairs of frequently
aligned words to have similar word embeddings. Thus, the use of this regularization term requires
to first obtain word-level alignments from parallel corpora. Zou et al. [10] use a similar approach,
with a different form for the regularizer and neural network language models as in [5]. In our work,
we specifically investigate whether a method that does not rely on word-level alignments can learn
comparably useful multilingual embeddings in the context of document classification.
Looking more generally at neural networks that learn multilingual representations of words or
phrases, we mention the work of Gao et al. [22] which showed that a useful linear mapping between
separately trained monolingual skip-gram language models could be learned. They too however
rely on the specification of pairs of words in the two languages to align. Mikolov et al. [11] also propose a method for training a neural network to learn useful representations of phrases, in the context
of a phrase-based translation model. In this case, phrase-level alignments (usually extracted from
word-level alignments) are required. Recently, Hermann and Blunsom [23], [24] proposed neural
network architectures and a margin-based training objective that, as in this work, does not rely on
word alignments. We will briefly discuss this work in the experiments section.
5
Experiments
The techniques proposed in this paper enable us to learn bilingual embeddings which capture crosslanguage similarity between words. We propose to evaluate the quality of these embeddings by using
them for the task of cross-language document classification. We followed closely the setup used by
Klementiev et al. [9] and compare with their method, for which word representations are publicly
available2 . The set up is as follows. A labeled data set of documents in some language X is available
to train a classifier, however we are interested in classifying documents in a different language Y
at test time. To achieve this, we leverage some bilingual corpora, which is not labeled with any
2
http://people.mmci.uni-saarland.de/?aklement/data/distrib/
5
document-level categories. This bilingual corpora is used to learn document representations that are
coherent between languages X and Y. The hope is thus that we can successfully apply the classifier
trained on document representations for language X directly to the document representations for
language Y. Following this setup, we performed experiments on 3 data sets of language pairs:
English/German (EN/DE), English/French (EN/FR) and English/Spanish (EN/ES).
5.1
Data
For learning the bilingual embeddings, we used sections of the Europarl corpus [25] which contains
roughly 2 million parallel sentences. We considered 3 language pairs. We used the same preprocessing as used by Klementiev et al. [9]. We tokenized the sentences using NLTK [26], removed
punctuations and lowercased all words. We did not remove stopwords.
As for the labeled document classification data sets, they were extracted from sections of the Reuters
RCV1/RCV2 corpora, again for the 3 pairs considered in our experiments. Following Klementiev
et al. [9], we consider only documents which were assigned exactly one of the 4 top level categories
in the topic hierarchy (CCAT, ECAT, GCAT and MCAT). These documents are also pre-processed
using a similar procedure as that used for the Europarl corpus. We used the same vocabularies as
those used by Klementiev et al. [9] (varying in size between 35, 000 and 50, 000).
For each pair of languages, our overall procedure for cross-language classification can be summarized as follows:
Train representation: Train bilingual word representations Wx and Wy on sentence pairs extracted from Europarl for languages X and Y. Optionally, we also use the monolingual documents
from RCV1/RCV2 to reinforce the monolingual embeddings (this choice is cross-validated). These
non-parallel documents can be used through the losses `(x) and `(y) (i.e. by reconstructing x from x
or y from y). Note that Klementiev et al. [9] also used this data when training word representations.
Train classifier: Train document classifier on the Reuters training set for language X , where documents are represented using the word representations Wx (see Section 3.2). As in Klementiev et al.
[9] we used an averaged perceptron trained for 10 epochs, for all the experiments.
Test-time classification: Use the classifier trained in the previous step on the Reuters test set for
language Y, using the word representations Wy to represent the documents.
We trained the following autoencoders3 : BAE-cr which uses reconstruction error based decoder
training (see Section 2.1) and BAE-tr which uses tree-based decoder training (see Section 2.2).
Models were trained for up to 20 epochs using the same data as described earlier. BAE-cr used
mini-batch (of size 20) stochastic gradient descent, while BAE-tr used regular stochastic gradient.
All results are for word embeddings of size D = 40, as in Klementiev et al. [9]. Further, to speed
up the training for BAE-cr we merged each 5 adjacent sentence pairs into a single training instance,
as described in Section 2.1. For all language pairs, the joint reconstruction ? was fixed to 1 and
the cross-lingual correlation factor ? to 4 for BAE-cr. For BAE-tr, none of these additional terms
were found to be particularly beneficial, so we set their weights to 0 for all tasks. The other hyperparameters were tuned to each task using a training/validation set split of 80% and 20% and using
the performance on the validation set of an averaged perceptron trained on the smaller training set
portion (notice that this corresponds to a monolingual classification experiment, since the general
assumption is that no labeled data is available in the test set language).
5.2
Comparison of the performance of different models
We now present the cross language classification results obtained by using the embeddings produced
by our two autoencoders. We also compare our models with the following approaches:
Klementiev et al.: This model uses word embeddings learned by a multitask neural network language model with a regularization term that encourages pairs of frequently aligned words to have
similar word embeddings. From these embeddings, document representations are computed as described in Section 3.2.
3
Our word representations and code are available at http://www.sarathchandar.in/crl.html
6
Table 1: Cross-lingual classification accuracy for 3 language pairs, with 1000 labeled examples.
BAE-tr
BAE-cr
Klementiev et al.
MT
Majority Class
EN ? DE
81.8
91.8
77.6
68.1
46.8
DE ? EN
60.1
74.2
71.1
67.4
46.8
EN ? FR
70.4
84.6
74.5
76.3
22.5
FR ? EN
61.8
74.2
61.9
71.1
25.0
EN ? ES
59.4
49.0
31.3
52.0
15.3
ES ? EN
60.4
64.4
63.0
58.4
22.2
Table 2: Example English words along with the closest words both in English (EN) and German
(DE), using the Euclidean distance between the embeddings learned by BAE-cr.
Word
Lang Nearest neighbors
Word
Lang Nearest neighbors
EN
january, march, october
EN
oil, supply, supplies, gas
january
oil
DE
januar, m?arz, oktober
DE
o? l, boden, befindet, ger?at
EN
president, i, mr, presidents
EN
microsoft, cds, insider
president
microsoft
DE
pr?asident, pr?asidentin
DE
microsoft, cds, warner
EN
said, told, say, believe
EN
market, markets, single
said
market
DE
gesagt, sagte, sehr, heute
DE
markt, marktes, m?arkte
MT: Here, test documents are translated to the language of the training documents using a standard
phrase-based MT system, MOSES4 which was trained using default parameters and a 5-gram language model on the Europarl corpus (same as the one used for inducing our bilingual embeddings).
Majority Class: Test documents are simply assigned the most frequent class in the training set.
For the EN/DE language pairs, we directly report the results from Klementiev et al. [9]. For the other
pairs (not reported in Klementiev et al. [9]), we used the embeddings available online and performed
the classification experiment ourselves. Similarly, we generated the MT baseline ourselves.
Table 1 summarizes the results. They were obtained using 1000 RCV training examples. We report
results in both directions, i.e. language X to Y and vice versa. The best performing method is always
either BAE-cr or BAE-tr, with BAE-cr having the best performance overall. In particular, BAE-cr
often outperforms the approach of Klementiev et al. [9] by a large margin.
We also mention the recent work of Hermann and Blunsom [23], who proposed two neural network
architectures for learning word and document representations using sentence-aligned data only. Instead of an autoencoder paradigm, they propose a margin-based objective that aims to make the
representation of aligned sentences closer than non-aligned sentences. While their trained embeddings are not publicly available, they report results for the EN/DE classification experiments, with
representations of the same size as here (D = 40) and trained on 500K EN/DE sentence pairs. Their
best model reaches accuracies of 83.7% and 71.4% respectively for the EN ? DE and DE ? EN
tasks. One clear advantage of our model is that unlike their model, it can use additional monolingual data. Indeed, when we train BAE-cr with 500k EN/DE sentence pairs, plus monolingual RCV
documents (which come at no additional cost), we get accuracies of 87.9% (EN ? DE) and 76.7%
(DE ? EN), still improving on their best model. If we do not use the monolingual data, BAE-cr?s
performance is worse but still competitive at 86.1% for EN ? DE and 68.8% for DE ? EN.
We also evaluate the effect of varying the amount of supervised training data for training the classifier. For brevity, we report only the results for the EN/DE pair, which are summarized in Figure 2.
We observe that BAE-cr clearly outperforms the other models at almost all data sizes. More importantly, it performs remarkably well at very low data sizes (100), suggesting it learns very meaningful
embeddings, though the method can still benefit from more labeled data (as in the DE ? EN case).
Table 2 also illustrates the properties captured within and across languages, for the EN/DE pair5 .
For a few English words, the words with closest word representations (in Euclidean distance) are
shown, for both English and German. We observe that words that form a translation pair are close,
but also that close words within a language are syntactically/semantically similar as well.
4
5
http://www.statmt.org/moses/
See also the supplementary material for a t-SNE visualization of the word representations.
7
Figure 2: Cross-lingual classification accuracy results, from EN ? DE (left), and DE ? EN (right).
The excellent performance of BAE-cr suggests that merging several sentences into single bags-ofwords can still yield good word embeddings. In other words, not only we do not need to rely
on word-level alignments, but exact sentence-level alignment is also not essential to reach good
performances. We experimented with the merging of 5, 25 and 50 adjacent sentences (see the
supplementary material). Generally speaking, these experiments also confirm that even coarser
merges can sometimes not be detrimental. However, for certain language pairs, there can be an
important decrease in performance. On the other hand, when comparing the performance of BAE-tr
with the use of 5-sentences merges, no substantial impact is observed.
6
Conclusion and Future Work
We presented evidence that meaningful bilingual word representations could be learned without
relying on word-level alignments or using fairly coarse sentence-level alignments. In particular, we
showed that even though our model does not use word level alignments, it is able to reach state-ofthe-art performance, even compared to a method that exploits word-level alignments. In addition, it
also outperforms a strong machine translation baseline.
For future work, we would like to investigate extensions of our bag-of-words bilingual autoencoder
to bags-of-n-grams, where the model would also have to learn representations for short phrases.
Such a model should be particularly useful in the context of a machine translation system. We
would also like to explore the possibility of converting our bilingual model to a multilingual model
which can learn common representations for multiple languages given different amounts of parallel
data between these languages.
Acknowledgement
We would like to thank Alexander Klementiev and Ivan Titov for providing the code for the classifier
and data indices. This work was supported in part by Google.
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4,717 | 5,271 | Pre-training of Recurrent Neural Networks via
Linear Autoencoders
Luca Pasa, Alessandro Sperduti
Department of Mathematics
University of Padova, Italy
{pasa,sperduti}@math.unipd.it
Abstract
We propose a pre-training technique for recurrent neural networks based on linear
autoencoder networks for sequences, i.e. linear dynamical systems modelling the
target sequences. We start by giving a closed form solution for the definition of
the optimal weights of a linear autoencoder given a training set of sequences. This
solution, however, is computationally very demanding, so we suggest a procedure
to get an approximate solution for a given number of hidden units. The weights
obtained for the linear autoencoder are then used as initial weights for the inputto-hidden connections of a recurrent neural network, which is then trained on the
desired task. Using four well known datasets of sequences of polyphonic music,
we show that the proposed pre-training approach is highly effective, since it allows
to largely improve the state of the art results on all the considered datasets.
1
Introduction
Recurrent Neural Networks (RNN) constitute a powerful computational tool for sequences modelling and prediction [1]. However, training a RNN is not an easy task, mainly because of the well
known vanishing gradient problem which makes difficult to learn long-term dependencies [2]. Although alternative architectures, e.g. LSTM networks [3], and more efficient training procedures,
such as Hessian Free Optimization [4], have been proposed to circumvent this problem, reliable and
effective training of RNNs is still an open problem.
The vanishing gradient problem is also an obstacle to Deep Learning, e.g., [5, 6, 7]. In that context,
there is a growing evidence that effective learning should be based on relevant and robust internal
representations developed in autonomy by the learning system. This is usually achieved in vectorial
spaces by exploiting nonlinear autoencoder networks to learn rich internal representations of input
data which are then used as input to shallow neural classifiers or predictors (see, for example, [8]).
The importance to start gradient-based learning from a good initial point in the parameter space has
also been pointed out in [9]. Relationship between autoencoder networks and Principal Component
Analysis (PCA) [10] is well known since late ?80s, especially in the case of linear hidden units [11,
12]. More recently, linear autoencoder networks for structured data have been studied in [13, 14, 15],
where an exact closed-form solution for the weights is given in the case of a number of hidden units
equal to the rank of the full data matrix.
In this paper, we borrow the conceptual framework presented in [13, 16] to devise an effective pretraining approach, based on linear autoencoder networks for sequences, to get a good starting point
into the weight space of a RNN, which can then be successfully trained even in presence of longterm dependencies. Specifically, we revise the theoretical approach presented in [13] by: i) giving
a simpler and direct solution to the problem of devising an exact closed-form solution (full rank
case) for the weights of a linear autoencoder network for sequences, highlighting the relationship
between the proposed solution and PCA of the input data; ii) introducing a new formulation of
1
the autoencoder learning problem able to return an optimal solution also in the case of a number
of hidden units which is less than the rank of the full data matrix; iii) proposing a procedure for
approximate learning of the autoencoder network weights under the scenario of very large sequence
datasets. More importantly, we show how to use the linear autoencoder network solution to derive a
good initial point into a RNN weight space, and how the proposed approach is able to return quite
impressive results when applied to prediction tasks involving long sequences of polyphonic music.
2
Linear Autoencoder Networks for Sequences
In [11, 12] it is shown that principal directions of a set of vectors xi ? Rk are related to solutions
obtained by training linear autoencoder networks
oi = Woutput Whidden xi , i = 1, . . . , n,
(1)
where Whidden ? Rp?k , Woutput ? Rk?p , p k, and the network is trained so to get oi = xi , ?i.
When considering a temporal sequence x1 , x2 , . . . , xt , . . . of input vectors, where t is a discrete time
index, a linear autoencoder can be defined by considering the coupled linear dynamical systems
xt
= Cyt
(3)
yt = Axt + Byt?1 (2)
yt?1
It should be noticed that eqs. (2) and (3) extend the linear transformation defined in eq. (1) by
introducing a memory term involving matrix B ? Rp?p . In fact, yt?1 is inserted in the right part
of equation (2) to keep track of the input history through time: this is done exploiting a state space
representation. Eq. (3) represents the decoding part of the autoencoder: when a state yt is multiplied
by C, the observed input xt at time t and state at time t ? 1, i.e. yt?1 , are generated. Decoding
can then continue from yt?1 . This formulation has been proposed, for example, in [17] where an
iterative procedure to learn weight matrices A and B, based on Oja?s rule, is presented. No proof
of convergence for the proposed procedure is however given. More recently, an exact closed-form
solution for the weights has been given in the case of a number of hidden units equal to the rank of
the full data matrix (full rank case) [13, 16]. In this section, we revise this result. In addition, we
give an exact solution also for the case in which the number of hidden units is strictly less than the
rank of the full data matrix.
The basic idea of [13, 16] is to look for directions of high variance into the state space of the
dynamical linear system (2). Let start by considering a single sequence x1 , x2 , . . . , xt , . . . , xn and
the state vectors of the corresponding induced state sequence collected as rows of a matrix Y =
T
[y1 , y2 , y3 , ? ? ? , yn ] . By using the initial condition y0 = 0 (the null vector), and the dynamical
linear system (2), we can rewrite the Y matrix as
? T
?? T
?
A
x1 0
0
0
??? 0
? xT xT
?
? AT BT
0
0
??? 0 ?
2
1
? T
? ? T 2T
?
T
T
?
?
? x3 x2
?
x
0
?
?
?
0
A
B
1
Y=?
??
?
? ..
?
..
..
..
..
.. ? ?
..
? .
?
?
?
.
.
.
.
.
.
T
T
T
T
T
T
T
n?1
xn xn?1 xn?2 ? ? ? x2 x1
A B
|
{z
}|
{z
}
?
?
n?s
where, given s = kn, ? ? R
is a data matrix collecting all the (inverted) input subsequences
(including the whole sequence) as rows, and ? is the parameter matrix of the dynamical system.
Now, we are interested in using a state space of dimension p n, i.e. yt ? Rp , such that as
much information as contained in ? is preserved. We start by factorizing ? using SVD, obtaining
? = V?UT where V ? Rn?n is an unitary matrix, ? ? Rn?s is a rectangular diagonal matrix
with nonnegative real numbers on the diagonal with ?1,1 ? ?2,2 ? ? ? ? ? ?n,n (the singular values),
and UT ? Rs?n is a unitary matrix.
It is important to notice that columns of UT which correspond to nonzero singular values, apart
some mathematical technicalities, basically correspond to the principal directions of data, i.e. PCA.
If the rank of ? is p, then only the first p elements of the diagonal of ? are not null, and the
T
above decomposition can be reduced to ? = V(p) ?(p) U(p) where V(p) ? Rn?p , ?(p) ? Rp?p ,
2
T
T
and U(p) ? Rp?n . Now we can observe that U(p) U(p) = I (where I is the identity matrix of
dimension p), since by definition the columns of U(p) are orthogonal, and by imposing ? = U(p) ,
we can derive ?optimal? matrices A ? Rp?k and B ? Rp?p for our dynamical system, which will
T
have corresponding state space matrix Y(p) = ?? = ?U(p) = V(p) ?(p) U(p) U(p) = V(p) ?(p) .
(p)
Thus, if we represent U(p) as composed of n submatrices Ui , each of size k ? p, the problem
reduces to find matrices A and B such that
? T
? ? (p) ?
A
U1
(p) ?
? AT BT
? ?
U2 ?
? T 2T
? ?
(p) ?
? A B
? ?
(p)
U3 ?
(4)
?=?
?=?
?=U .
?
? ?
..
.
?
?
? ?
?
.. ?
.
T
(p)
AT Bn?1
Un
The reason to impose ? = U(p) is to get a state space where the coordinates are uncorrelated so
to diagonalise the empirical sample covariance matrix of the states. Please, note that in this way
each state (i.e., row of the Y matrix) corresponds to a row of the data matrix ?, i.e. the unrolled
(sub)sequence read up to a given time t. If the rows of ? were vectors, this would correspond to
compute PCA, keeping only the fist p principal directions.
In the following, we demonstrate that there exists a solution to the above equation. We start
by observing that ? owns a special structure, i.e. given? = [?1 ?2 ? ? ? ?n ], where
?i ?
0
0
1?1
1?(n?1)
Rn?k , then for i = 1, . . . , n ? 1, ?i+1 = Rn ?i =
?i , and
I(n?1)?(n?1) 0(n?1)?1
Rn ?n = 0, i.e. the null matrix of size n ? k. Moreover, by singular value decomposition, we
(p) T
T
have ?i = V(p) ?(p) Ui , for i = 1, . . . , n. Using the fact that V(p) V(p) = I, and
(p)
(p)
combining the above equations, we get Ui+t = Ui Qt , for i = 1, . . . , n ? 1, and t =
?1
T
(p)
(p) (p)
1, . . . , n ? i, where Q = ?(p) V(p) RT
?
. Moreover, we have that Un Q = 0 since
nV
?1
(p) (p) (p) T T (p) (p) ?1
(p)
= (Rn ?n )T V(p) ?(p) . Thus, eq. (4) is satisfied by
Rn V ?
Un Q = Un ? V
| {z }
=0
(p) T
U1
T
A =
and B = Q . It is interesting to note that the original data ? can be recovered by
T
T
computing Y(p) U(p) = V(p) ?(p) U(p) = ?, which can be achieved by running the system
xt
AT
yt
=
yt?1
BT
AT
starting from yn , i.e.
is the matrix C defined in eq. (3).
BT
Finally, it is important to remark that the above construction works not only for a single sequence,
but also for a set of sequences of different length. For example, let consider the two sequences
(xa1 , xa2 , xa3 ) and (xb 1 , xb 2 ). Then, we have
? aT
?
"
#
x1
0
0
bT
x
0
1
? and ?b =
?a = ? xa2 T xa1 T 0
T
bT
x
xb1
aT
aT
aT
2
x3
x2
x1
?a
R4
, and R =
.
which can be collected together to obtain ? =
?b 02?1
R2 02?1
As a final remark, it should be stressed that the above construction only works if p is equal to the
rank of ?. In the next section, we treat the case in which p < rank(?).
2.1
Optimal solution for low dimensional autoencoders
T
? i = V(p) L(p) U(p) 6= ?i , and
When p < rank(?) the solution given above breaks down because ?
i
? i+1 6= Rn ?
? i . So the question is whether the proposed solutions for A and B still
consequently ?
hold the best reconstruction error when p < rank(?).
3
In this paper, we answer in negative terms to this question by resorting to a new formulation of our
(p)
problem where we introduce slack-like matrices Ei ? Rk?p , i = 1, . . . , n + 1 collecting the
reconstruction errors, which need to be minimised:
n+1
X
min
(p)
Q?Rp?p ,Ei
?
subject to :
?
?
?
?
?
?
?
(p)
i=1
(p)
U1 + E 1
(p)
(p)
U2 + E 2
(p)
(p)
U3 + E 3
..
.
(p)
(p)
kEi k2F
(p)
Un + E n
(p)
(p)
?
?
?
?
?
?
?
?Q = ?
?
?
? (p)
?
? Un + En(p)
?
(p)
En+1
?
?
?
?
?
?
?
?
?
U2 + E 2
(p)
(p)
U3 + E 3
..
.
(5)
Notice that the problem above is convex both in the objective function and in the constraints; thus
(p)
it only has global optimal solutions E?i and Q? , from which we can derive AT = U1 + E?1 and
T
?
T
(p)
(p)
B = Q . Specifically, when p = rank(?), Rs,k U is in the span of U and the optimal
T
(p)
solution is given by E?i = 0k?p ?i, and Q? = U(p) RT
, i.e. the solution we have already
s,k U
described. If p < rank(?), the optimal solution cannot have ?i, E?i = 0k?p . However, it is not
difficult to devise an iterative procedure to reach the minimum. Since in the experimental section we
do not exploit the solution to this problem for reasons that we will explain later, here we just sketch
(p)
such procedure. It helps to observe that, given a fixed Q, the optimal solution for Ei is given by
(p)
(p)
(p)
(p)
(p)
(p)
(p)
(p)
(p)
+
2
3
? ,E
? ,...,E
?
[E
1
2
n+1 ] = [U1 Q ? U2 , U1 Q ? U3 , U1 Q ? U4 , . . .] MQ
?
?
?Q ?Q2 ?Q3 ? ? ?
0
0
??? ?
? I
? 0
+
I
0
??? ?
?
?.
where MQ is the pseudo inverse of MQ = ?
0
I
??? ?
? 0
?
..
..
..
..
.
.
.
.
h
i
T
T
T
T T
? (p) = E
? (p) , E
? (p) , E
? (p) , ? ? ? , E
? n(p)
In general, E
can be decomposed into a component in the
1
2
3
?
?
span of U(p) and a component E(p) orthogonal to it. Notice that E(p) cannot be reduced, while
? (p) = U(p) + E(p) ? and taking
(part of) the other component can be absorbed into Q by defining U
h
i
T
T
T T
? = (U
? (p) )+ U
? (p) , U
? (p) , ? ? ? , U
? (p)T , E(p)
Q
.
n
2
3
n+1
? the new optimal values for E(p) are obtained and the process iterated till convergence.
Given Q,
i
3
Pre-training of Recurrent Neural Networks
Here we define our pre-training procedure for recurrent neural networks with one hidden layer of p
units, and O output units:
ot = ?(Woutput h(xt )) ? RO , h(xt ) = ?(Winput xt + Whidden h(xt?1 )) ? Rp
(6)
T
where Woutput ? RO?p , Whidden ? Rp?k , for a vector z ? Rm , ?(z) = [?(z1 ), . . . , ?(zm )] ,
?zi
.
and here we consider the symmetric sigmoid function ?(zi ) = 1?e
1+e?zi
The idea is to exploit the hidden state representation obtained by eqs. (2) as initial hidden state representation for the RNN described by eqs. (6). This is implemented by initialising the weight matrices
Winput and Whidden of (6) by using the matrices that jointly solve eqs. (2) and eqs. (3), i.e. A and
B (since C is function of A and B). Specifically, we initialize Winput with A, and Whidden with
B. Moreover, the use of symmetrical sigmoidal functions, which do give a very good approximation
of the identity function around the origin, allows a good transferring of the linear dynamics inside
4
RNN. For what concerns Woutput , we initialise it by using the best possible solution, i.e. the pseudoinverse of H times the target matrix T, which does minimise the output squared error. Learning
is then used to introduce nonlinear components that allow to improve the performance of the model.
More formally, let consider a prediction task where for each sequence sq ? (xq1 , xq2 , . . . , xqlq )
of length lq in the training set, a sequence tq of target vectors is defined, i.e. a training sequence is given by hsq , tq i ? h(xq1 , tq1 ), (xq2 , tq2 ), . . . , (xqlq , tqlq )i, where tqi ? RO . Given a trainPN
ing set with N sequences, let define the target matrix T ? RL?O , where L =
q=1 lq , as
1 1
1
2
?
N T
T = t1 , t2 , . . . , tl1 , t1 , . . . , tlN . The input matrix ? will have size L ? k. Let p be the desired number of hidden units for the recurrent neural network (RNN). Then the pre-training procedure can be defined as follows: i) compute the linear autoencoder for ? using p? principal direc?
?
?
tions, obtaining the optimal matrices A? ? Rp ?k and B? ? Rp ?p ; i) set Winput = A? and
?
Whidden = B ; iii) run the RNN over the training sequences, collecting the hidden activities vec?
tors (computed using symmetrical sigmoidal functions) over time as rows of matrix H ? RL?p ;
+
+
iv) set Woutput = H T, where H is the (left) pseudoinverse of H.
3.1
Computing an approximate solution for large datasets
In real world scenarios the application of our approach may turn difficult because of the size of
the data matrix. In fact, stable computation of principal directions is usually obtained by SVD decomposition of the data matrix ?, that in typical application domains involves a number of rows
and columns which is easily of the order of hundreds of thousands. Unfortunately, the computational complexity of SVD decomposition is basically cubic in the smallest of the matrix dimensions.
Memory consumption is also an important issue. Algorithms for approximate computation of SVD
have been suggested (e.g., [18]), however, since for our purposes we just need matrices V and ?
with a predefined number of columns (i.e. p), here we present an ad-hoc algorithm for approximate
computation of these matrices. Our solution is based on the following four main ideas: i) divide ?
in slices of k (i.e., size of input at time t) columns, so to exploit SVD decomposition at each slice
separately; ii) compute approximate V and ? matrices, with p columns, incrementally via truncated
SVD of temporary matrices obtained by concatenating the current approximation of V? with a new
slice; iii) compute the SVD decomposition of a temporary matrix via either its kernel or covariance
matrix, depending on the smallest between the number of rows and the number of columns of the
temporary matrix; iv) exploit QR decomposition to compute SVD decomposition.
Algorithm 1 shows in pseudo-code the main steps of our procedure. It maintains a temporary matrix
T which is used to collect incrementally an approximation of the principal subspace of dimension p
of ?. Initially (line 4) T is set equal to the last slices of ?, in a number sufficient to get a number
of columns larger than p (line 2). Matrices V and ? from the p-truncated SVD decomposition of
T are computed (line 5) via the K E C O procedure, described in Algorithm 2, and used to define a
new T matrix by concatenation with the last unused slice of ?. When all slices are processed, the
current V and ? matrices are returned. The K E C O procedure, described in Algorithm 2 , reduces
the computational burden by computing the p-truncated SVD decomposition of the input matrix
M via its kernel matrix (lines 3-4) if the number of rows of M is no larger than the number of
columns, otherwise the covariance matrix is used (lines 6-8). In both cases, the p-truncated SVD
decomposition is implemented via QR decomposition by the INDIRECT SVD procedure described in
Algorithm 3. This allows to reduce computation time when large matrices must be processed [19].
1
Finally, matrices V and S 2 (both kernel and covariance matrices have squared singular values of
M) are returned.
We use the strategy to process slices of ? in reverse order since, moving versus columns with larger
indices, the rank as well as the norm of slices become smaller and smaller, thus giving less and less
contribution to the principal subspace of dimension p. This should reduce the approximation error
cumulated by dropping the components from p + 1 to p + k during computation [20]. As a final
remark, we stress that since we compute an approximate solution for the principal directions of ?,
it makes no much sense to solve the problem given in eq. (5): learning will quickly compensate
for the approximations and/or sub-optimality of A and B obtained by matrices V and ? returned
by Algorithm 1. Thus, these are the matrices we have used for the experiments described in next
section.
5
Algorithm 1 Approximated V and ? with p components
1: function SVF OR B IG DATA(?, k, p)
2:
nStart = dp/ke
. Number of starting slices
3:
nSlice = (?.columns/k) ? nStart
. Number of remaining slices
4:
T = ?[:, k ? nSlice : ?.columns]
5:
V, ? =K E C O(T, p)
. Computation of V and ? for starting slices
6:
for i in REVERSED(range(nSlice)) do
. Computation of V and ? for remaining slices
7:
T = [?[:, i ? k:(i + 1) ? k], V?]
8:
V, ? =K E C O(T, p)
9:
end for
10:
return V, ?
11: end function
Algorithm 2 Kernel vs covariance computation Algorithm 3 Truncated SVD by QR
1: function K E C O(M, p)
1: function INDIRECT SVD(M, p)
2:
if M.rows <= ?.columns then
2:
Q, R =QR(M)
3:
K = MMT
3:
Vr , S, UT =SVD(R)
T
4:
V, Ssqr , U =INDIRECT SVD(K, p) 4:
V = QVr
5:
else
5:
S = S[1 : p, 1 : p]
6:
C = MT M
6:
V = V[1 : p, :]
7:
V, Ssqr , UT =INDIRECT SVD(C, p) 7:
UT = UT [:, 1 : p]
1
?
8:
return V, S, UT
8:
V = MUT Ssqr2
9: end function
9:
end if
1
2
10:
return V, Ssqr
11: end function
4
Experiments
In order to evaluate our pre-training approach, we decided to use the four polyphonic music sequences datasets used in [21] for assessing the prediction abilities of the RNN-RBM model. The
prediction task consists in predicting the notes played at time t given the sequence of notes played
till time t ? 1. The RNN-RBM model achieves state-of-the-art in such demanding prediction task.
As performance measure we adopted the accuracy measure used in [21] and described in [22]. Each
dataset is split in training set, validation set, and test set. Statistics on the datasets, including largest
sequence length, are given in columns 2-4 of Table 1. Each sequence in the dataset represents a song
having a maximum polyphony of 15 notes (average 3.9); each time step input spans the whole range
of piano from A0 to C8 and it is represented by using 88 binary values (i.e. k = 88).
Our pre-training approach (PreT-RNN) has been assessed by using a different number of hidden
units (i.e., p is set in turn to 50, 100, 150, 200, 250) and 5000 epochs of RNN training1 using the
Theano-based stochastic gradient descent software available at [23].
Random initialisation (Rnd) has also been used for networks with the same number of hidden units.
Specifically, for networks with 50 hidden units, we have evaluated the performance of 6 different
random initialisations. Finally, in order to verify that the nonlinearity introduced by the RNN is
actually useful to solve the prediction task, we have also evaluated the performance of a network
with linear units (250 hidden units) initialised with our pre-training procedure (PreT-Lin250).
To give an idea of the time performance of pre-training with respect to the training of a RNN, in
column 5 of Table 1 we have reported the time in seconds needed to compute pre-training matrices
c
c
(Pre-) (on Intel
Xeon
CPU E5-2670 @2.60GHz with 128 GB) and to perform training of a
RNN with p = 50 for 5000 epochs (on GPU NVidia K20). Please, note that for larger values of p,
the increase in computation time of pre-training is smaller than the increment in computation time
needed for training a RNN.
1
Due to early overfitting, for the Muse dataset we used 1000 epochs.
6
Dataset
Nottingham
Piano-midi.de
MuseData
JSB Chorales
Set
Training
(39165 ? 56408)
Test
Validation
Training
(70672 ? 387640)
Test
Validation
Training
(248479 ? 214192)
Test
Validation
Training
(27674 ? 22792)
Test
Validation
# Samples
195
Max length
641
170
173
87
1495
1229
4405
25
12
524
2305
1740
2434
25
135
229
2305
2523
259
77
76
320
289
(Pre-)Training Time
seconds
(226) 5837
p = 50
5000 epochs
seconds
(2971) 4147
p = 50
5000 epochs
seconds
(7338) 4190
p = 50
5000 epochs
seconds
(79) 6411
p = 50
5000 epochs
Model
RNN (w. HF)
RNN-RBM
PreT-RNN
PreT-Lin250
RNN (w. HF)
RNN-RBM
PreT-RNN
PreT-Lin250
RNN (w. HF)
RNN-RBM
PreT-RNN
PreT-Lin250
RNN (w. HF)
RNN-RBM
PreT-RNN
PreT-Lin250
ACC% [21]
62.93 (66.64)
75.40
75.23 (p = 250)
73.19
19.33 (23.34)
28.92
37.74 (p = 250)
16.87
23.25 (30.49)
34.02
57.57 (p = 200)
3.56
28.46 (29.41)
33.12
65.67 (p = 250)
38.32
Table 1: Datasets statistics including data matrix size for the training set (columns 2-4), computational times in seconds to perform pre-training and training for 5000 epochs with p = 50 (column
5), and accuracy results for state-of-the-art models [21] vs our pre-training approach (columns 6-7).
The acronym (w. HF) is used to identify an RNN trained by Hessian Free Optimization [4].
Training and test curves for all the models described above are reported in Figure 1. It is evident that
random initialisation does not allow the RNN to improve its performance in a reasonable amount of
epochs. Specifically, for random initialisation with p = 50 (Rnd 50), we have reported the average
and range of variation over the 6 different trails: different initial points do not change substantially
the performance of RNN. Increasing the number of hidden units allows the RNN to slightly increase
its performance. Using pre-training, on the other hand, allows the RNN to start training from a quite
favourable point, as demonstrated by an early sharp improvement of performances. Moreover, the
more hidden units are used, the more the improvement in performance is obtained, till overfitting is
observed. In particular, early overfitting occurs for the Muse dataset. It can be noticed that the linear
model (Linear) reaches performances which are in some cases better than RNN without pre-training.
However, it is important to notice that while it achieves good results on the training set (e.g. JSB and
Piano-midi), the corresponding performance on the test set is poor, showing a clear evidence of overfitting. Finally, in column 7 of Table 1, we have reported the accuracy obtained after validation on
the number of hidden units and number of epochs for our approaches (PreT-RNN and PreT-Lin250)
versus the results reported in [21] for RNN (also using Hessian Free Optimization) and RNN-RBM.
In any case, the use of pre-training largely improves the performances over standard RNN (with
or without Hessian Free Optimization). Moreover, with the exception of the Nottingham dataset,
the proposed approach outperforms the state-of-the-art results achieved by RNN-RBM. Large improvements are observed for the Muse and JSB datasets. Performance for the Nottingham dataset
is basically equivalent to the one obtained by RNN-RBM. For this dataset, also the linear model
with pre-training achieves quite good results, which seems to suggest that the prediction task for
this dataset is much easier than for the other datasets. The linear model outperforms RNN without
pre-training on Nottingham and JSB datasets, but shows problems with the Muse dataset.
5
Conclusions
We have proposed a pre-training technique for RNN based on linear autoencoders for sequences.
For this kind of autoencoders it is possible to give a closed form solution for the definition of the
?optimal? weights, which however, entails the computation of the SVD decomposition of the full
data matrix. For large data matrices exact SVD decomposition cannot be achieved, so we proposed
a computationally efficient procedure to get an approximation that turned to be effective for our
goals. Experimental results for a prediction task on datasets of sequences of polyphonic music
show the usefulness of the proposed pre-training approach, since it allows to largely improve the
state of the art results on all the considered datasets by using simple stochastic gradient descend for
learning. Even if the results are very encouraging the method needs to be assessed on data from
other application domains. Moreover, it is interesting to understand whether the analysis performed
in [24] on linear deep networks for vectors can be extended to recurrent architectures for sequences
and, in particular, to our method.
7
0.4
0.2
0.1
0
-0.1
Rnd 50 (6 trials)
Linear 250
Rnd 100
0
200
600
PreT 200
PreT 250
800
Nottingham
Test Set
1000
0.8
Epoch
0.7
0.7
0.6
0.6
0.5
0.5
Accuracy
Accuracy
PreT 50
PreT 150
PreT 100
400
Nottingham Training
Set
0.8
0.4
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0
0
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Epoch
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Epoch
Piano-Midi.de Training Set
Piano-Midi.de Test Set
0.55
0.4
0.5
0.35
Accuracy
0.45
0.4
0.3
0.35
0.25
0.3
0.25
0.2
0.15
0.2
0.15
0.1
0.1
0.05
0.05
0
0
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Epoch
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Epoch
Muse Dataset Training Set
Muse Dataset Test Set
0.7
0.6
0.6
0.5
0.5
0.4
Accuracy
Accuracy
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
200
400
600
800
1000
0
Epoch
200
400
600
800
1000
Epoch
JSB Chorales Training Set
JSB Chorales Test Set
0.8
0.7
0.7
0.6
0.6
0.5
0.5
Accuracy
-0.3
Rnd 150
Rnd 200
Rnd 250
Accuracy
-0.2
Accuracy
Accuracy
0.3
0.4
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0
0
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Epoch
0
500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Epoch
Figure 1: Training (left column) and test (right column) curves for the assessed approaches on the
four datasets. Curves are sampled at each epoch till epoch 100, and at steps of 100 epochs afterwards.
8
References
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9
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4,718 | 5,272 | Using Convolutional Neural Networks
to Recognize Rhythm Stimuli
from Electroencephalography Recordings
Sebastian Stober, Daniel J. Cameron and Jessica A. Grahn
Brain and Mind Institute, Department of Psychology, Western University
London, Ontario, Canada, N6A 5B7
{sstober,dcamer25,jgrahn}@uwo.ca
Abstract
Electroencephalography (EEG) recordings of rhythm perception might contain enough
information to distinguish different rhythm types/genres or even identify the rhythms
themselves. We apply convolutional neural networks (CNNs) to analyze and classify
EEG data recorded within a rhythm perception study in Kigali, Rwanda which comprises
12 East African and 12 Western rhythmic stimuli ? each presented in a loop for 32
seconds to 13 participants. We investigate the impact of the data representation and
the pre-processing steps for this classification tasks and compare different network
structures. Using CNNs, we are able to recognize individual rhythms from the EEG
with a mean classification accuracy of 24.4% (chance level 4.17%) over all subjects by
looking at less than three seconds from a single channel. Aggregating predictions for
multiple channels, a mean accuracy of up to 50% can be achieved for individual subjects.
1 Introduction
Musical rhythm occurs in all human societies and is related to many phenomena, such as the perception of a
regular emphasis (i.e., beat), and the impulse to move one?s body. It is a universal human phenomenon, but
differs between human cultures. The influence of culture on the processing of rhythm in the brain as well
as the brain mechanisms underlying musical rhythm are still not fully understood. In order to study these,
we recruited participants in East Africa and North America to test their ability to perceive and produce
rhythms derived from East African and Western music. Besides several behavioral tasks, which have
already been discussed in [1], the East African participants also underwent electroencephalography (EEG)
recording while listening to East African and Western musical rhythms thus enabling us to study the neural
mechanisms underlying rhythm perception.
Using two popular deep learning techniques ? stacked denoising autoencoders (SDAs) [2] and convolutional
neural networks (CNNs) [3] ? we already obtained encouraging early results for distinguishing East
African and Western stimuli in a binary classification task based on the recorded EEG [4]. In this paper, we
address the much harder classification problem of recognizing the 24 individual rhythms. In the following,
we will review related work in Section 2, describe the data acquisition and pre-processing in Section 3,
present our experimental findings in Section 4, and discuss further steps in Section 5.
2 Related work
How the brain responses to auditory rhythms has already been investigated in several studies using EEG
and magnoencephalography (MEG): Oscillatory neural activity in the gamma (20-60 Hz) frequency band
is sensitive to accented tones in a rhythmic sequence and anticipates isochronous tones [5]. Oscillations
in the beta (20-30 Hz) band increase in anticipation of strong tones in a non-isochronous sequence [6, 7, 8].
1
Another approach has measured the magnitude of steady state evoked potentials (SSEPs) (reflecting neural
oscillations entrained to the stimulus) while listening to rhythmic sequences [9, 10]. Here, enhancement
of SSEPs was found for frequencies related to the metrical structure of the rhythm (e.g., the frequency
of the beat). In contrast to these studies investigating the oscillatory activity in the brain, other studies
have used EEG to investigate event-related potentials (ERPs) in responses to tones occurring in rhythmic
sequences. This approach has been used to show distinct sensitivity to perturbations of the rhythmic pattern
vs. the metrical structure in rhythmic sequences [11], and to suggest that similar responses persist even
when attention is diverted away from the rhythmic stimulus [12]. Further, Will and Berg [13] observed
a significant increase in brain wave synchronization after periodic auditory stimulation with drum sounds
and clicks with repetition rates of 1?8Hz. Vlek et al. [14] already showed that imagined auditory accents
can be recognized from EEG. They asked ten subjects to listen to and later imagine three simple metric
patterns of two, three and four beats on top of a steady metronome click. Using logistic regression to
classify accented versus unaccented beats, they obtained an average single-trial accuracy of 70% for
perception and 61% for imagery. These results are very encouraging to further investigate the possibilities
for retrieving information about the perceived rhythm from EEG recordings.
Very recently, the potential of deep learning techniques for neuroimaging has been demonstrated for
functional and structural magnetic resonance imaging (MRI) data [15]. However, applications of deep
learning techniques within neuroscience and specifically for processing EEG recordings have been very
limited so far. Wulsin et al. [16] used deep belief nets (DBNs) to detect anomalies related to epilepsy
in EEG recordings of 11 subjects by classifying individual ?channel-seconds?, i.e., one-second chunks
from a single EEG channel without further information from other channels or about prior values. Their
classifier was first pre-trained layer by layer as an autoencoder on unlabelled data, followed by a supervised
fine-tuning with backpropagation on a much smaller labeled data set. They found that working on raw,
unprocessed data (sampled at 256Hz) led to a classification accuracy comparable to hand-crafted features.
Langkvist et al. [17] similarly employed DBNs combined with hidden Markov models (HMMs) to classify
different sleep stages. Their data for 25 subjects comprised EEG as well as recordings of eye movements
and skeletal muscle activity. Again, the data was segmented into one-second chunks. Here, a DBN on
raw data showed a classification accuracy close to one using 28 selected features.
3 Data acquisition & pre-processing
3.1 Stimuli
The African rhythm stimuli were derived from recordings of traditional East African music [18]. The
author (DC) composed the Western rhythmic stimuli. Rhythms were presented as sequences of sine
tones that were 100ms in duration with intensity ramped up/down over the first/final 50ms and a pitch
of either 375 or 500 Hz. All rhythms had a temporal structure of 12 equal units, in which each unit could
contain a sound or not. For each rhythmic stimulus, two individual rhythmic sequences were overlaid
whereby one sequence was played at the high pitch and the other at the low pitch. There were two groups
of three individual rhythmic sequences for each cultural type of rhythm as shown in Table 1. With three
combinations within each group and two possible pitch assignments, this resulted in six rhythmic stimuli
for each group, 12 per rhythm type and 24 in total.1 Finally, rhythmic stimuli could be played back at
one of two tempi, having a minimum inter-onset interval of either 180 or 240ms.
Furthermore, we also formed groups based on how these stimuli were created. These allowed a more coarse
classification with fewer classes. Ignoring the pitch assignments and thus considering the pairs [a,b] and [b,a]
as equivalent, 12 groups were formed. At the next level, the stimuli derived from the same of the four groups
of three sequences were grouped resulting in four groups of six stimuli. Finally, distinguishing East African
from Western stimuli resulted in the binary classification problem that we addressed in our earlier work.
3.2 Study description
Sixteen East African participants were recruited in Kigali, Rwanda (3 female, mean age: 23 years, mean
musical training: 3.4 years, mean dance training: 2.5 years). The participants first completed three
behavioral tasks: a rhythm discrimination task, a rhythm reproduction task, and a beat tapping task.
Afterward, thirteen subjects also participated in the EEG portion of the study. All participants were over
1
The 24 rhythm stimuli are available at http://dx.doi.org/10.6084/m9.figshare.1213903
2
Table 1: Rhythmic sequences in groups of three that pairings were based on. All ?x?s denote onsets. Larger,
bold ?X?s denote the beginning of a 12 unit cycle (downbeat).
Western Rhythms
1Xxxx xx
2X
x xx
3X xx xx
xx
Xxxx xx
x xX
x xx
xxxxX xx xx
East African Rhythms
1X xxxxx
2X x x x
3X x
x
xx
x x
xxxx
4X xx xx
x xX xx xx
x x
5Xxxx
xx x
Xxxx
xx x
6X xx xx xxxxX xx xx xxxx
xxxxX xxxxx
x xX
x x
x
X x x x
xxxx
x
4X xxx xxx xxX xxx xxx xx
5X xx xx xx xX xx xx xx x
6X xx xx x x X xx xx x x
the age of 18, had normal hearing, and had spent the majority of their lives in East Africa. They all gave
informed consent prior to participating and were compensated for their participation, as per approval by
the ethics boards at the Centre Hospitalier Universitaire de Kigali and the University of Western Ontario.
The participants were instructed to sit with eyes closed and without moving for the duration of the EEG
recording, and to maintain their attention on the auditory stimuli. All rhythms were repeated for 32 seconds,
presented in counterbalanced blocks (all East African rhythms then all Western rhythms, or vice versa),
and with randomized order within blocks. 12 rhythms of each type were presented ? all at the same tempo,
and each rhythm was preceded by 4 seconds of silence. EEG was recorded via a portable Grass EEG
system using 14 channels at a sampling rate of 400Hz and impedances were kept below 10k?.
3.3 Data pre-processing
EEG recordings are usually very noisy. They contain artifacts caused by muscle activity such as eye blinking
as well as possible drifts in the impedance of the individual electrodes over the course of a recording. Furthermore, the recording equipment is very sensitive and easily picks up interferences from the surroundings.
For instance, in this experiment, the power supply dominated the frequency band around 50Hz. All these
issues have led to the common practice to invest a lot of effort into pre-processing EEG data, often even manually rejecting single frames or channels. In contrast to this, we decided to put only little manual work into
cleaning the data and just removed obviously bad channels, thus leaving the main work to the deep learning
techniques. After bad channel removal, 12 channels remained for subjects 1?5 and 13 for subjects 6?13.
We followed the common practice in machine learning to partition the data into training, validation (or
model selection) and test sets. To this end, we split each 32s-long trial recording into three non-overlapping
pieces. The first T seconds after an optional offset were used for the validation set. The rationale behind
this was that we expected that the participants would need a few seconds in the beginning of each trial to
get used to the new rhythm. Thus, the data would be less suited for training but might still be good enough
to estimate the model accuracy on unseen data. The main part of each recording was used for training
and the remaining T seconds for testing. The time length T was tempo-dependent and corresponded to
the length of a single bar in the stimuli. Naturally, one would prefer segments that are as long as the 2-bar
stimuli. However, this would have reduced the amount of data left for training significantly and since only
the East African rhythm sequences 2 and 3 had differences between the first and second bar (cf. Table 1),
we only used 1 bar. With the optional offset, the data sets were aligned to start at the same position within
a bar.2 The specific values for the two tempi are listed in Table 2. Furthermore, we decided to process and
classify each EEG channel individually. Combining all 12 or 13 EEG channels in the analysis might allow
to detect spatial patterns and most likely lead to an increase of the classification performance. However,
this would increase the model complexity (number of parameters) by a factor of more than ten while at the
same time reducing the number of training and test examples by the same factor. Under these conditions,
the amount of data would not be sufficient to effectively train the CNN and lead to severe overfitting.
The data was finally converted into the input format required by the CNN to be learned.3 If the network
just took the raw EEG data, each waveform was normalized to a maximum amplitude of 1 and then split
into equally sized frames of length T matching the size of the network?s input layer. No windowing
2
With offset, the validation and test set would correspond to the same section of the stimuli for the fast tempo
whereas for the fast tempo, it would differ by 1 bar because of the odd number of bars in between.
3
Most of the processing was implemented through the librosa library available at https://github.com/
bmcfee/librosa/.
3
Table 2: Differences between slow and fast stimuli.
tempo participants beat length bar length T
bars optional offset training segment length
fast
slow
1?3, 7?9
4?6, 10?13
180ms
240ms
2160ms
2880ms
14.815
11.111
1760ms
320ms
27680ms - offset
26240ms - offset
function was applied and the hop size (controlling the overlap of consecutive windows) was either 24,
which corresponded to 60ms at the sampling rate of 400Hz, or the equivalent of T in samples. If the
network was designed to process the frequency spectrum, the processing involved:
1. computing the short-time Fourier transform (STFT) with given window length of 96 samples
and a hop size of 24 (This resulted in a new frequency spectrum vector every 60ms.),
2. computing the log amplitude,
3. scaling linearly to a maximum of 1 (per sequence),
4. (optionally) cutting of all frequency bins above the number requested by the network,
5. splitting the data into frames of length T (matching the network?s input dimensionality) with
a given hop size of 1 (60ms) or the equivalent of T .
Hops of 60ms were chosen as this equals to one fourth or one third of the beat length in the slow and
fast rhythms respectively. With this choice, we hoped to be able to pick up beat-related effects but also
to have a window size big enough for a sufficient frequency resolution in the spectrum. Including the
zero-frequency band, this resulted in 49 frequency bins up to 200Hz with a resolution of 4.17Hz. Using
the log amplitude in combination with the normalization had turned out to be the best approach in our
previous experiments trying to distinguish East African from Western stimuli [4].
4 Experiments
CNNs, as for instance described in [3], have a variety of structural parameters which need to be chosen
carefully. In general, CNNs are artificial neural networks (ANNs) with one or more convolutional layers.
In such layers, linear convolution operations are applied for local segments of the input followed by a nonlinear transformation and a pooling operation over neighboring segments. If the EEG data is represented as
waveform, the input has only one dimension (width) which corresponds to the time. If it is represented as
frequency spectrum, it has a second dimension (height) which corresponds to the frequency. The kernel for
each convolution operation is described by a weight matrix of a certain shape. Here, we only considered
the kernel width as free parameter and kept the height maximal. Multiple kernels can be applied in parallel
within the same layer whereby each corresponds to a different output channel of the layer. The stride parameter controls how much the kernels should advance on the input data between successive applications. Here,
we fixed this parameter at 1 resulting in a maximal overlap of consecutive input segments. Finally, the pooling parameter controls how many values of neighboring segments are aggregated using the max operation.
Like in our previous work, we used a DLSVM output layer as proposed in [19].4 This special kind of
output layer for classification uses the hinge loss as cost function and replaces the commonly applied
softmax. The convolutional layers applied the rectifier non-linearity f(x) = max(0,x) which does not
saturate like sigmoid functions and thus facilitates faster learning as proposed in [20]. The input length
in the time dimension was adapted to match the bar length T . All models were trained for 50 epochs using
stochastic gradient descent (SGD) (on mini-batches of size 100) with exponential decay of the learning
rate after each epoch and momentum. The best model was selected based on the accuracy on the validation
set. Furthermore, we applied dropout regularization [21]. In total, this resulted in four learning parameters
with value ranges derived from earlier experiments:
? the initial learning rate (between 0.001 and 0.01),
? the exponential learning rate decay per epoch (between 1.0 and 1.1),
? the initial momentum (between 0.0 and 0.5), and
? and the final momentum in the last epoch (between 0.0 and 0.99)
and three structural parameters for each convolutional layer
? the kernel width (between 1 and the input width for the layer),
? the number of channels (between 1 and 30), and
4
We used the experimental implementation for pylearn2 provided by Kyle Kastner at https:
//github.com/kastnerkyle/pylearn2/blob/svm_layer/pylearn2/models/mlp.py
4
? the pooling width (between 1 and 10).
In our previous work, we successfully applied CNNs with two convolutional layers to classify the perceived
rhythms into types (East African vs. Western) as well as to identify individual rhythms in a pilot experiment
[4]. However, we were only able to test a small number of manually tuned structural configurations,
leaving a considerable potential for further improvement. Here, we took a systematic approach for finding
good structural and learning parameters for the CNNs. To this end, we applied a Bayesian optimization
technique for hyper-parameter selection in machine learning algorithms, which has recently been described
by Snoek et al. [22] and has been implemented in Spearmint library.5 The basic idea is to treat the learning
algorithm?s generalization performance as a sample from a Gaussian process and select the next parameter
configuration to test based on the expected improvement. The authors showed that this way, the number
of experiment runs to minimize a given objective can be significantly reduced while surpassing the
performance of parameters chosen by human experts. We implemented6 our experiments using Theano
[23] and pylearn2 [24]. The computations were run on a dedicated 12-core workstation with two Nvidia
graphics cards ? a Tesla C2075 and a Quadro 2000.
We followed the common practice to optimize the performance on the validation set. Because the 24
classes we would like to predict were perfectly balanced, we chose the accuracy, i.e., the percentage of
correctly classified instances, as primary evaluation measure.7 Furthermore, ranking the 24 classes by their
corresponding network output values, we also computed the precision at rank 3 (prec.@3) and the mean
reciprocal rank (MRR) ? two commonly used information retrieval measures. The former corresponds to accuracy considering the top three classes in the ranking instead of just the first one. The latter is computed as:
|D|
1 X 1
MRR=
|D| i=1 ranki
(1)
where D is the set of test instances and ranki is the rank of the correct class for instance i. The value
range is (0,1] where the best value, 1, is obtained if the correct class is always ranked first.
4.1 Impact of pre-processing (subject 4)
At first, we analyzed the impact of the pre-processing on the performance of a model with a single
convolutional layer. For this, we only considered the recordings from subject 4 who were easiest to classify
in our earlier experiments. The exponential learning rate decay was fixed at 1.08 leaving three structural
and three learning parameters for the Bayesian optimization. Results are shown in Figure 1 (left).
Generally, CNNs using the frequency spectrum representation were faster. A possible reason could be
that the graphics cards performed better using two-dimensional kernels instead of long one-dimensional
ones. Furthermore, the search for good parameters was much harder for the waveform representation
because the value range for the kernel width was much wider ([1,1152] instead of [1,45]). Thus, the search
took much longer. For instance, using the large hop size, an accuracy of more than 20% was only achieved
after 208 runs for CNNs using waveform input with offset and after 47 runs without offset. Comparable
values were already obtained after 1 and 2 runs respectively for the CNNs with frequency spectrum input
and the values shown in Figure 1 (left) were obtained after 45 and 105 runs respectively. Consequently,
the frequency spectrum appeared to be the clearly preferable choice for the input representation.
With the small hop size of 60ms, a lot more training instances were generated because of the high overlap.
This slowed down learning by a factor of more than 10. Hence, fewer configurations could be tested within
the same time. Overall, the large hop size corresponding to 1 bar was favorable because of the significant
speed-up without an impact on accuracy. By using the offset in combination with the hop size of 1 bar,
all instances for training, validation and testing were aligned to the same position within a bar. This could
explain the increase in accuracy for this parameter combination together with the spectrum representation.
In combination with the waveform input, the inverse effect was observed. However, as it was generally
harder to find good solutions in this setting, it could be that testing more configurations eventually would
lead to the same result as for the spectrum.
5
https://github.com/JasperSnoek/spearmint
The code to run the experiments is available as supplementary material at http://dx.doi.org/10.6084/
m9.figshare.1213903
7
As the Bayesian optimization aims to minimize an objective, we let our learner report the misclassification rate
instead which is one minus the accuracy.
6
5
SVM
60ms
(60 runs)
offset
waveform
no 33.3% 233.7s
yes 34.8% 119.5s
33.7% 22.3s
33.0% 16.4s
1 bar
no 33.0% 12.7s
(300 runs) yes 24.7% 5.3s
33.3% 0.4s
35.8% 0.3s
60ms
no
yes
1 bar
no
yes
40
freq. spectrum
35
accuracy (%)
CNN
hop size
training did not finish
within 48 hours
11.1%
12.2%
30
25
20
15
10
5
22.2%
24.3%
0
0
10
20
30
40
50
number of frequency bins (4.167 Hz per bin)
Figure 1: Impact of pre-processing. Left: Classification accuracy and average epoch processing time for
different combinations of the pre-processing parameters. CNN structural and learning parameters were
obtained through Bayesian optimization for 300 runs for hop size 1 bar and 60 runs for hop size 60ms.
Processing times for CNNs were measured separately as single process using the Tesla C2075 graphics
card and averaged over 50 epochs. For comparison, SVM classification accuracies were obtained using
LIBSVM with polynomial kernel (degree 1?5). (Only the best values are shown.) Right: Impact of the
optional frequency bin cutoff on the accuracy.
For a comparison, we also trained support vector machine (SVM) classifiers using LIBSVM [25] on the
same pre-processed data. Here, training did not finish within 48 hours for the small hop size because of the
amount of training data. For waveform data, a polynomial kernel with degree 2 worked best, whereas for the
frequency spectrum, it was a polynomial kernel with degree 4. All values were significantly (more than 10%
absolute) below those obtained with a CNN. This shows using CNNs leads to a substantial improvement.
Next, we analyzed the impact of the optional frequency bin cutoff. To this end, we used the best
pre-processing parameter combination from the above comparison. This time, we fixed the momentum
parameters to an initial value of 0.5 and a final value of 0.99 as these clearly dominated within the best
configurations found so far. Instead, we did not fix the exponential learning rate decay. This resulted in
5 parameters to be optimized. We sampled the number of frequency bins from the range of [1,49] with
higher density for lower values and let the Bayesian optimization run 300 experiments for each value.
Results are shown in Figure 1 (right). A very significant accuracy increase can be observed between
12 and 15 bins which corresponds to a frequency band of 45.8?62.5 Hz in the high gamma range which
has been associated with beat perception, e.g., in [5]. The accuracy increase between 28 and 36 bins
(116?145 Hz) is hard to explain as EEG frequency ranges beyond 100 Hz have barely been studied so
far. Here, a further investigation of the learned patterns (reflected in the CNN kernels) could lead to more
insight. This analysis is still subject of ongoing research. The effect on the processing time was negligible.
Based on these findings, we chose the following pre-processing parameters for the remaining experiments:
The EEG data was represented as frequency spectrum using 49 bins. Input frames were obtained with a
hop size corresponding to the length of 1 bar, T , and with a offset to align all instances to the same position
within a bar.
4.2 One vs. two convolutional layers (all subjects)
Having determined the optimal pre-processing parameters for subject 4 and CNNs with a single
convolutional layer, we also used these settings to train individual models with one and two convolutional
layers for all subjects. This time, we allowed 500 runs of the Bayesian optimization to find the best
parameters in each setting. Additionally, we considered three groups of subjects. The ?fast? and ?slow?
group contained all subjects with the respective stimulus tempo (cf. Table 2) whereas the ?all? group
contained all 13 subjects. For the groups, we stopped the Bayesian optimization after 100 runs as there was
no more improvement and the processing time was much longer due to the bigger size of the combined
data sets. Results are shown in Table 3. Apart from the performance values for classifying individual
instances that correspond a segment from an EEG channel, we also aggregated all predictions from the
12 or 13 different channels of the same trial into one prediction by a simple majority vote. The obtained
accuracies are listed in Table 3 (right). Additionally, we computed the accuracies for the more coarse
variants of the classification problem with fewer classes (cf. Section 3.1).
6
Table 3: Structural parameters and performance values of the best CNNs with one or two convolutional
layers after Bayesian parameter optimization for each subject (500 runs) and the three subject groups (100
runs). Layer structure is written as [kernel shape] / pooling width x number of channels. (A more detailed
table can be found in the supplementary material.)
network structure
subject input 1st layer
2nd layer
1
2
3
4
5
6
7
8
9
10
11
12
13
19.1%
27.1%
21.9%
36.1%
18.1%
29.5%
23.1%
24.0%
21.8%
26.6%
26.6%
32.1%
20.2%
36.1%
46.5%
38.2%
63.5%
34.7%
48.1%
43.9%
44.2%
33.7%
51.0%
55.1%
60.9%
37.2%
0.34
0.42
0.36
0.55
0.33
0.45
0.40
0.41
0.36
0.44
0.45
0.51
0.36
25.0%
37.5%
20.8%
50.0%
16.7%
37.5%
33.3%
41.7%
25.0%
33.3%
33.3%
29.2%
25.0%
29.2%
37.5%
25.0%
62.5%
25.0%
41.7%
45.8%
41.7%
29.2%
33.3%
37.5%
33.3%
29.2%
58.3%
50.0%
45.8%
75.0%
41.7%
54.2%
54.2%
58.3%
58.3%
45.8%
41.7%
54.2%
50.0%
79.2%
87.5%
66.7%
83.3%
70.8%
75.0%
66.7%
91.7%
91.7%
66.7%
75.0%
83.3%
70.8%
mean (1 convolutional layer)
mean (2 convolutional layers)
24.4%
24.4%
46.4%
44.2%
0.41
0.40
30.8%
29.5%
36.5%
34.0%
51.6%
52.2%
74.7%
77.2%
fast
9.7%
9.5%
9.9%
9.1%
7.3%
7.2%
22.1%
21.6%
22.9%
24.3%
19.0%
18.4%
0.23
0.23
0.24
0.24
0.21
0.20
10.4%
11.8%
10.7%
10.1%
7.7%
8.7%
16.7%
19.4%
13.7%
13.1%
12.2%
12.5%
35.4%
38.9%
32.7%
31.5%
29.2%
31.4%
66.7%
67.4%
56.5%
58.9%
57.1%
55.4%
slow
all
33x49
33x49
33x49
45x49
45x49
45x49
33x49
33x49
33x49
45x49
45x49
45x49
45x49
33x49
33x49
45x49
45x49
33x49
33x49
[5x49]/3x16
[10x49]/1x22
[17x49]/1x30
[35x49]/1x30
[40x49]/2x30
[26x49]/5x30
[15x49]/1x13
[5x49]/2x21
[13x49]/2x21
[7x49]/1x30
[27x49]/1x30
[5x49]/5x30
[18x49]/10x21
[16x1]/5x12
channel mean (24 classes)
aggregated trial accuracy
accuracy prec.@3 MRR 24 classes 12 classes 4 classes 2 classes
[1x1]/10x30
[2x1]/2x24
[6x1]/4x30
[5x1]/10x30
[1x1]/6x30
[8x49]/1x22
[1x49]/1x30 [17x1]/1x30
[31x49]/1x30
[1x49]/10x23 [12x1]/5x27
[1x49]/1x30
[3x49]/9x22 [5x1]/5x18
As expected, models learned for groups of participants did not perform very well. Furthermore, the
classification accuracy varied a lot between subjects with the best accuracy (36.1% for subject 4) twice
as high as the worst (18.1% for subject 5). This was most likely due to strong individual differences in
the rhythm perception. But it might at least have been partly caused by the varying quality of the EEG
recordings. For instance, the signal was much noisier than usual for subject 5. For most subjects, the
aggregation per trial significantly increased the classification accuracy. Only in cases where the accuracy
for individual channels was low, such as for subject 5, the aggregation did not yield an improvement.
Overall, the performance of the simpler models with a single convolutional layer was on par with the
more complex ones ? and often even better. One possible reason for this could be that the models with
two convolutional layers had twice as many structural parameters and thus it was potentially harder to
find good configurations. Furthermore, with more weights to be learned and thus more degrees of freedom
to adapt, they were more prone to overfitting on this rather small data set. Figure 2 (left) visualizes the
confusion between the different rhythms for subject 4 where the best overall accuracy was achieved.8
Remarkably, only few of the East African rhythms were misclassified as Western (upper right quadrant)
and vice versa (lower left). For the East African music, confusion was mostly amongst neighbors (i.e.,
similar rhythms; upper left quadrant) ? especially rhythms based on sequences 2 and 3 that were the only
ones that cannot be captured correctly in a window of 1 bar ? whereas for the Western rhythms, there
were patterns indicating a strong perceived similarity between rhythm sequences 1 and 4. The accuracies
obtained for the classification tasks with fewer classes (cf. Table 3, right) paint a similar picture indicating
strong stimulus similarity as the main reason for confusion. In the mean confusion matrix, this effect is far
less pronounced. However, it can be observed in most of the confusion matrices for the individual subjects.
The results reported here still need to be taken with a grain of salt. Because of the study design, there is
only one trial session (of 32 seconds) per stimulus for each subject. Thus, there is the chance that the neural
networks learned to identify the individual trials and not the stimuli based on artifacts in the recordings that
only occurred sporadically throughout the experiment. Or there could have been brain processes unrelated
8
The respective confusion matrices for the models with two convolutional layers look very similar. They can be
found in the supplementary material together with the matrices for the other participants.
7
13
12
11
10
9
True label
8
7
6
5
4
3
2
1
0
[5, 4, 'a']
[6, 4, 'a']
[6, 5, 'a']
[2, 1, 'a']
[3, 2, 'a']
[3, 1, 'a']
[5, 6, 'a']
[2, 3, 'a']
[1, 2, 'a']
[1, 3, 'a']
[4, 5, 'a']
[4, 6, 'a']
[6, 4, 'w']
[2, 1, 'w']
[5, 4, 'w']
[2, 3, 'w']
[5, 6, 'w']
[3, 1, 'w']
[3, 2, 'w']
[4, 6, 'w']
[1, 3, 'w']
[1, 2, 'w']
[4, 5, 'w']
[6, 5, 'w']
subject 4 (labels in trial order)
Predicted label
13
12
11
10
9
8
7
6
5
4
3
2
1
[5, 4, 'a']
[6, 4, 'a']
[6, 5, 'a']
[2, 1, 'a']
[3, 2, 'a']
[3, 1, 'a']
[5, 6, 'a']
[2, 3, 'a']
[1, 2, 'a']
[1, 3, 'a']
[4, 5, 'a']
[4, 6, 'a']
[6, 4, 'w']
[2, 1, 'w']
[5, 4, 'w']
[2, 3, 'w']
[5, 6, 'w']
[3, 1, 'w']
[3, 2, 'w']
[4, 6, 'w']
[1, 3, 'w']
[1, 2, 'w']
[4, 5, 'w']
[6, 5, 'w']
[1, 2, 'a']
[1, 3, 'a']
[2, 1, 'a']
[2, 3, 'a']
[3, 1, 'a']
[3, 2, 'a']
[4, 5, 'a']
[4, 6, 'a']
[5, 4, 'a']
[5, 6, 'a']
[6, 4, 'a']
[6, 5, 'a']
[1, 2, 'w']
[1, 3, 'w']
[2, 1, 'w']
[2, 3, 'w']
[3, 1, 'w']
[3, 2, 'w']
[4, 5, 'w']
[4, 6, 'w']
[5, 4, 'w']
[5, 6, 'w']
[6, 4, 'w']
[6, 5, 'w']
True label
subject 4
[1, 2, 'a']
[1, 3, 'a']
[2, 1, 'a']
[2, 3, 'a']
[3, 1, 'a']
[3, 2, 'a']
[4, 5, 'a']
[4, 6, 'a']
[5, 4, 'a']
[5, 6, 'a']
[6, 4, 'a']
[6, 5, 'a']
[1, 2, 'w']
[1, 3, 'w']
[2, 1, 'w']
[2, 3, 'w']
[3, 1, 'w']
[3, 2, 'w']
[4, 5, 'w']
[4, 6, 'w']
[5, 4, 'w']
[5, 6, 'w']
[6, 4, 'w']
[6, 5, 'w']
0
Predicted label
Figure 2: Confusion matrices for the CNN with a single convolutional layer for subject 4. Labels contain
the ids of the high-pitched and low-pitch rhythm sequence (c.f. Table 1) and the rhythm type (?a? for
African, ?w? for Western). Left: Labels arranged such that most similar rhythms are close together. Right:
Labels in the order of the trials for this subject. More plots are provided in the supplementary material.
to rhythm perception that were only present during some of the trials. Re-arranging the labels within the confusion matrix such that they correspond to the order of the stimuli presentation (Figure 2, right) shows some
confusion between successive trials (blocks along the diagonal) which supports this hypothesis. Repeating
the experiment with multiple trials per stimulus for each subject should give more insights into this matter.
5 Conclusions
Distinguishing the rhythm stimuli used in this study is not easy as a listener. They are all presented in
the same tempo and comprise two 12/8 bars. Consequently, none of the participants scored more than
83% in the behavioral rhythm discrimination test. Considering this and the rather sub-par data quality of
the EEG recordings, the accuracies obtained for some of the participants are remarkable. They demonstrate
that perceived rhythms may be identified from EEG recorded during their auditory presentation using
convolutional neural networks that look only at a short segment of the signal from a single EEG channel
(corresponding to the length of a single bar of a two-bar stimulus).
We hope that our finding will encourage the application of deep learning techniques for EEG analysis
and stimulate more research in this direction. As a next step, we want to analyze the learned models
as they might provide some insight into the important underlying patterns within the EEG signals and
their corresponding neural processes. However, this is largely still an open problem. (As a first attempt,
visualizations of the kernel weight matrices and of input patterns producing the highest activations can
be found in the supplementary material.) We are also looking to correlate the classification performance
values with the subjects? scores in the behavioral part of the study.
The study is currently being repeated with North America participants and we are curious to see whether
we can replicate our findings. In particular, we hope to further improve the classification accuracy through
higher data quality of the new EEG recordings. Furthermore, we want to conduct a behavioral study to
obtain information about the perceived similarity between the stimuli. Finally, encouraged by our results,
we want to extend our focus by also considering more complex and richer stimuli such as audio recordings
of rhythms with realistic instrumentation instead of artificial sine tones.
Acknowledgments
This work was supported by a fellowship within the Postdoc-Program of the German Academic Exchange
Service (DAAD), by the Natural Sciences and Engineering Research Council of Canada (NSERC), through
the Western International Research Award R4911A07, and by an AUCC Students for Development Award.
8
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9
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4,719 | 5,273 | Neurons as Monte Carlo Samplers: Bayesian
Inference and Learning in Spiking Networks
Rajesh P.N. Rao
University of Washington
[email protected]
Yanping Huang
University of Washington
[email protected]
Abstract
We propose a spiking network model capable of performing both approximate
inference and learning for any hidden Markov model. The lower layer sensory
neurons detect noisy measurements of hidden world states. The higher layer neurons with recurrent connections infer a posterior distribution over world states
from spike trains generated by sensory neurons. We show how such a neuronal
network with synaptic plasticity can implement a form of Bayesian inference similar to Monte Carlo methods such as particle filtering. Each spike in the population
of inference neurons represents a sample of a particular hidden world state. The
spiking activity across the neural population approximates the posterior distribution of hidden state. The model provides a functional explanation for the Poissonlike noise commonly observed in cortical responses. Uncertainties in spike times
provide the necessary variability for sampling during inference. Unlike previous
models, the hidden world state is not observed by the sensory neurons, and the
temporal dynamics of the hidden state is unknown. We demonstrate how such
networks can sequentially learn hidden Markov models using a spike-timing dependent Hebbian learning rule and achieve power-law convergence rates.
1
Introduction
Humans are able to routinely estimate unknown world states from ambiguous and noisy stimuli,
and anticipate upcoming events by learning the temporal dynamics of relevant states of the world
from incomplete knowledge of the environment. For example, when facing an approaching tennis
ball, a player must not only estimate the current position of the ball, but also predict its trajectory
by inferring the ball?s velocity and acceleration before deciding on the next stroke. Tasks such as
these can be modeled using a hidden Markov model (HMM), where the relevant states of the world
are latent variables X related to sensory observations Z via a likelihood model (determined by the
emission probabilities). The latent states themselves evolve over time in a Markovian manner, the
dynamics being governed by a transition probabilities. In these tasks, the optimal way of combining such noisy sensory information is to use Bayesian inference, where the level of uncertainty for
each possible state is represented as a probability distribution [1]. Behavioral and neuropsychophysical experiments [2, 3, 4] have suggested that the brain may indeed maintain such a representation
and employ Bayesian inference and learning in a great variety of tasks in perception, sensori-motor
integration, and sensory adaptation. However, it remains an open question how the brain can sequentially infer the hidden state and learn the dynamics of the environment from the noisy sensory
observations.
Several models have been proposed based on populations of neurons to represent probability distribution [5, 6, 7, 8]. These models typically assume a static world state X. To get around this
limitation, firing-rate models [9, 10] have been proposed to used responses in populations of neurons to represent the time-varying posterior distributions of arbitrary hidden Markov models with
discrete states. For the continuous state space, similar models based on line attractor networks [11]
1
have been introduced for implementing the Kalman filter, which assumes all distributions are Gaussian and the dynamics is linear. Bobrowski et al. [12] proposed a spiking network model that can
compute the optimal posterior distribution in continuous time. The limitation of these models is that
model parameters (the emission and transition probabilities) are assumed to be known a priori. Deneve [13, 14] proposed a model for inference and learning based on the dynamics of a single neuron.
However, the maximum number of world state in her model is limited to two.
In this paper, we explore a neural implementation of HMMs in networks of spiking neurons that
perform approximate Bayesian inference similar to the Monte Carlo method of particle filtering [15].
We show how the time-varying posterior distribution P (Xt |Z1:t ) can be directly represented by
mean spike counts in sub-populations of neurons. Each model neuron in the neuron population
behaves as a coincidence detector, and each spike is viewed as a Monte Carlo sample of a particular
world state. At each time step, the probability of a spike in one neuron is shown to approximate
the posterior probability of the preferred state encoded by the neuron. Nearby neurons within the
same sub-population (analogous to a cortical column) encode the same preferred state. The model
thus provides a concrete neural implementation of sampling ideas previously suggested in [16, 17,
18, 19, 20]. In addition, we demonstrate how a spike-timing based Hebbian learning rule in our
network can implement an online version of the Expectation-Maximization(EM) algorithm to learn
the emission and transition matrices of HMMs.
2
Review of Hidden Markov Models
For clarity of notation, we briefly review the equations behind a discrete-time ?grid-based? Bayesian
filter for a hidden Markov model. Let the hidden state be {Xk ? X, k ? N} with dynamics
Xk+1 | (Xk = x0 ) ? f (x|x0 ), where f (x|x0 ) is the transition probability density, X is a discrete
state space of Xk , N is the set of time steps, and ??? denotes distributed according to. We focus
on estimating Xk by constructing its posterior distribution, based only on noisy measurements or
observations {Zk } ? Z where Z can be discrete or continuous. {Zk } are conditional independent
given {Xk } and are governed by the emission probabilities Zk | (Xk = x) ? g(z|x).
i
The posterior probability P (Xk = i|Z1:k ) = ?k|k
may be updated in two stages: a prediction stage
(Eq 1) and a measurement update (or correction) stage (Eq 2):
PX
j
i
P (Xk+1 = i | Z1:k ) =
?k+1|k
= j=1 ?k|k
f (xi |xj ),
(1)
i
P (Xk+1 = i | Z1:k+1 ) = ?k+1|k+1
=
i
?k+1|k
g(Zk+1 |xi )
PX
.
j
j
?
j=1 k+1|k g(Zk+1 |x )
(2)
This process is repeated for each time step. These two recursive equations above are the foundation
for any exact or approximate solution to Bayesian filtering, including well-known examples such as
Kalman filtering when the original continuous state space has been discretized into X bins.
3
Neural Network Model
We now describe the two-layer spiking neural network model we use (depicted in the central panel of
Figure 1(a)). The noisy observation Zk is not directly observed by the network, but sensed through
an array of Z sensory neurons, The lower layer consists of an array of sensory neurons, each of
which will be activated at time k if the observation Zk is in the receptive field. The higher layer
consists of an array of inference neurons, whose activities can be defined as:
s(k) = sgn(a(k) ? b(k))
(3)
where s(k) describes the binary response of an inference neuron at time k, the sign function
sgn(x) = 1 only when x > 0. a(k) represents the sum of neuron?s recurrent inputs, which is
determined by the recurrent weight matrix W among the inference neurons and the population responses sk?1 from the previous time step. b(k) represents the sum of feedforward inputs, which is
determined by the feed-forward weight matrix M as well as the activities in sensory neurons.
Note that Equation 3 defines the output of an abstract inference neuron which acts as a coincidence
detector and fires if and only if both recurrent and sensory inputs are received. In the supplementary
materials, we show that this abstract model neuron can be implemented using the standard leakyintegrate-and-fire (LIF) neurons used to model cortical neurons.
2
(a)
(b)
Figure 1: a. Spiking network model for sequential Monte Carlo Bayesian inference. b. Graphical
representation of spike distribution propagation
3.1
Neural Representation of Probability Distributions
Similar to the idea of grid-based filtering, we first divide the inference neurons into X subpopulations. s = {sil , i = 1, . . . X , l = 1, . . . , L}. We have sil (k) = 1 if there is a spike in the
l-th neuron of the i-th sub-population at time step k. Each sub-population of L neurons share the
same preferred world state, there being X such sub-populations representing each of X preferred
states. One can, for example, view such a neuronal sub-population as a cortical column, within
which neurons encode similar features [21].
Figure 1(a) illustrates how our neural network encodes a simple hidden Markov model with X =
Z = 1, . . . , 100. Xk = 50 is a static state and P (Zk |Xk ) is normally distributed. The network
utilizes 10,000 neurons for the Monte Carlo approximation, with each state preferred by a subpopulation of 100 neurons. At time k, the network observe Zk and the corresponding sensory
neuron whose receptive field contains Zk is activated and sends inputs to the inference neurons.
Combining with recurrent inputs from the previous time step, the responses in the inference neurons
are updated at each time step. As shown in the raster plot of Figure 1(a), the spikes across the entire
inference layer population form a Monte-Carlo approximation to the current posterior distribution:
nik|k :=
L
X
i
sil (k) ? ?k|k
(4)
l=1
where nik|k is the number of spiking neurons in the ith sub-population at time k, which can also be
PX
regarded as the instantaneous firing rate for sub-population i. Nk = i=1 nik|k is the total spike
count in the inference layer population. The set {nik|k } represents the un-normalized conditional
probabilities of Xk , so that P? (Xk = i|Z1:k ) = ? i = ni /Nk .
k|k
3.2
k|k
Bayesian Inference with Stochastic Synaptic Transmission
In this section, we assume the network is given the model parameters in a HMM and there is no
learning in connection weights in the network. To implement the prediction Eq 1 in a spiking
network, we initialize the recurrent connections between the inference neurons as the transition
probabilities: Wij = f (xj |xi )/CW , where CW is a scaling constant. We will discuss how our
network learns the HMM parameters from random initial synaptic weights in section 4.
We define the recurrent weight Wij to be the synaptic release probability between the i-th neuron
sub-population and the j-th neuron sub-population in the inference layer. Each neuron that spikes
at time step k will randomly evoke, with probability Wij , one recurrent excitatory post-synaptic
potential (EPSP) at time step k + 1, after some network delay. We define the number of recurrent
EPSPs received by neuron l in the j-th sub-population as ajl . Thus, ajl is the sum of Nk independent
(but not identically distributed) Bernoulli trials:
ajl (k + 1)
=
X X
L
X
i=1
il0 sil0 (k),
l0 =1
3
?l = 1 . . . L.
(5)
where P (il = 1) = Wij and P (il = 0) = 1 ? Wij . The sum ajl follows the so-called ?Poisson
binomial? distribution [22] and in the limit approaches the Poisson distribution:
X
Nk j
P (ajl (k + 1) ? 1) '
Wij nik|k =
(6)
?
CW k+1|k
i
The detailed analysis of the distribution of ail and the proof of equation 6 are provided in the supplementary materials.
The definition of model neuron in Eq 3 indicates that recurrent inputs alone are not strong enough
to make the inference neurons fire ? these inputs leave the neurons partially activated. We can
view these partially activated neurons as the proposed samples drawn from the prediction density
P (Xk+1 |Xk ). Let njk+1|k be the number of proposed samples in j-th sub-population, we have
E[njk+1|k |{nik|k }] = L
X
X
Wij nik|k = L
i=1
Nk j
? Var[njk+1|k |{nik|k }]
?
CW k+1|k
(7)
Thus, the prediction probability in equation 1 is represented by the expected number of neurons that
receive recurrent inputs.
When a new observation Zk+1 is received, the network will correct the prediction distribution based
on the current observation. Similar to rejection sampling used in sequential Monte Carlo algorithms [15], these proposed samples are accepted with a probability proportional to the observation
likelihood P (Zk+1 |Xk+1 ). We assume for simplicity that receptive fields of sensory neurons do not
overlap with each other (in the supplementary materials, we discuss the more general overlapping
case). Again we define the feedforward weight Mij to be the synaptic release probability between
sensory neuron i and inference neurons in the j-th sub-population. A spiking sensory neuron i
causes an EPSP in a neuron in the j-th sub-population with probability Mij , which is initialized
proportional to the likelihood:
P (bil (k + 1) ? 1) = g(Zk+1 |xi )/CM
i
(8)
j
where CM is a scaling constant such that Mij = g(Zk+1 = z | x )/CM .
Finally, an inference neuron fires a spike at time k + 1 if and only if it receives both recurrent and
sensory inputs. The corresponding firing probability is then the product of the probabilities of the
two inputs:P (sil (k + 1) = 1) = P (ail (k + 1) ? 1)P (bil (k + 1) ? 1)
PL
Let nik+1|k+1 = l=1 sil (k + 1) be the number of spikes in i-th sub-population at time k + 1, we
have
Nk
i
E[nik+1|k+1 |{nik|k }] = L
P (Zk+1 |Z1:k )?k+1|k+1
(9)
CW CM
Nk
i
g(Zk+1 |xi )?k+1|k
(10)
Var[nik+1|k+1 |{nik|k }] ' L
CW CM
Equation 9 ensures that the expected spike distribution at time k + 1 is a Monte Carlo approximation
to the updated posterior probability P (Xk+1 |Z1:k+1 ). It also determines how many neurons are
activated at time k + 1. To keep the number of spikes at different time steps relatively constant, the
scaling constant CM , CW and the number of neurons L could be of the same order of magnitude:
for example, CW = L = 10 ? N1 and CM (k + 1) = 10 ? Nk /N1 , resulting in a form of divisive
inhibition [23]. If the overall neural activity is weak at time k, then the global inhibition regulating
M is decreased to allow more spikes at time k + 1. Moreover, approximations in equations 6 and
N2
10 become exact when C 2k ? 0.
W
3.3
Filtering Examples
Figure 1(b) illustrates how the model network implements Bayesian inference with spike samples.
The top three rows of circles in the left panel in Figure 1(b) represent the neural activities in the
inference neurons, approximating respectively the prior, prediction, and posterior distributions in
the right panel. At time k, spikes (shown as filled circles) in the posterior population represent the
4
(a)
(b)
(c)
Figure 2: Filtering results for uni-modal (a) and bi-modal posterior distributions ((b) and (c) - see
text for details).
distribution P (Xk |Z1:k ). With recurrent weights W ? f (Xk+1 |Xk ), spiking neurons send EPSPs
to their neighbors and make them partially activated (shown as half-filled circles in the second row).
The distribution of partially activated neurons is a Monte-Carlo approximation to the prediction
distribution P (Xk+1 |Z1:k ). When a new observation Zk+1 arrives, the sensory neuron (filled circles
the bottom row) whose receptive field contains Zk+1 is activated, and sends feedforward EPSPs to
the inference neurons using synaptic weights M = g(Z|X). The inference neurons at time k +1 fire
only if they receive both recurrent and feedforward inputs. With the firing probability proportional to
the product of prediction probability P (Xk+1 |Z1:k ) and observation likelihood g(Zk+1 |Xk+1 ), the
spike distribution at time k + 1 (filled circles in the third row) again represents the updated posterior
P (Xk+1 |Z1:k+1 ).
We further tested the filtering results of the proposed neural network with two other example HMMs.
The first example is the classic stochastic volatility model, where X = Z = R. The transition model
of the hidden volatility variable f (Xk+1 |Xk ) = N (0.91Xk , 1.0), and the emission model of the
observed price given volatility is g(Zk |Xk ) = N (0, 0.25 exp(Xk )). The posterior distribution of
this model is uni-modal. In simulation we divided X into 100 bins, and initial spikes N1 = 1000.
We plotted the expected volatility with estimated standard deviation from the population posterior
distribution in Figure 2(a). We found that the neural network does indeed produce a reasonable
estimate of volatility and plausible confidence interval. The second example tests the network?s
ability to approximate bi-modal posterior distributions by comparing the time varying population
posterior distribution with the true one using heat maps (Figures 2(b) and 2(c)). The vertical axis
represents the hidden state and the horizontal axis represents time steps. The magnitude of the
probability is represented by the color. In this example, X = {1, . . . , 8} and there are 20 time steps.
3.4
Convergence Results and Poisson Variability
In this section, we discuss some convergence results for Bayesian filtering using the proposed spiking network and show our population estimator of the posterior probability is a consistent one. Let
ni
P?ki = Nk|k
be the population estimator of the true posterior probability P (Xk = i|Z1:k ) at time k.
k
Suppose the true distribution is known only at initial time k = 1: P? i = ? i . We would like to
1
1|1
investigate how the mean and variance of P?ki vary over time. We derived the updating equations for
mean and variance (see supplementary materials) and found two implications. First, the variance of
neural response is roughly proportional to the mean. Thus, rather than representing noise, Poisson
variability in the model occurs as a natural consequence of sampling and sparse coding. Second, the
variance Var[P?kj ] ? 1/N1 . Therefore Var[P?kj ] ? 0 as N1 ? ?, showing that P?kj is a consistent
j
estimator of ?k|k
. We tested the above two predictions using numerical experiments on arbitrary
HMMs, where we choose X = {1, 2, . . . 20}, Zk ? N (Xk ,P
5), the transition matrix f (xj |xi ) first
uniformly drawn from [0, 1], and then normalized to ensure j f (xj |xi ) = 1.
In Figures 3(a-c), each data point represents Var[P?kj ] along the vertical axis and E[P?kj ] ? E 2 [P?kj ]
along the horizontal axis, calculated over 100 trials with the same random transition matrix f , and
k = 1, . . . 10, j = 1, . . . 20. The solid lines represent a least squares power law fit to the data:
Var[P?kj ] = CV ? (E[P?kj ] ? E 2 [P?kj ])CE . For 100 different random transition matrices f , the means
5
?2
?4
10
10
y = 0.00355627 * x1.13
?3
10
y = 0.028804 * x1.2863
?5
10
k
Var[pjk]
?6
Var[pj ]
Var[pjk]
?4
10
?5
10
?5
10
10
?7
10
?6
?8
10
10
?7
10
?7
10 ?5
10
y = 0.000303182 * x1.037
?3
10
?9
?3
10
E[pj ] ? E2[pj ]
k
?1
10
0
10
?5
10
k
?3
10
E[pj ] ? E2[pj ]
k
?1
10
0
10
10 ?5
10
k
?4
10
?3
?2
10
10
E[pj ] ? E2[pj ]
k
(a)
(b)
(c)
(d)
(e)
(f)
?1
10
0
10
k
Figure 3: Variance versus Mean of estimator for different initial spike counts
of the exponential term CE were 1.2863, 1.13, and 1.037, with standard deviations 0.13, 0.08, and
0.03 respectively, for N1 = 100 and X = 4, 20, and 100. The mean of CE continues to approach
1 when X is increased, as shown in figure 3(d). Since Var[P?kj ] ? (E[P?kj ] ? E 2 [P?kj ]) implies
Var[njk|k ] ? E[njk|k ] (see supplementary material for derivation), these results verify the Poisson
variability prediction of our neural network.
The term CV represents the scaling constant for the variance. Figure 3(e) shows that the mean of
CV over 100 different transition matrices f (over 100 different trials with the same f ) is inversely
proportional to initial spike count N1 , with power law fit CV = 1.77N1?0.9245 . This indicates that
the variance of P?kj converges to 0 if N1 ? ?. The bias between estimated and true posterior
probability can be calculated as:
bias(f) =
X
K
1 X X ?i
i
(E[Pk ] ? ?k|k
)2
XK
i=1 k=1
The relationship between the mean of the bias (over 100 different f ) versus initial count N1 is shown
in figure 3(f). We also have an inverse proportionality between bias and N1 . Therefore, as the figure
j
shows, for arbitrary f , the estimator P?kj is a consistent estimator of ?k|k
.
4
On-line parameter learning
In the previous section, we assumed that the model parameters, i.e., the transition probabilities
f (Xk+1 |Xk ) and the emission probabilities g(Zk |Xk ), are known. In this section, we describe how
these parameters ? = {f, g} can be learned from noisy observations {Zk }. Traditional methods
to estimate model parameters are based on the Expectation-Maximization (EM) algorithm, which
maximizes the (log) likelihood of the unknown parameters log P? (Z1:k ) given a set of observations
collected previously. However, such an ?off-line? approach is biologically implausible because (1)
it requires animals to store all of the observations before learning, and (2) evolutionary pressures
dictate that animals update their belief over ? sequentially any time a new measurement becomes
available.
We therefore propose an on-line estimation method where observations are used for updating parameters as they become available and then discarded. We would like to find the parameters ? that
Pk
maximize the log likelihood: log P? (Z1:k ) = t=1 log P? (Zt |Zt?1 ). Our approach is based on recursively calculating the sufficient statistics of ? using stochastic approximation algorithms and the
6
Figure 4: Performance of the Hebbian Learning Rules.
Monte Carlo method, and employs an online EM algorithm obtained by approximating the expected
sufficient statistic T?(?k ) using the stochastic approximation (or Robbins-Monoro) procedure. Based
on the detailed derivations described in the supplementary materials, we obtain a Hebbian learning
rule for updating the synaptic weights based on the pre-synaptic and post-synaptic activities:
k
Mij
=
?k
Wijk
=
?k
njk|k
Nk
j
nk|k
n
? i (k)
k?1
+ (1 ? ?k
?P i
) ? Mij
? (k)
Nk
in
nik?1|k?1
Nk?1
?
njk|k
Nk
+ (1 ? ?k
nik?1|k?1
Nk?1
when njk|k > 0,
) ? Wijk?1
(11)
when nik?1|k?1 > 0, (12)
where n
? i (k) is the number of pre-synaptic spikes in the i-th sub-population of sensory neurons at
time k, ?k is the learning rate.
Learning both emission and transition probability matrices at the same time using the online EM
algorithm with stochastic approximation is in general very difficult because there are many local
minima in the likelihood function. To verify the correctness of our learning algorithms individually,
we first divide the learning process into two phases. The first phase involves learning the emission
probability g when the hidden world state is stationary, i.e., Wij = fij = ?ij . This corresponds to
learning the observation model of static objects at the center of gaze before learning the dynamics
f of objects. After an observation model g is learned, we relax the stationarity constraint, and allow
the spiking network to update the recurrent weights W to learn the arbitrary transition probability f .
Figure 4 illustrates the performance of learning rules (11) and (12) for a discrete HMM with X = 4
and Z = 12. X and Z values are spaced equally apart: X ? {1, . . . , 4} and Z ? { 23 , 1, 43 , . . . , 4 31 }.
The transition probability matrix f then involves 4?4 = 16 parameters and the emission probability
matrix g involves 12 ? 4 = 48 parameters.
In Figure 4(a), we examine the performance of learning rule (11) for the feedforward weights
M k , with fixed transition matrix. The true emission probability matrix has the
form g.j
=?
2
N (xj , ?Z
). The solid blue curve shows the mean square error (Frobenius norm)
M k ? g
F =
qP
k
k
2
ij (Mij ? gij ) between the learned feedforward weights M and the true emission probability matrix g over trials with different g,. The dotted lines show ? 1 standard deviation for MSE
based on 10 different trials. ?Z varied from trial to trial and was drawn uniformly between 0.2
and 0.4, representing different levels of observation noises. The initial spike distribution was uni0
form ni0|0 = nj0|0 , ?i, j = 1 . . . , X and the initial estimate Mi,j
= Z1 . The learning rate was set
to ?k = k1 , although a small constant learning rate such as ?k = 10?5 also gives rise to similar
learning results. A notable feature in Figure 4(a) is that the average MSE exhibits a fast powerlaw decrease. The red solid line in Figure 4(a) represents the power-law fit to the average MSE:
M SE(k) ? k ?1.1 . Furthermore, the standard deviation of MSE approaches zero as k grows large.
7
Figure 4(a) thus shows the asymptotic convergence of equation (11) irrespective of the ?Z of the
true emission matrix g.
We next examined the performance of learning rule 12 for the recurrent weights W k , given the
learned emission probability matrix g (the true transition probabilities f are unknown to the network). The initial estimator Wij0 = X1 . Similarly, Performance was evaluated by calculating the
qP
(W k ? fij )2 between the learned recurrent weight W k
mean square error
W k ? f
=
F
ij
ij
and the true f . Different randomly chosen transition matrices f were tested. When ?Z = 0.04, the
observation noise is 0.04
1/3 = 12% of the separation between two observed states. Hidden state identification in this case is relatively easy. The red solid line in figure 4(b) represents the power-law fit to
the average MSE: M SE(k) ? k ?0.36 . Similar convergence results can still be obtained for higher
?Z , e.g., ?Z = 0.4 (figure 4(c)). In this case, hidden state identification is much more difficult as
the observation noise is now 1.2 times the separation between two observed states. This difficulty
is reflected in a slower asymptotic convergence rate, with a power-law fit M SE(k) ? k ?0.21 , as
indicated by the red solid line in figure 4(c).
Finally, we show the results for learning both emission and transition matrices simultaneously in
figure 4(d,e). In this experiment, the true emission and transition matrices are deterministic, the
weight matrices are initialized as the sum of the true one and a uniformly random one: Wij0 ?
0
fij + and Mij
? gij + where is a uniform distributed noise between 0 and 1/NX . Although
the asymptotic convergence rate for this case is much slower, it still exhibits desired power-law
convergences in both M SEW (k) ? k ?0.02 and M SEM (k) ? k ?0.08 over 100 trials starting with
different initial weight matrices.
5
Discussion
Our model suggests that, contrary to the commonly held view, variability in spiking does not reflect ?noise? in the nervous system but captures the animal?s uncertainty about the outside world.
This suggestion is similar to some previous models [17, 19, 20], including models linking firing rate
variability to probabilistic representations [16, 8] but differs in the emphasis on spike-based representations, time-varying inputs, and learning. In our model, a probability distribution over a finite
sample space is represented by spike counts in neural sub-populations. Treating spikes as random
samples requires that neurons in a pool of identical cells fire independently. This hypothesis is supported by a recent experimental findings [21] that nearby neurons with similar orientation tuning and
common inputs show little or no correlation in activity. Our model offers a functional explanation
for the existence of such decorrelated neuronal activity in the cortex.
Unlike many previous models of cortical computation, our model treats synaptic transmission between neurons as a stochastic process rather than a deterministic event. This acknowledges the
inherent stochastic nature of neurotransmitter release and binding. Synapses between neurons usually have only a small number of vesicles available and a limited number of post-synaptic receptors
near the release sites. Recent physiological studies [24] have shown that only 3 NMDA receptors
open on average per release during synaptic transmission. These observations lend support to the
view espoused by the model that synapses should be treated as probabilistic computational units
rather than as simple scalar parameters as assumed in traditional neural network models.
The model for learning we have proposed builds on prior work on online learning [25, 26]. The
online algorithm used in our model for estimating HMM parameters involves three levels of approximation. The first level involves performing a stochastic approximation to estimate the expected
complete-data sufficient statistics over the joint distribution of all hidden states and observations.
Cappe and Moulines [26] showed that under some mild conditions, such an approximation produces
a consistent, asymptotically efficient estimator of the true parameters. The second approximation
comes from the use of filtered rather than smoothed posterior distributions. Although the convergence reported in the methods section is encouraging, a rigorous proof of convergence remains to
be shown. The asymptotic convergence rate using only the filtered distribution is about one third
the convergence rate obtained for the algorithms in [25] and [26], where the smoothed distribution
is used. The third approximation results from Monte-Carlo sampling of the posterior distribution.
As discussed in the methods section, the Monte Carlo approximation converges in the limit of large
numbers of particles (spikes).
8
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9
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multisensory:1 internal:1 support:1 tested:3 biol:1 |
4,720 | 5,274 | A framework for studying synaptic plasticity
with neural spike train data
Scott W. Linderman
Harvard University
Cambridge, MA 02138
Christopher H. Stock
Harvard College
Cambridge, MA 02138
Ryan P. Adams
Harvard University
Cambridge, MA 02138
[email protected]
[email protected]
[email protected]
Abstract
Learning and memory in the brain are implemented by complex, time-varying
changes in neural circuitry. The computational rules according to which synaptic
weights change over time are the subject of much research, and are not precisely
understood. Until recently, limitations in experimental methods have made it challenging to test hypotheses about synaptic plasticity on a large scale. However, as
such data become available and these barriers are lifted, it becomes necessary
to develop analysis techniques to validate plasticity models. Here, we present
a highly extensible framework for modeling arbitrary synaptic plasticity rules
on spike train data in populations of interconnected neurons. We treat synaptic weights as a (potentially nonlinear) dynamical system embedded in a fullyBayesian generalized linear model (GLM). In addition, we provide an algorithm
for inferring synaptic weight trajectories alongside the parameters of the GLM and
of the learning rules. Using this method, we perform model comparison of two
proposed variants of the well-known spike-timing-dependent plasticity (STDP)
rule, where nonlinear effects play a substantial role. On synthetic data generated
from the biophysical simulator NEURON, we show that we can recover the weight
trajectories, the pattern of connectivity, and the underlying learning rules.
1
Introduction
Synaptic plasticity is believed to be the fundamental building block of learning and memory in the
brain. Its study is of crucial importance to understanding the activity and function of neural circuits.
With innovations in neural recording technology providing access to the simultaneous activity of
increasingly large populations of neurons, statistical models are promising tools for formulating and
testing hypotheses about the dynamics of synaptic connectivity. Advances in optical techniques [1,
2], for example, have made it possible to simultaneously record from and stimulate large populations
of synaptically connected neurons. Armed with statistical tools capable of inferring time-varying
synaptic connectivity, neuroscientists could test competing models of synaptic plasticity, discover
new learning rules at the monosynaptic and network level, investigate the effects of disease on
synaptic plasticity, and potentially design stimuli to modify neural networks.
Despite the popularity of GLMs for spike data, relatively little work has attempted to model the
time-varying nature of neural interactions. Here we model interaction weights as a dynamical system
governed by parametric synaptic plasticity rules. To perform inference in this model, we use particle
Markov Chain Monte Carlo (pMCMC) [3], a recently developed inference technique for complex
time series. We use this new modeling framework to examine the problem of using recorded data to
distinguish between proposed variants of spike-timing-dependent plasticity (STDP) learning rules.
1
time
Figure 1: A simple network of four sparsely connected neurons whose synaptic weights are changing over time.
Here, the neurons have inhibitory self connections to mimic refractory effects, and are connected via a chain of
excitatory synapses, as indicated by the nonzero entries A1?2 , A2?3 , and A3?4 . The corresponding weights
of these synapses are strengthening over time (darker entries in W ), leading to larger impulse responses in the
firing rates and a greater number of induced post-synaptic spikes (black dots), as shown below.
2
Related Work
The GLM is a probabilistic model that considers spike trains to be realizations from a point process
with conditional rate ?(t) [4, 5]. From a biophysical perspective, we interpret this rate as a nonlinear
function of the cell?s membrane potential. When the membrane potential exceeds the spiking threshold potential of the cell, ?(t) rises to reflect the rate of the cell?s spiking, and when the membrane
potential decreases below the spiking threshold, ?(t) decays to zero. The membrane potential is
modeled as the sum of three terms: a linear function of the stimulus, I(t), for example a low-pass
filtered input current, the sum of excitatory and inhibitory PSPs induced by presynaptic neurons, and
n
a constant background rate. In a network of N neurons, let Sn = {sn,m }M
m=1 ? [0, T ] be the set of
observed spike times for neuron n, where T is the duration of the recording and Mn is the number
of spikes. The conditional firing rate of a neuron n can be written,
?
?
Z t
N M
n0
X
X
hn0 ?n (t ? sn0 ,m ) ? I[sn0 ,m < t]? , (1)
?n (t) = g ?bn +
kn (t ? ? ) ? I(? ) d? +
0
n0 =1 m=1
where bn is the background rate, the second term is a convolution of the (potentially vector-valued)
stimulus with a linear stimulus filter, kn (?t), and the third is a linear summation of impulse responses, hn0 ?n (?t), which preceding spikes on neuron n0 induce on the membrane potential of
neuron n. Finally, the rectifying nonlinearity g : R ? R+ converts this linear function of stimulus
and spike history into a nonnegative rate. While the spiking threshold potential is not explicitly
modeled in this framework, it is implicitly inferred in the amplitude of the impulse responses.
From this semi-biophysical perspective it is clear that one shortcoming of the standard GLM is that it
does not account for time-varying connectivity, despite decades of research showing that changes in
synaptic weight occur over a variety of time scales and are the basis of many fundamental cognitive
processes. This absence is due, in part, to the fact that this direct biophysical interpretation is not
warranted in most traditional experimental regimes, e.g., in multi-electrode array (MEA) recordings
where electrodes are relatively far apart. However, as high resolution optical recordings grow in
popularity, this assumption must be revisited; this is a central motivation for the present model.
There have been a few efforts to incorporate dynamics into the GLM. Stevenson and Koerding [6]
extended the GLM to take inter-spike intervals as a covariates and formulated a generalized bilinear
model for weights. Eldawlatly et al. [7] modeled the time-varying parameters of a GLM using a
dynamic Bayesian network (DBN). However, neither of these approaches accommodate the breadth
of synaptic plasticity rules present in the literature. For example, parametric STDP models with hard
2
bounds on the synaptic weight are not congruent with the convex optimization techniques used by
[6], nor are they naturally expressed in a DBN. Here we model time-varying synaptic weights as a
potentially nonlinear dynamical system and perform inference using particle MCMC.
Nonstationary, or time-varying, models of synaptic weights have also been studied outside the context of GLMs. For example, Petreska et al. [8] applied hidden switching linear dynamical systems models to neural recordings. This approach has many merits, especially in traditional MEA
recordings where synaptic connections are less likely and nonlinear dynamics are not necessarily
warranted. Outside the realm of computational neuroscience and spike train analysis, there exist a
number of dynamic statistical models, such as West et al. [9], which explored dynamic generalized
linear models. However, the types of models we are interested in for studying synaptic plasticity
are characterized by domain-specific transition models and sparsity structure, and until recently, the
tools for effectively performing inference in these models have been limited.
3
A Sparse Time-Varying Generalized Linear Model
In order to capture the time-varying nature of synaptic weights, we extend the standard GLM by first
factoring the impulse responses in the firing rate of Equation 1 into a product of three terms:
hn0 ?n (?t, t) ? An0 ?n Wn0 ?n (t) rn0 ?n (?t).
(2)
Here, An0 ?n ? {0, 1} is a binary random variable indicating the presence of a direct synapse
from neuron n0 to neuron n, Wn0 ?n (t) : [0, T ] ? R is a non stationary synaptic ?weight? trajectory
R ?associated with the synapse, and rn0 ?n (?t) is a nonnegative, normalized impulse response,
i.e. 0 rn0 ?n (? )d? = 1. Requiring rn0 ?n (?t) to be normalized gives meaning to the synaptic
weights: otherwise W would only be defined up to a scaling factor. For simplicity, we assume r(?t)
does not change over time, that is, only the amplitude and not the duration of the PSPs are timevarying. This restriction could be adapted in future work.
As is often done in GLMs, we model the normalized impulse responses as a linear combination of
basis functions. In order to enforce the normalization of r(?), however, we use a convex combination
of normalized, nonnegative basis functions. That is,
rn0 ?n (?t) ?
B
X
(n0 ?n)
?b
rb (?t),
b=1
R?
PB
(n0 ?n)
where 0 rb (? ) d? = 1, ?b and b=1 ?b
= 1, ?n, n0 . The same approach is used to model
the stimulus filters, kn (?t), but without the normalization and non-negativity constraints.
The binary random variables An0 ?n , which can be collected into an N ? N binary matrix A,
model the connectivity of the synaptic network. Similarly, the collection of weight trajectories {{Wn0 ?n (t)}}n0 ,n , which we will collectively refer to as W (t), model the time-varying synaptic weights. This factorization is often called a spike-and-slab prior [10], and it allows us to separate
our prior beliefs about the structure of the synaptic network from those about the evolution of synaptic weights. For example, in the most general case we might leverage a variety of random network
models [11] as prior distributions for A, but here we limit ourselves to the simplest network model,
the Erd?os-Renyi model. Under this model, each An0 ?n is an independent identically distributed
Bernoulli random variable with sparsity parameter ?.
Figure 1 illustrates how the adjacency matrix and the time-varying weights are integrated into the
GLM. Here, a four-neuron network is connected via a chain of excitatory synapses, and the synapses
strengthen over time due to an STDP rule. This is evidenced by the increasing amplitude of the
impulse responses in the firing rates. With larger synaptic weights comes an increased probability
of postsynaptic spikes, shown as black dots in the figure. In order to model the dynamics of the
time-varying synaptic weights, we turn to a rich literature on synaptic plasticity and learning rules.
3.1
Learning rules for time-varying synaptic weights
Decades of research on synapses and learning rules have yielded a plethora of models for the evolution of synaptic weights [12]. In most cases, this evolution can be written as a dynamical system,
dW (t)
= ` (W (t), {sn,m : sn,m < t} ) + (W (t), t),
dt
3
where ` is a potentially nonlinear learning rule that determines how synaptic weights change as a
function of previous spiking. This framework encompasses rate-based rules such as the Oja rule
[13] and timing-based rules such as STDP and its variants. The additive noise, (W (t), t), need not
be Gaussian, and many models require truncated noise distributions.
Following biological intuition, many common learning rules factor into a product of simpler functions.
For example, STDP (defined below) updates each synapse independently such that dWn0 ?n (t)/dt only depends on Wn0 ?n (t) and the presynaptic spike history Sn<t = {sn,m : sn,m < t}. Biologically speaking, this means that plasticity is local to the
synapse. More sophisticated rules allow dependencies among the columns of W . For example, the
incoming weights to neuron n may depend upon one another through normalization, as in the Oja
rule [13], which scales synapse strength according to the total strength of incoming synapses.
Extensive research in the last fifteen years has identified the relative spike timing between the preand postsynaptic neurons as a key component of synaptic plasticity, among other factors such as
mean firing rate and dendritic depolarization [14]. STDP is therefore one of the most prominent
learning rules in the literature today, with a number of proposed variants based on cell type and
biological plausibility. In the experiments to follow, we will make use of two of these proposed variants. First, consider the canonical STDP rule with a ?double-exponential? function parameterized
by ?? , ?+ , A? , and A+ [15], in which the effect of a given pair of pre-synaptic and post-synaptic
spikes on a weight may be written:
` (Wn0 ?n (t), Sn0 , Sn ) = I[t ? Sn ] `+ (Sn0 ; A+ , ?+ ) ? I[t ? Sn0 ] `? (Sn ; A? , ?? ),
(3)
X
X
`+ (Sn0 ; A+ , ?+ ) =
A+ e(t?sn0 ,m )/?+
`? (Sn ; A? , ?? ) =
A? e(t?sn,m )/?? .
sn0 ,m ?Sn0 <t
sn,m ?Sn<t
This rule states that weight changes only occur at the time of pre- or post-synaptic spikes, and that
the magnitude of the change is a nonlinear function of interspike intervals.
A slightly more complicated model known as the multiplicative STDP rule extends this by bounding
the weights above and below by Wmax and Wmin , respectively [16]. Then, the magnitude of the
weight update is scaled by the distance from the threshold:
` (Wn0 ?n (t), Sn0 , Sn ) = I[t ? Sn ] `?+ (Sn0 ; A+ , ?+ ) (Wmax ? Wn0 ?n (t)),
? I[t ? Sn0 ] `?? (Sn ; A? , ?? ) (Wn0 ?n (t) ? Wmin ).
(4)
Here, by setting `?? = min(`? , 1), we enforce that the synaptic weights always fall
within [Wmin , Wmax ]. With this rule, it often makes sense to set Wmin to zero.
Similarly, we can construct an additive, bounded model which is identical to the standard additive
STDP model except that weights are thresholded at a minimum and maximum value. In this model,
the weight never exceeds its set lower and upper bounds, but unlike the multiplicative STDP rule,
the proposed weight update is independent of the current weight except at the boundaries. Likewise,
whereas with the canonical STDP model it is sensible to use Gaussian noise for (t) in the bounded
multiplicative model we use truncated Gaussian noise to respect the hard upper and lower bounds
on the weights. Note that this noise is dependent upon the current weight, Wn0 ?n (t).
The nonlinear nature of this rule, which arises from the multiplicative interactions among the parameters, ?` = {A+ , ?+ , A? , ?? , Wmax , Wmax }, combined with the potentially non-Gaussian noise
models, pose substantial challenges for inference. However, the computational cost of these detailed
models is counterbalanced by dramatic expansions in the flexibility of the model and the incorporation of a priori knowledge of synaptic plasticity. These learning models can be interpreted as strong
regularizers of models that would otherwise be highly underdetermined, as there are N 2 weight trajectories and only N spike trains. In the next section we will leverage powerful new techniques for
Bayesian inference in order to capitalize on these expressive models of synaptic plasticity.
4
Inference via particle MCMC
The traditional approach to inference in the standard GLM is penalized maximum likelihood estimation. The log likelihood of a single conditional Poisson process is well known to be,
Z T
Mn
X
N
L ?n (t); {Sn }n=1 , I(t) = ?
?n (t) dt +
log (?n (sn,m )) ,
(5)
0
4
m=1
and the log likelihood of a population of non-interacting spike trains is simply the sum of
each of the log likelihoods for each neuron. The likelihood depends upon the parameters ?GLM = {bn , kn , {hn0 ?n (?t)}N
n0 =1 } through the definition of the rate function given in Equation 1. For some link functions g, the log likelihood is a concave function of ?GLM , and the MLE can
be found using efficient optimization techniques. Certain dynamical models, namely linear Gaussian latent state space models, also support efficient inference via point process filtering techniques
[17].
Due to the potentially nonlinear and non-Gaussian nature of STDP, these existing techniques are
not applicable here. Instead we use particle MCMC [3], a powerful technique for inference in time
series. Particle MCMC samples the posterior distribution over weight trajectories, W (t), the adjacency matrix A, and the model parameters ?GLM and ?` , given the observed spike trains, by combining particle filtering with MCMC. We represent the conditional distribution over weight trajectories
with a set of discrete particles. Let the instantaneous weights at (discretized) time t be represented
(p)
N ?N
by a set of P particles, {W t }P
and are assigned normalized parp=1 . The particles live in R
PP
1
ticle weights , ?p , which approximate the true distribution via Pr(W t ) ? p=1 ?p ?W (p) (W t ).
t
Particle filtering is a method of inferring a distribution over weight trajectories by iteratively propagating forward in time and reweighting according to how well the new samples explain the data. For
(p)
each particle W t at time t, we propagate forward one time step using the learning rule to obtain
(p)
a particle W t+1 . Then, using Equation 5, we evaluate the log likelihood of the spikes that occurred
in the window [t, t + 1) and update the weights. Since some of these particles may have very low
weights, after each step we resample the particles. After the T -th time step we are left with a set of
(p)
(p)
weight trajectories {(W 0 , . . . , W T )}P
p=1 , each associated with a particle weight ?p .
Particle filtering only yields a distribution over weight trajectories, and implicitly assumes that the
other parameters have been specified. Particle MCMC provides a broader inference algorithm for
both weights and other parameters. The idea is to interleave conditional particle filtering steps
that sample the weight trajectory given the current model parameters and the previously sampled
weights, with traditional Gibbs updates to sample the model parameters given the current weight
trajectory. This combination leaves the stationary distribution of the Markov chain invariant and
allows joint inference over weights and parameters. Gibbs updates for the remaining model parameters, including those of the learning rule, are described in the supplementary material.
Collapsed sampling of A and W (t) In addition to sampling of weight trajectories and model
parameters, particle MCMC approximates the marginal likelihood of entries in the adjacency matrix, A, integrating out the corresponding weight trajectory. We have, up to a constant,
Pr(An0 ?n | S, ?` , ?GLM , A?n0 ?n , W ?n0 ?n (t))
Z TZ ?
=
p(An0 ?n , Wn0 ?n (t) | S, ?` , ?GLM , A?n0 ?n , W ?n0 ?n (t)) dWn0 ?n (t) dt
0
??
"
#
T
P
Y
1 X (p)
?
?
?
Pr(An0 ?n ),
P p=1 t
t=1
where ?n0 ? n indicates all entries except for n0 ? n, and the particle weights are obtained by
running a particle filter for each assignment of An0 ?n . This allows us to jointly sample An?n0
and Wn?n0 (t) by first sampling An?n0 and then Wn?n0 (t) given An?n0 . By marginalizing out the
weight trajectory, our algorithm is able to explore the space of adjacency matrices more efficiently.
We capitalize on a number of other opportunities for computational efficiency as well. For example, if the learning rule factors into independent updates for each Wn0 ?n (t), then we can update
each synapse?s weight trajectory separately and reduce the particles to one-dimensional objects. In
our implementation, we also make use of a pMCMC variant with ancestor sampling [18] that significantly improves convergence. Any distribution may be used to propagate the particles forward;
using the learning rule is simply the easiest to implement and understand. We have omitted a number
of details in this description; for a thorough overview of particle MCMC, the reader should consult
[3, 18].
1
Note that the particle weights are not the same as the synaptic weights.
5
Figure 2: We fit time-varying weight trajectories to spike trains simulated from a GLM with two neurons
undergoing no plasticity (top row), an additive, unbounded STDP rule (middle), and a multiplicative, saturating
STDP rule (bottom row). We fit the first 50 seconds with four different models: MAP for an L1-regularized
GLM, and fully-Bayesian inference for a static, additive STDP, and multiplicative STDP learning rules. In all
cases, the correct models yield the highest predictive log likelihood on the final 10 seconds of the dataset.
5
Evaluation
We evaluated our technique with two types of synthetic data. First, we generated data from our
model, with known ground-truth. Second, we used the well-known simulator NEURON to simulate
driven, interconnected populations of neurons undergoing synaptic plasticity. For comparison, we
show how the sparse, time-varying GLM compares to a standard GLM with a group LASSO prior
on the impulse response coefficients for which we can perform efficient MAP estimation.
5.1
GLM-based simulations
As a proof of concept, we study a single synapse undergoing a variety of synaptic plasticity rules
and generating spikes according to a GLM. The neurons also have inhibitory self-connections to
mimic refractory effects. We tested three synaptic plasticity mechanisms: a static synapse (i.e., no
plasticity), the unbounded, additive STDP rule given by Equation 3, and the bounded, multiplicative
STDP rule given by Equation 4. For each learning rule, we simulated 60 seconds of spiking activity
at 1kHz temporal resolution, updating the synaptic weights every 1s. The baseline firing rates were
normally distributed with mean 20Hz and standard deviation of 5Hz. Correlations in the spike timing
led to changes in the synaptic weight trajectories that we could detect with our inference algorithm.
Figure 2 shows the true and inferred weight trajectories, the inferred learning rules, and the predictive
log likelihood on ten seconds of held out data for each of the three ground truth learning rules. When
the underlying weights are static (top row), MAP estimation and static learning rules do an excellent
6
mV
9
Figure 3: Evaluation of synapse detection on a 60 second spike train from a network of 10 neurons undergoing
synaptic plasticity with a saturating, additive STDP rule, simulated with NEURON. The sparse, time-varying
GLM with an additive rule outperforms the fully-Bayesian model with static weights, MAP estimation with L1
regularization, and simple thresholding of the cross-correlation matrix.
job of detecting the true weight whereas the two time-varying models must compensate by either
setting the learning rule as close to zero as possible, as the additive STDP does, or setting the
threshold such that the weight trajectory is nearly constant, as the multiplicative model does. Note
that the scales of the additive and multiplicative learning rules are not directly comparable since the
weight updates in the multiplicative case are modulated by how close the weight is to the threshold.
When the underlying weights vary (middle and bottom rows), the static models must compromise
with an intermediate weight. Though the STDP models are both able to capture the qualitative
trends, the correct model yields a better fit and better predictive power in both cases.
In terms of computational cost, our approach is clearly more expensive than alternative approaches
based on MAP estimation or MLE. We developed a parallel implementation of our algorithm to
capitalize on conditional independencies across neurons, i.e. for the additive and multiplicative
STDP rules we can sample the weights W ??n independently of the weights W ??n0 . On the two
neuron examples we achieve upward of 2 iterations per second (sampling all variables in the model),
and we run our model for 1000 iterations. Convergence of the Markov chain is assessed by analyzing
the log posterior of the samples, and typically stabilizes after a few hundred iterations. As we scale
to networks of ten neurons, our running time quickly increases to roughly 20 seconds per iteration,
which is mostly dominated by slice sampling the learning rule parameters. In order to evaluate the
conditional probability of a learning rule parameter, we need to sample the weight trajectories for
each synapse. Though these running times are nontrivial, they are not prohibitive for networks that
are realistically obtainable for optical study of synaptic plasticity.
5.2
Biophysical simulations
Using the biophysical simulator NEURON, we performed two experiments. First, we considered a
network of 10 sparsely interconnected neurons (28 excitatory synapses) undergoing synaptic plasticity according to an additive STDP rule. Each neuron was driven independently by a hidden
population of 13 excitatory neurons and 5 inhibitory neurons connected to the visible neuron with
probability 0.8 and fixed synaptic weights averaging 3.0 mV. The visible synapses were initialized
close to 6.0 mV and allowed to vary between 0.0 and 10.5 mV. The synaptic delay was fixed at
1.0 ms for all synapses. This yielded a mean firing rate of 10 Hz among visible neurons. Synaptic weights were recorded every 1.0 ms. These parameters were chosen to demonstrate interesting
variations in synaptic strength, and as we transition to biological applications it will be necessary to
evaluate the sensitivity of the model to these parameters and the appropriate regimes for the circuits
under study.
We began by investigating whether the model is able to accurately identify synapses from spikes, or
whether it is confounded by spurious correlations. Figure 3 shows that our approach identifies the
28 excitatory synapses in our network, as measured by ROC curve (Add. STDP AUC=0.99), and
outperforms static models and cross-correlation. In the sparse, time-varying GLM, the probability
of an edge is measured by the mean of A under the posterior, whereas in the standard GLM with
MAP estimation, the likelihood of an edge is measured by area under the impulse response.
7
mV
12
Figure 4: Analogously to Figure 2, a sparse, time-varying GLM can capture the weight trajectories and learning
rules from spike trains simulated by NEURON. Here an excitatory synapse undergoes additive STDP with a
hard upper bound on the excitatory postsynaptic current. The weight trajectory inferred by our model with the
same parametric form of the learning rule matches almost exactly, whereas the static models must compromise
in order to capture early and late stages of the data, and the multiplicative weight exhibits qualitatively different
trajectories. Nevertheless, in terms of predictive log likelihood, we do not have enough information to correctly
determine the underlying learning rule. Potential solutions are discussed in the main text.
Looking into the synapses that are detected by the time-varying model and missed by the static
model, we find an interesting pattern. The improved performance comes from synapses that decay
in strength over the recording period. Three examples of these synaptic weight trajectories are shown
in the right panel of Figure 3. The time-varying model assigns over 90% probability to each of the
three synapses, whereas the static model infers less than a 40% probability for each synapse.
Finally, we investigated our model?s ability to distinguish various learning rules by looking at a
single synapse, analogous to the experiment performed on data from the GLM. Figure 4 shows
the results of a weight trajectory for a synapse under additive STDP with a strict threshold on the
excitatory postsynaptic current. The time-varying GLM with an additive model captures the same
trajectory, as shown in the left panel. The GLM weights have been linearly rescaled to align with the
true weights, which are measured in millivolts. Furthermore, the inferred additive STDP learning
rule, in particular the time constants and relative amplitudes, perfectly match the true learning rule.
These results demonstrate that a sparse, time-varying GLM is capable of discovering synaptic weight
trajectories, but in terms of predictive likelihood, we still have insufficient evidence to distinguish
additive and multiplicative STDP rules. By the end of the training period, the weights have saturated
at a level that almost surely induces postsynaptic spikes. At this point, we cannot distinguish two
learning rules which have both reached saturation. This motivates further studies that leverage this
probabilistic model in an optimal experimental design framework, similar to recent work by Shababo
et al. [19], in order to conclusively test hypotheses about synaptic plasticity.
6
Discussion
Motivated by the advent of optical tools for interrogating networks of synaptically connected neurons, which make it possible to study synaptic plasticity in novel ways, we have extended the GLM
to model a sparse, time-varying synaptic network, and introduced a fully-Bayesian inference algorithm built upon particle MCMC. Our initial results suggest that it is possible to infer weight
trajectories for a variety of biologically plausible learning rules.
A number of interesting questions remain as we look to apply these methods to biological recordings. We have assumed access to precise spike times, though extracting spike times from optical
recordings poses inferential challenges of its own. Solutions like those of Vogelstein et al. [20]
could be incorporated into our probabilistic model. Computationally, particle MCMC could be replaced with stochastic EM to achieve improved performance [18], and optimal experimental design
could aid in the exploration of stimuli to distinguish between learning rules. Beyond these direct extensions, this work opens up potential to infer latent state spaces with potentially nonlinear dynamics
and non-Gaussian noise, and to infer learning rules at the synaptic or even the network level.
Acknowledgments This work was partially funded by DARPA YFA N66001-12-1-4219 and NSF IIS1421780. S.W.L. was supported by an NDSEG fellowship and by the NSF Center for Brains, Minds, and
Machines.
8
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design for mapping neural microcircuits. In Advances in Neural Information Processing Systems, pages
1304?1312, 2013.
[20] Joshua T Vogelstein, Brendon O Watson, Adam M Packer, Rafael Yuste, Bruno Jedynak, and Liam Paninski. Spike inference from calcium imaging using sequential Monte Carlo methods. Biophysical journal,
97(2):636?655, 2009.
9
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4,721 | 5,275 | Global Belief Recursive Neural Networks
Romain Paulus, Richard Socher?
MetaMind
Palo Alto, CA
{romain,richard}@metamind.io
Christopher D. Manning
Stanford University
353 Serra Mall
Stanford, CA 94305
[email protected]
Abstract
Recursive Neural Networks have recently obtained state of the art performance on
several natural language processing tasks. However, because of their feedforward
architecture they cannot correctly predict phrase or word labels that are determined by context. This is a problem in tasks such as aspect-specific sentiment
classification which tries to, for instance, predict that the word Android is positive
in the sentence Android beats iOS. We introduce global belief recursive neural
networks (GB-RNNs) which are based on the idea of extending purely feedforward neural networks to include one feedbackward step during inference. This
allows phrase level predictions and representations to give feedback to words. We
show the effectiveness of this model on the task of contextual sentiment analysis. We also show that dropout can improve RNN training and that a combination
of unsupervised and supervised word vector representations performs better than
either alone. The feedbackward step improves F1 performance by 3% over the
standard RNN on this task, obtains state-of-the-art performance on the SemEval
2013 challenge and can accurately predict the sentiment of specific entities.
1
Introduction
Models of natural language need the ability to compose the meaning of words and phrases in order
to understand complex utterances such as facts, multi-word entities, sentences or stories. There has
recently been a lot of work extending single word semantic vector spaces [27, 11, 15] to compositional models of bigrams [16, 29] or phrases of arbitrary length [25, 28, 24, 10]. Work in this
area so far has focused on computing the meaning of longer phrases in purely feedforward types
of architectures in which the meaning of the shorter constituents that are being composed is not
altered. However, a full treatment of semantic interpretation cannot be achieved without taking into
consideration that the meaning of words and phrases can also change once the sentence context is
observed. Take for instance the sentence in Fig. 1: The Android?s screen is better than the iPhone?s.
All current recursive deep learning sentiment models [26] would attempt to classify the phrase The
Android?s screen or than the iPhone?s, both of which are simply neutral. The sentiment of the overall sentence is undefined; it depends on which of the entities the user of the sentiment analysis cares
about. Generally, for many analyses of social media text, users are indeed most interested in the
sentiment directed towards a specific entity or phrase.
In order to solve the contextual classification problem in general and aspect-specific sentiment classification in particular, we introduce global belief recursive neural networks (GB-RNN). These models
generalize purely feedforward recursive neural networks (RNNs) by including a feedbackward step
at inference time. The backward computation uses the representations from both steps in its recursion and allows all phrases, to update their prediction based on the global context of the sentence.
Unlike recurrent neural networks or window-based methods [5] the important context can be many
?
Part of this research was performed while the author was at Stanford University.
1
?
-
0
Android
0
0
beats
iOS
Figure 1: Illustration of the problem of sentiment classification that uses only the phrase to be labeled
and ignores the context. The word Android is neutral in isolation but becomes positive in context.
words away from the phrase that is to be labeled. This will allow models to correctly classify that in
the sentence of Fig. 1, Android is described with positive sentiment and iOS was not. Neither was
possible to determine only from their respective phrases in isolation.
In order to validate the GB-RNN?s ability to contextually disambiguate sentiment on real text, we
use the Twitter dataset and annotations from Semeval Challenge 2013 Task 2.1 The GB-RNN outperforms both the standard RNN and all other baselines, as well the winner of the Sentiment competition of SemEval 2013, showing that it can successfully make use of surrounding context.
2
Related Work
Neural word vectors One common way to represent words is to use distributional word vectors
[27] learned via dimensionality reduction of large co-occurrence matrices over documents (as in
latent semantic analysis [13]), local context windows [15, 18] or combinations of both [11]. Words
with similar meanings are close to each other in the vector space. Since unsupervised word vectors computed from local context windows do not always encode task-specific information, such
as sentiment, word vectors can also be fine-tuned to such specific tasks [5, 24]. We introduce a
hybrid approach where some dimensions are obtained from an unsupervised model and others are
learned for the supervised task. We show that this performs better than both the purely supervised
and unsupervised semantic word vectors.
Recursive Neural Networks The idea of recursive neural networks (RNNs) for natural language
processing (NLP) is to train a deep learning model that can be applied to inputs of any length.
Unlike computer vision tasks, where it is easy to resize an image to a fixed number of pixels, natural sentences do not have a fixed size input. However, phrases and sentences have a grammatical
structure that can be parsed as a binary tree [22].
Following this tree structure, we can assign a fixed-length vector to each word at the leaves of
the tree, and combine word and phrase pairs recursively to create intermediate node vectors of the
same length, eventually having one final vector representing the whole sentence [19, 25]. Multiple
recursive combination functions have been explored, from linear transformation matrices to tensor
products [26]. In this work, we use the simple single matrix RNN to combine node vectors at each
recursive step.
Bidirectional-recurrent and bidirectional-recursive neural networks. Recurrent neural networks
are a special case of recursive neural networks that operate on chains and not trees. Unlike recursive
neural networks, they don?t require a tree structure and are usually applied to time series. In a recurrent neural network, every node is combined with a summarized representation of the past nodes
[8], and then the resulting combination will be forwarded to the next node. Bidirectional recurrent neural network architectures have also been explored [21] and usually compute representations
independently from both ends of a time series.
Bidirectional recursive models [12, 14], developed in parallel with ours, extend the definition of the
recursive neural netword by adding a backward propagation step, where information also flows from
the tree root back to the leaves. We compare our model to theirs theoretically in the model section,
and empirically in the experiments.
1
http://www.cs.york.ac.uk/semeval-2013/task2/
2
Figure 2: Propagation steps of the GB-RNN. Step 1 describes the standard RNN feedforward process, showing that the vector representation of ?Android? is independent of the rest of the document.
Step 2 computes additional vectors at each node (in red), using information from the higher level
nodes in the tree (in blue), allowing ?Android? and ?iOS? to have different representations given the
context.
[20] unfold the same autoencoder multiple times which gives it more representational power with
the same number of parameters. Our model is different in that it takes into consideration more
information at each step and can eventually make better local predictions by using global context.
Sentiment analysis. Sentiment analysis has been the subject of research for some time [4, 2, 3, 6,
1, 23]. Most approaches in sentiment analysis use ?bag of words? representations that do not take
the phrase structure into account but learn from word-level features. We explore our model?s ability
to determine contextual sentiment on Twitter, a social media platform.
3
Global Belief Recursive Neural Networks
In this section, we introduce a new model to compute context-dependent compositional vector representations of variable length phrases. These vectors are trained to be useful as features to classify
each phrase and word. Fig. 2 shows an example phrase computation that we will describe in detail
below. This section begins by motivating compositionality and context-dependence, followed by a
definition of standard recursive neural networks. Next, we introduce our novel global belief model
and hybrid unsupervised-supervised word vectors.
3.1
Context-Dependence as Motivation for Global Belief
A common simplifying assumption when mapping sentences into a feature vector is that word order
does not matter (?bag of words?). However, this will prevent any detailed understanding of language
as exemplified in Fig. 1, where the overall sentiment of the phrase ?Android beats iOS?, is unclear.
Instead, we need an understanding of each phrase which leads us to deep recursive models.
The first step for mapping a sentence into a vector space is to parse them into a binary tree structure
that captures the grammatical relationships between words. Such an input dependent binary tree then
determines the architecture of a recursive neural network which will compute the hidden vectors in a
bottom-up fashion starting with the word vectors. The resulting phrase vectors are given as features
to a classifier. This standard RDL architecture works well for classifying the inherent or contextindependent label of a phrase. For instance, it can correctly classify that a not so beautiful day is
negative in sentiment. However, not all phrases have an inherent sentiment as shown in Fig. 1.
The GB-RNN addresses this issue by propagating information from the root node back to the
leaf nodes as described below. There are other ways context can be incorporated such as with
bi-directional recurrent neural networks or with window-based methods. Both of these methods,
however, cannot incorporate information from words further away from the phrase to be labeled.
3.2
Standard Recursive Neural Networks
We first describe a simple recursive neural network that can be used for context-independent phraselevel classification. It can also be seen as the first step of a GB-RNN.
3
Assume, for now, that each word vector a ? Rn is obtained by sampling each element from a
uniform distribution: ai ? U(?0.001, 0.001). All these vectors are columns of a large embedding
matrix L ? Rn?|V | , where |V | is the size of the vocabulary. All word vectors are learned together
with the model.
For the example word vector sequence (abc) of Fig. 2, the RNN equations become:
b
a
p1 = f W
, p2 = f W
,
c
p1
(1)
where W ? Rn?2n is the matrix governing the composition and f the non-linear activation function. Each node vector is the given as input to a softmax classifier for a classification task such as
sentiment analysis.
3.3
GB-RNN: Global Belief Recursive Neural Networks
Our goal is to include contextual information in the recursive node vector representations. One
simple solution would be to just include the k context words to the left and right of each pair as in
[25]. However, this will only work if the necessary context is at most k words away. Furthermore,
in order to capture more complex linguistic phenomena it may be necessary to allow for multiple
words to compose the contextual shift in meaning. Instead, we will use the feedforward nodes from
a standard RNN architecture and simply move back down the tree. This can also be interpreted as
unfolding the tree and moving up its branches.
Hence, we keep the same Eq. 1 for computing the forward node vectors, but we introduce new
feedbackward vectors, denoted with a down arrow ? , at every level of the parse tree. Unlike the
feedforward vectors, which were computed with a bottom-up recursive function, feedbackward vectors are computed with a top-down recursive function. The backwards pass starts at the root node
and propagates all the way down to the single word vectors. At the root note, in our example the
node p2 , we have:
p?2 = f (V p2 ) ,
(2)
where V ? Vnd ?n so that all ?-node vectors are nd -dimensional. Starting from p?2 , we recursively
get ?-node vectors for every node as we go down the tree:
?
?
p2
p1
a
b
?
?
=
f
W
,
=
f
W
(3)
?
?
c?
p2
p1
p?1
where all ?-vectors, are nd -dimensional and hence W ? ? R(n+nd )?(n+nd ) is a new de-composition
matrix. Figure 2 step 2 illustrates this top-down recursive computation on our example. Once we
have both feedforward and feedbackward vectors for a given node, we concatenate them and employ
the standard softmax classifier
the final prediction. For instance, the classification for word
to make
a
a becomes: ya = softmax Wc
, where we fold the bias into the C-class classifier weights
a?
Wc ? RC?(n+1) .
At the root node, the equation for x?root could be replaced by simply copying x?root = xroot . But
there are two advantages of introducing a transform matrix V . First, it helps clearly differentiating features computed during the forward step and the backward step in multiplication with W ? .
Second, it allows to use a different dimension for the x? vectors, which reduces the number of parameters in the W ? and Wclass matrices, and adds more flexibility to the model. It also performs
better empirically.
3.4
Hybrid Word Vector Representations
There are two ways to initialize the word vectors that are given as inputs to the RNN models. The
simplest one is to initialize them to small random numbers as mentioned above and backpropagate
error signals into them in order to have them capture the necessary information for the task at hand.
This has the advantage of not requiring any other pre-training method and the vectors are sure to
capture domain knowledge. However, the vectors are more likely to overfit and less likely to generalize well to words that have not been in the (usually smaller) labeled training set. Another approach
4
Figure 3: Hybrid unsupervised-supervised vector representations for the most frequent 50 words
of the dataset. For each horizontal vector, the first 100 dimensions are trained on unlabeled twitter
messages, and the last dimensions are trained on labeled contextual sentiment examples.
is to use unsupervised methods that learn semantic word vectors such as [18]. One then has the
option to backpropagate task specific errors into these vectors or keep them at their initialization.
Backpropagating into them still has the potential disadvantage of hurting generalization apart from
slowing down training since it increases the number of parameters by a large amount (there are usually 100, 000 ? 50 many parameters in the embedding matrix L). Without propagating information
however one has to hope that the unsupervised method really captures all the necessary semantic
information which is often not the case for sentiment (which suffers from the antonym problem).
In this paper we propose to combine both ideas by representing each word as a concatenation of both
unsupervised vectors that are kept at their initialization during training and adding a small additional
vector into which we propagate the task specific error signal. This vector representation applies only
to the feedforward word vectors and shold not be confused with the combination of the feedwordard
and feedbackward node vectors in the softmax.
Figure 3.4 shows the resulting word vectors trained on unlabeled documents on one part (the first
100 dimensions), and trained on labeled examples on the other part (the remaining dimensions).
3.5
Training
The GB-RNN is trained by using backpropagation through structure [9]. We train the parameters by
optimizing the regularized cross-entropy error for labeled node vectors with mini-batched AdaGrad
[7]. Since we don?t have labels for every node of the training trees, we decided that unlabeled
nodes do not add an additional error during training. For all models, we use a development set to
cross-validate over regularization of the different weights, word vector size, mini-batch size, dropout
probability and activation function (rectified linear or logistic function).
We also applied the dropout technique to improve training with high dimensional word vectors.
Node vector units are randomly set to zero with a probability of 0.5 at each training step. Our
experiments show that applying dropout in this way helps differentiating word vector units and
hidden units, and leads to better performance. The high-dimensional hybrid word vectors that we
introduced previously have obtained a higher accuracy than other word vectors with the use of
dropout.
3.6
Comparison to Other Models
The idea of unfolding of neural networks is commonly used in autoencoders as well as in a recursive
setting [23], in this setting the unfolding is only used during training and not at inference time to
update the beliefs about the inputs.
Irsoy and Cardie [12] introduced a bidirectional RNN similar to ours. It employs the same standard
feedforward RNN, but a different computation for the backward ? vectors. In practice, their model is
defined by the same forward equations as ours. However, equation 3 which computes the backward
vectors is instead:
?
b
V b + Wlb? p?1
=f
(4)
? ?
c?
V c + Wrb
p1
5
Correct FUSION?s 5th General Meeting is tonight at 7 in ICS 213! Come out and carve pumpkins mid-quarter
with us!
Correct I would rather eat my left foot then to be taking the SATs tomorrow
Correct Special THANKS to EVERYONE for coming out to Taboo Tuesday With DST tonight! It was
FUN&educational!!! :) @XiEtaDST
Correct Tough loss for @statebaseball today. Good luck on Monday with selection Sunday
Correct I got the job at Claytons!(: I start Monday doing Sheetrock(: #MoneyMakin
Correct St Pattys is no big deal for me, no fucks are given, but Cinco De Mayo on the other hand .. thats my
2nd bday .
Incorrect ?@Hannah Sunder: The Walking Dead is just a great tv show? its bad ass just started to watch the
2nd season to catch up with the 3rd
Figure 4: Examples of predictions made by the GB-RNN for twitter documents. In this example,
red phrases are negative and blue phrases are positive. On the last example, the model predicted
incorrectly ?bad ass? as negative.
?
Where Wlb? and Wrb
are two matrices with dimensions nd ? nd . For a better comparison with our
model we rewrite Eq. 3 and make explicit the 4 blocks of W ? :
"
#
"
#!
?
?
?
Wlf
Wlb?
Wlf
p1 + Wlb? p?1
b
?
Let W =
, then
=f
,
(5)
?
?
?
? ?
c?
Wrf
Wrb
Wrf
p1 + Wrb
p1
?
?
?
?
where the dimensions of Wlf
and Wrf
are nd ? n, and the dimensions of Wld
and Wrd
are nd ? nd .
A closer comparison between Eqs. 4 and 5 reveals that both use a left and right forward transfor?
?
mation Wlf
p1 and Wrf
p1 , but the other parts of the sums differ. In the bidirectional-RNN, the
transformation of any children is defined by the forward parent and independent on its position (left
or right node). Whereas our GB-RNN makes uses of both the forward and backward parent node.
The intuition behind our choice is that using both nodes helps to push the model to disentangled
the children from their backward parent vector. We also note that our model does not use the forward node vector for computing the backward node vector, but we find this not necessary since the
softmax function already combines the two vectors.
Our model also has n ? nd more parameters to compute the feedbackward vectors than the
bidirectional-RNN. The W ? matrix of our model has 2n2d + 2n ? nd parameters, while the other
?
?
model has a total of 2n2d + n ? nd parameters with the Wlf
, Wrf
and V matrices. We show in the
next section that GB-RNN outperforms the bidirectional RNN in our experiments.
4
Experiments
We present a qualitative and quantitative analysis of the GB-RNN on a contextual sentiment classification task. The main dataset is provided by the SemEval 2013, Task 2 competition [17]. We
outperform the winners of the 2013 challenge, as well as several baseline and model ablations.
4.1
Evaluation Dataset
The SemEval competition dataset is composed of tweets labeled for 3 different sentiment classes:
positive, neutral and negative. The tweets in this dataset were split into a train (7862 labeled phrases),
development (7862) and development-test (7862) set. The final test set is composed of 10681 examples. Fig. 4 shows example GB-RNN predictions on phrases marked for classification in this dataset.
The development dataset consists only of tweets whereas the final evaluation dataset included also
short text messages (SMS in the tables below).
Tweets were parsed using the Stanford Parser [22] which includes tokenizing of negations (e.g.,
don?t becomes two tokens do and n?t). We constrained the parser to keep each phrase labeled by the
dataset inside its own subtree, so that each labeled example is represented by a single node and can
be classified easily.
6
Classifier
SVM
SVM
SVM
GB-RNN
Feature Sets
stemming, word cluster, SentiWordNet
score, negation
POS, lexicon, negations, emoticons,
elongated words, scores, syntactic dependency, PMI
punctuation, word n-grams, emoticons,
character n-grams, elongated words,
upper case, stopwords, phrase length,
negation, phrase position, large sentiment lexicons, microblogging features
parser, unsupervised word vectors (ensemble)
Twitter 2013 (F1)
85.19
SMS 2013 (F1)
88.37
87.38
85.79
88.93
88.00
89.41
88.40
Table 1: Comparison to the best Semeval 2013 Task 2 systems, their feature sets and F1 results on
each dataset for predicting sentiment of phrases in context. The GB-RNN obtains state of the art
performance on both datasets.
Model
Bigram Naive Bayes
Logistic Regression
SVM
RNN
Bidirectional-RNN (Irsoy and Cardie)
GB-RNN (best single model)
Twitter 2013
80.45
80.91
81.87
82.11
85.77
86.80
SMS 2013
78.53
80.37
81.91
84.07
84.77
87.15
Table 2: Comparison with baselines: F1 scores on the SemEval 2013 test datasets.
4.2 Comparison with Competition Systems
The first comparison is with several highly tuned systems from the SemEval 2013, Task 2 competition. The competition was scored by an average of positive and negative class F1 scores. Table 1
lists results for several methods, together with the resources and features used by each method. Most
systems used a considerable amount of hand-crafted features. In contrast, the GB-RNN only needs
a parser for the tree structure, unsupervised word vectors and training data. Since the competition
allowed for external data we outline below the additional training data we use. Our best model is an
ensemble of the top 5 GB-RNN models trained independently. Their predictions were then averaged
to produce the final output.
4.3 Comparison with Baselines
Next we compare our single best model to several baselines and model ablations. We used the same
hybrid word vectors with dropout training for the RNN, the bidirectional RNN and the GB-RNN.
The best models were selected by cross-validating on the dev set for several hyper-parameters (word
vectors dimension, hidden node vector dimension, number of training epochs, regularization parameters, activation function, training batch size and dropout probability) and we kept the models with
the highest cross-validation accuracy. Table 2 shows these results. The most important comparison
is against the purely feedforward RNN which does not take backward sentence context into account.
This model performs over 5% worse than the GB-RNN.
For the logistic regression and Bigram Naive Bayes classification, each labeled phrase was taken
as a separate example, removing the surrounding context. Another set of baselines used a context
window for classification as well as the entire tweet as input to the classifier.
Optimal performance for the single best GB-RNN was achieved by using vector sizes of 130 dimensions (100 pre-trained, fixed word vectors and 30 trained on sentiment data), a mini-batch size of
30, dropout with p = 0.5 and sigmoid non-linearity. In table 3, we show that the concatenation of
fixed, unsupervised vectors with additional randomly initialized, supervised vectors performs better
than both methods.
4.4 Model Analysis: Additional Training Data
Because the competition allowed the usage of arbitrary resources we included as training data labeled unigrams and bigrams extracted from the NRC-Canada system?s sentiment lexicon. Adding
these additional training examples increased accuracy by 2%. Although this lexicon helps reduc7
Word vectors
supervised word vectors
semantic word vectors
hybrid word vectors
dimension
15
100
100 + 34
Twitter 2013
85.15
85.67
86.80
SMS 2013
85.66
84.70
87.15
Table 3: F1 score comparison of word vectors on the SemEval 2013 Task 2 test dataset.
-
-
-
-
Chelski
-
-
+
-
+
want
-
this
+
Chelski
+
+
+
that it
-
-
so
bad
+
+
+
+
want +
this +
so
+
makes-
+
me
+
even
+
that +
it
-
bad
+
-
-
+
+
+
me
+
+
+
+
+
even
+
+
thinking+
+
we +
may
happier
-
+
-
+
beat
+
makes
+
+
thinking+
we may
happier
+
+
+
-
-
twice
them
+
+
+
+
in
+
+
beat
-
-
-
-
4
days
at
+
SB
+
+
+
+
twice
them
+
in
+
+
+
+
+
+
4
days
at
SB
Figure 5: Change in sentiment predictions in the tweet chelski want this so bad that it makes me even
happier thinking we may beat them twice in 4 days at SB between the RNN (left) and the GB-RNN
(right). In particular, we can see the change for the phrase want this so bad where it is correctly
predicted as positive with context.
ing the number of unknown tokens, it does not do a good job for training recursive composition
functions, because each example is short.
We also included our own dataset composed 176,311 noisily labeled tweets (using heuristics such as
smiley faces) as well as the movie reviews dataset from [26]. In both datasets the labels only denote
the context-independent sentiment of a phrase or full sentence. Hence, we trained the final model in
two steps: train the standard RNN, then train the full GB-RNN model on the smaller context-specific
competition data. Training the GB-RNN jointly in this fashion gave a 1% accuracy improvement.
5
Conclusion
We introduced global belief recursive neural networks, applied to the task of contextual sentiment
analysis. The idea of propagating beliefs through neural networks is a powerful and important piece
for interpreting natural language. The applicability of this idea is more general than RNNs and can
be helpful for a variety of NLP tasks such as word-sense disambiguation.
Acknowledgments
We thank the anonymous reviewers for their valuable comments.
References
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9
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4,722 | 5,276 | Deep Networks with Internal Selective Attention
through Feedback Connections
Marijn F. Stollenga?, Jonathan Masci? , Faustino Gomez, Juergen Schmidhuber
IDSIA, USI-SUPSI
Manno-Lugano, Switzerland
{marijn,jonathan,tino,juergen}@idsia.ch
Abstract
Traditional convolutional neural networks (CNN) are stationary and feedforward.
They neither change their parameters during evaluation nor use feedback from
higher to lower layers. Real brains, however, do. So does our Deep Attention
Selective Network (dasNet) architecture. DasNet?s feedback structure can dynamically alter its convolutional filter sensitivities during classification. It harnesses the power of sequential processing to improve classification performance,
by allowing the network to iteratively focus its internal attention on some of its
convolutional filters. Feedback is trained through direct policy search in a huge
million-dimensional parameter space, through scalable natural evolution strategies (SNES). On the CIFAR-10 and CIFAR-100 datasets, dasNet outperforms the
previous state-of-the-art model on unaugmented datasets.
1
Introduction
Deep convolutional neural networks (CNNs) [1] with max-pooling layers [2] trained by backprop
[3] on GPUs [4] have become the state-of-the-art in object recognition [5, 6, 7, 8], segmentation/detection [9, 10], and scene parsing [11, 12] (for an extensive review see [13]). These architectures consist of many stacked feedforward layers, mimicking the bottom-up path of the human
visual cortex, where each layer learns progressively more abstract representations of the input data.
Low-level stages tend to learn biologically plausible feature detectors, such as Gabor filters [14].
Detectors in higher layers learn to respond to concrete visual objects or their parts, e.g., [15]. Once
trained, the CNN never changes its weights or filters during evaluation.
Evolution has discovered efficient feedforward pathways for recognizing certain objects in the blink
of an eye. However, an expert ornithologist, asked to classify a bird belonging to one of two very
similar species, may have to think for more than a few milliseconds before answering [16, 17],
implying that several feedforward evaluations are performed, where each evaluation tries to elicit
different information from the image. Since humans benefit greatly from this strategy, we hypothesise CNNs can too. This requires: (1) the formulation of a non-stationary CNN that can adapt its
own behaviour post-training, and (2) a process that decides how to adapt the CNNs behaviour.
This paper introduces Deep Attention Selective Networks (dasNet) which model selective attention
in deep CNNs by allowing each layer to influence all other layers on successive passes over an image
through special connections (both bottom-up and top-down), that modulate the activity of the convolutional filters. The weights of these special connections implement a control policy that is learned
through reinforcement learning after the CNN has been trained in the usual way via supervised
learning. Given an input image, the attentional policy can enhance or suppress features over multiple passes to improve the classification of difficult cases not captured by the initially supervised
?
Shared first author.
1
training. Our aim is to let the system check the usefulness of internal CNN filters automatically,
omitting manual inspection [18].
In our current implementation, the attentional policy is evolved using Separable Natural Evolution
Strategies (SNES; [19]), instead of a conventional, single agent reinforcement learning method
(e.g. value iteration, temporal difference, policy gradients, etc.) due to the large number of parameters (over 1 million) required to control CNNs of the size typically used in image classification.
Experiments on CIFAR-10 and CIFAR100 [20] show that on difficult classification instances, the
network corrects itself by emphasising and de-emphasising certain filters, outperforming a previous
state-of-the-art CNN.
2
Maxout Networks
In this work we use the Maxout networks [7], combined with dropout [21], as the underlying model
for dasNet. Maxout networks represent the state-of-the-art for object recognition in various tasks and
have only been outperformed (by a small margin) by averaging committees of several convolutional
neural networks. A similar approach, which does not reduce dimensionality in favor of sparsity
in the representation has also been recently presented [22]. Maxout CNNs consist of a stack of
alternating convolutional and maxout layers, with a final classification layer on top:
Convolutional Layer. The input to this layer can be an image or the output of a previous layer,
consisting of c input maps of width m and height n: x ? Rc?m?n . The output consists of a set of c?
?
?
?
output maps: y ? Rc ?m ?n . The convolutional layer is parameterised by c ? c? filters of size k ? k.
?
We denote the filters by Fi,j ? Rk?k , where i and j are indexes of the input and output maps and ?
denotes the layer.
yj? =
i=c
?
?
?(xi ? Fi,j
)
(1)
i=0
where i and j index the input and output map respectively, ? is the convolutional operator, ? is an
element-wise nonlinear function, and ? is used to index the layer. The size of the output is determined
by the kernel size and the stride used for the convolution (see [7]).
Pooling Layer. A pooling layer is used to reduced the dimensionality of the output from a convolutional layer. The usual approach is to take the maximum value among non- or partially-overlapping
patches in every map, therefore reducing dimensionality along the height and width [2]. Instead, a
Maxout pooling layer reduces every b consecutive maps to one map, by keeping only the maximum
value for every pixel-position, where b is called the block size. Thus the map reduces c input maps
to c? = c/b output maps.
b
??1
?
yj,x,y
= max yj?b+i,x,y
i=0
?
?
(2)
?
where y ? ? Rc ?m ?n , and ? again is used to index the layer. The output of the pooling layer can
either be used as input to another pair of convolutional- and pooling layers, or form input to a final
classification layer.
Classification Layer. Finally, a classification step is performed. First the output of the last pooling
layer is flattened into one large vector ?x, to form the input to the following equations:
?
y?j? = max Fj?b+i
?x
(3)
v = ?(F ?+1 y?? )
(4)
i=0..b
where F ? ? RN ?|?x| (N is chosen), and ?(?) is the softmax activation function which produces the
class probabilities v. The input is projected by F and then reduced using a maxout, similar to the
pooling layer (3).
2
3
Reinforcement Learning
Reinforcement learning (RL) is a general framework for learning to make sequential decisions order
to maximise an external reward signal [23, 24]. The learning agent can be anything that has the
ability to act and perceive in a given environment.
At time t, the agent receives an observation ot ? O of the current state of the environment st ? S,
and selects an action, at ? A, chosen by a policy ? : O ? A, where S, O and A the spaces
of all possible states, observations, and action, respectively.1 The agent then enters state st+1 and
receives a reward rt ??
R. The objective is to find the policy, ?, that maximises the expected future
discounted reward, E[ t ? t rt ], where ? ? [0, 1] discounts the future, modeling the ?farsightedness?
of the agent.
In dasNet, both the observation and action spaces are real valued O = Rdim(O) , A = Rdim(A) .
Therefore, policy ?? must be represented by a function approximator, e.g. a neural network, parameterised by ?. Because the policies used to control the attention of the dasNet have state and
actions spaces of close to a thousand dimensions, the policy parameter vector, ?, will contain close
to a million weights, which is impractical for standard RL methods. Therefore, we instead evolve
the policy using a variant for Natural Evolution Strategies (NES; [25, 26]), called Separable NES
(SNES; [19]). The NES family of black-box optimization algorithms use parameterised probability
distributions over the search space, instead of an explicit population (i.e., a conventional ES [27]).
Typically, the distribution is a multivariate Gaussian parameterised by mean ? and covariance matrix
?. Each epoch a generation is sampled from the distribution, which is then updated the direction
of the natural gradient of the expected fitness of the distribution. SNES differs from standard NES
in that instead of maintaining the full covariance matrix of the search distribution, uses only the
diagonal entries. SNES is theoretically less powerful than standard NES, but is substantially more
efficient.
4
Deep Attention Selective Networks (dasNet)
The idea behind dasNet is to harness the power of sequential processing to improve classification
performance by allowing the network to iteratively focus the attention of its filters. First, the standard
Maxout net (see Section 2) is augmented to allow the filters to be weighted differently on different
passes over the same image (compare to equation 1):
yj?
=
a?j
i=c
?
?
?(xi ? Fi,j
),
(5)
i=0
where a?j is the weight of the j-th output map in layer ?, changing the strength of its activation,
before applying the maxout pooling operator. The vector a = [a00 , a01 , ? ? ? , a0c? , a10 , ? ? ? , a1c? , ? ? ? ]
represents the action that the learned policy must select in order to sequentially focus the attention
of the Maxout net on the most discriminative features in the image being processed. Changing action
a will alter the behaviour of the CNN, resulting in different outputs, even when the image x does
not change. We indicate this with the following notation:
vt = Mt (?, x)
(6)
where ? is the parameter vector of the policy, ?? , and vt is the output of the network on pass t.
Algorithm 1 describes the dasNet training algorithm. Given a Maxout net, M, that has already
been trained to classify images using training set, X, the policy, ?, is evolved using SNES to focus
the attention of M. Each pass through the while loop represents one generation of SNES. Each
generation starts by selecting a subset of n images from X at random.
Then each of the p samples drawn from the SNES search distribution (with mean ? and covariance
?) representing the parameters, ?i , of a candidate policy, ??i , undergoes n trials, one for each image
in the batch. During a trial, the image is presented to the Maxout net T times. In the first pass, t = 0,
the action, a0 , is set to ai = 1, ?i, so that the Maxout network functions as it would normally ?
1
In this work ? : O ? A is a deterministic policy; given an observation it will always output the same
action. However, ? could be extended to stochastic policies.
3
Algorithm 1 T RAIN DAS N ET (M, ?, ?, p, n)
1: while True do
2:
images ? N EXT BATCH(n)
3:
for i = 0 ? p do
4:
?i ? N(?, ?)
5:
for j = 0 ? n do
6:
a0 ? 1 {Initialise gates a with identity activation}
7:
for t = 0 ? T do
8:
vt = Mt (?i , xi )
9:
ot ? h(Mt )
10:
at+1 ? ??i (ot )
11:
end for
12:
Li = ??boost d log(vT )
13:
end for
14:
F[i] ? f (?i )
15:
?[i] ? ?i
16:
end for
17:
U PDATE SNES(F, ?) {Details in supplementary material.}
18: end while
the action has no effect. Once the image is propagated through the net, an observation vector, o0 , is
constructed by concatenating the following values extracted from M, by h(?):
1. the average activation of every output map Avg(yj ) (Equation 2), of each Maxout layer.
2. the intermediate activations y?j of the classification layer.
3. the class probability vector, vt .
While averaging map activations provides only partial state information, these values should still be
meaningful enough to allow for the selection of good actions. The candidate policy then maps the
observation to an action:
??i (o) = dim(A)?(? i ot ) = at ,
(7)
where ? ? Rdim(A)?dim(O) is the weight matrix of the neural network, and ? is the softmax. Note
that the softmax function is scaled by the dimensionality of the action space so that elements in the
action vector average to 1 (instead of regular softmax which sums to 1), ensuring that all network
outputs are positive, thereby keeping the filter activations stable.
On the next pass, the same image is processed again, but this time using the filter weighting, a1 .
This cycle is repeated until pass T (see figure 1 for a illustration of the process), at which time the
performance of the network is scored by:
Li = ??boost d log(vT )
vT = MT (?i , xi )
?boost =
{
?correct
?misclassified
if d = ?vT ??
otherwise,
(8)
(9)
(10)
where v is the output of M at the end of the pass T , d is the correct classification, and ?correct
and ?misclassif ied are constants. Li measures the weighted loss, where misclassified samples are
weighted higher than correctly classified samples ?misclassif ied > ?correct . This simple form of
boosting is used to focus on the ?difficult? misclassified images. Once all of the input images have
been processed, the policy is assigned the fitness:
cumulative score
f (?i ) =
z }| {
n
?
Li
i=1
4
regularization
z }| {
+ ?L2 ??i ?2
(11)
Classes
Classes
Softmax
Softmax
ma
era p
ge
s
error
Softmax
ma
era p
ge
av
av
s
Maps
Maps
Maps
gates
Filters
gates
Filters
ma
era p
ge
s
av
Filters
m
av ap
era
ge
s
policy
Maps
Maps
gates
Filters
Maps
gates
Filters
ag
B
RG
Observation Action
t=1
t=2
Im
Filters
e
ag
B
RG
Action
e
Im
t=T
Figure 1: The dasNet Network. Each image in classified after T passes through the network. After each forward propagation through the Maxout net, the output classification vector, the output of the second to last layer,
and the averages of all feature maps, are combined into an observation vector that is used by a deterministic
policy to choose an action that changes the weights of all the feature maps for the next pass of the same image.
After pass T , the output of the Maxout net is finally used to classify the image.
where ?L2 is a regularization parameter. Once all of the candidate policies have been evaluated,
SNES updates its distribution parameters (?, ?) according the natural gradient calculated from the
sampled fitness values, F. As SNES repeatedly updates the distribution over the course of many
generations, the expected fitness of the distribution improves, until the stopping criterion is met
when no improvement is made for several consecutive epochs.
5
Related Work
Human vision is still the most advanced and flexible perceptual system known. Architecturally,
visual cortex areas are highly connected, including direct connections over multiple levels and topdown connections. Felleman and Essen [28] constructed a (now famous) hierarchy diagram of
32 different visual cortical areas in macaque visual cortex. About 40% of all pairs of areas were
considered connected, and most connected areas were connected bidirectionally. The top-down
connections are more numerous than bottom-up connections, and generally more diffuse [29]. They
are thought to play primarily a modulatory role, while feedforward connections serve as directed
information carriers [30].
Analysis of response latencies to a newly-presented image lends credence to the theory that there are
two stages of visual processing: a fast, pre-attentive phase, due to feedforward processing, followed
by an attentional phase, due to the influence of recurrent processing [31]. After the feedforward pass,
we can recognise and localise simple salient stimuli, which can ?pop-out? [32], and response times
do not increase regardless of the number of distractors. However, this effect has only been conclusively shown for basic features such as colour or orientation; for categorical stimuli or faces, whether
there is a pop-out effect remains controversial [33]. Regarding the attentional phase, feedback connections are known to play important roles, such as in feature grouping [34], in differentiating a
foreground from its background, (especially when the foreground is not highly salient [35]), and
perceptual filling in [36]. Work by Bar et al. [37] supports the idea that top-down projections from
prefrontal cortex play an important role in object recognition by quickly extracting low-level spatial
frequency information to provide an initial guess about potential categories, forming a top-down
expectation that biases recognition. Recurrent connections seem to rely heavily on competitive inhibition and other feedback to make object recognition more robust [38, 39].
In the context of computer vision, RL has been shown to be able to learn saccades in visual scenes
to learn selective attention [40, 41], learn feedback to lower levels [42, 43], and improve face recognition [44, 45]. It has been shown to be effective for object recognition [46], and has also been
5
0.452
Method
Dropconnect [51]
Stochastic Pooling [52]
Multi-column CNN [5]
Maxout [7]
Maxout (trained by us)
dasNet
NiN [8]
NiN (augmented)
CIFAR-10
CIFAR-100
9.32%
15.13%
11.21%
9.38%
9.61%
9.22%
10.41%
8.81%
38.57%
34.54%
33.78%
35.68%
-
0.45
% Correct
Table 1: Classification results on CIFAR-10 and CIFAR-100
datasets. The error on the test-set is shown for several methods. Note that the result for Dropconnect is the average of
12 models. Our method improves over the state-of-the-art
reference implementation to which feedback connections are
added. The recent Network in Network architecture [8] has
better results when data-augmentation is applied.
0.448
0 steps
1 step
2 steps
3 steps
0.446
0.444
0.442
0.44
0
1
2 3 4 5 6 7 8 9
Number of steps evaluated
Figure 2: Two dasNets were trained on
CIFAR-100 for different values of T . Then
they were allowed to run for [0..9] iterations for each image. The performance
peeks at the number of steps that the network is trained on, after which the performance drops, but does not explode, showing
the dynamics are stable.
combined with traditional computer vision primitives [47]. Iterative processing of images using
recurrency has been successfully used for image reconstruction [48], face-localization [49] and
compression [50]. All these approaches show that recurrency in processing and an RL perspective can lead to novel algorithms that improve performance. However, this research is often applied to simplified datasets for demonstration purposes due to computation constraints, and are not
aimed at improving the state-of-the-art. In contrast, we apply this perspective directly to the known
state-of-the-art neural networks to show that this approach is now feasible and actually increases
performance.
6
Experiments on CIFAR-10/100
The experimental evaluation of dasNet focuses on ambiguous classification cases in the CIFAR-10
and CIFAR-100 data sets where, due to a high number of common features, two classes are often
mistaken for each other. These are the most interesting cases for our approach. By learning on top of
an already trained model, dasNet must aim at fixing these erroneous predictions without disrupting,
or forgetting, what has been learned. The CIFAR-10 dataset [20] is composed of 32 ? 32 colour
images split into 5?104 training and 104 testing samples, where each image is assigned to one of 10
classes. The CIFAR-100 is similarly composed, but contains 100 classes. The number of steps was
experimentally determined and fixed at T = 5; small enough to be computationally tractable while
still allowing for enough interaction. In all experiments we set ?correct = 0.005, ?misclassified = 1 and
?L2 = 0.005.
The Maxout network, M, was trained with data augmentation following global contrast normalization and ZCA normalization. The model consists of three convolutional maxout layers followed by
a fully connected maxout and softmax outputs. Dropout of 0.5 was used in all layers except the
input layer, and 0.2 for the input layer. The population size for SNES was set to 50. Training took
of dasNet took around 4 days on a GTX 560 Ti GPU, excluding the original time used to train M.
Table 1 shows the performance of dasNet vs. other methods, where it achieves a relative improvement of 6% with respect to the vanilla CNN. This establishes a new state-of-the-art result for this
challenging dataset, for unaugmented data. Figure 3 shows the classification of a cat-image from the
test-set. All output map activations in the final step are shown at the top. The difference in activations compared to the first step, i.e., the (de-)emphasis of each map, is shown on the bottom. On the
left are the class probabilities for each time-step. At the first step, the classification is ?dog?, and the
cat could indeed be mistaken for a puppy. Note that in the first step, the network has not yet received
any feedback. In the next step, the probability for ?cat? goes up dramatically, and subsequently drops
a bit in the following steps. The network has successfully disambiguated a cat from a dog. If we
investigate the filters, we see that in the lower layer emphasis changes significantly (see ?change
of layer 0?). Some filters focus more on surroundings whilst others de-emphasise the eyes. In the
6
layer 0
layer 1
layer 2
change of layer 0
change of layer 1
change of layer 2
airplane
automobile
bird
cat
deer
dog
frog
horse
ship
truck
1 2 3 4 5
Timesteps
class probabilities
Figure 3: The classification of a cat by the dasNet is shown. All output map activations in the final step
are shown on the top. Their changes relative to initial activations in the first step are shown at the bottom
(white = emphasis, black = suppression). The changes are normalised to show the effects more clearly. The
class probabilities over time are shown on the left. The network first classifies the image as a dog (wrong)
but corrects itself by emphasising its convolutional filters to see it is actually a cat. Two more examples are
included in the supplementary material.
second layer, almost all output maps are emphasised. In the third and highest convolutional layer,
the most complex changes to the network can be seen. At this level the positional correspondence
is largely lost, and the filters are known to code for ?higher level? features. It is in this layer that
changes are the most influential because they are closest to the final output layers.
It is hard to qualitatively analyze the effect of the alterations. If we compare each final activation in
layer 2 to its corresponding change (see Figure 3, right), we see that the activations are not simply
uniformly enhanced. Instead, complex suppression and enhancement patterns are found, increasing
and decreasing activation of specific pixels. Visualising what these high-level actually do is an open
problem in deep learning.
Dynamics To investigate the dynamics, a small 2-layer dasNet network was trained for different
values of T . Then they were evaluated by allowing them to run for [0..9] steps. Figure 2 shows
results of training dasNet on CIFAR-100 for T = 1 and T = 2. The performance goes up from the
vanilla CNN, peaks at the step = T as expected, and reduces but stays stable after that. So even
though the dasNet was trained using only a small number of steps, the dynamics stay stable when
these are evaluated for as many as 10 steps.
To verify whether the dasNet policy is actually making good use of its gates, we estimate their
information content in the following way: The gate values in the final step are used directly for
classification. The hypothesis is that if the gates are used properly, then their activation should
contain information that is relevant for classification. For this purpose, a dasNet that was trained with
T = 2. Then using only the final gate-values (so without e.g. the output of the classification layer),
a classification using 15-nearest neighbour and logistic regression was performed. This resulted in
a performance of 40.70% and 45.74% correct respectively, similar to the performance of dasNet,
confirming that they contain significant information.
7
Conclusion
DasNet is a deep neural network with feedback connections that are learned by through reinforcement learning to direct selective internal attention to certain features extracted from images. After
a rapid first shot image classification through a standard stack of feedforward filters, the feedback
can actively alter the importance of certain filters ?in hindsight?, correcting the initial guess via
additional internal ?thoughts?.
7
DasNet successfully learned to correct image misclassifications produced by a fully trained feedforward Maxout network. Its active, selective, internal spotlight of attention enabled state-of-the-art
results. Future research will also consider more complex actions that spatially focus on (or alter)
parts of observed images.
Acknowledgments
We acknowledge Matthew Luciw for his discussions and for providing a short literature review,
included in the Related Work section.
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4,723 | 5,277 | Iterative Neural Autoregressive Distribution
Estimator (NADE-k)
Tapani Raiko
Aalto University
Li Yao
Universit?e de Montr?eal
KyungHyun Cho
Universit?e de Montr?eal
Yoshua Bengio
Universit?e de Montr?eal,
CIFAR Senior Fellow
Abstract
Training of the neural autoregressive density estimator (NADE) can be viewed as
doing one step of probabilistic inference on missing values in data. We propose
a new model that extends this inference scheme to multiple steps, arguing that
it is easier to learn to improve a reconstruction in k steps rather than to learn to
reconstruct in a single inference step. The proposed model is an unsupervised
building block for deep learning that combines the desirable properties of NADE
and multi-prediction training: (1) Its test likelihood can be computed analytically,
(2) it is easy to generate independent samples from it, and (3) it uses an inference
engine that is a superset of variational inference for Boltzmann machines. The
proposed NADE-k is competitive with the state-of-the-art in density estimation
on the two datasets tested.
1
Introduction
Traditional building blocks for deep learning have some unsatisfactory properties. Boltzmann machines are, for instance, difficult to train due to the intractability of computing the statistics of the
model distribution, which leads to the potentially high-variance MCMC estimators during training
(if there are many well-separated modes (Bengio et al., 2013)) and the computationally intractable
objective function. Autoencoders have a simpler objective function (e.g., denoising reconstruction
error (Vincent et al., 2010)), which can be used for model selection but not for the important choice
of the corruption function. On the other hand, this paper follows up on the Neural Autoregressive
Distribution Estimator (NADE, Larochelle and Murray, 2011), which specializes previous neural
auto-regressive density estimators (Bengio and Bengio, 2000) and was recently extended (Uria et al.,
2014) to deeper architectures. It is appealing because both the training criterion (just log-likelihood)
and its gradient can be computed tractably and used for model selection, and the model can be
trained by stochastic gradient descent with backpropagation. However, it has been observed that the
performance of NADE has still room for improvement.
The idea of using missing value imputation as a training criterion has appeared in three recent papers. This approach can be seen either as training an energy-based model to impute missing values
well (Brakel et al., 2013), as training a generative probabilistic model to maximize a generalized
pseudo-log-likelihood (Goodfellow et al., 2013), or as training a denoising autoencoder with a masking corruption function (Uria et al., 2014). Recent work on generative stochastic networks (GSNs),
which include denoising auto-encoders as special cases, justifies dependency networks (Heckerman et al., 2000) as well as generalized pseudo-log-likelihood (Goodfellow et al., 2013), but have
the disadvantage that sampling from the trained ?stochastic fill-in? model requires a Markov chain
(repeatedly resampling some subset of the values given the others). In all these cases, learning
progresses by back-propagating the imputation (reconstruction) error through inference steps of the
model. This allows the model to better cope with a potentially imperfect inference algorithm. This
learning-to-cope was introduced recently in 2011 by Stoyanov et al. (2011) and Domke (2011).
1
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Figure 1: The choice of a structure for NADE-k is very flexible. The dark filled halves indicate that
a part of the input is observed and fixed to the observed values during the iterations. Left: Basic
structure corresponding to Equations (6?7) with n = 2 and k = 2. Middle: Depth added as in
NADE by Uria et al. (2014) with n = 3 and k = 2. Right: Depth added as in Multi-Prediction Deep
Boltzmann Machine by Goodfellow et al. (2013) with n = 2 and k = 3. The first two structures are
used in the experiments.
The NADE model involves an ordering over the components of the data vector. The core of the
model is the reconstruction of the next component given all the previous ones. In this paper we
reinterpret the reconstruction procedure as a single iteration in a variational inference algorithm,
and we propose a version where we use k iterations instead, inspired by (Goodfellow et al., 2013;
Brakel et al., 2013). We evaluate the proposed model on two datasets and show that it outperforms
the original NADE (Larochelle and Murray, 2011) as well as NADE trained with the order-agnostic
training algorithm (Uria et al., 2014).
2
Proposed Method: NADE-k
We propose a probabilistic model called NADE-k for D-dimensional binary data vectors x. We start
by defining p? for imputing missing values using a fully factorial conditional distribution:
Y
p? (xmis | xobs ) =
p? (xi | xobs ),
(1)
i?mis
where the subscripts mis and obs denote missing and observed components of x. From the conditional distribution p? we compute the joint probability distribution over x given an ordering o (a
permutation of the integers from 1 to D) by
p? (x | o) =
D
Y
p? (xod | xo<d ),
(2)
d=1
where o<d stands for indices o1 . . . od?1 .
The model is trained to minimize the negative log-likelihood averaged over all possible orderings o
L(?) = Eo?D! [Ex?data [? log p? (x | o)]] .
(3)
using an unbiased, stochastic estimator of L(?)
D
log p? (xo?d | xo<d )
(4)
D?d+1
by drawing o uniformly from all D! possible orderings and d uniformly from 1 . . . D (Uria et al.,
2014). Note that while the model definition in Eq. (2) is sequential in nature, the training criterion
(4) involves reconstruction of all the missing values in parallel. In this way, training does not involve
picking or following specific orders of indices.
?
L(?)
=?
In this paper, we define the conditional model p? (xmis | xobs ) using a deep feedforward neural
network with nk layers, where we use n weight matrices k times. This can also be interpreted as
running k successive inference steps with an n-layer neural network.
The input to the network is
vh0i = m Ex?data [x] + (1 ? m) x
(5)
where m is a binary mask vector indicating missing components with 1, and is an elementwise multiplication. Ex?data [x] is an empirical mean of the observations. For simplicity, we give
2
Figure 2: The inner working mechanism of NADE-k. The left most column shows the data vectors x,
the second column shows their masked version and the subsequent columns show the reconstructions
vh0i . . . vh10i (See Eq. (7)).
equations for a simple structure with n = 2. See Fig. 1 (left) for the illustration of this simple
structure.
In this case, the activations of the layers at the t-th step are
hhti = ?(Wvht?1i + c)
(6)
vhti = m ?(Vhhti + b) + (1 ? m) x
(7)
where ? is an element-wise nonlinearity, ? is a logistic sigmoid function, and the iteration index t
runs from 1 to k. The conditional probabilities of the variables (see Eq. (1)) are read from the output
vhki as
hki
p? (xi = 1 | xobs ) = vi .
(8)
Fig. 2 shows examples of how vhti evolves over iterations, with the trained model.
The parameters ? = {W, V, c, b} can be learned by stochastic gradient descent to minimize ?L(?)
?
in Eq. (3), or its stochastic approximation ?L(?)
in Eq. (4), with the stochastic gradient computed
by back-propagation.
Once the parameters ? are learned, we can define a mixture model by using a uniform probability
over a set of orderings O. We can compute the probability of a given vector x as a mixture model
1 X
pmixt (x | ?, O) =
p? (x | o)
(9)
|O|
o?O
with Eq. (2). We can draw independent samples from the mixture by first drawing an ordering o and
then sequentially drawing each variable using xod ? p? (xod | xo<d ). Furthermore, we can draw
samples from the conditional p(xmis | xobs ) easily by considering only orderings where the observed
indices appear before the missing ones.
Pretraining It is well known that training deep networks is difficult without pretraining, and in our
experiments, we train networks up to kn = 7 ? 3 = 21 layers. When pretraining, we train the model
to produce good reconstructions vhti at each step t = 1 . . . k. More formally, in the pretraining
phase, we replace Equations (4) and (8) by
L?pre (?) = ?
k
Y hti
1X
D
log
p? (xi | xo<d )
D ? d + 1 k t=1
i?o
(10)
?d
hti
hti
p? (xi = 1 | xobs ) = vi .
2.1
(11)
Related Methods and Approaches
Order-agnostic NADE The proposed method follows closely the order-agnostic version of
NADE (Uria et al., 2014), which may be considered as the special case of NADE-k with k = 1. On
the other hand, NADE-k can be seen as a deep NADE with some specific weight sharing (matrices
W and V are reused for different depths) and gating in the activations of some layers (See Equation
(7)).
3
Additionally, Uria et al. (2014) found it crucial to give the mask m as an auxiliary input to the
network, and initialized missing values to zero instead of the empirical mean (See Eq. (5)). Due to
these differences, we call their approach NADE-mask. One should note that NADE-mask has more
parameters due to using the mask as a separate input to the network, whereas NADE-k is roughly k
times more expensive to compute.
Probabilistic Inference Let us consider the task of missing value imputation in a probabilistic
latent variable model. We get the conditional probability of interest by marginalizing out the latent
variables from the posterior distribution:
Z
p(xmis | xobs ) =
p(h, xmis | xobs )dh.
(12)
h
Accessing the joint distribution p(h, xmis | xobs ) directly is often harder than alternatively updating
h and xmis based on the conditional distributions p(h | xmis , xobs ) and p(xmis | h).1 Variational
inference is one of the representative examples that exploit this.
In variational inference, a factorial distribution q(h, xmis ) = q(h)q(xmis ) is iteratively fitted to
p(h, xmis | xobs ) such that the KL-divergence between q and p
Z
p(h, xmis | xobs )
KL[q(h, xmis )||p(h, xmis | xobs )] = ?
q(h, xmis ) log
dhdxmis
(13)
q(h, xmis )
h,xmis
is minimized. The algorithm alternates between updating q(h) and q(xmis ), while considering the
other one fixed.
As an example, let us consider a restricted Boltzmann machine (RBM) defined by
p(v, h) ? exp(b> v + c> h + h> Wv).
(14)
We can fit an approximate posterior distribution parameterized as q(vi = 1) = v?i and q(hj = 1) =
? j to the true posterior distribution by iteratively computing
h
? ? ?(W?
h
v + c)
(15)
? ? m ?(W> h + b) + (1 ? m) v.
v
(16)
>
We notice the similarity to Eqs. (6)?(7): If we assume ? = ? and V = W , the inference in the
NADE-k is equivalent to performing k iterations of variational inference on an RBM for the missing
values (Peterson and Anderson, 1987). We can also get variational inference on a deep Boltzmann
machine (DBM) using the structure in Fig. 1 (right).
Multi-Prediction Deep Boltzmann Machine Goodfellow et al. (2013) and Brakel et al. (2013)
use backpropagation through variational inference steps to train a deep Boltzmann machine. This
is very similar to our work, except that they approach the problem from the view of maximizing
the generalized pseudo-likelihood (Huang and Ogata, 2002). Also, the deep Boltzmann machine
lacks the tractable probabilistic interpretation similar to NADE-k (See Eq. (2)) that would allow
to compute a probability or to generate independent samples without resorting to a Markov chain.
Also, our approach is somewhat more flexible in the choice of model structures, as can be seen in
Fig. 1. For instance, in the proposed NADE-k, encoding and decoding weights do not have to be
shared and any type of nonlinear activations, other than a logistic sigmoid function, can be used.
Product and Mixture of Experts One could ask what would happen if we would define an ensemble
likelihood along the line of the training criterion in Eq. (3). That is,
? log pprod (x | ?) ? Eo?D! [? log p(x | ?, o)] .
(17)
Maximizing this ensemble likelihood directly will correspond to training a product-of-experts
model (Hinton, 2000). However, this requires us to evaluate the intractable normalization constant
during training as well as in the inference, making the model not tractable anymore.
On the other hand, we may consider using the log-probability of a sample under the mixture-ofexperts model as the training criterion
? log pmixt (x | ?) = ? log Eo?D! [p(x | ?, o)] .
(18)
This criterion resembles clustering, where individual models may specialize in only a fraction of the
data. In this case, however, the simple estimator such as in Eq. (4) would not be available.
1
We make a typical assumption that observations are mutually independent given the latent variables.
4
Model
NADE 1HL(fixed order)
NADE 1HL
NADE 2HL
NADE-mask 1HL
NADE-mask 2HL
NADE-mask 4HL
EoNADE-mask 1HL(128 Ords)
EoNADE-mask 2HL(128 Ords)
Log-Prob.
-88.86
-99.37
-95.33
-92.17
-89.17
-89.60
-87.71
-85.10
Model
RBM (500h, CD-25)
DBN (500h+2000h)
DARN (500h)
DARN (500h, adaNoise)
NADE-5 1HL
NADE-5 2HL
EoNADE-5 1HL(128 Ords)
EoNADE-5 2HL(128 Ords)
Log-Prob.
? -86.34
? -84.55
? -84.71
? -84.13
-90.02
-87.14
-86.23
-84.68
Table 1: Results obtained on MNIST using various models and number of hidden layers (1HL
or 2HL). ?Ords? is short for ?orderings?. These are the average log-probabilities of the test set.
EoNADE refers to the ensemble probability (See Eq. (9)). From here on, in all figures and tables we
use ?HL? to denote the number of hidden layers and ?h? for the number of hidden units.
3
Experiments
We study the proposed model with two datasets: binarized MNIST handwritten digits and Caltech
101 silhouettes.
We train NADE-k with one or two hidden layers (n = 2 and n = 3, see Fig. 1, left and middle)
with a hyperbolic tangent as the activation function ?(?). We use stochastic gradient descent on
the training set with a minibatch size fixed to 100. We use AdaDelta (Zeiler, 2012) to adaptively
choose a learning rate for each parameter update on-the-fly. We use the validation set for earlystopping and to select the hyperparameters. With the best model on the validation set, we report the
log-probability computed on the test set. We have made our implementation available2 .
3.1
MNIST
We closely followed the procedure used by Uria et al. (2014), including the split of the dataset into
50,000 training samples, 10,000 validation samples and 10,000 test samples. We used the same
version where the data has been binarized by sampling.
We used a fixed width of 500 units per hidden layer. The number of steps k was selected among
{1, 2, 4, 5, 7}. According to our preliminary experiments, we found that no separate regularization
was needed when using a single hidden layer, but in case of two
hidden layers, we used weight
decay with the regularization constant in the interval e?5 , e?2 . Each model was pretrained for
1000 epochs and fine-tuned for 1000 epochs in the case of one hidden layer and 2000 epochs in the
case of two.
For both NADE-k with one and two hidden layers, the validation performance was best with k = 5.
The regularization constant was chosen to be 0.00122 for the two-hidden-layer model.
Results We report in Table 1 the mean of the test log-probabilities averaged over randomly selected
orderings. We also show the experimental results by others from (Uria et al., 2014; Gregor et al.,
2014). We denote the model proposed in (Uria et al., 2014) as a NADE-mask.
From Table 1, it is clear that NADE-k outperforms the corresponding NADE-mask both with the
individual orderings and ensembles over orderings using both 1 or 2 hidden layers. NADE-k with
two hidden layers achieved the generative performance comparable to that of the deep belief network
(DBN) with two hidden layers.
Fig. 3 shows training curves for some of the models. We can see that the NADE-1 does not perform
as well as NADE-mask. This confirms that in the case of k = 1, the auxiliary mask input is indeed
useful. Also, we can note that the performance of NADE-5 is still improving at the end of the
preallocated 2000 epochs, further suggesting that it may be possible to obtain a better performance
simply by training longer.
2
[email protected]:yaoli/nade k.git
5
120
?90
115
training cost
110
105
100
95
90
?94
end of pretrain
?96
NADE-mask 1HL
NADE-5 1HL
NADE-1 1HL
?98
end of pretrain
85
80
?92
testset log-probability
NADE-mask 1HL
NADE-5 1HL
NADE-1 1HL
0
500
1000
?100
200
1500
training epochs
400
600
(a)
800
1000
1200
1400
training epochs
1600
1800
2000
(b)
Figure 3: NADE-k with k steps of variational inference helps to reduce the training cost (a) and to
generalize better (b). NADE-mask performs better than NADE-1 without masks both in training and
test.
?87
?85
?90
?89
?90
?91
?92
?93
NADE-k 1HL
NADE-k 2HL
NADE-mask 1HL
NADE-mask 2HL
?94
?95
?96
1
2
4
5
trained with k steps of iterations
testset log-probability
testset log-probability
?88
?95
?100
?105
?110
?115
7
(a)
NADE-5 2HL
NADE-mask 2HL
0
5
10
15
perform k steps of iterations at test time
20
(b)
Figure 4: (a) The generalization performance of different NADE-k models trained with different k.
(b) The generalization performance of NADE-5 2h, trained with k=5, but with various k in test time.
Fig. 4 (a) shows the effect of the number of iterations k during training. Already with k = 2, we can
see that the NADE-k outperforms its corresponding NADE-mask. The performance increases until
k = 5. We believe the worse performance of k = 7 is due to the well known training difficulty of a
deep neural network, considering that NADE-7 with two hidden layers effectively is a deep neural
network with 21 layers.
At inference time, we found that it is important to use the exact k that one used to train the model.
As can be seen from Fig. 4 (b), the assigned probability increases up to the k, but starts decreasing
as the number of iterations goes over the k. 3
3.1.1
Qualitative Analysis
In Fig. 2, we present how each iteration t = 1 . . . k improves the corrupted input (vhti from Eq. (5)).
We also investigate what happens with test-time k being larger than the training k = 5. We can see
that in all cases, the iteration ? which is a fixed point update ? seems to converge to a point that is
in most cases close to the ground-truth sample. Fig. 4 (b) shows however that the generalization
performance drops after k = 5 when training with k = 5. From Fig. 2, we can see that the
reconstruction continues to be sharper even after k = 5, which seems to be the underlying reason
for this phenomenon.
3
In the future, one could explore possibilities for helping better converge beyond step k, for instance by
using costs based on reconstructions at k ? 1 and k even in the fine-tuning phase.
6
(b) Caltech-101 Silhouettes
(a) MNIST
Figure 5: Samples generated from NADE-k trained on (a) MNIST and (b) Caltech-101 Silhouettes.
(b)
(a)
Figure 6: Filters learned from NADE-5 2HL. (a) A random subset of the encodering filters. (b) A
random subset of the decoding filters.
From the samples generated from the trained NADE-5 with two hidden layers shown in Fig. 5 (a),
we can see that the model is able to generate digits. Furthermore, the filters learned by the model
show that it has learned parts of digits such as pen strokes (See Fig. 6).
3.1.2
Variability over Orderings
In Section 2, we argued that we can perform any inference task p(xmis | xobs ) easily and efficiently
by restricting the set of orderings O in Eq. (9) to ones where xobs is before xmis . For this to work
well, we should investigate how much the different orderings vary.
To measure the variability over orderings, we computed the variance of log p(x | o) for 128 randomly chosen orderings o with the trained NADE-k?s and NADE-mask with a single hidden layer.
For comparison, we computed the variance of log p(x | o) over the 10,000 test samples.
p
p
log p(x | o)
Eo,x [?]
Ex Varo [?]
Eo Varx [?]
Table 2: The variance of
NADE-mask 1HL -92.17
3.5
23.5
log p(x | o) over orderings o
NADE-5 1HL
-90.02
3.1
24.2
and over test samples x.
NADE-5 2HL
-87.14
2.4
22.7
In Table 2, the variability over the orderings is clearly much smaller than that over the samples.
Furthermore, the variability over orderings tends to decrease with the better models.
3.2
Caltech-101 silhouettes
We also evaluate the proposed NADE-k on Caltech-101 Silhouettes (Marlin et al., 2010), using
the standard split of 4100 training samples, 2264 validation samples and 2307 test samples. We
demonstrate the advantage of NADE-k compared with NADE-mask under the constraint that they
have a matching number of parameters. In particular, we compare NADE-k with 1000 hidden
units with NADE-mask with 670 hiddens. We also compare NADE-k with 4000 hidden units with
NADE-mask with 2670 hiddens.
We optimized the hyper-parameter k ? {1, 2, . . . , 10} in the case of NADE-k. In both NADE-k
and NADE-mask, we experimented without regularizations, with weight decays, or with dropout.
Unlike the previous experiments, we did not use the pretraining scheme (See Eq. (10)).
7
Table 3: Average log-probabilities of test samples of Caltech-101 Silhouettes. (?) The results are
from Cho et al. (2013). The terms in the parenthesis indicate the number of hidden units, the total
number of parameters (M for million), and the L2 regularization coefficient. NADE-mask 670h
achieves the best performance without any regularizations.
Model
RBM?
(2000h, 1.57M)
NADE-mask
(670h, 1.58M)
NADE-2
(1000h, 1.57M, L2=0.0054)
Test LL
-108.98
Model
RBM ?
(4000h, 3.14M)
NADE-mask
(2670h, 6.28M, L2=0.00106)
NADE-5
(4000h, 6.28M, L2=0.0068)
-112.51
-108.81
Test LL
-107.78
-110.95
-107.28
As we can see from Table 3, NADE-k outperforms the NADE-mask regardless of the number of
parameters. In addition, NADE-2 with 1000 hidden units matches the performance of an RBM with
the same number of parameters. Futhermore, NADE-5 has outperformed the previous best result
obtained with the RBMs in (Cho et al., 2013), achieving the state-of-art result on this dataset. We
can see from the samples generated by the NADE-k shown in Fig. 5 (b) that the model has learned
the data well.
4
Conclusions and Discussion
In this paper, we proposed a model called iterative neural autoregressive distribution estimator
(NADE-k) that extends the conventional neural autoregressive distribution estimator (NADE) and its
order-agnostic training procedure. The proposed NADE-k maintains the tractability of the original
NADE while we showed that it outperforms the original NADE as well as similar, but intractable
generative models such as restricted Boltzmann machines and deep belief networks.
The proposed extension is inspired from the variational inference in probabilistic models such as
restricted Boltzmann machines (RBM) and deep Boltzmann machines (DBM). Just like an iterative
mean-field approximation in Boltzmann machines, the proposed NADE-k performs multiple iterations through hidden layers and a visible layer to infer the probability of the missing value, unlike
the original NADE which performs the inference of a missing value in a single iteration through
hidden layers.
Our empirical results show that this approach of multiple iterations improves the performance of
a model that has the same number of parameters, compared to performing a single iteration. This
suggests that the inference method has significant effect on the efficiency of utilizing the model
parameters. Also, we were able to observe that the generative performance of NADE can come
close to more sophisticated models such as deep belief networks in our approach.
In the future, more in-depth analysis of the proposed NADE-k is needed. For instance, a relationship between NADE-k and the related models such as the RBM need to be both theoretically and
empirically studied. The computational speed of the method could be improved both in training (by
using better optimization algorithms. See, e.g., (Pascanu and Bengio, 2014)) and in testing (e.g. by
handling the components in chunks rather than fully sequentially). The computational efficiency of
sampling for NADE-k can be further improved based on the recent work of Yao et al. (2014) where
an annealed Markov chain may be used to efficiently generate samples from the trained ensemble.
Another promising idea to improve the model performance further is to let the model adjust its own
confidence based on d. For instance, in the top right corner of Fig. 2, we see a case with lots of missing values values (low d), where the model is too confident about the reconstructed digit 8 instead
of the correct digit 2.
Acknowledgements
The authors would like to acknowledge the support of NSERC, Calcul Qu?ebec, Compute Canada,
the Canada Research Chair and CIFAR, and developers of Theano (Bergstra et al., 2010; Bastien
et al., 2012).
8
References
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9
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4,724 | 5,278 | General Stochastic Networks for Classification
Matthias Z?ohrer and Franz Pernkopf
Signal Processing and Speech Communication Laboratory
Graz University of Technology
[email protected], [email protected]
Abstract
We extend generative stochastic networks to supervised learning of representations. In particular, we introduce a hybrid training objective considering a generative and discriminative cost function governed by a trade-off parameter ?. We use
a new variant of network training involving noise injection, i.e. walkback training, to jointly optimize multiple network layers. Neither additional regularization
constraints, such as `1, `2 norms or dropout variants, nor pooling- or convolutional layers were added. Nevertheless, we are able to obtain state-of-the-art performance on the MNIST dataset, without using permutation invariant digits and
outperform baseline models on sub-variants of the MNIST and rectangles dataset
significantly.
1
Introduction
Since 2006 there has been a boost in machine learning due to improvements in the field of unsupervised learning of representations. Most accomplishments originate from variants of restricted
Boltzmann machines (RBMs) [1], auto-encoders (AE) [2, 3] and sparse-coding [4, 5, 6]. Deep models in representation learning, also obtain impressive results in supervised learning problems, such
as speech recognition, e.g. [7, 8, 9] and computer vision tasks [10].
If no a-priori knowledge is modeled in the architecture, cf. convolutional layers or pooling layers
[11], generatively pre-trained networks are among the best when applied to supervised learning tasks
[12]. Usually, a generative representation is obtained through a greedy-layerwise training procedure
called contrastive divergence (CD) [1]. In this case, the network layer learns the representation from
the layer below by treating the latter as static input. Despite of the impressive results achieved with
CD, we identify two (minor) drawbacks when used for supervised learning: Firstly, after obtaining
a representation by pre-training a network, a new discriminative model is initialized with the trained
weights, splitting the training into two separate models. This seems to be neither biologically plausible, nor optimal when it comes to optimization, as carefully designed early stopping criteria have to
be implemented to prevent over- or under-fitting. Secondly, generative and discriminative objectives
might influence each other beneficially when combined during training. CD does not take this into
account.
In this work, we introduce a new training procedure for supervised learning of representations. In
particular we define a hybrid training objective for general stochastic networks (GSN), dividing the
cost function into a generative and discriminative part, controlled by a trade-off parameter ?. It turns
out that by annealing ?, when solving this unconstrained non-convex multi-objective optimization
problem, we do not suffer from the shortcomings described above. We are able to obtain stateof-the-art performance on the MNIST [13] dataset, without using permutation invariant digits and
significantly outperform baseline models on sub-variants of the MNIST and rectangle database [14].
Our approach is related to the generative-discriminative training approach of RBMs [15]. However
a different model and a new variant of network training involving noise injection, i.e. walkback
training [16, 17], is used to jointly optimize multiple network layers. Most notably, we did not
1
apply any additional regularization constraints, such as `1, `2 norms or dropout variants [12], [18],
unlocking further potential for possible optimizations. The model can be extended to learn multiple
tasks at the same time using jointly trained weights and by introducing multiple objectives. This
might also open a new prospect in the field of transfer learning [19] and multi-task learning [20]
beyond classification.
This paper is organized as follows: Section 2 presents mathematical background material i.e. the
GSN and a hybrid learning criterion. In Section 3 we empirically study the influence of hyper
parameters of GSNs and present experimental results. Section 4 concludes the paper and provides a
perspective on future work.
2
General Stochastic Networks
Recently, a new supervised learning algorithm called walkback training for generalized autoencoders (GAE) was introduced [16]. A follow-up study [17] defined a new network model ?
generative stochastic networks, extending the idea of walkback training to multiple layers. When
applied to image reconstruction, they were able to outperform various baseline systems, due to its
ability to learn multi-modal representations [17, 21]. In this paper, we extend the work of [17].
First, we provide mathematical background material for generative stochastic networks. Then, we
introduce modifications to make the model suitable for supervised learning. In particular we present
a hybrid training objective, dividing the cost into a generative and discriminative part. This paves
the way for any multi-objective learning of GSNs. We also introduce a new terminology, i.e. general stochastic networks, a model class including generative-, discriminative- and hybrid stochastic
network variants.
General Stochastic Networks for Unsupervised Learning
Restricted Boltzmann machines (RBM) [22] and denoising autoencoders (DAE) [3] share the following commonality; The input distribution P (X) is sampled to convergence in a Markov chain.
In the case of the DAE, the transition operator first samples the hidden state Ht from a corruption
distribution C(H|X), and generates a reconstruction from the parametrized model, i.e the density
P?2 (X|H).
Ht+1
P? 1
Ht+2
P?1
P?2
Xt+0
Ht+3
P?1
P?2
Xt+1
Ht+4
P?2
Xt+2
P? 1
P?1
P?2
Xt+3
Xt+4
Figure 1: DAE Markov chain.
The resulting DAE Markov chain, shown in Figure 1, is defined as
Ht+1 ? P?1 (H|Xt+0 ) and Xt+1 ? P?2 (X|Ht+1 ),
(1)
where Xt+0 is the input sample X, fed into the chain at time step 0 and Xt+1 is the reconstruction
of X at time step 1. In the case of a GSN, an additional dependency between the latent variables Ht
over time is introduced to the network graph. The GSN Markov chain is defined as follows:
Ht+1 ? P?1 (H|Ht+0 , Xt+0 ) and Xt+1 ? P?2 (X|Ht+1 ).
(2)
Figure 2 shows the corresponding network graph.
This chain can be expressed with deterministic functions of random variables f? ? {f?? , f?? }. In
particular, the density f? is used to model Ht+1 = f? (Xt+0 , Zt+0 , Ht+0 ), specified for some independent noise source Zt+0 , with the condition that Xt+0 cannot be recovered exactly from Ht+1 .
2
Ht+0
Ht+1
Ht+2
P?1
Ht+3
P?1
P?2
Xt+0
Ht+4
P?1
P?2
Xt+1
P?1
P?1
P?2
Xt+2
P? 2
Xt+3
Xt+4
Figure 2: GSN Markov chain.
We introduce f??i as a back-probable stochastic non-linearity of the form f??i = ?out + g(?in + a
?i )
with noise processes Zt ? {?in , ?out } for layer i. The variable a
?i is the activation for unit i, where
a
?i = W i Iti + bi with a weight matrix W i and bias bi , representing the parametric distribution. It is
embedded in a non-linear activation function g. The input Iti is either the realization xit of observed
sample Xti or the hidden realization hit of Hti . In general, f??i (Iti ) specifies an upward path in a GSN
i
for a specific layer i. In the case of Xt+1
= f??i (Zt+0 , Ht+1 ) we define f??i (Hti ) = ?out + g(?in + a
?i )
i
i
i T
i
as a downward path in the network i.e. a
? = (W ) Ht + b , using the transpose of the weight
matrix W i and the bias bi . This formulation allows to directly back-propagate the reconstruction log-likelihood P (X|H) for all parameters ? ? {W 0 , ..., W d , b0 , ..., bd } where d is the
number of hidden layers. In Figure 2 the GSN includes a simple hidden layer. This can be
extended to multiple hidden layers requiring multiple deterministic functions of random variables
f? ? {f??0 , ..., f??d , f??0 , ...f??d }.
Figure 3 visualizes the Markov chain for a multi-layer GSN, inspired by the unfolded computational
graph of a deep Boltzmann machine Gibbs sampling process.
3
Ht+3
3
Ht+4
f??2
f??2
2
Ht+2
f??1
f??1
f??1
f??1
1
Ht+3
f??0
f??2
2
Ht+4
f??1
1
Ht+2
f??0
f??0
f??2
2
Ht+3
f??1
1
Ht+1
f??0
f??2
f??0
f??1
1
Ht+4
f??0
f??0
f??0
f??0
0
Xt+0
0
Xt+1
0
Xt+2
0
Xt+3
0
Xt+4
Xt+0
0
Lt {Xt+1
, Xt+0 }
0
Lt {Xt+2
, Xt+0 }
0
Lt {Xt+3
, Xt+0 }
0
Lt {Xt+4
, Xt+0 }
Figure 3: GSN Markov chain with multiple layers and backprop-able stochastic units.
In the training case, alternatively even or odd layers are updated at the same time. The information
is propagated both upwards and downwards for K steps allowing the network to build higher order
representations. An example for this update process is given in Figure 3. In the even update (marked
2
1
0
0
1
= f??0 (Ht+1
) and Ht+2
=
in red) Ht+1
= f??0 (Xt+0
). In the odd update (marked in blue) Xt+1
?
?
?
?
2
1
1
0
2
3
0
1
2
1
f? (Ht+1 ) for k = 0. In the case of k = 1, Ht+2 = f? (Xt+1 ) + f? (Ht+2 ) and Ht+3 = f? (Ht+2 )
1
2
1
3
0
= f??0 (Ht+2
) and Ht+3
= f??1 (Ht+2
) + f??2 (Ht+3
) in the odd update.
in the even update and Xt+2
?
?
?
1
0
2
3
2
0
1
2
In case of k = 2, Ht+3 = f? (Xt+2 ) + f? (Ht+3 ) and Ht+4 = f? (Ht+3 ) in the even update and
0
1
2
1
3
Xt+3
= f??0 (Ht+3
) and Ht+4
= f??1 (Ht+3
) + f??2 (Ht+4
) in the odd update.
The cost function of a generative GSN can be written as:
C=
K
X
0
Lt {Xt+k
, Xt+0 },
k=1
3
(3)
Lt is a specific loss-function such as the mean squared error (MSE) at time step t. In general any
0
arbitrary loss function could be used (as long as they can be seen as a log-likelihood) [16]. Xt+k
0
is the reconstruction of the input Xt+0 at layer 0 after k steps. Optimizing the loss function by
building the sum over the costs of multiple corrupted reconstructions is called walkback training
[16, 17]. This form of network training leads to a significant performance boost when used for input
reconstruction. The network is able to handle multi-modal input representations and is therefore
considerably more favorable than standard generative models [16].
General Stochastic Networks for Supervised Learning
In order to make a GSN suitable for a supervised learning task we introduce the output Y to the
network graph. In this case L = log P (X) + log P (Y |X). Although the target Y is not fed into the
network, it is introduced as an additional cost term. The layer update-process stays the same.
3
Lt {Ht+1
, Yt+0 }
3
Lt {Ht+2
, Yt+0 }
3
Ht+3
3
Ht+4
f??2
f??2
2
Ht+2
f??1
f??1
f??0
f??1
f??1
1
Ht+3
f??0
f??2
2
Ht+4
f??1
1
Ht+2
f??0
f??2
2
Ht+3
f??1
1
Ht+1
f??0
f??2
f??0
f??1
1
Ht+4
f??0
f??0
f??0
f??0
0
Xt+0
0
Xt+1
0
Xt+2
0
Xt+3
0
Xt+4
Xt+0
0
Lt {Xt+1
, Xt+0 }
0
Lt {Xt+2
, Xt+0 }
0
Lt {Xt+3
, Xt+0 }
0
Lt {Xt+4
, Xt+0 }
Figure 4: GSN Markov chain for input Xt+0 and target Yt+0 with backprop-able stochastic units.
We define the following cost function for a 3-layer GSN:
C=
K
K
1?? X
? X
3
Lt {Xt+k , Xt+0 } +
Lt {Ht+k
, Yt+0 }
K
K ?d+1
k=1
k=d
|
{z
} |
{z
}
generative
discriminative
(4)
This is a non-convex multi-objective optimization problem, where ? weights the generative and
discriminative part of C. The parameter d specifies the number of network layers i.e. depth of the
network. Scaling the mean loss in (4) is not mandatory, but allows to equally balance both loss terms
with ? = 0.5 for input Xt+0 and target Yt+0 scaled to the same range. Again Figure 4 shows the
corresponding network graph for supervised learning with red and blue edges denoting the even and
odd network updates.
In general the hybrid objective optimization criterion is not restricted to hX, Y i, as additional input
and output terms could be introduced to the network. This setup might be useful for transfer-learning
[19] or multi-task scenarios [20], which is not discussed in this paper.
3
Experimental Results
In order to evaluate the capabilities of GSNs for supervised learning, we studied MNIST digits
[13], variants of MNIST digits [14] and the rectangle datasets [14]. The first database consists of
60.000 labeled training and 10.000 labeled test images of handwritten digits. The second dataset includes 6 variants of MNIST digits, i.e. { mnist-basic, mnist-rot, mnist-back-rand, mnist-back-image,
mnist-rot-back-image }, with additional factors of variation added to the original data. Each variant
includes 10.000 labeled training, 2000 labeled validation, and 50.000 labeled test images. The third
dataset involves two subsets, i.e. { rectangle, rectangle-image }. The dataset rectangle consists of
4
1000 labeled training, 200 labeled validation, and 50.000 labeled test images. The dataset rectangleimage includes 10.000 labeled train, 2000 labeled validation and 50.000 labeled test images.
In a first experiment we focused on the multi-objective optimization problem defined in (4). Next we
evaluated the number of walkback steps in a GSN, necessary for convergence. In a third experiment
we analyzed the influence of different Gaussian noise settings during walkback training, improving
the generalization capabilities of the network. Finally we summarize classification results for all
datasets and compare to baseline systems [14].
3.1
Multi-Objective Optimization in a Hybrid Learning Setup
In order to solve the non-convex multi-objective optimization problem, variants of stochastic gradient descent (SGD) can be used. We applied a search over fixed ? values on all problems. Furthermore, we show that the use of an annealed ? factor, during training works best in practice.
In all experiments a three layer GSN, i.e. GSN-3, with 2000 neurons in each layer, randomly initialized with small Gaussian noise, i.e. 0.01 ? N (0, 1), and an MSE loss function for both inputs and
targets was used. Regarding optimization we applied SGD with a learning rate ? = 0.1, a momentum term of 0.9 and a multiplicative annealing factor ?n+1 = ?n ? 0.99 per epoch n for the learning
rate. A rectifier unit [23] was chosen as activation function. Following the ideas of [24] no explicit
sampling was applied at the input and output layer. In the test case the zero-one loss was computed
averaging the network?s output over k walkback steps.
Analysis of the Hybrid Learning Parameter ?
Concerning the influence of the trade-off parameter ?, we tested fixed ? values in the range
? ? {0.01, 0.1, 0.2, ..., 0.9, 0.99}, where low values emphasize the discriminative part in the objective and vice versa. Walkback training with K = 6 steps using zero-mean pre- and postactivation Gaussian noise with zero mean and variance ? = 0.1 was performed for 500 training epochs. In a more dynamic scenario ?n=1 = 1 was annealed by ?n+1 = ?n ? ? to reach
?n=500 ? {0.01, 0.1, 0.2, ..., 0.9, 0.99} within 500 epochs, simulating generative pre-training to a
certain extend.
Figure 5: Influence of dynamic and static ? on MNIST variants basic (left), rotated (middle) and
background (right) where ? denotes the training-, 4 the validation- and 5 the test-set. The dashed
line denotes the static setup, the bold line the dynamic setup.
Figure 5 compares the results of both GSNs, using static and dynamic ? setups on the MNIST
variants basic, rotated and background. The use of a dynamic i.e. annealed ?n=500 = 0.01, achieved
the best validation and test error in all experiments. In this case, more attention was given to the
generative proportion P (X) of the objective (4) in the early stage of training. After approximately
400 epochs discriminative training i.e. fine-tuning, dominates. This setup is closely related to DBN
training, where emphasis is on optimizing P (X) at the beginning of the optimization, whereas
P (Y |X) is important at the last stages. In case of the GSN, the annealed ? achieves a more smooth
transition by shifting the weight in the optimization criterion from P (X) to P (Y |X) within one
model.
5
Analysis of Walkback Steps K
In a next experiment we tested the influence of K walkback steps for GSNs. Figure 6 shows the
results for different GSNs, trained with K ? {6, 7, 8, 9, 10} walkback steps and annealed ? with
? = 0.99. In all cases the information was at least propagated once up and once downwards in the
d = 3 layer network using fixed Gaussian pre- and post-activation noise with ? = 0 and ? = 0.1.
Figure 6: Evaluating the number of walkback steps on MNIST variants basic (left), rotated (middle)
and background (right) where ? denotes the training-, 4 the validation- and 5 the test-set.
Figure 6 shows that increasing the walkback steps, does not improve the generalization capabilities
of the used GSNs. The setup K = 2 ? d is sufficient for convergence and achieves the best validation
and test result in all experiments.
Analysis of Pre- and Post-Activation Noise
Injecting noise during the training process of GSNs serves as a regularizer and improves the generalization capabilities of the model [17]. In this experiment the influence of Gaussian pre- and
post-activation noise with ? = 0 and ? ? {0.05, 0.1, 0.15, 0.2, 0.25, 0.3} and deactivated noise
during training, was tested on a GSN-3 trained for K = 6 walkback steps. The trade-off factor
? was annealed with ? = 0.99. Figure 7 summarizes the results of the different GSNs for the
MNIST variants basic, rotated and background. Setting ? = 0.1 achieved the best overall result
on the validation- and test-set for all three experiments. In all other cases the GSNs either over- or
underfitted the data.
Figure 7: Evaluating noise injections during training on MNIST variants basic (left), rotated (middle)
and background (right) where ? denotes the training-, 4 the validation- and 5 the test-set.
3.2
MNIST results
Table 1 presents the average classification error of three runs of all MNIST variation datasets obtained by a GSN-3, using fixed Gaussian pre- and post-activation noise with ? = 0, ? = 0.1 and
K = 6 walkback steps. The hybrid learning parameter ? was annealed with ? = 0.99 and ?n=1 = 1.
A small grid test was performed in the range of N ? d with N ? {1000, 2000, 3000} neurons per
layer for d ? {1, 2, 3} layers to find the optimal network configuration.
6
Dataset
SVMrbf
SVMpoly NNet
DBN-1
SAA-3
DBN-3
GSN-3
mnist-basic
3.03
?0.15
3.69
?0.17
4.69
?0.19
3.94
?0.17
3.46
?0.16
3.11
?0.15
2.40
?0.04
mnist-rot*
11.11
?0.28
15.42
?0.32
18.11
?0.34
10.30
?0.27
10.30
?0.27
14.69
?0.31
8.66
?0.08
mnist-back-rand
14.58
?0.31
16.62
?0.33
20.04
?0.35
9.80
?0.26
11.28
?0.28
6.73
?0.22
9.38
?0.03
mnist-back-image
22.61
?0.37
24.01
?0.37
27.41
?0.39
16.15
?0.32
23.00
?0.37
16.31
?0.32
16.04
?0.04
mnist-rot-back-image*
55.18
?0.44
2.15
?0.13
56.41
?0.43
2.15
?0.13
62.16
?0.43
7.16
?0.23
47.39
?0.44
4.71
?0.19
51.93
?0.44
2.41
?0.13
52.21
?0.44
2.60
?0.14
43.86
?0.05
2.04
?0.04
24.04
?0.37
24.05
?0.37
33.20
?0.41
23.69
?0.37
24.05
?0.37
22.50
?0.37
22.10
?0.03
rectangles
rectangles-image
Table 1: MNIST variations and recangle results [14]; For datasets marked by (*) updated results are
shown [25].
Table 1 shows that a three layer GSN clearly outperforms all other models, except for the MNIST
random-background dataset. In particular, when comparing the GSN-3 to the radial basis function
support vector machine (SVMrbf), i.e. the second best model on MNIST basic, the GSN-3 achieved
an relative improvement of 20.79% on the test set. On the MNIST rotated dataset the GSN-3 was
able to beat the second best model i.e. DBN-1, by 15.92% on the test set. On the MNIST rotatedbackground there is an relative improvement of 7.25% on the test set between the second best model,
i.e. DBN-1, and the GSN-3. All results are statistically significant. Regarding the number of model
parameters, although we cannot directly compare the models in terms of network parameters, it is
worth to mention that a far smaller grid test was used to generate the results for all GSNs, cf. [14].
When comparing the classification error of the GSN-3 trained without noise, obtained in the previous
experiments (7) with Table 1, the GSN-3 achieved the test error of 2.72% on the MNIST variant
basic, outperforming all other models on this task. On the MNIST variant rotated, the GSN-3 also
outperformed the DBN-3, obtaining a test error of 11.2%. This indicates that not only the Gaussian
regularizer in the walkback training improves the generalization capabilities of the network, but also
the hybrid training criterion of the GSN.
Table 2 lists the results for the MNIST dataset without additional affine transformations applied to
the data i.e. permutation invariant digits. A three layer GSN achieved the state-of-the-art test error
of 0.80%.
Network
Result
Rectifier MLP + dropout [12]
DBM [26]
Maxout MLP + dropout [27]
MP-DBM [28]
Deep Convex Network [29]
Manifold Tangent Classifier [30]
DBM + dropout [12]
GSN-3
1.05%
0.95%
0.94%
0.91%
0.83%
0.81%
0.79%
0.80%
Table 2: MNIST results.
7
It might be worth noting that in addition to the noise process in walkback training, no other regularizers, such as `1, `2 norms and dropout variants [12], [18] were used in the GSNs. In general ? 800
training epochs with early-stopping are necessary for GSN training.
All simulations1 were executed on a GPU with the help of the mathematical expression compiler
Theano [31].
4
Conclusions and Future Work
We have extended GSNs for classification problems. In particular we defined an hybrid multiobjective training criterion for GSNs, dividing the cost function into a generative and discriminative
part. This renders the need for generative pre-training unnecessary. We analyzed the influence of
the objective?s trade-off parameter ? empirically, showing that by annealing ? we outperform a
static choice of ?. Furthermore, we discussed effects of noise injections and sampling steps during
walkback training. As a conservative starting point we restricted the model to use only rectifier
units. Neither additional regularization constraints, such as `1, `2 norms or dropout variants [12],
[18], nor pooling- [11, 32] or convolutional layers [11] were added. Nevertheless, the GSN was
able to outperform various baseline systems, in particular a deep belief network (DBN), a multi
layer perceptron (MLP), a support vector machine (SVM) and a stacked auto-associator (SSA), on
variants of the MNIST dataset. Furthermore, we also achieved state-of-the-art performance on the
original MNIST dataset without permutation invariant digits. The model not only converges faster
in terms of training iterations, but also show better generalization behavior in most cases. Our
approach opens a wide field of new applications for GSNs. In future research we explore adaptive
noise injection methods for GSNs and non-convex multi-objective optimization strategies.
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4,725 | 5,279 | Improved Multimodal Deep Learning
with Variation of Information
Kihyuk Sohn, Wenling Shang and Honglak Lee
University of Michigan Ann Arbor, MI, USA
{kihyuks,shangw,honglak}@umich.edu
Abstract
Deep learning has been successfully applied to multimodal representation learning problems, with a common strategy to learning joint representations that are
shared across multiple modalities on top of layers of modality-specific networks.
Nonetheless, there still remains a question how to learn a good association between data modalities; in particular, a good generative model of multimodal data
should be able to reason about missing data modality given the rest of data modalities. In this paper, we propose a novel multimodal representation learning framework that explicitly aims this goal. Rather than learning with maximum likelihood, we train the model to minimize the variation of information. We provide a
theoretical insight why the proposed learning objective is sufficient to estimate the
data-generating joint distribution of multimodal data. We apply our method to restricted Boltzmann machines and introduce learning methods based on contrastive
divergence and multi-prediction training. In addition, we extend to deep networks
with recurrent encoding structure to finetune the whole network. In experiments,
we demonstrate the state-of-the-art visual recognition performance on MIR-Flickr
database and PASCAL VOC 2007 database with and without text features.
1
Introduction
Different types of multiple data modalities can be used to describe the same event. For example,
images, which are often represented with pixels or image descriptors, can also be described with
accompanying text (e.g., user tags or subtitles) or audio data (e.g., human voice or natural sound).
There have been several applications of multimodal learning from multiple domains such as emotion [13] and speech [10] recognition with audio-visual data, robotics applications with visual and
depth data [15, 17, 32, 23], or medical applications with visual and temporal data [26]. These data
from multiple sources are semantically correlated, and sometimes provide complementary information to each other. In order to exchange such information, it is important to capture a high-level
association between data modalities with a compact set of latent variables. However, learning associations between multiple heterogeneous data distributions is a challenging problem.
A naive approach is to concatenate the data descriptors from different sources of input to construct a
single high-dimensional feature vector and use it to solve a unimodal representation learning problem. Unfortunately, this approach has been unsuccessful since the correlation between features in
each data modality is much stronger than that between data modalities [21]. As a result, the learning
algorithms are easily tempted to learn dominant patterns in each data modality separately while giving up learning patterns that occur simultaneously in multiple data modalities. To resolve this issue,
deep learning methods, such as deep autoencoders [9] or deep Boltzmann machines (DBM) [24],
have been used to this problem [21, 27], with a common strategy to learning joint representations that
are shared across multiple modalities at the higher layer of the deep network after learning layers of
modality-specific networks. The rationale is that the learned features may have less within-modality
correlation than raw features, and this makes it easier to capture patterns across data modalities. Despite the promise, there still remains a challenging question how to learn a good association between
multiple data modalities that can effectively deal with missing data modalities in the testing time.
One necessary condition of being a good generative model of multimodal data is to have an ability
to predict or reason about missing data modalities given partial observation. To this end, we propose
1
a novel multimodal representation learning framework that explicitly aims this goal. The key idea
is to minimize the information distance between data modalities through the shared latent representations. More concretely, we train the model to minimize the variation of information (VI), an
information theoretic measure that computes the distance between random variables, i.e., multiple
data modalities. Note that this is in contrast to the previous approaches on multimodal deep learning, which are based on maximum (joint) likelihood (ML) learning [21, 27]. We provide an intuition
how our method could be more effective in learning the joint representation of multimodal data than
ML learning, and show theoretical insights why the proposed learning objective is sufficient to estimate the data-generating joint distribution of multimodal data. We apply the proposed framework to
multimodal restricted Boltzmann machine (MRBM). We introduce two learning algorithms, based
on contrastive divergence [19] and multi-prediction training [6]. Finally, we extend to multimodal
deep recurrent neural network (MDRNN) for unsupervised finetuning of whole network. In experiments, we demonstrate the state-of-the-art visual recognition performance on MIR-Flickr database
and PASCAL VOC 2007 database with and without text features.
2
Multimodal Learning with Variation of Information
In this section, we propose a novel training objective based on the VI. We make a comparison to
the ML objective, a typical learning objective for training models of multimodal data, to give an
insight how our proposal outperforms the baseline. Finally, we establish a theorem showing that the
proposed learning objective is sufficient to obtain a good generative model that fully recovers the
joint data-generating distribution of multimodal data.
Notation. We use uppercase letters X, Y to denote random variables, lowercase letters x, y for
realizations. Let PD be the data-generating distribution and P? the model distribution parameterized by ?. For presentation clarity, we slightly abuse the notation for Q to denote conditional
(Q(x|y), Q(y|x)), marginal (Q(x), Q(y)), as well as joint distributions (Q(x, y)) that are derived
from the joint distribution Q(x, y). The type of distribution for Q should be clear from the context.
2.1
Minimum Variation of Information Learning
Motivated from the necessary condition of good generative models to reason about the missing data
modality, it seems natural to learn to maximize the amount of information that one data modality has
about the others. We quantify such an amount of information between data modalities using variation
of information (VI). The VI is an information theoretic measure that computes the information
distance between two random variables (e.g., data modalities), and is written as follows:1
(1)
VIQ (X, Y ) = ?EQ(X,Y ) log Q(X|Y ) + log Q(Y |X)
where Q(X, Y ) = P? (X, Y ) is any joint distribution on random variables (X, Y ) parametrized by
?. Informally, VI is small when the conditional LLs Q(X|Y ) and Q(Y |X) are ?peaked?, meaning
that X has low entropy conditioned on Y and vice versa. Following the intuition, we define new
multimodal learning criteria, a minimum variation of information (MinVI) learning, as follows:
(2)
MinVI: min? LVI (?), LVI (?) = ?EPD (X,Y ) log P? (X|Y ) + log P? (Y |X)
Note the difference in LVI (?) that we take the expectation over PD in LVI (?). Furthermore, we
observe that the MinVI objective can be decomposed into a sum of two negative conditional LLs.
This indeed well aligns with our initial motivation about reasoning missing data modality. In the
following, we provide a more insight of our MinVI objective in relation to the ML objective, which
is a standard learning objective in generative models.
2.2 Relation to Maximum Likelihood Learning
The ML objective function can be written as a minimization of the negative LL (NLL) as follows:
ML: min? LNLL (?), LNLL (?) = ?EPD (X,Y ) log P? (X, Y ) ,
(3)
and we can show that the NLL objective function is reformulated as follows:
2LNLL (?) = KL (PD (X)kP? (X)) + KL (PD (Y )kP? (Y )) +
|
{z
}
(a)
EPD (X) KL (PD (Y |X)kP? (Y |X)) + EPD (Y ) KL (PD (X|Y )kP? (X|Y )) + C,
|
{z
}
(4)
(b)
1
In practice, we use finite samples of the training data and use a regularizer (e.g., l2 regularizer) to avoid
overfitting to the finite sample distribution.
2
where C is a constant which is irrelevant to ?. Note that (b) is equivalent to LVI (?) in Equation (2)
up to a constant. We provide a full derivation of Equation (4) in supplementary material.
Ignoring the constant, the NLL objective is composed of four terms of KL divergence. Since KL
divergence is non-negative and is 0 only when two distributions match, the ML learning in Equation (3) can be viewed as a distribution matching problem involving (a) marginal likelihoods and (b)
conditional likelihoods. Here, we argue that (a) is more difficult to optimize than (b) because there
are often too many modes in the marginal distribution. Compared to that, the number of modes can
be dramatically reduced in the conditional distribution since the conditioning variables may restrict
the support of random variable effectively. Therefore, (a) may become a dominant factor to be minimized during the optimization process and as a trade-off, (b) will be easily compromised, which
makes it difficult to learn a good association between data modalities. On the other hand, the MinVI
objective focuses on modelling the conditional distributions (Equation (4)), which is arguably easier to optimize. Indeed, similar argument has been made for generalized denoising autoencoders
(DAEs) [1] and generative stochastic networks (GSNs) [2], which focus on learning the transition
? where X
? is a corrupted version of data X, or P? (X|H), where H can be
operators (e.g., P? (X|X),
arbitrary latent variables) to bypass an intractable problem of learning density model P? (X).
2.3
Theoretical Results
Bengio et al. [1, 2] proved that learning transition operators of DAEs or GSNs is sufficient to learn
a good generative model that estimates a data-generating distribution. Under similar assumptions,
we establish a theoretical result that we can obtain a good density estimator for joint distribution
of multimodal data by learning the transition operators derived from the conditional distributions of
one data modality given the other. In multimodal learning framework, the transition operators TnX
and TnY with model distribution P?n (X, Y ) are defined
P for Markov chains of data modalities X and
Y , respectively. Specifically, TnX (x[t]|x[t ? 1]) = y?Y P?n (x[t]|y) P?n (y|x[t ? 1]) and TnY is
defined in a similar way. Now, we formalize the theorem as follows:
Theorem 2.1 For finite state space X , Y, if, ?x ? X , ?y ? Y, P?n (?|y) and P?n (?|x) converges in
probability to PD (?|y) and PD (?|x), respectively, and TnX and TnY are ergodic Markov chains, then,
as the number of examples n ? ?, the asymptotic distribution ?n (X) and ?n (Y ) converge to datagenerating marginal distributions PD (X) and PD (Y ), respectively. Moreover, the joint probability
distribution P?n (x, y) converges to PD (x, y) in probability.
The proof is provided in supplementary material. The theorem ensures that the MinVI objective
can lead to a good generative model estimating the joint data-generating distribution of multimodal
data. The theorem holds under two assumptions, consistency of density estimators and ergodicity
of transition operators. The ergodicity of transition operators are satisfied for wide variety of neural
networks, such as an RBM or DBM. 2 The consistency assumption is more difficult to satisfy and
the aforementioned deep energy-based models nor RNN may not satisfy the condition due to the
approximated posteriors using factorized distribution. Probably, deep networks that allow exact
posterior inference, such as stochastic feedforward neural networks [20, 29], could be a better model
in our multimodal learning framework, but we leave this as a future work.
3
Application to Multimodal Deep Learning
In this section, we describe the MinVI learning in multimodal deep learning framework. To overview
our pipeline, we use the commonly used network architecture that consists of layers of modalityspecific deep networks followed by a layer of neural network that jointly models the multiple modalities [21, 27]. The network is trained in two steps: In layer-wise pretraining, each layer of modalityspecific deep network is trained using restricted Boltzmann machines (RBMs). For the top-layer
shared network, we train MRBM with MinVI objective (Section 3.2). Then, we finetune the whole
deep network by constructing multimodal deep recurrent neural network (MDRNN) (Section 3.3).
3.1
Restricted Boltzmann Machines for Multimodal Learning
The restricted Boltzmann machine (RBM) is an undirected graphical model that defines the distribution of visible units using hidden units. For multimodal input, we define the joint distribution of
2
For energy-based models like RBM and DBM, it is straightforward to see that every state has non-zero
probability and can be reached from any other state. However, the mixing of the chain might be slow in practice.
3
multimodal RBM (MRBM) [21, 27] as P (x, y, h) =
E(x, y, h) = ?
Nx X
K
X
x
xi Wik
hk ?
i=1 k=1
Ny K
X
X
1
Z
exp ?E(x, y, h) with the energy function:
y
yj Wjk
hk ?
j=1 k=1
K
X
bk hk ?
Nx
X
i=1
k=1
cxi xi ?
Ny
X
cyj yj ,
(5)
j=1
where Z is the normalizing constant, x ? {0, 1}Nx , y ? {0, 1}Ny are the binary visible (i.e.,
observation) variables of multimodal input, and h ? {0, 1}K are the binary hidden (i.e., latent)
variables. W x ? RNx ?K defines the weights between x and h, and W y ? RNy ?K defines the
weights between y and h. cx ? RNx , cy ? RNy , and b ? RK are bias vectors corresponding to x,
y, and h, respectively. Note that the MRBM is equivalent to an RBM whose visible variables are
constructed by concatenating the visible variables of multiple input modalities, i.e., v = [x ; y].
Due to bipartite structure, variables in the same layer are conditionally independent given the variables of the other layer, and the conditional probabilities are written as follows:
X
X y
x
P (hk = 1 | x, y) = ?
Wik
xi +
Wjk yj + bk ,
(6)
i
P (xi = 1 | h) = ?
X
j
x
Wik
hk
X y
+ cxi , P (yj = 1 | h) = ?
Wjk hk + cyj ,
k
(7)
k
1
. Similarly to the standard RBM, the MRBM can be trained to maximize
where ?(x) = 1+exp(?x)
the joint LL (log P (x, y)) using stochastic gradient descent (SGD) while approximating the gradient
with contrastive divergence (CD) [8] or persistent CD (PCD) [30]. In our case, however, we train
the MRBM in MinVI criteria. We will discuss the inference and training algorithms in Section 3.2.
When we have access to all data modalities, we can use Equation (6) for exact posterior inference.
On the other hand, when some of the input modalities are missing, the inference is intractable, and
we resort to the variational method. For example, when we are given
x but no y, the true posterior can
Q Q
be approximated with a fully factorized distribution Q(y, h) = j k Q(yj )Q(hk ) by minimizing
the KL Q(y, h)kP? (y, h|x) . This leads to the following fixed-point equations:
X
X y
X y
x
?k = ?
? k + cy ,
h
Wik
xi +
Wjk y?j + bk , y?j = ?
Wjk h
(8)
j
i
j
k
? k = Q(hk ) and y?j = Q(yj ). The variational inference proceeds by alternately updating the
where h
? and y? that are initialized with all 0?s.
mean-field parameters h
3.2
Training Algorithms
CD-PercLoss. As in Equation (2), the objective function can be decomposed into two conditional
LLs, and the MRBM with MinVI objective can be trained equivalently by training the two conditional RBMs (CRBMs) while sharing the weights. Since the objective functions are the sum of
two conditional LLs, we compute the (approximate) gradient of each CRBM separately using CDPercLoss [19] and accumulate them to update parameters.3
Multi-Prediction. We found a few practical issues of CD-PercLoss training: First, the gradient
estimates are inaccurate. Second, there exists a difference between encoding process of training and
testing, especially when the unimodal query (e.g., one of the data modality is missing) is considered for testing. As an alternative objective, we propose multi-prediction (MP) training of MRBM
in MinVI criteria. The MP training was originally proposed to train deep Boltzmann machines
(DBMs) [6] as an alternative to the stochastic approximation procedure learning [24]. The idea is to
train the model good at predicting any subset of input variables given the rest of them by constructing
the recurrent network with encoding function derived from the variational inference problem.
The MP training can be adapted to train MRBM with MinVI objective with some modifications.
For example, the CRBM with an objective log P (y|x) can be trained by randomly selecting the
subset of variables to be predicted only from the target modality y, but the conditioning modality x
3
In CD-PercLoss learning, we run separate Gibbs chains for different conditioning variables and select the
negative particles with the lowest free energy among sampled particles. We refer [19] for further details.
4
Wx(1)
x=hx(0)
Wx(2)
hx(1)
Wx(3)
hx(2)
Wy(3)
h(3)
Wy(2)
hy(2)
Wy(1)
hy(1)
y=hy(0)
Figure 1: An instance of MDRNN with target y given x. Multiple iterations of bottom-up updates
(y ? h(3) ; Equation (11)) and top-down updates (h(3) ? y; Equation (13)) are performed. The
arrow indicates encoding direction.
is assumed to be given in all cases. Specifically, given an arbitrary subset s ? {1, ? ? ? , Ny } drawn
from the independent Bernoulli distribution PS , the MP algorithm predicts ys = {yj : j ? s} given
x and y\s = {yj : j ?
/ s} through the iterative encoding function derived from fixed-point equations
?k = ?
h
X
x
Wik
xi +
i
X
j?s
y
Wjk
y?j +
X
X y
y
? k + cy , j ? s,
Wjk
yj + bk , y?j = ?
Wjk h
j
j ?s
/
(9)
k
which is a solution to the variationalQ
inference
Q problem minQ KL Q(ys , h)kP? (ys , h|x, y\s ) with
factorized distribution Q(ys , h) = j?s k Q(yj )Q(hk ). Note that Equation (9) is similar to the
Equation (8) except that only yj , j ? s are updated. Using an iterative encoding function, the
network parameters are trained using SGD while computing the gradient by backpropagating the
error between the prediction and the ground truth of ys through the derived recurrent network. The
MP formulation (e.g., encoding function) of the CRBM with log P (x|y) can be derived similarly,
and the gradients are simply the addition of two gradients that are computed individually.
We have two additional hyper parameters, the number of mean-field updates and the sampling ratio
of a subset s to be predicted from the target data modality. In our experiments, it was sufficient to
use 10 ? 20 iterations until convergence. We used the sampling ratio of 1 (i.e., all the variables in
the target data modality are to be predicted) since we are already conditioned on one data modality,
which is sufficient to make a good prediction of variables in the target data modality.
3.3
Finetuning Multimodal Deep Network with Recurrent Neural Network
Motivated from the MP training of MRBM, we propose multimodal deep recurrent neural network
(MDRNN) that tries to predict the target modality given the input modality through the recurrent
encoding function, which iteratively performs a full pass of bottom-up and top-down encoding from
bottom-layer visible variables to top-layer joint representation back to bottom-layer through the
modality-specific deep networks. We show an instance of L = 3 layer MDRNN in Figure 1, and the
encoding functions are written as follows:4
(l)
x,(l)> (l?1)
x,(l)
x ? h(L?1)
:
h
=
?
W
h
+
b
, l = 1, ? ? ? , L ? 1 (10)
x
x
x
(l)
y ? h(L?1)
:
hy
= ? W y,(l)> h(l?1)
+ by,(l) , l = 1, ? ? ? , L ? 1 (11)
y
y
h(L?1)
, h(L?1)
? h(L) : h(L) = ? W x,(L)> h(L?1)
+ W y,(L)> h(L?1)
+ b(L)
(12)
x
y
x
y
(l?1)
y,(l?1)
h(L) ? y : hy
= ? W y,(l) h(l)
, l = L, ? ? ? , 1
(13)
y +b
(0)
(0)
where hx = x and hy = y. The visible variables of the target modality are initialized with 0?s.
In other words, in the initial bottom-up update, we compute h(L) only from x while setting y = 0
using Equation (10),(11),(12). Then, we run multiple iterations of top-down (Equation (13)) and
bottom-up updates (Equation (11), (12)). Finally, we compute the gradient by backpropagating the
reconstruction error of target modality through the network.
4
There could be different ways of constructing MDRNN; for instance, one can construct the RNN with
DBM-style mean-field updates. In our empirical evaluation, however, running full pass of bottom-up and topdown updates performed the best, and DBM-style updates didn?t give competitive results.
5
Ground Truth
Query
ML (PCD)
MinVI (CDPercLoss)
MinVI (MP)
Figure 2: Visualization of samples with inferred missing modality. From top to bottom, we visualize
ground truth, left or right halves of digits, generated samples with inferred missing modality using
MRBM with ML objective, MinVI objective using CD-PercLoss and MP training methods.
Input modalities at test time
ML (PCD)
MinVI (CD-PercLoss)
MinVI (MP)
Left+Right
1.57%
1.71%
1.73%
Left
14.98%
9.42%
6.58%
Right
18.88%
11.02%
7.27%
Table 1: Test set handwritten digit recognition errors of MRBMs trained with different objectives
and learning algorithms. Linear SVM was used for classification with joint feature representations.
4
Experiments
4.1
Toy Example on MNIST
In our first experiment, we evaluate the proposed learning algorithm on the MNIST handwritten
digit recognition dataset [16]. We consider left and right halves of the digit images as two input
modalities and report the recognition performance with different combinations of input modalities
at the test time, such as full (left + right) or missing (left or right) data modalities. We compare
the performance of the MRBM trained with 1) ML objective using PCD [30], or MinVI objectives
with 2) CD-PercLoss or 3) MP training. The recognition errors are provided in Table 1. Compared
to ML training, the recognition errors for unimodal queries are reduced by more than a half with
MP training of MinVI objective. For multimodal queries, the model trained with ML objective
performed the best, although the performance gain was incremental. CD-PercLoss training of MinVI
objective also showed significant improvement over ML training, but the errors were not as low
as those obtained with MP training. We believe that, although it is an approximation of MinVI
objective, the exact gradient for MP algorithm makes learning more efficient than CD-PercLoss.
For the rest of the paper, we focus on MP training method.
In Figure 2, we visualize the generated samples conditioned on one input modality (e.g., left or right
halves of digits). There are many samples generated by the models with MinVI objective that look
clearly better than those generated by the model with ML objective.
4.2
MIR-Flickr Database
In this section, we evaluate our methods on MIR-Flickr database [11], which is composed of 1 million examples of image and their user tags collected from the social photo-sharing website Flickr.5
Among those, 25000 examples are annotated with 24 potential topics and 14 regular topics, which
leads to 38 classes in total with distributed class membership. The topics include object categories
such as dog, flower, and people, or scenic concepts such as sky, sea, and night.
We used the same visual and text features as in [27].6 Specifically, the image feature is 3857 dimensional vector composed of Pyramid Histogram of Words (PHOW) features [3], GIST [22], and
MPEG-7 descriptors [18]. We preprocessed the image features to have zero mean and unit variance for each dimension across all examples. The text feature is a word count vector of 2000 most
frequent tags. The number of tags varies from 0 to 72, with 5.15 tags per example in average.
Following the experimental protocol [12, 27], we randomly split the labeled data into 15000 for
training and 10000 for testing, and used 5000 from training set for validation. We iterate the procedure for 5 times and report the mean average precision (mAP) over 38 classes.
Model Architecture. As used in [27], the network is composed of [3857, 1024, 1024] variables
for visual pathway, [2000, 1024, 1024] variables for text pathway, and 2048 variables for top-layer
MRBM. As described in Section 3, we pretrain the modality-specific deep networks in a greedy
5
6
http://www.flickr.com
http://www.cs.toronto.edu/?nitish/multimodal/index.html
6
layerwise manner, and finetune the whole network by initializing MDRNN with the pretrained network. Specifically, we used gaussian RBM for the bottom layer of visual pathway and binary RBM
for text pathway.7 The intermediate layers are trained with binary RBMs, and the top-layer MRBM
is trained using MP training algorithm. For the layer-wise pretraining of RBMs, we used PCD [30]
to approximate gradient. Since our algorithm requires both data modalities during the training, we
excluded examples with too sparse or no tags from unlabeled dataset and used about 750K examples with at least 2 tags. After unsupervised training, we extract joint feature representations of the
labeled training data and use them to train multiclass logistic regression classifiers.
Recognition Tasks. For recognition tasks,
Model
Multimodal query
we train multiclass logistic regression classiAutoencoder
0.610
fiers using joint representations as input feaMultimodal DBM [27]
0.609
tures. Depending on the availability of data
Multimodal DBM? [28]
0.641
modalities at testing time, we evaluate the perMK-SVM [7]
0.623
formance using multimodal queries (i.e., both
TagProp [31]
0.640
MDRNN
0.686 ? 0.003
visual and text data are available) and unimodal
queries (i.e., visual data is available while the
Model
Unimodal query
text data is missing). The summary results are
Autoencoder
0.495
Multimodal DBM [27]
0.531
in Table 2. We report the test set mAPs of our
0.530
MK-SVM [7]
proposed model and compared to other methMDRNN
0.607 ? 0.005
ods. The proposed MDRNN outperformed the
previous state-of-the-art in multimodal queries
Table 2: Test set mAPs on MIR-Flickr database.
by 4.5% in mAP. The performance improveWe implemented autoencoder following the dement becomes more significant for unimodal
scription in [21]. Multimodal DBM? is supervised
queries, achieving 7.6% improvement in mAP
finetuned model. See [28] for details.
over the best published result. As we used the
same input features in [27], the results suggest that our proposed algorithm learns better representations shared across multiple modalities.
To take a closer look into our model, we performed additional control experiment. In particular, we
explore the benefit of recurrent encoding network structure of MDRNN. We compare the performance of the models with different number of mean-field iterations.8 We report the validation set
mAPs of models with different number of iterations (0 ? 10) in Table 3. For multimodal query, the
MDRNN with 10 iterations improves the recognition performance by only 0.8% compared to the
model with 0 iterations. However, the improvement becomes significant for unimodal query, achieving 5.0% performance gain. In addition, we note that the largest improvement was made when we
have at least one iteration (from 0 to 1 iteration, 3.4% gain; from 1 to 10 iteration, 1.6% gain). This
suggests that the most crucial factor of improvement comes from the inference with reconstructed
missing data modality (e.g., text features), and the quality of inferred missing modality improves as
we increase the number of iterations.
# iterations
Multimodal query
Unimodal query
0
0.677
0.557
1
0.678
0.591
2
0.679
0.599
3
0.680
0.602
5
0.682
0.605
10
0.685
0.607
Table 3: Validation set mAPs on MIR-Flickr database with different number of mean-field iterations.
Retrieval Tasks. We perform retrieval tasks using multimodal and unimodal input queries. Following the experimental setting in [27], we select 5000 image-text pairs from the test set to form a
database and use 1000 disjoint set of examples from the test set as queries. For each query example,
we compute the relevance score to the data points as a cosine similarity of joint representations.
The binary relevance label between query and the data points are determined 1 if any of the 38
class labels are overlapped. Our proposed model achieves 0.633 mAP with multimodal query and
0.638 mAP with unimodal query. This significantly outperforms the performance of multimodal
DBM [27], which reported 0.622 mAP with multimodal query and 0.614 mAP with unimodal query.
7
We assume text features as binary, which is different from [27] where they modeled using replicatedsoftmax RBM [25]. The rationale is that the tags are not likely to be assigned more than once for single image.
8
In [21], they proposed the ?video-only? deep autoencoder whose objective is to predict audio data and
reconstruct video data when only video data is given as an input during the training. Our baseline model
(MDRNN with 0 iterations) is similar, but different since we don?t have a reconstruction training objective.
7
night, city, river,
dark, buildings, skyline
night, long exposure,
reflection, buildings,
massachusetts, boston
sunset, explore, sun
sunset, platinumphoto,
trees, silhouette
toys
lego
skyline, indiana, 1855mm
city, lights, buildings,
fireworks, skyscrapers
nikon, night, d80, asia,
skyline, hongkong, harbour
sunset, sol, searchthebest,
atardecer, nubes, abigfave
sunset
canon, naturesfinest, 30d
diy, robot
toy, plastic,
kitty, miniature
lego
Figure 3: Retrieval results with multimodal queries. The leftmost image-text pairs are multimodal
query samples and those in the right side of the bar are retrieved samples with the highest similarities
to the query sample from the database. We include more results in supplementary material.
4.3
PASCAL VOC 2007
We evaluate the proposed algorithm on PASCAL VOC 2007 database. The original dataset doesn?t
contain user tags, but Guillaumin et al. [7] has collected the user tags from Flickr website.9
Motivated from the success of convolutional neural networks (CNNs) on large-scale visual object
recognition [14], we used the DeCAF7 features [5] as an input features for visual pathway, where
DeCAF7 is 4096 dimensional feature extracted from the CNN trained on ImageNet [4]. For text
features, we used the vocabulary of size 804 suggested by [7]. For unsupervised feature learning of
MDRNN, we used unlabeled data of MIR-Flickr database while converting the text features using
the new vocabulary from PASCAL database. The network architecture used in this experiment is as
follows: [4096, 1536, 1536] variables for the visual pathway, [804, 512, 1536] variables for the text
pathway, and 2048 variables for top-layer joint network.
Following the standard practice, we report the mAP over 20 object classes. The performance improvement of our proposed method was significant, achieving 81.5% mAP with multimodal queries
and 76.2% mAP with unimodal queries, whereas the performance of baseline model was 74.5%
mAP with multimodal queries (DeCAF7 + Text) and 74.3% mAP with unimodal queries (DeCAF7 ).
5
Conclusion
Motivated from the property of good generative models of multimodal data, we proposed a novel
multimodal deep learning framework based on variation of information. The minimum variation of
information objective enables to learn a good shared representations of multiple heterogeneous data
modalities with a better prediction of missing input modality. We demonstrated the effectiveness
of our proposed method on multimodal RBM and its deep extensions and showed state-of-the-art
recognition performance on MIR-Flickr database and competitive performance on PASCAL VOC
2007 database with multimodal (visual + text) and unimodal (visual only) queries.
Acknowledgments
This work was supported in part by ONR N00014-13-1-0762, Toyota, and Google Faculty Research
Award.
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9
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4,726 | 528 | Direction Selective Silicon Retina
that uses N uIl Inhibition
Ronald G. Benson and Tobi Delbriick
Computation and Neural Systems Program, 139-74
California Institute of Technology
Pasadena CA 91125
email: [email protected] and [email protected]
Abstract
Biological retinas extract spatial and temporal features in an attempt to
reduce the complexity of performing visual tasks. We have built and tested
a silicon retina which encodes several useful temporal features found in vertebrate retinas. The cells in our silicon retina are selective to direction,
highly sensitive to positive contrast changes around an ambient light level,
and tuned to a particular velocity. Inhibitory connections in the null direction perform the direction selectivity we desire. This silicon retina is
on a 4.6 x 6.8mm die and consists of a 47 x 41 array of photoreceptors.
1
INTRODUCTION
The ability to sense motion in the visual world is essential to survival in animals.
Visual motion processing is indispensable; it tells us about predators and prey, our
own motion and image stablization on the retina. Many algorithms for performing
early visual motion processing have been proposed [HK87] [Nak85]. A key salient
feature of motion is direction selectivity, ie the ability to detect the direction of
moving features. We have implemented Barlow and Levick's model, [BHL64], which
hypothesizes inhibition in the null direction to accomplish direction selectivity.
In contrast to our work, Boahen, [BA91], in these proceedings, describes a silicon
retina that is specialized to do spatial filtering of the image. Mahowald, [Mah91],
describes a silicon retina that has surround interactions and adapts over mulitiple
time scales. Her silicon retina is designed to act as an analog preprocessor and
756
Direction Selective Silicon Retina that uses Null Inhibition
Pixels inhibit to the left
Preferred
.Null ~
Preferred dPrection
~Photoreceptor
L~ DS cell
(a)
. Inhibition
(b)
Figure 1: Barlow and Levick model of direction selectivity (DS). (a) Shows
how two cells are connected in an inhibitory fashion and (b) a mosaic of such
cells.
so the gain of the output stage is rather low. In addition there is no rectification
into on- and off-pathways. This and earlier work on silicon early vision systems
have stressed spatial processing performed by biological retinas at the expense of
temporal processing.
The work we describe here and the work described by Delbriick, [DM9l], emphasizes
temporal processing. Temporal differentiation and separation of intensity changes
into on- and off-pathways are important computations performed by vertebrate
retinas. Additionally, specialized vertebrate retinas, [BHL64], have cells which are
sensitive to moving stimuli and respond maximally to a preferred direction; they
have almost zero response in the opposite or null direction. We have designed and
tested a silicon retina that models these direction selective velocity tuned cells.
These receptors excite cells which respond to positive contrast changes only and
are selective for a particular direction of stimuli. Our silicon retina may be useful
as a preprocessor for later visual processing and certainly as an enhancement for
the already existing spatial retinas. It is a striking demonstration of the perceptual
saliency of contrast changes and directed motion in the visual world.
2
INHIBITION IN THE NULL DIRECTION
Barlow and Levick, [BHL64]' described a mechanism for direction selectivity found
in the rabbit retina which postulates inhibitory connections to achieve the desired
direction selectivity. Their model is shown in Figure l(a). As a moving edge
passes over the photoreceptors from left to right, the left photoreceptor is excited
first, causing its direction selective (DS) cell to fire. The right photoreceptor fires
when the edge reaches it and since it has an inhibitory connection to the left DS
cell, the right photoreceptor retards further output from the left DS cell. If an edge
is moving in the opposite or null direction (right to left), the activity evoked in the
right photoreceptor completely inhibits the left DS cell from firing, thus creating a
direction selective cell.
757
758
Benson and Delbriick
Inhibition from right
Inhibition to left
Ir
Q
r
Photoreceptor
~
Preferred Direction
DS cell
Figure 2: Photoreceptor and direction selective (DS) cell. The output of the
high-gain, adaptive photoreceptor is fed capacitively to the input of the DS
cell. The output of the photoreceptor sends inhibition to the left. Inhibition
from the right photoreceptors connect to the input of the DS cell.
In the above explanation with the edge moving in the preferred direction (left to
right), as the edge moves faster, the inhibition from leading photoreceptors truncates
the output of the DS cell ever sooner. In fact, it is this inhibitory connection which
leads to velocity tuning in the preferred direction.
By tiling these cells as shown in Figure l(b), it is possible to obtain an array of
directionally tuned cells. This is the architecture we used in our chip. Direction
selectivity is inherent in the connections of the mosaic, ie the hardwiring of the
inhibitory connections leads to directionally tuned cells.
3
PIXEL OPERATION
A pixel consists of a photoreceptor, a direction selective (DS) cell and inhibition to
and from other pixels as shown in Figure 2. The photoreceptor has high-gain and
is adaptive [Mah91, DM91]. The output from this receptor, Vp , is coupled into the
DS cell which acts as a rectifying gain element, [MS91], that is only sensitive to
positive-going transitions due to increases in light intensity at the receptor input.
Additionally, the output from the photoreceptor is capacitively coupled to the inhibitory synapses which send their inhibition to the left and are coupled into the
DS cell of the neighboring cells.
A more detailed analysis of the DS cell yields several insights into this cell's functionality. A step increase of 6. V at Vp , caused by a step increase in light intensity
incident upon the phototransistor, results in a charge injection of Cc 6. V at Vi. This
charge is leaked away by QT at a rate IT, set by voltage VT. Hence, to first order,
the output pulse width T is simply
T
= Cc 6.V.
IT
There is also a threshold minimum step input size that will result in enough change
Direction Selective Silicon Retina that uses Null Inhibition
1.6
-..
>
1.2
Output
'-'
~
<Il
0.8
= 0.4
c.
<Il
0
~
~
0.0
Input intensity
0 40 80 120 160 200
Time (msec)
Figure 3: Pixel response to intensity step. Bottom trace is intensity; top trace
is pixel output.
in Vi to pull Vout all the way to ground. This threshold is set by Cc and the gain of
the photoreceptor.
When the input to the rectifying gain element is not a step, but instead a steady
increase in voltage, the current lin flowing into node Vi is
= CcVp.
When this current exceeds IT there is a net increase in the voltage Vi, and the
output Vout will quickly go low. The condition lin = IT defines the threshold
limit for stimuli detection, i.e. input stimuli resu~ting in an lin < IT are not
perceptible to the pixel. For a changing intensity I, the adaptive photoreceptor
stage outputs a voltage Vp proportional to j / I, where I is the input light intensity.
This photoreceptor behavior means that the pixel threshold will occur at whatever
j / I causes Cc Vp to exceed the constant current I r .
lin
The inhibitory synapses (shown as Inhibition from right in Figure 2) provide additionalleakage from Vi resulting in a shortened response width from the DS cell.
This analysis suggests that a characterization of the pixel should investigate both
the response amplitude, measured as pulse width versus input intensity step size,
and the response threshold, measured with temporal intensity contrast. In the next
section we show such measurements.
4
CHARACTERIZATION OF THE PIXEL
We have tested both an isolated pixel and a complete 2-dimensional retina of 47 x 41
pixels. Both circuits were fabricated in a 2J.tm p-well CMOS double poly process
available through the MOSIS facility. The retina is scanned out onto a monitor using
a completely integrated on-chip scanner[MD91]. The only external components are
a video amplifier and a crystal.
We show a typical response of the isolated pixel to an input step of intensity in
Figure 3. In response to the input step increase of intensity, the pixel output goes
low and saturates for a time set by the bias Vr in Figure 2. Eventually the pixel
recovers and the output returns to its quiescent level. In response to the step
decrease of intensity there is almost no response as seen in Figure 3.
759
760
Benson and Delbriick
~~
16O
UIII
rIJ
!,120
..c=
....be
80
=
III
/
- 40
III
rIJ
::l
A..
1.8
Step Contrast
(a)
2.2
Temporal Frequency (Hz)
(b)
Figure 4: (a) Pulse width of response as function of input contrast step size.
The abscissa is measured in units of ratio-intensity, i.e., a value of 1 means
no intensity step, a value of 1.1 means a step from a normalized intensity of 1
to a normalized intensity of 1.1, and so forth. The different curves show the
response at different absolute light levels; the number in the figure legend is
the log of the absolute intensity. (b) Receptor threshold measurements. At
each temporal frequency, we determined the minimum necessary amplitude of
triangular intensity variations to make the pixel respond. The different curves
were taken at different background intensity levels, shown to the left of each
curve. For example, the bottom curve was taken at a background level of 1
unit of intensity; at 8 Hz, the threshold occurred at a variation of 0.2 units of
intensity.
The output from the pixel is essentially quantized in amplitude, but the resulting
pulse has a finite duration related to the input intensity step. The analysis in
Section 3 showed that the output pulse width, T, should be linear in the input
intensity contrast step. In Figure 4{ a), we show the measured pulse-width as a
function of input contrast step. To show the adaptive nature of the receptor, we
did this same measurement at several different absolute intensity levels.
Our silicon retina sees some features of a moving image and not others. Detection
of a moving feature depends on its contrast and velocity. To characterize this
behavior, we measured a receptor's thresholds for intensity variations, as a function
of temporal frequency.
These measurements are shown in Figure 4(b); the curves define "zones of visibility"; if stimuli lie below a curve, they are visible, if they fall above a curve they
are not. (The different curves are for different absolute intensity levels.) For low
temporal frequencies stimuli are visible only if they are high contrast; at higher
temporal frequencies, but still below the photoreceptor cutoff frequency, lower contrast stimuli are visible. Simply put, if the input image has low contrast and is
slowly moving, it is not seen. Only high contrast or quickly moving features are
salient stimuli. More precisely, for temporal frequencies below the photoreceptor
cutoff frequency, the threshold occurs at a constant value of the temporal intensity
contrast j / I.
Direction Selective Silicon Retina that uses Null Inhibition
761
Preferred
L
R
Inhib
Preferred
fN uu -
Photoreceptors
"'----- DS
Excitatio
L
-
R
- - - - Inhib
' - - - - - DS
(a)
0.1 sec
(b)
Figure 5: (a) shows the basic connectivity of the tested cell. (b) top trace is
the response due to an edge moving in the preferred direction (left to right).
(b) bottom trace is the response due to an edged moving in the null direction
(right to left).
5
NULL DIRECTION INHIBITION PROPERTIES
We performed a series of tests to characterize the inhibition for various orientations
and velocities. The data in Figure 5(b) shows the outputs of two photo receptors,
the inhibitory signal and the output of a DS cell. The top panel in Figure 5(b) shows
the outputs in the preferred direction and the bottom panel shows them in the null
direction. Notice that the out pu t of the left photoreceptor (L in Figure 5 (b) top
panel) precedes the right (R). The output of the DS cell is quite pronounced, but is
truncated by the inhibition from the right photoreceptor. On the other hand, the
bottom panel shows that the output of the DS cell is almost completely truncated
by the inhibitory input.
A DS cell receives most inhibition when the stimulus is travelling exactly in the null
direction. As seen in Figure 6(a) as the angle of stimulus is rotated, the maximum
response from the DS cell is obtained when the stimulus is moving in the preferred
direction (directly opposite to the null direction). As the bar is rotated toward the
null direction, the response of the cell is reduced due to the increasing amount of
inhibition received from the neighboring photo receptors.
If a bar is moving in the preferred direction with varying velocity, there is a velocity,
Vmaz , for which the DS cell responds maximally as shown in Figure 6(b). As the
bar is moved faster than Vmaz , inhibition arrives at the cell sooner, thus truncating
the response. As the cell is moved slower than Vmaz, less input is provided to the
DS cell as described in Section 3. In the null direction (negative in Figure 6(b?) the
cell does not respond, as expected, until the bar is travelling fast enough to beat
the inhibition's onset (recall delay from Figure 5).
In Figure 7 we show the response of the entire silicon retina to a rotating fan. When
the fan blades are moving to the left the retina does not respond, but when moving
to the right, note the large response. Note the largest response when the blades are
moving exactly in the preferred direction.
762
Benson and Delbruck
160
-;;-120
8
~
Q)
rn
? 80
0..
~
~ 40
-0.8 -0.4 0.0
0.4
0.8
Velocity (arbitrary units)
(a)
(b)
Figure 6: (a) polar plot which shows the pixels are directionally tuned.
(b) shows velocity tuning of the DS cell (positive velocities are in the preferred direction).
(a)
(b)
Figure 7: (a) Rotating fan used as stimulus to the retina. (b) Output of the
retina.
Direction Selective Silicon Retina that uses Null Inhibition
6
CONCLUSION
We have designed and tested a silicon retina that detects temporal changes in an
image. The salient image features are sufficiently high contrast stimuli, relatively
fast increase in intensity (measured with respect to the recent past history of the
intensity), direction and velocity of moving stimuli. These saliency measures result
in a large compression of information, which will be useful in later processing stages.
Acknowledgments
Our thanks to Carver Mead and John Hopfield for their guidance and encouragement, to the Office of Naval Research for their support under grant NAV N0001489-J-1675, and, of course, to the MOSIS fabrication service.
References
[BA91]
K. Boahen and A. Andreou. A contrast sensitive silicon retina with reciprocal synapses. In S. Hanson J. Moody and R. Lippmann, editors,
Advances in Neural Information Processing Systems, Volume 4. Morgan
Kaufmann, Palo Alto, CA, 1991.
[BHL64] H.B. Barlow, M.R. Hill, and W.R. Levick. Retinal ganglion cells responding selectively to direction and speed of image motion in the rabbit. J.
Physiol., 173:377-407, 1964.
[DM91] T. Delbriick and Carver Mead. Silicon adaptive photoreceptor array that
computes temporal intensity derivatives. In Proc. SPIE 1541, volume
1541-12, pages 92-99, San Diego, CA, July 1991. Infrared Sensors: Detectors, Electronics, and Signal Processing.
E. Hildreth and C. Koch. The analysis of visual motion: From computational theory to neuronal mechanisms. Annual Review in Neuroscience,
10:477-533, 1987.
[Mah91] M.A. Mahowald. Silicon retina with adaptive photoreceptor. In SPIE
Technical Symposia on Optical Engineering and Photonics in Aerospace
Sensing, Orlando, FL, April 1991. Visual Information Processing: From
Neurons to Chips.
[MD91] C.A. Mead and T. Delbriick. Scanners for use in visualizing analog VLSI
circuitry. Analog Integrated Circuits and Signal Processing, 1:93-106, 1991.
[MS91] C.A. Mead and R. Sarpeshkar. An axon circuit. Internal Memo, Physics
of Computation Laboratory, Caltech, 1991.
[HK87]
[Nak85] K. Nakayama. Biological image motion processing: A review.
Research, 25(5):625-660, 1985.
Vision
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4,727 | 5,280 | Restricted Boltzmann machines modeling human
choice
Makoto Otsuka
IBM Research - Tokyo
[email protected]
Takayuki Osogami
IBM Research - Tokyo
[email protected]
Abstract
We extend the multinomial logit model to represent some of the empirical phenomena that are frequently observed in the choices made by humans. These phenomena include the similarity effect, the attraction effect, and the compromise
effect. We formally quantify the strength of these phenomena that can be represented by our choice model, which illuminates the flexibility of our choice model.
We then show that our choice model can be represented as a restricted Boltzmann
machine and that its parameters can be learned effectively from data. Our numerical experiments with real data of human choices suggest that we can train our
choice model in such a way that it represents the typical phenomena of choice.
1
Introduction
Choice is a fundamental behavior of humans and has been studied extensively in Artificial Intelligence and related areas. The prior work suggests that the choices made by humans can significantly
depend on available alternatives, or the choice set, in rather complex but systematic ways [13]. The
empirical phenomena that result from such dependency on the choice set include the similarity effect, the attraction effect, and the compromise effect. Informally, the similarity effect refers to the
phenomenon that a new product, S, reduces the share of a similar product, A, more than a dissimilar
product, B (see Figure 1 (a)). With the attraction effect, a new dominated product, D, increases the
share of the dominant product, A (see Figure 1 (b)). With the compromise effect, a product, C, has
a relatively larger share when two extreme products, A and B, are in the market than when only
one of A and B is in the market (see Figure 1 (c)). We call these three empirical phenomena as the
typical choice phenomena.
However, the standard choice model of the multinomial logit model (MLM) and its variants cannot
represent at least one of the typical choice phenomena [13]. More descriptive models have been
proposed to represent the typical choice phenomena in some representative cases [14, 19]. However,
it is unclear when and to what degree the typical choice phenomena can be represented. Also, no
algorithms have been proposed for training these descriptive models from data.
S
A
A
A
D
C
B
(a) Similarity
B
B
(b) Attraction
(c) Compromise
Figure 1: Choice sets that cause typical choice phenomena.
1
We extend the MLM to represent the typical choice phenomena, which is our first contribution.
We show that our choice model can be represented as a restricted Boltzmann machine (RBM). Our
choice model is thus called the RBM choice model. An advantage of this representation as an RBM
is that training algorithms for RBMs are readily available. See Section 2.
We then formally define the measure of the strength for each typical choice phenomenon and quantify the strength of each typical choice phenomenon that the RBM choice model can represent. Our
analysis not only gives a guarantee on the flexibility of the RBM choice model but also illuminates
why the RBM choice model can represent the typical choice phenomena. These definitions and
analysis constitute our second contribution and are presented in Section 3.
Our experiments suggest that we can train the RBM choice model in such a way that it represents
the typical choice phenomena. We show that the trained RBM choice model can then adequately
predict real human choice on the means of transportation [2]. These experimental results constitute
our third contribution and are presented in Section 4.
2
Choice model with restricted Boltzmann machine
We extend the MLM to represent the typical choice phenomena. Let I be the set of items. For A ?
X ? I, we study the probability that an item, A, is selected from a choice set, X . This probability
is called the choice probability. The model of choice, equipped with the choice probability, is called
a choice model. We use A, B, C, D, S, or X to denote an item and X , Y, or a set such as {A, B} to
denote a choice set.
For the MLM, the choice probability of A from X can be represented by
?(A|X )
p(A|X ) = P
,
X?X ?(X|X )
(1)
where we refer to ?(X|X ) as the choice rate of X from X . The choice rate of the MLM is given by
?MLM (X|X ) = exp(bX ),
(2)
where bX can be interpreted as the attractiveness of X. One could define bX through uX , the
vector of the utilities of the attributes for X, and ?, the vector of the weight on each attribute (i.e.,
bX ? ??uX ). Observe that ?MLM (X|X ) is independent of X as long as X ? X . This independence
causes the incapability of the MLM in representing the typical choice phenomena.
We extend the choice rate of (2) but keep the choice probability in the form of (1). Specifically, we
consider the following choice rate:
Y
k
?(X|X ) ? exp(bX )
1 + exp TXk + UX
,
(3)
k?K
where we define
TXk ?
X
TYk .
(4)
Y ?X
k
Our choice model has parameters, bX , TXk , UX
for X ? X , k ? K, that take values in (??, ?).
Equation (3) modifies exp(bX ) by multiplying factors. Each factor is associated with an index, k,
k
and has parameters, TXk and UX
, that depend on k. The set of these indices is denoted by K.
We now show that our choice model can be represented as a restricted Boltzmann machine (RBM).
This means that we can use existing algorithms for RBMs to learn the parameters of the RBM choice
model (see Appendix A.1).
An RBM consists of a layer of visible units, i ? V, and a layer of hidden units, k ? H. A visible unit,
i, and a hidden unit, k, are connected with weight, Wik . The units within each layer are disconnected
from each other. Each unit is associated with a bias. The bias of a visible unit, i, is denoted by bvis
i .
The bias of a hidden unit, k, is denoted by bhid
k . A visible unit, i, is associated with a binary variable,
zi , and a hidden unit, k, is associated with a binary variable, hk , which takes a value in {0, 1}.
For a given configuration of binary variables, the energy of the RBM is defined as
XX
hid
E? (z, h) ? ?
zi Wik hk + bvis
i zi + bk hk ,
i?V k?H
2
(5)
...
Hidden
k
...
UAk
TXk
Choice set
...
...
X
...
A
Selected item
...
bA
Figure 2: RBM choice model
where ? ? {W, bvis , bhid } denotes the parameters of the RBM. The probability of realizing a particular configuration of (z, h) is given by
exp(?E? (z, h))
P? (z, h) ? P P
.
(6)
0
0
z0
h0 exp(?E? (z , h ))
P
P
The summation with respect to a binary vector (i.e., z0 or h0 ) denotes the summation over all of
the possible binary vectors of a given length. The length of z 0 is |V|, and the length of h0 is |H|.
The RBM choice model can be represented as an RBM having the structure in Figure 2. Here, the
layer of visible units is split into two parts: one for the choice set and the other for the selected item.
The corresponding binary vector is denoted by z = (v, w). Here, v is a binary vector associated
with the part for the choice set. Specifically, v has length |I|, and vX = 1 denotes that X is in the
k
choice set. Analogously, w has length |I|, and wA = 1 denotes that A is selected. We use TX
to
denote the weight between a hidden unit, k, and a visible unit, X, for the choice set. We use UAk to
denote the weight between a hidden unit, k, and a visible unit, A, for the selected item. The bias is
zero for all of the hidden units and for all of the visible units for the choice set. The bias for a visible
unit, A, for the selected item is denoted by bA . Finally, let H = K.
The choice rate (3) of the RBM choice model can then be represented by
X
?(A|X ) =
exp ?E? v X , wA , h ,
(7)
h
X
where we define the binary vectors, v , wA , such that viX = 1 iff i ? X and wjA = 1 iff j = A.
Observe that the right-hand side of (7) is
!
X
X
X X
X
X
A
k
k
exp(?E? ((v , w ), h)) =
exp
TX hk +
UA hk + bA
(8)
h
X?X
h
=
exp(bA )
h
=
exp(bA )
k
XY
Y
exp
k
TXk
+ UAk hk
(9)
k
X
exp
TXk + UAk hk ,
(10)
k hk ?{0,1}
which is equivalent to (3).
The RBM choice model assumes that one item from a choice set is selected. In the context of the
RBM, this means that wA = 1 for only one A ? X ? I. Using (6), our choice probability (1) can
be represented by
P
P? ((v X , wA ), h)
hP
.
(11)
p(A|X ) = P
X
X
X?X
h P? ((v , w ), h)
This is the conditional probability of realizing the configuration, (v X , wA ), given that the realized
configuration is either of the (v X , wX ) for X ? X . See Appendix A.2for an extension of the RBM
choice model.
3
Flexibility of the RBM choice model
In this section, we formally study the flexibility of the RBM choice model. Recall that ?(X|X ) in
(3) is modified from ?MLM (X|X ) in (2) by a factor,
k
1 + exp TXk + UX
,
(12)
3
for each k in K, so that ?(X|X ) can depend on X through TXk . We will see how this modification
allows the RBM choice model to represent each of the typical choice phenomena.
The similarity effect refers to the following phenomenon [14]:
p(A|{A, B}) > p(B|{A, B})
p(A|{A, B, S}) < p(B|{A, B, S}).
and
(13)
Motivated by (13), we define the strength of the similarity effect as follows:
Definition 1. For A, B ? X , the strength of the similarity effect of S on A relative to B with X is
defined as follows:
(sim)
?A,B,S,X
?
p(A|X ) p(B|X ? {S})
.
p(B|X ) p(A|X ? {S})
(14)
(sim)
When ?A,B,S,X = 1, adding S into X does not change the ratio between p(A|X ) and p(B|X ).
(sim)
Namely, there is no similarity effect. When ?A,B,S,X > 1, we can increase
(sim)
of ?A,B,S,X
(sim)
?A,B,S,X <
p(B|X )
p(A|X )
by a factor
by the addition of S into X . This corresponds to the similarity effect of (13). When
1, this ratio decreases by an analogous factor. We will study the strength of this (rather
general) similarity effect without the restriction that S is ?similar? to A (see Figure 1 (a)).
Because p(X|X ) has a common denominator for X = A and X = B, we have
(sim)
?A,B,S,X
=
?(A|X ) ?(B|X ? {S})
.
?(B|X ) ?(A|X ? {S})
(15)
The MLM cannot represent the similarity effect, because the ?MLM (X|X ) in (2) is independent of
X . For any choice sets, X and Y, we must have
?MLM (A|X )
?MLM (B|X )
=
?MLM (A|Y)
.
?MLM (B|Y)
(16)
The equality (16) is known as the independence from irrelevant alternatives (IIA).
The RBM choice model can represent an arbitrary strength of the similarity effect. Specifically, by
? into K of (3), we can set ?(A|X ?{S}) at an arbitrary value without affecting
adding an element, k,
?(A|X )
the value of ?(B|Y), ?B 6= A, for any Y. We prove the following theorem in Appendix C:
Theorem 1. Consider an RBM choice model where the choice rate of X from X is given by (2). Let
?
?(X|X
) be the corresponding choice rate after adding k? into K. Namely,
?
?
k
?
.
(17)
?(X|X
) = ?(X|X ) 1 + exp TXk + UX
Consider an item A ? X and an item S 6? X . For any c ? (0, ?) and ? > 0, we can then choose
?
?
T?k and U?k such that
?(B|Y)
?
?(A|X
? {S})
?
c=
;
?>
? 1 , ?Y, B s.t. B 6= A.
(18)
?
?(B|Y)
?(A|X
)
By (15) and Theorem 1, the strength of the similarity effect after adding k? into K is
(sim)
??A,B,S,X =
?
?
?(A|X
)
?(B|X
? {S})
1 ?(B|X ? {S})
.
?
?
?
c
?(B|X )
?(A|X ? {S})
?(B|X )
(19)
? indeed allows
Because c can take an arbitrary value in (0, ?), the additional factor, (12) with k = k,
(sim)
??
to take any positive value without affecting the value of ?(B|Y), ?B 6= A, for any Y. The
A,B,S,X
first part of (18) guarantees that this additional factor does not change p(X|Y) for any X if A ?
/ Y.
Note that what we have shown is not limited to the similarity effect of (13). The RBM choice model
can represent an arbitrary phenomenon where the choice set affects the ratio of the choice rate.
4
According to [14], the attraction effect is represented by
p(A|{A, B}) < p(A|{A, B, D}).
(20)
MLM
The MLM cannot represent
the
attraction
effect,
because
the
?
(X|Y)
in
(2)
is
independent
P
P
of Y, and we must have X?X ?MLM (X|X ) ? X?Y ?MLM (X|Y) for X ? Y, which in turn
implies the regularity principle: p(X|X ) ? p(X|Y) for X ? Y.
Motivated by (20), we define the strength of the attraction effect as the magnitude of the change in
the choice probability of an item when another item is added into the choice set. Formally,
Definition 2. For A ? X , the strength of the attraction effect of D on A with X is defined as follows:
p(A|X ? {D})
(att)
?A,D,X ?
.
(21)
p(A|X )
(att)
When there is no attraction effect, adding D into X can only decrease p(A|X ); hence, ?A,D,X ? 1.
(att)
The standard definition of the attraction effect (20) implies ?A,D,X > 1. We study the strength of
this attraction effect without the restriction that A ?dominates? D (see Figure 1 (b)).
We prove the following theorem in Appendix C:
Theorem 2. Consider the two RBM choice models in Theorem 1. The first RBM choice model has
the choice rate given by (3), and the second RBM choice model has the choice rate given by (17).
Let p(?|?) denote the choice probability for the first RBM choice model and p?(?|?) denote the choice
probability for the second RBM choice model. Consider an item A ? X and an item D 6? X . For
?
?
any r ? (p(A|X ? {D}), 1/p(A|X )) and ? > 0, we can choose T?k , U?k such that
?(B|Y)
p?(A|X ? {D})
?
r=
;
?>
? 1 , ?Y, B s.t. B 6= A.
(22)
?(B|Y)
p?(A|X )
We expect that the range, (p(A|X ? {D}), 1/p(A|X )), of r in the theorem covers the attraction
effect in practice. Also, this range is the widest possible in the following sense. The factor (12) can
only increase ?(X|Y) for any X, Y. The form of (1) then implies that, to decrease p(A|Y), we must
increase ?(X|Y) for X 6= A. However, increasing ?(X|Y) for X 6= A is not allowed due to the
? can only increase
second part of (22) with ? ? 0. Namely, the additional factor, (12) with k = k,
p(A|Y) for any Y under the condition of the second part of (22). The lower limit, p(A|X ? {D}),
is achieved when p?(A|X ) ? 1, while keeping p?(A|X ? {D}) ? p(A|X ? {D}). The upper limit,
1/p(A|X ), is achieved when p?(A|X ? {D}) ? 1, while keeping p?(A|X ) ? p(A|X ).
According to [18], the compromise effect is formally represented by
p(C|{A, B, C})
p(C|{A, B, C})
X
X
> p(C|{A, C}) and
> p(C|{B, C}). (23)
p(X|{A, B, C})
p(X|{A, B, C})
X?{A,C}
X?{B,C}
The MLM cannot represent the compromise effect, because the ?MLM (X|Y) in (2) is independent
of Y, which in turn makes the inequalities in (23) equalities.
Motivated by (23), we define the strength of the compromise effect as the magnitude of the change
in the conditional probability of selecting an item, C, given that either C or another item, A, is
selected when yet another item, B, is added into the choice set. More precisely, we also exchange
the roles of A and B, and study the minimum magnitude of those changes:
Definition 3. For a choice set, X , and items, A, B, C, such that A, B, C ? X , let
qAC (C|X )
?A,B,C,X ?
,
(24)
qAC (C|X \ {B})
where, for Y such that A, C ? Y, we define
p(C|Y)
qAC (C|Y) ? P
.
(25)
X?{A,C} p(X|Y)
The strength of the compromise effect of A and B on C with X is then defined as
(com)
?A,B,C,X
?
min {?A,B,C,X , ?B,A,C,X } .
5
(26)
Here, we do not have the restriction that C is a ?compromise? between A and B (see Figure 1 (c)).
In Appendix C:we prove the following theorem:
Theorem 3. Consider a choice set, X , and three items, A, B, C ? X . Consider the two RBM choice
(com)
models in Theorem 2. Let ??A,B,C,X be defined analogously to (26) but with p?(?|?). Let
q ? max {qAC (C|X \ {B}), qBC (C|X \ {A})}
q ? min {qAC (C|X ), qBC (C|X )} .
Then, for any r ? (q, 1/q) and ? > 0, we can choose T?k , U?k such that
?(X|Y)
?
(com)
? 1 , ?Y, X s.t. X 6= C.
r = ??A,B,C,X ;
?>
?(X|Y)
(27)
(28)
(29)
We expect that the range of r in the theorem covers the compromising effect in practice. Also,
this range is best possible in the sense analogous to what we have discussed with the range in
? can only increase p(C|Y) for any Y
Theorem 2. Because the additional factor, (12) with k = k,
under the condition of the second part of (29), it can only increase qXC (C|Y) for X ? {A, B}. The
lower limit, q, is achieved when qXC (C|X \ {X}) ? 1, while keeping qXC (C|X ) approximately
unchanged, for X ? {A, B}. The upper limit, 1/q, is achieved when qXC (C|X ) ? 1, while
keeping qXC (C|X \ {X}) approximately unchanged, for X ? {A, B}.
4
Numerical experiments
We now validate the effectiveness of the RBM choice model in predicting the choices made by
humans. Here we use the dataset from [2], which is based on the survey conducted in Switzerland,
where people are asked to choose a means of transportation from given options. A subset of the
dataset is used to train the RBM choice model, which is then used to predict the choice in the
remaining dataset. In Appendix B.2,we also conduct an experiment with artificial dataset and show
that the RBM choice model can indeed be trained to represent each of the typical choice phenomena.
This flexibility in the representation is the basis of the predictive accuracy of the RBM choice model
to be presented in this section. All of our experiments are run on a single core of a Windows PC
with main memory of 8 GB and Core i5 CPU of 2.6 GHz.
The dataset [2] consists of 10,728 choices that 1,192 people have made from a varying choice set.
For those who own a car, the choice set has three items: a train, a maglev, and a car. For those who
do not own a car, the choice set consists of a train and a maglev. The train can operate at the interval
of 30, 60, or 120 minutes. The maglev can operate at the interval of 10, 20, or 30 minutes. The
trains (or maglevs) with different intervals are considered to be distinct items in our experiment.
Figure 3 (a) shows the empirical choice probability for each choice set. Each choice set consists of
a train with a particular interval (blue, shaded) and a maglev with a particular interval (red, mesh)
possibly with a car (yellow, circles). The interval of the maglev varies as is indicated at the bottom
of the figure. The interval of the train is indicated at the left side of the figure. For each combination
of the intervals of the train and the maglev, there are two choice sets, with or without a car.
We evaluate the accuracy of the RBM choice model in predicting the choice probability for an
arbitrary choice set, when the RBM choice model is trained with the data of the choice for the
remaining 17 choice sets (i.e., we have 18 test cases). We train the RBM choice model (or the
MLM) by the use of discriminative training with stochastic gradient descent using the mini-batch
of size 50 and the learning rate of ? = 0.1 (see Appendix A.1).Each run of the evaluation uses the
entire training dataset 50 times for training, and the evaluation is repeated five times by varying the
initial values of the parameters. The elements
independently with samples
p
p of T and U are initialized
from the uniform distribution on [?10/ max(|I|, |K|), ?10/ max(|I|, |K|)], where |I| = 7 is
the number of items under consideration, and |K| is the number of hidden nodes. The elements of b
are initialized with samples from the uniform distribution on [?1, 1].
Figure 3 (b) shows the Kullback-Leibler (KL) divergence between the predicted distribution of the
choice and the corresponding true distribution. The dots connected with a solid line show the the
6
Train120
0.20
0.15
0.10
0.05
0.00
Maglev10
Maglev20
Maglev30
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Train60
Train30
Train60
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Maglev10
Maglev20
Maglev30
Car
Maglev10
Maglev20
0
1
2
4
8
Number of hidden units
16
(b) Error
Train120
Train120
Train60
Train30
(a) Dataset
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Test
0.25
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Car
Average KL divergence
Train60
Train30
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1.0
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Train30
Train60
Train120
Maglev10
Maglev20
Maglev30
Car
Maglev10
Maglev20
Maglev30
(d) MLM
Figure 3: Dataset (a), the predictive error of the RBM choice model against the number of hidden
units (b), and the choice probabilities learned by the RBM choice model (c) and the MLM (d).
average KL divergence over all of the 18 test cases and five runs with varying initialization. The
average KL divergence is also evaluated for training data and is shown with a dashed line. The
confidence interval represents the corresponding standard deviation. The wide confidence interval is
largely due to the variance between test instances (see Figure 4 in the appendix.The horizontal axis
shows the number of the hidden units in the RBM choice model, where zero hidden units correspond
to the MLM. The average KL divergence is reduced from 0.12 for the MLM to 0.02 for the RBM
choice model with 16 hidden units, an improvement by a factor of six.
Figure 3 (c)-(d) shows the choice probabilities given by (a) the RBM choice model with 16 hidden
units and (b) the MLM, after these models are trained for the test case where the choice set consists
of the train with 30-minute interval (Train30) and the maglev with 20-minute interval (Maglev20).
Observe that the RBM choice model gives the choice probabilities that are close to the true choice
probabilities shown in Figure 3 (a), while the MLM has difficulty in fitting these choice probabilities.
Taking a closer look at Figure 3 (a), we can observe that the MLM is fundamentally incapable of
learning this dataset. For example, Train30 is more popular than Maglev20 for people who do not
own cars, while the preference is reversed for car owners (i.e., the attraction effect). The attraction
effect can also be seen for the combination of Maglev30 and Train60. As we have discussed in
Section 3, the MLM cannot represent such attraction effects, but the RBM choice model can.
5
Related work
We now review the prior work related to our contributions. We will see that all of the existing
choice models either cannot represent at least one of the typical choice phenomena or do not have
systematic training algorithms. We will also see that the prior work has analyzed choice models
with respect to whether those choice models can represent typical choice phenomena or others but
only in specific cases of specific strength. On the contrary, our analysis shows that the RBM choice
model can represent the typical choice phenomena for all cases of the specified strength.
A majority of the prior work on the choice model is about the MLM and its variants such as the
hierarchical MLM [5], the multinomial probit model [6], and, generally, random utility models [17].
7
In particular, the attraction effect cannot be represented by these variants of the MLM [13]. In
general, when the choice probability depends only on the values that are determined independently
for each item (e.g., the models of [3, 7]), none of the typical choice phenomena can be represented
[18]. Recently, Hruschka has proposed a choice model based on an RBM [9], but his choice model
cannot represent any of the typical choice phenomena, because the corresponding choice rate is
independent of the choice set. It is thus nontrivial how we use the RBM as a choice model in such a
way that the typical choice phenomena can be represented. In [11], a hierarchical Bayesian choice
model is shown to represent the attraction effect in a specific case.
There also exist choice models that have been numerically shown to represent all of the typical
choice phenomena for some specific cases. For example, sequential sampling models, including
the decision field theory [4] and the leaky competing accumulator model [19], are meant to directly
mimic the cognitive process of the human making a choice [12]. However, no paper has shown an
algorithm that can train a sequential sampling model in such a way that the trained model exhibits the
typical choice phenomena. Shenoy and Yu propose a hierarchical Bayesian model to represent the
three typical choice phenomena [16]. Although they perform inferences of the posterior distributions
that are needed to compute the choice probabilities with their model, they do not show how to train
their model to fit the choice probabilities to given data. Their experiments show that their model
represents the typical choice phenomena in particular cases, where the parameters of the model are
set manually. Rieskamp et al. classify choice models according to whether a choice model can never
represent a certain phenomenon or can do so in some cases to some degree [13]. The phenomena
studied in [13] are not limited to the typical choice phenomena, but they list the typical choice
phenomena as the ones that are robust and significant. Also, Otter et al. exclusively study all of the
typical choice phenomena [12].
Luce is a pioneer of the formal analysis of choice models, which however is largely qualitative [10].
For example, Lemma 3 of [10] can tell us whether a given choice model satisfies the IIA in (16)
for all cases or it violates the IIA for some cases to some degree. We address the new question of
to what degree a choice model can represent each of the typical choice phenomena (e.g., to what
degree the RBM choice model can violate the IIA).
Finally, our theorems can be contrasted with the universal approximation theorem of RBMs, which
states that an arbitrary distribution can be approximated arbitrarily closely with a sufficient number
of hidden units [15, 8]. This is in contrast to our theorems, which show that a single hidden unit
suffices to represent the typical choice phenomena of the strength that is specified in the theorems.
6
Conclusion
The RBM choice model is developed to represent the typical choice phenomena that have been
reported frequently in the literature of cognitive psychology and related areas. Our work motivates a
new direction of research on using RBMs to model such complex behavior of humans. Particularly
interesting behavior includes the one that is considered to be irrational or the one that results from
cognitive biases (see e.g. [1]). The advantages of the RBM choice model that are demonstrated in
this paper include their flexibility in representing complex behavior and the availability of effective
training algorithms.
The RBM choice model can incorporate the attributes of the items in its parameters. Specifically,
one can represent the parameters of the RBM choice model as functions of uX , the attributes of
X ? I analogously to the MLM, where bX can be represented as bX = ? ? uX as we have discussed
after (2). The focus of this paper is in designing the fundamental structure of the RBM choice model
and analyzing its fundamental properties, and the study about the RBM choice model with attributes
will be reported elsewhere. Although the attributes are important for generalization of the RBM
model to unseen items, our experiments suggest that the RBM choice model, without attributes, can
learn the typical choice phenomena from a given choice set and generalize it to unseen choice sets.
Acknowledgements
A part of this research is supported by JST, CREST.
8
References
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9
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4,728 | 5,281 | Deep Fragment Embeddings for Bidirectional Image
Sentence Mapping
Andrej Karpathy
Armand Joulin
Li Fei-Fei
Department of Computer Science, Stanford University, Stanford, CA 94305, USA
{karpathy,ajoulin,feifeili}@cs.stanford.edu
Abstract
We introduce a model for bidirectional retrieval of images and sentences through
a deep, multi-modal embedding of visual and natural language data. Unlike previous models that directly map images or sentences into a common embedding
space, our model works on a finer level and embeds fragments of images (objects) and fragments of sentences (typed dependency tree relations) into a common space. We then introduce a structured max-margin objective that allows our
model to explicitly associate these fragments across modalities. Extensive experimental evaluation shows that reasoning on both the global level of images and
sentences and the finer level of their respective fragments improves performance
on image-sentence retrieval tasks. Additionally, our model provides interpretable
predictions for the image-sentence retrieval task since the inferred inter-modal
alignment of fragments is explicit.
1
Introduction
There is significant value in the ability to associate natural language descriptions with images. Describing the contents of images is useful for automated image captioning and conversely, the ability
to retrieve images based on natural language queries has immediate image search applications. In
particular, in this work we are interested in training a model on a set of images and their associated
natural language descriptions such that we can later rank a fixed set of withheld sentences given an
image query, and vice versa.
This task is challenging because it requires detailed understanding of the content of images, sentences and their inter-modal correspondence. Consider an example sentence query, such as ?A dog
with a tennis ball is swimming in murky water? (Figure 1). In order to successfully retrieve a corresponding image, we must accurately identify all entities, attributes and relationships present in the
sentence and ground them appropriately to a complex visual scene.
Our primary contribution is in formulating a structured, max-margin objective for a deep neural network that learns to embed both visual and language data into a common, multimodal space. Unlike
previous work that embeds images and sentences, our model breaks down and embeds fragments of
images (objects) and fragments of sentences (dependency tree relations [1]) in a common embedding space and explicitly reasons about their latent, inter-modal correspondences. Extensive empirical evaluation validates our approach. In particular, we report dramatic improvements over state of
the art methods on image-sentence retrieval tasks on Pascal1K [2], Flickr8K [3] and Flickr30K [4]
datasets. We make our code publicly available.
2
Related Work
Image Annotation and Image Search. There is a growing body of work that associates images and
sentences. Some approaches focus on the direction of describing the contents of images, formulated
either as a task of mapping images to a fixed set of sentences written by people [5, 6], or as a task of
automatically generating novel captions [7, 8, 9, 10, 11, 12]. More closely related to our motivation
are methods that allow natural bi-drectional mapping between the two modalities. Socher and FeiFei [13] and Hodosh et al. [3] use Kernel Canonical Correlation Analysis to align images and
sentences, but their method is not easily scalable since it relies on computing kernels quadratic in
1
Figure 1: Our model takes a dataset of
images and their sentence descriptions
and learns to associate their fragments.
In images, fragments correspond to object detections and scene context. In sentences, fragments consist of typed dependency tree [1] relations.
number of images and sentences. Farhadi et al. [5] learn a common meaning space, but their method
is limited to representing both images and sentences with a single triplet of (object, action, scene).
Zitnick et al. [14] use a Conditional Random Field to reason about spatial relationships in cartoon
scenes and their relation to natural language descriptions. Finally, joint models of language and
perception have also been explored in robotics settings [15].
Multimodal Representation Learning. Our approach falls into a general category of learning
from multi-modal data. Several probabilistic models for representing joint multimodal probability
distributions over images and sentences have been developed, using Deep Boltzmann Machines [16],
log-bilinear models [17], and topic models [18, 19]. Ngiam et al. [20] described an autoencoder
that learns audio-video representations through a shared bottleneck layer. More closely related to
our task and approach is the work of Frome et al. [21], who introduced a model that learns to
map images and words to a common semantic embedding with a ranking cost. Adopting a similar
approach, Socher et al. [22] described a Dependency Tree Recursive Neural Network that puts
entire sentences into correspondence with visual data. However, these methods reason about the
image only on the global level using a single, fixed-sized representation from the top layer of a
Convolutional Neural Network as a description for the entire image. In contrast, our model explicitly
reasons about objects that make up a complex scene.
Neural Representations for Images and Natural Language. Our model is a neural network
that is connected to image pixels on one side and raw 1-of-k word representations on the other.
There have been multiple approaches for learning neural representations in these data domains. In
Computer Vision, Convolutional Neural Networks (CNNs) [23] have recently been shown to learn
powerful image representations that support state of the art image classification [24, 25, 26] and
object detection [27, 28]. In language domain, several neural network models have been proposed
to learn word/n-gram representations [29, 30, 31, 32, 33, 34], sentence representations [35] and
paragraph/document representations [36].
3
Proposed Model
Learning and Inference. Our task is to retrieve relevant images given a sentence query, and conversely, relevant sentences given an image query. We train our model on a set of N images and N
corresponding sentences that describe their content (Figure 2). Given this set of correspondences,
we learn the weights of a neural network with a structured loss to output a high score when a compatible image-sentence pair is fed through the network, and low score otherwise. Once the training is
complete, all training data is discarded and the network is evaluated on a withheld set of images and
sentences. The evaluation scores all image-sentence pairs in the test set, sorts the images/sentences
in order of decreasing score and records the location of a ground truth result in the list.
Fragment Embeddings. Our core insight is that images are complex structures that are made
up of multiple entities that the sentences make explicit references to. We capture this intuition
directly in our model by breaking down both images and sentences into fragments and reason about
their alignment. In particular, we propose to detect objects as image fragments and use sentence
dependency tree relations [1] as sentence fragments (Figure 2).
Objective. We will compute the representation of both image and sentence fragments with a neural
network, and interpret the top layer as high-dimensional vectors embedded in a common multimodal space. We will think of the inner product between these vectors as a fragment compatibility
score, and compute the global image-sentence score as a fixed function of the scores of their respective fragments. Intuitively, an image-sentence pair will obtain a high global score if the sentence
fragments can each be confidently matched to some fragment in the image. Finally, we will learn
the weights of the neural networks such that the true image-sentence pairs achieve a score higher
(by a margin) than false image-sentence pairs.
2
Figure 2: Computing the Fragment and image-sentence similarities. Left: CNN representations (green) of
detected objects are mapped to the fragment embedding space (blue, Section 3.2). Right: Dependency tree
relations in the sentence are embedded (Section 3.1). Our model interprets inner products (shown as boxes)
between fragments as a similarity score. The alignment (shaded boxes) is latent and inferred by our model
(Section 3.3.1). The image-sentence similarity is computed as a fixed function of the pairwise fragment scores.
We first describe the neural networks that compute the Image and Sentence Fragment embeddings.
Then we discuss the objective function, which is composed of the two aforementioned objectives.
3.1 Dependency Tree Relations as Sentence Fragments
We would like to extract and represent the set of visually identifiable entities described in a sentence.
For instance, using the example in Figure 2, we would like to identify the entities (dog, child)
and characterise their attributes (black, young) and their pairwise interactions (chasing). Inspired
by previous work [5, 22] we observe that a dependency tree of a sentence provides a rich set of
typed relationships that can serve this purpose more effectively than individual words or bigrams.
We discard the tree structure in favor of a simpler model and interpret each relation (edge) as an
individual sentence fragment (Figure 2, right shows 5 example dependency relations). Thus, we
represent every word using 1-of-k encoding vector w using a dictionary of 400,000 words and map
every dependency triplet (R, w1 , w2 ) into the embedding space as follows:
W e w1
s = f WR
+ bR .
(1)
W e w2
Here, We is a d ? 400, 000 matrix that encodes a 1-of-k vector into a d-dimensional word vector
representation (we use d = 200). We fix We to weights obtained through an unsupervised objective
described in Huang et al. [34]. Note that every relation R can have its own set of weights WR and
biases bR . We fix the element-wise nonlinearity f (.) to be the Rectified Linear Unit (ReLU), which
computes x ? max(0, x). The size of the embedded space is cross-validated, and we found that
values of approximately 1000 generally work well.
3.2 Object Detections as Image Fragments
Similar to sentences, we wish to extract and describe the set of entities that images are composed of.
Inspired by prior work [7], as a modeling assumption we observe that the subject of most sentence
descriptions are attributes of objects and their context in a scene. This naturally motivates the use of
objects and the global context as the fragments of an image. In particular, we follow Girshick et al.
[27] and detect objects in every image with a Region Convolutional Neural Network (RCNN). The
CNN is pre-trained on ImageNet [37] and finetuned on the 200 classes of the ImageNet Detection
Challenge [38]. We use the top 19 detected locations and the entire image as the image fragments
and compute the embedding vectors based on the pixels Ib inside each bounding box as follows:
v = Wm [CNN?c (Ib )] + bm ,
(2)
where CNN(Ib ) takes the image inside a given bounding box and returns the 4096-dimensional
activations of the fully connected layer immediately before the classifier. The CNN architecture is
identical to the one described in Girhsick et al. [27]. It contains approximately 60 million parameters
?c and closely resembles the architecture of Krizhevsky et al [25].
3.3 Objective Function
We are now ready to formulate the objective function. Recall that we are given a training set of N
images and corresponding sentences. In the previous sections we described parameterized functions
that map every sentence and image to a set of fragment vectors {s} and {v}, respectively. All
parameters of our model are contained in these two functions. As shown in Figure 2, our model
3
Figure 3: The two objectives for a
batch of 2 examples. Left: Rows represent fragments vi , columns sj . Every square shows an ideal scenario of
yij = sign(viT sj ) in the MIL objective. Red boxes are yij = ?1.
Yellow indicates members of positive bags that happen to currently
be yij = ?1. Right: The scores
are accumulated with Equation 6 into
image-sentence score matrix Skl .
then interprets the inner product viT sj between an image fragment vi and a sentence fragment sj as
a similarity score, and computes the image-sentence similarity as a fixed function of the scores of
their respective fragments.
We are motivated by two criteria in designing the objective function. First, the image-sentence
similarities should be consistent with the ground truth correspondences. That is, corresponding
image-sentence pairs should have a higher score than all other image-sentence pairs. This will
be enforced by the Global Ranking Objective. Second, we introduce a Fragment Alignment
Objective that explicitly learns the appearance of sentence fragments (such as ?black dog?) in the
visual domain. Our full objective is the sum of these two objectives and a regularization term:
C(?) = CF (?) + ?CG (?) + ?||?||22 ,
(3)
where ? is a shorthand for parameters of our neural network (? = {We , WR , bR , Wm , bm , ?c }) and
? and ? are hyperparameters that we cross-validate. We now describe both objectives in more detail.
3.3.1 Fragment Alignment Objective
The Fragment Alignment Objective encodes the intuition that if a sentence contains a fragment
(e.g.?blue ball?, Figure 3), at least one of the boxes in the corresponding image should have a high
score with this fragment, while all the other boxes in all the other images that have no mention of
?blue ball? should have a low score. This assumption can be violated in multiple ways: a triplet
may not refer to anything visually identifiable in the image. The box that the triplet refers to may
not be detected by the RCNN. Lastly, other images may contain the described visual concept but
its mention may omitted in the associated sentence descriptions. Nonetheless, the assumption is
still satisfied in many cases and can be used to formulate a cost function. Consider an (incomplete)
Fragment Alignment Objective that assumes a dense alignment between every corresponding image
and sentence fragments:
XX
C0 (?) =
max(0, 1 ? yij viT sj ).
(4)
i
j
Here, the sum is over all pairs of image and sentence fragments in the training set. The quantity viT sj
can be interpreted as the alignment score of visual fragment vi and sentence fragment sj . In this
incomplete objective, we define yij as +1 if fragments vi and sj occur together in a corresponding
image-sentence pair, and ?1 otherwise. Intuitively, C0 (?) encourages scores in red regions of Figure
3 to be less than -1 and scores along the block diagonal (green and yellow) to be more than +1.
Multiple Instance Learning extension. The problem with the objective C0 (?) is that it assumes
dense alignment between all pairs of fragments in every corresponding image-sentence pair. However, this is hardly ever the case. For example, in Figure 3, the ?boy playing? triplet refers to only
one of the three detections. We now describe a Multiple Instance Learning (MIL) [39] extension
of the objective C0 that attempts to infer the latent alignment between fragments in corresponding
image-sentence pairs. Concretely, for every triplet we put image fragments in the associated image into a positive bag, while image fragments in every other image become negative examples.
Our precise formulation is inspired by the mi-SVM [40], which is a simple and natural extension
of a Support Vector Machine to a Multiple Instance Learning setting. Instead of treating the yij as
constants, we minimize over them by wrapping Equation 4 as follows:
4
CF (?) = min C0 (?)
yij
s.t.
X yij + 1
? 1 ?j
2
i?p
(5)
j
yij = ?1 ?i, j s.t. mv (i) 6= ms (j) and yij ? {?1, 1}
Here, we define pj to be the set of image fragments in the positive bag for sentence fragment j.
mv (i) and ms (j) return the index of the image and sentence (an element of {1, . . . , N }) that the
fragments vi and sj belong to. Note that the inequality simply states that at least one of the yij
should be positive for every sentence fragment j (i.e. at least one green box in every column in
Figure 3). This objective cannot be solved efficiently [40] but a commonly used heuristic is to set
yij = sign(viT sj ). If the constraint is not satisfied for any positive bag (i.e. all scores were below
zero), the highest-scoring item in the positive bag is set to have a positive label.
3.3.2 Global Ranking Objective
Recall that the Global Ranking Objective ensures that the computed image-sentence similarities are
consistent with the ground truth annotation. First, we define the image-sentence alignment score to
be the average thresholded score of their pairwise fragment scores:
XX
1
Skl =
max(0, viT sj ).
(6)
|gk |(|gl | + n) i?g j?g
k
l
Here, gk is the set of image fragments in image k and gl is the set of sentence fragments in sentence
l. Both k, l range from 1, . . . , N . We truncate scores at zero because in the mi-SVM objective, scores
greater than 0 are considered correct alignments and scores less than 0 are considered to be incorrect
alignments (i.e. false members of a positive bag). In practice, we found that it was helpful to add
a smoothing term n, since short sentences can otherwise have an advantage (we found that n = 5
works well and that this setting is not very sensitive). The Global Ranking Objective then becomes:
i
X
XhX
max(0, Slk ? Skk + ?) .
(7)
CG (?) =
max(0, Skl ? Skk + ?) +
k
l
l
|
{z
}
rank images
|
{z
rank sentences
}
Here, ? is a hyperparameter that we cross-validate. The objective stipulates that the score for true
image-sentence pairs Skk should be higher than Skl or Slk for any l 6= k by at least a margin of ?.
3.4 Optimization
We use Stochastic Gradient Descent (SGD) with mini-batches of 100, momentum of 0.9 and make
20 epochs through the training data. The learning rate is cross-validated and annealed by a fraction
of ?0.1 for the last two epochs. Since both Multiple Instance Learning and CNN finetuning benefit
from a good initialization, we run the first 10 epochs with the fragment alignment objective C0
and CNN weights ?c fixed. After 10 epochs, we switch to the full MIL objective CF and begin
finetuning the CNN. The word embedding matrix We is kept fixed due to overfitting concerns. Our
implementation runs at approximately 1 second per batch on a standard CPU workstation.
4
Experiments
Datasets. We evaluate our image-sentence retrieval performance on Pascal1K [2], Flickr8K [3] and
Flickr30K [4] datasets. The datasets contain 1,000, 8,000 and 30,000 images respectively and each
image is annotated using Amazon Mechanical Turk with 5 independent sentences.
Sentence Data Preprocessing. We did not explicitly filter, spellcheck or normalize any of the
sentences for simplicity. We use the Stanford CoreNLP parser to compute the dependency trees
for every sentence. Since there are many possible relation types (as many as hundreds), due to
overfitting concerns and practical considerations we remove all relation types that occur less than
1% of the time in each dataset. In practice, this reduces the number of relations from 136 to 16 in
Pascal1K, 170 to 17 in Flickr8K, and from 212 to 21 in Flickr30K. Additionally, all words that are
not found in our dictionary of 400,000 words [34] are discarded.
Image Data Preprocessing. We use the Caffe [41] implementation of the ImageNet Detection
RCNN model [27] to detect objects in all images. On our machine with a Tesla K40 GPU,
the RCNN processes one image in approximately 25 seconds. We discard the predictions for
200 ImageNet detection classes and only keep the 4096-D activations of the fully connect layer
immediately before the classifier at all of the top 19 detected locations and from the entire image.
5
Model
Random Ranking
Socher et al. [22]
kCCA. [22]
DeViSE [21]
SDT-RNN [22]
Our model
R@1
4.0
23.0
21.0
17.0
25.0
39.0
Pascal1K
Image Annotation
R@5 R@10 Mean r
9.0
12.0
71.0
45.0
63.0
16.9
47.0
61.0
18.0
57.0
68.0
11.9
56.0
70.0
13.4
68.0
79.0
10.5
R@1
1.6
16.4
16.4
21.6
25.4
23.6
Image Search
R@5 R@10
5.2
10.6
46.6
65.6
41.4
58.0
54.6
72.4
65.2
84.4
65.2
79.8
Mean r
50.0
12.5
15.9
9.5
7.0
7.6
Table 1: Pascal1K ranking experiments. R@K is Recall@K (high is good). Mean r is the mean rank (low is
good). Note that the test set only consists of 100 images.
Model
Random Ranking
Socher et al. [22]
DeViSE [21]
SDT-RNN [22]
Fragment Alignment Objective
Global Ranking Objective
(?) Fragment + Global
? ? Images: Fullframe Only
? ? Sentences: BOW
? ? Sentences: Bigrams
Our model (? + MIL)
* Hodosh et al. [3]
* Our model (? + MIL)
R@1
0.1
4.5
4.8
6.0
7.2
5.8
12.5
5.9
9.1
8.7
12.6
8.3
9.3
Flickr8K
Image Annotation
R@5 R@10 Med r
0.6
1.1
631
18.0
28.6
32
16.5
27.3
28
22.7
34.0
23
21.9
31.8
25
21.8
34.8
20
29.4
43.8
14
19.2
27.3
34
25.9
40.7
17
28.5
41.0
16
32.9
44.0
14
21.6
30.3
34
24.9
37.4
21
R@1
0.1
6.1
5.9
6.6
5.9
7.5
8.6
5.2
6.9
8.5
9.7
7.6
8.8
Image Search
R@5 R@10
0.5
1.0
18.5
29.0
20.1
29.6
21.6
31.7
20.0
30.3
23.4
35.0
26.7
38.7
17.6
26.5
22.4
34.0
25.2
37.0
29.6
42.5
20.7
30.1
27.9
41.3
Med r
500
29
29
25
26
21
17
32
23
20
15
38
17
Table 2: Flickr8K experiments. R@K is Recall@K (high is good). Med r is the median rank (low is good).
The starred evaluation criterion (*) in [3] is slightly different since it discards 4,000 out of 5,000 test sentences.
Evaluation Protocol for Bidirectional Retrieval. For Pascal1K we follow Socher et al. [22] and
use 800 images for training, 100 for validation and 100 for testing. For Flickr datasets we use
1,000 images for validation, 1,000 for testing and the rest for training (consistent with [3]). We
compute the dense image-sentence similarity Skl between every image-sentence pair in the test set
and rank images and sentences in order of decreasing score. For both Image Annotation and Image
Search, we report the median rank of the closest ground truth result in the list, as well as Recall@K
which computes the fraction of times the correct result was found among the top K items. When
comparing to Hodosh et al. [3] we closely follow their evaluation protocol and only keep a subset
of N sentences out of total 5N (we use the first sentence out of every group of 5).
4.1 Comparison Methods
SDT-RNN. Socher et al. [22] embed a fullframe CNN representation with the sentence representation from a Semantic Dependency Tree Recursive Neural Network (SDT-RNN). Their loss matches
our global ranking objective. We requested the source code of Socher et al. [22] and substituted the
Flickr8K and Flick30K datasets. To better understand the benefits of using our detection CNN and
for a more fair comparison we also train their method with our CNN features. Since we have multiple objects per image, we average representations from all objects with detection confidence above a
(cross-validated) threshold. We refer to the exact method of Socher et al. [22] with a single fullframe
CNN as ?Socher et al?, and to their method with our combined CNN features as ?SDT-RNN?.
DeViSE. The DeViSE [21] source code is not publicly available but their approach is a special case
of our method with the following modifications: we use the average (L2-normalized) word vectors
as a sentence fragment, the average CNN activation of all objects above a detection threshold (as
discussed in case of SDT-RNN) as an image fragment and only use the global ranking objective.
4.2 Quantitative Evaluation
Our model outperforms previous methods. Our full method consistently outperforms previous
methods on Flickr8K (Table 2) and Flickr30K (Table 3) datasets. On Pascal1K (Table 1) the
SDT-RNN appears to be competitive on Image Search.
Fragment and Global Objectives are complementary. As seen in Tables 2 and 3, both objectives
perform well independently, but benefit from the combination. Note that the Global Objective
performs consistently better, possibly because it directly minimizes the evaluation criterion (ranking
6
Model
Random Ranking
DeViSE [21]
SDT-RNN [22]
Fragment Alignment Objective
Global Ranking Objective
(?) Fragment + Global
Our model (? + MIL)
Our model + Finetune CNN
R@1
0.1
4.5
9.6
11
11.5
12.0
14.2
16.4
Flickr30K
Image Annotation
R@5 R@10 Med r
0.6
1.1
631
18.1
29.2
26
29.8
41.1
16
28.7
39.3
18
33.2
44.9
14
37.1
50.0
10
37.7
51.3
10
40.2
54.7
8
R@1
0.1
6.7
8.9
7.6
8.8
9.9
10.2
10.3
Image Search
R@5 R@10
0.5
1.0
21.9
32.7
29.8
41.1
23.8
34.5
27.6
38.4
30.5
43.2
30.8
44.2
31.4
44.5
Med r
500
25
16
22
17
14
14
13
Table 3: Flickr30K experiments. R@K is Recall@K (high is good). Med r is the median rank (low is good).
Figure 4: Qualitative Image Annotation results. Below each image we show the top 5 sentences (among a set
of 5,000 test sentences) in descending confidence. We also show the triplets for the top sentence and connect
each to the detections with the highest compatibility score (indicated by lines). The numbers next to each triplet
indicate the matching fragment score. We color a sentence green if it correct and red otherwise.
cost), while the Fragment Alignment Objective only does so indirectly.
Extracting object representations is important. Using only the global scene-level CNN representation as a single fragment for every image leads to a consistent drop in performance, suggesting
that a single fullframe CNN alone is inadequate in effectively representing the images. (Table 2)
Dependency tree relations outperform BoW/bigram representations. We compare to a simpler
Bag of Words (BoW) baseline to understand the contribution of dependency relations. In BoW
baseline we iterate over words instead of dependency triplets when creating bags of sentence
fragments (set w1 = w2 in Equation1). As can be seen in the Table 2, this leads to a consistent drop
in performance. This drop could be attributed to a difference between using one word or two words
at a time, so we also compare to a bigram baseline where the words w1 , w2 in Equation 1 refer to
consecutive words in a sentence, not nodes that share an edge in the dependency tree. Again, we
observe a consistent performance drop, which suggests that the dependency relations provide useful
structure that the neural network takes advantage of.
Finetuning the CNN helps on Flickr30K. Our end-to-end Neural Network approach allows us to
backpropagate gradients all the way down to raw data (pixels or 1-of-k word encodings). In particular, we observed additional improvements on the Flickr30K dataset (Table 3) when we finetune the
CNN. Training the CNN improves the validation error for a while but the model eventually starts to
overfit. Thus, we found it critical to keep track of the validation error and freeze the model before it
overfits. We were not able to improve the validation performance on Pascal1K and Flickr8K datasets
and suspect that there is an insufficient amount of training data.
4.3
Qualitative Experiments
Interpretable Predictions. We show some example sentence retrieval results in Figure 4. The
alignment in our model is explicitly inferred on the fragment level, which allows us to interpret the
scores between images and sentences. For instance, in the last image it is apparent that the model
retrieved the top sentence because it erroneously associated a mention of a blue person to the blue
flag on the bottom right of the image.
Fragment Alignment Objective trains attribute detectors. The detection CNN is trained to
predict one of 200 ImageNet Detection classes, so it is not clear if the representation is powerful
enough to support learning of more complex attributes of the objects or generalize to novel classes.
To see whether our model successfully learns to predict sentence triplets, we fix a triplet vector and
7
Figure 5: We fix a triplet and retrieve the highest scoring image fragments in the test set. Note that ball, person
and dog are ImageNet Detection classes but their visual properties (e.g. soccer, standing, snowboarding, black)
are not. Jackets and rocky scenes are not ImageNet Detection classes. Find more in supplementary material.
search for the highest scoring boxes in the test set. Qualitative results shown in Figure 5 suggest
that the model is indeed capable of generalizing to more fine-grained subcategories (such as ?black
dog?, ?soccer ball?) and to out of sample classes such as ?rocky terrain? and ?jacket?.
Limitations. Our model is subject to multiple limitations. From a modeling perspective, the use of
edges from a dependency tree is simple, but not always appropriate. First, a single complex phrase
that describes a single visual entity can be split across multiple sentence fragments. For example,
?black and white dog? is parsed as two relations (CONJ, black, white) and (AMOD, white, dog).
Conversely, there are many dependency relations that don?t have a clear grounding in the image (for
example ?each other?). Furthermore, phrases such as ?three children playing? that describe some
particular number of visual entiries are not modeled. While we have shown that the relations give
rise to more powerful representations than words or bigrams, a more careful treatment of sentence
fragments will likely lead to further improvements. On the image side, the non-maximum suppression in the RCNN can sometimes detect, for example, multiple people inside one person. Since the
model does not take into account any spatial information associated with the detections, it is hard
for it to disambiguate between two distinct people or spurious detections of one person.
5
Conclusions
We addressed the problem of bidirectional retrieval of images and sentences. Our neural network
learns a multi-modal embedding space for fragments of images and sentences and reasons about
their latent, inter-modal alignment. We have shown that our model significantly improves the retrieval performance on image sentence retrieval tasks compared to previous work. Our model also
produces interpretable predictions. In future work we hope to develop better sentence fragment
representations, incorporate spatial reasoning, and move beyond bags of fragments.
Acknowledgments. We thank Justin Johnson and Jon Krause for helpful comments and discussions.
We gratefully acknowledge the support of NVIDIA Corporation with the donation of the GPUs used
for this research. This research is supported by an ONR MURI grant, and NSF ISS-1115313.
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9
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4,729 | 5,282 | Recursive Context Propagation Network for Semantic
Scene Labeling
Abhishek Sharma
University of Maryland
College Park, MD
[email protected]
Oncel Tuzel
Ming-Yu Liu
Mitsubishi Electric Research Labs (MERL)
Cambridge, MA
{oncel,mliu}@merl.com
Abstract
We propose a deep feed-forward neural network architecture for pixel-wise semantic scene labeling. It uses a novel recursive neural network architecture for
context propagation, referred to as rCPN. It first maps the local visual features
into a semantic space followed by a bottom-up aggregation of local information
into a global representation of the entire image. Then a top-down propagation of
the aggregated information takes place that enhances the contextual information of
each local feature. Therefore, the information from every location in the image is
propagated to every other location. Experimental results on Stanford background
and SIFT Flow datasets show that the proposed method outperforms previous approaches. It is also orders of magnitude faster than previous methods and takes
only 0.07 seconds on a GPU for pixel-wise labeling of a 256 ? 256 image starting
from raw RGB pixel values, given the super-pixel mask that takes an additional
0.3 seconds using an off-the-shelf implementation.
1
Introduction
Semantic labeling aims at getting pixel-wise dense labeling of an image in terms of semantic concepts such as tree, road, sky, water, foreground objects etc. Mathematically, the problem can be
framed as a mapping from a set of nodes arranged on a 2D grid (pixels) to the semantic categories.
Typically, this task is broken down into two steps - feature extraction and inference. Feature extraction involves retrieving descriptive information useful for semantic labeling under varying illumination and view-point conditions. These features are generally color, texture or gradient based and
extracted from a local patch around each pixel. Inference step consists of predicting the labels of
the pixels using the extracted features. The rich diversity in the appearance of even simple concepts
(sky, water, grass) makes the semantic labeling very challenging. Surprisingly, human performance
is almost close to perfect on this task. This striking difference of performance has been a heated field
of research in vision community. Past experiences and recent research [1, 2, 3] have conclusively
established that the ability of humans to utilize the information from the entire image is the main
reason behind the large performance gap. Interestingly, [2, 3] have shown that human performance
in labeling a small local region (super-pixel) is worse than a computer when both are looking at only
that region of the image. Motivated from these observations, increasingly sophisticated inference
algorithms have been developed to utilize the information from the entire image. Conditional Random Fields (CRFs) [4] and Structured Support Vector Machines (SVMs) [5] are among the most
successful and widely used algorithms for inference.
We model the semantic labeling task as a mapping from the set of all pixels in an image I to the
corresponding label image Y. We have several design considerations: (1) the mapping should be
evaluated fast, (2) it should utilize the entire image such that every location influences the labeling of
every other location, (3) mapping parameters should be learned from the training data, (4) it should
scale to different image sizes. In addition, good generalization requires limiting the capacity of
1
Semantic labels:
SKY
WATER
BLDG
BOAT
TREE
Figure 1: Conceptual illustration of recursive context propagation network (rCPN). rCPN recursively aggregates contextual information from local neighborhoods to the entire image and then
disseminates global context information back to individual local features. In this example, starting
from confusion between boat and building, the propagated context information helps resolve the
confusion by using the feature of the water segment.
the mapping while still utilizing the entire image information at once. For example, a simple fullyconnected-linear mapping from I to Y requires 4 Trillion parameters for an image of size 256?256,
but it will fail to generalize under practical conditions of limited training data.
Considering the requirements discussed above, we designed the mapping as a single feed-forward
neural network with carefully controlled capacity by parameter sharing. All the network parameters
are learned from the data and the feed-forward structure allows fast inference. The proposed network
can be functionally partitioned into two sub-networks: local feature extraction and recursive context
propagation.
As the name implies, local-feature extraction refers to the extraction of pixel- or region-wise visual
features for semantic labeling. We used the multi scale convolutional neural network (Multi-CNN)
architecture proposed in [6] to get pixel-wise features. Convolutional structure with shared parameters brings down the number of parameters for local feature extraction.
We propose a novel recursive context propagation network (rCPN), which, starting from the local
features, recursively aggregates contextual information from local neighborhoods up to the entire
image and then disseminates the aggregated information back to individual local features for better
semantic classification. rCPN is a recursive neural network with shared parameters through the parse
tree hierarchy. A conceptual illustration of this network is given in Figure 1. The scene consists of
three segments corresponding to a boat, a tree and a water/sky region. The nodes of the graph
(formed by a binary parse tree and its inversion) represent semantic description of the segments.
The distributions on the left are probable label distributions for the adjacent segments based on their
appearance. Initially (at the bottom), the boat can be confused as a white building, while looking
only at the bottom-left segment. The rCPN recursively combines two segment descriptions and
produces the semantic description of the combined segment. For example, as the tree is combined
with the boat, the belief that the combined segment includes a building increased since usually they
appear together in the images. Similarly, after we merge the water/sky segment description with
this segment description, the probability of the boat increased since the simultaneous occurrence
of water and building is rare. The middle node in the graph (root node of the segmentation tree)
2
corresponds to the semantic description of the entire image. After all the segment descriptions are
merged into a single holistic description of the entire image, this information is propagated to the
local regions. It is achieved by recursive updates of the semantic descriptions of the segments given
the descriptions of their parent segments. Finally, contextually enhanced descriptions of the leaf
nodes are used to label the segments. Note that, rCPN propagates segment semantic descriptions but
not the label distributions shown in the illustration.
Our work is influenced by Socher et al.?s work [7] that learns a non-linear mapping from feature
space to a semantic space, termed as semantic mapping. It is learned by optimizing a structure
prediction cost on the ground-truth parse trees of training images or sentences. Next, a classifier
is learned on the semantic mappings of the individual local features from the training images. At
test time, local features are projected to the semantic space using the learned semantic mapping
followed by classification. Therefore, only the information contained in each individual local feature
is used for labeling. In contrast, we use recursive bottom-top-bottom paths on randomly generated
parse trees to propagate contextual information from local regions to all other regions in the image.
Therefore, our approach uses entire image information for labeling each local region. Please see
experiments section for detailed comparison.
The main contributions of the proposed approach are:
? The proposed model is scalable. It is a combination of a CNN and a recursive neural network which is trained without using any human-designed features. In addition, convolution
and recursive structure allows scaling to arbitrary image sizes while still utilizing the entire
image content at once.
? We achieved state-of-the-art labeling accuracy on two important benchmarks while being
an order of magnitude faster than the existing methods due to feed-forward operations. It
takes only 0.07 seconds on GPU and 0.8 seconds on CPU for pixel-wise semantic labeling
of a 256 ? 256 image, with a given super-segmentation mask, that can be computed using
an off-the-shelf algorithm within 0.3 second.
? Proposed rCPN module can be used in conjunction with pre-computed features to propagate context information through the structure of an image (see experiments section) and
potentially for other structured prediction tasks.
2
Semantic labeling architecture
In this section we describe our semantic labeling architecture and discuss the design choices for
practical considerations. An illustration of this architecture is shown in Figure 2. The input image is
fed to a CNN, FCN N , which extracts local features per pixel. We then use a super-pixel tessellation
of the input image and average pool the local features within the same super-pixel. Following, we
use the proposed rCPN to recursively propagate the local information throughout the image using a
parse tree hierarchy and finally label the super-pixels.
2.1
Local feature extraction
We used the multi scale CNN architecture proposed in Farabet et al. [6] for extracting per pixel local
features. This network has three convolutional stages which are organized as 8 ? 8 ? 16 conv ?
2 ? 2 maxpool ? 7 ? 7 ? 64 conv ? 2 ? 2 maxpool ? 7 ? 7 ? 256 conv configuration, each maxpooling is non-overlapping. After each convolution we apply a rectified linear (ReLU) nonlinearity.
Unlike [6], we do not preprocess the input raw RGB images other than scaling it between 0 to 1,
and centering by subtracting 0.5. Tied filters are applied separately at three scales of the Gaussian
pyramid. The final feature maps at lower scales are spatially scaled up to the size of the feature map
at the highest scale and concatenated to get 256 ? 3 = 768 dimensional features per pixel. The
obtained pixel features are fed to a Softmax classifier for final classification. Please refer to [6] for
more details. After training, we drop the final Softmax layer and use the 768 dimensional features
as local features.
Note that the 768 dimensional concatenated output feature map is still 1/4th of the height and width
of the input image due to the max-pooling operations. To obtain the input size per pixel feature map
we either (1) shift the input image by one pixel on a 4 ? 4 grid to get 16 output feature maps that are
3
?
?
??
????
????
+
?1
????
??
????
??
????
?2
??
?12
????
????
?34
????
????
?????
????
?1
????
?12
SKY
BLDG
TREE
?34
??
WATER
BOAT
????
super-pixels
Local Feature Extraction
Recursive Context Propagation Network (rCPN)
Figure 2: Overview of semantic scene labeling architecture
combined to get the full-resolution image, or (2) scale-up each feature map by a factor of 4 using
bilinear interpolation. We refer to the later strategy as fast feature map generation in experiments
section.
Super-pixel representation: Although it is possible to do per pixel classification using the rCPN,
learning such a model is computationally intensive and the resulting network is too deep to propagate the gradients efficiently due to recursion. To reduce the complexity, we utilize a super-pixel
segmentation algorithm [8], which provides the desired number of super-pixels per image. This algorithm uses pairwise color similarity together with an entropy rate criteria to produce homogenous
super-pixels with roughly equal sizes. We average pool the local features within the same superpixel and retrieve s local features, {vi }i=1...s , one per super-pixel. In our experiments we used
s = 100 super-pixels per image.
2.2
Recursive context propagation network
rCPN consists of four neural networks: Fsem maps local features to the semantic space in which
the local information is propagated to other segments; Fcom recursively aggregates local information from smaller segments to larger segments through a parse tree hierarchy to capture a holistic
description of the image; Fdec recursively disseminates the holistic description to smaller segments
using the same parse tree; and Flab classifies the super-pixels utilizing the contextually enhanced
features.
Parse tree synthesis: Both for training and inference, the binary parse trees that are used for propagating information through the network are synthesized at random. We used a simple agglomerative
algorithm to synthesize the trees by combining sub-trees (starting from a single node) according to
the neighborhood information. To reduce the complexity and avoid degenerate solutions, the synthesis algorithm favors roughly balanced parse trees by greedily selecting sub-trees with smaller
heights at random. Note that, we use parse trees only as a tool to propagate the contextual information throughout the image. Therefore, we are not limited to the parse trees that represent an accurate
hierarchical segmentation of the image unlike [9, 7].
Semantic mapping network is a feed-forward neural network which maps the local features to the
dsem dimensional semantic vector space
xi = Fsem (vi ; ?sem ),
(1)
where ?sem is the model parameter. The aim of the semantic features is to capture a joint representation of the local features and the context, and being able to propagate this information through a
parse tree hierarchy to other super-pixels.
Combiner network is a recursive neural network which recursively maps the semantic features of
two child nodes (super-pixels) in the parse tree to obtain the semantic feature of the parent node
(combination of the two child nodes)
xi,j = Fcom ([xi , xj ]; ?com ).
4
(2)
Intuitively, combiner network attempts to aggregate the semantic content of the children nodes such
that the parent node becomes representative of its children. The information is recursively aggregated bottom-up from super-pixels to the root node through the parse tree. The semantic features of
the root node correspond to the holistic description of the entire image.
Decombiner network is a recursive neural network which recursively disseminates the context
information from the parent nodes to the children nodes throughout the parse tree hierarchy. This
network maps the semantic features of the child node and its parent to the contextually enhanced
feature of the child node
? i = Fdec ([?
x
xi,j , xi ]; ?dec ).
(3)
Since we start from the root feature of the entire image and apply the decombiner network top-down
recursively until we reach the super-pixel features, every super-pixel feature contains the contextual
information aggregated from the entire image. Therefore, it is influenced by every other super-pixel
in the image.
Labeler network is the final feed forward network which maps the contextually enhanced semantic
features (?
xi ) of each super-pixel to one of the semantic category labels
yj = Flab (?
xi ; ?lab ).
(4)
Contextually enhanced features contain both local and global context information, thereby leading
to better classification.
Side information: It is possible to input information to the recursive networks not only at the leaf
nodes but also at any level of the parse tree. The side information can encode the static knowledge
about the parse tree nodes and is not recurred through the tree. In our implementation we used
average x and y locations of the nodes and their sizes as the side information.
3
Learning
Proposed labeling architecture is a feed-forward neural network that can be trained end-to-end.
However, the recursion makes the depth of the neural network too deep for an efficient joint training.
Therefore, we first learn the CNN parameters (?CN N ) using the raw image and the ground truth per
pixel labels. The trained CNN model is used to extract super-pixel features followed by training of
rCPN (?rCP N = [?sem , ?com , ?dec , ?lab ]) to predict the ground truth super-pixel labels.
Feature extractor CNN is trained on a GPU using a publicly available implementation CAFFE [10].
In order to avoid over-fitting we used data augmentation and dropout. All the training images were
flipped horizontally to get twice the original images. We used dropout in the last layer with dropout
ratio equal to 0.5. Standard back-propagation for CNN is used with stochastic gradient descent
update scheme on mini-batches of 6 images, with weight decay (? = 5 ? 10?5 ) and momentum
(? = 0.9). It typically took 6-8 hours of training on a GPU as compared to 3-5 days training on a
CPU as reported in [6]. We found that simply using RGB images with ReLU units and dropout gave
slightly better pixel-wise accuracy as compared to [6].
rCPN parameters are trained using back-propagation through structure [11], which back-propagates
the error through the parse tree, from Flab to Fsem . The basic idea is to split the error message
at each node and propagate it to the children nodes. Limited memory BFGS [12] with line-search
is used for parameter updates using publicly available implementation1 . From each super-pixel we
obtained 5 different features by average pooling a random subset of pixels within the super-pixel (as
opposed to average pooling all the pixels), and used a different random parse tree for each set of
random feature, thus we increased our training data by a factor of 5. It typically took 600 to 1000
iterations for complete training.
4
Experiments
We extensively tested the proposed model on two widely used datasets for semantic scene labeling Stanford background [13] and SIFT Flow [14]. Stanford background dataset contains 715 color
images of outdoor scenes, it has 8 classes and the images are approximately 240 ? 320 pixels.
1
http://www.di.ens.fr/?mschmidt/Software/minFunc.html
5
We used the 572 train and 143 test image split provided by [7] for reporting the results. SIFT
Flow contains 2688, 256 ? 256 color images with 33 semantic classes. We experimented with the
train/test (2488/200) split provided by the authors of [15]. We have used three evaluation metrics Per pixel accuracy (PPA): ratio of the correct pixels to the total pixels in the test images; Mean
class accuracy (MCA): mean of the category-wise pixel accuracy; Time per image (Time): time
required to label an input image starting from the raw image input, we report our time on both GPU
and CPU.
The local feature extraction through Multi-CNN [6] encodes contextual information due to large
field of view (FOV); the FOV for 1, 1/2 and 1/4 scaled input images is 47 ? 47, 94 ? 94 and
188 ? 188 pixels, respectively. Therefore, we designed the experiments under single and multi scale
settings to assess rCPN?s contribution. Mutli-CNN + rCPN refers to the case where feature maps
from all the three scales (1,1/2 and 1/4), 3 ? 256 = 768 dimensional local feature, for each pixel
are used. Single-CNN + rCPN refers to the case where only the 256 feature maps corresponding
to the original resolution image are used. Evidently, the amount of contextual information in the
local features of Single-CNN is significantly lesser than that of Multi-CNN because of smaller FOV.
All the individual modules in rCPN, Fsem , Fcom , Fdec and Flab , are single layer neural networks
with ReLU non-linearity and dsem = 60 for all the experiments. We used 20 randomly generated
parse trees for each image and used voting for the final super-pixel labels. We did not optimize
these hyper-parameters and believe that parameter-tuning can further increase the performance. The
baseline is two-layer neural network with 60 hidden neurons classifier with Single-CNN or MultiCNN features of super-pixels and referred to as Multi/Single-CNN + Plain NN.
4.1
SIFT Flow dataset
We used 100 super-pixels per image obtained by method of [8]. The result on SIFT Flow database is
shown in Table 1. From the comparison it is clear that we outperform all the other previous methods
on pixel accuracy while being an order of magnitude faster. Farabet et al. [6] improved the mean
class accuracy by training a model based on the balanced class frequency. Since some of the classes
in SIFT Flow dataset are under represented, the class accuracies for them are very low. Therefore,
following [6], we also trained a balanced rCPN model that puts more weights on the errors for rare
classes as compared to the dominant ones, referred to as Multi-CNN + rCPN balanced. Smoothed
inverse frequency of the pixels of each category is used as the weights. Balanced training helped
improve our mean class accuracy from 33.6 % to 48.0 %, which is still slightly worse than [6] (48.0
% vs 50.8 %), but our pixel accuracy is higher (75.5 % vs 72.3 %). Multi-CNN + rCPN performed
better than Single-CNN + rCPN and both performed significantly better than Plain NN approaches,
because the later approaches do not utilize global contextual information. We also observed that the
relative improvement over Plain NN was more with Single-CNN features which uses less context
information than that of Multi-CNN.
4.2
Stanford background dataset
We used publicly available super-pixels provided by [7] with our CNN based local features to obtain
super-pixel features. A comparison of our results with previous approaches on Stanford background
database is shown in Table 2. We outperform previous approaches on all the performance metrics.
Interestingly, we observe that Single-CNN + rCPN performs better than Multi-CNN + rCPN for
pixel accuracy. We believe that it is due to over-fitting on high-dimensional Multi-CNN features
and relatively smaller training data size with only 572 images. Once again the improvement due to
rCPN over plain NN is more prominent in the case of Single-CNN features.
Model analysis: In this section, we analyze the performance of individual components of the proposed model. First, we use rCPN with hand-designed features to evaluate the performance of context
model alone, beyond the learned local features using CNN. We utilize the visual features and superpixels used in semantic mapping and CRF labeling framework [7, 13], and trained our rCPN module.
The results are presented in Table 3. We see that rCPN module significantly improves upon the existing context models, namely a CRF model used in [13] and semantic space proposed in [7]. In
addition, CNN based visual features improve over the hand-designed features.
Next, we analyze the performance of combiner and decombiner networks separately. To evaluate combiner network in isolation, we first obtain the semantic mapping (xi ) of each super-pixel?s
6
Table 1: SIFT Flow result
Table 2: Stanford background result
Method
PPA
MCA
Tighe, [15]
Liu, [14]
Singh, [16]
Eigen, [17]
Farabet, [6]
(Balanced), [6]
Tighe, [18]
Pinheiro, [19]
Single-CNN +
Plain NN
Multi-CNN +
Plain NN
Single-CNN +
rCPN
Multi-CNN +
rCPN
Multi-CNN +
rCPN Balanced
Multi-CNN +
rCPN Fast
77.0
76.7
79.2
77.1
78.5
72.3
78.6
77.7
30.1
NA
33.8
32.5
29.6
50.8
39.2
29.8
Time (s)
CPU/GPU
8.4 / NA
31 / NA
20 / NA
16.6 / NA
NA / NA
NA / NA
? 8.4 / NA
NA / NA
72.8
25.5
5.1/0.5
76.3
32.1
13.1/1.4
77.2
25.5
5.1/0.5
79.6
33.6
13.1/1.4
75.5
48.0
13.1/1.4
79.5
33.4
1.1/0.37
Method
PPA
MCA
Gould, [13]
Munoz, [20]
Tighe, [15]
Kumar, [21]
Socher, [7]
Lempitzky, [9]
Singh, [16]
Farabet, [6]
Eigen, [17]
Pinheiro, [19]
Single-CNN +
Plain NN
Multi-CNN +
Plain NN
Single-CNN +
rCPN
Multi-CNN +
rCPN
Multi-CNN +
rCPN Fast
76.4
76.9
77.5
79.4
78.1
81.9
74.1
81.4
75.3
80.2
NA
NA
NA
NA
NA
72.4
62.2
76.0
66.5
69.9
Time (s)
CPU/GPU
30 to 600 / NA
12 / NA
4 / NA
? 600 / NA
NA / NA
? 60 / NA
20 / NA
60.5 / NA
16.6 / NA
10.6 / NA
80.1
69.7
5.1/0.5
80.9
74.4
13.1/1.4
81.9
73.6
5.1/0.5
81.0
78.8
13.1/1.4
80.9
78.8
1.1/0.37
Table 3: Stanford hand-designed local feature
Method
PPA
2-layer NN [7]
76.1
CRF [13]
76.4
Semantic space [7]
78.1
proposed rCPN
81.4
visual feature using rCPN?s Fsem and append to it the root feature of the entire image to obtain
xcom
= [xi , xroot ]. Then we train a separate Softmax classifier on xcom
. This resulted in better peri
i
formance for both Single-scale (PPA: 80.4 & MCA: 71.5) and Multi-scale (PPA: 80.8 & MCA: 79.1)
CNN feature settings over (Single/Multi)-CNN + Plain NN. As was previously shown in Table 2,
decombiner network further improves this model.
Computation speed: Our fast method (Section 2.1) takes only 0.37 seconds (0.3 for super-pixel
segmentation, 0.06 for feature extraction and 0.01 for rCPN and labeling) to label a 256 ? 256 image
starting from the raw RGB image on a GTX Titan GPU and 1.1 seconds on a Intel core i7 CPU. In
both of the experiments the performance loss is negligible using the fast method. Interestingly, the
time bottleneck of our approach on a GPU is the super-pixel segmentation time.
Several typical labeling results on Stanford background dataset using the proposed semantic scene
labeling algorithm are shown in Figure 3.
5
Related Work
Scene labeling has two broad categories of approaches - non-parametric and model-based. Recently,
many non-parametric approaches for natural scene parsing have been proposed [15, 14, 16, 17, 18].
The underlying theme is to find similar looking images to the query image from a database of pixelwise labeled images, followed by super-pixel label transfer from the retrieved images to the query
image. Finally, a structured prediction model such as CRF is used to integrate contextual information
to obtain the final labeling. These approaches mainly differ in the retrieval of candidate images or
super-pixels, transfer of label from the retrieved candidates to the query image, and the form of the
structured prediction model used for final labeling. They are based on nearest-neighbor retrieval that
introduces a performance/accuracy tradeoff. The variations present in natural scene images are large
and it is very difficult to cover this entire space of variation with a reasonable size database, which
limits the accuracy of these methods. On the other extreme, a very large database would require
large retrieval-time, which limits the scalability of these methods.
7
sky
tree
road
grass
water
bldg
mntn
fig obj
Figure 3: Typical labeling results on Stanford background dataset using our method
Model-based approaches learn the appearance of semantic categories and relations among them
using a parametric model. In [13, 20, 2, 3, 22], CRF models are used to combine unary potentials
devised through the visual features extracted from super-pixels with the neighborhood constraints.
The differences are mainly in terms of the visual features, unary potentials and the structure of
the CRF model. Lempitsky et al. [9] have formulated a joint-CRF on multiple levels of an image
segmentation hierarchy to achieve better results than a flat-CRF on the image super-pixels only.
Socher et al. [7] learnt a mapping from the visual features to a semantic space followed by a twolayer neural network for classification. Better use of contextual information, with the same superpixels and features, increased the performance on Stanford background dataset from CRF based
method of Gould et al. to semantic mapping of Socher et al. to the proposed work (76.4% ?
78.1% ? 81.4%). It indicates that neural network based models have the potential to learn more
complicated interactions than a CRF. Moreover, they have the advantage of being extremely fast, due
to the feed-forward nature. Farabet et al. [6] proposed to learn the visual features from image/label
training pairs using a multi-scale convolutional neural network. They reported state-of-the-art results
on various datasets using gPb, purity-cover and CRF on top of their learned features. Pinheiro et al.
[19] extended their work by feeding in the per-pixel predicted labels using a CNN classifier to the
next stage of the same CNN classifier. However, their propagation structure is not adaptive to the
image content and only propagating label information did not improve much over the prior work.
Similar to these methods, we also make use of the Multi-CNN module to extract local features in
our pipeline. However, our novel context propagation network shows that propagating semantic
representation bottom-up and top-down using a parse three hierarchy is a more effective way to
aggregate global context information. Please see Tables 1 and 2 for a detailed comparison of our
method with the methods discussed above.
CRFs model the joint distribution of the output variables given the observations and can include
higher order potentials in addition to the unary potentials. Higher order potentials allow these models to represent the dependencies between the output variables, which is important for structured
prediction tasks. On the downside, except for a few exceptions such as non-loopy models, inference
in these models is NP-Hard that can be only approximately solved and is time consuming. Moreover,
parameter learning procedures that are tractable usually limit the form of the potential functions to
simple forms such as linear models. In contrast, in our model, we can efficiently learn complex
relations between a single output variable and all the observations from an image, allowing a large
context to be considered effectively. Additionally, the inference procedure is a simple feed-forward
pass that can be performed very fast. However, the form of our function is still a unary term and our
model cannot represent higher order output dependencies. Our model can also be used to obtain the
unary potential for a structured inference model.
6
Conclusion
We introduced a novel deep neural network architecture, which is a combination of a convolutional
neural network and recursive neural network, for semantic scene labeling. The key contribution
is the recursive context propagation network, which effectively propagates contextual information
from one location of the image to other locations in a feed-forward manner. This structure led to
the state-of-the-art semantic scene labeling results on Stanford background and SIFT Flow datasets
with very fast processing speed. Next we plan to scale-up our model for recently introduced large
scale learning task [23].
8
References
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9
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4,730 | 5,283 | Multiscale Fields of Patterns
John G. Oberlin
Brown University
Providence, RI 02906
john [email protected]
Pedro F. Felzenszwalb
Brown University
Providence, RI 02906
[email protected]
Abstract
We describe a framework for defining high-order image models that can be used
in a variety of applications. The approach involves modeling local patterns in a
multiscale representation of an image. Local properties of a coarsened image reflect non-local properties of the original image. In the case of binary images local
properties are defined by the binary patterns observed over small neighborhoods
around each pixel. With the multiscale representation we capture the frequency
of patterns observed at different scales of resolution. This framework leads to
expressive priors that depend on a relatively small number of parameters. For inference and learning we use an MCMC method for block sampling with very large
blocks. We evaluate the approach with two example applications. One involves
contour detection. The other involves binary segmentation.
1
Introduction
Markov random fields are widely used as priors for solving a variety of vision problems such as
image restoration and stereo [5, 8]. Most of the work in the area has concentrated on low-order
models involving pairs of neighboring pixels. However, it is clear that realistic image priors need to
capture higher-order properties of images.
In this paper we describe a general framework for defining high-order image models that can be used
in a variety of applications. The approach involves modeling local properties in a multiscale representation of an image. This leads to a natural low-dimensional representation of a high-order model.
We concentrate on the problem of estimating binary images. In this case local image properties can
be captured by the binary patterns in small neighborhoods around each pixel.
We define a Field of Patterns (FoP) model using an energy function that assigns a cost to each 3x3
pattern observed in an image pyramid. The cost of a pattern depends on the scale where it appears.
Figure 1 shows a binary image corresponding to a contour map from the Berkeley segmentation
dataset (BSD) [12, 2] and a pyramid representation obtained by repeated coarsening. The 3x3 patterns we observe after repeated coarsening depend on large neighborhoods of the original image.
These coarse 3x3 patterns capture non-local image properties. We train models using a maximumlikelihood criteria. This involves selecting pattern costs making the expected frequency of patterns
in a random sample from the model match the average frequency of patterns in the training images.
Using the pyramid representation the model matches frequencies of patterns at each resolution.
In practice we use MCMC methods for inference and learning. In Section 3 we describe an MCMC
sampling algorithm that can update a very large area of an image (a horizontal or vertical band of
pixels) in a single step, by combining the forward-backward algorithm for one-dimensional Markov
models with a Metropolis-Hastings procedure.
We evaluated our models and algorithms on two different applications. One involves contour detection. The other involves binary segmentation. These two applications require very different image
priors. For contour detection the prior should encourage a network of thin contours, while for bi1
(a)
(b)
(c)
Figure 1: (a) Multiscale/pyramid representation of a contour map. (b) Coarsest image scaled up
for better visualization, with a 3x3 pattern highlighted. The leftmost object in the original image
appears as a 3x3 ?circle? pattern in the coarse image. (c) Patches of contour maps (top) that coarsen
to a particular 3x3 pattern (bottom) after reducing their resolution by a factor of 8.
nary segmentation the prior should encourage spatially coherent masks. In both cases we can design
effective models using maximum-likelihood estimation.
1.1
Related Work
FRAME models [24] and more recently Fields of Experts (FoE) [15] defined high-order energy
models using the response of linear filters. FoP models are closely related. The detection of 3x3
patterns at different resolutions corresponds to using non-linear filters of increasing size. In FoP we
have a fixed set of pre-defined non-linear filters that detect common patterns at different resolutions.
This avoids filter learning, which leads to a non-convex optimization problem in FoE.
A restricted set of 3x3 binary patterns was considered in [6] to define priors for image restoration.
Binary patterns were also used in [17] to model curvature of a binary shape. There has been recent
work on inference algorithms for CRFs defined by binary patterns [19] and it may be possible to
develop efficient inference algorithms for FoP models using those techniques.
The work in [23] defined a variety of multiresolution models for images based on a quad-tree representation. The quad-tree leads to models that support efficient learning and inference via dynamic
programming, but such models also suffer from artifacts due to the underlying tree-structure. The
work in [7] defined binary image priors using deep Boltzmann machines. Those models are based
on a hierarchy of hidden variables that is related to our multiscale representation. However in our
case the multiscale representation is a deterministic function of the image and does not involve extra
hidden variables as [7]. The approach we take to define a multiscale model is similar to [9] where
local properties of subsampled signals where used to model curves.
One of our motivating applications involves detecting contours in noisy images. This problem has a
long history in computer vision, going back at least to [16], who used a type of Markov model for
detecting salient contours. Related approaches include the stochastic completion field in [22, 21],
spectral methods [11], the curve indicator random field [3], and the more recent work in [1].
2
Fields of Patterns (FoP)
Let G = [n] ? [m] be the grid of pixels in an n by m image. Let x = {x(i, j) | (i, j) ? G} be a
hidden binary image and y = {y(i, j) | (i, j) ? G} be a set of observations (such as a grayscale or
color image). Our goal is to estimate x from y.
We define p(x|y) using an energy function that is a sum of two terms,
1
exp(?E(x, y)) E(x, y) = EFoP (x) + Edata (x, y)
p(x|y) =
Z(y)
2
(1)
It is sometimes useful to think of EFoP (x) as a model for binary images and Edata (x, y) as a data
model even though technically there is no such distinction in a conditional model.
2.1
Singlescale FoP Model
The singlescale FoP model is one of the simplest energy models that can capture the basic properties
of contour maps or other images that contain thin objects. We use x[i, j] to denote the binary pattern
defined by x in the 3x3 window centered at pixel (i, j), treating values outside of the image as 0. A
singlescale FoP model is defined by the local patterns in x,
X
EFoP (x) =
V (x[i, j]).
(2)
(i,j)?G
Here V is a potential function assigning costs (or energies) to binary patterns. Note that there are 512
possible binary patterns in a 3x3 window. We can make the model invariant to rotations and mirror
symmetries by tying parameters together. The resulting model has 102 parameters (some patterns
have more symmetries than others) and can be learned from smaller datasets. We used invariant
models for all of the experiments reported in this paper.
2.2
Multiscale FoP Model
To capture non-local statistics we look at local patterns in a multiscale representation of x. For a
model with K scales let ?(x) = x0 , . . . , xK?1 be an image pyramid where x0 = x and xk+1 is
a coarsening of xk . Here xk is a binary image defined over a grid G k = [n/2k ] ? [m/2k ]. The
coarsening we use in practice is defined by a logical OR operation,
xk+1 (i, j) = xk (2i, 2j) ? xk (2i + 1, 2j) ? xk (2i, 2j + 1)k ? xk (2i + 1, 2j + 1)
(3)
This particular coarsening maps connected objects at one scale of resolution to connected objects at
the next scale, but other coarsenings may be appropriate in different applications.
A multiscale FoP model is defined by the local patterns in ?(x),
EFoP (x) =
K?1
X
X
V k (xk [i, j]).
(4)
k=0 (i,j)?G k
This model is parameterized by K potential functions V k ., one for each scale in the pyramid ?(x).
In many applications we expect the frequencies of a 3x3 pattern to be different at each scale. The
potential functions can encourage or discourage specific patterns to occur at specific scales.
Note that ?(x) is a deterministic function and the pyramid representation does not introduce new
random variables. The pyramid simply defines a convenient way to specify potential functions over
large regions of x. A single potential function in a multiscale model can depend on a large area of
x due to the coarsenings. For large enough K (proportional to log of the image size) the Markov
blanket of a pixel can be the whole image.
While the experiments in Section 5 use the conditional modeling approach specified by Equation
(1), we can also use EFoP to define priors over binary images. Samples from these priors illustrate the information that is captured by a FoP model, specially the added benefit of the multiscale
representation. Figure 2 shows samples from FoP priors trained on contour maps of natural images.
The empirical studies in [14] suggest that low-order Markov models can not capture the empirical
length distribution of contours in natural images. A multiscale FoP model can control the size
distribution of objects much better than a low-order MRF. After coarsening the diameter of an object
goes down by a factor of approximately two, and eventually the object is mapped to a single pixel.
The scale at which this happens can be captured by a 3x3 pattern with an ?on? pixel surrounded by
?off? pixels (this assumes there are no other objects nearby). Since the cost of a pattern depends on
the scale at which it appears we can assign a cost to an object that is based loosely upon its size.
2.3
Data Model
Let y be an input image and ?(y) be an image pyramid computed from y. Our data models are
defined by sums over pixels in the two pyramids ?(x) and ?(y). In our experiments y is a graylevel
3
(a)
(b)
(c)
Figure 2: (a) Examples of training images T extracted from the BSD. (b) Samples from a singlescale
FoP prior trained on T . (c) Samples from a multiscale FoP prior trained on T . The multiscale model
is better at capturing the lengths of contours and relationships between them.
image with values in {0, . . . , M ? 1}. The pyramid ?(y) is defined in analogy to ?(x) except that
we use a local average for coarsening instead of the logical OR,
y k+1 (i, j) = b(y k (2i, 2j) + y k (2i + 1, 2j) + y k (2i, 2j + 1) + y k (2i + 1, 2j + 1))/4c
(5)
The data model is parameterized by K vectors D0 , . . . , DK?1 ? RM
Edata (x, y) =
K?1
X
X
xk (i, j)Dk (y k (i, j))
(6)
k=0 (i,j)?G k
Here Dk (y k (i, j)) is an observation cost incurred when xk (i, j) = 1. There is no need to include an
observation cost when xk (i, j) = 0 because only energy differences affect the posterior p(x|y).
We note that it would be interesting to consider data models that capture complex relationships
between local patterns in ?(x) and ?(y). For example a local maximum in y k (i, j) might give
evidence for xk (i, j) = 1, or a particular 3x3 pattern in xk [i, j].
2.4
Log-Linear Representation
The energy function E(x, y) of a FoP model can be expressed by a dot product between a vector of
model parameters w and a feature vector ?(x, y). The vector ?(x, y) has one block for each scale.
In the k-th block we have: (1) 512 (or 102 for invariant models) entries counting the number of
times each 3x3 pattern occurs in xk ; and (2) M entries counting the number of times each possible
value for y(i, j) occurs where xk (i, j) = 1. The vector w specifies the cost for each pattern in each
scale (V k ) and the parameters of the data model (Dk ). We then have that E(x, y) = w ? ?(x, y).
This log-linear form is useful for learning the model parameters as described in Section 4.
3
Inference with a Band Sampler
In inference we have a set of observations y and want to estimate x. We use MCMC methods [13]
to draw samples from p(x|y) and estimate the posterior marginal probabilities p(x(i, j) = 1|y).
Sampling is also used for learning model parameters as described in Section 4.
In a block Gibbs sampler we repeatedly update x by picking a block of pixels B and sampling new
values for xB from p(xB |y, xB ). If the blocks are selected appropriately this defines a Markov chain
with stationary distribution p(x|y).
We can implement a block Gibbs sampler for a multiscale FoP model by keeping track of the image
pyramid ?(x) as we update x. To sample from p(xB |y, xB ) we consider each possible configuration
4
for xB . We can efficiently update ?(x) to reflect a possible configuration for xB and evaluate the
terms in E(x, y) that depend on xB . This takes O(K|B|) time for each configuration for xB . This
in turn leads to an O(K|B|2|B| ) time algorithm for sampling from p(xB |y, xB? ). The running time
can be reduced to O(K2|B| ) using Gray codes to iterate over configurations for xB .
Here we define a band sampler that updates all pixels in a horizontal or vertical band of x in a single
step. Consider an n by m image x and let B be a horizontal band of pixels with h rows. Since
|B| = mh a straightforward implementation of block sampling for B is completely impractical.
However, for an Ising model we can generate samples from p(xB |y, xB ) in O(m22h ) time using
the forward-backward algorithm for Markov models. We simply treat each column of B as a single
variable with 2h possible states. A similar idea can be used for FoP models.
Let S be a state space where a state specifies a joint configuration of binary values for the pixels in
a column of B. Note that |S| = 2h . Let z1 , . . . , zm be a representation of xB in terms of the state
of each column. For a singlescale FoP model the distribution p(z1 , . . . , zn |y, xB? ) is a 2nd-order
Markov model. This allows for efficient sampling using forward weights computed via dynamic
programming. Such an algorithm takes O(m23h ) time to generate a sample from p(xB |y, xB ),
which is efficient for moderate values of h.
In a multiscale FoP model the 3x3 patterns in the upper levels of ?(x) depend on many columns of
B. This means p(z1 , . . . , zn |xB? ) is no longer 2nd-order. Therefore instead of sampling xB directly
we use a Metropolis-Hastings approach. Let p be a multiscale FoP model we would like to sample
from. Let q be a singlescale FoP model that approximates p. Let x be the current state of the
Markov chain and x0 be a proposal generated by the singlescale band sampler for q. We accept x0
with probability min(1, ((p(x0 |y)q(x|y))/(p(x|y)q(x0 |y)))). Efficient computation of acceptance
probabilities can be done using the pyramid representations of x and y. For each proposal we update
?(x) to ?(x0 ) and compute the difference in energy due to the change under both p and q.
One problem with the Metropolis-Hastings approach is that if proposals are rejected very often the
resulting Markov chain mixes slowly. We can avoid this problem by noting that most of the work
required to generate a sample from the proposal distribution involves computing forward weights
that can be re-used to generate other samples. Each step of our band sampler for a multiscale FoP
model picks a band B (horizontal or vertical) and generates many proposals for xB , accepting each
one with the appropriate acceptance probability. As long as one of the proposals is accepted the
work done in computing forward weights is not wasted.
4
Learning
We can learn models using maximum-likelihood and stochastic gradient descent. This is similar to
what was done in [24, 15, 20]. But in our case we have a conditional model so we maximize the
conditional likelihood of the training examples.
Let T = {(x1 , yi ), . . . , (xN , yN )} be a training set with N examples. We define the training objective using the negative log-likelihood of the data plus a regularization term. The regularization
ensures no pattern is too costly. This helps the Markov chains used during learning and inference to
mix reasonably fast. Let L(xi , yi ) = ? log p(xi |yi ). The training objective is given by
N
X
?
O(w) = ||w||2 +
L(xi , yi ).
(7)
2
i=1
This objective is convex and
?O(w) = ?w +
N
X
?(xi , yi ) ? Ep(x|yi ) [?(x, yi )].
(8)
i=1
Here Ep(x|yi ) [?(x, yi )] is the expectation of ?(x, yi ) under the posterior p(x|yi ) defined by the
current model parameters w. A stochastic approximation to the gradient ?O(w) can be obtained
by sampling x0i from p(x|yi ). Let ? be a learning rate. In each stochastic gradient descent step we
sample x0i from p(x|yi ) and update w as follows
N
X
w := w ? ?(?w +
?(xi , yi ) ? ?(x0i , yi )).
(9)
i=1
5
To sample the x0i we run N Markov chains, one for each training example, using the band sampler
from Section 3. After each model update we advance each Markov chain for a small number of steps
using the latest model parameters to obtain new samples x0i .
5
Applications
To evaluate the ability of FoP to adapt to different problems we consider two different applications.
In both cases we estimate hidden binary images x from grayscale input images y. We used ground
truth binary images obtained from standard datasets and synthetic observations. For the experiments
described here we generate y by sampling a value y(i, j) for each pixel independently from a normal
distribution with standard deviation ?y and mean ?0 or ?1 , depending on x(i, j),
y(i, j) ? N (?x(i,j) , ?y2 ).
(10)
We have also done experiments with more complex data models but the results we obtained were
similar to the results described here.
5.1
Contour Detection
The BSD [12, 2] contains images of natural scenes and manual segmentations of the most salient
objects in those images. We used one manual segmentation for each image in the BSD500. From
each image we generated a contour map x indicating the location of boundaryes between segments
in the image. To generate the observations y we used ?0 = 150, ?1 = 100 and ?y = 40 in Equation
(10). Our training and test sets each have 200 examples. We first trained a 1-scale FoP model. We
then trained a 4-level FoP model using the 1-level model as a proposal distribution for the band
sampler (see Section 3). Training each model took 2 days on a 20-core machine. During training
and testing we used the band sampler with h = 3 rows. Inference involves estimating posterior
marginal probabilities for each pixel by sampling from p(x|y). Inference on each image took 20
minutes on an 8-core machine.
For comparison we implemented a baseline technique using linear filters. Following [10] we used the
second derivative of an elongated Gaussian filter together with its Hilbert transform. The filters had
an elongation factor of 4 and we experimented with different values for the base standard deviation
?b of the Gaussian. The sum of squared responses of both filters defines an oriented energy map. We
evaluated the filters at 16 orientations and took the maximum response at each pixel. We performed
non-maximum suppression along the dominant orientations to obtain a thin contour map.
Figure 3 illustrates our results on 3 examples from the test set. Results on more examples are available in the supplemental material. For the FoP models we show the posterior marginal probabilities
p(x(i, j) = 1|y). The darkness of a pixel is proportional to the marginal probability. The FoP
models do a good job suppressing noise and localizing the contours. The multiscale FoP model in
particular gives fairly clean results despite the highly noisy inputs. The baseline results at lower ?b
values suffer from significant noise, detecting many spurious edges. The baseline at higher ?b values
suppresses noise at the expense of having poor localization and missing high-curvature boundaries.
For a quantitative evaluation we compute precision-recall curves for the different models by thresholding the estimated contour maps at different values. Figure 4 shows the precision-recall curves.
The average precision (AP) was found by calculating the area under the precision-recall curves. The
1-level FoP model AP was 0.73. The 4-level FoP model AP was 0.78. The best baseline AP was
0.18 obtained with ?b = 1. We have also done experiments using lower observation noise levels ?y .
With low observation noise the 1-level and 4-level FoP results become similar and baseline results
improve significantly approaching the FoP results.
5.2
Binary Segmentation
For this experiment we obtained binary images from the Swedish Leaf Dataset [18]. We focused on
the class of Rowan leaves because they have complex shapes. Each image defines a segmentation
mask x. To generate the observations y we used ?0 = 150, ?1 = 100 and ?y = 100 in Equation
(10). We used a higher ?y compared to the previous experiment because the 2D nature of masks
makes it possible to recover them under higher noise. We used 50 examples for training and 25
6
Contour map x
Observation y
Baseline ?b = 1
Baseline ?b = 4
FoP 1
FoP 4
Figure 3: Contour detection results. Top-to-bottom: Hidden contour map x, input image y, output
of oriented filter baseline with ?b = 1 and ?b = 4, output of 1-level and 4-level FoP model.
examples for testing. We trained FoP models with the same procedure and parameters used for the
contour detection experiment. For a baseline, we used graph-cuts [5, 4] to perform MAP inference
with an Ising model. We set the data term using our knowledge of the observation model and picked
the pairwise discontinuity cost minimizing the per-pixel error rate in the test set.
Figure 5 illustrates the results of the different methods. Results on other images are available in the
supplemental material. The precision-recall curves are in Figure 4. Graph-cuts yields a precisionrecall point, with precision 0.893 and recall 0.916. The 1-level FoP model has a higher precision of
0.915 at the same recall. The 4-level FoP model raises the precision to 0.929 at the same recall. The
7
(a) Contour detection
(b) Binary segmentation
Figure 4: (a) Precision-recall curves for the contour detection experiment. (b) Precision-recall curves
for the segmentation experiment (the graph-cuts baseline yields a single precision-recall point).
Mask x
Observation y
Graph-cuts
FoP 1
FoP 4
Figure 5: Binary segmentation examples. The 4-level FoP model does a better job recovering pixels
near the object boundary and the stem of the leaves.
differences in precision are small because they are due to pixels near the object boundary but those
are the hardest pixels to get right. There are clear differences that can be seen by visual inspection.
6
Conclusion
We described a general framework for defining high-order image models. The idea involves modeling local properties in a multiscale representation of an image. This leads to a natural lowdimensional parameterization for high-order models that exploits standard pyramid representations
of images. Our experiments demonstrate the approach yields good results on two applications that
require very different image priors, illustrating the broad applicability of our models. An interesting
direction for future work is to consider FoP models for non-binary images.
Acknowledgements
We would like to thank Alexandra Shapiro for helpful discussions and initial experiments related to
this project. This material is based upon work supported by the National Science Foundation under
Grant No. 1161282.
8
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9
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4,731 | 5,284 | Weakly-supervised Discovery of
Visual Pattern Configurations
Hyun Oh Song
Yong Jae Lee*
Stefanie Jegelka
*
University of California, Berkeley
Trevor Darrell
University of California, Davis
Abstract
The prominence of weakly labeled data gives rise to a growing demand for object detection methods that can cope with minimal supervision. We propose an
approach that automatically identifies discriminative configurations of visual patterns that are characteristic of a given object class. We formulate the problem as a
constrained submodular optimization problem and demonstrate the benefits of the
discovered configurations in remedying mislocalizations and finding informative
positive and negative training examples. Together, these lead to state-of-the-art
weakly-supervised detection results on the challenging PASCAL VOC dataset.
1
Introduction
The growing amount of sparsely and noisily labeled image data demands robust detection methods
that can cope with a minimal amount of supervision. A prominent example of this scenario is the
abundant availability of labels at the image level (i.e., whether a certain object is present or absent
in the image); detailed annotations of the exact location of the object are tedious and expensive and,
consequently, scarce. Learning methods that can handle image-level labels circumvent the need
for such detailed annotations and therefore have the potential to effectively use the vast textually
annotated visual data available on the Web. Moreover, if the detailed annotations happen to be noisy
or erroneous, such weakly supervised methods can even be more robust than fully supervised ones.
Motivated by these developments, recent work has explored learning methods that decreasingly
rely on strong supervision. Early ideas for weakly supervised detection [11, 32] paved the way
by successfully learning part-based object models, albeit on simple object-centric datasets (e.g.,
Caltech-101). Since then, a number of approaches [21, 26, 29] have aimed at learning models from
more realistic and challenging data sets that feature large intra-category appearance variations and
background clutter. These approaches typically generate multiple candidate regions and retain the
ones that occur most frequently in the positively-labeled images. However, due to intra-category
variations and deformations, the identified (single) patches often correspond to only a part of the
object, such as a human face instead of the entire body. Such mislocalizations are a frequent problem
for weakly supervised detection methods.
Mislocalization and too large or too small bounding boxes are problematic in two respects. First,
detection is commonly phrased as multiple instance learning (MIL) and solved by non-convex optimization methods that alternatingly guess the location of the objects as positive examples (since
the true location is unknown) and train a detector based on those guesses. This procedure is heavily
affected by the initial localizations. Second, the detector is often trained in stages; in each stage one
adds informative ?hard? negative examples to the training data. If we are not given accurate true
object localizations in the training data, these hard examples must be derived from the detections
inferred in earlier rounds. The higher the accuracy of the initial localizations, the more informative
is the augmented training data ? and this is key to the accuracy of the final learned model.
In this work, we address the issue of mislocalizations by identifying characteristic, discriminative
configurations of multiple patches (rather than a single one). This part-based approach is motivated
1
by the observation that automatically discovered single ?discriminative? patches often correspond
to object parts. In addition, while background patches (e.g., of water or sky) can also occur throughout the positive images, they will re-occur in arbitrary rather than ?typical? configurations. We
develop an effective method that takes as input a set of images with labels of the form ?the object is
present/absent?, and automatically identifies characteristic part configurations of the given object.
To identify such configurations, we use two main criteria. First, useful patches are discriminative,
i.e., they occur in many positively-labeled images, and rarely in the negatively labeled ones. To identify such patches, we use a discriminative covering formulation similar to [29]. Second, the patches
should represent different parts, i.e., they may be close but should not overlap too much. In covering
formulations, one may rule out overlaps by saying that for two overlapping regions, one ?covers?
the other, i.e., they are treated as identical and picking one is as good as picking both. But identity is
a transitive relation, and the density of possible regions in detection would imply that all regions are
identical, strongly discouraging the selection of more than one part per image. Partial covers face
the problem of scale invariance. Hence, we instead formulate an independence constraint. This second criterion ensures that we select regions that may be close but are non-redundant and sufficiently
non-overlapping. We show that this constrained selection problem corresponds to maximizing a
submodular function subject to a matroid intersection constraint, which leads to approximation algorithms with theoretical worst-case bounds. Given candidate parts identified by these two criteria,
we effectively find frequently co-occurring configurations that take into account relative position,
scale, and viewpoint.
We demonstrate multiple benefits of the discovered configurations. First, we observe that configurations of patches can produce more accurate spatial coverage of the full object, especially when the
most discriminative pattern corresponds to an object part. Second, any overlapping region between
co-occurring visual patterns is likely to cover a part (but not the full) of the object of interest. Thus,
they can be used to generate mis-localized positives as informative hard negatives for training (see
white boxes in Figure 3), which can further reduce localization errors at test time.
In short, our main contribution is a weakly-supervised object detection method that automatically
discovers frequent configurations of discriminative visual patterns to train robust object detectors.
In our experiments on the challenging PASCAL VOC dataset, we find the inclusion of our discriminative, automatically detected configurations to outperform all existing state-of-the-art methods.
2
Related work
Weakly-supervised object detection. Object detectors have commonly been trained in a fullysupervised manner, using tight bounding box annotations that cover the object of interest (e.g., [10]).
To reduce laborious bounding box annotation costs, recent weakly-supervised approaches [3, 4, 11,
21, 26, 29, 32] use image-level object-presence labels with no information on object location.
Early efforts [11, 32] focused on simple datasets that have a single prominent object in each image
(e.g., Caltech-101). More recent approaches [21, 26, 29] work with the more challenging PASCAL
dataset that contains multiple objects in each image and large intra-category appearance variations.
Of these, Song et al. [29] achieve state-of-the-art results by finding discriminative image patches
that occur frequently in the positive images but rarely in the negative images, using deep Convolutional Neural Network (CNN) features [17] and a submodular cover formulation. We build on their
approach to identify discriminative patches. But, contrary to [29] which assumes patches to contain
entire objects, we assume patches to contain either full objects or merely object parts, and automatically piece together those patches to produce better full-object estimates. To this end, we change
the covering formulation and identify patches that are both representative and explicitly mutually
different. This leads to more robust object estimates and further allows our system to intelligently
select ?hard negatives? (mislocalized objects), both of which improve detection performance.
Visual data mining. Existing approaches discover high-level object categories [14, 7, 28], mid-level
patches [5, 16, 24], or low-level foreground features [18] by grouping similar visual patterns (i.e.,
images, patches, or contours) according to their texture, color, shape, etc. Recent methods [5, 16]
use weakly-supervised labels to discover discriminative visual patterns. We use related ideas, but
formulate the problem as a submodular optimization over matroids, which leads to approximation
algorithms with theoretical worst-case guarantees. Covering formulations have also been used in
2
[1, 2], but after running a trained object detector. An alternative discriminative approach is to use
spectral methods [34].
Modeling co-occurring visual patterns. It is known that modeling the spatial and geometric relationship between co-occurring visual patterns (objects or object-parts) often improves visual recognition performance [8, 18, 10, 11, 19, 23, 27, 24, 32, 33]. Co-occurring patterns are usually represented as doublets [24], higher-order constellations [11, 32] or star-shaped models [10]. Among
these, our work is most inspired by [11, 32], which learn part-based models with weak supervision. We use more informative deep CNN features and a different formulation, and show results on
more difficult datasets. Our work is also related to [19], which discovers high-level object compositions (?visual phrases? [8]), but with ground-truth bounding box annotations. In contrast, we aim to
discover part compositions to represent full objects and do so with less supervision.
3
Approach
Our goal is to find a discriminative set of patches that co-occur in the same configuration in many
positively-labeled images. We address this goal in two steps. First, we find a set of patches that are
discriminative; i.e., they occur frequently in positive images and rarely in negative images. Second,
we efficiently find co-occurring configurations of pairs of such patches. Our approach easily extends
beyond pairs; for simplicity and to retain configurations that occur frequently enough, we here
restrict ourselves to pairs.
Discriminative candidate patches. For identifying discriminative patches, we begin with a construction similar to that of Song et al. [29]. Let P be the set of positively-labeled images. Each
image I contains candidate boxes {bI,1 , . . . , bI,m } found via selective search [30]. For each bI,i , we
find its closest matching neighbor bI 0 ,j in each other image I 0 (regardless of the image label). The
K closest of those neighbors form the neighborhood N (bI,i ); the remaining ones are discarded.
Discriminative patches have neighborhoods mainly within images in P, i.e., if B(P) is the set of all
patches from images in P, then |N (b) ? B(P)| ? K. To identify a small, diverse and representative
set of such patches, like [29], we construct a bipartite graph G = (U, V, E), where both U and V
contain copies of B(P). Each patch b ? V is connected to the copy of its nearest neighbors in U (i.e.,
N (b) ? B(P)). These will be K or fewer, depending on whether the K nearest neighbors of b occur
in B(P) or in negatively-labeled images. The most representative patches maximize the covering
function
F (S) = |?(S)|,
(1)
where ?(S) = {u ? U | (b, u) ? E for some b ? S} ? U is the neighborhood of S ? V in the
bipartite graph. Figure 1 shows a cartoon illustration. The function F is monotone and submodular,
and the C maximizing elements (for a given C) can be selected greedily [20].
However, if we aim to find part configurations, we must select multiple, jointly informative patches
per image. Patches selected to merely maximize coverage can still be redundant, since the most
frequently occurring ones are often highly overlapping. A straightforward modification would be
to treat highly overlapping patches as identical. This identification would still admit a submodular
cover model as in Equation (1). But, in our case, the candidate patches are very densely packed in
the image, and, by transitivity, we would have to make all of them identical. In consequence, this
would completely rule out the selection of more than one patch in an image and thereby prohibit the
discovery of any co-occurring configurations.
Instead, we directly constrain our selection such that no two patches b, b0 ? V can be picked whose
neighborhoods overlap by more than a fraction ?. By overlap, we mean that the patches in the
neighborhoods of b, b0 overlap significantly (they need not be identical). This notion of diversity is
reminiscent of NMS and similar to that in [5], but we here phrase and analyze it as a constrained
submodular optimization problem. Our constraint can be expressed in terms of a different graph
GC = (V, EC ) with nodes V. In GC , there is an edge between b and b0 if their neighborhoods overlap
prohibitively, as illustrated in Figure 1. Our family of feasible solutions is
M = {S ? V | ? b, b0 ? S there is no edge (b, b0 ) ? EC }.
(2)
In other words, M is the family of all independent sets in GC . We aim to maximize
maxS?V F (S) s.t. S ? M.
3
(3)
V
U
Figure 1: Left: bipartite graph G that defines the utility function F and identifies discriminative
patches; right: graph GC that defines the diversifying independence constraints M. We may pick
C1 (yellow) and C3 (green) together, but not C2 (red) with any of those.
This problem is NP-hard. We solve it approximately via the following greedy algorithm. Begin with
S 0 = ?, and, in iteration t, add b ? argmaxb?V\S |?(b) \ ?(S t?1 )|. As we add b, we delete all of
b?s neighbors in GC from V. We continue until V = ?. If the neighborhoods of any b, b0 are disjoint
but contain overlapping elements (?(b) ? ?(b0 ) = ? but there exist u ? ?(b) and u0 ? ?(b0 ) that
overlap), then this algorithm amounts to the following simplified scheme: we first sort all b ? V in
non-increasing order by their degree ?(b), i.e., their number of neighbors in B(P), and visit them in
this order. We always add the currently highest b in the list to S, then delete it from the list, and with
it all its immediate (overlapping) neighbors in GC . The following lemma states an approximation
factor for the greedy algorithm, where ? is the maximum degree of any node in GC .
Lemma 1. The solution Sg returned by the greedy algorithm is a 1/(? + 2) approximation for
1
Problem (2): F (Sg ) ? ?+2
F (S ? ). If ?(b) ? ?(b0 ) = ? for all b, b0 ? V, then the worst-case
approximation factor is 1/(? + 1).
The proof relies on phrasing M as an intersection of matroids.
Definition 1 (Matroid). A matroid (V, Ik ) consists of a ground set V and a family Ik ? 2V of
?independent sets? that satisfy three axioms: (1) ? ? Ik ; (2) downward closedness: if S ? Ik then
T ? Ik for all T ? S; and (3) the exchange property: if S, T ? Ik and |S| < |T |, then there is an
element v ? T \ S such that S ? {v} ? Ik .
Proof. (Lemma 1) We will argue that Problem (2) is the problem of maximizing a monotone submodular function subject to the constraint that the solution lies in the intersection of ? + 1 matroids.
With this insight, the approximation factor of the greedy algorithm for submodular F follows from
[12] and that for non-intersecting ?(b) from [15], since in the latter case the problem is that of
finding a maximum weight vector in the intersection of ? + 1 matroids.
It remains to argue that M is an intersection of matroids. Our matroids will be partition matroids
(over the ground set V) whose independent sets are of the form Ik = {S | |S ? e| ? 1, for all e ?
Ek }. To define those, we partition the edges in GC into disjoint sets Ek , i.e., no two edges in Ek
share a common node. The Ek can be found by an edge coloring ? one Ek and Ik for each color k.
By Vizing?s theorem [31], we need at most ?+1 colors. The matroid Ik demands that for each edge
e ? Ek , we may only select one of its adjacent nodes. All matroids together say that for any edge
e ? E, we may only select one of the adjacent nodes, and that is the constraint in Equation (2), i.e.
T?+1
M = k=1 Ik . We do not ever need to explicitly compute Ek and Ik ; all we need to do is check
membership in the intersection, and this is equivalent to checking whether a set S is an independent
set in GC , which is achieved implicitly via the deletions in the algorithm.
From the constrained greedy algorithm, we obtain a set S ? V of discriminative patches. Together
with its neighborhood ?(b), each patch b ? V forms a representative cluster. Figure 2 shows some
example patches derived from the labels ?aeroplane? and ?motorbike?. The discovered patches
intuitively look like ?parts? of the objects, and are frequent but sufficiently different.
Finding frequent configurations. The next step is to find frequent configurations of co-occurring
clusters, e.g., the head patch of a person on top of the torso patch, or a bicycle with visible wheels.
4
Figure 2: Examples of discovered patch ?clusters? for aeroplane, motorbike, and cat. The discovered
patches intuitively look like object parts, and are frequent but sufficiently different.
A ?configuration? consists of patches from two clusters Ci , Cj , their relative location, and their
viewpoint and scale. In practice, we give preference to pairs that by themselves are very relevant
and maximize a weighted combination of co-occurrence count and coverage max{?(Ci ), ?(Cj )}.
All possible configurations of all pairs of patches amount to too many to explicitly write down and
count. Instead, we follow an efficient procedure for finding frequent configurations. Our approach
is inspired by [19], but does not require any supervision. We first find configurations that occur in at
least two images. To do so, we consider each pair of images I1 , I2 that have at least two co-occurring
clusters. For each correspondence of cluster patches across the images, we find a corresponding
transform operation (translation, scale, viewpoint change). This results in a point in a 4D transform
space, for each cluster correspondence. We quantize this space into B bins. Our candidate configurations will be pairs of cluster correspondences ((bI1 ,1 , bI2 ,1 ), (bI1 ,2 , bI2 ,2 )) ? (Ci ?Ci )?(Cj ?Cj )
that fall in the same bin, i.e., share the same transform and have the same relative location. Between
a given pair of images, there can be multiple such pairs of correspondences. We keep track of those
via a multi-graph GP = (P, EP ) that has a node for each image I ? P. For each correspondence
((bI1 ,1 , bI2 ,1 ), (bI1 ,2 , bI2 ,2 )), we draw an edge (I1 , I2 ) and label it by the clusters Ci , Cj and the
common relative position. As a result, there can be multiple edges (I1 , Ij ) in GP with different edge
labels.
The most frequently occurring configuration can now be read out by finding the largest connected
component in GP induced by retaining only edges with the same label. We use the largest component(s) as the characteristic configurations for a given image label (object class). If the component
is very small, then there is not enough information to determine co-occurrences, and we simply use
the most frequent single cluster. The final single ?correct? localization will be the smallest bounding
box that contains the full configuration.
Discovering mislocalized hard negatives. Discovering frequent configurations can not only lead
to better localization estimates of the full object, but they can also be used to generate mislocalized
estimates as ?hard negatives? when training the object detector. We exploit this idea as follows.
Let b1 , b2 be a discovered configuration within a given image. These patches typically constitute
co-occurring parts or a part and the full object. Our foreground estimate is the smallest box that
includes both b1 and b2 . Hence, any region within the foreground estimate that does not overlap
simultaneously with both b1 and b2 will capture only a fragment of the foreground object. We extract
the four largest such rectangular regions (see white boxes in Figure 3) as hard negative examples.
Specifically, we parameterize any rectangular region with [xl , xr , y t , y b ], i.e., its x-left, x-right,
y-top, and y-bottom coordinate values. Let the bounding box of bi (i = 1, 2) be [xli , xri , yit , yib ],
the foreground estimate be [xlf , xrf , yft , yfb ], and let xl = max(xl1 , xl2 ), xr = min(xr1 , xr2 ), y t =
max(y1t , y2t ), y b = min(y1b , y2b ). We generate four hard negatives: [xlf , xl , yfb , yft ], [xr , xrf , yfb , yft ],
[xlf , xrf , yft , y t ], [xlf , xrf , y b , yfb ]. If either b1 or b2 is very small in size relative to the foreground, the
resulting hard negatives can have high overlap with the foreground, which will introduce undesirable
noise (false negatives) when training the detector. Thus, we shrink any hard negative that overlaps
with the foreground estimate by more than 50%, until its overlap is 50% (we adjust the boundary
that does not coincide with any of the foreground estimation boundaries).
5
Figure 3: Automatically discovered foreground estimation box (magenta), hard negative (white),
and the patch (yellow) that induced the hard negative. Note that we are only showing the largest one
out of (up to) four hard negatives per image.
Note that simply taking arbitrary rectangular regions that overlap with the foreground estimation box
by some threshold will not always generate useful hard negatives (as we show in the experiments).
If the overlap threshold is too low, the selected regions will be uninformative, and if the overlap
threshold is too high, the selected regions will cover too much of the foreground. Our approach
selects informative hard negatives more robustly by ruling out the overlapping region between the
configuration patches, which is very likely be part of the foreground object but not the full object.
Mining positives and training the detector. While the discovered configurations typically lead
to better foreground localization, their absolute count can be relatively low compared to the total
number of positive images. This is due to inaccuracies in the initial patch discovery stage: for a
frequent configuration to be discovered, both of its patches must be found accurately. Thus, we also
mine additional positives from the set of remaining positive images P 0 that did not produce any of
the discovered configurations.
To do so, we train an initial object detector, using the foreground estimates derived from our discovered configurations as positive examples, and the corresponding discovered hard negative regions as
negatives. In addition, we mine negative examples in negative images as in [10]. We run the detector
on all selective search regions in P 0 and retain the region in each image with the highest detection
score as an additional positive training example. Our final detector is trained on this augmented
training data, and iteratively improved by latent SVM (LSVM) updates (see [10, 29] for details).
4
Experiments
In this section, we analyze: (1) detection performance of the models trained with the discovered
configurations, and (2) impact of the discovered hard negatives on detection performance.
Implementation details. We employ a recent region based detection framework [13, 29] and use the
same fc7 features from the CNN model [6] on region proposals [30] throughout the experiments. For
discriminative patch discovery, we use K = |P|/2, ? = K/20. For correspondence detection, we
discretize the 4D transform space of {x: relative horizontal shift, y: relative vertical shift, s: relative
scale, p: relative aspect ratio} with ?x = 30 px, ?y = 30 px, ?s = 1 px/px, ?p = 1 px/px.
We chose this binning scheme by examining a few qualitative examples so that scale and aspect
ratio agreement between the two paired instances are more strict, while their translation agreement
is more loose, in order to handle deformable objects. More details regarding the transform space
binning can be found in [22].
Discovered configurations. Figure 5 shows the discovered configurations (solid green and yellow
boxes) and foreground estimates (dashed magenta boxes) that have high degree in graph GP for all
20 classes in the PASCAL dataset. Our method consistently finds meaningful combinations such
as a wheel and body of bicycles, face and torso of people, locomotive basement and upper body
parts of trains/buses, and window and body frame of cars. Some failures include cases where the
algorithm latches onto different objects co-occurring in consistent configurations such as the lamp
and sofa combination (right column, second row from the bottom in Figure 5).
Weakly-supervised object detection. Following the evaluation protocol of the PASCAL VOC
dataset, we report detection results on the PASCAL test set using detection average precision. For a
direct comparison with the state-of-the-art weakly-supervised object detection method [29], we do
not use the extra instance level annotations such as pose, difficult, truncated and restrict the supervision to the image-level object presence annotations. Table 1 compares our detection results against
two baseline methods [25, 29] on the full dataset. Our method improves detection performance on
15 of the 20 classes. It is worth noting that our method yields significant improvement on the person
6
aero bike bird boat btl bus car cat chr cow tble dog horse mbk pson plnt shp sofa train tv mAP
[25] 13.4 44.0 3.1 3.1 0.0 31.2 43.9 7.1 0.1 9.3 9.9 1.5 29.4 38.3 4.6 0.1 0.4 3.8 34.2 0.0 13.9
[29] 27.6 41.9 19.7 9.1 10.4 35.8 39.1 33.6 0.6 20.9 10.0 27.7 29.4 39.2 9.1 19.3 20.5 17.1 35.6 7.1 22.7
ours1 31.9 47.0 21.9 8.7 4.9 34.4 41.8 25.6 0.3 19.5 14.2 23.0 27.8 38.7 21.2 17.6 26.9 12.8 40.1 9.2 23.4
ours2 36.3 47.6 23.3 12.3 11.1 36.0 46.6 25.4 0.7 23.5 12.5 23.5 27.9 40.9 14.8 19.2 24.2 17.1 37.7 11.6 24.6
Table 1: Detection average precision (%) on full PASCAL VOC 2007 test set. ours1 : before latent
updates. ours2 : after latent updates
w/o hard negatives
neighboring hard negatives
discovered hard negatives
ours + SVM
22.5
22.2
23.4
ours + LSVM
23.7
23.9
24.6
Table 2: Effect of our hard negative examples on full PASCAL VOC 2007 test set.
class, which is arguably the most important category in the PASCAL dataset. Figure 4 shows some
example high scoring detections on the test set. Our method produces more complete detections
since it is trained on better localized instances of the object-of-interest.
Figure 4: Example detections on test set. Green: our method, red: [29]
Impact of discovered hard negatives. To analyze the effect of our discovered hard negatives, we
compare to two baselines: (1) not adding any negative examples from positives images, and (2)
adding image regions around the foreground estimate, as conventionally implemented in fully supervised object detection algorithms [9, 13]. For the latter, we use the criterion from [13], where
all image regions in positive images with overlap score (intersection over union with respect to any
foreground region) less than 0.3 are used as ?neighboring? negative image regions on positive images. Table 2 shows the effect of our hard negative examples on detection mean average precision for
all classes (mAP). We also added neighboring negative examples to [29], but this decreases its mAP
from 20.3% to 20.2% (before latent updates) and from 22.7% to 21.8% (after latent updates). These
experiments show that adding neighboring negative regions does not lead to noticeable improvement over not adding any negative regions from positive images, while adding our automatically
discovered hard negative regions improves detection performance more substantially.
Conclusion. We developed a weakly-supervised object detection method that discovers frequent
configurations of discriminative visual patterns. We showed that the discovered configurations provide more accurate spatial coverage of the full object and provide a way to generate useful hard
negatives. Together, these lead to state-of-the-art weakly-supervised detection results on the challenging PASCAL VOC dataset.
Acknowledgement. This work was supported in part by DARPA?s MSEE and SMISC programs, by NSF awards IIS-1427425, IIS-1212798, IIS-1116411, and by
support from Toyota.
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Figure 5: Example configurations that have high degree in graph GP . The solid green and yellow boxes show the discovered discriminative visual parts, and the dashed magenta box shows the
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9
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4,732 | 5,285 | Encoding High Dimensional Local Features by Sparse
Coding Based Fisher Vectors
Lingqiao Liu1 , Chunhua Shen1,2 , Lei Wang3 , Anton van den Hengel1,2 , Chao Wang3
1
School of Computer Science, University of Adelaide, Australia
2
ARC Centre of Excellence for Robotic Vision
3
School of Computer Science and Software Engineering, University of Wollongong, Australia
Abstract
Deriving from the gradient vector of a generative model of local features, Fisher
vector coding (FVC) has been identified as an effective coding method for image classification. Most, if not all, FVC implementations employ the Gaussian
mixture model (GMM) to characterize the generation process of local features.
This choice has shown to be sufficient for traditional low dimensional local features, e.g., SIFT; and typically, good performance can be achieved with only a few
hundred Gaussian distributions. However, the same number of Gaussians is insufficient to model the feature space spanned by higher dimensional local features,
which have become popular recently. In order to improve the modeling capacity
for high dimensional features, it turns out to be inefficient and computationally
impractical to simply increase the number of Gaussians.
In this paper, we propose a model in which each local feature is drawn from a
Gaussian distribution whose mean vector is sampled from a subspace. With certain approximation, this model can be converted to a sparse coding procedure and
the learning/inference problems can be readily solved by standard sparse coding
methods. By calculating the gradient vector of the proposed model, we derive
a new fisher vector encoding strategy, termed Sparse Coding based Fisher Vector Coding (SCFVC). Moreover, we adopt the recently developed Deep Convolutional Neural Network (CNN) descriptor as a high dimensional local feature
and implement image classification with the proposed SCFVC. Our experimental evaluations demonstrate that our method not only significantly outperforms
the traditional GMM based Fisher vector encoding but also achieves the state-ofthe-art performance in generic object recognition, indoor scene, and fine-grained
image classification problems.
1
Introduction
Fisher vector coding is a coding method derived from the Fisher kernel [1] which was originally proposed to compare two samples induced by a generative model. Since its introduction to computer
vision [2], many improvements and variants have been proposed. For example, in [3] the normalization of Fisher vectors is identified as an essential step to achieve good performance; in [4] the spatial
information of local features is incorporated; in [5] the model parameters are learned through a endto-end supervised training algorithm and in [6] multiple layers of Fisher vector coding modules are
stacked into a deep architecture. With these extensions, Fisher vector coding has been established
as the state-of-the-art image classification approach.
Almost all of these methods share one common component: they all employ Gaussian mixture
model (GMM) as the generative model for local features. This choice has been proved effective in
modeling standard local features such as SIFT, which are often of low dimension. Usually, using a
1
mixture of a few hundred Gaussians has been sufficient to guarantee good performance. Generally
speaking, the distribution of local features can only be well captured by a Gaussian distribution
within a local region due to the variety of local feature appearances and thus the number of Gaussian
mixtures needed is essentially determined by the volume of the feature space of local features.
Recently, the choice of local features has gone beyond the traditional local patch descriptors such as
SIFT or SURF [7] and higher dimensional local features such as the activation of a pre-trained deep
neural-network [8] or pooled coding vectors from a local region [9, 10] have demonstrated promising performance. The higher dimensionality and rich visual content captured by those features make
the volume of their feature space much larger than that of traditional local features. Consequently, a
much larger number of Gaussian mixtures will be needed in order to model the feature space accurately. However, this would lead to the explosion of the resulted image representation dimensionality
and thus is usually computationally impractical.
To alleviate this difficulty, here we propose an alternative solution. We model the generation process
of local features as randomly drawing features from a Gaussian distribution whose mean vector is
randomly drawn from a subspace. With certain approximation, we convert this model to a sparse
coding model and leverage an off-the-shelf solver to solve the learning and inference problems.
With further derivation, this model leads to a new Fisher vector coding algorithm called Sparse
Coding based Fisher Vector Coding (SCFVC). Moreover, we adopt the recently developed Deep
Convolutional Neural Network to generate regional local features and apply the proposed SCFVC
to these local features to build an image classification system.
To demonstrate its effectiveness in encoding the high dimensional local feature, we conduct a series
of experiments on generic object, indoor scene and fine-grained image classification datasets, it is
shown that our method not only significantly outperforms the traditional GMM based Fisher vector
coding in encoding high dimensional local features but also achieves state-of-the-art performance in
these image classification problems.
2
2.1
Fisher vector coding
General formulation
Given two samples generated from a generative model, their similarity can be evaluated by using a
Fisher kernel [1]. The sample can take any form, including a vector or a vector set, as long as its generation process can be modeled. For a Fisher vector based image classification approach, the sample
is a set of local features extracted from an image which we denote it as X = {x1 , x2 , ? ? ? , xT }.
Assuming each xi is modeled by a p.d.f P (x|?) and is drawn i.i.d, in Fisher kernel a sample X can
be described by the gradient vector over the model parameter ?
X
GX
?? log P (xi |?).
(1)
? = ?? log P (X|?) =
i
T ?1 X
GX
G? ,
? F
The Fisher kernel is then defined as K(X, Y) =
where F is the information matrix
X XT
and is defined as F = E[G? G? ]. In practice, the role of the information matrix is less significant
and is often omitted for computational simplicity [3]. As a result, two samples can be directly
compared by the linear kernel of their corresponding gradient vectors which are often called Fisher
vectors. From a bag-of-features model perspective, the evaluation of the Fisher kernel for two
images can be seen as first calculating the gradient or Fisher vector of each local feature and then
performing sum-pooling. In this sense, the Fisher vector calculation for each local feature can be
seen as a coding method and we call it Fisher vector coding in this paper.
2.2
GMM based Fisher vector coding and its limitation
To implement the Fisher vector coding framework introduced above, one needs to specify the distribution P (x|?). In the literature, most, if not all, works choose GMM to model the generation
process of x, which can be described as follows:
? Draw a Gaussian model N (?k , ?k ) from the prior distribution P (k), k = 1, 2, ? ? ? , m .
? Draw a local feature x from N (?k , ?k ).
2
Generally speaking, the distribution of x resembles a Gaussian distribution only within a local region
of feature space. Thus, for a GMM, each of Gaussian essentially models a small partition of the
feature space and many of them are needed to depict the whole feature space. Consequently, the
number of mixtures needed will be determined by the volume of the feature space. For the commonly
used low dimensional local features, such as SIFT, it has been shown that it is sufficient to set the
number of mixtures to few hundreds. However, for higher dimensional local features this number
may be insufficient. This is because the volume of feature space usually increases quickly with the
feature dimensionality. Consequently, the same number of mixtures will result in a coarser partition
resolution and imprecise modeling.
To increase the partition resolution for higher dimensional feature space, one straightforward solution is to increase the number of Gaussians. However, it turns out that the partition resolution
increases slowly (compared to our method which will be introduced in the next section) with the
number of mixtures. In other words, much larger number of Gaussians will be needed and this will
result in a Fisher vector whose dimensionality is too high to be handled in practice.
3
3.1
Our method
Infinite number of Gaussians mixture
Our solution to this issue is to go beyond a fixed number of Gaussian distributions and use an
infinite number of them. More specifically, we assume that a local feature is drawn from a Gaussian
distribution with a randomly generated mean vector. The mean vector is a point on a subspace
spanned by a set of bases (which can be complete or over-complete) and is indexed by a latent
coding vector u. The detailed generation process is as follows:
? Draw a coding vector u from a zero mean Laplacian distribution P (u) =
1
2?
exp(? |u|
? ).
? Draw a local feature x from the Gaussian distribution N (Bu, ?),
where the Laplace prior for P (u) ensures the sparsity of resulting Fisher vector which can be helpful
for coding. Essentially, the above process resembles a sparse coding model. To show this relationship, let?s first write the marginal distribution of x:
Z
Z
P (x) =
P (x, u|B)du =
P (x|u, B)P (u)du.
(2)
u
u
The above formulation involves an integral operator which makes the likelihood evaluation difficult.
To simplify the calculation, we use the point-wise maximum within the integral term to approximate
the likelihood, that is,
P (x) ? P (x|u? , B)P (u? ).
u? = argmax P (x|u, B)P (u)
(3)
u
2
2
By assumming that ? = diag(?12 , ? ? ? , ?m
) and setting ?12 = ? ? ? = ?m
= ? 2 as a constant. The
logarithm of P (x) is written as
log(P (x|B)) = min
u
1
kx ? Buk22 + ?kuk1 ,
?2
(4)
which is exactly the objective value of a sparse coding problem. This relationship suggests that we
can learn the model parameter B and infer the latent variable u by using the off-the-shelf sparse
coding solvers.
One question for the above method is that compared to simply increasing the number of models
in traditional GMM, how much improvement is achieved by increasing the partition resolution. To
answer this question, we designed an experiment to compare these two schemes. In our experiment,
the partition resolution is roughly measured by the average distance (denoted as d ) between a feature
and its closest mean vector in the GMM or the above model. The larger d is, the lower the partition
resolution is. The comparison is shown in Figure 1. In Figure 1 (a), we increase the dimensionality
3
3
2.55
2.8
2.5
2.6
2.45
2.4
d
d
of local features 1 and for each dimension we calculate d in a GMM model with 100 mixtures. As
seen, d increases quickly with the feature dimensionality. In Figure 1 (b), we try to reduce d by
introducing more mixture distributions in GMM model. However, as can be seen, d drops slowly
with the increase in the number of Gaussians. In contrast, with the proposed method, we can achieve
much lower d by using only 100 bases. This result illustrates the advantage of our method.
2.2
2.4
2.35
GMM
GMM with 100 mixtures
2
1.8
100
Proposed model (with 100 bases)
2.3
200
300
400
500
600
700
800
900
1000
Dimensionality of regional local features
2.25
100
200
300
400
500
600
700
800
900
1000
Number of Gaussian mixtures
(a)
(b)
Figure 1: Comparison of two ways to increase the partition resolution. (a) For GMM, d (the average distance between a local feature and its closest mean vector) increases with the local feature
dimensionality. Here the GMM is fixed at 100 Gaussians. (b) d is reduced in two ways (1) simply
increasing the number of Gaussian distributions in the mixture. (2) using the proposed generation
process. As can be seen, the latter achieves much lower d even with a small number of bases.
3.2
Sparse coding based Fisher vector coding
Once the generative model of local features is established, we can readily derive the corresponding
Fisher coding vector by differentiating its log likelihood, that is,
? 12 kx ? Bu? k22 + ?ku? k1
? log(P (x|B))
= ?
?B
?B
u? = argmax P (x|u, B)P (u).
C(x) =
(5)
u
Note that the differentiation involves u? which implicitly interacts with B. To calculate this term,
we notice that the sparse coding problem can be reformulated as a general quadratic programming
problem by defining u+ and u? as the positive and negative parts of u, that is, the sparse coding
problem can be rewritten as
min
u+ ,u?
kx ? B(u+ ? u? )k22 + ?1T (u+ + u? )
s.t. u+ ? 0
0
+
u? ? 0
(6)
? T
By further defining u = (u , u ) , log(P (x|B)) can be expressed in the following general form,
1 T
T
log(P (x|B)) = L(B) = max
u0 v(B) ? u0 P(B)u0 ,
u0
2
(7)
where P(B) and v(B) are a matrix term and a vector term depending on B respectively. The
derivative of L(B) has been studied in [11]. According to the Lemma 2 in [11], we can differentiate
L(B) with respect to B as if u0 did not depend on B. In other words, we can firstly calculate u0 or
(x|B))
equivalently u? by solving the sparse coding problem and then obtain the Fisher vector ? log(P
?B
as
? ?12 kx ? Bu? k22 + ?ku? k1
= (x ? Bu? )u? T .
?B
(8)
1
This is achieved by performing PCA on a 4096-dimensional CNN regional descriptor. For more details
about the feature we used, please refer to Section 3.4
4
Table 1: Comparison of results on Pascal VOC 2007. The lower part of this table lists some results
reported in the literature. We only report the mean average precision over 20 classes. The average
precision for each class is listed in Table 2.
Methods
mean average precision
Comments
SCFVC (proposed)
GMMFVC
76.9%
73.8%
single scale, no augmented data
single scale, no augmented data
CNNaug-SVM [8]
CNN-SVM [8]
NUS [13]
GHM [14]
AGS [15]
77.2%
73.9%
70.5%
64.7%
71.1%
with augmented data, use CNN for whole image
no augmented data.use CNN for whole image
-
Note that the Fisher vector expressed in Eq. (8) has an interesting form: it is simply the outer product
between the sparse coding vector u? and the reconstruction residual term (x ? Bu? ). In traditional
sparse coding, only the kth dimension of a coding vector uk is used to indicate the relationship
between a local feature x and the kth basis. Here in the sparse coding based Fisher vector, the
coding value uk multiplied by the reconstruction residual is used to capture their relationship.
3.3
Pooling and normalization
From the i.i.d assumption in Eq. (1), the Fisher vector of the whole image is 2
X ? log(P (xi |B))
X
? log(P (I|B))
=
=
(xi ? Bu?i )u?i > .
?B
?B
xi ?I
(9)
xi ?I
This is equivalent to performing the sum-pooling for the extracted Fisher coding vectors. However,
it has been observed [3, 12] that the image signature obtained using sum-pooling tends to overemphasize the information from the background [3] or bursting visual words [12]. It is important to
apply normalization when sum-pooling is used. In this paper, we apply intra-normalization [12] to
normalize
the pooled Fisher vectors. More specifically, we apply l2 normalization to the subvectors
P
(x
? Bu?i )u?i,k ?k, where k indicates the kth dimension of the sparse coding u?i . Besides
i
xi ?I
intra-normalization, we also utilize the power normalization as suggested in [3].
3.4
Deep CNN based regional local features
Recently, the middle-layer activation of a pre-trained deep CNN has been demonstrated to be a
powerful image descriptor [8, 16]. In this paper, we employ this descriptor to generate a number
of local features for an image. More specifically, an input image is first resized to 512?512 pixels
and regions with the size of 227?227 pixels are cropped with the stride 8 pixels. These regions
are subsequently feed into the deep CNN and the activation of the sixth layer is extracted as local
features for these regions. In our implementation, we use the Caffe [17] package which provides a
deep CNN pre-trained on ILSVRC2012 dataset and its 6-th layer is a 4096-dimensional vector. This
strategy has demonstrated better performance than directly using deep CNN features for the whole
image recently [16].
Once regional local features are extracted, we encoded them using the proposed SCFVC method and
generate an image level representation by sum-pooling and normalization. Certainly, our method is
open to the choice of other high-dimensional local features. The reason for choosing deep CNN
features in this paper is that by doing so we can demonstrate state-of-the-art image classification
performance.
4
Experimental results
We conduct experimental evaluation of the proposed sparse coding based Fisher vector coding
(SCFVC) on three large datasets: Pascal VOC 2007, MIT indoor scene-67 and Caltech-UCSD Birds2
the vectorized form of
? log(P (I|B))
?B
is used as the image representation.
5
Table 2: Comparison of results on Pascal VOC 2007 for each of 20 classes. Besides the proposed
SCFVC and the GMMFVC baseline, the performance obtained by directly using CNN as global
feature is also compared.
SCFVC
GMMFVC
CNN-SVM
SCFVC
GMMFVC
CNN-SVM
aero
bike
bird
boat
bottle
bus
car
cat
chair
cow
89.5
87.1
88.5
84.1
80.6
81.0
83.7
80.3
83.5
83.7
79.7
82.0
43.9
42.8
42.0
76.7
72.2
72.5
87.8
87.4
85.3
82.5
76.1
81.6
60.6
58.6
59.9
69.6
64.0
58.5
table
dog
horse
mbike
person
plant
sheep
sofa
train
TV
72.0
66.9
66.5
77.1
75.1
77.8
88.7
84.9
81.8
82.1
81.2
78.8
94.4
93.1
90.2
56.8
53.1
54.8
71.4
70.8
71.1
67.7
66.2
62.6
90.9
87.9
87.2
75.0
71.3
71.8
Table 3: Comparison of results on MIT-67. The lower part of this table lists some results reported
in the literature.
Methods
Classification Accuracy
Comments
SCFVC (proposed)
GMMFVC
68.2%
64.3%
with single scale
with single scale
MOP-CNN [16]
VLAD level2 [16]
CNN-SVM [8]
FV+Bag of parts [19]
DPM [20]
68.9%
65.5%
58.4%
63.2%
37.6%
with three scales
with single best scale
use CNN for whole image
-
200-2011. These are commonly used evaluation benchmarks for generic object classification, scene
classification and fine-grained image classification respectively. The focus of these experiments is
to examine that whether the proposed SCFVC outperforms the traditional Fisher vector coding in
encoding high dimensional local features.
4.1
Experiment setup
In our experiments, the activations of the sixth layer of a pre-trained deep CNN are used as regional
local features. PCA is applied to further reduce the regional local features from 4096 dimensions to
2000 dimensions. The number of Gaussian distributions and the codebook size for sparse coding is
set to 100 throughout our experiments unless otherwise mentioned.
For the sparse coding, we use the algorithm in [18] to learn the codebook and perform the coding
vector inference. For all experiments, linear SVM is used as the classifier.
4.2
Main results
Pascal-07 Pascal VOC 2007 contains 9963 images with 20 object categories which form 20 binary
(object vs. non-object) classification tasks. The use of deep CNN features has demonstrated the
state-of-the-art performance [8] on this dataset. In contrast to [8], here we use the deep CNN features as local features to model a set of image regions rather than as a global feature to model the
whole image. The results of the proposed SCFVC and traditional Fisher vector coding, denoted as
GMMFVC, are shown in Table 1 and Table 2. As can be seen from Table 1, the proposed SCFVC
leads to superior performance over the traditional GMMFVC and outperforms GMMFVC by 3%.
By cross-referencing Table 2, it is clear that the proposed SCFVC outperforms GMMFVC in all
of 20 categories. Also, we notice that the GMMFVC is merely comparable to the performance of
directly using deep CNN as global features, namely, CNN-SVM in Table 1. Since both the proposed
SCFVC and GMMFVC adopt deep CNN features as local features, this observation suggests that
the advantage of using deep CNN features as local features can only be clearly demonstrated when
the appropriate coding method, i.e. the proposed SCFVC is employed. Note that to further boost the
6
Table 4: Comparison of results on Birds-200 2011. The lower part of this table lists some results
reported in the literature.
Methods
Classification Accuracy
Comments
SCFVC (proposed)
GMMFVC
66.4%
61.7%
with single scale
with single scale
CNNaug-SVM [8]
CNN-SVM [8]
DPD+CNN+LogReg [21]
DPD [22]
61.8%
53.3%
65.0%
51.0%
with augmented data, use CNN for the whole image
no augmented data, use CNN as global features
use part information
-
69
Classification Accuracy %
68
67
66
65
64
63
SCFV
GMMFV
62
61
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Dimensionality of regional local features
Figure 2: The performance comparison of classification accuracy vs. local feature dimensionality
for the proposed SCFVC and GMMFVC on MIT-67.
performance, one can adopt some additional approaches like introducing augmented data or combining multiple scales. Some of the methods compared in Table 1 have employed these approaches
and we have commented this fact as so inform readers that whether these methods are directly comparable to the proposed SCFVC. We do not pursue these approaches in this paper since the focus of
our experiment is to compare the proposed SCFVC against traditional GMMFVC.
MIT-67 MIT-67 contains 6700 images with 67 indoor scene categories. This dataset is quite challenging because the differences between some categories are very subtle. The comparison of classification results are shown in Table 3. Again, we observe that the proposed SCFVC significantly outperforms traditional GMMFVC. To the best of our knowledge, the best performance on this dataset
is achieved in [16] by concatenating the features extracted from three different scales. The proposed
method achieves the same performance only using a single scale. We also tried to concatenate the
image representation generated from the proposed SCFVC with the global deep CNN feature. The
resulted performance can be as high as 70% which is by far the best performance achieved on this
dataset.
Birds-200-2011 Birds-200-2011 contains 11788 with 200 different birds species, which is a commonly used benchmark for fine-grained image classification. The experimental results on this
dataset are shown in Table 4. The advantage of SCFVC over GMMFVC is more pronounced on
this dataset: SCFVC outperforms GMMFVC by over 4%. We also notice two interesting observations: (1) GMMFVC even achieves comparable performance to the method of using the global
deep CNN feature with augmented data, namely, CNNaug-SVM in Table 4. (2) Although we do
not use any parts information (of birds), our method outperforms the result using parts information
(DPD+CNN+LogReg in Table 4). These two observations suggest that using deep CNN features
as local features is better for fine-grained problems and the proposed method can further boost its
advantage.
7
Table 5: Comparison of results on MIT-67 with three different settings: (1) 100-basis codebook with
1000 dimensional local features, denoted as SCFV-100-1000D (2) 400 Gaussian mixtures with 300
dimensional local features, denoted as GMMFV-400-300D (3) 1000 Gaussian mixtures with 100
dimensional local features denoted as GMMFV-1000-100D. They have the same/similar total image
representation dimensionality.
SCFV-100-1000D
68.1%
4.3
GMMFV-400-300D
64.0%
GMMFV-1000-100D
60.8%
Discussion
In the above experiments, the dimensionality of local features is fixed to 2000. But how about the
performance comparison between the proposed SCFV and traditional GMMFV on lower dimensional features? To investigate this issue, we vary the dimensionality of the deep CNN features from
100 to 2000 and compare the performance of the two Fisher vector coding methods on MIT-67. The
results are shown in Figure 2. As can be seen, for lower dimensionality (like 100), the two methods
achieve comparable performance and in general both methods benefit from using higher dimensional
features. However, for traditional GMMFVC, the performance gain obtained from increasing feature dimensionality is lower than that obtained by the proposed SCFVC. For example, from 100 to
1000 dimensions, the traditional GMMFVC only obtains 4% performance improvement while our
SCFVC achieves 7% performance gain. This validates our argument that the proposed SCFVC is
especially suited for encoding high dimensional local features.
Since GMMFVC works well for lower dimensional features, how about reducing the higher dimensional local features to lower dimensions and use more Gaussian mixtures? Will it be able to achieve
comparable performance to our SCFVC which uses higher dimensional local features but a smaller
number of bases? To investigate this issue, we also evaluate the classification performance on MIT67 using 400 Gaussian mixtures with 300-dimension local features and 1000 Gaussian mixtures with
100-dimension local features. Thus the total dimensionality of these two image representations will
be similar to that of our SCFVC which uses 100 bases and 1000-dimension local features. The comparison is shown in Table 5. As can be seen, the performance of these two settings are much inferior
to the proposed one. This suggests that some discriminative information may have already been lost
after the PCA dimensionality reduction and the discriminative power can not be re-boosted by simply introducing more Gaussian distributions. This verifies the necessity of using high dimensional
local features and justifies the value of the proposed method.
In general, the inference step in sparse coding can be slower than the membership assignment in
GMM model. However, the computational efficiency can be greatly improved by using an approximated sparse coding algorithm such as learned FISTA [23] or orthogonal matching pursuit [10].
Also, the proposed method can be easily generalized to several similar coding models, such as local
linear coding [24]. In that case, the computational efficiency is almost identical (or even faster if
approximated k-nearest neighbor algorithms are used) to the traditional GMMFVC.
5
Conclusion
In this work, we study the use of Fisher vector coding to encode high-dimensional local features.
Our main discovery is that traditional GMM based Fisher vector coding is not particular well suited
to modeling high-dimensional local features. As an alternative, we proposed to use a generation
process which allows the mean vector of a Gaussian distribution to be chosen from a point in a
subspace. This model leads to a new Fisher vector coding method which is based on sparse coding
model. Combining with the activation of the middle layer of a pre-trained CNN as high-dimensional
local features, we build an image classification system and experimentally demonstrate that the
proposed coding method is superior to the traditional GMM in encoding high-dimensional local
features and can achieve state-of-the-art performance in three image classification problems.
Acknowledgements This work was in part supported by Australian Research Council grants
FT120100969, LP120200485, and the Data to Decisions Cooperative Research Centre. Correspondence should be addressed to C. Shen (email: [email protected]).
8
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9
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4,733 | 5,286 | Self-Adaptable Templates for Feature Coding
Xavier Boix1,2? Gemma Roig1,2? Salomon Diether1 Luc Van Gool1
1
Computer Vision Laboratory, ETH Zurich, Switzerland
2
LCSL, Massachusetts Institute of Technology & Istituto Italiano di Tecnologia, Cambridge, MA
{xboix,gemmar}@mit.edu
{boxavier,gemmar,sdiether,vangool}@vision.ee.ethz.ch
Abstract
Hierarchical feed-forward networks have been successfully applied in object
recognition. At each level of the hierarchy, features are extracted and encoded,
followed by a pooling step. Within this processing pipeline, the common trend is
to learn the feature coding templates, often referred as codebook entries, filters, or
over-complete basis. Recently, an approach that apparently does not use templates
has been shown to obtain very promising results. This is the second-order pooling
(O2P) [1, 2, 3, 4, 5]. In this paper, we analyze O2P as a coding-pooling scheme.
We find that at testing phase, O2P automatically adapts the feature coding templates to the input features, rather than using templates learned during the training
phase. From this finding, we are able to bring common concepts of coding-pooling
schemes to O2P, such as feature quantization. This allows for significant accuracy
improvements of O2P in standard benchmarks of image classification, namely
Caltech101 and VOC07.
1
Introduction
Many object recognition schemes, inspired from biological vision, are based on feed-forward hierarchical architectures, e.g. [6, 7, 8]. In each level in the hierarchy, the algorithms can be usually
divided into the steps of feature coding and spatial pooling. The feature coding extracts similarities
between the set of input features and a set of templates (the so called filters, over-complete basis or
codebook), and then, the similarity responses are transformed using some non-linearities. Finally,
the spatial pooling extracts one single vector from the set of transformed responses. The specific architecture of the network (e.g. how many layers), and the specific algorithms for the coding-pooling
at each layer are usually set for a recognition task and dataset, cf. [9].
Second-order Pooling (O2P) is an alternative algorithm to the aforementioned coding-pooling
scheme. O2P has been introduced in medical imaging to analyze magnetic resonance images [1, 2],
and lately, O2P achieved state-of-the-art in some of the traditional computer vision tasks [3, 4, 5, 10].
A surprising fact of O2P is that it is formulated without feature coding templates [5]. This is in contrast to the common coding-pooling schemes, in which the templates are learned during a training
phase, and at testing phase, the templates remain fixed to the learned values.
Motivated by the intriguing properties of O2P, in this paper we try to re-formulate O2P as a codingpooling scheme. In doing so, we find that O2P actually computes similarities to feature coding
templates as the rest of the coding-pooling schemes. Yet, what remains uncommon of O2P, is that
the templates are ?recomputed? for each specific input, rather than being fixed to learned values. In
O2P, the templates are self-adapted to the input, and hence, they do not require learning.
From our formulation, we are able to bring common concepts of coding-pooling schemes to O2P,
such as feature quantization. This allows us to achieve significant improvements of the accuracy
?
Both first authors contributed equally.
1
of O2P for image classification. We report experiments on two challenging benchmarks for image
classification, namely Caltech101 [11], and VOC07 [12].
2
Preliminaries
In this Section, we introduce O2P as well as several coding-pooling schemes, and identify some
common terminology in the literature. This will serve as a basis for the new formulation of O2P,
that we introduce in the following section.
The algorithms that we analyze in this section are usually part of a layer of a hierarchical network
for object recognition. The input to these algorithms is a set of feature vectors that come from the
output of the previous layer, or from the raw image. Let {xi }N be the set of input feature vectors
to the algorithm, which is the set of N feature vectors, xi ? RM , indexed by i ? {1, . . . , N }.
The output of the algorithm is a single vector, which we denote as y, and it may have a different
dimensionality than the input vectors.
In the following subsections, we present the algorithms and terminology of template-based methods,
and then, we introduce the formulation of O2P that appears in the literature that apparently does not
use templates.
2.1
Coding-Pooling based on Evaluating Similarities to Templates
Template-based methods are build upon similarities between the input vectors and a set of templates.
Depending on the terminology of each algorithm, the templates may be denoted as filters, codebook,
or over-complete basis. From now on, we will refer to all of them as templates. We denote the set
of templates as {bk ? RM }P . In this paper, bk and the input feature vectors xi have the same
dimensionality, M . The set of templates is fixed to learned values during the training phase. There
are many possible learning algorithms, but analyzing them is not necessary here.
The algorithms that are interesting for our purposes, start by computing a similarity measure between
the input feature vectors {xi }N and the templates {bk }P . Let ?(xi , bk ) be the similarity function,
which depends on each algorithm. We define ? i as the vector that contains the similarities of xi to
the set of templates {bk }, and ? ? RM ?P the matrix whose columns are the vectors ? i , i.e.
?ki = ?(xi , bk ).
(1)
Once ? is computed, the algorithms that we analyze apply some non-linear transformation to ?, and
then, the resulting responses are merged together, with the so called pooling operation. The pooling
consists on generating one single response value for each template. We denote as gk (?) the function
that includes both the non-linear transformation and the pooling operation, where gk : RM ?P ? R.
We include both operations in the same function, but in the literature it is usually presented as two
separate steps. Finally, the output vector y is built using {gk (?)}P , {bk }P and {xi }N , depending
on the algorithm. It is also quite common to concatenate the outputs of neighboring regions to
generate the final output of the layer.
We now show how the presented terminology is applied to some methods based on evaluating similarities to templates, namely assignment-based methods and Fisher Vector. In the sequel, these
algorithms will be a basis to reformulate O2P.
Assignment-based Methods The popular Bag-of-Words and some of its variants fall into this
category, e.g. [13, 14, 15]. These methods consist on assigning each input vector xi to a set of
templates (the so called vector quantization), and then, building a histogram of the assignments,
which corresponds to the average pooling operation.
We now present them using our terminology. After computing the similarities to the templates, ?
(usually based on `2 distance), gk (?) computes both the vector quantization and the pooling. Let
s be the number of templates to which each input vector is assigned, and let ? 0i be the resulting
assignment vector of xi (i.e. ? 0i is the result of applying vector quantisation on xi ). ? 0i has s entries
set to 1 and the rest to 0, that indicate the assignment. Finally, gkP
(?) also computes the pooling for
the assignments corresponding to the template k, i.e. gk (?) = N1 i<N ? 0ki . The final output vector
is the concatenation of the resulting pooling of the different templates, y = (g1 (?), . . . , gP (?)).
2
Fisher Vectors It uses the first and second order statistics of the similarities between the features
and the templates [16]. Fisher Vector builds two vectors for each template bk , which are
1 X
1 X
(1)
(2)
?k =
?ki (bk ? xi ) ?k =
?ki (bk ? xi )2 ? Ck ,
(2)
Ak
Bk
i<N
i<N
1
1
exp ? (xi ? bk )t Dk (xi ? bk ) .
(3)
where ?ki =
Zk
2
Ak , Bk , Ck are learned constants, Zk a normalization factor and Dk is a learned constant matrix of
the model. Note that in Eq. (3), ?ki is a similarity between the feature vector xi and the template bk .
(1)
(2)
(1)
(2)
The final output vector is y = (?1 , ?1 . . . , ?P , ?P ). For further details we refer the reader
to [16].
We use our terminology to do a very simple re-write of the terms. We define gk (?) and bF
k (we use
the super-index F to indicate that are from Fisher vectors, and different from bk ) as
1
(1)
(2)
(1)
(2)
(? , ?k ).
(4)
gk (?) = k(?k , ?k )k2 , bF
k =
gk (?) k
We can see the templates of Fisher vectors, bF
k , are obtained from computing some transformations
to the original learned template bk , which involve the input set of features {xi }. gk (?) is the norm
(1)
(2)
of (?k , ?k ), which gives an idea of the importance of each template in {xi }, similarly to gk (?)
in assignment-based methods. Note that bF
k and gk (?) are related to only one fixed template, bk .
F
The final output vector becomes y = (g1 (?)bF
1 , . . . , gP (?)bP ).
2.2
Second-Order Pooling
Second-order Pooling (O2P) was introduced in medical imaging to describe the voxels produced in
diffusion tensor imaging [1], and to process tensor fields [2, 17]. O2P starts by building a correlation
matrix from the set of feature (column) vectors {xi ? RM }N , i.e.
1 X
xi xti ,
(5)
K=
N
i<N
where xti
M ?M
is the transpose vector of xi , and K ? R
is a square matrix. K is a symmetric positive
definite (SPD) matrix, and contains second-order statistics of {xi }. The set of SPD matrices form
a Riemannian manifold, and hence, the conventional operations in the Euclidean space can not be
used. Several metrics have been proposed for SPD matrices, and the most celebrated is the LogEuclidean metric [17]. Such metric consists of mapping the SPD matrices to the tangent space by
using the logarithm of the matrix, log(K). In the tangent space, the standard Euclidean metrics can
be used.
The logarithm of an SPD matrix can be computed in practice by applying the logarithm individually
to each of the eigenvalues of K [18]. Thus, the final output vector for O2P can be written as
!
X
t
y = vec (log(K)) = vec
log(?k )ek ek ,
(6)
k<M
where ek are the eigenvectors of K, and ?k the corresponding eigenvalues. The vec(?) operator
vectorizes log(K).
In Eq. (6), apparently, there are no similarities to a set of templates. The absence of templates makes
O2P look quite different from template-based methods. Recently, O2P achieved state-of-the-art
results in some computer vision tasks, e.g. in object detection [3], semantic segmentation [5, 10],
and for patch description [4]. Both reasons, motivates us to further analyze O2P in relation to
template-based methods.
3
Self-Adaptability of the Templates
In this section, we introduce a formulation that relates O2P and template-based methods. The new
formulation is based on comparing two final representation vectors, rather than defining how the
3
final vector y is built. We denote hyr , ys i as the inner product between yr and ys , which are the
final representation vectors from two sets of input feature vectors, {xri }N and {xsi }N , respectively,
where we use the superscripts r and s to indicate the respective representation for each set. It will
become clear during this section why we analyze hyr , ys i instead of y.
We divide the analysis in three subsections. In subsection 3.1, we re-write the formulation of the
template-based methods of Section 2 with the inner product hyr , ys i. In subsection 3.2, we do the
same for O2P, and this unveils that O2P is also based on evaluating similarities to templates. In
subsection 3.3, we analyze the characteristics of the templates in O2P, which have the particularity
that are self-adapted to the input.
3.1
Re-Formulation of Template-Based Methods
We re-write a generic formulation for the template-based methods described in Section 2 with the
inner product between two final output vectors. The algorithms of Section 2 can be expressed as
XX
hyr , ys i =
gk (? r )gq (? s )S(brk , bsq ),
(7)
k<P q<P
where ?ki = ?(xi , bk ),
and S(u, v) is a similarity function between the templates that depends on each algorithm. Recall
that gk (?) is a function that includes the non-linearities and the pooling of the similarities between
the input feature vectors and the the templates. To see how Eq. (7) arises naturally from the algorithms of Section 2, we now analyze them in terms of this formulation.
Assignment-Based Methods
as
The inner product between two final output vectors can be written
hyr , ys i =(g1 (? r ), . . . , gP (? r ))t (g1s (? s ), . . . , gPs (? s )) =
X
XX
=
gk (? r )gk (? s ) =
gk (? r )gq (? s )I(brk = bsq ),
(8)
k<P q<P
k<P
where the last step introduces an outer summation, and the indicator function I(?) eliminates the
unnecessary cross terms. Comparing this last equation to Eq. (7), we can identify that S(brk , bsq ) is
the indicator function (returns 1 when brk = bsq , and 0 otherwise).
Fisher Vectors The inner product between two final Fisher Vectors is
r
rF t
s
sF
s
sF
hyr , ys i =(g1 (? r )brF
1 , . . . , gP (? )bP ) (g1 (? )b1 , . . . , gP (? )bP )
XX
sF
=
gk (? r )gq (? s )I(brk = bsq )hbrF
k , bq i.
(9)
k<P q<P
The indicator function appears for the same reason as in Assignment-Based Methods. The final
sF
templates for each set of input vectors, brF
k , bk , respectively, are compared with each other with
rF t sF
rF
sF
t sF
the similarity (bk ) bq . Thus, S(bk , bq ) in Eq. (7) is equal to I(brk = bsq )(brF
k ) bq .
3.2
O2P as Coding-Pooling based on Template Similarities
We now re-formulate O2P, in the same way as we did for template-based methods in the previous
subsection. This will allow relating O2P to template-based methods, and show that O2P also uses
similarities to templates.
We re-write the definition of O2P in Eq. (6) with hyr , ys i. Using the property vec(A)t vec(B) =
tr(At B), where tr(?) is the trace function of a matrix, hyr , ys i becomes (in the supplementary
material we do the full derivation)
hyr , ys i = hvec (log(Kr )) , vec (log(Ks ))i =
X X
=
log(?rk ) log(?sq )herk , esq i2 ,
(10)
k<M q<M
where ek etk is a square matrix, and the eigenvectors, {erk }M and {esk }M , are compared all against
each other with herk , esq i2 . Going back to the generic formulation of template-based methods in
4
Method
Assignment-based
Fisher Vectors
O2P
S(brk , bsq )
?ki = ?(xi , bk ) templates
I(brk = bsq )
hxi , bk i
fixed
r
s
sF
sF
I(bk = bq )hbk , bP i
Eq. (3)
fixed/adapted
hbrk , bsq i2
hxi , bk i2
self-adapted
gk (?)
P 0
1
i ?ki
N
(1)
(2)
k(?k , P
?k )k2
log N1 i ?ki
Table 1: Summary Table of the elements of our formulation for Assignment-based methods, Fisher
Vectors and O2P.
Eq. (7), we can see that the similarity function between the templates, S(erk , esq ), can be identified in
O2P as herk , esq i2 . Also, note that in O2P the sums go over M , which is the number of eigenvectors,
and in Eq. (7), go over P , which is the number of templates. Finally, gk (?) in Eq. (7) corresponds
to log(?k ) in O2P.
At this point, we have expressed O2P in a similar way as template-based methods. Yet, we still have
to find the similarity between the input feature vectors and the templates. For that purpose, we use
the definition of eigenvalues and eigenvectors, i.e. ?k ek = Kek , and also that tr(ek etk ) = 1 (the
t
eigenvectors are orthonormal). Then,
P wet can derive the following equivalence: ?k = ?k tr(ek ek ) =
1
t
tr(Kek ek ). Replacing K by N i xi xi , we find that the eigenvalues, ?k , can be written using the
similarity between the input vectors, xi , and the eigenvectors, ek :
1 X
1 X
tr((xi xti )(ek etk )) =
hxi , ek i2 .
(11)
?k =
N i
N i
Finally, we can integrate all the above derivations in Eq. (10), and we obtain that
X X
hyr , ys i =
gk (? r )gq (? s )herk , esq i2 ,
(12)
k<M q<M
where gk (?) = log(?k ) = log
1 X
?ki
N
i<N
2
and ?ki = ?(xi , ek ) = hxi , ek i .
!
,
(13)
(14)
We can see by analyzing Eq. (12) that this equation takes the same form as the general equation
of template-based methods in Eq. (7). Note that the eigenvectors take the same role as the set of
templates, i.e. bk = ek and P = M . Also, observe that S(brk , bsq ) is the square of the inner product
between eigenvectors, ?(xi , bk ) is the square of the inner product between the input vectors and the
eigenvectors, and the pooling operation is the logarithm of the average of the similarities. In Table 1
we summarize the corresponding elements of all the described methods.
3.3
Self-Adaptative Templates
We define self-adaptative templates as templates that only depend on the input set of feature vectors, and are not fixed to predefined values. This is the case in O2P, because the templates in O2P
correspond to the eigenvectors computed from the set of input feature vectors. The templates in
O2P are not fixed to values learned during the training phase. Interestingly, the final templates in
F
Fisher Vectors, bF
k , are also partially self-adapted to the input vectors. Note that bk are obtained by
modifying the fixed learned templates, bk , with the input feature vectors.
Finally, note that in O2P the number of templates is equal to the dimensionality of the input feature
vectors. Thus, in O2P the number of templates can not be increased without changing the input
vectors? length, M . This begs the following question: do M templates allow for sufficient generalization for object recognition for any set of input vectors? We analyze this question in the next
section.
4
Application: Quantization for O2P
We observe in the experiments section that the performance of O2P degrades when the number of
vectors in the set of input features increases. It is reasonable that M templates are not sufficient
when the number of different vectors in {xi }N increases, specially when they are very different
5
Algorithm 1: Sparse Quantization in O2P
Input: {xi }N , k
Output: y
foreach i = {1, . . . , N } do
? i ? Set k highest values of xi to its vector entry: xi , and the rest to 0
x
end
P
?ix
? ti
K = N1 i x
y = vec(log(K))
from each other. We now introduce an algorithm to increase the robustness of O2P to the variability
of the input vectors.
We quantize the input feature vectors, {xi }, before computing O2P. Quantization may discard details, and hence, reduce the variability among vectors. In the experiments section it is reported
that this allows preventing the degradation of performance in object recognition, when the number
of input feature vectors increases. The quantization algorithm that we use is sparse quantization
(SQ) [15, 19], because SQ does not change the dimensionality of the feature vector. Also, SQ is fast
to compute, and does not increase the computational cost of O2P.
Sparse Quantization for O2P For the quantization of {xi } we use SQ, which is a quantization
to the set of k-sparse vectors. Let Rqk be the set of k-sparse vectors, i.e. {s ? Rq : ksk0 ? k}.
Also, we define Bqk = {0, 1}qk = {s ? {0, 1}q : ksk0 = k}, which is the set of binary vectors
with k elements set to one and (q ? k) set to zero. The cardinality of |Bqk | is equal to kq . The
quantization of a vector v ? Rq into a codebook {ci } is a mapping of v to the closest element in
? ? = arg minv? ?{ci } k?
? ? is the quantized vector v. In the case of SQ, the
{ci }, i.e. v
v ? vk2 , where v
codebook {ci } contains the set of k-sparse vectors. These may be any of the previously introduced
types: Rqk , Bqk . An important advantage of SQ over a general quantization is that it can be computed
much more efficiently. The naive way to compute a general quantization is to evaluate the nearest
neighbor of v in {ci }, which may be costly to compute for large codebooks and high-dimensional
v. In contrast, SQ can be computed by selecting the k higher values of the set {vi }, i.e. for SQ into
Rqk , v?i = vi if i is one of the k-highest entries of vector v, and 0 otherwise. For SQ into Bqk , the
dimension indexed by the k-highest are set to 1 instead of vi , and 0 otherwise. (We refer the reader
to [15, 19] for a more detailed explanation on SQ).
In Algorithm 1 we depict the implementation of SQ in O2P, which highlights its simplicity. The
computational cost of SQ is negligible compared to the cost of computing O2P. We use the set of
k-sparse vectors in RM
k for SQ, which worked best in practice, as shown in the following.
5
Experiments
In this section, we analyze O2P in image classification from dense sampled SIFT descriptors. This
setup is common in image classification, and it allows direct comparison to previous works on O2P.
We report results on the Caltech101 [11] and VOC07 [12] datasets, using the standard evaluation
benchmarks, which are the mean average precision accuracy across all classes.
5.1
Implementation Details
We use the standard pipeline for image classification. We never use flipped or blurred images to
extend the training set.
Pipeline. For Caltech101, the image is re-sized to take a maximum height and width of 300
pixels, which is the standard resizing protocol for this dataset. For VOC07 the size of the images
remains the same as the original. We extract SIFT [8] from patches on a regular grid, at different
scales. In Caltech 101, we extract them at every 8 pixels and at the scales of 16, 32 and 48 pixels
diameter. In VOC07, SIFT is sampled at each 4 pixels and at the scales of 12, 24 and 36 pixels
diameter. O2P is computed using the SIFT descriptors as input, and using spatial pyramids. In
6
Caltech101, we generate the pooling regions dividing the image in 4 ? 4, 2 ? 2 and 1 ? 1 regions,
and in VOC07 in 3 ? 1, 2 ? 2 and 1 ? 1 regions. To generate the final descriptor for the whole
image, we concatenate the descriptors for each pooled region. We apply the power normalization to
the final feature dimensions, sign(x)|x|3/4 , that was shown to work well in practice [5]. Finally, we
use a linear one-versus-rest SVM classifier for each class with the parameter C of the SVM set to
1000. We use the LIBLINEAR library for the SVM[20].
Other Feature Codings. As a sanity check of our results, we replace O2P with the Bag-ofWords [13] baseline, without changing any of the parameters. In Caltech101, we replace the average
pooling of Bag-of-Words by max-pooling (without normalization) as it performs better. The codebook is learned by randomly picking a set of patches as codebook entries, which was shown to work
well for the encodings we are evaluating [14]. We use a codebook of 8192 entries, since with more
entries the performance does not increase significantly, but the computational cost does.
5.2
Results on Caltech101
We use 3 random splits of 30 images per class for training and the rest for testing. In Fig. 1a, results
are shown for different spatial pyramid configurations, as well as different levels of quantization.
Note that SQ with k = 128 is not introducing any quantization, as SIFT features are 128 dimensional
vectors. Note that using SQ increases the performance more than 5% compared to when not using
SQ (k = 128), when using only the first level of the pyramid. For the other levels of the pyramid,
there is less improvement with SQ. This is in accordance with the observation that in smaller regions
there are less SIFT vectors, the variability is smaller, and the limited amount of templates is able to
better capture the meaningful information than in bigger regions. We can also see that for small k
of SQ, the performance degrades due to the introduction of too much quantization.
We also run experiments with Bag-of-Words with max-pooling (74.8%), and O2P without SQ
(76.52%), and both of them are surpassed by O2P with SQ (78.63%). In [5], O2P accuracy is
reported to be 79.2% with SIFT descriptor (we do not compare to their version of enriched SIFT,
since all our experiments are with normal SIFT). We inspected the code of [5], and we found that
the difference of accuracy mainly comes from using a more drastic resizing of the image, that takes
a maximum of 100 pixels of width and height (usually in the literature it is 300 pixels). Note that resizing is another way of discarding information, and hence, O2P may benefit from that. We confirm
this by resizing the image back to 300 pixels in [5]?s code, and the accuracy is 77.1%, similar to the
one that we report without SQ in our code. The accuracy is not exactly the same due to differences
in the SIFT parameters in [5]. Also, we tested SQ in [5]?s code with the resizing to a maximum of
100 pixels, and the accuracy increased to 79.45%, which is higher than reported in [5], and close to
state-of-the-art results using SIFT descriptors (80.3%) [21].
5.3
Results on VOC07
In Fig. 1b, we run the same experiment as in Caltech101. Note that the impact of SQ is even more
evident than in Caltech101. In Table 2 we report the per-class accuracy, in addition to the mean
average precision reported in Fig. 1b. We follow the evaluation procedure as described in [12].
With the full pyramid, when we use SQ the accuracy increases from 18.81% to 50.97%. In contrast to Caltech101, O2P with SQ performance is similar to our implementation of Bag-of-Words
(51.14%). Thus, under adverse conditions for O2P, i.e. images with high variability such as in
VOC07 and with a high number of input vectors, we can use SQ and obtain huge improvements of
the O2P?s accuracy. The best reported results [22] in VOC07 are around 10% better than O2P with
SQ, yet we obtain more than 30% improvement from the baseline.
6
Conclusions
We found that O2P can be posed as a coding-pooling scheme based on evaluating similarities to templates. The templates of O2P self-adapt to the input, while the rest of the analyzed methods do not.
In practice, our formulation was used to improve the performance of O2P in image classification.
We are currently analyzing self-adaptative templates in deep hierarchical networks.
7
1 pyr.
1+2 pyr.
1+2+3 pyr.
1+2+3 pyr. w/o SQ
Caltech 101
PASCAL VOC 2007
50.97%
78.63%
Mean average precision
Mean accuracy
0.8
76.52%
0.75
75.55%
0.7
0.65
65.14%
0.6
0.55
SQ selected in val. set
5
20
40
60
80
100
128
0.5
49.09%
0.4
41.20%
0.3
0.2
0.1
18.81%
5
20
40
60
80
100
Sparse Quantization
Sparse Quantization
(a)
(b)
128
35
14
33
27
4
74
28
69
61
12
56
17
56
43
8
19
7
18
20
6
44
9
41
28
7
41
21
37
30
12
78
55
74
66
49
35
7
36
33
5
Average
50
26
47
40
22
TV/Monitor
52
14
47
38
10
Sofa
Potted Plant
Sheep
69
40
68
58
29
Train
Motorbike
Person
51
19
50
37
8
Horse
23
6
20
15
6
63
18
62
58
9
Cow
Dinning Table
Dog
45
12
41
32
11
Chair
Bottle
Bus
53
9
50
41
7
Car
Bird
Boat
72
34
71
66
21
Cat
Bicycle
3 Pyr. O2P + SQ
3 Pyr. O2P w/o SQ
2 Pyr. O2P + SQ
1 Pyr. O2P + SQ
1 Pyr. O2P w/o SQ
Aeroplane
Figure 1: Results for different numbers of non-zero entries of SQ. Note that SQ at k = 128 is not
introducing any quantization, since SIFT features are 128 dimensional vectors. (a) Caltech 101
(using 30 images per class for training), (b) VOC07.
50
10
51
37
7
67
16
66
56
9
45
12
44
36
9
50.97
18.81
49.09
41.20
12.53
Table 2: PASCAL VOC 2007 classification results. The average score provides the per-class average. We report results for O2P, with and without SQ, with the first plus second plus third levels of
pyramids (3 Pyr.), O2P with SQ with the first plus second levels of pyramids (2 Pyr.), and O2P with
and without SQ only with the first level of pyramids (1 Pyr.).
Acknowledgments: We thank the ERC for support from AdG VarCity.
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[13] G. Csurka, C. R. Dance, L. Fan, J. Willamowski, and C. Bray, ?Visual categorization with bags
of keypoints,? in Workshop on Statistical Learning in Computer Vision, ECCV, 2004.
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vector quantization,? in ICML, 2011.
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in ECCV, 2012.
[16] J. Sanchez, F. Perronnin, T. Mensink, and J. Verbeek, ?Image classification with the fisher
vector: Theory and practice,? IJCV, 2013.
[17] V. Arsigny, P. Fillard, X. Pennec, and N. Ayache, ?Geometric means in a novel vector space
structure on symmetric positive-definite matrices,? Journal on matrix analysis and applications, 2007.
[18] R. Bhatia, Positive definite matrices. Princeton University Press, 2009.
[19] X. Boix, M. Gygli, G. Roig, and L. Van Gool, ?Sparse quantization for patch description,? in
CVPR, 2013.
[20] R. E. Fan, K. W. Chang, C. J. Hsieh, X. R. Wang, and C. J. Lin, ?LIBLINEAR: A library for
large linear classification,? JMLR, 2008.
[21] O. Duchenne, A. Joulin, and J. Ponce, ?A graph-matching kernel for object categorization,? in
ICCV, 2011.
[22] X. Zhou, K. Yu, T. Zhang, and T. S. Huang, ?Image classification using super-vector coding of
local image descriptors,? in ECCV, 2010.
9
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4,734 | 5,287 | Learning From Weakly Supervised Data by The
Expectation Loss SVM (e-SVM) algorithm
Junhua Mao
Department of Statistics
University of California, Los Angeles
[email protected]
Jun Zhu
Department of Statistics
University of California, Los Angeles
[email protected]
Alan Yuille
Department of Statistics
University of California, Los Angeles
[email protected]
Abstract
In many situations we have some measurement of confidence on ?positiveness?
for a binary label. The ?positiveness? is a continuous value whose range is a
bounded interval. It quantifies the affiliation of each training data to the positive
class. We propose a novel learning algorithm called expectation loss SVM (eSVM) that is devoted to the problems where only the ?positiveness? instead of a
binary label of each training sample is available. Our e-SVM algorithm can also
be readily extended to learn segment classifiers under weak supervision where the
exact positiveness value of each training example is unobserved. In experiments,
we show that the e-SVM algorithm can effectively address the segment proposal
classification task under both strong supervision (e.g. the pixel-level annotations
are available) and the weak supervision (e.g. only bounding-box annotations are
available), and outperforms the alternative approaches. Besides, we further validate this method on two major tasks of computer vision: semantic segmentation
and object detection. Our method achieves the state-of-the-art object detection
performance on PASCAL VOC 2007 dataset.
1
Introduction
Recent work in computer vision relies heavily on manually labeled datasets to achieve satisfactory
performance. However, the detailed hand-labelling of datasets is expensive and impractical for large
datasets such as ImageNet [6]. It is better to have learning algorithms that can work with data that
has only been weakly labelled, for example by putting a bounding box around an object instead of
segmenting it or parsing it into parts.
In this paper we present a learning algorithm called expectation loss SVM (e-SVM). It requires
a method that can generate a set of proposals for the true label (e.g., the exact silhouette of the
object). But this set of proposals may be very large, each proposal may be only partially correct
(the correctness can be quantified by a continues value between 0 and 1 called ?positiveness?), and
several proposals may be required to obtain the correct label. In the training stage, our algorithm
can deal with the strong supervised case where the positiveness of the proposals are observed, and
can easily extend to the weakly supervised case by treating the positiveness as latent variables. In
the testing stage, it will predict the label for each proposal and provide a confidence score.
There are some alternative approaches for this problem, such as Support Vector Classification (SVC)
and Support Vector Regression (SVR). For the SVC algorithm, because this is not a standard binary
1
Annotations
Segment Proposals
IoU Ratios
Test images
0.79
0.02
...
0
Train
e-SVM
...
latent
latent
...
latent
Classifiers
Confidence of
class ?dog?: 3.49
...
0.25
-2.76
Figure 1: The illustration of our algorithm. In the training process, the e-SVM model can handle
two types of annotations: pixel level (strong supervision) and bounding box (weak supervision)
annotations. For pixel level annotations, we set the positiveness of the proposal as IoU overlap
ratios with the groundtruth and train classifiers using basic e-SVM. For bounding box annotations,
we treat the positiveness as latent variables and use latent e-SVM to train classifiers. In the testing
process, the e-SVM will provide each segment proposal a class label and a confidence score. (Best
viewed in color)
classification problem, one might need to binarize the positiveness using ad-hoc heuristics to determine a threshold, which degrades its performance [18]. To address this problem, previous works
usually used SVR [4, 18] to train the class confidence prediction models in segmentic segmentation. However, it is also not a standard regression problem since the value of positiveness belongs
to a bounded interval [0, 1]. We compare our e-SVM to these two related methods in the segment proposal confidence prediction problem. The positiveness of each segment proposal is set as
the Intersection over Union (IoU) overlap ratio between the proposal and the pixel level instance
groundtruth. We test our algorithm under two types of scenarios with different annotations: the
pixel level annotations (positiveness is observed) and the bounding box annotations (positiveness
is unobserved). Experiments show that our model outperforms SVC and SVR in both scenarios.
Figure 1 illustrates the framework of our algorithm.
We further validate our approach on two fundamental computer vision tasks: (i) semantic segmentation, and (ii) object detection. Firstly, we consider semantic segmentation. There has recently been
impressive progress at this task using rich appearance cues. Segments are extracted from images
[1, 3, 4, 12], appearance cues are computed for each segment [5, 21, 25], and classifiers are trained
using groundtruth pixel labeling [18]. Methods of this type are almost always among the winners
of the PASCAL VOC segmentation challenge [5]. But all these methods rely on datasets which
have been hand-labelled at the pixel level. For this application we generate the segment proposals
using CPMC segments [4]. The positiveness of each proposal is set as the Intersection over Union
(IoU) overlap ratio. We show that appearance cues learnt by e-SVM, using either the bounding
box annotations or pixel level annotations, are more effective than those learnt with SVC and SVR
on PASCAL VOC 2011 [9] segmentation dataset. Our algorithm is also flexible enough to utilize
additional bounding box annotations to further improve the results.
Secondly, we address object detection by exploiting the effectiveness of segmentation cues and coupling them to existing object detection methods. For this application, the data is only weakly labeled
because the groundtruth for object detection is typically specified by bounding boxes (e.g. PASCAL
VOC [8, 9] and Imagenet [6]), which means that pixel level groundtruth is not available. We use
either CPMC or super-pixels as methods for producing segment proposals. IoU is again used to represent the positiveness of the proposals. We test our approach on the PASCAL dataset using, as our
base detector, the Regions with CNN features (RCNN) [14] (currently state of the art on PASCAL
and outperforms previous works by a large margin). This method first used selective search method
[24] to extract candidate bounding boxes. For each candidate bounding box, it extracts features by
deep networks [16] learned on Imagenet dataset and fine-tuned on PASCAL. We couple our appearance cues to this system by simple concatenating our spatial confidence map features based on the
trained e-SVM classifiers and the deep learning features, and then train a linear SVM. We show that
this simple approach yields an average improvement of 1.5 percent on per-class average precision
(AP).
We note that our approach is general. It can use any segment proposal detectors, any image features,
and any classifiers. When applied to object detection it could use any base detector, and we could
couple the appearance cues with the base detector in many different ways (we choose the simplest).
2
In addition, it can handle other classification problems where only the ?positiveness? of the samples
instead of binary labels are available.
2
Related work on weakly supervised learning and weighted SVMs
We have introduced some of the most relevant works published recently for semantic segmentation
or object detection. In this section, we will briefly review related work of weakly supervised learning methods for segment classification, and discuss the connection to instance weighted SVM in
literature.
The problem settings for most previous works generally assumed that they only get a set of accompanying words of an image or a set of image level labeling, which is different from the problem
settings in this paper. Multiple Instance Learning (MIL) [7, 2] was adopted to solve these problems
[20, 22]. MIL handles cases where at least one positive instance is present in a positive bag and
only the labels of a set of bags are available. Vezhnevets et.al. [26] proposed a Multi-Image Model
(MIM) to solve this problem and showed that MIL in [22] is a special case of MIM. Later, [26]
developed MIM to a generalized MIM and used it as their segmentation model. Recently, Liu et.al.
[19] presented a weakly-supervised dual clustering approach to handle this task.
Our weakly supervised problem setting is in the middle between these settings and the strong supervision case (i.e. the full pixel level annotations are available). It is also very important and useful
because bounding box annotations of large-scale image dataset are already available (e.g. Imagenet
[6]) while the pixel level annotations of large datasets are still hard to obtain. This weakly supervised problem cannot be solved by MIL. We cannot assume that at least one ?completely? positive
instance (i.e. a CPMC segment proposals) is present in a positive bag (i.e. a groundtruth instance)
since most of the proposals will contain both foreground pixels and background pixels. We will
show how our e-SVM and its latent extension address this problem in the next sections.
In machine learning literature, the weighted SVM (WSVM) methods [23, 27, ?] also use an instancedependent weight on the cost of each example, and can improve the robustness of model estimation
[23], alleviate the effect of outliers [27], leverage privileged information [17] or deal with unbalanced classification problems. The difference between our e-SVM and WSVMs mainly lies in that
it weights labels instead of data points, which leads to each example contributing both to the costs
of positive and negative labels. Although the loss function of e-SVM model is different from those
of WSVMs, it can be effortlessly solved by any standard SVM solver (e.g., LibLinear [10]) like
those used in WSVMs. This is an advantage because it does not require a specific solver for the
implementation of our e-SVM.
3
The expectation loss SVM model
In this section, we will first describe the basic formulation of our expectation loss SVM model
(e-SVM) in section 3.1 when the positiveness of each segment proposal is observed. Then, in section 3.2, a latent e-SVM model is introduced to handle the weak supervision situation where the
positiveness of each segment proposal is unobserved.
3.1
The basic e-SVM model
We are given a set of training images D. Using some segmentation method (we adopt CPMC [4]
in this work), we can generate a set of foreground segment proposals {S1 , S2 , . . . , SN } from these
images. For each segment Si , we extract feature xi , xi ? Rd .
Suppose the pixelwise annotations are available for all the groundtruth instances in D. For each
object class, we can calculate the IoU ratio ui (ui ? [0, 1]) between each segment Si and the
groundtruth instances labeling, and set the positiveness of Si as ui (although positiveness can be
some functions of IoU ratio, for simplicity, we just set it as IoU and use ui to represent the positiveness in the following paragraphs). Because many foreground segments overlap partially with
the groundtruth instances (i.e. 0 < ui < 1), it is not a standard binary classification problem for
training. Of course, we can define a threshold ?b and treat all the segments whose ui ? ?b as positive
examples and the segments whose ui < ?b as negative examples. In this way, this problem is transferred to a Support Vector Classification (SVC) problem. But it needs some heuristics to determine
?b and its performance is only partially satisfactory [18].
3
To address this issue, we proposed our expectation loss SVM model as an extension of the classical
SVC models. In this model, we treat the label Yi of each segment as an unobserved random variable.
Yi ? {?1, +1}. Given xi , we assume that Yi follows a Bernoulli distribution. The probability of
Yi = 1 given xi (i.e. the success probability of the Bernoulli distribution) is denoted as ?i . We
assume that ?i is a function of the positiveness ui , i.e. ?i = g(ui ). In the experiment, we simply set
?i = ui .
Similar to the traditional linear SVC problem, we adopt a linear function as the prediction function:
F (xi ) = wT xi + b. For simplicity, we denote [w b] as w, [xi 1] as xi and F (xi ) = wT xi in the
remaining part of the paper. The loss function of our e-SVM is the expectation over the random
variables Yi :
N
1
1 X
L(w) =?w ? wT w +
EY [max(0, 1 ? Yi wT xi )]
2
N i=1 i
N
1
1 X +
[l ? Pr(Yi = +1|xi ) + li? ? Pr(Yi = ?1|xi )]
=?w ? wT w +
2
N i=1 i
(1)
N
1
1 X +
=?w ? wT w +
{l ? g(ui ) + li? ? [1 ? g(ui )]}
2
N i=1 i
where li+ = max(0, 1 ? wT xi ) and li? = max(0, 1 + wT xi ).
Given the pixelwise groundtruth annotations, g(ui ) is known. From Equation 1, we can see that it
is equivalent to ?weight? each sample with a function of its positiveness. The standard linear SVM
solver is used to solve this model with loss function of L(w). In the experiments, we show that
the performance of our e-SVM is much better than SVC and slightly better than Support Vector
Regression (SVR) in the segment classification task.
3.2
The latent e-SVM model
One of the advantage of our e-SVM model is that we can easily extend it to the situation where only
bounding box annotations are available (this type of labeling is of most interest in the paper). Under
this weakly supervised setting, we cannot obtain the exact value of the positiveness (IoU) ui for each
segment. Instead, ui will be treated as a latent variable which will be determined by minimizing the
following loss function:
N
1 X +
1
L(w, u) = ?w ? wT w +
{l ? g(ui ) + li? ? [1 ? g(ui )]} + ?R ? R(u)
2
N i=1 i
(2)
where u denotes {ui }i=1,...,N . R(u) is a regularization term for u. We can see that the loss function
in Equation 1 is a special case of that in Equation 2 by setting u as constant and ?R equal to 0.
When u is fixed, L(w, u) is a standard linear SVM loss, which is convex with respect to w. When
w is fixed, L(w, u) is also a convex function if R(u) is a convex function with respect to u. The IoU
between a segment Si and groundtruth bounding boxes, denoted as ubb
i , can serve as an initialization
for ui . We can iteratively fix u and w, and solve the two convex optimization problems until it
converges. The pseudo-code for the optimization algorithm is shown in Algorithm 1.
Algorithm 1 The optimization for training latent e-SVM
Initialization:
1: u(cur) ? ubb ;
Process:
2: repeat
3:
w(new) ? arg minw L(w, u(cur) );
4:
u(new) ? arg minu L(w(new) , u);
5:
u(cur) ? u(new) ;
6: until Converge
4
If we do not add any regularization term on u (i.e. set ?R = 0), u will become 0 or 1 in the
optimization step in line 4 of algorithm 1 because the loss function becomes a linear function with
respect to u when w is fixed. It turns to be similar to a latent SVM and can lead the algorithm to
stuck in the local minimal as shown in the experiments. The regularization term will prevent this
situation under the assumption that the true value of u should be around ubb .
There are a lot of different designs of the regularization term R(u). In practice, we use the following
one based on the cross entropy between two Bernoulli distributions with success probability ubb
i and
ui respectively.
R(u) = ?
N
1 X bb
[u ? log(ui ) + (1 ? ubb
i ) ? log(1 ? ui )]
N i=1 i
N
1 X
=?
DKL [Bern(ubb
i )||Bern(ui )] + C
N i=1
(3)
where C is a constant value with respect to u. DKL (.) represents the KL distance between two
Bernoulli distributions. This regularization term is a convex function with respect to u and achieves
its minimal when u = ubb . It is a strong regularization term since its value increases very fast when
u 6= ubb .
4
4.1
Visual Tasks
Semantic segmentation
We can easily apply our e-SVM model to the semantic segmentation task with the framework proposed by Carreira et al. [5]. Firstly, CPMC segment proposals [4] are generated and the secondorder pooling features [5] are extracted from each segment. Then we train the segment classifiers
using either e-SVM or latent e-SVM according to whether the groundtruth pixel-level annotations
are available. In the testing stage, the CPMC segments are sorted based on their confidence scores
output by the trained classifiers. The top ones will be selected to produce the predicted semantic
label map.
4.2
Object detection
For the task of object detection, we can only acquire bounding-box annotations instead of pixel-level
labeling. Therefore, it is natural to apply our latent e-SVM in this task to provide complementary
information for the current object detection system.
In the state-of-the-art object detection systems [11, 13, 24, 14], the window candidates of foreground
object are extracted from images and the confidence scores are predicted on them. Window candidates are extracted either by sliding window approaches (used in e.g. the deformable part-based
model [11, 13]) or most recently, the Selective Search method [24] (used in e.g. the Region Convolutional Neural Networks [14]). This method lowers down the number of window candidates
compared to the traditional sliding window approach.
Original Image
Confidence Map
e-SVM
classifiers
Pooling in
each bins
Mapping
segment
confidence
to pixels
(a)
Features
(c)
(b)
Figure 2: The illustration of our spatial confidence map features for window candidates based on
e-SVM. The confidence scores of the segments are mapped to pixels to generate a pixel-level confidence map. We will divide a window candidate into m ? m spatial bins and pool the confidence
scores of the pixels in each bin. It leads to a m ? m dimensional feature.
5
It is not easy to directly incorporate confidence scores of the segments into these object detection
systems based on window candidates. The difficulty lies in two aspects. First, only some of the
segments are totally inside a window candidate or totally outside the window candidate. It might be
hard to calculate the contribution of the confidence score of a segment that only partially overlaps
with a window candidate. Second, the window candidates (even the groundtruth bounding boxes)
will contain some of the background regions. Some regions (e.g. the regions near the boundary of
the window candidates) will have higher probability to be the background region than the regions in
the center. Treating them equally will harm the accuracy of the whole detection system.
In order to solve these issues, we propose a new spatial confidence map feature. Given an image and
a set of window candidates, we first calculate the confidence scores of all the segments in the image
using the learned e-SVM models. The confidence score for a segment S is denoted as CfdScore(S).
For each pixel, the confidence score is set as the maximum confidence score of all the segments
that contain this pixel. CfdScore(p) = max?S,p?S CfdScore(S). In this way, we can handle the
difficulty of partial overlapping between segments and candidate windows. For the second difficulty,
we divide each candidate window into M = m ? m spatial bins and pool the confidence scores of
the pixels in each bin. Because the classifiers are trained with the one-vs-all scheme, our spatial
confidence map feature is class-specific. It leads to a (M ? K)-dimensional feature for each candidate window, where K refers to the total number of object classes. After that, we encode it by
additive kernels approximation mapping [25] and obtain the final feature representation of candidate
windows. The feature generating process is illustrated in Figure 2. In the testing stage, we can
concatenate this segment feature with the features from other object detection systems.
5
Experiments
In this section, we first evaluate the performance of e-SVM method on segment proposal classification, by using two new evaluation criterions for this task. After that, we apply our method to two
essential tasks in computer vision: semantic segmentation and object detection. For semantic segmentation task, we test the proposed eSVM and latent eSVM on two different scenarios (i.e., with
pixel-level groundtruth label annotation and with only bounding-box object annotation) respectively.
For object detection task, we combine our confidence map feature with the state-of-the-art object detection system, and show our method can obtain non-trivial improvement on detection performance.
5.1
Performance evaluation on e-SVM
We use PASCAL VOC 2011 [9] segmentation dataset in this experiment. It is a subset of the whole
PASCAL 2011 datasets with 1112 images in the training set and 1111 images in the validation set,
with 20 foreground object classes in total. We use the official training set and validation set for
training and testing respectively. Similar to [5], we extract 150 CPMC [4] segment proposals for
each image and compute the second-order pooling features on each segment. Besides, we use the
same sequential pasting scheme [5] as the inference algorithm in testing.
5.1.1
Evaluation criteria
In literature [5], the supervised learning framework of segment-based prediction model either regressed the overlapping value or converted it to a binary classification problem via a threshold value, and evaluate the performance by certain task-specific criterion (i.e., the pixel-wise accuracy used
for semantic segmentation). In this paper, we adopt a direct performance evaluation criteria for the
segment-wise target class prediction task, which is consistent with the learning problem itself and
not biased to particular tasks. Unfortunately, we have not found any work on this sort of direct
performance evaluation, and thus introduce two new evaluation criteria for this purpose. We first
briefly describe them as follows:
Threshold Average Precision Curve (TAPC) Although the ground-truth target value (i.e., the
overlap rate of segment and bounding box) is a real value in the range of [0, 1], we can transform
original prediction problem to a series of binary problems, each of which is conducted by thresholding the original groundtruth overlap rate. Thus, we calculate the Precison-Recall Curve as well
as AP on each of binary classification problem, and compute the mean AP w.r.t. different threshold
values as a performance measurement for the segment-based class confidence prediction problem.
6
e-SVM
SVR
SVC-0.0
SVC-0.2
SVC-0.4
SVC-0.6
SVC-0.8
TAPC
36.69
35.23
22.48
33.96
35.62
32.57
26.73
e-SVM
SVR
SVC-0
SVC-0.2
SVC-0.4
SVC-0.6
SVC-0.8
NDCG
0.8750
0.8652
0.8153
0.8672
0.8656
0.8485
0.8244
TAPC
TAPC
38
36
34
32
30
28
26
24
22
20
33.00
0.8700
29.50
0.8500
26.00
0.8300
22.50
0.8100
19.00
0.7900
15.50
0.7700
0.7500
12.00
e-SVM
SVR
L-eSVM
SVC-0.0 SVC-0.2 SVC-0.4 SVC-0.6 SVC-0.8
SVR
SVC-0.0 SVC-0.2 SVC-0.4 SVC-0.6 SVC-0.8
NDCG
NDCG
0.8800
0.8700
0.8700
0.8500
0.8600
0.8300
0.8500
0.8400
0.8100
0.8300
0.7900
0.8200
0.7700
0.8100
0.7500
0.8000
e-SVM
SVR
L-eSVM
SVC-0 SVC-0.2 SVC-0.4 SVC-0.6 SVC-0.8
(a) Using pixel level annotations
SVR
SVC-0 SVC-0.2 SVC-0.4 SVC-0.6 SVC-0.8
(b) Using bounding box annotations
Figure 3: Performance evaluation and comparison to SVC and SVR
Normalized Discounted Cumulative Gain (NDCG) [15] Considering that a higher confidence
value is expected to be predicted for the segment with higher overlap rate, we think this prediction
problem can be treated as a ranking problem, and thus we use the Normalized Discounted Cumulative Gain (NDCG), which is common performance measurement for ranking problem, as another
kind of performance evaluation criterion in this paper.
5.1.2
Comparisons to SVC and SVR
Based on the TAPC and NDCG introduced above, we evaluate the performance of our e-SVM model
on PASCAL VOC 2011 segmentation dataset, and compare the results to two common methods (i.e.
SVC and SVR) in literature. Note that we test the SVC?s performance with a variety of binary classification problems, each of which are trained by using different threshold values (e.g., 0, 0.2, 0.4,
0.6 and 0.8 as shown in figure 3). In figure 3 (a) and (b), we show the experimental results w.r.t. the
model/classifier trained with clean pixel-wise object class labels and weakly-labelled bounding-box
annotation, respectively. For both cases, we can see that our method obtains consistently superior
performance than SVC model for all different threshold values. Besides, we can see that the TAPC
and NDCG of our method are higher than those of SVR, which is a popular regression model for
continuously valued target variable based on the max-margin principle.
5.2
Results of semantic segmentation
For the semantic segmentation task, we test our e-SVM model with PASCAL VOC 2011 segmtation
dataset using training set for training and validation set for testing. We evaluate the performance
under two different data annotation settings, i.e., training with pixel-wise semantic class label maps
and object bounding-box annotations. The accuracy w.r.t. these two settings are 36.8% and 27.7%
respectively, which are comparable to the results of the state-of-the-art segment confidence prediction model (i.e., SVR) [5] used in semantic segmentation task.
5.3
Results of object detection
As mentioned in Section 4.2, one of the natural applications of our e-SVM method is the object
detection task. Most recently, Girshick et.al [14] presented a Regions with CNN features method
(RCNN) using the Convolutional Neural Network pre-trained on the ImageNet Dataset [6] and finetuned on the PASCAL VOC datasets. They achieved a significantly improvement over the previous
state-of-the-art algorithms (e.g. Deformable Part-based Model (DPM) [11])and push the detection
7
RCNN
Ours
Gain
RCNN (bb)
Ours (bb)
Gain
RCNN
Ours
Gain
RCNN (bb)
Ours (bb)
Gain (bb)
plane
64.1
63.7
-0.4
68.1
70.4
2.3
table
45.8
47.8
2.0
54.5
56.4
1.9
bike
69.2
70.2
1.0
72.8
74.2
1.4
dog
55.8
57.9
2.1
61.2
62.9
1.8
bird
50.4
51.9
1.5
56.8
59.1
2.3
horse
61.0
61.2
0.3
69.1
69.3
0.2
boat
41.2
42.5
1.3
43.0
44.7
1.6
motor.
66.8
67.5
0.8
68.6
69.9
1.4
bottle
33.2
33.4
0.2
36.8
38.0
1.2
person
53.9
54.9
1.0
58.7
59.6
0.9
bus
62.8
63.2
0.4
66.3
67.2
1.0
plant
30.9
34.5
3.7
33.4
35.6
2.2
car
70.5
71.3
0.8
74.2
74.6
0.3
sheep
53.3
55.8
2.5
62.9
64.6
1.7
cat
61.8
62.0
0.2
67.6
69.0
1.3
sofa
49.2
51.0
1.8
51.1
53.2
2.1
chair
32.4
34.7
2.3
34.4
36.7
2.3
train
56.9
58.4
1.6
62.5
64.3
1.8
cow
58.4
58.7
0.2
63.5
64.3
0.8
tv
64.1
65.0
0.9
64.8
65.5
0.7
Average
54.1
55.3
1.2
58.5
60.0
1.5
Table 1: Detection results on PASCAL 2007. ?bb? means the result after applying bounding box
regression. Gain means the improved AP of our system compared to RCNN under the same settings
(both with bounding box or without). The better results in the comparisons are bold.
performance into a very high level (The average AP is 58.5 with boundary regularization on PASCAL VOC 2007).
A question arises: can we further improve their performance? The answer is yes. In our method,
we first learn the latent e-SVM models based on the object bounding-box annotation, and calculate
the spatial confidence map features as in section 4.2. Then we simply concatenate them with RCNN
the features to train object classifiers on candidate windows. We use PASCAL VOC 2007 dataset
in this experiment. As shown in table 1, our method can improve the average AP by 1.2 before
applying bounding boxes regression. For some categories that the original RCNN does not perform
well, such as potted plant, the gain of AP is up to 3.65. After applying bounding box regression for
both RCNN and our algorithm, the gain of performance is 1.5 on average.
In the experiment, we set m = 5 and adopt average pooling on the pixel level confidence scores
within each spatial bin. We also modified the bounding box regularization method used in [14] by
augmenting the fifth layer features with additive kernels approximation methods [25]. It will lead to
a slightly improved performance.
In summary, we achieved an average AP of 60.0, which is 1.5 higher than the best known results
(the original RCNN with bounding box regression) of this dataset. Please note that we only use the
annotations on PASCAL VOC 2007 to train the e-SVM classifiers and have not considered context.
The results are expected to be further improved if the data in ImageNet is used.
6
Conclusion
We present a novel learning algorithm call e-SVM that can well handle the situation in which the
labels of training data are continuous values whose range is a bounded interval. It can be applied
to segment proposal classification task and can be easily extended to learn segment classifiers under
weak supervision (e.g. only bounding box annotations are available). We apply this method on
two major tasks of computer vision (i.e., semantic segmentation and object detection), and obtain
the state-of-the-art object detection performance on PASCAL VOC 2007 dataset. We believe that,
with the ever growing size of datesets, it is increasingly important to learn segment classifiers under
weak supervision to reduce the amount of labeling required. In future work, we will consider using
the bounding box annotation from large datasets, such as ImageNet, to further improve semantic
segmentation performance on PASCAL VOC.
Acknowledgements. We gratefully acknowledge funding support from the National Science Foundation (NSF) with award CCF-1317376, and from the National Institute of Health NIH Grant
5R01EY022247-03. We also thank the NVIDIA Corporation for providing GPUs in our experiments.
8
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9
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4,735 | 5,288 | Predicting Useful Neighborhoods
for Lazy Local Learning
Kristen Grauman
University of Texas at Austin
[email protected]
Aron Yu
University of Texas at Austin
[email protected]
Abstract
Lazy local learning methods train a classifier ?on the fly? at test time, using only
a subset of the training instances that are most relevant to the novel test example.
The goal is to tailor the classifier to the properties of the data surrounding the test
example. Existing methods assume that the instances most useful for building the
local model are strictly those closest to the test example. However, this fails to
account for the fact that the success of the resulting classifier depends on the full
distribution of selected training instances. Rather than simply gathering the test
example?s nearest neighbors, we propose to predict the subset of training data that
is jointly relevant to training its local model. We develop an approach to discover
patterns between queries and their ?good? neighborhoods using large-scale multilabel classification with compressed sensing. Given a novel test point, we estimate
both the composition and size of the training subset likely to yield an accurate local
model. We demonstrate the approach on image classification tasks on SUN and
aPascal and show its advantages over traditional global and local approaches.
1
Introduction
Many domains today?vision, speech, biology, and others?are flush with data. Data availability, combined with recent large-scale annotation efforts and crowdsourcing developments, have
yielded labeled datasets of unprecedented size. Though a boon for learning approaches, large labeled datasets also present new challenges. Beyond the obvious scalability concerns, the diversity
of the data can make it difficult to learn a single global model that will generalize well. For example,
a standard binary dog classifier forced to simultaneously account for the visual variations among
hundreds of dog breeds may be ?diluted? to the point it falls short in detecting new dog instances.
Furthermore, with training points distributed unevenly across the feature space, the model capacity
required in any given region of the space will vary. As a result, if we train a single high capacity
learning algorithm, it may succeed near parts of the decision boundary that are densely populated
with training examples, yet fail in poorly sampled areas of the feature space.
Local learning methods offer a promising direction to address these challenges. Local learning is an
instance of ?lazy learning?, where one defers processing of the training data until test time. Rather
than estimate a single global model from all training data, local learning methods instead focus on a
subset of the data most relevant to the particular test instance. This helps learn fine-grained models
tailored to the new input, and makes it possible to adjust the capacity of the learning algorithm to
the local properties of the data [5]. Local methods include classic nearest neighbor classification as
well as various novel formulations that use only nearby points to either train a model [2, 3, 5, 13, 29]
or learn a feature transformation [8, 9, 15, 25] that caters to the novel input.
A key technical question in local learning is how to determine which training instances are relevant
to a test instance. All existing methods rely on an important core assumption: that the instances most
useful for building a local model are those that are nearest to the test example. This assumption is
well-motivated by the factors discussed above, in terms of data density and intra-class variation.
1
Furthermore, identifying training examples solely based on proximity has the appeal of permitting
specialized similarity functions (whether learned or engineered for the problem domain), which can
be valuable for good results, especially in structured input spaces.
On the other hand, there is a problem with this core assumption. By treating the individual nearness
of training points as a metric of their utility for local training, existing methods fail to model how
those training points will actually be employed. Namely, the relative success of a locally trained
model is a function of the entire set or distribution of the selected data points?not simply the
individual pointwise nearness of each one against the query. In other words, the ideal target subset
consists of a set of instances that together yield a good predictive model for the test instance.
Based on this observation, we propose to learn the properties of a ?good neighborhood? for local
training. Given a test instance, the goal is to predict which subset of the training data should be
enlisted to train a local model on the fly. The desired prediction task is non-trivial: with a large
labeled dataset, the power set of candidates is enormous, and we can observe relatively few training
instances for which the most effective neighborhood is known. We show that the problem can be
cast in terms of large-scale multi-label classification, where we learn a mapping from an individual
instance to an indicator vector over the entire training set that specifies which instances are jointly
useful to the query. Our approach maintains an inherent bias towards neighborhoods that are local,
yet makes it possible to discover subsets that (i) deviate from a strict nearest-neighbor ranking and
(ii) vary in size.
The proposed technique is a general framework to enhance local learning. We demonstrate its impact
on image classification tasks for computer vision, and show its substantial advantages over existing
local learning strategies. Our results illustrate the value in estimating the size and composition of
discriminative neighborhoods, rather than relying on proximity alone.
2
Related Work
Local learning algorithms Lazy local learning methods are most relevant to our work. Existing methods primarily vary in how they exploit the labeled instances nearest to a test point. One
strategy is to identify a fixed number of neighbors most similar to the test point, then train a model
with only those examples (e.g., a neural network [5], SVM [29], ranking function [3, 13], or linear
regression [2]). Alternatively, the nearest training points can be used to learn a transformation of
the feature space (e.g., Linear Discriminant Analysis); after projecting the data into the new space,
the model is better tailored to the query?s neighborhood properties [8, 9, 15, 25]. In local selection
methods, strictly the subset of nearby data is used, whereas in locally weighted methods, all training
points are used but weighted according to their distance [2]. All prior methods select the local neighborhood based on proximity, and they typically fix its size. In contrast, our idea is to predict the set
of training instances that will produce an effective discriminative model for a given test instance.
Metric learning The question ?what is relevant to a test point?? also brings to mind the metric
learning problem. Metric learning methods optimize the parameters of a distance function so as to
best satisfy known (dis)similarity constraints between training data [4]. Most relevant to our work
are those that learn local metrics; rather than learn a single global parameterization, the metric varies
in different regions of the feature space. For example, to improve nearest neighbor classification,
in [11] a set of feature weights is learned for each individual training example, while in [26, 28]
separate metrics are trained for clusters discovered in the training data.
Such methods are valuable when the data is multi-modal and thus ill-suited by a single global metric.
Furthermore, one could plug a learned metric into the basic local learning framework. However,
we stress that learning what a good neighbor looks like (metric learning?s goal) is distinct from
learning what a good neighborhood looks like (our goal). Whereas a metric can be trained with
pairwise constraints indicating what should be near or far, jointly predicting the instances that ought
to compose a neighborhood requires a distinct form of learning, which we tackle in this work.
Hierarchical classification For large multi-class problems, hierarchical classification approaches
offer a different way to exploit ?locality? among the training data. The idea is to assemble a tree of
decision points, where at each node only a subset of labels are considered (e.g., [6, 12, 21]). Such
methods are valuable for reducing computational complexity at test time, and broadly speaking they
share the motivation of focusing on finer-grained learning tasks to improve accuracy. However,
2
otherwise the work is quite distant from our problem. Hierarchical methods precompute groups of
labels to isolate in classification tasks, and apply the same classifiers to all test instances; lazy local
learning predicts at test time what set of training instances are relevant for each novel test instance.
Weighting training instances Our problem can be seen as deciding which training instances to
?trust? most. Various scenarios call for associating weights with training instances such that some
influence the learned parameters more than others. For example, weighted instances can reflect label
confidences [27], help cope with imbalanced training sets [24], or resist the influence of outliers [20].
However, unlike our setting, the weights are given at training time and they are used to create a
single global model. Methods to estimate the weights per example arise in domain adaptation,
where one aims to give more weight to source domain samples distributed most like those in the
target domain [14, 17, 18]. These are non-local, offline approaches, whereas we predict useful
neighborhoods in an online, query-dependent manner. Rather than close the mismatch between a
source and target domain, we aim to find a subset of training data amenable to a local model.
Active learning Active learning [23] aims to identify informative unlabeled training instances,
with the goal of minimizing labeling effort when training a single (global) classifier. In contrast, our
goal is to ignore those labeled training points that are irrelevant to a particular novel instance.
3
Approach
We propose to predict the set of training instances which, for a given test example, are likely to
compose an effective neighborhood for local classifier learning. We use the word ?neighborhood?
to refer to such a subset of training data?though we stress that the optimal subset need not consist
of strictly rank-ordered nearest neighbor points.
Our approach has three main phases: (i) an offline stage where we generate positive training neighborhoods (Sec. 3.1), (ii) an offline stage where we learn a mapping from individual examples to their
useful neighborhoods (Sec. 3.2), and (iii) an online phase where we apply the learned model to infer
a novel example?s neighborhood, train a local classifier, and predict the test label (Sec. 3.3).
3.1
Generating training neighborhoods
Let T = {(x1 , c1 ), . . . , (xM , cM )} denote the set of M category-labeled training examples. Each
xi ? <d is a vector in some d-dimensional feature space, and each ci ? {1, . . . , C} is its target
category label. Given these examples, we first aim to generate a set of training neighborhoods, N =
{(xn1 , yn1 ), . . . , (xnN , ynN )}. Each training neighborhood (xni , yni ) consists of an individual
instance xni paired with a set of training instance indices capturing its target ?neighbors?, the latter
being represented as a M -dimensional indicator vector yni . If yni (j) = 1, this means xj appears
in the target neighborhood for xni . Otherwise, yni (j) = 0. Note that the dimensionality of this
target indicator vector is M , the number of total available training examples. We will generate N
such pairs, where typically N M .
As discussed above, there are very good motivations for incorporating nearby points for local learning. Indeed, we do not intend to eschew the ?locality? aspect of local learning. Rather, we start from
the premise that points near to a query are likely relevant?but relevance is not necessarily preserved
purely by their rank order, nor must the best local set be within a fixed radius of the query (or have a
fixed set size). Instead, we aim to generalize the locality concept to jointly estimate the members of
a neighborhood such that taken together they are equipped to train an accurate query-specific model.
With these goals in mind, we devise an empirical approach to generate the pairs (xni , yni ) ? N .
The main idea is to sample a series of candidate neighborhoods for each instance xni , evaluate their
relative success at predicting the training instance?s label, and record the best candidate.
Specifically, for instance xni , we first compute its proximity to the M ? 1 other training images in
the feature space. (We simply apply Euclidean distance, but a task-specific kernel or learned metric
could also be used here.) Then, for each of a series of possible neighborhood sizes {k1 , . . . , kK },
we sample a neighborhood of size k from among all training images, subject to two requirements:
(i) points nearer to xni are more likely to be chosen, and (ii) the category label composition within
the neighborhood set is balanced. In particular, for each possible category label 1, . . . , C we sample
k
C training instances without replacement, where the weight associated with an instance is inversely
3
related to its (normalized) distance to xni . We repeat the sampling S times for each value of k,
yielding K ? S candidates per instance xni .
Next, for each of these candidates, we learn a local model. Throughout we employ linear support
vector machine (SVM) classifiers, both due to their training efficiency and because lower capacity
models are suited to the sparse, local datasets under consideration; however, kernelized/non-linear
models are also possible.1 Note that any number of the K ? S sampled neighborhoods may yield a
classifier that correctly predicts xni ?s category label cni . Thus, to determine which among the successful classifiers is best, we rank them by their prediction confidences. Let pks (xni ) = P (cni |xni )
be the posterior estimated by the s-th candidate classifier for neighborhood size k, as computed via
Platt scaling using the neighborhood points. To automatically select the best k for instance xni , we
average these posteriors across all samples per k value, then take the one with the highest probabilPS
ity: k ? = arg max S1 s=1 pks (xni ). The averaging step aims to smooth the estimated probability
k
using the samples for that value of k, each of which favors near points but varies in its composition.
Finally, we obtain a single neighborhood pair (xni , yni ), where yni is the indicator vector for the
?
neighborhood sampled with size k ? having the highest posterior pks .
In general we can expect higher values of S and denser samplings of k to provide best results, though
at a correspondingly higher computational cost during this offline training procedure.
3.2
Learning to predict neighborhoods with compressed sensing
With the training instance-neighborhood pairs in hand, we next aim to learn a function capturing
their relationships. This function must estimate the proper neighborhood for novel test instances.
We are faced with a non-trivial learning task. The most straightforward approach might be to learn
a binary decision function for each xi ? T , trained with all xnj for which ynj (i) = 1 as positives.
However, this approach has several problems. First, it would require training M binary classifiers,
and in the applications of interest M ?the number of all available category-labeled examples?may
be very large, easily reaching the millions. Second, it would fail to represent the dependencies
between the instances appearing in a single training neighborhood, which ought to be informative
for our task. Finally, it is unclear how to properly gather negative instances for such a naive solution.
Instead, we pose the learning task as a large-scale multi-label classification problem. In multi-label
classification, a single data point may have multiple labels. Typical examples include image and web
page tagging [16, 19] or recommending advertiser bid phrases [1]. In our case, rather than predict
which labels to associate with a novel example, we want to predict which training instances belong
in its neighborhood. This is exactly what is encoded by the target indicator vectors defined above,
yni . Furthermore, we want to exploit the fact that, compared to the number of all labeled training
images, the most useful local neighborhoods will contain relatively few examples.
Therefore, we adopt a large-scale multi-label classification approach based on compressed sensing [19] into our framework. With it, we can leverage sparsity in the high-dimensional target neighborhood space to efficiently learn a prediction function that jointly estimates all useful neighbors.
First, for each of the N training neighborhoods, we project its M -dimensional neighborhood vector yni to a lower-dimensional space using a random transformation: zni = ? yni , where ? is a
D ? M random matrix, and D denotes the compressed indicators? dimensionality. Then, we learn
regression functions to map the original features to these projected values zn1 , . . . , znN as targets.
That is, we obtain a series of D M regression functions f1 , . . . , fD minimizing the loss in the
compressed indicator vector space. Given a novel instance xq , those same regression functions are
applied to map to the reduced space, [f1 (xq ), . . . , fD (xq )]. Finally, we predict the complete indicator vector by recovering the M -dimensional vector using a standard reconstruction algorithm from
the compressed sensing literature.
We employ the Bayesian multi-label compressed sensing framework of [19], since it unifies the
regression and sparse recovery stages, yielding accurate results for a compact set of latent variables.
Due to compressed sensing guarantees, an M -dimensional indicator vector with l nonzero entries
can be recovered efficiently using D = O(l log Ml ) [16].
1
In our experiments the datasets have binary labels (C = 2); in the case of C > 2 the local model must be
multi-class, e.g., a one-versus-rest SVM.
4
3.3
Inferring the neighborhood for a novel example
All processing so far is performed offline. At test time, we are given a novel example xq , and must
predict its category label. We first predict its neighborhood using the compressed sensing approach
overviewed in the previous section, obtaining the M -dimensional vector y?q . The entries of this
vector are real-valued, and correspond to our relative confidence that each category-labeled instance
xi ? T belongs in xq ?s neighborhood.
Past multi-label classification work focuses its evaluation on the precision of (a fixed number of) the
top few most confident predictions and the raw reconstruction error [16, 19], and does not handle
the important issue of how to truncate the values to produce hard binary decisions. In contrast, our
setting demands that we extract both the neighborhood size estimate as well as the neighborhood
composition from the estimated real-valued indicator vector.
To this end, we perform steps paralleling the training procedure defined in Sec. 3.1, as follows. First,
we use the sorted confidence values in y?q to generate a series of candidate neighborhoods of sizes
varying from k1 to kK , each time ensuring balance among the category labels. That is, for each
k, we take the Ck most confident training instances per label. Recall that all M training instances
referenced by y?q have a known category label among 1, . . . , C. Analogous to before, we then apply
each of the K candidate predicted neighborhoods in turn to train a local classifier. Of those, we
return the category label prediction from the classifier with the most confident decision value.
Note that this process automatically selects the neighborhood size k to apply for the novel input. In
contrast, existing local learning approaches typically manually define this parameter and fix it for
all test examples [5, 8, 13, 15, 29]. Our results show that approach is sub-optimal; not only does the
most useful neighborhood deviate from the strict ranked list of neighbors, it also varies in size.
We previously explored an alternative approach for inference, where we directly used the confidences in y?q as weights in an importance-weighted SVM. That is, for each query, we trained a model
with all M data points, but modulated their influence according to the soft indicator vector y?q , such
that less confident points incurred lower slack penalties. However, we found that approach inferior,
likely due to the difficulty in validating the slack scale factor for all training instances (problematic
in the local learning setting) as well as the highly imbalanced datasets we tackle in the experiments.
3.4
Discussion
While local learning methods strive to improve accuracy over standard global models, their lazy use
of training data makes them more expensive to apply. This is true of any local approach that needs to
compute distances to neighbors and train a fresh classifier online for each new test example. In our
case, using Matlab, the run-time for processing a single novel test point can vary from 30 seconds to
30 minutes. It is dominated by the compressed sensing reconstruction step, which takes about 80%
of the computation time and is highly dependent on the complexity of the trained model. One could
improve performance by using approximate nearest neighbor methods to sort T , or pre-computing
a set of representative local models. We leave these implementation improvements as future work.
The offline stages of our algorithm (Secs. 3.1 and 3.2) require about 5 hours for datasets with M =
14, 000, N = 2, 000, d = 6, 300, and D = 2, 000. The run-time is dominated by the SVM
evaluation of K ? S candidate training neighborhoods on the N images, which could be performed
in parallel. The compressed sensing formulation is quite valuable for efficiency here; if we were to
instead naively train M independent classifiers, the offline run-time would be on the order of days.
We found that building category-label balance into the training and inference algorithms was crucial
for good results when dealing with highly imbalanced datasets. Earlier versions of our method
that ignored label balance would often predict neighborhoods with only the same label as the query.
Local methods typically handle this by reverting to a nearest neighbor decision. However, as we will
see below, this can be inferior to explicitly learning to identify a local and balanced neighborhood,
which can be used to build a more sophisticated classifier (like an SVM).
Finally, while our training procedure designates a single neighborhood as the prediction target, it is
determined by a necessarily limited sample of candidates (Sec. 3.1). Our confidence ranking step
accounts for the differences between those candidates that ultimately make the same label prediction.
Nonetheless, the non-exhaustive training samples mean that slight variations on the target vectors
5
may be equally good in practice. This suggests future extensions to explicitly represent ?missing?
entries in the indicator vector during training or employ some form of active learning.
4
Experiments
We validate our approach on an array of binary image classification tasks on public datasets.
Datasets We consider two challenging datasets with visual attribute classification tasks. The SUN
Attributes dataset [22] (SUN) contains 14,340 scene images labeled with binary attributes of various types (e.g., materials, functions, lighting). We use all images and randomly select 8 attribute
categories. We use the 6,300-dimensional HOG 2 ? 2 features provided by the authors, since they
perform best for this dataset [22]. The aPascal training dataset [10] contains 6,440 object images
labeled with attributes describing the objects? shapes, materials, and parts. We use all images and
randomly select 6 attribute categories. We use the base features from [10], which include color,
texture, edges, and HOG. We reduce their dimensionality to 200 using PCA. For both datasets, we
treat each attribute as a separate binary classification task (C = 2).
Implementation Details For each attribute, we compose a test set of 100 randomly chosen images
(balanced between positives and negatives), and use all other images for T . This makes M =
14, 240 for SUN and M = 6, 340 for aPascal. We use N = 2, 000 training neighborhoods for
both, and set D = {2000, 1000} for SUN and aPascal, roughly 15% of their original label indicator
lengths. Generally higher values of D yield better accuracy (less compression), but for a greater
expense. We fix the number of samples S = 100, and consider neighborhood sizes from k1 = 50
and kK = 500, in increments of 10 to 50.
Baselines and Setup We compare to the following methods: (1) Global: for each test image, we
apply the same global classifier trained with all M training images; (2) Local: for each test image,
we apply a classifier trained with only its nearest neighbors, as measured with Euclidean distance on
the image features. This baseline considers a series of k values, like our method, and independently
selects the best k per test point according to the confidence of the resulting local classifiers (see
Sec. 3.3). (3) Local+ML: same as Local, except the Euclidean distance is replaced with a learned
metric. We apply the ITML metric learning algorithm [7] using the authors? public code.
Global represents the default classification approach, and lets us gauge to what extent the classification task requires local models at all (e.g., how multi-modal the dataset is). The two Local baselines
represent the standard local learning approach [3, 5, 13, 15, 25, 29], in which proximal data points
are used to train a model per test case, as discussed in Sec. 2. By using proximity instead of y?q to
define neighborhoods, they isolate the impact of our compressed sensing approach.
All results reported for our method and the Local baselines use the automatically selected k value
per test image (cf. Sec. 3.3), unless otherwise noted. Each local method independently selects its
best k value. All methods use the exact same image features and train linear SVMs, with the cost
parameter cross-validated based on the Global baseline. To ensure the baselines do not suffer from
the imbalanced data, we show results for the baselines using both balanced (B) and unbalanced (U)
training sets. For the balanced case, for Global we randomly downsample the negatives and average
results over 10 such runs, and for Local we gather the nearest k2 neighbors from each class.
SUN Results The SUN attributes are quite challenging classification tasks. Images within the
same attribute exhibit wide visual variety. For example, the attribute ?eating? (see Fig. 1, top right) is
positive for any images where annotators could envision eating occurring, spanning from an restaurant scene, to home a kitchen, to a person eating, to a banquet table close-up. Furthermore, the
attribute may occupy only a portion of the image (e.g., ?metal? might occupy any subset of the
pixels). It is exactly this variety that we expect local learning may handle well.
Table 1 shows the results on SUN. Our method outperforms all baselines for all attributes. Global
benefits from a balanced training set (B), but still underperforms our method (by 6 points on average). We attribute this to the high intra-class variability of the dataset. Most notably, conventional
Local learning performs very poorly?whether or not we enforce balance. (Recall that the test sets
are always balanced, so chance is 0.50.) Adding metric learning to local (Local+ML) improves
things only marginally, likely because the attributes are not consistently localized in the image. We
also implemented a local metric learning baseline that clusters the training points then learns a met6
Global
B
U
0.80 0.60
0.73 0.55
0.69 0.59
0.77 0.56
0.64 0.57
0.70 0.54
0.78 0.77
0.60 0.67
Attribute
hiking
eating
exercise
farming
metal
still water
clouds
sunny
Local
B
U
0.51 0.56
0.50 0.50
0.50 0.53
0.51 0.54
0.50 0.50
0.51 0.53
0.70 0.74
0.65 0.67
Local+ML
B
U
0.55 0.65
0.50 0.51
0.50 0.53
0.52 0.57
0.50 0.51
0.51 0.52
0.74 0.75
0.62 0.60
Ours
Local
0.85
0.78
0.74
0.83
0.67
0.76
0.80
0.73
0.53
0.50
0.50
0.51
0.50
0.50
0.65
0.59
Local+ML
k = 400
0.53
0.50
0.50
0.51
0.50
0.50
0.74
0.57
Ours
0.89
0.79
0.75
0.81
0.67
0.71
0.79
0.72
Ours
Fix-k*
0.89
0.82
0.77
0.88
0.70
0.81
0.84
0.78
Table 1: Accuracy (% of correctly labeled images) for the SUN dataset. B and U refers to balanced and
unbalanced training data, respectively. All local results to left of double line use k values automatically selected
per method and per test instance; all those to the right use a fixed k for all queries. See text for details.
Local
?clouds?
Local
Ours
Ours
Local
?sunny?
Local
Ours
Local
?farming?
Ours
?exercise?
Ours
Local
?eating?
Ours
?hiking?
Figure 1: Example neighborhoods using visual similarity alone (Local) and compressed sensing inference
(Ours) on SUN. For each attribute, we show a positive test image and its top 5 neighbors. Best viewed on pdf.
ric per cluster, similar to [26, 28], then proceeds as Local+ML. Its results are similar to those of
Local+ML (see Supp. file).
The results left of the double bar correspond to auto-selected k values per query, which averaged
k = 106 with a standard deviation of 24 for our method; see Supp. file for per attribute statistics. The
rightmost columns of Table 1 show results when we fix k for all the local methods for all queries,
as is standard practice.2 Here too, our gain over Local is sizeable, assuring that Local is not at any
disadvantage due to our k auto-selection procedure.
The rightmost column, Fix-k*, shows our results had we been able to choose the optimal fixed k
(applied uniformly to all queries). Note this requires peeking at the test labels, and is something
of an upper bound. It is useful, however, to isolate the quality of our neighborhood membership
confidence estimates from the issue of automatically selecting the neighborhood size. We see there
is room for improvement on the latter.
Our method is more expensive at test time than the Local baseline due to the compressed sensing
reconstruction step (see Sec. 3.4). In an attempt to equalize that factor, we also ran an experiment
where the Local method was allowed to check more candidate k values than our method. Specifically, it could generate as many (proximity-based) candidate neighborhoods at test time as would fit
in the run-time required by our approach, where k ranges from 20 up to 6,000 in increments of 10.
Preliminary tests, however, showed that this gave no accuracy improvement to the baseline. This
indicates our method?s higher computational overhead is warranted.
Despite its potential to handle intra-class variations, the Local baseline fails on SUN because the
neighbors that look most similar are often negative, leading to near-chance accuracy. Even when
we balance its local neighborhood by label, the positives it retrieves can be quite distant (e.g., see
?exercise? in Fig. 1). Our approach, on the other hand, combines locality with what it learned about
2
We chose k = 400 based on the range where the Local baseline had best results.
7
Attribute
wing
wheel
plastic
cloth
furry
shiny
Global
B
U
0.69 0.76
0.84 0.86
0.67 0.71
0.74 0.72
0.80 0.80
0.72 0.77
Local
B
U
0.58 0.67
0.61 0.71
0.50 0.60
0.70 0.67
0.58 0.75
0.56 0.67
Local+ML
B
U
0.59 0.67
0.62 0.69
0.50 0.54
0.72 0.68
0.60 0.71
0.57 0.64
Ours
0.71
0.78
0.64
0.72
0.81
0.72
Local
0.50
0.54
0.50
0.69
0.54
0.52
Local+ML
k = 400
0.53
0.63
0.50
0.65
0.63
0.55
Ours
0.66
0.74
0.54
0.64
0.72
0.62
Ours
Fix-k*
0.78
0.81
0.67
0.77
0.82
0.73
Table 2: Accuracy (% of correctly labeled images) for the aPascal dataset, formatted as in Table 1
useful neighbor combinations, attaining much better results. Altogether, our gains over both Local
and Local+ML?20 points on average?support our central claim that learning what makes a good
neighbor is not equivalent to learning what makes a good neighborhood.
Figure 1 shows example test images and the top 5 images in the neighborhoods produced by both
Local and our approach. We stress that while Local?s neighbors are ranked based on visual similarity,
our method?s ?neighborhood? uses visual similarity only to guide its sampling during training, then
directly predicts which instances are useful. Thus, purer visual similarity in the retrieved examples
is not necessarily optimal. We see that the most confident neighborhood members predicted by
our method are more often positives. Relying solely on visual similarity, Local can retrieve less
informative instances (e.g., see ?farming?) that share global appearance but do not assist in capturing
the class distribution. The attributes where the Local baseline is most successful, ?sunny? and
?cloudy?, seem to differ from the rest in that (i) they exhibit more consistent global image properties,
and (ii) they have many more positives in the dataset (e.g., 2,416 positives for ?sunny? vs. only 281
for ?farming?). In fact, this scenario is exactly where one would expect traditional visual ranking for
local learning to be adequate. Our method does well not only in such cases, but also where image
nearness is not a good proxy for relevance to classifier construction.
aPascal Results Table 2 shows the results on the aPascal dataset. Again we see a clear and consistent advantage of our approach compared to the conventional Local baselines, with an average
accuracy gain of 10 points across all the Local variants. The addition of metric learning again
provides a slight boost over local, but is inferior to our method, again showing the importance of
learning good neighborhoods. On average, the auto-selected k values for this dataset were 144 with
a standard deviation of 20 for our method; see Supp. file for per attribute statistics.
That said, on this dataset Global has a slight advantage over our method, by 2.7 points on average.
We attribute Global?s success on this dataset to two factors: the images have better spatial alignment
(they are cropped to the boundaries of the object, as opposed to displaying a whole scene as in
SUN), and each attribute exhibits lower visual diversity (they stem from just 20 object classes, as
opposed to 707 scene classes in SUN). See Supp. file. For this data, training with all examples
is most effective. While this dataset yields a negative result for local learning on the whole, it is
nonetheless a positive result for the proposed form of local learning, since we steadily outperform
the standard Local baseline. Furthermore, in principle, our approach could match the accuracy of
the Global method if we let kK = M during training; in that case our method could learn that for
certain queries, it is best to use all examples. This is a flexibility not offered by traditional local
methods. However, due to run-time considerations, at the time of writing we have not yet verified
this in practice.
5
Conclusions
We proposed a new form of lazy local learning that predicts at test time what training data is relevant
for the classification task. Rather than rely solely on feature space proximity, our key insight is to
learn to predict a useful neighborhood. Our results on two challenging image datasets show our
method?s advantages, particularly when categories are multi-modal and/or its similar instances are
difficult to match based on global feature distances alone. In future work, we plan to explore ways
to exploit active learning during training neighborhood generation to reduce its costs. We will also
pursue extensions to allow incremental additions to the labeled data without complete retraining.
Acknowledgements We thank Ashish Kapoor for helpful discussions. This research is supported
in part by NSF IIS-1065390.
8
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9
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4,736 | 5,289 | A Unified Semantic Embedding:
Relating Taxonomies and Attributes
Sung Ju Hwang?
Disney Research
Pittsburgh, PA
[email protected]
Leonid Sigal
Disney Research
Pittsburgh, PA
[email protected]
Abstract
We propose a method that learns a discriminative yet semantic space for object
categorization, where we also embed auxiliary semantic entities such as supercategories and attributes. Contrary to prior work, which only utilized them as side information, we explicitly embed these semantic entities into the same space where
we embed categories, which enables us to represent a category as their linear combination. By exploiting such a unified model for semantics, we enforce each category to be generated as a supercategory + a sparse combination of attributes, with
an additional exclusive regularization to learn discriminative composition. The
proposed reconstructive regularization guides the discriminative learning process
to learn a model with better generalization. This model also generates compact semantic description of each category, which enhances interoperability and enables
humans to analyze what has been learned.
1
Introduction
Object categorization is a challenging problem that requires drawing boundaries between groups of
objects in a seemingly continuous space. Semantic approaches have gained a lot of attention recently
as object categorization became more focused on large-scale and fine-grained recognition tasks and
datasets. Attributes [1, 2, 3, 4] and semantic taxonomies [5, 6, 7, 8] are two popular semantic sources
which impose certain relations between the category models, including a more recently introduced
analogies [9] that induce even higher-order relations between them. While many techniques have
been introduced to utilize each of the individual semantic sources for object categorization, no unified model has been proposed to relate them.
We propose a unified semantic model where we can learn to place categories, supercategories, and
attributes as points (or vectors) in a hypothetical common semantic space, and taxonomies provide
specific topological relationships between these semantic entities. Further, we propose a discriminative learning framework, based on dictionary learning and large margin embedding, to learn each
of these semantic entities to be well separated and pseudo-orthogonal, such that we can use them to
improve visual recognition tasks such as category or attribute recognition.
However, having semantic entities embedded into a common space is not enough to utilize the
vast number of relations that exist between the semantic entities. Thus, we impose a graph-based
regularization between the semantic embeddings, such that each semantic embedding is regularized
by sparse combination of auxiliary semantic embeddings. This additional requirement imposed on
the discriminative learning model would guide the learning such that we obtain not just the optimal
model for class discrimination, but to learn a semantically plausible model which has a potential to
be more robust and human-interpretable; we call this model Unified Semantic Embedding (USE).
?
Now at Ulsan National Institute of Science and Technology in Ulsan, South Korea
1
Figure 1: Concept: We regularize each category
to be represented by its supercategory + a sparse
combination of attributes, where the regularization
parameters are learned. The resulting embedding
model improves the generalization ability by the
specific relations between the semantic entities, and
also is able to compactly represent a novel category
in this manner. For example, given a novel category
tiger, our model can describe it as a striped feline.
The observation we make to draw the relation between the categories and attributes, is that a category
can be represented as the sum of its supercategory + the category-specific modifier, which in many
cases can be represented by a combination of attributes. Further, we want the representation to be
compact. Instead of describing a dalmatian as a domestic animal with a lean body, four legs, a
long tail, and spots, it is more efficient to say it is a spotted dog (Figure 1). It is also more exact
since the higher-level category dog contains all general properties of different dog breeds, including
indescribable dog-specific properties, such as the shape of the head, and its posture.
This exemplifies how a human would describe an object, to efficiently communicate and understand
the concept. Such decomposition of a category into attributes+supercategory can hold for categories
at any level. For example, supercategory feline can be described as a stalking carnivore.
With the addition of this new generative objective, our goal is to learn a discriminative model that
can be compactly represented as a combination of semantic entities, which helps learn a model that is
semantically more reasonable. We want to balance between these two discriminative and generative
objectives when learning a model for each object category. For object categories that have scarce
training examples, we can put more weight on the generative part of the model.
Contributions: Our contributions are threefold: (1) We show a multitask learning formulation for
object categorization that learns a unified semantic space for supercategories and attributes, while
drawing relations between them. (2) We propose a novel sparse-coding based regularization that
enforces the object category representation to be reconstructed as the sum of a supercategory and a
sparse combination of attributes. (3) We show from the experiments that the generative learning with
the sparse-coding based regularization helps improve object categorization performance, especially
in the one or few-shot learning case, by generating semantically plausible predictions.
2
Related Work
Semantic methods for object recognition. For many years, vision researchers have sought to
exploit external semantic knowledge about the object to incorporate semantics into learning of the
model. Taxonomies, or class hierarchies were the first to be explored by vision researchers [5, 6], and
were mostly used to efficiently rule out irrelevant category hypotheses leveraging class hierarchical
structure [8, 10]. Attributes are visual or semantic properties of an object that are common across
multiple categories, mostly regarded as describable mid-level representations. They have been used
to directly infer categories [1, 2], or as additional supervision to aid the main categorization problem
in the multitask learning framework [3]. While many methods have been proposed to leverage either
of these two popular types of semantic knowledge, little work has been done to relate the two, which
our paper aims to address.
Discriminative embedding for object categorization. Since the conventional kernel-based multiclass SVM does not scale due to its memory and computational requirements for today?s large-scale
classification tasks, embedding-based methods have gained recent popularity. Embedding-based
methods perform classification on a low dimensional shared space optimized for class discrimination. Most methods learn two linear projections, for data instances and class labels, to a common
lower-dimensional space optimized by ranking loss. Bengio et al. [10] solves the problem using
stochastic gradient, and also provides a way to learn a tree structure which enables one to efficiently
predict the class label at the test time. Mensink et al. [11] eliminated the need of class embedding by
replacing them with the class mean, which enabled generalization to new classes at near zero cost.
There are also efforts in incorporating semantic information into the learned embedding space.
Weinberger et al. [7] used the taxonomies to preserve the inter-class similarities in the learned space,
2
in terms of distance. Akata et al. [4] used attributes and taxonomy information as labels, replacing
the conventional unit-vector based class representation with more structured labels to improve on
zero-shot performance. One most recent work in this direction is DEVISE [12], which learns embeddings that maximize the ranking loss, as an additional layer on top of the deep network for both
images and labels. However, these models impose structure only on the output space, and structure
on the learned space is not explicitly enforced, which is our goal.
Recently, Hwang et al. [9] introduced one such model, which regularizes the category quadruplets,
that form an analogy, to form a parallelogram. Our goal is similar, but we explore a more general
compositional relationship, which we learn without any manual supervision.
Multitask learning. Our work can be viewed as a multitask learning method, since we relate
each model for different semantic entities by learning both the joint semantic space and enforcing
geometric constraints between them. Perhaps the most similar work is [13], where the parameter
of each model is regularized while fixing the parameter for its parent-level models. We use similar
strategy but instead of enforcing sharing between the models, we simply learn each model to be
close to its approximation obtained using higher-level (more abstract) concepts.
Sparse coding. Our method to approximate each category embedding as a sum of its direct supercategory plus a sparse combination of attributes, is similar to the objective of sparse coding. One
work that is specifically relevant to ours is Mairal et al. [14], where the learning objective is to reduce both the classification and reconstruction error, given class labels. In our model, however, the
dictionary atoms are also discriminatively learned with supervision, and are assembled to be a semantically meaningful combination of a supercategory + attributes, while [14] learns the dictionary
atoms in an unsupervised way.
3
Approach
We now explain our unified semantic embedding model, which learns a discriminative common
low-dimensional space to embed both the images and semantic concepts including object categories,
while enforcing relationships between them using semantic reconstruction.
Suppose that we have a d-dimensional image descriptor and m-dimensional vector describing labels
associated with the instances, including category labels at different semantic granularities and attributes. Our goal then is to embed both images and the labels onto a single unified semantic space,
where the images are associated with their corresponding semantic labels.
To formally state the problem, given a training set D that has N labeled examples, i.e. D =
d
{xi , yi }N
i=1 , where xi ? R denotes image descriptors and yi ? {1, . . . , m} are their labels associated with m unique concepts, we want to embed each xi as zi , and each label yi as uyi in the
de -dimensional space, such that the similarity between zi and uyi , S(zi , uyi ) is maximized.
One way to solve the above problem is to use regression, using S(zi , uyi ) = ?kzi ? uyi k22 . That is,
we estimate the data embedding zi as zi = W xi , and minimize their distances to the correct label
embeddings uyi ? Rm where the dimension for yi is set to 1 and every other dimension is set to 0:
m X
N
X
min
kW xi ? uyi k22 + ?kW k2F .
(1)
W
c=1 i=1
The above ridge regression will project each instance close to its correct embedding. However, it
does not guarantee that the resulting embeddings are well separated. Therefore, most embedding
methods for categorization add in discriminative constraints which ensure that the projected instances have higher similarity to their own category embedding than to others. One way to enforce
this is to use large-margin constraints on distance: kW xi ?uyi k22 +1 ? kW xi ?uc k22 +?ic , yi 6= c
which can be translated into to the following discriminative loss:
X
LC (W , U , xi , yi ) =
[1 + kW xi ? uyi k22 ? kW xi ? uc k22 ]+ , ?c 6= yi ,
(2)
c
where U is the columwise concatenation of each label embedding vector, such that uj denotes jth
column of U . After replacing the generative loss in the ridge regression formula with the discriminative loss, we get the following discriminative learning problem:
N
X
(3)
min
LC (W , U , xi , yi ) + ?kW k2F + ?kU k2F , yi ? {1, . . . , m},
W ,U
i
3
where ? regularizes W and U from shooting to infinity. This is one of the most common objective
used for learning discriminative category embeddings for multi-class classification [10, 7], while
ranking loss-based [15] models have been also explored for LC . Bilinear model on a single variable
W has been also used in Akata et al. [4], which uses structured labels (attributes) as uyi .
3.1
Embedding auxiliary semantic entities.
Now we describe how we embed the supercategories and attributes onto the learned shared space.
Supercategories. While our objective is to better categorize entry level categories, categories in
general can appear at different semantic granularities. For example, a zebra could be both an equus,
and an odd-toed ungulate. To learn the embeddings for the supercategories, we map each data
instance to be closer to its correct supercategory embedding than to its siblings: kW xi ?us k22 +1 ?
kW xi ? uc k22 + ?sc , ?s ? Pyi and c ? Ss where Pyi denotes the set of superclasses at all levels
for class s, and Ss is the set of its siblings. The constraints can be translated into the following loss
term:
X X
[1 + kW xi ? us k22 ? kW xi ? uc k22 ]+ .
(4)
LS (W , U , xi , yi ) =
s?Pyi c?Ss
Attributes. Attributes can be considered normalized basis vectors for the semantic space, whose
combination represents a category. Basically, we want to maximize the correlation between the
projected instance that possess the attribute, and its correct attribute embedding, as follows:
LA (W , U , xi , yi ) = 1 ?
X
(W xi )T yia ua , kua k2 ? 1, yia ? {0, 1}, ?a ? Ayi ,
(5)
a
where Ac is the set of all attributes for class c and ua is an embedding vector for an attribute a.
3.2
Relationship between the categories, supercategories, and attributes
Simply summing up all previously defined loss functions while adding {us } and {ua } as additional columns of U will result in a multi-task formulation that implicitly associate the semantic
entities, through the shared data embedding W . However, we want to further utilize the relationships between the semantic entities, to explicitly impose structural regularization on the semantic
embeddings U . One simple and intuitive relation is that an object class can be represented as the
combination of its parent level category plus a sparse combination of attributes, which translates into
the following constraint:
uc = up + U A ?c , c ? Cp , k?c k0 ?1 , ?c 0, ?c ? {1, . . . , C},
(6)
A
where U is the aggregation of all attribute embeddings {ua }, Cp is the set of children classes for
class p, ?1 is the sparsity parameter, and C is the number of categories. We require ? to be nonnegative, since it makes more sense and more efficient to describe an object with attributes that it
might have, rather than describing it by attributes that it might not have.
We rewrite Eq. 7 into a regularization term as follows, replacing the `0 -norm constraints with `1 norm regularizations for tractable optimization:
C
X
R(U , B) =
kuc ? up ? U A ?c k22 + ?2 k?c + ?o k22 ,
c
(7)
c ? Cp , o ? Pc ? Sc , 0 ?c ?1 , ?c ? {1, . . . , C},
where B is the matrix whose jth column vector ?j is the reconstruction weight for class j, Sc is the
set of all sibling classes for class c, and ?2 is the parameters to enforce exclusivity.
The exclusive regularization term is used to prevent the semantic reconstruction ?c for class c from
fitting to the same attributes fitted by its parents and siblings. This is because attributes common
across parent and child, and between siblings, are less discriminative. This regularization is especially useful for discrimination between siblings, which belong to the same superclass and only
differ by the category-specific modifier. By generating unique semantic decomposition for each
class, we can better discriminate between any two categories using a semantic combination of discriminatively learned auxiliary entities.
4
With the sparsity regularization enforced by ?1 , the simple sum of the two weights will prevent the
two (super)categories from having high weight for a single attribute, which will let each category
embedding to fit to exclusive attribute set. This, in fact, is the exclusive lasso regularizer introduced
in [16], except for the nonnegativity constraint on ?c , which makes the problem easier to solve.
3.3
Unified semantic embeddings for object categorization
After augmenting the categorization objective in Eq. 3 with the superclass and attributes loss and the
sparse-coding based regularization in Eq. 7, we obtain the following multitask learning formulation
that jointly learns all the semantic entities along with the sparse-coding based regularization:
min
N
X
W ,U ,B
kwj k22
LC (W , U , xi , yi ) + ?1 (LS (W , U , xi , yi ) + LA (W , U , xi , yi )) + ?2 R(U , B);
i=1
?
?, kuk k22
(8)
? ?, 0 ?c ?1 ?j ? {1, . . . , d}, ?k ? {1, . . . , m}, ?c ? {1, . . . , C + S},
where S is the number of supercategories, wj is W ?s jth column, and ?1 and ?2 are parameters to
balance between the main and auxiliary tasks, and discriminative and generative objective.
Eq. 8 could be also used for knowledge transfer when learning a model for a novel set of categories,
by replacing U A in R(U , B) with U S , learned on class set S to transfer the knowledge from.
3.4
Numerical optimization
Eq. 8 is not jointly convex in all variables, and has both discriminative and generative terms. This
problem is similar to the problem in [14], where the objective is to learn the dictionary, sparse
coefficients, and classifier parameters together, and can be optimized using a similar alternating
optimization, while each subproblem differs. We first describe how we optimize for each variable.
Learning of W and U . The optimization of both embedding models are similar, except for the
reconstructive regularization on U . and the main bottleneck lies in the minimization of the O(N m)
large-margin losses. Since the losses are non-differentiable, we solve the problems using stochastic
subgradient method. Specifically, we implement the proximal gradient algorithm in [17], handling
the `-2 norm constraints with proximal operators.
Learning B. This is similar to the sparse coding problem, but simpler. We use projected gradient
t+ 1
method, where at each iteration t, we project the solution of the objective ?c 2 for category c to `-1
norm ball and nonnegative orthant, to obtain ?ct that satisfies the constraints.
Alternating optimization. We decompose Eq. 8 to two convex problems: 1) Optimization of the
data embedding W and approximation parameter B (Since the two variable do not have direct link
between them) , and 2) Optimization of the category embedding U . We alternate the process of
optimizing each of the convex problems while fixing the remaining variables, until the convergence
criterion 1 is met, or the maximum number of iteration is reached.
Run-time complexity. Training: Optimization of W and U using proximal stochastic gradient [17], have time complexities of O(de d(k + 1)) and O(de (dk + C)) respectively. Both terms are
dominated by the gradient computation for k(k N ) sampled constraints, that is O(de dk). Outer
loop for alternation converges within 5-10 iterations depending on . Test: Test time complexity is
exactly the same as in LME, which is O(de (C + d)).
4
Results
We validate our method for multiclass categorization performance on two different datasets generated from a public image collection, and also test for knowledge transfer on few-shot learning.
4.1
Datasets
We use Animals with Attributes dataset [1], which consists of 30, 475 images of 50 animal classes,
with 85 class-level attributes 2 . We use the Wordnet hierarchy to generate supercategories. Since
kW t+1 ? W t k2 + kU t+1 ? U t k2 + kB t+1 ? B t k2 <
Attributes are defined on color (black, orange), texture (stripes, spots), parts (longneck, hooves), and other
high-level behavioral properties (slow, hibernate, domestic) of the animals
1
2
5
there is no fixed training/test split, we use {30,30,30} random split for training/validation/test. We
generate the following two datasets using the provided features. 1) AWA-PCA: We compose a 300dimensional feature vectors by performing PCA on each of 6 types of features provided, including
SIFT, rgSIFT, SURF, HoG, LSS, and CQ to have 50 dimensions per each feature type, and concatenating them. 2) AWA-DeCAF: For the second dataset, we use the provided 4096-D DeCAF features
[18] obtained from the layer just before the output layer of a deep convolutional neural network.
4.2
Baselines
We compare our proposed method against multiple existing embedding-based categorization approaches, that either do not use any semantic information, or use semantic information but do not
explicitly embed semantic entities. For non-semantics baselines, we use the following: 1)Ridge
Regression: A linear regression with `-2 norm (Eq. 1). 2) NCM: Nearest mean classifier from [11],
which uses the class mean as category embeddings (uc = x?c ). We use the code provided by the
authors3 . 3) LME: A base large-margin embedding (Eq. 3) solved using alternating optimization.
For implicit semantic baselines, we consider two different methods. 4) LMTE: Our implementation
of the Weinberger et al. [7], which enforces the semantic similarity between class embeddings as
distance constraints [7], where U is regularized to preserve the pairwise class similarities from a
given taxonomy. 5-7) ALE, HLE, AHLE: Our implementation of the attribute label embedding in
Akata et al. [4], which encodes the semantic information by representing each class with structured
labels that indicate the class? association with superclasses and attributes. We implement variants
that use attributes (ALE), leaf level + superclass labels (HLE), and both (AHLE) labels.
For our models, we implement multiple variants to analyze the impact of each semantic entity and
the proposed regularization. 1) LME-MTL-S: The multitask semantic embedding model learned
with supercategories. 2) LME-MTL-A: The multitask embedding model learned with attributes. 3)
USE-No Reg.: The unified semantic embedding model learned using both attributes and supercategories, without semantic regularization. 4) USE-Reg: USE with the sparse coding regularization.
For parameters, the projection dimension de = 50 for all our models. 4 For other parameters, we
find the optimal value by cross-validation on the validation set. We set ?1 = 1 that balances the main
and auxiliary task equally, and search for ?2 for discriminative/generative tradeoff, in the range of
{0.01, 0.1, 0.2 . . . , 1, 10}, and set `-2 norm regularization parameter ? = 1. For sparsity parameter
?1 , we set it to select on average several (3 or 4) attributes per class, and for disjoint parameter ?2 ,
we use 10?1 , without tuning for performance.
No
semantics
Implicit
semantics
Explicit
semantics
USE
Method
Ridge Regression
NCM [11]
LME
LMTE [7]
ALE [4]
HLE [4]
AHLE [4]
LME-MTL-S
LME-MTL-A
USE-No Reg.
USE-Reg.
1
19.31 ? 1.15
18.93 ? 1.71
19.87 ? 1.56
20.76 ? 1.64
15.72 ? 1.14
17.09 ? 1.09
16.65 ? 0.47
20.77 ? 1.41
20.65 ? 0.83
21.07 ? 1.53
21.64 ? 1.02
Flat hit @ k (%)
2
28.34 ? 1.53
29.75 ? 0.92
30.47 ? 1.56
30.71 ? 1.35
25.63 ? 1.44
27.52 ? 1.20
26.55 ? 0.77
32.09 ? 1.67
31.51 ? 0.72
31.59 ? 1.57
32.69 ? 0.83
5
44.17 ? 2.33
47.33 ? 1.60
48.07 ? 1.06
47.76 ? 2.25
43.42 ? 1.67
45.49 ? 0.61
43.05 ? 1.22
50.94 ? 1.21
49.40 ? 0.62
50.11 ? 1.51
52.04 ? 1.02
Hierarchical precision @ k (%)
2
5
28.95 ? 0.54
39.39 ? 0.17
30.81 ? 0.53
43.43 ? 0.53
30.98 ? 0.62
42.63 ? 0.56
31.05 ? 0.71
43.13 ? 0.29
29.26 ? 0.50
43.71 ? 0.34
30.51 ? 0.48
44.76 ? 0.20
29.49 ? 0.89
43.41 ? 0.65
33.71 ? 0.94
45.73 ? 0.71
31.69 ? 0.49
43.47 ? 0.23
33.67 ? 0.55
45.41 ? 0.43
33.37 ? 0.74
47.17 ? 0.91
Table 1: Multiclass classification performance on AWA-PCA dataset (300-D PCA features).
4.3
Multiclass categorization
We first evaluate the suggested multitask learning framework for categorization performance. We
report the average classification performance and standard error over 5 random training/test splits
in Table 1 and 2, using both flat hit@k, which is the accuracy for the top-k predictions made, and
hierarchical precision@k from [12], which is a precision the given label is correct at k, at all levels.
Non-semantic baselines, ridge regression and NCM, were outperformed by our most basic LME
model. For implicit semantic baselines, ALE-variants underperformed even the ridge regression
3
4
http://staff.science.uva.nl/?tmensink/code.php
Except for ALE variants where de =m, the number of semantic entities.
6
No
semantics
Implicit
semantics
Explicit
semantics
USE
Method
Ridge Regression
NCM [11]
LME
LMTE [7]
ALE [4]
HLE [4]
AHLE [4]
LME-MTL-S
LME-MTL-A
USE-No Reg.
USE-Reg.
1
38.39 ? 1.48
43.49 ? 1.23
44.76 ? 1.77
38.92 ? 1.12
36.40 ? 1.03
33.56 ? 1.64
38.01 ? 1.69
45.03 ? 1.32
45.55 ? 1.71
45.93 ? 1.76
46.42 ? 1.33
Flat hit @ k (%)
2
48.61 ? 1.29
57.45 ? 0.91
58.08 ? 2.05
49.97 ? 1.16
50.43 ? 1.92
45.93 ? 2.56
52.07 ? 1.19
57.73 ? 1.75
58.60 ? 1.76
59.37 ? 1.32
59.54 ? 0.73
5
62.12 ? 1.20
75.48 ? 0.58
75.11 ? 1.48
63.35 ? 1.38
70.25 ? 1.97
64.66 ? 1.77
71.53 ? 1.41
74.43 ? 1.26
74.97 ? 1.15
74.97 ? 1.15
76.62 ? 1.45
Hierarchical precision @ k (%)
2
5
38.51 ? 0.61
41.73 ? 0.54
45.25 ? 0.52
50.32 ? 0.47
44.84 ? 0.98
49.87 ? 0.39
38.67 ? 0.46
41.72 ? 0.45
42.52 ? 1.17
52.46 ? 0.37
46.11 ? 2.65
56.79 ? 2.05
44.43 ? 0.66
54.39 ? 0.55
46.05 ? 0.89
51.08 ? 0.36
44.23 ? 0.95
48.52 ? 0.29
47.13 ? 0.62
51.04 ? 0.46
47.39 ? 0.82
53.35 ? 0.30
Table 2: Multiclass classification performance on AWA-DeCAF dataset (4096-D DeCAF features).
baseline with regard to the top-1 classification accuracy 5 , while they improve upon the top-2 recognition accuracy and hierarchical precision. This shows that hard-encoding structures in the label
space do not necessarily improve the discrimination performance, while it helps to learn a more
semantic space. LMTE makes substantial improvement on 300-D features, but not on DeCAF features.
Explicit embedding of semantic entities using our method improved both the top-1 accuracy and
the hierarchical precision, with USE variants achieving the best performance in both. Specifically,
adding superclass embeddings as auxiliary entities improves the hierarchical precision, while using
attributes improves the flat top-k classification accuracy. USE-Reg, especially, made substantial
improvements on flat hit and hierarchical precision @ 5, which shows the proposed regularization?s
effectiveness in learning a semantic space that also discriminates well.
Category
Otter
Skunk
Deer
Moose
Equine
Primate
Ground-truth attributes
Supercategory + learned attributes
An animal that swims, fish, water, new world, small, flippers,
furry, black, brown, tail, . . .
A musteline mammal that is quadrapedal, flippers, furry,
ocean
An animal that is smelly, black, stripes, white, tail, furry,
ground, quadrapedal, new world, walks, . . .
A musteline mammal that has stripes
An animal that is brown, fast, horns, grazer, forest,
quadrapedal, vegetation, timid, hooves, walks, . . .
A deer that has spots, nestspot, longneck, yellow, hooves
An animal that has horns, brown, big, quadrapedal, new
world, vegetation, grazer, hooves, strong, ground,. . .
A deer that is arctic, stripes, black
N/A
N/A
An odd-toed ungulate, that is lean and active
An animal, that has hands and bipedal
Table 3: Semantic description generated using ground truth attributes labels and learned semantic decomposition of each categorys. For ground truth labels, we show top-10 ranked by their human-ranked relevance. For
our method, we rank the attributes by their learned weights. Incorrect attributes are colored in red.
4.3.1
Qualitative analysis
Besides learning a space that is both discriminative and generalizes well, our method?s main advantage, over existing methods, is its ability to generate compact, semantic descriptions for each
category it has learned. This is a great caveat, since in most models, including the state-of-the
art deep convolutional networks, humans cannot understand what has been learned; by generating
human-understandable explanation, our model can communicate with the human, allowing understanding of the rationale behind the categorization decisions, and to possibly allow feedback for
correction.
To show the effectiveness of using supercategory+attributes in the description, we report the learned
reconstruction for our model, compared against the description generated by its ground-truth attributes in Table 3. The results show that our method generates compact description of each category, focusing on its discriminative attributes. For example, our method select attributes such as
flippers for otter, and stripes for skunk, instead of attributes common and nondescriminative such as
tail. Note that some attributes that are ranked less relevant by humans were selected for their discriminativity, e.g., yellow for dear and black for moose, both of which human annotators regarded
5
We did extensive parameter search for the ALE variants.
7
placental
ungulate
horns
hooves
longneck
carnivore
pads
stalker
paws
aquatic
plankton
ocean
swims
rodent
plankton
g.ape
feline
plankton
yellow
orange
canine
longneck
whale
longleg
plains
fields
primate
plankton
hands
bipedal
even?toed
hunter
dog
strainteeth
toughskin
longneck
bear
musteln procyonid
pinnpd
dolphin
plankton strainteethlongneck
walks
planktonbaleen
longneck longneck toughskin
ground
longneck
strainteethplankton strainteeth
stalker
sheperd
cat
strainteeth
toughskin
hairless
big cat
toughskin
longneck
big
ruminant
plankton
meatteeth
hunter
bovid
hooves
horns
grazer
odd?toed ungulate
plankton
meatteeth
hunter
domestic
plankton
longneck
toughskin
equine
hunter
meatteeth
small
deer
muscle
bovine
small
meatteeth
s
es
irl
ha
s in
s
rn ou hsk ds
ho ulb ug pa e
b to ic iv
:
t ct
w h
co bus low es y a
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pi op ripe in usk me ws
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hi a: ugh ngl talk ds c
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ze e: s: tive tic p ts
rs ro ac es po
ho oce in om w s ly
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Si ian tro lbo
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Pe cat ert fis
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b es ts e eth alk
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te : l s s s o
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gr pb le: ive me tee
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nd
hu wh : a ttee r bu ks ha
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bl lp me raz ll w se
do e: s g ma in
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w : t in h
r
k. us nta oug ous aze
r u t lb r c
g i
al o :
w : m ee l bu ns est
al nz da or m
se pa ipe : h do
im b ey up eth l
ch lla: nk gro tte da e
ri mo st ea ipe tiv
go er ore s m r b nac
id : f ld lke s i
sp se fie sta lk s
a
ou rce ds w ipe
m fie pa hes str
t: l: tc ds ss
ra irre pa pa irle
s ha
u r:
h
sq ste wim lys h bus
m : s f us s
ha er ton n b usk
av nk to n t
be pla ank kto els
t: pl an n
ba it: pl tun
:
bb nt on
ra ha nkt
ep la
el e: p
ol
m
Figure 2: Learned discriminative attributes association on the AWA-PCA dataset. Incorrect attributes are
colored in red.
AWA?DeCAF
70
40
60
35
30
No transfer
AHLE
USE
USE?Reg.
25
20
15
0
2
4
6
8
Number of training examples
10
Accuracy (%)
Accuracy (%)
AWA?PCA
45
Class
Humpback
whale
Leopard
50
40
No transfer
AHLE
USE
USE?Reg.
30
20
0
2
4
6
8
Number of training examples
10
Learned decomposition
A baleen whale, with plankton, flippers, blue, skimmer, arctic
A big cat that is orange, claws, black
An even-toed ungulate, that is gray,
Hippopotamus
bulbous, water, smelly, hands
A primate, that is mountains, strong,
Chimpanzee
stalker, black
A domestic cat, that is arctic, nestspot,
Persian Cat
fish, bush
Figure 3: Few-shot experiment result on the AWA dataset, and generated semantic decompositions.
as brown. Further, our method selects discriminative attributes for each supercategory, while there
is no provided attribute label for supercategories.
Figure 2 shows the discriminative attributes disjointly selected at each node on the class hierarchy.
We observe that coarser grained categories fit to attributes that are common throughout all its children (e.g. pads, stalker and paws for carnivore), while the finer grained categories fit to attributes
that help for finer-grained distinctions (e.g. orange for tiger, spots for leopard, and desert for lion).
4.4 One-shot/Few-shot learning
Our method is expected to be especially useful for few-shot learning, by generating a richer description than existing methods, that approximate the new input category using either trained categories
or attributes. For this experiment, we divide the 50 categories into predefined 40/10 training/test
split, and compare with the knowledge transfer using AHLE. We assume that no attribute label is
provided for test set. For AHLE, and USE, we regularize the learning of W with W S learned
on training class set S by adding ?2 kW ? W S k22 , to LME (Eq. 3). For USE-Reg we use the
reconstructive regularizer to regularize the model to generate semantic decomposition using U S .
Figure 3 shows the result, and the learned semantic decomposition of each novel category. While all
methods make improvements over the no-transfer baseline, USE-Reg achieves the most improvement, improving two-shot result on AWA-DeCAF from 38.93% to 49.87%, where USE comes in
second with 44.87%. Most learned reconstructions look reasonable, and fit to discriminative traits
that help to discriminate between the test classes, which in this case are colors; orange for leopard,
gray for hippopotamus, blue for humpback whale, and arctic (white) for Persian cat.
5
Conclusion
We propose a unified semantic space model that learns a discriminative space for object categorization, with the help of auxiliary semantic entities such as supercategories and attributes. The auxiliary
entities aid object categorization both indirectly, by sharing a common data embedding, and directly,
by a sparse-coding based regularizer that enforces the category to be generated by its supercategory
+ a sparse combination of attributes. Our USE model improves both the flat-hit accuracy and hierarchical precision on the AWA dataset, and also generates semantically meaningful decomposition
of categories, that provides human-interpretable rationale.
8
References
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9
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4,737 | 529 | Neural Computing with Small Weights
Kai-Yeung Siu
Dept. of Electrical & Computer Engineering
University of California, Irvine
Irvine, CA 92717
J ehoshua Bruck
IBM Research Division
Almaden Research Center
San Jose, CA 95120-6099
Abstract
An important issue in neural computation is the dynamic range of weights
in the neural networks. Many experimental results on learning indicate
that the weights in the networks can grow prohibitively large with the
size of the inputs. Here we address this issue by studying the tradeoffs
between the depth and the size of weights in polynomial-size networks
of linear threshold elements (LTEs). We show that there is an efficient
way of simulating a network of LTEs with large weights by a network
of LTEs with small weights. In particular, we prove that every depth-d,
polynomial-size network of LTEs with exponentially large integer weights
can be simulated by a depth-(2d + 1), polynomial-size network of LTEs
with polynomially bounded integer weights. To prove these results, we
use tools from harmonic analysis of Boolean functions. Our technique is
quite general, it provides insights to some other problems. For example,
we are able to improve the best known results on the depth of a network
of linear threshold elements that computes the COM PARISO N, SUM
and PRO DU CT of two n-bits numbers, and the MAX 1M U M and the
SORTING of n n-bit numbers.
1
Introduction
The motivation for this work comes from the area of neural networks, where a
linear threshold element is the basic processing element. Many experimental results
on learning have indicated that the magnitudes of the coefficients in the threshold
elements grow very fast with the size of the inputs and therefore limit the practical
use of the network. One natural question to ask is the following: How limited
944
Neural Computing with Small Weights
is the computational power of the network if we restrict ourselves to threshold
elements with only "small" growth in the coefficients? We answer this question by
showing that we can trade-off an exponential growth with a polynomial growth in
the magnitudes of coefficients by increasing the depth of the network by a factor of
almost two and a polynomial growth in the size.
Linear Threshold Functions: A linear threshold function f(X) is a Boolean
function such that
f(X)
= sgn(F(X? = {_II
where
if F(X) > 0
if F(X) < 0
n
F(X) =
2:=
Wi . Xi
+ Wo
i=l
Throughout this paper, a Boolean function will be defined as f : {I, _I}n --+
{I, -I}; namely, 0 and 1 are represented by 1 and -1, respectively. Without loss
of generality, we can assume F(X):/; 0 for all X E {I,-I}n. The coefficients Wi
are commonly referred to as the weights of the threshold function. We denote the
class of all linear threshold functions by LT1 ?
---
LT1 functions: In this paper, we shall study a subclass of LT1 which we denote
by
is characterized by
1 . Each function f(X) = sgn(L:~=l Wi' Xi + wo) in
IT
IT1
the property that the weights Wi are integers and bounded by a polynomial in n,
i.e. IWil ~ n C for some constant c > O.
Threshold Circuits: A threshold circuit [5, 10] is a Boolean network in which
every gate computes an LT1 function. The size of a threshold circuit is the number
of LT1 elements in the circuit. Let LTk denote the class of threshold circuits of
depth k with the size bounded by a polynomial in the number of inputs. We define
LTk similarly except that we allow each gate in LTk to compute an LTI function.
-
---
---
Although the definition of (LTd linear threshold function allows the weights to be
real numbers, it is known [12] that we can replace each of the real weights by integers
of O( n log n) bits, where n is the number of input Boolean variables. So in the rest
of the paper, we shall assume without loss of generality that all weights are integers.
However, this still allows the magnitudes of the weights to increase exponentially
fast with the size of the inputs. It is natural to ask if this is necessary. In other
words, is there a linear threshold function that must require exponentially large
weights? Since there are 2n(n~) linear threshold functions in n variables [8, 14, 15],
there exists at least one which requires O(n 2 ) bits to specify the weights. By the
pigeonhole principle, at least one weight of such a function must need O(n) bits,
and thus is exponentially large in magnitude. i.e.
-
L TI ~ LT1
The above result was proved in [9] using a different method by explicitly constructing
an LT1 function and proving that it is not in LT1 . In the following section, we
shall show that the COMPARISON function (to be defined later) also requires
exponentially large weights. We will refer to this function later on in the proof of
-
945
946
Siu and Bruck
our main results. Main Results: The fact that we can simulate a linear threshold
function with exponentially large weights in a 'constant' number oflayers of elements
with 'small' weights follows from the results in [3] and [11]. Their results showed
that the sum of n n-bit numbers is computable in a constant number of layers of
'counting' gates, which in turn can be simulated by a constant number of layers of
threshold elements with 'small' weights. However, it was not explicitly stated how
many layers are needed in each step of their construction and direct application of
their results would yield a constant such as 13. In this paper, we shall reduce the
constant to 3 by giving a more 'depth'-efficient algorithm and by using harmonic
analysis of Boolean functions [1,2,6]. We then generalize this result to higher depth
circuits and show how to simulate a threshold circuit of depth-d and exponentially
large weights in a depth-(2d + 1) threshold circuit of 'small' weights, i.e. LTd ~
fr2d+l.
As another application of harmonic analysis, we also show that the
COM P ARISON and ADDITION of two n-bit numbers is computable with only
two layers of elements with 'small' weights, while it was only known to be computable in 3 layers [5]. We also indicate how our 'depth'-efficient algorithm can be
applied to show that the product of two n-bit numbers can be computed in LT4 .
In addition, we show that the MAXIMUM and SORTING ofn n-bit numbers
can be computed in fr3 and LT4 , respectively.
--
2
Main Results
Definition: Let X = (Xl, ... , Xn), Y = (YI, ... , Yn) E {I, _l}n. We consider X
and Y as two n-bit numbers representing E?=l Xi? 2' and E?=l Yi . 2i , respectively.
The COMPARISON function is defined as
C(X, Y) = 1 iff X
In other words,
~ Y
n
C(X, Y) =
sgn{L:: 2i(Xi - yd + I}
i=l
Lemma 1
COMPARISON
-
tt LTI
On the other hand, using harmonic analysis [2], we can show the following:
Lemma 2
COMPARISON E
m
Spectral representation of Boolean functions: Recently, harmonic analysis
has been found to be a powerful tool in studying the computational complexity of
Boolean functions [1, 2, 7]. The idea is that every Boolean function f : {I, _1}n -+
{I, -I} can be represented as a polynomial over the field of rational numbers as
follows:
f(X)
aa xa
= L
aE{O,l}n
Neural Computing with Small Weights
h
were
al
X a = x al
1 x2
an ?
.?. Xn
Such representation is unique and the coefficients of the polynomial,
{a, l}n}, are called the spectral coefficients of f.
{aal Q E
We shall define the Ll spectral norm of f to be
IIfll
=
~ laal?
ae{O,I}n
The proof of Lemma 2 is based on the spectral techniques developed in [2]. Using
probabilistic arguments, it was proved in [2] that if a Boolean function has Ll spec.tral norm which is polynomially bounded, then the function is computable in LT2 ?
We observe (together with Noga Alon) that the techniques in [2] can be generalized
to show that any Boolean function with polynomially bounded Ll spectral norm
can even be closely approximated by a sparse polynomial. This observation is crucial
when we extend our result from a single element to networks of elements with large
weights.
Lemma 3 Let f(X) : {I, _l}n --+ {I, -I} such that IIfll
for any k > 0, there exists a sparse polynomial
F(X)
= N1 2:'.:: wa Xa
~
n C for some c. Then
such that
aes
IF(X) - f(X)1 ~ n- k ,
where Wa and N are integers, S c {O, l}n, the size of S, Wa and N are all bounded
by a polynomial in n. Hence, f(X) E
2?
LT
As a consequence of this result, Lemma 2 follows since it can be shown that
COM PARISON has a polynQmially bounded Ll spectral norm.
Now we are ready to state our main results. Although most linear threshold functions require exponentially large weights, we can always simulate them by 3 layers
of
elements.
in
Theorem 1
-
LTI ~ LT3
The result stated in Theorem 1 implies that a depth-d threshold circuit with exponentially large weights can be simulated by a depth-3d threshold circuit with
polynomially large weights. Using the result of Lemma 3, we can actually obtain a
more depth-efficient simulation.
Theorem 2
As another consequence of Lemma 3, we have the following :
947
948
Siu and Bruck
Corollary 1 Let /1 (X), ... , fm(X) be functions with polynomially bounded Ll spectral norms, and g(/1 (X), ... , fm(X? be an fi\ function with fi(X) 's as inputs,
I.e.
m
g(/1(X), ... , fm(X?
= sgn(2: Wdi(X) + wo)
i=l
Then 9 can be expressed as a sign of a sparse polynomial in X with polynomially
many number of monomial terms xcr 's and polynomially bounded integer coefficients. Hence 9 E LT2.
---
If all LTI functions have polynomially bounded Ll spectral norms, then it would
follow that LTI C iT 2 ? However, even the simple MAJORITY function does not
have a polynomially bounded Ll spectral norm. We shall prove this fact via the
following theorem. (As in Lemma 3, by a sparse polynomial we mean a polynomial
with only polynomially many monomial terms xcr's).
Theorem 3 The
iT l
function MAJORITY:
n
sgn(2: X i)
i=l
cannot be approximated by a sparse polynomial with an error o( n -1).
Other applications of the harmonic analysis techniques and the results of Lemma 3
yields the following theorems:
Theorem 4
Let x, y be two n-bit numbers. Then
ADDITION(x, y) E
m
---
Theorem 5 The product of two n-bit integers can be computed in LT4 ?
---
Theorem 6 The MAX I MU M of n n-bit numbers can be computed in LT3.
Theorem 7 The SORTING ofn n-bit numbers can be computed in
3
IT4 .
Concluding Remarks
Our main result indicates that for networks of linear threshold elements, we can
trade-off arbitrary real weights with polynomially bounded integer weights, at the
expense of a polynomial increase in the size and a factor of almost two in the depth of
the network. The proofs of the results in this paper can be found in [13]. We would
like to mention that our results have recently been improved by Goldmann, Hastad
and Razborov [4]. They showed that any polynomial-size depth-d network oflinear
threshold elements with arbitrary weights can be simulated by a polynomial-size
depth-( d + 1) network with "small" (polynomially bounded integer) weights. While
our construction can be made explicit, only the existence of the simulation result is
proved in [4]; it is left as an open problem in [4] if there is an explicit construction
of their results.
Neural Computing with Small Weights
Acknowledgements
This work was done while Kai-Yeung Siu was a research student associate at IBM
Almaden Research Center and was supported in part by the Joint Services Program
at Stanford University (US Army, US Navy, US Air Force) under Contract DAAL0388-C-0011, and the Department of the Navy (NAVELEX), NASA Headquarters,
Center for Aeronautics and Space Information Sciences under Grant NAGW-419S6.
References
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949
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4,738 | 5,290 | Zero-Shot Recognition with Unreliable Attributes
Kristen Grauman
University of Texas at Austin
Austin, TX 78701
[email protected]
Dinesh Jayaraman
University of Texas at Austin
Austin, TX 78701
[email protected]
Abstract
In principle, zero-shot learning makes it possible to train a recognition model
simply by specifying the category?s attributes. For example, with classifiers for
generic attributes like striped and four-legged, one can construct a classifier for
the zebra category by enumerating which properties it possesses?even without
providing zebra training images. In practice, however, the standard zero-shot
paradigm suffers because attribute predictions in novel images are hard to get
right. We propose a novel random forest approach to train zero-shot models that
explicitly accounts for the unreliability of attribute predictions. By leveraging
statistics about each attribute?s error tendencies, our method obtains more robust
discriminative models for the unseen classes. We further devise extensions to handle the few-shot scenario and unreliable attribute descriptions. On three datasets,
we demonstrate the benefit for visual category learning with zero or few training
examples, a critical domain for rare categories or categories defined on the fly.
1
Introduction
Visual recognition research has achieved major successes in recent years using large datasets and
discriminative learning algorithms. The typical scenario assumes a multi-class task where one has
ample labeled training images for each class (object, scene, etc.) of interest. However, many realworld settings do not meet these assumptions. Rather than fix the system to a closed set of thoroughly
trained object detectors, one would like to acquire models for new categories with minimal effort
and training examples. Doing so is essential not only to cope with the ?long-tailed? distribution of
objects in the world, but also to support applications where new categories emerge dynamically?for
example, when a scientist defines a new phenomenon of interest to be detected in her visual data.
Zero-shot learning offers a compelling solution. In zero-shot learning, a novel class is trained via
description?not labeled training examples [10, 18, 8]. In general, this requires the learner to have
access to some mid-level semantic representation, such that a human teacher can define a novel
unseen class by specifying a configuration of those semantic properties. In visual recognition, the
semantic properties are attributes shared among categories, like black, has ears, or rugged. Supposing the system can predict the presence of any such attribute in novel images, then adding a new
category model amounts to defining its attribute ?signature? [8, 3, 18, 24, 19]. For example, even
without labeling any images of zebras, one could build a zebra classifier by instructing the system
that zebras are striped, black and white, etc. Interestingly, computational models for attribute-based
recognition are supported by the cognitive science literature, where researchers explore how humans
conceive of objects as bundles of attributes [25, 17, 5].
So, in principle, if we could perfectly predict attribute presence1 , zero-shot learning would offer
an elegant solution to generating novel classifiers on the fly. The problem, however, is that we
can?t assume perfect attribute predictions. Visual attributes are in practice quite difficult to learn
1
and have an attribute vocabulary rich enough to form distinct signatures for each category of interest
1
accurately?often even more so than object categories themselves. This is because many attributes
are correlated with one another (given only images of furry brown bears, how do we learn furry and
brown separately? [6]), and abstract linguistic properties can have very diverse visual instantiations
(compare a bumpy road to a bumpy rash). Thus, attribute-based zero-shot recognition remains in the
?proof of concept? realm, in practice falling short of alternate transfer methods [23].
We propose an approach to train zero-shot models that explicitly accounts for the unreliability of
attribute predictions. Whereas existing methods take attribute predictions at face value, our method
during training acknowledges the known biases of the mid-level attribute models. Specifically,
we develop a random forest algorithm that, given attribute signatures for each category, exploits
the attribute classifiers? receiver operating characteristics to select discriminative and predictable
decision nodes. We further generalize the idea to account for unreliable class-attribute associations.
Finally, we extend the solution to the ?few-shot? setting, where a small number of category-labeled
images are also available for training.
We demonstrate the idea on three large datasets of object and scene categories, and show its clear
advantages over status quo models. Our results suggest the valuable role attributes can play for
low-cost visual category learning, in spite of the inherent difficulty in learning them reliably.
2
Related Work
Most existing zero-shot models take a two-stage classification approach: given a novel image, first its
attributes are predicted, then its class label is predicted as a function of those attributes. For example,
in [3, 18, 30], each unseen object class is described by a binary indicator vector (?signature?) over its
attributes; a new image is mapped to the unseen class with the signature most similar to its attribute
predictions. The probabilistic Direct Attribute Prediction (DAP) method [8] takes a similar form, but
adds priors for the classes and attributes and computes a MAP prediction of the unseen class label.
A topic model variant is explored in [31]. The DAP model has gained traction and is often used in
other work [23, 19, 29]. In all of the above methods, as in ours, training an unseen class amounts to
specifying its attribute signature. In contrast to our approach, none of the existing methods account
for attribute unreliability when learning an unseen category. As we will see in the results, this has a
dramatic impact on generalization.
We stress that attribute unreliability is distinct from attribute strength. The former (our focus) pertains to how reliable the mid-level classifier is, whereas the latter pertains to how strongly an image
exhibits an attribute (e.g., as modeled by relative [19] or probabilistic [8] attributes). PAC bounds
on the tolerable error for mid-level classifiers are given in [18], but that work does not propose a
solution to mitigate the influence of their uncertainty.
While the above two-stage attribute-based formulation is most common, an alternative zero-shot
strategy is to exploit external knowledge about class relationships to adapt classifiers to an unseen
class. For example, an unseen object?s classifier can be estimated by combining the nearest existing classifiers (trained with images) in the ImageNet hierarchy [23, 14], or by combining classifiers
based on label co-occurrences [13]. In a similar spirit, label embeddings [1] or feature embeddings [4] can exploit semantic information for zero-shot predictions. Unlike these models, we focus
on defining new categories through language-based description (with attributes). This has the advantage of giving a human supervisor direct control on the unseen class?s definition, even if its
attribute signature is unlike that observed in any existing trained model.
Acknowledging that attribute classifiers are often unreliable, recent work abandons purely semantic
attributes in favor of discovering mid-level features that are both detectable and discriminative for
a set of class labels [11, 22, 26, 15, 30, 27, 1]. However, there is no guarantee that the discovered
features will align with semantic properties, particularly ?nameable? ones. This typically makes
them inapplicable to zero-shot learning, since a human supervisor can no longer define the unseen
class with concise semantic terms. Nonetheless, one can attempt to assign semantics post-hoc (e.g.,
[30]). We demonstrate that our method can benefit zero-shot learning with such discovered (pseudo)attributes as well.
Our idea for handling unreliable attributes in random forests is related to fractional tuples for handling missing values in decision trees [21]. In that approach, points with missing values are distributed down the tree in proportion to the observed values in all other data. Similar concepts are
explored in [28] to handle features represented as discrete distributions and in [16] to propagate
2
instances with soft node memberships. Our approach also entails propagating training instances in
proportion to uncertainty. However, our zero-shot scenario is distinct, and, accordingly, the training
and testing domains differ in important ways. At training time, rather than build a decision tree from
labeled data points, we construct each tree using the unseen classes? attribute signatures. Then, at
test time, the inputs are attribute classifier predictions. Furthermore, we show how to propagate both
signatures and data points through the tree simultaneously, which makes it possible to account for
inter-dependencies among the input dimensions and also enables a few-shot extension.
3
Approach
Given a vocabulary of M visual attributes, each unseen class k is described in terms of its attribute
signature Ak , which is an M -dimensional vector where Ak (i) gives the association of attribute
i with class k.2 Typically the association values would be binary?meaning that the attribute is
always present/absent in the class?but they may also be real-valued when such fine-grained data
is available. We model each unseen class with a single signature (e.g., whales are big and gray).
However, it is straightforward to handle the case where a class has a multi-modal definition (e.g.,
whales are big and gray OR whales are big and black), by learning a zero-shot model per ?mode?.
Whether the attribute vocabulary is hand-designed [8, 3, 19, 29, 23] or discovered [30, 11, 22], our
approach assumes it is expressive enough to discriminate between the categories.
Suppose there are K unseen classes of interest, for which we have no training images. Our zero-shot
method takes as input the K attribute signatures and a dataset of images labeled with attributes, and
produces a classifier for each unseen class as output. At test time, the goal is to predict which unseen
class appears in a novel image.
In the following, we first describe the initial stage of building the attribute classifiers (Sec. 3.1).
Then we introduce a zero-shot random forest trained with attribute signatures (Sec. 3.2). Next we
explain how to augment that training procedure to account for attribute unreliability (Sec. 3.2.2) and
signature uncertainty (Sec. 3.2.3). Finally, we present an extension to few-shot learning (Sec. 3.3).
3.1
Learning the attribute vocabulary
As in any attribute-based zero-shot method [3, 8, 18, 23, 19, 7, 29], we first must train classifiers to
predict the presence or absence of each of the M attributes in novel images. Importantly, the images
used to train the attribute classifiers may come from a variety of objects/scenes and need not contain
any instances of the unseen categories. The fact that attributes are shared across category boundaries
is precisely what allows zero-shot learning.
We train one SVM per attribute, using a training set of images xi (represented with standard descriptors) with binary M -dimensional label vectors yi , where yi (m) = 1 indicates that attribute m
is present in xi . Let a
?m (x) denote the Platt probability score from the m-th such SVM applied to
test input x.
3.2
Zero-shot random forests
Next we introduce our key contribution: a random forest model for zero-shot learning.
3.2.1
Basic formulation: Signature random forest
First we define a basic random forest training algorithm for the zero-shot setting. The main idea is
to train an ensemble of decision trees using attribute signatures?not image descriptors or vectors
of attribute predictions. In the zero-shot setting, this is all the training information available. Later,
at test time, we will have an image in hand, and we will apply the trained random forest to estimate
its class posteriors.
Recall that the k-th unseen class is defined by its attribute signature Ak ? <M . We treat each such
signature as the lone positive ?exemplar? for its class, and discriminatively train random forests to
distinguish all the signatures, A1 , . . . , AK . We take a one-versus-all approach, training one forest
for each unseen class. So, when training class k, the K ? 1 other class signatures are the negatives.
2
We use ?class? and ?category? to refer to an object or scene, e.g., zebra or beach, and ?attribute? to refer
to a property, e.g., striped or sunny. ?Unseen? means we have no training images for that class.
3
For each class, we build an ensemble of decision trees in a breadth-first manner. Each tree is learned
by recursively splitting the signatures into subsets at each node, starting at the root. Let In denote
an indicator vector of length K that records which signatures appear at node n. For the root node,
all K signatures are present, so we have In = [1, . . . , 1]. Following the typical random forest
protocol [2], the training instances are recursively split according to a randomized test; it compares
one dimension of the signature against a threshold t, then propagates each one to the left child l
or right child r depending on the outcome, yielding indicator vectors Il and Ir . Specifically, if
In (k) = 1, then if Ak (m) > t, we have Ir (k) = 1. Otherwise, Ir (k) = 0. Further, Il = In ? Ir .
Thus, during training we must choose two things at each node: the query attribute m and the threshold t, represented jointly as the split (m, t). We sample a limited number of (m, t) combinations3
and choose the one that maximizes the expected information gain IGbasic :
`
?
IGbasic (m, t) = H(pIn ) ? P (Ai (m) > t|In (i) = 1) H(pIl ) + P (Ai (m) ? t|In (i) = 1) H(pIr ) (1)
?
?
kIl k1
kIr k1
= H(pIn ) ?
(2)
H(pIl ) +
H(pIr ) ,
kIn k1
kIn k1
P
where H(p) = ? i p(i) log2 p(i) is the entropy of a distribution p. The 1-norm on an indicator
vector I sums up the occurrences I(k) of each signature, which for now are binary, I(k) ? {0, 1}.
Since we are training a zero-shot forest to discriminate class k from the rest, the distribution over
class labels at node n is a length-2 vector:
P
In (k)
i6=k In (i)
p In =
,
.
(3)
kIn k1
kIn k1
We grow each tree in the forest to a fixed, maximum depth, terminating a branch prematurely if less
than 5% of training samples have reached a node on it. We learn J = 100 trees per forest.
Given a novel test image xtest , we compute its predicted attribute signature a
?(xtest ) =
[?
a1 (xtest ), . . . , a
?M (xtest )] by applying the attribute SVMs. Then, to predict the posterior for
class k, we use a
?(xtest ) to traverse to a leaf node in each tree of k?s forest. Let Pkj (`) denote
the fraction of positive training instances at a leaf node ` in tree j of the forest for class k. Then
P
P (k|?
a(xtest )) = J1 j Pkj (`), the average of the posteriors across the ensemble.
If we somehow had perfect attribute classifiers, this basic zero-shot random forest (in fact, one such
tree alone) would be sufficient. Next, we show how to adapt the training procedure defined so far to
account for their unreliability.
3.2.2
Accounting for attribute prediction unreliability
While our training ?exemplars? are the true attribute signatures for each unseen class, the test images will have only approximate estimates of the attributes they contain. We therefore augment the
zero-shot random forest to account for this unreliability during training. The main idea is to generalize the recursive splitting procedure above such that a given signature can pursue multiple paths
down the tree. Critically, those paths will be determined by the false positive/true positive rates of
the individual attribute predictors. In this way, we expand each idealized training signature into a
distribution in the predicted attribute space. Essentially, this preemptively builds in the appropriate
?cushion? of expected errors when choosing discriminative splits.
Implementing this idea requires two primary extensions to the formulation in Sec. 3.2.1: (i) we
inject attribute validation data and its associated attribute classification error statistics into the tree
formation process, and (ii) we redefine the information gain to account for the partial propagation
of training signatures. We explain each of these components in turn next.
First, in addition to signatures, at each node we maintain a set of validation data in order to gauge
the error tendencies of each attribute classifier. For the experiments in this paper (Sec 4), our method
reserves some attribute classifier training data for this purpose. Denote this set of attribute-labeled
images as DV . During random forest training, this data is recursively propagated down the tree
following each split once it is chosen. Let DV (n) ? DV denote the set of validation data inherited
at node n. At the root, DV (n) = DV .
3
With binary Ai (m), all 0 < t < 1 are equivalent in Sec 3.2.1. Selecting t becomes important in Sec 3.2.2.
4
With validation data thus injected, we can estimate the test-time receiver operating characteristic
(ROC)4 for an attribute classifier at any node in the tree. For example, the estimated false positive
rate at node n for attribute m at threshold t is FP(n, m, t) = Pn (?
am (x) > t | y(m) = 0), which is
the fraction of examples in DV (n) for which the attribute m is absent, but the SVM predicts it to be
present at threshold t. Here, y(m) denotes the m-th attribute?s label for image x.
For any node n, let In0 be a real-valued indicator vector, such that In0 (k) ? [0, 1] records the fractional
occurrence of the training signature for class k at node n. At the root node, In0 (k) = 1, ?k. For a
split (m, t) at node n, a signature Ak splits into the right and left child nodes according to its ROC
for attribute m at the operating point specified by t. In particular, we have:
Ir0 (k) = In0 (k)Pn (?
am (x) > t | y(m) = Ak (m)), and Il0 (k) = In0 (k)Pn (?
am (x) ? t | y(m) = Ak (m)),
(4)
where x ? DV (n) . When Ak (m) = 1, the probability terms are TP(n, m, t) and FN(n, m, t)
respectively; when Ak (m) = 0, they are FP(n, m, t) and TN(n, m, t). In this way, we channel all
predicted negatives to the left child node. In contrast, a naive random forest (RF) trained on signatures assumes ideal attribute classifiers and channels all ground truth negatives?i.e., true negatives
and false positives?through the left node.
To illustrate the meaning of this fractional propagation, consider a class ?elephant? known to have
the attribute ?gray?. If the ?gray? attribute classifier fires only on 60% of the ?gray? samples in the
validation set, i.e., TP=0.6, then only 0.6 fraction of the ?elephant? signature is passed on to the
positive (i.e., right) node. This process repeats through more levels until fractions of the single ?elephant? signature have reached all leaf nodes. Thus, a single class signature emulates the estimated
statistics of a full training set of class-labeled instances with attribute predictions.
We stress two things about the validation data propagation. First, the data in DV is labeled by
attributes only; it has no unseen class labels and never features in the information gain computation.
Its only role is to estimate the ROC values. Second, the recursive sub-selection of the validation data
is important to capture the dependency of TP/FP rates at higher level splits. For example, if we were
to select split (m, t) at the root, then the fractional signatures pushed to the left child must all have
A(m) < t, meaning that for a candidate split (m, s) at the left child, where s > t, the correct TP and
FP rates are both 0. This is accounted for when we use DV (n) to compute the ROC, but would not
have been, had we just used DV . Thus, our formulation properly accounts for dependencies between
attributes when selecting discriminative thresholds, an issue not addressed by existing methods for
missing [21] or probabilistically distributed features [28].
Next, we redefine the information gain. When building a zero-shot tree conscious of attribute unreliability, we choose the split maximizing the expected information gain according to the fractionally
propagated signatures (compare to Eqn. (2)):
0
kIl k1
kIr0 k1
IGzero (m, t) = H(pIn0 ) ?
H(pIl0 ) + 0 H(pIr0 ) .
(5)
kIn0 k1
kIn k1
The distribution pIz0 , z ? {l, r} is computed as in Eqn. (3). For full pseudocode and a schematic
illustration of our method, please see supp.
The discriminative splits under this criterion will be those that not only distinguish the unseen classes
but also persevere (at test time) as a strong signal in spite of the attribute classifiers? error tendencies. This means the trees will prefer both reliable attributes that are discriminative among the
classes, as well as less reliable attributes coupled with intelligently selected operating points that
remain distinctive. Furthermore, they will omit splits that, though highly discriminative in terms of
idealized signatures, were found to be ?unlearnable? among the validation data. For example, in
the extreme case, if an attribute classifier cannot distinguish positives and negatives, meaning that
TPR=FPR, then the signatures of all classes are equally likely to propagate to the left or right, i.e.,
Ir0 (k)/In0 (k) = Ir0 (j)/In0 (j) and Il0 (k)/In0 (k) = Il0 (j)/In0 (j) for all k, j, which yields an information gain of 0 in Eqn. (5) (see supp). Thus, our method, while explicitly making the best of imperfect
attribute classification, inherently prefers more learnable attributes.
4
The ROC captures the true positive (TP) vs. false positive (FP) rates (equivalently the true negative (TN)
and false negative (FN) rates) as a function of a decision value threshold.
5
The proposed approach produces unseen category classifiers with zero category-labeled images.
The attribute-labeled validation data is important to our solution?s robustness. If that data perfectly
represented the true attribute errors on images from the unseen classes (which we cannot access,
of course, because images from those classes appear only at test time), then our training procedure
would be equivalent to building a random forest on the test samples? attribute classifier outputs.
3.2.3
Accounting for class signature uncertainty
Beyond attribute classifier unreliability, our framework can also deal with another source of zeroshot uncertainty: instances of a class often deviate from class-level attribute signatures. To tackle
this, we redefine the soft indicators Ir0 and Il0 in Eqn. 4, appending a term to account for annotation
noise. Please see supp. for details.
3.3
Extending to few-shot random forests
Our approach also admits a natural extension to few-shot training. Extensions of zero-shot models
to the few-shot setting have been attempted before [31, 26, 14, 1]. In this case, we are given not only
attribute signatures, but also a dataset DT consisting of a small number of images with their class
labels. We essentially use the signatures A1 , . . . , AK as a prior for selecting good tree splits that
also satisfy the traditional training examples. The information gain on the signatures is as defined in
Sec. 3.2.2, while the information gain on the training images, for which we can compute classifier
outputs, uses the standard measure defined in Sec. 3.2.1. Using some notation shortcuts, for few-shot
training we recursively select the split that maximizes the combined information gain:
IGf ew (m, t) = ? IGzero (m, t){A1 , . . . , AK } + (1 ? ?) IGbasic (m, t){DT },
(6)
where ? controls the role of the signature-based prior. Intuitively, we can expect lower values of ? to
suffice as the size of DT increases, since with more training examples we can more precisely learn
the class?s appearance. This few-shot extension can be interpreted as a new way to learn random
forests with descriptive priors.
4
Experiments
Datasets and setup We use three datasets: (1) Animals with Attributes (AwA) [8] (M = 85
attributes, K = 10 unseen classes, 30,475 total images), (2) aPascal/aYahoo objects (aPY) [3] (M =
65, K = 12, 15,339 images) (3) SUN scene attributes (SUN) [20] (M = 102, K = 10, 14,340
images). These datasets capture a wide array of categories (animals, indoor and outdoor scenes,
household objects, etc.) and attributes (parts, affordances, habitats, shapes, materials, etc.). The
attribute-labeled images originate from 40, 20, and 707 ?seen? classes in each dataset, respectively;
we use the class labels solely to map to attribute annotations. We use the unseen class splits specified
in [9] for AwA and aPY, and randomly select the 10 unseen classes for SUN (see supp.). For all three,
we use the features provided with the datasets, which include color histograms, SIFT, PHOG, and
others (see [9, 3, 20] for details).
Following [8], we train attribute SVMs with combined ?2 -kernels, one kernel per feature channel,
and set C = 10. Our method reserves 20% of the attribute-labeled images as ROC validation data,
then pools it with the remaining 80% to train the final attribute classifiers. We stress that our method
and all baselines have access to exactly the same amount of attribute-labeled data.
We report results as mean and standard error measured over 20 random trials. Based on crossvalidation, we use tree depths of (AwA-9, aPY-6, SUN-8), and generate (#m, #t) tests per node
(AwA-(10,7), aPY-(8,2), SUN-(4,5)). When too few validation points (< 10 positives or negatives)
reach a node n, we revert to computing statistics over the full validation set DV rather than DV (n).
Baselines In addition to several state-of-the-art published results and ablated variants of our
method, we also compare to two baselines: (1) SIGNATURE RF: random forests trained on classattribute signatures as described in Sec. 3.2.1, without an attribute uncertainty model, and (2) DAP:
Direct Attribute Prediction [8, 9], which is a leading attribute-based zero-shot object recognition
method widely used in the literature [8, 3, 18, 30, 8, 23, 19, 29].5
5
We use the authors? code: http://attributes.kyb.tuebingen.mpg.de/
6
Uniform noise levels
Attribute?specific noise levels
100
100
ours
signature?RF
DAP
80
accuracy(%)
accuracy(%)
80
60
40
20
0
60
40
ours
signature?RF
DAP
20
0
0.5
1
1.5
2
2.5
3
3.5
0
4
0
noise level ?
0.5
1
1.5
2
2.5
3
3.5
4
mean noise level ?
Figure 1: Zero-shot accuracy on AwA as a function of attribute uncertainty, in controlled noise
scenarios.
Method/Dataset
DAP
SIGNATURE - RF
OURS W / O ROC PROP, SIG UNCERTAINTY
OURS W / O SIG UNCERTAINTY
OURS
OURS + TRUE ROC
AwA
40.50
36.65 ? 0.16
39.97 ? 0.09
41.88 ? 0.08
43.01 ? 0.07
54.22 ? 0.03
aPY
18.12
12.70 ? 0.38
24.25 ? 0.18
24.79 ? 0.11
26.02 ? 0.05
33.54 ? 0.07
SUN
52.50
13.20 ? 0.34
47.46 ? 0.29
56.18 ? 0.27
56.18 ? 0.27
66.65 ? 0.31
Table 1: Zero-shot learning accuracy on all three datasets. Accuracy is percentage of correct category predictions on unseen class images, ? standard error.
4.1
Zero-shot object and scene recognition
Controlled noise experiments Our approach is designed to overcome the unreliability of attribute
classifiers. To glean insight into how it works, we first test it with controlled noise in the test images?
attribute predictions. We start with hypothetical perfect attribute classifier scores a
?m (x) = Ak (m)
for x in class k, then progressively add noise to represent increasing errors in the predictions. We
examine two scenarios: (1) where all attribute classifiers are equally noisy, and (2) where the average
noise level varies per attribute. See supp. for details on the noise model.
Figure 1 shows the results using AwA. By definition, all methods are perfectly accurate with zero
noise. Once the attributes are unreliable (i.e., noise > 0), however, our approach is consistently
better. Furthermore, our gains are notably larger in the second scenario where noise levels vary
per attribute (right plot), illustrating how our approach properly favors more learnable attributes
as discussed in Sec. 3.2.2. In contrast, SIGNATURE - RF is liable to break down with even minor
imperfections in attribute prediction. These results affirm that our method benefits from both (1)
estimating and accounting for classifier noisiness and (2) avoiding uninformative attribute classifiers.
Real unreliable attributes experiments Next we present the key zero-shot results for our method
applied to three challenging datasets using over 250 real attribute classifiers. Table 1 shows the
results. Our method significantly outperforms the existing DAP method [9]. This is an important
result: DAP is today the most commonly used model for zero-shot object recognition, whether using
this exact DAP formulation [8, 23, 19, 29] or very similar non-probabilistic variants [3, 30]. Note that
our approach beats DAP despite the fact we use only 80% of the attribute-labelled images to train
attribute classifiers. This indicates that modeling how good/bad the attribute classifiers are is even
more important than having better attribute classifiers. Furthermore, this demonstrates that modeling
only the confidence of an attribute?s presence in a test image (which DAP does) is inadequate; our
idea to characterize their error tendencies during training is valuable.
Our substantial improvements over SIGNATURE - RF also confirm it is imperative to model attribute
classifier unreliability. Our gains over DAP are especially large on SUN and aPY, which have fewer
positive training samples per attribute, leading to less reliable attribute classifiers?exactly where
our method is needed most. On AwA too, we outperform DAP on 7 out of 10 categories, with
largest gains on ?giant panda?(10.2%),?whale seal?(9.4%) and ?persian cat?(7.4%), classes that are
very different from the train classes. Further, if we repeat the experiment on AwA reducing to 500
randomly chosen images for attribute training, our overall accuracy gain over DAP widens to 8 points
(28.0 ? 0.9 vs. 20.42).
7
58
50 shot (our prior)
100 shot (our prior)
200 shot (our prior)
50 shot (baseline)
100 shot (baseline)
200 shot (baseline)
56
accuracy(%)
54
52
50
Method
Lampert et al. [8]
Yu and Aloimonos [31]
Rohrbach et al. [24]
Kankuekul et al. [7]
Yu et al. [30]
OURS (named attributes)
OURS (discovered attributes)
48
46
44
42
40
38
36
0
0.2
0.4
0.6
0.8
1
1.2
lambda
Accuracy
40.5
40.0
35.7
32.7
48.3
43.0 ? 0.07
48.7 ? 0.09
(b) Zero-shot vs. state of the art
(a) Few-shot. Stars denote selected ?.
Figure 2: (a) Few-shot results. (b) Zero-shot results on AwA compared to the state of the art.
Table 1 also helps isolate the impact of two components of our method: the model of signature
uncertainty (see OURS W / O SIG UNCERTAINTY), and the recursive propagation of validation data
(see OURS W / O ROC PROP, SIG UNCERTAINTY). For the latter, we further compute TPR/FPRs
globally on the full validation dataset DV rather than for node-specific subsets DV (n). We see both
aspects contribute to our full method?s best performance (see OURS). Finally, OURS + TRUE ROC
provides an ?upper bound? on the accuracy achievable with our method for these datasets; this is the
result attainable were we to use the unseen class images as validation data DV . This also points to
an interesting direction for future work: to better model expected error rates on images with unseen
attribute combinations. Our initial attempts in this regard included focusing validation data on seen
class images with signatures most like those of the unseen classes, but the impact was negligible.
Figure 2b compares our method against all published results on AwA, using both named and discovered attributes. When using standard AwA named attributes, our method comfortably outperforms
all prior methods. Further, when we use the discovered attributes from [30], it performs comparably
to their attribute decoding method, achieving the state-of-the-art on AwA. This result was obtained
using a generalization of our method to handle the continuous attribute strength signatures of [30].
4.2
Few-shot object and scene recognition
Finally, we demonstrate our few-shot extension. Figure 2a shows the results, as a function of both the
amount of labeled training images and the prior-weighting parameter ? (cf. Sec 3.3).6 When ? = 0,
we rely solely on the training images DT ; when ? = 1, we rely solely on the attribute signatures i.e.,
zero-shot learning. As a baseline, we compare to a method that uses solely the few training images
to learn the unseen classes (dotted lines). We see the clear advantage of our attribute signature prior
for few-shot random forest training. Furthermore, we see that, as expected, the optimal ? shifts
towards 0 as more samples are added. Still, even with 200 training images in DT , the prior plays
a role (e.g., the best ? = 0.3 on blue curve). The star per curve indicates the ? value our method
selects automatically with cross-validation.
5
Conclusion
We introduced a zero-shot training approach that models unreliable attributes?both due to classifier
predictions and uncertainty in their association with unseen classes. Our results on three challenging
datasets indicate the method?s promise, and suggest that the elegance of zero-shot learning need
not be abandoned in spite of the fact that visual attributes remain very difficult to predict reliably.
Further, our idea is applicable to other uses of semantic mid-level concepts for higher tasks e.g.,
poselets for action recognition [12], discriminative mid-level patches for location recognition [27]
etc., and in domains outside computer vision. In future work, we plan to develop extensions to
accommodate inter-attribute correlations in the random forest tests and multi-label random forests
to improve scalability for many unseen classes.
Acknowledgements: We thank Christoph Lampert and Felix Yu for helpful discussions and sharing
their code. This research is supported in part by NSF IIS-1065390 and ONR ATL.
6
These are for AwA; see supp. for similar results on the other two datasets.
8
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9
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4,739 | 5,291 | Articulated Pose Estimation by a Graphical Model
with Image Dependent Pairwise Relations
Alan Yuille
University of California, Los Angeles
Los Angeles, CA 90024
[email protected]
Xianjie Chen
University of California, Los Angeles
Los Angeles, CA 90024
[email protected]
Abstract
We present a method for estimating articulated human pose from a single static
image based on a graphical model with novel pairwise relations that make adaptive use of local image measurements. More precisely, we specify a graphical
model for human pose which exploits the fact the local image measurements can
be used both to detect parts (or joints) and also to predict the spatial relationships
between them (Image Dependent Pairwise Relations). These spatial relationships
are represented by a mixture model. We use Deep Convolutional Neural Networks
(DCNNs) to learn conditional probabilities for the presence of parts and their spatial relationships within image patches. Hence our model combines the representational flexibility of graphical models with the efficiency and statistical power of
DCNNs. Our method significantly outperforms the state of the art methods on the
LSP and FLIC datasets and also performs very well on the Buffy dataset without
any training.
1
Introduction
Articulated pose estimation is one of the fundamental challenges in computer vision. Progress in
this area can immediately be applied to important vision tasks such as human tracking [2], action
recognition [25] and video analysis.
Most work on pose estimation has been based on graphical model [8, 6, 27, 1, 10, 2, 4]. The graph
nodes represent the body parts (or joints), and the edges model the pairwise relationships between
the parts. The score function, or energy, of the model contains unary terms at each node which
capture the local appearance cues of the part, and pairwise terms defined at the edges which capture
the local contextual relations between the parts. Recently, DeepPose [23] advocates modeling pose
in a holistic manner and captures the full context of all body parts in a Deep Convolutional Neural
Network (DCNN) [12] based regressor.
In this paper, we present a graphical model with image dependent pairwise relations (IDPRs). As
illustrated in Figure 1, we can reliably predict the relative positions of a part?s neighbors (as well as
the presence of the part itself) by only observing the local image patch around it. So in our model
the local image patches give input to both the unary and pairwise terms. This gives stronger pairwise
terms because data independent relations are typically either too loose to be helpful or too strict to
model highly variable poses.
Our approach requires us to have a method that can extract information about pairwise part relations,
as well as part presence, from local image patches. We require this method to be efficient and to
share features between different parts and part relationships. To do this, we train a DCNN to output
1
Lower Arm:
Upper Arm:
Elbow:
Wrist:
Figure 1: Motivation. The local image measurements around a part, e.g., in an image patch, can reliably
predict the relative positions of all its neighbors (as well as detect the part). Center Panel: The local image
patch centered at the elbow can reliably predict the relative positions of the shoulder and wrist, and the local
image patch centered at the wrist can reliably predict the relative position of the elbow. Left & Right Panels: We
define different types of pairwise spatial relationships (i.e., a mixture model) for each pair of neighboring parts.
The Left Panel shows typical spatial relationships the elbow can have with its neighbors, i.e., the shoulder and
wrist. The Right Panel shows typical spatial relationships the wrist can have with its neighbor, i.e., the elbow.
estimates for the part presence and spatial relationships which are used in our unary and pairwise
terms of our score function. The weight parameters of different terms in the model are trained using
Structured Supported Vector Machine (S-SVM) [24]. In summary, our model combines the representational flexibility of graphical models, including the ability to represent spatial relationships,
with the data driven power of DCNNs.
We perform experiments on two standard pose estimation benchmarks: LSP dataset [10] and FLIC
dataset [20]. Our method outperforms the state of the art methods by a significant margin on both
datasets. We also do cross-dataset evaluation on Buffy dataset [7] (without training on this dataset)
and obtain strong results which shows the ability of our model to generalize.
2
The Model
The Graphical Model and its Variables: We represent human pose by a graphical model G =
(V, E) where the nodes V specify the positions of the parts (or joints) and the edges E indicates
which parts are spatially related. For simplicity, we impose that the graph structure forms a K?node
tree, where K = |V|. The positions of the parts are denoted by l, where li = (x, y) specifies the
pixel location of part i, for i ? {1, . . . , K}. For each edge in the graph (i, j) ? E, we specify a
discrete set of spatial relationships indexed by tij , which corresponds to a mixture of different spatial
relationships (see Figure 1). We denote the set of spatial relationships by t = {tij , tji |(i, j) ? E}.
The image is written as I. We will define a score function F (l, t|t) as follows as a sum of unary and
pairwise terms.
Unary Terms: The unary terms give local evidence for part i ? V to lie at location li and is based
on the local image patch I(li ). They are of form:
U (li |I) = wi ?(i|I(li ); ?),
(1)
where ?(.|.; ?) is the (scalar-valued) appearance term with ? as its parameters (specified in the next
section), and wi is a scalar weight parameter.
Image Dependent Pairwise Relational (IDPR) Terms: These IDPR terms capture our intuition
that neighboring parts (i, j) ? E can roughly predict their relative spatial positions using only local
information (see Figure 1). In our model, the relative positions of parts i and j are discretized
into several types tij ? {1, . . . , Tij } (i.e., a mixture of different relationships) with corresponding
t
mean relative positions rijij plus small deformations which are modeled by the standard quadratic
2
deformation term. More formally, the pairwise relational score of each edge (i, j) ? E is given by:
R(li , lj , tij , tji |I) =
+
t
t
hwijij , ?(lj ? li ? rijij )i + wij ?(tij |I(li ); ?)
,
t
t
hwjiji , ?(li ? lj ? rjiji )i + wji ?(tji |I(lj ); ?)
(2)
where ?(?l = [?x, ?y]) = [?x ?x2 ?y ?y 2 ]| are the standard quadratic deformation features,
?(.|.; ?) is the Image Dependent Pairwise Relational (IDPR) term with ? as its parameters (specified
t
t
in the next section), and wijij , wij , wjiji , wji are the weight parameters. The notation h., .i specifies
dot product and boldface indicates a vector.
The Full Score: The full score F (l, t|I) is a function of the part locations l, the pairwise relation
types t, and the input image I. It is expressed as the sum of the unary and pairwise terms:
X
X
F (l, t|I) =
U (li |I) +
R(li , lj , tij , tji |I) + w0 ,
(3)
i?V
(i,j)?E
where w0 is a scalar weight on constant 1 (i.e., the bias term).
t
t
The model consists of three sets of parameters: the mean relative positions r = {rijij , rjiji |(i, j) ? E}
of different pairwise relation types; the parameters ? of the appearance terms and IDPR terms; and
t
t
the weight parameters w (i.e., wi , wijij , wij , wjiji , wji and w0 ). See Section 4 for the learning of
these parameters.
2.1
Image Dependent Terms and DCNNs
The appearance terms and IDPR terms depend on the image patches. In other words, a local image
patch I(li ) not only gives evidence for the presence of a part i, but also about the relationship tij
between it and its neighbors j ? N(i), where j ? N(i) if, and only if, (i, j) ? E. This requires
us to learn distribution for the state variables i, tij conditioned on the image patches I(li ). In order
to specify this distribution we must define the state space more precisely, because the number of
pairwise spatial relationships varies for different parts with different numbers of neighbors (see
Figure 1), and we need also consider the possibility that the patch does not contain a part.
We define c to be the random variable which denotes which part is present c = i for i ? {1, ..., K}
or c = 0 if no part is present (i.e., the background). We define mcN(c) to be the random variable that
determines the spatial relation types of c and takes values in McN(c) . If c = i has one neighbor j
(e.g., the wrist), then MiN(i) = {1, . . . , Tij }. If c = i has two neighbors j and k (e.g., the elbow),
then MiN(i) = {1, . . . , Tij } ? {1, . . . , Tik }. If c = 0, then we define M0N(0) = {0}.
The full space is represented as:
PK
S = ?K
c=0 {c} ? McN(c)
(4)
The size of the space is |S| = c=0 |McN(c) |. Each element in this space corresponds to a part with
all the types of its pairwise relationships, or the background.
We use DCNN [12] to learn the conditional probability distribution p(c, mcN(c) |I(li ); ?). DCNN is
suitable for this task because it is very efficient and enables us to share features. See section 4 for
more details.
We specify the appearance terms ?(.|.; ?) and IDPR terms ?(.|.; ?) in terms of p(c, mcN(c) |I(li ); ?)
by marginalization:
?(i|I(li ); ?) = log(p(c = i|I(li ); ?))
?(tij |I(li ); ?) = log(p(mij = tij |c = i, I(li ); ?))
2.2
(5)
(6)
Relationship to other models
We now briefly discuss how our method relates to standard models.
Pictorial Structure: We recover pictorial structure models [6] by only allowing one relationship
type (i.e., Tij = 1). In this case, our IDPR term conveys no information. Our model reduces to
3
standard unary and (image independent) pairwise terms. The only slight difference is that we use
DCNN to learn the unary terms instead of using HOG filters.
Mixtures-of-parts: [27] describes a model with a mixture of templates for each part, where each
template is called a ?type? of the part. The ?type? of each part is defined by its relative position with
respect to its parent. This can be obtained by restricting each part in our model to only predict the
relative position of its parent (i.e., Tij = 1, if j is not parent of i). In this case, each part is associated
with only one informative IDPR term, which can be merged with the appearance term of each part
to define different ?types? of part in [27]. Also this method does not use DCNNs.
Conditional Random Fields (CRFs): Our model is also related to the conditional random field
literature on data-dependent priors [18, 13, 15, 19]. The data-dependent priors and unary terms
are typically modeled separately in the CRFs. In this paper, we efficiently model all the image
dependent terms (i.e. unary terms and IDPR terms) together in a single DCNN by exploiting the fact
the local image measurements are reliable for predicting both the presence of a part and the pairwise
relationships of a part with its neighbors.
3
Inference
To detect the optimal configuration for each person, we search for the configurations of the locations
l and types t that maximize the score function: (l? , t? ) = arg maxl,t F (l, t|I). Since our relational
graph is a tree, this can be done efficiently via dynamic programming.
Let K(i) be the set of children of part i in the graph (K(i) = ?, if part i is a leaf), and Si (li |I) be
maximum score of the subtree rooted at part i with part i located at li . The maximum score of each
subtree can be computed as follow:
X
Si (li |I) = U (li |I) +
max (R(li , lk , tik , tki |I) + Sk (lk |I))
(7)
k?K(i)
lk ,tik ,tki
Using Equation 7, we can recursively compute the overall best score of the model, and the optimal
configuration of locations and types can be recovered by the standard backward pass of dynamic
programming.
Computation: Since our pairwise term is a quadratic function of locations, li and lj , the max
operation over lk in Equation 7 can be accelerated by using the generalized distance transforms [6].
The resulting approach is very efficient, taking O(T 2 LK) time once the image dependent terms are
computed, where T is the number of relation types, L is the total number of locations, and K is the
total number of parts in the model. This analysis assumes that all the pairwise spatial relationships
have the same number of types, i.e., Tij = Tji = T, ?(i, j) ? E.
The computation of the image dependent terms is also efficient. They are computed over all the
locations by a single DCNN. Applying DCNN in a sliding fashion is inherently efficient, since the
computations common to overlapping regions are naturally shared [22].
4
Learning
Now we consider the problem of learning the model parameters from images with labeled part
locations, which is the data available in most of the human pose datasets [17, 7, 10, 20]. We derive
type labels tij from part location annotations and adopt a supervised approach to learn the model.
Our model consists of three sets of parameters: the mean relative positions r of different pairwise
relation types; the parameters ? of the image dependent terms; and the weight parameters w. They
are learnt separately by the K-means algorithm for r, DCNN for ?, and S-SVM for w.
Mean Relative Positions and Type Labels: Given the labeled positive images {(In , ln )}N
n=1 , let
dij be the relative position from part i to its neighbor j. We cluster the relative positions over the
training set {dnij }N
n=1 to get Tij clusters (in the experiments Tij = 11 for all pairwise relations).
Each cluster corresponds to a set of instances of part i that share similar spatial relationship with
its neighbor part j. Thus we define each cluster as a pairwise relation type tij from part i to j in
t
our model, and use the center of each cluster as the mean relative position rijij associated with each
4
type. In this way, the mean relative positions of different pairwise relation types are learnt, and the
type label tnij for each training instance is derived based on its cluster index. We use K-means in our
experiments by setting K = Tij to do the clustering.
Parameters of Image Dependent Terms: After deriving type labels, each local image patch I(ln )
centered at an annotated part location is labeled with category label cn ? {1, . . . , K}, that indicates which part is present, and also the type labels mncn N(cn ) that indicate its relation types with
all its neighbors. In this way, we get a set of labelled patches {I(ln ), cn , mncn N(cn ) }KN
n=1 from positive images (each positive image provides K part patches), and also a set of background patches
{I(ln ), 0, 0} sampled from negative images.
Given the labelled part patches and background patches, we train a multi-class DCNN classifier by
standard stochastic gradient descent using softmax loss. The DCNN consists of five convolutional
layers, 2 max-pooling layers and three fully-connected layers with a final |S| dimensions softmax
output, which is defined as our conditional probability distribution, i.e., p(c, mcN(c) |I(li ); ?). The
architecture of our network is summarized in Figure 2.
Weight Parameters: Each pose in the positive image is now labeled with annotated part locations
and derived type labels: (In , ln , tn ). We use S-SVM to learn the weight parameters w. The structure
prediction problem is simplified by using 0 ? 1 loss, that is all the training examples either have all
dimensions of its labels correct or all dimensions of its labels wrong. We denote the former ones
as pos examples, and the later ones as neg examples. Since the full score function (Equation 3) is
linear in the weight parameters w, we write the optimization function as:
X
1
min hw, wi + C
max(0, 1 ? yn hw, ?(In , ln , tn )i),
(8)
w 2
n
where yn ? {1, ?1}, and ?(In , ln , tn ) is a sparse feature vector representing the n-th example
and is the concatenation of the image dependent terms (calculated from the learnt DCNN), spatial
deformation features, and constant 1. Here yn = 1 if n ? pos, and yn = ?1 if n ? neg.
5
Experiment
This section introduces the datasets, clarifies the evaluation metrics, describes our experimental
setup, presents comparative evaluation results and gives diagnostic experiments.
5.1
Datasets and Evaluation Metrics
We perform our experiments on two publicly available human pose estimation benchmarks: (i)
the ?Leeds Sports Poses? (LSP) dataset [10], that contains 1000 training and 1000 testing images
from sport activities with annotated full-body human poses; (ii) the ?Frames Labeled In Cinema?
(FLIC) dataset [20] that contains 3987 training and 1016 testing images from Hollywood movies
with annotated upper-body human poses. We follow previous work and use the observer-centric
annotations on both benchmarks. To train our models, we also use the negative training images from
the INRIAPerson dataset [3] (These images do not contain people).
We use the most popular evaluation metrics to allow comparison with previous work. Percentage
of Correct Parts (PCP) is the standard evaluation metric on several benchmarks including the LSP
dataset. However, as discussed in [27], there are several alternative interpretations of PCP that can
lead to severely different results. In our paper, we use the stricter version unless otherwise stated,
that is we evaluate only a single highest-scoring estimation result for one test image and a body part
is considered as correct if both of its segment endpoints (joints) lie within 50% of the length of the
ground-truth annotated endpoints (Each test image on the LSP dataset contains only one annotated
person). We refer to this version of PCP as strict PCP.
On the FLIC dataset, we use both strict PCP and the evaluation metric specified with it [20]: Percentage of Detected Joints (PDJ). PDJ measures the performance using a curve of the percentage
of correctly localized joints by varying localization precision threshold. The localization precision
threshold is normalized by the scale (defined as distance between left shoulder and right hip) of each
ground truth pose to make it scale invariant. There are multiple people in the FLIC images, so each
5
7
5
3
3
3
dense
+
dropout
9
128
128
x3
6
54
4096
x5
3
4
3
dense
9x
9
8
128
dense
+
dropout
conv
9x
9
9
x1
128
3
conv
9x
9x
18
36
32
3
3
conv
3
conv
+
norm
+
pool
3
5
7
conv
+
norm
+
pool
OR
4096
|S|
Figure 2: Architectures of our DCNNs. The size of input patch is 36 ? 36 pixels on the LSP dataset, and
54 ? 54 pixels on the FLIC dataset. The DCNNs consist of five convolutional layers, 2 max-pooling layers
and three fully-connected (dense) layers with a final |S| dimensions output. We use dropout, local response
normalization (norm) and overlapping pooling (pool) described in [12].
ground truth person is also annotated with a torso detection box. During evaluation, we return a
single highest-scoring estimation result for each ground truth person by restricting our neck part to
be localized inside a window defined by the provided torso box.
5.2
Implementation detail
Data Augmentation: Our DCNN has millions of parameters, while only several thousand of training images are available. In order to reduce overfitting, we augment the training data by rotating
the positive training images through 360? . These images are also horizontally flipped to double the
training images. This increases the number of training examples of body parts with different spatial
relationships with its neighbors (See the elbows along the diagonal of the Left Panel in Figure 1).
We hold out random positive images as a validation set for the DCNN training. Also the weight
parameters w are trained on this held out set to reduce overfitting to training data.
Note that our DCNN is trained using local part patches and background patches instead of the whole
images. This naturally increases the number of training examples by a factor of K (the number of
parts). Although the number of dimensions of the DCNN final output also increases linearly with the
number of parts, the number of parameters only slightly increase in the last fully-connected layer.
This is because most of the parameters are shared between different parts, and thus we can benefit
from larger K by having more training examples per parameter. In our experiments, we increase K
by adding the midway points between annotated parts, which results in 26 parts on the LSP dataset
and 18 parts on the FLIC dataset. Covering a person by more parts also reduces the distance between
neighboring parts, which is good for modeling foreshortening [27].
Graph Structure: We define a full-body graph structure for the LSP dataset, and a upper-body
graph structure for the FLIC dataset respectively. The graph connects the annotated parts and their
midways points to form a tree (See the skeletons in Figure 5 for clarification).
Settings: We use the same number of types for all pairs of neighbors for simplicity. We set it as
11 on all datasets (i.e., Tij = Tji = 11, ?(i, j) ? E), which results in 11 spatial relation types
for the parts with one neighbor (e.g., the wrist), 112 spatial relation types for the parts with two
neighbors (e.g., the elbow), and so forth (recall Figure 1). The patch size of each part is set as
36 ? 36 pixels on the LSP dataset, and 54 ? 54 pixels on the FLIC dataset, as the FLIC images are
of higher resolution. We use similar DCNN architectures on both datasets, which differ in the first
layer because of different input patch sizes. Figure 2 visualizes the architectures we used. We use
the Caffe [9] implementation of DCNN in our experiments.
5.3
Benchmark results
We show strict PCP results on the LSP dataset in Table 1, and on the FLIC dataset in Table 2. We
also show PDJ results on the FLIC dataset in Figure 3. As is shown, our method outperforms state
of the art methods by a significant margin on both datasets (see the captions for detailed analysis).
Figure 5 shows some estimation examples on the LSP and FLIC datasets.
6
Method
Ours
Torso
92.7
Head
87.8
U.arms
69.2
L.arms
55.4
U.legs
82.9
L.legs
77.0
Mean
75.0
Pishchulin et al. [16]
Ouyang et al. [14]
DeepPose* [23]
Pishchulin et al. [15]
Eichner&Ferrari [4]
Yang&Ramanan [26]
88.7
85.8
87.5
86.2
84.1
85.6
83.1
78.1
80.1
77.1
61.5
63.3
56
54.2
56.5
52.5
44.9
46.6
38
33.9
37.4
35.9
78.8
76.5
77
75.7
74.3
69.5
73.4
72.2
71
68.0
69.3
65.6
69.2
68.6
62.9
64.3
60.8
Table 1: Comparison of strict PCP results on the LSP dataset. Our method improves on all parts by a significant
margin, and outperforms the best previously published result [16] by 5.8% on average. Note that DeepPose uses
Person-Centric annotations and is trained with an extra 10,000 images.
MODEC[20]
U.arms
97.0
L.arms
86.8
84.4
52.1
Mean
91.9
68.3
Table 2: Comparison of strict PCP results on the
FLIC dataset. Our method significantly outperforms
MODEC [20].
5.4
Wrists
0.8
Percentage of Detected Joints (PDJ)
Method
Ours
Percentage of Detected Joints (PDJ)
Elbows
1
0.9
MODEC: 75.5%
DeepPose: 91.0%
Ours: 94.9%
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.05
0.1
0.15
Normalized Precision Threshold
0.2
1
0.9
0.8
MODEC: 57.9%
DeepPose: 80.9%
Ours: 92.0%
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.05
0.1
0.15
Normalized Precision Threshold
0.2
Figure 3: Comparison of PDJ curves of elbows and
wrists on the FLIC dataset. The legend shows the
PDJ numbers at the threshold of 0.2.
Diagnostic Experiments
We perform diagnostic experiments to show the cross-dataset generalization ability of our model,
and better understand the influence of each term in our model.
Cross-dataset Generalization: We directly apply the trained model on the FLIC dataset to the
official test set of Buffy dataset [7] (i.e., no training on the Buffy dataset), which also contains
upper-body human poses. The Buffy test set includes a subset of people whose upper-body can be
detected. We get the newest detection windows from [5], and compare our results to previously
published work on this subset.
Most previous work was evaluated with the official evaluation toolkit of Buffy, which uses a less
strict PCP implementation1 . We refer to this version of PCP as Buffy PCP and report it along with the
strict PCP in Table 3. We also show the PDJ curves in Figure 4. As is shown by both criterions, our
method significantly outperforms the state of the arts, which shows the good generalization ability
of our method. Also note that both DeepPose [23] and our method are trained on the FLIC dataset.
Compared with Figure 3, the margin between our method and DeepPose significantly increases in
Figure 4, which implies that our model generalizes better to the Buffy dataset.
L.arms
89.0
84.1
Mean
92.9
89.3
97.8
94.3
95.3
93.2
93.2
68.6
57.5
63.0
60.6
60.3
83.2
75.9
79.2
76.9
76.8
Elbows
Table 3: Cross-dataset PCP results on Buffy test subset. The PCP numbers are Buffy PCP unless otherwise stated. Note that our method is trained on the
FLIC dataset.
Wrists
1
0.9
0.8
Percentage of Detected Joints (PDJ)
Yang[27]
Yang[27] strict
Sapp[21]
FLPM[11]
Eichner[5]
U.arms
96.8
94.5
Percentage of Detected Joints (PDJ)
Method
Ours*
Ours* strict
Yang: 80.4%
MODEC: 77.0%
DeepPose*: 83.4%
Ours*: 93.2%
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.05
0.1
0.15
Normalized Precision Threshold
0.2
1
0.9
0.8
Yang: 57.4%
MODEC: 58.8%
DeepPose*: 64.6%
Ours*: 89.4%
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.05
0.1
0.15
Normalized Precision Threshold
0.2
Figure 4: Cross-dataset PDJ curves on Buffy test
subset. The legend shows the PDJ numbers at the
threshold of 0.2. Note that both our method and
DeepPose [23] are trained on the FLIC dataset.
1
A part is considered correctly localized if the average distance between its endpoints (joints) and groundtruth is less than 50% of the length of the ground-truth annotated endpoints.
7
Method
Unary-Only
No-IDPRs
Torso
56.3
87.4
Head
66.4
74.8
U.arms
28.9
60.7
L.arms
15.5
43.0
U.legs
50.8
73.2
L.legs
45.9
65.1
Mean
40.5
64.6
Full Model
92.7
87.8
69.2
55.4
82.9
77.0
75.0
Table 4: Diagnostic term analysis strict PCP results on the LSP dataset. The unary term alone is still not
powerful enough to get good results, even though it?s trained by a DCNN classifier. No-IDPRs method, whose
pairwise terms are not dependent on the image (see Terms Analysis in Section 5.4), can get comparable performance with the state-of-the-art, and adding IDPR terms significantly boost our final performance to 75.0%.
Terms Analysis: We design two experiments to better understand the influence of each term in
our model. In the first experiment, we use only the unary terms and thus all the parts are localized
independently. In the second experiment, we replace the IDPR terms with image independent priors
(i.e., in Equation 2, wij ?(tij |I(li ); ?) and wji ?(tji |I(lj ); ?) are replaced with scalar prior terms
t
t
bijij and bjiji respectively), and retrain the weight parameters along with the new prior terms. In this
case, our pairwise relational terms do not depend on the image, but instead is a mixture of Gaussian
deformations with image independent biases. We refer to the first experiment as Unary-Only and the
second one as No-IDPRs, short for No IDPR terms. The experiments are done on the LSP dataset
using identical appearance terms for fair comparison. We show strict PCP results in Table 4. As is
shown, all terms in our model significantly improve the performance (see the caption for detail).
6
Conclusion
We have presented a graphical model for human pose which exploits the fact the local image measurements can be used both to detect parts (or joints) and also to predict the spatial relationships
between them (Image Dependent Pairwise Relations). These spatial relationships are represented
by a mixture model over types of spatial relationships. We use DCNNs to learn conditional probabilities for the presence of parts and their spatial relationships within image patches. Hence our
model combines the representational flexibility of graphical models with the efficiency and statistical power of DCNNs. Our method outperforms the state of the art methods on the LSP and FLIC
datasets and also performs very well on the Buffy dataset without any training.
Figure 5: Results on the LSP and FLIC datasets. We show the part localization results along with the graph
skeleton we used in the model. The last row shows some failure cases, which are typically due to large foreshortening, occlusions and distractions from clothing or overlapping people.
7
Acknowledgements
This research has been supported by grants ONR MURI N000014-10-1-0933, ONR N00014-12-10883 and ARO 62250-CS. The GPUs used in this research were generously donated by the NVIDIA
Corporation.
8
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4,740 | 5,292 | Exact Post Model Selection Inference for Marginal
Screening
Jason D. Lee
Computational and Mathematical Engineering
Stanford University
Stanford, CA 94305
[email protected]
Jonathan E. Taylor
Department of Statistics
Stanford University
Stanford, CA 94305
[email protected]
Abstract
We develop a framework for post model selection inference, via marginal screening, in linear regression. At the core of this framework is a result that characterizes the exact distribution of linear functions of the response y, conditional on
the model being selected (?condition on selection" framework). This allows us
to construct valid confidence intervals and hypothesis tests for regression coefficients that account for the selection procedure. In contrast to recent work in
high-dimensional statistics, our results are exact (non-asymptotic) and require no
eigenvalue-like assumptions on the design matrix X. Furthermore, the computational cost of marginal regression, constructing confidence intervals and hypothesis testing is negligible compared to the cost of linear regression, thus making
our methods particularly suitable for extremely large datasets. Although we focus
on marginal screening to illustrate the applicability of the condition on selection
framework, this framework is much more broadly applicable. We show how to
apply the proposed framework to several other selection procedures including orthogonal matching pursuit and marginal screening+Lasso.
1
Introduction
Consider the model
yi = ?(xi ) + i , i ? N (0, ? 2 I),
(1)
p
where ?(x) is an arbitrary function, and xi ? R . Our goal is to perform inference on
(X T X)?1 X T ?, which is the best linear predictor of ?. In the classical setting of n > p , the
least squares estimator ?? = (X T X)?1 X T y is a commonly used estimator for (X T X)?1 X T ?.
Under the linear model assumption ? = X? 0 , the exact distribution of ?? is
?? ? N (? 0 , ? 2 (X T X)?1 ).
(2)
Using the normal distribution, we can test the hypothesis H0 : ?j0 = 0 and form confidence intervals
for ?j0 using the z-test.
However in the high-dimensional p > n setting, the least squares estimator is an underdetermined
problem, and the predominant approach is to perform variable selection or model selection [4].
There are many approaches to variable selection including AIC/BIC, greedy algorithms such as
forward stepwise regression, orthogonal matching pursuit, and regularization methods such as the
Lasso. The focus of this paper will be on the model selection procedure known as marginal screening, which selects the k most correlated features xj with the response y.
Marginal screening is the simplest and most commonly used of the variable selection procedures
[9, 21, 16]. Marginal screening requires only O(np) computation and is several orders of magnitude
1
faster than regularization methods such as the Lasso; it is extremely suitable for extremely large
datasets where the Lasso may be computationally intractable to apply. Furthermore, the selection
properties are comparable to the Lasso [8].
Since marginal screening utilizes the response variable y, the confidence intervals and statistical
tests based on the distribution in (2) are not valid; confidence intervals with nominal 1 ? ? coverage
may no longer cover at the advertised level:
Pr ?j0 ? C1?? (x) < 1 ? ?.
Several authors have previously noted this problem including recent work in [13, 14, 15, 2]. A major
line of work [13, 14, 15] has described the difficulty of inference post model selection: the distribution of post model selection estimates is complicated and cannot be approximated in a uniform
sense by their asymptotic counterparts.
In this paper, we describe how to form exact confidence intervals for linear regression coefficients
post model selection. We assume the model (1), and operate under the fixed design matrix X
setting. The linear regression coefficients constrained to a subset of variables S is linear in ?,
eTj (XST XS )?1 XST ? = ? T ? for some ?. We derive the conditional distribution of ? T y for any
vector ?, so we are able to form confidence intervals for regression coefficients.
In Section 2 we discuss related work on high-dimensional statistical inference, and Section 3 introduces the marginal screening algorithm and shows how z intervals may fail to have the correct
coverage properties. Section 4 and 5 show how to represent the marginal screening selection event
as constraints on y, and construct pivotal quantities for the truncated Gaussian. Section 6 uses these
tools to develop valid confidence intervals, and Section 7 evaluates the methodology on two real
datasets.
Although the focus of this paper is on marginal screening, the ?condition on selection" framework,
first proposed for the Lasso in [12], is much more general; we use marginal screening as a simple and
clean illustration of the applicability of this framework. In Section 8, we discuss several extensions
including how to apply the framework to other variable/model selection procedures and to nonlinear
regression problems. Section 8 covers 1) marginal screening+Lasso, a screen and clean procedure
that first uses marginal screening and cleans with the Lasso, and orthogonal matching pursuit (OMP).
2
Related Work
Most of the theoretical work on high-dimensional linear models focuses on consistency. Such results
establish, under restrictive assumptions on X, the Lasso ?? is close to the unknown ? 0 [19] and
selects the correct model [26, 23, 11]. We refer to the reader to [4] for a comprehensive discussion
about the theoretical properties of the Lasso.
There is also recent work on obtaining confidence intervals and significance testing for penalized Mestimators such as the Lasso. One class of methods uses sample splitting or subsampling to obtain
confidence intervals and p-values [24, 18]. In the post model selection literature, the recent work of
[2] proposed the POSI approach, a correction to the usual t-test confidence intervals by controlling
the familywise error rate for all parameters in any possible submodel. The POSI methodology is
extremely computationally intensive and currently only applicable for p ? 30.
A separate line of work establishes the asymptotic normality of a corrected estimator obtained by
? z,
?inverting? the KKT conditions [22, 25, 10]. The corrected estimator ?b has the form ?b = ?? + ???
? is an approximate inverse to the Gram matrix
where z? is a subgradient of the penalty at ?? and ?
X T X. The two main drawbacks to this approach are 1) the confidence intervals are valid only when
the M-estimator is consistent, and thus require restricted eigenvalue conditions on X, 2) obtaining
? and 3) the method is specific to regularized
? is usually much more expensive than obtaining ?,
?
estimators, and does not extend to marginal screening, forward stepwise, and other variable selection
methods.
Most closely related to our work is the ?condition on selection" framework laid out in [12] for the
Lasso. Our work extends this methodology to other variable selection methods such as marginal
screening, marginal screening followed by the Lasso (marginal screening+Lasso) and orthogonal
matching pursuit. The primary contribution of this work is the observation that many model selection
2
methods, including marginal screening and Lasso, lead to ?selection events" that can be represented
as a set of constraints on the response variable y. By conditioning on the selection event, we can
characterize the exact distribution of ? T y. This paper focuses on marginal screening, since it is
the simplest of variable selection methods, and thus the applicability of the ?condition on selection
event" framework is most transparent. However, this framework is not limited to marginal screening
and can be applied to a wide a class of model selection procedures including greedy algorithms such
as orthogonal matching pursuit. We discuss some of these possible extensions in Section 8, but leave
a thorough investigation to future work.
A remarkable aspect of our work is that we only assume X is in general position, and the test is exact,
meaning the distributional results are true even under finite samples. By extension, we do not make
any assumptions on n and p, which is unusual in high-dimensional statistics [4]. Furthermore, the
computational requirements of our test are negligible compared to computing the linear regression
coefficients.
3
Marginal Screening
Let X ? Rn?p be the design matrix, y ? Rn the response variable, and assume the model
yi = ?(xi ) + i , i ? N (0, ? 2 I). We will assume that X is in general position and has unit norm
columns. The algorithm estimates ?? via Algorithm 1. The marginal screening algorithm chooses
Algorithm 1 Marginal screening algorithm
1:
2:
3:
4:
Input: Design matrix X, response y, and model size k.
Compute |X T y|.
Let S? be the index of the k largest entries of |X T y|.
Compute ??S? = (XST? XS? )?1 XST? y
the k variables with highest absolute dot product with y, and then fits a linear model over those k
variables. We will assume k ? min(n, p). For any fixed subset of variables S, the distribution of
??S = (XST XS )?1 XST y is
??S ? N (XST XS )?1 XST ?, ? 2 (XST XS )?1
(3)
?
We will use the notation ?j?S
:= (?S? )j , where j is indexing a variable in the set S. The z-test
intervals for a regression coefficient are
C(?, j, S) := ??j?S ? ?z1??/2 (XST XS )jj , ??j?S + ?z1??/2 (XST XS )jj
(4)
?
and each interval has 1 ? ? coverage, meaning Pr ?j?S
? C(?, j, S) = 1 ? ?. However if S? is
chosen using a model selection procedure that depends on y, the distributional
result (3)no longer
?
? < 1 ? ?.
holds and the z-test intervals will not cover at the 1 ? ? level, and Pr ?j?S? ? C(?, j, S)
3.1
Failure of z-test confidence intervals
We will illustrate empirically that the z-test intervals do not cover at 1 ? ? when S? is chosen by
marginal screening in Algorithm 1. For this experiment we generated X from a standard normal
with n = 20 and p = 200. The signal vector is 2 sparse with ?10 , ?20 = SNR, y = X? 0 + , and
? N (0, 1). The confidence intervals were constructed for the k = 2 variables selected by the
marginal screening algorithm. The z-test intervals were constructed via (4) with ? = .1, and the
adjusted intervals were constructed using Algorithm 2. The results are described in Figure 1.
4
Representing the selection event
Since Equation (3) does not hold for a selected S? when the selection procedure depends on y, the
z-test intervals are not valid. Our strategy will be to understand the conditional distribution of y
3
Coverage Proportion
1
0.9
0.8
0.7
Adjusted
Z test
0.6
0.5
0.4
?1
0
log10 SNR
1
Figure 1: Plots of the coverage proportion across a range of SNR (log-scale). We see that the
coverage proportion of the z intervals can be far below the nominal level of 1 ? ? = .9, even at
SNR =5. The adjusted intervals always have coverage proportion .9. Each point represents 500
independent trials.
and contrasts (linear functions of y) ? T y, then construct inference conditional on the selection event
? We will use E(y)
?
E.
to represent a random variable, and E to represent an element of the range of
?
?
E(y). In the case of marginal screening, the selection event E(y)
corresponds to the set of selected
?
variables S and signs s:
n
o
?
E(y)
= y : sign(xTi y)xTi y > ?xTj y for all i ? S? and j ? S?c
n
o
= y : s?i xTi y > ?xTj y and s?i xTi y ? 0 for all i ? S? and j ? S?c
n
o
? s?)y ? b(S,
? s?)
= y : A(S,
(5)
? s?) and vector b(S,
? s?)1 . We will use the selection event E
? and the selected
for some matrix A(S,
?
variables/signs pair (S, s?) interchangeably since they are in bijection.
F
The space Rn is partitioned by the selection events, Rn =
(S,s) {y : A(S, s)y ?
2
b(S,
s)}
.
The
vector
y
can
be
decomposed
with
respect
to
the
partition
as follows y =
P
y
1
(A(S,
s)y
?
b(S,
s)).
S,s
Theorem 4.1. The distribution of y conditional on the selection event is a constrained Gaussian,
d
?
y|{E(y)
= E} = z {A(S, s)z ? b}, z ? N (?, ? 2 I).
Proof. The event
E is in bijection with a pair (S, s), and y is unconditionally Gaussian. Thus the
conditional y {A(S, s)y ? b(S, s)} is a Gaussian constrained to the set {A(S, s)y ? b(S, s)}.
5
Truncated Gaussian test
This section summarizes the recent tools developed in [12] for testing contrasts3 ? T y of a constrained Gaussian y. The results are stated without proof and the proofs can be found in [12]. The
primary result is a one-dimensional pivotal quantity for ? T ?. This pivot relies on characterizing the
distribution of ? T y as a truncated normal. The key step to deriving this pivot is the following lemma:
Lemma 5.1. The conditioning set can be rewritten in terms of ? T y as follows:
{Ay ? b} = {V ? (y) ? ? T y ? V + (y), V 0 (y) ? 0}
1
b can be taken to be 0 for marginal screening, but this extra generality is needed for other model selection
methods.
2
It is also possible to use a coarser partition, where each element of the partition only corresponds to a
subset of variables S. See [12] for details.
3
A contrast of y is a linear function of the form ? T y.
4
where
?=
A??
? T ??
(6)
bj ? (Ay)j + ?j ? T y
j: ?j <0
?j
(7)
bj ? (Ay)j + ?j ? T y
.
j: ?j >0
?j
(8)
V ? = V ? (y) = max
V + = V + (y) = min
V 0 = V 0 (y) = min bj ? (Ay)j
(9)
j: ?j =0
Moreover, (V + , V ? , V 0 ) are independent of ? T y.
[a,b]
Theorem 5.2. Let ?(x) denote the CDF of a N (0, 1) random variable, and let F?,?2 denote the
CDF of T N (?, ?, a, b), i.e.:
?((x ? ?)/?) ? ?((a ? ?)/?)
[a,b]
F?,?2 (x) =
.
(10)
?((b ? ?)/?) ? ?((a ? ?)/?)
[V ? ,V + ]
Then F?T ?, ?T ?? (? T y) is a pivotal quantity, conditional on {Ay ? b}:
[V ? ,V + ]
F?T ?, ?T ?? (? T y) {Ay ? b} ? Unif(0, 1)
where V
?
(11)
+
and V are defined in (7) and (8).
300
1
250
0.8
200
0.6
150
0.4
100
0.2
50
0
0
0.2
0.4
0.6
0.8
0
1
empirical cdf
Unif(0,1) cdf
0
0.2
0.4
0.6
0.8
1
[V ? ,V + ]
Figure 2: Histogram and qq plot of F?T ?, ?T ?? (? T y) where y is a constrained Gaussian. The
distribution is very close to Unif(0, 1), which is in agreement with Theorem 5.2.
6
Inference for marginal screening
In this section, we apply the theory summarized in Sections 4 and 5 to marginal screening. In
particular, we will construct confidence intervals for the selected variables.
To summarize the developments so far, recall that our model (1) says that y ? N (?, ? 2 I).
?
The distribution of interest is y|{E(y)
= E}, and by Theorem 4.1, this is equivalent to
y|{A(S, s)z ? b(S, s)}, where y ? N (?, ? 2 I). By applying Theorem 5.2, we obtain the pivotal
quantity
[V ? ,V + ]
?
F?T ?, ?2 ||?||2 (? T y) {E(y)
= E} ? Unif(0, 1)
(12)
2
for any ?, where V ? and V + are defined in (7) and (8).
In this section, we describe how to form confidence intervals for the components of ?S?? =
(XST? XS? )?1 XST? ?. The best linear predictor of ? that uses only the selected variables is ?S?? , and
?? ? = (X T X ? )?1 X T y is an unbiased estimate of ? ? . If we choose
S
?
S
S
?
S
?j =
?
S
T
?1 T
((XS? XS? ) XS? ej )T ,
5
(13)
?
th
then ?jT ? = ?j?
? , so the above framework provides a method for inference about the j variable in
S
?
the model S.
6.1
Confidence intervals for selected variables
?
Next, we discuss how to obtain confidence intervals for ?j?
The standard way
?.
S
to
obtain an interval is to invert a pivotal quantity [5].
In other words, since
[V ? ,V + ]
? = E} = ? one can define a (1 ? ?) (conditional)
Pr ?2 ? F? ? , ?2 ||?j ||2 (?jT y) ? 1 ? ?2 {E
?
j?S
?
confidence interval for ?j,
? as
E
n
?
?o
[V ? ,V + ]
.
x : ? Fx, ?2 ||?j ||2 (?jT y) ? 1 ?
2
2
(14)
In fact, F is monotone decreasing in x, so to find its endpoints, one need only solve for the root of a
smooth one-dimensional function. The monotonicity is a consequence of the fact that the truncated
Gaussian distribution is a natural exponential family and hence has monotone likelihood ratio in ?
[17].
We now formalize the above observations in the following result, an immediate consequence of
Theorem 5.2.
? s?)) and U? = U? (?j , (S,
? s?))
Corollary 6.1. Let ?j be defined as in (13), and let L? = L? (?j , (S,
be the (unique) values satisfying
?
?
[V ? ,V + ]
[V ? ,V + ]
FL? , ?2 ||?j ||2 (?jT y) = 1 ?
FU? , ?2 ||?j ||2 (?jT y) =
(15)
2
2
?
?
Then [L? , U? ] is a (1 ? ?) confidence interval for ?j?
? , conditional on E:
S
?
?
(16)
P ?j?
? ? [L? , U? ] {E = E} = 1 ? ?.
S
?
?
Proof. The confidence region of ?j?
? is the set of ?j such that the test of H0 : ?j?S
? accepts at the
S
[V ? ,V + ]
1 ? ? level. The function Fx, ?2 ||?j ||2 (?jT y) is monotone in x, so solving for L? and U? identify the
most extreme values where H0 is still accepted. This gives a 1 ? ? confidence interval.
Next, we establish the unconditional coverage of the constructed confidence intervals and the false
coverage rate (FCR) control [1].
?
Corollary 6.2. For each j ? S,
?
j
j
Pr ?j?
(17)
? ? [L? , U? ] = 1 ? ?.
S
j j
Furthermore, the FCR of the intervals [L? , U? ] j?E? is ?.
Proof. By (16), the conditional coverage of the confidence intervals are 1 ? ?. The coverage holds
?
for every element of the partition {E(y)
= E}, so
X
? = E} Pr(E
? = E)
Pr ? ? ? ? [L? , U? ] {E
Pr ? ? ? ? [Lj , U j ] =
j?S
?
?
j?S
E
X
? = E) = 1 ? ?.
=
(1 ? ?) Pr(E
E
Remark 6.3. We would like to emphasize that the previous Corollary shows that the constructed
?
confidence intervals are unconditionally valid. The conditioning on the selection event E(y)
=E
was only for mathematical convenience to work out the exact pivot. Unlike standard z-test intervals,
?
the coverage target, ?j?
? , and the interval [L? , U? ] are random. In a typical confidence interval
S
only the interval is random; however in the post-selection inference setting, the selected model is
random, so both the interval and the target are necessarily random [2].
We summarize the algorithm for selecting and constructing confidence intervals below.
6
Algorithm 2 Confidence intervals for selected variables
1: Input: Design matrix X, response y, model size k.
2: Use Algorithm 1 to select a subset of variables S? and signs s? = sign(X T? y).
? s?) and b = b(S,
? s?) using (5). Let ?j = (X T )? ej .
3: Let A = A(S,
?
S
S
4: Solve for Lj? and U?j using Equation (15) where V ? and V + are computed via (7) and (8) using
the A, b, and ?j previously defined.
?
5: Output: Return the intervals [Lj? , U?j ] for j ? S.
7
Experiments
In Figure 1, we have already seen that the confidence intervals constructed using Algorithm 2 have
exactly 1 ? ? coverage proportion. In this section, we perform two experiments on real data where
the linear model does not hold, the noise is not Gaussian, and the noise variance is unknown.
7.1
Diabetes dataset
The diabetes dataset contains n = 442 diabetes patients measured on p = 10 baseline variables [6].
The baseline variables are age, sex, body mass index, average blood pressure, and six blood serum
measurements, and the response y is a quantitative measure of disease progression measured one
yk
year after the baseline. Since the noise variance ? 2 is unknown, we estimate it by ? 2 = ky??
n?p ,
Coverage Proportion
1
0.8
0.6
Z?test
Adjusted
Nominal
0.4
0.2
0.6
0.8
1??
1
Figure 3: Plot of 1 ? ? vs the coverage proportion for diabetes dataset. The nominal curve is the
line y = x. The coverage proportion of the adjusted intervals agree with the nominal coverage level,
but the z-test coverage proportion is strictly below the nominal level. The adjusted intervals perform
well, despite the noise being non-Gaussian, and ? 2 unknown.
where y? = X ?? and ?? = (X T X)?1 X T y. For each trial we generated new responses y?i = X ?? + ?,
and ? is bootstrapped from the residuals ri = yi ? y?i . We used marginal screening to select k = 2
variables, and then fit linear regression on the selected variables. The adjusted confidence intervals
were constructed using Algorithm 2 with the estimated ? 2 . The nominal coverage level is varied
across 1 ? ? ? {.5, .6, .7, .8, .9, .95, .99}. From Figure 3, we observe that the adjusted intervals
always cover at the nominal level, whereas the z-test is always below. The experiment was repeated
2000 times.
7.2
Riboflavin dataset
Our second data example is a high-throughput genomic dataset about riboflavin (vitamin B2) production rate [3]. There are p = 4088 variables which measure the log expression level of different
genes, a single real-valued response y which measures the logarithm of the riboflavin production
rate, and n = 71 samples. We first estimate ? 2 by using cross-validation [20], and apply marginal
screening with k = 30, as chosen in [3]. We then use Algorithm 2 to identify genes significant at
7
? = 10%. The genes identified as significant were YCKE_at, YOAB_at, and YURQ_at. After
using Bonferroni to control for FWER, we found YOAB_at remained significant.
8
Extensions
The purpose of this section is to illustrate the broad applicability of the condition on selection framework. For expository purposes, we focused the paper on marginal screening where the framework
is particularly easy to understand. In the rest of this section, we show how to apply the framework
to marginal screening+Lasso, and orthogonal matching pursuit. This is a non-exhaustive list of
selection procedures where the condition on selection framework is applicable, but we hope this incomplete list emphasizes the ease of constructing tests and confidence intervals post-model selection
via conditioning.
8.1
Marginal screening + Lasso
The marginal screening+Lasso procedure was introduced in [7] as a variable selection method for
k
the ultra-high dimensional setting of p = O(en ). Fan et al. [7] recommend applying the marginal
screening algorithm with k = n ? 1, followed by the Lasso on the selected variables. This is a
two-stage procedure, so to properly account for the selection we must encode the selection event
of marginal screening followed by Lasso. This can be done by representing the two stage selection
as a single event. Let (S?m , s?m ) be the variables and signs selected by marginal screening, and the
(S?L , z?L ) be the variables and signs selected by Lasso [12]. In Proposition 2.2 of [12], it is shown
how to encode the Lasso selection event (S?L , z?L ) as a set of constraints {AL y ? bL } 4 , and in
Section 4 we showed how to encode the marginal screening selection event (S?m , s?m ) as a set of
constraints {Am y ? bm }. Thus the selection event of marginal screening+Lasso can be encoded
as {AL y ? bL , Am y ? bm }. Using these constraints, the hypothesis test and confidence intervals
described in Algorithm 2 are valid for marginal screening+Lasso.
8.2
Orthogonal Matching Pursuit
Orthogonal matching pursuit (OMP) is a commonly used variable selection method. At each iteration, OMP selects the variable most correlated with the residual r, and then recomputes the residual
using the residual of least squares using the selected variables. Similar to Section 4, we can represent
the OMP selection event as a set of linear constraints on y.
?
E(y)
= y : sign(xTpi ri )xTpi ri > ?xTj ri , for all j 6= pi and all i ? [k]
= {y : s?i xTpi (I ? XS?i?1 XS??
i?1
s?i xTpi (I ? XS?i?1 XS??
i?1
)y > ?xTj (I ? XS?i?1 XS??
)y and
i?1
)y > 0, for all j 6= pi , and all i ? [k] }
The selection event encodes that OMP selected a certain variable and the sign of the correlation of
that variable with the residual, at steps 1 to k. The primary difference between the OMP selection
event and the marginal screening selection event is that the OMP event also describes the order at
which the variables were chosen.
9
Conclusion
Due to the increasing size of datasets, marginal screening has become an important method for
fast variable selection. However, the standard hypothesis tests and confidence intervals used in
linear regression are invalid after using marginal screening to select important variables. We have
described a method to form confidence intervals after marginal screening. The condition on selection
framework is not restricted to marginal screening, and also applies to OMP and marginal screening
+ Lasso. The supplementary material also discusses the framework applied to non-negative least
squares.
4
The Lasso selection event is with respect to the Lasso optimization problem after marginal screening.
8
References
[1] Yoav Benjamini and Daniel Yekutieli. False discovery rate?adjusted multiple confidence intervals for
selected parameters. Journal of the American Statistical Association, 100(469):71?81, 2005.
[2] Richard Berk, Lawrence Brown, Andreas Buja, Kai Zhang, and Linda Zhao. Valid post-selection inference. Annals of Statistics, 41(2):802?837, 2013.
[3] Peter B?hlmann, Markus Kalisch, and Lukas Meier. High-dimensional statistics with a view toward
applications in biology. Statistics, 1, 2014.
[4] Peter Lukas B?hlmann and Sara A van de Geer. Statistics for High-dimensional Data. Springer, 2011.
[5] George Casella and Roger L Berger. Statistical inference, volume 70. Duxbury Press Belmont, CA, 1990.
[6] Bradley Efron, Trevor Hastie, Iain Johnstone, and Robert Tibshirani. Least angle regression. The Annals
of statistics, 32(2):407?499, 2004.
[7] Jianqing Fan and Jinchi Lv. Sure independence screening for ultrahigh dimensional feature space. Journal
of the Royal Statistical Society: Series B (Statistical Methodology), 70(5):849?911, 2008.
[8] Christopher R Genovese, Jiashun Jin, Larry Wasserman, and Zhigang Yao. A comparison of the lasso
and marginal regression. The Journal of Machine Learning Research, 98888:2107?2143, 2012.
[9] Isabelle Guyon and Andr? Elisseeff. An introduction to variable and feature selection. The Journal of
Machine Learning Research, 3:1157?1182, 2003.
[10] Adel Javanmard and Andrea Montanari. Confidence intervals and hypothesis testing for high-dimensional
regression. arXiv preprint arXiv:1306.3171, 2013.
[11] Jason Lee, Yuekai Sun, and Jonathan E Taylor. On model selection consistency of penalized m-estimators:
a geometric theory. In Advances in Neural Information Processing Systems, pages 342?350, 2013.
[12] Jason D Lee, Dennis L Sun, Yuekai Sun, and Jonathan E Taylor. Exact inference after model selection
via the lasso. arXiv preprint arXiv:1311.6238, 2013.
[13] Hannes Leeb and Benedikt M P?tscher. The finite-sample distribution of post-model-selection estimators
and uniform versus nonuniform approximations. Econometric Theory, 19(1):100?142, 2003.
[14] Hannes Leeb and Benedikt M P?tscher. Model selection and inference: Facts and fiction. Econometric
Theory, 21(1):21?59, 2005.
[15] Hannes Leeb and Benedikt M P?tscher. Can one estimate the conditional distribution of post-modelselection estimators? The Annals of Statistics, pages 2554?2591, 2006.
[16] Jeff Leek.
Prediction:
the lasso vs just using the top 10 predictors.
http://simplystatistics.tumblr.com/post/18132467723/
prediction-the-lasso-vs-just-using-the-top-10.
[17] Erich L. Lehmann and Joseph P. Romano. Testing Statistical Hypotheses. Springer, 3 edition, 2005.
[18] Nicolai Meinshausen, Lukas Meier, and Peter B?hlmann. P-values for high-dimensional regression. Journal of the American Statistical Association, 104(488), 2009.
[19] Sahand N Negahban, Pradeep Ravikumar, Martin J Wainwright, and Bin Yu. A unified framework
for high-dimensional analysis of m-estimators with decomposable regularizers. Statistical Science,
27(4):538?557, 2012.
[20] Stephen Reid, Robert Tibshirani, and Jerome Friedman. A study of error variance estimation in lasso
regression. arXiv preprint arXiv:1311.5274, 2013.
[21] Virginia Goss Tusher, Robert Tibshirani, and Gilbert Chu. Significance analysis of microarrays applied
to the ionizing radiation response. Proceedings of the National Academy of Sciences, 98(9):5116?5121,
2001.
[22] Sara van de Geer, Peter B?hlmann, and Ya?acov Ritov. On asymptotically optimal confidence regions and
tests for high-dimensional models. arXiv preprint arXiv:1303.0518, 2013.
[23] M.J. Wainwright. Sharp thresholds for high-dimensional and noisy sparsity recovery using `1 -constrained
quadratic programming (lasso). 55(5):2183?2202, 2009.
[24] Larry Wasserman and Kathryn Roeder.
37(5A):2178, 2009.
High dimensional variable selection.
Annals of statistics,
[25] Cun-Hui Zhang and S Zhang. Confidence intervals for low-dimensional parameters with highdimensional data. arXiv preprint arXiv:1110.2563, 2011.
[26] P. Zhao and B. Yu. On model selection consistency of lasso. 7:2541?2563, 2006.
9
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4,741 | 5,293 | On Iterative Hard Thresholding Methods for
High-dimensional M-Estimation
Prateek Jain?
Ambuj Tewari?
Purushottam Kar?
Microsoft Research, INDIA
?
University of Michigan, Ann Arbor, USA
{prajain,t-purkar}@microsoft.com, [email protected]
?
Abstract
The use of M-estimators in generalized linear regression models in high dimensional settings requires risk minimization with hard L0 constraints. Of the known
methods, the class of projected gradient descent (also known as iterative hard
thresholding (IHT)) methods is known to offer the fastest and most scalable solutions. However, the current state-of-the-art is only able to analyze these methods
in extremely restrictive settings which do not hold in high dimensional statistical models. In this work we bridge this gap by providing the first analysis for
IHT-style methods in the high dimensional statistical setting. Our bounds are tight
and match known minimax lower bounds. Our results rely on a general analysis
framework that enables us to analyze several popular hard thresholding style algorithms (such as HTP, CoSaMP, SP) in the high dimensional regression setting.
Finally, we extend our analysis to the problem of low-rank matrix recovery.
1
Introduction
Modern statistical estimation is routinely faced with real world problems where the number of parameters p handily outnumbers the number of observations n. In general, consistent estimation of
parameters is not possible in such a situation. Consequently, a rich line of work has focused on
models that satisfy special structural assumptions such as sparsity or low-rank structure. Under
these assumptions, several works (for example, see [1, 2, 3, 4, 5]) have established that consistent
estimation is information theoretically possible in the ?n p? regime as well.
The question of efficient estimation, however, is faced with feasibility issues since consistent estimation routines often end-up solving NP-hard problems. Examples include sparse regression which
requires loss minimization with sparsity constraints and low-rank regression which requires dealing
with rank constraints which are not efficiently solvable in general [6].
Interestingly, recent works have demonstrated that these hardness results can be avoided by assuming
certain natural conditions over the loss function being minimized such as restricted strong convexity
(RSC) and restricted strong smoothness (RSS). The estimation routines proposed in these works
typically make use of convex relaxations [5] or greedy methods [7, 8, 9] which do not suffer from
infeasibility issues.
Despite this, certain limitations have precluded widespread use of these techniques. Convex
relaxation-based methods typically suffer from slow rates as they solve non-smooth optimization
problems apart from being hard to analyze in terms of global guarantees. Greedy methods, on the
other hand, are slow in situations with non-negligible sparsity or relatively high rank, owing to their
incremental approach of adding/removing individual support elements.
Instead, the methods of choice for practical applications are actually projected gradient (PGD) methods, also referred to as iterative hard thresholding (IHT) methods. These methods directly project
1
the gradient descent update onto the underlying (non-convex) feasible set. This projection can be
performed efficiently for several interesting structures such as sparsity and low rank. However, traditional PGD analyses for convex problems viz. [10] do not apply to these techniques due to the
non-convex structure of the problem.
An exception to this is the recent work [11] that demonstrates that PGD with non-convex regularization can offer consistent estimates for certain high-dimensional problems. However, the work in [11]
is only able to analyze penalties such as SCAD, MCP and capped L1 . Moreover, their framework
cannot handle commonly used penalties such as L0 or low-rank constraints.
Insufficiency of RIP based Guarantees for M-estimation. As noted above, PGD/IHT-style methods have been very popular in literature for sparse recovery and several algorithms including Iterative
Hard Thresholding (IHT) [12] or GraDeS [13], Hard Thresholding Pursuit (HTP) [14], CoSaMP
[15], Subspace Pursuit (SP) [16], and OMPR(`) [17] have been proposed. However, the analysis
of these algorithms has traditionally been restricted to settings that satisfy the Restricted Isometry
property (RIP) or incoherence property. As the discussion below demonstrates, this renders these
analyses inaccessible to high-dimensional statistical estimation problems.
All existing results analyzing these methods require the condition number of the loss function, restricted to sparse vectors, to be smaller than a universal constant. The best known such constant is
due to the work of [17] that requires a bound on the RIP constant ?2k ? 0.5 (or equivalently a bound
1+?2k
1??2k ? 3 on the condition number). In contrast, real-life high dimensional statistical settings,
wherein pairs of variables can be arbitrarily correlated, routinely require estimation methods to performunder arbitrarily
large condition numbers. In particular if two variates have a covariance matrix
1
1?
like
, then the restricted condition number (on a support set of size just 2) of the sam1?
1
ple matrix cannot be brought down below 1/ even with infinitely many samples. In particular when
< 1/6, none of the existing results for hard thresholding methods offer any guarantees. Moreover,
most of these analyses consider only the least squares objective. Although recent attempts have
been made to extend this to general differentiable objectives [18, 19], the results continue to require
that the restricted condition number be less than a universal constant and remain unsatisfactory in a
statistical setting.
Overview of Results. Our main contribution in this work is an analysis of PGD/IHT-style methods
in statistical settings. Our bounds are tight, achieve known minmax lower bounds [20], and hold
for arbitrary differentiable, possibly even non-convex functions. Our results hold even when the
underlying condition number is arbitrarily large and only require the function to satisfy RSC/RSS
conditions. In particular, this reveals that these iterative methods are indeed applicable to statistical
settings, a result that escaped all previous works.
Our first result shows that the PGD/IHT methods achieve global convergence if used with a relaxed
projection step. More formally, if the optimal parameter is s? -sparse and the problem satisfies
RSC and RSS constraints ? and L respectively (see Section 2), then PGD methods offer global
convergence so long as they employ projection to an s-sparse set where s ? 4(L/?)2 s? . This
gives convergence rates that are identical to those of convex relaxation and greedy methods for the
Gaussian sparse linear model. We then move to a family of efficient ?fully corrective? methods and
show as before, that for arbitrary functions satisfying the RSC/RSS properties, these methods offer
global convergence.
Next, we show that these results allow PGD-style methods to offer global convergence in a variety
of statistical estimation problems such as sparse linear regression and low rank matrix regression.
Our results effortlessly extend to the noisy setting as a corollary and give bounds similar to those of
[21] that relies on solving an L1 regularized problem.
Our proofs are able to exploit that even though hard-thresholding is not the prox-operator for any
convex prox function, it still provides strong contraction when projection is performed onto sets of
sparsity s s? . This crucial observation allows us to provide the first unified analysis for hard
thresholding based gradient descent algorithms. Our empirical results confirm our predictions with
respect to the recovery properties of IHT-style algorithms on badly-conditioned sparse recovery
problems, as well as demonstrate that these methods can be orders of magnitudes faster than their
L1 and greedy counterparts.
2
Organization. Section 2 sets the notation and the problem statement. Section 3 introduces the
PGD/IHT algorithm that we study and proves that the method guarantees recovery assuming the
RSC/RSS property. We also generalize our guarantees to the problem of low-rank matrix regression.
Section 4 then provides crisp sample complexity bounds and statistical guarantees for the PGD/IHT
estimators. Section 5 extends our analysis to a broad family of compressive sensing algorithms that
include the so-called fully-corrective hard thresholding methods and provide similar results for them
as well. We present some empirical results in Section 6 and conclude in Section 7.
2
Problem Setup and Notations
High-dimensional Sparse Estimation. Given data points X = [X1 , . . . , Xn ]T , where Xi ? Rp ,
and the target Y = [Y1 , . . . , Yn ]T , where Yi ? R, the goal is to compute an s? -sparse ? ? s.t.,
? ? = arg
min
?,k?k0 ?s?
f (?).
(1)
P
Typically, f can be thought of as an empirical risk function i.e. f (?) = n1 i `(hXi , ?i, Yi ) for some
loss function ` (see examples in Section 4). However, for our analysis of PGD and other algorithms,
we need not assume any other property of f other than differentiability and the following two RSC
and RSS properties.
Definition 1 (RSC Property). A differentiable function f : Rp ? R is said to satisfy restricted
strong convexity (RSC) at sparsity level s = s1 + s2 with strong convexity constraint ?s if the
following holds for all ?1 , ?2 s.t. k?1 k0 ? s1 and k?2 k0 ? s2 :
?s
f (?1 ) ? f (?2 ) ? h?1 ? ?2 , ?? f (?2 )i +
k?1 ? ?2 k22 .
2
Definition 2 (RSS Property). A differentiable function f : Rp ? R is said to satisfy restricted
strong smoothness (RSS) at sparsity level s = s1 + s2 with strong convexity constraint Ls if the
following holds for all ?1 , ?2 s.t. k?1 k0 ? s1 and k?2 k0 ? s2 :
f (?1 ) ? f (?2 ) ? h?1 ? ?2 , ?? f (?2 )i +
Ls
k?1 ? ?2 k22 .
2
Low-rank Matrix Regression. Low-rank matrix regression is similar to sparse estimation as presented above except that each data point is now a matrix i.e. Xi ? Rp1 ?p2 , the goal being to estimate
a low-rank matrix W ? Rp1 ?p2 that minimizes the empirical loss function on the given data.
W ? = arg
min
W,rank(W )?r
f (W ).
(2)
For this problem the RSC and RSS properties for f are defined similarly as in Definition 1, 2 except
that the L0 norm is replaced by the rank function.
3
Iterative Hard-thresholding Method
In this section we study the popular projected gradient descent (a.k.a iterative hard thresholding)
method for the case of the feasible set being the set of sparse vectors (see Algorithm 1 for pseudocode). The projection operator Ps (z), can be implemented efficiently in this case by projecting
z onto the set of s-sparse vectors by selecting the s largest elements (in magnitude) of z. The standard projection property implies that kPs (z) ? zk22 ? k? 0 ? zk22 for all k? 0 k0 ? s. However, it
turns out that we can prove a significantly stronger property of hard thresholding for the case when
k? 0 k0 ? s? and s? s. This property is key to analysing IHT and is formalized below.
Lemma 1. For any index set I, any z ? RI , let ? = Ps (z). Then for any ? ? ? RI such that
k? ? k0 ? s? , we have
|I| ? s ?
k? ? zk22 .
k? ? zk22 ?
|I| ? s?
See Appendix A for a detailed proof.
Our analysis combines the above observation with the RSC/RSS properties of f to provide geometric
convergence rates for the IHT procedure below.
3
Algorithm 1 Iterative Hard-thresholding
1: Input: Function f with gradient oracle, sparsity level s, step-size ?
2: ? 1 = 0, t = 1
3: while not converged do
4:
? t+1 = Ps (? t ? ??? f (? t )), t = t + 1
5: end while
6: Output: ? t
Theorem 1. Let f have RSC and RSS parameters given by L2s+s? (f ) = L and ?2s+s? (f ) = ?
2 ?
2
. Also let ? ? =
s and ? = 3L
respectively. Let Algorithm 1 be invoked with f , s ? 32 L
?
0
f (? )
arg min?,k?k0 ?s? f (?). Then, the ? -th iterate of Algorithm 1, for ? = O( L
? ? log( )) satisfies:
f (? ? ) ? f (? ? ) ? .
Proof. (Sketch) Let S t = supp(? t ), S ? = supp(? ? ), S t+1 = supp(? t+1 ) and I t = S ? ?S t ?S t+1 .
Using the RSS property and the fact that supp(? t ) ? I t and supp(? t+1 ) ? I t , we have:
L
f (? t+1 ) ? f (? t ) ? h? t+1 ? ? t , g t i + k? t+1 ? ? t k22 ,
2
L t+1
2
1
= k?I t ? ?It t +
? g t t k2 ?
kg t t k2 ,
2
3L I 2 2L I 2
?1 L
|I t | ? s
1
1
? ? t
? k?I?t ? ?It t + ? gIt t k22 ?
(kgIt t \(S t ?S ? ) k22 + kgSt t ?S ? k22 ),
?
2 |I | ? s
L
2L
(3)
where ?1 follows from an application of Lemma 1 with I = I t and the Pythagoras theorem. The
above equation has three critical terms. The first term can be bounded using the RSS condition.
1
kgSt t ?S ? k22 bounds the third term
Using f (? t ) ? f (? ? ) ? hgSt t ?S ? , ? t ? ? ? i ? ?2 k? t ? ? ? k22 ? 2?
in (3). The second term is more interesting as in general elements of gSt ? can be arbitrarily small.
However, elements of gIt t \(S t ?S ? ) should be at least as large as gSt ? \S t+1 as they are selected by
hard-thresholding. Combining this insight with bounds for gSt ? \S t+1 and with (3), we obtain the
theorem. See Appendix A for a detailed proof.
3.1
Low-rank Matrix Regression
We now generalize our previous analysis to a projected gradient descent (PGD) method for low-rank
matrix regression. Formally, we study the following problem:
min f (W ), s.t., rank(W ) ? s.
(4)
W
The hard-thresholding projection step for low-rank matrices can be solved using SVD i.e.
P Ms (W ) = Us ?s VsT ,
where W = U ?V is the singular value decomposition of W . Us , Vs are the top-s singular vectors
(left and right, respectively) of W and ?s is the diagonal matrix of the top-s singular values of W .
To proceed, we first note a property of the above projection similar to Lemma 1.
Lemma 2. Let W ? Rp1 ?p2 be a rank-|I t | matrix and let p1 ? p2 . Then for any rank-s? matrix
W ? ? Rp1 ?p2 we have
|I t | ? s
kP Ms (W ) ? W k2F ? t
kW ? ? W k2F .
(5)
|I | ? s?
T
Proof. Let W = U ?V T be the singular value decomposition of W . Now, kP Ms (W ) ? W k2F =
P|I t |
2
2
i=s+1 ?i = kPs (diag(?)) ? diag(?)k2 , where ?1 ? ? ? ? ? ?|I t | ? 0 are the singular values of
W . Using Lemma 1, we get:
|I t | ? s
|I t | ? s
?
2
kP Ms (W ) ? W k2F ? t
k?
?
diag(?)k
?
kW ? ? W k2F ,
(6)
2
|I | ? s?
|I t | ? s?
P
where the last step uses the von Neumann?s trace inequality (T r(A ? B) ? i ?i (A)?i (B)).
4
The following result for low-rank matrix regression immediately follows from Lemma 4.
Theorem 2. Let f have RSC and RSS parameters given by L2s+s? (f ) = L and ?2s+s? (f ) = ?.
Replace the projection operator Ps in Algorithm 1 with its matrix counterpart P Ms as defined in (5).
2 ?
2
s , ? = 3L
. Also let W ? = arg minW,rank(W )?s? f (W ).
Suppose we invoke it with f, s ? 32 L
?
0
f (W )
Then the ? -th iterate of Algorithm 1, for ? = O( L
) satisfies:
? ? log(
f (W ? ) ? f (W ? ) ? .
Proof. A proof progression similar to that of Theorem 1 suffices. The only changes that need to be
made are: firstly Lemma 2 has to be invoked in place of Lemma 1. Secondly, in place of considering vectors restricted to a subset of coordinates viz. ?S , gIt , we would need to consider matrices
restricted to subspaces i.e. WS = US UST W where US is a set of singular vectors spanning the
range-space of S.
4
High Dimensional Statistical Estimation
This section elaborates on how the results of the previous section can be used to give guarantees for
IHT-style techniques in a variety of statistical estimation problems. We will first present a generic
convergence result and then specialize it to various settings. Suppose we have a sample of data
points Z1:n and a loss function L(?; Z1:n ) that depends on a parameter ? and the sample. Then we
can show the following result. (See Appendix B for a proof.)
Theorem 3. Let ?? be any s? -sparse vector. Suppose L(?; Z1:n ) is differentiable and satisfies RSC and RSS at sparsity level s + s? with parameters ?s+s? and Ls+s? respectively, for
2
L2s+s?
s ? 32 ?2s+s
s? . Let ? ? be the ? -th iterate of Algorithm 1 for ? chosen as in Theorem 1
?
and ? be the function value error incurred by Algorithm 1. Then we have
s
?
? Z1:n )k?
2
2 s + s? k?L(?;
?
?
k? ? ? k2 ?
+
.
?s+s?
?s+s?
Note that the result does not require the loss function to be convex. This fact will be crucially used
later. We now apply the above result to several statistical estimation scenarios.
? Xi i + ?i where
Sparse Linear Regression. Here Zi = (Xi , Yi ) ? Rp ? R and Yi = h?,
2
?i ? N (0, ? ) is label noise. The empirical loss is the usual least squares loss i.e. L(?; Z1:n ) =
1
2
n kY ? X?k2 . Suppose X1:n are drawn i.i.d. from a sub-Gaussian distribution with covariance
? with ?jj ? 1 for all j. Then [22, Lemma 6] immediately implies that RSC and RSS at
p
sparsity level k hold, with probability at least 1 ? e?c0 n , with ?k = 12 ?min (?) ? c1 k log
and
n
p
?
Lk = 2?max (?) + c1 k log
(c
,
c
are
universal
constants).
So
we
can
set
k
=
2s
+
s
and
if
0
1
n
n > 4c1 k log p/?min (?) then we have ?k ? 41 ?min (?) and Lk ? 2.25?max (?) which means that
2 ?
Lk /9?k ? ?(?) := ?max (?)/?min (?). Thus it is enough to choose
q s = 2592?(?) s and ap? Z1:n )k? = kX T ?/nk? ? 2? log p with probability at least
ply Theorem 3. Note that k?L(?;
n
1?c2 p?c3 (c2 , c3 are universal constants). Putting everything together, we have the following bound
with high probability:
r
r
?(?)
s? log p
?
k?? ? ? k2 ? 145
?
+2
,
?min (?)
n
?min (?)
where is the function value error incurred by Algorithm 1.
Noisy and Missing Data. We now look at cases with feature noise as well. More specifically,
? i ?s that are corrupted versions of Xi ?s. Two models of noise are
assume that we only have access to X
? i = Xi +Wi where Wi ? N (0, ?W ), and b) (missing
popular in literature [21]: a) (additive noise) X
?
data) X is an R?{?}-valued matrix obtained by independently, with probability ? ? [0, 1), replacing
each entry in X with ?. For the case of additive noise (missing data can be handled similarly),
? i , Yi ) and L(?; Z1:n ) = 1 ? T ??
? ? ?? T ? where ?
?=X
? T X/n
? ? ?W and ?? = X
? T Y /n are
Zi = (X
2
5
Algorithm 2 Two-stage Hard-thresholding
1: Input: function f with gradient oracle, sparsity level s, sparsity expansion level `
2: ? 1 = 0, t = 1
3: while not converged do
4:
g t = ?? f (? t ), S t = supp(? t )
5:
Z t = S t ? (largest ` elements of |gSt t |)
6:
? t = arg min?,supp(?)?Z t f (?)
// fully corrective step
t
t
e
7:
? = Ps (? )
8:
? t+1 = arg min?,supp(?)?supp(?et ) f (?), t = t + 1
// fully corrective step
9: end while
10: Output: ? t
unbiased estimators of ? and ?T ?? respectively. [21, Appendix A, Lemma 1] implies that RSC, RSS
at sparsity level k hold, with failure probability exponentially small in n, with ?k = 12 ?min (?) ?
(k?k2 +k?W k2 )2
op
op
, 1) log p.
k? (p)/n and Lk = 1.5?max (?) + k? (p)/n for ? (p) = c0 ?min (?) max(
2
(?)
?min
Thus for k = 2s + s? and n ? 4k? (p)/?min (?) we have Lk /?k ? 7?(?). Note that L(?; Z1:n )
is non-convex but we can still apply Theorem 3 with s = 1568?(?)2 s? because RSC, RSS hold.
? Z1:n )k? ?
Using thephigh probability upper bound (see [21, Appendix A, Lemma 2]) k?L(?;
?
c1 ?
? k?k2 log p/n gives us the following
r
r
?
?(?)
?
? 2 s log p + 2
k?? ? ? k2 ? c2
?
? k?k
?min (?)
n
?min (?)
q
where ?
? = k?W k2op + k?k2op (k?W kop + ?) and is the function value error in Algorithm 1.
5
Fully-corrective Methods
In this section, we study a variety of ?fully-corrective? methods. These methods keep the optimization objective fully minimized over the support of the current iterate. To this end, we first prove a
fundamental theorem for fully-corrective methods that formalizes the intuition that for such methods, a large function value should imply a large gradient at any sparse ? as well. This result is similar
to Lemma 1 of [17] but holds under RSC/RSS conditions (rather than the RIP condition as in [17]),
as well as for the general loss functions. See Appendix C for a detailed proof.
Lemma 3. Consider a function f with RSC parameter given by L2s+s? (f ) = L and RSS parameter
given by ?2s+s? (f ) = ?. Let ? ? = arg min?,k?k0 ?s? f (?) with S ? = supp(? ? ). Let S t ? [p] be
any subset of co-ordinates s.t. |S t | ? s. Let ? t = arg min?,supp(?)?S t f (?). Then, we have:
2?(f (? t ) ? f (? ? )) ? kgSt t ?S ? k22 ? ?2 k?St t \S ? k22
Two-stage Methods. We will, for now, concentrate on a family of two-stage fully corrective methods that contains popular compressive sensing algorithms like CoSaMP and Subspace Pursuit (see
Algorithm 2 for pseudocode). These algorithms have thus far been analyzed only under RIP conditions for the least squares objective. Using our analysis framework developed in the previous
sections, we present a generic RSC/RSS-based analysis for general two-stage methods for arbitrary
loss functions. Our analysis shall use the following key observation that the the hard thresholding
step in two stage methods does not increase the objective function a lot.
We defer the analysis of partial hard thresholding methods to a later version of the paper. This family
includes the OMPR(`) method [17], which is known to provide the best known RIP guarantees in
the compressive sensing setting. Using our proof techniques, we can show that this method offers
geometric convergence rates in the statistical setting as well.
Lemma 4. Let Zt ? [n] and |Zt | ? q. Let ? t = arg min?,supp(?)?Zt f (?) and ?bt = Pq (? t ).
Then, the following holds:
`
L
? (f (? t ) ? f (? ? )).
f (?bt ) ? f (? t ) ? ?
? s + ` ? s?
6
0
0.1
0.2
0.3
Noise level (sigma)
0.4
(a)
100
50
0
0.5
1
1.5
2
Dimensionality (p)
2.5
2
10
0
HTP
GraDeS
L1
FoBa
10
?2
10
?3
10
0
100
4
x 10
(b)
200 300 400
Sparsity (s*)
500
Support Recovery Error
20
150
10
HTP
GraDeS
L1
FoBa
Runtime (sec)
40
Runtime (sec)
Support Recovery Error
HTP
GraDeS
L1
FoBa
60
0
4
200
80
40
CoSaMP
HTP
GraDeS
30
20
10
0
80
100 120 140 160
Projected Sparsity (s)
(c)
(d)
Figure 1: A comparison of hard thresholding techniques (HTP) and projected gradient methods
(GraDeS) with L1 and greedy methods (FoBa) on sparse noisy linear regression tasks. 1(a) gives
the number of undiscovered elements from supp(? ? ) as label noise levels are increased. 1(b) shows
the variation in running times with increasing dimensionality p. 1(c) gives the variation in running
times (in logscale) when the true sparsity level s? is increased keeping p fixed. HTP and GraDeS are
clearly much more scalable than L1 and FoBa. 1(d) shows the recovery properties of different IHT
methods under large condition number (? = 50) setting as the size of projected set is increased.
Proof. Let v t = ?? f (? t ). Then, using the RSS property we get:
?2 L
L
|`|
?1 L
f (?bt ) ? f (? t ) ? h?bt ? ? t , v t i + k?bt ? ? t k22 = k?bt ? ? t k22 ?
kw ? ? t k22 ,
2
2
2 |s + ` ? s? |
(7)
t
= 0 and by
where w is any vector such that wZt = 0 and kwk0 ? s? . ?1 follows by observing vZ
t
t
?
b
noting that supp(? ) ? Zt . ?2 follows by Lemma 1 and the fact that kwk0 ? s . Now, using the
RSC property and the fact that ?? f (? t ) = 0, we have:
?
kw ? ? t k22 ? f (? t ) ? f (w) ? f (? t ) ? f (? ? ).
(8)
2
The result now follows by combining (7) and (8).
Theorem 4. Let f have RSC and RSS parameters given by ?2s+s? (f ) = ? and L2s+` (f ) =
2
L2 ?
?
?
L resp. Call Algorithm 2 with f , ` ? s? and s ? 4 L
?2 ` + s ? ` ? 4 ?2 s . Also let ? =
0
f (? )
arg min?,k?k0 ?s? f (?). Then, the ? -th iterate of Algorithm 2, for ? = O( L
? ? log( ) satisfies:
f (? ? ) ? f (? ? ) ? .
See Appendix C for a detailed proof.
6
Experiments
We conducted simulations on high dimensional sparse linear regression problems to verify our predictions. Our experiments demonstrate that hard thresholding and projected gradient techniques can
not only offer recovery in stochastic setting, but offer much more scalable routines for the same.
Data: Our problem setting is identical to the one described in the previous section. We fixed a
parameter vector ?? by choosing s? random coordinates and setting them randomly to ?1 values.
? Xi i + ?i where
Data samples were generated as Zi = (Xi , Yi ) where Xi ? N (0, Ip ) and Yi = h?,
2
?
?i ? N (0, ? ). We studied the effect of varying dimensionality p, sparsity s , sample size n and
label noise level ? on the recovery properties of the various algorithms as well as their run times.
We chose baseline values of p = 20000, s? = 100, ? = 0.1, n = fo ? s? log p where fo is the
oversampling factor with default value fo = 2. Keeping all other quantities fixed, we varied one of
the quantities and generated independent data samples for the experiments.
Algorithms: We studied a variety of hard-thresholding style algorithms including HTP [14],
GraDeS [13] (or IHT [12]), CoSaMP [15], OMPR [17] and SP [16]. We compared them with a
standard implementation of the L1 projected scaled sub-gradient technique [23] for the lasso problem and a greedy method FoBa [24] for the same.
7
Evaluation Metrics: For the baseline noise level ? = 0.1, we found that all the algorithms were
able to recover the support set within an error of 2%. Consequently, our focus shifted to running
times for these experiments. In the experiments where noise levels were varied, we recorded, for
each method, the number of undiscovered support set elements.
Results: Figure1 describes the results of our experiments in graphical form. For sake of clarity
we included only HTP, GraDeS, L1 and FoBa results in these graphs. Graphs for the other algorithms CoSaMP, SP and OMPR can be seen in the supplementary material. The graphs indicate that
whereas hard thresholding techniques are equally effective as L1 and greedy techniques for recovery in noisy settings, as indicated by Figure1(a), the former can be much more efficient and scalable
than the latter. For instance, as Figure1(b), for the base level of p = 20000, HTP was 150? faster
than the L1 method. For higher values of p, the runtime gap widened to more than 350?. We also
note that in both these cases, HTP actually offered exact support recovery whereas L1 was unable to
recover 2 and 4 support elements respectively.
Although FoBa was faster than L1 on Figure1(b) experiments, it was still slower than HTP by 50?
and 90? for p = 20000 and 25000 respectively. Moreover, due to its greedy and incremental
nature, FoBa was found to suffer badly in settings with larger true sparsity levels. As Figure 1(c)
indicates, for even moderate sparsity levels of s? = 300 and 500, FoBa is 60 ? 75? slower than
HTP. As mentioned before, the reason for this slowdown is the greedy approach followed by FoBa:
whereas HTP took less than 5 iterations to converge for these two problems, FoBa spend 300 and
500 iterations respectively. GraDeS was found to offer much lesser run times in comparison being
slower than HTP by 30 ? 40? for larger values of p and 2 ? 5? slower for larger values of s? .
Experiments on badly conditioned problems. We also ran experiments to verify the performance
of IHT algorithms in high condition number setting. Values of p, s? and ? were kept at baseline
? we selected s? /2 random coordinates from
levels. After selecting the optimal parameter vector ?,
?
its support and s /2 random coordinates outside its support and constructed a covariance matrix
with heavy correlations between these chosen coordinates. The condition number of the resulting
matrix was close to 50. Samples were drawn from this distribution and the recovery properties of
the different IHT-style algorithms was observed as the projected sparsity levels s were increased.
Our results (see Figure 1(d)) corroborate our theoretical observation that these algorithms show
a remarkable improvement in recovery properties for ill-conditioned problems with an enlarged
projection size.
7
Discussion and Conclusions
In our work we studied iterative hard thresholding algorithms and showed that these techniques
can offer global convergence guarantees for arbitrary, possibly non-convex, differentiable objective
functions, which nevertheless satisfy Restricted Strong Convexity/Smoothness (RSC/RSM) conditions. Our results apply to a large family of algorithms that includes existing algorithms such as
IHT, GraDeS, CoSaMP, SP and OMPR. Previously the analyses of these algorithms required stringent RIP conditions that did not allow the (restricted) condition number to be larger than universal
constants specific to these algorithms.
Our basic insight was to relax this stringent requirement by running these iterative algorithms with
an enlarged support size. We showed that guarantees for high-dimensional M-estimation follow
seamlessly from our results by invoking results on RSC/RSM conditions that have already been
established in the literature for a variety of statistical settings. Our theoretical results put hard
thresholding methods on par with those based on convex relaxation or greedy algorithms. Our
experimental results demonstrate that hard thresholding methods outperform convex relaxation and
greedy methods in terms of running time, sometime by orders of magnitude, all the while offering
competitive or better recovery properties.
Our results apply to sparsity and low rank structure, arguably two of the most commonly used
structures in high dimensional statistical learning problems. In future work, it would be interesting
to generalize our algorithms and their analyses to more general structures. A unified analysis for
general structures will probably create interesting connections with existing unified frameworks
such as those based on decomposability [5] and atomic norms [25].
8
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9
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4,742 | 5,294 | A Safe Screening Rule for Sparse Logistic Regression
Jiayu Zhou
Arizona State University
Tempe, AZ 85287
[email protected]
Jie Wang
Arizona State University
Tempe, AZ 85287
[email protected]
Jun Liu
SAS Institute Inc.
Cary, NC 27513
[email protected]
Peter Wonka
Arizona State University
Tempe, AZ 85287
[email protected]
Jieping Ye
Arizona State University
Tempe, AZ 85287
[email protected]
Abstract
The `1 -regularized logistic regression (or sparse logistic regression) is a widely
used method for simultaneous classification and feature selection. Although many
recent efforts have been devoted to its efficient implementation, its application to
high dimensional data still poses significant challenges. In this paper, we present a
fast and effective sparse logistic regression screening rule (Slores) to identify the
?0? components in the solution vector, which may lead to a substantial reduction
in the number of features to be entered to the optimization. An appealing feature
of Slores is that the data set needs to be scanned only once to run the screening and
its computational cost is negligible compared to that of solving the sparse logistic
regression problem. Moreover, Slores is independent of solvers for sparse logistic regression, thus Slores can be integrated with any existing solver to improve
the efficiency. We have evaluated Slores using high-dimensional data sets from
different applications. Experiments demonstrate that Slores outperforms the existing state-of-the-art screening rules and the efficiency of solving sparse logistic
regression can be improved by one magnitude.
1
Introduction
Logistic regression (LR) is a popular and well established classification method that has been widely
used in many domains such as machine learning [4, 7], text mining [3, 8], image processing [9, 15],
bioinformatics [1, 13, 19, 27, 28], medical and social sciences [2, 17] etc. When the number of
feature variables is large compared to the number of training samples, logistic regression is prone
to over-fitting. To reduce over-fitting, regularization has been shown to be a promising approach.
Typical examples include `2 and `1 regularization. Although `1 regularized LR is more challenging
to solve compared to `2 regularized LR, it has received much attention in the last few years and
the interest in it is growing [20, 25, 28] due to the increasing prevalence of high-dimensional data.
The most appealing property of `1 regularized LR is the sparsity of the resulting models, which is
equivalent to feature selection.
In the past few years, many algorithms have been proposed to efficiently solve the `1 regularized
LR [5, 12, 11, 18]. However, for large-scale problems, solving the `1 regularized LR with higher
accuracy remains challenging. One promising solution is by ?screening?, that is, we first identify
the ?inactive? features, which have 0 coefficients in the solution and then discard them from the
optimization. This would result in a reduced feature matrix and substantial savings in computational
cost and memory size. In [6], El Ghaoui et al. proposed novel screening rules, called ?SAFE?,
to accelerate the optimization for a class of `1 regularized problems, including LASSO [21], `1
1
regularized LR and `1 regularized support vector machines. Inspired by SAFE, Tibshirani et al.
[22] proposed ?strong rules? for a large class of `1 regularized problems, including LASSO, elastic
net, `1 regularized LR and more general convex problems. In [26], Xiang et al. proposed ?DOME?
rules to further improve SAFE rules for LASSO based on the observation that SAFE rules can be
understood as a special case of the general ?sphere test?. Although both strong rules and the sphere
tests are more effective in discarding features than SAFE for solving LASSO, it is worthwhile to
mention that strong rules may mistakenly discard features that have non-zero coefficients in the
solution and the sphere tests are not easy to be generalized to handle the `1 regularized LR. To the
best of our knowledge, the SAFE rule is the only screening test for the `1 regularized LR that is
?safe?, that is, it only discards features that are guaranteed to be absent from the resulting models.
In this paper, we develop novel screening rules, called
?Slores?, for the `1 regularized LR. The proposed screening tests detect inactive features by estimating an upper
bound of the inner product between each feature vector
and the ?dual optimal solution? of the `1 regularized LR, which is unknown. The more accurate the estimation
is, the more inactive features can be detected. An accurate estimation of such an upper bound turns out to be
quite challenging. Indeed most of the key ideas/insights
behind existing ?safe? screening rules for LASSO heavily rely on the least square loss, which are not applicable
for the `1 regularized LR case due to the presence of the
logistic loss. To this end, we propose a novel framework Figure 1: Comparison of Slores, strong
to accurately estimate an upper bound. Our key techni- rule and SAFE on the prostate cancer
cal contribution is to formulate the estimation of an upper data set.
bound of the inner product as a constrained convex optimization problem and show that it admits
a closed form solution. Therefore, the estimation of the inner product can be computed efficiently.
Our extensive experiments have shown that Slores discards far more features than SAFE yet requires
much less computational efforts. In contrast with strong rules, Slores is ?safe?, i.e., it never discards
features which have non-zero coefficients in the solution.
To illustrate the effectiveness of Slores, we compare Slores, strong rule and SAFE on a data set of
prostate cancer along a sequence of 86 parameters equally spaced on the ?/?max scale from 0.1 to
0.95, where ? is the parameter for the `1 penalty and ?max is the smallest tuning parameter [10] such
that the solution of the `1 regularized LR is 0 [please refer to Eq. (1)]. The data matrix contains 132
patients with 15154 features. To measure the performance of different screening rules, we compute
the rejection ratio which is the ratio between the number of features discarded by screening rules
and the number of features with 0 coefficients in the solution. Therefore, the larger the rejection
ratio is, the more effective the screening rule is. The results are shown in Fig. 1. We can see that
Slores discards far more features than SAFE especially when ?/?max is large while the strong rule
is not applicable when ?/?max ? 0.5. We present more results and discussions to demonstrate the
effectiveness of Slores in Section 6. For proofs of the lemmas, corollaries, and theorems, please
refer to the long version of this paper [24].
2
Basics and Motivations
In this section, we briefly review the basics of the `1 regularized LR and then motivate the general
screening rules via the KKT conditions. Suppose we are given a set of training samples {xi }m
i=1
and the associate labels b ? <m , where xi ? <p and bi ? {1, ?1} for all i ? {1, . . . , m}. The `1
regularized logistic regression is:
m
1 X
? i i ? bi c)) + ?k?k1 ,
min
log(1 + exp(?h?, x
?,c m
i=1
(LRP? )
? i = bi xi , and ? > 0 is the
where ? ? <p and c ? < are the model parameters to be estimated, x
? i and the j th
tuning parameter. We denote by X ? <m?p the data matrix with the ith row being x
j
? .
column being x
2
Let C = {? ? <m : ?i ? (0, 1), i = 1, . . . , m} and f (y) = y log(y) + (1 ? y) log(1 ? y) for
y ? (0, 1). The dual problem of (LRP? ) [24] is given by
(
)
m
1 X
? T ?k? ? m?, h?, bi = 0, ? ? C .
min g(?) =
f (?i ) : kX
(LRD? )
?
m i=1
To simplify notations, we denote the feasible set of problem (LRD? ) as F? , and let (??? , c?? ) and
??? be the optimal solutions of problems (LRP? ) and (LRD? ) respectively. In [10], the authors have
shown that for some special choice of the tuning parameter ?, both of (LRP? ) and (LRD? ) have
closed form solutions. In fact, let P = {i : bi = 1}, N = {i : bi = ?1}, and m+ and m? be the
cardinalities of P and N respectively. We define
? T ??
?max = 1 kX
k? ,
(1)
?max
m
where
(
[??? max ]i
=
m?
m ,
m+
m ,
if i ? P,
if i ? N ,
i = 1, . . . , m.
(2)
([?]i denotes the ith component of a vector.) Then, it is known [10] that ??? = 0 and ??? = ??? max
whenever ? ? ?max . When ? ? (0, ?max ], it is known that (LRD? ) has a unique optimal solution
[24]. We can now write the KKT conditions of problems (LRP? ) and (LRD? ) as
?
if [??? ]j > 0,
?m?,
?
j
? i ? ?m?,
h?? , x
(3)
if [??? ]j < 0, j = 1, . . . , p.
?
[?m?, m?], if [??? ]j = 0.
In view of Eq. (3), we can see that
? j i| < m? ? [??? ]j = 0.
|h??? , x
(R1)
?j i
|h??? , x
?j
In other words, if
< m?, then the KKT conditions imply that the coefficient of x in the
solution ??? is 0 and thus the j th feature can be safely removed from the optimization of (LRP? ).
However, for the general case in which ? < ?max , (R1) is not applicable since it assumes the
knowledge of ??? . Although it is unknown, we can still estimate a region A? which contains ??? . As
? j i| < m?, we can also conclude that [??? ]j = 0 by (R1). In other words,
a result, if max??A? |h?, x
(R1) can be relaxed as
? j ) := max |h?, x
? j i| < m? ? [??? ]j = 0.
T (??? , x
(R10 )
??A?
In this paper, (R10 ) serves as the foundation for constructing our screening rules, Slores. From
? j ) are more aggressive in discarding
(R10 ), it is easy to see that screening rules with smaller T (??? , x
? ?j
features. To give a tight estimation of T (?? , x ), we need to restrict the region A? which includes
?j )
??? as small as possible. In Section 3, we show that the estimation of the upper bound T (??? , x
can be obtained via solving a convex optimization problem. We show in Section 4 that the convex
optimization problem admits a closed form solution and derive Slores in Section 5 based on (R10 ).
3
Estimating the Upper Bound via Solving a Convex Optimization Problem
? j ) of |h??? , x
? j i|. In
In this section, we present a novel framework to estimate an upper bound T (??? , x
the subsequent development, we assume a parameter ?0 and the corresponding dual optimal ??? 0 are
given. In our Slores rule to be presented in Section 5, we set ?0 and ??? 0 to be ?max and ??? max given
? j ) as a constrained convex optimization
in Eqs. (1) and (2). We formulate the estimation of T (??? , x
problem in this section, which will be shown to admit a closed form solution in Section 4.
?i
1
1
1
4
For the dual function g(?), it follows that [?g(?)]i = m
log( 1??
), [?2 g(?)]i,i = m
?i (1??i ) ? m .
i
4
Since ?2 g(?) is a diagonal matrix, it follows that ?2 g(?) m
I, where I is the identity matrix.
4
Thus, g(?) is strongly convex with modulus ? = m [16]. Rigorously, we have the following lemma.
Lemma 1. Let ? > 0 and ?1 , ?2 ? F? , then
a).
g(?2 ) ? g(?1 ) ? h?g(?1 ), ?2 ? ?1 i +
2
m k?2
? ?1 k22 .
(4)
b). If ?1 6= ?2 , the inequality in (4) becomes a strict inequality, i.e., ??? becomes ?>?.
3
Given ? ? (0, ?0 ], it is easy to see that both of ??? and ??? 0 belong to F?0 . Therefore, Lemma 1 can
be a useful tool to bound ??? with the knowledge of ??? 0 . In fact, we have the following theorem.
Theorem 2. Let ?max ? ?0 > ? > 0, then the following holds:
i
mh ? ?
(5)
a).
k??? ? ??? 0 k22 ?
g ?0 ??0 ? g(??? 0 ) + 1 ? ??0 h?g(??? 0 ), ??? 0 i
2
b). If ??? 6= ??? 0 , the inequality in (5) becomes a strict inequality, i.e., ??? becomes ?<?.
Theorem 2 implies that ??? is inside a ball centred at ??? 0 with radius
r h
r=
m
2
g
? ?
?0 ??0
? g(??? 0 ) + (1 ?
i
?
?
?
?0 )h?g(??0 ), ??0 i
.
(6)
Recall that to make our screening rules more aggressive in discarding features, we need to get a tight
? j ) of |h??? , x
? j i| [please see (R10 )]. Thus, it is desirable to further restrict the
upper bound T (??? , x
possible region A? of ??? . Clearly, we can see that
h??? , bi = 0
(7)
???
?j i
h??? 0 , x
since
is feasible for problem (LRD? ). On the other hand, we call the set I?0 = {j :
|m?0 |, j = 1, . . . , p} the ?active set? of ??? 0 . We have the following lemma for the active set.
=
? j i| =
Lemma 3. Given the optimal solution ??? of problem (LRD? ), the active set I? = {j : |h??? , x
m?, j = 1, . . . , p} is not empty if ? ? (0, ?max ].
Since ?0 ? (0, ?max ], we can see that I?0 is not empty by Lemma 3. We pick j0 ? I?0 and set
? ? = sign(h??? 0 , x
? j0 i)?
x
xj0 .
(8)
It follows that h?
x? , ??? 0 i = m?0 . Due to the feasibility of ??? for problem (LRD? ), ??? satisfies
? ? i ? m?.
h??? , x
(9)
As a result, Theorem 2, Eq. (7) and (9) imply that ??? is contained in the following set:
? ? i ? m?}.
A??0 := {? : k? ? ??? 0 k22 ? r2 , h?, bi = 0, h?, x
? j i| ? max??A?? |h?, x
? j i|. Therefore, (R10 ) implies that if
Since ??? ? A??0 , we can see that |h??? , x
0
? j ; ??? 0 )
T (??? , x
? j i|
:= max |h?, x
??A?
?
(UBP)
0
? j can be discarded from the optimization
is smaller than m?, we can conclude that [??? ]j = 0 and x
? j ) with T (??? , x
? j ; ??? 0 ) and A??0
of (LRP? ). Notice that, we replace the notations A? and T (??? , x
?
? j ; ??? 0 ), (R10 )
to emphasize their dependence on ??0 . Clearly, as long as we can solve for T (??? , x
would be an applicable screening rule to discard features which have 0 coefficients in ??? . We give a
closed form solution of problem (UBP) in the next section.
4
Solving the Convex Optimization Problem (UBP)
In this section, we show how to solve the convex optimization problem (UBP) based on the standard
Lagrangian multiplier method. We first transform problem (UBP) into a pair of convex minimization
problem (UBP0 ) via Eq. (11) and then show that the strong duality holds for (UBP0 ) in Lemma 6. The
strong duality guarantees the applicability of the Lagrangian multiplier method. We then give the
closed form solution of (UBP0 ) in Theorem 8. After we solve problem (UBP0 ), it is straightforward
to compute the solution of problem (UBP) via Eq. (11).
Before we solve (UBP) for the general case, it is worthwhile to mention a special case in which
xj ,bi
? j ? h?kbk
P?
xj = x
2 b = 0. Clearly, P is the projection operator which projects a vector onto the
2
orthogonal complement of the space spanned by b. In fact, we have the following theorem.
Theorem 4. Let ?max ? ?0 > ? > 0, and assume ??? 0 is known. For j ? {1, . . . , p}, if P?
xj = 0,
? ?j ?
then T (?? , x ; ??0 ) = 0.
4
Because of (R10 ), we immediately have the following corollary.
Corollary 5. Let ? ? (0, ?max ) and j ? {1, . . . , p}. If P?
xj = 0, then [??? ]j = 0.
For the general case in which P?
xj 6= 0, let
? j ; ??? 0 ) := max h?, +?
? j ; ??? 0 ) := max h?, ??
T+ (??? , x
xj i, T? (??? , x
xj i.
??A?
?
??A?
?
0
(10)
0
Clearly, we have
? j ; ??? 0 ) = max{T+ (??? , x
? j ; ??? 0 ), T? (??? , x
? j ; ??? 0 )}.
T (??? , x
(11)
Therefore, we can solve problem (UBP) by solving the two sub-problems in (10).
Let ? ? {+1, ?1}. Then problems in (10) can be written uniformly as
? j ; ??? 0 ) = max h?, ??
xj i.
T? (??? , x
??A?
?
(UBPs )
0
To make use of the standard Lagrangian multiplier method, we transform problem (UBPs ) to the
following minimization problem:
? j ; ??? 0 ) = min h?, ???
?T? (??? , x
xj i
??A?
?
(UBP0 )
0
j
by noting that max??A?? h?, ??
x i = ? min??A?? h?, ???
xj i.
0
0
Lemma 6. Let ?max ? ?0 > ? > 0 and assume ??? 0 is known. The strong duality holds for problem
(UBP0 ). Moreover, problem (UBP0 ) admits an optimal solution in A??0 .
Because the strong duality holds for problem (UBP0 ) by Lemma 6, the Lagrangian multiplier method
is applicable for (UBP0 ). In general, we need to first solve the dual problem and then recover the
? ? are defined by
optimal solution of the primal problem via KKT conditions. Recall that r and x
Eq. (6) and (8) respectively. Lemma 7 derives the dual problems of (UBP0 ) for different cases.
Lemma 7. Let ?max ? ?0 > ? > 0 and assume ??? 0 is known. For j ? {1, . . . , p} and P?
xj 6= 0,
? = ???
let x
xj . Denote
n
o
x,P?
x? i
U1 = {(u1 , u2 ) : u1 > 0, u2 ? 0} and U2 = (u1 , u2 ) : u1 = 0, u2 = ? hP?
.
kP?
x? k2
2
a). If
hP?
x,P?
x? i
kP?
xk2 kP?
x ? k2
max
(u1 ,u2 )?U1
? (?1, 1], the dual problem of (UBP0 ) is equivalent to:
? i ? 21 u1 r2 .
g?(u1 , u2 ) = ? 2u1 1 kP?
x + u2 P?
x? k22 + u2 m(?0 ? ?) + h??? 0 , x
(UBD0 )
Moreover, g?(u1 , u2 ) attains its maximum in U1 .
b). If
hP?
x,P?
x? i
kP?
xk2 kP?
x ? k2
= ?1, the dual problem of (UBP0 ) is equivalent to:
(
max
(u1 ,u2 )?U1 ?U2
g?(u1 , u2 ) =
g?(u1 , u2 ),
kP?
xk2
? kP?
x? k2 m?,
if (u1 , u2 ) ? U1 ,
if (u1 , u2 ) ? U2 .
(UBD00 )
We can now solve problem (UBP0 ) in the following theorem.
Theorem 8. Let ?max ? ?0 > ? > 0, d =
? = ???
and P?
xj 6= 0, let x
xj .
a). If
hP?
x,P?
x? i
kP?
xk2 kP?
x ? k2
m(?0 ??)
rkP?
x? k2
and assume ??? 0 is known. For j ? {1, . . . , p}
? d, then
? j ; ??? 0 ) = rkP?
? i;
T? (??? , x
xk2 ? h??? 0 , x
5
(12)
b). If
hP?
x,P?
x? i
kP?
xk2 kP?
x ? k2
< d, then
? j ; ??? 0 ) = rkP?
? i,
T? (??? , x
x + u?2 P?
x? k2 ? u?2 m(?0 ? ?) ? h??? 0 , x
(13)
where
u?2 =
?
?a1 + ?
,
2a2
? 4
kP?
x k2 (1 ?
?
a2 =
d2 ),
a1 = 2hP?
x, P?
x ikP?
x? k22 (1 ? d2 ),
? 2
a0 = hP?
x, P?
x i ? d2 kP?
xk22 kP?
x? k22 ,
? = a21 ? 4a2 a0 = 4d2 (1 ? d2 )kP?
x? k42 (kP?
xk22 kP?
x? k22 ? hP?
x, P?
x? i2 ).
(14)
Notice that, although the dual problems of (UBP0 ) in Lemma 7 are different, the resulting upper
? j ; ??? 0 ) can be given by Theorem 8 in a uniform way. The tricky part is how to deal
bound T? (??? , x
with the extremal cases in which
5
hP?
x,P?
x? i
kP?
xk2 kP?
x ? k2
? {?1, +1}.
The proposed Slores Rule for `1 Regularized Logistic Regression
Using (R10 ), we are now ready to construct the screening rules for the `1 Regularized Logistic
Regression. By Corollary 5, we can see that the orthogonality between the j th feature and the
? j from the resulting model. For the general case in which
response vector b implies the absence of x
? j ; ??? 0 ) = max{T+ (??? , x
? j ; ??? 0 ), T? (??? , x
? j ; ??? 0 )} < m?,
P?
xj 6= 0, (R10 ) implies that if T (??? , x
then the j th feature can be discarded from the optimization of (LRP? ). Notice that, letting ? = ?1,
? j ; ??? 0 ) have been solved by Theorem 8. Rigorously, we have the
? j ; ??? 0 ) and T? (??? , x
T+ (??? , x
following theorem.
Theorem 9 (Slores). Let ?0 > ? > 0 and assume ??? 0 is known.
1. If ? ? ?max , then ??? = 0;
2. If ?max ? ?0 > ? > 0 and either of the following holds:
(a) P?
xj = 0,
? j ; ??? 0 ) : ? = ?1} < m?,
(b) max{T? (??? , x
?
then [?? ]j = 0.
Based on Theorem 9, we construct the Slores rule as summarized below in Algorithm 1.
Notice that, the output R of Slores is the indices
?
of the features that need to be entered to the Algorithm 1 R = Slores(X, b, ?, ?0 , ??0 )
optimization. As a result, suppose the output
Initialize R := {1, . . . , p};
if ? ? ?max then
of Algorithm 1 is R = {j1 , . . . , jk }, we can
set R = ?;
substitute the full matrix X in problem (LRP? )
else
j1
j
k
? ) and
with the sub-matrix XR = (?
x ,...,x
for j = 1 to p do
just solve for [??? ]R and c?? .
j
if P?
x = 0 then
remove j from R;
? j ; ??? 0 ) : ? = ?1} < m?
else if max{T? (??? , x
then
remove j from R;
end if
end for
end if
Return: R
On the other hand, Algorithm 1 implies that
Slores needs five inputs. Since X and b come
with the data and ? is chosen by the user, we only need to specify ??? 0 and ?0 . In other words,
we need to provide Slores with a dual optimal solution of problem (LRD? ) for an arbitrary parameter. A natural choice is by setting
?0 = ?max and ??? 0 = ??? max given by Eq. (1)
and Eq. (2) in closed form.
6
Experiments
We evaluate our screening rules using the newgroup data set [10] and Yahoo web pages data sets
[23]. The newgroup data set is cultured from the data by Koh et al. [10]. The Yahoo data sets include 11 top-level categories, each of which is further divided into a set of subcategories. In
6
our experiment we construct five balanced binary classification datasets from the topics of Computers, Education, Health, Recreation, and Science. For each topic, we choose samples from
one subcategory as the positive class and randomly sample an equal number of samples from the
rest of subcategories as the negative class. The statistics of the data sets are given in Table 1.
Table 1: Statistics of the test data sets.
We compare the performance of Slores and the
Data
set
m
p
no. nonzeros
strong rule which achieves state-of-the-art pernewsgroup
11269 61188
1467345
formance for `1 regularized LR. We do not inComputers
216
25259
23181
clude SAFE because it is less effective in disEducation
254
20782
28287
carding features than strong rules and requires
Health
228
18430
40145
much higher computational time [22]. Fig. 1
Recreation
370
25095
49986
has shown the performance of Slores, strong
Science
222
24002
37227
rule and SAFE. We compare the efficiency of
the three screening rules on the same prostate
cancer data set in Table 2. All of the screen- Table 2: Running time (in seconds) of Slores,
ing rules are tested along a sequence of 86 pa- strong rule, SAFE and the solver.
rameter values equally spaced on the ?/?max
Slores Strong Rule
SAFE
Solver
scale from 0.1 to 0.95. We repeat the procedure
0.37
0.33
1128.65
10.56
100 times and during each time we undersample 80% of the data. We report the total running time of the three screening rules over the 86 values
of ?/?max in Table 2. For reference, we also report the total running time of the solver1 . We observe
that the running time of Slores and strong rule is negligible compared to that of the solver. However,
SAFE takes much longer time even than the solver.
In Section 6.1, we evaluate the performance of Slores and strong rule. Recall that we use the rejection ratio, i.e., the ratio between the number of features discarded by the screening rules and the
number of features with 0 coefficients in the solution, to measure the performance of screening rules.
Note that, because no features with non-zero coefficients in the solution would be mistakenly discarded by Slores, its rejection ratio is no larger than one. We then compare the efficiency of Slores
and strong rule in Section 6.2.
The experiment settings are as follows. For each data set, we undersample 80% of the date and
run Slores and strong rules along a sequence of 86 parameter values equally spaced on the ?/?max
scale from 0.1 to 0.95. We repeat the procedure 100 times and report the average performance and
running time at each of the 86 values of ?/?max . Slores, strong rules and SAFE are all implemented
in Matlab. All of the experiments are carried out on a Intel(R) (i7-2600) 3.4Ghz processor.
6.1
Comparison of Performance
In this experiment, we evaluate the performance of the Slores and the strong rule via the rejection
ratio. Fig. 2 shows the rejection ratio of Slores and strong rule on six real data sets. When ?/?max >
0.5, we can see that both Slores and strong rule are able to identify almost 100% of the inactive
features, i.e., features with 0 coefficients in the solution vector. However, when ?/?max ? 0.5,
strong rule can not detect the inactive features. In contrast, we observe that Slores exhibits much
stronger capability in discarding inactive features for small ?, even when ?/?max is close to 0.1.
Taking the data point at which ?/?max = 0.1 for example, Slores discards about 99% inactive
features for the newsgroup data set. For the other data sets, more than 80% inactive features are
identified by Slores. Thus, in terms of rejection ratio, Slores significantly outperforms the strong
rule. Moreover, the discarded features by Slores are guaranteed to have 0 coefficients in the solution.
But strong rule may mistakenly discard features which have non-zero coefficients in the solution.
6.2
Comparison of Efficiency
We compare efficiency of Slores and the strong rule in this experiment. The data sets for evaluating
the rules are the same as Section 6.1. The running time of the screening rules reported in Fig. 3
includes the computational cost of the rules themselves and that of the solver after screening. We
plot the running time of the screening rules against that of the solver without screening. As indicated
by Fig. 2, when ?/?max > 0.5, Slores and strong rule discards almost 100% of the inactive features.
1
In this paper, the ground truth is computed by SLEP [14].
7
(a) newsgroup
(b) Computers
(c) Education
(d) Health
(e) Recreation
(f) Science
Figure 2: Comparison of the performance of Slores and strong rules on six real data sets.
(a) newsgroup
(b) Computers
(c) Education
(d) Health
(e) Recreation
(f) Science
Figure 3: Comparison of the efficiency of Slores and strong rule on six real data sets.
As a result, the size of the feature matrix involved in the optimization of problem (LRP? ) is greatly
reduced. From Fig. 3, we can observe that the efficiency is improved by about one magnitude on
average compared to that of the solver without screening. However, when ?/?max < 0.5, strong
rule can not identify any inactive features and thus the running time is almost the same as that of the
solver without screening. In contrast, Slores is still able to identify more than 80% of the inactive
features for the data sets cultured from the Yahoo web pages data sets and thus the efficiency is
improved by roughly 5 times. For the newgroup data set, about 99% inactive features are identified
by Slores which leads to about 10 times savings in running time. These results demonstrate the
power of the proposed Slores rule in improving the efficiency of solving the `1 regularized LR.
7
Conclusions
In this paper, we propose novel screening rules to effectively discard features for `1 regularized
LR. Extensive numerical experiments on real data demonstrate that Slores outperforms the existing
state-of-the-art screening rules. We plan to extend the framework of Slores to more general sparse
formulations, including convex ones, like group Lasso, fused Lasso, `1 regularized SVM, and nonconvex ones, like `p regularized problems where 0 < p < 1.
8
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9
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4,743 | 5,295 | Generalized Dantzig Selector:
Application to the k-support norm
Soumyadeep Chatterjee?
Sheng Chen?
Arindam Banerjee
Dept. of Computer Science & Engg.
University of Minnesota, Twin Cities
{chatter,shengc,banerjee}@cs.umn.edu
Abstract
We propose a Generalized Dantzig Selector (GDS) for linear models, in which any
norm encoding the parameter structure can be leveraged for estimation. We investigate both computational and statistical aspects of the GDS. Based on conjugate
proximal operator, a flexible inexact ADMM framework is designed for solving
GDS. Thereafter, non-asymptotic high-probability bounds are established on the
estimation error, which rely on Gaussian widths of the unit norm ball and the error
set. Further, we consider a non-trivial example of the GDS using k-support norm.
We derive an efficient method to compute the proximal operator for k-support
norm since existing methods are inapplicable in this setting. For statistical analysis, we provide upper bounds for the Gaussian widths needed in the GDS analysis,
yielding the first statistical recovery guarantee for estimation with the k-support
norm. The experimental results confirm our theoretical analysis.
1 Introduction
The Dantzig Selector (DS) [3, 5] provides an alternative to regularized regression approaches such as
Lasso [19, 22] for sparse estimation. While DS does not consider a regularized maximum likelihood
approach, [3] has established clear similarities between the estimates from DS and Lasso. While
norm regularized regression approaches have been generalized to more general norms [14, 2], the
literature on DS has primarily focused on the sparse L1 norm case, with a few notable exceptions
which have considered extensions to sparse group-structured norms [11].
In this paper, we consider linear models of the form y = X? ? + w, where y ? Rn is a set of
observations, X ? Rn?p is a design matrix with i.i.d. standard Gaussian entries, and w ? Rn
is i.i.d. standard Gaussian noise. For any given norm R(?), the parameter ? ? is assumed to be
structured in terms of having a low value of R(? ? ). For this setting, we propose the following
Generalized Dantzig Selector (GDS) for parameter estimation:
!
"
?? = argmin R(?) s.t. R? XT (y ? X?) ? ?p ,
(1)
??Rp
where R? (?) is the dual norm of R(?), and ?p is a suitable constant. If R(?) is the L1 norm, (1)
reduces to standard DS [5]. A key novel aspect of GDS is that the constraint is in terms of the dual
norm R? (?) of the original structure inducing norm R(?). It is instructive to contrast GDS with
the recently proposed atomic norm based estimation framework [6] which, unlike GDS, considers
constraints based on the L2 norm of the error ?y ? X??2 .
In this paper, we consider both computational and statistical aspects of the GDS. For the L1 -norm
Dantzig selector, [5] proposed a primal-dual interior point method since the optimization is a linear
program. DASSO and its generalization proposed in [10, 9] focused on homotopy methods, which
?
Both authors contributed equally.
1
provide a piecewise linear solution path through a sequential simplex-like algorithm. However, none
of the algorithms above can be immediately extended to our general formulation. In recent work,
the Alternating Direction Method of Multipliers (ADMM) has been applied to the L1 -norm Dantzig
selection problem [12, 21], and the linearized version in [21] proved to be efficient. Motivated
by such results for DS, we propose a general inexact ADMM [20] framework for GDS where the
primal update steps, interestingly, turn out respectively to be proximal updates involving R(?) and
its convex conjugate, the indicator of R? (x) ? ?p . As a result, by Moreau decomposition, it suffices
to develop efficient proximal update for either R(?) or its conjugate. On the statistical side, we
establish non-asymptotic high-probability bounds on the estimation error ??? ? ? ? ?2 . Interestingly,
the bound depends on the Gaussian width of the unit norm ball of R(?) as well as the Gaussian width
of intersection of error cone and unit sphere [6, 16].
As a non-trivial example of the GDS framework, we consider estimation using the recently proposed
k-support norm [1, 13]. We show that proximal operators for k-support norm can be efficiently
computed in O(p log p + log k log(p ? k)), and hence the estimation can be done efficiently. Note
that existing work [1, 13] on k-support norm has focused on the proximal operator for the square of
the k-support norm, which is not directly applicable in our setting. On the statistical side, we provide
upper bounds for the Gaussian widths of the unit norm ball and the error cone as needed in the GDS
framework, yielding the first statistical recovery guarantee for estimation with the k-support norm.
The rest of the paper is organized as follows: We establish general optimization and statistical
recovery results for GDS for any norm in Section 2. In Section 3, we present efficient algorithms
and estimation error bounds for the k-support norm. We present experimental results in Section 4
and conclude in Section 5. All technical analysis and proofs can be found in [7].
2 General Optimization and Statistical Recovery Guarantees
The problem in (1) is a convex program, and a suitable choice of ?p ensures that the feasible set
is not empty. We start the section with an inexact ADMM framework for solving problems of the
form (1), and then present bounds on the estimation error establishing statistical consistency of GDS.
2.1 General Optimization Framework using Inexact ADMM
For convenience, we temporarily drop the subscript p of ?p . We let A = XT X, b = XT y, and
define the set C? = {v : R? (v) ? ?}. The optimization problem is equivalent to
min R(?)
?,v
s.t. b ? A? = v, v ? C? .
(2)
Due to the nonsmoothness of both R and R? , solving (2) can be quite challenging and a generally
applicable algorithm is Alternating Direction Method of Multipliers (ADMM) [4]. The augmented
Lagrangian function for (2) is given as
?
LR (?, v, z) = R(?) + ?z, A? + v ? b? + ||A? + v ? b||22 ,
(3)
2
where z is the Lagrange multiplier and ? controls the penalty introduced by the quadratic term. The
iterative updates of the variables (?, v, z) in standard ADMM are given by
? k+1 ? argmin LR (?, vk , zk ) ,
(4)
vk+1 ? argmin LR (? k+1 , v, zk ) ,
(5)
zk+1 ? zk + ?(A? k+1 + vk+1 ? b) .
(6)
?
v?C?
Note that update (4) amounts to a norm regularized least squares problem for ?, which can be
computationally expensive. Thus we use an inexact update for ? instead, which can alleviate the
computational cost and lead to a quite simple algorithm. Inspired by [21, 20], we consider a simpler
subproblem for the ?-update which minimizes
%
? $%
%A? k + vk ? b%2 +
L#kR (?, vk , zk ) = R(?) + ?zk , A? + vk ? b? +
2
2
(7)
%2 (
' ?%
&
k
T
k
k
k
2 ? ? ? , A (A? + v ? b) + %? ? ? %2 ,
2
2
Algorithm 1 ADMM for Generalized Dantzig Selector
Input: A = XT X, b = XT y, ?, ?
Output: Optimal ?? of (1)
1: Initialize (?, v, z)
2: while not converged do
!
3:
? k+1 ? prox 2R ? k ? ?2 AT (A? k + vk ? b +
??
!
k"
4:
vk+1 ? prox C b ? A? k+1 ? z?
?
5:
zk+1 ? zk + ?(A? k+1 + vk+1 ? b)
6: end while
"
zk
? )
where ? is a user-defined parameter. L#kR (?, vk , zk ) can be viewed as an approximation of
LR (?, vk , zk ) with the quadratic term linearized at ? k . Then the update (4) is replaced by
? k+1 ? argmin L#kR (?, vk , zk )
?
)
*
! k 2 T
zk "%
2R(?) 1 %
%2
%
k
k
+ %? ? ? ? A (A? + v ? b + ) % .
= argmin
??
2
?
?
2
?
(8)
Similarly the update of v in (5) can be recast as
1%
zk %2
vk+1 ? argmin LR (? k+1 , v, zk ) = argmin %v ? (b ? A? k+1 ? )%2 .
?
v?C?
v?C? 2
(9)
In fact, the updates of both ? and v are to compute certain proximal operators [15]. In general, the
proximal operator proxh (?) of a closed proper convex function h : Rp ? R ? {+?} is defined as
+1
,
?w ? x?22 + h(w) .
proxh (x) = argmin
2
w?Rp
Hence it is easy to see that (8) and (9) correspond to prox 2R (?) and prox
??
C? (?) is the indicator function of set C? given by
)
0
if x ? C?
(x)
=
.
C?
+?
if otherwise
C?
(?), respectively, where
In Algorithm 1, we provide our general ADMM for the GDS. For the ADMM to work, we need two
subroutines that can efficiently compute the proximal operators for the functions in Line 3 and 4
respectively. The simplicity of the proposed approach stems from the fact that we in fact need only
one subroutine, for any one of the functions, since the functions are conjugates of each other.
Proposition 1 Given ? > 0 and a norm R(?), the two functions, f (x) = ?R(x) and g(x) =
are convex conjugate to each other, thus giving the following identity,
x = proxf (x) + proxg (x) .
C? (x)
(10)
Proof: The Proposition 1 simply follows from the definition of convex conjugate and dual norm,
and (10) is just Moreau decomposition provided in [15].
The decomposition enables conversion of the two types of proximal operator to each other at negligible cost (i.e., vector subtraction). Thus we have the flexibility in Algorithm 1 to focus on the
proximal operator that is efficiently computable, and the other can be simply obtained through (10).
Remark on convergence: Note that Algorithm 1 is a special case of inexact Bregman ADMM proposed in [20], which matches the case of linearizing quadratic penalty term by using B??? (?, ?k ) =
?
1
2
2 ?? ? ?k ?2 as Bregman divergence. In order to converge, the algorithm requires 2 to be larger than
T
the spectral radius of A A, and the convergence rate is O(1/T ) according to Theorem 2 in [20].
3
2.2 Statistical Recovery for Generalized Dantzig Selector
Our goal is to provide non-asymptotic bounds on ??? ? ? ? ?2 between the true parameter ? ? and
? = ?? ? ? ? . For any set ? ? Rp , we
the minimizer ?? of (1). Let the error vector be defined as ?
would measure the size of this set using its Gaussian width [17, 6], which is defined as ?(?) =
Eg [supz?? ?g, z?] , where g is a vector of i.i.d. standard Gaussian entries. We also consider the
? defined as
error cone TR (? ? ), generated by the set of possible error vectors ? and containing ?,
TR (? ? ) := cone {? ? Rp : R(? ? + ?) ? R(? ? )} .
(11)
Note that this set contains a restricted set of directions and does not in general span the entire space
of Rp . With these definitions, we obtain our main result.
Theorem 1 Suppose that both design matrix X and noise w consists of i.i.d. Gaussian entries with
zero mean variance 1 and X has normalized columns, i.e. ?X(j) ?2 = 1, j = 1, . . . , p. If we solve
the problem (1) with
.
?p ? cE R? (XT w) ,
(12)
where c > 1 is a constant, then, with probability at least (1 ? ?1 exp(??2 n)), we have
/
4 R(? ? )?p
?
?
,
?? ? ? ?2 ?
(?n ? ?(TR (? ? ) ? Sp?1 ))
(13)
where ?(TR (? ? ) ? Sp?1 ) is the Gaussian width of the intersection of the error cone TR (? ? ) and the
unit spherical shell S?p?1 , and ?n is the expected length of a length n i.i.d. standard Gaussian vector
n
with ?n+1
< ?n < n, and ?1 , ?2 > 0 are constants.
Remark: The choice of ?p is also intimately connected to the notion of Gaussian width. Note that
for X with unit length columns, XT w = z is an i.i.d. standard Gaussian vector. Therefore the right
hand side of (12) can be written as
0
1
- ? T .
E R (X w) = E
sup ?u, z? = ? ({u : R(u) ? 1}) ,
(14)
u: R(u)?1
which is the Gaussian width of the unit ball of the norm R(?).
Example: L1 -norm Dantzig Selector When R(?) is chosen to be L1 norm, the dual norm is the
L? norm, and (1) is reduced to the standard DS, given by
?? = argmin ???1
??Rp
s.t. ?XT (y ? X?)?? ? ? .
We know that prox????1 (?) is given by the elementwise soft-thresholding operation
.
prox????1 (x) i = sign(xi ) ? max(0, |xi | ? ?) .
(15)
(16)
Based on Proposition 1, the ADMM updates in Algorithm 1 can be instantiated as
!
2
zk "
? k+1 ? prox 2???1 ? k ? AT (A? k + vk ? u + ) ,
??
?
?
k
!
z
zk "
vk+1 ? (u ? A? k+1 ? ) ? prox????1 u ? A? k+1 ?
,
?
?
zk+1 ? zk + ?(A? k+1 + vk+1 ? u) ,
where the update of v leverages the decomposition (10). Similar updates were used in [21] for
L1 -norm Dantzig selector.
For statistical recovery, we assume that ? ? is s-sparse, i.e., contains s non-zero entries, and that
?? ? ?2 = 1, so that ?? ? ?1 ? s. It was shown in [6] that!the" Gaussian width of the -set (TL1 (? ?.) ?
Sp?1 ) is upper bounded as ?(TL1 (? ? )?Sp?1 )2 ? 2s log ps + 54 s. Also note that E R? (XT w) =
4
E[?XT w?? ] ? log p, since XT w is a vector of i.i.d. standard Gaussian entries [5]. Therefore, if
we solve (15) with ?p = 2 log p, then
5
34
/
? ? log p
32??
s
log
p
1
?
2
??? ? ? ?2 ? $
(17)
( =O
! "
n
?n ? 2s log ps + 54 s
with high probability, which agrees with known results for DS [3, 5].
3 Dantzig Selection with k-support norm
We first introduce some notations. Given any ? ? Rp , let |?| denote its absolute-valued counterpart
and ? ? denote the permutation of ? with its elements arranged in decreasing order. In previous
work [1, 13], the k-support norm has been defined as
?
?
? 9
?
9
???sp
=
min
?v
?
:
supp(v
)
?
I,
v
=
?
,
(18)
I 2
I
I
k
? (k)
?
(k)
I?G
I?G
where G (k) denotes the set of subsets of {1, . . . , p} of cardinality at most k. The unit ball of this
norm is the set Ck = conv {? ? Rp : ???0 ? k, ???2 ? 1} . The dual norm of the k-support norm
is given by
3 k
51
,
+
9 ?2 2
sp?
|?|i
.
(19)
???k = max ??G ?2 : G ? G (k) =
i=1
Note that k = 1 gives the L1 norm and its dual norm is L? norm. The k-support norm was
proposed in order to overcome some of the empirical shortcomings of the elastic net [23] and the
(group)-sparse regularizers. It was shown in [1] to behave similarly as the?elastic net in the sense
that the unit norm ball of the k-support norm is within a constant factor of 2 of the unit elastic net
ball. Although multiple papers have reported good empirical performance of the k-support norm on
selecting correlated features, where L1 regularization fails, there exists no statistical analysis of the
k-support norm. Besides, current computational methods consider square of k-support norm in their
formulation, which might fail to work out in certain cases.
In the rest of this section, we focus on GDS with R(?) = ???sp
k given as
?? = argmin ???sp
k
??Rp
?
?XT (y ? X?)?sp
? ?p .
k
s.t.
(20)
For the indicator function C? (?) of the dual norm, we present a fast algorithm for computing its
proximal operator by exploiting the structure of its solution, which can be directly plugged in Algorithm 1 to solve (20). Further, we prove statistical recovery bounds for k-support norm Dantzig
selection, which hold even for a high-dimensional scenario, where n < p.
3.1 Computation of Proximal Operator
?
In order to solve (20), either prox????sp
(?) or prox C (?) for ? ? ?sp
should be efficiently comk
k
?
putable. Existing methods [1, 13] are inapplicable to our scenario since they compute the proximal
operator for squared k-support norm, from which prox C (?) cannot be directly obtained. In Theo?
rem 2, we show that prox C (?) can be efficiently computed, and thus Algorithm 1 is applicable.
?
?
?
? ?, then w? = prox C (x) = x. If ?x?sp
> ?,
Theorem 2 Given ? > 0 and x ? Rp , if ?x?sp
k
k
?
=r
=
s
?
? 2
define Asr = i=s+1 |x|i , Bs = i=1 (|x|i ) , in which 0 ? s < k and k ? r ? p, and construct
the nonlinear equation of ?,
>
?2
1+?
2
(k ? s)Asr
? ?2 (1 + ?)2 + Bs = 0 .
(21)
r ? s + (k ? s)?
5
Let ?sr be given by
)
nonnegative root of (21) if s > 0 and the root exists
0
otherwise
.
(22)
Then the proximal operator w? = proxIC (x) is given by
?
?
?
1
?
if 1 ? i ? s?
?
1+? ? r? |x|i
?
? 2 ?s2 ?B
?
s?
if s? < i ? r? and ?s? r? = 0
k?s?
|w? |?i =
As ? r ?
?
?
if s? < i ? r? and ?s? r? > 0
?
r ? ?s? +(k?s? )?s? r?
?
?
?
|x|i
if r? < i ? p
,
(23)
?sr =
where the indices s? and r? with computed |w? |? satisfy the following two inequalities:
|w? |?s? > |w? |?k ,
|x|?r? +1
|w? |?k
(24)
|x|?r?
?
<
.
(25)
There might be multiple pairs of (s, r) satisfying the inequalities (24)-(25), and we choose the pair
with the smallest ?|x|? ? |w|? ?2 . Finally, w? is obtained by sign-changing and reordering |w? |? to
conform to x.
Remark: The nonlinear equation (21) is quartic, for which we can use general formula to get all the
roots [18]. In addition, if it exists, the nonnegative root is unique, as shown in the proof [7].
Theorem 2 indicates that computing prox C (?) requires sorting of entries in |x| and a two?
dimensional search of s? and r? . Hence the total time complexity is O(p log p + k(p ? k)). However, a more careful observation can particularly reduce the search complexity from O(k(p ? k)) to
O(log k log(p ? k)), which is motivated by Theorem 3.
Theorem 3 In search of (s? , r? ) defined in Theorem 2, there can be only one r? for a given candidate
s? of s? , such that the inequality (25) is satisfied. Moreover if such r? exists, then for any r < r?, the
? ?k violates the first part of (25), and for r > r?, |w|
? ?k violates the second part of (25).
associated |w|
On the other hand, based on the r?, we have following assertion of s? ,
?
? > s? if r? does not exist
? ?k satisfies (24) .
? s? if r? exists and corresponding |w|
(26)
s?
?
? ?k violates (24)
< s? if r? exists but corresponding |w|
Based on Theorem 3, the accelerated search procedure for finding (s? , r? ) is to execute a twodimensional binary search, and Algorithm 2 gives the details. Therefore the total time complexity
becomes O(p log p + log k log(p ? k)). Compared with previous proximal operators for squared
k-support norm, this complexity is better than that in [1], and roughly the same as the one in [13].
3.2 Statistical Recovery Guarantees for k-support norm
The analysis of the generalized Dantzig Selector for k-support norm consists of addressing two key
challenges. First, note that Theorem 1 requires an appropriate choice of ?p . Second, one needs
to compute the Gaussian width of the subset of the error set TR (? ? ) ? Sp?1 . For the k-support
norm, we can get upper bounds to both of these quantities. We start by defining some notation. Let
G ? ? G (k) be the set of groups intersecting with the support of ? ? , and let S be the union of groups
in G ? , such that s = |S|. Then, we have the following bounds which are used for choosing ?p , and
bounding the Gaussian width.
Theorem 4 For the k-support norm Generalized Dantzig Selection problem (20), we obtain
B2
A4
$ ep (
.
E R? (XT w) ? k
+1
2 log
k
A4
( ? B2 C s D
$
CsD
+2 + k ?
+s.
2k log p ? k ?
?(TR (? ? ) ? Sp?1 )2 ?
k
k
6
(27)
(28)
Algorithm 2 Algorithm for computing prox
?
C?
(?) of ? ? ?sp
k
Input: x, k, ?
Output: w? = prox C (x)
?
sp?
1: if ?x?k ? ? then
2:
w? := x
3: else
4:
l := 0, u := k ? 1, and sort |x| to get |x|?
5:
while l ? u do
? based on (23)
6:
s? := ?(l + u)/2?, and binary search for r? that satisfies (25) and compute w
7:
if r? does not exist then
8:
l := s? + 1
9:
else if r? exists and (24) is satisfied then
? l := s? + 1
10:
w? := w,
11:
else if r? exists but (24) is not satisfied then
12:
u := s? ? 1
13:
end if
14:
end while
15: end if
Our analysis technique for these bounds follows [16]. Similar results were obtained in [8] in the
context of calculating norms of Gaussian vectors, and our bounds are of the same order as those
(2
$2
! "
+
1
2 log ep
, and under the assumptions of Theorem 1, we
of [8]. Choosing ?p = 2k
k
obtain the following result on the error bound for the minimizer of (20).
Corollary 1 Suppose that we obtain i.i.d. Gaussian measurements X, and the noise w is i.i.d.
N (0, 1). If we solve (20) with ?p chosen as above. Then, with high probability, we obtain
$2
?2
/
! " ? (
! " ? ?
+ k
8?? ??sp
2k log ep
sk log kp + sk
k
k
? .
?
??? ? ? ? ?2 ?
= O?
(29)
(?n ? ?(TR (? ? ) ? Sp?1 ))
n
Remark The error bound provides a natural interpretation for the two special cases of the k-support
norm, viz. k = 1 and k = p. First, for kA= 1 the k-support
norm is exactly the same as the L1 norm,
B
2
s log p
and the error bound obtained will be O
, the same as known results of DS, and shown in
n
Section 2.2. Second, for k = p, the k-support norm is equal to the L2 norm, and the error cone
"
!/ (11)
sp
is then simply a half space (there is no structural constraint) and the error bound scales as O
n .
4 Experimental Results
On the optimization side, we focus on the efficiency of different proximal operators related to ksupport norm. On the statistical side, we concentrate on the behavior and performance of GDS with
k-support norm. All experiments are implemented in MATLAB.
4.1 Efficiency of Proximal Operator
We tested four proximal operators related to k-support norm, which are normal prox C (?) in The?
1
orem 2 and the accelerated one in Theorem 3, prox 2?
2 (?) in [1], and prox ? ???2 (?) in [13].
(???sp
k )
2
?
The dimension p of vector varied from 1000 to 10000, and the ratio p/k = {200, 100, 50, 20}. As
illustrated in Figure 1, the speedup of accelerated prox C (?) is considerable compared with the
?
1
normal one and prox 2?
2 (?). It is also slightly better than the prox ? ???2 (?).
(???sp
k )
2
?
4.2 Statistical Recovery on Synthetic Data
Data generation We fixed p = 600, and ? ? = (10, . . . , 10, 10, . . . , 10, 10, . . . , 10, 0, 0, . . . , 0)
I JK L I JK L I JK L I JK L
10
10
10
570
throughout the experiment, in which nonzero entries were divided equally into three groups. The
design matrix X were generated from a normal distribution such that the entries in the same group
7
p / k = 200
p / k = 100
?2
?1
?2
?2
?3
?3
?4
?4
?4
10000
5000
p
10000
0
?1
?3
5000
p
1
0
log(time)
?1
p / k = 20
1
0
log(time)
0
log(time)
p / k = 50
1
log(time)
1
?1
?2
?3
5000
p
10000
?4
5000
p
10000
Figure 1: Efficiency of proximal operators. Diamond: proxIC (?) in Theorem 2, Square: prox
?
1 (???sp )2
k
2?
(?)
in [1], Downward-pointing triangle: prox ? ???2 (?) in [13], Upward-pointing triangle: accelerated proxIC (?)
?
2
?
in Theorem 3. For each (p, k), 200 vectors are randomly generated for testing. Time is measured in seconds.
have the same mean sampled from N (0, 1). X was normalized afterwards. The response vector y
was given by y = X? ? + 0.01 ? N (0, 1). The number of samples n is specified later.
ROC curves with different k We fixed n = 400 to obtain the ROC plot for k = {1, 10, 50} as
shown in Figure 2(a). ?p ranged from 10?2 to 103 . Small k gets better ROC curve.
60
k=1
k = 10
k = 50
?2 error : ??? ? ?? ?2
0.8
0.7
Mean of error
40
0.6
TPR
2.5
k=1
k = 10
k = 50
50
?2 error : ??? ? ?? ?2
1
0.9
0.5
1.5
30
0.4
20
0.3
0.2
2
1
0.5
10
0.1
0
0
0.2
0.4
FPR
0.6
(a) ROC curves
0.8
1
0
0
50
100
150
n
200
(b) L2 error vs. n
250
300
0
0
5
10
15
20
25
30
35
40
k
(c) L2 error vs. k
Figure 2: (a) The true positive rate reaches 1 quite early for k = 1, 10. When k = 50, the ROC gets worse
due to the strong smoothing effect introduced by large k. (b) For each k, the L2 error is large when the sample
is inadequate. As n increases, the error decreases dramatically for k = 1, 10 and becomes stable afterwards,
while the decrease is not that significant for k = 50 and the error remains relatively large. (c) Both mean and
standard deviation of L2 error are decreasing as k increases until it exceeds the number of nonzero entries in
?? , and then the error goes up for larger k.
L2 error vs. n We investigated how the L2 error ??? ? ? ? ?2 of Dantzig selector changes as the
number of samples increases, where k = {1, 10, 50} and n = {30, 60, 90, . . . , 300}. k = 1, 10 give
small errors when n is sufficiently large.
L2 error vs. k We also looked at the L2 error with different k. We again fixed n = 400 and
varied k from 1 to 39. For each k, we repeated the experiment 100 times, and obtained the mean and
standard deviation plot in Figure 2(c). The result shows that the k-support-norm GDS with suitable
k outperforms the L1 -norm DS (i.e. k = 1) when correlated variables present in data.
5 Conclusions
In this paper, we introduced the GDS, which generalizes the standard L1 -norm Dantzig selector to
estimation with any norm, such that structural information encoded in the norm can be efficiently
exploited. A flexible framework based on inexact ADMM is proposed for solving the GDS, which
only requires one of conjugate proximal operators to be efficiently solved. Further, we provide a
unified statistical analysis framework for the GDS, which utilizes Gaussian widths of certain restricted sets for proving consistency. In the non-trivial example of k-support norm, we showed that
the proximal operators used in the inexact ADMM can be computed more efficiently compared to
previously proposed variants. Our statistical analysis for the k-support norm provides the first result
of consistency of this structured norm. Further, experimental results provided sound support to the
theoretical development in the paper.
Acknowledgements
The research was supported by NSF grants IIS-1447566, IIS-1422557, CCF-1451986, CNS1314560, IIS-0953274, IIS-1029711, and by NASA grant NNX12AQ39A.
8
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4,744 | 5,296 | Parallel Feature Selection inspired by Group Testing
Yingbo Zhou?
Utkarsh Porwal?
CSE Department
SUNY at Buffalo
{yingbozh, utkarshp}@buffalo.edu
Ce Zhang
CS Department
University of Wisconsin-Madison
[email protected]
Hung Ngo
CSE Department
SUNY at Buffalo
[email protected]
Christopher R?e
CS Department
Stanford University
[email protected]
XuanLong Nguyen
EECS Department
University of Michigan
[email protected]
Venu Govindaraju
CSE Department
SUNY at Buffalo
[email protected]
Abstract
This paper presents a parallel feature selection method for classi?cation that scales
up to very high dimensions and large data sizes. Our original method is inspired
by group testing theory, under which the feature selection procedure consists of a
collection of randomized tests to be performed in parallel. Each test corresponds
to a subset of features, for which a scoring function may be applied to measure
the relevance of the features in a classi?cation task. We develop a general theory providing suf?cient conditions under which true features are guaranteed to
be correctly identi?ed. Superior performance of our method is demonstrated on
a challenging relation extraction task from a very large data set that have both
redundant features and sample size in the order of millions. We present comprehensive comparisons with state-of-the-art feature selection methods on a range of
data sets, for which our method exhibits competitive performance in terms of running time and accuracy. Moreover, it also yields substantial speedup when used
as a pre-processing step for most other existing methods.
1
Introduction
Feature selection (FS) is a fundamental and classic problem in machine learning [10, 4, 12]. In
classi?cation, FS is the following problem: Given a universe U of possible features, identify a
subset of features F ? U such that using the features in F one can build a model to best predict
the target class. The set F not only in?uences the model?s accuracy, its computational cost, but also
the ability of an analyst to understand the resulting model. In applications, such as gene selection
from micro-array data [10, 4], text categorization [3], and ?nance [22], U may contain hundreds of
thousands of features from which one wants to select only a small handful for F .
While the overall goal is to have an FS method that is both computationally ef?cient and statistically
sound, natural formulations of the FS problem are known to be NP-hard [2]. For large scale data,
scalability is a crucial criterion, because FS often serves not as an end but a means to other sophisticated subsequent learning. In reality, practitioners often resort to heuristic methods, which can
broadly be categorized into three types: wrapper, embedded, and ?lter [10, 4, 12]. In the wrapper
method, a classi?er is used as a black-box to test on any subset of features. In ?lter methods no
classi?er is used; instead, features are selected based on generic statistical properties of the (labeled)
?
* denotes equal contribution
1
data such as mutual information and entropy. Embedded methods have built in mechanisms for FS
as an integral part of the classi?er training. Devising a mathematically rigorous framework to explain and justify FS heuristics is an emerging research area. Recently Brown et al. [4] considered
common FS heuristics using a formulation based on conditional likelihood maximization.
The primary contribution of this paper is a new framework for parallelizable feature selection, which
is inspired by the theory of group testing. By exploiting parallelism in our test design we obtain a
FS method that is easily scalable to millions of features and samples or more, while preserving
useful statistical properties in terms of classi?cation accuracy, stability and robustness. Recall that
group testing is a combinatorial search paradigm [7] in which one wants to identify a small subset of
?positive items? from a large universe of possible items. In the original application, items are blood
samples of WWII draftees and an item is positive if it is infected with syphilis. Testing individual
blood sample is very expensive; the group testing approach is to distribute samples into pools in
a smart way. If a pool is tested negative, then all samples in the pool are negative. On the other
hand, if a pool is tested positive then at least one sample in the pool is positive. We can think of
the FS problem in the group testing framework: there is a presumably small, unknown subset F of
relevant features in a large universe of N features. Both FS and group testing algorithms perform
the same basic operation: apply a ?test? to a subset T of the underlying universe; this test produces
a score, s(T ), that is designed to measure the quality of the features T (or return positive/negative
in the group testing case). From the collection of test scores the relevant features are supposed to
be identi?ed. Most existing FS algorithms can be thought of as sequential instantiations in this
framework1 : we select the set T to test based on the scores of previous tests. For example, let X =
(X1 , . . . , XN ) be a collection of features (variables) and Y be the class label. In the joint mutual
information (JMI) method [25], the feature set T is grown sequentially by adding one feature at each
iteration. The?
next feature?s score, s(Xk ), is de?ned relative to the set of features already selected in
T : s(Xk ) = Xj ?T I(Xk , Xj ; Y ). As each such scoring operation takes a non-negligible amount
of time, a sequential method may take a long time to complete.
A key insight is that group testing needs not be done sequentially. With a good pooling design, all
the tests can be performed in parallel in which we determine the pooling design without knowing
any pool?s test outcome. From the vector of test outcomes, one can identify exactly the collection
of positive blood samples. Parallel group testing, commonly called non-adaptive group testing
(NAGT) is a natural paradigm and has found numerous applications in many areas of mathematics,
computer Science, and biology [18]. It is natural to wonder whether a ?parallel? FS scheme can be
designed for machine learning in the same way NAGT was possible: all feature sets T are speci?ed
in advance, without knowing the scores of any other tests, and from the ?nal collection of scores the
features are identi?ed. This paper initiates a mathematical investigation of this possibility.
At a high level, our parallel feature selection (PFS) scheme has three inter-related components: (1)
the test design indicates the collection of subsets of features to be tested, (2) the scoring function
s : 2[N ] ? R that assigns a score to each test, and (3) the feature identi?cation algorithm that
identi?es the ?nal selected feature set from the test scores. The design space is thus very large. Every
combination of the three components leads to a new PFS scheme.2 We argue that PFS schemes are
preferred over sequential FS for two reasons:
1. scalability, the tests in a PFS schem can be performed in parallel, and thus the scheme can
be scaled to large datasets using standard parallel computing techniques, and
2. stability, errors in individual trials do not affect PFS methods as dramatically as sequential
methods. In fact, we will show in this paper that increasing the number of tests improves
the accuracy of our PFS scheme.
We propose and study one such PFS approach. We show that our approach has comparable (and
sometimes better) empirical quality compared to previous heuristic approaches while providing
sound statistical guarantees and substantially improved scalability.
Our technical contributions We propose a simple approach for the ?rst and the third components
of a PFS scheme. For the second component, we prove a suf?cient condition on the scoring function
under which the feature identi?cation algorithm we propose is guaranteed to identify exactly the set
1
A notable exception is the MIM method, which is easily parallelizable and can be regarded as a special
implementation of our framework
2
It is important to emphasize that this PFS framework is applicable to both ?lter and wrapper approaches.
In the wrapper approach, the score s(T ) might be the training error of some classi?er, for instance.
2
of original (true) features. In particular, we introduce a notion called C-separability, which roughly
indicates the strength of the scoring function in separating a relevant feature from an irrelevant
feature. We show that when s is C-separable and we can estimate s, we are able to guarantee exact
recovery of the right set of features with high probability. Moreover, when C > 0, the number of
tests can be asymptotically logarithmic in the number of features in U .
In theory, we provide suf?cient conditions (a Na??ve Bayes assumption) according to which one can
obtain separable scoring functions, including the KL divergence and mutual information (MI). In
practice, we demonstrate that MI is separable even when the suf?cient condition does not hold,
and moreover, on generated synthetic data sets, our method is shown recover exactly the relevant
features. We proceed to provide a comprehensive evaluation of our method on a range of real-world
data sets of both large and small sizes. It is the large scale data sets where our method exhibits
superior performance. In particular, for a huge relation extraction data set (TAC-KBP) that has
millions redundant features and samples, we outperform all existing methods in accuracy and time,
in addition to generating plausible features (in fact, many competing methods could not ?nish the
execution). For the more familiar NIPS 2013 FS Challenge data, our method is also competitive
(best or second-best) on the two largest data sets. Since our method hinges on the accuracy of score
functions, which is dif?cult achieve for small data, our performance is more modest in this regime
(staying in the middle of the pack in terms of classi?cation accuracy). Nonetheless, we show that our
method can be used as a preprocessing step for other FS methods to eliminate a large portion of the
feature space, thereby providing substantial computational speedups while retaining the accuracy of
those methods.
2
Parallel Feature Selection
The general setting Let N be the total number of input features. For each subset T ? [N ] :=
{1, . . . , N }, there is a score s(T ) normalized to be in [0, 1] that assesses the ?quality? of features in
T . We select a collection of t tests, each of which is a subset T ? [N ] such that from the scores
of all tests we can identify the unknown subset F of d relevant variables that are most important
to the classi?cation task. We encode the collection of t tests with a binary matrix A = (aij ) of
dimension t ? N , where aij = 1 iff feature j belongs to test i. Corresponding to each row i of A is
a ?test score? si = s({j | aij = 1}) ? [0, 1]. Specifying A is called test design, identifying F from
the score vector (si )i?[t] is the job of the feature identi?cation algorithm. The scheme is inherently
parallel because all the tests must be speci?ed in advance and executed in parallel; then the features
are selected from all the test outcomes.
Test design and feature identi?cation Our test design and feature identi?cation algorithms are
extremely simple. We construct the test matrix A randomly by putting a feature in the test with
probability p (to be chosen later). Then, from the test scores we rank the features and select d
top-ranked features. The ranking function is de?ned as follows. Given a t ? N test matrix A, let
aj denote its jth column. The dot-product ?aj , s? is the total score of all the tests that feature j
participates in. We de?ne ?(j) = ?aj , s? to be the rank of feature j with respect to the test matrix
A and the score function s.
The scoring function The crucial piece stiching together the entire scheme is the scoring function. The following theorem explains why the above test design and feature identi?cation strategy
make sense, as long as one can choose a scoring function s that satis?es a natural separability property. Intuitively, separable scoring functions require that adding more hidden features into a test set
increase its score.
De?nition 2.1 (Separable scoring function). Let C ? 0 be a real number. The score function
s : 2[N ] ? [0, 1] is said to be C-separable if the following property holds: for every f ? F and
f? ?
/ F , and for every T ? [N ] ? {f, f?}, we have s(T ? {f }) ? s(T ? {f?}) ? C.
In words, with a separable scoring function adding a relevant feature should be better than adding
an irrelevant feature to a given subset T of features. Due to space limination, the proofs of the
following theorem, propositions, and corollaries can be found in the supplementary materials. The
essence of the idea is that, when s can separate relevant features from irrelevant features, with high
probability a relevant feature will be ranked higher than an irrelevant feature. Hoeffding?s inequality
is then used to bound the number of tests.
3
Theorem 2.2. Let A be the random t ? N test matrix obtained by setting each entry to be 1 with
probability p ? [0, 1] and 0 with probability 1 ? p. If the scoring function s is C-separable, then the
expected rank of a feature in F is at least the expected rank of a feature not in F .
Furthermore, if C > 0, then for any ? ? (0, 1), with probability at least 1 ? ? every feature in F
has rank higher than every feature not in F , provided that the number of tests t satis?es
?
?
d(N ? d)
2
t? 2 2
log
.
(1)
C p (1 ? p)2
?
By setting p = 1/2 in the above theorem, we obtain the following. It is quite remarkable that,
assuming we can estimate the scores accurately, we only need about O(log N ) tests to identify F .
Corollary 2.3. Let C > 0 be a constant such that there is a C-separable scoring function s. Let
d = |F |, where F is the set of hidden features. Let ? ? (0, 1) be an arbitrary constant. Then, there
is a distribution of t ? N test matrices A with t = O(log(d(N ? d)/?)) such that, by selecting a
test matrix randomly from the distribution, the d top-ranked features are exactly the hidden features
with probability at least 1 ? ?.
Of course, in reality estimating the scores accurately is a very dif?cult problem, both statistically
and computationally, depending on what the scoring function is. We elaborate more on this point
below. But ?rst, we show that separable scoring functions exist, under certain assumption about the
underlying distribution.
Suf?cient conditions for separable scoring functions We demonstrate the existence of separable
scoring functions given some suf?cient conditions on the data. In practice, loss functions such as
classi?cation error and other surrogate losses may be used as scoring functions. For binary classi?cation, information-theoretic quantities such as Kullback-Leibler divergence, Hellinger distance and
the total variation ? all of which special cases of f -divergences [5, 1] ? may also be considered.
For multi-class classi?cation, mutual information (MI) is a popular choice.
The data pairs (X, Y ) are assumed to be iid samples from a joint distribution P (X, Y ). The following result shows that under the so-called ?naive Bayes? condition, i.e., all components of random
vector X are conditionally independent given label variable Y , the Kullback-Leibler distance is a
separable scoring function in a binary classi?cation setting:
Proposition 2.4. Consider the binary classi?cation setting, i.e., Y ? {0, 1} and assume that the
naive Bayes condition holds. De?ne score function to be the Kullback-Leibler divergence:
s(T ) := KL(P (X T |Y = 0)||P (X T |Y = 1)).
Then s is a separable scoring function. Moreover, s is C-separable, where C := minf ?F s(f ).
Proposition 2.5. Consider the multi-class classi?cation setting, and assume that the naive Bayes
condition holds. Moreover, for any pair f ? F and f? ?
/ F , the following holds for any T ?
[N ] ? {f, f?}
I(Xf ; Y ) ? I(Xf ; X T ) ? I(Xf?; Y ) ? I(Xf?; X T ).
Then, the MI function s(T ) := I(XT ; Y ) is a separable scoring function.
We note the naturalness of the condition so required, as quantity I(Xf ; Y ) ? I(Xf ; XT ) may be
viewed as the relevance of feature f with respect to the label Y , subtracted by the redundancy with
other existing features T . If we assume further that X f? is independent of both X T and the label Y ,
and there is a positive constant C such that I(Xf ; Y ) ? I(Xf ; XT ) ? C for any f ? F , then s(T )
is obviously a C-separable scoring function. It should be noted that the naive Bayes conditions are
suf?cient, but not necessary for a scoring function to be C-separable.
Separable scoring functions for ?lters and wrappers. In practice, information-based scoring
functions need to be estimated from the data. Consistent estimators of scoring functions such as KL
divergence (more generally f -divergences) and MI are available (e.g., [20]). This provides the theoretical support for applying our test technique to ?lter methods: when the number of training data is
suf?ciently large, a consistent estimate of a separable scoring function must also be a separable scoring function. On the other hand, a wrapper method uses a classi?cation algorithm?s performance as
a scoring function for testing. Therefore, the choice of the underlying (surrogate) loss function plays
a critical role. The following result provides the existence of loss functions which induce separable
scoring functions for the wrapper method:
4
T
Proposition 2.6. Consider the binary classi?cation setting, and let P0T :=
? P (X T |Y = 0), P1 :=
P (X T |Y = 1). Assume that an f -divergence of the form: s(T ) = ?(dP0T /dP1T )dP1T is a
separable scoring function for some convex function ? : R+ ? R. Then there exists a surrogate
loss function l : R ? R ? R+ under which the minimum l-risk: Rl (T ) := inf g E [l(Y, g(X T ))] is
also a separable scoring function. Here the in?mum is taken over all measurable classi?er functions
g acting on feature input X T , E denotes expectation with respect to the joint distribution of X T
and Y .
This result follows from Theorem 1 of [19], who established a precise correspondence between f divergences de?ned by convex ? and equivalent classes of surrogate losses l. As a consequence,
if the Hellinger distance between P0T and P1T is separable, then the wrapper method using the
Adaboost classi?er corresponds to a separable scoring function. Similarly, a separable KullbackLeibler divergence implies that of a logistic regression based wrapper; while a separable variational
distance implies that of a SVM based wrapper.
3
Experimental results
3.1 Synthetic experiments
In this section, we synthetically illustrate that separable scoring functions exist and our PFS framework is sound beyond the Na??ve Bayes assumption (NBA). We ?rst show that MI is C-separable for
large C even when the NBA is violated. The NBA was only needed in Propositions 2.4 and 2.5 in
order for the proofs to go through. Then, we show that our framework recovers exactly the relevant
features for two common classes of input distributions.
We generate 1, 000 data points from two separated 2-D Gaussians with the same covariance matrix but different means, one centered
at (?2, ?2) and the other at (2, 2). We start
with the identity covariance matrix, and gradually change the off diagonal element to ?0.999,
representing highly correlated features. Then,
we add 1,000 dimensional zero mean Gaussian
noise with the same covariance matrix, where
the diagonal is 1 and the off-diagonal elements
increases from 0 gradually to 0.999. We then
calculate the MI between two features and the Figure 1: Illustration of MI as a separable scorclass label, and the two features are selected in ing function for the case of statistically dependent
three settings: 1) the two genuine dimensions; features. The top left point shows the scores for
2) one of the genuine feature and one from the the 1st setting; the middle points shows the scores
noisy dimensions; 3) two random pair from the for the 2nd setting; and the bottom points shows
noisy dimensions. The MI that we get from the scores for the 3rd setting.
these three conditions is shown in Figure 1. It is clear from this ?gure MI is a separable scoring
function, despite the fact that the NBA is violated.
We also synthetically evaluated our entire PFS idea, using two multinomials and two Gaussians to
generate two binary classi?cation task data. Our PFS scheme is able to capture exactly the relevant
features in most cases. Details are in the supplementary material section due to lack of space.
3.2 Real-world data experiment results
This section evaluates our approach in terms of accuracy, scalability, and robustness accross a range
of real-world data sets: small, medium, and large. We will show that our PFS scheme works very
well on medium and large data sets; because, as was shown in Section 3.1, with suf?cient data to
estimate test scores, we expect our method to work well in terms of accuracy. On the small datasets,
our approach is only competitive and does not dominate existing approaches, due to the lack of data
to estimate scores well. However, we show that we can still use our PFS scheme as a pre-processing
step to ?lter down the number of dimensions; this step reduces the dimensionality, helps speed up
existing FS methods from 3-5 times while keeps their accuracies.
3.2.1 The data sets and competing methods
Large: TAC-KBP is a large data set with the number of samples and dimensions in the millions3 ;
its domain is on relation extraction from natural language text. Medium: GISETTE and MADE3
http://nlp.cs.qc.cuny.edu/kbp/2010/
5
LON are two largest data sets from the NIPS 2003 feature selection challenge4 , with the number of
dimensions in the thousands. Small: Colon, Leukemia, Lymph, NCI9, and Lung are chosen from
the small Micro-array datasets [6], along with the UCI datasets5 . These sets typically have a few
hundreds to a few thousands variables, with only tens of data samples.
We compared our method with various baseline methods including mutual information
maximization[14] (MIM), maximum relevancy minimum redundancy[21] (MRMR), conditional
mutual information maximization[9] (CMIM), joint mutual information[25] (JMI), double input
symmetrical relevance[16] (DISR), conditional infomax feature extraction[15] (CIFE), interaction
capping[11] (ICAP), fast correlation based ?lter[26] (FCBF), local learning based feature selection
[23] (LOGO), and feature generating machine [24] (FGM).
3.2.2
Accuracy
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Figure 2: Result from different methods on TAC-KBP dataset. (a) Precision/Recall of different
methods; (b) Top-5 keywords appearing in the Top-20 features selected by our method. Dotted lines
in (a) are FGM (or MIM) with our approach as pre-processing step.
Accuracy results on large data set. As shown in Figure 2(a), our method dominates both MIM
and FGM. Given the same precision, our method achieves 2-14? higher recall than FGM, and 1.22.4? higher recall than MIM. Other competitors do not ?nish execution in 12 hours. We compare the
top-features produced by our method and MIM, and ?nd that our method is able to extract features
that are strong indicators only when they are combined with other features, while MIM, which tests
features individually, ignores this type of combination. We then validate that the features selected
by our method makes intuitive sense. For each relation, we select the top-20 features and report the
keyword in these features.6 As shown in Figure 2(b), these top-features selected by our method are
good indicators of each relation. We also observe that using our approach as the pre-processing step
improves the quality of FGM signi?cantly. In Figure 2(a) (the broken lines), we run FGM (MIM)
on the top-10K features produced by our approach. We see that running FGM with pre-processing
achieves up to 10? higher recall given the same precision than running FGM on all 1M features.
Accuracy results on medium data sets Since the focus of the evaluation is to analyze the ef?cacy
of feature selection approaches, we employed the same strategy as Brown et al.[4] i.e. the ?nal
classi?cation is done using k-nearest neighbor classi?er with k ?xed to three, and applied Euclidean
distance7 .
We denote our method by Fk (and Wk ), where F denotes ?lter (and W denotes wrapper method).
k denotes the number of tests (i.e. let N be the dimension of data, then the total number of tests is
kN ). We bin each dimension of the data into ?ve equal distanced bins when the data is real valued,
otherwise the data is not processed8 . MI is used as the scoring function for ?lter method, and loglikelihood is used for scoring the wrapper method. The wrapper we used is logistic regression9 .
4
5
6
http://www.nipsfsc.ecs.soton.ac.uk/datasets/
http://archive.ics.uci.edu/ml/
Following the syntax used by Mintz et al. [17], if a feature has the form [?poss wif e ?prop of ], we report
the keyword as wife in Figure 2(b).
7
The classi?er for FGM is linear support vector machine (SVM), since it optimized for the SVM criteria.
8
For SVM based method, the real valued data is not processed, and all data is normalized to have unit length.
9
The logistic regressor used in wrapper is only to get the testing scores, the ?nal classi?cation scheme is
still k-NN.
6
(a)
(b)
Figure 3: Result from real world datasets: a) curve showing the ratio between the errors of various
methods applied on original data and on ?ltered data, where a large portion of the dimension is
?ltered out (value larger than one indicates performance improvement); b) the speed up we get by
applying our method as a pre-processing method on various methods across different datasets, the
?at dashed line indicates the location where the speed up is one.
For GISETTE we select up to 500 features and for MADELON we select up to 100 features. To get
the test results, we use the features according to the smallest validation error for each method, and
the results on test set are illustrated in table 4.
Table 1: Test set balanced error rate (%) from different methods on NIPS datasets
Datasets
Best
Perf.
2nd Best
Perf.
3rd Best
Perf.
Median
Perf.
Ours
(F3 )
Ours
(W3 )
Ours
(F10 )
Ours
(W10 )
GISETTE
MADELON
2.15
10.61
3.06
11.28
3.09
12.33
3.86
25.92
4.85
22.61
2.72
10.17
4.69
18.39
2.89
10.50
Accuracy results on the small data sets. As expected, due to the lack of data to estimate scores,
our accuracy performance is average for this data set. Numbers can be found in the supplementary
materials. However, as suggested by theorem A.3 (in supplementary materials), our method can also
be used as a preprocessing step for other feature selection method to eliminate a large portion of the
features. In this case, we use the ?lter methods to ?lter out e + 0.1 of the input features, where e is
the desired proportion of the features that one wants to reserve.
Using our method as preprocessing step achieves 3-5 times speedup as compare to the time spend
by original methods that take multiple passes through the datasets, and keeps or improves the performance in most of the cases (see ?gure 3 a and b). The actual running time can be found in
supplementary materials.
Scalability
???????????????
3.2.3
3600000
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360000
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?????? ??????
3600000
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?????????????????
3600000
?
360000
?
360000
?
36000
?
36000
?
3600
?
3600
?
??????????????
???????????????
36000
?
???????????
3600
?
360
?
1
?
10
?
100
?
???????????
1000
?
360
?
10000
?
100000
?
1000000
?
??????????????
360
?
10000
?
100000
?
1000000
?
??????????????
Figure 4: Scalability Experiment of Our Approach
We validate that our method is able to run on large-scale data set ef?ciently, and the ability to take
advantage of parallelism is the key to its scalability.
7
Experiment Setup Given the TAC-KBP data set, we report the execution time by varying the
degree of parallelism, number of features, and number of examples. We ?rst produce a series of
data sets by sub-sampling the original data set with different number examples ({104 , 105 , 106 })
and number of features ({104 , 105 , 106 }). We also try different degree of parallelism by running
our approach using a single thread, 4-threads on a 4-core CPU, 32 threads on a single 8-CPU (4core/CPU) machine, and multiple machines available in the national Open Science Grid (OSG).
For each combination of number of features, number of examples, and degree of parallelism, we
estimate the throughput as the number of tests that we can run in 1 second, and estimate the total
running time accordingly. We also ran our largest data set (106 rows and 106 columns) on OSG and
report the actual run time.
Degree of Parallelism Figure 4(a) reports the (estimated) run time on the largest data set (106
rows and 106 columns) with different degree of parallelism. We ?rst observe that running our
approach requires non-trivial amount of computational resources?if we only use a single thread, we
need about 400 hours to ?nish our approach. However, the running time of our approach decreases
linearly with the number of cores that we used. If we run our approach on a single machine with 32
cores, it ?nishes in just 11 hours. This linear speed-up behavior allows our approach to scale to very
large data set?when we run our approach on the national Open Science Grid, we observed that our
approach is able to ?nish in 2.2 hours (0.7 hours for actual execution, and 1.5 hours for scheduling
overhead).
The Impact of Number of Features and Number of Examples Figure 4(b,c) report the run time
with different number of features and number of examples, respectively. In Figure 4(b), we ?x the
number of examples to be 105 , and vary the number of features, and in Figure 4(c), we ?x the number
of features to be 106 and vary the number of examples. We see that as the number of features or the
number of examples increase, our approach uses more time; however, the running time never grows
super-linearly. This behavior implies the potential of our approach to scale to even larger data sets.
3.2.4
Stability and robustness
Our method exhibits several robustness properties. In particular, the proof of Theorem 2.2 suggests
that as the number of tests are increased the performance also improves. Therefore, in this section
we empirically evaluate this observation. We picked four datasets: KRVSKP, Landset, Splice and
Waveform from the UCI datasets and both NIPS datasets.
(a)
(b)
(c)
(d)
Figure 5: Change of performance with respect of number of tests on several UCI datasets with (a)
?lter and (b) wrapper methods; and (c) GISETTE and (d) MADELON datasets.
The trend is pretty clear as can be observed from ?gure 5. The performance of both wrapper and
?lter methods improves as we increase the number of tests, which can be attributed to the increase of
robustness against inferior estimates for the test scores as the number of tests increases. In addition,
apart from MADELON dataset, the performance converges fast, normally around k = 10 ? 15.
Additional stability experiments can be found in the supplementary materials, where we evaluate
ours and other methods in terms of consistency index.
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9
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4,745 | 5,297 | Spectral k-Support Norm Regularization
Andrew M. McDonald, Massimiliano Pontil, Dimitris Stamos
Department of Computer Science
University College London
{a.mcdonald,m.pontil,d.stamos}@cs.ucl.ac.uk
Abstract
The k-support norm has successfully been applied to sparse vector prediction
problems. We observe that it belongs to a wider class of norms, which we call the
box-norms. Within this framework we derive an efficient algorithm to compute
the proximity operator of the squared norm, improving upon the original method
for the k-support norm. We extend the norms from the vector to the matrix setting
and we introduce the spectral k-support norm. We study its properties and show
that it is closely related to the multitask learning cluster norm. We apply the norms
to real and synthetic matrix completion datasets. Our findings indicate that spectral k-support norm regularization gives state of the art performance, consistently
improving over trace norm regularization and the matrix elastic net.
1
Introduction
In recent years there has been a great deal of interest in the problem of learning a low rank matrix
from a set of linear measurements. A widely studied and successful instance of this problem arises
in the context of matrix completion or collaborative filtering, in which we want to recover a low
rank (or approximately low rank) matrix from a small sample of its entries, see e.g. [1, 2]. One
prominent method to solve this problem is trace norm regularization: we look for a matrix which
closely fits the observed entries and has a small trace norm (sum of singular values) [3, 4, 5]. Besides
collaborative filtering, this problem has important applications ranging from multitask learning, to
computer vision and natural language processing, to mention but a few.
In this paper, we propose new techniques to learn low rank matrices. These are inspired by the notion
of the k-support norm [6], which was recently studied in the context of sparse vector prediction and
shown to empirically outperform the Lasso [7] and Elastic Net [8] penalties. We note that this
norm can naturally be extended to the matrix setting and its characteristic properties relating to the
cardinality operator translate in a natural manner to matrices. Our approach is suggested by the
observation that the k-support norm belongs to a broader class of norms, which makes it apparent
that they can be extended to spectral matrix norms. Moreover, it provides a link between the spectral
k-support norm and the cluster norm, a regularizer introduced in the context of multitask learning
[9]. This result allows us to interpret the spectral k-support norm as a special case of the cluster
norm and furthermore adds a new perspective of the cluster norm as a perturbation of the former.
The main contributions of this paper are threefold. First, we show that the k-support norm can
be written as a parametrized infimum of quadratics, which we term the box-norms, and which are
symmetric gauge functions. This allows us to extend the norms to orthogonally invariant matrix
norms using a classical result by von Neumann [10]. Second, we show that the spectral box-norm
is essentially equivalent to the cluster norm, which in turn can be interpreted as a perturbation of
the spectral k-support norm, in the sense of the Moreau envelope [11]. Third, we use the infimum
framework to compute the box-norm and the proximity operator of the squared norm in O(d log d)
time. Apart from improving on the O(d(k + log d)) algorithm in [6], this method allows one to use
optimal first order optimization algorithms [12] with the cluster norm. Finally, we present numerical
1
experiments which indicate that the spectral k-support norm shows a significant improvement in
performance over regularization with the trace norm and the matrix elastic net, on four popular
matrix completion benchmarks.
The paper is organized as follows. In Section 2 we recall the k-support norm, and define the boxnorm. In Section 3 we study their properties, we introduce the corresponding spectral norms, and
we observe the connection to the cluster norm. In Section 4 we compute the norm and we derive
a fast method to compute the proximity operator. Finally, in Section 5 we report on our numerical
experiments. The supplementary material contains derivations of the results in the body of the paper.
2
Preliminaries
In this section, we recall the k-support norm and we introduce the box-norm and its dual. The ksupport norm k ? k(k) was introduced in [6] as the norm whose unit ball is the convex hull of the
set of vectors of cardinality at most k and `2 -norm no greater than one. The authors show that the
k-support norm can be written as the infimal convolution [11]
8
9
<X
=
X
kwk(k) = inf
kvg k2 : vg 2 Rd , supp(vg ) ? g,
v g = w , w 2 Rd ,
(1)
:
;
g2Gk
g2Gk
where Gk is the collection of all subsets of {1, . . . , d} containing at most k elements, and for any
v 2 Rd , the set supp(v) = {i : vi 6= 0} denotes the support of v. When used as a regularizer,
the norm encourages vectors w to be a sum of a limited number of vectors with small support. The
k-support norm is a special case of the group lasso with overlap [13], where the cardinality of the
support sets is at most k. Despite the complicated form of the primal norm, the dual norm has a
simple formulation, namely the `2 -norm of the k largest components of the vector
v
u k
uX
kuk?,(k) = t (|u|#i )2 , u 2 Rd ,
(2)
i=1
where |u|# is the vector obtained from u by reordering its components so that they are non-increasing
in absolute value [6]. The k-support norm includes the `1 -norm and `2 -norm as special cases. This
is clear from the dual norm since for k = 1 and k = d, it is equal to the `1 -norm and `2 -norm,
respectively. We note that while definition (1) involves a combinatorial number of variables, [6]
observed that the norm can be computed in O(d log d).
We now define the box-norm, and in the following section we will show that the k-support norm is
a special case of this family.
Pd
Definition 2.1. Let 0 ? a ? b and c 2 [ad, bd] and let ? = {? 2 Rd : a ? ?i ? b, i=1 ?i ? c}.
The box-norm is defined as
v
u
d
u
X
wi2
kwk? = t inf
, w 2 Rd .
(3)
?2? i=1 ?i
This formulation will be fundamental in deriving the proximity operator in Section 4.1. Note that
we may assume without loss of generality that b = 1, as by rescaling we obtain an equivalent norm,
however we do not explicitly fix b in the sequel.
s
d
P
Proposition 2.2. The norm (3) is well defined and the dual norm is kuk?,? = sup
?i u2i .
?2? i=1
The result holds true in the more general case that ? is a bounded convex subset of the strictly
positive orthant (for related results see [14, 15, 16, 17, 18, 19] and references therein). In this
paper we limit ourselves to the box constraints above. In particular we note that the constraints are
invariant with respect to permutation of the components of ?, and as we shall see this property is
key to extend the norm to matrices.
2
3
Properties of the Norms
In this section, we study the properties of the vector norms, and we extend the norms to the matrix
setting. We begin by deriving the dual box-norm.
Proposition 3.1. The dual box-norm is given by
q
kuk?,? = akuk22 + (b a)kuk2?,(k) + (b a)(? k)(|u|#k+1 )2 ,
(4)
where ? =
c da
b a
and k is the largest integer not exceeding ?.
We see from (4) that the dual norm decomposes into two `2 -norms plus a residual term, which
vanishes if ? = k, and for the rest of this paper we assume this holds, which loses little generality.
Note that setting a = 0, b = 1, and c = k 2 {1, . . . , d}, the dual box-norm (4) is the `2 -norm of the
largest k components of u, and we recover the dual k-support norm in equation (2). It follows that
the k-support norm is a box-norm with parameters a = 0, b = 1, c = k.
The following infimal convolution interpretation of the box-norm provides a link between the boxnorm and the k-support norm, and illustrates the effect of the parameters.
Proposition 3.2. If 0 < a ? b and c = (b a)k + da, for k 2 {1, . . . , d}, then
8
9
v
<X u
=
2
2
X
X
X
vg,i
vg,i
u
t
kwk? = inf
+
: v g 2 Rd ,
vg = w .
(5)
:
;
b
a
g2Gk
i2g
i2g
/
g2Gk
Notice that if b = 1, then as a tends to zero, we obtain the expression of the k-support norm (1),
recovering in particular the support constraints. If a is small and positive, the support constraints
are not imposed, however effectively most of the weight for each vg tends to be concentrated on
supp(g). Hence, Proposition 3.2 suggests that the box-norm regularizer will encourage vectors w
whose dominant components are a subset of a union of a small number of groups g 2 Gk .
The previous results have characterized the k-support norm as a special case of the box-norm. Conversely, the box-norm can be seen as a perturbation of the k-support norm with a quadratic term.
Proposition 3.3. Let k?k? be the box-norm on Rd with parameters 0 < a < b and c = k(b a)+da,
for k 2 {1, . . . , d}, then
?
1
1
2
kwk? = min
kw zk22 +
kzk2(k) .
(6)
d
a
b a
z2R
Consider the regularization problem minw2Rd kXw yk22 + kwk2? , with data X and response y.
Using Proposition 3.3 and setting w = u + z, we see that this problem is equivalent to
?
min kX(u + z) yk22 + kuk22 +
kzk2(k) .
a
b a
u,z2Rd
Furthermore, if (?
u, z?) solves this problem then w
?=u
? + z? solves problem (6). The solution w
? can
therefore be interpreted as the superposition of a vector which has small `2 norm, and a vector which
has small k-support norm, with the parameter a regulating these two components. Specifically, as
a tends to zero, in order to prevent the objective from blowing up, u
? must also tend to zero and we
recover k-support norm regularization. Similarly, as a tends to b, z? vanishes and we have a simple
ridge regression problem.
3.1
The Spectral k-Support Norm and the Spectral Box-Norm
We now turn our focus to the matrix norms. For this purpose, we recall that a norm k ? k on Rd?m is
called orthogonally invariant if kW k = kU W V k, for any orthogonal matrices U 2 Rd?d and
V 2 Rm?m . A classical result by von Neumann [10] establishes that a norm is orthogonally
invariant if and only if it is of the form kW k = g( (W )), where (W ) is the vector formed by
the singular values of W in nonincreasing order, and g is a symmetric gauge function, that is a norm
which is invariant under permutations and sign changes of the vector components.
3
Lemma 3.4. If ? is a convex bounded subset of the strictly positive orthant in Rd which is invariant
under permutations, then k ? k? is a symmetric gauge function.
In particular, this readily applies to both the k-support norm and box-norm. We can therefore extend
both norms to orthogonally invariant norms, which we term the spectral k-support norm and the
spectral box-norm respectively, and which we write (with some abuse of notation) as kW k(k) =
k (W )k(k) and kW k? = k (W )k? . We note that since the k-support norm subsumes the `1 and
`2 -norms for k = 1 and k = d respectively, the corresponding spectral k-support norms are equal
to the trace and Frobenius norms respectively. We first characterize the unit ball of the spectral
k-support norm.
Proposition 3.5. The unit ball of the spectral k-support norm is the convex hull of the set of matrices
of rank at most k and Frobenius norm no greater than one.
Referring to the unit ball characterization of the k-support norm, we note that the restriction on the
cardinality of the vectors whose convex hull defines the unit ball naturally extends to a restriction
on the rank operator in the matrix setting. Furthermore, as noted in [6], regularization using the
k-support norm encourages vectors to be sparse, but less so that the `1 -norm. In matrix problems, as
the extreme points of the unit ball have rank k, Proposition 3.5 suggests that the spectral k-support
norm for k > 1 should encourage matrices to have low rank, but less so than the trace norm.
3.2
Cluster Norm
We end this section by briefly discussing the cluster norm, which was introduced in [9] as a convex
relaxation of a multitask clustering problem. The norm is defined, for every W 2 Rd?m , as
r
kW kcl =
inf tr(S 1 W > W )
(7)
S2Sm
where Sm = {S 2 Rm?m , S ? 0 : aI
S
bI, tr S = c}, and 0 < a ? b. In [9] the authors
state that the cluster norm of W equals the box-norm of the vector formed by the singular values of
W where c = (b a)k + da. Here we provide a proof of this result. Denote by i (?) the eigenvalues
of a matrix which we write in nonincreasing order 1 (?)
???
2 (?)
d (?). Note that if ?i are
the eigenvalues of S then ?i = d i+1 (S 1 ). We have that
tr(S
1
W > W ) = tr(S
1
U ?2 U > )
m
X
1
d i+1 (S
) i (W > W ) =
i=1
d
X
2
i (W )
i=1
?i
where we have used the inequality [20, Sec. H.1.h] for S 1 , W > W ? 0. Since this inequality is
attained whenever S = U Diag(?)U , where U are the eigenvectors of W > W , we see that kW kcl =
k (W )k? , that is, the cluster norm coincides with the spectral box-norm. In particular, we see that
the spectral k-support norm is a special case of the cluster norm, where we let a tend to zero, b = 1
and c = k. Moreover, the methods to compute the norm and its proximity operator described in the
following section can directly be applied to the cluster norm.
As in the case of the vector norm (Proposition 3.3), the spectral box-norm or cluster norm can be
written as a perturbation of spectral k-support norm with a quadratic term.
Proposition 3.6. Let k ? k? be a matrix box-norm with parameters a, b, c and let k =
1
kW k2? = min kW
Z a
Zk2F +
1
b
a
c da
b a .
Then
kZk2(k) .
In other words, this result shows that the cluster norm can be seen as the Moreau envelope [11] of a
spectral k-support norm.
4
Computing the Norms and their Proximity Operator
In this section, we compute the norm and the proximity operator of the squared norm by explicitly
solving the optimization problem in (3). We begin with the vector norm.
4
Theorem 4.1. For every w 2 Rd it holds that
kwk2? =
1
1
1
kwQ k22 + kwI k21 + kwL k22 ,
b
p
a
where wQ = (|w|#1 , . . . , |w|#q ), wI = (|w|#q+1 , . . . , |w|#d ` ), wL = (|w|#d
` are the unique integers in {0, . . . , d} that satisfy q + ` ? d,
|wq |
b
p=c
qb
d `
1 X
|wq+1 |
|wi | >
,
p i=q+1
b
|wd ` |
a
(8)
#
`+1 , . . . , |w|d ),
d `
1 X
|wd `+1 |
|wi | >
,
p i=q+1
a
and q and
(9)
`a and we have defined |w0 | = 1 and |wd+1 | = 0.
Proof. (Sketch) We need to solve the optimization problem
inf
?
?X
d
i=1
d
X
wi2
: a ? ?i ? b,
?i ? c .
?i
i=1
(10)
We assume without loss of generality that the wi are ordered nonincreasing in absolute values, and
it follows that at the optimum the ?i are also ordered nonincreasing. We further assume that wi 6= 0
for all i and c ? db, so the sum constraint will be tight at the optimum. The Lagrangian is given by
!
d
d
X
wi2
1 X
L(?, ?) =
+ 2
?i c
?
?
i=1 i
i=1
Pd
where 1/?2 is a strictly positive multiplier to be chosen such that S(?) := i=1 ?i (?) = c. We
can then solve the original problem by minimizing the Lagrangian over the constraint ? 2 [a, b]d .
Due to the decoupling effect of the multiplier we can solve the simplified problem componentwise,
obtaining the solution
?i = ?i (?) = min(b, max(a, ?|wi |))
(11)
where S(?) = c. The minimizer has the form ? = (b, . . . , b, ?q+1 , . . . , ?d ` , a, . . . , a), where q, `
Pd `
are determined by the value of ?. From S(?) = c we get ? = p/( i=q+1 |wi |). The value
of the norm in (8) follows by substituting ? into the objective. Finally, by construction we have
?q b > ?q+1 and ?d ` > a ?d `+1 , which give rise to the conditions in (9).
Theorem 4.1 suggests two methods for computing the box-norm. First we find ? such that S(?) = c;
this value uniquely determines ? in (11), and the norm follows by substitution into (10). Alternatively, we identify q and ` that jointly satisfy (9) and we compute the norm using (8). Taking
advantage of the structure of ? in the former method leads to a computation time that is O(d log d).
Theorem 4.2. The computation of the box-norm can be completed in O(d log d) time.
The k-support norm is a special case of the box-norm, and as a direct corollary of Theorem 4.1 and
Theorem 4.2, we recover [6, Proposition 2.1].
4.1
Proximity Operator
Proximal gradient methods can be used to solve optimization problems of the form minw f (w) +
g(w), where f is a convex loss function with Lipschitz continuous gradient, > 0 is a regularization parameter, and g is a convex function for which the proximity operator can be computed
efficiently, see [12, 21, 22] and references therein. The proximity operator of g with parameter ? > 0
is defined as
?
1
prox?g (w) = argmin
kx wk2 + ?g(x) : x 2 Rd .
2
We now use the infimum formulation of the box-norm to derive the proximity operator of the squared
norm.
5
Algorithm 1 Computation of x = prox
2
k?k2?
Require: parameters a, b, c, .
n
od
2d
b+
1. Sort points ?i i=1 = a+
|wj | , |wj |
2.
3.
4.
5.
j=1
(w).
such that ?i ? ?i+1 ;
Identify points ?i and ?i+1 such that S(?i ) ? c and S(?i+1 ) c by binary search;
Find ?? between ?i and ?i+1 such that S(?? ) = c by linear interpolation;
Compute ?i (?? ) for i = 1, . . . , d;
wi
Return xi = ??ii+
for i = 1, . . . , d.
Theorem 4.3. The proximity operator of the square of the box-norm at point w 2 Rd with parameter
?d w d
?1 w 1
2 is given by prox k?k2 (w) = ( ?1 + , . . . , ?d + ), where
?
2
?i = ?i (?) = min(b, max(a, ?|wi |
))
(12)
Pd
and ? is chosen such that S(?) := i=1 ?i (?) = c. Furthermore, the computation of the proximity
operator can be completed in O(d log d) time.
The proof follows a similar reasoning to the proof of Theorem 4.1. Algorithm 1 illustrates the
computation of the proximity operator for the squared box-norm in O(d log d) time. This includes
the k-support as a special case, where we let a tend to zero, and set b = 1 and c = k, which
improves upon the complexity of the O(d(k + log d)) computation provided in [6], and we illustrate
the improvement empirically in Table 1.
4.2
Proximity Operator for Orthogonally Invariant Norms
The computational considerations outlined above can be naturally extended to the matrix setting by
using von Neumann?s trace inequality (see, e.g. [23]). Here we comment on the computation of the
proximity operator, which is important for our numerical experiments in the following section. The
proximity operator of an orthogonally invariant norm k ? k = g( (?)) is given by
proxk?k (W ) = U diag(proxg ( (W )))V > , W 2 Rm?d ,
where U and V are the matrices formed by the left and right singular vectors of W (see e.g. [24,
Prop 3.1]). Using this result we can employ proximal gradient methods to solve matrix regularization
problems using the squared spectral k-support norm and spectral box-norm.
5
Numerical Experiments
In this section, we report on the statistical performance of the spectral regularizers in matrix completion experiments. We also offer an interpretation of the role of the parameters in the box-norm
and we empirically verify the improved performance of the proximity operator computation (see
Table 1). We compare the trace norm (tr) [25], matrix elastic net (en) [26], spectral k-support (ks)
and the spectral box-norm (box). The Frobenius norm, which is equal to the spectral k-support
norm for k = d, performed considerably worse than the trace norm and we omit the results here.
We report test error and standard deviation, matrix rank (r) and optimal parameter values for k and
a, which were determined by validation, as were the regularization parameters. When comparing
performance, we used a t-test to determine statistical significance at a level of p < 0.001. For the
optimization we used an accelerated proximal gradient method (FISTA), see e.g. [12, 21, 22], with
the percentage change in objective as convergence criterion, with a tolerance of 10 5 for the simulated datasets and 10 3 for the real datasets. As is typical with spectral regularizers we found that
the spectrum of the learned matrix exhibited a rapid decay to zero. In order to explicitly impose a
low rank on the solution we included a final step where we hard-threshold the singular values of the
final matrix below a level determined by validation. We report on both sets of results below.
5.1
Simulated Data
Matrix Completion. We applied the norms to matrix completion on noisy observations of low rank
matrices. Each m ? m matrix was generated as W = AB > + E, where A, B 2 Rm?r , r ? m, and
6
Table 1: Comparison of proximity operator algorithms for the k-support norm (time in s), k = 0.05d.
Algorithm 1 is the method in [6], Algorithm 2 is our method.
d
Alg. 1
Alg. 2
1,000
2,000
4,000
8,000
16,000
32,000
0.0443
0.0011
0.1567
0.0016
0.5907
0.0026
2.3065
0.0046
9.0080
0.0101
35.6199
0.0181
0.03
5
k value
a value
4
0.02
0.01
3
2
0
2
4
6
SNR
8
1
10
Figure 1: Impact of signal to noise on a.
2
4
6
8
true rank
10
Figure 2: Impact of matrix rank on k.
the entries of A, B and E are i.i.d. standard Gaussian. We set m = 100, r 2 {5, 10} and we sampled
uniformly a percentage ? 2 {10%, 20%, 30%} of the entries for training, and used a fixed 10% for
validation. The error was measured as ktrue predictedk2 /ktruek2 [5] and averaged over 100 trials.
The results are summarized in Table 2. In the thresholding case, all methods recovered the rank of
the true noiseless matrix. The spectral box-norm generated the lowest test errors in all regimes, with
the spectral k-support a close second, in particular in the thresholding case. This suggests that the
non zero parameter a in the spectral box-norm counteracted the noise to some extent.
Role of Parameters. In the same setting we investigated the role of the parameters in the boxnorm. As previously discussed, parameter b can be set to 1 without loss of generality. Figure 1
shows the optimal value of a chosen by validation for varying signal to noise ratios (SNR), keeping
k fixed. We see that for greater noise levels (smaller SNR), the optimal value for a increases. While
for a > 0, the recovered solutions are not sparse, as we show below this can still lead to improved
performance in experiments, in particular in the presence of noise. Figure 2 shows the optimal value
of k chosen by validation for matrices with increasing rank, keeping a fixed. We notice that as the
rank of the matrix increases, the optimal k value increases, which is expected since it is an upper
bound on the sum of the singular values.
Table 2: Matrix completion on simulated data sets, without (left) and with (right) thresholding.
dataset
norm
test error
r
k
rank 5 tr
?=10% en
ks
box
0.8184 (0.03)
0.8164 (0.03)
0.8036 (0.03)
0.7805 (0.03)
rank 5 tr
?=20% en
ks
box
0.4085 (0.03) 23
0.4081 (0.03) 23
0.4031 (0.03) 21 3.1
0.3898 (0.03) 100 1.3
rank 10 tr
?=20% en
ks
box
0.6356 (0.03)
0.6359 (0.03)
0.6284 (0.03)
0.6243 (0.03)
rank 10 tr
?=30% en
ks
box
dataset
a
20
20
16 3.6
87 2.9 1.7e-2
norm
test error
r
k
a
rank 5 tr
?=10% en
ks
box
0.7799 (0.04)
0.7794 (0.04)
0.7728 (0.04)
0.7649 (0.04)
5
5
5 4.23
5 3.63 8.1e-3
9e-3
rank 5 tr
?=20% en
ks
box
0.3449 (0.02)
0.3445 (0.02)
0.3381 (0.02)
0.3380 (0.02)
5
5
5 2.97
5 3.28 1.9e-3
27
27
24 4.4
89 1.8
9e-3
rank 10 tr
?=20% en
ks
box
0.6084 (0.03)
0.6074 (0.03)
0.6000 (0.03)
0.6000 (0.03)
10
10
10 5.02
10 5.22 1.9e-3
0.3642 (0.02) 36
0.3638 (0.002 36
0.3579 (0.02) 33 5.0
0.3486 (0.02) 100 2.5
9e-3
rank 10 tr
?=30% en
ks
box
0.3086 (0.02)
0.3082 (0.02)
0.3025 (0.02)
0.3025 (0.02)
10
10
10 5.13
10 5.16
7
3e-4
Table 3: Matrix completion on real data sets, without (left) and with (right) thresholding.
dataset
norm test error
r
k
dataset
a
norm test error
r
k
a
MovieLens tr
100k
en
? = 50%
ks
box
0.2034
0.2034
0.2031
0.2035
87
87
102 1.00
943 1.00 1e-5
MovieLens tr
100k
en
? = 50%
ks
box
0.2017 13
0.2017 13
0.1990 9 1.87
0.1989 10 2.00 1e-5
MovieLens tr
1M
en
? = 50%
ks
box
0.1821 325
0.1821 319
0.1820 317 1.00
0.1817 3576 1.09 3e-5
MovieLens tr
1M
en
? = 50%
ks
box
0.1790
0.1789
0.1782
0.1777
17
17
17 1.80
19 2.00 1e-6
Jester 1
20 per line
tr
en
ks
box
0.1787
0.1787
0.1764
0.1766
98
98
84 5.00
100 4.00 1e-6
Jester 1
20 per line
tr
en
ks
box
0.1752
0.1752
0.1739
0.1726
11
11
11 6.38
11 6.40 2e-5
Jester 3
8 per line
tr
en
ks
box
0.1988
0.1988
0.1970
0.1973
49
49
46 3.70
100 5.91 1e-3
Jester 3
8 per line
tr
en
ks
box
0.1959
0.1959
0.1942
0.1940
3
3
3 2.13
3 4.00 8e-4
5.2
Real Data
Matrix Completion (MovieLens and Jester). In this section we report on matrix completion on
real data sets. We observe a percentage of the (user, rating) entries of a matrix and the task is to predict the unobserved ratings, with the assumption that the true matrix has low rank. The datasets we
considered were MovieLens 100k and MovieLens 1M (http://grouplens.org/datasets/movielens/),
which consist of user ratings of movies, and Jester 1 and Jester 3 (http://goldberg.berkeley.edu/jesterdata/), which consist of users and ratings of jokes (Jester 2 showed essentially identical performance
to Jester 1). Following [4], for MovieLens we uniformly sampled ? = 50% of the available entries
for each user for training, and for Jester 1 and Jester 3 we sampled 20, respectively 8, ratings per
user, and we used 10% for validation. The error was measured as normalized mean absolute error,
ktrue predictedk2
#observations/(rmax rmin ) , where rmin and rmax are lower and upper bounds for the ratings [4]. The
results are outlined in Table 3. In the thresholding case, the spectral box and k-support norms had
the best performance. In the absence of thresholding, the spectral k-support showed slightly better
performance. Comparing to the synthetic data sets, this suggests that in the absence of noise the
parameter a did not provide any benefit. We note that in the absence of thresholding our results for
the trace norm on MovieLens 100k agreed with those in [3].
6
Conclusion
We showed that the k-support norm belongs to the family of box-norms and noted that these can
be naturally extended from the vector to the matrix setting. We also provided a connection between
the k-support norm and the cluster norm, which essentially coincides with the spectral box-norm.
We further observed that the cluster norm is a perturbation of the spectral k-support norm, and we
were able to compute the norm and its proximity operator. Our experiments indicate that the spectral
box-norm and k-support norm consistently outperform the trace norm and the matrix elastic net on
various matrix completion problems. With a single parameter to validate, compared to two for the
spectral box-norm, our results suggest that the spectral k-support norm is a powerful alternative to
the trace norm and the elastic net, which has the same number of parameters. In future work, we
would like to study the application of the norms to clustering problems in multitask learning [9],
in particular the impact of centering. It would also be valuable to derive statistical inequalities and
Rademacher complexities for these norms.
Acknowledgements
We would like to thank Andreas Maurer, Charles Micchelli and especially Andreas Argyriou for
useful discussions. Part of this work was supported by EPSRC Grant EP/H027203/1.
8
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9
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4,746 | 5,298 | Beta-Negative Binomial Process and Exchangeable
Random Partitions for Mixed-Membership Modeling
Mingyuan Zhou
IROM Department, McCombs School of Business
The University of Texas at Austin, Austin, TX 78712, USA
[email protected]
Abstract
The beta-negative binomial process (BNBP), an integer-valued stochastic process,
is employed to partition a count vector into a latent random count matrix. As the
marginal probability distribution of the BNBP that governs the exchangeable random partitions of grouped data has not yet been developed, current inference for
the BNBP has to truncate the number of atoms of the beta process. This paper
introduces an exchangeable partition probability function to explicitly describe
how the BNBP clusters the data points of each group into a random number of
exchangeable partitions, which are shared across all the groups. A fully collapsed Gibbs sampler is developed for the BNBP, leading to a novel nonparametric
Bayesian topic model that is distinct from existing ones, with simple implementation, fast convergence, good mixing, and state-of-the-art predictive performance.
1
Introduction
For mixture modeling, there is a wide selection of nonparametric Bayesian priors, such as the Dirichlet process [1] and the more general family of normalized random measures with independent increments (NRMIs) [2, 3]. Although a draw from an NRMI usually consists of countably infinite atoms
that are impossible to instantiate in practice, one may transform the infinite-dimensional problem
into a finite one by marginalizing out the NRMI. For instance, it is well known that the marginalization of the Dirichlet process random probability measure under multinomial sampling leads to
the Chinese restaurant process [4, 5]. The general structure of the Chinese restaurant process is
broadened by [5] to the so called exchangeable partition probability function (EPPF) model, leading
to fully collapsed inference and providing a unified view of the characteristics of various nonparametric Bayesian mixture-modeling priors. Despite significant progress on EPPF models in the past
decade, their use in mixture modeling (clustering) is usually limited to a single set of data points.
Moving beyond mixture modeling of a single set, there has been significant recent interest in mixedmembership modeling, i.e., mixture modeling of grouped data x1 , . . . , xJ , where each group xj =
{xji }i=1,mj consists of mj data points that are exchangeable within the group. To cluster the mj
data points in each group into a random, potentially unbounded number of partitions, which are
exchangeable and shared across all the groups, is a much more challenging statistical problem.
While the hierarchical Dirichlet process (HDP) [6] is a popular choice, it is shown in [7] that a wide
variety of integer-valued stochastic processes, including the gamma-Poisson process [8, 9], betanegative binomial process (BNBP) [10, 11], and gamma-negative binomial process (GNBP), can all
be applied to mixed-membership modeling. However, none of these stochastic processes are able
to describe their marginal distributions that govern the exchangeable random partitions of grouped
data. Without these marginal distributions, the HDP exploits an alternative representation known as
the Chinese restaurant franchise [6] to derive collapsed inference, while fully collapsed inference is
available for neither the BNBP nor the GNBP.
1
The EPPF provides a unified treatment to mixture modeling, but there is hardly a unified treatment
to mixed-membership modeling. As the first step to fill that gap, this paper thoroughly investigates
the law of the BNBP that governs its exchangeable random partitions of grouped data. As directly
deriving the BNBP?s EPPF for mixed-membership modeling is difficult, we first randomize the
group sizes {mj }j and derive the joint distribution of {mj }j and their random partitions on a shared
list of exchangeable clusters; we then derive the marginal distribution of the group-size count vector
m = (m1 , . . . , mJ )T , and use Bayes? rule to further arrive at the BNBP?s EPPF that describes the
prior distribution of a latent column-exchangeable random count matrix, whose jth row sums to mj .
The general method to arrive at an EPPF for mixed-membership modeling using an integer-valued
stochastic process is an important contribution. We make several additional contributions: 1) We
derive a prediction rule for the BNBP to simulate exchangeable random partitions of grouped data
governed by its EPPF. 2) We construct a BNBP topic model, derive a fully collapsed Gibbs sampler that analytically marginalizes out not only the topics and topic weights, but also the infinitedimensional beta process, and provide closed-form update equations for model parameters. 3) The
straightforward to implement BNBP topic model sampling algorithm converges fast, mixes well,
and produces state-of-the-art predictive performance with a compact representation of the corpus.
1.1
Exchangeable Partition Probability Function
Let ?m = {A1 , . . . , Al } denote a random partition of the set [m] = {1, 2, . . . , m}, where there
are l partitions and each element i ? [m] belongs to one and only one set Ak from ?m . If P (?m =
{A1 , . . . , Al }|m) depends only on the number and sizes of the Ak ?s, regardless of their order, then
it is called an exchangeable partition probability function (EPPF) of ?m . An EPPF of ?m is an
EPPF of ? := (?1 , ?2 , . . .) if P (?m |n) = P (?m |m) does not depend on n, where P (?m |n)
denotes the marginal partition probability for [m] when it is known the sample size is n. Such a
constraint can also be expressed as an addition rule for the EPPF [5]. In this paper, the addition rule
is not required and the proposed EPPF is allowed to be dependent on the group sizes (or sample
size if the number of groups is one). Detailed discussions about sample size dependent EPPFs
can be found in [12]. We generalize the work of [12] to model the partition of a count vector into a
latent column-exchangeable random count matrix. A marginal sampler for ?-stable Poisson-Kigman
mixture models (but not mixed-membership models) is proposed in [13], encompassing a large class
of random probability measures and their corresponding EPPFs of ?. Note that the BNBP is not
within that class and both the BNBP?s EPPF and perdition rule are dependent on the group sizes.
1.2
Beta Process
The beta process B ? BP(c, B0 ) is a completely random measure defined on the product space
[0, 1] ? ?, with a concentration parameter c > 0 and a finite and continuous base measure B0 over
a complete separable metric space ? [14, 15] . We define the L?evy measure of the beta process as
?(dpd?) = p?1 (1 ? p)c?1 dpB0 (d?).
(1)
A draw from B ? BP(c, B0 ) can be represented as a countably infinite sum as B =
P
?
k=1 pk ??k , ?k ? g0 , where ?0 = B0 (?) is the mass parameter and g0 (d?) = B0 (d?)/?0
is the base distribution. The beta process is unique in that the beta distribution
is not infinitely
P
divisible, and its measure on a Borel set A ? ?, expressed as B(A) =
k:?k ?A pk , could be
larger than P
one and hence clearly not a beta random variable. In this paper we will work with
Q(A) = ? k:?k ?A ln(1 ? pk ), defined as a logbeta random variable, to analyze model properties
and derive closed-form Gibbs sampling update equations. We provide these details in the Appendix.
2
Exchangeable Cluster/Partition Probability Functions for the BNBP
The integer-valued beta-negative binomial process (BNBP) is defined as
Xj |B ? NBP(rj , B), B ? BP(c, B0 ),
(2)
where for the jth group rj is the negative binomialP
dispersion parameter and Xj |B ? NBP(rj , B)
is a negative binomial process such that Xj (A) = k:?k ?A njk , njk ? NB(rj , pk ) for each Borel
set A ? ?. The negative binomial distribution n ? NB(r, p) has probability mass function (PMF)
n
r
fN (n) = ?(n+r)
n!?(r) p (1 ? p) for n ? Z, where Z = {0, 1, . . .}. Our definition of the BNBP follows
2
those of [10, 7, 11], where for inference [10, 7] used finite truncation and [11] used slice sampling.
There are two recent papers [16, 17] that both marginalize out the beta process from the negative
binomial process, with the predictive structures of the BNBP described as the negative binomial
Indian buffet process (IBP) [16] and ?ice cream? buffet process [17], respectively. Both processes
are also related to the ?multi-scoop? IBP of [10], and they all generalize the binary-valued IBP [18].
Different from these two papers on infinite random count matrices, this paper focuses on generating
a latent column-exchangeable random count matrix, each of whose row sums to a fixed observed
integer. This paper generalizes the techniques developed in [17, 12] to define an EPPF for mixedmembership modeling and derive truncation-free fully collapsed inference.
The BNBP by nature is an integer-valued stochastic process as Xj (A) is a random count for each
Borel set A ? ?. As the negative binomial process is also a gamma-Poisson mixture process, we
can augment (2) as a beta-gamma-Poisson process as
Xj |?j ? PP(?j ), ?j |rj , B ? ?P[rj , B/(1 ? B)], B ? BP(c, B0 ),
where Xj |?j ? PP(?j ) is a Poisson process such thatPXj (A) ? Pois[?j (A)], and ?j |B ?
?P[rj , B/(1?B)] is a gamma process such that ?j (A) = k:?k ?A ?jk , ?jk ? Gamma[rj , pk /(1?
pk )], for each Borel set A ? ?. The mixed-membership-modeling potentials of the BNBP become
clear under this augmented representation. The Poisson process provides a bridge to link count modeling and mixture modeling [7], since Xj ? PP(?j ) can be equivalently generated by first drawing
a total random count mj := Xj (?) ? Pois[?j (?)] and then assigning this random count to disjoint
disjoint Borel sets of ? using a multinomial distribution.
2.1
Exchangeable Cluster Probability Function
In mixed-membership modeling, the size of each group is observed rather being random, thus although the BNBP?s augmented representation is instructive, it is still unclear how exactly the integervalued stochastic process leads to a prior distribution on exchangeable random partitions of grouped
data. The first step for us to arrive at such a prior distribution is to build a sample size dependent
model that treats the number of data points to be clustered (partitioned) in each group as random.
Below we will first derive an exchangeable cluster probability function (ECPF) governed by the
BNBP to describe the joint distribution of the random group sizes and their random partitions over a
random, potentially unbounded number of exchangeable clusters shared across all the groups. Later
we will show how to derive the EPPF from the ECPF using Bayes? rule.
Pmj
Lemma 1. Denote ?k (zji ) as a unit point mass at zji = k, njk = i=1
?k (zji ), and Xj (A) =
P
k:?k ?A njk as the number of data points in group j assigned to the atoms within the Borel set
A ? ?. The Xj ?s generated via the group size dependent model as
P?
?
zji ? k=1 ?jjk
(?) ?k , mj ? Pois(?j (?)),
?j ? ?P[rj , B/(1 ? B)], B ? BP(c, B0 )
(3)
is equivalent in distribution to the Xj ?s generated from a BNBP as in (2).
Proof. With B =
P?
pk ??k and ?j =
P?
k=1 ?jk ??k , the joint distribution of the cluster indices
Qmj
?jz
P? ji
z j = (zj1 , . . . , zjmj ) given ?j and mj can be expressed as p(z j |?j , mj ) = i=1
=
0 =1 ?jk0
k
Q? njk
1
P
mj
k=1 ?jk , which is not in a fully factorized form. As mj is linked to the total random
( ?
k=1 ?jk )
mass ?j (?) with a Poisson distribution, we have the joint likelihood of z j and mj given ?j as
Q? n
(4)
f (z j , mj |?j ) = f (z j |?j , mj )Pois(mj , ?j (?)) = m1j ! k=1 ?jkjk e??jk ,
k=1
which is fully factorized and hence amenable to marginalization. Since ?jk ? Gamma[rj , pk /(1 ?
pk )], we can marginalize ?jk out analytically as f (z j , mj |rj , B) = E?j [f (z j , mj |?j )], leading to
Q? ?(njk +rj ) njk
f (z j , mj |rj , B) = m1j ! k=1 ?(r
pk (1 ? pk )rj .
(5)
j)
Multiplying the above equation with a multinomial coefficient transforms the prior distribution
for the categorical random variables z j to the prior distribution for a random count vector as
Q?
m !
f (nj1 , . . . , nj? |rj , B) = Q? jnjk ! f (z j , mj |rj , B) = k=1 NB(njk ; rj , pk ). Thus in the prior,
k=1
3
for each group, the sample size dependent model in ( 3)
njk ? NB(rj , pk ) random number of
Pdraws
?
data points independently
at
each
atom.
With
X
:=
n
j
k=1 jk ??k , we have Xj |B ? NBP(rj , B)
P
such that Xj (A) = k:?k ?A njk , njk ? NB(rj , pk ).
The Lemma below provides a finite-dimensional distribution obtained by marginalizing out the
infinite-dimensional beta process from the BNBP. The proof is provided in the Appendix.
Lemma 2 (ECPF). The exchangeable cluster probability function (ECPF) of the BNBP, which describes the joint distribution of the random count vector m := (m1 , . . . , mJ )T and its exchangeable
random partitions z = (z11 , . . . , zJmJ ), can be expressed as
h
i
K
? J e??0 [?(c+r? )??(c)] QKJ
?(njk +rj )
?(n?k )?(c+r? ) QJ
QJ
f (z, m|r, ?0 , c) = 0
,
(6)
k=1
j=1
?(c+n?k +r? )
?(rj )
m !
j=1
j
where KJ is the number of observed points of discontinuity for which n?k > 0, r := (r1 , . . . , rJ )T ,
PJ
PJ
r? := j=1 rj , n?k := j=1 njk , and mj ? Z is the random size of group j.
2.2
Exchangeable Partition Probability Function and Prediction Rule
Having the ECPF does not directly lead to the EPPF for the BNBP, as an EPPF describes the distribution of the exchangeable random partitions of the data groups whose sizes are observed rather than
being random. To arrive at the EPPF, first we organize z into a random count matrix NJ ? ZJ?KJ ,
whose jth row represents the partition of the mj data points into the KJ shared exchangeable clusters and whose order of these KJ nonzero columns is chosen uniformly at random from one of the
KJ ! possible permutations, then we obtain a prior distribution on a BNBP random count matrix as
QJ
m !
f (NJ |r, ?0 , c) = K1J ! j=1 QKJ j
f (z, m|r, ?0 , c)
=
k=1 njk !
KJ ?? [?(c+r? )??(c)]
0
?0 e
KJ !
QKJ
?(n?k )?(c+r? )
k=1 ?(c+n?k +r? )
?(njk +rj )
j=1 njk !?(rj ) .
QJ
(7)
As described in detail in [17], although the matrix prior does not appear to be simple, direct calculation shows that this random count matrix has KJ ? Pois {?0 [?(c + r? ) ? ?(c)]} independent and
identically distributed (i.i.d.) columns that can be generated via
n:k ? DirMult(n?k , r1 , . . . , rJ ), n?k ? Digam(r? , c),
(8)
where n:k := (n1k , . . . , nJk )T is the count vector for the kth column (cluster),
the Dirichlet-multinomial (DirMult) distribution [19] has PMF DirMult(n:k |n?k , r) =
?(njk +rj )
?(r? ) QJ
QJn?k !
, and the digamma distribution [20] has PMF Digam(n|r, c) =
j=1
?(rj )
n ! ?(n?k +r? )
j=1
jk
?(r+n)?(c+r)
1
?(c+r)??(c) n?(c+n+r)?(r) ,
where n = 1, 2, . . .. Thus in the prior, the BNBP generates a Poisson
random number of clusters, the size of each cluster follows a digamma distribution, and each cluster
is further partitioned into the J groups using a Dirichlet-multinomial distribution [17].
With both the ECPF and random count matrix prior governed by the BNBP, we are ready to derive
both the EPPF and prediction rule, given in the following two Lemmas, with proofs in the Appendix.
P
Lemma 3 (EPPF). Let PK n:k =m denote a summation over all sets of count vectors with
k=1
PJ
PK
n
=
m,
where
m
=
:k
?
k=1
j=1 mj and n?k ? 1. The group-size dependent exchangeable
partition probability function (EPPF) governed by the BNBP can be expressed as
K
QKJ h ?(n?k )?(c+r? ) QJ ?(njk +rj ) i
?0 J
QJ
f (z|m, r, ?0 , c) = P
m?
K 0 =1
mj !
j=1
0
?0K
K0!
k=1
?(c+n?k +r? )
QK 0
P
PK 0
k0 =1
n:k0 =m
j=1
?(rj )
?(n?k0 )?(c+r? )
k0 =1 ?(c+n?k0 +r? )
?(njk0 +rj )
j=1 njk0 !?(rj )
QJ
,
(9)
which is a function of the cluster sizes {njk }k=1,KJ , regardless of the orders of the indices k?s.
Although the EPPF is fairly complicated, one may derive a simple prediction rule, as shown below,
to simulate exchangeable random partitions of grouped data governed by this EPPF.
Lemma 4 (Prediction Rule). With y ?ji representing the variable y that excludes the contribution
of xji , the prediction rule of the BNBP group size dependent model in (3) can be expressed as
?
?ji
?ji
?k
? n?ji
(n?ji
jk + rj ), for k = 1, . . . , KJ ;
c+n?k +r?
?ji
P (zji = k|z , m, r, ?0 , c) ?
(10)
? ?0
?ji
r
,
if
k
=
K
+
1.
j
J
c+r?
4
(a) ri = 1
25 18 1 3
Group
4
6
8
1
48
4
45
1
48
1
49
1
1
50
2
5 1 1
2
1
1
2 11
2
1
10 33 12
3
4
5
Partition
6
7
8
1
33 9 3 4
8 34
1
49
1
28 14
6 20 26 1
2
8
1 1
15 14 13
1
4 1
1 1 2
8 17 17 7
10
10 17 18 13
2
12
14
1
1
1
1
1
1
1
1 1
16 18 12
1
1
1
6 13 29 7
1 2
1 1
2
2 2 1 1 1
1
1
22 21 6
1
6
1 1
1
10 31 3
1
1
2
4
26 9 9 1 2
2 19 24 2
4 21 14 9
1 3
33 8 3 3 1
49
10
1 1
4 11 18 1 13 3
1
1
1
39 5 1 1 2
2
1
1
3 7
2 37 2
49
(c) ri = 100
(b) ri = 10
1
23
Group
27
Group
49
2
2
5
Partition
10
15
Partition
Figure 1:
Random draws from the EPPF that governs the BNBP?s exchangeable random partitions of
10 groups (rows), each of which has 50 data points. The parameters of the EPPF are set as c = 2,
?0 = ?(c+P 12rj )??(c) , and (a) rj = 1, (b) rj = 10, or (c) rj = 100 for all the 10 groups. The jth row
j
of each matrix, which sums to 50, represents the partition of the mj = 50 data points of the jth group over
a random number of exchangeable clusters, and the kth column of each matrix represents the kth nonempty
cluster in order of appearance in Gibbs sampling (the empty clusters are deleted).
2.3
Simulating Exchangeable Random Partitions of Grouped Data
While the EPPF (9) is not simple, the prediction rule (10) clearly shows that the probability of
selecting k is proportional to the product of two terms: one is related to the kth cluster?s overall
popularity across groups, and the other is only related to the kth cluster?s popularity at that group
and that group?s dispersion parameter; and the probability of creating a new cluster is related to ?0 ,
c, r? and rj . The BNBP?s exchangeable random partitions of the group-size count vector m, whose
prior distribution is governed by (9), can be easily simulated via Gibbs sampling using (10).
Running Gibbs sampling using (10) for 2500 iterations and displaying the last sample, we show in
Figure 1 (a)-(c) three distinct exchangeable random partitions of the same group-size count vector,
under three different parameter settings. It is clear that the dispersion parameters {rj }j play a
critical role in controlling how overdispersed the counts are: the smaller the {rj }j are, the more
overdispersed the counts in each row and those in each column are. This is unsurprising as in the
BNBP?s prior, we have njk ? NB(rj , pk ) and n:k ? DirMult(n?k , r1 , . . . , rJ ). Figure 1 suggests
that it is important to infer rj rather than setting them in a heuristic manner or fixing them.
3
Beta-Negative Binomial Process Topic Model
With the BNBP?s EPPF derived, it becomes evident that the integer-valued BNBP also governs a
prior distribution for exchangeable random partitions of grouped data. To demonstrate the use of
the BNBP, we apply it to topic modeling [21] of a document corpus, a special case of mixture
modeling of grouped data, where the words of the jth document xj1 , . . . , xjmj constitute a group
xj (mj words in document j), each word xji is an exchangeable group member indexed by vji in
a vocabulary with V unique terms. We choose the base distribution as a V dimensional Dirichlet
distribution as g0 (?) = Dir(?; ?, . . . , ?), and choose a multinomial distribution to link a word to a
topic (atom). We express the hierarchical construction of the BNBP topic model as
P?
?
xji ? Mult(?zji ), ?k ? Dir(?, . . . , ?), zji ? k=1 ?jjk
(?) ?k , mj ? Pois(?j (?)),
B
?j ? ?P rj , 1?B
, rj ? Gamma(a0 , 1/b0 ), B ? BP(c, B0 ), ?0 ? Gamma(e0 , 1/f0 ). (11)
Pmj
Let n
:=
i=1 ?v (xji )?k (zji ). Multiplying (4) and the data likelihood f (xj |z j , ?) =
QV vjk
Q?
nvjk
(?
)
, where ? = (?1 , . . . , ?? ), we have f (xj , z j , mj |?, ?j ) =
v=1Q k=1 vk
Q?
V
Q
? QV
v=1 nvjk !
k=1
v=1 Pois(nvjk ; ?vk ?jk ). Thus the BNBP topic model can also be conk=1
mj !
sidered as an infinite Poisson factor model [10], where the term-document
word count matrix
P?
(mvj )v=1:V, j=1:J is factorized under the Poisson likelihood as mvj =
k=1 nvjk , nvjk ?
Pois(?vk ?jk ), whose likelihood f ({nvjk }v,k |?, ?j ) is different from f (xj , z j , mj |?, ?j ) up to
a multinomial coefficient.
QJ
The full conditional likelihood f (x, z, m|?, ?) =
j=1 f (xj , z j , mj |?, ?j ) can be further
nQ
o Q? QJ ?njk e??jk
Q
?
V
nv?k
k=1
jk
Qj=1
expressed as f (x, z, m|?, ?) =
?
, where the
J
k=1
v=1 ?vk
m !
j=1
j
marginalization of ? from the first right-hand-side term is the product of Dirichlet-multinomial distributions and the second right-hand-side term leads to the ECPF. Thus we have a fully marginalized
5
QKJ h ?(V ?) QV ?(nv?k +?) i
likelihood as f (x, z, m|?0 , c, r) = f (z, m|?0 , c, r) k=1
. Directly
v=1
?(n?k +V ?)
?(?)
applying Bayes? rule to this fully marginalized likelihood, we construct a nonparametric Bayesian
fully collapsed Gibbs sampler for the BNBP topic model as
?
?ji
?
?k
n?ji
?ji
? ?+nvji?ji
?k
?
? (n?ji
?ji
jk + rj ), for k = 1, . . . , KJ ;
?ji
V
?+n
c+n
+r
?
?k
?k
P (zji = k|x, z , ?0 , m, c, r) ?
(12)
?
?ji
? 1 ? ?0 ? r ,
if
k
=
K
+
1.
j
J
V
c+r?
In the Appendix we include all the other closed-form Gibbs sampling update equations.
3.1
Comparison with Other Collapsed Gibbs Samplers
One may compare the collapsed Gibbs sampler of the BNBP topic model with that of latent Dirichlet
allocation (LDA) [22], which, in our notation, can be expressed as
P (zji = k|x, z ?ji , m, ?, K) ?
?+n?ji
v ?k
ji
V ?+n?ji
?k
? (n?ji
jk + ?), for k = 1, . . . , K,
(13)
where the number of topics K and the topic proportion Dirichlet smoothing parameter ? are both
tuning parameters. The BNBP topic model is a nonparametric Bayesian algorithm that removes the
need to tune these parameters. One may also compare the BNBP topic model with the HDP-LDA
[6, 23], whose direct assignment sampler in our notation can be expressed as
? ?+n?ji
vji ?k
?ji
?
+ ??
rk ), for k = 1, . . . , KJ?ji ;
?ji ? (njk
P (zji = k|x, z ?ji , m, ?, r? ) ? V ?+n?k
(14)
?1
?ji
r? ),
if k = KJ + 1;
V ? (??
e j ? DP(?, G),
e
where ? is the concentration parameter for the group-specific Dirichlet processes ?
e k ) and r?? = G(?\D
e
e
and r?k = G(?
J ) are the measures of the globally shared Dirichlet process G ?
e
DP(?0 , G0 ) over the observed points of discontinuity and absolutely continuous space, respectively.
Comparison between (14) and (12) shows that distinct from the HDP-LDA that combines a topic?s
global and local popularities in an additive manner as (n?ji
rk ), the BNBP topic model comjk + ??
bines them in a multiplicative manner as
as the product of n?ji
?k and
n?ji
jk +rj
c+n?ji
?k +r?
n?ji
?k
c+n?ji
?k +r?
? (n?ji
jk + rj ). This term can also be rewritten
, the latter of which represents how much the jth document
contributes to the overall popularity of the kth topic. Therefore, the BNBP and HDP-LDA have distinct mechanisms to automatically shrink the number of topics. Note that while the BNBP sampler
in (12) is fully collapsed, the direct assignment sampler of the HDP-LDA in (14) is only partially
e nor the concentration parameter ? are
collapsed as neither the globally shared Dirichlet process G
e (but still
marginalized out. To derive a collapsed sampler for the HDP-LDA that marginalizes out G
not ?), one has to use the Chinese restaurant franchise [6], which has cumbersome book-keeping as
each word is indirectly linked to its topic via a latent table index.
4
Example Results
We consider the JACM1 , PsyReview2 , and NIPS123 corpora, restricting the vocabulary to terms that
occur in five or more documents. The JACM corpus includes 536 documents, with V = 1539 unique
terms and 68,055 total word counts. The PsyReview corpus includes 1281 documents, with V =
2566 and 71,279 total word counts. The NIPS12 corpus includes 1740 documents, with V = 13, 649
and 2,301,375 total word counts. To evaluate the BNBP topic model4 and its performance relative to
that of the HDP-LDA, which are both nonparametric Bayesian algorithms, we randomly choose 50%
1
http://www.cs.princeton.edu/?blei/downloads/
http://psiexp.ss.uci.edu/research/programs data/toolbox.htm
3
http://www.cs.nyu.edu/?roweis/data.html
4
Matlab code available in http://mingyuanzhou.github.io/
2
6
Number of topics
(a) HDP?LDA, JACM
(b) BNBP Topic Model, JACM
1000
1000
100
100
10
10
0
500
1000
1500
0
Number of topics
(c) HDP?LDA, PsyReview
500
1000
1000
100
100
10
1500
10
0
500
1000
1500
0
(e) HDP?LDA, NIPS12
Number of topics
1000
(d) BNBP Topic Model, PsyReview
500
1000
1500
(f) BNBP Topic Model, NIPS12
1000
1000
100
100
10
10
0
500
1000
Gibbs sampling iteration
1500
0
500
1000
Gibbs sampling iteration
1500
Figure 2: The inferred number of topics KJ for the first 1500 Gibbs sampling iterations for the (a) HDP-LDA
and (b) BNBP topic model on JACM. (c)-(d) and (e)-(f) are analogous plots to (a)-(c) for the PsyReview and
NIPS12 corpora, respectively. From bottom to top in each plot, the red, blue, magenta, black, green, yellow,
and cyan curves correspond to the results for ? = 0.50, 0.25, 0.10, 0.05, 0.02, 0.01, and 0.005, respectively.
of the words in each document as training, and use the remaining ones to calculate per-word heldout
perplexity. We set the hyperparameters as a0 = b0 = e0 = f0 = 0.01. We consider 2500 Gibbs
sampling iterations and collect the last 1500 samples. In each iteration, we randomize the ordering
of the words. For each collected sample, we draw the topics (?k |?) ? Dir(? + n1?k , . . . , ? +
nJ?k ), and the topics weights (?jk |?) ? Gamma(njk + rj , pk ) for the BNBP and topic proportions
(? k |?) ? Dir(nj1 + ??
r1 , . . . , njKJ + ??
rKJ ) for the HDP, with which the per-word perplexity is
P P
(s) (s)
P P test
s
k ?vk ?jk
1
, where s ? {1, . . . , S} is the index
computed as exp ? mtest v j mvj ln P P
P
(s) (s)
??
s
v
k
?vk ?jk
of a collected MCMC sample, mtest
vj is the number of test words at term v in document j, and
P P
mtest = v j mtest
.
The
final
results
are averaged over five random training/testing partitions.
vj
The evaluation method is similar to those used in [24, 25, 26, 10]. Similar to [26, 10], we set the
topic Dirichlet smoothing parameter as ? = 0.01, 0.02, 0.05, 0.10, 0.25, or 0.50. To test how the
algorithms perform in more extreme settings, we also consider ? = 0.001, 0.002, and 0.005. All
algorithms are implemented with unoptimized Matlab code. On a 3.4 GHz CPU, the fully collapsed
Gibbs sampler of the BNBP topic model takes about 2.5 seconds per iteration on the NIPS12 corpus
when the inferred number of topics is around 180. The direct assignment sampler of the HDP-LDA
has comparable computational complexity when the inferred number of topics is similar. Note that
when the inferred number of topics KJ is large, the sparse computation technique for LDA [27, 28]
may also be used to considerably speed up the sampling algorithm of the BNBP topic model.
We first diagnose the convergence and mixing of the collapsed Gibbs samplers for the HDP-LDA
and BNBP topic model via the trace plots of their samples. The three plots in the left column of
Figures 2 show that the HDP-LDA travels relatively slowly to the target distributions of the number
of topics, often reaching them in more than 300 iterations, whereas the three plots in the right column
show that the BNBP topic model travels quickly to the target distributions, usually reaching them
in less than 100 iterations. Moreover, Figures 2 shows that the chains of the HDP-LDA are taking
in small steps and do not traverse their distributions quickly, whereas the chains of the BNBP topic
models mix very well locally and traverse their distributions relatively quickly.
A smaller topic Dirichlet smoothing parameter ? generally supports a larger number of topics, as
shown in the left column of Figure 3, and hence often leads to lower perplexities, as shown in
the middle column of Figure 3; however, an ? that is as small as 0.001 (not commonly used in
practice) may lead to more than a thousand topics and consequently overfit the corpus, which is
particularly evident for the HDP-LDA on both the JACM and PsyReview corpora. Similar trends
are also likely to be observed on the larger NIPS2012 corpus if we allow the values of ? to be
even smaller than 0.001. As shown in the middle column of Figure 3, for the same ?, the BNBP
topic model, usually representing the corpus with a smaller number of topics, often have higher
perplexities than those of the HDP-LDA, which is unsurprising as the BNBP topic model has a
multiplicative control mechanism to more strongly shrink the number of topics, whereas the HDP
has a softer additive shrinkage mechanism. Similar performance differences have also been observed
7
(a)
(b)
0
0.01
0.1
Topic Dirichlet parameter ?
(d)
0.5
Heldout perplexity
Number of topics K
2
10
0
0.01
0.1
Topic Dirichlet parameter ?
280
260
240
0.5
10
1000
10
BNBP Topic Model
HDP?LDA
0
0.5
900
1000
0.01
0.1
Topic Dirichlet parameter ?
1000
900
800
0.5
10
2200
2000
2000
1800
1600
1400
1200
0.01
0.1
Topic Dirichlet parameter ?
100
Number of topics K
1000 2000
(i)
2200
1000
100
Number of topics K
(f)
1100
(h)
2
0.01
0.1
Topic Dirichlet parameter ?
300
1200
(g)
Heldout perplexity
Number of topics K
0.01
0.1
Topic Dirichlet parameter ?
(e)
1100
800
0.5
10
10
260
1200
BNBP Topic Model
HDP?LDA
4
280
240
10
10
300
Heldout perplexity
4
320
Heldout perplexity
Heldout perplexity
Number of topics K
2
10
10
(c)
320
BNBP Topic Model
HDP?LDA
Heldout perplexity
4
10
0.5
1800
1600
1400
1200
1000
10
100
Number of topics K
1000
Figure 3: Comparison between the HDP-LDA and BNBP topic model with the topic Dirichlet smoothing parameter ? ? {0.001, 0.002, 0.005, 0.01, 0.02, 0.05, 0.10, 0.25, 0.50}. For the JACM corpus: (a) the posterior
mean of the inferred number of topics KJ and (b) per-word heldout perplexity, both as a function of ?, and (c)
per-word heldout perplexity as a function of the inferred number of topics KJ ; the topic Dirichlet smoothing
parameter ? and the number of topics KJ are displayed in the logarithmic scale. (d)-(f) Analogous plots to
(a)-(c) for the PsyReview corpus. (g)-(i) Analogous plots to (a)-(c) for the NIPS12 corpus, where the results of
? = 0.002 and 0.001 for the HDP-LDA are omitted.
in [7], where the HDP and BNBP are inferred under finite approximations with truncated block
Gibbs sampling. However, it does not necessarily mean that the HDP-LDA has better predictive
performance than the BNBP topic model. In fact, as shown in the right column of Figure 3, the
BNBP topic model?s perplexity tends to be lower than that of the HDP-LDA if their inferred number
of topics are comparable and the ? is not overly small, which implies that the BNBP topic model is
able to achieve the same predictive power as the HDP-LDA, but with a more compact representation
of the corpus under common experimental settings. While it is interesting to understand the ultimate
potentials of the HDP-LDA and BNBP topic model for out-of-sample prediction by setting the
? to be very small, a moderate ? that supports a moderate number of topics is usually preferred
in practice, for which the BNBP topic model could be a preferred choice over the HDP-LDA, as
our experimental results on three different corpora all suggest that the BNBP topic model could
achieve lower-perplexity using the same number of topics. To further understand why the BNBP
topic model and HDP-LDA have distinct characteristics, one may view them from a count-modeling
perspective [7] and examine how they differently control the relationship between the variances and
means of the latent topic usage count vectors {(n1k , . . . , nJk )}k .
We also find that the BNBP collapsed Gibbs sampler clearly outperforms the blocked Gibbs sampler
of [7] in terms of convergence speed, computational complexity and memory requirement. But a
blocked Gibbs sampler based on finite truncation [7] or adaptive truncation [11] does have a clear
advantage that it is easy to parallelize. The heuristics used to parallelize an HDP collapsed sampler
[24] may also be modified to parallelize the proposed BNBP collapsed sampler.
5
Conclusions
A group size dependent exchangeable partition probability function (EPPF) for mixed-membership
modeling is developed using the integer-valued beta-negative binomial process (BNBP). The exchangeable random partitions of grouped data, governed by the EPPF of the BNBP, are strongly influenced by the group-specific dispersion parameters. We construct a BNBP nonparametric Bayesian
topic model that is distinct from existing ones, intuitive to interpret, and straightforward to implement. The fully collapsed Gibbs sampler converges fast, mixes well, and has state-of-the-art predictive performance when a compact representation of the corpus is desired. The method to derive the
EPPF for the BNBP via a group size dependent model is unique, and it is of interest to further investigate whether this method can be generalized to derive new EPPFs for mixed-membership modeling
that could be introduced by other integer-valued stochastic processes, including the gamma-Poisson
and gamma-negative binomial processes.
8
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9
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4,747 | 5,299 | The Infinite Mixture of Infinite Gaussian Mixtures
Bartek Rajwa
Bindley Bioscience Center
Purdue University
W. Lafayette, IN 47907
[email protected]
Halid Z. Yerebakan
Department of
Computer and Information Science
IUPUI
Indianapolis, IN 46202
[email protected]
Murat Dundar
Department of
Computer and Information Science
IUPUI
Indianapolis, IN 46202
[email protected]
Abstract
Dirichlet process mixture of Gaussians (DPMG) has been used in the literature for
clustering and density estimation problems. However, many real-world data exhibit cluster distributions that cannot be captured by a single Gaussian. Modeling
such data sets by DPMG creates several extraneous clusters even when clusters are
relatively well-defined. Herein, we present the infinite mixture of infinite Gaussian mixtures (I2 GMM) for more flexible modeling of data sets with skewed and
multi-modal cluster distributions. Instead of using a single Gaussian for each cluster as in the standard DPMG model, the generative model of I2 GMM uses a single
DPMG for each cluster. The individual DPMGs are linked together through centering of their base distributions at the atoms of a higher level DP prior. Inference
is performed by a collapsed Gibbs sampler that also enables partial parallelization. Experimental results on several artificial and real-world data sets suggest
the proposed I2 GMM model can predict clusters more accurately than existing
variational Bayes and Gibbs sampler versions of DPMG.
1
Introduction
The traditional approach to fitting a Gaussian mixture model onto the data involves using the wellknown expectation-maximization algorithm to estimate component parameters [7]. The major limitation of this approach is the need to define the number of clusters in advance. Although there are
several ways to predict the number of clusters in a data set in an offline manner, these techniques
are in general suboptimal as they decouple the two interdependent tasks: predicting the number of
clusters and predicting model parameters.
Dirichlet process mixture of Gaussians (DPMG), also known as the infinite Gaussian mixture model
(IGMM), is a Gaussian mixture model (GMM) with a Dirichlet process (DP) prior defined over
mixture components [8]. Unlike traditional mixture modeling, DPMG predicts the number of clusters while simultaneously performing model inference. In the DPMG model the number of clusters
can arbitrarily grow to better accommodate data as needed. DPMG in general works well when
the clusters are well-defined with Gaussian-like distributions. When the distributions of clusters are
heavy-tailed, skewed, or multi-modal multiple mixture components per cluster may be needed for
more accurate modeling of cluster data. Since there is no dependency structure in DPMG to asso1
ciate mixture components with clusters, additional mixture components produced during inference
are all treated as independent clusters. This results in a suboptimal clustering of underlying data.
We propose the infinite mixture of IGMMs (I2 GMM) for more accurate clustering of data sets exhibiting skewed and multi-modal cluster distributions. The underlying generative model of I2 GMM
employs a different DPMG for each cluster data. A dependency structure is imposed across individual DPMGs through centering of their base distibutions at one of the atoms of the higher level DP.
This way individual cluster data are modeled by lower level DPs using one DPMG for each cluster
and atoms defining the base distributions of individual clusters and cluster proportions are modeled
by the higher level DP. Our model allows sharing of the covariance matrices across mixture components of the same DPMG. The data model, which is conjugate to the base distributions of both
higher and lower level DPs, makes obtaining closed form solutions of posterior predictive distributions possible. We use a collapsed Gibbs sampler scheme for inference. Each scan of the Gibbs
sampler involves two loops. One that iterates over individual data instances to sample component
indicator variables and another one that iterates over components to sample cluster indicator variables. Conditioned on the cluster indicator variables, component indicator variables can be sampled
in a parallel fashion, which significantly speeds up inference under certain circumstances.
2
Related Work
Dependent Dirichlet processes (DDP) have been studied in the literature for modeling collection
of distributions that vary in time, in spatial region, in covariate space, or in grouped data settings
(images, documents, biological samples). Previous work most related to the current work involves
studies that investigate DDP in grouped data settings.
Teh et al. uses a hierarchical DP (HDP) prior over the base distributions of individual DP models to
introduce a sharing mechanism that allows for sharing of atoms across multiple groups [15]. When
each group data is modeled by a different DPMG this allows for sharing of the same mean vector
and covariance matrix across multiple groups. Such a dependency may potentially be useful in a
multi-group setting. However, when all data are contained in a single group as in the current study
sharing the same mixture component across multiple cluster distributions leads to shared mixture
components being statistically unidentifiable.
The HDP-RE model by Kim & Smyth [10] and transformed DP by Sudderth et al. [14] relaxes the
exact sharing imposed by HDP to have a dependency structure between multiple groups that allow
for components to share perturbed copies of atoms. Although such a sharing mechanism may be
useful for modeling random variations in component parameters across multiple groups, it is not
very useful for clustering data sets with skewed and multi-modal distributions. Both HDP-RE and
transformed DP still model each group data by a single DPMG and suffer from the same drawbacks
as DPMG when clustering data sets with skewed and multi-modal distributions.
The nested Dirichlet Pricess (nDP) by Rodriguez et al. [13] is a DP whose base distribution is in
turn another DP. This model is introduced for modeling multi-group data sets where groups share
not just individual mixture components as in HDP but the entire mixture model defined by a DPMG.
nDP can be adapted to single group data sets with multiple clusters but with the restriction that
each DPMG is shared only once to ensure identifiability. Such a restriction practically eliminates
dependencies across DPMGs modeling different clusters and would not offer clustering property at
the group level.
Unlike existing work which creates dependencies across multiple DPMG through exact or perturbed
sharing of mixture components or through sharing of the entire mixture model, proposed I2 GMM
model associates each cluster with a distinct atom of the higher level DP through centering of the
base distribution of the corresponding DPMG at that atom. Thus, the higher level DP defines metaclusters whereas lower level DPs model actual cluster data. Mixture components associated with
the same DPMG have their own mean vectors but share the same covariance matrix. Apart from
preserving the conjugacy of the data model covariance sharing across mixture components of the
same DPMG allows for identification of clusters that differ in cluster shapes even when they are not
well separated by their means.
2
3
Dirichlet Process Mixture
Dirichlet process is a distribution over discrete distributions. It is parameterized by a concentration
parameter ? and a base distribution H denoted by DP (?H). Each probability mass in a sample
discrete distribution is called as atom. According to the stick-breaking construction of DP [9], each
sample from a DP can be considered as a collection of countably infinite number of atoms. In this
representation base distribution is a prior over the locations of the atoms and concentration parameter affects the distribution of the atom weights, i.e., stick lengths. Another popular characterization
of DP includes the Chinese restaurant process (CRP) [3] which we utilize during model inference.
Discrete nature of its samples makes DP suitable as a prior distribution over mixture weights in
mixture models. Although samples from DP are defined by an infinite dimensional discrete distribution, the posterior distribution conditioned on a finite data always uses finite number of mixture
components.
We denote each data instance by xi ? Rd where i ? {1, ..., n}, n is the total number of data
instances. For each instance, ?i indicates the set of parameters from which the instance is sampled.
For the Gaussian data model ?i = {?i , ?i } where ?i denotes the mean vector and ?i the covariance
matrix. The generative model of the Dirichlet Process Gaussian Mixture is given by (1).
xi ? p(xi |?i )
?i ? G
G ? DP (?H)
(1)
Owing to the discreteness of the distribution G, ?i ?s corresponding to different instances will not be
all distinct. It is this property of DP that offers clustering over ?i and in turn over data instances.
Choosing H from a family of distributions conjugate to the Gaussian distribution produces a closedform solution for the posterior predictive distribution of DPMG. The bivariate prior over the atoms
of G is defined in (2).
H
=
N IW (?0 , ?0 , ?0 , m) = N (?|?0 ,
?
) ? W ?1 (?|?0 , m)
?0
(2)
where ?0 is the prior mean and ?0 is a scaling constant that controls the deviation of the mean
vectors from the prior mean. The parameter ?0 is the scaling matrix and m is degrees of freedom.
The posterior predictive distribution for a Gaussian data model and NIW prior can be obtained
by integrating out ? and ? analytically. Integrating out ? and ? leaves us with the component
indicator variables ti for each instance xi as the only random variables in the state space. Using the
CRP representation of DP, ti ?s can be sampled as in (3).
?p(xi )
if k = K + 1
?i
p(ti = k|X, t ) ?
(3)
?i
? ?i
n?i
if k ? K
k p(xi |Ak , x
k )
? k ) denote the posterior predictive distributions for an empty and ocwhere p(xi ) and p(xi |Ak , x
cupied component, respectively, both of which are multivariate Student-t distributions. X and t
denote the sets of all data instances and their corresponding indicator variables, respectively. nk is
? k are the scatter matrix and sample mean
the number of data instances in component k. Ak and x
for component k, respectively. The superscript ?i notation indicates the exclusion of the effect of
instance i from the corresponding variable. Inference for DPMG can also be performed using the
stick-breaking representation of DP with the actual inference performed either by a Gibbs sampler
or through variational Bayes [5, 11].
4
The Infinite Mixture of Infinite Gaussian Mixture Models
When modeling data sets containing skewed and multi-modal clusters, DPMG tends to produce
multiple components for each cluster. Owing to the single-layer structure of DPMG, no direct associations among different components of the same cluster can be made. As a result of this limitation
all components are treated as independent clusters resulting in a situation where the number of clusters are overpredicted and the actual cluster data are split into multiple subclusters. A more flexible
model for clustering data sets with skewed and multi-modal clusters can be obtained using a two3
layer generative model as in (4).
xi
?i
Gj
Hj
?
?
?
=
(?j , ?j ) ?
G ?
H =
N (xi |?i , ?j )
Gj
DP (?Hj )
N (?j , ?j /?1 )
(4)
G
DP (?H)
N IW (?0 , ?0 , ?0 , m)
In this model, top layer DP generates cluster-specific parameters ?j and ?j according to the base
distribution H and concentration parameter ?. These parameters in turn define the base distributions
Hj of the bottom layer DPs. Since each Hj is representing a different cluster, Hj ?s can be considered
as meta-clusters from which mixture components of the corresponding cluster are generated. In this
model both the number of clusters and the number of mixture components within a cluster can
be potentially infinite hence the name I2 GMM. The top layer DP models the number of clusters,
their sizes, and the base distribution of the bottom layer DPs whereas each bottom layer DP models
the number of components in a cluster and their sizes. Allowing atom locations in the bottom
layer DPGMs to be different than their corresponding cluster atom provides the flexibility to model
clusters that cannot be effectively modeled by a single Gaussian. The scaling parameter ?1 adjusts
within cluster scattering of the component mean vectors whereas the scaling parameter ?0 adjusts
between cluster scattering of the cluster-specific mean vectors. Expressing both H and Hj ?s as
functions of ?j not only preserves the conjugacy of the model but also allows for sharing of the
same covariance matrix across mixture components of the same cluster.
Posterior inference for the proposed model in (4) can be performed by a collapsed Gibbs sampler
n
by iteratively sampling component indicator variables t = {ti }i=1 of data instances and cluster
K
indicator variables c = {ck }k=1 of mixture components. When sampling ti we restrict sampling
with components whose cluster indicator variables are equal to cti in addition to a new component.
The conditional distribution for sampling ti can be expressed by the following equation.
?p(xi )
if k = K + 1
p(ti = k|X, t?i , c) ?
(5)
?i
?i
?
n?i
p(x
|A
,
x
,
S
)
if
k : ck = cti
i
c
k
k
k
k
? ` , n` }`:c` =ck . When sampling component indicator variables, owing to the
where Sck = {A` , x
dependency among data instances, removing a data instance from a component not only affect the
parameters of the components it belongs to but also the corresponding cluster parameters. Technically speaking the parameters of both the component and corresponding cluster has to be updated
for exact inference. However, updating cluster parameters for every data instance removed will significantly slow down inference. For practical purposes we only update component parameters and
assume that removing a single data instance does not significantly change cluster parameters. The
conditional distribution for sampling ck can be expressed by the following equation.
Q
? Q
if j = J + 1
?k
i:ti =k p(xi )
p(ck = j|X, t, c ) ?
(6)
mj i:ti =k p(xi |Sj ) if j ? J
? ` , n` }`:c` =j , J is the number of clusters, and mj is the number of mixture
where Sj = {A` , x
components assigned to cluster j. Next, we discuss the derivation of the component-level posterior
? ?i
predictive distributions, i.e., p(xi |A?i
k ,x
k , Sck ), which can be obtained by evaluating the integral
in (7).
Z Z
?i
?
? ?i
p(xi |A?i
,
x
,
S
)
=
p(xi |?k , ?ck )p(?k , ?ck |A?i
(7)
c
k
k
k
k ,x
k , Sck )??k ??ck
To evaluate the integral in (7) we need the posterior distribution of the component parameters,
? ?i
namely p(?k , ?ck |A?i
k ,x
k , Sck ), which is proportional to
? ?i
p(?k , ?ck |A?i
k ,x
k , Sck ) ?
=
? ?i
p(?k , ?ck , A?i
k ,x
k |Sck )
?i
p(?
xk |?k , ?ck )p(A?i
k |?ck )p(?k |?ck , Sck )p(?k |Sck )
4
(8)
where
p(?
x?i
k |?k , ?ck ) =
p(A?i
=
k |?ck )
p(?k |?ck , Sck ) =
p(?ck |Sck )
=
?
?
=
?
?
=
?1
N ?k , (n?i
?ck
k )
W ?ck , n?i
k ?1
? ?
N (?,
? ?1 ?cP
)
k
P
?1
W
?
+
0
`:c` =ck A` , m +
`:c` =ck (n` ? 1)
P
n` ?1
? ` +?0 ?0
`:c` =ck (n` +?1 ) x
P
n` ?1
`:c` =ck (n` +?1 ) +?0
P
n ?
( `:c =c (n `+?1 ) +?0 )?1
`
k
1
`
P
n` ?1
`:c` =ck (n` +?1 ) +?0 +?1
? ?i
Once we substitute p(?k , ?ck |A?i
k ,x
k , Sck ) into (7) and evaluate the integral we obtain
?i
?i
? k , Sck ) in the form of a multivariate Student-t distribution.
p(xi |Ak , x
? v)
? ?i
? ?,
p(xi |Ak?i , x
k , Sck ) = stu ? t(?,
(9)
? and the degrees of freedom (v) are given below.
The location vector (?
?), the scale matrix (?),
Location vector:
?=
?
? ?i
?
n?i
??
k x
k +?
?i
nk + ?
?
(10)
Scale matrix:
?=
?
?0 +
P
`:c` =ck
A` + A?i
k +
n?i
?
k ?
(?
x?i
k
n?i
?
k +?
? x?i
? T
? ?)(?
k ? ?)
(?
?+n?i
k )v
(?
?+n?i
k +1)
(11)
Degrees of freedom:
v =m+
X
(n` ? 1) + n?i
k ?d+1
(12)
`:c` =ck
? ?i
The cluster-level posterior predictive distributions can be readily obtained from p(xi |A?i
k ,x
k , Sck )
? k , and nk from (10)-(12). Similarly, posterior predictive distribution for an empty
by dropping Ak , x
? k , and nk .
component/cluster can be obtained by dropping Sck from (10)-(12) in addition to Ak , x
Thanks to the two-layer structure of the proposed model, the inference for I2 GMM can be partially
parallelized. Conditioned on the cluster indicator variables, component indicator variables for data
instances in the same cluster can be sampled independent of the data instances in other clusters.
The amount of actual speed up that can be achieved by parallelization depends on multiple factors
including the number of clusters, cluster sizes, and how fast the other loop that iterates over cluster
indicator variables can be run.
5
Experiments
We evaluate the proposed I2 GMM model on five different data sets and compare its performance
against three different versions of DPMG in terms of clustering accuracy and run time.
5.1
Data Sets
Flower formed by Gaussians: We generated a flower-shaped two-dimensional artificial data set
using a different Gaussian mixture model for each of the four different parts (petals, stem, and two
leaves) of the flower. Each part is considered as a separate cluster. Although covariance matrices
are same for all Gaussian components within a mixture they do differ between mixtures to create
clusters of different shapes. Petals are formed by a mixture of nine Gaussians sharing a spherical
covariance. Stem is formed by a mixture of four Gaussians sharing a diagonal covariance. Each leaf
is formed by a mixture of two Gaussians sharing a full covariance. There are a total of seventeen
Gaussian components, four clusters, and 17,000 instances (1000 instances per component) in this
data set. Scatter plot of this data set is shown in Fig 1a.
Lymphoma: Lymphoma data set is one of the data sets used in the FlowCAP (Flow Cytometry Critical Assessment of Population Identification Methods) 2010 competition [1]. This data set consists
5
of thirty sub-data sets each generated from a lymph node biopsy sample of a patient using a flow
cytometer. Flow cytometry is a single-cell screening, analysis, and sorting technology that plays a
crucial role in research and clinical immunology, hematology, and oncology. The cellular phenotypes are defined in FC by combinations of morphological features (measured by elastic light scatter)
and abundances of surface and intracellular markers revealed by fluorescently labeled antibodies. In
the lymphoma data set each of the sub-data set contains thousands of instances with each instance
representing a cell by a five-dimensional feature vector. For each sub-data set cell populations are
manually gated by experts. Each sub-data has between two to four cell populations, i.e., clusters.
Owing to the intrinsic mechanical and optical limitations of a flow cytometer, distributions of cell
populations in the FC data end up being heavy-tailed or skewed, which makes their modeling by a
single Gaussian highly impractical [12]. Although clusters in this data set are relatively well-defined
accurate modeling of cell distributions is a challenge due to skewed nature of distributions.
Rare cell populations: This data set is a small subset of one of the data sets used in the FlowCAP
2012 competition [1]. The data set contains about 279,546 instances with each instance characterizing a white blood cell in a six-dimensional feature space. There are three clusters manually labeled
by experts. This is an interesting data set for two reasons. First, clusters are highly unbalanced in
terms of the number of instances belonging to each cluster. Two of the clusters, which are highly
significant for measuring immunological response of the patient, are extremely rare. The ratios of
the number of instances available from each of the two rare classes to the total number of instances
are 0.0004 and 0.0005, respectively. Second, the third cluster, which contains all cells not belonging to one of the two rare-cell populations, has a distribution that is both skewed and multi-modal
making it extremely challenging to recover its distribution as a single cluster.
Hyperspectral imagery: This data set is a flightline over a university campus. The hyperspectral
data provides image data in 126 spectral bands in the visible and infrared regions. A total of 21,518
pixels from eight different land cover types are manually labeled. Some of the land cover types
such as roof tops have multi-modal distributions. Cluster sizes are also relatively unbalanced with
pixels belonging to roof tops constituting about one half of the labeled pixels. To reduce run time the
dimensionality is reduced by projecting the original data onto its first thirty principal components.
The data with reduced dimensionality is used in all experiments.
Letter recognition: This is a benchmark data set available through the UCI machine learning repository [4]. There are twenty six well-balanced clusters (one for each letter) in this data set.
2
1
0
?1
?2
?3
?4
?5
?6
?7
?3
?2
?1
0
1
2
3
(b) I2 GMM
(a) True Clusters
4
4
2
2
0
0
?2
?2
?4
?4
?6
?6
?8
?8
2
1
0
?1
?2
?3
?4
?5
?6
?7
?10
?4
?3
?2
?1
0
(c) VB
1
2
3
4
?10
?4
?3
?2
?1
0
1
(d) KD-VB
2
3
4
?3
?2
?1
0
1
2
3
(e) ColGibbs
Figure 1: Clusters predicted by I2 GMM, VB, KD-VB, and ColGibbs on the flower data set. Black
contours in the first figure indicate distributions of individual Gaussian components forming the
flower. Each color refers to a different cluster. Points denote data instances.
6
Table 1: Micro and macro F1 scores produced by I2 GMM, VB, KD-VB, and ColGibbs on the five
data sets. For each data set the first line includes micro F1 scores and the second line macro F1
scores. Numbers in parenthesis indicate standard deviations across ten repetitions. Results for the
lyphoma data set are the average of results from thirty sub-data sets.
Data set
I2 GMM
I2 GMMp
VB
KD-VB
ColGibbs
Flower
0.975 (0.032) 0.991 (0.003) 0.640 (0.087)
0.584
0.525 (0.010)
0.982 (0.015) 0.990 (0.002) 0.643 (0.059)
0.639
0.611 (0.009)
Lymphoma
0.920 (0.016) 0.922 (0.020) 0.454 (0.056)
0.819
0.634 (0.034)
0.847 (0.021) 0.847 (0.022) 0.509 (0.044)
0.762
0.656 (0.029)
Rare classes
0.487 (0.031) 0.493 (0.020) 0.182 (0.015)
0.353
0.234 (0.059)
0.756 (0.012) 0.756 (0.010) 0.441 (0.032)
0.472
0.638 (0.023)
Hyperspectral
0.624 (0.017) 0.626 (0.021) 0.433 (0.031)
0.554
0.427 (0.024)
0.667 (0.018) 0.661 (0.012) 0.580 (0.034)
0.380
0.596 (0.020)
Letter Recognition 0.459 (0.015) 0.467 (0.017) 0.420 (0.015)
0.267
0.398 (0.018)
0.460 (0.015) 0.467 (0.017) 0.420 (0.015)
0.267
0.399 (0.018)
5.2
Benchmark Models and Evaluation Metric
We compare the performance of the proposed I2 GMM model with three different versions of DPMG.
These include the collapsed Gibbs sampler version (ColGibbs) discussed in Section 3, the variational
Bayes version (VB) introduced in [5], and the KD-tree based accelerated variational Bayes version
(KD-VB) introduced in [11]. For I2 GMM and ColGibbs we used our own implementations develR
oped in C++. For VB and KD-VB we used existing MATLAB
(Natick,
MA) implementations 1 .
In order to see the effect of parallelization over execution times we ran the proposed technique in
two modes: parallelized (I2 GMMp) and unparallelized (I2 GMM).
All data sets are scaled to have unit variance for each feature. The ColGibbs model has five free
parameters (?, ?0 , m, ?0 , ?0 ), I2 GMM model has two more parameters (?1 , ?) than ColGibbs. We
use vague priors with ? and ? by fixing their value to one. We set m to the minimum feasible value,
which is d+2, to achieve maximum degrees of freedom in the shape of the covariance matrices. The
prior mean ?0 is set to the mean of the entire data. The scale matrix ?0 is set to I/s, where I is the
identity matrix. This leaves the scaling constant s of ?0 , ?0 , and ?1 as the three free parameters. We
use s = 150/(d(logd)), ?0 = 0.05, and ?1 = 0.5 in experiments with all five data sets described
above.
Micro and macro F1 scores are used as performance measures for comparing clustering accuracy of
these four techniques. As one-to-many matchings are expected between true and predicted clusters,
the F1 score for a true cluster is computed as the maximum of the F1 scores for all predicted clusters.
The Gibbs sampler for ColGibbs and I2 GMM are run for 1500 sweeps. The first 1000 samples are
ignored as burn-in and eleven samples drawn with fifty sweeps apart are saved for final evaluation.
We used an approach similar to the one proposed in [6] for matching cluster labels across different
samples. The mode of cluster labels computed across ten samples are assigned as the final cluster
label for each data instance. ColGibbs and I2 GMM use stochastic sampling whereas VB use a
random initialization stage. Thus, these three techniques may produce results that vary from one
run to other on the same data set. Therefore we repeat each experiment ten times and report average
results of ten repetitions for these three techniques.
5.3
Results and Discussion
Micro and macro F1 produced by the four techniques on all five data sets are reported in Table 1. On
the flower data set I2 GMM achieves almost perfect micro and macro F1 scores and correctly predicts
the true number of clusters. The other three techniques produce several extraneous clusters which
lead to poor F1 scores. Clusters predicted by each of the four techniques are shown in Fig. 1. As
expected ColGibbs identify distributions of individual Gaussian components as clusters as opposed
to the actual clusters formed by mixtures of Gaussians. The piece-wise linear cluster boundaries
1
https://sites.google.com/site/kenichikurihara/academic-software/
variational-dirichlet-process-gaussian-mixture-model
7
Table 2: Execution times for I2 GMM, I2 GMMp, VB, KD-VB, and ColGibbs in seconds on the
five data sets. Numbers in parenthesis indicate standard deviations across ten repetitions. For the
lymphoma data set results reported are average run-time per sub-data set.
Data set
I2 GMM
I2 GMMp
VB
KD-VB
ColGibbs
Flower
54 (2)
41 (4)
1 (0.2)
7
59 (1)
Lymphoma
119 (4)
85 (4)
51 (10)
3
63 (3)
Rare classes
9,738 (349) 5,034 (220) 2171 (569)
16
7,250 (182)
Hyperspectral
5,385 (109) 3,456 (174) 582 (156)
2
7,455 (221)
Letter Recognition
1545 (63)
953 (26)
122 (22)
12
2,785 (123)
obtained by VB and KD-VB, splitting original clusters into multiple subclusters, can be explained
by simplistic model assumptions and approximations that characterize variational Bayes algorithms.
On the lymphoma data set the proposed I2 GMM model achieves an average micro and macro F1
scores of 0.920 and 0.848, respectively. These values are not only significantly higher than corresponding F1 scores produced by the other three techniques but also on par with the best performing
techniques in the FlowCAP 2010 competition [2]. Results for thirty individual sub-data sets in the
lymphoma data set are available in the supplementary document. A similar trend is also observed
with the other three real-world data sets as I2 GMM achieves the best F1 score among the four techniques. Between I2 GMM and ColGibbs, I2 GMM consistently generates less number of clusters
across all data sets as expected. Overall, among the three different versions of DPMG that differ in
the inference algorithm used, there is no clear consensus across five data sets as to which version
predicts clusters more accurately. However, the proposed I2 GMM model which extends DPMG to
skewed and multi-modal clusters, clearly stands out as the most accurate model on all five data sets.
Run time results included in Table 2 favors variational Bayes techniques over the Gibbs samplerbased ones as expected. Despite longer run times, significantly higher F1 scores achieved on data
sets with diverse characteristics suggest that I2 GMM can be preferred over DPMG for more accurate
clustering. Results also suggest that I2 GMM can benefit from parallelization. The actual amount of
improvement in execution time depend on data characteristics as well as how fast the unparallelized
loop can be run. The largest gain by parallelization is obtained on the rare classes data set which
offered almost two-fold increase by parallelization on an eight-core workstation.
6
Conclusions
We introduced I2 GMM for more effective clustering of multivariate data sets containing skewed
and multi-modal clusters. The proposed model extends DPMG to introduce dependencies between
components and clusters by a two-layer generative model. Unlike standard DPMG where each
cluster is modeled by a single Gaussian, I2 GMM offers the flexibility to model each cluster data
by a mixture of potentially infinite number of components. Results on experiments with real and
artificial data sets favor I2 GMM over variational Bayes and collapsed Gibbs sampler versions of
DPMG in terms of clustering accuracy. Although execution time can be improved by sampling
component indicator variables in parallel, the amount of speed up that can be gained is limited with
the execution time of the sampling of the cluster indicator variables. As most time consuming part of
this task is the sequential computation of likelihoods for data instances, significant gains in execution
time can be achieved by parallelizing the computation of likelihoods. I2 GMM is implemented in
C++. The source files and executables are available on the web. 2
Acknowledgments
This research was sponsored by the National Science Foundation (NSF) under Grant Number IIS1252648 (CAREER), by the National Institute of Biomedical Imaging and Bioengineering (NIBIB)
under Grant Number 5R21EB015707, and by the PhRMA Foundation (2012 Research Starter Grant
in Informatics). The content is solely the responsibility of the authors and does not represent the
official views of NSF, NIBIB or PhRMA.
2
https://github.com/halidziya/I2GMM
8
References
[1] FlowCAP - flow cytometry: Critical assessment of population identification methods. http:
//flowcap.flowsite.org/.
[2] N. Aghaeepour, G. Finak, FlowCAP Consortium, DREAM Consortium, H. Hoos, T. R. Mosmann, R. Brinkman, R. Gottardo, and R. H. Scheuermann. Critical assessment of automated
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cytometric data analysis. Proc Natl Acad Sci U S A, 106(21):8519?24, 2009.
[13] A. Rodriguez, D. B. Dunson, and A. E. Gelfand. The nested Dirichlet process. Journal of The
American Statistical Association, 103:1131?1154, 2008.
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9
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evaluate:3 |
4,748 | 53 | 338
The Connectivity Analysis of Simple Association
- orHow Many Connections Do You Need!
Dan Hammerstrom *
Oregon Graduate Center, Beaverton, OR 97006
ABSTRACT
The efficient realization, using current silicon technology, of Very Large Connection
Networks (VLCN) with more than a billion connections requires that these networks exhibit
a high degree of communication locality. Real neural networks exhibit significant locality,
yet most connectionist/neural network models have little. In this paper, the connectivity
requirements of a simple associative network are analyzed using communication theory.
Several techniques based on communication theory are presented that improve the robustness of the network in the face of sparse, local interconnect structures. Also discussed are
some potential problems when information is distributed too widely.
INTRODUCTION
Connectionist/neural network researchers are learning to program networks that exhibit a broad range of cognitive behavior. Unfortunately, existing computer systems are limited in their ability to emulate such networks efficiently. The cost of emulating a network,
whether with special purpose, highly parallel, silicon-based architectures, or with traditional
parallel architectures, is directly proportional to the number of connections in the network.
This number tends to increase geometrically as the number of nodes increases. Even with
large, massively parallel architectures, connections take time and silicon area. Many existing neural network models scale poorly in learning time and connections, precluding large
implementations.
The connectivity 'costs of a network are directly related to its locality. A network
exhibits locality 01 communication 1 if most of its processing elements connect to other physically adjacent processing elements in any reasonable mapping of the elements onto a planar
surface. There is much evidence that real neural networks exhibit locality2. In this paper,
a technique is presented for analyzing the effects of locality on the process of association.
These networks use a complex node similar to the higher-order learning units of Maxwell et
al. 3
NETWORK MODEL
The network model used in this paper is now defined (see Figure 1).
Definition 1: A recursive neural network, called a c-graph is a graph structure,
r( V,E, e), where:
?
There is a set of CNs (network nodes), V, whose outputs can take a range of positive
real values, Vi, between 0 and 1. There are N. nodes in the set.
?
There is a set of codons, E, that can take a range of positive real values, eij (for
codon j of node i), between 0 and 1. There are Ne codons dedicated to each CN (the
output of each codon is only used by its local CN), so there are a total of Ne N. codons
in the network. The fan-in or order of a codon is Ie. It is assumed that leis the
same for each codon, and Ne is the same for each CN.
*This work was supported in part by the Semiconductor Research Corporation contract no. 86-10-097, and
jointly by the Office of Naval Research and Air Force Office of Scientific Research, ONR contract no. NOOO14 87 K
0259.
? American Institute of Physics 1988
339
Ie
codon j
Figure 1 - A ON
?
Cijk E C is a set of connections of ONs to codons, 1<i ,k<N. and 1<j <Ne , Cijk can
take two values {O,l} indicating the existence of a connection from ON k to codon j
of ON i . 0
Definition 2: The value of ON i is
Vi
=
F[8+~eijl
(1)
J-l
The function, F, is a continuous non-linear, monotonic function, such as the sigmoid function. 0
Definition 9: Define a mapping, D(i,j,x)_y, where x is an input vector to rand y is
the Ie element input vector of codon j of ON i. That is, y has as its elements those elements of Zk of x where Cijk=1, \;/ k. 0
The D function indicates the subset of x seen by codon j of ON i. Different input vectors may map to the same codon vectors, e.g., D(i,j,x)-y and D(i,j,Zj-y, where x~7.
Definition 4: The codon values eij are determined as follows. Let X( m) be input vector
m of the M learned input vectors for ON i. For codon eij of ON i, let Tij be the set of I cdimensional vectors such that lij(m)E T ij , and D(i,j,X(m))-lij(m). That is, each vector,
lij( m) in Tij consists of those subvectors of X( m) that are in codon ii's receptive field.
The variable 1 indexes the L ( i ,i) vectors of T ij . The number of distinct vectors in Tij
may be less than the total number of learned vectors (L(i,j)<M). Though the X(m) are
distinct, the subsets, lij(m), need not be, since there is a possible many to one mapping of
the x vectors onto each vector lij.
Let Xl be the subset of vectors where vi=l (ON i is supposed to output a 1), and
those vectors where vi=O, then define
0#(/) - .izeof {D(i,i ,Z'( m)) "
.,-q}
for q=O,1, and \;/ m that map to this I. That is, ni~(I) is the number of
.xo be
(2)
x vectors that map
340
into "Iij{l) where
tlj-O
and ni}{I) is the number of 7 vectors that map into "Iii (I), where
tI;-1.
The compreaaion of a codon for a vector "Iii(1) then is defined as
n.1.(/)
He.?( I) = _ _I.:....;;J- ' - - IJ
(3)
nj}(I)+nj~(I)
(Hqj(l)=O when both nt, nO-O.) The output of codon
I),
eii' is the maximum-likelyhood
decoding
(4)
Where He indicates the likely hood of t l j - l when a vector 7 that maps to , is input, and'
is that vector 1'(') where min[d.(1'('),y)] \I I, D(i,j,7)-V, and 7 is the current input vector. In other words, , is that vector (of the set of subset learned vectors that codon ij
receives) that is closest (using distance measure d.) to V (the subset input vector). 0
The output of a codon is the "most-likely" output according to its inputs. For example, when there is no code compression at a codon, eji-1, if the "closest" (in terms of some
measure of vector distance, e.g. Hamming distance) subvector in the receptive field of the
codon belongs to a learned vector where the CN is to output a 1. The codons described here
are very similar to those proposed by Marr 4 and implement ne!'Lrest-neighbor classification.
It is assumed that codon function is determined statically prior to network operation, that
is, the desired categories have already been learned.
To measure performance, network capacity is used.
Definition 5: The input noiae, Or, is the average d. between an input vector and the
closest (minimum d.) learned vector, where d. is a measure of the "difference" between two
vectors - for bit vectors this can be Hamming distance. The output noise, 0 0 , is the average
distance between network output and the learned output vector associated with the closest
learned input vector. The in/ormation gain, Gr , is just
Gt
=-10.[ ~~
I
(5)
o
Definition 6: The capacity of a network is the maximum number of learned vectors such
that the information gain, Gr , is strictly positive (>0). 0
COMMUNICATION ANALOGY
Consider a single connection network node, or CN. (The remainder of this paper will
be restricted to a single CN.) Assume that the CN output value space is restricted to two
values, 0 and 1. Therefore, the CN must decide whether the input it sees belongs to the
class of "0" codes, those codes for which it remains off, or the class of "I" codes, those codes
for which it becomes active. The inputs it sees in its receptive field constitute a subset of
the input vectors (the D( ... ) function) to the network. It is also assumed that the CN is an
ideal I-NN (Nearest Neighbor) classifier or feature detector. That is, given a particular set
of learned vectors, the CN will classify an arbitrary input according to the class of the
nearest (using d. as a measure of distance) learned vector. This situation is equivalent to
the case where a single CN has a single codon whose receptive field size is equivalent to that
of the CN.
Imagine a sender who wishes to send one bit of information over a noisy channel. The
sender has a probabilistic encoder that choses a code word (learned vector) according to
some probability distribution. The receiver knows this code set, though it has no knowledge
of which bit is being sent. Noise is added to the code word during its transmission over the
341
channel, which is analogous to applying an input vector to a network's inputs, where the
vector lies within some learned vector's region. The "noise" is represented by the distance
( d,,) between the input vector and the associated learned vector.
The code word sent over the channel consists of those bits that are seen in the receptive field of the ON being modeled. In the associative mapping of input vectors to output
vectors, each ON must respond with the appropriate output (0 or 1) for the associated
learned output vector. Therefore, a ON is a decoder that estimates in which class the
received code word belongs. This is a classic block encoding problem, where increasing the
field size is equivalent to increasing code length. As the receptive field size increases, the
performance of the decoder improves in the presence of noise. Using communication theory
then, the trade-off between interconnection costs as they relate to field size and the functionality of a node as it relates to the correctness of its decision making process (output
errors) can be characterized.
As the receptive field size of a node increases, so does the redundancy of the input,
though this is dependent on the particular codes being used for the learned vectors, since
there are situations where increasing the field size provides no additional information.
There is a point of diminishing returns, where each additional bit provides ever less reduction in output error. Another factor is that interconnection costs increase exponentially
with field size. The result of these two trends is a cost performance measure that has a single global maximum value. In other words, given a set of learned vectors and their probabilities, and a set of interconnection costs, a "best" receptive field size can be determined,
beyond which, increasing connectivity brings diminishing returns.
SINGLE CODON, WITH NO CODE COMPRESSION
A single neural element with a single codon and with no code compression can be
modelled exactly as a communication channel (see Figure 2). Each network node is assumed
to have a single codon whose receptive field size is equal to that of the receptive field size of
the node.
sender
I I
encoder
~
nOIsy
I
I
Ch.nne11~1 : ~
transmitter
receiver
I
decoder
ON
Figure 2 - A Transmission Channel
recelver
342
The operation of the channel is as follows. A bit is input into the channel encoder,
which selects a random code of length N and transmits that code over the channel. The
receiver then, using nearest neighbor classification, decides if the original message was either
a 0 or a 1.
Let M be the number of code words used by the encoder. The rate* then indicates the
density of the code space.
Definition 7: The rate, R, of a communication channel is
R = 10gM
-
(6)
N
o
The block length, N, corresponds directly to the receptive field size of the codon, i.e.,
N=/e. The derivations in later sections use a related measure:
Definition 8: The code utilization, b, is the number of learned vectors assigned to a particular code or
(7)
b can be written in terms of R
b
=
2N (R-l)
(8)
As b approaches 1, code compression increases. b is essentially unbounded, since M may be
significantly larger than 2N. 0
The decode error (information loss) due to code compression is a random variable that
depends on the compression rate and the a priori probabilities, therefore, it will be different
with different learned vector sets and codons within a set. As the average code utilization
for all codons approaches 1, code compression occurs more often and codon decode error is
unavoidable.
Let Zi be the vector output of the encoder, and the input to the channel, where each
element of Zi is either a 1 or o. Let Vi be the vector output of the channel, and the input to
the decoder, where each element is either a 1 or a o. The Noisy Channel Coding Theorem is
now presented for a general case, where the individual M input codes are to be distinguished. The result is then extended to a CN, where, even though M input codes are
used, the ON need only distinguish those codes where it must output a 1 from those where it
must output a o. The theorem is from Gallager (5.6.1)5. Random codes are assumed
throughout.
Theorem 1: Let a discrete memoryless channel have transition probabilities PNU/k)
and, for any positive integer N and positive number R, consider the ensemble of (N,R)
block codes in which each letter of each code word is independently selected according to
fe
l
the probability assignment Q(k). Then, for each message m, l<m< NR
and all p,
O<p<l, the ensemble average probability of decoding error using maximum-likelyhood
decoding satisfies
(9)
where
?In the definitions given here and the theorems below, the notation of Gall ager 6 is used. Many of the
definitions and theorems are also from Gallager.
343
Eo(p,Q)=-ln~ [ ~1 Q(k)PU/kp!p ]
i-il
l+P
(10)
k-il
o
These results are now adjusted ror our special case.
Theorem 2: For a single CN, the average channel error rate ror random code vectors is
Pc.,.~2q(l-q )Pe ? m
where q=Q(k)
\I k
(11)
is the probability or an input vector bit being a 1. 0
These results cover a wide range or models. A more easily computable expression can
be derived by recognizing some or the restrictions inherent in the CN model. First, assume
that all channel code bits are equally likely, that is, \I k, Q( k )=q, that the error model is
the Binary Symmetric Channel (BSC), and that the errors are identically distributed and
independent - that is, each bit has the same probability, f, or being in error, independent
or the code word and the bit position in the code word.
A simplified version or the above theorem can be derived. Maximizing P gives the
tightest bounds:
Pc.,.
< 0.5 O$p~l
maxPe(p)
(12)
where (letting codon input be the block length, N = I c)
P,(p)
:'> eXP{-f,IE,(P)-PR1}
(13)
The minimum value or this expression is obtained when p=1 (for q=0.5):
Eo; -log 2 [
(o.sV,+O.SVl-,)'
1
(14)
SINGLE-CODON WITH CODE COMPRESSION
Unfortunately, the implementation complexity of a codon grows exponentially with the
size or the codon, which limits its practical size. An alternative is to approximate single
codon function of a single CN with many smaller, overlapped codons. The goal is to maintain performance and reduce implementation costs, thus improving the cost/performance of
the decoding process. As codons get smaller, the receptive field size becomes smaller relative
to the number of CNs in the network. When this happens there is codon compression, or
vector alia6ing, that introduces its own errors into the decoding process due to information
loss. Networks can overcome this error by using multiple redundant codons (with overlapping receptive fields) that tend to correct the compression error.
Compression occurs when two code words requiring different decoder output share the
same representation (within the receptive field or the codon) . The following theorem gives
the probability of incorrect codon output with and without compression error.
Theorem 9: For a BSC model where q=0.5, the codon receptive field is Ic, the code utilization is b, and the channel bits are selected randomly and independently, the probability
of a codon decoding error when b > 1 is approximately
Pc.,.
< (l-f)"Pc- [1-(I-f)"
]0.5
where the expected compression error per codon is approximated by
(15)
344
Pc = 0.5
(16)
and from equations 13-14, when 6<1
P,,,, < exp { -
j, [-log [
[(O .?V.+O .?Vl-' J'
I-RI}
(17)
Proof is given in Hammerstrom6 . 0
As 6 grows, Pc approaches 0.5 asymptotically. Thus, the performance of a single codon
degrades rapidly in the presence of even small amounts of compression.
MULTIPLE CODONS WITH CODE COMPRESSION
The use or mUltiple small codons is more efficient than a few large codons, but there
are some fundamental performance constraints. When a codon is split into two or more
smaller codons (and the original receptive field is subdivided accordingly), there are several
effects to be considered. First, the error rate of each new codon increases due to a decrease
in receptive field size (the codon's block code length). The second effect is that the code
utilization, II, will increase for each codon, since the same number of learned vectors is
mapped into a smaller receptive field. This change also increases the error rate per codon
due to code compression. In fact, as the individual codon receptive fields get smaller,
significant code compression occurs. For higher-order input codes, there is an added error
that occurs when the order of the individual codons is decreased (since random codes are
being assumed, this effect is not considered here). The third effect is the mass action of
large numbers of codons. Even though individual codons may be in error, if the majority
are correct, then the ON will have correct output. This effect decreases the total error rate.
Assume that each ON has more than one codon, c>1. The union of the receptive fields
for these codons is the receptive field for the ON with no no restrictions on the degree of
overlap of the various codon receptive fields within or between ONs. For a ON with a large
number of codons, the codon overlap will generally be random and uniformly distributed.
Also assume that the transmission errors seen by different receptive fields are independent.
Now consider what happens to a codon's compression error rate (ignoring transmission
error for the time being) when a codon is replaced by two or more smaller co dons covering
the same receptive field. This replacement process can continue until there are only 1..
codons, which, incidentally, is analogous to most current neural models. For a multiple
codon ON, assume that each codon votes a 1 or o. The summation unit then totals this
information and outputs a 1 if the majority of codons vote for a 1, etc.
Theorem 4: The probability of a ON error due to compression error is
1
Pc = "'\7?';
00
J
21r c!2-cp.-l!2
V cP.(i-p.)
where
Pc
J.2
e 2 dy
(18)
is given in equation 16 and q=0.5.
Pc incorporates the two effects of moving to mUltiple smaller codons and adding more
codons. Using equation 17 gives the total error probability (per bit), PeN:
(19)
Proof is in Hammerstrom6 . 0
345
For networks that perform association as defined in this paper, the connection weights
rapidly approach a single uniform value as the size of the network grows. In information
theoretic terms, the information content of those weights approaches zero as the compression increases. Why then do simple non-conjunctive networks (1-codon equivalent) work at
alI? In the next section I define connectivity cost constraints and show that the answer to
the first question is that the general associative structures defined here do not scale costeffectively and more importantly that there are limits to the degree of distribution of information.
CONNECTIVITY COSTS
It is much easier to assess costs if some implementation medium is assumed. I have
chosen standard silicon, which is a two dimensional surface where ON's and codons take up
surface area according to their receptive field sizes. In addition, there is area devoted to
the metal lines that interconnect the ONs. A specific VLSI technology need not be assumed,
since the comparisons are relative, thus keeping ONs, codons, and metal in the proper proportions, according to a standard metal width, m. (which also includes the inter-metal
pitch). For the analyses performed here, it is assumed that
levels of metal are possible.
m,
In the previous section I established the relationship of network performance , in terms
of the transmission error rate, E, and the network capacity, M. In this section I present an
implementation cost, which is total silicon area, A. This figure can then be used to derive a
cost/performance figure that can be used to compare such factors as codon size and receptive field size. There are two components to the total area: A ON , the area of a ON, and
AMI, the area of the metal interconnect between ONs. AON consists of the silicon area
requirements of the codons for all ONs. The metal area for local, intra-ON interconnect is
considered to be much smaller than that of the codons themselves and of that of the more
global, inter-ON interconnect, and is not considered here. The area per ON is roughly
m.
AON = cfeme(-)
2
(20)
m,
where me is the maximum number of vectors that each codon must distinguish, for 6>1,
me = 2".
Theorem 5: Assume a rectangular, un6ounded* grid of ONs (all ONs are equi-distant
from their four nearest neighbors), where each ON has a bounded receptive field of its nON
nearest ONs, where "ON is the receptive field size for the ON, nON =
C~e
,
where c is the
number of codons, and R is the intra-ON redundancy, that is, the ratio of inputs to
synapses (e.g., when R=l each ON input is used once at the ON, when R=2 each input is
used on the average at two sites). The metal area required to support each ON's receptive
field is (proof is giving by Hammerstrom6 ):
AMI = [
----w-+
"ON3
3"ON
2
~
+9"ON
21 [ m.j2
m,
(21)
The total area per ON, A, then is
?Another implementation IItrategy ill to place &II eNII along a diagonal, which givell n 2 area. However, thill
technique only works ror a bounded number or eNII and when dendritic computation can be lipread over a large
area, which limits the range or p08llible eN implementationll. The theorem IItated here covers an infinite plane or
eNII each with a bounded receptive Held.
346
(22)
o
Even with the assumption of maximum locality, the total metal interconnect area
increases as the cube of the per CN receptive field size!
SINGLE CN SIMULATION
What do the bounds tell us about CN connectivity requirements? From simulations,
increasing the CN's receptive field size improves the performance (increases capacity), but
there is also an increasing cost, which increases faster than the performance! Another
observation is that redundancy is quite effective as a means for increasing the effectiveness
of a CN with constrained connectivity. (There are some limits to R, since it can reach a
point where the intra-CN connectivity approaches that of inter-CN for some situations.)
With a fixed nON, increasing cost-effectiveness (A 1m) is possible by increasing both order
and redundancy.
In order to verify the derived bounds, I also wrote a discrete event simulation of a CN,
where a random set of learned vectors were chosen and the CN's codons were programmed
according to the model presented earlier. Learned vectors were chosen randomly and subjected to random noise, L The CN then attempted to categorize these inputs into two
major groups (CN output = 1 and CN output = 0). For the most part the analytic bounds
agreed with the simulation, though they tended to be optimistic in slightly underestimating
the error. These differences can be easily explained by the simplifying assumptions that
were made to make the analytic bounds mathematically tractable.
DISTRmUTED VS. LOCALIZED
Throughout this paper, it has been tacitly assumed that representations are distributed
across a number of CNs, and that any single CN participates in a number of representations. In a local representation each CN represents a single concept or feature . It is the distribution of representation that makes the CN's decode job difficult, since it is the cause of
the code compression problem.
There has been much debate in the connectionist/neuromodelling community as to the
advantages and disadvantages of each approach; the interested reader is referred to Hinton7 , Baum et al. 8, and BallardQ ? Some of the results derived here are relevant to this
debate. A1s the distribution of representation increases, the compression per CN increases
accordingly. It was shown above that the mean error in a codon's response quickly
approaches 0.5, independent of the input noise . This result also holds at the CN level. For
each individual CN, this error can be offset by adding more codons, but this is expensive
and tends to obviate one of the arguments in favor of distributed representations, that is,
the multi-use advantage, where fewer CNs are needed because of more complex, redundant
encodings. A1s the degree of distribution increases, the required connectivity and the code
compression increases, so the added information that each codon adds to its CN's decoding
process goes to zero (equivalent to all weights approaching a uniform value) .
SUMMARY AND CONCLUSIONS
In this paper a single CN (node) performance model was developed that was based on
Communication Theory. Likewise, an implementation cost model was derived .
The communication model introduced the codon as a higher-order decoding element
and showed that for small codons (much less than total CN fan-in, or convergence) code
compression, or vector aliasing, within the codon's receptive field is a severe problem for
347
large networks. As code compression increases, the information added by any individual
codon to the CN's decoding task rapidly approaches zero .
The cost model showed that for 2-dimensional silicon, the area required for inter-node
metal connectivity grows as the cube of a CN's fan-in.
The combination of these two trends indicates that past a certain point, which is
highly dependent on the probability structure of the learned vector space, increasing the
fan-in of a CN (as is done, for example, when the distribution of representation is increased)
yields diminishing returns in terms of total cost-performance. Though the rate of diminishing returns can be decreased by the use of redundant, higher-order connections.
The next step is to apply these techniques to ensembles of nodes (CNs) operating in a
competitive learning or feature extraction environment.
REFERENCES
[I]
J. Bailey, "A VLSI Interconnect Structure for Neural Networks," Ph.D.
Dissertation, Department of Computer SciencejEngineering, OGC. In Preparation.
[2]
V. B. Mountcastle, "An Organizing Principle for Cerebral Function: The Unit
Module and the Distributed System," in The Mindful Brain, MIT Press, Cambridge,
MA,1977.
[3]
T. Maxwell, C . L. Giles, Y . C. Lee and H. H. Chen, "Transformation Invariance
Using High Order Correlations in Neural Net Architectures," Proceeding8
International Con! on SY8tem8, Man, and Cybernetic8, 1986.
[4]
D. Marr, "A Theory for Cerebral Neocortex," Proc. Roy. Soc . London, vol.
176(1970), pp . 161-234.
[5]
R. G. Gallager, Information Theory and Reliable Communication, John Wiley and
Sons, New York, 1968.
[6]
D. Hammerstrom, "A Connectivity Analysis of Recursive, Auto-Associative
Connection Networks," Tech. Report CS/E-86-009, Dept. of Computer
SciencejEngineering, Oregon Graduate Center, Beaverton, Oregon, August 1986.
[7]
G. E . Hinton, "Distributed Representations," Technical Report CMU-CS-84-157,
Computer Science Dept., Carnegie-Mellon University, Pittsburgh, PA 15213, 1984.
[8]
E. B. Baum, J. Moody and F . Wilczek, "Internal Representations for Associative
Memory," Technical Report NSF-ITP-86-138, Institute for Theoretical Physics,
Santa Barbara, CA, 1986.
[9]
D. H . Ballard, "Cortical Connections and Parallel Processing: Structure and
Function," Technical Report 133, Computer Science Department, Rochester, NY,
January 1985.
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4,749 | 530 | The Effective Number of Parameters:
An Analysis of Generalization and Regularization
in Nonlinear Learning Systems
John E. Moody
Department of Computer Science, Yale University
P.O. Box 2158 Yale Station, New Haven, CT 06520-2158
Internet: [email protected], Phone: (203)432-1200
Abstract
We present an analysis of how the generalization performance (expected
test set error) relates to the expected training set error for nonlinear learning systems, such as multilayer perceptrons and radial basis functions. The
principal result is the following relationship (computed to second order)
between the expected test set and tlaining set errors:
(1)
e,
Here, n is the size of the training sample
u;f f is the effective noise
variance in the response variable( s), ,x is a regularization or weight decay
parameter, and Peff(,x) is the effective number of parameters in the nonlinear model. The expectations ( ) of training set and test set errors are
taken over possible training sets and training and test sets e' respectively. The effective number of parameters Peff(,x) usually differs from the
true number of model parameters P for nonlinear or regularized models;
this theoretical conclusion is supported by Monte Carlo experiments. In
addition to the surprising result that Peff(,x) ;/; p, we propose an estimate
of (1) called the generalized prediction error (GPE) which generalizes well
established estimates of prediction risk such as Akaike's F P E and AI C,
Mallows Cp, and Barron's PSE to the nonlinear setting.!
e
lCPE and Peff(>") were previously introduced in Moody (1991).
847
848
Moody
1
Background and Motivation
Many of the nonlinear learning systems of current interest for adaptive control,
adaptive signal processing, and time series prediction, are supervised learning systems of the regression type. Understanding the relationship between generalization
performance and training error and being able to estimate the generalization performance of such systems is of crucial importance. We will take the prediction risk
(expected test set error) as our measure of generalization performance.
2
Learning from Examples
Consider a set of n real-valued input/output data pairs ~(n) = {~i = (xi, yi); i =
1, ... , n} drawn from a stationary density 3(~). The observations can be viewed as
being generated according to the signal plus noise model 2
(2)
where yi is the observed response (dependent variable), Xl are the independent
variables sampled with input probability density O( x), Ei is independent, identicaIIydistributed (iid) noise sampled with density ~(E) having mean 0 and variance (72,3
and J.t(x) is the conditional mean, an unknown function. From the signal plus noise
perspective, the density 3(~) = 3(x, y) can be represented as the product of two
components, the conditional density w(ylx) and the input density O(x):
3(x, y)
w(ylx) O(x)
~(y - J.t(x? O(x)
(3)
The learning problem is then to find an estimate jJ,(x) of the conditional mean J.t(x)
on the basis of the training set ~(n).
In many real world problems, few a priori assumptions can be made about the
functional form of J.t(x). Since a parame~ric function class is usually not known,
one must resort to a nonparametric regression approach, whereby one constructs an
estimate jJ,(x) f(x) for J.t(x) from a large class of functions F known to have good
approximation properties (for example, F could be all possible radial basis function networks and multilayer perceptrons). The class of approximation functions is
usually the union of a countable set of subclasses (specific network architectures)4
A C F for which the elements of each subclass f(w, x) E A are continuously
parametrized by a set of p
p( A) weights w = {WCX; 0:
1, ... , p}. The task of
finding the estimate f( x) thus consists of two problems: choosing the best architecture A and choosing the best set of weights given the architecture. Note that in
=
=
=
w
2The assumption of additive noise ( which is independent of x is a standard assumption
and is not overly restrictive. Many other conceivable signal/noise models can be transformed into this form. For example, the multiplicative model y = /L(x)(l + () becomes
y' = /L'(x) + (' for the transformed variable y' = log(y).
3Note that we have made only a minimal assumption about the noise (, that it is has
finite variance (T2 independent of x. Specifically, we do not need to make the assumption
that the noise density <I>(() is of known form (e.g. gaussian) for the following development.
4For example, a "fully connected two layer perceptron with five internal units".
The Effective Number of Parameters
the nonparametric setting, there does not typically exist a function f( w'" , x) E F
with a finite number of parameters such that f(w"', x)
I1(X) for arbitrary l1(x).
For this reason, the estimators ji( x) = f( x) will be biased estimators of 11( x). 5
w,
=
The first problem (finding the architecture A) requires a search over possible architectures (e.g. network sizes and topologies), usually starting with small architectures and then considering larger ones. By necessity, the search is not usually
exhaustive and must use heuristics to reduce search complexity. (A heuristic search
procedure for two layer networks is presented in Moody and Utans (1992).)
The second problem (finding a good set of weights for f(w,x)) is accomplished by
minimizing an objective function:
(4)
WA = argminw U(A, w, e(n)) .
The objective function U consists of an error function plus a regularizer:
(5)
U(A, w,e(n)) = nEtrain(W,e(n)) + A S(w)
Here, the error Etrain(W,e(n)) measures the "distance" between the target response
values yi and the fitted values f(w,xi):
n
'.
.)] ,
Etrain(W,e(n)) = ~1 "
6 E[y"f(w,x'
(6)
i=l
and S( w) is a regularization or weight-decay function which biases the solution
toward functions with a priori "desirable" characteristics, such as smoothness. The
parameter A ~ 0 is the regularization or weight decay parameter and must itself be
optimized. 6
The most familiar example of an objective function uses the squared error 7
E[yi,f(w, xi)] = [yi - f(w,x i )]2 and a weight decay term:
n
U(A,w,~(n)) = L(yi - f(w,x i ))2
i=l
p
+ A Lg(w CY )
(7)
cy=l
The first term is the sum of squared errors (SSE) of the model f (w, x) with resp ect
to the training data, while the second term penalizes either small, medium, or
large weights, depending on the form of g(wCY). Two common examples of weight
decay functions are the ridge regression form g( wCY) = (w CY )2 (which penalizes large
weights) and the Rumelhart form g(w CY ) = (w CY )2/[(wO)2 + (w CY )2] (which penalizes
weights of intermediate values near wO).
J
5By biased, we mean that the mean squared bias is nonzero: MSB = p(x)((/:t(x))elL(x))2dx > o. Here, p(x) is some positive weighting function on the input space and
()e denotes an expected valued taken over possible training sets ?(n). For unbiasedness
(MSB = 0) to occur, there must exist a set of weights w* such that f(w"', x) = IL(X),
and the learned weights ill must be "close to" w*. For "near unbiasedness", we must have
w* argminwMSB(w) such that (MSB(w?)::::: 0) and ill "close to" w*.
6The optimization of..x will be discussed in Moody (1992).
7 Other error functions, such as those used in generalized linear models (see for example
McCullagh and NeIder 1983) or robust statistics (see for example Huber 1981) are more
appropriate than the squared error if the noise is known to be non-gaussian or the data
contains many outliers.
=
849
850
Moody
An example of a regularizer which is not explicitly a weight decay term is:
S(w) =
1
dxO(x)IIOxxf(w, x)112
.
(8)
This is a smoothing term which penalizes functional fits with high curvature.
3
Prediction Risk
=
With l1(x) f( w[c;( n)], x) denoting an estimate of the true regression function J.t(x)
trained on a data set c;( n), we wish to estimate the prediction risk P, which is the
exp ected error of 11( x) in predicting future data. In principle, we can either define
P for models l1(x) trained on arbitrary training sets of size n sampled from the
unknown density w(ylx )O( x) or for training sets of size n with input density equal
to the empirical density defined by the single training set available:
1 n
O'(x)
= -n L
8(x - xi) .
(9)
i=1
For such training sets, the n inputs xi are held fixed, but the response variables yi
are sampled with the conditional densities w(ylx i ). Since O'(x) is known, but O(x)
is generally not known a priori, we adopt the latter approach.
For a large ensemble of such training sets, the expected training set error is 8
(f...;n( A)),
/
~t
f[Y;, I(
J~ t.
\
w[~( n)], X;)])
E
1=1
f[lI ,J(
w[~( n)], x;)]
{g
wMx; )dll }
(10)
For a future exemplar (x,z) sampled with density w(zlx)O(x), the prediction risk
P is defined as:
P =
Jf[z,J(w[~(n)]'x)lw(zlx)n(x) {g
W(Y;IX;)d Y;} dzdx
(11)
Again, however, we don't assume that O(x) is known, so computing (11) is not
possible.
Following Akaike (1970), Barron (1984), and numerous other authors (see Eubank
1988), we can define the prediction risk P as the expected test set error for test sets
of size n e'(n)
{c;i,
(xi,zi); i
1, ... ,n} having the empirical input density
0' (x). This expected test set error has form:
=
(f.".(A)),<,
=
/
=
~ tf[i,J(w[~(n)l,x;)l)
J! t.
\
(12)
EE'
1=1
f[z; ,J(
w[~( n)], x;)I
{g
w(y; Ix; )w( z; Ix;)dy; dz; }
8Following the physics convention, we use angled brackets ( ) to denote expected values.
The subscripts denote the random variables being integrated over.
The Effective Number of Parameters
We take (12) as a proxy for the true prediction risk P.
In order to compute (12), it will not be necessary to know the precise functional
form of the noise density ~(f). Knowing just the noise variance (T2 will enable an
exact calculation for linear models trained with the SSE error and an approximate
calculation correct to second order for general nonlinear models. The results of
these calculations are presented in the next two sections.
4
The Expected Test Set Error for Linear Models
The relationship between expected training set and expected test set errors for linear
models trained using the SSE error function with no regularizer is well known in
statistics (Akaike 1970, Barron 1984, Eubank 1988). The exact relation for test and
training sets with density (9):
(13)
As pointed out by Barron (1984), (13) can also apply approximately to the case of
a nonlinear model f( w, x) trained by minimizing the sum of squared errors SSE.
This approximation can be arrived at in two ways. First, the model few, x) can be
treated as locally linear in a neighborhood of w. This approximation ignores the
hessian and higher order shape of f( w, x) in parameter space. Alternatively, the
model f( w, x) can be assumed to be locally quadratic in parameter space wand
unbiased.
However, the extension of (13) as an approximate relation for nonlinear models
breaks down if any of the following situations hold:
The SSE error function is not used. (For example, one may use a robust error
measure (Huber 1981) or log likelihood error measure instead.)
A regularization term is included in the objective function. (This introduces bias.)
The locally linear approximation for few, x) is not good.
The unbiasedness assumption for few, x) is incorrect.
5
The Expected Test Set Error for Nonlinear Models
For neural network models, which are typically nonparametric (thus biased) and
highly nonlinear, a new relationship is needed to replace (13). We have derived
such a result correct to second order for nonlinear models:
(14)
This result differs from the classical result (13) by the appearance of Pelf ()..) (the
effective number of parameters), (T;1f (the effective noise variance in the response
variable( s?, and a dependence on ).. (the regularization or weight decay parameter).
A full derivation of (14) will be presented in a longer paper (Moody 1992). The
result is obtained by considering the noise terms fi for both the training and test
851
852
Moody
sets as perturbations to an idealized model fit to noise free data. The perturbative
expansion is computed out to second order in the fi s subject to the constraint that
the estimated weights w minimize the perturbed objective function. Computing
expectation values and comparing the expansions for expected test and training
errors yields (14). It is important to re-emphasize that deriving (14) does not
require knowing the precise form of the noise density ~(f). Only a knowledge of u 2
is assumed.
The effective number of parameters Peff(>') usually differs from the true number
of model parameters P and depends upon the amount of model bias, model nonlinearity, and on our prior model preferences (eg. smoothness) as determined by
the regularization parameter A and the form of our regularizer. The precise form of
Peff(A) is
Peff(A)
1",
_
= trC = -2.
L..J1iaUaJTf3i
,
(15)
laf3
where C is the generalized influence matrix which generalizes the standard influence
or hat matrix of linear regression, 1ia is the n x p matrix of derivatives of the training
error function
1ia
=-88 . -88 nE(w,e(n)) ,
yl
wa
(16)
and U;;J is the inverse of the hessian of the total objective function
Uaf3
8
8
= 8w a 8wf3 U(A, w, e(n))
(17)
In the general case that u 2 (x) varies with location in the input space x, the effective
noise variance u;ff is a weighted average of the noise variances u 2 {xi). For the
uniform signal plus noise model model we have described above, u;f f = u 2 ?
6
The Effects of Regularization
In the neural network community, the most commonly used regularizers are weight
decay functions. The use of weight decay is motivated by the intuitive notion that
it removes unnecessary weights from the model. An analysis of Peff{A) with weight
decay (A > 0) confirms this intuitive notion. Furthermore, whenever u 2 > 0 and
n < 00, there exists some Aoptimal > 0 such that the expected test set error (12) is
minimized. This is because weight decay methods yield models with lower model
variance, even though they are biased. These effects will be discussed further in
Moody (1992).
For models trained with squared error ~SSE) and quadratic weight decay g(w a ) =
(w a )2, the assumptions of unbiasedness or local linearizability lead to the following
expression for Peff{A) which we call the linearized effective number of parameters
Plin{A):
(18)
9S trictly speaking, a model with quadratic weight decay is unbiased only if the "true"
weights are o.
The Effective Number of Parameters
mplied. Li nearized. and Full P-effectiv e
Linearized
. - -----.--~--
.... ..
,
'II(
~,
Full
..0 ?
~"
,
t
~
'"K
ImpJi d
E
~--
:i
~ ~-
1\'
u
::=> ,
i"
,.
,.
,.
1-
I.
"
Weight Decay Parameter (Lambda)
Figure 1: The full Peff(~) (15) agrees with the implied Pimp(~) (19) to within
exp erimental error, whereas the linearized Plin (~) (18) does not (except for very
large ~). These results verify the significance of (14) and (15) for nonlinear models.
Here,
",01
is the a th eigenvalue of the P x P matrix
J{
= TtT, with T
as defined in
(16).
The form of Pelf(~) can be computed easily for other weight decay functions, such
as the Rumelhart form g(w Ol ) = (w Ol )2/[(wO)2 + (w Ol )2]. The basic result for all
weight decay or regularization functions , however, is that Peff (~) is a decreasing
P and Pelf(oo)
0, as is evident in the special case
function of ~ with Pelf(O)
(18). If the model is nonlinear and biased, then Pelf (0) generally differs from p.
=
7
=
Testing the Theory
To test the result (14) in a nonlinear context, we computed the full Pej j(A) (15),
the linearized Plin(~) (18), and the implied number of parameters Pimp (A) (19) for a
nonlinear test problem. The value of Pimp (~) is obtained by computing the expected
training and test errors for an ensemble of training sets of size n with known noise
variance u 2 and solving for Pelf (~) in equation (14):
(19)
The """s indicate Monte Carlo estimates based on computations using a finite ensemble (10 in our experiments) of training sets. The test problem was to fit training
sets of size 50 generated as a sum of three sigmoids plus noise, with the noise sampled from the uniform density. The model architecture f(w , x) was also a sum of
three sigmoids and the weights w were estimated by minimizing (7) with quadratic
weight decay. See figure 1.
853
854
Moody
8
G PE: An Estimate of Prediction Risk for Nonlinear
Systems
A number of well established, closely related criteria for estimating the prediction
risk for linear or linearizable models are available. These include Akaike's F P E
(1970), Akaike's AlC (1973) Mallow's Cp (1973), and Barron's PSE (1984). (See
also Akaike (1974) and Eubank (1988).) These estimates are all based on equation
(13).
The generalized prediction error (G P E) generalizes the classical estimators F P E,
AIC, Cp, and PSE to the nonlinear setting by estimating (14) as follows:
-.
()
( ) = PGPE = &train n
GPE>'
+ 2u~2eff Peff(>')
n
.
(20)
The estimation process and the quality of the resulting GP E estimates will be
described in greater detail elsewhere.
Acknowledgements
The author wishes to thank Andrew Barron and Joseph Chang for helpful conversations.
This research was supported by AFOSR grant 89-0478 and ONR grant N00014-89-J-1228.
References
H. Akaike. (1970). Statistical predictor identification. Ann. Inst. Stat. Math., 22:203.
H. Akaike. (1973). Information theory and an extension of the maximum likelihood
principle. In 2nd Inti. Symp. on Information Theory, Akademia Kiado, Budapest, 267.
H. Akaike. (1974). A new look at the statistical model identification.
Auto. Control, 19:716-723.
IEEE Trans.
A. Barron. (1984). Predicted squared error: a criterion for automatic model selection. In
Self-Organizing Methods in Modeling, S. Farlow, ed., Marcel Dekker, New York.
R. Eubank. (1988). Spline Smoothing and Nonparametric Regression. Marcel Dekker,
New York.
P. J. Huber. (1981). Robust Statistics. Wiley, New York.
C. L. Mallows. (1973). Some comments on Cpo Technometrics 15:661-675.
P. McCullagh and J.A. NeIder. (1983). Generalized Linear Models. Chapman and Hall,
New York.
J. Moody. (1991). Note on Generalization, Regularization, and Architecture Selection
in Nonlinear Learning Systems. In B.H. Juang, S.Y. Kung, and C.A. Kamm, editors,
Neural Networks for Signal Processing, IEEE Press, Piscataway, N J.
J. Moody. (1992). Long version of this paper, in preparation.
J. Moody and J. Utans. (1992). Principled architecture selection for neural networks:
application to corporate bond rating prediction. In this volume.
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4,750 | 5,300 | Capturing Semantically Meaningful Word
Dependencies with an Admixture of Poisson MRFs
David I. Inouye
Pradeep Ravikumar
Inderjit S. Dhillon
Department of Computer Science
University of Texas at Austin
{dinouye,pradeepr,inderjit}@cs.utexas.edu
Abstract
We develop a fast algorithm for the Admixture of Poisson MRFs (APM) topic
model [1] and propose a novel metric to directly evaluate this model. The APM
topic model recently introduced by Inouye et al. [1] is the first topic model that
allows for word dependencies within each topic unlike in previous topic models
like LDA that assume independence between words within a topic. Research in
both the semantic coherence of a topic models [2, 3, 4, 5] and measures of model
fitness [6] provide strong support that explicitly modeling word dependencies?as
in APM?could be both semantically meaningful and essential for appropriately
modeling real text data. Though APM shows significant promise for providing
a better topic model, APM has a high computational complexity because O(p2 )
parameters must be estimated where p is the number of words ([1] could only
provide results for datasets with p = 200). In light of this, we develop a parallel alternating Newton-like algorithm for training the APM model that can handle p = 104 as an important step towards scaling to large datasets. In addition,
Inouye et al. [1] only provided tentative and inconclusive results on the utility
of APM. Thus, motivated by simple intuitions and previous evaluations of topic
models, we propose a novel evaluation metric based on human evocation scores
between word pairs (i.e. how much one word ?brings to mind? another word [7]).
We provide compelling quantitative and qualitative results on the BNC corpus
that demonstrate the superiority of APM over previous topic models for identifying semantically meaningful word dependencies. (MATLAB code available at:
http://bigdata.ices.utexas.edu/software/apm/)
1
Introduction and Related Work
In standard topic models such as LDA [8, 9], the primary representation for each topic is simply
a list of top 10 or 15 words. To understand a topic, a person must manually consider many of the
possible 10
relationships and attempt to
2 pairwise relationships as well as possibly larger m-wise
10
infer abstract meaning from this list of words. Of all the 2 pairwise relationships probably a
very small number of them are direct relationships. For example, a topic with the list of words
?money?, ?fund?, ?exchange? and ?company? can be understood as referring to investment but this
can only be inferred from a very high-level human abstraction of meaning. This problem has given
rise to research on automatically labeling topics with a topic word or phrase that summarizes the
topic [10, 11, 12]. [13] propose to evaluate topic models by randomly replacing a topic word with
a random word and evaluating whether a human can identify the intruding word. The intuition for
this metric is that the top words of a good topic will be related, and therefore, a person will be
able to easily identify the word that does not have any relationship to the other words. [2, 3, 5]
compute statistics related to Pointwise Mutual Information for all pairs of top words in a topic
and attempt to correlate this with human judgments. All of these metrics suggest that capturing
1
semantically meaningful relationships between pairs of words is fundamental to the interpretability
and usefulness of topic models as a document summarization and exploration tool.
In light of these metrics, [1] recently proposed a topic model called Admixture of Poisson MRFs
(APM) that relaxes the independence assumption for the topic distributions and explicitly models
word dependencies. This can be motivated in part by [6] who investigated whether the Multinomial (i.e. independent) assumption of word-topic distributions actually fits real-world text data.
Somewhat unsurprisingly, [6] found that the Multinomial assumption was often violated and thus
gives evidence that models with word dependencies?such as APM?may be a fundamentally more
appropriate model for text data.
Previous research in topic modeling has implicitly uncovered this issue with model misfit by finding
that models with 50, 100 or even 500 topics tend to perform better on semantic coherence experiments than smaller models with only 10 or 20 topics [4]. Though using more topics may allow topic
models to ignore the issue of word dependencies, using more topics can make the coherence of a
topic model more difficult as suggested by [4] who found that using 100 or 500 topics did not significantly improve the coherence results over 50 topics. Intuitively, a topic model with a much smaller
number of topics (e.g. 5 or 10) is easier to comprehend. For instance, if training on newspaper text,
the number of topics could roughly correspond to the number of sections in a newspaper such as
news, weather and sports. Or, if modeling an encyclopedia, the top-level topics could be art, history,
science, and society. Thus, rather than using more topics, APM opens the way for a promising topic
model that can overcome this model misfit issue while only using a small number of topics.
Even though APM shows promise for being a significantly more powerful and more realistic topic
model than previous models, the original paper acknowledged the significant computational complexity. Instead of needing to fit O(k(n + p)) parameters, APM needs to estimate O(k(n + p2 ))
parameters. [1] suggested that by using a sparsity prior (i.e. `1 regularization of the likelihood), this
computational complexity could be reduced. However, [1] could only produce some quantitative results on a very small dataset with only 200 words. In addition, the quantitative results from [1] were
tentative and inconclusive on whether APM could actually perform better than LDA in coherence
experiments.
Therefore, in this paper, we seek to answer two major open questions regarding APM: 1) Is there an
algorithm that can overcome the computational complexity of APM and handle real-world datasets?
2) Does the APM model actually capture more semantically interesting concepts that were not possible with previous topic models? We answer the first question by developing a parallel alternating
algorithm whose independent subproblems are solved using a Newton-like algorithm similar to the
algorithms developed for sparse inverse covariance estimation [14]. As in [14], this new APM
algorithm exploits the sparsity of the solution to significantly reduce the computational time for
computing the approximate Newton direction. However, unlike [14], the APM model is solving for
k Poisson MRFs simultaneously whereas [14] is only solving for a single Gaussian MRF. Another
difference from [14] is that the whole algorithm can be easily parallelized up to min(n, p).
For the second question about the semantic utility of APM, we develop a novel evaluation metric that
more directly evaluates the APM model against human judgments of semantic relatedness?a notion
called evocation introduced by [7]. Intuitively, the idea is that humans seek to understand traditional
topic models by looking at the list of top words. They will implicitly attempt to find how these
words are related and extract some more abstract meaning that generalizes the set of words. Thus,
this evaluation metric attempts to explicitly score how well pairs of words capture some semantically
meaningful word dependency. Previous research has evaluated topic models using word similarity
measures [4]. However, our work is different from [4] in three significant ways: 1) our metrics use
evocation rather than similarity (e.g. antonyms should have high evocation but low similarity), 2) we
evaluate top individual word pairs instead of rough aggregate statistics, and 3) we evaluate a topic
model that directly captures word dependencies (i.e. APM). We demonstrate that APM substantially
outperforms other topic models in both quantitative and qualitative ways.
2
Background on Admixture of Poisson MRFs (APM)
Admixtures The general notion of admixtures introduced by [1] generalizes many previous topic
models including PLSA [15], LDA [8], and the Spherical Admixture Model (SAM) [16]. Admix2
tures have also been known as mixed membership models [17]. In contrast to mixture distributions
which assume that each observation is drawn from 1 of k component distributions, admixture distributions assume that each observation is drawn from an admixed distribution whose parameters are
a mixture of component parameters. As examples of admixtures, PLSA and LDA are admixtures of
Multinomials whereas SAM is an admixture of Von-Mises Fisher distributions. In addition, because
of the connections between Poissons and Multinomials, PLSA and LDA can be seen as admixtures
of independent Poisson distributions [1].
Poisson MRFs (PMRF) Yang et al. [18] introduced a multivariate generalization of the Poisson
that assumes that the conditional distributions are univariate Poisson which is similar to a Gaussian
MRF whose conditionals are Gaussian (unlike a Guassian MRF, however, the marginals are not
univariate Poisson). A PMRF can be parameterized by a node vector ? and an edge matrix ? whose
non-zeros encode the direct dependencies between words: PrPMRF (x | ?, ?) = exp ? T x+xT ?x?
Pp
s=1 ln(xs !) ? A (?, ?) , where A (?, ?) is the log partition function needed for normalization.
This formulation needs to be slightly modified to allow for positive edges using the ideas from [19].
The log partition function can be approximated
Ppby using the pseudo log-likelihood instead of the
true likelihood, which means that A (?, ?) ? s=1 exp(?s + xT ?s ). The reader should note that
because this is an MRF distribution, all the properties of MRFs apply to PMRFs including that a
word is independent of all other words given the value of its neighbors. For example, in a chain
graph, all the variables are correlated with each other but they have a much simpler dependency
structure that can be encoded with O(n) parameters. Therefore, PMRFs more directly and succinctly
capture the dependencies between words as opposed to other simple statistics such as covariance or
pointwise mutual information.
Admixture of Poisson MRFs (APM) Inouye et al. [1] essentially constructed a new admixture model by using Poisson MRFs as the topic-word distributions instead of the usual Multinomial as in LDA. This allows for word dependencies within each topic. For example, if the word
?classification? appears in a document, ?supervised? is more likely to appear than in general documents. Given the admixture weights vector for a document the likelihood of a document
is sim
Pk
Pk
ply: PrAPM (x | w, ? 1...k , ?1...k ) = PrPMRF x | ? = j=1 wj ? j , ? = j=1 wj ?j (please see
Appendix A for notational conventions used throughout the paper). Inouye et al. [1] define a
Dirichlet(?) prior on the admixture weights and a conjugate prior with hyperparameter ? on the
PMRF parameters which can be easily incorporated as pseudo counts. For our experiments as described in Sec. 4.1, we set ? = 1 (i.e. a uniform prior on admixture weights) and ? = {0, 1}.
3
Parallel Alternating Newton-like Algorithm for APM
In the original APM paper [1], parameters were estimated by maximizing the joint approximate
posterior over all variables.1 Instead of maximizing jointly over all parameters, we split the problem
into alternating convex optimization problems. Let us denote the likelihood part (i.e. the smooth
part) of the optimization function as g(W, ? 1...k , ?1...k ) and the non-smooth `1 regularization term
as h where the full negative posterior is defined as f = g + h. The smooth part of the approximate
posterior can be written as:
g=?
n p
k
k
X
i
1 XX h X
wij xis (?sj + xTi ?js ) ? exp
wij (?sj + xTi ?js ) ,
n i=1 s=1 j=1
j=1
(1)
where xi is the word-count vector for the ith document, wi is the admixture weight vector for the
ith document, and ? j and ?j are the PMRF parameters for the jth component (see Appendix B for
derivation). By writing g in this form, it is straightforward to see that even though the whole optimization problem is not convex because of the interaction between the admixture weights w and the
PMRF parameters, the problem is convex if either the admixture weights W or the component parameters ? 1...k , ?1...k are held fixed. To simplify the notation in the following sections, we combine
1
This posterior approximation was based on the pseudo-likelihood while ignoring the symmetry constraint
so that nodewise regression parameters are independent. This leads to an overcomplete parameterization for
APM. For an overview of composite likelihood methods, see [20]. For a comparison of pseudo-likelihood
versus nodewise regressions, see [21].
3
the node (which is analogous to an intercept term in regression) and edge parameters by defining
zi = [1 xTi ]T , ?js = [?sj (?js )T ]T and ?s = [?1s ?2s ? ? ? ?ks ].
Thus, we can alternate between optimizing two similar optimization problems where one has a nonsmooth `1 regularization and the other has the constraint that wi must lie on the simplex ?k :
arg min
?1 ,?2 ,??? ,?p
arg min
w1 ,w2 ,??? ,wn ??k
p
p
n
i X
X
1 Xh
s s
T s
?
tr(? ? ) ?
exp(zi ? wi ) +
?kvec(?s )\1 k1
n s=1
s=1
i=1
?
p
n
i
X
1 Xh T
?i wi ?
exp(ziT ?s wi )) ,
n i=1
s=1
(2)
(3)
where ?i and ?s are constants in the optimization that can be computed from the data matrix X and
the other parameters that are being held fixed (see Alg. 2 in Appendix D for computation of ?s ).
This alternating scheme is analogous to Alternating Least Squares (ALS) for Non-negative Matrix
Factorization (NMF) [22] and EM-like algorithms such as k-means. By writing the optimization as
in Eq. 2 and Eq. 3, we also expose the simple independence between the subproblems because they
are simple summations. Thus, we can easily parallelize both optimization problems upto min(n, p)
with little overhead and simple changes to the code?in our MATLAB implementation, we only
changed a for loop to a parfor loop.
3.1
Newton-like Algorithms for Subproblems
For each of the subproblems, we develop Newton-like optimization algorithms. For the component
PMRFs, we borrow several important ideas from [14] including fixed and free sets of variables for
the `1 regularized optimization problem. The overall idea is to construct a quadratic approximation
around the current solution and approximately optimize this simpler function to find a step direction.
Usually, finding the Newton direction requires computing the Hessian for all the optimization variables but because of the `1 regularization, we only need to focus on variables that might be non-zero.
This set of free variables, denoted F, can be simply determined from the gradient and current iterate
[14]. Since usually there is only a small number of free variables compared to fixed variables (i.e. ?
is large enough), we can simply run coordinate descent on these free variables and only implicitly
calculate Hessian information as needed in each coordinate descent step. After finding an approximate Newton direction, we find a step size that satisfies the Armijo rule and then update the iterate
(see Alg. 2 in Appendix D).
We also employed a similar Newton-like algorithm for estimating the admixture weights. Instead of
the `1 regularization term, however, this subproblem has the constraint that the admixture weights
wi must lie on the simplex so that each document can be properly interpreted as a convex mixture
of over topic parameters. For this constraint, we used a dual-coordinate descent algorithm to find
the approximate Newton direction as in [23].
Finally, we put both subproblem algorithms together and alternate between the two (see Alg. 1 in
Appendix D). For tracing through different ? parameters, ? is initially set to ? so that the model
trains an independent APM model first. Then, the initial ? = ?max is found by computing the largest
gradient of the final independent iteration. Every time the alternating algorithm converges, the value
of ? is decreased so that a set of models is trained for decreasing values of ?.
3.2
Timing Results
We conducted two main timing experiments to show that the algorithm can be efficiently parallelized
and the algorithm can scale to reasonably large datasets. For the parallel timing experiment, we used
the BNC corpus described in Sec. 4.1 (n = 4049, p = 1646) and fixed k = 5, ? = 8 and a total of
30 alternating iterations. For the large data experiment, we used a Wikipedia dataset formed from a
recent Wikipedia dump by choosing the top 10k words neglecting stop words and then selecting the
longest documents. We ran several main iterations of the algorithm with this dataset while fixing
the parameters k = 5 and ? = 0.5. All timing experiments were conducted on the TACC Maverick
system with Intel Xeon E5-2680 v2 Ivy Bridge CPUs (2.80 GHz), 20 CPUs per node, and 12.8 GB
memory per CPU (https://www.tacc.utexas.edu/).
4
The parallel timing results can be seen in Fig. 1 (left) which shows that the algorithm does have
almost linear speedup when parallelizing across multiple workers. Though we only had access to a
single computer with 20 processors, substantially more speed up could be obtained by using more
processors on a distributed computing system. This simple parallelism makes this algorithm viable
for much larger datasets. The timing results for the Wikipedia can be seen in Fig. 1 (right). These
results give an approximate computational complexity of O(np2 ) which show that the proposed
algorithm has the potential for scaling to datasets where n is O(105 ) and p is O(104 ). The O(p2 )
comes from the fact that there are p subproblems and each subproblem needs to calculate the gradient
which is O(p) as well as approximate the Newton direction for a subset of the variables. The
first iteration takes longer because the initial parameter values are na??vely set to 0 whereas future
iterations start from reasonable initial value.
APM Training Time on Wikipedia Dataset
Parallel Speedup on BNC Dataset
4
Perfect Speedup
15
1st Iter.
Time (hrs)
Speedup
20
Actual Speedup
10
5
2.2
5
10
15
# of MATLAB Workers
20
2.2
1
1
0
3.4
3.1
2
0
0
Avg. Next 3 Iter.
3
0.6
n = 20,000
p = 5,000
# of Words = 50M
n = 100,000
p = 5,000
# of Words = 133M
n = 20,000
p = 10,000
# of Words = 57M
Figure 1: (left) The speedup on the BNC dataset shows that the algorithm scales approximately linearly with the number of workers because the subproblems are all independent. (right) The timing
results on the Wikipedia dataset show that the algorithm scales to larger datasets and has a computational complexity of approximately O(np2 ).
4
Evocation Metric
Boyd-Graber et al. [7] introduced the notion of evocation which denotes the idea of which words
?evoke? or ?bring to mind? other words. There can be many types of evocation including the following examples from [7]: [rose - flower] (example), [brave - noble] (kind), [yell - talk] (manner),
[eggs - bacon] (co-occurence), [snore - sleep] (setting), [wet - desert] (antonymy), [work - lazy] (exclusivity), and [banana - kiwi] (likeness). This is distinctive from word similarity or synonymy since
two words can have very different meanings but ?bring to mind? the other word (e.g. antonyms).
This notion of word relatedness is a much simpler but potentially more semantically meaningful and
interpretable than word similarity. For instance, ?work? and ?lazy? do not have similar meanings
but are related through the semantic meanings of the words. Another difference is that?unlike word
semantic similarity? words that generally appear in very different contexts yet mean the same thing
would probably not have a high evocation score. For example, ?networks? and ?graphs? both have a
definition that means a set of nodes and edges yet usually one word is chosen in a particular context.
Recent work in evaluating topic models [2, 3, 4, 5] formulate automated metrics based on automatically scoring all pairs of top words and noticing that they correlate with human judgment of overall
topic coherence. All of these metrics are based on the common assumption that a person should
be able to understand a topic by understanding the abstract semantic connections between the word
pairs. Thus, evocation is a reasonable notion for evaluating topic modeling because it directly evaluates the level of semantic connection between word pairs. In addition, this new evocation metric
provides a way to explicitly evaluate the edge matrices of APM, which would be ignored in previous
metrics because explicit word dependencies are not modeled in other topic models.
We now formally define our evocation metric. Given human-evaluated scores for a subset of word
pairs H and the corresponding weights given by a topic model for this subset of word pairs M, let
us define ?M (j) to be an ordering of the word pairs induced by M such that M?(1) ? M?(2) ?
? ? ? ? M?(|H|) . Then, the top-m evocation metric is simply:
m
X
Evocm (M, H) =
H?M(j) .
(4)
j=1
Note that the scaling of M is inconsequential because M is only needed to define an ordering or
? = ? exp(M) would yield the same evocation score for all scalar
ranking of H. For example, M
5
values ? > 0 because the ordering would be maintained. Essentially, M merely induces an ordering
of the word pairs and the evocation score is the sum of the human scores for these top m word pairs.
For APM, the word pair weights come primarily from the PMRF edge matrices ?1...k ?the PMRF
node vectors are only used to provide an ordering if there are not enough non-zeros in the edge
matrices. For the other Multinomial-based topic models which do not have parameters explicitly
associated with word-pairs, we can compute the most likely word pairs in a topic by multiplying
their corresponding marginal probabilities. This weighting corresponds to the probability that two
independent draws from the topic distribution produce the word pair and thus is the most obvious
choice for Multinomial-based topic models.
Since this metric only gives a way to evaluate one topic, we consider two ways of determining
Pk 1
j
the overall evocation score for the whole topic model: Evoc-1 =
j=1 k Evocm (M , H) and
Pk 1 j
Evoc-2 = Evocm ( j=1 k M , H). In words, these are ?average evocation of topics? and ?evocation of average topic? respectively. Evoc-1 measures whether all or at least most topics capture
meaningful word associations since it can be affected by uninteresting topics. Evoc-2 is reasonable
for measuring whether the topic model as a whole is capturing word semantics even if some of the
topics are not capturing interesting word associations. This second measure has some relation to
the word similarity measure of topic coherence in [4]. However, [4] uses similarity rather than evocation, does not directly evaluate top individual word pairs and does not evaluate any models with
word dependencies such as APM.
4.1
Experimental Setup
Human-Scored Evocation Dataset The original human-scored evocation dataset was produced
by a set of trained undergraduates in which 1,000 words were hand selected primarily based on
their frequency and usage in the British National Corpus (BNC) [7]. From the possible pairwise
evaluations, approximately 10% of the word pairs were randomly selected to be manually scored by
a set of trained undergraduates. The second dataset was constructed by predicting the pairs of words
that were likely to have a high evocation using a standard machine learning classifier. This new set of
pairs was scored using Amazon MTurk (mturk.com) by using the original dataset as a control [24].
Though these scores are between synsets?which are a word, part-of-speech and sense triplet?, we
mapped all of the synsets to word, part-of-speech pairs since that is the only information we have
for the BNC corpus. This led to a total of 1646 words. In addition, though the evocation dataset has
scores for directed relationships (i.e. word1 ? word2 could have a different score than word2 ?
word1), we averaged these two scores because the directionality of the relationship is not modeled
by APM or any other topic model.
BNC Corpus Because the evocation dataset was based on the BNC corpus, we used the BNC corpus for our experiments. We processed the BNC corpus by lemmatizing each word using the WordNetLemmatizer included in the nltk package (nltk.org) and then attaching the part-of-speech,
which is already included in the BNC corpus. We only retained the counts for the 1646 words that
occurred in the human-scored datasets but processed all 4049 documents in the corpus.
APM Model Parameters We trained APM on the BNC corpus with several different parameter
settings including various ? and ? parameter settings. We also trained two particular APM models
denoted APM-LowReg and APM-HeldOut. APM-LowReg uses a very small regularization parameter so that almost all edges are non-zero. APM-HeldOut automatically selects a reasonable value
for ? based on the likelihood of a held-out set of the documents. Thus, the APM-HeldOut model
does not require a user-specified ? parameter but?as seen in the following sections?still performs
reasonably well even compared to the APM model in which many different parameter settings are
attempted. In addition, the APM-HeldOut can stop the training early when the model begins to overfit the data rather than tracing through all the ? parameters?this could lead to a significant gain in
model training time. The authors suggest that APM-HeldOut is a simple baseline model for future
comparison if a user does not want to specify ?.
Other Models For comparison, we trained five other models: Correlated Topic Models (CTM),
Hierarchical Dirichlet Process (HDP), Latent Dirichlet Allocation (LDA), Replicated Softmax
(RSM), and a na??ve random baseline (RND). CTM models correlations between topics [25]. HDP
6
is a non-parametric Bayesian model that selects the number of topics based input data and hyperparameters [26]. The standard topic model LDA was trained using MALLET [27]. LDA was trained
for at least 5,000 iterations and HDP was trained for at least 300 iterations since HDP is computationally expensive. RSM is an undirected topic model based on Restricted Boltzmann Machines
(RBM) [28]. The random model is merely the expected evocation score if edges are ranked at random. We ran a full factorial experimental setting where all the combinations of a set of parameter
values were trained to give a fair comparison between models (see Appendix C for a summary of
parameter values). All these comparison models only indirectly model dependencies between words
through the latent variables since the topic distributions are Multinomials whereas APM can directly
model the dependencies between words since the topic distributions are Poisson MRFs.
Selecting Best Parameters We randomly split the human scores into a 50% tuning split and 50%
testing split. Note that we have a tuning split rather than a training split because the model training
algorithms are unsupervised (i.e. they never see the human scores) so the only supervision occurs in
selecting the final model parameters (i.e. during the tuning phase). Therefore, we selected the final
parameters based on the tuning split and computed the final evocation scores on the test split. Thus,
even when selecting the parameter settings, the modeling process never sees the test data.
4.2
Main Results
The Evoc-1 and Evoc-2 scores with m = 50 for all models can be seen in Fig. 2.2 For Evoc-1, the
APM models significantly outperform all other models for a small number of topics and even capture
many semantically meaningful word pairs with a single topic. For higher number of topics, the APM
models seem to perform only competitively with previous topic models. It seems that APM-LowReg
performs better with a small number of topics whereas APM-HeldOut?which generally chooses a
relatively high ??seems more robust for large number of topics. These trends are likely caused
because there is a relatively small number of documents (n = 4049) so the APM-LowReg begins to
significantly overfit the data as the number of topics increases whereas APM-HeldOut does not seem
to overfit as much. For all the APM models, the degradation in performance as the number of topics
increases is most likely caused by the fact that a Poisson MRF with O(p2 ) parameters is a much
more flexible distribution than a Multinomial, and thus, fewer topics are needed to appropriately
model the data. These results also give some evidence that APM can succinctly model the data with
a much smaller number of topics than is needed for independent topic models; this succinctness
could be particularly helpful for the interpretability and intuitions of topic models.
Evocation (m= 50)
APM
APM-LowReg
APM-HeldOut
CTM
HDP
LDA
RSM
RND
1600
1400
1200
1000
800
600
400
200
0
k=1
3
5
10
25
50
Evoc-1 (Avg. Evoc. of Topics)
k=1
3
5
10
25
50
Evoc-2 (Evoc. of Avg. Topic)
Figure 2: Both Evoc-1 scores (left) and Evoc-2 scores (right) demonstrate that APM usually significantly outperforms other topic models in capturing meaningful word pairs.
For the Evoc-2 score, the APM models?including the APM-HeldOut model which automatically
determines a ? from the data?significantly outperform previous topic models even for a large number of topics. This supports the idea that APM only needs a small number of topics to capture many
of the semantically meaningful word dependencies. Thus, when increasing the number of topics
beyond 5, the performance does not decrease as in Evoc-1. It is likely that this discrepancy is caused
by the fact that many of the edges are concentrated in a small number of topics even when the number of topics is 10 or 25. As expected because of previous research in topic models, most other topic
2
For simplicity and comparability, we grouped HDP into the topic number that was closest to its discovered
number of topics because HDP can select a variable number of topics.
7
models perform slightly better with a larger number of topics. Though it is possible that using 100
or 500 topics for these topic models might give an evocation score better than APM with 5 topics,
this would only enforce the idea that APM can perform better or at least competitively with previous
topic models while only using a comparatively small number of topics.
Qualitative Analysis of Top 20 Word Pairs for Best LDA and APM Models To validate the
intuition of using evocation as an human-standard evaluation metric, we present the top 20 word
pairs for the best standard topic model?in this case LDA?and the best APM model for the Evoc-2
metric as seen in Table 1. The best performing LDA model was trained with 50 topics, ? = 1 and
? = 0.0001. The best APM model was the APM-LowReg model trained with only 5 topics and a
small regularization parameter ? = 0.05. It is important to note that the best model for LDA has
50 topics while the best model for APM only has 5 topics. As before, this reinforces the theme that
APM can capture more semantically meaningful word pairs with a smaller number of topics than
previous topic models.
1:=Top
APM
(right)
LDALDA
Evocation
Evocation
ofTable
Avg.
of Avg.
Graph
Graph
967
= 96720 words for LDA (left) and
APM
APM
Evocation
Evocation
of Avg.
of Avg.
Graph
Graph
= 1627
= 1627
RankRank
Evoc.
Evoc.
1
2
3
4
5
6
7
8
9
10
1 38
2 0
3 13
4 69
5 0
6 82
7 38
8 35
9 7
10 38
EdgeEdge
RankRank
Evoc.
Evoc.
38 woman.n
woman.n
??
man.n
man.n 11
0 woman.n
woman.n
??
wife.n
wife.n
12
13
train.n
train.n
??
car.n
car.n
13
69 school.n
school.n
??
class.n
class.n 14
0
drive.v
drive.v
??
car.n
car.n
15
82 teach.v
teach.v
??
school.n
school.n 16
38 engine.n
engine.n
??
car.n
car.n
17
35 publish.v
publish.v
??
book.n
book.n 18
7 religious.a
religious.a
??
church.n
church.n 19
38 state.n
state.n
??
government.n
government.n
20
EdgeEdge
RankRank
Evoc.
Evoc.
11 0 0
car.n
car.n
??
bus.n
bus.n
12 31 31
year.n
year.n
??
day.n
day.n
13 25 25
car.n
car.n
??
seat.n
seat.n
14 50 50 teach.v
teach.v
??
student.n
student.n
15 0 0
tell.v
tell.v
??
get.v
get.v
16 38 38
wife.n
wife.n
??
man.n
man.n
17100 100
run.v
run.v
??
car.n
car.n
18 0 0
give.v
give.v
??
get.v
get.v
19 16 16 paper.n
paper.n
??
book.n
book.n
20 19 19 white.a
white.a
??
black.a
black.a
1
2
3
4
5
6
7
8
9
10
1 13
2 60
3 13
4 50
5 38
6 75
7 57
8 13
9 7
10 97
EdgeEdge
RankRank
Evoc.
Evoc.
13 smoke.v
smoke.v
??
cigarette.n
cigarette.n11
60
love.v
love.v
??
love.n
love.n
12
13
eat.v
eat.v
??
food.n
food.n 13
50 west.n
west.n
??
east.n
east.n
14
38 south.n
south.n
??
north.n
north.n 15
75
iron.n
iron.n
??
steel.n
steel.n 16
57question.n
question.n
??
answer.n
answer.n 17
13
boil.v
boil.v
??
potato.n
potato.n 18
7 religious.a
religious.a
??
church.n
church.n 19
97husband.n
husband.n
??
wife.n
wife.n
20
11 72
12 28
13 25
14 0
15 35
16 0
17 19
18 41
19 33
20 7
EdgeEdge
72
aunt.n
aunt.n
??
uncle.n
uncle.n
28
tea.n
tea.n
??
coffee.n
coffee.n
25operational.a
operational.a
??
aircraft.n
aircraft.n
0competition.n
competition.n
??
compete.v
compete.v
35 green.n
green.n
??
green.a
green.a
0
fox.n
fox.n
??
animal.n
animal.n
19 smoke.n
smoke.n
??
fire.n
fire.n
41 wine.n
wine.n
??
drink.v
drink.v
33 troop.n
troop.n
??
force.n
force.n
7
lock.n
lock.n
??
key.n
key.n
One interesting example is that LDA finds two word pairs [woman.n - wife.n] and [wife.n - man.n]
that capture some semantic notion of marriage. However, APM directly captures this semantic
meaning with [husband.n - wife.n]. APM also finds more intuitive verb-noun relationships that are
closely tied semantically and portray a particular context: [smoke.v - cigarette.n], [eat.v - food.n],
[boil.v - potato.n], and [drink.v - wine.n] whereas LDA tends to select less interesting verb-noun
relationships such as [run.v - car.n]. In addition, APM finds multiple semantically coherent yet high
level word pairs such as [iron.n - steel.n], [question.n - answer.n], and [aunt.n - uncle.n], whereas
LDA finds several low-level edges such as [year.n - day.n] and [tell.v - get.v]. These overall trends
become even more evident when looking at the top 50 edges as can be found in the Appendix E.
Both the quantitative evaluation metrics (i.e. Evoc-1 and Evoc-2) as well as a qualitative exploration
of the top word pairs give strong evidence that APM can succinctly capture both more interesting
and higher-level semantic concepts through word dependencies than independent topic models.
5
Conclusion and Future Work
We motivated the need for more expressive topic models that consider word dependencies?such
as APM?by considering previous work on topic model evaluation metrics. We overcame the significant computational barrier of APM by providing a fast alternating Newton-like algorithm which
can be easily parallelized. We proposed a new evaluation metric based on human evocation scores
that seeks to measure whether a model is capturing semantically meaningful word pairs. Finally,
we presented compelling quantitative and qualitative measures showing the superiority of APM in
capturing semantically meaningful word pairs. In addition, this metric suggests new evaluations
of topic models based on evaluating top word pairs rather than top words. One drawback with the
current human-scored data is that only a small portion of the word pairs have been scored. Thus,
one extension is to dynamically collect more human scores as needed for evaluation. This work also
opens the door for exciting new word-semantic applications for APM such as Word Sense Induction
using topic models [29], keyword expansion or suggestion, document summarization, and document
visualization because APM is capturing semantically meaningful relationships between words.
Acknowledgments
D. Inouye was supported by the NSF Graduate Research Fellowship via DGE-1110007. P. Ravikumar acknowledges support from ARO via W911NF-12-1-0390 and NSF via IIS-1149803, IIS1447574, and DMS-1264033. I. Dhillon acknowledges support from NSF via CCF-1117055.
8
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9
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4,751 | 5,301 | Dynamic Rank Factor Model for Text Streams
Shaobo Han?, Lin Du?, Esther Salazar and Lawrence Carin
Duke University, Durham, NC 27708
{shaobo.han, lin.du, esther.salazar, lcarin}@duke.edu
Abstract
We propose a semi-parametric and dynamic rank factor model for topic modeling, capable of (i) discovering topic prevalence over time, and (ii) learning contemporary multi-scale dependence structures, providing topic and word correlations as a byproduct. The high-dimensional and time-evolving ordinal/rank observations (such as word counts), after an arbitrary monotone transformation, are
well accommodated through an underlying dynamic sparse factor model. The
framework naturally admits heavy-tailed innovations, capable of inferring abrupt
temporal jumps in the importance of topics. Posterior inference is performed
through straightforward Gibbs sampling, based on the forward-filtering backwardsampling algorithm. Moreover, an efficient data subsampling scheme is leveraged
to speed up inference on massive datasets. The modeling framework is illustrated
on two real datasets: the US State of the Union Address and the JSTOR collection
from Science.
1
Introduction
Multivariate longitudinal ordinal/count data arise in many areas, including economics, opinion polls,
text mining, and social science research. Due to the lack of discrete multivariate distributions supporting a rich enough correlation structure, one popular choice in modeling correlated categorical
data employs the multivariate normal mixture of independent exponential family distributions, after
appropriate transformations. Examples include the logistic-normal model for compositional data
[1], the Poisson log-normal model for correlated count data [2], and the ordered probit model for
multivariate ordinal data [3]. Moreover, a dynamic Bayesian extension of the generalized linear
model [4] may be considered, for capturing the temporal dependencies of non-Gaussian data (such
as ordinal data). In this general framework, the observations are assumed to follow an exponential family distribution, with natural parameter related to a conditionally Gaussian dynamic model
[5], via a nonlinear transformation. However, these model specifications may still be too restrictive
in practice, for the following reasons: (i) Observations are usually discrete, non-negative and with
a massive number of zero values and, unfortunately, far from any standard parametric distributions
(e.g., multinomial, Poisson, negative binomial and even their zero-inflated variants). (ii) The number
of contemporaneous series can be large, bringing difficulties in sharing/learning statistical strength
and in performing efficient computations. (iii) The linear state evolution is not truly manifested after
a nonlinear transformation, where positive shocks (such as outliers and jumps) are magnified and
negative shocks are suppressed; hence, handling temporal jumps (up and down) is a challenge for
the above models.
We present a flexible semi-parametric Bayesian model, termed dynamic rank factor model (DRFM),
that does not suffer these drawbacks. We first reduce the effect of model misspecification by modeling the sampling distribution non-parametrically. To do so, we fit the observed data only after
some implicit monotone transformation, learned automatically via the extended rank likelihood [6].
Second, instead of treating panels of time series as independent collections of variables, we analyze
them jointly, with the high-dimensional cross-sectional dependencies estimated via a latent factor
?
contributed equally
1
model. Finally, by avoiding nonlinear transformations, both smooth transitions and sudden changes
(?jumps?) are better preserved in the state-space model, using heavy-tailed innovations.
The proposed model offers an alternative to both dynamic and correlated topic models [7, 8, 9],
with additional modeling facility of word dependencies, and improved ability to handle jumps. It
also provides a semi-parametric Bayesian treatment of dynamic sparse factor model. Further, our
proposed framework is applicable in the analysis of multiple ordinal time series, where the innovations follow either stationary Gaussian or heavy-tailed distributions.
2
Dynamic Rank Factor Model
We perform analysis of multivariate ordinal time series. In the most general sense, such ordinal
variables indicate a ranking of responses in the sample space, rather than a cardinal measure [10].
Examples include real continuous variables, discrete ordered variables with or without numerical
scales or, more specially, counts, which can be viewed as discrete variables with integer numeric
scales. Our goal is twofold: (i) discover the common trends that govern variations in observations,
and (ii) extract interpretable patterns from the cross-sectional dependencies.
Dependencies among multivariate non-normal variables may be induced through normally distributed latent variables. Suppose we have P ordinal-valued time series yp,t , p = 1, . . . , P ,
t = 1, . . . , T . The general framework contains three components:
yp,t ? g(zp,t ), zp,t ? p(? t ), ? t ? q(? t?1 ),
(1)
where g(?) is the sampling distribution, or marginal likelihood for the observations, the latent variable zp,t is modeled by p(?) (assumed to be Gaussian) with underlying system parameters ? t , and
q(?) is the system equation representing Markovian dynamics for the time-evolving parameter ? t .
In order to gain more model flexibility and robustness against misspecification, we propose a semiparametric Bayesian dynamic factor model for multiple ordinal time series analysis. The model is
based on the extended rank likelihood [6], allowing the transformation from the latent conditionally
Gaussian dynamic model to the multivariate observations, treated non-parametrically.
Extended rank likelihood (ERL): There exist many approaches for dealing with ordinal data, however, they all have some restrictions. For continuous variables, the underlying normality assumption
could be easily violated without a carefully chosen deterministic transformation. For discrete ordinal variables, an ordered probit model, with cut points, becomes computationally expensive if the
number of categories is large. For count variables, a multinomial model requires finite support on
the integer values. Poisson and negative binomial models lack flexibility from a practical viewpoint,
and often lead to non-conjugacy when employing log-normal priors.
Being aware of these issues, a natural candidate for consideration is the ERL [6]. With appropriate
monotone transformations learned automatically from data, it offers a unified framework for handling both continuous [11] and discrete ordinal variables. The ERL depends only on the ranks of the
observations (zero values in observations are further restricted to have negative latent variables),
zp,t ? D(Y ) ? {z p,t ? R : yp,t < yp0 ,t0 ? zp,t < zp0 ,t0 , and zp,t ? 0 if yp,t = 0}.
(2)
In particular, this offers a distribution-free approach, with relaxed assumptions compared to parametric models, such as Poisson log-normal [12]. It also avoids the burden of computing nuisance
parameters in the ordered probit model (cut points). The ERL has been utilized in Bayesian Gaussian
copula modeling, to characterize the dependence of mixed data [6]. In [13] a low-rank decomposition of the covariance matrix is further employed and efficient posterior sampling is developed in
[14]. The proposed work herein can be viewed as a dynamic extension of that framework.
2.1
Latent sparse dynamic factor model
In the forthcoming text, G(?, ?) denotes a gamma distribution with shape parameter ? and rate
parameter ?, TN(l,u) (?, ? 2 ) denotes a univariate truncated normal distribution within the interval
(l, u), and N+ (0, ? 2 ) is the half-normal distribution that only has non-negative support.
Assume z t ? N (0, ?t ), where ?t is usually a high-dimensional (P ? P ) covariance matrix.
To reduce the number of parameters, we assume a low rank factor model decomposition of the
covariance matrix ?t = ?V t ?T + R such that
z t = ?st + t , t ? N (0, R), R = I P .
(3)
2
Common trends (importance of topics) are captured by a low-dimensional factor score parameter
st . We assume autoregressive dynamics on sk,t ? AR(1|(?k , ?k,t )) with heavy-tailed innovations,
sk,t = ?k sk,t?1 + ?k,t , 0 < ?k < 1, ?k,t ? TPBN(e, f, ?), ? 1/2 ? C + (0, h),
(4)
where ?k,t follows the three-parameter beta mixture of normal TPBN(e, f, ?) distribution [15]. Parameter e controls the peak around zero, f controls the heaviness on the tails, and ? controls the
global sparsity with a half-Cauchy prior [16]. This prior encourages smooth transitions in general,
while jumps are captured by the heavy tails. The conjugate hierarchy may be equivalently represented as
?k,t ? N (0, ?k,t ), ?k,t ? G(e, ?k,t ), ?k,t ? G(f, ?) ? ? G(1/2, ?), ? ? G(1/2, h2 ).
Truncated normal priors are employed on ?k , ?k ? TN(0,1) (?0 , ?02 ), and assume s0,k ? N (0, ?s2 ).
Note that the extended rank likelihood is scale-free; therefore, we do not need to include a redundant
intercept parameter in (3). For the same reason, we set R = I P .
Model identifiability issues: Although
p the covariance matrix ?t is not identifiable [10], the related
correlation matrix C t = ?[i,j],t / ?[i,i],t ?[j,j],t , (i, j = 1, . . . , P ) may be identified, using the
parameter expansion technique [3, 13]. Further, the rank K in the low-rank decomposition of ?t is
also not unique. For the purpose of brevity, we do not explore this uncertainty here, but the tools
developed in the Bayesian factor analysis literature [17, 18, 19] can be easily adopted.
Identifiability is a key concern for factor analysis. Conventionally, for fixed K, a full-rank, lowertriangular structure in ? ensures identifiability [20]. Unfortunately, this assumption depends on the
ordering of variables. As a solution, we add nonnegative and sparseness constraints on the factor
loadings, to alleviate the inherit ambiguity, while also improving interpretability. Also, we add a
Procrustes post-processing step [21] on the posterior samples, to reduce this indeterminacy.
The nonnegative and (near) sparseness constraints are imposed by the following hierarchy,
1/2
?p,k ? N+ (0, lp,k ) lp,k ? G(a, up,k ), up,k ? G(b, ?k ), ?k ? C + (0, d).
(5)
Integrating out lp,k and up,k , we obtain a half-TPBN prior ?p,k ? TPBN+ (a, b, ?k ). The columnwise shrinkage parameters ?k enable factors to be of different sparsity levels [22]. We set hyperparameters a = b = e = f = 0.5, d = P , h = 1, ?s2 = 1. For weakly informative priors, we set
? = ? = 0.01; ?0 = 0.5, ?02 = 10.
2.2
Extension to handle multiple documents
nt
t
At each time point t we may have a corpus of documents {y nt t }N
nt =1 , where y t is a P -dimensional
observation vector, and Nt denotes the number of documents at time t. The model presented in
Section 2.1 is readily extended to handle this situation. Specifically, at each time point t, for each
nt
, is
document nt , the ERL representation for word count p, denoted by yp,t
nt
nt
yp,t
= g zp,t
, p = 1, . . . , P, t = 1, . . . , T, nt = 1, . . . , Nt ,
where z nt t ? RP and P is the vocabulary size. We assume a latent factor model for z nt t such that
z nt t = ?bnt t + nt t , nt t ? N (0, I P ), bnt t ? N (st , ?), ? = diag(?), ?k?1 ? G(?, ?),
?K
where ? ? RP
is the topic-word loading matrix, representing the K topics as columns of ?.
+
The factor score vector bnt t ? RK is the topic usage for each document y nt t , corresponding to locations in a low-dimensional RK space. The other parts of the model remain unchanged. The latent
trajectory s1:T represents the common trends for the K topics. Moreover, through the forward filtering backward sampling (FFBS) algorithm [23, 24], we also obtain time-evolving topic correlation
matrices ?t ? RK?K and word dependencies matrices C t ? RP ?P , offering a multi-scale graph
representation, a useful tool for document visualization.
2.3
Comparison with admixture topic models
Many topic models are unified in the admixture framework [25],
!
K
X
P (y n |w, ?) = P y n ?n =
wk,n ?k ,
Base
Admix
(6)
k=1
where y n is the P -dimensional observation vector of word counts in the n th document, and P denotes the vocabulary size. Traditionally, y n is generated from an admixture of base distributions, wn
is the admixture weight (topic proportion for document n), and ?k is the canonical parameter (word
3
distribution for topic k), which denotes the location of the kth topic on the P -1 dimensional simplex.
For example, latent Dirichlet allocation (LDA) [26] assumes the base distribution to be multinomial,
with ?k ? Dir(?0 ), wn ? Dir(? 0 ). The correlated topic model (CTM) [8] modifies the topic distribution, with wn ? Logistic Normal(?, ?). The dynamic topic model (DTM) [7] analyzes document collections in a known chronological order. In order to incorporate the state space model, both
the topic proportion and the word distribution are changed to logistic normal, with isotropic covariance matrices wt ? Logistic Normal(wt?1 , ? 2 I K ) and ?k,t ? Logistic Normal(?k,t?1 , vI P ),
respectively. To overcome the drawbacks of multinomial base, spherical topic models [27] assume
the von Mises-Fisher (vMF) distribution as its base distribution, with ?k ? vMF(?, ?) lying on a
unit P -1 dimensional sphere. Recently in [25] the base and word distribution are both replaced with
Poisson Markov random fields (MRFs), which characterizes word dependencies.
We present here a semi-parametric factor model formulation,
!
K
X
P(y n |s, ?) , P z n ? D(Y ) ?n =
sk,n ?k ,
(7)
k=1
with y n defined as above, ?k ? RP
+ is a vector of nonnegative weights, indicating the P vocabulary usage in each individual topics k, and sn ? RK is the topic usage. Note that the extended
rank likelihood does not depend on any assumptions about the data marginal distribution, making it
appropriate for a broad class of ordinal-valued observations, e.g., term frequency-inverse document
frequency (tf-idf) or rankings, beyond word counts. However, the proposed model here is not an
admixture model, as the topic usage is allowed to be either positive or negative.
The DRFM framework has some appealing advantages: (i) It is more natural and convenient to incorporate with sparsity, rank selection, and state-space model; (ii) it provides topic-correlations and
word-dependences as a byproduct; and (iii) computationally, this model is tractable and often leads
to locally conjugate posterior inference. DRFM has limitations. Since the marginal distributions
are of unspecified types, objective criteria (e.g. perplexity) is not directly computable. This makes
quantitative comparisons to other parametric baselines developed in the literature very difficult.
3
Conjugate Posterior Inference
Let ? = {?, S, L, U , ?, ?, ?, ? , ?, ?, ?} denote the set of parameters in basic model, and let Z be
the augmented data (from the ERL). We use Gibbs sampling to approximate the joint posterior distribution p(Z, ?|Z ? R(Y )). The algorithm alternates between sampling p(Z|?, Z ? R(Y )) and
p(?|Z, Z ? R(Y )) (reduced to p(?|Z)). The derivation of the Gibbs sampler is straightforward,
and for brevity here we only highlight the sampling steps for Z, and the forward filtering backward
sampling (FFBS) steps for the trajectory s1:T . The Supplementary Material contains further details
for the inference.
PK
? Sampling zp,t : p(zp,t |?, Z ? R(Y ), Z ?p,?t ) ? TN[zp,t ,zp,t ] ( k=1 ?p,k sk,t , 1), where zp,t =
max{zp0 ,t0 : yp0 ,t0 < yp,t } and zp,t = min{zp0 ,t0 : yp0 ,t0 > yp,t }.
This conditional sampling scheme is widely used in [6, 10, 13]. In [14] a novel Hamiltonian Monte
Carlo (HMC) approach has been developed recently, for a Gaussian copula extended rank likelihood
model, where ranking is only within each row of Z. This method simultaneously samples a column
vector of z i conditioned on other columns Z ?i , with higher computation but better mixing.
? Sampling st : we have the state model st |st?1 ? N (Ast?1 , Qt ), and the observation model
z t |st ? N (?st , R),1 where A = diag(?), Qt = diag(? t ), R = I P . for t = 1, . . . , T
1. Forward Filtering: beginning at t = 0 with s0 ? N (0, ?s2 I K ), for all t = 1, . . . , T , we
find the on-line posteriors at t, p(st |z 1:t ) = N (mt , V t ), where mt = V t {?T R?1 z t +
?1
T ?1
H ?1
?]?1 , and H t = Qt + AV t?1 AT .
t Amt?1 }, V t = [H t + ? R
2. Backward Sampling: starting from N (f
mt , Ve t ), the backward smoothing density, i.e., the
e t?1 ), where
conditional distribution of st?1 given st , is p(st?1 |st , z 1:(t?1) ) = N (e
?t?1 , ?
T ?1
?1
?1
T ?1
?1
e
e
e t?1 = ?t?1 {A Qt st + V t?1 mt?1 }, ?t?1 = (V t?1 + A Qt A) .
?
There exist different variants of FFBS schemes (see [28] for a detailed comparison); the method we
choose here enjoys fast decay in autocorrelation and reduced computation time.
1
For brevity, we omit the dependencies on ? in notation
4
3.1
Time-evolving topic and word dependencies
We also have the backward recursion density at t ? 1, p(st?1 |z 1:T ) = N (f
mt?1 , Ve t?1 ), where
T ?1
?1
T ?1 e
e
e
e
e
e
ft?1 = ?t?1 (A Qt m
ft + V t?1 mt?1 ) and V t?1 = ?t?1 + ?t?1 A Qt V t Q?1
m
t A?t?1 .
We perform inference on the K ? K time-evolving topic dependences in s1:T , using the posterior
p
covariances {Ve 1:T } (with topic correlation matrices ?1:T , ?[r,s],t = V[r,s],t / V[r,r],t V[s,s],t , r, s =
1, . . . , K), and further obtain the P ? P time-evolving word dependencies capsuled in {?1:T }
with ?t = ?Ve t ?T + I P . Essentially, this can be viewed as a dynamic Gaussian copula model,
et ? N (0, C t ), where g(?) is a non-decreasing function of a univariate marginal
yp,t = g(e
zp,t ), z
likelihood and C t (t = 1, . . . , T ) is the correlation matrix capturing the multivariate dependence.
We obtain a posterior distribution for C 1:T as a byproduct, without having to estimate the nuisance
parameters in marginal likelihoods g(?). This decoupling strategy resembles the idea of copula
models.
3.2
Accelerated MCMC via document subsampling
For large-scale datasets, recent approaches efficiently reduce the computational load of Monte Carlo
Markov chain (MCMC) by data subsampling [29, 30]. We borrow this idea of subsampling documents when considering a large corpora (e.g., in our experiments, we consider analysis of articles
in the magazine Science, composed of 139379 articles from years 1880 to 2002, and a vocabulary
size 5855). In our model, the augmented data z nt t (nt = 1, . . . , Nt ) for each document is relatively
expensive to sample. One simple method is random document sampling without replacement. However, by treating all likelihood contributions symmetrically, this method leads to a highly inefficient
MCMC chain with poor mixing [29].
Alternatively, we adopt the probability proportional-to-size (PSS) sampling scheme in [30], i.e.,
sampling the documents with inclusion probability proportional to the likelihood contributions. For
each MCMC iteration, the sub-sampling procedure for documents at time t is designed as follows:
? Step 1: Given a small subset Vt ? {1, . . . , Nt } of chosen documents, only sample {z dt } for all
d ? Vt and compute the augment log-likelihood contributions (with B t integrated out) `Vt (z dt ) =
e where R
e = ???T + I P . Note that, only a K-dimensional matrix inversion is
N (?st , R),
e ?1 = I P ? ?(??1 + ?T ?)T ?T .
required, by using the Woodbury matrix inversion formula R
? Step 2: Similar to [30], we use a Gaussian process [31] to predict the log-likelihood for
the remaining documents `Vtc (z dt ) = K(Vtc , Vt )K(Vt , Vt )?1 `Vt (z dt ), where K is a Nt ?
j
i
Nt squared-exponential
kernel,
which denotes the similarity of documents: K(y t , y t ) =
?f2 exp ?||y it ? y jt ||2 /(2s2 ) , i, j = 1, . . . , Nt , ?f2 = 1, s = 1.
P
ed = wd / d0 wd0 .
? Step 3: Calculate the inclusion probability wd ? exp [`(z dt )], d = 1, . . . , Nt , w
? Step 4: Sampling the next subset Vt of pre-specified size |Vt | with inclusion probability w
ed , and
store it for the use of the next MCMC iteration.
In practice, this adaptive design allows MCMC to run more efficiently on a full dataset of large
scale, often mitigating the need to do parallel MCMC implementation. Future work could also consider nonparametric function estimation subject to monotonicity constraint, e.g. Gaussian process
projections recently developed in [32].
4
Experiments
Different from DTM [7] , the proposed model has the jumps directly at the level of the factor scores
(no exponentiation or normalization needed), and therefore it proved more effective in uncovering
jumps in factor scores over time. Demonstrations of this phenomenon in a synthetic experiment are
detailed in the Supplementary Material. In the following, we present exploratory data analysis on
two real examples, demonstrating the ability of the proposed model to infer temporal jumps in topic
importance, and to infer correlations across topics and words.
4.1
Case Study I: State of the Union dataset
The State of the Union dataset contains the transcripts of T = 225 US State of the Union addresses,
from 1790 to 2014. We take each transcript as a document, i.e., we have one document per year.
5
After removing stop words, and removing terms that occur fewer than 3 times in one document and
less than 10 times overall, we have P = 7518 unique words. The observation yp,t corresponds to
the frequency of word p of the State of the Union transcript from year t.
We apply the proposed DRFM setting and learned K = 25 topics. To better understand the temporal
dynamic per topic, six topics are selected and the posterior mean of their latent trajectories sk,1:T
are shown in Figure 1 (with also the top 12 most probable words associated with each of the topics).
A complete table with all 25 learned topics and top 12 words is provided in the Supplementary
Material. The learned trajectory associated with every topic indicates different temporal patterns
across all the topics. Clearly, we can identify jumps associated with some key historical events. For
instance, for Topic 10, we observe a positive jump in 1846 associated with the Mexican-American
war. Topic 13 is related with the Spanish-American war of 1898, with a positive jump in that year.
In Topic 24, we observe a positive jump in 1914, when the Panama Canal was officially opened
(words Panana and canal are included). In Topic 18, the positive jumps observed from 1997 to
1999 seem to be associated with the creation of the State Children?s Health Insurance Program in
1997. We note that the words for this topic are explicitly related with this issue. Topic 25 appears to
be related to banking; the significant spike around 1836 appears to correspond to the Second Bank
of the United States, which was allowed to go out of existence, and end national banking that year.
In 1863 Congress passed the National Banking Act, which ended the ?free-banking? period from
1836-1863; note the spike around 1863 in Topic 25.
Topic 10
4
4
Topic 17
2
2
0
0
6
Topic 13
4
6
Topic 18
4
2
2
0
0
10
6
Topic 24
4
0
0
-5
1800
1850
Topic#10
Mexico
Government
Texas
United
War
Mexican
Army
Territory
Country
Peace
Policy
Lands
1900
Topic#13
Government
United
Islands
Commission
Island
Cuba
Spain
Act
General
Military
International
Officiers
1950
Topic 25
5
2
1800
2000
Topic#24
United
Treaty
Isthmus
Public
Panama
Law
Territory
America
Canal
Service
Banks
Colombia
Topic#17
Jobs
Country
Tax
American
Economy
Deficit
Americans
Energy
Businesses
Health
Plan
Care
1850
1900
Topic#18
Children
America
Americans
Care
Tonight
Support
Century
Health
Working
Challenge
Security
Families
1950
2000
Topic#25
Government
Public
Banks
Bank
Currency
Money
United
Federal
American
National
Duty
Institutions
Figure 1: (State of the Union dataset) Above: Time evolving from 1790 to 2014 for six selected
topics. The plotted values represent the posterior means. Below: Top 12 most probable words
associated with the above topics.
Our modeling framework is able to capture dynamic patterns of topics and word correlations. To
illustrate this, we select three years (associated with some meaningful historical events) and analyze
their corresponding topic and word correlations. Figure 2 (first row) shows graphs of the topic
correlation matrices, in which the nodes represent topics and the edges indicate positive (green) and
negative (red) correlations (we show correlations with absolute value larger than 0.01). We notice
that Topics 11 and 22 are positively correlated with those years. Some of the most probable words
associated with each of them are: increase, united, law and legislation (for Topic 11) and war,
Mexico, peace, army, enemy and military (for Topic 22). We also are interested in understanding
the time-varying correlation between words. To do so, and for the same years as before, in Figure 2
(second row) we plot the dendrogram associated with the learned correlation matrix for words. In
the plots, different colors indicate highly correlated word clusters defined by cutting the branches off
the dendrogram. Those figures reveal different sets of highly correlated words for different years. By
6
1846
Mexican-American War
T6
T5
T2
T22
T11
T5
T14
T13
T7
T14
T18
T18
T20
T12
T10
T25
T9
T18
T21
T19
T16
T10
T23
T21
T21
T1
tre
ex cu b asu
pe rr an r
nd en k y
itu cys
re
dofisca s
milllliars l
o
n
b
illion
constitut
ion
union
president
n
tio om
nareeed s
f re ion
f at rgy
nene
nt
lth
heeavelopmse
d ogram
pr
ic
nom
eco
program
country
general
pow
er
auth
pub ority
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ize en e
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te an
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tes
noilvers
s ndve
boser
re
secretary
n
attentpioort
re ne
ju er
b se
num
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inc l essor r
sin ab a
bu l w
servicey
secrettiaron
atterneporet
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mb e
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incrog omms
p n ra
og
ec ro
p
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n
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rt
re o
ent
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sub laws
es r
sin bo al
bu lation ion
t
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de
ve
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ed
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d er
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fore
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country
general
autho
pub rity
li
fe
g de c
p ov ral
u re ern
co nio side men
ns n nt t
tit
ut
io
n
T1
T9
T24
nation
labor
bu
lawsiness
ta
c x
jobare
tocnhilds
a
i
m
am e ghren
er ric t
ic an
a s
T1
m
bil illio
p wlio n
mil eacearn
forcitary
armes
island y
cubas
island
mexico
T23
T12
T2
T16
n go
silotesld
v
resbonder
trea erves
sury
b
anks
curren
cy
policy
administration
ent
pm
develo ra
m
og
pr
omic
ecoongramsh
pr earltgy
h e
en turesal
i c
nd fis llars
pe
do
ex
T16
T15
T15
T15
T24
T9
T20
T17
T23
T8
T11
T7
T20
T4
T22
T2
T24
T25
T11
T22
T14
T3
T17
T10
T12
T4
T8
T19
T3
T8
T3
T5
T13
T13
T6
T6
T4
T19
T17
2003
Iraq War
service
billion
fis
ex cal
millipenditure
s
o
d
o
r lla n
b es rs
si on erve
gonotlverds
e
ld s
T25 T7
1929
Economic Depression
Figure 2: (State of the Union dataset) First row: Inferred correlations between topics for some
specific years associated with some meaningful historical events. Green edges indicate positive
correlations and red edges indicate negative correlations. Second row: Learned dendrogram based
upon the correlation matrix between the top 10 words associated with each topic (we display 80
unique words in total).
inspecting all the words correlation, we noticed that the set of words {government, federal, public,
power, authority, general, country} are highly correlated across the whole period.
4.2
Case Study II: Analysis of Science dataset
We analyze a collection of scientific documents from the JSTOR Science journal [7]. This dataset
contains a collection of 139379 documents from 1880 to 2002 (T = 123), with approximately 1100
documents per year. After removing terms that occurred fewer than 25 times, the total vocabulary
size is P = 5855. We learn K = 50 topics from the inferred posterior distribution, for brevity and
simplicity, we only show 20 of them. We handle about 2700 documents per iteration (subsampling
rate: 2%). Table 1 shows the 20 selected topics and the top 10 most probable words associated with
each of them. By inspection, we notice that those topics are related with specific fields in science.
For instance, Topic 2 is more related to ?scientific research?, Topic 10 to ?natural resources?, and
Topic 15 to ?genetics?. Figure 3 shows the time-varying trend for some specific words, zbp,1:T , which
reveals the importance of those words across time. Finally, Figure 4 shows the correlation between
the selected 20 topics. For instance, in 1950 and 2000, topic 9 (related to mouse, cells, human,
transgenic) and topic 17 (related to virus, rna, tumor, infection) are highly correlated.
DNA
RNA
Gene
2.5
2
1
Cancer
Patients
Nuclear
0.7
0.6
0.8
1.5
Astronomy
Psychology
Brain
0.5
0.6
1
0.4
0.5
0.4
0
0.2
0.3
0.2
?0.5
0.1
0
?1
1880
1900
1920
1940
1960
1980
2000
1880
1900
1920
1940
1960
1980
2000
0
1880
1900
1920
1940
1960
1980
2000
Figure 3: (Science dataset) the inferred latent trend for variable zbp,1:T associated with words.
7
1900
1950
T14
T14
T8
T15
T12
T9
2000
T11
T11
T13
T20
T19
T18
T20
T17
T19
T20
T13
T3
T7
T15
T16
T15
T17
T2
T6
T5
T9
T4
T10
T16
T10
T5
T18
T3
T10
T4
T19
T12
T8
T9
T1
T1
T17
T6
T5
T2
T4
T2
T11
T16
T13
T12
T7
T3
T6
T14
T18
T8
T1
T7
1.0
0.5
0.0
?0.5
?1.0
Figure 4: (Science dataset) Inferred correlations between topics for some specific years. Green
edges indicate positive correlations and red edges indicate negative correlations.
Table 1: Selected 20 topics associated with the analysis of the Science dataset and top 10 most
probable words.
Topic#1
cells
cell
normal
two
growth
development
tissue
body
egg
blood
Topic#11
system
nuclear
new
systems
power
cost
computer
fuel
coal
plant
5
Topic#2
research
national
government
support
federal
development
new
program
scientific
basic
Topic#12
energy
theory
temperature
radiation
atoms
surface
atomic
mass
atom
time
Topic#3
field
magnetic
solar
energy
spin
state
electron
quantum
temperature
current
Topic#13
association
science
meeting
university
american
society
section
president
committee
secretary
Topic#4
animals
brain
neurons
activity
response
rats
control
fig
effects
days
Topic#14
protein
proteins
cell
membrane
amino
sequence
binding
acid
residues
sequences
Topic#5
energy
oil
percent
production
fuel
total
growth
states
electricity
coal
Topic#15
human
genome
sequence
chromosome
gene
genes
map
data
sequences
genetic
Topic#6
university
professor
college
president
department
research
institute
director
society
school
Topic#16
professor
university
society
department
college
president
director
american
appointed
medical
Topic#7
science
scientific
new
scientists
human
men
sciences
knowledge
meeting
work
Topic#17
virus
rna
viruses
particles
tumor
mice
disease
viral
human
infection
Topic#8
work
research
scientific
laboratory
made
university
results
science
survey
department
Topic#18
energy
electron
state
fig
two
structure
reaction
laser
high
temperature
Topic#9
mice
mouse
type
wild
fig
cells
human
transgenic
animals
mutant
Topic#19
stars
mass
star
temperature
solar
gas
data
density
surface
galaxies
Topic#10
water
surface
temperature
soil
pressure
sea
plants
solution
plant
air
Topic#20
rna
fig
mrna
protein
site
sequence
splicing
synthesis
trna
rnas
Discussion
We have proposed a DRFM framework that could be applied to a broad class of applications such
as: (i) dynamic topic model for the analysis of time-stamped document collections; (ii) joint analysis of multiple time series, with ordinal valued observations; and (iii) multivariate ordinal dynamic
factor analysis or dynamic copula analysis for mixed type of data. The proposed model is a semiparametric methodology, which offers modeling flexibilities and reduces the effect of model misspecification. However, as the marginal likelihood is distribution-free, we could not calculate the
model evidence or other evaluation metrics based on it (e.g. held-out likelihood). As a consequence,
we are lack of objective evaluation criteria, which allow us to perform formal model comparisons.
In our proposed setting, we are able to perform either retrospective analysis or multi-step ahead
forecasting (using the recursive equations derived in the FFBS algorithm). Finally, our inference
framework is easily adaptable for using sequential Monte Carlo (SMC) methods [33] allowing online learning.
Acknowledgments
The research reported here was funded in part by ARO, DARPA, DOE, NGA and ONR. The authors
are grateful to Jonas Wallin, Lund University, Sweden, for providing efficient package on simulation
of the GIG distribution.
8
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4,752 | 5,302 | A provable SVD-based algorithm for learning topics
in dominant admixture corpus
Trapit Bansal?, C. Bhattacharyya??
Department of Computer Science and Automation
Indian Institute of Science
Bangalore -560012, India
[email protected]
[email protected]
Ravindran Kannan
Microsoft Research
India
[email protected]
Abstract
Topic models, such as Latent Dirichlet Allocation (LDA), posit that documents
are drawn from admixtures of distributions over words, known as topics. The
inference problem of recovering topics from such a collection of documents drawn
from admixtures, is NP-hard. Making a strong assumption called separability, [4]
gave the first provable algorithm for inference. For the widely used LDA model,
[6] gave a provable algorithm using clever tensor-methods. But [4, 6] do not learn
topic vectors with bounded l1 error (a natural measure for probability vectors).
Our aim is to develop a model which makes intuitive and empirically supported
assumptions and to design an algorithm with natural, simple components such as
SVD, which provably solves the inference problem for the model with bounded l1
error. A topic in LDA and other models is essentially characterized by a group of
co-occurring words. Motivated by this, we introduce topic specific Catchwords,
a group of words which occur with strictly greater frequency in a topic than any
other topic individually and are required to have high frequency together rather
than individually. A major contribution of the paper is to show that under this
more realistic assumption, which is empirically verified on real corpora, a singular value decomposition (SVD) based algorithm with a crucial pre-processing step
of thresholding, can provably recover the topics from a collection of documents
drawn from Dominant admixtures. Dominant admixtures are convex combination
of distributions in which one distribution has a significantly higher contribution
than the others. Apart from the simplicity of the algorithm, the sample complexity
has near optimal dependence on w0 , the lowest probability that a topic is dominant, and is better than [4]. Empirical evidence shows that on several real world
corpora, both Catchwords and Dominant admixture assumptions hold and the proposed algorithm substantially outperforms the state of the art [5].
1
Introduction
Topic models [1] assume that each document in a text corpus is generated from an ad-mixture of
topics, where, each topic is a distribution over words in a Vocabulary. An admixture is a convex
combination of distributions. Words in the document are then picked in i.i.d. trials, each trial has a
multinomial distribution over words given by the weighted combination of topic distributions. The
problem of inference, recovering the topic distributions from such a collection of documents, is
provably NP-hard. Existing literature pursues techniques such as variational methods [2] or MCMC
procedures [3] for approximating the maximum likelihood estimates.
?
http://mllab.csa.iisc.ernet.in/tsvd
1
Given the intractability of the problem one needs further assumptions on topics to derive polynomial
time algorithms which can provably recover topics. A possible (strong) assumption is that each
document has only one topic but the collection can have many topics. A document with only one
topic is sometimes referred as a pure topic document. [7] proved that a natural algorithm, based
on SVD, recovers topics when each document is pure and in addition, for each topic, there is a set
of words, called primary words, whose total frequency in that topic is close to 1. More recently,
[6] show using tensor methods that if the topic weights have Dirichlet distribution, we can learn
the topic matrix. Note that while this allows non-pure documents, the Dirichlet distribution gives
essentially uncorrelated topic weights.
In an interesting recent development [4, 5] gave the first provable algorithm which can recover topics
from a corpus of documents drawn from admixtures, assuming separability. Topics are said to be
separable if in every topic there exists at least one Anchor word. A word in a topic is said to be an
Anchor word for that topic if it has a high probability in that topic and zero probability in remaining
topics. The requirement of high probability in a topic for a single word is unrealistic.
Our Contributions: Topic distributions, such as those learnt in LDA, try to model the cooccurrence of a group of words which describes a theme. Keeping this in mind we introduce the
notion of Catchwords. A group of words are called Catchwords of a topic, if each word occurs
strictly more frequently in the topic than other topics and together they have high frequency. This
is a much weaker assumption than separability. Furthermore we observe, empirically, that posterior
topic weights assigned by LDA to a document often have the property that one of the weights is
significantly higher than the rest. Motivated by this observation, which has not been exploited by
topic modeling literature, we suggest a new assumption. It is natural to assume that in a text corpus,
a document, even if it has multiple themes, will have an overarching dominant theme. In this paper
we focus on document collections drawn from dominant admixtures. A document collection is said
to be drawn from a dominant admixture if for every document, there is one topic whose weight is
significantly higher than the other topics and in addition, for every topic, there is a small fraction of
documents which are nearly purely on that topic. The main contribution of the paper is to show that
under these assumptions, our algorithm, which we call TSVD, indeed provably finds a good approximation in total l1 error to the topic matrix. We prove a bound on the error of our approximation
which does not grow with dictionary size d, unlike [5] where the error grows linearly with d.
Empirical evidence shows that on semi-synthetic corpora constructed from several real world
datasets, as suggested by [5], TSVD substantially outperforms the state of the art [5]. In particular it is seen that compared to [5] TSVD gives 27% lower error in terms of l1 recovery on 90% of
the topics.
Problem Definition: d, k, s will denote respectively, the number of words in the dictionary, number of topics and number of documents. d, s are large,
P whereas, k is to be thought of as much
smaller. Let Sk = {x = (x1 , x2 , . . . , xk ) : xl ? 0; l xl = 1}. For each topic, there is a fixed
vector in Sk giving the probability of each word in that topic. Let M be the d ? k matrix with these
vectors as its columns.
Documents are picked in i.i.d. trials.
To pick document j, one first picks a k-vector
W1j , W2j , . . . , Wkj of topic weights according to a fixed distribution on Sk . Let P?,j = MW?,j
be the weighted combination of the topic vectors. Then the m words of the document are picked in
i.i.d. trials; each trial picks a word according to the multinomial distribution with P?,j as the probabilities. All that is given as data is the frequency of words in each document, namely, we are given
the d ? s matrix A, where Aij = Number of occurrencesmof word i in Document j . Note that E(A|W) = P,
where, the expectation is taken entry-wise.
In this paper we consider the problem of finding M given A.
2
Previous Results
In this section we review literature related to designing provable algorithms for topic models. For an
overview of topic models we refer the reader to the excellent survey of [1]. Provable algorithms for
recovering topic models was started by [7]. Latent Semantic Indexing (LSI) [8] remains a successful
method for retrieving similar documents by using SVD. [7] showed that one can recover M from a
2
collection of documents, with pure topics, by using SVD based procedure under the additional Primary Words assumption. [6] showed that in the admixture case, if one assumes Dirichlet distribution
for the topic weights, then, indeed, using tensor methods, one can learn M to l2 error provided some
added assumptions on numerical parameters like condition number are satisfied.
The first provably polynomial time algorithm for admixture corpus was given in [4, 5]. For a topic
l, a word i is an anchor word if: Mi,l ? p0 and Mi,l0 = 0 ?l0 6= l.
Theorem 2.1 [4] If every topic has an anchor word, there is a polynomial time algorithm that
? such that with high probability,
returns an M
6
4
k X
d
X
k log d
k
?
|Mil ? Mil | ? d? provided s ? Max O
,
,O
6
4
2
2
a ? p0 ? m
? 2 a2
i=1
l=1
where, ? is the condition number of E(W W T ), a is the minimum expected weight of a topic and m
is the number of words in each document.
Note that the error grows linearly in the dictionary size d, which is often large. Note also the
dependence of s on parameters p0 , which is, 1/p60 and on a, which is 1/a4 . If, say, the word ?run? is
an anchor word for the topic ?baseball? and p0 = 0.1, then the requirement is that every 10 th word
in a document on this topic is ?run?. This seems too strong to be realistic. It would be more realistic
to ask that a set of words like - ?run?, ?hit?, ?score?, etc. together have frequency at least 0.1 which
is what our catchwords assumption does.
3
Learning Topics from Dominant Admixtures
Informally, a document is said to be drawn from a Dominant Admixture if the document has one
dominant topic. Besides its simplicity, we show empirical evidence from real corpora to demonstrate
that topic dominance is a reasonable assumption. The Dominant Topic assumption is weaker than
the Pure Topic assumption. More importantly, SVD based procedures proposed by [7] will not
apply. Inspired by the Primary Words assumption we introduce the assumption that each topic has a
set of Catchwords which individually occur more frequently in that topic than others. This is again
a much weaker assumption than both Primary Words and Anchor Words assumptions and can be
verified experimentally. In this section we establish that by applying SVD on a matrix, obtained by
thresholding the word-document matrix, and subsequent k-means clustering can learn topics having
Catchwords from a Dominant Admixture corpus.
3.1
Assumptions: Catchwords and Dominant admixtures
Let ?, ?, ?, ?, ?0 be non-negative reals satisfying: ? + ? ? (1 ? ?)?,
? + 2? ? 0.5,
? ? 0.08
Dominant topic Assumption (a) For j = 1, 2, . . . , s, document j has a dominant topic l(j) such
that Wl(j),j ? ? and Wl0 j ? ?, ?l0 6= l(j).
(b) For each topic l, there are at least ?0 w0 s documents in each of which topic l has weight at least
1 ? ?.
Catchwords Assumption: There are k disjoint P
sets of words - S1 , S2 , . . . , Sk such that with ?
defined in (5), ?i ? Sl , ?l0 6= l, Mil0 ? ?Mil ,
i?Sl Mil ? p0 ,
20
?i ? Sl , m? 2 ?Mil ? 8 ln
.
(1)
?w0
Part (b) of the Dominant Topic Assumption is in a sense necessary for ?identifiability? - namely for
the model to have a set of k document vectors so that every document vector is in the convex hull of
these vectors. The Catchwords assumption is natural to describe a theme as it tries to model a unique
group of words which is likely to co-occur when a theme is expressed. This assumption is close to
topics discovered by LDA like models, which try to model co-occurence of words. If ?, ? ? ?(1),
then, the assumption (1) says Mil ? ?? (1/m). In fact if Mil ? o(1/m), we do not expect to see
word i (in topic l), so it cannot be called a catchword at all.
3
A slightly different (but equivalent) description of the model will be useful to keep in mind. What
is fixed (not stochastic) are the matrices M and the distribution of the weight matrix W. To pick
document j, we can first pick the dominant topic l in document j and condition the distribution of
W?,j on this being the dominant topic. One could instead also think of W?,j being picked from a
Pk
mixture of k distributions. Then, we let Pij = l=1 Mil Wlj and pick the m words of the document
in i.i.d multinomial trials as before. We will assume that
Tl = {j : l is the dominant topic in document j} satisfies |Tl | = wl s,
where, wl is the probability of topic l being dominant. This is only approximately valid, but the
error is small enough that we can disregard it.
For ? ? {0, 1, 2, . . . , m}, let pi (?, l) be the probability that j ? Tl and Aij = ?/m and qi (?, l) the
corresponding ?empirical probability?:
Z
m
pi (?, l) =
Pij? (1 ? Pij )m?? Prob(W?,j | j ? Tl ) Prob(j ? Tl ), where, P?,j = MW?,j .
?
W?,j
(2)
1
qi (?, l) = |{j ? Tl : Aij = ?/m}| .
(3)
s
Note that pi (?, l) is a real number, whereas, qi (?, l) is a random variable with E(qi (?, l)) = pi (?, l).
We need a technical assumption on the pi (?, l) (which is weaker than unimodality).
No-Local-Min Assumption: We assume that pi (?, l) does not have a local minimum, in the sense:
pi (?, l) > Min(pi (? ? 1, l), pi (? + 1, l)) ? ? ? {1, 2, . . . , m ? 1}.
(4)
The justification for this assumption is two-fold. First, generally, Zipf?s law kind of behaviour where
the number of words plotted against relative frequencies declines as a power function has often been
observed. Such a plot is monotonically decreasing and indeed satisfies our assumption. But for
Catchwords, we do not expect this behaviour - indeed, we expect the curve to go up initially as the
relative frequency increases, then reach a maximum and then decline. This is a unimodal function
and also satisfies our assumption.
Relative sizes of parameters: Before we close this section, a discussion on the values of the parameters is in order. Here, s is large. For asymptotic analysis, we can think of it as going to infinity.
1/w0 is also large and can be thought of as going to infinity. [In fact, if 1/w0 ? O(1), then, intuitively, we see that there is no use of a corpus of more than constant size - since our model has
i.i.d. documents, intuitively, the number of samples we need should depend mainly on 1/w0 ]. m is
(much) smaller, but need not be constant.
c refers to a generic constant independent of m, s, 1/w0 , ?, ?; its value may be different in different
contexts.
3.2
The TSVD Algorithm
Existing SVD based procedures for clustering on raw word-document matrices fail because the
spread of frequencies of a word within a topic is often more (at least not significantly less) than the
gap between the word?s frequencies in two different topics. Hypothetically, the frequency for the
word run, in the topic Sports, may range upwards of 0.01, say. But in other topics, it may range
from, say, 0 to 0.005. The success of the algorithm will lie on correctly identifying the dominant
topics such as sports by identifying that the word run has occurred with high frequency. In this
example, the gap (0.01-0.005) between Sports and other topics is less than the spread within Sports
(1.0-0.01), so a 2-clustering approach (based on SVD) will split the topic Sports into two. While
this is a toy example, note that if we threshold the frequencies at say 0.01, ideally, sports will be all
above and the rest all below the threshold, making the succeeding job of clustering easy.
There are several issues in extending beyond the toy case. Data is not one-dimensional. We will use
different thresholds for each word; word i will have a threshold ?i /m. Also, we have to compute
?i /m. Ideally we would not like to split any Tl , namely, we would like that for each l and and each
i, either most j ? Tl have Aij > ?i /m or most j ? Tl have Aij ? ?i /m. We will show that
4
our threshold procedure indeed achieves this. One other nuance: to avoid conditioning, we split
the data A into two parts A(1) and A(2) , compute the thresholds using A(1) and actually do the
thresholding on A(2) . We will assume that the intial A had 2s columns, so each part now has s
columns. Also, T1 , T2 , . . . , Tk partitions the columns of A(1) as well as those of A(2) . The columns
of thresholded matrix B are then clustered, by a technique we call Project and Cluster, namely,
we project the columns of B to its k?dimensional SVD subspace and cluster in the projection.
The projection before clustering has recently been proven [9] (see also [10]) to yield good starting
cluster centers. The clustering so found is not yet satisfactory. We use the classic Lloyd?s k-means
algorithm proposed by [12]. As we will show, the partition produced after clustering, {R1 , . . . , Rk }
of A(2) is close to the partition induced by the Dominant Topics, {T1 , . . . , Tk }. Catchwords of topic
l are now (approximately) identified as the most frequently occurring words in documents in Rl .
Finally, we identify nearly pure documents in Tl (approximately) as the documents in which the
catchwords occur the most. Then we get an approximation to M?,l by averaging these nearly pure
documents. We now describe the precise algorithm.
3.3
Topic recovery using Thresholded SVD
Threshold SVD based K-means (TSVD)
?
1 ?p0 ?0 ?p0 ?
? , .
? = Min
,
900c20 k 3 m 640m k
(5)
1. Randomly partition the columns of A into two matrices A(1) and A(2) of s columns each.
2. Thresholding
(a) Compute Thresholds on A(1) For each i, let ?i be the highest value of ? ?
(1)
(1)
?
?
{0, 1, 2, . . . , m} such that |{j : Aij > m
}| ? w20 s; |{j : Aij = m
}| ? 3?w0 s.
(?
(2)
?i if Aij > ?i /m and ?i ? 8 ln(20/?w0 )
(b) Do the thresholding on A(2) : Bij =
.
0
otherwise
3. SVD Find the best rank k approximation B(k) to B.
4. Identify Dominant Topics
(a) Project and Cluster Find (approximately) optimal k-means clustering of the columns
of B(k) .
(b) Lloyd?s Algorithm Using the clustering found in Step 4(a) as the starting clustering,
apply Lloyd?s k-means algorithm to the columns of B (B, not B(k) ).
(c) Let R1 , R2 , . . . , Rk be the k?partition of [s] corresponding to the clustering after
Lloyd?s. //*Will prove that Rl ? Tl *//
5. Identify Catchwords
(2)
(a) For each i, l, compute g(i, l) = the (b?0 w0 s/2c)th highest element of {Aij : j ? Rl }.
4
0
, where, ? =
(b) Let Jl = i : g(i, l) > Max m?
2 ln(20/?w0 ), Maxl0 6=l ? g(i, l )
1?2?
.
(1+?)(?+?)
P
(2)
6. Find Topic Vectors Find the b?0 w0 s/2c highest i?Jl Aij among all j ? [s] and return
? ?,l to M?,l .
the average of these A?,j as our approximation M
Theorem 3.1 Main Theorem Under the Dominant Topic, Catchwords and No-Local-Min assump? so that
tions, the algorithm succeeds with high probability in finding an M
6 2
2
X
? il | ? O(k?), provided 1 s ? ?? 1 k m + m k + d
|Mil ? M
.
w0 ?2 p20
?20 ? 2 ?p0
?0 ? 2
i,l
1
The superscript ? hides a logarithmic factor in dsk/?fail , where, ?fail > 0 is the desired upper bound on the
probability of failure.
5
A note on the sample complexity is in order. Notably, the dependence of s on w0 is best possible
(namely s ? ?? (1/w0 )) within logarithmic factors, since, if we had fewer than 1/w0 documents, a
topic which is dominant with probability only w0 may have none of the documents in the collection.
The dependence of s on d needs to be at least d/?0 w0 ? 2 : to see this, note that we only assume
that there are r = O(?0 w0 s) nearly pure documents on each topic. Assuming we ?
can find
? this
set (the algorithm approximately does), their average has standard deviation of about Mil / r in
coordinate i. If topic vector M?,l has O(d) entries, each of?size O(1/d), to get an approximation
of M?,l to l1 error ?, we need the per coordinate error 1/ dr to be at most ?/d which implies
s ? d/?0 w0 ? 2 . Note that to get comparable error in [4], we need a quadratic dependence on d.
There is a long sequence of Lemmas to prove the theorem. To improve the readability of the paper
we relegate the proofs to supplementary material [14]. The essence of the proof lies in proving
that the clustering step correctly identifies the partition induced by the dominant topics. For this,
we take advantage of a recent development on the k?means algorithm from [9] [see also [10]],
where, it is shown that under a condition called the Proximity Condition, Lloyd?s k means algorithm
starting with the centers provided by the SVD-based algorithm, correctly identifies almost all the
documents? dominant topics. We prove that indeed the Proximity Condition holds. This calls for
machinery from Random Matrix theory (in particular bounds on singular values). We prove that the
singular values of the thresholded word-document matrix are nicely bounded. Once the dominant
topic of each document is identified, we are able to find the Catchwords for each topic. Now, we
rely upon part (b.) of the Dominant Topic assumption : that is there is a small fraction of nearly Pure
Topic-documents for each topic. The Catchwords help isolate the nearly pure-topic documents and
hence find the topic vectors. The proofs are complicated by the fact that each step of the algorithm
induces conditioning on the data ? for example, after clustering, the document vectors in one cluster
are not independent anymore.
4
Experimental Results
We compare the thresholded SVD based k-means (TSVD2 ) algorithm 3.3 with the algorithms of
[5], Recover-KL and Recover-L2, using the code made available by the authors3 . We observed
the results of Recover-KL to be better than Recover-L2, and report here the results of Recover-KL
(abbreviated R-KL), full set of results can be found in supplementary section 5. We first provide
empirical support for the algorithm assumptions in Section 3.1, namely the dominant topic and the
catchwords assumption. Then we show on 4 different semi-synthetic data that TSVD provides as
good or better recovery of topics than the Recover algorithms. Finally on real-life datasets, we show
that the algorithm performs as well as [5] in terms of perplexity and topic coherence.
Implementation Details: TSVD parameters (w0 , ?, ?0 , ?) are not known in advance for real corpus. We tested empirically for multiple settings and the following values gave the best performance.
Thresholding parameters used were: w0 = k1 , ? = 61 . For finding the catchwords, ? = 1.1, ?0 = 13
in step 5. For finding the topic vectors (step 6), taking the top 50% (?0 w0 = k1 ) gave empirically
better results. The same values were used on all the datasets tested. The new algorithm is sensitive
to the initialization of the first k-means step in the projected SVD space. To remedy this, we run 10
independent random initializations of the algorithm with K-Means++ [13] and report the best result.
Datasets: We use four real word datasets in the experiments. As pre-processing steps we removed
standard stop-words, selected the vocabulary size by term-frequency and removed documents with
less than 20 words. Datasets used are: (1) NIPS4 : Consists of 1,500 NIPS full papers, vocabulary
of 2,000 words and mean document length 1023. (2) NYT4 : Consists of a random subset of 30,000
documents from the New York Times dataset, vocabulary of 5,000 words and mean document length
238. (3) Pubmed4 : Consists of a random subset of 30,000 documents from the Pubmed abstracts
dataset, vocabulary of 5,030 words and mean document length 58. (4) 20NewsGroup5 (20NG):
Consist of 13,389 documents, vocabulary of 7,118 words and mean document length 160.
2
Resources available at: http://mllab.csa.iisc.ernet.in/tsvd
http://www.cs.nyu.edu/?halpern/files/anchor-word-recovery.zip
4
http://archive.ics.uci.edu/ml/datasets/Bag+of+Words
5
http://qwone.com/?jason/20Newsgroups
3
6
Corpus
s
k
NIPS
NYT
Pubmed
20NG
1500
30000
30000
13389
50
50
50
20
% s with Dominant
Topics (? = 0.4)
56.6%
63.7%
62.2%
74.1%
% s with Pure
Topics (? = 0.05)
2.3%
8.5%
5.1%
39.5%
% Topics
with CW
96%
98%
78%
85%
CW Mean
Frequency
0.05
0.07
0.05
0.06
Table 1: Algorithm Assumptions. For dominant topic assumption, fraction of documents with satisfy
the assumption for (?, ?) = (0.4, 0.3) are shown. % documents with almost pure topics (? = 0.05,
i.e. 95% pure) are also shown. Last two columns show results for catchwords (CW) assumption.
4.1
Algorithm Assumptions
To check the dominant topic and catchwords assumptions, we first run 1000 iterations of Gibbs
sampling on the real corpus and learn the posterior document-topic distribution ({W.,j }) for each
document in the corpus (by averaging over 10 saved-states separated by 50 iterations after the 500
burn-in iterations). We will use this posterior document-topic distribution as the document generating distribution to check the two assumptions.
Dominant topic assumption: Table 1 shows the fraction of the documents in each corpus which
satisfy this assumption with ? = 0.4 (minimum probability of dominant topic) and ? = 0.3 (maximum probability of non-dominant topics). The fraction of documents which have almost pure topics
with highest topic weight at least 0.95 (? = 0.05) is also shown. The results indicate that the dominant topic assumption is well justified (on average 64% documents satisfy the assumption) and there
is also a substantial fraction of documents satisfying almost pure topic assumption.
Catchwords assumption: We first find a k-clustering of the documents {T1 , . . . , Tk } by assigning
all documents which have highest posterior probability for the same topic into one cluster. Then
we use step 5 of TSVD (Algorithm 3.3) to find the set of catchwords for each topic-cluster, i.e.
1
{S1 , . . . , Sk }, with the parameters: 0 w0 = 3k
, ? = 2.3 (taking into account constraints in Section
3.1, ? = 0.4, ? = 0.3, ? = 0.05, ? = 0.07). Table 1 reports the fraction of topics with non-empty
set of catchwords and the average per topic frequency of the catchwords6 . Results indicate that
most topics on real data contain catchwords (Table 1, second-last column). Moreover, the average
per-topic frequency of the group of catchwords for that topic is also quite high (Table 1, last column).
4.2
Empirical Results
Semi-synthetic Data: Following [5], we generate semi-synthetic corpora from LDA model trained
by MCMC, to ensure that the synthetic corpora retain the characteristics of real data. Gibbs sampling7 is run for 1000 iterations on all the four datasets and the final word-topic distribution is used
to generate varying number of synthetic documents with document-topic distribution drawn from a
symmetric Dirichlet with hyper-parameter 0.01. For NIPS, NYT and Pubmed we use k = 50 topics,
for 20NewsGroup k = 20, and mean document lengths of 1000, 300, 100 and 200 respectively. Note
that the synthetic data is not guaranteed to satisfy dominant topic assumption for every document
(on average about 80% documents satisfy the assumption for value of (?, ?) tested in Section 4.1).
? ) for the semiTopic Recovery on Semi-synthetic Data: We learn the word-topic distribution (M
synthetic corpora using TSVD and the Recover algorithms of [5]. Given these learned topic dis? by
tributions and the original data-generating distribution (M ), we align the topics of M and M
bipartite matching and evaluate the l1 distance between each pair of topics. We report the average
of l1 error across topics (called l1 reconstruction-error [5]) in Table 2 for TSVD and Recover-KL
(R-KL). TSVD has smaller error on most datasets than the R-KL algorithm. We observed performance of TSVD to be always better than Recover-L2 (see supplement Table 1 for full results). Best
performance is observed on NIPS which has largest mean document length, indicating that larger
m leads to better recovery. Results on 20NG are slightly worse than R-KL for smaller sample size,
but performance improves for larger number of documents. While the error-values in Table 2 are
6
P
k
1
k
7
1
l=1 |Tl |
P
i?Sl
P
j?Tl
Aij
Dirichlet hyperparameters used: document-topic = 0.03 and topic-word = 1
7
Corpus
NIPS
40
30
NIPS
20
Number of Topics
10
0
0.00 0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.20
NYT
Algorithm
R?KL
TSVD
Pubmed
40
20NG
30
20
10
0
0.00 0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.20
NYT
L1 Reconstruction Error
Figure 1: Histogram of l1 error across
topics (40,000 documents). TSVD(blue,
solid border) gets smaller error on most
topics than R-KL(green, dashed border).
Documents
R-KL
TSVD
40,000
50,000
60,000
40,000
50,000
60,000
40,000
50,000
60,000
40,000
50,000
60,000
0.308
0.308
0.311
0.332
0.326
0.328
0.120
0.114
0.110
0.208
0.206
0.200
0.115 (62.7%)
0.145 (52.9%)
0.131 (57.9%)
0.288 (13.3%)
0.280 (14.1%)
0.284 (13.4%)
0.124 (-3.3%)
0.113 (0.9%)
0.106 (3.6%)
0.195 (6.3%)
0.185 (10.2%)
0.194 (3.0%)
Table 2: l1 reconstruction error on various semi-synthetic
datasets. Brackets in the last column give percent improvement over R-KL (best performing Recover algorithm).
Full results in supplementary.
averaged values across topics, Figure 1 shows that TSVD algorithm achieves much better topic recovery for majority of the topics (>90%) on most datasets (overall average improvement of 27%,
full results in supplement Figure 1).
Topic Recovery on Real Data: To evaluate perplexity [2] on real data, the held-out sets consist of 350 documents for NIPS, 10000 documents for NYT and Pubmed, and 6780 documents for
20NewsGroup. TSVD achieved perplexity measure of 835 (NIPS), 1307 (Pubmed), 1555 (NYT),
2390 (20NG) while Recover-KL achieved 754 (NIPS), 1188 (Pubmed), 1579 (NYT), 2431 (20NG)
(refer to supplement Table 2 for complete results). TSVD gives comparable perplexity with RecoverKL, results being slightly better on NYT and 20NewsGroup which are larger datasets with moderately high mean document lengths. We also find comparable results on Topic Coherence [11] (see
Table 2 in supplementary for topic coherence results and Table 3 for list of top words of topics).
Summary: We evaluated the proposed algorithm, TSVD, rigorously on multiple datasets with respect to the state of the art [5] (Recover-KL and Recover-L2), following the evaluation methodology
of [5]. In Table 2 we show that the l1 reconstruction error for the new algorithm is small and on
average 19.6% better than the best results of the Recover algorithms [5]. In Figure 1, we show that
TSVD achieves significantly better recover on majority of the topics. We also demonstrate that on
real datasets the algorithm achieves comparable perplexity and topic coherence to Recover algorithms. Moreover, we show on multiple real world datasets that the algorithm assumptions are well
justified in practice.
Conclusion
Real world corpora often exhibits the property that in every document there is one topic dominantly
present. A standard SVD based procedure will not be able to detect these topics, however TSVD,
a thresholded SVD based procedure, as suggested in this paper, discovers these topics. While SVD
is time-consuming, there have been a host of recent sampling-based approaches which make SVD
easier to apply to massive corpora which may be distributed among many servers. We believe that
apart from topic recovery, thresholded SVD can be applied even more broadly to similar problems,
such as matrix factorization, and will be the basis for future research.
Acknowledgements TB was supported by a Department of Science and Technology (DST) grant.
References
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2012.
8
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[14] Supplementary material
9
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4,753 | 5,303 | Learning a Concept Hierarchy from
Multi-labeled Documents
1
Viet-An Nguyen1?, Jordan Boyd-Graber2 , Philip Resnik1,3,4 , Jonathan Chang5
2
5
Computer Science, 3 Linguistics, 4 UMIACS
Computer Science
Facebook
Univ. of Maryland, College Park, MD
Univ. of Colorado, Boulder, CO Menlo Park, CA
[email protected]
Jordan.Boyd.Graber [email protected]
[email protected]
@colorado.edu
Abstract
While topic models can discover patterns of word usage in large corpora, it is
difficult to meld this unsupervised structure with noisy, human-provided labels,
especially when the label space is large. In this paper, we present a model?Label
to Hierarchy (L 2 H)?that can induce a hierarchy of user-generated labels and the
topics associated with those labels from a set of multi-labeled documents. The
model is robust enough to account for missing labels from untrained, disparate
annotators and provide an interpretable summary of an otherwise unwieldy label
set. We show empirically the effectiveness of L 2 H in predicting held-out words
and labels for unseen documents.
1
Understanding Large Text Corpora through Label Annotations
Probabilistic topic models [4] discover the thematic structure of documents from news, blogs, and
web pages. Typical unsupervised topic models such as latent Dirichlet allocation [7, LDA] uncover
topics from unannotated documents. In many settings, however, documents are also associated
with additional data, which provide a foundation for joint models of text with continuous response
variables [6, 48, 27], categorical labels [37, 18, 46, 26] or link structure [9].
This paper focuses on additional information in the form of multi-labeled data, where each document
is tagged with a set of labels. These data are ubiquitous. Web pages are tagged with multiple
directories,1 books are labeled with different categories or political speeches are annotated with
multiple issues.2 Previous topic models on multi-labeled data focus on a small set of relatively
independent labels [25, 36, 46]. Unfortunately, in many real-world examples, the number of labels?
from hundreds to thousands?is incompatible with the independence assumptions of these models.
In this paper, we capture the dependence among the labels using a learned tree-structured hierarchy.
Our proposed model, L 2 H?Label to Hierarchy?learns from label co-occurrence and word usage to
discover a hierarchy of topics associated with user-generated labels. We show empirically that L 2 H
can improve over relevant baselines in predicting words or missing labels in two prediction tasks.
L 2 H is designed to explicitly capture the relationships among labels to discover a highly interpretable
hierarchy from multi-labeled data. This interpretable hierarchy helps improve prediction performance
and also provides an effective way to search, browse and understand multi-labeled data [17, 10, 8, 12].
?
Part of this work was done while the first author interned at Facebook.
Open Directory Project (http://www.dmoz.org/)
2
Policy Agenda Codebook (http://policyagendas.org/)
1
1
2
L 2 H:
Capturing Label Dependencies using a Tree-structured Hierarchy
Discovering a topical hierarchy from text has been the focus of much topic modeling research. One
popular approach is to learn an unsupervised hierarchy of topics. For example, hLDA [5] learns
an unbounded tree-structured hierarchy of topics from unannotated documents. One drawback of
hLDA is that documents only are associated with a single path in the topic tree. Recent work
relaxing this restriction include TSSB [1], nHDP [30], nCRF [2] and S H LDA [27]. Going beyond
tree structure, PAM [20] captures the topic hierarchy using a pre-defined DAG, inspiring more flexible
extensions [19, 24]. However, since only unannotated text is used to infer the hierarchical topics,
it usually requires an additional step of topic labeling to make them interpretable. This difficulty
motivates work leveraging existing taxonomies such as HSLDA [31] and hLLDA [32].
A second active area of research is constructing a taxonomy from multi-labeled data. For example,
Heymann and Garcia-Molina [17] extract a tag hierarchy using the tag network centrality; similar work has been applied to protein hierarchies [42]. Hierarchies of concepts have come from
seeded ontologies [39], crowdsourcing [29], and user-specified relations [33]. More sophisticated
approaches build domain-specific keyword taxonomies with adapting Bayesian Rose Trees [21].
These approaches, however, concentrate on the tags, ignoring the content the tags describe.
In this paper, we combine ideas from these two lines of research and introduce L 2 H, a hierarchical
topic model that discovers a tree-structured hierarchy of concepts from a collection of multi-labeled
documents. L 2 H takes as input a set of D documents {wd }, each tagged with a set of labels ld .
The label set L contains K unique, unstructured labels and the word vocabulary size is V . To
learn an interpretable taxonomy, L 2 H associates each label?a user-generated word/phrase?with a
topic?a multinomial distribution over the vocabulary?to form a concept, and infers a tree-structured
hierarchy to capture the relationships among concepts. Figure 1 shows the plate diagram for L 2 H,
together with its generative process.
?
?
?
??? , ???
??? , ???
??
????,?
??
????,?
????,?
??
??
???
?
???
?
1. Create label graph G and draw a uniform spanning tree T from G (? 2.1)
2. For each node k ? [1, K] in T
(a) If k is the root, draw background topic ?k ? Dir(?u)
(b) Otherwise, draw topic ?k ? Dir(???(k) ) where ?(k) is node k?s
parent.
3. For each document d ? [1, D] having labels ld
(a) Define L0d and L1d using T and ld (cf. ? 2.2)
(b) Draw ?d0 ? Dir(L0d ? ?) and ?d1 ? Dir(L1d ? ?)
(c) Draw a stochastic switching variable ?d ? Beta(?0 , ?1 )
(d) For each token n ? [1, Nd ]
i. Draw set indicator xd,n ? Bern(?d )
x
ii. Draw topic indicator zd,n ? Mult(?d d,n )
iii. Draw word wd,n ? Mult(?zd,n )
Figure 1: Generative process and the plate diagram notation of L 2 H.
2.1
Generating a labeled topic hierarchy
We assume an underlying directed graph G = (E, V) in which each node is a concept consisting of
(1) a label?observable user-generated input, and (2) a topic?latent multinomial distribution over
words.3 The prior weight of a directed edge from node i to node k is the fraction of documents tagged
with label k which are also tagged with label i: ti,k = Di,k /Dj . We also assume an additional
Background node. Edges to the Background node have prior zero weight and edges from the
Background node to node i have prior weight troot,i = Di /maxk Dk . Here, Di is the number of
documents tagged with label i, and Di,k is the number of documents tagged with both labels i and k.
The tree T is a spanning tree generated from G. The probability
of a tree given the graph G is thus
Q
the product of all its edge prior weights p(T | G) = e?E te . To capture the intuition that child
nodes in the hierarchy specialize the concepts of their parents, we model the topic ?k at each node
3
In this paper, we use node when emphasizing the structure discovered by the model. Each node corresponds
to a concept which consists of a label and a topic.
2
k using a Dirichlet distribution whose mean is centered at the topic of node k?s parent ?(k), i.e.,
?k ? Dir(???(k) ). The topic at the root node is drawn from a symmetric Dirichlet ?root ? Dir(?u),
where u is a uniform distribution over the vocabulary [1, 2]. This is similar to the idea of ?backoff? in
language models where more specific contexts inherit the ideas expressed in more general contexts;
i.e., if we talk about ?pedagogy? in education, there?s a high likelihood we?ll also talk about it in
university education [22, 41].
2.2
Generating documents
As in LDA, each word in a document is generated by one of the latent topics. L 2 H, however, also
uses the labels and topic hierarchy to restrict the topics a document uses. The document?s label set ld
identifies which nodes are more likely to be used. Restricting tokens of a document in this way?to
be generated only from a subset of the topics depending the document?s labels?creates specific,
focused, labeled topics [36, Labeled LDA].
Unfortunately, ld is unlikely to be an exhaustive enumeration: particularly when the label set is large,
users often forget or overlook relevant labels. We therefore depend on the learned topology of the
hierarchy to fill in the gaps of what users forget by expanding ld into a broader set, L1d , which is the
union of nodes on the paths from the root node to any of the document?s label nodes. We call this
the document?s candidate set. The candidate set also induces a complementary set L0d ? L \ L1d
(illustrated in Figure 2). Previous approaches such as LPAM [3] and Tree labeled LDA [40] also
leverage the label hierarchy to expand the original label set. However, these previous models require
that the label hierarchy is given rather than inferred as in our L 2 H.
0
1
3
2
4
5
6
Figure 2: Illustration of the candidate label set: given a document d
having labels ld = {2, 4} (double-circled nodes), the candidate label
set of d consists of nodes on all the paths from the root node to node
2 and node 4. L1d = {0, 1, 2, 4} and L0d = {3, 5, 6}. This allows
an imperfect label set to induce topics that the document should be
associated with even if they weren?t explicitly enumerated.
L 2 H replaces Labeled LDA?s absolute restriction to specific topics to a soft preference. To achieve this,
each document d has a switching variable ?d drawn from Beta(?0 , ?1 ), which effectively decides
how likely tokens in d are to be generated from L1d versus L0d . Token n in document d is generated by
first flipping the biased coin ?d to choose the set indicator xd,n . Given xd,n , the label zd,n is drawn
x
from the corresponding label distribution ?d d,n and the token is generated from the corresponding
topic wd,n ? Mult(?zd,n ).
3
Posterior Inference
Given a set of documents with observed words {wd } and labels {ld }, inference finds the posterior
distribution over the latent variables. We use a Markov chain Monte Carlo (MCMC) algorithm to
perform posterior inference, in which each iteration after initialization consists of the following
steps: (1) sample a set indicator xd,n and topic assignment zd,n for each token, (2) sample a word
distribution ?k for each node k in the tree, and (3) update the structure of the label tree.
Initialization: With the large number of labels, the space of hierarchical structures that MCMC
needs to explore is huge. Initializing the tree-structure hierarchy is crucial to help the sampler focus
on more important regions of the search space and help the sampler converge. We initialize the
hierarchy with the maximum a priori probability tree by running Chu-Liu/Edmonds? algorithm to
find the maximum spanning tree on the graph G starting at the background node.
Sampling assignments xd,n and zd,n : For each token, we need to sample whether it was generated
from the label set or not, xd,n . We choose label set i with probability
node in the chosen set i with probability
?d,n
Nd,k
+?
?d,n
Cd,i +?|Lid |
?d,n
Cd,i
+?i
?d,n
Cd,?
+?0 +?1
and we sample a
? ?k,wd,n . Here, Cd,i is the number of times
tokens in document d are assigned to label set i; Nd,k is the number of times tokens in document d
3
are assigned to node k. Marginal counts are denoted by ?, and ?d,n denotes the counts excluding the
assignment of token wd,n .
After we have the label set, we can sample the topic assignment. This is more efficient than sampling
jointly, as most tokens are in the label set, and there are a limited number of topics in the label set.
The probability of assigning node k to zd,n is
p(xd,n = i, zd,n = k | x?d,n , z ?d,n , ?, Lid ) ?
?d,n
Cd,i
+ ?i
?d,n
Cd,?
+ ?0 + ?1
?
?d,n
Nd,k
+?
?d,n
Cd,i
+ ?|Lid |
? ?k,wd,n
(1)
Sampling topics ?: As discussed in Section 2.1, topics on each path in the hierarchy form a
cascaded Dirichlet-multinomial chain where the multinomial ?k at node k is drawn from a Dirichlet
distribution with the mean vector being the topic ??(k) at the parent node ?(k). Given assignments
of tokens to nodes, we need to determine the conditional probability of a word given the token. This
can be done efficiently in two steps: bottom-up smoothing and top-down sampling [2].
? k,v of node k propagated from its children.
? Bottom-up smoothing: This step estimates the counts M
This can be approximated efficiently by using the minimal/maximal path assumption [11, 44]. For
? k,v if Mk0 ,v > 0.
the minimal path assumption, each child node k 0 of k propagates a value of 1 to M
0
? k,v .
For the maximal path assumption, each child node k of k propagates the full count Mk0 ,v to M
? k,v for each node from leaf to root, we sample the word
? Top-down sampling: After estimating M
distributions top-down using its actual counts mk , its children?s propagated counts m
? k and its
parent?s word distribution ??(k) as ?k ? Dir(mk + m
? k + ???(k) ).
Updating tree structure T : We update the tree structure by looping through each non-root
node, proposing a new parent node and either accepting or rejecting the proposed parent using
the Metropolis-Hastings algorithm. More specifically, given a non-root node k with current parent i,
we propose a new parent node j by sampling from all incoming nodes of k in graph G, with probability
proportional to the corresponding edge weights. If the proposed parent node j is a descendant of
k, we reject the proposal
to avoid a cycle.
If it is not a descendant, we accept the proposed move
Q(i?k) P (j?k)
with probability min 1, Q(j?k) P (i?k) , where Q and P denote the proposal distribution and the
model?s joint distribution respectively, and i ? k denotes the case where i is the parent of k.
Since we sample the proposed parent using the edge weights, the proposal probability ratio is
Q(i ? k)
ti,k
=
Q(j ? k)
tj,k
(2)
The joint probability of L 2 H?s observed and latent variables is:
P =
Y
e?E
p(e | G)
D
Y
p(xd | ?)p(zd | xd , ld , ?)p(wd | zd , ?)
d=1
K
Y
p(?l | ??(l) , ?)p(?root | ?)
(3)
l=1
When node k changes its parent from node i to j, the candidate set L1d changes for any document d
that is tagged with any label in the subtree rooted at k. Let 4k denote the subtree rooted at k and
D4k = {d | ?l ? 4k ? l ? ld } the set of documents whose candidate set might change when k?s
parent changes. Canceling unchanged quantities, the ratio of the joint probabilities is:
P (j ? k)
tj,k
=
P (i ? k)
ti,k
K
Y p(zd | j ? k) p(xd | j ? k) p(wd | j ? k) Y
p(?l | j ? k)
p(zd | i ? k) p(xd | i ? k) p(wd | i ? k)
p(?l | i ? k)
d?D4k
(4)
l=1
We now expand each factor in Equation 4. The probability of node assignments zd for document d is
computed by integrating out the document-topic multinomials ?d0 and ?d1 (for the candidate set and its
inverse):
Y
Y ?(Nd,l + ?)
?(?|Lxd |)
p(zd | xd , L0d , L1d ; ?) =
(5)
x
?(Cd,x + ?|Ld |)
?(?)
x
l?Ld
x?{0,1}
4
Similarly, we compute the probability of xd for each document d, integrating out ?d ,
p(xd | ?) =
?(?0 + ?1 )
?(Cd,? + ?0 + ?1 )
Y
x?{0,1}
?(Cd,x + ?i )
?(?x )
(6)
Since we explicitly sample the topic ?l at each node l, we need to re-sample all topics for the case that
QK
l | j?k)
j is the parent of i to compute the ratio l=1 p(?
p(?l | i?k) . Given the sampled ?, the word likelihood is
QNd
p(wd | zd , ?) = n=1
?zd,n ,wd,n .
However, re-sampling the topics for the whole hierarchy for every node proposal is inefficient. To
avoid that, we keep all ??s fixed and approximate the ratio as:
R
K
Y p(wd | j ? k) Y
? k | ?k ) p(?k | ?j ) d?k
p(mk + m
p(?l | j ? k)
?
? R k
(7)
? k | ?k ) p(?k | ?i ) d?k
p(wd | i ? k)
p(?l | i ? k)
p(mk + m
?k
d?D4k
l=1
? k is the word counts propagated from children of k.
where mk is the word counts at node k and m
Since ? is fixed and the node assignments z are unchanged, the word likelihoods cancel out except
for tokens assigned at k or any of its children. The integration in Equation 7 is
Z
? k | ?k ) p(?k | ?j ) d?k =
p(mk + m
?k
V
Y
? k,v + ??i,v )
?(?)
?(Mk,v + M
(8)
? k,? + ?)
?(??i,v )
?(Mk,? + M
v=1
Using Equations 2 and 4, we can compute the Metropolis-Hastings acceptance probability.
4
Experiments: Analyzing Political Agendas in U.S. Congresses
In our experiments, we focus on studying political attention in the legislative process, of interest to
both computer scientists [13, 14] and political scientists [15, 34]. GovTrack provides bill text from
the US Congress, each of which is assigned multiple political issues by the Congressional Research
Service. Examples of Congressional issues include Education, Higher Education, Health, Medicare, etc. To evaluate the effectiveness of L 2 H, we evaluate on two computational tasks: document
modeling?measuring perplexity on a held-out set of documents?and multi-label classification. We
also discuss qualitative results based on the label hierarchy learned by our model.
Data: We use the text and labels from GovTrack for the 109th through 112th Congresses (2005?
2012). For both quantitative tasks, we perform 5-fold cross-validation. For each fold, we perform
standard pre-processing steps on the training set including tokenization, removing stopwords, stemming, adding bigrams, and filtering using TF - IDF to obtain a vocabulary of 10,000 words (final
statistics in Figure 3).4 After building the vocabulary from training documents, we discard all
out-of-vocabulary words in the test documents. We ignore labels associated with fewer than 100 bills.
4.1
Document modeling
In the first quantitative experiment, we focus on the task of predicting the words in held-out test
documents, given their labels. This is measured by perplexity, a widely-used evaluation metric [7, 45].
To compute perplexity, we follow the ?estimating ?? method described in Wallach et al. [45, Sec.
5.1] and split each test document d into wdTE1 and wdTE2 . During training, we estimate all topics?
? During test, first we run Gibbs sampling using the learned topics
distributions over the vocabulary ?.
TE 1
on wd to estimate the topic proportions ??dTEfor each test document d. Then, we compute the
P
TE
? )
log(p(wd 2 | ld ,??dTE ,?)
perplexity on the held-out words wdTE2 as exp ? d
where N TE2 is the total
N TE2
number of tokens in wdTE2 .
4
We find bigram candidates that occur at least ten times in the training set and use a ?2 test to filter out those
having a ?2 value less than 5.0. We then treat selected bigrams as single word types in the vocabulary.
5
Setup We compare our proposed model L 2 H with the following methods:
? LDA [7]: unsupervised topic model with a flat topic structure. In our experiments, we set the
number of topics of LDA equal to the number of labels in each dataset.
? L - LDA [36]: associates each topic with a label, and a document is generated using the topics
associated with the document?s labels only.
? L 2 F (Label to Flat structure): a simplified version of L 2 H with a fixed, flat topic structure. The
major difference between L 2 F and L - LDA is that L 2 F allows tokens to be drawn from topics that are
not in the document?s label set via the use of the switching variable (Section 2.2). Improvements
of L 2 H over L 2 F show the importance of the hierarchical structure.
For all models, the number of topics is the number of labels in the dataset. We run for 1,000 iterations
on the training data with a burn-in period of 500 iterations. After the burn-in period, we store ten
sets of estimated parameters, one after every fifty iterations. During test time, we run ten chains
using these ten learned models on the test data and compute the perplexity after 100 iterations. The
perplexity of each fold is the average value over the ten chains [28].
Number of bills
Congress
Figure 3: Dataset statistics
109
110
250
111
112
?
300
Perplexity
10000
14034 13673 12274
5000 13067
0
109 110 111 112
Number of labels
400
300
200 418
375
243
205
100
0
109 110 111 112
?
LDA
?
?
200
? L?LDA
L2F
L2H
150
Figure 4: Perplexity on held-out documents, averaged over 5 folds
Results: Figure 4 shows the perplexity of the four models averaged over five folds on the four
datasets. LDA outperforms the other models with labels since it can freely optimize the likelihood
without additional constraints. L - LDA and L 2 F are comparable. However, L 2 H significantly outperforms both L - LDA and L 2 F. Thus, if incorporating labels into a model, learning an additional topic
hierarchy improves predictive power and generalizability of L - LDA.
4.2
Multi-label Classification
Multi-label classification is predicting a set of labels for a test document given its text [43, 23, 47].
The prediction is from a set of pre-defined K labels and each document can be tagged with any
of the 2K possible subsets. In this experiment, we use M 3 L?an efficient max-margin multi-label
classifier [16]?to study how features extracted from our L 2 H improve classification.
We use F1 as the evaluation metric. The F1 score is first computed for each document d as F1 (d) =
2?P (d)?R(d)
P (d)+R(d) , where P (d) and R(d) are the precision and recall for document d. After F1 (d) is
computed for all documents, the overall performance can be summarized by micro-averaging and
macro-averaging to obtain Micro-F1 and Macro-F1 respectively. In macro-averaging, F1 is first
computed for each document using its own confusion matrix and then averaged. In micro-averaging,
on the other hand, only a single confusion matrix is computed for all documents, and the F1 score is
computed based on this single confusion matrix [38].
Setup We use the following sets of features:
? TF: Each document is represented by a vector of term frequency of all word types in the vocabulary.
? TF - IDF: Each document is represented by a vector ?dTFIDF of TF - IDF of all word types.
? L - LDA & TF - IDF: Ramage et al. [35] combine L - LDA features and TF - IDF features to improve the
performance on recommendation tasks. Likewise, we extract a K-dimensional vector ??dL - LDA and
combine with TF - IDF vector ?dTFIDF to form the feature vector of L - LDA & TF - IDF.5
5
We run L - LDA on train for 1,000 iterations and ten models after 500 burn-in iterations. For each model, we
sample assignments for all tokens using 100 iterations and average over chains to estimate ??dL - LDA .
6
? L 2 H & TF - IDF: Similarly, we combine TF - IDF with the features ??dL 2 H = {??d0 , ??d1 } extracted using
L 2 H (same MCMC setup as L - LDA ).
One complication for L 2 H is the candidate label set L1d , which is not observed during test time. Thus,
during test time, we estimate L1d using TF - IDF. Let Dl be the set of documents tagged with label
l. For each l, we compute a TF - IDF vector ?lTFIDF = avgd?Dl ?dTFIDF . Then for each document d, we
generate the k nearest labels using cosine similarity, and add them to the candidate label set L1d of
d. Finally, we expand this initial set by adding all labels on the paths from the root of the learned
hierarchy to any of the k nearest labels (Figure 2). We explored different values of k ? {3, 5, 7, 9},
with similar results; the results in this section are reported with k = 5.
109
110
111
112
109
110
0.65
?
?
112
?
?
Micro F1
0.60
Macro F1
111
0.6
0.55
0.50
TF
?
L?LDA & TFIDF
L2H & TFIDF
?
0.5
?
? TFIDF
?
0.45
Figure 5: Multi-label classification results. The results are averaged over 5 folds.
Results Figure 5 shows classification results. For both Macro-F 1 and Micro-F 1, TF - IDF, L LDA & TF - IDF and L 2 H & TF - IDF significantly outperform TF . Also, L - LDA & TF - IDF performs better
than TF - IDF, which is consistent with Ramage et al. (2010) [35].
L 2 H & TF - IDF performs better than L - LDA & TF - IDF, which in turn performs better than TF - IDF. This
shows that features extracted from L 2 H are more predictive than those extracted from L - LDA, and
both improve classification. The improvements of L 2 H & TF - IDF and L - LDA & TF - IDF over TF - IDF are
clearer for Macro-F 1 compared with Micro-F 1. Thus, features from both topic models help improve
prediction, regardless of the frequencies of their tagged labels.
4.3
Learned label hierarchy: A taxonomy of Congressional issues
Terrorism
Military operations and
strategy
intellig, intellig_commun,
afghanistan, nation_intellig,
guantanamo_bai, qaeda,
central_intellig, detent, pakistan,
interrog, defens_intellig, detaine,
armi, air_forc, none, navi,
addit_amount, control_act,
emerg_deficit, fund_appropri,
balanc_budget, terror_pursuant,
transfer_author,marin_corp
International affairs
libya, unit_nation, intern_religi,
bahrain, religi_freedom,
religi_minor, freedom_act, africa,
violenc, secur_council,
benghazi, privileg_resolut, hostil,
Foreign aid and
international relief
International law and
treaties
Human rights
fund_appropri, foreign_assist,
remain_avail, regular_notif,
intern_develop, relat_program,
unit_nation, pakistan,
foreign_oper, usaid, prior_act
foreign_assist, intern_develop,
vessel, foreign_countri, sanit,
appropri_congression,
develop_countri, violenc, girl,
defens_articl, export
traffick, russian_feder,
traffick_victim, prison, alien,
visa, nation_act, victim, detent,
human_traffick, corrupt, russian,
foreign_labor, sex_traffick,
Military personnel and
dependents
Department of Defense
Asia
Int'l organizations &
cooperation
Middle East
coast_guard, vessel, command,
special_select, sexual_violenc,
academi, sexual_harass, navi,
former_offic, gulf_coast, haze,
port, marin, marin_debri
air_forc, militari_construct,
author_act, armi, nation_defens,
navi, militari_depart, aircraft,
congression_defens, command,
sexual_assault, activ_duti
china, vietnam, taiwan, republ,
chines, sea, north_korea,
tibetan, north_korean, refuge,
south_china, intern_religi, tibet,
enterpris, religi_freedom
export, arm_export, control_act,
foreign_assist, cuba,
defens_articl, foreign_countri,
foreign_servic, export_administr,
author_act, munit_list
syria, israel, iran, enterpris_fund,
unit_nation, egypt, palestinian,
cypru, tunisia, hezbollah,
lebanon, republ, hama, syrian,
violenc, weapon,
Armed forces and
national security
Department of
Homeland Security
Europe
Latin America
Sanctions
Religion
cemeteri, nation_guard, dog,
service_memb,
homeless_veteran, funer,
medic_center, militari_servic,
arlington_nation, armi, guard
cybersecur, inform_secur,
inform_system, cover_critic,
critic_infrastructur,
inform_infrastructur,
cybersecur_threat,
republ, belaru, turkei, nato,
holocaust_survivor, north_atlant,
holocaust, european_union,
albania, jew, china, macedonia,
treati_organ, albanian, greec
border_protect, haiti,
merchandis, evas, tariff_act,
cover_merchandis, export,
custom_territori, custom_enforc,,
countervail_duti, intern_trade
iran, sanction, syria,
comprehens_iran, north_korea,
financi_institut, presid_determin,
islam_republ, foreign_person,
weapon, iran_sanction
unit_nation, israel, iaea, harass,
syria, iran, peacekeep_oper,
regular_budget, unrwa,
palestinian, refuge, durban, bulli,
secur_council
Figure 6: A subtree in the hierarchy learned by L 2 H. The subtree root International Affairs is a child
node of the Background root node.
To qualitatively analyze the hierarchy learned by our model, Figure 6 shows a subtree whose root
is about International Affairs, obtained by running L 2 H on bills in the 112th U.S. Congress. The
learned topic at International Affairs shows the focus of 112th Congress on the Arab Spring?a
revolutionary wave of demonstrations and protests in Arab countries like Libya, Bahrain, etc. The
concept is then split into two distinctive aspects of international affairs: Military and Diplomacy.
7
We are working with domain experts to formally evaluate the learned concept hierarchy. A political
scientist (personal communication) comments:
The international affairs topic does an excellent job of capturing the key distinction
between military/defense and diplomacy/aid. Even more impressive is that it then
also captures the major policy areas within each of these issues: the distinction
between traditional military issues and terrorism-related issues, and the distinction
between thematic policy (e.g., human rights) and geographic/regional policy.
5
Conclusion
We have presented L 2 H, a model that discovers not just the interaction between overt labels and
the latent topics used in a corpus, but also how they fit together in a hierarchy. Hierarchies are a
natural way to organize information, and combining labels with a hierarchy provides a mechanism
for integrating user knowledge and data-driven summaries in a single, consistent structure. Our
experiments show that L 2 H yields interpretable label/topic structures, that it can substantially improve
model perplexity compared to baseline approaches, and that it improves performance on a multi-label
prediction task.
Acknowledgments
We thank Kristina Miler, Ke Zhai, Leo Claudino, and He He for helpful discussions, and thank
the anonymous reviewers for insightful comments. This research was supported in part by NSF
under grant #1211153 (Resnik) and #1018625 (Boyd-Graber and Resnik). Any opinions, findings,
conclusions, or recommendations expressed here are those of the authors and do not necessarily
reflect the view of the sponsor.
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9
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4,754 | 5,304 | On a Theory of Nonparametric Pairwise Similarity
for Clustering: Connecting Clustering to
Classification
Yingzhen Yang1 Feng Liang1 Shuicheng Yan2 Zhangyang Wang1 Thomas S. Huang1
1
University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
{yyang58,liangf,zwang119,t-huang1}@illinois.edu
2
National University of Singapore, Singapore, 117576
[email protected]
Abstract
Pairwise clustering methods partition the data space into clusters by the pairwise
similarity between data points. The success of pairwise clustering largely depends on the pairwise similarity function defined over the data points, where kernel similarity is broadly used. In this paper, we present a novel pairwise clustering
framework by bridging the gap between clustering and multi-class classification.
This pairwise clustering framework learns an unsupervised nonparametric classifier from each data partition, and search for the optimal partition of the data by
minimizing the generalization error of the learned classifiers associated with the
data partitions. We consider two nonparametric classifiers in this framework, i.e.
the nearest neighbor classifier and the plug-in classifier. Modeling the underlying data distribution by nonparametric kernel density estimation, the generalization error bounds for both unsupervised nonparametric classifiers are the sum of
nonparametric pairwise similarity terms between the data points for the purpose
of clustering. Under uniform distribution, the nonparametric similarity terms induced by both unsupervised classifiers exhibit a well known form of kernel similarity. We also prove that the generalization error bound for the unsupervised plugin classifier is asymptotically equal to the weighted volume of cluster boundary
[1] for Low Density Separation, a widely used criteria for semi-supervised learning and clustering. Based on the derived nonparametric pairwise similarity using
the plug-in classifier, we propose a new nonparametric exemplar-based clustering
method with enhanced discriminative capability, whose superiority is evidenced
by the experimental results.
1
Introduction
Pairwise clustering methods partition the data into a set of self-similar clusters based on the pairwise similarity between the data points. Representative clustering methods include K-means [2]
which minimizes the within-cluster dissimilarities, spectral clustering [3] which identifies clusters
of more complex shapes lying on low dimensional manifolds, and the pairwise clustering method
[4] using message-passing algorithm to inference the cluster labels in a pairwise undirected graphical model. Utilizing pairwise similarity, these pairwise clustering methods often avoid estimating
complex hidden variables or parameters, which is difficult for high dimensional data.
However, most pairwise clustering methods assume that the pairwise similarity is given [2, 3], or
they learn a more complicated similarity measure based on several given base similarities [4]. In
this paper, we present a new framework for pairwise clustering where the pairwise similarity is
derived as the generalization error bound for the unsupervised nonparametric classifier. The un1
supervised classifier is learned from unlabeled data and the hypothetical labeling. The quality of
the hypothetical labeling is measured by the associated generalization error of the learned classifier, and the hypothetical labeling with minimum associated generalization error bound is preferred.
We consider two nonparametric classifiers, i.e. the nearest neighbor classifier (NN) and the plug-in
classifier (or the kernel density classifier). The generalization error bounds for both unsupervised
classifiers are expressed as sum of pairwise terms between the data points, which can be interpreted
as nonparametric pairwise similarity measure between the data points. Under uniform distribution,
both nonparametric similarity measures exhibit a well known form of kernel similarity. We also
prove that the generalization error bound for the unsupervised plug-in classifier is asymptotically
equal to the weighted volume of cluster boundary [1] for Low Density Separation, a widely used
criteria for semi-supervised learning and clustering.
Our work is closely related to discriminative clustering methods by unsupervised classification,
which search for the cluster boundaries with the help of unsupervised classifier. For example, [5]
learns a max-margin two-class classifier to group unlabeled data in an unsupervised manner, known
as unsupervised SVM whose theoretical property is further analyzed in [6]. Also, [7] learns the
kernel logistic regression classifier, and uses the entropy of the posterior distribution of the class
label by the classifier to measure the quality of the learned classifier. More recent work presented in
[8] learns an unsupervised classifier by maximizing the mutual information between cluster labels
and the data, and the Squared-Loss Mutual Information is employed to produce a convex optimization problem. Although such discriminative methods produce satisfactory empirical results, the
optimization of complex parameters hampers their application in high-dimensional data. Following
the same principle of unsupervised classification using nonparametric classifiers, we derive nonparametric pairwise similarity and eliminate the need of estimating complicated parameters of the
unsupervised classifer. As an application, we develop a new nonparametric exemplar-based clustering method with the derived nonparametric pairwise similarity induced by the plug-in classifier, and
our new method demonstrates better empirical clustering results than the existing exemplar-based
clustering methods.
It should be emphasized that our generalization bounds are essentially different from the literature. As nonparametric classification methods, the generalization properties of the nearest neighbor
classifier (NN) and the plug-in classifier are extensively studied. Previous research focuses on the
average generalization error of the NN [9, 10], which is the average error of the NN over all the
random training data sets, or the excess risk of the plug-in classifier [11, 12]. In [9], it is shown that
the average generalization error of the NN is bounded by twice of the Bayes error. Assuming that
the class of the regression functions has a smooth parameter ?, [11] proves that the excess risk of
??
the plug-in classifier converges to 0 of the order n 2?+d where d is the dimension of the data. [12]
further shows that the plug-in classifier attains faster convergence rate of the excess risk, namely
1
n? 2 , under some margin assumption on the data distribution. All these generalization error bounds
depend on the unknown Bayes error. By virtue of kernel density estimation and generalized kernel density estimation [13], our generalization bounds are represented mostly in terms of the data,
leading to the pairwise similarities for clustering.
2
Formulation of Pairwise Clustering by Unsupervised Nonparametric
Classification
The discriminative clustering literature [5, 7] has demonstrated the potential of multi-class classification for the clustering problem. Inspired by the natural connection between clustering and
classification, we model the clustering problem as a multi-class classification problem: a classifier
is learned from the training data built by a hypothetical labeling, which is a possible cluster labeling.
The optimal hypothetical labeling is supposed to be the one such that its associated classifier has the
minimum generalization error bound. To study the generalization bound for the classifier learned
from the hypothetical labeling, we define the concept of classification model. Given unlabeled data
{xl }nl=1 , a classification model MY is constructed for any hypothetical labeling Y = {yl }nl=1 as
below:
Definition 1. The classification
model corresponding
to the hypothetical labeling Y = {yl }nl=1
(
)
Q
is defined as MY = S, PXY , {?i , fi }i=1 , F . S = {xl , yl }nl=1 are the labeled data by the
2
hypothetical labeling, and S are assumed to be i.i.d. samples drawn from the joint distribution PXY = PX|Y PY , where (X, Y ) is a random couple, X ? IRd represents the data and
Y ? {1, 2, ..., Q} is the class label of X, Q is the number of classes determined by the hypothetical
(i)
labeling. Furthermore, PXY is specified by {? (i) , f (i) }Q
is the class prior for
i=1 as follows: ?
(i)
class i, i.e. Pr [Y = i] = ? ; the conditional distribution PX|Y =i has probabilistic density function f (i) , i = 1, . . . , Q. F is a classifier trained using the training data S. The generalization error
of the classification model MY is defined as the generalization error of the classifier F in MY .
In this paper, we study two types of classification models with the nearest neighbor classifier and
the plug-in classifier respectively, and derive their generalization error bounds as sum of pairwise
similarity between the data. Given a specific type of classification model, the optimal hypothetical
labeling corresponds to the classification model with minimum generalization error bound. The
optimal hypothetical labeling also generates a data partition where the sum of pairwise similarity
between the data from different clusters is minimized, which is a common criteria for discriminative
clustering.
In the following text, we derive the generalization error bounds for the two types of classification
models. Before that, we introduce more notations and assumptions for the classification model.
Denote by PX the induced marginal distribution of X, and f is the probabilistic density function of
Q
?
PX which is a mixture of Q class-conditional densities: f =
? (i) f (i) . ? (i) (x) is the regression
i=1
(i)
(i)
function of Y on X = x, i.e. ? (i) (x) = Pr [Y = i |X = x ] = ? ff(x)(x) . For the sake of the
consistency of the kernel density estimators used in the sequel, there are further assumptions on
the marginal density and class-conditional densities in the classification model for any hypothetical
labeling:
(A) f is bounded from below, i.e. f ? fmin > 0
(i)
(B) {f (i) } is bounded from above, i.e. f (i) ? fmax , and f (i) ? ??,ci , 1 ? i ? Q.
where ??,c is the class of H?older-? smooth functions with H?older constant c:
??,c , {f : IRd ? IR | ?x, y, |f (x) ? f (y)| ? c?x ? y?? }, ? > 0
It follows from assumption (B) that f ? ??,c where c =
?
? (i) ci . Assumption (A) and (B) are
i
mild. The upper bound for the density functions is widely required for the consistency of kernel
density estimators [14, 15]; H?older-? smoothness is required to bound the bias of such estimators,
and it also appears in [12] for estimating the excess risk of the plug-in classifier. The lower bound
for the marginal density is used to derive the consistency of the estimator of the regression function
? (i) (Lemma 2) and the consistency of the generalized kernel density estimator (Lemma 3). We
denote by PX the collection of marginal distributions that satisfy assumption (A), and denote by
PX|Y the collection of class-conditional distributions that satisfy assumption (B). We then define
the collection of joint distributions PXY that PXY belongs to, which requires the marginal density
and class-conditional densities satisfy assumption (A)-(B):
PXY , {PXY | PX ? PX , {PX|Y =i } ? PX|Y , min{? (i) } > 0}
i
(1)
Given the joint distribution PXY , the generalization error of the classifier F learned from the training data S is:
R (FS ) , Pr [(X, Y ) : F (X) ?= Y ]
(2)
Nonparametric kernel density estimator (KDE) serves as the primary tool of estimating the underlying probabilistic density functions in our generalization analysis, and we introduce the KDE of f
as below:
n
1?
Khn (x ? xl )
f?n,hn (x) =
n
where Kh (x) =
1
e
(2?)d/2
?x?2
? 2
1
K
hd
(x)
h
(3)
l=1
is the isotropic Gaussian kernel with bandwidth h and K (x) ,
. We have the following VC property of the Gaussian kernel K. Define the class
3
of functions
(
F , {K
t??
h
)
, t ? IRd , h ?= 0}
(4)
The VC property appears in [14, 15, 16, 17, 18], and it is proved that F is a bounded VC class of
measurable functions with respect to the envelope function F such that |u| ? F for any u ? F (e.g.
d
F ? (2?)? 2 ). It follows that? there exist positive numbers A and v such that for every probability
measure P on IRd for which F 2 dP < ? and any 0 < ? < 1,
(
) ( A )v
N F , ???L2 (P ) , ? ?F ?L2 (P ) ?
?
(5)
(
)
? ? is defined as the minimal number of open d-balls
?
where N T , d,
of radius ? required to cover
(
)
T in the metric space T , d? . A and v are called the VC characteristics of F.
The VC property of K is required for the consistency of kernel density estimators shown in
Lemma 2. Also, we adopt the kernel estimator of ? (i) below
n
?
(i)
??n,hn (x) =
l=1
Khn (x ? xl )1I{yl =i}
nf?n,hn (x)
(6)
Before stating Lemma 2, we introduce several frequently used quantities throughout this paper. Let
L, C > 0 be constants which only depend on the VC characteristics of the Gaussian kernel K. We
define
f0 ,
Q
?
(i)
? (i) fmax
?02 , ?K?22 f0
(7)
i=1
Also, for all positive numbers ? ? C and ? > 0, we define
E?2 ,
log (1 + ?/4L)
?L? 2
(8)
Based on Corollary 2.2 in [14], Lemma 2 and Lemma 3 in the Appendix (more complete version
in the supplementary) show the strong consistency (almost sure uniformly convergence) of several
(i)
kernel density estimators, i.e. f?n,hn , {?
?n,hn } and the generalized kernel density estimator, and they
form the basis for the derivation of the generalization error bounds for the two types of classification
models.
3
Generalization Bounds
We derive the generalization error bounds for the two types of classification models with the nearest
neighbor classifier and the plug-in classifier respectively. Substituting these kernel density estimators for the corresponding true density functions, Theorem 1 and Theorem 2 present the generalization error bounds for the classification models with the plug-in classifier and the nearest neighbor
classifier. The dominant terms of both bounds are expressed as sum of pairwise similarity depending solely on the data, which facilitates the application of clustering. We also show the connection
between the error bound for the plug-in classifier and Low Density Separation in this section. The
detailed proofs are included in the supplementary.
3.1
Generalization Bound for the Classification Model with Plug-In Classifier
The plug-in classifier resembles the Bayes classifier while it uses the kernel density estimator of the
regression function ? (i) instead of the true ? (i) . It has the form
(i)
PI (X) = arg max ??n,hn (X)
(9)
1?i?Q
(i)
where ??n,hn is the nonparametric kernel estimator of the regression function ? (i) by (6). The
generalization capability of the plug-in classifier has been studied by the literature[11, 12]. Let
4
F ? be the Bayes classifier, it is proved that the excess risk of PIS , namely IES R (PIS ) ? R (F ? ),
??
converges to 0 of the order n 2?+d under some complexity assumption on the class of the regression
functions with smooth parameter ? that {? (i) } belongs to [11, 12]. However, this result cannot be
used to derive the generalization error bound for the plug-in classifier comprising of nonparametric
pairwise similarities in our setting.
We show the upper bound for the generalization error of PIS in Lemma 1.
Lemma 1. For any PXY ? PXY , there exists a n0 which depends on ?0 and VC characteristics
E? 2
of K, when n > n0 , with probability greater than 1 ? 2QLhn
plug-in classifier satisfies
R (PIS ) ?
PI
Rn
=
PI
Rn
+O
?
?
( log h?1
n
nhdn
0
+ h?n
, the generalization error of the
)
(10)
[
]
(i)
(j)
IEX ??n,hn (X) ??n,hn (X)
(11)
log h?1
n
nhd
n
? 0, ??n,hn is the kernel
i,j=1,...,Q,i?=j
where E?2 is defined by (8), hn is chosen such that hn ? 0,
(i)
(i)
estimator of the regression function. Moreover, the equality in (10) holds when ??n,hn ?
1 ? i ? Q.
1
Q
for
Based on Lemma 1, we can bound the error of the plug-in classifier from above by RnPI . Theorem 1
then gives the bound for the error of the plug-in classifier in the corresponding classification model
using the generalized kernel density estimator in Lemma 3. The bound has a form of sum of pairwise
similarity between the data from different classes.
Theorem
1. (Error of) the Plug-In Classifier) Given the classification model MY =
(
S, PXY , {?i , fi }Q
i=1 , PI such that PXY ? PXY , there exists a n1 which depends on ?0 , ?1 and
E? 2
the VC characteristics of K, when n > n1 , with probability greater than 1 ? 2QLhn
the generalization error of the plug-in classifier satisfies
? n (PIS ) + O
R (PIS ) ? R
? n (PIS ) =
where R
1
n2
?
l,m
(
?
)
log h?1
n
+ h?n
d
nhn
?K?22 fmax
,
fmin
?lm Glm,?2hn , ?12 =
0
E? 2
? QLhn
1
,
(12)
?lm = 1I{yl ?=ym } is a class indicator
function and
Glm,h = Gh (xl , xm ) , Gh (x, y) =
E?2 is defined by (8), hn is chosen such that hn ? 0,
estimator of f defined by (3).
Kh (x ? y)
1
1
2
2
f?n,h
(x)f?n,h
(y)
log h?1
n
nhd
n
,
(13)
? 0, f?n,hn is the kernel density
(?
)
log h?1
?
n
? n is the dominant term determined solely by the data and the excess error O
R
+
h
n
nhd
n
goes to 0 with infinite n. In the following subsection, we show the close connection between the
error bound for the plug-in classifier and the weighted volume of cluster boundary, and the latter is
proposed by [1] for Low Density Separation.
3.1.1
Connection to Low Density Separation
Low Density Separation [19], a well-known criteria for clustering, requires that the cluster boundary
should pass through regions of low density. It has been extensively studied in unsupervised learning
and semi-supervised learning [20, 21, 22]. Suppose the data {xl }nl=1 lies on a domain ? ? Rd .
Let f be the probability density function on ?, S be the cluster boundary which separates ? into
two parts S1 and S2 . Following the Low Density Separation assumption, [1] suggests that the
5
?
cluster boundary S with low weighted volume
f (s)ds should be preferable. [1] also proves that
S
a particular type of cut function converges to the weighted volume of S. Based on their study, we
obtain the following result relating the error of the plug-in classifier to the weighted volume of the
cluster boundary.
Corollary 1. Under the assumption of Theorem 1, for any kernel bandwidth sequence {hn }?
n=1
1
such that lim hn = 0 and hn > n? 4d+4 , with probability 1,
n??
?
lim
n??
? ?
Rn (PIS ) =
2hn
?
f (s)ds
(14)
S
3.2
Generalization Bound for the Classification Model with Nearest Neighbor Classifier
Theorem 2 shows the generalization error bound for the classification model with nearest neighbor
classifier (NN), which has a similar form as (12).
(
)
Theorem 2. (Error of the NN) Given the classification model MY = S, PXY , {?i , fi }Q
i=1 , NN
d
such that PXY ? PXY and the support of PX is bounded by [?M0 , M0 ] , there exists a n0 which
depends on ?0 and VC characteristics of K, when n > n0 , with probability greater than 1 ?
E? 2
2QLhn
0
? (2M0 )d ndd0 e?n
1?dd0
fmin
, the generalization error of the NN satisfies:
? n (NNS ) + c0
R (NNS ) ? R
? n (NN) =
where R
1
n
?
?
)
( log h?1
(? )?
n
?
?d0 ?
+
h
d n
+O
n
nhdn
(15)
Hlm,hn ?lm ,
1?l<m?n
Hlm,hn = Khn (xl ? xm )
(
?
Vl
f?n,hn (x) dx
f?n,hn (xl )
?
+
f?n,hn (x) dx )
,
f?n,hn (xm )
Vm
(16)
E?2 is defined by (8), d0 is a constant such that dd0 < 1, f?n,hn is the kernel density estimator of
log h?1
f defined by (3) with the kernel bandwidth hn satisfying hn ? 0, nhdn ? 0, Vl is the Voronoi
n
cell associated with xl , c0 is a constant, ?lm = 1I{yl ?=ym } is a class indicator function such that
?lm = 1 if xl and xm belongs to different classes, and 0 otherwise. Moreover, the equality in (15)
1
holds when ? (i) ? Q
for 1 ? i ? Q.
Glm,?2hn in (13) and Hlm,hn in (16) are the new pairwise similarity functions induced by the plugin classifier and the nearest neighbor classifier respectively. According to the proof of Theorem 1 and
Theorem 2, the kernel density estimator f? can be replaced by the true density f in the denominators
of (13) and (16), and the conclusions of Theorem 1 and 2 still hold. Therefore, both Glm,?2hn and
Hlm,hn are equal to ordinary Gaussian kernels (up to a scale) with different kernel bandwidth under
uniform distribution, which explains the broadly used kernel similarity in data clustering from an
angle of supervised learning.
4
Application to Exemplar-Based Clustering
We propose a nonparametric exemplar-based clustering algorithm using the derived nonparametric
pairwise similarity by the plug-in classifier. In exemplar-based clustering, each xl is associated with
a cluster indicator el (l ? {1, 2, ...n} , el ? {1, 2, ...n}), indicating that xl takes xel as the cluster
n
exemplar. Data from the same cluster share the same cluster exemplar. We define e , {el }l=1 .
Moreover, a configuration of the cluster indicators e is consistent iff el = l when em = l for any
l, m ? 1..n, meaning that xl should take itself as its exemplar if any xm take xl as its exemplar. It is
required that the cluster indicators e should always be consistent. Affinity Propagation (AP) [23], a
representative of the exemplar-based clustering methods, solves the following optimization problem
min
e
n
?
Sl,el
s.t.
l=1
6
e is consistent
(17)
Sl,el is the dissimilarity between xl and xel , and note that Sl,l is set to be nonzero to avoid the trivial
minimizer of (17).
Now we aim to improve the discriminative capability of the exemplar-based clustering (17) using
the nonparametric pairwise similarity derived by the unsupervised plug-in classifier. As mentioned
? is evaluated by the generalization error bound for
before, the quality of the hypothetical labeling y
? with minimum
the nonparametric plug-in classifier trained by Sy? , and the hypothetical labeling y
? n (PIS ) = arg miny? ? ?lm G ?
associated error bound is preferred, i.e. arg miny? R
lm, 2hn where
l,m?
?lm Glm,?2hn also
?lm = 1Iy?l ?=y?m and Glm,?2hn is defined in (13). By Lemma 3, minimizing
l,m
enforces minimization of the weighted volume of cluster boundary asymptotically. To avoid the
trivial clustering where all the data are grouped into a single cluster, we use the sum of within)
(
n
?
cluster dissimilarities term
exp ?Glel ,?2hn to control the size of clusters. Therefore, the
l=1
objective function of our pairwise clustering method is below:
? (e) =
n
?
(
)
)
?(
exp ?Glel ,?2hn + ?
??lm Glm,?2hn + ?lm (el , em )
l=1
(18)
l,m
where ?lm is a function to enforce the consistency of the cluster indicators:
{
?lm (el , em ) =
? em = l, el ?= l or el = m, em ?= m
,
0 otherwise
? is a balancing parameter. Due to the form of (18), we construct a pairwise Markov Random
Field (MRF) representing the unary term ul and the pairwise term ??lm Glm,?2hn + ?lm as the data
likelihood and prior respectively. The variables e are modeled as nodes and the unary term and
pairwise term in (18) are modeled as potential functions in the pairwise MRF. The minimization of
the objective function is then converted to a MAP (Maximum a Posterior) problem in the pairwise
MRF. (18) is minimized by Max-Product Belief Propagation (BP).
The computational complexity of our clustering algorithm is O(T EN ), where E is the number of
edges in the pairwise MRF, T is the number of iterations of message passing in the BP algorithm.
We call our new algorithm Plug-In Exemplar Clustering (PIEC), and compare it to representative
exemplar-based clustering methods, i.e. AP and Convex Clustering with Exemplar-Based Model
(CEB) [24], for clustering on three real data sets from UCI repository, i.e. Iris, Vertebral Column
(VC) and Breast Tissue (BT). We record the average clustering accuracy (AC) and the standard
deviation of AC for all the exemplar-based clustering methods when they produce the correct number
of clusters for each data set with different values of hn and ?, and the results are shown in Table 1.
Although AP produces better clustering accuracy on the VC data set, PIEC generates the correct
cluster numbers for much more times. The dash in Table 1 indicates that the corresponding clustering
method cannot produce the correct cluster number. The default value for the{kernel bandwidth} hn is
h?n , which is set as the variance of the pairwise distance between data points ?xl ? xm ?l<m . The
default value for the balancing parameter ? is 1. We let hn = ?h?n , ? varies between [0.2, 1] and
? varies between [0.2, 1.9] with step 0.2 and 0.05 respectively, resulting in 170 different parameter
settings. We also generate the same number of parameter settings for AP and CEB.
Table 1: Comparison Between Exemplar-Based Clustering Methods. The number in the bracket is
the number of times when the corresponding algorithm produces correct cluster numbers.
Data sets
AP
CEB
PIEC
5
Iris
0.8933 ? 0.0138 (16)
0.6929 ? 0.0168 (15)
0.9089 ? 0.0033 (15)
VC
0.6677 (14)
0.4748 ? 0.0014 (5)
0.5263 ? 0.0173 (35)
BT
0.4906 (1)
0.3868 ? 0.08 (2)
0.6585 ? 0.0103 (5)
Conclusion
We propose a new pairwise clustering framework where nonparametric pairwise similarity is derived by minimizing the generalization error unsupervised nonparametric classifier. Our framework
bridges the gap between clustering and multi-class classification, and explains the widely used kernel similarity for clustering. In addition, we prove that the generalization error bound for the unsupervised plug-in classifier is asymptotically equal to the weighted volume of cluster boundary for
7
Low Density Separation. Based on the derived nonparametric pairwise similarity using the plug-in
classifier, we propose a new nonparametric exemplar-based clustering method with enhanced discriminative capability compared to the exiting exemplar-based clustering methods.
Appendix
Lemma 2. (Consistency of Kernel Density Estimator) Let the kernel bandwidth hn of the Gaussian
log h?1
kernel K be chosen such that hn ? 0, nhdn ? 0. For any PX ? PX , there exists a n0 which
n
E? 2
depends on ?0 and VC characteristics of K, when n > n0 , with probability greater than 1 ? Lhn
over the data {xl },
?
fn,hn (x) ? f (x)
=O
?
?
( log h?1
n
nhdn
+ h?n
0
)
(19)
where f?n,hn is the kernel density estimator of f . Furthermore, for any PXY ? PXY , when n > n0 ,
E? 2
then with probability greater than 1 ? 2Lhn
over the data {xl },
0
(i)
?n,hn (x) ? ? (i) (x)
?
?
=O
?
( log h?1
n
nhdn
+ h?n
)
(20)
for each 1 ? i ? Q.
Lemma 3. (Consistency of the Generalized Kernel Density Estimator) Suppose f is the probabilistic
density function of PX ? PX . Let g be a bounded function defined on X and g ? ??,g0 , 0 < gmin ?
g ? gmax , and e = fg . Define the generalized kernel density estimator of e as
e?n,h ,
n
1 ? Kh (x ? xl )
n
g (xl )
(21)
l=1
Let ?g2 =
?K?22 fmax
.
2
gmin
There exists ng which depends on ?g and the VC characteristics of K, When
E?2
n > ng , with probability greater than 1 ? Lhn
g
over the data {xl },
??
en,hn (x) ? e (x)?? = O
where hn is chosen such that hn ? 0,
log h?1
n
nhd
n
?
( log h?1
n
nhdn
+ h?n
)
(22)
? 0.
Sketch of proof: For fixed h ?= 0, we consider the class of functions
Fg , {
K
( t?? )
h
g (?)
, t ? IRd }
It can be verified that Fg is also a bounded VC class with the envelope function Fg =
(
) ( A )v
N Fg , ???L2 (P ) , ? ?Fg ?L2 (P ) ?
?
F
,
gmin
and
(23)
Then (22) follows from similar argument in the proof of Lemma 2 and Corollary 2.2 in [14].
The generalized kernel density estimator (21) is also used in [13] to estimate the Laplacian PDF
Distance between two probabilistic density functions, and the authors only provide the proof of
pointwise weak consistency of this estimator in [13]. Under mild conditions, our Lemma 3 and
Lemma 2 show the strong consistency of the generalized kernel density estimator and the traditional
kernel density estimator under the same theoretical framework of the VC property of the kernel.
Acknowledgements. This material is based upon work supported by the National Science Foundation under Grant No. 1318971.
8
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4,755 | 5,305 | Robust Bayesian Max-Margin Clustering
Changyou Chen?
Jun Zhu?
Xinhua Zhang]
?
Dept. of Electrical and Computer Engineering, Duke University, Durham, NC, USA
?
State Key Lab of Intelligent Technology & Systems; Tsinghua National TNList Lab;
?
Dept. of Computer Science & Tech., Tsinghua University, Beijing 100084, China
]
Australian National University (ANU) and National ICT Australia (NICTA), Canberra, Australia
[email protected]; [email protected]; [email protected]
Abstract
We present max-margin Bayesian clustering (BMC), a general and robust framework that incorporates the max-margin criterion into Bayesian clustering models,
as well as two concrete models of BMC to demonstrate its flexibility and effectiveness in dealing with different clustering tasks. The Dirichlet process max-margin
Gaussian mixture is a nonparametric Bayesian clustering model that relaxes the
underlying Gaussian assumption of Dirichlet process Gaussian mixtures by incorporating max-margin posterior constraints, and is able to infer the number of
clusters from data. We further extend the ideas to present max-margin clustering topic model, which can learn the latent topic representation of each document
while at the same time cluster documents in the max-margin fashion. Extensive
experiments are performed on a number of real datasets, and the results indicate
superior clustering performance of our methods compared to related baselines.
1
Introduction
Existing clustering methods fall roughly into two categories. Deterministic clustering directly optimises some loss functions, while Bayesian clustering models the data generating process and infers the clustering structure via Bayes rule. Typical deterministic methods include the well known
kmeans [1], nCut [2], support vector clustering [3], Bregman divergence clustering [4, 5], and the
methods built on the very effective max-margin principle [6?9]. Although these methods can flexibly incorporate constraints for better performance, it is challenging for them to finely capture hidden
regularities in the data, e.g., automated inference of the number of clusters and the hierarchies underlying the clusters. In contrast, Bayesian clustering provides favourable convenience in modelling
latent structures, and their posterior distributions can be inferred in a principled fashion. For example, by defining a Dirichlet process (DP) prior on the mixing probability of Gaussian mixtures,
Dirichlet process Gaussian mixture models [10] (DPGMM) can infer the number of clusters in the
dataset. Other priors on latent structures include the hierarchical cluster structure [11?13], coclustering structure [14], etc. However, Bayesian clustering is typically difficult to accommodate
external constraints such as max-margin. This is because under the standard Bayesian inference
designing some informative priors (if any) that satisfy these constraints is highly challenging.
To address this issue, we propose Bayesian max-margin clustering (BMC), which allows maxmargin constraints to be flexibly incorporated into a Bayesian clustering model. Distinct from the
traditional max-margin clustering, BMC is fully Bayesian and enables probabilistic inference of the
number of clusters or the latent feature representations of data. Technically, BMC leverages the
regularized Bayesian inference (RegBayes) principle [15], which has shown promise on supervised
learning tasks, such as classification [16, 17], link prediction [18], and matrix factorisation [19],
where max-margin constraints are introduced to improve the discriminative power of a Bayesian
1
model. However, little exploration has been devoted to the unsupervised setting, due in part to the
absence of true labels that makes it technically challenging to enforce max-margin constraints. BMC
constitutes a first extension of RegBayes to the unsupervised clustering task. Note that distinct from
the clustering models using maximum entropy principle [20, 21] or posterior regularisation [22],
BMC is more general due to the intrinsic generality of RegBayes [15].
We demonstrate the flexibility and effectiveness of BMC by two concrete instantiations. The first is
Dirichlet process max-margin Gaussian mixture (DPMMGM), a nonparametric Bayesian clustering
model that relaxes the Gaussian assumption underlying DPGMM by incorporating max-margin constraints, and is able to infer the number of clusters in the raw input space. To further discover latent
feature representations, we propose the max-margin clustering topic model (MMCTM). As a topic
model, it performs max-margin clustering of documents, while at the same time learns the latent
topic representation for each document. For both DPMMGM and MMCTM, we develop efficient
MCMC algorithms by exploiting data augmentation techniques. This avoids imposing restrictive
assumptions such as in variational Bayes, thereby facilitating the inference of the true posterior. Extensive experiments demonstrate superior clustering performance of BMC over various competitors.
2
Regularized Bayesian Inference
We first briefly overview the principle of regularised Bayesian inference (RegBayes) [15].
The motivation of RegBayes is to enrich the posterior of a probabilistic model by incorporating additional constraints, under an information-theoretical optimisation formulation. Formally, suppose a
probabilistic model has latent variables ?, endowed with a prior p(?) (examples of ? will be clear
soon later). We also have observations X := {x1 , ? ? ? , xn }, with xi ? Rp . Let p(X|?) be the
likelihood. Then, posterior inference via the Bayes? theorem is equivalent to solving the following
optimisation problem [15]:
inf
q(?)?P
KL(q(?) || p(?)) ? E??q(?) [log p(X|?)]
(1)
where P is the space of probability distribution1 , q(?) is the required posterior (here and afterwards
we will drop the dependency on X for notation simplicity). In other words, the Bayesian posterior
p(?|X) is identical to the optimal solution to (1). The power of RegBayes stems in part from
the flexibility of engineering P, which typically encodes constraints imposed on q(?), e.g., via
expectations of some feature functions of ? (and possibly the data X). Furthermore, the constraints
can be parameterised by some auxiliary variable ?. For example, ? may quantify the extent to which
the constraints are violated, then it is penalised in the objective through a function U . To summarise,
RegBayes can be generally formulated as
inf KL(q(?) || p(?))? E??q(?) [log p(X|?)]+ U (?)
?,q(?)
s.t. q(?) ? P(?).
(2)
To distinguish from the standard Bayesian posterior, the optimal q(?) is called post-data posterior.
Under mild regularity conditions, RegBayes admits a generic representation theorem to characterise
the solution q(?) [15]. It is also shown to be more general than the conventional Bayesian methods,
including those methods that introduce constraints on a prior. Such generality is essential for us
to develop a Bayesian framework of max-margin clustering. Note that like many sophisticated
Bayesian models, posterior inference remains as a key challenge of developing novel RegBayes
models. Therefore, one of our key technical contributions is on developing efficient and accurate
algorithms for BMC, as detailed below.
3
Robust Bayesian Max-margin Clustering
For clustering, one key assumption of our model is that X forms a latent cluster structure. In particular, let each cluster be associated with a latent projector ?k ? Rp , which is included in ? and has
prior distribution subsumed in p(?). Given any distribution q on ?, we then define the compatibility
score of xi with respect to cluster k by using the marginal distribution on ?k (as ?k ? ?):
Fk (xi ) = Eq(?k ) ?kT xi = Eq(?) ?kT xi .
(3)
1
In theory, we also require that q is absolutely continuous with respect to p to make the KL-divergence well
defined. The present paper treats this constraint as an implicit assumption for clarity.
2
For each example xi , we introduce a random variable yi valued in Z+ , which denotes its cluster
assignment and is also included in ?. Inspired by conventional multiclass SVM [7, 23], we utilize
P(?) in RegBayes (2) to encode the max-margin constraints based on Fk (xi ), with the slack variable
? penalised via their sum in U (?). This amounts to our Bayesian max-margin clustering (BMC):
X
inf L(q(?)) + 2c
?i
(4)
?i ?0,q(?)
s.t.
i
Fyi (xi ) ? Fk (xi ) ? ` I(yi 6= k) ? ?i ,
?i, k
where L(q(?)) = KL(q(?)||p(?)) ? E??q(?) [log p(X|?)] measures the KL divergence between
q and the original Bayesian posterior p(?|X) (up to a constant); I(?) = 1 if ? holds true, and
0 otherwise; ` > 0 is a constant scalar of margin. Note we found that the commonly adopted
balance constraints in max-margin clustering models [6] either are unnecessary or do not help in our
framework. We will address this issue in specific models.
Clearly by absorbing the slack variables ?, the optimisation problem (4) is equivalent to
X
inf L(q(?)) + 2c
max 0, max E??q(?) [?ik ]
q(?)
k:k6=yi
i
(5)
where ?ik := ` I(yi 6= k) ? (?yi ? ?k )T xi . Exact solution to (5) is hard to compute. An alternative
approach is to approximate the posterior by assuming independence between random variables, e.g.
variational inference. However, this is usually slow and susceptible to local optimal. In order to
obtain an analytic optimal distribution q that facilitates efficient Bayesian inference, we resort to the
technique of Gibbs classifier [17] which approximates (in fact, upper bounds due to the convexity
of max function) the second term in (5) by an expected hinge loss, i.e., moving the expectation out
of the max. This leads to our final formulation of BMC:
X
inf L(q(?)) + 2c
E??q(?) max 0, max ?ik .
(6)
q(?)
k:k6=yi
i
Problem (6) is still much more challenging than existing RegBayes models [17], which are restricted
to supervised learning with two classes only. Specifically, BMC allows multiple clusters/classes in
an unsupervised setting, and the latent cluster membership yi needs to be inferred. This complicates
the model and brings challenges for posterior inference, as addressed below. In a nutshell, our
inference algorithms rely on two key steps by exploring data augmentation techniques. First, in order
to tackle the multi-class case, we introduce auxiliary variables si := arg maxk:k6=yi ?ik . Applying
standard derivations in calculus of variation [24] and augmenting the model with {si }, we obtain an
analytic form of the optimal solution to (6) by augmenting ? (refer to Appendix A for details):
Y
q(?, {si }) ? p(?|X)
exp(?2c max(0, ?isi )) .
(7)
i
Second, since the max term in (7) obfuscates efficient sampling, we apply the augmentation technique introduced by [17], which showed that q(?, {si }) is identical to the marginal distribution of
the augmented post-data posterior
Y
??i (?i |?),
(8)
q(?, {si }, {?i }) ? p(?|X)
i
?1
?i 2
where ??i (?i |?) :=
exp
+ c?isi )2 . Here ?i is an augmented variable for xi that has
an generalised inverse Gaussian distribution [25] given ? and xi .
?1
2?i (?i
Note that our two steps of data augmentation are exact and incur no approximation. With the augmented variables ({si }, {?i }), we can develop efficient sampling algorithms for the augmented
posterior q(?, {si }, {?i }) without restrictive assumptions, thereby allowing us to approach the true
target posterior q(?) by dropping the augmented variables. The details will become clear soon in
our subsequent clustering models.
4
Dirichlet Process Max-margin Gaussian Mixture Models
In (4), we have left unspecified the prior p(?) and the likelihood p(X|?). This section presents an
instantiation of Bayesian nonparametric clustering for non-Gaussian data. We will present another
instantiation of max-margin document clustering based on topic models in next section.
3
w
yi
?k
v
?k
?, S
?0
?
N
?k
?k
K
?1
?0
xi
v
?
?
?
T
yi
?
?k
r, m
?t
?i
zil
K
wil
Ni
?
D
Figure 1: Left: Graphical model of DPMMGM. The part excluding ?k and v corresponds to
DPGMM. Right: Graphical model of MMCTM. The one excluding {?k } and the arrow between
yi and wil corresponds to CTM.
Here a convenient model of p(X, ?) is mixture of Gaussian. Let the mean and variance of the k-th
cluster component be ?k and ?k . In a nonparametric setting, the number of clusters is allowed to
be infinite, and the cluster yi that each data point belongs to is drawn from a Dirichlet process [10].
n
To summarize, the latent variables are ? = {?k , ?k , ?k }?
k=1 ? {yi }i=1 . The prior p(?) is specified
as: ?k and ?k employ a standard Normal-inverse Wishart prior [26]:
?k ? N (?k ; m, (r?k )?1 ), and ?k ? IW(?k ; S, ?).
(9)
+
yi ? Z has a Dirichlet process prior with parameter ?. ?k follows a normal prior with mean 0 and
variance vI, where I is the identity matrix. The likelihood p(xi |?) is N (xi ; ?yi , (r?yi )?1 ), i.e.
independent of ?k . The max-margin constraints take effects in the model via ??i ?s in (8). Note this
model of p(?, X), apart from ?k , is effectively the Dirichlet process Gaussian mixture model [10]
(DPGMM). Therefore, we call our post-data posterior q(?, {si }, {?i }) in (8) as Dirichlet process
max-margin Gaussian mixture model (DPMMGM). The hyperparameters include m, r, S, ?, ?, v.
Interpretation as a generalised DP mixture The formula of the augmented post-data posterior
in (8) reveals that, compared with DPGMM, each data point is associated with an additional factor
??i (?i |?). Thus we can interpret DPMMGM as a generalised DP mixture with Normal-inversed
Wishart-Normal as the base distribution, and a generalised pseudo likelihood that is proportional to
f (xi , ?i |yi , ?y , ?y , {?k }) := N (xi ; ?y , (r?y )?1 )??i (?i |?) .
(10)
i
i
i
i
To summarise, DPMMGM employs the following generative process with the graphical model
shown in Fig. 1 (left):
(?k , ?k , ?k ) ? N ?k ; m, (r?k )?1 ? IW (?k ; S, ?) ? N (?k ; 0, vI) , k = 1, 2, ? ? ?
w ? Stick-Breaking(?),
yi |w ? Discrete(w),
i ? [n]
(xi , ?i )|yi , {?k , ?k , ?k } ' f (xi , ?i |yi , ?yi , ?yi , {?k }).
i ? [n]
Here [n] := {1, ? ? ? , n} is the set of integers up to n and ' means that (xi , ?i ) is generative from a
distribution that is proportional to f (?). Since this normalisation constant is shared by all samples
xi , there is no need to deal with it by posterior inference. Another benefit of this interpretation
is that it allows us to use existing techniques for non-conjugate DP mixtures to sample the cluster
indicators yi efficiently, and to infer the number of clusters in the data. This approach is different
from previous work on RegBayes nonparametric models where truncated approximation is used to
deal with the infinite dimensional model space [15, 18]. In contrast, our method does not rely on any
approximation. Note that DPMMGM does not need the complicated class balance constraints [6]
because the Gaussians in the pseudo likelihood would balance the clusters to some extent.
Posterior inference Posterior inference for DPMMGM can be done by efficient Gibbs sampling. We integrate out the infinite dimension vector w, so the variables needed to be sampled
are {?k , ?k , ?k }k ? {yi , si , ?i }i . Conditional distributions are derived in Appendix B. Note that we
use an extension of the Reused Algorithm [27] to jointly sample (yi , si ), which allows it to allocate
to empty clusters in Bayesian nonparametric setting. The time complexity is almost the same as
DPGMM except for the additional step to sample ?k , with cost O(p3 ). So it would be necessary to
put the constraints on a subspace (e.g., by projection) of the original feature space when p is high.
4
5
Max-margin Clustering Topic Model
Although many applications exhibit clustering structures in the raw observed data which can be
effectively captured by DPMMGM, it is common that such regularities are more salient in terms
of some high-level but latent features. For example, topic distributions are often more useful than
word frequency in the task of document clustering. Therefore, we develop a max-margin clustering
topic model (MMCTM) in the framework of BMC, which allows topic discovery to co-occur with
document clustering in a Bayesian and max-margin fashion. To this end, the latent Dirichlet allocation (LDA) [28] needs to be extended by introducing a cluster label into the model, and define each
cluster as a mixture of topic distributions. This cluster-based topic model [29] (CTM) can then be
used in concert with BMC to enforce large margin between clusters in the posterior q(?).
Let V be the size of the word vocabulary, T be the number of topics, and K be the number of clusters,
1N be a N -dimensional one vector. Then the generative process of CTM for the documents goes as:
1. For each topic t, generate its word distribution ?t : ?t |? ? Dir(?1V ).
2. Draw a base topic distribution ?0 : ?0 |?0 ? Dir(?0 1T ). Then for each cluster k, generate its
topic distribution mixture ?k : ?k |?1 , ?0 ? Dir(?1 ?0 ).
3. Draw a base cluster distribution ?: ?|? ? Dir(?1K ). Then for each document i ? [D]:
? Generate a cluster label yi and a topic distribution ?i : yi |? ? Discrete(?), ?i |?, ?yi ?
Dir(??yi ).
? Generate the observed words wil : zil ? Discrete(?i ), wil ? Discrete(?zil ), ? l ? [Ni ].
Fig. 1 (right) shows the structure. We then augment CTM with max-margin constraints, and get
the same posterior as in Eq. (7), with the variables ? corresponding to {?t }Tt=1 ? {?k , ?k }K
k=1 ?
D,Ni
D
{?0 , ?} ? {?i , yi }i=1 ? {zil }i=1,l=1 .
Compared with the raw word space which is normally extremely high-dimensional and sparse, it
is more reasonable to characterise the clustering structure in the latent feature space?the empirical
latent topic distributions as in the MedLDA [16]. Specifically, we summarise the topic distribution
PNi
of document i by xi ? RT , whose t-th element is N1i l=1
I(zil = t). Then the compatibility score
for document i with respect to cluster k is defined similar to (3) as Fk (xi ) = Eq(?) ?kT xi . Note,
however, the expectation is also taken over xi since it is not observed.
Posterior inference To achieve fast mixing, we integrate out {?t }Tt=1 ? {?0 , ?} ? {?k }K
k=1 ?
D,Ni
D
D
K
{?i }i=1 in the posterior, thus ? = {yi }i=1 ? {?k }k=1 ? {zil }i=1,l=1 . The integration is straightforward by the Dirichlet-Multinomial conjugacy. The detailed form of the posterior and the conditional
distributions are derived in Appendix C. By extending CTM with max-margin, we note that many
of the the sampling formulas are extension of those in CTM [29], with additional sampling for ?k ,
thus the sampling can be done fairly efficiently.
Dealing with vacuous solutions Different from DPMMGM, the max-margin constraints in MMCTM do not interact with the observed words wil , but with the latent topic representations xi (or zil )
that are also inferred from the model. This easily makes the latent representation zi ?s collapse into a
single cluster, a vacuous solution plaguing many other unsupervised learning methods as well. One
remedy is to incorporate the cluster balance constraints into the model [7]. However, this does not
help in our Bayesian setting because apart from significant increase in computational cost, MCMC
often fails to converge in practice2 . Another solution is to morph the problem into a weakly semisupervised setting, where we assign to each cluster a few documents according to their true label
(we will refer to these documents as landmarks), and sample the rest as in the above unsupervised
setting. These ?labeled examples? can be considered as introducing constraints that are alternative
to the balance constraints. Usually only a very small number of labeled documents are needed, thus
barely increasing the cost in training and labelling. We will focus on this setting in experiment.
6
6.1
Experiments
Dirichlet Process Max-margin Gaussian Mixture
2
We observed the cluster sizes kept bouncing with sampling iterations. This is probably due to the highly
nonlinear mapping from observed word space to the feature space (topic distribution), making the problem
multi-modal, i.e., there are multiple optimal topic assignments in the post-data posterior (8). Also the balance
constraints might weaken the max-margin constraints too much.
5
We first show the distinction between our DPMMGM and DPGMM
by running both models on the non-Gaussian half-rings data set [30].
There are a number of hyperparameters to be determined, e.g.,
(?, r, S, ?, v, c, `); see Section 4. It turns out the cluster structure is
insensitive to (?, r, S, ?), and so we use a standard sampling method
to update ? [31], while r, ?, S are sampled by employing Gamma,
truncated Poisson, inverse Wishart priors respectively, as is done
in [32]. We set v = 0.01, c = 0.1, ` = 5 in this experiment. Note that
the clustering structure is sensitive to the values of c and `, which will
be studied below. Empirically we find that DPMMGM converges
much faster than DPGMM, both converging well within 200 iterations (see Appendix D.4 for examples). In Fig. 2, the clustering
structures demonstrate clearly that DPMMGM relaxes the Gaussian
assumption of the data distribution, and correctly finds the number of Figure 2: An illustration
clusters based on the margin boundary, whereas DPGMM produces of DPGMM (up) and DPMa too fragmented partition of the data for the clustering task.
MGM (bottom).
0
5
0
5
0
5
0
5
0
5
0
5
0
5
0
5
0
5
0
5
Parameter sensitivity We next study the sensitivity of hyperparameters c and `, with other hyperparameters sampled during inference as above. Intuitively the impact of these parameters is as follows.
c controls the weight that the max-margin constraint places on the posterior. If there were no other
constraint, the max-margin constraint would drive the data points to collapse into a single cluster.
As a result, we expect that a larger value of the weight c will result in fewer clusters. Similarly,
increasing the value of ` will lead to a higher loss for any violation of the constraints, thus driving
the data points to collapse as well. To test these implications, we run DPMMGM on a 2-dimensional
synthetic dataset with 15 clusters [33]. We vary c and ` to study how the cluster structures change
with respect to these parameter settings. As can be observed from Fig. 3, the results indeed follow
our intuition, providing a mean to control the cluster structure in applications.
8
8
8
8
8
6
6
6
6
6
4
4
4
4
4
2
2
2
2
2
0
0
0
0
0
8
8
8
8
8
6
6
6
6
6
4
4
4
4
4
2
2
2
2
2
(a) c :5e-6, ` :5e-1
(b) c :5e-4, ` :5e-1
(c) c :5e-3, ` :5e-1
(d) c :5e-2, ` :5e-1
(e) c :5e-1, ` :5e-1
8
8
8
8
8
6
6
6
6
6
4
4
4
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4
2
2
2
2
2
0
0
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0
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8
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6
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4
4
4
4
2
2
2
2
2
(f) c :5e-3, ` :5e-4
(g) c :5e-3, ` :5e-2
(h) c :5e-3, ` :5e-1
(i) c :5e-3, ` :2
(j) c :5e-3, ` :5
Figure 3: Clustering structures with varied ` and c: (first row) fixed ` and increasing c; (second row)
fixed c and increasing `. Lines are ??s. Clearly the number cluster decreases with growing c and `.
Real Datasets. As other clustering models, we test DPMMGM on ten real datasets (small to moderate sizes) from the UCI repository [34]. Scaling up to large dataset is an interesting future. The
first three columns of Table 1 list some of the statistics of these datasets (we used random subsets of
the three large datasets ? Letter, MNIST, and Segmentation).
A heuristic approach for model selection. Model selection is generally hard for unsupervised
clustering. Most existing algorithms simply fix the hyperparameters without examining their impacts
on model performance [10, 35]. In DPMMGM, the hyperparameters c and ` are critical to clustering
quality since they control the number of clusters. Without training data in our setting they can not
be set using cross validation. Moreover, they are not feasible to be estimated use Bayesian sampling
as well because they are not parameters from a proper Bayesian model. we thus introduce a timeefficient heuristic approach to selecting appropriate values. Suppose the dataset is known to have
K clusters. Our heuristic goes as follows. First initialise c and ` to 0.1. Then at each iteration,
we compare the inferred number of clusters with K. If it is larger than K (otherwise we do the
converse), we choose c or ` randomly, and increase its value by nu , where u is a uniform random
variable in [0, 1] and n is the number of iterations so far. According to the parameter sensitivity
studied above, increasing c or ` tends to decrease the number of clusters, and the model eventually
6
Dataset
Glass
Half circle
Iris
Letter
MNIST
Satimage
Segment?n
Vehicle
Vowel
Wine
Data property
n
p
K
214
10
7
300
2
2
150
4
3
1000 16 10
1000 784 10
4435 36
6
1000 19
7
846
18
4
990
10 11
178
13
3
kmeans
0.37?0.04
0.43?0.00
0.72?0.08
0.33?0.01
0.50?0.01
0.57?0.06
0.52?0.03
0.10?0.00
0.42?0.01
0.84?0.01
nCut
0.22?0.00
1.00?0.00
0.61?0.00
0.04?0.00
0.38?0.00
0.55?0.00
0.34?0.00
0.14?0.00
0.44?0.00
0.46?0.00
NMI
DPGMM
0.37?0.05
0.49?0.02
0.73?0.00
0.19?0.09
0.55?0.03
0.21?0.05
0.23?0.09
0.02?0.02
0.28?0.03
0.56?0.02
DPMMGM
0.46?0.01
0.67?0.02
0.73?0.00
0.38?0.04
0.56?0.01
0.51?0.01
0.61?0.05
0.14?0.00
0.39?0.02
0.90?0.02
DPMMGM?
0.45?0.01
0.51?0.07
0.73?0.00
0.23?0.04
0.55?0.02
0.30?0.00
0.52?0.10
0.05?0.00
0.41?0.02
0.59?0.01
Table 1: Comparison for different methods on NMI scores. K: true number of clusters.
stabilises due to the stochastic decrement by nu . We denote the model learned from this heuristic
as DPMMGM. In the case where the true number of clusters is unknown, we can still apply this
strategy, except that the number of clusters K needs to be first inferred from DPGMM. This method
is denoted as DPMMGM? .
Comparison. We measure the quality of clustering results by using the standard normalised mutual
information (NMI) criterion [36]. We compare our DPMMGM with the well established KMeans,
nCut and DPGMM clustering methods3 . All experiments are repeated for five times with random
initialisation. The results are shown in Table 1. Clearly DPMMGM significantly outperforms other
models, achieving the best NMI scores. DPMMGM? , which is not informed of the true number of
clusters, still obtains reasonably high NMI scores, and outperforms the DPGMM model.
6.2
Max-margin Clustering Topic Model
Datasets. We test the MMCTM model on two document datasets: 20NEWS and Reuters-R8 . For
the 20NEWS dataset, we combine the training and test datasets used in [16], which ends up with 20
categories/clusters with roughly balanced cluster sizes. It contains 18,772 documents in total with a
vocabulary size of 61,188. The Reuters-R8 dataset is a subset of the Reuters-21578 dataset4 , with of
8 categories and 7,674 documents in total. The size of different categories is biased, with the lowest
number of documents in a category being 51 while the highest being 2,292.
Comparison We choose L ? {5, 10, 15, 20, 25} documents randomly from each category as the
landmarks, use 80% documents for training and the rest for testing. We set the number of topics
(i.e., T ) to 50, and set the Dirichlet prior in Section 5 to ? = 0.1, ? = 0.01, ? = ?0 = ?1
= 10, as clustering quality is not sensitive to them. For the other hyperparameters related to the
max-margin constraints, e.g., v in the Gaussian prior for ?, the balance parameter c, and the cost
parameter `, instead of doing cross validation which is computationally expensive and not helpful
for our scenario with few labeled data, we simply set v = 0.1, c = 9, ` = 0.1. This is found to
be a good setting and denoted as MMCTM. To test the robustness of this setting, we vary c over
{0.1, 0.2, 0.5, 0.7, 1, 3, 5, 7, 9, 15, 30, 50} and keep v = ` = 0.1 (` and c play similar roles and so
varying one is enough). We choose the best performance out of these parameter settings, denoted
as MMCTM? , which can be roughly deemed as the setting for the optimal performance. We compared MMCTM with state-of-the-art SVM and semi-supervised SVM (S3VM) models. They are
efficiently implemented in [37], and the related parameters are chosen by 5-fold cross validation.
As in [16], raw word frequencies are used as input features. We also compare MMCTM with a
Bayesian baseline?cluster based topic model (CTM) [29], the building block of MMCTM without
the max-margin constraints. Note we did not compare with the standard MedLDA [16] because it
is supervised. We measure the performance by cluster accuracy, which is the proportion of correctly clustered documents. To accelerate MMCTM, we simply initialise it with CTM, and find it
converges surprisingly fast in term of accuracy, e.g., usually within 30 iterations (refer to Appendix
3
We additionally show some comparison with some existing max-margin clustering models in Appendix D.2
on two-cluster data because their code only deals with the case of two clusters. Our method performs best.
4
Downloaded from csmining.org/index.php/r52-and-r8-of-reuters-21578.html.
7
5
10
15
20
25
17.22? 4.2
24.50? 4.5
22.76? 4.2
26.07? 7.2
27.20? 1.5
5
10
15
20
25
41.27? 16.7
42.63? 7.4
39.67? 9.9
58.24? 8.3
51.93? 5.9
MMCTM
MMCTM?
37.13? 2.9 39.36? 3.2
46.99? 2.4 47.91? 2.8
52.80? 1.2 52.49? 1.4
56.10? 1.5 54.44? 2.1
59.15? 1.4 57.45? 1.7
Reuters-R8
56.70? 1.9
54.92? 1.6
55.06? 2.7
56.62? 2.2
55.70? 2.4
57.86? 0.9
56.56? 1.3
57.80? 2.2
59.70? 1.4
61.92? 3.0
78.12? 1.1
80.69? 1.2
83.25? 1.7
85.66? 1.0
84.95? 0.1
79.18? 4.1
80.04? 5.3
85.48? 2.1
82.92? 1.7
86.56? 2.5
80.86? 2.9
83.48? 1.0
86.86? 2.5
83.82? 1.6
88.12? 0.5
SVM
S3VM
20NEWS
78.51? 2.3
79.15? 1.2
81.87? 0.8
73.95? 2.0
82.39? 1.8
Accuracy (%)
CTM
training test
60
40
20
0
10 20 30 50 70 100
Number of topics (#topic)
(a) 20NEWS dataset
Accuracy (%)
L
80
60
40
20
0
10 20 30 50 70 100
Number of topics
(b) Reuters-R8 dataset
Table
2: Clustering acc. (in %). Bold means significantly
different. Figure 4: Accuracy vs. #topic aaaaaaaaaaaaaaaa
60
60
40
40
20
20
0
0
?20
?20
?40
?40
?60
?60
Figure
5: 2-D
tSNE
20NEWS
MMCTM
and 0CTM
viewed
in
?40
?30 (left)
?20
?10
10 (right).
20
30 Best
40
50
?50
?40
?30
?20
?10embedding
0
10
20on 30
40
50 for?50
color. See Appendix D.3 for the results on Reuters-R8 datasets.
D.5). The accuracies are shown in Table 2, and we can see that MMCTM outperforms other models
(also see Appendix D.4), except for SVM when L = 20 on the Reuters-R8 dataset. In addition,
MMCTM performs almost as well as using the optimal parameter setting (MMCTM? ).
Sensitivity to the number of topics (i.e., T ). Note the above experiments simply set T = 50. To
validate the affect of T , we varied T from 10 to 100, and the corresponding accuracies are plotted
In Fig. 4 for the two datasets. In both cases, T = 50 seems to be a good parameter value.
Cluster embedding. We finally plot the clustering results by embedding them into the 2dimensional plane using tSNE [38]. In Fig. 5, it can be observed that compared to CTM, MMCTM
generates well separated clusters with much larger margin between clusters.
7
Conclusions
We propose a robust Bayesian max-margin clustering framework to bridge the gap between maxmargin learning and Bayesian clustering, allowing many Bayesian clustering algorithms to be directly equipped with the max-margin criterion. Posterior inference is done via two data augmentation techniques. Two models from the framework are proposed for Bayesian nonparametric maxmargin clustering and topic model based document clustering. Experimental results show our models significantly outperform existing methods with competitive clustering accuracy.
Acknowledgments
This work was supported by an Australia China Science and Research Fund grant (ACSRF-06283)
from the Department of Industry, Innovation, Climate Change, Science, Research and Tertiary Education of the Australian Government, the National Key Project for Basic Research of China (No.
2013CB329403), and NSF of China (Nos. 61322308, 61332007). NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program.
8
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9
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4,756 | 5,306 | On Integrated Clustering and Outlier Detection
Linsey Pang
University of Sydney
[email protected]
Lionel Ott
University of Sydney
[email protected]
Sanjay Chawla
University of Sydney
[email protected]
Fabio Ramos
University of Sydney
[email protected]
Abstract
We model the joint clustering and outlier detection problem using an extension
of the facility location formulation. The advantages of combining clustering and
outlier selection include: (i) the resulting clusters tend to be compact and semantically coherent (ii) the clusters are more robust against data perturbations and (iii)
the outliers are contextualised by the clusters and more interpretable. We provide a
practical subgradient-based algorithm for the problem and also study the theoretical properties of algorithm in terms of approximation and convergence. Extensive
evaluation on synthetic and real data sets attest to both the quality and scalability
of our proposed method.
1
Introduction
Clustering and outlier detection are often studied as separate problems [1]. However, it is natural
to consider them simultaneously. For example, outliers can have a disproportionate impact on the
location and shape of clusters which in turn can help identify, contextualize and interpret the outliers.
Pelillo [2] proposed a game theoretic definition of clustering algorithms which emphasis the need for
methods that require as little information as possible while being capable of dealing with outliers.
The area of ?robust statistics? studies the design of statistical methods which are less sensitive to the
presence of outliers [3]. For example, the median and trimmed mean estimators are less sensitive
to outliers than the mean. Similarly, versions of Principal Component Analysis (PCA) have been
proposed [4, 5, 6] which are more robust against model mis-specification and outliers. An important
primitive in the area of robust statistics is the notion of Minimum Covariance Determinant (MCD):
Given a set of n multivariate data points and a parameter `, the objective is to identify a subset of
points which minimizes the determinant of the variance-covariance matrix over all subsets of size
n ? `. The resulting variance-covariance matrix can be integrated into the Mahalanobis distance and
used as part of a chi-square test to identify multivariate outliers [7].
In the theoretical computer science literature, similar problems have been studied in the context
of clustering and facility location. For example, Chen [8] has considered and proposed a constant
factor approximation algorithm for the k-median with outliers problem: Given n data points and
parameters k and `, the objective is to remove a set of ` points such that the cost of k-median
clustering on the remaining n ? ` points is minimized. Our model is similar to the one proposed by
Charikar et. al. [9] who have used a primal-dual formulation to derive an approximation algorithm
for the facility location with outlier problem.
More recently, Chawla and Gionis [10] have proposed k-means--, a practical and scalable algorithm
for the k-means with outlier problem. k-means-- is a simple extension of the k-means algorithm and
is guaranteed to converge to a local optima. However, the algorithm inherits the weaknesses of the
1
classical k-means algorithm. These are: (i) the requirement of setting the number of clusters k and
(ii) initial specification of the k centroids. It is well known that the choice of k and initial set of
centroids can have a disproportionate impact on the result.
In this paper we model clustering and outlier detection as an integer programming optimization task
and then propose a Lagrangian relaxation to design a scalable subgradient-based algorithm. The
resulting algorithm discovers the number of clusters and requires as input: the distance (discrepancy)
between pairs of points, the cost of creating a new cluster and the number ` of outliers to select.
The remainder of the paper is structured as follows. In Section 2 we formally describe the problem as an integer program. In Section 3, we describe the Lagrangian relaxation and details of the
subgradient algorithm. The approximation properties of the relaxation and the convergence of the
subgradient algorithm are discussed in Section 4. Experiments on synthetic and real data sets are
the focus of Section 5 before concluding with Section 6. The supplementary section derives an extension of the affinity propagation algorithm [11] to detect outliers (APOC) - which will be used for
comparison.
2
Problem Formulation
The Facility Location with Outliers (FLO) problem is defined as follows [9]. Given a set of data
points with distances D = {dij }, the cluster creation costs ci and the number of outliers `, we define
the task of clustering and outlier detection as the problem of finding the assignments to the binary
exemplar indicators yj , outlier indicators oi and point assignments xij that minimizes the following
objective function:
X
XX
FLO ? min
cj yj +
dij xij ,
(1)
j
i
j
subject to xij ? yj
X
oi +
xij = 1
X
(2)
(3)
j
oi = `
(4)
i
xij , yj , oi ? {0, 1}.
In order to obtain a valid solution a set of constraints have been imposed:
?
?
?
?
(5)
points can only be assigned to valid exemplars Eq. (2);
every point must be assigned to exactly one other point or declared an outlier Eq. (3);
exactly ` outliers have to be selected Eq. (4);
only integer solutions are allowed Eq. (5).
These constraints describe the facility location problem with outlier detection. This formulation will
allow the algorithm to select the number of clusters automatically and implicitly defines outliers as
those points whose presence in the dataset has the biggest negative impact on the overall solution.
The problem is known to be NP-hard and while approximation algorithms have been proposed, when
distances are assumed to be a metric, there is no known algorithm which is practical, scalable, and
comes with solution guarantees [9]. For example, a linear relaxation of the problem and a solution
using a linear programming solver is not scalable to large data sets as the number of variables is
O(n2 ). In fact we will show that the Lagrangian relaxation of the problem is exactly equivalent to a
linear relaxation and the corresponding subgradient algorithm scales to large data sets, has a small
memory footprint, can be easily parallelized, and does not require access to a linear programming
solver.
3
Lagrangian Relaxation of FLO
The Lagrangian relaxation is based on the following recipe and observations: (i) relax (or dualize)
?tough? constraints of the original FLO problem by moving them to the objective; (ii) associate
2
a Lagrange multiplier (?) with the relaxed constraints which intuitively captures the price of constraints not being satisfied; (iii) For any non-negative ?, FLO(?) is a lower-bound on the FLO
problem. As a function of ?, FLO(?) is a concave but non-differentiable; (iv) Use a subgradient
algorithm to maximize FLO(?) as a function of ? in order to close the gap between the primal and
the dual.
P
More specifically, we relax the constraint oi + j xij = 1 for each i and associate a Lagrange
multiplier ?i with each constraint. Rearranging the terms yields:
X
X
XX
FLO(?) = min
(1 ? oi )?i +
cj yj +
(dij ? ?i )xij .
(6)
|
i
{z
}
outliers
subject to xij ? yi
X
oi = `
|
j
i
j
{z
clustering
}
(7)
(8)
i
0 ? xij , yj , oi ? {0, 1}
?i, j
(9)
We can now solve the relaxed problem with a heuristic finding valid assignments that attempt to
minimize Eq. (6) without optimality guarantees [12]. The Lagrange multipliers ? act as a penalty
incurred for constraint violations which we try to minimize. From Eq. (6) we see that the penalty
influences two parts: outlier selection and clustering. The heuristic starts by selecting good outliers
by designating the ` points with largest ? as outliers, as this removes a large part of the penalty. For
the remaining N ? ` points clustering assignments are found by setting xij = 0 for all pairs for
which dij ? ?i ? 0. To select the exemplars we compute:
X
?j = cj +
(dij ? ?i ),
(10)
i:dij ??i <0
which represents the amortized cost of selecting point j as exemplar and assigning points to it. Thus,
if ?j < 0 we select point j as an exemplar and set yj = 1, otherwise we set yj = 0. Finally, we set
xij = yj if dij ? ?i < 0. From this complete assignment found by the heuristic we compute a new
subgradient st and update the Lagrangian multipliers ?t as follows:
X
sti = 1 ?
xij ? oi
(11)
j
?ti
= max(?it?1 + ?t si , 0),
(12)
where ?t is the step size at time t computed as
?t = ?0 pow(?, t) ? ? (0, 1),
(13)
where pow(a, b) = ab . To obtain the final solution we repeat the above steps until the changes
become small enough, at which point we extract a feasible solution. This is guaranteed to converge
if a step function is used for which the following holds [12]:
lim
n??
n
X
t=1
?t = ?
and
lim ?t = 0.
t??
(14)
A high level algorithm description is given in Algorithm 1.
4
Analysis of Lagrangian Relaxation
In this section, we analyze the solution obtained from using the Lagrangian relaxation (LR) method.
Our analysis will have two parts. In the first part, we will show that the Lagrangian relaxation is
exactly equivalent to solving the linear relaxation of the FLO problem. Thus if FLO(IP), FLO(LP)
and FLO(LR) are the optimal value of integer program, linear relaxation and linear programming
solution respectively, we will show that FLO(LR) = FLO(LP). In the second part, we will analyze
the convergence rate of the subgradient method and the impact of outliers.
3
Algorithm 1: LagrangianRelaxation()
Initialize ?0 , x0 , t
while not converged do
st ? ComputeSubgradient(xt?1 )
?t ? ComputeLambda(st )
xt ? FLO(?t )
(solve via heuristic)
t?t+1
end
?
?
?
?
?
?
?
?
?
?
?
A=?
?
?
?
?
?
?
?
?
?
?
?
?1
1
?1
0
?1
1
0
0
?1
1
1
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
Figure 1: Visualization of the building blocks of the A matrix. The top left is a n2 ? n2 identity
matrix which is followed by n row stacked blocks of n ? n negative identity matrices. To the right
of those is another n2 ? n block of zeros. The final row in the block matrix consists of n2 + n zeros
followed by n ones.
4.1
Quality of the Lagrangian Relaxation
P
2
Consider the constraint set L = {(x, y, o) ? Zn +2n |xij ? yj ? i oi ? ` ? i, j}. Then it is well
known that the optimal value of FLO(LR) of the Lagrangian relaxation is equal to the cost of the
following optimization problem [12]:
min
X
cj yj +
j
oi +
X
XX
i
xij dij
(15)
j
xij = 1
(16)
j
conv(L)
(17)
where conv(L) is the convex hull of the set L. We now show that L is integral and therefore
X
2
conv(L) = {(x, y, o) ? Rn +2n |xij ? yj ?
oi ? ` ? i, j}
i
This in turn will imply that FLO(LR) = FLO(LP). In order to show that L is integral, we will establish
that that the constraint matrix corresponding to the set L is totally unimodular (TU). For completeness, we recall several important definitions and theorems from integer program theory [12]:
Definition 1. A matrix A is totally unimodular if every square submatrix of A, has determinant in
the set {?1, 0, 1}.
n
Proposition 1. Given a linear program: min{cT x : Ax ? b, x ? R+
}, let b be the set of integer
vectors for which the problem instance has finite value. Then the optimal solution has integral
solutions if A is totally unimodular.
An equivalent definition of total unimodularity (TU) and often easier to establish is captured in the
following theorem.
Theorem 1. Let A be a matrix. Then A is TU iff for any subset of rows X of A, there exists a
coloring of rows of X, with 1 or -1 such that the weighted sum of every column (while restricting
the sum to rows in X) is -1, 0 or 1.
We are now ready to state and prove the main theorem in this section.
4
Theorem 2. The matrix corresponding to the constraint set L is totally unimodular.
Proof. We need to consider the constraints
xij ? yj ? i, j
n
X
i=1
(18)
oi ? `
(19)
We can express the above constraints in the form Au = b where u is the vector:
T
u = [x11 , . . . , x1n , . . . , xn1 , . . . , xnn , y1 , . . . , yn , o1 , . . . , on ]
The block matrix A is of the form:
I
A=
0
B
0
0
1
(20)
(21)
Here I is an n2 ? n2 identity matrix, B is stack of n matrices of size n ? n where each element of
the stack is a negative identity matrix, and 1 is an 1 ? n block of 10 s. See Figure 1 for a detailed
visualization.
Now to prove that A is TU, we will use Theorem 1. Take any subset X of rows of A. Whether we
color the rows of X by 1 or -1, the column sum (within X) of a column of I will be in {?1, 0, 1}.
A similar argument holds for columns of the block matrix 1. Now consider the submatrix B. We
can express X as
X = ?ni=1,i?B(X,:) Xi
(22)
where each Xi = {r ? X|X(r, i) = ?1}. Given that B is a stack of negative diagonal matrices,
Xi ? Xj = ? for i 6= j. Now consider a column j of B. If Xj has even number of ?10 s, then split
the elements of Xj evenly and color one half as 1 and the other as ?1. Then the sum of column j
(for rows in X) will be 0. On the other hand, if another set of rows Xk has odd number of ?1, color
the rows of Xk alternatively with 1 and ?1. Since Xj and Xk are disjoint their colorings can be
carried out independently. Then the sum of column j will be 1 or ?1. Thus we satisfy the condition
of Theorem 1 and conclude that A is TU.
4.2
Convergence of Subgradient Method
As noted above, the langrangian dual is given by max{FLO(?)|? ? 0}. Furthermore, we use a
gradient ascent method to update the ??s as [?ti ]ni=1 = max(?t?1
+ ?t si , 0) where sti = 1 ?
i
P
j xij ? oi and ?t is the step-size.
Now, assuming that the norm of the subgradients are bounded, i.e., ksk2 ? G and the distance
between the initial point and the optimal set, k?1 ? ?? k2 ? R, it is known that [13]:
Pt
R2 + G2 i=1 ?i2
t
?
|Z(? ) ? Z(? )| ?
Pt
2 i=1 ?i
This can be used to show that to obtain accuracy (for any step size), the number of iterations is
lower bounded by O(RG/2 ), We examine the impact of integrating clustering and outliers on the
convergence rate. We make the following observations:
P
Observation 1. At a given iteration t and for a given data point i, if oti = 1 then j xtij = 0 and
sti = 0 and therefore ?t+1
= ?ti .
i
Observation 2. At a given iteration tP
and for a given data point i, if oti = 0 and the point i is
assigned to exactly one exemplar, then j xtij = 1 and therefore sti = 0 and ?t+1
= ?ti .
i
In conjunction with the algorithm for solving FLO(?) and the above observations we can draw
important conclusions regarding the behavior of the algorithm including (i) the ? values associated
with outliers will be relatively larger and stabilize earlier and (ii) the ? values of the exemplars will
be relatively smaller and will take longer to stabilize.
5
5
Experiments
In this section we evaluate the proposed method on both synthetic and real data and compare it to
other methods. We first present experiments using synthetic data to show quantitative analysis of
the methods in a controlled environment. Then, we present clustering and outlier results obtained
on the MNIST image data set.
We compare our Langrangian Relaxation (LR) based method to two other methods, k-means-- and
an extension of affinity propagation [11] to outlier clustering (APOC) whose details can be found in
the supplementary material. Both LR and APOC require a cost for creating clusters. We obtain this
value as ? ? median(dij ), i.e. the median of all distances multiplied by a scaling factor ? which typically is in the range [1, 30]. The initial centroids required by k-means-- are found using k-means++
[14] and unless specified otherwise k-means-- is provided with the correct number of clusters k.
5.1
Synthetic Data
We use synthetic datasets for controlled performance evaluation and comparison between the different methods. The data is generated by randomly sampling k clusters with m points, each from
d-dimensional normal distributions N (?, ?) with randomly selected ? and ?. To these clusters
we add ` additional outlier points that have a low probability of belonging to any of the selected
clusters. The distance between points is computed using the Euclidean distance. We focus on 2D
distributions as they are more challenging then higher dimensional data due to the separability of
the data.
To assess the performance of the methods we use the following three metrics:
1. Normalized Jaccard index, measures how accurately a method selects the ground truth outliers. It is a coefficient computed between selected outliers O and ground-truth outliers O? .
The final coefficient is normalized with regards to the best possible coefficient obtainable
in the following way:
J(O, O? ) =
|O ? O? | min(|O|, |O? |)
/
.
|O ? O? | max(|O|, |O? |)
(23)
2. Local outlier factor [15] (LOF) measures the outlier quality of a point. We compute the
ratio between the average LOF of O and O? , which indicates the quality of the set of
selected outliers.
3. V-Measure [16] indicates the quality of the overall clustering solution. The outliers are
considered as an additional class for this measure.
For the Jaccard index and V-Measure a value of 1 is optimal, while for the LOF factor a larger value
is better.
Since the number of outliers `, required by all methods, is typically not known exactly we explore
how its misspecification affects the results. We generate 2D datasets with 2000 inliers and 200
outliers and vary the number of outliers ` selected by the methods. The results in Figure 2 show
that in general none of the methods fail completely if the value of ` is misspecified. Looking at the
Jaccard index, which indicates the percentage of true outliers selected, we see that if ` is smaller
then the true number of outliers all methods pick only outliers. When ` is greater then the true
number of outliers we can see a that LR and APOC improve with larger ` while k-means-- does only
sometimes. This is due to the formulation of LR which selects the largest outliers, which APOC
does to some extent as well. This means that if some outliers are initially missed they are more
likely to be selected if ` is larger then the true number of outliers. Looking at the LOF ratio we
can see that selecting more outliers then present in the data set reduces the score somewhat but not
dramatically, which provides the method with robustness. Finally, V-Measure results show that the
overall clustering results remain accurate, even if the number of outliers is misspecified.
We experimentally investigate the quality of the solution by comparing with the results obtained by
solving the LP relaxation using CPLEX. This comparison indicates what quality can be typically expected from the different methods. Additionally, we can evaluate the speed of these approximations.
We evaluate 100 datasets, consisting of 2D Gaussian clusters and outliers, with varying number of
6
0
100 200 300 400
Selected Outliers (`)
V-Measure
0.5
k-means-APOC
LR
1
1
LOF Ratio
Jaccard Index
1
0.5
0
100 200 300 400
Selected Outliers (`)
0.5
0
100 200 300 400
Selected Outliers (`)
2,000
20
10
APOC
LR
Time (s)
APOC
LR
30
Time (s)
Speedup
Figure 2: The impact of number of outliers specified (`) on the quality of the clustering and outlier detection performance. LR and APOC perform similarly with more stability and better outlier
choices compared to k-means--. We can see that overestimating ` is more detrimental to the overall
performance, as indicated by the LOF Ratio and V-Measure, then underestimating it.
1,000
0
0
500
1,000 1,500 2,000
Data Points
(a) Speedup over LP
100
10?2
?4
0
5,000
Data Points
(b) Total Runtime
10,000
10
APOC
LR
0
5,000
10,000
Data Points
(c) Time per Iteration
Figure 3: The graphs shows how the number of points influences different measures. In (a) we
compare the speedup of both LR and APOC over LP. (b) compares the total runtime needed to solve
the clustering problem for LR and APOC . Finally, (c) plots the time required (on a log scale) for a
single iteration for LR and APOC.
points. On average LR obtains 94% ? 5% of the LP objective value, APOC obtains an energy that is
95% ? 4% of the optimal solution found by LP and k-means--, with correct k, obtains 86% ? 12% of
the optimum. These results reinforce the previous analysis; LR and APOC perform similarly while
outperforming k-means--. Next we look at the speed-up of LR and APOC over LP. Figure 3 a) shows
both methods are significantly faster with the speed-up increasing as the number of points increases.
Overall for a small price in quality the two methods obtain a significantly faster solution. k-means-outperforms the other two methods easily with regards to speed but has neither the accuracy nor the
ability to infer the number of clusters directly from the data.
Next we compare the runtime of LR and APOC. Figure 3 b) shows the overall runtime of both
methods for varying number of data points. Here we observe that APOC is faster then LR, however,
by observing the time a single iteration takes, shown in Figure 3 c), we see that LR is much faster
on a per iteration basis compared to APOC. In practice LR requires several times the number of
iterations of APOC, which is affected by the step size function used. Using a more sophisticated
method of computing the step size will provide large gains to LR. Finally, the biggest difference
between LR and APOC is that the latter requires all messages and distances to be held in memory.
This obviously scales poorly for large datasets. Conversely, LR computes the distances at runtime
and only needs to store indicator vectors and a sparse assignment matrix, thus using much less
memory. This makes LR amenable to processing large scale datasets. For example, with single
precision floating point numbers, dense matrices and 10 000 points APOC requires around 2200 MB
of memory while LR only needs 370 MB. Further gains can be obtained by using sparse matrices
which is straight forward in the case of LR but complicated for APOC.
5.2
MNIST Data
The MNIST dataset, introduced by LeCun et al. [17], contains 28 ? 28 pixel images of handwritten
digits. We extract features from these images by representing them as 768 dimensional vectors which
is reduced to 25 dimensions using PCA. The distance between these vectors is computed using the
L2 norm. In Figure 4 we show exemplary results obtained when processing 10 000 digits with the
7
(a) Digit 1
(b) Digit 4
(c) Outliers
Figure 4: Each row in (a) and (b) shows a different appearance of a digit captured by a cluster. The
outliers shown in (c) tend to have heavier then usual stroke, are incomplete or are not recognizable
as a digit.
Table 1: Evaluation of clustering results of the MNIST data set with different cost scaling values ?
for LR and APOC as well as different settings for k-means--. We can see that increasing the cost
results in fewer clusters but as a trade off reduces the homogeneity of the clusters.
LR
?
V-Measure
Homogeneity
Completeness
Clusters
APOC
k-means--
5
15
25
15
n.a.
n.a.
0.52
0.78
0.39
120
0.67
0.74
0.61
13
0.54
0.65
0.46
27
0.53
0.72
0.42
51
0.51
0.50
0.52
10
0.58
0.75
0.47
40
LR method with ? = 5 and ` = 500. Each row in Figure 4 a) and b) shows examples of clusters
representing the digits 1 and 4, respectively. This illustrates how different the same digit can appear
and the separation induced by the clusters. Figure 4 c) contains a subset of the outliers selected by
the method. These outliers have different characteristics that make them sensible outliers, such as:
thick stroke, incomplete, unrecognizable or ambiguous meaning.
To investigate the influence the cluster creation cost has we run the experiment with different values
of ?. In Table 1 we show results for LR with values of cost scaling factor ? = {5, 15, 25}, APOC
with ? = 15 and k-means-- with k = {10, 40}. We can see that LR obtains the best V-Measure score
out of all methods with ? = 15. The homogeneity and completeness scores reflect this as well, while
homogeneity is similar to other settings the completeness value is much better. Looking at APOC we
see that it struggles to obtain the same quality as LR. In the case of k-means-- we can observed how
providing the algorithm with the actual number of clusters results in worse performance compared
to a larger number of clusters which highlights the advantage of methods capable of automatically
selecting the number of clusters from the data.
6
Conclusion
In this paper we presented a novel approach to joint clustering and outlier detection formulated
as an integer program. The method only requires pairwise distances and the number of outliers
as input and detects the number of clusters directly from the data. Using a Lagrangian relaxation
of the problem formulation, which is solved using a subgradient method, we obtain a method that
is provably equivalent to a linear programming relaxation. Our proposed algorithm is simple to
implement, highly scalable, and has a small memory footprint. The clusters and outliers found by
the algorithm are meaningful and easily interpretable.
8
References
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9
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4,757 | 5,307 | Convex Optimization Procedure for Clustering:
Theoretical Revisit
Changbo Zhu
Department of Electrical and Computer Engineering
Department of Mathematics
National University of Singapore
[email protected]
Chenlei Leng
Department of Statistics
University of Warwick
[email protected]
Huan Xu
Department of Mechanical Engineering
National University of Singapore
[email protected]
Shuicheng Yan
Department of Electrical and Computer Engineering
National University of Singapore
[email protected]
Abstract
In this paper, we present theoretical analysis of SON ? a convex optimization
procedure for clustering using a sum-of-norms (SON) regularization recently proposed in [8, 10, 11, 17]. In particular, we show if the samples are drawn from two
cubes, each being one cluster, then SON can provably identify the cluster membership provided that the distance between the two cubes is larger than a threshold
which (linearly) depends on the size of the cube and the ratio of numbers of samples in each cluster. To the best of our knowledge, this paper is the first to provide
a rigorous analysis to understand why and when SON works. We believe this may
provide important insights to develop novel convex optimization based algorithms
for clustering.
1
Introduction
Clustering is an important problem in unsupervised learning that deals with grouping observations
(data points) appropriately based on their similarities or distances [20]. Many clustering algorithms have been proposed in literature, including K-means, spectral clustering, Gaussian mixture models and hierarchical clustering, to solve problems with respect to a wide range of cluster
shapes. However, much research has pointed out that these methods all suffer from instabilities
[3, 20, 16, 15, 13, 19]. Taking K-means as an example, the formulation of K-means is NP-hard and
the typical way to solve it is the Lloyd?s method, which requires randomly initializing the clusters.
However, different initialization may lead to significantly different final cluster results.
1.1
A Convex Optimization Procedure for Clustering
Recently, Lindsten et al. [10, 11], Hocking et al. [8] and Pelckmans et al. [17] proposed the
following convex optimization procedure for clustering, which is termed as SON by Lindsten et al.
[11] (Also called Clusterpath by Hocking et al. [8]),
X
? = arg min kA ? Xk2 + ?
X
kXi? ? Xj? k2 .
(1)
F
X?Rn?p
i<j
Here A is a given data matrix of dimension n ? p where each row is a data point, ? is a tunable
parameter to determine the number of clusters, k ? kF denotes the Frobenius norm and Xi? denotes
the ith row of X.
1
The main idea of the algorithm is that if the i-th sample and the j-th sample belong to the same
? i? and X
? j? should be equal. Intuitively, this is due to the fact that the second term is a
cluster, then X
? to be the same, and can be seen as a generalization
regularization term that enforces the rows of X
? From another
of the fused Lasso penalty [18]. In particular, this penalty seeks to fuse the rows of X.
point of view, the regularization term can be seen as an `1,2 norm, i.e., the sum of `2 norm. Such a
norm is known to encourage block sparse (in this case row-sparse) solutions [1]. Thus, it is expected
? i? ? X
? j? = 0.
that for many (i, j) pairs, X
Mathematically, given c disjoint clusters {C1 , C2 , ? ? ? , Cc } with Ci ? Rp for i = 1, 2, ? ? ? , c, we
define the Cluster Membership Matrix of a given data matrix A as the following.
Definition 1. Given a data matrix A of dimension n ? p, for j = 1, 2, ? ? ? , c, set Ij = {i | Ai? ?
Cj , 1 ? i ? n}. We say that a matrix X of dimension n ? p is a Cluster Membership Matrix of A
if
Xi? = Xj?
if i ? Ik , j ? Ik and 1 ? k ? c
Xi? 6= Xj?
if i ? Im , j ? Il , 1 ? m ? c, 1 ? l ? c and m 6= l.
? of Problem (1) is a Cluster Membership Matrix
Given a data matrix A, if the optimal solution X
?
of A, then we can determine the cluster membership by simply grouping the identical rows of X
together. We say that SON successfully recovers the cluster membership of A in this case.
Notice that unlike previous approaches, SON does not suffer from the instability issue since it is a
strictly convex optimization problem and the solution is fixed once a data matrix A is given. Moreover, SON can easily be adapted to incorporate a priori knowledge of the clustering membership.
For example, if we have prior knowledge about which points are more likely to be in the same
cluster,
P we can appropriately weight the regularization term, i.e., change the regularization term to
? i<j ?ij kXi? ? Xj? k2 for some ?ij > 0.
The main contribution of this paper is to provide theoretic analysis of SON, in particular to derive
sufficient conditions when SON successfully recovers the clustering membership. We show that
if there are two clusters, each of which is a cube, then SON succeeds provided that the distance
between the cubes is larger than a threshold value that depends on the cube size and the ratio of
number of samples drawn in each cluster. Thus, the intuitive argument about why SON works is
made rigorous and mathematically solid. To the best of our knowledge, this is the first attempt to
theoretically quantify why and when SON succeeds.
Related Work: we briefly review the related works on SON. Hocking et al. [8] proposed SON,
arguing that it can be seen as a generalization of hierarchical clustering, and presented via numerical
simulations several situations in which SON works while K-means and average linkage hierarchical
clustering fail. They also developed R package called ?clusterpath? which can be used to solve
Problem (1). Independently, Lindsten et al. [10, 11] derived SON as a convex relaxation of Kmeans clustering. In the algorithmic aspect, Chi et al. [6] developed two methods to solve Problem
(1), namely, Alternating Direction Method of Multipliers (ADMM) and alternating minimization
algorithm (AMA). Marchetti et al. [14] generalized SON to the high-dimensional and noisy cases.
Yet, in all these works, no attempt has been made to study rigorously why and when SON succeeds.
Notation: in this paper, matrices are denoted by upper case boldface letters (e.g. A, B), sets are
denoted by blackboard bold characters (e.g. R, I, C) and operators are denoted by Fraktur characters
(e.g. D, M). Given a matrix A, we use Ai? to denote its ith row, and A?j to denote its jth column.
Its (i, j)th entry is denoted by Ai,j . Two norms are used: we use k ? kF to denote the Frobenius
norm and k ? k2 to denote the l2 norm of a vector. The space spanned by the rows of A is denoted
by Row(A). Moreover, given a matrix A of dimension n ? p and a function f : Rp 7? Rq , we use
the notation f (A) to denote the matrix whose ith row is f (Ai? ).
2
Main Result
In this section we present our main theoretic result ? a provable guarantee when SON succeeds in
identifying cluster membership.
2
2.1
Preliminaries
We first define some operators that will be frequently used in the remainder of the paper.
Definition 2. Given any two matrices E of dimension n1 ? p and F of dimension n2 ? p, define
the difference operator D1 on E, D2 on the two matrices E, F and D on the matrix constructed by
concatenating E and F vertically as
?
?
?
?
E1? ? E2?
E1? ? F1?
? E1? ? F2? ?
? E1? ? E3? ?
?
?
?
?
..
..
?
?
?
?
?
?
?
?
.
.
?
?
?
?
!
? E1? ? Fn2 ? ?
? E1? ? En1 ? ?
D1 (E)
? E ?F ?
?
? E ?E
E
2?
3?
? , D2 (E, F) = ? 2?
1? ? and D(
D1 (F)
.
)=
D1 (E) = ?
?
?
?
?
..
..
F
?
?
?
?
D
(E,
F)
2
.
.
?
?
?
?
?E ?F ?
?
? E ?E
2?
n1 ?
n2 ? ?
? 2?
?
?
?
?
?
?
..
..
?
?
?
?
.
.
E(n1 ?1)? ? En1 ?
En1 ? ? Fn2 ?
In words, the operator D1 calculates the difference between every two rows of a matrix and lists the
results in the order indicated in the definition. Similarly, given two matrices E and F, the operator
D2 (E, F) calculates the difference of any two rows between E and F, one from E and the other from
F. We also define the following average operation which calculates the mean of the row vectors.
Definition 3. Given any matrix E of dimension n ? p, define the average operator on E as
M(E) =
n
1 X
(
Ei? ).
n i=1
Definition 4. A matrix E is called column centered if M(E) = 0.
2.2
Theoretical Guarantees
Our main result essentially says that when there are two clusters, each of which is a cube, and
they are reasonably separated away from each other, then SON successfully recovers the cluster
membership. We now make this formal. For i = 1, 2, suppose Ci ? Rp is a cube with center
(?i1 , ?i2 , ? ? ? , ?ip ) and edge length si = 2(?i1 , ?i2 , ? ? ? , ?ip ) , i.e.,
Ci = [?i1 ? ?i1 , ?i1 + ?i1 ] ? ? ? ? ? [?ip ? ?ip , ?ip + ?ip ].
Definition 5. The distance d1,2 between cubes C1 and C2 is
d1,2 , inf{kx ? yk2 | x ? C1 , y ? C2 }.
Definition 6. The weighted size w1,2 with respect to C1 , C2 , n1 and n2 is defined as
2n2 (n1 ? 1)
2n1 (n2 ? 1)
w1,2 = max
+ 1 ks1 k2 ,
+ 1 ks2 k2 .
n21
n22
Theorem 1. Given a column centered data matrix A of dimension n ? p, where each row is arbitrarily picked from either cube C1 or cube C2 and there are totally ni rows chosen from Ci for
i = 1, 2, if w1,2 < d1,2 , then by choosing the parameter ? ? R such that w1,2 < n2 ? < d1,2 , we
have the following:
1. SON can correctly determine the cluster membership of A;
2. Rearrange the rows of A such that
?
?
Ai1?
1
? Ai2? ?
A
?
?
i
A=
and
A
=
? .. ? ,
A2
? . ?
Aini ?
3
(2)
where for i = 1, 2 and j = 1, 2, ? ? ? , ni , Aij? = (Aij,1 , Aij,2 , ? ? ? , Aij,p ) ? Ci . Then, the
? of Problem (1) is given by
optimal solution X
?
n?
? n2
1
?
D2 (A1 , A2 ) ,
if Ai? ? C1 ;
1
2
n1 +n2
2kM(D2 (A ,A ))k2 M
?
Xi? =
n?
1
2
? ? n1
if Ai? ? C2 .
n1 +n2 1 ? 2kM(D2 (A1 ,A2 ))k2 M D2 (A , A ) ,
The theorem essentially states that we need d1,2 to be large and w1,2 to be small for correct determination of the cluster membership of A. This is indeed intuitive. Notice that d1,2 is the distance
between the cubes and w1,2 is a constant that depends on the size of the cube as well as the ratio
between the samples in each cube. Obviously, if the cubes are too close with each other, i.e., d1,2 is
small, or if the sizes of the clusters are too big compared to their distance, it is difficult to determine
the cluster membership correctly. Moreover, when n1 n2 or n1 n2 , w1,2 is large, and the
theorem states that it is difficult to determine the cluster membership. This is also well expected,
since in this case one cluster will be overwhelmed by the other, and hence determining where the
data points are chosen from becomes problematic.
The assumption in Theorem 1 that the data matrix A is column centered can be easily relaxed, using
the following proposition which states that the result of SON is invariant to any isometry operation.
Definition 7. An isometry of Rn is a function f : Rn ? Rn that preserves the distance between
vectors, i.e.,
kf (u) ? f (w)k2 = ku ? wk2 , ? u, w ? Rn .
Proposition 1. (Isometry Invariance) Given a data matrix A of dimension n ? p where each row
is chosen from some cluster Ci , i = 1, 2, ? ? ? , c, and f (?) an isometry of Rp , we have
X
? = arg min kA ? Xk2 + ?
X
kXi? ? Xj? k2
F
n?p
X?R
i<j
? = arg min kf (A) ?
??f (X)
X?Rn?p
Xk2F
+?
X
kXi? ? Xj? k2 .
i<j
This further implies that if SON successfully determines the cluster membership of A, then it also
successfully determines the cluster membership of f (A).
3
Kernelization
SON can be easily kernelized as we show in this section. In the kernel clustering setup, instead
of clustering {Ai? } such that points within a cluster are closer in the original space, we want to
cluster {Ai? } such that points within a cluster are closer in the feature space. Mathematically, this
means we map Ai? to a Hilbert space H (the feature space) by the feature mapping function ?(?)
and perform clustering on {?(Ai? )}.
Notice that we can write Problem (1) in terms of the inner product hAi? , Aj? i , hAi? , Xj? i and
hXi? , Xj? i. Thus, for SON in the feature space, we only need to replace all these inner products
by h?(Ai? ), ?(Aj? )i , h?(Ai? ), Xj? i and hXi? , Xj? i. Thus, SON in the feature space can be formulated as
n
X
? = arg min
X
(h?(Ai? ), ?(Ai? )i ? 2 h?(Ai? ), Xi? i + hXi? , Xi? i)
n?q
X?R
i=1
+?
Xq
(3)
hXi? , Xi? i ? 2 hXi? , Xj? i + hXj? , Xj? i.
i<j
We have the following representation theorem about the optimal solution of (3).
Theorem 2. (Representation Theorem) Each row of the optimal solution of Problem (3) can be
written as a linear combination of rows of A, i.e.,
? i? =
X
n
X
aij ?(Aj? ).
j=1
4
Thus, to solve SON in the feature space reduces to finding the optimal weight {aij }. Define the
kernel function as K(x, y) = h?(x), ?(y)i. Then Problem (3) is equivalent to
!
n
n
n X
n
X
X
X
K(Ai? , Ai? ) ? 2
min
aik K(Ai? , Ak? ) +
aik ail K(Ak? , Al? )
{aij }
i=1
k=1
k=1 l=1
v
u n n
X uX
X
t
+?
K(Ak? , Al? )(aik ail ? 2aik ajl + ajk ajl ),
i<j
(4)
k=1 l=1
which is a second order cone program since the kernel is positive semi-definite. Notice that this
implies that solving SON in the feature space only requires knowing the kernel function rather than
the feature mapping ?(?).
4
Proof
We sketch the proof of Theorem 1 here. The detailed proof is given in the supplementary material.
4.1
Preliminaries
We first introduce some notations useful in the proof. We use In to denote an identity matrix of
dimension n ? n and use 1m?n to denote a matrix of dimension m ? n with all entries being 1.
Similarly, we use 0m?n to denote a matrix of dimension m ? n with all entries being 0.
We now define some special matrices. Let Hn denote a matrix of dimension (n ? 1) ? n which
is constructed by concatenating 1(n?1)?1 and ?In?1 horizontally, i.e., Hn = (1(n?1)?1 ?
In?1 ). For i = 1, 2, ? ? ? , n ? 2, we first concatenate matrices Hn?i and 0(n?1?i)?i horizontally to form a matrix (0(n?1?i)?i Hn?i ). Then, we construct Rn by concatenating
{Hn , (0(n?2)?1 Hn?1 ), ? ? ? , (01?(n?2) H2 )} vertically, i.e.,
?
?
Hn
?0(n?2)?1 Hn?1 ?
?0
?
(n?3)?2 Hn?2 ? .
Rn , ?
?
?
.
?
?
..
01?(n?2) H2
We concatenate m copies of ?In vertically to form a new matrix and denote it by Wmn?n . Let
Gm,n,i denote an m ? n dimensional matrix where the entries of the ith column all equal 1 and
all the other entries equal 0, i.e., Gm,n,i , (0m?(i?1) 1m?1 0m?(n?i) ). Then, we concatenate
{Gm,n,1 , Gm,n,2 , ? ? ? , Gm,n,n } vertically and denote it by Smn?n , i.e.,
?
?
?
?
?In
Gm,n,1
??In ?
? Gm,n,2 ?
?
?
?
Wmn?n , ?
? ... ? , Smn?n , ? ... ? .
?In
Finally, set ? ,
?
Rn1 ?1
? 0n
?
( 22 )?(n1 ?1)
S(n1 ?1)n2 ?(n1 ?1)
4.2
I(n1 ?1)
2
0(n2 )?(n1 ?1)
2
2
0(n1 ?1)n2 ?(n1 ?1)
2
Gm,n,n
0(n1 ?1)?(n2 )
2
2
I(n2 )
2
0(n1 ?1)n2 ?(n2 )
2
0(n1 ?1)?n2
2
R n2
W(n1 ?1)n2 ?n2
?
0(n1 ?1)?(n1 ?1)n2
2
0(n2 )?(n1 ?1)n2 ?
?.
2
I(n1 ?1)n2
Proof sketch of Theorem 1
The proof of Theorem 1 is based on the idea of ?lifting?. That is, we project Problem (1) into a
higher dimensional space (in particular, from n rows to n(n ? 1)/2 rows), which then allows us to
separate the regularization term into the sum of l2 norm of each row. Although this brings additional
5
linear constraints to the formulation, it facilitates the analysis. In the following, we divide the proof
into 3 steps and explain the main idea of each step.
Step 1: In this step, we derive an equivalent form of Problem (1) and give optimality conditions.
For convenience, set B(1,2) = D2 (A1 , A2 ), B1 = D1 (A1 ), B2 = D1 (A2 ) and V = {y ?
n
R( 2 ) | ?y = 0}. The following lemmas show that we can lift the original problem into an equivalent
problem that is easier to analyze.
? of problem (1) is
Lemma 1. If the data matrix A is column centered, then the optimal solution X
?
?
also column centered. Further more, set B = D(A) and Y = D(X), we have
n(n?1)
2
X 1
? i? k2 .
kBi? ? Y
2
n
i=1
? 2 =
kA ? Xk
F
Lemma 2. Given a column centered data matrix A, set B = D(A) and S = {Z ?
n
? is the optimal solution to Problem (1) iff
R( 2 )?p | ?Z?j = 0, 1 ? j ? p}. Then, X
n(n?1)
2
? = arg min
D(X)
Y?S
X
i=1
1
( kBi? ? Yi? k22 + ? kYi? k2 ).
n
(5)
? is the membership matrix of A by solving Problem (5). ComThus, we can determine whether X
pared to Problem (1), Problem (5) is more amenable to analyze as it is the sum of separable equations. That is, for i = 1, 2, ? ? ? , n(n?1)
, we can minimize each n1 kBi? ? Yi? k22 + ? kYi? k2 individ2
ually with the additional constraint ?Y = 0. Following standard convex analysis (Page 303 of [2]),
? and ?
? are an optimal primal and dual solution pair of Problem (5) if and only if
Y
and
? ?j ? V, (?
? ?j )T ? V? , j = 1, 2, ? ? ? , p,
Y
(6)
1
n
2
T
?
?
Yi? ? arg minp ( kBi? ? yk2 + ?kyk2 ? y?i? ), i = 1, 2, ? ? ? ,
.
y?R n
2
(7)
? Since A is constructed by concatenating matrices A1 and
Step 2: In this step, we construct ?.
? to be concatenated by two matrices vertically. Due to the fact that
A2 vertically, we also expect X
?
?
? and ?
? as the following
Y = D(X), for 1 ? l ? p, we write Y
? ?1 ?
? ?1 ?
??l
Y?l
? ?l = ? ?
? ?l = ? Y
? 2 ? and Y
?2 ?
?
?l
?l
(1,2)
(1,2)
?
?
??l
Y?l
n
?i , Y
? i ? R( 2i ) for i = 1, 2 and ?
? (1,2) , Y
? (1,2) ? Rn1 n2 , which are determined below.
where ?
?l
?l
?l
?l
? ?l )T ? V? is equivaBy the structure of ?, after some algebraic operations, it can be shown that (?
lent to the following equalities that hold,
(1,2)
? 1 = ?ST
?
RTn1 ?
?l
n1 n2 ?n1 ??l
(1,2)
? 2 = ?WT
?
RTn2 ?
?l
n1 n2 ?n2 ??l
,
.
(8)
? (1,2) . Set
We now construct ?
2 (1,2)
(1,2)
? (1,2)
M B
? Bm?
, 1 ? m ? n1 n2 .
?
m? =
n
(9)
? (1,2) is now fixed, we can bound the right hand sides of the two equalities in (8). In order to
Since ?
? 1 and ?
? 2 , we need the following lemma.
bound the entries of ?
?l
?l
n
P
Lemma 3. Given cn ? Rn , i.e., cn = (c1 , c2 , ? ? ? , cn )T , such that
ci = 0 and ?b ? R, |ci | ? b,
i=1
then ?x ? R
n(n?1)
2
, such that kxk? ?
2
nb
and RTn x = cn .
6
n1 = 25 & n2 = 50
n1 = n2 = 25
n1 = 25 & n2 = 75
40
Theoretical bounds w1,2
Empirical performance d?1,2
Theoretical bounds w1,2
Empirical performance d?1,2
30
Theoretical bounds w1,2
Empirical performance d?1,2
60
30
20
40
20
10
0
20
10
2
4
6
8
10
0
2
4
6
ksk2
ksk2
8
10
0
2
4
6
8
10
ksk2
Figure 1: Theoretical bounds and empirical performance. This figure illustrates the case in which
n1 , n2 are constants and ksk2 is increasing.
Then, because we can bound the right hand sides of the two equalities of (8), by using Lemma 3, we
?1,?
? 2 satisfying (8) such that the following holds
can show that there exist ?
?l
?l
(n1 ? 1)
2
? 2 k? ? 2 (n1 ) (n2 ? 1) (4?2l ).
(n2 )
(4?1l ) and k?
?l
n
n21
n
n22
? of dimension n ? p such that
To summarize this step, we have constructed ?
2
? 1 2
? ,?
? satisfies (10), 1 ? l ? p,
??
?l
?l
2 (1,2)
(1,2)
??
? (1,2)
M B
? Bm?
, 1 ? m ? n1 n2 .
m? =
n
? 1 k? ?
k?
?l
? Set
Step 3: Finally, we construct Y.
?
?1 = Y
? 2 = 0, 1 ? l ? p,
?
?Y
?l
?l
?
? (1,2)
?
?
?Ym?
=
n?
1?
2kM B(1,2) k2
!
(10)
M B(1,2) , 1 ? m ? n1 n2 .
? and Y
? constructed, it is easy to checked that conditions
Choosing w1,2 < n2 ? < d1,2 , according to ?
? and Y
? are an optimal primal and dual solution pair of Problem (5).
(6) and (7) are satisfied. So ?
5
Experiments
We now report some numerical experimental results. The empirical performance of SON has been
reported in numerous works [8, 10, 11]. It has been shown that SON outperforms traditional clustering methods like K-means in many situations. As such, we do not reproduce such results. Instead,
we conduct experiments to validate our theoretic results.
Recall that Theorem 1 states that when samples are drawn from two cubes, SON guarantees to
successfully recover the cluster membership if the distance between cubes is larger than a threshold
which is linear to the cube size ksi k and the ratio between n1 and n2 . To validate this, we randomly
draw a data matrix A where each row belongs to one of the two cubes, and find numerically the
largest distance d?1,2 between the cubes where the cluster membership is not correctly recovered.
Clearly, d?1,2 provides an empirical estimator of the minimal distance needed to successfully recover
the cluster membership. We compare the theoretic bound w1,2 with the empirical performance d?1,2
to validate our theorem. The specific procedures of the experiments are as follows.
1. Choose two cubes C1 and C2 from space Rp with size s1 = 2(?11 , ?12 , ? ? ? , ?1p ) and
s2 = 2(?21 , ?22 , ? ? ? , ?2p ), and the distance between C1 and C2 is d.
2. Choose arbitrarily n1 points from C1 and n2 points from C2 and form the data matrix Ad
of dimension n ? p. Repeat and sample m data matrices {Ad1 , Ad2 , ? ? ? , Adm }.
7
ks1 k2 = ks2 k2 = 2
ks1 k2 = ks2 k2 = 1
ks1 k2 = ks2 k2 = 3
20
50
Theoretical bounds w1,2
Empirical performance d?1,2
Theoretical bounds w1,2
Empirical performance d?1,2
30
15
20
Theoretical bounds w1,2
Empirical performance d?1,2
40
30
10
20
10
5
0
10
2
4
6
8
n2
n1
0
2
4
6
n2
n1
8
0
2
4
6
8
n2
n1
Figure 2: Theoretical bounds and empirical performance. This figure illustrates the case in which
ks1 k2 ,ks2 k2 are constants and the ratio nn12 is increasing.
3. Repeat for different d. Set
d?1,2 = max{d|?1 ? j ? m s.t. SON fails to determine the cluster membership of Adj }.
4. Repeat for different cube sizes ks1 k2 and ks2 k2 .
5. Repeat for different sample numbers n1 and n2 .
In the experiments, we focus on the samples chosen from R2 , i.e., p = 2, and use synthetic data to
obtain the empirical performance. The results are shown in Figure 1 and 2. Figure 1 presents the
situation where n1 and n2 are fixed and the cube sizes are increasing. In particular,
the two cubes are
?
both of size l ? l, i.e., both with edge length (l, l). Thus we have ksk2 = 2l. Clearly, we can see
that the empirical performance and the theoretical bounds are both linearly increasing with respect
to ksk2 , which implies that our theoretical results correctly predict how the performance of SON
depends on ksk2 . Figure 2 presents the situation in which ksk1 and ksk2 are fixed, while the ratio
n2
n1 is changing. Again, we observe that both the empirical performance and the theoretical bounds
are linearly increasing with respect to nn21 , which implies that our theoretical bounds w1,2 predict the
correct relation between the performance of SON and nn12 .
6
Conclusion
In this paper, we provided theoretical analysis for the recently presented convex optimization procedure for clustering, which we term as SON. We showed that if all samples are drawn from two
clusters, each being a cube, then SON is guaranteed to successfully recover the cluster membership
provided that the distance between the two cubes is greater than the ?weighted size? ? a term that
linearly depends on the cube size and the ratio between the numbers of the samples in each cluster.
Such linear dependence is also observed in our numerical experiment, which demonstrates (at least
qualitatively) the validity of our results.
The main thrust of this paper is to explore using techniques from high-dimensional statistics, in
particular regularization methods that extract low-dimensional structures such as sparsity or lowrankness, to tackle clustering problems. These techniques have recently been successfully applied to
graph clustering and subspace clustering [4, 7, 12, 5, 9], but not so much to distance-based clustering
tasks with the only exception of SON, to the best of our knowledge. This paper is the first attempt
to provide a rigorous analysis to derive sufficient conditions when SON succeeds. We believe this
not only helps to understand why SON works in practice as shown in previous works [8, 10, 11], but
also provides important insights to develop novel algorithms based on high-dimensional statistics
tools for clustering tasks.
Acknowledgments
The work of H. Xu was partially supported by the Ministry of Education of Singapore through
AcRF Tier Two grant R-265-000-443-112. This work is also partially supported by the grant from
Microsoft Research Asia with grant number R-263-000-B13-597.
8
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9
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4,758 | 5,308 | Greedy Subspace Clustering
Dohyung Park
Dept. of Electrical and Computer Engineering
The University of Texas at Austin
[email protected]
Constantine Caramanis
Dept. of Electrical and Computer Engineering
The University of Texas at Austin
[email protected]
Sujay Sanghavi
Dept. of Electrical and Computer Engineering
The University of Texas at Austin
[email protected]
Abstract
We consider the problem of subspace clustering: given points that lie on or near
the union of many low-dimensional linear subspaces, recover the subspaces. To
this end, one first identifies sets of points close to the same subspace and uses the
sets to estimate the subspaces. As the geometric structure of the clusters (linear
subspaces) forbids proper performance of general distance based approaches such
as K-means, many model-specific methods have been proposed. In this paper,
we provide new simple and efficient algorithms for this problem. Our statistical
analysis shows that the algorithms are guaranteed exact (perfect) clustering performance under certain conditions on the number of points and the affinity between
subspaces. These conditions are weaker than those considered in the standard
statistical literature. Experimental results on synthetic data generated from the
standard unions of subspaces model demonstrate our theory. We also show that
our algorithm performs competitively against state-of-the-art algorithms on realworld applications such as motion segmentation and face clustering, with much
simpler implementation and lower computational cost.
1
Introduction
Subspace clustering is a classic problem where one is given points in a high-dimensional ambient
space and would like to approximate them by a union of lower-dimensional linear subspaces. In
particular, each subspace contains a subset of the points. This problem is hard because one needs to
jointly find the subspaces, and the points corresponding to each; the data we are given are unlabeled.
The unions of subspaces model naturally arises in settings where data from multiple latent phenomena are mixed together and need to be separated. Applications of subspace clustering include motion
segmentation [23], face clustering [8], gene expression analysis [10], and system identification [22].
In these applications, data points with the same label (e.g., face images of a person under varying
illumination conditions, feature points of a moving rigid object in a video sequence) lie on a lowdimensional subspace, and the mixed dataset can be modeled by unions of subspaces. For detailed
description of the applications, we refer the readers to the reviews [10, 20] and references therein.
There is now a sizable literature on empirical methods for this particular problem and some statistical analysis as well. Many recently proposed methods, which perform remarkably well and have
theoretical guarantees on their performances, can be characterized as involving two steps: (a) finding a ?neighborhood? for each data point, and (b) finding the subspaces and/or clustering the points
given these neighborhoods. Here, neighbors of a point are other points that the algorithm estimates
to lie on the same subspace as the point (and not necessarily just closest in Euclidean distance).
1
Algorithm
What is guaranteed
Subspace
condition
SSC [4, 16]
LRR [14]
SSC-OMP [3]
TSC [6, 7]
LRSSC [24]
Correct neighborhoods
Exact clustering
Correct neighborhoods
Exact clustering
Correct neighborhoods
NSN+GSR
NSN+Spectral
Conditions for:
Fully random model
log(n/d)
O( log(nL) )
None
No intersection
No intersection
None
None
d
p
=
d
p
d
p
1
= O( log(nL)
)
1
= O( log(nL)
)
Exact clustering
None
Exact clustering
None
d
p
d
p
-
log n
= O( log(ndL)
)
log n
= O( log(ndL)
)
Semi-random
pmodel
log(n/d)
max a? = O( log(nL) )
1
max a? = O( log(nL)
)
q
log n
max a? = O( (log dL)?log(ndL)
)
-
Table 1: Subspace clustering algorithms with theoretical guarantees. LRR and SSC-OMP have only
deterministic guarantees, not statistical ones. In the two standard statistical models, there are n data
points on each of L d-dimensional subspaces in Rp . For the definition of max a?, we refer the
readers to Section 3.1.
Our contributions: In this paper we devise new algorithms for each of the two steps above; (a) we
develop a new method, Nearest Subspace Neighbor (NSN), to determine a neighborhood set for each
point, and (b) a new method, Greedy Subspace Recovery (GSR), to recover subspaces from given
neighborhoods. Each of these two methods can be used in conjunction with other methods for the
corresponding other step; however, in this paper we focus on two algorithms that use NSN followed
by GSR and Spectral clustering, respectively. Our main result is establishing statistical guarantees
for exact clustering with general subspace conditions, in the standard models considered in recent
analytical literature on subspace clustering. Our condition for exact recovery is weaker than the
conditions of other existing algorithms that only guarantee correct neighborhoods1 , which do not
always lead to correct clustering. We provide numerical results which demonstrate our theory. We
also show that for the real-world applications our algorithm performs competitively against those
of state-of-the-art algorithms, but the computational cost is much lower than them. Moreover, our
algorithms are much simpler to implement.
1.1
Related work
The problem was first formulated in the data mining community [10]. Most of the related work in
this field assumes that an underlying subspace is parallel to some canonical axes. Subspace clustering for unions of arbitrary subspaces is considered mostly in the machine learning and the computer
vision communities [20]. Most of the results from those communities are based on empirical justification. They provided algorithms derived from theoretical intuition and showed that they perform
empirically well with practical dataset. To name a few, GPCA [21], Spectral curvature clustering
(SCC) [2], and many iterative methods [1, 19, 26] show their good empirical performance for subspace clustering. However, they lack theoretical analysis that guarantees exact clustering.
As described above, several algorithms with a common structure are recently proposed with both
theoretical guarantees and remarkable empirical performance. Elhamifar and Vidal [4] proposed an
algorithm called Sparse Subspace Clustering (SSC), which uses `1 -minimization for neighborhood
construction. They proved that if the subspaces have no intersection2 , SSC always finds a correct
neighborhood matrix. Later, Soltanolkotabi and Candes [16] provided a statistical guarantee of the
algorithm for subspaces with intersection. Dyer et al. [3] proposed another algorithm called SSCOMP, which uses Orthogonal Matching Pursuit (OMP) instead of `1 -minimization in SSC. Another
algorithm called Low-Rank Representation (LRR) which uses nuclear norm minimization is proposed by Liu et al. [14]. Wang et al. [24] proposed an hybrid algorithm, Low-Rank and Sparse Subspace Clustering (LRSSC), which involves both `1 -norm and nuclear norm. Heckel and B?olcskei [6]
presented Thresholding based Subspace Clustering (TSC), which constructs neighborhoods based
on the inner products between data points. All of these algorithms use spectral clustering for the
clustering step.
The analysis in those papers focuses on neither exact recovery of the subspaces nor exact clustering
in general subspace conditions. SSC, SSC-OMP, and LRSSC only guarantee correct neighborhoods which do not always lead to exact clustering. LRR guarantees exact clustering only when
1
2
By correct neighborhood, we mean that for each point every neighbor point lies on the same subspace.
By no intersection between subspaces, we mean that they share only the null point.
2
the subspaces have no intersections. In this paper, we provide novel algorithms that guarantee exact
clustering in general subspace conditions. When we were preparing this manuscript, it is proved
that TSC guarantees exact clustering under certain conditions [7], but the conditions are stricter than
ours. (See Table 1)
1.2
Notation
There is a set of N data points in Rp , denoted by Y = {y1 , . . . , yN }. The data points are lying on
or near a union of L subspaces D = [L
i=1 Di . Each subspace Di is of dimension di which is smaller
than p. For each point yj , wj denotes the index of the nearest subspace. Let Ni denote the number
PN
of points whose nearest subspace is Di , i.e., Ni = j=1 Iwj =i . Throughout this paper, sets and
subspaces are denoted by calligraphic letters. Matrices and key parameters are denoted by letters
in upper case, and vectors and scalars are denoted by letters in lower case. We frequently denote
the set of n indices by [n] = {1, 2, . . . , n}. As usual, span{?}
Pn denotes a subspace spanned by a
set of vectors. For example, span{v1 , . . . , vn } = {v : v = i=1 ?i vi , ?1 , . . . , ?n 2 R}. ProjU y
is defined as the projection of y onto subspace U . That is, ProjU y = arg minu2U ky uk2 . I{?}
L
denotes the indicator function which is one if the statement is true and zero otherwise. Finally,
denotes the direct sum.
2
Algorithms
We propose two algorithms for subspace clustering as follows.
? NSN+GSR : Run Nearest Subspace Neighbor (NSN) to construct a neighborhood matrix
W 2 {0, 1}N ?N , and then run Greedy Subspace Recovery (GSR) for W .
? NSN+Spectral : Run Nearest Subspace Neighbor (NSN) to construct a neighborhood matrix W 2 {0, 1}N ?N , and then run spectral clustering for Z = W + W > .
2.1
Nearest Subspace Neighbor (NSN)
NSN approaches the problem of finding neighbor points most likely to be on the same subspace in
a greedy fashion. At first, given a point y without any other knowledge, the one single point that is
most likely to be a neighbor of y is the nearest point of the line span{y}. In the following steps, if
we have found a few correct neighbor points (lying on the same true subspace) and have no other
knowledge about the true subspace and the rest of the points, then the most potentially correct point
is the one closest to the subspace spanned by the correct neighbors we have. This motivates us to
propose NSN described in the following.
Algorithm 1 Nearest Subspace Neighbor (NSN)
Input: A set of N samples Y = {y1 , . . . , yN }, The number of required neighbors K, Maximum
subspace dimension kmax .
Output: A neighborhood matrix W 2 {0, 1}N ?N
yi
yi /kyi k2 , 8i 2 [N ]
. Normalize magnitudes
for i = 1, . . . , N do
. Run NSN for each data point
Ii
{i}
for k = 1, . . . , K do
. Iteratively add the closest point to the current subspace
if k ? kmax then
U
span{yj : j 2 Ii }
end if
j?
arg maxj2[N ]\Ii kProjU yj k2
Ii
Ii [ {j ? }
end for
Wij
Ij2Ii or yj 2U , 8j 2 [N ]
. Construct the neighborhood matrix
end for
NSN collects K neighbors sequentially for each point. At each step k, a k-dimensional subspace U
spanned by the point and its k 1 neighbors is constructed, and the point closest to the subspace is
3
newly collected. After k kmax , the subspace U constructed at the kmax th step is used for collecting neighbors. At last, if there are more points lying on U , they are also counted as neighbors. The
subspace U can be stored in the form of a matrix U 2 Rp?dim(U ) whose columns form an orthonormal basis of U . Then kProjU yj k2 can be computed easily because it is equal to kU > yj k2 . While
a naive implementation requires O(K 2 pN 2 ) computational cost, this can be reduced to O(KpN 2 ),
and the faster implementation is described in Section A.1. We note that this computational cost is
much lower than that of the convex optimization based methods (e.g., SSC [4] and LRR [14]) which
solve a convex program with N 2 variables and pN constraints.
NSN for subspace clustering shares the same philosophy with Orthogonal Matching Pursuit (OMP)
for sparse recovery in the sense that it incrementally picks the point (dictionary element) that is
the most likely to be correct, assuming that the algorithms have found the correct ones. In subspace
clustering, that point is the one closest to the subspace spanned by the currently selected points, while
in sparse recovery it is the one closest to the residual of linear regression by the selected points. In
the sparse recovery literature, the performance of OMP is shown to be comparable to that of Basis
Pursuit (`1 -minimization) both theoretically and empirically [18, 11]. One of the contributions of
this work is to show that this high-level intuition is indeed born out, provable, as we show that NSN
also performs well in collecting neighbors lying on the same subspace.
2.2
Greedy Subspace Recovery (GSR)
Suppose that NSN has found correct neighbors for a data point. How can we check if they are
indeed correct, that is, lying on the same true subspace? One natural way is to count the number
of points close to the subspace spanned by the neighbors. If they span one of the true subspaces,
then many other points will be lying on the span. If they do not span any true subspaces, few points
will be close to it. This fact motivates us to use a greedy algorithm to recover the subspaces. Using
the neighborhood constructed by NSN (or some other algorithm), we recover the L subspaces. If
there is a neighborhood set containing only the points on the same subspace for each subspace, the
algorithm successfully recovers the unions of the true subspaces exactly.
Algorithm 2 Greedy Subspace Recovery (GSR)
Input: N points Y = {y1 , . . . , yN }, A neighborhood matrix W 2 {0, 1}N ?N , Error bound ?
? = [L D
?
Output: Estimated subspaces D
?1 , . . . , w
?N
l=1 l . Estimated labels w
yi
yi /kyi k2 , 8i 2 [N ]
. Normalize magnitudes
Wi
Top-d{yj : Wij = 1}, 8i 2 [N ] . Estimate a subspace using the neighbors for each point
I
[N ]
while I =
6 ; do
. Iteratively pick the best subspace estimates
PN
i?
arg maxi2I j=1 I{kProjWi yj k2 1 ?}
?l
? i?
D
W
I
I \ {j : kProjWi? yj k2 1 ?}
end while
w
?i
arg maxl2[L] kProjD? l yi k2 , 8i 2 [N ]
. Label the points using the subspace estimates
Recall that the matrix W contains the labelings of the points, so that Wij = 1 if point i is assigned
to subspace j. Top-d{yj : Wij = 1} denotes the d-dimensional principal subspace of the set of
vectors {yj : Wij = 1}. This can be obtained by taking the first d left singular vectors of the
matrix whose columns are the vector in the set. If there are only d vectors in the set, Gram-Schmidt
orthogonalization will give us the subspace. As in NSN, it is efficient to store a subspace Wi in
the form of its orthogonal basis because we can easily compute the norm of a projection onto the
subspace.
Testing a candidate subspace by counting the number of near points has already been considered in
the subspace clustering literature. In [25], the authors proposed to run RANdom SAmple Consensus
(RANSAC) iteratively. RANSAC randomly selects a few points and checks if there are many other
points near the subspace spanned by the collected points. Instead of randomly choosing sample
points, GSR receives some candidate subspaces (in the form of sets of points) from NSN (or possibly
some other algorithm) and selects subspaces in a greedy way as specified in the algorithm above.
4
3
Theoretical results
We analyze our algorithms in two standard noiseless models. The main theorems present sufficient
conditions under which the algorithms cluster the points exactly with high probability. For simplicity
of analysis, we assume that every subspace is of the same dimension, and the number of data points
on each subspace is the same, i.e., d , d1 = ? ? ? = dL , n , N1 = ? ? ? = NL . We assume that d
is known to the algorithm. Nonetheless, our analysis can extend to the general case.
3.1
Statistical models
We consider two models which have been used in the subspace clustering literature:
? Fully random model: The subspaces are drawn iid uniformly at random, and the points are
also iid randomly generated.
? Semi-random model: The subspaces are arbitrarily determined, but the points are iid randomly generated.
Let Di 2 Rp?d , i 2 [L] be a matrix whose columns form an orthonormal basis of Di . An important
measure that we use in the analysis is the affinity between two subspaces, defined as
s
Pd
>
2 i,j
kDi Dj kF
k=1 cos ?k
p
a?(i, j) ,
=
2 [0, 1],
d
d
where ?ki,j is the kth principal angle between Di and Dj . Two subspaces Di and Dj are identical if
and only if a?(i, j) = 1. If a?(i, j) = 0, every vector on Di is orthogonal to any vectors on Dj . We
also define the maximum affinity as
max a? ,
max
i,j2[L],i6=j
a?(i, j) 2 [0, 1].
There are N = nL points, and there are n points exactly lying on each subspace. We assume that
each data point yi is drawn iid uniformly at random from Sp 1 \ Dwi where Sp 1 is the unit sphere
in Rp . Equivalently,
yi = D wi x i ,
xi ? Unif(Sd
1
),
8i 2 [N ].
As the points are generated randomly on their corresponding subspaces, there are no points lying on
an intersection of two subspaces, almost surely. This implies that with probability one the points are
clustered correctly provided that the true subspaces are recovered exactly.
3.2
Main theorems
The first theorem gives a statistical guarantee for the fully random model.
Theorem 1 Suppose L d-dimensional subspaces and n points on each subspace are generated in
the fully random model with n polynomial in d. There are constants C1 , C2 > 0 such that if
?
n
ne ?2
d
C2 log n
> C1 log
,
<
,
(1)
d
d
p
log(ndL 1 )
3L
then with probability at least 1
, NSN+GSR3 clusters the points exactly. Also, there are
1
0
0
other constants C1 , C2 > 0 such that if (1) with C1 and C2 replaced by C10 and C20 holds then
NSN+Spectral4 clusters the points exactly with probability at least 1 13L . e is the exponential
constant.
3
4
NSN with K = kmax = d followed by GSR with arbitrarily small ?.
NSN with K = kmax = d.
5
Our sufficient conditions for exact clustering explain when subspace clustering becomes easy or
difficult, and they are consistent with our intuition. For NSN to find correct neighbors, the points on
the same subspace should be many enough so that they look like lying on a subspace. This condition
is spelled out in the first inequality of (1). We note that the condition holds even when n/d is a
constant, i.e., n is linear in d. The second inequality implies that the dimension of the subspaces
should not be too high for subspaces to be distinguishable. If d is high, the random subspaces are
more likely to be close to each other, and hence they become more difficult to be distinguished.
However, as n increases, the points become dense on the subspaces, and hence it becomes easier to
identify different subspaces.
Let us compare our result with the conditions required for success in the fully random model in the
existing literature. In [16], it is required for SSC to have correct neighborhoods that n should be
superlinear in d when d/p fixed. In [6, 24], the conditions on d/p becomes worse as we have more
points. On the other hand, our algorithms are guaranteed exact clustering of the points, and the
sufficient condition is order-wise at least as good as the conditions for correct neighborhoods by the
existing algorithms (See Table 1). Moreover, exact clustering is guaranteed even when n is linear in
d, and d/p fixed.
For the semi-random model, we have the following general theorem.
Theorem 2 Suppose L d-dimensional subspaces are arbitrarily chosen, and n points on each
subspace are generated in the semi-random model with n polynomial in d. There are constants
C1 , C2 > 0 such that if
s
?
n
ne ?2
C2 log n
> C1 log
, max a? <
.
(2)
d
d
log(dL 1 ) ? log(ndL 1 )
then with probability at least 1
3L
1
, NSN+GSR5 clusters the points exactly.
In the semi-random model, the sufficient condition does not depend on the ambient dimension p.
When the affinities between subspaces are fixed, and the points are exactly lying on the subspaces,
the difficulty of the problem does not depend on the ambient dimension. It rather depends on
max a?, which measures how close the subspaces are. As they become closer to each other, it
becomes more difficult to distinguish the subspaces. The second inequality of (2) explains this intuition. The inequality also shows that if we have more data points, the problem becomes easier to
identify different subspaces.
Compared with other algorithms, NSN+GSR is guaranteed exact clustering, and more importantly,
the condition on max a? improves as n grows. This remark is consistent with the practical performance of the algorithm which improves as the number of data points increases, while the existing guarantees of otherpalgorithms are not. In [16], correct neighborhoods in SSC are guaranteed if max a? = O( log(n/d)/ log(nL)). In [6], exact clustering of TSC is guaranteed if
max a? = O(1/ log(nL)). However, these algorithms perform empirically better as the number of
data points increases.
4
Experimental results
In this section, we empirically compare our algorithms with the existing algorithms in terms of
clustering performance and computational time (on a single desktop). For NSN, we used the fast
implementation described in Section A.1. The compared algorithms are K-means, K-flats6 , SSC,
LRR, SCC, TSC7 , and SSC-OMP8 . The numbers of replicates in K-means, K-flats, and the K5
NSN with K = d 1 and kmax = d2 log de followed by GSR with arbitrarily small ?.
K-flats is similar to K-means. At each iteration, it computes top-d principal subspaces of the points with
the same label, and then labels every point based on its distances to those subspaces.
7
The MATLAB codes for SSC, LRR, SCC, and TSC are obtained from http://www.cis.
jhu.edu/?ehsan/code.htm,
https://sites.google.com/site/guangcanliu,
and
http://www.math.duke.edu/?glchen/scc.html,
http://www.nari.ee.ethz.ch/
commth/research/downloads/sc.html, respectively.
8
For each data point, OMP constructs a neighborhood for each point by regressing the point on the other
points up to 10 4 accuracy.
6
6
Ambient dimension (p)
SSC
SSC?OMP
LRR
TSC
NSN+Spectral
NSN+GSR
1
50
50
50
50
50
50
35
35
35
35
35
35
20
20
20
20
20
20
10
10
10
10
10
10
5
5
5
0.8
0.6
0.4
0.2
5
5
2 4 6 8 10
5
2 4 6 8 10
2 4 6 8 10
2 4 6 8 10
2 4 6 8 10
Number of points per dimension for each subspace (n/d)
2 4 6 8 10
0
Figure 1: CE of algorithms on 5 random d-dimensional subspaces and n random points on each
subspace. The figures shows CE for different numbers of n/d and ambient dimension p. d/p is
fixed to be 3/5. Brighter cells represent that less data points are clustered incorrectly.
Ambient dimension (p)
l1?minimization (SSC)
OMP (SSC?OMP)
Nuclear norm min. (LRR)
Nearest neighbor (TSC)
NSN
1
50
50
50
50
50
35
35
35
35
35
20
20
20
20
20
10
10
10
10
10
5
5
5
5
5
0.8
0.6
0.4
0.2
2
4
6
8
10
2
4
6 8 10
2 4 6 8 10
2 4 6
Number of points per dimension for each subspace (n/d)
8
10
2
4
6
8
10
0
Figure 2: NSE for the same model parameters as those in Figure 1. Brighter cells represent that
more data points have all correct neighbors.
100?dim ambient space, five 10?dim subspaces
100?dim ambient space, 10?dim subspaces, 20 points/subspace
5
5
3
4
Time (sec)
Time (sec)
4
l1?minimization (SSC)
OMP (SSC?OMP)
Nuclear norm min. (LRR)
Thresholding (TSC)
NSN
2
1
0
20
3
2
1
40
60
80
Number of data points per subspace (n)
0
100
5
10
15
20
Number of subspaces (L)
25
Figure 3: Average computational time of the neighborhood selection algorithms
means used in the spectral clustering are all fixed to 10. The algorithms are compared in terms of
Clustering error (CE) and Neighborhood selection error (NSE), defined as
(CE) = min
?2?L
N
1 X
I(wi 6= ?(w
?i )),
N i=1
(NSE) =
N
1 X
I(9j : Wij 6= 0, wi 6= wj )
N i=1
where ?L is the permutation space of [L]. CE is the proportion of incorrectly labeled data points.
Since clustering is invariant up to permutation of label indices, the error is equal to the minimum
disagreement over the permutation of label indices. NSE measures the proportion of the points
which do not have all correct neighbors.9
4.1
Synthetic data
We compare the performances on synthetic data generated from the fully random model. In Rp ,
five d-dimensional subspaces are generated uniformly at random. Then for each subspace n unitnorm points are generated iid uniformly at random on the subspace. To see the agreement with the
theoretical result, we ran the algorithms under fixed d/p and varied n and d. We set d/p = 3/5 so
that each pair of subspaces has intersection. Figures 1 and 2 show CE and NSE, respectively. Each
error value is averaged over 100 trials. Figure 1 indicates that our algorithm clusters the data points
better than the other algorithms. As predicted in the theorems, the clustering performance improves
9
For the neighborhood matrices from SSC, LRR, and SSC-OMP, the d points with the maximum weights
are regarded as neighbors for each point. For TSC, the d nearest neighbors are collected for each point.
7
L
2
3
Algorithms
Mean CE (%)
Median CE (%)
Avg. Time (sec)
Mean CE (%)
Median CE (%)
Avg. Time (sec)
K-means
19.80
17.92
26.10
20.48
-
K-flats
13.62
10.65
0.80
14.07
14.18
1.89
SSC
1.52
0.00
3.03
4.40
0.56
5.39
LRR
2.13
0.00
3.42
4.03
1.43
4.05
SCC
2.06
0.00
1.28
6.37
0.21
2.16
SSC-OMP(8)
16.92
12.77
0.50
27.96
30.98
0.82
TSC(10)
18.44
16.92
0.50
28.58
29.67
1.15
NSN+Spectral(5)
3.62
0.00
0.25
8.28
2.76
0.51
Table 2: CE and computational time of algorithms on Hopkins155 dataset. L is the number of
clusters (motions). The numbers in the parentheses represent the number of neighbors for each
point collected in the corresponding algorithms.
L
2
3
5
10
Algorithms
Mean CE (%)
Median CE (%)
Avg. Time (sec)
Mean CE (%)
Median CE (%)
Avg. Time (sec)
Mean CE (%)
Median CE (%)
Avg. Time (sec)
Mean CE (%)
Median CE (%)
Avg. Time (sec)
K-means
45.98
47.66
62.55
63.54
73.77
74.06
82.68
82.97
-
SSC
1.77
0.00
37.72
5.77
1.56
49.45
4.79
2.97
74.91
9.43
8.75
157.5
K-flats
37.62
39.06
15.78
45.81
47.92
27.91
55.51
56.25
52.90
62.72
62.89
134.0
SSC-OMP
4.45
1.17
0.45
6.35
2.86
0.76
8.93
5.00
1.41
15.32
17.11
5.26
TSC
11.84
1.56
0.33
20.02
15.62
0.60
11.90
33.91
1.17
39.48
39.45
3.17
NSN+Spectral
1.71
0.78
0.78
3.63
3.12
3.37
5.81
4.69
5.62
9.82
9.06
14.73
Table 3: CE and computational time of algorithms on Extended Yale B dataset. For each number of
clusters (faces) L, the algorithms ran over 100 random subsets drawn from the overall 38 clusters.
as the number of points increases. However, it also improves as the dimension of subspaces grows in
contrast to the theoretical analysis. We believe that this is because our analysis on GSR is not tight.
In Figure 2, we can see that more data points obtain correct neighbors as n increases or d decreases,
which conforms the theoretical analysis.
We also compare the computational time of the neighborhood selection algorithms for different
numbers of subspaces and data points. As shown in Figure 3, the greedy algorithms (OMP, Thresholding, and NSN) are significantly more scalable than the convex optimization based algorithms
(`1 -minimization and nuclear norm minimization).
4.2
Real-world data : motion segmentation and face clustering
We compare our algorithm with the existing ones in the applications of motion segmentation and
face clustering. For the motion segmentation, we used Hopkins155 dataset [17], which contains
155 video sequences of 2 or 3 motions. For the face clustering, we used Extended Yale B dataset
with cropped images from [5, 13]. The dataset contains 64 images for each of 38 individuals in
frontal view and different illumination conditions. To compare with the existing algorithms, we
used the set of 48 ? 42 resized raw images provided by the authors of [4]. The parameters of the
existing algorithms were set as provided in their source codes.10 Tables 2 and 3 show CE and average
computational time.11 We can see that NSN+Spectral performs competitively with the methods with
the lowest errors, but much faster. Compared to the other greedy neighborhood construction based
algorithms, SSC-OMP and TSC, our algorithm performs significantly better.
Acknowledgments
The authors would like to acknowledge NSF grants 1302435, 0954059, 1017525, 1056028 and
DTRA grant HDTRA1-13-1-0024 for supporting this research. This research was also partially
supported by the U.S. Department of Transportation through the Data-Supported Transportation
Operations and Planning (D-STOP) Tier 1 University Transportation Center.
10
As SSC-OMP and TSC do not have proposed number of parameters for motion segmentation, we found
the numbers minimizing the mean CE. The numbers are given in the table.
11
The LRR code provided by the author did not perform properly with the face clustering dataset that we
used. We did not run NSN+GSR since the data points are not well distributed in its corresponding subspaces.
8
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9
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4,759 | 5,309 | Graph Clustering With Missing Data : Convex
Algorithms and Analysis
Ramya Korlakai Vinayak, Samet Oymak, Babak Hassibi
Department of Electrical Engineering
California Institute of Technology, Pasadena, CA 91125
{ramya, soymak}@caltech.edu, [email protected]
Abstract
We consider the problem of finding clusters in an unweighted graph, when the
graph is partially observed. We analyze two programs, one which works for dense
graphs and one which works for both sparse and dense graphs, but requires some a
priori knowledge of the total cluster size, that are based on the convex optimization
approach for low-rank matrix recovery using nuclear norm minimization. For
the commonly used Stochastic Block Model, we obtain explicit bounds on the
parameters of the problem (size and sparsity of clusters, the amount of observed
data) and the regularization parameter characterize the success and failure of the
programs. We corroborate our theoretical findings through extensive simulations.
We also run our algorithm on a real data set obtained from crowdsourcing an
image classification task on the Amazon Mechanical Turk, and observe significant
performance improvement over traditional methods such as k-means.
1
Introduction
Clustering [1] broadly refers to the problem of identifying data points that are similar to each other.
It has applications in various problems in machine learning, data mining [2, 3], social networks [4?
6], bioinformatics [7, 8], etc. In this paper we focus on graph clustering [9] problems where the data
is in the form of an unweighted graph. Clearly, to observe the entire graph on n nodes requires n2
measurements. In most practical scenarios this is infeasible and we can only expect to have partial
observations. That is, for some node pairs we know whether there exists an edge between them
or not, whereas for the rest of the node pairs we do not have this knowledge. This leads us to the
problem of clustering graphs with missing data.
Given the adjacency matrix of an unweighted graph, a cluster is defined as a set of nodes that are
densely connected to each other when compared to the rest of the nodes. We consider the problem of
identifying such clusters when the input is a partially observed adjacency matrix. We use the popular
Stochastic Block Model (SBM) [10] or Planted Partition Model [11] to analyze the performance of
the proposed algorithms. SBM is a random graph model where the edge probability depends on
whether the pair of nodes being considered belong to the same cluster or not. More specifically, the
edge probability is higher when both nodes belong to the same cluster. Further, we assume that each
entry of the adjacency matrix of the graph is observed independently with probability r. We will
define the model in detail in Section 2.1.
1.1
Clustering by Low-Rank Matrix Recovery and Completion
The idea of using convex optimization for clustering has been proposed in [12?21]. While each of
these works differ in certain ways, and we will comment on their relation to the current paper in
Section 1.3, the common approach they use for clustering is inspired by recent work on low-rank
matrix recovery and completion via regularized nuclear norm (trace norm) minimization [22?26].
1
In the case of unweighted graphs, an ideal clustered graph is a union of disjoint cliques. Given
the adjacency matrix of an unweighted graph with clusters (denser connectivity inside the clusters
compared to outside), we can interpret it as an ideal clustered graph with missing edges inside the
clusters and erroneous edges in between clusters. Recovering the low-rank matrix corresponding to
the disjoint cliques is equivalent to finding the clusters.
We will look at the following well known convex program which aims to recover and complete the
low-rank matrix (L) from the partially observed adjacency matrix (Aobs ):
Simple Convex Program:
minimize kLk? + ?kSk1
(1.1)
subject to
1 ? Li,j ? 0 for all i, j ? {1, 2, . . . n}
(1.2)
L,S
obs
L
obs
+S
=A
obs
(1.3)
where ? ? 0 is the regularization parameter, k.k? is the nuclear norm (sum of the singular values
of the matrix), and k.k1 is the l1 -norm (sum of absolute values of the entries of the matrix). S is
the sparse error matrix that accounts for the missing edges inside the clusters and erroneous edges
outside the clusters on the observed entries. Lobs and Sobs denote entries of L and S that correspond
to the observed part of the adjacency matrix.
Program 1.1 is very simple and intuitive. Further, it does not require any information other than
the observed part of the adjacency matrix. In [13], the authors analyze Program 1.1 without the
constraint (1.2). While dropping (1.2) makes the convex program less effective, it does allow [13] to
make use of low-rank matrix completion results for its analysis. In [16] and [21], the authors analyze
Program 1.1 when the entire adjacency matrix is observed. In [17], the authors study a slightly more
general program, where the regularization parameter is different for the extra edges and the missing
edges. However, the adjacency matrix is completely observed.
It is not difficult to see that, when the edge probability inside the cluster is p < 1/2, that (as n ? ?)
Program 1.1 will return L0 = 0 as the optimal solution (since if the cluster is not dense enough it is
more costly to complete the missing edges). As a result our analysis of Program 1.1, and the main
result of Theorem 1, assumes p > 1/2. Clearly, there are many instances of graphs we would like
to cluster where p < 1/2. If the total size of the cluster region (i.e, the total number of edges in
the cluster, denoted by |R|) is known, then the following convex program can be used, and can be
shown to work for p < 1/2 (see Theorem 2).
Improved Convex Program:
minimize kLk? + ?kSk1
(1.4)
L,S
subject to
1 ? Li,j ? Si,j ? 0 for all i, j ? {1, 2, . . . n}
Li,j = Si,j whenever
sum(L) ? |R|
Aobs
i,j
=0
(1.5)
(1.6)
(1.7)
As before, L is the low-rank matrix corresponding to the ideal cluster structure and ? ? 0 is the
regularization parameter. However, S is now the sparse error matrix that accounts only for the
missing edges inside the clusters on the observed part of adjacency matrix. [16] and [19] study
programs similar to Program 1.4 for the case of a completely observed adjacency matrix. In [19],
the constraint 1.7 is a strict equality. In [15] the authors analyze a program close to Program 1.4 but
without the l1 penalty.
If R is not known, it is possible to solve Problem 1.4 for several values of R until the desired
performance is obtained. Our empirical results reported in Section 3, suggest that the solution is not
very sensitive to the choice of R.
1.2
Our Contributions
? We analyze the Simple Convex Program 1.1 for the SBM with partial observations. We provide
explicit bounds on the regularization parameter as a function of the parameters of the SBM, that
2
characterizes the success and failure conditions of Program 1.1 (see results in Section 2.2). We
show that clusters that are either too small or too sparse constitute the bottleneck. Our analysis is
helpful in understanding the phase transition from failure to success for the simple approach.
? We also analyze the Improved Convex Program 1.4. We explicitly characterize the conditions on
the parameters of the SBM and the regularization parameter for successfully recovering clusters
using this approach (see results in Section 2.3).
? Apart from providing theoretical guarantees and corroborating them with simulation results (Section 3), we also apply Programs 1.1 and 1.4 on a real data set (Section 3.3) obtained by crowdsourcing an image labeling task on Amazon Mechanical Turk.
1.3
Related Work
In [13], the authors consider the problem of identifying clusters from partially observed unweighted graphs. For the SBM with partial observations, they analyze Program 1.1 without constraint
p (1.2), and show that under certain conditions, the minimum cluster size must be at least
O( n(log(n))4 /r) for successful recovery of the clusters. Unlike our analysis, the exact requirement on the cluster size is not known (since the constant of proportionality is not known). Also
they do not provide conditions under which the approach fails to identify the clusters. Finding the
explicit bounds on the constant of proportionality is critical to understanding the phase transition
from failure to successfully identifying clusters.
In [14?19], analyze convex programs similar to the Programs
1.1 and 1.4 for the SBM and show
?
that the minimum cluster size should be at least O( n) for successfully recovering the clusters.
However, the exact requirement on the cluster size is not known. Also, they do not provide explicit
conditions for failure, and except for [16] they do not address the case when the data is missing.
In contrast, we consider the problem of clustering with missing data. We explicitly characterize
the constants by providing bounds on the model parameters that decide if Programs 1.1 and 1.4
can successfully identify clusters. Furthermore, for Program 1.1, we also explicitly characterize the
conditions under which the program fails.
In [16], the authors extend their results to partial observations by scaling the edge probabilities by r
(observation probability), which will not work for r < 1/2 or 1/2 < p < 1/2r in Program 1.1 . [21]
analyzes Program 1.1 for the SBM and provides conditions for success and failure of the program
when the entire adjacency matrix is observed. The dependence on the number of observed entries
emerges non-trivially in our analysis. Further, [21] does not address the drawback of Program 1.1,
which is p > 1/2, whereas in our work we analyze Program 1.4 that overcomes this drawback.
2
2.1
Partially Observed Unweighted Graph
Model
Definition 2.1 (Stochastic Block Model). Let A = AT be the adjacency matrix of a graph on n
nodes with K disjoint clusters of size ni each, i = 1, 2, ? ? ? , K. Let 1 ? pi ? 0, i = 1, ? ? ? , K and
1 ? q ? 0. For l > m,
1 w.p. pi , if both nodes l, m are in the same cluster i.
Al,m =
(2.1)
1 w.p. q, if nodes l, m are not in the same cluster.
If pi > q for each i, then we expect the density of edges to be higher inside the clusters compared to
outside. We will say the random variable Y has a ?(r, ?) distribution, for 0 ? ?, r ? 1, written as
Y ? ?(r, ?), if
?
?1, w.p. r?
Y = 0, w.p. r(1 ? ?)
?
?, w.p. (1 ? r)
where ? denotes unknown.
Definition 2.2 (Partial Observation Model). Let A be the adjacency matrix of a random graph
generated according to the Stochastic Block Model of Definition 2.1. Let 0 < r ? 1. Each entry of
3
the adjacency matrix A is observed independently with probability r. Let Aobs denote the observed
adjacency matrix. Then for l > m: (Aobs )l,m ? ?(r, pi ) if both the nodes l and m belong to the
same cluster i. Otherwise, (Aobs )l,m ? ?(r, q).
2.2
Results : Simple Convex Program
Let [n] = {1, 2, ? ? ? , n}. Let R be the union of regions induced by the clusters and Rc = [n] ? [n] ?
PK
PK
R its complement. Note that |R| = i=1 n2i and |Rc | = n2 ? i=1 n2i . Let nmin := min ni ,
1?i?K
pmin := min pi and nmax := max ni .
1?i?K
1?i?K
The following definitions are important to describe our results.
? Define Di := ni r (2pi ? 1) as the effective density of cluster i and Dmin = min Di .
1?i?K
? ?succ
?
q
PK
:= max 2r ni 2( 1r ? 1) + 4 (q(1 ? q) + pi (1 ? pi )) and ?fail := i=1
1?i?K
n2i
n
p
? q1
?1
? ??1
rq(n ? ?fail ).
succ := 2r n
r ? 1 + 4q(1 ? q) + ?succ and ?fail :=
We note that the thresholds, ?succ and ?fail depend only the parameters of the model. Some simple
algebra shows that ?succ < ?fail .
Theorem 1 (Simple Program). Consider a random graph generated according to the Partial ObserK
vation Model of Definition (2.2) with K disjoint clusters of sizes {ni }K
i=1 , and probabilities {pi }i=1
1
0
0
and q, such that pmin > 2 > q > 0. Given > 0, there exists positive constants c1 , c2 such that,
1. If ? ? (1 + )?fail , then Program 1.1 fails to correctly recover the clusters with probability
1 ? c01 exp(?c02 |Rc |).
2. If 0 < ? ? (1 ? )?succ ,
? If Dmin ? (1 + ) ?1 , then Program 1.1 succeeds in correctly recovering the clusters with
probability 1 ? c01 n2 exp(?c02 nmin ).
? If Dmin ? (1 ? ) ?1 , then Program 1.1 fails to correctly recover the clusters with probability
1 ? c01 exp(?c02 nmin ).
Discussion:
1. Theorem 1 characterizes the success and failure of Program 1.1 as a function of the regularization
parameter ?. In particular, if ? > ?fail , Program 1.1 fails with high probability. If ? < ?succ ,
Program 1.1 succeeds with high probability if and only if Dmin > ?1 . However, Theorem 1 has
nothing to say about ?succ < ? < ?fail .
? q 1
n
2. Small Cluster Regime: When nmax = o(n), we have ??1
=
2r
succ
r ? 1 + 4q(1 ? q) .
For simplicity let pi = p, ? i, which yields Dmin = nmin r(2p ? 1). Then Dmin > ??1
succ implies,
s
?
2 n
1
nmin >
? 1 + 4q(1 ? q) ,
(2.2)
2p ? 1
r
giving a lower bound on the minimum cluster size that is sufficient for success.
2.3
Results: Improved Convex Program
The following definitions are critical to describe our results.
? i := ni r (pi ? q) as the effective density of cluster i and D
? min = min D
? i.
? Define D
1?i?K
? ??succ
?
q
:= 2 max r ni (1 ? pi )( 1r ? 1 + pi ) + (1 ? q)( 1r ? 1 + q)
1?i?K
4
Observation Probability (r)
Observation Probability (r)
1
Success
Failure
0.8
0.6
0.4
0.2
0.6
0.7
0.8
0.9
1
Edge Probability inside the cluster (p)
(a)
1
Success
Failure
0.8
0.6
0.4
0.2
50
100
150
Minimum Cluster Size
200
(b)
Figure 1: Region of success (white region) and failure (black region) of Program 1.1 with ? =
1.01D?1
min . The solid red curve is the threshold for success (? < ?succ ) and the dashed green line
which is the threshold for failure (? > ?fail ) as predicted by Theorem 1.
? ?1 := 2r?n
? ?
succ
q
( 1r ? 1 + q)(1 ? q) + ??succ .
? succ depends only on the parameters of the model.
We note that the threshold, ?
Theorem 2 (Improved Program). Consider a random graph generated according to the Partial
Observation Model of Definition 2.2, with K disjoint clusters of sizes {ni }K
i=1 , and probabilities
0
0
{pi }K
i=1 and q, such that pmin > q > 0. Given > 0, there exists positive constants c1 , c2 such
1
?
?
that: If 0 < ? ? (1 ? )?succ and Dmin ? (1 + ) ? , then Program 1.4 succeeds in recovering the
clusters with probability 1 ? c01 n2 exp(?c02 nmin ).
Discussion:1
1. Theorem 2 gives a sufficient condition for the success of Program 1.4 as a function of ?. In
? ?1 < ? < ?
? succ .
particular, for any ? > 0, we succeed if D
min
q
1
? ?1 = 2r?n
2. Small Cluster Regime: When nmax = o(n), we have ?
succ
r ? 1 + q (1 ? q). For
? min = nmin r(p ? q). Then D
? min > ?
? ?1 implies,
simplicity let pi = p, ? i, which yields D
succ
s
?
2 n
1
? 1 + q (1 ? q),
(2.3)
nmin >
p?q
r
which gives a lower bound on the minimum cluster size that is sufficient for success.
3. (p, q) as a function of n: We now briefly discuss the regime in which cluster sizes are large
(i.e. O(n)) and we are interested in the parameters (p, q) as a function of n that allows proposed
approaches to be successful. Critical to Program 1.4 is the constraint (1.6): Li,j = Si,j when
obs
Aobs
). With missing data,
i,j = 0 (which is the only constraint involving the adjacency A
obs
Ai,j = 0 with probability r(1 ? p) inside the clusters and r(1 ? q) outside the clusters. Defining
p? = rp + 1 ? r and q? = rq + 1 ? r, the number of constraints in (1.6) becomes statistically
equivalent to those of a fully observed graph where p and q are replaced by p? and q?. Consequently,
for a fixed r > 0, from (2.3), we require p ? p ? q & O( ?1n ) for success. However, setting
the unobserved entries to 0, yields Ai,j = 0 with probability 1 ? rp inside the clusters and
1 ? rq outside the clusters. This is equivalent to a fully observed graph where p and q are
replaced by rp and rq. In this case, we can allow p ? O( n1 ) for success which is order-wise
better, and matches the results in McSherry [27]. Intuitively, clustering a fully observed graph
with parameters p? = rp + 1 ? r and q? = rq + 1 ? r is much more difficult than one with rp
and rq, since the links are more noisy in the former case. Hence, while it is beneficial to leave
the unobserved entries blank in Program 1.1, for Program 1.4 it is in fact beneficial to set the
unobserved entries to 0.
1
The proofs for Theorems 1 and 2 are provided in the supplementary material.
5
Probability of Success
Observation Probability (r)
1
Success
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1
Edge Probability inside the cluster (p)
1
0.5
Simple
Improved
0
0.2
0.4
0.6
0.8
1
Edge Probability inside the clusters (p)
(b) Comparison range of edge probability p for Simple Program 1.1 and Improved Program 1.4.
(a) Region of success (white region) and failure
? succ .
(black region) of Program 1.4 with ? = 0.49?
The solid red curve is the threshold for success
? min > ??1 ) as predicted by Theorem 2.
(D
Figure 2: Simulation results for Improved Program.
3
Experimental Results
We implement Program 1.1 and 1.4 using the inexact augmented Lagrange method of multipliers [28]. Note that this method solves the Program 1.1 and 1.4 approximately. Further, the numerical
imprecisions will prevent the entries of the output of the algorithms from being strictly equal to 0 or
1. We use the mean of all the entries of the output as a hard threshold to round each entry. That is,
if an entry is less than the threshold, it is rounded to 0 and to 1 otherwise. We compare the output
of the algorithm after rounding to the optimal solution (L0 ), and declare success if the number of
wrong entries is less than 0.1%.
Set Up: We consider at an unweighted graph on n = 600 nodes with 3 disjoint clusters. For
simplicity the clusters are of equal size n1 = n2 = n3 , and the edge probability inside the clusters
are same p1 = p2 = p3 = p. The edge probability outside the clusters is fixed, q = 0.1. We generate
the adjacency matrix randomly according to the Stochastic Block Model 2.1 and Partial Observation
Model 2.2. All the results are an average over 20 experiments.
3.1
Simulations for Simple Convex Program
Dependence between r and p: In the first set of experiments we keep n1 = n2 = n3 = 200, and
vary p from 0.55 to 1 and r from 0.05 to 1 in steps of 0.05.
Dependence between nmin and r: In the second set of experiments we keep the edge probability
inside the clusters fixed, p = 0.85. The cluster size is varied from nmin = 20 to nmin = 200 in steps
of 20 and r is varied from 0.05 to 1 in steps of 0.05.
In both the experiments, we set the regularization parameter ? = 1.01D?1
min , ensuring that Dmin >
1/?, enabling us to focus on observing the transition around ?succ and ?fail . The outcome of the
experiments are shown in the Figures 1a and 1b. The experimental region of success is shown in
white and the region of failure is shown in black. The theoretical region of success is about the solid
red curve (? < ?succ ) and the region of failure is below dashed green curve (? > ?fail ). As we can
see the transition indeed occurs between the two thresholds ?succ and ?fail .
3.2
Simulations for Improved Convex Program
We keep the cluster size, n1 = n2 = n3 = 200 and vary p from 0.15 to 1 and r from 0.05 to 1 in
? succ , ensuring that ? < ?
? succ , enabling
steps of 0.05. We set the regularization parameter, ? = 0.49?
?
us to focus on observing the condition of success around Dmin . The outcome of this experiment is
shown in the Figure 2a. The experimental region of success is shown in white and region of failure
is shown in black. The theoretical region of success is above solid red curve.
Comparison with the Simple Convex Program: In this experiment, we are interested in observing
the range of p for which the Programs 1.1 and 1.4 work. Keeping the cluster size n1 = n2 = n3 =
6
Matrix Recovered by Simple Program Matrix Recovered by Improved Program
Ideal Clusters
0.5
1
50
50
100
100
150
150
200
200
250
250
300
300
350
350
400
400
1.5
200
(a)
300
400
150
200
250
300
350
400
450
400
450
400
450
400
450
1
1.5
50
100
150
200
250
300
350
Clusters Identified from Simple Program
0.5
1
50
100
150
200
250
300
350
Clusters Identified from Improved Program
0.5
1
450
100
100
Clusters identifyed by k?means on A
1.5
450
50
0.5
1.5
100
200
300
400
(b)
50
100
150
200
250
300
350
(c) Comparing with k-means clustering.
Figure 3: Result of using (a) Program 1.1 (Simple) and (b) Program 1.4 (Improved) on the real data
set. (c) Comparing the clustering output after running Program 1.1 and Program 1.4 with the output
of applying k-means clustering directly on A (with unknown entries set to 0).
200 and r = 1, we vary the edge probability inside the clusters from p = 0.15 to p = 1 in steps
of 0.05. For each instance of the adjacency matrix, we run both Program 1.1 and 1.4. We plot the
probability of success of both the algorithms in Figure 2b. As we can observe, Program 1.1 starts
succeeding only after p > 1/2, whereas for Program 1.4 it starts at p ? 0.35.
3.3
Labeling Images: Amazon MTurk Experiment
Creating a training dataset by labeling images is a tedious task. It would be useful to crowdsource
this task instead. Consider a specific example of a set of images of dogs of different breeds. We want
to cluster them such that the images of dogs of the same breed are in the same cluster. One could
show a set of images to each worker, and ask him/her to identify the breed of dog in each of those
images. But such a task would require the workers to be experts in identifying the dog breeds. A
relatively reasonable task is to ask the workers to compare pairs of images, and for each pair, answer
whether they
think the dogs in the images are of the same breed or not. If we have n images, then
there are n2 distinct pairs of images, and it will pretty quickly become unreasonable to compare all
possible pairs. This is an example where we could obtain a subset of the data and try to cluster the
images based on the partial observations.
Image Data Set: We used images of 3 different breeds of dogs : Norfolk Terrier (172 images), Toy
Poodle (151 images) and Bouvier des Flandres (150 images) from the Standford Dogs Dataset [29].
We uploaded all the 473 images of dogs on an image hosting server (we used imgur.com).
MTurk Task: We used Amazon Mechanical Turk [30] as the platformfor crowdsourcing. For
each worker, we showed 30 pairs of images chosen randomly from the n2 possible pairs. The task
assigned to the worker was to compare each pair of images, and answer whether they think the dogs
belong to the same breed or not. If the worker?s response is a ?yes?, then there we fill the entry of
the adjacency matrix corresponding to the pair as 1, and 0 if the answer is a ?no?.
Collected Data: We recorded around 608 responses. We were able to fill 16, 750 out of 111, 628
entries in A. That is, we observed 15% of the total number of entries. Compared with true answers
(which we know a priori), the answers given by the workers had around 23.53% errors (3941 out of
16750). The empirical parameters for the partially observed graph thus obtained is shown Table 1.
?
We ran Program 1.1 and Program 1.4 with regularization parameter,
? = 1/ n. Further, for Pro
gram 1.4, we set the size of the cluster region, R to 0.125 times n2 . Figure 3a shows the recovered
matrices. Entries with value 1 are depicted by white and 0 is depicted by black. In Figure 3c we
compare the clusters output by running the k-means algorithm directly on the adjacency matrix
A (with unknown entries set to 0) to that obtained by running k-means algorithm on the matrices
recovered after running Program 1.1 (Simple Program) and Program 1.4 (Improved Program) respectively. The overall error with k-means was 40.8% whereas the error significantly reduced to
15.86% and 7.19% respectively when we used the matrices recoverd from Programs 1.1 and 1.4
respectively (see Table 2). Further, note that for running the k-means algorithm we need to know
the exact number of clusters. A common heuristic is to identify the top K eigenvalues that are much
7
Table 1: Empirical Parameters from the real data.
Table 2: Number of miss-classified images
Params
n
K
n1
n2
n3
Value
473
3
172
151
150
Params
r
q
p1
p2
p3
Value
0.1500
0.1929
0.7587
0.6444
0.7687
Clusters?
K-means
Simple
Improved
1
39
9
1
2
150
57
29
3
4
8
4
Total
193
74
34
larger than the rest. In Figure 4 we plot the sorted eigenvalues for the adjacency matrix A and the
recovered matrices. We can see that the top 3 eigen values are very easily distinguished from the
rest for the matrix recovered after running Program 1.4.
A sample of the data is shown in Figure 5. We observe that factors such as color, grooming, posture,
face visibility etc. can result in confusion while comparing image pairs. Also, note that the ability
of the workers to distinguish the dog breeds is neither guaranteed nor uniform. Thus, the edge
probability inside and outside clusters are not uniform. Nonetheless, Programs 1.1 and Program 1.4,
especially Program 1.4, are quite successful in clustering the data with only 15% observations.
30
300
300
A
Simple
20
200
10
100
100
0
0
0
?10
0
200
400
600
?100
0
200
200
400
600
?100
0
Improved
200
400
600
Figure 4: Plot of sorted eigen values for (1) Adjacency matrix with unknown entries filled by 0, (2)
Recovered adjacency matrix from Program 1.1, (3) Recovered adjacency matrix from Program 1.4
Norfolk Terrier
Toy Poodle
Bouvier des Flandres
Figure 5: Sample images of three breeds of dogs that were used in the MTurk experiment.
The authors thank the anonymous reviewers for their insightful comments. This work was supported
in part by the National Science Foundation under grants CCF-0729203, CNS-0932428 and CIF1018927, by the Office of Naval Research under the MURI grant N00014-08-1-0747, and by a grant
from Qualcomm Inc. The first author is also supported by the Schlumberger Foundation Faculty for
the Future Program Grant.
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9
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4,760 | 531 | Node Splitting: A Constructive Algorithm for
Feed-Forward Neural Networks
Mike Wynne-Jones
Research Initiative in Pattern Recognition
St. Andrews Road, Great Malvern
WR14 3PS, UK
[email protected]
Abstract
A constructive algorithm is proposed for feed-forward neural networks,
which uses node-splitting in the hidden layers to build large networks from
smaller ones. The small network forms an approximate model of a set of
training data, and the split creates a larger more powerful network which is
initialised with the approximate solution already found. The insufficiency
of the smaller network in modelling the system which generated the data
leads to oscillation in those hidden nodes whose weight vectors cover regions in the input space where more detail is required in the model. These
nodes are identified and split in two using principal component analysis,
allowing the new nodes t.o cover the two main modes of each oscillating
vector. Nodes are selected for splitting using principal component analysis
on the oscillating weight vectors, or by examining the Hessian matrix of
second derivatives of the network error with respect to the weight.s. The
second derivat.ive method can also be applied to the input layer, where it
provides a useful indication of t.he relative import.ances of parameters for
the classification t.ask. Node splitting in a standard Multi Layer Percept.ron is equivalent to introducing a hinge in the decision boundary to allow
more detail to be learned. Initial results were promising, but further evaluation indicates that the long range effects of decision boundaries cause
the new nodes to slip back to the old node position, and nothing is gained.
This problem does not occur in networks of localised receptive fields such
as radial basis functions or gaussian mixtures, where the t.echnique appears
to work well.
1072
Node Splitting: A Contructive Algorithm for Feed-Forward Neural Networks
1
Introduction
To achieve good generalisation in neural networks and other techniques for inferring
a model from data, we aim to match the number of degrees of freedom of the model
to that of the system generating the data. With too small a model we learn an
incomplete solution, while too many free parameters capture individual training
samples and noise.
Since the optimum size of network is seldom known in advance, there are two alternative ways of finding it. The constructive algorithm aims to build an approximate
model, and then add new nodes to learn more detail, thereby approaching the optimum network size from below. Pruning algorithms, on the other hand, start with
a network which is known to be too big, and then cut out nodes or weights which
do not contribute to the model. A review of recent techniques [\VJ91a] has led the
author to favour the constructive approach, since pruning still requires an estimate
of the optimum size, and the initial large net.works can take a long time t.o train.
Constructive algorithms offer fast training of the initial small networks, with the
network size and training slowness reflecting the amount of information already
learned. The best approach of all would be a constructive algorithm which also
allowed the pruning of unnecessary nodes or weights from the net.work.
The constructive algorithm trains a net.work until no further detail of the training
data can be learned, and then adds new nodes to t.he network. New nodes can be
added with random weights, or with pre-determined weight.s. Random weights are
likely to disrupt the approximate solut.ion already found, and are unlikely to be
initially placed in parts of the weight space where they can learn something useful,
although encouraging results have been reported in t.his ar~a.[Ash89] This problem
is likely to be accentuated in higher dimensional spaces. Alt.ernatively, weights can
be pre-determined by measurements on the performance of the seed network, and
this is the approach adopted here. One node is turned into two, each wit.h half the
output weight. A divergence is introduced in the weights into the nodes which is
sufficient for them behave independently in future training without disrupting the
approximate solution already found.
2
Node-Splitting
A network is trained using standard techniques until no furt.her improvement on
training set performance is achieved. Since we begin with a small network, we have
an approximate model of the data, which captures the dominant properties of the
generating system but lacks detail. We now freeze the weight.s in the network, and
calculate the updates which would be made them, using simple gradient descent,
by each separate t.raining pattern. Figure 1 shows t.he frozen vector of weights into
a single hidden node, and the scatter of proposed updates around the equilibrium
posit.ion.
The picture shows the case of a hidden node where there is one clear direction
of oscillation. This might be caused by two clusters of data within a class, each
trying to use the node in its own area of the input space, or by a decision boundary
pulled clockwise by some patterns and anti clockwise by others. If the oscillation
is strong, either in its exhibition of a clear direction or in comparison with other
1073
1074
Wynne-Jones
New Node
#1 --~U(
Weight Update
Vectors
Figure 1: A hidden node weight vector and updates proposed hy individual t.raining
patterns
nodes in the same layer, then the node is split in two. The new nodes are placed
one standard deviation either side of the old position. \Vhile this divergence gives
the nodes a push in the right direction, allowing them t.o continue to diverge in later
t.raining, the overall effect on the network is small. In most cases t.here is very little
degradation in performance as a result of the split.
The direction and size of oscillation are calculated by principal component analysis of the weight updates. By a traditional method, we are required to make a
cova.riance matrix of the weight updat.es for the weight vector int.o each node:
c
= L6w6wT
(1)
p
where p is the number of patterns. The mat.rix is then decomposed to a set of eigenvalues and eigenvectors; the largest. eigenvalue is the variance of oscillation and the
corresponding eigenvector is it.s direction. Suitable techniques for performing this
decomposition include Singular Value Dewmposition and Householder Reduction.
[Vet86] A much more suit.able way of calculating the principal components of a
stream of continuous measurements such as weight updat.es is iterative est.imation.
An est.imate is stored for each required principal component. vector, and the estimat.es are updated using each sample. [Oja83, San89] By Oja's method, the scalar
product of t.he current sample vector wit.h each current est.imate of the eigenvectors
is used as a mat.ching coefficient., M. The matching coefficient is used to re-estima.te
the eigenvalues and eigenvectors, in conjunction wit.h a gain term). which decays
as the number of patterns seen increases. The eigenvectors are updated by a proportion )'M of the current sample, and t.he eigenvalues hy ).lU 2 . The trace (sum of
eigenvalues) can also be est.imated simply as the mean of the traces (sum of diagonal
elements) of t.he individual sample covariance mat.rices. The principal component
vectors are renormalised and orthogonalised after every few updat.es. This algorithm
is of order n, the number of eigenvalues required, for the re-estimation, and O(n2)
for the orthogonalisation; the matrix decomposition method can take exponential
Node Splitting: A Contructive Algorithm for Feed-Forward Neural Networks
time, and is always much slower in practice.
In a recent paper on Ateiosis Networks, Hanson introduced stochastic weights in the
multi layer perceptron, with the aim of avoiding local minima in training.[Han90]
A sample was taken from a gaussian distribution each time a weight was used;
the mean was updated by gradient descent, and the variance reflected the network
convergence. The variance was allowed to decay with time, so that the network
would approach a deterministic state, but was increased in proportion to the updates
made to the mean. \Vhile the network wa.g far from convergence these updates were
large, and the variance remained large. Node splitting wa.g implemented in this
system, in nodes where the variances on the weights were large compared with the
means. In such cases, two new nodes were created with the weights one standard
deviation either side of the old mean: one SD is added to all weights to one node,
and subtracted for all weights to the other. Preliminary results were promising, but
there appear to be two problems with this approach for node-splitting. First, the
splitting criterion is not good: a useless node with all weights close to zero could
have comparatively large variances on the weights owing to noise. This node would
be split indefinit.ely. Secondly and more interestingly, the split is made wit.hout
regard to the correlations in sign between the weight updates, shown as dots in the
scatter plot.s of figure 2. In figure 2a, Meiosis would correctly place new nodes in the
positions marked with crosses, while in figure 2b, the new nodes would he placed
in completely the wrong places. This problem does not occur in the node splitting
scheme based on principal component analysis.
(a)
(b)
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? ?? ?
?? ?
?? ?
??
............
-.~
..
..... ....
~
X
.. .
X
Figure 2: Meiosis networks split correctly if the weight. updates are correlated in
sign (a), but fail when they are not (b).
3
Selecting nodes for splitting
Node splitting is carried out in t.he direct.ion of maximum variance of the scatter plot
of weight updates proposed by individual training samples. The hidden layer nodes
most likely t.o benefit from splitting are those for which the non-spherical nature
1075
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Wynne-Jones
of the scatter plot is most pronounced. In later implementations this criterion was
measured by comparing the largest eigenvalue with the sum of the eigenvalues,
both these quantities being calculated by the iterative method. This is less simple
in cases where there are a number of dominant directions of variance; the scatter
plot might, for example be a four dimensional disk in a ten dimensional space, and
hence present the possibility of splitt.ing one node into eight. It is hoped that these
more complicat.ed splits will be the suhject of further research.
An alternative approach in determining the need of nodes to be split, in comparison
with other nodes in the same layer, is to use the second derivat.ives of t.he network
error with respect to a parameter of the nodes which is normalised across all nodes
in a given layer of the network. Such a parameter wa.c;; proposed by Mozer and
Smolensky in [Sm089]: a multiplicative gat.ing function is applied to the outputs of
the nodes, with its gating parameter set to one. Small incrempnt.s in this parameter
can be used to characterise the error surface around the unity value, with the result
that derivatives are normalised a.cross all nodes in a given layer of the network.
Mozer and Smolensky rpplaced the sum squared error crit.erion with a modulus error criterion to preserve non-zero gradients close to the local minimum reached in
training; we prefer to characterise the t.rue error surface by mpans of second derivat.ives, which can be calculated by repeated use of the chain rule (hackpropagat.ion).
Backpropagat.ion of second derivat.ivps has previously been rpport.ed in [So190] and
[Hea90].
Since a high curvat.ure error minimum in t.he space of t.he gat.ing parampt.er for a
particular nocie indicat.es st.eep gradipnt.s surrounding thp minimum, it is t.hese nodes
which exhibit. t.he great.est instability in their weight-space position. In t.he weight
space, if the curvat.ure is high only in cert.ain directions, we have the situat.ion in
figure 1, where the node is oscillating, and is in need of splitt.ing. If the curvature is
high in all directions in comparison with other nodes, the network is highly sensitive
to changes in t.he node or it.s weights, and again it will benefit from splitting.
At t.he ot.her end of the scale of curvat.ure sensitivity, a node or weight wit.h very low
curvat.ure is one to which t.he network error is quit.e insensit.ive, and the parameter
is a suitable candidate for pruning. This scheme has previously been used for weight
pruning by Le Cun, Denker et a1. [SoW 0] , and offers the pot.ential for an int.egrated
syst.em of splitting and pruning - a truly adapt.ive net.work archit.ecture.
3.1
Applying the sensitivity measnre to inpnt nodes
In a.ddit.ion to using t.he ga.ting parameter sensit.ivit.y to select nodes for pruning,
Mozer and Smolensky mention the possibility of using it on the input nodes to
indicate those inputs to which the c1a.<;sification is most sensitive. This has been
implemented in our syst.em wit.h the second derivat.ive sensitivity measure, and applied to a large financial classification prohlem supplied by THORN El\JI Research.
The analysis was carried out. on the 78-dimensional dat.a, and the input sensitivities
varied over several orders of magnit.ude. The inputs were grouped into four sets according to sensitivit.y, and MLPs of 10 hidden nodes were trained on each subset of
the dat.a. \Vhile the low sensitivit.y groups failed to learn anyt.hing at all, t.he higher
sensit.ivit.y groups quickly attained a reasonable classification rat.e. Ident.ification of
useless inputs leads t.o greatly increased training speed in fut.ure analysis, and can
Node Splitting: A Contructive Algorithm for Feed-Forward Neural Networks
yield valuable economies in future data collection. This work is reported in more
detail in [WJ91b].
4
Evaluation in Multi Layer Percept ron networks
Despite the promising results from initial evaluations, further testing showed that
the splitter technique was often unable to improve on the performance of the network used as a seed for the first split. These test were carried out on a number of
different classification problems, where large numbers of hidden nodes were already
known to be required, and with a number of different splitting criteria. Prolonged
experimentation and consideration of this failure lead to the hypothesis that a split
might be made to correct some miscla.<;sified patterns in one region of the input
space but, owing to the long range effects of MLP decision boundaries, the changed
positions of the planes might cause a much greater number of misclassifications
elsewhere. These would tend to cause the newly creat.ed nodes to slip back to the
position of the node from which they were created, with no overall benefit. This
possibility was tested hy re-implementing the splitter technique in a gaussian mixture modeling system, which uses a network of localised receptive fields, and hence
does not have the long range effects which occurred in the multi layer perceptron.
5
Implementation of the splitter in a Gaussian Mixture
Model, and the results
The Gaussian Mixt.ures Model [Cox91] is a clustering algorithm, which attempts
to model the distribution of a points in a data set. It consists of a numher of
mult.ivariate gaussian dist.rihut.ions in different posit.ions in t.he input space, and
wit.h different variances in different direct.ions. The responses of t.hese recept.ive
fields (humps) are weighted and summed together; the weights are calculated to
sat.isfy the PDF const.raint. that t.he responses should sum to one over the data set.
For the experiment.s on node splitting, the variance was the same in all direct.ions for
a particular bump, leading to a model which is a sum of weight.ed spherical gaussian
distribut.ions of different sizes and in different. positions. The model is t.rained by
gradient ascent in the likelihood of the model fitting the data, which leads t.o a
set of learning rules for re-estimat.ing the weights, then t.he cent.re positions of the
recept.ive fields, then their variances.
For t.he splitter, a small model is trained until nothing more can be learned, and
the paramet.ers are frozen. The training set is run t.hrough once more, and the
updat.es are calculated which each pattern attempts to make to the centre position
of each receptive field. The first principal component and trace of these updates are
calculated by the iterative met.hod, and any nodes for which t.he principal component
variance is a large proportion of the trace is split in two.
The algorithm is quick to converge, and is slowed down only a. lit.tle by the oV('fhead
of computing the principal component and trace. Figure 3 shows the application of
t.he gaussian mixture splitter to modelling a circle and an enclosing annulus; in the
circle (a) there is no dominant. principa.l component direction in the data ('Overed by
the receptive field of each node (shown at. one st.anda.rd deviation by a circle), while
1077
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Wynne-Jones
in (b) three nodes are clearly insufficient to model the annulus, and one has just
undergone a split. (c) shows the same data set. and model a little later in t.raining
after a number of splits have taken place. The technique has been evaluated on a
number of other simple problems, with no negat.ive results to date.
Figure 3: Gaussian mixt.ure model with node-splitting applied to a circle and surrounding annulus
6
Conclusions
The split.ter t.echnique based on taking the principal component. of the influences
on hidden nodes in a network, ha.g been shown to be useful in the multi layer
perceptron in only a very limited number of cases. The split in this kind of net.work
corresponds to a hinge in the decision boundary, which corrects the errors for which
it was calculated, but usually caused for more errors in other parts of the input
space. This problem does not occur in networks of localised receptive fields such
as radial ba."is funct.ions of gaussian mixture distributions, where it appears to
work very well. Further studies will include splitting nodes into more than two, in
cases where there is more than one dominant principal component, and applying
node-split.t.ing to different. modelling algorithms, and to gaussian mixtures in hidden
markov models for speech recognition.
The analysis of the sensit.ivity of the net.work error to individual nodes gives an
ordered list which can be used for both splitting and pruning in the same network,
although splitting does not generally work in the MLP. This measure has been
demonstrated in t.he input layer, to identify which network inputs are more or less
useful in the classification t.ask.
Acknowledgements
The author is greatly indebted to John Bridle and Steve Luttrell of RSRE, Neil
Thacker of Sheffield University, and colleagues in the Research Initiative in Pattern
Node Splitting: A Contructive Algorithm for Feed-Forward Neural Networks
Recognition and its member companies for helpful comments and advice; also to
David Bounds of Aston University and RIPR for advice and encouragement.
References
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4,761 | 5,310 | Dimensionality Reduction with Subspace Structure
Preservation
Ifeoma Nwogu
Department of Computer Science
SUNY Buffalo
Buffalo, NY 14260
[email protected]
Devansh Arpit
Department of Computer Science
SUNY Buffalo
Buffalo, NY 14260
[email protected]
Venu Govindaraju
Department of Computer Science
SUNY Buffalo
Buffalo, NY 14260
[email protected]
Abstract
Modeling data as being sampled from a union of independent subspaces has been
widely applied to a number of real world applications. However, dimensionality reduction approaches that theoretically preserve this independence assumption
have not been well studied. Our key contribution is to show that 2K projection
vectors are sufficient for the independence preservation of any K class data sampled from a union of independent subspaces. It is this non-trivial observation that
we use for designing our dimensionality reduction technique. In this paper, we
propose a novel dimensionality reduction algorithm that theoretically preserves
this structure for a given dataset. We support our theoretical analysis with empirical results on both synthetic and real world data achieving state-of-the-art results
compared to popular dimensionality reduction techniques.
1
Introduction
A number of real world applications model data as being sampled from a union of independent
subspaces. These applications include image representation and compression [7], systems theory
[13], image segmentation [16], motion segmentation [14], face clustering [8, 6] and texture segmentation [9], to name a few. Dimensionality reduction is generally used prior to applying these
methods because most of these algorithms optimize expensive loss functions like nuclear norm, `1
regularization, e.t.c. Most of these applications simply apply off-the-shelf dimensionality reduction
techniques or resize images (in case of image data) as a pre-processing step.
The union of independent subspace model can be thought of as a generalization of the traditional
approach of representing a given set of data points using a single low dimensional subspace (e.g.
Principal Component Analysis). For the application of algorithms that model data at hand with this
independence assumption, the subspace structure of the data needs to be preserved after dimensionality reduction. Although a number of existing dimensionality reduction techniques [11, 4, 1, 5] try
to preserve the spacial geometry of any given data, no prior work has tried to explicitly preserve the
independence between subspaces to the best of our knowledge.
In this paper, we propose a novel dimensionality reduction technique that preserves independence
between multiple subspaces. In order to achieve this, we first show that for any two disjoint subspaces with arbitrary dimensionality, there exists a two dimensional subspace such that both the
1
subspaces collapse to form two lines. We then extend this non-trivial idea to multi-class case and
show that 2K projection vectors are sufficient for preserving the subspace structure of a K class
dataset. Further, we design an efficient algorithm that finds the projection vectors with the aforementioned properties while being able to handle corrupted data at the same time.
2
Preliminaries
Let S1 , S2 . . . SK be K subspaces in Rn . We say that these K subspaces are independent if there
does not exist any non-zero vector in Si which is a linear combination of vectors in the other K ? 1
subspaces. Let the columns of the matrix Bi ? Rn?d denote the support of the ith subspace of d
dimensions. Then any vector in this subspace can be represented as x = Bi w ?w ? Rd . Now we
define the notion of margin between two subspaces.
Definition 1 (Subspace Margin) Subspaces Si and Sj are separated by margin ?ij if
?ij =
max
u?Si ,v?Sj
hu, vi
kuk2 kvk2
(1)
Thus margin between any two subspaces is defined as the maximum dot product between two unit
vectors (u, v), one from either subspace. Such a vector pair (u, v) is known as the principal vector
pair between the two subspaces while the angle between these vectors is called the principal angle.
With these definitions of independent subspaces and margin, assume that we are given a dataset
which has been sampled from a union of independent linear subspaces. Specifically, each class in
this dataset lies along one such independent subspace. Then our goal is to reduce the dimensionality
of this dataset such that after projection, each class continues to lie along a linear subspace and that
each such subspace is independent of all others. Formally, let X = [X1 , X2 . . . , XK ] be a K class
dataset in Rn such that vectors from class i (x ? Xi ) lie along subspace Si . Then our goal is to
? i := {P T x : x ? Xi }
find a projection matrix (P ? Rn?m ) such that the projected data vectors X
?
?
(i ? {1 . . . K}) are such that data vectors Xi belong to a linear subspace (Si in Rm ). Further, each
subspace S?i (i ? {1 . . . K}) is independent of all others.
3
Proposed Approach
In this section, we propose a novel subspace learning approach applicable to labeled datasets that
theoretically guarantees independent subspace structure preservation. The number of projection
vectors required by our approach is not only independent of the size of the dataset but is also fixed,
depending only on the number of classes. Specifically, we show that for any K class labeled dataset
with independent subspace structure, only 2K projection vectors are required for structure preservation.
The entire idea of being able to find a fixed number of projection vectors for the structure preservation of a K class dataset is motivated by theorem 2. This theorem states a useful property of any
pair of disjoint subspaces.
Theorem 2 Let unit vectors v1 and v2 be the ith principal vector pair for any two disjoint subspaces
S1 and S2 in Rn . Let the columns of the matrix P ? Rn?2 be any two orthonormal vectors in the
span of v1 and v2 . Then for all vectors x ? Sj , P T x = ?tj (j ? {1, 2}), where ? ? R depends on
x and tj ? R2 is a fixed vector independent of x. Further,
tT
1 t2
kt1 k2 kt2 k2
= v1T v2
Proof: We use the notation (M )j to denote the j th column vector of matrix M for any arbitrary
matrix M . We claim that tj = P T vj (j ? {1, 2}). Also, without any loss of generality, assume that
(P )1 = v1 . Then in order to prove theorem 2, it suffices to show that ?x ? S1 , (P )T2 x = 0. By
symmetry, ?x ? S2 , P T x will also lie along a line in the subspace spanned by the columns of P .
Let the columns of B1 ? Rn?d1 and B2 ? Rn?d2 be the support of S1 and S2 respectively, where d1
and d2 are the dimensionality of the two subspaces. Then we can represent v1 and v2 as v1 = B1 w1
and v2 = B2 w2 for some w1 ? Rd1 and w2 ? Rd2 . Let B1 w be any arbitrary vector in S1 where
2
(a) Independent subspaces in 3
dimensions
(b) Subspaces after projection
Figure 1: A three dimensional example of the application of theorem 2. See text in section 3 for
details.
w ? Rd1 . Then we need to show that T := (B1 w)T (P )2 = 0?w. Notice that,
T = (B1 w)T (B2 w2 ? (w1T B1T B2 w2 )B1 w1 )
= wT (B1T B2 w2 ? (w1T B1T B2 w2 )w1 ) ?w
(2)
Let U SV T be the svd of B1T B2 . Then w1 and w2 are the ith columns of U and V respectively, and
v1T v2 is the ith diagonal element of S if v1 and v2 are the ith principal vectors of S1 and S2 . Thus,
T = wT (U SV T w2 ? Sii (U )i )
(3)
= wT (Sii (U )i ? Sii (U )i ) = 0
Geometrically, this theorem says that after projection on the plane (P ) defined by any one of the
principal vector pairs between subspaces S1 and S2 , both the entire subspaces collapse to just two
lines such that points from S1 lie along one line while points from S2 lie along the second line.
Further, the angle that separates these lines is equal to the angle between the ith principal vector pair
between S1 and S2 if the span of the ith principal vector pair is used as P .
We apply theorem 2 on a three dimensional example as shown in figure 1. In figure 1 (a), the first
subspace (y-z plane) is denoted by red color while the second subspace is the black line in x-y axis.
Notice that for this setting, the x-y plane (denoted by blue color) is in the span of the 1st (and only)
principal vector pair between the two subspaces. After projection of both the entire subspaces onto
the x-y plane, we get two lines (figure 1 (b)) as stated in the theorem.
Finally, we now show that for any K class dataset with independent subspace structure, 2K projection vectors are sufficient for structure preservation.
n
Theorem 3 Let X = {x}N
i=1 be a K class dataset in R with Independent Subspace structure. Let
n?2K
P = [P1 . . . PK ] ? R
be a projection matrix for X such that the columns of the matrix Pk ?
Rn?2 consists
of
orthonormal
vectors in the span of any principal vector pair between subspaces
P
Sk and j6=k Sj . Then the Independent Subspace structure of the dataset X is preserved after
projection on the 2K vectors in P .
Before stating the proof of this theorem, we first state lemma 4 which we will use later in our proof.
This lemma states that if two vectors are separated by a non-zero angle, then after augmenting these
vectors with any arbitrary vectors, the new vectors remain separated by some non-zero angle as
well. This straightforward idea will help us extend the two subspace case in theorem 2 to multiple
subspaces.
Lemma 4 Let x1 , y1 be any two fixed vectors of same dimensionality with respect to each other
xT y
such that kx1 k12 ky11 k2 = ? where ? ? [0, 1). Let x2 , y2 be any two arbitrary vectors of same
dimensionality with respect to each other. Then there exists a constant ?? ? [0, 1) such that vectors
0T 0
y
x0 = [x1 ; x2 ] and y 0 = [y1 ; y2 ] are also separated such that kx0xk2 ky
? ?? .
0k
2
Proof of theorem 3:
3
Algorithm 1 Computation of projection matrix P
INPUT: X,K,?, itermax
for k=1 to K do
?
w2? ? random vector in RNk
while iter < itermax or ? not converged do
? w?
X
w1? ? maxw1 kXk w1 ? kX? kkw?2k2 k2 + ?kw1 k2
2
w1? ? w1? /norm(w1? )
X w?
? k w2 k2 + ?kw2 k2
w2? ? maxw2 k kXkkw?1k2 ? X
1
w2? ? w2? /norm(w2? )
? k w? )
? ? (Xk w1? )T (X
2
end while
? k w? ]
Pk ? [Xk w1? , X
2
end for
P ? ? [P1 . . . PK ]
OUTPUT: P ?
P
For the proof of theorem 3, it suffices to show that data vectors from subspaces Sk and j6=k Sj
(for any k ? {1 . . . K}) are
P separated by margin less than 1 after projection using P . Let x and y
be any vectors in Sk and j6=k Sj respectively and the columns of the matrix Pk be in the span of
the ith (say) principal vector pair between these subspaces. Using theorem 2, the projected vectors
by an angle equal to the the angle between the ith principal vector
PkT x and PkT y are separated
P
pair between Sk and j6=k Sj . Let the cosine of this angle be ?. Then, using lemma 4, the added
dimensions in the vectors PkT x and PkT y to form the vectors P T x and P T y are also separated by
some margin ?? < 1. As the same argument holds for vectors from all classes, the Independent
Subspace Structure of the dataset remains preserved after projection.
For any two disjoint subspaces, theorem 2 tells us that there is a two dimensional plane in which
the entire projected subspaces form two lines. It can be argued that after adding arbitrary valued
finite dimensions to the basis of this plane, the two projected subspaces will also remain disjoint
(see proof of theorem 3). Theorem 3 simply applies this argument to each subspace and the sum of
the remaining subspaces one at a time. Thus for K subspaces, we get 2K projection vectors.
Finally, our approach projects data to 2K dimensions which could be a concern if the original
feature dimension itself is less than 2K. However, since we are only concerned with data that has
underlying independent subspace assumption, notice that the feature dimension must be at least
K. This is because each class must lie on at least 1 dimension which is linearly independent of
others. However, this is too strict an assumption and it is straight forward to see that if we relax this
assumption to 2 dimensions for each class, the feature dimensions are already at 2K.
3.1
Implementation
A naive approach to finding projection vectors (say for a binary class case) would be to compute
the SVD of the matrix X1T X2 , where the columns of X1 and X2 contain vectors from class 1 and
class 2 respectively. For large datasets this would not only be computationally expensive but also
be incapable of handling noise. Thus, even though theorem 3 guarantees the structure preservation
of the dataset X after projection using P as specified, this does not solve the problem of dimensionality reduction. The reason is that given a labeled dataset sampled from a union of independent
subspaces, we do not have any information about the basis or even the dimensionality of the underlying subspaces. Under these circumstances, constructing the projection matrix P as specified
in theorem 3 itself becomes a problem. To solve this problem, we propose
P an algorithm that tries
to find the underlying principal vector pair between subspaces Sk and j6=k Sj (for k = 1 to K)
given the labeled dataset X. The assumption behind this attempt is that samples from each subspace
(class) are not heavily corrupted and that the underlying subspaces are independent.
Notice that we are not specifically interested in a particular principal vector pair between any two
subspaces for the computation of the projection matrix. This is because we have assumed independent subspaces and so each principal vector pair is separated by some margin ? < 1. Hence we
4
need an algorithm that computes any arbitrary principal vector pair, given data from two independent subspaces. These vectors can then be used to form one of the K submatrices in P as specified
in theorem 3 . ForPcomputing the submatrix Pk , we need to find a principal vector pair between
subspaces Sk and j6=k Sj . In terms of dataset X, we estimate the vector pair using data in Xk
? k where X
? k := X \ {Xk }. We repeat this process for each class to finally form the entire
and X
?
matrix P . Our approach is stated in algorithm 1. For each class k, the idea is to start with a random
? k and find the vector in Xk closest to this vector. Then fix this vector and
vector in the span of X
? k . Repeating this process till the convergence of the cosine between
search of the closest vector in X
these 2 vectors leads to a principal vector pair. In order to estimate the closest vector from opposite subspace, we have used a quadratic program in 1 that minimizes the reconstruction error of the
fixed vector (of one subspace) using vectors from the opposite subspace. The regularization in the
optimization is to handle noise in data.
3.2
Justification
The definition 1 for margin ? between two subspaces S1 and S2 can be equivalently expressed as
1 ? ? = min
w1 ,w2
1
kB1 w1 ? B2 w2 k2 s.t. kB1 w1 k2 = 1, kB2 w2 k2 = 1
2
(4)
where the columns of B1 ? Rn?d1 and B2 ? Rn?d2 are the basis of the subspaces S1 and S2
respectively such that B1T B1 and B2T B2 are both identity matrices.
Proposition 5 Let B1 ? Rn?d1 and B2 ? Rn?d2 be the basis of two disjoint subspaces S1 and
S2 . Then for any principal vector pair (ui , vi ) between the subspaces S1 and S2 , the corresponding
vector pair (w1 ? Rd1 ,w2 ? Rd2 ), s.t. ui = B1 w1 and vi = B2 w2 , is a local minima to the
objective in equation (4).
Proof: The Lagrangian function for the above objective is:
L(w1 , w2 , ?) =
1
1 T T
w B B1 w1 + w2T B2T B2 w2 ?w1T B1T B2 w2 +?1 (kB1 w1 k2 ?1)+?2 (kB2 w2 k2 ?1)
2 1 1
2
(5)
Then setting the gradient w.r.t. w1 to zero we get
?w1 L = (1 + ?1 )w1 ? B1T B2 w2 = 0
(6)
Let U SV T be the SVD of B1T B2 and w1 and w2 be the ith columns of U and V respectively. Then
equation (6) becomes
?w L = (1 + ?1 )w1 ? U SV T w2
= (1 + ?1 )w1 ? Sii w1 = 0
(7)
Thus the gradient w.r.t. w1 is zero when ?1 = 1 ? Sii . Similarly, it can be shown that the gradient
w.r.t. w2 is zero when ?2 = 1 ? Sii . Thus the gradient of the Lagrangian L is 0 w.r.t. both w1 and
w2 for every corresponding principal vector pair. Thus vector pair (w1 , w2 ) corresponding to any
of the principal vector pairs between subspaces S1 and S2 is a local minima to the objective 4.
Since (w1 , w2 ) corresponding to any principal vector pair between two disjoint subspaces form a
local minima to the objective given by equation (4), one can alternatively minimize equation (4)
w.r.t. w1 and w2 and reach one of the local minima. Thus, by assuming independent subspace
structure for all the K classes in algorithm 1 and setting ? to zero, it is straight forward to see that
the algorithm yields a projection matrix that satisfies the criteria specified by theorem 3.
Finally, real world data do not strictly satisfy the independent subspace assumption in general and
even a slight corruption in data may easily lead to the violation of this independence. In order to
tackle this problem, we add a regularization (? > 0) term while solving for the principal vector
pair in algorithm 1. If we assume that the corruption is not heavy, reconstructing a sample using
vectors belonging to another subspace would require a large coefficient over those vectors. The
regularization avoids reconstructing data from one class using vectors from another class that are
slightly corrupted by assigning such vectors small coefficients.
5
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(a) Data projected
using Pa
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(b) Data projected
using Pb
?0.9
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(c)
?0.4
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0
(d)
Figure 3: Four different pairs of classes from
the Extended Yale dataset B projected onto
two dimensional subspaces using proposed
approach. See section 4.1.1 for details.
Figure 2: Qualitative comparison between
(a) true projection matrix and (b) projection
matrix from the proposed approach on high
dimensional synthetic two class data. See
section 4.1.1 for details.
3.3
?0.2
?0.25
Complexity
Solving algorithm 1 requires solving an unconstrained quadratic program within a while-loop. Assume that we run this while loop for T iterations and that we use conjugate gradient descent to solve
the quadratic program in each iteration. Also, it is known that for any matrix A ? Ra?b and vector
b ? Ra , conjugate gradient applied to a problem of the form
arg minkAx ? bk2
(8)
w
?
takes time O(ab K), where K is the condition number of AT A. Thus it is straight forward to
see that the?time required to compute the projection matrix for a K class problem in our case is
O(KT nN K), where n is the dimensionality of feature space, N is the total number of samples
and K is the condition number of the matrix (XkT Xk + ?I). Here I is the identity matrix.
4
Empirical Analysis
In this section, we present empirical evidence to support our theoretical analysis of our subspace
learning approach. For real world data, we use the following datasets:
1. Extended Yale dataset B [3]: It consists of ? 2414 frontal face images of 38 individuals (K = 38)
with 64 images per person. These images were taken under constrained but varying illumination
conditions.
2. AR dataset [10]: This dataset consists of more than 4000 frontal face images of 126 individuals
with 26 images per person. These images were taken under varying illumination, expression and
facial disguise. For our experiments, similar to [15], we use images from 100 individuals (K =
100) with 50 males and 50 females. We further use only 14 images per class which correspond to
illumination and expression changes. This corresponds to 7 images from Session 1 and rest 7 from
Session 2.
3. PIE dataset [12]: The pose, illumination, and expression (PIE) database is a subset of CMU PIE
dataset consisting of 11, 554 images of 68 people (K = 68).
We crop all the images to 32 ? 32, and concatenate all the pixel intensity to form our feature vectors.
Further, we normalize all data vectors to have unit `2 norm.
6
(a) Yale dataset B
(b) AR dataset
(c) PIE dataset
Figure 4: Multi-class separation after projection using proposed approach for different
datasets. See section 4.1.2 for details.
4.1 Qualitative Analysis
4.1.1 Two Subspaces-Two Lines
We test both the claim of theorem 2 and the quality of approximation achieved by algorithm 1 in
this section. We perform these tests on both synthetic and real data.
1. Synthetic Data: We generate two random subspaces in R1000 of dimensionality 20 and 30 (notice
that these subspaces will be independent with probability 1). We randomly generate 100 data vectors
from each subspace and normalize them to have unit length. We then compute the 1st principal
vector pair between the two subspaces using their basis vectors by performing SVD of B1T B2 ,
where B1 and B2 are the basis of the two subspaces. We orthonormalize the vector pair to form the
projection matrix Pa . Next, we use the labeled dataset of 200 points generated to form the projection
matrix Pb by applying algorithm 1. The entire dataset of 200 points is then projected onto Pa and Pb
separately and plotted in figure 2. The green and red points denote data from either subspace. The
results not only substantiate our claim in theorem 2 but also suggest that the proposed algorithm for
estimating the projection matrix is a good approximation.
2. Real Data: Here we use Extended Yale dataset B for analysis. Since we are interested in projection of two class data in this experimental setup, we randomly choose 4 different pairs of classes
from the dataset and use the labeled data from each pair to generate the two dimensional projection
matrix (for that pair) using algorithm 1. The resulting projected data from the 4 pairs can be seen in
figure 3. As is evident from the figure, the projected two class data for each pair approximately lie
along two different lines.
4.1.2
Multi-class separability
We analyze the separation between the K classes of a given K-class dataset after dimensionality
reduction. First we compute the projection matrix for that dataset using our approach and project the
data. Second, we compute the top principal vector for each class separately from the projected data.
This gives us K vectors. Let the columns of the matrix Z ? R2K?K contain these vectors. Then
in order to visualize inter-class separability, we simply take the dot product of the matrix Z with
itself, i.e. Z T Z. Figure 4 shows this visualization for the three face datasets. The diagonal elements
represent self-dot product; thus the value is 1 (white). The off-diagonal elements represent interclass dot product and these values are consistently small (dark) for all the three datasets reflecting
between class separability.
4.2
Quantitative Analysis
In order to evaluate theorem 3, we perform a classification experiment on all the three real world
datasets mentioned above after projecting the data vectors using different dimensionality reduction
techniques. We compare our quantitative results against PCA, Linear discriminant analysis (LDA),
Regularized LDA and Random Projections (RP) 1 . We make use of sparse coding [15] for classification.
1
We also used LPP (Locality Preserving Projections) [4], NPE (Neighborhood Preserving Embedding) [5],
and Laplacian Eigenmaps [1] for dimensionality reduction on Extended Yale B dataset. However, because
the best performing of these reduction techniques yielded a result of only 73% compared to the close to 98%
accuracy from our approach, we do not report results from these methods.
7
For Extended Yale dataset B, we use all 38 classes for evaluation with 50% ? 50% train-test split 1
and 70% ? 30% train-test split 2. Since our method is randomized, we perform 50 runs of computing the projection matrix using algorithm 1 and report the mean accuracy with standard deviation.
Similarly for RP, we generate 50 different random matrices and then perform classification. Since
all other methods are deterministic, there is no need for multiple runs.
Table 1: Classification Accuracy on Extended Yale dataset B with 50%-50% train-test split.
See section 4.2 for details.
Method
Ours
PCA LDA Reg-LDA
RP
dim
76
76
37
37
76
acc
98.06 ? 0.18 92.54 83.68
95.77
93.78 ? 0.48
Table 2: Classification Accuracy on Extended Yale dataset B with 70%-30% train-test split.
See section 4.2 for details.
Method
Ours
PCA LDA Reg-LDA
RP
dim
76
76
37
37
76
acc
99.45 ? 0.20 93.98 93.85
97.47
94.72 ? 0.66
Table 3: Classification Accuracy on AR dataset. See section 4.2 for details.
Method
Ours
PCA LDA Reg-LDA
RP
dim
200
200
99
99
200
acc
92.18 ? 0.08 85.00
88.71
84.76 ? 1.36
Table 4: Classification Accuracy on a subset of CMU PIE dataset. See section 4.2 for details.
Method
Ours
PCA LDA Reg-LDA
RP
dim
136
136
67
67
136
acc
93.65 ? 0.08 87.76 86.71
92.59
90.46 ? 0.93
Table 5: Classification Accuracy on a subset of CMU PIE dataset. See section 4.2 for details.
Method
Ours
PCA LDA Reg-LDA
RP
dim
20
20
9
9
20
acc
99.07 ? 0.09 97.06 95.88
97.25
95.03 ? 0.41
For AR dataset, we take the 7 images from Session 1 for training and the 7 images from Session 2
for testing. The results are shown in table 3. The result using LDA is not reported because we found
that the summed within class covariance was degenerate and hence LDA was not applicable. It can
be clearly seen that our approach significantly outperforms other dimensionality reduction methods.
Finally for PIE dataset, we perform experiments on two different subsets. First, we take all the
68 classes and for each class, we randomly choose 25 images for training and 25 for testing. The
performance for this subset is shown in table 4. Second, we take only the first 10 classes of the
dataset and of all the 170 images per class, we randomly split the data into 70% ? 30% train-test set.
The performance for this subset is shown in table 5.
Evidently, our approach consistently yields the best performance on all the three datasets compared
to other dimensionality reduction methods.
5
Conclusion
We proposed a theoretical analysis on the preservation of independence between multiple subspaces.
We show that for K independent subspaces, 2K projection vectors are sufficient for independence
preservation (theorem 3). This result is motivated from our observation that for any two disjoint
subspaces of arbitrary dimensionality, there exists a two dimensional plane such that after projection, the entire subspaces collapse to just two lines (theorem 2). Resulting from this analysis, we
proposed an efficient iterative algorithm (1) that tries to exploit these properties for learning a projection matrix for dimensionality reduction that preserves independence between multiple subspaces.
Our empirical results on three real world datasets yield state-of-the-art results compared to popular
dimensionality reduction methods.
8
References
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9
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4,762 | 5,311 | Distributed Balanced Clustering via Mapping
Coresets
Aditya Bhaskara
Google NYC
[email protected]
MohammadHossein Bateni
Google NYC
[email protected]
Vahab Mirrokni
Google NYC
[email protected]
Silvio Lattanzi
Google NYC
[email protected]
Abstract
Large-scale clustering of data points in metric spaces is an important problem in
mining big data sets. For many applications, we face explicit or implicit size constraints for each cluster which leads to the problem of clustering under capacity
constraints or the ?balanced clustering? problem. Although the balanced clustering problem has been widely studied, developing a theoretically sound distributed
algorithm remains an open problem. In this paper we develop a new framework
based on ?mapping coresets? to tackle this issue. Our technique results in first
distributed approximation algorithms for balanced clustering problems for a wide
range of clustering objective functions such as k-center, k-median, and k-means.
1
Introduction
Large-scale clustering of data points in metric spaces is an important problem in mining big data
sets. Many variants of such clustering problems have been studied spanning, for instance, a wide
range of `p objective functions including the k-means, k-median, and k-center problems. Motivated
by a variety of big data applications, distributed clustering has attracted significant attention over the
literature [11, 4, 5]. In many of these applications, an explicit or implicit size constraint is imposed
for each cluster; e.g., if we cluster the points such that each cluster fits on one machine, the size
constraint is enforced by the storage constraint on each machine. We refer to this as balanced clustering. In the setting of network location problems, these are referred to as capacitated clustering
problems [6, 16, 17, 10, 3]. The distributed balanced clustering problem is also well-studied and
several distributed algorithms have been developed for it in the context of large-scale graph partitioning [21, 20]1 . Despite this extensive literature, none of the distributed algorithms developed for
the balanced version of the problem have theoretical approximation guarantees. The present work
presents the first such distributed algorithms for a wide range of balanced clustering problems with
provable approximation guarantees. To acheive this goal, we develop a new technique based on
mapping coresets.
A coreset for a set of points in a metric space is a subset of these points with the property that an
approximate solution to the whole point-set can be obtained given the coreset alone. An augmented
concept for coresets is the notion of composable coresets which have the following property: for a
collection of sets, the approximate solution to the union of the sets in the collection can be obtained
given the union of the composable coresets for the point sets in the collection. This notion was
1
A main difference between the balanced graph partitioning problems and balanced clustering problems
considered here is that in the graph partitioning problems a main objective function is to minimize the cut
function.
1
MapReduce model
Problem
Approximation
L-balanced k-center
O(1)
k-clustering in `p
O(p)
L-balanced k-clustering in `p
(O(p),2)
Streaming model
Problem
Approximation
L-balanced k-center
O(1)
k-clustering in `p
O(p)
L-balanced k-clustering in `p
(O(p),2)
Rounds
O(1)
O(1)
O(1)
Passes
O(1)
O(1)
O(1)
Table 1: Our contributions, all results hold for k < n1/2? , for constant > 0. We notice that for
the L-balanced k-clustering (p) general we get a bicriteria optimization (we can potentially open 2k
centers in our solutions).
formally defined in a recent paper by Indyk et al [14]. In this paper, we augment the notion of
composable coresets further, and introduce the concept of mapping coresets. A mapping coreset is a
coreset with an additional mapping of points in the original space to points in the coreset. As we will
see, this will help us solve balanced clustering problems for a wide range of objective functions and
a variety of massive data processing applications, including streaming algorithms and MapReduce
computations. Roughly speaking, this is how a mapping coreset is used to develop a distributed
algorithm for the balanced clustering problems: we first partition the data set into several blocks in
a specific manner. We then compute a coreset for each block. In addition, we compute a mapping
of points in the original space to points in the coreset. Finally, we collect all these coresets, and then
solve the clustering problem for the union of the coresets. We can them use the (inverse) map to get
back a clustering for the original points.
Our Contributions. In this paper, we introduce a framework for solving distributed clustering
problems. Using the concept of mapping coresets as described above, our framework applies to
balanced clustering problems, which are much harder than their unrestricted counterparts in terms
of approximation.
The rough template of our results is the following: given a single machine ?-approximation algorithm for a clustering problem (with or without balance constraints), we give a distributed algorithm
for the problem that has an O(?) approximation guarantee. Our results also imply streaming algorithms for such clustering problems, using sublinear memory and constant number of passes. More
precisely, we consider balanced clustering problems with an `p objective. For specific choice of p, it
captures the commonly used k-center, k-median and k-means objectives. Our results are also very
robust?for instance, bicriteria approximations (violating either the number of clusters or the cluster
sizes) on a single machine can be used to give distributed bicriteria approximation algorithms, with
a constant loss in the cost. This is particularly important for balanced versions of k-median and
k-means, for which we know of constant factor approximation to the cost only if we allow violating
one of the constraints. (Moreover, mild violation might not be terribly bad in certain applications,
as long as we obtain small cost.)
Finally, other than presenting the first distributed approximations for balanced clustering, our general framework also implies constant-factor distributed approximations for a general class of uncapacitated clustering problems (for which we are not aware of distributed algorithms with formal
guarantees). We summarize our new results in Table 1.
Related Work. The notion of coresets has been introduced in [2]. In this paper, we use the term
coresets to refer to an augmented notion of coresets, referred to as ?composable coresets? [14].
The notion of (composable) coresets are also related to the concept of mergeable summaries that
have been studied in the literature [1]. The main difference between the two is that aggregating
mergeable summaries does not increase the approximation error, while in the case of coresets the
error amplifies. The idea of using coresets has been applied either explicitly or implicitly in the
streaming model [12, 2] and in the MapReduce framework [15, 18, 5, 14]. However, none of the
previous work applies these ideas for balanced clustering problems.
2
There has been a lot of work on designing efficient distributed algorithms for clustering problems in
metric spaces. A formal computation model for the MapReduce framework has been introduced by
Karloff et al. [15]. The first paper that studied clustering problems in this model is by Ene et al. [11],
where the authors prove that one can use an ? approximation algorithm for the k-center or k-median
problem to obtain a 4? + 2 and a 10? + 3 approximation respectively for the k-center or k-median
problems in the MapReduce model. Subsequently Bahmani et al. [4] showed how to implement kmeans++ efficiently in the MapReduce model. Finally, very recently, Balcan et al. [5] demonstrate
how one can use an ? approximation algorithm for the k-means or k-median problem to obtain
coresets in the distributed (and MapReduce) setting. They however do not consider the balanced
clustering problems or the general set of clustering problems with the `p objective function.
The literature of clustering in the streaming model is also very rich. The first paper we are aware
of is due to Charikar et al. [7], who study the k-center problem in the classic streaming setting.
Subsquently Guha et al. [12] give the first single pass constant approximation algorithm to the kmedian problem. Following up on this, the memory requirements and the approximation factors of
their result were further improved by Charikar et al. in [8].
Finally, capacitated (or balanced) clustering is well studied in approximation algorithms [6, 16,
9], with constant factors known in some cases and only bicriteria in others. Our results may be
interpreted as saying that the capacity constraints may be a barrier to approximation, but are not a
barrier to parallelizability. This is the reason our approximation guarantees are bicriteria.
2
Preliminaries
In all the problems we study, we will denote by (V, d) the metric space we are working with. We
will denote n = |V |, the number of points in V . We will also write duv as short hand for d(u, v).
Given points u, v, we assume we have an oracle access to duv (or can compute it, as in geometric
settings). Formally, a clustering C of a set of points V is a collection of sets C1 , C2 , . . . , Cr which
partition V . Each cluster Ci has a center vi , and we define the ?`p cost? of this clustering as
!1/p
XX
p
costp (C) :=
d(v, vi )
.
(1)
i
v?Ci
When p is clear from the context, we will simply refer to this quantity as the cost of the clustering
and denote it cost(C).
Let us now define the L-balanced k-clustering problem with `p cost.
Definition 1 (L-balanced k-clustering (p)). Given (V, d) and a size bound L, find a clustering C of
V which has at most k clusters, at most L points in each cluster, and cluster centers v1 , . . . , vk so
as to minimize costp (C), the `p cost defined in Eq. (1).
The case p = 1 is the capacitated k-median and with p = ? is also known as the capacitated
k-center problem (with uniform capacities).
Definition 2 (Mapping and mapping cost). Given a multiset S and a set V , we call
P a bijective function f : V ? S a mapping from V to S and we define the cost of a mapping as v?V d(v, f (v))p .
Definition 3 (Clustering and optimal solution). Given a clustering problem P with an `p objective,
we define OP TP as the cost of the optimal solution to P.
3
Mapping coreset framework
The main idea behind our distributed framework is a new family of coresets that help in dealing with
balanced clustering.
Definition 4 (?-mapping coreset). Given a set of points V , a ?-mapping coreset for a clustering
problem P consists of a multiset S with elements from V , and a mapping from V to S such that the
total cost of the mapping is upper bounded by ? ? OP TPp . We define the size of a ?-mapping coreset
as the number of distinct elements in S.
Note that our definition does not prescribe the size of the mapping coreset ? this can be a parameter
we choose. We now define the composability of coresets.
3
Definition 5 (Composable ?-mapping coreset). Given disjoint sets of points V1 , V2 , . . . , Vm , and
corresponding ?-mapping coresets S1 , S2 , . . . , Sm , the coresets are said to be composable if we have
that ?i Si is a 2p ?-mapping coreset for ?i Vi (the overall map is the union of those for V1 , . . . , Vm ).
Remark. The non-trivial aspect of showing that coresets compose comes from the fact that we
compare the cost of mapping to the cost of OPTP on the union of Vi (which we need to show is not
too small). Our main theorem is now the following
Theorem 1. Let V be a set of points and suppose L, k, p ? 1 are parameters. Then for any
U ? V , there exists an algorithm that takes U as input, and produces a 2p -mapping coreset for the
2
?
L-balanced k-clustering (p) problem for U . The size of this coreset is O(k),
and the algorithm
uses space that is quadratic in |U |. Furthermore, for any partition V1 , V2 , . . . , Vr of V , the mapping
coresets produced by the algorithm on V1 , V2 , . . . , Vr compose.
Clustering via ?-mapping coresets
3.1
The theorem implies a simple general framework for distributed clustering:
1. Split the input into m chunks arbitrarily (such that each chunk fits on a machine), and
compute a (composable) 2p -mapping coreset for each of the chunks. For each point in the
coreset, assign a multiplicity equal to the number of points mapped to it (including itself).
2. Gather all the coresets (and multiplicities of their points) into one machine, and compute a
k-clustering of this multiset.
3. Once clusters for the points (and their copies) are found, we can ?map back?, and find a
clustering of the original points.
The idea is that in each chunk, the size of the coreset will be small, thus the union of the coresets
is small (and hence fits on one machine). The second step requires care: the clustering algorithm
should work when the points have associated multiplicities, and use limited memory. This is captured as follows.
Definition 6 (Space-efficient algorithm). Given an instance (V, d) for a k-clustering problem in
which V has N distinct points, each with some multiplicity, a sequential ?-approximation algorithm
is called space-efficient if the space used by the algorithm is O(N 2 ? poly(k)).
The framework itself is a very natural one, thus the key portions are the step of finding the mapping
coresets that (a) have small mapping cost and (b) compose well on arbitrary partition of the input,
and that of finding space efficient algorithms. Sections 4 and 5 give details of these two steps.
Further, because the framework is general, we can apply many ?levels? of it. This is illustrated
below in Section 3.2.
To prove the correctness of the framework, we also need to prove that moving from the original
points in a chunk to a coreset with multiplicities (as described in (1)) does not affect us too much in
the approximation. We prove this using a general theorem:
Theorem 2. Let f : V 7? S be a bijection. Let C be any clustering of V , and let C 0 denote the clustering of S obtained by applying a bijection f to the clustering C. Then there exists
P a choice of centers for C 0 such that cost(C 0 )p ? 22p?1 (cost(C)p + ?total ), where ?total denotes v?V d(v, f (v))p .
In our case, if we consider the set of points in the coreset with multiplicities, the mapping gives a
bijection, thus the above theorem applies in showing that the cost of clustering is not much more than
the ?mapping cost? given by the bijection. The theorem can also be used in the opposite direction,
as will be crucial in obtaining an approximation guarantee.
Preserving balanced property. The above theorem allows us to move back and forth (algorithmically) between clusterings of V and (the coreset with multiplicities) S as long as there is a small-cost
mapping. Furthermore, since f is a bijection, we have the property that if the clustering was balanced
in V , the corresponding one in S will be balanced as well, and vice versa.
Putting things together. Let us now see how to use the theorems to obtain approximation guarantees. Suppose we have a mapping f from V to the union of the coresets of the chunks (called
2
? is used to hide a logarithmic factor.
Here and elsewhere below, O(?)
4
S, which is a multi set), with total mapping cost ?total . Suppose also that we have an ? spaceefficient approximation algorithm for clustering S. Now we can use the Theorem 2 to show that in
S, there exists a clustering whose cost, raised to the p-th power, is at most 22p?1 (cost(C)p + ?total ).
This means that the approximation algorithm on S gives a clustering of cost (to the pth power)
? 22p?1 ?p (cost(C)p + ?total ). Finally, using Theorem 2 in the opposite direction, we can map
back the clusters from S to V and get a an upper bound on the clustering cost (to the pth power) of
22p?1 (22p?1 ?p (cost(C)p + ?total ) + ?total ). But now using Theorem 1, we know that for the f in our
algorithm, ?total ? 2p cost(C)p . So plugging this into the bound above, and after some manipulations
(and taking pth roots) we obtain that the cost of the final clustering is ? 32?cost(C). The details of
this calculation can be found in the supplementary material.
Remark. The approximation ratio (i.e., 32?) seems quite pessimistic. In our experiments, we have
observed (if we randomly partition the points initially) that the constants are much better (often at
most 1.5). The slack in our analysis arises mainly because of Theorem 2, in which the worst case in
the analysis is very unlikely to occur in practice.
3.2
Mapping Coresets for Clustering in MapReduce
The above distributed algorithm can be placed in the formal model for MapReduce introduced by
Karloff et al. [15].
The model has two main restrictions, one on the total number of machines and another on the
memory available on each machine. In particular, given an input of size N , and a sufficiently
small ? > 0, in the model there are N 1?? machines, each with N 1?? memory available for the
computation. As a result, the total amount of memory available to the entire system is O(N 2?2? ).
In each round a computation is executed on each machine in parallel and then the outputs of the
computation are shuffled between the machines.
In this model the efficiency of an algorithm is measured by the number of the ?rounds? of MapReduce
in the algorithm. A class of algorithms of particular interest are the ones that run in a constant
number of rounds. This class of algorithms are denoted MRC 0 .
The high level idea is to use coreset construction and a sequential space-efficient ?-approximation
algorithm (as outlined above). Unfortunately, this approach does not work as such in the MapReduce model because both the coreset construction algorithm, and the space-efficient algorithm, require memory quadratic in the size of their input. Therefore we perform multiple ?levels? of our
framework.
Given an instance (V, d), the MapReduce algorithm proceeds as follows:
1. Partition the points arbitrarily into 2n(1+?)/2 sets.
2. Compute the composable 2p -mapping coreset on each of the machines (in parallel) to obtain
e
f and the multisets S1 , S2 , . . . , S2n(1+?)/2 , each with roughly O(k)
distinct points.
3. Partition the computed coreset again into n1/4 sets.
4. Compute composable 2p -mapping coresets on each of the machines (in parallel) to obtain
e
f 0 , and multisets S10 , S20 , . . . , Sn0 1/4 , each with O(k)
distinct points.
5. Merge all the S10 , S20 , . . . , Sn0 1/4 on a single machine and compute a clustering using the
sequential space-efficient ?-approximation algorithm.
6. Map back the points in S10 , S20 , . . . , Sn0 1/4 to the points in S1 , S2 , . . . , S2n(1+?)/2 using the
function f 0?1 and obtain a clustering of the points in S1 , S2 , . . . , S2n(1+?)/2 .
7. Map back the points in S1 , S2 , . . . , S2n(1+?)/2 to the points in V using the function f ?1 and
thus obtain a clustering of the initial set of points.
Note that if k < n1/4? , for constant > ?, at every step of the MapReduce, the input size on
each machine is bounded by n(1??)/2 and thus we can run our coreset reduction and a space-efficient
algorithm (in which we think of the poly(k) as constant ? else we need minor modification). Furthermore if n1/4? ? k < n(1?)/2 , for constant > ?, we can exploit the trade-off between number
of rounds and approximation factor to get a similar result (refer to the supplement for details).
5
Figure 1: We split the input into m parts, compute mapping coresets for each part, and aggregate
them. We then compute a solution to this aggregate and map the clustering back to the input.
We are now ready to state our main theorem in the MapReduce framework:
Theorem 3. Given an instance (V, d) for a k-clustering problem, with |V | = n and a sequential
space-efficient ? approximation algorithm for the (L-balanced) k-clustering (p) problem, there exists a MapReduce algorithm that runs in O(1) rounds and obtains an O(?) approximation for the
(L-balanced) k-clustering (p) problem, for L, p ? 1 and 0 < k < n(1?)/2 (constant > 0).
The previous theorem combined with the results of Section 5 gives us the results presented in Table 1. Furthermore it is possible to extend this approach to obtain streaming algorithms via the same
techniques. We defer the details of this to the supplementary material.
4
Coresets and Analysis
We now come to the proof of our main result?Theorem 1. We give an algorithm to construct
coresets, and then show that coresets constructed this way compose.
Constructing composable coresets.
Suppose we are given a set of points V . We first show how to select a set of points S that are
close to each vertex in V , and use this set as a coreset with a good mapping f . The selection of
S uses a modification of the algorithm of Lin and Vitter [19] for k-median. We remark that any
approximation algorithm for k-median with `p objective can be used in place of the linear program
(as we did in our experiments, for p = ?, in which a greedy farthest point traversal can be used).
Consider a solution (x, y) to the following linear programming (LP) relaxation:
XX
min
d(u, v)p xuv
subject to
u
v
X
xuv = 1
for all u
(every u assigned to a center)
v
xuv ? yv
X
yu ? k
for all u, v
(assigned only to center)
(at most k centers)
u
0 ? xuv , yu ? 1
for all u, v.
In the above algorithms, we can always treat p ? log n, and in particular the case p = ?, as
p = log n. This introduces only negligible error in our computations but make them tractable. More
specifically, when working with p = log n, the power operators do not increase the size of the input
by more than a factor log n.
6
Rounding We perform a simple randomized rounding with weights scaled up by O(log n): round
each yu to 1 with a probability equal to min{1, yu (4 log n)/}. Let us denote this probability by yu0 ,
and the set of ?centers? thus obtained, by S. We prove the following (proof in the supplement)
Lemma 4. With probability (1?1/n), the set S of selected centers satisfies the following properties.
1. Each vertex has a relatively close selected center. In particular, for every u ? V , there is a
h
i1/p
P
center opened at distance at most (1 + ) v d(u, v)p xuv
.
2. Not too many centers are selected; i.e., |S| <
8k log n
.
Mapping and multiplicity. Once we have a set S of centers, we map every v ? V the center closest
to it, i.e., f (v) = arg mins?S d(v, s). If ms points in V are mapped to some s ? S, we set its
multiplicty to ms . This defines a bijection from V to the resulting multiset.
Composability of the coresets.
We now come to the crucial step, the proof of composability for the mapping coresets constructed
earlier, i.e., the ?furthermore? part of Theorem 1.
To show this, we consider any vertex sets V1 , V2 , . . . , Vm , and mapping coresets S1 , S2 , . . . , Sm
obtained by the rounding algorithm above. We have to prove that the total moving cost is at most
(1 + )2p OP TP , where the optimum value is for the instance ?i Vi . We denote by LP (Vi ) the
optimum value of the linear program
above, when the set of points involved is Vi . Finally, we write
P
?v := d(v, fv )p , and ?total := v?V ?v . We now have:
Lemma 5. Let LPi denote the objective value of the optimum solution to LP (Vi ), i.e., the LP
relaxation written earlier when only vertices in Vi are considered. Then we have
X
?total ? (1 + )
LPi .
i
The proof follows directly from Lemma 4 and the definition of f . The next lemma is crucial: it
shows that LP (V ) cannot be too small. The proof is deferred to the supplement.
P
Lemma 6. In the notation above, we have i LPi ? 2p ? LP (V ).
The two lemmas imply that the total mapping cost is at most (1 + )2p OP TP , because LP (V ) is
clearly ? OP TP . This completes the proof of Theorem 1.
5
Space efficient algorithms on a single machine
Our framework ultimately reduces distributed computation to a sequential computation on a compressed instance. For this, we need to adapt the known algorithms on balanced k-clustering, in order
to handle compressed instances. We now give a high level overview and defer the details to the
supplementary material.
For balanced k-center, we modify the linear programming (LP) based algorithm of [16], and its analysis to deal with compressed instances. This involves the following trick: if we have a compressed
instance with N points, since there are only k centers to open, at most k ?copies? of each point are
candidate centers. We believe this trick can be applied more generally to LP based algorithms.
For balanced k-clustering with other `p objectives (even p = 1), it is not known how to obtain
constant factor approximation algorithms (even without the space efficient restriction). Thus we
consider bicriteria approximations, in which we respect the cluster size constraints, but have up to
2k clusters. This can be done for all `p objectives as follows: first solve the problem approximately
without enforcing the balanced constraint, then post-process the clusters obtained. If a cluster contains ni points for ni > L, then subdivide the cluster into dni /Le many clusters. The division
should be done carefully (see supplement).
The post-processing step only involves the counts of the vertices in different clusters, and hence can
be done in a space efficient manner. Thus the crucial part is to find the ?unconstrained? k-clustering
in a space efficient way. For this, the typical algorithms are either based on local search (e.g., due
7
Graph
US
World
Relative size of
sequential instance
0.33%
0.1%
Relative increase
in radius
+52%
+58%
Table 2: Quality degradation due to the two-round approach.
Figure 2: Scalability of parallel implementation.
to [13]), or based on rounding linear programs. The former can easily be seen to be space efficient
(we only need to keep track of the number of centers picked at each location). The latter can be
made space efficient using the same trick we use for k-center.
6
Empirical study
In order to gauge its practicality, we implement our algorithm. We are interested in measuring its
scalability in addition to the effect of having several rounds on the quality of the solution.
In particular, we compare the quality of the solution (i.e., the maximum radius from the k-center
objective) produced by the parallel implementation to that of the sequential one-machine implementation of the farthest seed heuristic. In some sense, our algorithm is a parallel implementation of
this algorithm. However, the instance is too big for the sequential algorithm to be feasible. As a
result, we run the sequential algorithm on a small sample of the instance, hence a potentially easier
instance.
Our experiments deal with two instances to test this effect: the larger instance is the world graph
with hundreds of millions of nodes, and the smaller one is the graph of US road networks with tens
of millions of nodes. Each node has the coordinate locations, which we use to compute great-circle
distances?the closest distance between two points on the surface of the earth. We always look for
1000 clusters, and run our parallel algorithms on a few hundred machines.
Table 2 shows that the quality of the solution does not degrade substantially if we use the tworound algorithm, more suited to parallel implementation. The last column shows the increase in the
maximum radius of clusters due to computing the k-centers in two rounds as described in the paper.
Note that the radius increase numbers quoted in the table are upper bounds since the sequential
algorithm could only be run on a simpler instance. In reality, the quality reduction may be even less.
1
subset of the actual
In case of the US Graph, the sequential algorithm was run on a random 300
1
graph, whereas a random 1000 subset was used for the World Graph.
We next investigate how the running time of our algorithm scales with the size of the instance. We
focus on the bigger instance (World Graph) and once again take its random samples of different
sizes (10% up to 100%). This yields to varying instance sizes, but does not change the structure of
the problem significantly, and is perfect for measuring scalability. Figure 2 shows the increase in
running time is sublinear. In particular, a ten-fold increase in instance size only leads to a factor 3.6
increase in running time.
8
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9
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4,763 | 5,312 | Zeta Hull Pursuits:
Learning Nonconvex Data Hulls
?
Yuanjun Xiong? Wei Liu? Deli Zhao Xiaoou Tang?
Information Engineering Department, The Chinese University of Hong Kong, Hong Kong
?
IBM T. J. Watson Research Center, Yorktown Heights, New York, USA
Advanced Algorithm Research Group, HTC, Beijing, China
{yjxiong,xtang}@ie.cuhk.edu.hk [email protected] deli [email protected]
Abstract
Selecting a small informative subset from a given dataset, also called column sampling, has drawn much attention in machine learning. For incorporating structured
data information into column sampling, research efforts were devoted to the cases
where data points are ?tted with clusters, simplices, or general convex hulls. This
paper aims to study nonconvex hull learning which has rarely been investigated in
the literature. In order to learn data-adaptive nonconvex hulls, a novel approach
is proposed based on a graph-theoretic measure that leverages graph cycles to
characterize the structural complexities of input data points. Employing this measure, we present a greedy algorithmic framework, dubbed Zeta Hulls, to perform
structured column sampling. The process of pursuing a Zeta hull involves the
computation of matrix inverse. To accelerate the matrix inversion computation
and reduce its space complexity as well, we exploit a low-rank approximation to
the graph adjacency matrix by using an ef?cient anchor graph technique. Extensive experimental results show that data representation learned by Zeta Hulls can
achieve state-of-the-art accuracy in text and image classi?cation tasks.
1
Introduction
In the era of big data, a natural idea is to select a small subset of m samples Ce = {xe1 , . . . , xem }
from a whole set of n data points X = {x1 , . . . , xn } such that the selected points Ce can capture
the underlying properties or structures of X . Then machine learning and data mining algorithms can
be carried out with Ce instead of X , thereby leading to signi?cant reductions in computational and
space complexities. Let us write the matrix forms of Ce and X as C = [xe1 , . . . , xem ] ? Rd?m
and X = [x1 , . . . , xn ] ? Rd?n , respectively. Here d is the dimensions of input data points. In
other words, C is a column subset selection of X. The task of selecting C from X is also called by
column sampling in the literature, and maintains importance in a variety of ?elds besides machine
learning, such as signal processing, geoscience and remote sensing, and applied mathematics. This
paper concentrates on solving the column sampling problem by means of graph-theoretic methods.
Existing methods in column sampling fall into two main categories according to their objectives: 1)
approximate the data matrix X, and 2) discover the underlying data structures. For machine learning
methods using kernel or similar ?N-Body? techniques, the Nystr?om matrix approximation is usually
applied to approximate large matrices. Such circumstances include fast training of nonlinear kernel
support vector machines (SVM) in the dual form [30], spectral clustering [8], manifold learning [25],
etc. Minimizing a relative approximation error is typically harnessed as the objective of column sampling, by which the most intuitive solution is to perform uniform sampling [30]. Other non-uniform
sampling schemes choose columns via various criteria, such as probabilistic samplings according
to diagonal elements of a kernel matrix [7], reconstruction errors [15], determinant measurements
[1], cluster centroids [33], and statistical leverage scores [21]. On the other hand, column sampling
1
may be cast into a combinatorial optimization problem, which can be tackled by using greedy strategies in polynomial time [4] and boosted by using advanced sampling strategies to further reduce the
relative approximation error [14].
From another perspective, we are aware that data points may form some interesting structures. Understanding these structures has been proven bene?cial to approximate or represent data inputs [11].
One of the most famous algorithms for dimensionality reduction, Non-negative Matrix Factorization
(NMF) [16], learns a low-dimensional convex hull from data points through a convex relaxation [3].
This idea was extended to signal separation by pursuing a convex hull with a maximized volume
[27] to enclose input data points. Assuming that vertices are equally distant, the problem of ?tting
a simplex with a maximized volume to data reduces to a simple greedy column selection procedure
[26]. The simplex ?tting approach demonstrated its success in face recognition tasks [32]. Parallel research in geoscience and remote sensing is also active, where the vertices of a convex hull
are coined as endmembers or extreme points, leading to a classic ?N-Finder? algorithm [31]. The
above approaches tried to learn data structures that are usually characterized by convexity. Hence,
they may fail to reveal the intrinsic data structures when the distributions of data points are diverse,
e.g., data being on manifolds or concave structures. Probabilistic models like Determinantal Point
Process (DPP) [13] measure data densities, so they are likely to overcome the convexity issue. However, few previous work accessed structural information of possibly nonconvex data for column
sampling/subset selection tasks.
This paper aims to address the issue of learning nonconvex structures of data in the case where
the data distributions can be arbitrary. More speci?cally, we learn a nonconvex hull to encapsulate
the data structure. The on-hull points tightly enclose the dataset but do not need to form a convex
set. Thus, nonconvex hulls can be more adaptive to capture practically complex data structures.
Akin to convex hull learning, our proposed approach also extracts extreme points from an input
dataset. To complete this task, we start with exploring the property of graph cycles in a neighborhood
graph built over the input data points. Using cycle-based measures to characterize data structures
has been proven successful in clustering data of multiple types of distributions [34]. To induce a
measure of structural complexities stemming from graph cycles, we introduce the Zeta Function
which applies the integration of graph cycles to model the linkage properties of the neighborhood
graph. The key advantage of the Zeta function is uniting both global and local connection properties
of the graph. As such, we are able to learn a hull which encompasses almost all input data points
but is not necessary to be convex. With structural complexities captured in the form of the Zeta
function, we present a leave-one-out strategy to ?nd the extreme points. The basic idea is that
removing the on-hull points only has weak impact on structural complexities of the graph. The
decision of removal will be based on extremeness of a data point. Our model, dubbed Zeta Hulls, is
derived by computing and analyzing the extremeness of data points. The greedy pursuit of the Zeta
Hull model requires the computation of the inversion of a matrix obtained from the graph af?nity
matrix, which is computationally prohibitive for massive-scale data. To accelerate such a matrix
manipulation, we employ the Anchor Graph [18] technique in the sense that the original graph can
be approximated with respect to the anchors originating from a randomly sampled data subset. Our
model is testi?ed through extensive experiments on toy data and real-world text and image datasets.
Experimental results show that in terms of unsupervised data representation learning, the Zeta Hull
based methods outperform the state-of-the-art methods used in convex hull learning, clustering,
matrix factorization, and dictionary learning.
2
Nonconvex Hull Learning
To elaborate on our approach, we ?rst introduce and de?ne the point extremeness. It measures the
degree of a data point being prone to lie on or near a nonconvex hull by virtue of a neighborhood
graph drawn from an input dataset. As an intuitive criterion, the data point with strong connections
in the graph should have the low point extremeness. To obtain the extremeness measure, we need to
explore the underlying structure of the graph, where graph cycles are employed.
2.1
Zeta Function and Structural Complexity
We model graph cycles by means of a sum-product rule and then integrate them using a Zeta function. There are many variants of original Riemann Zeta Function, one of which is specialized in
2
(b) Remaining Graph
(a) Original Graph
Figure 1: An illustration of pursuing on-hull points using the graph measure. (a) shows a point set
with a k-nearest neighbor graph. Points in red are ones lying on the hull of the point set, e.g., the
points we tend to select by the Zeta Hull Pursuit algorithm. (b) shows the remaining point set and the
graph after removing the on-hull points together with their corresponding edges. We observe that
the removal of the on-hull (i.e., ?extreme?) points yields little impact on the structural complexity
of the graph.
weighted adjacency graphs. Applying the theoretical results of Zeta functions provides us a powerful tool for characterizing structural complexities of graphs. The numerical description of graph
structures will play a critical role in column sampling/subset selection tasks.
Formally, given a graph G(X , E) with n nodes being data points in X = {xi }ni=1 , let the n ? n
matrix W denote the weighted adjacency (or af?nity) matrix of the graph G built over the dataset X .
Usually the graph af?nities are calculated with a proper distance metric. To be generic, we assume
that G is directed. Then an edge leaving from xi to xj is denoted as eij . A path of length from
xi to xj is de?ned as P (i, j, ) = {ehk tk }k=1 with h1 = i and t = j. Note that the nodes in
this path can be duplicate. A graph cycle, as a special case of paths of length , is also de?ned as
? = P (i, i, ) (i = 1, . . . , n). The sum-product path af?nity ? for all -length cycles can then
?1
be computed by ? = ? ?? ?? = ? ?? wt?1 h1 k=1 whk tk , where ? denotes the set of all
possible cycles of length and whk tk denotes the (hk , tk )-entry of W, i.e., the af?nity from node
xhk to node xtk . The edge et?1 h1 is the last edge that closes the cycle. The computed compound
af?nity ? provides a measure for all cyclic connections of length . Then we integrate such af?nities
for the cycles of lengths being from one to in?nity to derive the graph Zeta function as follows,
?
z
?z (G) = exp
,
(1)
?
=1
where z is a constant. We only consider the situation where z is real-valued. The Zeta function
in Eq. (1) has been proven to enjoy a closed form. Its convergence is also guaranteed when z <
1/?(W), where ?(W) is the spectral radius of W. These lead to Theorem 1 [23].
Theorem 1. Let I be the identity matrix and ?(W) be the spectral radius of the matrix W, respectively. If 0 < z < 1/?(W), then ?z (G) = 1/ det(I ? zW).
Note that W can be asymmetric, implying that ?i can be complex. In this case, Theorem 1 still
holds. Theorem 1 indicates that the graph Zeta function we formulate in Eq. (1) provides a closedform expression for describing the structural complexity of a graph. The next subsection will give
the de?nition of the point extremeness by analyzing the structural complexity.
2.2
Zeta Hull Pursuits
From now on, for simplicity we use G = ?z (G) to represent the structural complexity of the original
graph G. To measure the point extremeness numerically, we perform a leave-one-out strategy in the
sense that each point in C is successively left out and the variation of G is investigated. This is a
natural way to pursue extreme points, because if a point xj lies on the hull it has few communications
with the other points. After removing this point and its corresponding edges, the reductive structural
complexity of the remaining graph G/xj , which we denote as G/xj , will still be close to G . Hence,
the point extremeness ?xj is modeled as the relative change of the structural complexity G , that is
G
?xj = G/x
. Now we have the following theorem.
j
Theorem 2. Given G and G/xj as in Theorem 1, the point extremeness measure ?xj of point xj
satis?es ?xj = (I ? zW)?1
(jj) , i.e., the point extremeness measure of point xj is equal to the j-th
diagonal entry of the matrix (I ? zW)?1 .
3
Algorithm 1 Zeta Hull Pursuits
Input: A dataset X , the number m of data points to be selected, and free parameters z, ? and k.
Output: The hull of sampled columns Ce := Cm+1 .
Initialize: construct W, C1 ? ?, X1 = X , c1 = 0, and W1 = W
for i = 1 to m do
?xj := (I ? zWi )?1
(jj) , for xj ? Xi
xei := arg minxj ?Xi (?xj + ?i e
j Wci )
Ci+1 := Ci ? xei
ci+1 := ci + eei
Xi+1 := Ci /xei
Wi+1 := Wi with the ei -th row and column removed
end for
According to previous analysis, the data point with a small ?xj tends to be on the hull and therefore
has a strong extremeness. To seek the on-hull points, we need to select a subset of m points Ce =
{xe1 , . . . , xem } from X such that they have the strongest point extremenesses. We formulate this
goal into the following optimization problem:
Ce = arg min g(C) + ?c Wc,
(2)
C?X
where c is a selection vector with m nonzero elements cei = 1 (i = 1, . . . , m), and g(C) is the
function which measures
mthe impact on the structural complexity after removing the extracted points.
In our case, g(C) = i=1 ?xci . The second term in Eq. (2) is a regularization term enforcing that
the selected data points do not intersect with each other. It will enable the selection process to have
a better representative capability. The parameter ? controls the extent of the regularization.
Naively solving the combinatorial optimization problem in Eq. (2) requires exponential time. By
adopting a greedy strategy, we can solve this optimization problem in an iterative manner and with
a feasible time complexity. Speci?cally, in each iteration we extract one point from the current data
set and add it to the subset of the selected points. Sticking to this greedy strategy, we will attain the
desired m on-hull points after m iterations. In the i-th iteration, we extract the point xei according
to the criterion
?
xei = arg min ?xj + e
Wci?1 ,
(3)
i j
xj ?Xi?1
where ej is the j-th standard basis vector, and ci?1 is the selection vector according to i ? 1 selected
points before the i-th iteration.
We name our algorithm Zeta Hull Pursuits in order to emphasize that we use the Zeta function to
pursue the nonconvex data hull. Algorithm 1 summarizes the Zeta Hull Pursuits algorithm.
3
Zeta Hull Pursuits via Anchors
Algorithm 1 is applicable to small to medium-scale data X due to its cubical time complexity and
quadratic space complexity with respect to the data size |X |. Here we propose a scalable algorithm
facilitated by a reasonable prior to tackle the nonconvex hull learning problem ef?ciently. The idea is
to build a low-rank approximation to the graph adjacency matrix W with a small number of sampled
data points, namely anchor points. We resort to the Anchor Graph technique [18], which has been
successfully applied to handle large-scale hashing[20] and semi-supervised learning problems.
3.1
Anchor Graphs
The anchor graph framework is an elegant way to approximate neighborhood graphs. It ?rst chooses
a subset of l anchor points U = {uj }lj=1 from X . Then for each data point in X , its s nearest anchors
in U are sought, thereby forming an s-nearest anchor graph. The anchor graph theory assumes that
the original graph af?nity matrix W can be reconstructed from the anchor graph with a small number
of anchors (l n). Anchor points can be selected by random sampling or a rough clustering
process. Many algorithms are available to embed a data point to its s nearest anchor points, as
?
suggested in
we adopt the simplest approach to build the anchor embedding matrix H;
[18]. Here
2
2
? ij = exp ?dij /? , uj ? {s nearest anchors of xi } , where dij is the distance from data
say, h
0,
otherwise
4
Algorithm 2
Anchor-based Zeta Hull Pursuits
Input: A dataset X , the number m of data points to be sampled, the number l of anchors, the
number s of nearest anchors, and a free parameter z.
Output: The hull of sampled columns Ce := Cm+1 .
Initialize: construct H, X1 = X , C1 = ?, and H1 = H
for i = 1 to m do
perform SVD to obtain Hi := U?VT
l
?2j
2
?xj := z k=1 1?z?
2 (Ujk ) , for xj ? Xi
k
?
xei := arg minxj ?Xi (?xj +
i hj ht )
xt ?Ci
Ci+1 := Ci ? xei
Xi+1 := Xi /xei
Hi+1 := Hi with the ei -th row removed
end for
point xi to anchor uj , and ? is a parameter controlling the bandwidth of the exponential function.
? is then normalized so that its every row sums to one. In doing so, we can approximate
The matrix H
? = H?
? ?1 H
? , where ? is a diagonal matrix whose i-th
the af?nity matrix of the original graph as W
? As a result, all matrix manipulations
diagonal element is equal to the sum of the i-th column of H.
upon the original graph af?nity matrix W can be approximated by substituting the anchor graph
? for W.
af?nity matrix W
3.2
Extremeness Computation via Anchors
Note that the computation of the point extremeness for ?xj depends on the diagonal elements of
(I ? zW)?1 . Using the anchor graph technique, we can write (I ? zW)?1 = (I ? zHH )?1 ,
? ? 12 . Thus we have the following theorem that enables an ef?cient computation of
where H = H?
?xj . The proof is detailed in the supplementary material.
Theorem 3. Let the singular vector decomposition of H be H = U?V , where ? =
l
?2k
2
diag(?1 , . . . , ?l ). If H H is not singular, then ??1
xj = 1 + z
k=1 1?z?2 (Ujk ) , where U =
HV??1 and Ujk denotes the (i, j)-th entry of U.
k
Theorem 3 reveals that the major computation of ?xj will reduce to the eigendecomposition of a
much smaller matrix H H, which results in a direct acceleration of the Zeta hull pursuit process.
At the same time,
term of Eq. (3) encountered in the i-th iteration can be estimated by
the second
1
h
h
,
where hj denotes the j-th row of H and ci?1 is the selection vector
e
j
t
j Wci = i
xt ?Ci
of the extracted point set before the i-th iteration. These lead to the Anchor-based Zeta Hull Pursuits
algorithm shown in Algorithm 2.
3.3
Downdating SVD
In Algorithm 2, the singular value decomposition dominates the total time cost. We notice that
reusing information in previous iterations can save the computation time. The removal of one row
from H is equivalent to a rank-one modi?cation to the original matrix. Downdating SVD [10] was
proposed to handle this operation. Given the diagonal singular value matrix ?i and the point xei
chosen in the i-th iteration, the singular value matrix ?i+1 for the next iteration can be calculated
1
by the eigendecomposition of an l ? l matrix D derived from ?i , where D = (I ? 1+?
h ei h
ei )?i ,
2
2
2
and ? + hei 2 = 1. The decomposition of D can be ef?ciently performed in O(l ) time [10].
Then the computation of Ui+1 is achieved by a multiplication of Ui with an l ? l matrix produced
by the decomposition operation on D, which permits a natural parallelism. Consequently, we can
further accelerate Algorithm 2 by using a parallel computing scheme.
3.4
Complexity Analysis
We now analyze the complexities of Algorithms 1 and 2. For Algorithm 1, the most time-consuming
step is to solve the matrix inverse of n ? n size, which costs a time complexity of O(n3 ). The
overall time complexity is thus O(mn3 ) for extracting m points. In the implementation we can use
5
(a) m = 20, ZHP
(b) m = 40, ZHP
(c) m = 80, ZHP
(d) m = 200, ZHP
(e) m = 20, A-ZHP
(f) m = 40, A-ZHP
(g) m = 80, A-ZHP
(h) m = 200, A-ZHP
(i) m = 40, Leverage Score
(j) m = 40, Simplex
(k) m = 40, CUR
(l) m = 40, K-medoids
Figure 2: Zeta hull pursuits on the two-moon toy dataset. We select m data points from the dataset
with various methods. In the sub-?gures, blue dots are data points. The selected samples are surrounded with red circles. The caption of each sub-?gure describes the number of selected points m
and the method used to select those data points. First two rows shows the results of our algorithms
with different m. The third row illustrates the comparisons with other methods when m = 40. For
the leverage score approach, we follow the steps in [21].
the sparse matrix computation to reduce the constant factor [5]. For Algorithm 2, the most timeconsuming step is to perform SVD over H, so the overall time complexity is O(mnl2 ). Leveraging
downdating SVD, we only need to calculate the full SVD of H once in O(nl2 ) time and iteratively
update the decomposition in O(l2 ) time per iteration. The matrix multiplication operation then
dominates the total time cost. Also, it can be parallelized using a multi-core CPU or a modern GPU,
resulting in a very small constant factor in the time complexity. Since l is usually less than 10% of n,
Algorithm 2 is orders of magnitude faster than Algorithm 1. For cases where l needs to be relatively
large (20% of n for example), the computational cost will not show a considerable increase since H
is usually a very sparse matrix.
4
Experiments
The Zeta Hull model aims at learning the structures of dataset. We evaluate how well our model
achieves this goal by performing classi?cation experiments. For simplicity, we abbreviate our algorithms as follows: the original Zeta Hull Pursuit algorithm (Algorithm 1), ZHP and its anchor
version (Algorithm 2), A-ZHP. To compare with the state-of-the-art, we choose some renowned
methods: K-medoids, CUR matrix factorization (CUR) [29], simplex volume maximization (Simplex) [26], sparse dictionary learning (DictLearn) [22] and convex non-negative matrix factorization
(C-NMF) [6]. Basically, we use the extracted data points to learn a representation for each data
point in an unsupervised manner. Classi?cation is done by feeding the representation into a classi?er. The representation will be built in two ways: 1) the sparse coding [22] and 2) the locality
simplex coding [26]. To differentiate our algorithms from the original anchor graph framework, we
conduct a set of experiments using the left singular vectors of the anchor embedding matrix H as
the representation. In these experiments, anchors used in the anchor graph technique are randomly
selected from the training set. To compare with existing low-dimension embedding approaches, we
run the Large-Scale Manifold method [24] using the same number of landmarks as that of extracted
points.
4.1 Toy Dataset
First we illustrate our algorithms on a toy dataset. The dataset, commonly known as ?the two
moons?, consists of 2000 data points on the 2D plane which are manifold-structured and comprise
nonconvex distributions. This experiment on the two moons provides illustrative results of our
algorithms in the presence of nonconvexity. We select different numbers of column subsets m =
{20, 40, 80, 200} and compare with various other methods. A visualization of the results is shown
in Figure 2. We can see that our algorithms can extract the nonconvex hull of the data cloud more
accurately.
4.2 Text and Image Datasets
For the classi?cation experiments in this section, we derive the two types of data representations (the
sparse coding and the local simplex coding) from the points/columns extracted by compared meth6
Table 1: Classi?cation error rates in percentage (%) on texts (TDT2 and Newsgroups) and handwritten number datasets (MNIST). The numbers in bold font highlight best results under the settings.
In this table, ?SC? refers to the results using the sparse coding to form the representation, while
?LSC? refers to the results using local simplex coding. The cells with ?-? indicate that the ZHP
method is too expensive to be performed under the associated settings. The ?Anchor Graph? refers
to the additional experiments using the original anchor graph framework [18].
Methods
ZHP
A-ZHP
Simplex [26]
DictLearn [22]
C-NMF [6]
CUR [29]
K-medoids [12]
Anchor Graph [18]
TDT2
m = 500
m = 1000
SC
LSC
SC
LSC
2.31
2.52
3.79
3.73
4.83
6.82
9.14
1.97
2.68
1.73
5.62
3.46
3.73
7.87
0.48
0.96
1.51
2.57
2.31
1.52
4.69
1.53
2.08
1.77
1.18
2.07
2.37
3.73
5.81
2.68
Newsgroups
m = 500
m = 1000
SC
LSC
SC
LSC
11.79
10.77
13.55
10.41
9.51
10.76
11.68
11.83
15.32
11.44
19.73
12.02
12.32
7.1
6.58
8.16
8.04
6.72
9.63
7.72
7.42
12.38
9.47
19.67
10.04
8.76
MNIST
m = 500
m = 2000
SC
LSC
SC
LSC
3.45
3.07
5.79
5.79
3.16
3.16
5.07
5.27
10.13
10.13
9.28
9.28
3.17
1.43
2.27
1.36
3.01
3.79
2.72
1.19
1.51
2.11
3.04
5.27
2.31
2.33
Table 2: Recognition error rates in percentage (%) on object and face datasets. We select L samples
for each class in the training set for training or forming the gallery. The numbers in bold font
highlight best results under the settings. In this table, ?SC? refers to the results using the sparse
coding to form the representation, while ?LSC? refers to the results using local simplex coding. The
?Raw Feature? refers to the experiments conducted on the raw features vectors. The face recognition
process is described in Sec. (4.2).
Methods
A-ZHP
Simplex [26]
DictLearn [22]
C-NMF [6]
CUR [21]
K-medoids [12]
Anchor Graph [18]
Large Manifold [24]
Raw Feature [28]
Caltech101
d = 21504, L = 30
m = 500
m = 1000
SC
LSC
SC
LSC
25.77
26.82
29.83
26.16
26.95
29.73
30.66
27.83
29.74
28.77
27.82
27.64
26.32
28.71
23.13
25.81
26.83
25.18
26.73
29.51
28.72
27.62
26.16
26.81
26.09
25.73
25.15
27.92
Caltech101
d = 5120, L = 30
m = 500
m = 1000
SC
LSC
SC
LSC
29.61
28.95
32.43
29.66
29.15
31.83
32.57
31.13
31.69
32.57
29.85
29.63
30.53
32.67
26.7
25.62
26.59
30.62
27.47
28.93
29.67
31.15
28.73
30.72
31.13
28.97
28.28
28.14
30.19
31.18
MultiPIE
d = 2000, L = 30
m = 500
m = 2000
SC
LSC
SC
LSC
14.2
15.8
20.8
17.5
21.9
19.8
20.8
19.9
19.6
20.4
21.3
29.7
11.3
13.7
19.7
14.8
21.6
17.7
19.6
17.7
18.5
19.9
20.7
25.4
17.6
31.4
14.4
30.1
27.6
ods. By measuring the performance of applying these representations to solving the classi?cation
tasks, we can evaluate the representative power of the compared point/column selection methods.
The sparse coding is widely used for obtaining the representation for classi?cation. Here a standard
1 -regularized projection algorithm (LASSO) [22] is adopted to learn the sparse representation from
the extracted data points. LASSO will deliver a sparse coef?cient vector, which is applied as the
representation of the data point. We use ?SC? to indicate the related results in Table 1 and Table 2.
The local simplex coding reconstructs one data point as a convex combination of a set of nearest
exemplar points, which form local simplexes [26]. Imposing this convex reconstruction constraint
leads to non-negative combination coef?cients. The sparse coef?cients vector will be used as data
representation. ?LSC? indicates the related results in Table 1 and Table 2.
The classi?cation pipeline is as follows. After extracting m points/columns from the training set,
all data points will be represented with these selected points using the two approaches above. Then
we feed the representations into a linear SVM for the training and testing. The better classi?cation
accuracy will reveal the stronger representative power of the column selection algorithm. In all
experiments, the parameter z is ?xed at 0.05 to guarantee the convergence of the Zeta function. We
?nd that ?nal results are robust to z once the convergence is guaranteed. For the A-ZHP algorithm,
the parameter s is ?xed at 10 and the number of anchor points l is set as 10% of the training set
size. The bandwidth parameter ? of the exponential function is tuned on the training set to obtain a
reasonable anchor embedding.
The classi?cation of text contents relies on the informative representation of the plain words or sentences. Two text datasets are adopted for classi?cation, i.e. the TDT2 dataset and the Newsgroups
dataset [2]. In experiments, a subset of TDT2 is used (TDT2-30). It has 9394 samples from 30
classes. Each feature vector is of 36771 dimensions and normalized into unit length. The training
set contains 6000 samples randomly selected from the dataset and rest of the samples are used for
7
testing. The parameter m is set to be 500 and 1000 on this dataset. The Newsgroups dataset contains 18846 samples from 20 classes. The training set contains 11314, while the testing set has 7532.
The two sets are separated in advance [2] and ordered in time sequence to be more challenging for
classi?ers. The parameter m is set to be 500 and 1000 on this dataset. The classi?cation results are
reported in Table 1.
For object and face recognition tasks we conduct experiments under three classic scenarios, the
hand-written digits classi?cation, the image recognition, and the human face recognition. Related
experimental results are reported in Table 1 and Table 2.
The MNIST dataset serves as a standard benchmark for machine learning algorithms. It contains 10
classes of images corresponding to hand-written numbers from 0 to 9. The training set has 60000
images and the testing set has 10000 images. Each sample is a 784-dimensional vector.
The Caltech101 dataset [17] is a widely used benchmark for object recognition systems. It consists
of images from 102 classes of objects (101 object classes and one background class). We randomly
select 30 labeled images from every class for training the classi?er and 3000 images for testing.
The recognition rates averaged over all classes are reported. Every image is processed into a feature
vector of 21504 dimensions by the method in [28]. We also conduct experiment on a feature subset
of the top 5000 dimensions (Caltech101-5k). In this experiment, m is set to be 500 and 1000.
On-hull points are extracted on the training set.
The MultiPIE human face dataset is a widely applied benchmark for face recognition [9]. We follow
a standard gallery-probe protocol of face recognition. The testing set is divided into the gallery set
and the probe set. The identity predication of a probe image comes from its nearest neighbor of
Euclidean distance in the gallery. We randomly select 30, 000 images of 200 subjects as the training
set for learning the data representation. Then we pick out 3000 images of the other 100 subjects
(L = 30) to form the gallery set and 6000 images as the probes. The head poses of all these faces
are between ?15 degrees. Each face image is processed into a vector of 5000 dimensions using the
local binary pattern descriptor and PCA. We vary the parameter m from 500 to 2000 to evaluate the
in?uence of number of sampled points.
Discussion. For the experiments on these high-dimensional datasets, the methods based on the
Zeta Hull model outperform most compared methods and also show promising performance improvements over raw data representation. When the number of extracted points grows, the resulting
classi?cation accuracy increases. This corroborates that the Zeta Hull model can effectively capture
intrinsic structures of given datasets. More importantly, the discriminative information is preserved
through learning these Zeta hulls. The representation yielded by the Zeta Hull model is sparse and of
manageable dimensionality (500-2000), which substantially eases the workload of classi?er training. This property is also favorable for tackling other large-scale learning problems. Due to the
graph-theoretic measure that uni?es the local and global connection properties of a graph, the Zeta
Hull model leads to better data representation compared against existing graph-based embedding
and manifold learning methods. For the comparison with the Large-Scale Manifold method [24]
on the MultiPIE dataset, we ?nd that even using 10K landmarks, its accuracy is still inferior to our
methods relying on the Zeta Hull model. We also notice that noise may also affect the quality of Zeta
hulls. This dif?culty can be circumvented by running a number of well-established outlier removal
methods such as [19].
5
Conclusion
In this paper, we proposed a geometric model, dubbed Zeta Hulls, for column sampling through
learning nonconvex hulls of input data. The Zeta Hull model was built upon a novel graph-theoretic
measure which quanti?es the point extremeness to unify local and global connection properties of
individual data point in an adjacency graph. By means of the Zeta function de?ned on the graph,
the point extremeness measure amounts to the diagonal elements of a matrix related to the graph
adjacency matrix. We also reduced the time and space complexities for computing a Zeta hull by
incorporating an ef?cient anchor graph technique. A synthetic experiment ?rst showed that the Zeta
Hull model can detect appropriate hulls for non-convexly distributed data. The extensive real-world
experiments conducted on benchmark text and image datasets further demonstrated the superiority
of the Zeta Hull model over competing methods including convex hull learning, clustering, matrix
factorization, and dictionary learning.
Acknowledgement This research is partially supported by project #MMT-8115038 of the Shun
Hing Institute of Advanced Engineering, The Chinese University of Hong Kong.
8
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support:1 evaluate:3 tested:1 correlated:1 |
4,764 | 5,313 | The Bayesian Case Model: A Generative Approach
for Case-Based Reasoning and Prototype
Classification
Been Kim, Cynthia Rudin and Julie Shah
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
{beenkim, rudin, julie a shah}@csail.mit.edu
Abstract
We present the Bayesian Case Model (BCM), a general framework for Bayesian
case-based reasoning (CBR) and prototype classification and clustering. BCM
brings the intuitive power of CBR to a Bayesian generative framework. The BCM
learns prototypes, the ?quintessential? observations that best represent clusters in
a dataset, by performing joint inference on cluster labels, prototypes and important features. Simultaneously, BCM pursues sparsity by learning subspaces, the
sets of features that play important roles in the characterization of the prototypes.
The prototype and subspace representation provides quantitative benefits in interpretability while preserving classification accuracy. Human subject experiments
verify statistically significant improvements to participants? understanding when
using explanations produced by BCM, compared to those given by prior art.
1 Introduction
People like to look at examples. Through advertising, marketers present examples of people we
might want to emulate in order to lure us into making a purchase. We might ignore recommendations
made by Amazon.com and look instead at an Amazon customer?s Listmania to find an example of a
customer like us. We might ignore medical guidelines computed from a large number of patients in
favor of medical blogs where we can get examples of individual patients? experiences.
Numerous studies have demonstrated that exemplar-based reasoning, involving various forms of
matching and prototyping, is fundamental to our most effective strategies for tactical decisionmaking ([26, 9, 21]). For example, naturalistic studies have shown that skilled decision makers
in the fire service use recognition-primed decision making, in which new situations are matched to
typical cases where certain actions are appropriate and usually successful [21]. To assist humans in
leveraging large data sources to make better decisions, we desire that machine learning algorithms
provide output in forms that are easily incorporated into the human decision-making process.
Studies of human decision-making and cognition provided the key inspiration for artificial intelligence Case-Based Reasoning (CBR) approaches [2, 28]. CBR relies on the idea that a new situation
can be well-represented by the summarized experience of previously solved problems [28]. CBR
has been used in important real-world applications [24, 4], but is fundamentally limited, in that it
does not learn the underlying complex structure of data in an unsupervised fashion and may not
scale to datasets with high-dimensional feature spaces (as discussed in [29]).
In this work, we introduce a new Bayesian model, called the Bayesian Case Model (BCM), for
prototype clustering and subspace learning. In this model, the prototype is the exemplar that is most
representative of the cluster. The subspace representation is a powerful output of the model because
we neither need nor want the best exemplar to be similar to the current situation in all possible ways:
1
for instance, a moviegoer who likes the same horror films as we do might be useful for identifying
good horror films, regardless of their cartoon preferences. We model the underlying data using a
mixture model, and infer sets of features that are important within each cluster (i.e., subspace). This
type of model can help to bridge the gap between machine learning methods and humans, who use
examples as a fundamental part of their decision-making strategies.
We show that BCM produces prediction accuracy comparable to or better than prior art for standard
datasets. We also verify through human subject experiments that the prototypes and subspaces
present as meaningful feedback for the characterization of important aspects of a dataset. In these
experiments, the exemplar-based output of BCM resulted in statistically significant improvements
to participants? performance of a task requiring an understanding of clusters within a dataset, as
compared to outputs produced by prior art.
2 Background and Related Work
People organize and interpret information through exemplar-based reasoning, particularly when they
are solving problems ([26, 7, 9, 21]). AI Cased-Based Reasoning approaches are motivated by
this insight, and provide example cases along with the machine-learned solution. Studies show
that example cases significantly improve user confidence in the resulting solutions, as compared to
providing the solution alone or by also displaying a rule that was used to find the solution [11].
However, CBR requires solutions (i.e. labels) for previous cases, and does not learn the underlying
structure of the data in an unsupervised fashion. Maintaining transparency in complex situations
also remains a challenge [29]. CBR models designed explicitly to produce explanations [1] rely on
the backward chaining of the causal relation from a solution, which does not scale as complexity
increases. The cognitive load of the user also increases with the complexity of the similarity measure
used for comparing cases [14]. Other CBR models for explanations require the model to be manually
crafted in advance by experts [25].
Alternatively, the mixture model is a powerful tool for discovering cluster distributions in an unsupervised fashion. However, this approach does not provide intuitive explanations for the learned
clusters (as pointed out in [8]). Sparse topic models are designed to improve interpretability by reducing the number of words per topic [32, 13]. However, using the number of features as a proxy for
interpretability is problematic, as sparsity is often not a good or complete measure of interpretability
[14]. Explanations produced by mixture models are typically presented as distributions over features. Even users with technical expertise in machine learning may have a difficult time interpreting
such output, especially when the cluster is distributed over a large number of features [14].
Our approach, the Bayesian Case Model (BCM), simultaneously performs unsupervised clustering
and learns both the most representative cases (i.e., prototypes) and important features (i.e., subspaces). BCM preserves the power of CBR in generating interpretable output, where interpretability
comes not only from sparsity but from the prototype exemplars.
In our view, there are at least three widely known types of interpretable models: sparse linear
classifiers ([30, 8, 31]); discretization methods, such as decision trees and decision lists (e.g.,
[12, 32, 13, 23, 15]); and prototype- or case-based classifiers (e.g., nearest neighbors [10] or a supervised optimization-based method [5]). (See [14] for a review of interpretable classification.) BCM is
intended as the third model type, but uses unsupervised generative mechanisms to explain clusters,
rather than supervised approaches [16] or by focusing myopically on neighboring points [3].
3 The Bayesian Case Model
Intuitively, BCM generates each observation using the important pieces of related prototypes. The
model might generate a movie profile made of the horror movies from a quintessential horror movie
watcher, and action movies from a quintessential action moviegoer.
BCM begins with a standard discrete mixture model [18, 6] to represent the underlying structure
of the observations. It augments the standard mixture model with prototypes and subspace feature
indicators that characterize the clusters. We show in Section 4.2 that prototypes and subspace feature
indicators improve human interpretability as compared to the standard mixture model output. The
graphical model for BCM is depicted in Figure 1.
2
?, c
N
q
ps
?s
?s
S
?
?i
zij
xij
F
N
Figure 1: Graphical model for the Bayesian Case Model
We start with N observations, denoted by x = {x1 , x2 , . . . , xN }, with each xi represented as a random mixture over clusters. There are S clusters, where S is assumed to be known in advance. (This
assumption can easily be relaxed through extension to a non-parametric mixture model.) Vector ?i
are the mixture weights over these clusters for the ith observation xi , ?i ? RS+ . Each observation
has P features, and we denote the j th feature of the ith observation as xij . Each feature j of the
observation xi comes from one of the clusters, the index of the cluster for xij is denoted by zij and
the full set of cluster assignments for observation-feature pairs is denoted by z. Each zij takes on the
value of a cluster index between 1 and S. Hyperparameters q, ?, c, and ? are assumed to be fixed.
The explanatory power of BCM results from how the clusters are characterized. While a standard
mixture model assumes that each cluster take the form of a predefined parametric distribution (e.g.,
normal), BCM characterizes each cluster by a prototype, ps , and a subspace feature indicator, ?s .
Intuitively, the subspace feature indicator selects only a few features that play an important role in
identifying the cluster and prototype (hence, BCM clusters are subspace clusters). We intuitively
define these latent variables below.
Prototype, ps : The prototype ps for cluster s is defined as one observation in x that maximizes
p(ps |?s , z, x), with the probability density and ?s as defined below. Our notation for element j of
ps is psj . Since ps is a prototype, it is equal to one of the observations, so psj = xij for some i.
Note that more than one maximum may exist per cluster; in this case, one prototype is arbitrarily
chosen. Intuitively, the prototype is the ?quintessential? observation that best represents the cluster.
Subspace feature indicator ?s : Intuitively, ?s ?turns on? the features that are important for characterizing cluster s and selecting the prototype, ps . Here, ?s ? {0, 1}P is an indicator variable that
is 1 on the subset of features that maximizes p(?s |ps , z, x), with the probability for ?s as defined
below. Here, ?s is a binary vector of size P , where each element is an indicator of whether or not
feature j belongs to subspace s.
The generative process for BCM is as follows: First, we generate the subspace clusters. A subspace cluster can be fully described by three components: 1) a prototype, ps , generated by sampling
uniformly over all observations, 1 . . . N ; 2) a feature indicator vector, ?s , that indicates important
features for that subspace cluster, where each element of the feature indicator (?sj ) is generated
according to a Bernoulli distribution with hyperparameter q; and 3) the distribution of feature outcomes for each feature, ?s , for subspace s, which we now describe.
Distribution of feature outcomes ?s for cluster s: Here, ?s is a data structure wherein each ?row?
?sj is a discrete probability distribution of possible outcomes for feature j. Explicitly, ?sj is a vector
of length Vj , where Vj is the number of possible outcomes of feature j. Let us define ? as a vector
of the possible outcomes of feature j (e.g., for feature ?color?, ? = [red, blue, yellow]), where ?v
represents a particular outcome for that feature (e.g., ?v = blue). We will generate ?s so that it
mostly takes outcomes from the prototype ps for the important dimensions of the cluster. We do this
by considering the vector g, indexed by possible outcomes v, as follows:
gpsj ,?sj ,? (v) = ?(1 + c1[wsj =1 and psj =?v ] ),
where c and ? are constant hyperparameters that indicate how much we will copy the prototype in
order to generate the observations. The distribution of feature outcomes will be determined by g
through ?sj ? Dirichlet(gpsj ,?sj ,? ). To explain at an intuitive level: First, consider the irrelevant
dimensions j in subspace s, which have wsj = 0. In that case, ?sj will look like a uniform distribu3
tion over all possible outcomes for features j; the feature values for the unimportant dimensions are
generated arbitrarily according to the prior. Next, consider relevant dimensions where wsj = 1. In
this case, ?sj will generally take on a larger value ? + c for the feature value that prototype ps has on
feature j, which is called ?v . All of the other possible outcomes are taken with lower probability ?.
As a result, we will be more likely to select the outcome ?v that agrees with the prototype ps . In the
extreme case where c is very large, we can copy the cluster?s prototype directly within the cluster?s
relevant subspace and assign the rest of the feature values randomly.
An observation is then a mix of different prototypes, wherein we take the most important pieces of
each prototype. To do this, mixture weights ?i are generated according to a Dirichlet distribution,
parameterized by hyperparameter ?. From there, to select a cluster and obtain the cluster index zij
for each xij , we sample from a multinomial distribution with parameters ?i . Finally, each feature for
an observation, xij , is sampled from the feature distribution of the assigned subspace cluster (?zij ).
(Note that Latent Dirichlet Allocation (LDA) [6] also begins with a standard mixture model, though
our feature values exist in a discrete set that is not necessarily binary.) Here is the full model, with
hyperparameters c, ?, q, and ?:
?sj ? Bernoulli(q) ?s, j
ps ? Uniform(1, N ) ?s
?sj ? Dirichlet(gpsj ,?sj ,? ) ?s, j
where gpsj ,?sj ,? (v) = ?(1 + c1[wsj =1 and psj =?v ] )
?i ? Dirichlet(?) ?i
zij ? Multinomial(?i ) ?i, j
xij ? Multinomial(?zij j ) ?i, j.
Our model can be readily extended to different similarity measures, such as standard kernel methods
or domain specific similarity measures, by modifying the function g. For example, we can use the
least squares loss i.e., for fixed threshold ?, gpsj ,?sj ,? (v) = ?(1 + c1[wsj =1 and (psj ??v )2 ??] ); or,
more generally, gpsj ,?sj ,? (v) = ?(1 + c1[wsj =1 and ?(psj ,?v )??] ).
In terms of setting hyperparameters, there are natural settings for ? (all entries being 1). This
means that there are three real-valued parameters to set, which can be done through cross-validation,
another layer of hierarchy with more diffuse hyperparameters, or plain intuition. To use BCM for
classification, vector ?i is used as S features for a classifier, such as SVM.
3.1 Motivating example
This section provides an illustrative example for prototypes, subspace feature indicators and subspace clusters, using a dataset composed of a mixture of smiley faces. The feature set for a smiley
face is composed of types, shapes and colors of eyes and mouths. For the purpose of this example,
assume that the ground truth is that there are three clusters, each of which has two features that are
important for defining that cluster. In Table 1, we show the first cluster, with a subspace defined by
the color (green) and shape (square) of the face; the rest of the features are not important for defining
the cluster. For the second cluster, color (orange) and eye shape define the subspace. We generated
240 smiley faces from BCM?s prior with ? = 0.1 for all entries, and q = 0.5, ? = 1 and c = 50.
Data in assigned to cluster
LDA
Top 3 words and probabilities
1
0.26
0.23
0.24
0.27
color (
) and shape (
are important.
)
color (
) and eye (
are important.
)
eye (
) and mouth (
are important.
)
0.16
3
0.35
BCM
Subspaces
0.12
2
0.26
Prototype
0.15
Table 1: The mixture of smiley faces for LDA and BCM
4
BCM works differently to Latent Dirichlet Allocation (LDA) [6], which presents its output in a very
different form. Table 1 depicts the representation of clusters in both LDA (middle column) and BCM
(right column). This dataset is particularly simple, and we chose this comparison because the two
most important features that both LDA and BCM learn are identical for each cluster. However, LDA
does not learn prototypes, and represents information differently. To convey cluster information
using LDA (i.e., to define a topic), we must record several probability distributions ? one for each
feature. For BCM, we need only to record a prototype (e.g., the green face depicted in the top row,
right column of the figure), and state which features were important for that cluster?s subspace (e.g.,
shape and color). For this reason, BCM is more succinct than LDA with regard to what information
must be recorded in order to define the clusters. One could define a ?special? constrained version
of LDA with topics having uniform weights over a subset of features, and with ?word? distributions
centered around a particular value. This would require a similar amount of memory; however, it loses
information, with respect to the fact that BCM carries a full prototype within it for each cluster.
A major benefit of BCM over LDA is that the ?words? in each topic (the choice of feature values) are
coupled and not assumed to be independent ? correlations can be controlled depending on the choice
of parameters. The independence assumption of LDA can be very strong, and this may be crippling
for its use in many important applications. Given our example of images, one could easily generate
an image with eyes and a nose that cannot physically occur on a single person (perhaps overlapping).
BCM can also generate this image, but it would be unlikely, as the model would generally prefer to
copy the important features from a prototype.
BCM performs joint inference on prototypes, subspace feature indicators and cluster labels for observations. This encourages the inference step to achieve solutions where clusters are better represented by prototypes. We will show that this is beneficial in terms of predictive accuracy in Section 4.1. We will also show through an experiment involving human subjects that BCM?s succinct
representation is very effective for communicating the characteristics of clusters in Section 4.2.
3.2 Inference: collapsed Gibbs sampling
We use collapsed Gibbs sampling to perform inference, as this has been observed to converge
quickly, particularly in mixture models [17]. We sample ?sj , zij , and ps , where ? and ? are integrated out. Note that we can recover ? by simply counting the number of feature values assigned
to each subspace. Integrating out ? and ? results in the following expression for sampling zij :
p(zij = s|zi?j , x, p, ?, ?, ?) ?
g(psj , ?sj , ?) + n(s,?,j,xij )
?/S + n(s,i,?j,?)
,
?P
?+n
s g(psj , ?sj , ?) + n(s,?,j,?)
(1)
where n(s,i,j,v) = 1(zij = s, xij = v). In other words, if xij takes feature value v for feature j
and is assigned to cluster s, then n(s,i,j,v) = 1, or 0 otherwise. Notation n(s,?,j,v) is the number of
times that the j th feature of an observation
takes feature value v and that observation is assigned to
P
subspace cluster s (i.e., n(s,?,j,v) = i 1(zij = s, xij = v)). Notation n(s,?,j,?) means sum over
i and v. We use n(s,i,?j,v) to denote a count that does not include the feature j. The derivation is
similar to the standard collapsed Gibbs sampling for LDA mixture models [17].
Similarly, integrating out ? results in the following expression for sampling ?sj :
?
B(g(psj , 1, ?) + n(s,?,j,?) )
?
?
?q ?
B(g(psj , 1, ?))
p(?sj = b|q, psj , ?, ?, x, z, ?) ?
B(g(p
?
sj , 0, ?) + n(s,?,j,?) )
?
?1 ? q ?
B(g(psj , 0, ?))
b=1
(2)
b = 0,
where B is the Beta function and comes from integrating out ? variables, which are sampled from
Dirichlet distributions.
4 Results
In this section, we show that BCM produces prediction accuracy comparable to or better than LDA
for standard datasets. We also verify the interpretability of BCM through human subject experiments
involving a task that requires an understanding of clusters within a dataset. We show statistically
5
(a) Accuracy and standard deviation
with SVM
(b) Unsupervised accuracy
for BCM
(c) Sensitivity analysis for BCM
Figure 2: Prediction test accuracy reported for the Handwritten Digit [19] and 20 Newsgroups
datasets [22]. (a) applies SVM for both LDA and BCM, (b) presents the unsupervised accuracy
of BCM for Handwritten Digit (top) and 20 Newsgroups (bottom) and (c) depicts the sensitivity
analysis conducted for hyperparameters for Handwritten Digit dataset. Datasets were produced by
randomly sampling 10 to 70 observations of each digit for the Handwritten Digit dataset, and 100450 documents per document class for the 20 Newsgroups dataset. The Handwritten Digit pixel
values (range from 0 to 255) were rescaled into seven bins (range from 0 to 6). Each 16-by-16 pixel
picture was represented as a 1D vector of pixel values, with a length of 256. Both BCM and LDA
were randomly initialized with the same seed (one half of the labels were incorrect and randomly
mixed), The number of iterations was set at 1,000. S = 4 for 20 Newsgroups and S = 10 for
Handwritten Digit. ? = 0.01, ? = 1, c = 50, q = 0.8.
significant improvements in objective measures of task performance using prototypes produced by
BCM, compared to output of LDA. Finally, we visually illustrate that the learned prototypes and subspaces present as meaningful feedback for the characterization of important aspects of the dataset.
4.1 BCM maintains prediction accuracy.
We show that BCM output produces prediction accuracy comparable to or better than LDA, which
uses the same mixture model (Section 3) to learn the underlying structure but does not learn explanations (i.e., prototypes and subspaces). We validate this through use of two standard datasets:
Handwritten Digit [19] and 20 Newsgroups [22]. We use the implementation of LDA available from
[27], which incorporates Gibbs sampling, the same inference technique used for BCM.
Figure 2a depicts the ratio of correctly assigned cluster labels for BCM and LDA. In order to compare the prediction accuracy with LDA, the learned cluster labels are provided as features to a support vector machine (SVM) with linear kernel, as is often done in the LDA literature on clustering [6]. The improved accuracy of BCM over LDA, as depicted in the figures, is explained in part
by the ability of BCM to capture dependencies among features via prototypes, as described in Section 3. We also note that prediction accuracy when using the full 20 Newsgroups dataset acquired
by LDA (accuracy: 0.68? 0.01) matches that reported previously for this dataset when using a combined LDA and SVM approach [33]. Also, LDA accuracy for the full Handwritten Digit dataset
(accuracy: 0.76 ? 0.017) is comparable to that produced by BCM using the subsampled dataset (70
samples per digit, accuracy: 0.77 ? 0.03).
As indicated by Figure 2b, BCM achieves high unsupervised clustering accuracy as a function of
iterations. We can compute this measure for BCM because each cluster is characterized by a prototype ? a particular data point with a label in the given datasets. (Note that this is not possible for
LDA.) We set ? to prefer each ?i to be sparse, so only one prototype generates each observation,
6
Figure 3: Web-interface for the human subject experiment
and we use that prototype?s label for the observation. Sensitivity analysis in Figure 2c indicates that
the additional parameters introduced to learn prototypes and subspaces (i.e., q, ? and c) are not too
sensitive within the range of reasonable choices.
4.2 Verifying the interpretability of BCM
We verified the interpretability of BCM by performing human subject experiments that incorporated
a task requiring an understanding of clusters within a dataset. This task required each participant
to assign 16 recipes, described only by a set of required ingredients (recipe names and instructions
were withheld), to one cluster representation out of a set of four to six. (This approach is similar
to those used in prior work to measure comprehensibility [20].) We chose a recipe dataset1 for this
task because such a dataset requires clusters to be well-explained in order for subjects to be able to
perform classification, but does not require special expertise or training.
Our experiment incorporated a within-subjects design, which allowed for more powerful statistical
testing and mitigated the effects of inter-participant variability. To account for possible learning
effects, we blocked the BCM and LDA questions and balanced the assignment of participants into
the two ordering groups: Half of the subjects were presented with all eight BCM questions first,
while the other half first saw the eight LDA questions. Twenty-four participants (10 females, 14
males, average age 27 years) performed the task, answering a total of 384 questions. Subjects were
encouraged to answer the questions as quickly and accurately as possible, but were instructed to take
a 5-second break every four questions in order to mitigate the potential effects of fatigue.
Cluster representations (i.e., explanations) from LDA were presented as the set of top ingredients
for each recipe topic cluster. For BCM we presented the ingredients of the prototype without the
name of the recipe and without subspaces. The number of top ingredients shown for LDA was set as
the number of ingredients from the corresponding BCM prototype and ran Gibbs sampling for LDA
with different initializations until the ground truth clusters were visually identifiable.
Using explanations from BCM, the average classification accuracy was 85.9%, which was statistically significantly higher (c2 (1, N = 24) = 12.15, p ? 0.001) than that of LDA, (71.3%). For
both LDA and BCM, each ground truth label was manually coded by two domain experts: the first
author and one independent analyst (kappa coefficient: 1). These manually-produced ground truth
labels were identical to those that LDA and BCM predicted for each recipe. There was no statistically significant difference between BCM and LDA in the amount of time spent on each question
(t(24) = 0.89, p = 0.37); the overall average was 32 seconds per question, with 3% more time spent
on BCM than on LDA. Subjective evaluation using Likert-style questionnaires produced no statistically significant differences between reported preferences for LDA versus BCM. Interestingly, this
suggests that participants did not have insight into their superior performance using output from
BCM versus that from LDA.
1
Computer Cooking Contest: http://liris.cnrs.fr/ccc/ccc2014/
7
Prototype (Recipe names)
Ingredients ( Subspaces )
Herbs and Tomato in Pasta
basil, garlic, Italian seasoning, oil
pasta pepper salt, tomato
Generic chili recipe
Microwave brownies
beer chili powder cumin, garlic, meat, oil, onion, pepper, salt,
tomato
baking powder
sugar, butter,
Spiced-punch
chocolate chopped pecans, eggs,
flour, salt, vanilla
cinnamon stick,
lemon juice
orange juice
pineapple juice
sugar, water, whole cloves
(b) Recipe dataset
(a) Handwritten Digit dataset
Figure 4: Learned prototypes and subspaces for the Handwritten Digit and Recipe datasets.
Overall, the experiment demonstrated substantial improvement to participants? classification accuracy when using BCM compared with LDA, with no degradation to other objective or subjective
measures of task performance.
4.3 Learning subspaces
Figure 4a illustrates the learned prototypes and subspaces as a function of sampling iterations for the
Handwritten Digit dataset. For the later iterations, shown on the right of the figure, the BCM output
effectively characterizes the important aspects of the data. In particular, the subspaces learned by
BCM are pixels that define the digit for the cluster?s prototype.
Interestingly, the subspace highlights the absence of writing in certain areas. This makes sense: For
example, one can define a ?7? by showing the absence of pixels on the left of the image where the
loop of a ?9? might otherwise appear. The pixels located where there is variability among digits of
the same cluster are not part of the defining subspace for the cluster.
Because we initialized randomly, in early iterations, the subspaces tend to identify features common
to the observations that were randomly initialized to the cluster. This is because ?s assigns higher
likelihood to features with the most similar values across observations within a given cluster. For
example, most digits ?agree? (i.e., have the same zero pixel value) near the borders; thus, these are
the first areas that are refined, as shown in Figure 4a. Over iterations, the third row of Figure 4a
shows how BCM learns to separate the digits ?3? and ?5,? which tend to share many pixel values in
similar locations. Note that the sparsity of the subspaces can be customized by hyperparameter q.
Next, we show results for BCM using the Computer Cooking Contest dataset in Figure 4b. Each prototype consists of a set of ingredients for a recipe, and the subspace is a set of important ingredients
that define that cluster, highlighted in red boxes. For instance, BCM found a ?chili? cluster defined
by the subspace ?beer,? ?chili powder,? and ?tomato.? A recipe called ?Generic Chili Recipe? was
chosen as the prototype for the cluster. (Note that beer is indeed a typical ingredient in chili recipes.)
5 Conclusion
The Bayesian Case Model provides a generative framework for case-based reasoning and prototypebased modeling. Its clusters come with natural explanations; namely, a prototype (a quintessential
exemplar for the cluster) and a set of defining features for that cluster. We showed the quantitative
advantages in prediction quality and interpretability resulting from the use of BCM. Exemplar-based
modeling (nearest-neighbors, case-based reasoning) has historical roots dating back to the beginning
of artificial intelligence; this method offers a fresh perspective on this topic, and a new way of
thinking about the balance of accuracy and interpretability in predictive modeling.
8
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9
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4,765 | 5,314 | Consistency of Spectral Partitioning of Uniform
Hypergraphs under Planted Partition Model
Debarghya Ghoshdastidar
Ambedkar Dukkipati
Department of Computer Science & Automation
Indian Institute of Science
Bangalore ? 560012, India
{debarghya.g,ad}@csa.iisc.ernet.in
Abstract
Spectral graph partitioning methods have received significant attention from both
practitioners and theorists in computer science. Some notable studies have been
carried out regarding the behavior of these methods for infinitely large sample size
(von Luxburg et al., 2008; Rohe et al., 2011), which provide sufficient confidence
to practitioners about the effectiveness of these methods. On the other hand, recent
developments in computer vision have led to a plethora of applications, where the
model deals with multi-way affinity relations and can be posed as uniform hypergraphs. In this paper, we view these models as random m-uniform hypergraphs
and establish the consistency of spectral algorithm in this general setting. We develop a planted partition model or stochastic blockmodel for such problems using
higher order tensors, present a spectral technique suited for the purpose and study
its large sample behavior. The analysis reveals that the algorithm is consistent for
m-uniform hypergraphs for larger values of m, and also the rate of convergence
improves for increasing m. Our result provides the first theoretical evidence that
establishes the importance of m-way affinities.
1
Introduction
The central theme in approaches like kernel machines [1] and spectral clustering [2, 3] is the use
of symmetric matrices that encode certain similarity relations between pairs of data instances. This
allows one to use the tools of matrix theory to design efficient algorithms and provide theoretical
analysis for the same. Spectral graph theory [4] provides classic examples of this methodology,
where various hard combinatorial problems pertaining to graphs are relaxed to problems of matrix
theory. In this work, we focus on spectral partitioning, where the aim is to group the nodes of a graph
into disjoint sets using the eigenvectors of the adjacency matrix or the Laplacian operator. A statistical framework for this partitioning problem is the planted partition or stochastic blockmodel [5].
Here, one assumes the existence of an unknown map that partitions the nodes of a random graph,
and the probability of occurrence of any edge follows the partition rule. In a recent work, Rohe et
al. [6] studied normalized spectral clustering under the stochastic blockmodel and proved that, for
this method, the fractional number of misclustered nodes goes to zero as the sample size grows.
However, recent developments in signal processing, computer vision and statistical modeling have
posed numerous problems, where one is interested in computing multi-way similarity functions that
compute similarity among more than two data points. A few applications are listed below.
Example 1. In geometric grouping, one is required to cluster points sampled from a number of
geometric objects or manifolds [7]. Usually, these objects are highly overlapping, and one cannot
use standard distance based pairwise affinities to retrieve the desired clusters. Hence, one needs to
construct multi-point similarities based on the geometric structure. A special case is the subspace
clustering problem encountered in motion segmentation [7], face clustering [8] etc.
1
Example 2. The problem of point-set matching [9] underlies several problems in computer vision
including image registration, object recognition, feature tracking etc. The problem is often formulated as finding a strongly connected component in a uniform hypergraph [9, 10], where the strongly
connected component represents the correct matching. This formulation has the flavor of the standard problem of detecting cliques in random graphs.
Both of the above problems are variants of the classic hypergraph partitioning problem, that arose in
the VLSI community [11] in 1980s, and has been an active area of research till date [12]. Spectral approaches for hypergraph partitioning also exist in the literature [13, 14, 15], and various definitions of
the hypergraph Laplacian matrix has been proposed based on different criteria. Recent studies [16]
suggest an alternative representation of uniform hypergraphs in terms of the ?affinity tensor?. Tensors have been popular in machine learning and signal processing for a considerable time (see [17]),
and have even found use in graph partitioning and detecting planted partitions [17, 18]. But their
role in hypergraph partitioning have been mostly overlooked in the literature. Recently, techniques
have emerged in computer vision that use such affinity tensors in hypergraph partitioning [8, 9].
This paper provides the first consistency result on uniform hypergraph partitioning by analyzing the
spectral decomposition of the affinity tensor. The main contributions of this work are the following.
(1) We propose a planted partition model for random uniform hypergraphs similar to that of
graphs [5]. We show that the above examples are special cases of the proposed partition model.
(2) We present a spectral technique to extract the underlying partitions of the model. This method
relies on a spectral decomposition of tensors [19] that can be computed in polynomial time, and
hence, it is computationally efficient than the tensorial approaches in [10, 8].
(3) We analyze the proposed approach and provide almost sure bounds on the number of misclustered nodes. Our analysis reveals that the presented method is consistent almost surely in the grouping problem and for detection of a strongly connected component, whenever one uses m-way affinities for any m ? 3 and m ? 4, respectively. The derived rate of convergence also shows that the
use of higher order affinities lead to a faster decay in the number of misclustered nodes.
(4) We numerically demonstrate the performance of the approach on benchmark datasets.
2
Planted partitions in random uniform hypergraphs
We describe the planted partition model for an undirected unweighted graph. Let ? : {1, . . . , n} ?
{1, . . . , k} be an (unknown) partition of n nodes into k disjoint groups, i.e., ?i = ?(i) denotes the
partition in which node-i belongs. We also define an assignment matrix Zn ? {0, 1}n?k such that
(Zn )ij = 1 if j = ?i , and 0 otherwise. For some unknown symmetric matrix B ? [0, 1]k?k , the
random graph on the n nodes contains the edge (i, j) with probability B?i ?j . Let the symmetric
matrix An ? {0, 1}n?n be a realization of the affinity matrix of the random graph on n nodes. The
aim is to identify Zn given the matrix An . In some cases, one also needs to estimate the entries in
B. One can hope to achieve this goal for the following reason: If An ? Rn?n contains the expected
values of the entries in An conditioned on B and ?, then one can write An as An = Zn BZnT [6].
Thus, if one can find An , then this relation can be used to find Zn .
We generalize the partition model to uniform hypergraphs. A hypergraph is a structure on n nodes
with multi-way connections or hyperedges. Formally, each hyperedge in an undirected unweighted
hypergraph is a collection of an arbitrary number of vertices. A special case is that of m-uniform
hypergraph, where each hyperedge contains exactly m nodes. One can note that a graph is a 2uniform hypergraph. An often cited example of uniform hypergraph is as follows [10]. Let the
nodes be representative of points in an Euclidean space, where a hyperedge exists if the points
are collinear. For m = 2, we obtain a complete graph that does not convey enough information
about the nodes. However, for m = 3, the constructed hypergraph is a union of several connected
components, each component representing a set of collinear points. The affinity relations of an muniform hypergraph can be represented in the form of an mth -order tensor An ? {0, 1}n?n?...?n ,
which we call an affinity tensor. The entry (An )i1 ...im = 1 if there exists a hyperedge on nodes
i1 , . . . , im . One can observe that the tensor is symmetric, i.e., invariant under any permutation of
indices. In some works [16], the tensor is scaled by a factor of 1/(m ? 1)! for certain reasons.
Let ? and Zn be as defined above, and B ? [0, 1]k?...?k be an mth -order k-dimensional symmetric
tensor. The random m-uniform hypergraph on the n nodes is constructed such that a hyperedge
occurs on nodes i1 , . . . , im with probability B?i1 ...?im . If An is a random affinity tensor of the
2
hypergraph, our aim is to find Zn or ? from An . Notice that if An ? Rn?...?n contains the
expected values of the entries in An , then one can write the entries in An as
(An )i1 ...im = B?i1 ...?im =
k
X
Bj1 ...jm (Zn )i1 j1 . . . (Zn )im jm .
(1)
j1 ,...,jm =1
The subscript n in the above terms emphasizes their dependence on the number of nodes. We now
describe how two standard applications in computer vision can be formulated as the problem of
detecting planted partitions in uniform hypergraphs.
2.1
Subspace clustering problem
In motion segmentation [7, 20] or illumination invariant face clustering [8], the data belong to a
high dimensional space. However, the instances belonging to each cluster approximately span a
low-dimensional subspace (usually, of dimension 3 or 4). Here, one needs to check whether m
points approximate such a subspace, where this information is useful only when m is larger than the
dimension of the underlying subspace of interest. The model can be represented as an m-uniform
hypergraph, where a hyperedge occurs on m nodes whenever they approximately span a subspace.
The partition model for this problem is similar to the standard four parameter blockmodel [6]. The
number of partitions is k, and each partition contains s nodes, i.e., n = ks. There exists probabilities
p ? (0, 1] and q ? [0, p) such that any set of m vectors span a subspace with probability p if all m
vectors belong to the same group, and with probability q if they come from different groups. Thus,
the tensor B has the form Bi...i = p for all i = 1, . . . , k, and Bi1 ...im = q for all the other entries.
2.2
Point set matching problem
We consider a simplified version of the matching problem [10], where one is given two sets of
points of interest, each of size s. In practice, these points may come from two different images
of the same object or scene, and the goal is to match the corresponding points. One can see that
there are s2 candidate matches. However, if one considers m correct matches then certain properties
are preserved. For instance, let i1 , . . . , im be some points from the first image, and i01 , . . . , i0m be
the corresponding points in the second image, then the angles or ratio of areas of triangles formed
among these points are more or less preserved [9]. Thus, the set of matches (i1 , i01 ), . . . , (im , i0m )
have a certain connection, which is usually not present if the matches are not exact.
The above model is an m-uniform hypergraph on n = s2 nodes, each node representing a candidate match, and a hyperedge is formed if properties (like preservation
of angles) is satisfied by m
?
candidate matches. Here, one can see that there are only s = n correct matches, which have a
large number of hyperedges among them, whereas very few hyperedges
may
? be present for other
?
combinations. Thus, the partition model has two groups of size n and (n ? n), respectively. For
p, q ? [0, 1], p q, p denotes the probability of a hyperedge among m correct matches and for any
other m candidates, there is a hyperedge with probability q. Thus, if the first partition is the strongly
connected component, then we have B ? [0, 1]2?...?2 with B1...1 = p and Bi1 ...im = q otherwise.
3
Spectral partitioning algorithm and its consistency
Before presenting the algorithm, we provide some background on spectral decomposition of tensors.
In the related literature, one can find a number of significantly different characterizations of the
spectral properties of tensors. While the work in [16] builds on a variational characterization, De
Lathauwer et al. [19] provide an explicit decomposition of a tensor in the spirit of the singular
value decomposition of matrices. The second line of study is more appropriate for our work since
our analysis significantly relies on the use of Davis-Kahan perturbation theorem [21] that uses an
explicit decomposition, and has been often used to analyze spectral clustering [2, 6].
The work in [19] provides a way of expressing any mth -order n-dimensional symmetric tensor,
An , as a mode-k product [19] of a certain core tensor with m orthonormal matrices, where each
bn ? {0, 1}n?nm?1 ,
orthonormal matrix is formed from the orthonormal left singular vectors of A
3
whose entries, for all i = 1, . . . , n and j = 1, . . . , nm?1 , are defined as
bn )ij = (An )i i ...i ,
(A
1 2
m
if i = i1 and j = 1 +
m
X
(il ? 1)nl?2 .
(2)
l=2
bn , often called the mode-1 flattened matrix, forms a key component of the
The above matrix A
bn contain inforpartitioning algorithm. Later, we show that the leading k left singular vectors of A
mation about the true partitions in the hypergraph. It is easier to work with the symmetric matrix
bn A
bTn ? Rn?n , whose eigenvectors correspond to the left singular vectors of A
bn . The
Wn = A
spectral partitioning algorithm is presented in Algorithm 1, which is quite similar to the normalized
spectral clustering [2]. Such a tensor based approach was first studied in [7] for geometric grouping. Subsequent improvements of the algorithm were proposed in [22, 20]. However, we deviate
from these methods as we do not normalize the rows of the eigenvector matrix. The method in [9]
also uses the largest eigenvector of the flattened matrix for the point set matching problem. This is
computed via tensor power iterations. To keep the analysis simple, we do not use such iterations.
The complexity of Algorithm 1 is O(nm+1 ), which can be significantly improved using sampling
techniques as in [7, 9, 20]. The matrix Dn is used for normalization as in spectral clustering.
Algorithm 1 Spectral partitioning of m-uniform hypergraph
bn using (2).
1. From the mth -order affinity tensor An , construct A
bn A
bT , and Dn ? Rn?n be diagonal with (Dn )ii = Pn (Wn )ij .
2. Let Wn = A
n
j=1
?1/2
3. Set Ln = Dn
?1/2
Wn Dn
.
4. Compute leading k orthonormal eigenvectors of Ln , denoted by matrix Xn ? Rn?k .
5. Cluster the rows of Xn into k clusters using k-means clustering.
6. Assign node-i of hypergraph to j th partition if ith row of Xn is grouped in j th cluster.
An alternative technique of using eigenvectors of Laplacian matrix is often preferred in graph parbn , in
titioning [3], and has been extended to hypergraphs [13, 15]. Unlike the flattened matrix, A
Algorithm 1, such Laplacians do not preserve the spectral properties of a higher-order structure such
as the affinity tensor that accurately represents the affinities of the hypergraph. Hence, we avoid the
use of hypergraph Laplacian.
3.1
Consistency of above algorithm
We now comment on the error incurred by Algorithm 1. For this, let Mn be the set of nodes that
are incorrectly clustered by Algorithm 1. It is tricky to formalize the definition of Mn in clustering
problems. We follow the definition of Mn given in [6] that requires some details of the analysis
and hence, a formal definition is postponed till Section 4. In addition, we need the following terms.
b ?
The analysis depends on the tensor B ? [0, 1]k?...?k of the underlying random model. Let B
k?k
k?km?1
be the flattening of tensor B using (2). We also define a matrix Cn ? R
as
[0, 1]
b nT Zn )?(m?1) B
b T (ZnT Zn )1/2 ,
Cn = (ZnT Zn )1/2 B(Z
(3)
where (ZnT Zn )?(m?1) is the (m ? 1)-times Kronecker product of ZnT Zn with itself. Use of such
Kronecker product is quite common in tensor decompositions (see [19]). Observe that the positive
semi-definite matrix Cn contains information regarding the connectivity of clusters (stored in B)
and the cluster sizes (diagonal entries of ZnT Zn ). Let ?k (Cn ) be the smallest eigenvalue of Cn ,
which is non-negative. In addition, define Dn ? Rn?n as the expectation of the diagonal matrix
Dn . One can see that (Dn )ii ? nm for all i = 1, . . . , n. Let Dn and Dn be the smallest and largest
values in Dn . Also, let S n and S n be the sizes of the smallest and largest partitions, respectively.
We have the following bound on the number of misclustered nodes.
Theorem 1. If there exists N such that for all n > N ,
r
?k (Cn ) 2nm?1
2
m
>0
and
Dn ? n (m ? 1)!
?n :=
?
,
Dn
log n
Dn
4
m?1
and if (log n)3/2 = o ?n n 2 , then the number of misclustered nodes
|Mn | = O
S n (log n)2 nm+1
?n2 D2n
almost surely.
The above result is too general to provide conclusive remarks about consistency of the algorithm.
Hence, we focus on two examples, precisely the ones described in Sections 2.1 and 2.2. However,
without loss of generality, we assume here that q > 0 since otherwise, the problem of detecting the
partitions is trivial (at least for reasonably large n) as we can construct the partitions only based on
the presence of hyperedges. The following results are proved in the appendix. The proofs mainly
depend on computation of ?k (Cn ), which can be derived for the first example, while for the second,
it is enough to work with a lower bound of ?k (Cn ). Further, in the first example, we make the result
general by allowing the number of clusters, k, to grow with n under certain conditions.
Corollary 2. Consider the setting
1 of subspace
clustering described in Section 2.1. If the number
?1
of clusters k satisfy k = O n 2m (log n)
, then the conditions in Theorem 1 are satisfied and
2m?1
2
3?2m
k
(log n)
(log n)
|Mn | = O
=
O
almost surely. Hence, for m > 2, |Mn | ? 0
1
nm?2
nm?3+ 2m
|Mn |
a.s. as n ? ?, i.e., the algorithm is consistent. For m = 2, we can only conclude
? 0 a.s.
n
From the above result, it is evident that the rate of convergence improves as m increases, indicating
that, ignoring practical considerations, one should prefer the use of higher order affinities. However, the condition of number of clusters becomes more strict in such cases. We note here that our
result and conditions are quite similar to those given in [6] for the case of four-parameter blockmodel. Thus, Algorithm 1 is comparable to spectral clustering [6]. Next, we consider the setting of
Section 2.2.
Corollary 3. For the problem of point set matching
described
in Section 2.2, the conditions in
(log n)2
Theorem 1 are satisfied for m ? 3 and |Mn | = O
a.s. Hence, for m > 3, |Mn | ? 0
nm?3
|Mn |
? 0 a.s.
a.s. as n ? ?, i.e., the algorithm is consistent. For m = 3, we can only conclude
n
The above result shows, theoretically, why higher order matching provides high accuracy in practice [9]. It also suggests that increase in the order of tensor will lead to a better convergence rate.
We note that the following result does not
? hold for graphs (m = 2). In Corollary 3, we used the fact
that the smaller partition is of size s = n. The result can be made more general in terms of s, i.e.,
for m > 4, if s ? 3p
q 3 eventually, then Algorithm 1 is consistent.
Before providing the detailed analysis (proof of Theorem 1), we briefly comment on the model
considered here. In Section 2, we have followed the lines of [6] to define the model with An =
Zn BZnT . However, this would mean that the diagonal entries in An are non-negative, and hence,
there is a non-zero probability of formation of self loops that is not common in practice. The same
issue exists for hypergraphs. To avoid this, one can add a correction term to An so that the entries
with repeated indices become zero. Under this correction, conditions in Theorem 1 should not
change significantly. This is easy to verify for graphs, but it is not straightforward for hypergraphs.
4
Analysis of partitioning algorithm
In this section, we prove Theorem 1. The result follows from a series of lemmas. The proof requires
cn be the flattening of the tensor An defined in (1). Then we can
defining certain terms. Let A
T ?(m?1)
c
b
write An = Zn B(Zn )
, where (ZnT )?(m?1) is (m ? 1)-times Kronecker product of ZnT
with itself. Along with the definitions in Section 3, let Wn ? Rn?n be the expectation of Wn , and
T
?1/2
?1/2
cn A
cn + Pn , where Pn is
Ln = Dn Wn Dn . One can see that Wn can be written as Wn = A
cn . The proof contains the following steps:
a diagonal matrix defined in terms of the entries in A
(1) For any fixed n, we show that if ?n > 0 (stated in Theorem 1), the leading k orthonormal
5
eigenvectors of Ln has k distinct rows, where each row is a representative of a partition.
(2) Since, Ln is not the expectation of Ln , we derive a bound on the Frobeniusqnorm of their
difference. The bound holds almost surely for all n if eventually Dn ? nm (m ? 1)! log2 n .
(3) We use a version of Davis-Kahan sin-? theorem given in [6] that
surely bounds the
almost
m?1
difference in the leading eigenvectors of Ln and Ln if (log n)3/2 = o ?n n 2 .
(4) Finally, we rely on [6, Lemma 3.2], which holds in our case, to define the set of misclustered
nodes Mn , and its size is bounded almost surely using the previously derived bounds.
We now present the statements for the above results. The proofs can be found in the appendix.
Lemma 4. Fix n and let ?n be as defined in Theorem 1. If ?n > 0, then there exists ?n ? Rk?k such
that the columns of Zn ?n are the leading k orthonormal eigenvectors of Ln . Moreover, for nodes i
and j, ?i = ?j if and only if the ith and j th rows of Zn ?n are identical.
Thus, clustering the rows of Zn ?n into k clusters will provide the true partitions, and the cluster
centers will precisely be these rows. The condition ?n > 0 is required to ensure that the eigenvalues
corresponding to the columns of Zn ?n are strictly greater than other eigenvalues. The requirement
of a positive eigen-gap is essential for analysis of any spectral partitioning method [2, 23]. Next, we
focus on deriving the upper bound for kLn ? Ln kF .
q
Lemma 5. If there exists N such that Dn ? nm (m ? 1)! log2 n for all n > N , then
kLn ? Ln kF ?
4n
m+1
2
log n
,
Dn
(4)
almost surely.
The condition in the above result implies that each vertex is reasonably connected to other vertices
of the hypergraph, i.e., there are no outliers. It is easy to satisfy this condition in the stated examples
as Dn ? q 2 nm and
hence, it holds for all q > 0. Under the condition, one can also see that the
bound in (4) is O
(log n)3/2
n
m?1
2
and hence goes to zero as n increases. Note that in Lemma 4, ?n > 0
need not hold for all n, but if it holds eventually, then we can choose N such that the conditions in
Lemmas 4 and 5 both hold for all n > N . Under such a case, we use the Davis-Kahan perturbation
theorem [21] as stated in [6, Theorem 2.1] to claim the following.
Lemma
6. Let Xn ? Rn?k contain the leading k orthonormal eigenvectorsq
of Ln . If (log n)3/2 =
o ?n n
m?1
2
2
log n
and there exists N such that ?n > 0 and Dn ? nm (m ? 1)!
then there exists an orthonormal (rotation) matrix On ? R
kXn ? Zn ?n On kF ?
16n
k?k
for all n > N ,
such that
m+1
2
log n
,
?n D n
(5)
almost surely.
m?1
is crucial as it ensures that the difference in eigenvalues
The condition (log n)3/2 = o ?n n 2
of Ln and Ln decays much faster than the eigen-gap in Ln . This condition requires the eigen-gap
(lower bounded by ?n ) to decay at a relatively slow rate, and is necessary for using [6, Theorem 2.1].
The bound (5) only says that rows of Xn converges to some rotation of the rows of Zn ?n . However,
this is not an issue since the k-means algorithm is expected to perform well as long as the rows of
Xn corresponding to each partition are tightly clustered, and the k clusters are well-separated. Now,
let z1 , . . . , zn be the rows of Zn , and let ci be the center of the cluster in which ith row of Xn is
grouped for each i ? {1, . . . , n}. We use a key result from [6] that is applicable in our setting.
Lemma 7. [6, Lemma 3.2] For the matrix On from Lemma 6, if kci ? zi ?n On k2 < ? 1 , then
2S n
kci ? zi ?n On k2 < kci ? zj ?n On k2 for all zj 6= zi .
This result hints that one may use the definition of correct clustering as follows. Node-i is correctly
clustered if its center ci is closer to zi ?n On than the rows corresponding to other partitions. A sufficient condition to satisfy this definition is kci ? zi ?n On k2 < ? 1 . Hence, the set of misclustered
2S n
nodes is defined as [6]
(
Mn =
1
i ? {1, . . . , n} : kci ? zi ?n On k2 ? p
2S n
6
)
.
(6)
It is easy to see that if Mn is empty, i.e., all nodes satisfy the condition kci ? zi ?n On k2 < ? 1
2S n
,
then the clustering leads to true partitions, and does not incur any error. Hence, for statements, where
|Mn | is small (at least compared to n), one can always use such a definition for misclustered nodes.
The next result provides a simple bound on |Mn |, that immediately leads to Theorem 1.
Lemma 8. If the k-means algorithm achieves its global optimum, then the set Mn satisfies
|Mn | ? 8S n kXn ? Zn ?n On k2F .
(7)
In practice, k-means algorithm tries to find a local minimum, and hence, one should run this step
with multiple initializations to achieve a global minimum. However, empirically we found that
good performance is achieved even if we use a single run of k-means. From above lemma, it is
straightforward to arrive at Theorem 1 by using the bound in Lemma 6.
5
5.1
Experiments
Validation of Corollaries 2 and 3
We demonstrate the claims of Corollaries 2 and 3, where we stated that for higher order tensors, the
number of misclustered nodes decays to zero at a faster rate. We run Algorithm 1 on both the models
of subspace clustering and point-set matching, varying the number of nodes n, the results for each n
being averaged over 10 trials. For the clustering model (Section 2.1), we choose p = 0.6, q = 0.4,
and consider two cases of k = 2 and 3 cluster problems. Figure 1 (top row) shows that in this model,
the number of errors eventually decreases for all m, even m = 2. This observation is similar to the
one in [6]. However, the decrease is much faster for m = 3, where accurate partitioning is often
observed for n ? 100. We also observe that error rises for larger k, thus validating the dependence
of the bound on k. A similar inference can be drawn from Figure 1 (second row) for the matching
?
problem (Section 2.2), where we use p = 0.9, q = 0.1 and the number of correct matches as n.
5.2
Motion Segmentation on Hopkins 155 dataset
We now turn to practical applications, and test the performance of Algorithm 1 in motion segmentation. We perform the experiments on the Hopkins 155 dataset [24], which contains 120 videos with
2 independent affine motions. Figure 1 (third row) shows two cases, where Algorithm 1 correctly
clusters the trajectories into their true groups. We used 4th -order tensors in the approach, where the
bn is tackled by using only 500 uniformly sampled columns of A
bn for comlarge dimensionality of A
puting Wn . We also compare the performance of Algorithm 1, averaged over 20 runs, with some
standard approaches. The results for other methods have been taken from [20]. We observe that Algorithm 1 performs reasonably well, while the best performance is obtained using Sparse Grassmann
Clustering (SGC) [20], which is expected as SGC is an iterative improvement of Algorithm 1.
5.3
Matching point sets from the Mpeg-7 shape database
We now consider a matching problem using points sampled from images in Mpeg-7 database [25].
This problem has been considered in [10]. We use 70 random images, one from each shape class.
Ten points were sampled from the boundary of each shape, which formed one point set. The other
set of points was generated by adding Gaussian noise of variance ? 2 to the original points and then
using a random affine transformation on the points. In Figure 1 (last row), we compare performance
of Algorithm 1 with the methods in [9, 10], which have been shown to outperform other methods.
We use 4-way similarities based on ratio of areas of two triangles. We show the variation in the
number of correctly detected matches and the F1-score for all methods as ? increases from 0 to
0.2. The results show that Algorithm 1 is quite robust compared to [10] in detecting true matches.
However, Algorithm 1 does not use additional post-processing as in [9], and hence, allows high
number of false positives that reduces F1-score, whereas [9, 10] show similar trends in both plots.
6
Concluding remarks
In this paper, we presented a planted partition model for unweighted undirected uniform hypergraphs. We devised a spectral approach (Algorithm 1) for detecting the partitions from the affinity
7
The plots show variation in the
number (left) and fraction (right)
of misclustered nodes as n increases in k = 2 and 3 cluster problems for 2 and 3-uniform
hypergraphs. Black lines are for
m = 2 and red for m = 3. Solid
lines for k = 2, and dashed lines
for k = 3.
The plots show variation in number (left) and fraction (right) of
incorrect matches as n increases
in matching problem for 2 and
3-uniform hypergraphs. Black
lines are for m = 2 and red for
m = 3.
Percentage error in clustering
LSA
4.23 %
SCC
2.89 %
LRR-H
2.13 %
LRSC
3.69 %
SSC
1.52 %
SGC
1.03 %
Algorithm 1
1.83 %
Figure 1: First row: Number of misclustered nodes in clustering problem as n increases.
Second row: Number of misclustered nodes in matching problem as n increases.
Third row: Grouping two affine motions with Algorithm 1 (left), and performance comparison of
Algorithm 1 with other methods (right).
Fourth row: Variation in number of correct matches detected (left) and F1-score (middle) as noise
level, ? increases. (right) A pair of images where Algorithm 1 correctly matches all sampled points.
tensor of the corresponding random hypergraph. The above model is appropriate for a number of
problems in computer vision including motion segmentation, illumination-invariant face clustering,
point-set matching, feature tracking etc. We analyzed the approach to provide an almost sure upper
bound on the number of misclustered nodes (c.f. Theorem 1). Using this bound, we conclude that
for the problems of subspace clustering and point-set matching, Algorithm 1 is consistent for m ? 3
and m ? 4, respectively. To the best of our knowledge, this is the first theoretical study of the above
problems in a probabilistic setting, and also the first theoretical evidence that shows importance of
m-way affinities.
Acknowledgement
D. Ghoshdastidar is supported by Google Ph.D. Fellowship in Statistical Learning Theory.
8
References
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9
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4,766 | 5,315 | Projecting Markov Random Field Parameters for
Fast Mixing
Justin Domke
NICTA, The Australian National University
[email protected]
Xianghang Liu
NICTA, The University of New South Wales
[email protected]
Abstract
Markov chain Monte Carlo (MCMC) algorithms are simple and extremely powerful techniques to sample from almost arbitrary distributions. The flaw in practice
is that it can take a large and/or unknown amount of time to converge to the stationary distribution. This paper gives sufficient conditions to guarantee that univariate
Gibbs sampling on Markov Random Fields (MRFs) will be fast mixing, in a precise sense. Further, an algorithm is given to project onto this set of fast-mixing
parameters in the Euclidean norm. Following recent work, we give an example use
of this to project in various divergence measures, comparing univariate marginals
obtained by sampling after projection to common variational methods and Gibbs
sampling on the original parameters.
1 Introduction
Exact inference in Markov Random Fields (MRFs) is generally intractable, motivating approximate
algorithms. There are two main classes of approximate inference algorithms: variational methods
and Markov chain Monte Carlo (MCMC) algorithms [13].
Among variational methods, mean-field approximations [9] are based on a ?tractable? family of
distributions, such as the fully-factorized distributions. Inference finds a distribution in the tractable
set to minimize the KL-divergence from the true distribution. Other methods, such as loopy belief
propagation (LBP), generalized belief propagation [14] and expectation propagation [10] use a less
restricted family of target distributions, but approximate the KL-divergence. Variational methods
are typically fast, and often produce high-quality approximations. However, when the variational
approximations are poor, estimates can be correspondingly worse.
MCMC strategies, such as Gibbs sampling, simulate a Markov chain whose stationary distribution is
the target distribution. Inference queries are then answered by the samples drawn from the Markov
chain. In principle, MCMC will be arbitrarily accurate if run long enough. The principal difficulty
is that the time for the Markov chain to converge to its stationary distribution, or the ?mixing time?,
can be exponential in the number of variables.
This paper is inspired by a recent hybrid approach for Ising models [3]. This approach minimizes
the divergence from the true distribution to one in a tractable family. However, the tractable family
is a ?fast mixing? family where Gibbs sampling is guaranteed to quickly converge to the stationary
distribution. They observe that an Ising model will be fast mixing if the spectral norm of a matrix
containing the absolute values of all interactions strengths is controlled. An algorithm projects
onto this fast mixing parameter set in the Euclidean norm, and projected gradient descent (PGD)
can minimize various divergence measures. This often leads to inference results that are better
than either simple variational methods or univariate Gibbs sampling (with a limited time budget).
However, this approach is limited to Ising models, and scales poorly in the size of the model, due to
the difficulty of projecting onto the spectral norm.
1
The principal contributions of this paper are, first, a set of sufficient conditions to guarantee that
univariate Gibbs sampling on an MRF will be fast-mixing (Section 4), and an algorithm to project
onto this set in the Euclidean norm (Section 5). A secondary contribution of this paper is considering
an alternative matrix norm (the induced ?-norm) that is somewhat looser than the spectral norm,
but more computationally efficient. Following previous work [3], these ideas are experimentally
validated via a projected gradient descent algorithm to minimize other divergences, and looking at
the accuracy of the resulting marginals. The ability to project onto a fast-mixing parameter set may
also be of independent interest. For example, it might be used during maximum likelihood learning
to ensure that the gradients estimated through sampling are more accurate.
2 Notation
We consider discrete pairwise MRFs with n variables, where the i-th variable takes values in
{1, ..., Li }, E is the set of edges, and ? are the potentials on each edge. Each edge in E is an ordered
pair (i, j) with i ? j. The parameters are a set of matrices ? := {?ij |?ij ? RLi ?Lj , ?(i, j) ? E}.
When i > j, and (j, i) ? E, we let ?ij denote the transpose of ?ji . The corresponding distribution is
?
?
#
p(x; ?) = exp ?
?ij (xi , xj ) ? A(?)? ,
(1)
(i,j)?E
where A(?) := log
&
x exp
'&
(
ij
?
(x
,
x
)
is the log-partition function, and ?ij (xi , xj )
i
j
(i,j)?E
denotes the entry in the xi -th row and xj -th column of ?ij . It is easy to show that any parametrization
of a pairwise MRF can be converted into this form. ?Self-edges? (i, i) can be included in E if one
wishes to explicitly represent univariate terms.
It is sometimes convenient to work with the exponential family representation
p(x; ?) = exp{f (x) ? ? ? A(?)},
(2)
where f (x) is the sufficient statistics for configuration x. If these are indicator functions for all
configurations of all pairs in E, then the two representations are equivalent.
3 Background Theory on Rapid Mixing
This section reviews background on mixing times that will be used later in the paper.
Definition 1. Given two finite
& distributions p and q, the total variation distance ? ? ?T V is defined
as ?p(X) ? q(X)?T V = 21 x |p(X = x) ? q(X = x)|.
Next, one must define a measure of how fast a Markov chain converges to the stationary distribution.
Let the state of the Markov chain after t iterations be X t . Given a constant ?, this is done by finding
some number of iterations ? (?) such that the induced distribution p(X t |X 0 = x) will always have a
distance of less than ? from the stationary distribution, irrespective of the starting state x.
Definition 2. Let {X t } be the sequence of random variables corresponding to running Gibbs sampling on a distribution p. The mixing time ? (?) is defined as ? (?) = min{t : d(t) < ?}, where
d(t) = maxx ?P(X t |X 0 = x) ? p(X)?T V is the maximum distance at time t when considering all
possible starting states x.
Now, we are interested in when Gibbs sampling on a distribution p can be shown to have a fast
mixing time. The central property we use is the dependency of one variable on another, defined
informally as how much the conditional distribution over Xi can be changed when all variables
other than Xj are the same.
Definition 3. Given a distribution p, the dependency matrix R is defined by
Rij = ? max ? ?p(Xi |x?i ) ? p(Xi |x??i )?T V .
x,x :x?j =x?j
Here, the constraint x?j = x??j indicates that all variables in x and x? are identical except xj . The
central result on rapid mixing is given by the following Theorem, due to Dyer et al. [5], generalizing
the work of Hayes [7]. Informally, it states that if ?R? < 1 for any sub-multiplicative norm ? ? ?,
then mixing will take on the order of n ln n iterations, where n is the number of variables.
2
Theorem 4. [5, Lemma 17] If ? ? ? is any sub-multiplicative matrix norm and ||R|| < 1, the mixing
time of univariate!Gibbs sampling
on a system with n variables with random updates is bounded by
"
?1n ? ?1T
n
n?
? (?) ? 1??R? ln
.
?
Here, ?1n ? denotes the same matrix norm applied to a matrix of ones of size n ? 1, and similarly
for 1Tn . In particular, if ? ? ? induced by a vector p-norm, then ?1n ? ?1Tn ? = n.
Since this result is true for a variety of norms, it is natural to ask, for a given matrix R, which norm
will give the strongest result. It can be shown that for symmetric matrices (such as the dependency
matrix), the spectral norm ? ? ?2 is always superior.
Theorem 5. [5, Lemma 13] If A is a symmetric matrix and ? ? ? is any sub-multiplicative norm,
then ?A?2 ? ?A?.
Unfortunately, as will be discussed below, the spectral norm can be more computationally expensive
than other norms. As such, we will also consider the use of the ?-norm ? ? ?? . This leads to
additional looseness in the bound in general, but is limited in some cases. In particular if R = rG
where G is the adjacency matrix for some regular graph with degree d, then for all induced p-norms,
?R? = rd, since ?R? = maxx?=0 ?Rx?/?x| = r maxx?=0 ?Gx?/?x? = r?Go?/?o? = rd, where
o is a vector of ones. Thus, the extra looseness from using, say, ? ? ?? instead of ? ? ?2 will tend to
be minimal when the graph is close to regular, and the dependency is close to a constant value. For
irregular graphs with highly variable dependency, the looseness can be much larger.
4 Dependency for Markov Random Fields
In order to establish that Gibbs sampling on a given MRF will be fast mixing, it is necessary to
compute (a bound on) the dependency matrix R, as done in the following result. The proof of this
result is fairly long, and so it is postponed to the Appendix. Note that it follows from several bounds
on the dependency that are tighter, but less computationally convenient.
Theorem 6. The dependency matrix for a pairwise Markov random field is bounded by
1 ij
ij
Rij (?) ? max ???a
? ??b
?? .
a,b 2
ij
Here, ??a
indicates the a?th column of ?ij . Note that the MRF can include univariate terms as selfedges with no impact on the dependency bound, regardless of the strength of the univariate terms. It
can be seen easily that from the definition of R (Definition 3), for any i the entry Rii for self-edges
(i, i) should always be zero. One can, without loss of generality, set each column of ?ii to be the
same, meaning that Rii = 0 in the above bound.
5 Euclidean Projection Operator
The Euclidean distance between two MRFs parameterized respectively by ? and ? is ?? ? ??2 :=
#
ij
ij 2
(i,j)?E ?? ? ? ?F . This section considers projecting a given vector ? onto the fast mixing set
or, formally, finding a vector ? with minimum Euclidean distance to ?, subject to the constraint
that a norm ? ? ?? applied to the bound on the dependency matrix R is less than some constant c.
Euclidean projection is considered because, first, it is a straightforward measure of the closeness
between two parameters and, second, it is the building block of the projected gradient descent for
projection in other distance measures. To begin with, we do not specify the matrix norm ? ? ?? , as it
could be any sub-multiplicative norm (Section 3).
Thus, in principle, we would like to find ? to solve
projc (?) := argmin ?? ? ??2 .
(3)
?:?R(?)?? ?c
Unfortunately, while convex, this optimization turns out to be somewhat expensive to solve, due to
a lack of smoothness Instead, we introduce a matrix Z, and constrain that Zij ? Rij (?), where
Rij (?) is the bound on dependency in Thm 6 (as an equality). We add an extra quadratic term
3
??Z ? Y ?2F to the objective, where Y is an arbitrarily given matrix and ? > 0 is trade-off between
the smoothness and the closeness to original problem (3). The smoothed projection operator is
projC (?, Y ) := argmin ?? ? ??2 + ??Z ? Y ?2F , C = {(?, Z) : Zij ? Rij (?), ?Z?? ? c}. (4)
(?,Z)?C
If ? = 0, this yields a solution that is identical to that of Eq. 3. However, when ? = 0, the objective
in Eq. 4 is not strongly convex as a function of Z, which results in a dual function which is nonsmooth, meaning it must be solved with a method like subgradient descent, with a slow convergence
rate. In general, of course, the optimal point of Eq. 4 is different to that of Eq. 3. However,
the main usage of the Euclidean projection operator is the projection step in the projected gradient
descent algorithm for divergence minimization. In these tasks the smoothed projection operator can
be directly used in the place of the non-smoothed one without changing the final result. In situations
when the exact Euclidean projection is required, it can be done by initializing Y1 arbitrarily and
repeating (?k+1 , Yk+1 ) ? projC (?, Yk ), for k = 1, 2, . . . until convergence.
5.1 Dual Representation
Theorem 7. Eq. 4 has the dual representation
maximize
?,?,?,?
g(?, ?, ?, ?)
,
(5)
subject to ?ij (a, b, c) ? 0, ?ij (a, b, c) ? 0, ?(i, j) ? E, a, b, c
where
g(?, ?, ?, ?) = min h1 (Z; ?, ?, ?, ?) + min h2 (?; ?, ?)
Z
?
h1 (Z; ?, ?, ?, ?) = ?tr(Z?T ) + I(?Z?? ? c) + ??Z ? Y ?2F
# ij
1 ! !"
ij
h2 (?; ?, ?) = ?? ? ??2 +
?ij (a, b, c) ? ?ij (a, b, c) (?c,a
? ?c,b
),
2
i,j?E a,b,c
$
? ij +
?
in which ?ij := ?ij Dij + ?
:=
a,b,c ?ij (a, b, c) + ?ij (a, b, c), where ?ij
%
?ij
if (i, j) ? E
, and D is an indicator matrix with Dij = 0 if (i, j) ? E or (j, i) ? E,
??ij if (j, i) ? E
and Dij = 1 otherwise. The dual variables ?ij and ?ij are arrays of size Lj ? Li ? Li for all pairs
(i, j) ? E while ? and ? are of size n ? n.
The proof of this is in the Appendix. Here, I(?) is the indicator function with I(x) = 0 when x is
true and I(x) = ? otherwise.
Being a smooth optimization problem with simple bound constraints, Eq. 5 can be solved with
LBFGS-B [2]. For a gradient-based method like this to be practical, it must be possible to quickly
evaluate g and its gradient. This is complicated by the fact that g is defined in terms of the minimization of h1 with respect to Z and h2 with respect to ?. We discuss how to solve these problems
now. We first consider the minimization of h2 . This is a quadratic function of ? and can be solved
?
analytically via the condition that ??
h2 (?; ?, ?) = 0. The closed form solution is
&
'
!
!
!
1 !
ij
ij
?c,a = ?c,a ?
?ij (a, b, c) ?
?ij (b, a, c) ?
?ij (a, b, c) +
?ij (b, a, c)
4
b
b
b
b
?(i, j) ? E, 1 ? a, c ? m.. The time complexity is linear in the size of ?.
Minimizing h1 is more involved. We assume to start that there exists an algorithm to quickly project
a matrix onto the set {Z : ?Z?? ? c}, i.e. to solve the optimization problem of
min ?Z ? A?2F .
(6)
?Z?? ?c
Then, we observe that arg minZ h1 is equal to
arg min ?tr(Z?T ) + I(?Z?? ? c) + ??Z ? Y ?2F = arg min ?Z ? (Y +
Z
?Z?? ?c
4
1
?)?2F .
2?
For different norms ? ? ?? , the projection algorithm will be different and can have a large impact on
efficiency. We will discuss in the followings sections the choices of ? ? ?? and an algorithm for the
?-norm.
Finally, once h1 and h2 have been solved, the gradient of g is (by Danskin?s theorem [1])
?g
?g
= ? Dij Z?ij ,
=Z?ji ? Z?ij ,
??ij
??ij
?g
1
?g
ij
= (??ij ? ??c,b
) ? Z?ij ,
= ? ??ij (a,b,c) g,
??ij (a, b, c) 2 c,a
??ij (a, b, c)
where Z? and ?? represent the solutions to the subproblems.
5.2 Spectral Norm
When ? ??? is set to the spectral norm, i.e. the largest singular value of a matrix, the projection in Eq.
6 can be performed by thresholding the singular values of A [3]. Theoretically, using spectral norm
will give a tighter bound on Z than other norms (Section 3). However, computing a full singular
value decomposition can be impractically slow for a graph with a large number of variables.
5.3 ?-norm
!
Here, we consider setting ? ? ?? to the ?-norm, ?A?? = maxi j |Aij |, which measures the
maximum l1 norm of the rows of A. This norm has several computational advantages. Firstly, to
project a matrix onto a ?-norm ball {A : ?A? ? ? c}, we can simply project each row ai of the
matrix onto the l1 -norm ball {a : ?a?1 ? c}. Duchi et al. [4] provide a method linear in the number
of nonzeros in a and logarithmic in the length of a. Thus, if Z is an n ? n, matrix, Eq. 6 for the
?-norm can be solved in time n2 and, for sufficiently sparse matrices, in time n log n.
A second advantage of the ?-norm is that (unlike the spectral norm) projection in Eq. 6 preserves
the sparsity of the matrix. Thus, one can disregard the matrix D and dual variables ? when solving
the optimization in Theorem 7. This means that Z itself can be represented sparsely, i.e. we only
need variables for those (i, j) ? E. These simplifications significantly improve the efficiency of
projection, with some tradeoff in accuracy.
6 Projection in Divergences
In this section, we want to find a distribution p(x; ?) in the fast mixing family closest to a target
distribution p(x; ?) in some divergence D(?, ?). The choice of divergence depends on convenience
of projection, the approximate family and the inference task. We will first present a general algorithmic framework based on projected gradient descent (Algorithm 1), and then discuss the details
of several previously proposed divergences [11, 3].
6.1 General algorithm framework for divergence minimization
The problem of projection in divergences is formulated as
min D(?, ?),
??C?
(7)
D(?, ?) is some divergence measure, and C? := {? : ?Z, s.t.(?, Z) ? C}, where C is the feasible set
in Eq. 4. Our general strategy for this is to use projected gradient descent to solve the optimization
min D(?, ?),
(?,Z)?C
(8)
using the joint operator to project onto C described in Section 5.
For different divergences, the only difference in projection algorithm is the evaluation of the gradient
?? D(?, ?). It is clear that if (?? , Z ? ) is the solution of Eq. 8, then ?? is the solution of 7.
6.2 Divergences
5
Algorithm 1 Projected gradient descent for divergence projection
Initialize (?1 , Z1 ), k ? 1.
repeat
?? ? ?k ? ??? D(?, ?k )
(?k+1 , Zk+1 ) ? projC (?? , Zk )
k ?k+1
until convergence
In this section, we will discuss the different choices of divergences and corresponding projection algorithms.
Grid, Attractive only
0.7
0.6
6.2.1 KL-divergence
Marginal Error
KL-divergence KL(???)
:=
p(x;?)
is arguably the
x p(x; ?) log p(x;?)
optimal divergence for marginal inference
because it strives to preserve the marginals
of p(x; ?) and p(x; ?). However, projection
in KL-divergence is intractable here because
the evaluation of the gradient ?? KL(???)
requires the marginals of distribution ?.
The
!
0.5
0.3
0.2
0.1
0
0
6.2.2 Piecewise KL-divergence
0.5
1
1.5
2
2.5
3
Interaction Strength
Edge density = 0.50, Mixed
3.5
4
3.5
4
0.45
LBP
TRW
Mean?Field
Original Parameters
Euclidean
Piecewise KL(?||?)
KL(?||?)
0.4
0.35
0.3
Marginal Error
One tractable surrogate of KL(???) is
the piecewise KL-divergence [3] defined over some tractable subgraphs.
Here, D(?, ?) := maxT ?T KL(?T ??T ),
where T is a set of low-treewidth subgraphs. The gradient can be evaluated as
?? D(?, ?) = ?? KL(?T ? ??T ? ) where
T ? = arg maxT ?T KL(?T ??T ). For any
T in T , KL(?T ??T ) and its gradient can be
evaluated by the junction-tree algorithm.
0.4
LBP
TRW
Mean?Field
Original Parameters
Euclidean
Piecewise KL(?||?) (TW 1)
Piecewise KL(?||?) (TW 2)
KL(?||?)
0.25
0.2
0.15
0.1
0.05
0
0
6.2.3 Reversed KL-divergence
0.5
1
1.5
2
2.5
Interaction Strength
3
The ?reversed? KL-divergence KL(???) is
minimized by mean-field methods.
In Figure 1: Mean univariate marginal error on 16 ? 16
general KL(???) is inferior to KL(???) grids (top) with attractive interactions and medianfor marginal inference since it tends to density random graphs (bottom) with mixed interacunderestimate the support of the distri- tions, comparing 30k iterations of Gibbs sampling afbution [11]. Still, it often works well ter projection (onto the l? norm) to variational methin practice. ?? KL(???)
can computed ods. The6 original parameters also show a lower curve
!
as ?"? KL(???) =
p(x;
?)(? ? ?) ? with 10 samples.
x
#
f (x) f (x) ? ?(?) , which can be approximated by samples generated from p(x; ?) [3]. In implementation, we maintain a ?pool? of samples,
each of which is updated by a single Gibbs step after each iteration of Algorithm 1.
7 Experiments
The experiments below take two stages: first, the parameters are projected (in some divergence) and
then we compare the accuracy of sampling with the resulting marginals. We focus on this second
aspect. However, we provide a comparison of the computation time for various projection algorithms
in Table 1, and when comparing the accuracy of sampling with a given amount of time, provide two
6
curves for sampling with the original parameters, where one curve has an extra amount of sampling
effort roughly approximating the time to perform projection in the reversed KL divergence.
7.1 Synthetic MRFs
Interaction strength = 2.00, Attractive Only
0.55
LBP
TRW
Mean?Field
Original Parameters
Euclidean
Piecewise KL(?||?) (TW 1)
Piecewise KL(?||?) (TW 2)
KL(?||?)
0.5
Our first experiment follows that of [8, 3]
in evaluating the accuracy of approximation
methods in marginal inference. In the experiments, we approximate randomly generated
MRF models with rapid-mixing distributions
using the projection algorithms described previously. Then, the marginals of fast mixing
approximate distributions are estimated by running a Gibbs chain on each distribution. These
are compared against exact marginals as computed by the junction tree algorithm. We use
the mean absolute difference of the marginals
|p(Xi = 1) ? q(Xi = 1)| as the accuracy measure. We compare to Naive mean-field (MF),
Gibbs sampling on original parameters (Gibbs),
and Loopy belief propagation (LBP). Many
other methods have been compared against a
similar benchmark [6, 8].
0.45
Marginal Error
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05 0
10
1
2
10
10
3
4
5
10
10
Number of Samples
6
10
10
Edge density = 0.50, Interaction strength = 3.00, Mixed
0.5
LBP
TRW
Mean?Field
Original Parameters
Euclidean
Piecewise KL(?||?)
KL(?||?)
0.45
0.4
Marginal Error
0.35
0.3
While our methods are for general MRFs, we
0.25
test on Ising potentials because this is a stan0.2
dard benchmark. Two graph topologies are
used: two-dimensional 16 ? 16 grids and 10
0.15
node random graphs, where each edge is in0.1
dependently present with probability pe ?
0.05
{0.3, 0.5, 0.7}. Node parameters ?i are uni10
10
10
10
10
10
form from [?dn , dn ] with fixed field strength
Number of Samples
dn = 1.0. Edge parameters ?ij are uniform
from [?de , de ] or [0, de ] to obtain mixed or at- Figure 2: Examples of the accuracy of obtained
tractive interactions respectively, with interac- marginals vs. the number of samples. Top:
tion strengths de ? {0, 0.5, . . . , 4}. Figure 1 Grid graphs. Bottom: Median-Density Random
shows the average marginal error at different graphs.
interaction strengths. Error bars show the standard error normalized by the number of samples, which can be interpreted as a 68.27% confidence
interval. We also include time-accuracy comparisons in Figure 2. All results are averaged over 50
random trials. We run Gibbs long enough ( 106 samples) to get a fair comparison in terms of running
time.
0
1
2
3
4
5
Except where otherwise stated, parameters are projected onto the ball {? : ?R(?)?? ? c}, where
c = 2.5 is larger than the value of c = 1 suggested by the proofs above. Better results are obtained
by using this larger constraint set, presumably because of looseness in the bound. For piecewise
projection, grids use simple vertical and horizontal chains of treewidth either one or two. For random
graphs, we randomly generate spanning trees until all edges are covered. Gradient descent uses a
fixed step size of ? = 0.1. A Gibbs step is one ?systematic-scan? pass over all variables between.
The reversed KL divergence maintains a pool of 500 samples, each of which is updated by a single
Gibbs step in each iteration.
We wish to compare the trade-off between computation time and accuracy represented by the choice
between the use of the ? and spectral norms. We measure the running time on 16 ? 16 grids in
Table 1, and compare the accuracy in Figure 3.
The appendix contains results for a three-state Potts model on an 8 ? 8 grid, as a test of the multivariate setting. Here, the intractable divergence KL(???) is included for reference, with the projection
computed with the help of the junction tree algorithm for inference.
7
Table 1: Running times on 16 ? 16 grids with attractive interactions. Euclidean projection converges
in around 5 LBFGS-B iterations. Piecewise projection (with a treewidth of 1) and reversed KL
projection use 60 gradient descent steps. All results use a single core of a Intel i7 860 processor.
Gibbs
Euclidean
Piecewise
Reversed-KL
30k Steps 106 Steps l? norm l2 norm l? norm l2 norm l? norm l2 norm
de = 1.5 0.67s
22.42s
1.50s
25.63s 12.87s 45.26s 13.13s 66.81s
de = 3.0 0.67s
22.42s
3.26s 164.34s 20.73s 211.08s 20.12s 254.25s
7.2 Berkeley binary image denoising
Grid, Mixed
Marginal Error
This experiment evaluates various methods
0.35
Euclidean SP
for denoising binary images from the BerkePiecewise KL(?||?) (TW 1) SP
0.3
ley segmentation dataset downscaled from
Piecewise KL(?||?) (TW 2) SP
KL(?||?) SP
300 ? 200 to 120 ? 80. The images are
Euclidean Inf
0.25
binarized by setting Yi = 1 if pixel i is above
Piecewise KL(?||?) (TW 1) Inf
Piecewise KL(?||?) (TW 2) Inf
the average gray scale in the image, and
0.2
KL(?||?) Inf
Yi = ?1. The noisy image X is created by
0.15
i 1.25
setting: Xi = Yi2+1 i (1 ? t1.25
) + 1?Y
,
i
2 ti
in which ti is sampled uniformly from [0, 1].
0.1
For inference purposes, the conditional
0.05
distribution
as P (Y |X)
?
! " Y is modeled
#
"
?
exp ? ij Yi Yj + 2 i (2Xi ? 1)Yi ,
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Interaction Strength
where the pairwise strength ? > 0 encourages
smoothness.
On this attractive-only Ising
potential, the Swendsen-Wang method [12] Figure 3: The marginal error using ?-norm promixes rapidly, and so we use the resulting jection (solid lines) and spectral-norm projection
samples to estimate the ground truth. The (dotted lines) on 16x16 Ising grids.
parameters ? and ? are heuristically chosen to
be 0.5 and 0.7 respectively.
Marginal Error
Figure 4 shows the decrease of average
marginal error. To compare running time, Euclidean and K(???) projection cost approximately the same as sampling 105 and 4.8 ? 105
samples respectively. Gibbs sampling on the
original parameter converges very slowly. Sampling the approximate distributions from our
projection algorithms converge quickly in less
than 104 samples.
0.5
LBP
Mean?Field
Original Parameter
Euclidean
Piecewise KL(?||?) (TW 1)
KL(?||?)
0.45
Marginal Error
0.4
0.35
0.3
0.25
0.2
0.15
8 Conclusions
0.1 0
10
1
10
2
10
3
10
4
10
5
10
6
10
We derived sufficient conditions on the parameNumber of samples
ters of an MRF to ensure fast-mixing of univariate Gibbs sampling, along with an algorithm to Figure 4: Average marginal error on the Berkeley
project onto this set in the Euclidean norm. As segmentation dataset.
an example use, we explored the accuracy of
samples obtained by projecting parameters and
then sampling, which is competitive with simple variational methods as well as traditional Gibbs
sampling. Other possible applications of fast-mixing parameter sets include constraining parameters during learning.
Acknowledgments
NICTA is funded by the Australian Government through the Department of Communications and
the Australian Research Council through the ICT Centre of Excellence Program.
8
References
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4,767 | 5,316 | Blossom Tree Graphical Models
John Lafferty
Department of Statistics
Department of Computer Science
University of Chicago
Zhe Liu
Department of Statistics
University of Chicago
Abstract
We combine the ideas behind trees and Gaussian graphical models to form a new
nonparametric family of graphical models. Our approach is to attach nonparanormal ?blossoms?, with arbitrary graphs, to a collection of nonparametric trees.
The tree edges are chosen to connect variables that most violate joint Gaussianity.
The non-tree edges are partitioned into disjoint groups, and assigned to tree nodes
using a nonparametric partial correlation statistic. A nonparanormal blossom is
then ?grown? for each group using established methods based on the graphical
lasso. The result is a factorization with respect to the union of the tree branches
and blossoms, defining a high-dimensional joint density that can be efficiently estimated and evaluated on test points. Theoretical properties and experiments with
simulated and real data demonstrate the effectiveness of blossom trees.
1
Introduction
Let p? (x) be a probability density on Rd corresponding to a random vector X = (X1 , . . . , Xd ). The
undirected graph G = (V, E) associated with p? has d = |V | vertices corresponding to X1 , . . . , Xd ,
and missing edges (i, j) 6? E whenever Xi and Xj are conditionally independent given the other
variables. The undirected graph is a useful way of exploring and modeling the distribution.
In this paper we are concerned with building graphical models for continuous variables, under
weaker assumptions than those imposed by existing methods. If p? (x) > 0 is strictly positive,
the Hammersley-Clifford theorem implies that the density has the form
!
Y
X
?
p (x) ?
?C (xC ) = exp
fC (xC ) .
(1.1)
C?C
C?C
In this expression, C denotes the set of cliques in the graph, and ?C (xC ) = exp(fC (xC )) > 0
denotes arbitrary potential functions. This represents a very large and rich set of nonparametric
graphical models. The fundamental difficulty is that it is in general intractable to compute the
normalizing constant. A compromise must be made to achieve computationally tractable inference,
typically involving strong assumptions on the functions fC , on the graph G = {C}, or both.
The default model for graphical modeling of continuous data is the multivariate Gaussian. When
the Gaussian has covariance matrix ?, the graph is encoded in the sparsity pattern of the precision
matrix ? = ??1 . Specifically, edge (i, j) is missing if and only if ?ij = 0. Recent work has
focused on sparse estimates of the precision matrix [8, 10]. In particular, an efficient algorithm for
computing the estimator using a graphical version of the lasso is developed in [3]. The nonparanormal [5], a form of Gaussian copula, weakens the Gaussian assumption by imposing Gaussianity
on the transformed random vector f (X) = (f1 (X1 ), . . . , fd (Xd )), where each fj is a monotonic
function. This allows arbitrary single variable marginal probability distributions in the model [5].
1
Both the Gaussian graphical model and the nonparanormal maintain tractable inference without
placing limitations on the independence graph. But they are limited in their ability to flexibly model
the bivariate and higher order marginals. At another extreme, forest-structured graphical models
permit arbitrary bivariate marginals, but maintain tractability by restricting to acyclic graphs. An
nonparametric approach based on forests and trees is developed in [7] as a nonparametric method
for estimating the density in high-dimensional settings. However, the ability to model complex
independence graphs is compromised.
In this paper we bring together the Gaussian, nonparanormal, and forest graphical models, using
what we call blossom tree graphical models. Informally, a blossom tree consists of a forest of trees,
and a collection of subgraphs?the blossoms?possibly containing many cycles. The vertex sets
of the blossoms are disjoint, and each blossom contains at most one node of a tree. We estimate
nonparanormal graphical models over the blossoms, and nonparametric bivariate densities over the
branches (edges) of the trees. Using the properties of the nonparanormal, these components can
be combined, or factored, to give a valid joint density for X = (X1 , . . . , Xd ). The details of our
construction are given in Section 2. We develop an estimation procedure for blossom tree graphical models, including an algorithm for selecting tree branches, partition the remaining vertices into
potential blossoms, and then estimating the graphical structures of the blossoms. Since an objective is to relax the Gaussian assumption, our criterion for selecting tree branches is deviation from
Gaussianity. Toward this end, we use the negentropy, showing that it has strong statistical properties
in high dimensions. In order to partition the nodes into blossoms, we employ a nonparametric partial correlation statistic. We use a data-splitting scheme to select the optimal blossom tree structure
based on held-out risk.
In the following section, we present the details of our method, including definitions of blossom tree
graphs, the associated family of graphical models, and our estimation methods. In Sections 3 and 4,
we present experiments with simulated and real data. Finally, we conclude in Section 5. Statistical
properties, detailed proofs, and further experimental results are collected in a supplement.
2
Blossom Tree Graphs and Estimation Methods
To unify the Gaussian, nonparanormal and forest graphical models we make the following definition.
Definition 2.1. A blossom tree on a node set V = {1, 2, . . . , d} is a graph G = (V, E), together
with a decomposition of the edge set E as E = F ? {?B?B B} satisfying the following properties:
1. F is acyclic;
2. V (B) ? V (B 0 ) = ?, for B, B 0 ? B with B 6= B 0 , where V (B) denotes the vertex set of
B.
3. |V (B) ? V (F )| ? 1 for each B ? B;
S
4. V (F ) ? B V (B) = V .
The subgraphs B ? B are called blossoms. The unique node ?(B) ? V (B) ? V (F ), which may be
empty, is called the pedicel of the blossom. The set of pedicels is denoted P(F ) ? V (F ).
Property 1 says that the set of edges F forms a union of trees?a forest. Property 2 says that distinct
blossoms share no vertices or edges in common. Property 3 says that each blossom is connected to at
most one tree node. Property 4 says that every node in the graph is either in a tree or a blossom. Note
that the blossoms are not required to be connected, but must have at most one vertex in common
with the forest?this is the pedicel node.
2
(a) blossom tree
(b) violation
(c) blossom tree
(d) violation
Figure 1: Four graphs, two blossom trees. The tree edges are colored blue, the blossom edges are
colored black, and pedicels are orange. Graphs (a) and (c) correspond to blossom trees. Graphs (b)
and (d) violate the restriction that each blossom has only a single pedicel, or attachment to a tree.
Suppose that p(x) = p(x1 , . . . , xd ) is the density of a distribution that has an independence graph
given by a blossom tree F ? {?B B}. Then from the blossom tree properties we have that
p(x) = p(XV (F ) )
Y
B?B
= p(XV (F ) )
Y
B?B
= p(XV (F ) )
p(XV (B) | XV (F ) )
(2.1)
p(XV (B) | X?(B) )
(2.2)
Y p(XV (B) )
p(X?(B) )
(2.3)
B?B
=
Y
(s,t)?F
=
Y
(s,t)?F
p(Xs , Xt )
p(Xs )p(Xt )
p(Xs , Xt )
p(Xs )p(Xt )
Y
p(Xs )
s?V (F )
Y
s?V (F )\P(F )
Y p(XV (B) )
p(X?(B) )
(2.4)
B?B
p(Xs )
Y
p(XV (B) ).
(2.5)
B?B
The first equality follows from disjointness of the blossoms. The second equality follows from the
existence of a single pedicel node attaching the blossom to a tree. The fourth equality follows from
the standard factorization of forests, and the last equality follows from the fact that each non-empty
pedicel for a blossom is unique. We call the set of distributions that factor in this way the family of
blossom tree graphical models.
A key property of the nonparanormal [5] is that the single node marginal probabilities p(Xs ) are
arbitrary. This property allows us to form graphical models where each blossom distribution satisfies
XV (B) ? NPN(?B , ?B , fB ), while enforcing that the single node marginal of the pedicel ?(B)
agrees with the marginals of this node defined by the forest. This allows us to define and estimate
distributions that are consistent with the factorization (2.5).
Let X (1) , . . . , X (n) be n i.i.d. Rd -valued data vectors sampled from p? (x) where X (l) =
(l)
(l)
(X1 , . . . , Xd ). Our goal is to derive a method for high-dimensional undirected graph estimation and density estimation, using a family of semiparametric estimators based on the blossom tree
structure. Let FB denote the blossom tree structure F ? {?B B}. Our estimation procedure is the
following.
First, randomly partition the data X (1) , . . . , X (n) into two sets D1 and D2 of sample size n1 and n2 .
Then apply the following steps.
1. Using D1 , estimate the bivariate densities p? (xi , xj ) using kernel density estimation. Also,
estimate the covariance ?ij for each pair of variables. Apply Kruskal?s algorithm on the
estimated pairwise negentropy matrix to construct a family of forests {Fb(k) } with k =
0, . . . , d ? 1 edges;
2. Using D1 , for each forest Fb(k) obtained in Step 1, build the blossom tree-structured graph
(k)
FbBb . The forest structure Fb(k) is modeled by nonparametric kernel density estimators,
b (k) is modeled by the graphical lasso or nonparanormal. A family of
while each blossom B
i
3
graphs is obtained by computing regularization paths for the blossoms, using the graphical
lasso.
(b
k)
3. Using D2 , choose FbBb from this family of blossom tree models that maximizes the heldout log-likelihood.
The details of each step are presented below.
2.1 Step 1: Construct A Family of Forests
In information theory and statistics, negentropy is used as a measure of distance to normality. The
negentropy is zero for Gaussian densities and is always nonnegative. The negentropy between variables Xi and Xj is defined as
J(Xi ; Xj ) = H(?(xi , xj )) ? H(p? (xi , xj )),
(2.6)
where H(?) denotes the differential entropy of a density, and ?(xi , xj ) is an Gaussian density with
the same mean and covariance matrix as p? (xi , xj ).
Kruskal?s algorithm [4] is a greedy algorithm to find a maximum weight spanning tree of a weighted
graph. At each step it includes an edge connecting the pair of nodes with the maximum weight
among all unvisited pairs, if doing so does not form a cycle. The algorithm also results in the best
k-edge weighted forest after k < d edges have been included.
In our setting, we define the weight w(i, j) of nodes i and j as the negentropy between Xi and Xj ,
and use Kruskal?s algorithm to build the maximum weight spanning forest Fb(k) with k edges where
k < d. In such a way, the pairs of nodes that are less likely to be a bivariate Gaussian are included
in the forest and then are modeled nonparametrically.
Since the true density p? is unknown, we replace the population negentropy J(Xi ; Xj ) by the estimate
b pn (xi , xj )),
Jbn (Xi ; Xj ) = H(?bn (xi , xj )) ? H(b
(2.7)
1
1
1
where ?bn1 (xi , xj ) is an estimate of the Gaussian density ?(xi , xj ) for Xi and Xj using D1 ,
b
pbn1 (xi , xj ) is a bivariate kernel density estimate for Xi and Xj , and H(?)
denotes the empirical
ij
b ij
differential entropy. In particular, let ? be the covariance matrix of Xi and Xj . Denote ?
n1 as the
empirical covariance matrix of Xi and Xj based on D1 , then the plug-in estimate
1
b ij
(2.8)
H(?bn1 (xi , xj )) = 1 + log(2?) + logdet(?
n1 ).
2
Let K(?) be a univariate kernel function. Then given an evaluation point (xi , xj ), the bivariate kernel
(l)
(l)
density estimate for (Xi , Xj ) based on observations {Xi , Xj }l?D1 is given by
!
!
(l)
(l)
Xj ? xj
1 X
1
Xi ? xi
pbn1 (xi , xj ) =
K
K
,
(2.9)
n1
h2i h2j
h2i
h2j
l?D1
where h2i and h2j are bandwidth parameters for (Xi , Xj ). To compute the empirical differential
b pn (xi , xj )), we numerically evaluate a two-dimensional integral.
entropy H(b
1
h
i
Once the estimated negentropy matrix Jbn1 (Xi ; Xj )
is obtained, we apply Kruskal?s algorithm
d?d
to construct a family of forests {Fb(k) }k=0...d?1 .
2.2 Step 2: Build and Model the Blossom Tree Graphs
Suppose that we have a forest-structured graph F with |V (F )| < d vertices. Then for each remaining non-forest node, we need to determine which blossom it belongs to. We exploit the following
basic fact.
4
Proposition 2.1. Suppose that X ? p? is a density for a blossom tree graphical model with forest
F . Let i 6? V (F ) and s ? V (F ). Then node i is not in a blossom attached to tree node s if and only
if
Xi ?? Xs | Xt for some node t ? V (F ) such that (s, t) ? E(F ).
(2.10)
We use this property, together with a measure of partial correlation, in order to partition the nonforest nodes into blossoms. Partial correlation measures the degree of association between two
random variables, with the effect of a set of controlling random variables removed. Traditionally,
the partial correlation between variables Xi and Xs given a controlling variable Xt is the correlation
between the residuals i\t and s\t resulting from the linear regression of Xi with Xt and of Xs
with Xt , respectively. However, if the underlying joint Gaussian or nonparanormal assumption is
not satisfied, linear regression cannot remove all of the effects of the controlling variable.
We thus use a nonparametric version of partial correlation. Following [1], suppose Xi = g(Xt )+i\t
and Xs = h(Xt )+s\t , for certain functions g and h such that E(i\t | Xt ) = 0 and E(s\t | Xt ) =
0. Define the nonparametric partial correlation as
.q
?is?t = E(i\t s\t )
E(2i\t ) E(2s\t ).
(2.11)
It is shown in [1] that if Xi ?
? Xs | Xt , then ?is?t = 0. We thus conclude the following.
Proposition 2.2. If ?is?t 6= 0 for all t such that (s, t) ? E(F ), node i is in a blossom attached to
node s.
(l)
(l)
(l)
(l)
(l)
Let gb and b
h be local polynomial estimators of g and h, and b
i\t = Xi ? gb(Xt ), b
s\t = Xs ?
(l)
b
h(X ) for any l ? D1 , then an estimate of ?is?t is given by
t
?bis?t =
X
l?D1
.s X
(l) (l)
(b
i\t b
s\t )
(l)
(b
i\t )2
l?D1
X
(l)
(b
s\t )2 .
(2.12)
l?D1
Based on Proposition 2.2, for each forest Fb(k) obtained in Step 1, we then assign each non-forest
node i to the blossom with the pedicel given by
sbi = argmax
min
b(k) )}
b(k) ) {t: (s,t)?E(F
s?V (F
|b
?is?t |.
(2.13)
(k)
After iterating over all non-forest nodes, we obtain a blossom tree-structured graph FbBb . Then
the forest structure is nonparametrically modeled by the bivariate and univariate kernel density estimations, while each blossom is modeled with the graphical lasso or nonparanormal. In particular,
when k = 0 that there is no forest node, our method is reduced to modeling the entire graph by the
graphical lasso or nonparanormal.
Alternative testing procedures based on nonparametric partial correlations could be adopted for partitioning nodes into blossoms. However, such methods may have large computational cost, and low
power for small sample sizes.
Note that while each non-forest node is associated with a pedicel in this step, after graph estimation
for the blossoms, the node may well become disconnected from the forest.
2.3 Step 3: Optimize the Blossom Tree Graphs
(d?1)
The full blossom tree graph FbBb
obtained in Steps 1 and 2 might result in overfitting in the density
estimate. Thus we need to choose an optimal graph with the number of forest edges k ? d ? 1.
Besides, the tuning parameters involved in the fitting of each blossom by the graphical lasso or
nonparanormal also induce a bias-variance tradeoff.
5
(k)
To optimize the blossom tree structures over {FbBb }k=0...d?1 , we choose the complexity parameter
b
k as the one that maximizes the log-likelihood on D2 , using the factorization (2.5):
?
(l)
(l)
X
Y
pbn1 (Xi , Xj )
1
b
log ?
?
k = argmax
(l)
(l)
k?{0,...,d?1} n2 l?D
bn1 (Xi )b
pn1 (Xj )
b(k) ) p
2
(i,j)?E(F
?
k
Y
Y
(k)
?
(l)
(2.14)
pbn1 (Xs(l) )
?bni1 X b(k) ? ,
b(k) )\P(F
b(k) )
s?V (F
(k)
?
i=1
V (Bi
)
where ?bni1 is the density estimate for blossoms by the graphical lasso or nonparanormal, with the
(k)
tuning parameter ?i selected to maximize the held-out log-likelihood. That is, the complexity of
each blossom is also optimized on D2 .
Thus the final blossom tree density estimator is given by
p b(k)
b (x) =
Fb
B
Y
b
b(k)
(i,j)?E(F
)
pbn1 (xi , xj )
pbn1 (xi )b
pn1 (xj )
Y
b
b
b(k)
b(k)
s?V (F
)\P(F
)
pbn1 (Xs(l) )
b
k
Y
i=1
b
(k)
?
?bni1 (x b(k)
b ).
Bi
(2.15)
In practice, Step 3 can be carried out simultaneously with Steps 1 and 2. Whenever a new edge is
added to the current forest in Kruskal?s algorithm, the blossoms are re-constructed and re-modeled.
Then the held-out log-likelihood of the obtained density estimator can be immediately computed.
In addition, since there are no overlapping nodes between different blossoms, the sparsity tuning
parameters are selected separately for each blossom, which reduces the computational cost considerably.
3
Analysis of Simulated Data
Here we present numerical results based on simulations. We compare the blossom tree density
estimator with the graphical lasso [3] and forest density estimator [7]. To evaluate the performance
of these estimators, we compute and compare the log-likelihood of each method on held-out data.
We simulate high-dimensional data which are consistent with an undirected graph. We generate multivariate non-Gaussian data using a sequence of mixtures of two Gaussian distributions with contrary
correlation and equal weights. Then a subset of variables are chosen to generate the blossoms that
are distributed as multivariate Gaussians. In dimensional d = 80, we sample n1 = n2 = 400 data
points from this synthetic distribution.
A typical run showing the held-out log-likelihood and estimated graphs is provided in Figures 2 and
3. The term ?trunk? is used to represent the edge added to the forest in a blossom tree graph. We can
see that the blossom tree density estimator is superior to other methods in terms of generalization
performance. In particular, the graphical lasso is unable to uncover the edges that are generated
nonparametrically. This is expected, since different blossoms have zero correlations among each
other and are thus regarded as independent by the algorithm of graphical lasso. For the modeling of
the variables that are contained in a blossom and are thus multivariate Gaussian distributed, there is
an efficiency loss in the forest density estimator, compared to the graphical lasso. This illustrates the
advantage of blossom tree density estimator. As is seen from the number of selected edges by each
method shown in Figure 2, the blossom tree density estimator selects a graph with a similar sparsity
pattern as the true graph.
4
Analysis of Cell Signalling Data
We analyze a flow cytometry dataset on d = 11 proteins from [9]. A subset of n = 853 cells were
chosen. A nonparanormal transformation was estimated and the observations, for each variable,
6
80
60
50
40
true
glasso
glasso
30
Number of selected edges
70
?108
?109
?110
?113
20
?112
Held out log?likelihood
?111
true
0
20
40
60
80
0
Number of trunks
20
40
60
80
Number of trunks
Figure 2: Results on simulations. Left: Held-out log-likelihood of the graphical lasso (horizontal
dotted line), forest density estimator (horizontal dashed line), and blossom tree density estimator
(circles); Right: Number of selected edges by these methods. The horizontal solid line indicates the
number of edges in the true graph, and the solid triangle indicates the best blossom tree graph. The
first circle for blossom tree refers to the 1-trunk case.
true
true
glasso
(a) true
(b) glasso
glasso
forest
forest
(c) forest
forest?blossom
forest?blossom
(d) blossom tree
Figure 3: Results on simulations. Graph (a) corresponds to the true graph. Graphs (b), (c) and (d)
forest
forest?blossom
forest?blossom
correspond
toforest
the estimated graphs
by the graphical
lasso, forest density estimator, and blossom tree
density estimator, respectively. The tree edges are colored red, and the blossom edges are colored
black.
were replaced by their respective normal scores, subject to a Winsorized truncation [5]. We study
the associations among the proteins using the graphical lasso, forest density estimator, and blossom
tree forest density estimator. The maximum held-out log-likelihood for glasso, forest, and blossom
tree are -14.3, -13.8, and -13.7, respectively. We can see that blossom tree is slighter better than
forest in terms of the generalization performance, both of which outperform glasso. Results of
estimated graphs are provided in Figures 4. When the maximum of held-out log-likelihood curve
is achieved, glasso selects 28 edges, forest selects 7 edges, and blossom tree selects 10 edges. The
two graphs uncovered by forest and blossom tree agree on most edges, although the latter contains
cycles.
5
Conclusion
We have proposed a combination of tree-based graphical models and Gaussian graphical models to
form a new nonparametric approach for high dimensional data. Blossom tree models relax the normality assumption and increase statistical efficiency by modeling the forest with nonparametric kernel density estimators and modeling each blossom with the graphical lasso or nonparanormal. Our
experimental results indicate that this method can be a practical alternative to standard approaches
to graph and density estimation.
7
(a) graph reported in [9]
(b) glasso
(c) forest
(d) blossom tree
Figure 4: Results on cell signalling data. Graph (a) refers to the fitted graph reported in [9]. Graphs
(b), (c) and (d) correspond to the estimated graphs by the graphical lasso, forest density estimator,
and blossom tree density estimator, respectively.
Acknowledgements
Research supported in part by NSF grant IIS-1116730, AFOSR grant FA9550-09-1-0373, ONR
grant N000141210762, and an Amazon AWS in Education Machine Learning Research grant.
References
[1] Wicher Bergsma. A note on the distribution of the partial correlation coefficient with nonparametrically estimated marginal regressions. arXiv:1101.4616, 2011.
[2] T. Tony Cai, Tengyuan Liang, and Harrison H. Zhou. Law of log determinant of sample
covariance matrix and optimal estimation of differential entropy for high-dimensional gaussian
distributions. arXiv:1309.0482, 2013.
[3] Jerome H. Friedman, Trevor Hastie, and Robert Tibshirani. Sparse inverse covariance estimation with the graphical lasso. Biostatistics, 9(3):432?441, 2008.
[4] Joseph B. Kruskal. On the shortest spanning subtree of a graph and the traveling salesman
problem. In Proceedings of the American Mathematical Society, volume 7, pages 48?50,
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[5] Han Liu, John Lafferty, and Larry Wasserman. The nonparanormal: Semiparametric estimation
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[6] Han Liu, Larry Wasserman, and John D. Lafferty. Exponential concentration for mutual information estimation with application to forests. In Advances in Neural Information Processing
Systems (NIPS), 2012.
[7] Han Liu, Min Xu, Haijie Gu, Anupam Gupta, John Lafferty, and Larry Wasserman. Forest
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[9] Karen Sachs, Omar Perez, Dana Pe?er, Douglas A. Lauffenburger, and Garry P. Nolan.
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8
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4,768 | 5,317 | Distributed Parameter Estimation
in Probabilistic Graphical Models
Yariv D. Mizrahi1 Misha Denil2 Nando de Freitas2,3,4
1
University of British Columbia, Canada
2
University of Oxford, United Kingdom
3
Canadian Institute for Advanced Research
4
Google DeepMind
[email protected]
{misha.denil,nando}@cs.ox.ac.uk
Abstract
This paper presents foundational theoretical results on distributed parameter estimation for undirected probabilistic graphical models. It introduces a general
condition on composite likelihood decompositions of these models which guarantees the global consistency of distributed estimators, provided the local estimators
are consistent.
1
Introduction
Undirected probabilistic graphical models, also known as Markov Random Fields (MRFs), are a
natural framework for modelling in networks, such as sensor networks and social networks [24, 11,
20]. In large-scale domains there is great interest in designing distributed learning algorithms to
estimate parameters of these models from data [27, 13, 19]. Designing distributed algorithms in
this setting is challenging because the distribution over variables in an MRF depends on the global
structure of the model.
In this paper we make several theoretical contributions to the design of algorithms for distributed
parameter estimation in MRFs by showing how the recent works of Liu and Ihler [13] and of Mizrahi
et al. [19] can both be seen as special cases of distributed composite likelihood. Casting these two
works in a common framework allows us to transfer results between them, strengthening the results
of both works.
Mizrahi et al. introduced a theoretical result, known as the LAP condition, to show that it is possible
to learn MRFs with untied parameters in a fully-parallel but globally consistent manner. Their result
led to the construction of a globally consistent estimator, whose cost is linear in the number of cliques
as opposed to exponential as in centralised maximum likelihood estimators. While remarkable, their
results apply only to a specific factorisation, with the cost of learning being exponential in the size of
the factors. While their factors are small for lattice-MRFs and other models of low degree, they can
be as large as the original graph for other models, such as fully-observed Boltzmann machines [1]. In
this paper, we introduce the Strong LAP Condition, which characterises a large class of composite
likelihood factorisations for which it is possible to obtain global consistency, provided the local
estimators are consistent. This much stronger condition enables us to construct linear and globally
consistent distributed estimators for a much wider class of models than Mizrahi et al., including
fully-connected Boltzmann machines.
Using our framework we also show how the asymptotic theory of Liu and Ihler applies more generally to distributed composite likelihood estimators. In particular, the Strong LAP Condition provides
a sufficient condition to guarantee the validity of a core assumption made in the theory of Liu and
Ihler, namely that each local estimate for the parameter of a clique is a consistent estimator of the
1
1
2
3
1
2
3
1
2
3
1
2
3
4
5
6
4
5
6
4
5
6
4
5
6
7
8
9
7
8
9
7
8
9
7
8
9
Figure 1: Left: A simple 2d-lattice MRF to illustrate our notation. For node j = 7 we have N (xj ) =
{x4 , x8 }. Centre left: The 1-neighbourhood of the clique q = {x7 , x8 } including additional edges
(dashed lines) present in the marginal over the 1-neighbourhood. Factors of this form are used by
the LAP algorithm of Mizrahi et. al. Centre right: The MRF used by our conditional estimator of
Section 5 when using the same domain as Mizrahi et. al. Right: A smaller neighbourhood which
we show is also sufficient to estimate the clique parameter of q.
corresponding clique parameter in the joint distribution. By applying the Strong LAP Condition to
verify the assumption of Liu and Ihler, we are able to import their M-estimation results into the LAP
framework directly, bridging the gap between LAP and consensus estimators.
2
Background
Our goal is to estimate the D-dimensional parameter vector ? of an MRF with the following Gibbs
density or mass function:
X
1
p(x | ?) =
exp(
E(xc | ? c ))
(1)
Z(?)
c
Here c 2 C is an index over the cliques of an undirected graph G = (V, E), E(xc | ? c ) is known as
the energy or Gibbs potential, and Z(?) is a normalizing term known as the partition function.
When E(xc | ? c ) = ? Tc c (xc ), where c (xc ) is a local sufficient statistic derived from the values
of the local data vector xc , this model is known as a maximum entropy or log-linear model. In
this paper we do not restrict ourselves to a specific form for the potentials, leaving them as general
functions; we require only that their parameters are identifiable. Throughout this paper we focus
on the case where the xj ?s are discrete random variables, however generalising our results to the
continuous case is straightforward.
The j-th node of G is associated with the random variable xj for j = 1, . . . , M , and the edge connecting nodes j and k represents the statistical interaction between xj and xk . By the HammersleyClifford Theorem [10], the random vector x satisfies the Markov property with respect to the graph
G, i.e., p(xj |x j ) = p(xj |xN (xj ) ) for all j where x j denotes all variables in x excluding xj ,
and xN (xj ) are the variables in the neighbourhood of node j (variables associated with nodes in G
directly connected to node j).
2.1
Centralised estimation
The standard approach to parameter estimation in statistics is through maximum likelihood, which
chooses parameters ? by maximising
LM L (?) =
N
Y
n=1
p(xn | ?)
(2)
(To keep the notation light, we reserve n to index the data samples. In particular, xn denotes the
n-th |V|-dimensional data vector and xmn refers to the n-th observation of node m.)
This estimator has played a central role in statistics as it has many desirable properties including
consistency, efficiency and asymptotic normality. However, applying maximum likelihood estimation to an MRF is generally intractable since computing the value of log LM L and its derivative
require evaluating the partition function, and an expectation over the model, respectively. Both of
these values involve a sum over exponentially many terms.
2
To surmount this difficulty it is common to approximate p(x | ?) as a product over more tractable
terms. This approach is known as composite likelihood and leads to an objective of the form
LCL (?) =
N Y
I
Y
f i (xn , ? i )
(3)
n=1 i=1
where ? i denote the (possibly shared) parameters of each composite likelihood factor f i .
Composite likelihood estimators are both well studied and widely applied [6, 14, 12, 7, 16, 2, 22,
4, 21]. In practice the f i terms are chosen to be easy to compute, and are typically local functions,
depending only on some local region of the underlying graph G.
An early and influential variant of composite likelihood is pseudo-likelihood (PL) [3], where
f i (x, ? i ) is chosen to be the conditional distribution of xi given its neighbours,
LP L (?) =
N Y
M
Y
n=1 m=1
p(xmn | xN (xm )n , ? m )
(4)
Since the joint distribution has a Markov structure with respect to the graph G, the conditional
distribution for xm depends only on its neighbours, namely xN (xm ) . In general more statistically
efficient composite likelihood estimators can be obtained by blocking, i.e. choosing the f i (x, ? i ) to
be conditional or marginal likelihoods over blocks of variables, which may be allowed to overlap.
Composite likelihood estimators are often divided into conditional and marginal variants, depending
on whether the f i (x, ? i ) are formed from conditional or marginal likelihoods. In machine learning
the conditional variant is quite popular [12, 7, 16, 15, 4] while the marginal variant has received less
attention. In statistics, both the marginal and conditional variants of composite likelihood are well
studied (see the comprehensive review of Varin et. al. [26]).
An unfortunate difficulty with composite likelihood is that the estimators cannot be computed in
parallel, since elements of ? are often shared between the different factors. For a fixed value of ?
the terms of log LCL decouple over data and over blocks of the decomposition; however, if ? is not
fixed then the terms remain coupled.
2.2
Consensus estimation
Seeking greater parallelism, researchers have investigated methods for decoupling the sub-problems
in composite likelihood. This leads to the class of consensus estimators, which perform parameter
estimation independently in each composite likelihood factor. This approach results in parameters
that are shared between factors being estimated multiple times, and a final consensus step is required
to force agreement between the solutions from separate sub-problems [27, 13].
Centralised estimators enforce sub-problem agreement throughout the estimation process, requiring
many rounds of communication in a distributed setting. Consensus estimators allow sub-problems
to disagree during optimisation, enforcing agreement as a post-processing step which requires only
a single round of communication.
Liu and Ihler [13] approach distributed composite likelihood by optimising each term separately
!
N
Y
i
? = arg max
?
f i (xAi ,n , ? )
(5)
i
?
i
i
n=1
where Ai denotes the group of variables associated with block i, and ? i is the corresponding set of
parameters. In this setting the sets i ? V are allowed to overlap, but the optimisations are carried
out independently, so multiple estimates for overlapping parameters are obtained. Following Liu
and Ihler we have used the notation ? i = ? i to make this interdependence between factors explicit.
i
? into a degenerate estiThe analysis of this setting proceeds by embedding each local estimator ?
i
i
i
?
?
mator ? for the global parameter vector ? by setting ? c = 0 for c 2
/ i . The degenerate estimators
are combined into a single non-degenerate global estimate using different consensus operators, e.g.
?i.
weighted averages of the ?
3
The analysis of Liu and Ihler assumes that for each sub-problem i and for each c 2
i
p
? )c ! ? c
(?
i
i
(6)
i.e., each local estimate for the parameter of clique c is a consistent estimator of the corresponding
clique parameter in the joint distribution. This assumption does not hold in general, and one of the
contributions of this work is to give a general condition under which this assumption holds.
The analysis of Liu and Ihler [13] considers the case where the local estimators in Equation 5 are arbitrary M -estimators [25], however their experiments address only the case of pseudo-likelihood. In
Section 5 we prove that the factorisation used by pseudo-likelihood satisfies Equation 6, explaining
the good results in their experiments.
2.3
Distributed estimation
Consensus estimation dramatically increases the parallelism of composite likelihood estimates by
relaxing the requirements on enforcing agreement between coupled sub-problems. Recently Mizrahi
et. al. [19] have shown that if the composite likelihood factorisation is constructed correctly then
consistent parameter estimates can be obtained without requiring a consensus step.
In the LAP algorithm of Mizrahi et al. [19] the domain of each composite likelihood factor (which
they call the auxiliary MRF) is constructed by surrounding each maximal clique q with the variables
in its 1-neighbourhood
[
Aq =
c
c\q6=;
which contains all of the variables of q itself as well as the variables with at least one neighbour in
q; see Figure 1 for an example. For MRFs of low degree the sets Aq are small, and consequently
maximum likelihood estimates for parameters of MRFs over these sets can be obtained efficiently.
The parametric form of each factor in LAP is chosen to coincide with the marginal distribution over
Aq .
The factorisation of Mizrahi et al. is essentially the same as in Equation 5, but the domain of each
term is carefully selected, and the LAP theorems are proved only for the case where f i (xAq , ? q ) =
p(xAq , ? q ).
As in consensus estimation, parameter estimation in LAP is performed separately and in parallel for
each term; however, agreement between sub-problems is handled differently. Instead of combining
parameter estimates from different sub-problems, LAP designates a specific sub-problem as authoritative for each parameter (in particular the sub-problem with domain Aq is authoritative for the
parameter ? q ). The global solution is constructed by collecting parameters from each sub-problem
for which it is authoritative and discarding the rest.
In order to obtain consistency for LAP, Mizrahi et al. [19] assume that both the joint distribution and
each composite likelihood factor are parametrised using normalized potentials.
Definition 1. A Gibbs potential E(xc |? c ) is said to be normalised with respect to zero if E(xc |? c ) =
0 whenever there exists t 2 c such that xt = 0.
A perhaps under-appreciated existence and uniqueness theorem [9, 5] for MRFs states that there
exists one and only one potential normalized with respect to zero corresponding to a Gibbs distribution. This result ensures a one to one correspondence between Gibbs distributions and normalised
potential representations of an MRF.
The consistency of LAP relies on the following observation. Suppose we have a Gibbs distribution
p(xV | ?) that factors according to the clique system C, and suppose that the parametrisation is
chosen so that the potentials are normalised with respect to zero. For a particular clique of interest
q, the marginal over xAq can be written as follows (see Appendix A for a detailed derivation)
p(xAq | ?) =
1
exp( E(xq | ? q )
Z(?)
4
X
c2Cq \{q}
E(xc | ? V\q ))
(7)
where Cq denotes the clique system of the marginal, which in general includes cliques not present in
the joint. The same distribution can also be written in terms of different parameters ?
X
1
p(xAq | ?) =
exp( E(xq | ?q )
E(xc | ?c ))
(8)
Z(?)
c2Cq \{q}
which are also assumed to be normalised with respect to zero. As shown in Mizrahi et. al. [19], the
uniqueness of normalised potentials can be used to obtain the following result.
Proposition 2 (LAP argument [19]). If the parametrisations of p(xV | ?) and p(xAq | ?) are chosen to be normalized with respect to zero, and if the parameters are identifiable with respect to the
potentials, then ? q = ?q .
This proposition enables Mizrahi et. al. [19] to obtain consistency for LAP under the standard
smoothness and identifiability assumptions for MRFs [8].
3
Contributions of this paper
The strength of the results of Mizrahi et al. [19] is to show that it is possible to perform parameter
estimation in a completely distributed way without sacrificing global consistency. They prove that
through careful design of a composite likelihood factorisation it is possible to obtain estimates for
each parameter of the joint distribution in isolation, without requiring even a final consensus step
to enforce sub-problem agreement. Their weakness is that the LAP algorithm is very restrictive,
requiring a specific composite likelihood factorisation.
The strength of the results of Liu and Ihler [13] is that they apply in a very general setting (arbitrary
M -estimators) and make no assumptions about the underlying structure of the MRF. On the other
hand they assume the convergence in Equation 6, and do not characterise the conditions under which
this assumption holds.
The key to unifying these works is to notice that the specific decomposition used in LAP is chosen
essentially to ensure the convergence of Equation 6. This leads to our development of the Strong
LAP Condition and an associated Strong LAP Argument, which is a drop in replacement for the LAP
argument of Mizrahi et al. and holds for a much larger range of composite likelihood factorisations
than their original proof allows.
Since the purpose of the Strong LAP Condition is to guarantee the convergence of Equation 6, we
are able to import the results of Liu and Ihler [13] into the LAP framework directly, bridging the
gap between LAP and consensus estimators. The same Strong LAP Condition also provides the
necessary convergence guarantee for the results of Liu and Ihler to apply.
Finally we show how the Strong LAP Condition can lead to the development of new estimators, by
developing a new distributed estimator which subsumes the distributed pseudo-likelihood and gives
estimates that are both consistent and asymptotically normal.
4
Strong LAP argument
In this section we present the Strong LAP Condition, which provides a general condition under
which the convergence of Equation 6 holds. This turns out to be intimately connected to the structure
of the underlying graph.
Definition 3 (Relative Path Connectivity). Let G = (V, E) be an undirected graph, and let A be a
given subset of V. We say that two nodes i, j 2 A are path connected with respect to V \ A if there
exists a path P = {i, s1 , s2 , . . . , sn , j} =
6 {i, j} with none of the sk 2 A. Otherwise, we say that
i, j are path disconnected with respect to V \ A.
in
For a given A ? V we partition the clique system of G into two parts, CA
that contains all of the
out
in
cliques that are a subset of A, and CA = C \ CA that contains the remaining cliques of G. Using
this notation we can write the marginal distribution over xA as
X
X
X
1
p(xA | ?) =
exp(
E(xc | ? c ))
exp(
E(xc | ? c ))
(9)
Z(?)
out
in
x
c2CA
V\A
5
c2CA
(b)
(a)
1
3
6
4
2
j
k
5
i
(c)
2
3
(d)
2
3
0
1
4
0
1
4
5
2
3
0
1
4
5
5
Figure 2: (a) Illustrating the concept of relative path connectivity. Here, A = {i, j, k}. While (k, j)
are path connected via {3, 4} and (k, i) are path connected via {2, 1, 5}, the pair (i, j) are path
disconnected with respect to V \ A. (b)-(d) Illustrating the difference between LAP and Strong LAP.
(b) Shows a star graph with q highlighted. (c) Shows Aq required by LAP. (d) Shows an alternative
neighbourhood allowed by Strong LAP. Thus, if the root node is a response variable and the leafs
are covariates, Strong LAP states we can estimate each parameter separately and consistently.
P
P
Up to a normalisation constant, xV\A exp(
out E(xc | ? c )) induces a Gibbs density (and
c2CA
therefore an MRF) on A, which we refer to as the induced MRF. (For example, as illustrated in
Figure 1 centre-left, the induced MRF involves all the cliques over the nodes 4, 5 and 9.) By the
Hammersley-Clifford theorem this MRF has a corresponding graph which we refer to as the induced
graph and denote GA . Note that the induced graph does not have the same structure as the marginal,
it contains only edges which are created by summing over xV\A .
Remark 4. To work in the general case, we assume throughout that that if an MRF contains the
path {i, j, k} then summing over j creates the edge (i, k) in the marginal.
Proposition 5. Let A be a subset of V, and let i, j 2 A. The edge (i, j) exists in the induced graph
GA if and only if i and j are path connected with respect to V \ A.
Proof. If i and j are path connected then there is a path P = {i, s1 , s2 , . . . , sn , j} =
6 {i, j} with
none of the sk 2 A. Summing over sk forms an edge (sk 1 , sk+1 ). By induction, summing over
s1 , . . . , sn forms the edge (i, j).
If i and j are path disconnected with respect to V \ A then summing over any s 2 V \ A cannot
form the edge (i, j) or i and j would be path connected through the path {i, s, j}. By induction, if
the edge (i, j) is formed by summing over s1 , . . . , sn this implies that i and j are path connected via
{i, s1 , . . . , sn , j}, contradicting the assumption.
Corollary 6. B ? A is a clique in the induced graph GA if and only if all pairs of nodes in B are
path connected with respect to V \ A.
Definition 7 (Strong LAP condition). Let G = (V, E) be an undirected graph and let q 2 C be a
clique of interest. We say that a set A such that q ? A ? V satisfies the strong LAP condition for q
if there exist i, j 2 q such that i and j are path-disconnected with respect to V \ A.
Proposition 8. Let G = (V, E) be an undirected graph and let q 2 C be a clique of interest. If
Aq satisfies the Strong LAP condition for q then the joint distribution p(xV | ?) and the marginal
p(xAq | ?) share the same normalised potential for q.
Proof. If Aq satisfies the Strong LAP Condition for q then by Corollary 6 the induced MRF contains
no potential for q. Inspection of Equation 9 reveals that the same E(xq | ? q ) appears as a potential
in both the marginal and the joint distributions. The result follows by uniqueness of the normalised
potential representation.
We now restrict our attention to a set Aq which satisfies the Strong LAP Condition for a clique of
interest q. The marginal over p(xAq | ?) can be written as in Equation 9 in terms of ?, or in terms of
auxiliary parameters ?
X
1
p(xAq | ?) =
exp(
E(xc | ?c ))
(10)
Z(?)
c2Cq
Where Cq is the clique system over the marginal. We will assume both parametrisations are normalised with respect to zero.
Theorem 9 (Strong LAP Argument). Let q be a clique in G and let q ? Aq ? V. Suppose p(xV | ?)
and p(xAq | ?) are parametrised so that their potentials are normalised with respect to zero and the
parameters are identifiable with respect to the potentials. If Aq satisfies the Strong LAP Condition
for q then ? q = ?q .
6
Proof. From Proposition 8 we know that p(xV | ?) and p(xAq | ?) share the same clique potential
for q. Alternatively we can write the marginal distribution as in Equation 10 in terms of auxiliary
variables ?. By uniqueness, both parametrisations must have the same normalised potentials.
Since the potentials are equal, we can match terms between the two parametrisations. In particular
since E(xq | ? q ) = E(xq | ?q ) we see that ? q = ?q by identifiability.
4.1
Efficiency and the choice of decomposition
Theorem 9 implies that distributed composite likelihood is consistent for a wide class of decompositions of the joint distribution; however it does not address the issue of statistical efficiency.
This question has been studied empirically in the work of Meng et. al. [17, 18], who introduce a
distributed algorithm for Gaussian random fields and consider neighbourhoods of different sizes.
Meng et. al. find the larger neighbourhoods produce better empirical results and the following theorem confirms this observation.
Theorem 10. Let A be set of nodes which satisfies the Strong LAP Condition for q. Let ??A be the
ML parameter estimate of the marginal over A. If B is a superset of A, and ??B is the ML parameter
estimate of the marginal over B. Then (asymptotically):
|?q
(??B )q | ? |?q
(??A )q |.
Proof. Suppose that |?q (??B )q | > |?q (??A )q |. Then the estimates ??A over the various subsets A
of B improve upon the ML estimates of the marginal on B. This contradicts the Cramer-Rao lower
bound achieved by the ML estimate of the marginal on B.
In general the choice of decomposition implies a trade-off in computational and statistical efficiency.
Larger factors are preferable from a statistical efficiency standpoint, but increase computation and
decrease the degree of parallelism.
5
Conditional LAP
The Strong LAP Argument tells us that if we construct composite likelihood factors using marginal
distributions over domains that satisfy the Strong LAP Condition then the LAP algorithm of Mizrahi
et. al. [19] remains consistent. In this section we show that more can be achieved.
Once we have satisfied the Strong LAP Condition we know it is acceptable to match parameters
between the joint distribution p(xV | ?) and the auxiliary distribution p(xAq | ?). To obtain a consistent LAP algorithm from this correspondence all that is required is to have a consistent estimate
of ?q . Mizrahi et. al. [19] achieve this by applying maximum likelihood estimation to p(xAq | ?),
but any consistent estimator is valid.
We exploit this fact to show how the Strong LAP Argument can be applied to create a consistent
conditional LAP algorithm, where conditional estimation is performed in each auxiliary MRF. This
allows us to apply the LAP methodology to a broader class of models. For some models, such as
large densely connected graphs, we cannot rely on the LAP algorithm of Mizrahi et. al. [19]. For
example, for a restricted Boltzmann machine (RBM) [23], the 1-neighbourhood of any pairwise
clique includes the entire graph. Hence, the complexity of LAP is exponential in the size of V.
However, it is linear for conditional LAP, without sacrificing consistency.
Theorem 11. Let q be a clique in G and let xj 2 q ? Aq ? V. If Aq satisfies the Strong LAP
Condition for q then p(xV | ?) and p(xj | xAq \{xj } , ?) share the same normalised potential for q.
Proof. We can write the conditional distribution of xj given Aq \ {xj } as
p(xAq | ?)
p(xj | xAq \{xj } , ?) = P
xj p(xAq | ?)
(11)
Both the numerator and the denominator of Equation 11 are Gibbs distributions, and can therefore
be expressed in terms of potentials over clique systems.
7
Since Aq satisfies the Strong LAP Condition for qPwe know that p(xAq | ?) and p(xV | ?) have the
same potential for q. Moreover, the domain of xj p(xAq | ?) does not include q, so it cannot
contain a potential for q. We conclude that the potential for q in p(xj | xAq \{xj } , ?) must be shared
with p(xV | ?).
Remark 12. There exists a Gibbs representation normalised with respect to zero for
p(xj | xAq \{xj } , ?). Moreover, the clique potential for q is unique in that representation.
Existence in the above remark is an immediate result of the the existence of normalized representation both for the numerator and denominator of Equation 11, and the fact that difference
of normalised potentials is a normalized potential. For uniqueness, first note that p(xAq | ?) =
p(xj | xAq \{xj } , ?)p(xAq \{xj } , ?) The variable xj is not part of p(xAq \{xj } , ?) and hence this distribution does not contain the clique q. Suppose there were two different normalised representations
for the conditional p(xj | xAq \{xj } , ?). This would then imply two normalised representations for
the joint, which contradicts the fact that the joint has a unique normalized representation.
We can now proceed as in the original LAP construction from Mizrahi et al. [19]. For a clique of
interest q we find a set Aq which satisfies the Strong LAP Condition for q. However, instead of
creating an auxiliary parametrisation of the marginal we create an auxiliary parametrisation of the
conditional in Equation 11.
X
1
p(xj | xAq \{xj } , ?) =
exp(
E(xc | ?c ))
(12)
Zj (?)
c2CAq
From Theorem 11 we know that E(xq | ?q ) = E(xq | ? q ). Equality of the parameters is also
obtained, provided they are identifiable.
Corollary 13. If Aq satisfies the Strong LAP Condition for q then any consistent estimator of ?q in
p(xj | xAq \{xj } , ?) is also a consistent estimator of ? q in p(xV | ?).
5.1
Connection to distributed pseudo-likelihood and composite likelihood
Theorem 11 tells us that if Aq satisfies the Strong LAP Condition for q then to estimate ? q in
p(xV | ?) it is sufficient to have an estimate of ?q in p(xj | xAq \{xj } , ?) for any xj 2 q. This tells
us that it is sufficient to use pseudo-likelihood-like conditional factors, provided that their domains
satisfy the Strong LAP Condition. The following remark completes the connection by telling us
that the Strong LAP Condition is satisfied by the specific domains used in the pseudo-likelihood
factorisation.
Remark 14. Let q = {x1 , x2 , .., xm } be a clique of interest, with 1-neighbourhood Aq = q [
{N (xi )}xi 2q . Then for any xj 2 q, the set q [ N (xj ) satisfies the Strong LAP Condition for q.
Moreover, q [ N (xj ) satisfies the Strong LAP Condition for all cliques in the graph that contain xj .
Importantly, to estimate every unary clique potential we need to visit each node in the graph. However, to estimate pairwise clique potentials, visiting all nodes is redundant because the parameters of
each pairwise clique are estimated twice. If a parameter is estimated more than once it is reasonable
from a statistical standpoint to apply a consensus operator to obtain a single estimate. The theory of
Liu and Ihler tells us that the consensus estimates are consistent and asymptotically normal, provided
Equation 6 is satisfied. In turn, the Strong LAP Condition guarantees the convergence in Equation 6.
We can go beyond pseudo-likelihood and consider either marginal or conditional factorisations over
larger groups of variables. Since the asymptotic results of Liu and Ihler [13] apply to any distributed composite likelihood estimator where the convergence of Equation 6 holds, it follows that
any distributed composite likelihood estimator where each factor satisfies the Strong LAP Condition
(including LAP and the conditional composite likelihood estimator from Section 5) immediately
gains asymptotic normality and variance guarantees as a result of their work and ours.
6
Conclusion
We presented foundational theoretical results for distributed composite likelihood. The results provide us with sufficient conditions to apply the results of Liu and Ihler to a broad class of distributed
estimators. The theory also led us to the construction of a new globally consistent estimator, whose
complexity is linear even for many densely connected graphs. We view extending these results to
model selection, tied parameters, models with latent variables, and inference tasks as very important
avenues for future research.
8
References
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9
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4,769 | 5,318 | Elementary Estimators for Graphical Models
Eunho Yang
IBM T.J. Watson Research Center
[email protected]
Aur?elie C. Lozano
IBM T.J. Watson Research Center
[email protected]
Pradeep Ravikumar
University of Texas at Austin
[email protected]
Abstract
We propose a class of closed-form estimators for sparsity-structured graphical
models, expressed as exponential family distributions, under high-dimensional
settings. Our approach builds on observing the precise manner in which the classical graphical model MLE ?breaks down? under high-dimensional settings. Our estimator uses a carefully constructed, well-defined and closed-form backward map,
and then performs thresholding operations to ensure the desired sparsity structure.
We provide a rigorous statistical analysis that shows that surprisingly our simple
class of estimators recovers the same asymptotic convergence rates as those of the
`1 -regularized MLEs that are much more difficult to compute. We corroborate
this statistical performance, as well as significant computational advantages via
simulations of both discrete and Gaussian graphical models.
1
Introduction
Undirected graphical models, also known as Markov random fields (MRFs), are a powerful class
of statistical models, that represent distributions over a large number of variables using graphs, and
where the structure of the graph encodes Markov conditional independence assumptions among the
variables. MRFs are widely used in a variety of domains, including natural language processing [1],
image processing [2, 3, 4], statistical physics [5], and spatial statistics [6]. Popular instances of
this class of models include Gaussian graphical models (GMRFs) [7, 8, 9, 10], used for modeling
continuous real-valued data, and discrete graphical models including the Ising model where each
variable takes values in a discrete set [10, 11, 12]. In this paper, we consider the problem of highdimensional estimation, where the number of variables p may exceed the number of observations n.
In such high-dimensional settings, it is still possible to perform consistent estimation by leveraging
low-dimensional structure. Sparse and group-sparse structural constraints, where few parameters (or
parameter groups) are non-zero, are particularly pertinent in the context of MRFs as they translate
into graphs with few edges.
A key class of estimators for learning graphical models has thus been based on maximum likelihood
estimators (MLE) with sparsity-encouraging regularization. For the task of sparse GMRF estimation, the state-of-the-art estimator minimizes the Gaussian negative log-likelihood regularized by the
`1 norm of the entries (or the off-diagonal entries) of the concentration matrix (see [8, 9, 10]). Strong
statistical guarantees for this estimator have been established (see [13] and references therein). The
resulting optimization problem is a log-determinant program, which can be solved in polynomial
time with interior point methods [14], or by co-ordinate descent algorithms [9, 10]. In a computationally simpler approach for sparse GMRFs, [7] proposed the use of neighborhood selection,
which consists of estimating conditional independence relationships separately for each node in the
graph, via `1 -regularized linear regression, or LASSO [15]. They showed that the procedure can
1
consistently recover the sparse GMRF structure even under high-dimensional settings. The neighborhood selection approach has also been successfully applied to discrete Markov random fields.
In particular, for binary graphical models, [11] showed that consistent neighborhood selection can
be performed via `1 -regularized logistic regression. These results were generalized to general discrete graphical models (where each variable can take m
2 values) by [12] through node-wise
multi-class logistic regression with group sparsity. A related regularized convex program to solve
for sparse GMRFs is the CLIME estimator [16], which reduces the estimation problem to solving
linear programs. Overall, while state of the art optimization methods have been developed to solve
all of these regularized (and consequently non-smooth) convex programs, their iterative approach is
very expensive for large scale problems. Indeed, scaling such regularized convex programs to very
large scale settings has attracted considerable recent research and attention.
In this paper, we investigate the following leading question: ?Can one devise simple estimators
with closed-form solutions that are yet consistent and achieve the sharp convergence rates of the
aforementioned regularized convex programs?? This question was originally considered in the
context of linear regression by [17] and to which they had given a positive answer. It is thus natural
to wonder whether an affirmative response can be provided for the more complicated statistical
modeling setting of MRFs as well.
Our key idea is to revisit the vanilla MLE for estimating a graphical model, and consider where it
?breaks down? in the case of high-dimensions. The vanilla graphical model MLE can be expressed
as a backward mapping [18] in an exponential family distribution that computes the model parameters corresponding to some given (sample) moments. There are however two caveats with this
backward mapping: it is not available in closed form for many classes of models, and even if it were
available in closed form, it need not be well-defined in high-dimensional settings (i.e. could lead to
unbounded model parameter estimates). Accordingly, we consider the use of carefully constructed
proxy backward maps that are both available in closed-form, and well-defined in high-dimensional
settings. We then perform simple thresholding operations on these proxy backward maps to obtain
our final estimators. Our class of algorithms is thus both computationally practical and highly scalable. We provide a unified statistical analysis of our class of algorithms for graphical models arising
from general exponential families. We then instantiate our analysis for the specific cases of GMRFs
and DMRFs, and show that the resulting algorithms come with strong statistical guarantees achieving near-optimal convergence rates, but doing so computationally much faster than the regularized
convex programs. These surprising results are confirmed via simulation for both GMRFs and DMRFs. There has been considerable recent interest in large-scale statistical model estimation, and in
particular, in scaling these to very large-scale settings. We believe our much simpler class of closedform graphical model estimators have the potential to be estimators of choice in such large-scale
settings, particularly if it attracts research on optimizing and scaling its closed-form operations.
2
Background and Problem Setup
Since most popular graphical model families can be expressed as exponential families (see [18]), we
consider general exponential family distributions for a random variable X 2 Rp :
n
o
P(X; ?) = exp h?, (X)i A(?)
(1)
where ? 2 ? ? Rd is the canonical parameter to be estimated, (X) denotes the sufficient statistics
with feature function : Rp 7! Rd , and A(?) is the log-partition function.
An alternative parameterization of the exponential family, to the canonical parameterization above,
def
is via the vector of ?mean parameters? ?(?) = E? [ (X)], which are the moments of the sufficient
statistics (X) under the exponential family distribution. We denote the set of all possible moments
by the moment polytope: M = {? : 9 distribution p s.t. Ep ( ) = ?}, which consist of moments
of the sufficient statistics under all possible distributions. The problem of computing the mean
parameters ?(?) 2 M given the canonical parameters ? 2 ? constitutes the key machine learning
problem of inference in graphical models (expressed in exponential family form (1)). Let us denote
this computation via a so-called forward mapping A : ? 7! M. By properties of exponential family
distributions, the forward mapping A can actually be expressed in terms of the first derivative of
the log-partition function A(?): A : ? 7! ? = rA(?). It can be shown that this map is injective
(one-to-one with its range) if the exponential family is minimal. Moreover, it is onto the interior of
2
M, denoted by Mo . Thus, for any mean parameter ? 2 Mo , there exists a canonical parameter
?(?) 2 ? such that E?(?) [ (X)] = ?. Unless the exponential family is minimal, the corresponding
canonical parameter ?(?) however need not be unique. Thus while there will always exist a so-called
backward mapping A? : Mo 7! ?, that computes the canonical parameters corresponding to given
moments, it need not be unique. A candidate backward map can be constructed via the conjugate of
the log-partition function A? (?) = sup?2? h?, ?i A(?): A? : ? 7! ? = rA? (?).
2.1
High-dimensional Graphical Model Selection
We focus on the high-dimensional setting, where the number of variables p may greatly exceed
the sample size n. Under such high-dimensional settings, it is now well understood that consistent
estimation is possible if structural constraints are imposed on the model parameters ?. In this paper,
we focus on the structural constraint of sparsity, for which the `1 norm is known to be well-suited.
Given n samples {X (i) }ni=1 from P(X; ?? ) that belongs to an exponential family (1), a popular
class of M -estimators for recovering the sparse model parameter ?? is the `1 -regularized maximum
likelihood estimators:
minimize h ?, b i A(?) + n k?k1
(2)
?
Pn
where b := n1 i=1 (X (i) ) is the empirical mean of the sufficient statistics. Since the log partition
function A(?) in (1) is convex, the problem (2) is convex as well.
We now briefly review the two most popular examples of exponential families in the context of
graphical models.
Gaussian Graphical Models. Consider a random vector X = (X1 , . . . , Xp ) with associated pvariate Gaussian distribution N (X|?, ?), mean vector ? and covariance matrix ?. The probability
density function of X can be formulated as an instance of (1):
? 1
?
P(X|?, ?) = exp
hh?, XX > ii + h?, Xi A(?, ?)
(3)
2
where hhA, Bii denotes the trace inner product tr(A B T ). Here, the canonical parameters are the
precision matrix ? and a vector ?, with domain ? := {(?, ?) 2 Rp ?Rp?p : ? 0, ? = ?T }. The
corresponding moment parameters of the graphical model distribution are given by the mean ? =
E? [X], and the covariance matrix ? = E? [XX T ] of the Gaussian. The forward map A : (?, ?) 7!
(?, ?) computing these from the canonical parameters can be written as: ? = ? 1 and ? = ? 1 ?.
The moment polytope can be seen to be given by M = {(?, ?) 2 Rp ? Rp?p : ? ??T ? 0, ? ?
0}, with interior Mo = {(?, ?) 2 Rp ?Rp?p : ? ??T 0, ? 0}. The corresponding backward
map A? : (?, ?) 7! (?, ?) for (?, ?) 2 Mo can be computed as: ? = ? 1 and ? = ? 1 ?.
Without loss of generality, assume that ? = 0 (and hence ? = 0). Then the set of non-zero entries
in the precision matrix ? corresponds to the set of edges in an associated Gaussian Markov random
field (GMRF). In cases where the graph underlying the GMRF has relatively few edges, it thus
makes sense to impose `1 regularization on the off-diagonal entries of ?. Given n i.i.d. random
vectors X (i) 2 Rp sampled from N (0, ?? ), the corresponding `1 -regularized maximum likelihood
estimator (MLE) is given by:
minimize hh?, Sii
? 0
log det ? +
n k?k1,off
Pn
(i)
where S is the sample covariance matrix defined as
i=1 X
P
n
1
(i)
i=1 X , and k ? k1,off is the off-diagonal element-wise `1 norm.
n
(4)
,
X X (i)
X
>
, X :=
Discrete Graphical Models. Let X = (X1 , . . . , Xp ) be a random vector where each variable Xi
takes values in a discrete set X of cardinality m. Given a graph G = (V, E), a pairwise Markov
random field over X is specified via nodewise functions ?s : X 7! R for all s 2 V , and pairwise
functions ?st : X ? X 7! R for
(s, t) 2 E, as
n all
o
P
P
P(X) = exp
?
(X
)
+
?
(X
,
X
)
A(?)
.
(5)
s
s
st
s
t
s2V
(s,t)2E
This family of probability distributions can be represented using the so-called overcomplete representations [18] as follows. For each random variable Xs and j 2 {1, . . . , m}, define nodewise
3
indicators I[Xs = j] equal to 1 if Xs = j and 0 otherwise. Then pairwise MRFs in (5) can be
rewritten as ?
X
X
P(X) = exp
?s;j I[Xs = j] +
?st;jk I[Xs = j, Xt = k] A(?)
(6)
s2V ;j2[m]
(s,t)2E;j,k2[m]
for a set of parameters ? := {?s;j , ?st;jk : s, t 2 V ; (s, t) 2 E; j, k 2 [m]}. Given these sufficient
statistics, the mean/moment parameters are given by the moments ?s;j := E? I[Xs = j] =
P(Xs = j; ?), and ?st;jk := E? I[Xs = j, Xt = k] = P(Xs = j, Xt = k; ?), which precisely
correspond to nodewise and pairwise marginals of the discrete graphical model. Thus, the forward
mapping A : ? 7! ? would correspond to the inference task of computing nodewise and pairwise
marginals of the discrete graphical model given the canonical parameters. A backward mapping
A? : ? 7! ? corresponds to computing a set of canonical parameters such that the corresponding
graphical model distribution would yield the given set of nodewise and pairwise marginals. The
moment polytope in this case consists of the set of all nodewise and pairwise marginals of any
distribution over the random vector X, and hence is termed the marginal polytope; it is typically
intractable to characterize in high-dimensions [18].
Given n i.i.d. samples from an unknown distribution (6) with parameter ?? , one could consider
b
estimating the graphical model structure with an `1 -regularized MLE: ?b 2 minimizeP
? h?, i +
A(?) + k?k1,E , where k ? k1,E is the `1 norm of the edge-parameters: k?k1,E =
s6=t k?st k,
and where we have collated the edgewise parameters {?st;jk }m
for
an
edge
(s,
t)
2
E
into the
j,k=1
vector ?st . However, there is an critical caveat to actually computing this regularized MLE: the
computation of the log-partition function A(?) is intractable (see [18] for details). To overcome this
issue, one might consider instead the following class of M -estimators, discussed in [19]:
?b 2 minimize h?, bi + B(?) + k?k1,E .
?
(7)
Here B(?) is a variational approximation to the log-partition function A(?) of the form: B(?) =
sup?2L h?, ?i B ? (?), where L is a tractable bound on the marginal polytope M, and B ? (?) is a
tractable approximation to the graphical model entropy A? (?). An example of such approximation,
?
which
we shall P
later leverage in this paper, is the tree-reweighted entropy [20] given by Btrw
(?) =
P
s Hs (?s )
st ?st Ist (?st ), where Hs (?s ) is the entropy for node s, Ist (?st ) is the mutual
information for an edge (s, t), and ?st denote the edge-weights that lie in a so-called spanning tree
polytope. If all ?st are set to 1, this boils down to the Bethe approximation [21].
3
Closed-form Estimators for Graphical Models
The state-of-the-art `1 -regularized MLE estimators discussed in the previous section enjoy strong
statistical guarantees but involve solving difficult non-smooth programs. Scaling them to very largescale problems is thus an important and challenging ongoing research area.
In this paper we tackle the scalability issue at the source by departing from regularized MLE approaches and proposing instead a family of closed-form estimators for graphical models.
Elem-GM Estimation:
(8)
minimize k?k1
?
s. t. ?
B ? ( b)
where B ? (?) is the proxy of backward mapping A? , and
1
n
?
n
is a regularization parameter as in (2).
One of the most important properties of (8) is that the estimator is available in closed-form: ?b =
S n B ? ( b) , where [S (u)]i = sign(ui ) max(|ui |
, 0) is the element-wise soft-thresholding
function. This can be shown by the fact that the optimization problem (8) is decomposable into
independent element-wise sub-problems, where each sub-problem corresponds to soft-thresholding.
To get some intuition on our approach, let us first revisit classical MLE estimators for graphical models as in (1), and see where they ?break down? in a high-dimensional setting: minimize? h ?, b i
A(?). By the stationary condition of this optimization problem, the MLE estimator can be simply
expressed as a backward mapping A? ( b). There are two caveats here in high-dimensional settings.
4
The first is that this backward mapping need not have a simple closed-form, and is typically intractable to compute for a large number of variables p. The second is that the backward mapping is
well-defined only for mean parameters that are in the interior Mo of the marginal polytope, whereas
the sample moments b might well lie on the boundary of the marginal polytope. We will illustrate
these two caveats in the next two examples.
Our key idea is to use instead a well-defined proxy function B ? (?) in lieu of the MLE backward
map A? (?) so that B ? ( b) is both well-defined under high-dimensional settings, as well as with a
simple closed-form. The optimization problem (8) seeks an estimator with minimum complexity in
terms of regularizer k ? k1 while being close enough to some ?initial estimator? B ? ( b) in terms of
element-wise `1 norm; ensuring that the final estimator has the desired sparse structure.
3.1
Strong Statistical Guarantees of Closed-form Estimators
We now provide a statistical analysis of estimators in (8) under the following structural constraint:
(C-Sparsity) The ?true? canonical exponential family parameter ?? is exactly sparse with k nonzero elements indexed by the support set S. All other elements in S c are zeros.
Theorem 1. Consider any graphical model in (1) with sparse canonical parameter ?? as stated in
(C-Sparsity). Suppose we solve (8) setting the constraint bound n such that n
?? B ? ( b) 1 .
(A) Then the optimal solution ?b satisfies the following error bounds:
p
?b ?? 1 ? 2 n , k?b ?? k2 ? 4 k n , and
?b ?? 1 ? 8k n .
(B) The support set of the estimate ?b correctly excludes all true zero coordinates of ?? . Moreover,
under the additional assumption that mins2S |?s? |
3 ?? B ? ( b) 1 , it correctly includes all
non-zero coordinates of ?? .
Remarks. Theorem 1 is a non-probabilistic result, and holds deterministically for any selection of
?
n and any selection of B (?). We would then use a probabilistic analysis when we applying the
theorem to specific distributional settings and choices of the backward map B ? (?).
We note that while the theorem analyses the case of sparsity structured parameters, our class of
estimators as well as analyses can be seamlessly extended to more general structures (such as group
sparsity and low rank), by substituting appropriate regularization functions in (8).
A key ingredient in our class of closed-form estimators is the proxy backward map B ? ( b). The
conditions of the theorem require that this backward map has to be carefully constructed in order
for the error bounds and sparsistency guarantees to hold. In the following sections, we will see how
to precisely construct such backward maps B ? (?) for specific problem instances, and then derive the
corresponding consequences of our abstract theorem as corollaries.
4
Closed-form Estimators for Inverse Covariance Estimation in Gaussian
Graphical Models
In this section, we derive a class of closed-form estimators for the multivariate Gaussian setting
in Section 2.1. From our discussion of Gaussian graphical models in Section 2.1, the backward
mapping from moments to the canonical parameters can be simply computed as A? (?) = ? 1 , but
only provided ? 2 Mo := {? 2 Rp?p : ?
0}. However, given the sample covariance, we
cannot just compute the MLE as A? (S) = S 1 since the sample covariance matrix is rank-deficient
and hence does not belong the Mo under high-dimensional settings where p > n.
In our estimation framework (8), we thus use an alternative backward mapping B ? (?) via a thresholding operator. Specifically, for any matrix M 2 Rp?p , we consider the family of thresholding operators T? (M ) : Rp?p ! Rp?p with thresholding parameter ?, defined as [T? (M )]ij := ?? (Mij )
where ?? (?) is an element-wise thresholding operator. Soft-thresholding is a natural option, however,
along the lines of [22], we can use arbitrary sparse thresholding operators satisfying the conditions:
(C-Thresh) For any input a 2 R, (i) |?? (a)| ? |a|, (ii) |?? (a)| = 0 for |a| ? ?, and finally (iii)
|?? (a) a| ? ?.
5
As long as T? (S) is invertible (which we shall examine in section 4.1), we can define B ? (S) :=
[T? (S)] 1 and obtain the following class of estimators:
Elem-GGM Estimation:
(9)
minimize k?k1,off
?
s. t. ?
[T? (S)]
1
1,off
?
n
where k ? k1,off is the off-diagonal element-wise `1 norm as the dual of k ? k1,off .
Comparison with related work.
Note that [16] suggest a Dantzig-like estimator :
minimize? k?k1 s. t. kS? Ik1 ? n where both k ? k1 and k ? k1 are entry-wise (`1 and `1 ,
respectively) norms for a matrix. This estimator applies penalty functions even for the diagonal elements so that the problem can be decoupled into multiple but much simpler optimization problems.
It still requires solving p linear programs with 2p linear constraints for each. On the other hand, the
estimator from (9) has a closed-form solution as long as T? (S) is invertible.
4.1
Convergence Rates for Elem-GGM
In this section we derive a corollary of theorem 1 for Elem-GGM. A prerequisite is to show that
B ? (S) := [T? (S)] 1 is well-defined and ?well-behaved?. The following conditions define a broad
class of Gaussian graphical models that satisfy this requirement.
(C-MinInf?) The true canonical parameter ?? of (3) has bounded induced operator norm such that
?
1
|||?? |||1 := supw6=02Rp k?kwkwk
? ?1 .
1
(C-Sparse?) The true covariance matrix ?? := (?? ) 1 is ?approximately sparse? along the lines
?
of Bickel and Levina [23]: for some positive constant
D for all diagonal entries, and
PpD, ?ii? ?
moreover, for some 0 ? q < 1 and c0 (p), maxi j=1 |?ij |q ? c0 (p). If q = 0, then this
condition boils down to ?? being sparse. We additionally require inf w6=02Rp
k?? wk1
kwk1
?2 .
Now we are ready to utilize Theorem 1 and derive the convergence rates for our Elem-GGM (9).
Corollary 1. Consider Gaussian graphical models (3) where the true parameter ?? has k non-zero
off-diagonal elements, and the conditions in (C-MinInf?) and (C-Sparse?) hold. Suppose that we
solve the optimization problem q
in (9) with a generalized
thresholding operator satisfying (C-Thresh)
q
10? log p0
log p0
0
and setting ? := 16(maxi ?ii )
:= a
n for p := max{n, p}. Furthermore, suppose
q n
0
b of
also that we select n := 4??12 a lognp . Then, as long as n > c3 log p0 , any optimal solution ?
(9) satisfies
b
?
??
1,off
?
8?1 a
?2
r
log p0
,
n
with probability at least 1
b
?
??
F
?
16?1 a
?2
c1 exp( c2 log p0 ).
r
k log p0
,
n
b
?
??
1,off
?
32?1 a
k
?2
r
log p0
n
We remark that the rates in Corollary 1 are asymptotically the same as those for?standard
`?1 regularq
k log p0
?
b
ized MLE estimators in (4); for instance, [13] show that |||?MLE ? |||F = O
. This is
n
remarkable given the simplicity of Elem-GGM.
5
Closed-form Estimators for Discrete Markov Random Fields
We now specialize our class of closed-form estimators (8) to the setting of discrete Markov random
fields described in Section 2.1. In this case, computing the backward mapping A? is non-trivial and
typically intractable if the graphical structure has loops [18]. Therefore, we need an approximation
of the backward map A? , for which we will leverage the tree-reweighted variational approximation
? b
discussed in Section 2.1. Consider the following map ?? := Btrw
( ), where
bst;jk
??s;j = log bs;j , and ??st;jk = ?st log
(10)
bs;j bt;k
Pn
Pn
where bs;j = n1 i=1 I[Xs,i = j] and bst;jk = n1 i=1 I[Xs,i = j]I[Xt,i = k] are the empirical
?
moments of the sufficient statistics in (6) (we define 0/0 := 1). It was shown in [20] that Btrw
(?)
6
satisfies the following property: the (pseudo)marginals computed by performing tree-reweighted
? b
variational inference with the parameters ?? := Btrw
( ) yield the sufficient statistics b. In other
?
words, the approximate backward map Btrw
computes an element in the pre-image of the approximate forward map given by tree-reweighted variational inference. Since tree-reweighted variational
?
inference approximates the true marginals well in practice, the map Btrw
(?) is thus a great candidate
for as an approximate backward map.
As an alternative to the `1 regularized approximate MLE estimators (7), we thus obtain the
?
following class of estimators using Btrw
(?) as an instance of (8):
Elem-DMRF Estimation:
(11)
minimize k?k1,E
?
? b
Btrw
( )
s. t. ?
1,E
?
n
where k ? k1,E is the maximum absolute value of edge-parameters as a dual of k ? k1,E .
? b
Note that given the empirical means of sufficient statistics, Btrw
( ) can usually be obtained easily,
without the need of explicitly specifying the log-partition function approximation B(?) in (7).
5.1
Convergence Rates for Elem-DRMF
We now derive the convergence rates of Elem-DRMF for the case where B ? (?) is selected as in
(10) following the tree reweighed approximation [20]. Let ?? be the ?true? marginals (or mean
parameters) from the true log-partition function and true canonical parameter ?? : ?? = A(?? ). We
shall require that the approximation Btrw (?) be close enough to the true A(?) in terms of backward
mapping. In addition we assume that true marginal distributions are strictly positive.
(C-LogPartition)
??
?
Btrw
(?? )
1,E
? ?.
(C-Marginal) For all s 2 V and j 2 [m], the true singleton marginal ??s;j := E?? I[Xs = j] =
P(Xs = j; ?? ) satisfies ?min < ??s;j for some strictly positive constant ?min 2 (0, 1). Similarly,
for all s, t 2 V and all j, k 2 [m], ??st;jk satisfies ?min < ??st;jk .
Now we are ready to utilize Theorem 1 to derive the convergence rates for our closed-form estimator
?
(11) when ?? has k non-zero pairwise parameters ?st
, where we recall the notatation that ?st :=
m
{?st;jk }j,k=1 is a collation of the edgewise parameters for edge (s, t). We also define k?kq,E :=
P
( s6=t k?st kq )1/q , for q 2 {1, 2, 1}.
Corollary 2. Consider discrete Markov random fields (6) when the true parameter ?? has actually
k non-zero pair-wise parameters, and the conditions in (C-LogPartition) and (C-Marginal) also
?
hold in these discrete MRFs. Suppose that we solve the optimization problem in (11) with Btrw
(?)
set as q
(10) using tree reweighed approximation. Furthermore, suppose also that we select n :=
4c2 log p
? + c1 logn p for some positive constant c1 depending only on ?min . Then, as long as n > 1?2 ,
min
there are universal positive constants (c2 , c3 ) such that any optimal solution ?b of (11) satisfies
k?b
?
? k1,E ? 2? + 2c1
r
log p b
, k?
n
with probability at least 1
6
?
p
? k2,E ? 4 k? + 4c1
c2 exp( c3 log p0 ).
r
k log p b
, k?
n
?
? k1,E ? 8k? + 8c1 k
r
log p
n
Experiments
In this section, we report a set of synthetic experiments corroborating our theoretical results on both
Gaussian and discrete graphical models.
Gaussian Graphical Models We now corroborate Corollary 1, and furthermore, compare our
estimator with the `1 regularized MLE in terms of statistical performance with respect to the
b ?? kq for q 2 {2, 1}, as well as in terms of computational performance.
parameter error k?
To generate true inverse covariance matrices ?? with a random sparsity structure, we follow the
procedure described in [25, 24]. We first generate a sparse matrix U whose non-zero entries are set
to ?1 with equal probabilities. ?? is then set to U > U and then a diagonal term is added to ensure
7
Table 1: Performance of our Elem-GM vs. state of the art QUIC algorithm [24] solving (4) under
two different regimes: (Left) (n, p) = (800, 1600), (Right) (n, p) = (5000, 10000).
Elem-GM
QUIC
K
0.01
0.02
0.05
0.1
0.5
1
2
3
4
Time(sec)
<1
<1
<1
<1
2575.5
1009
272.1
78.1
28.7
`F (off)
6.36
6.19
5.91
6
12.74
7.30
6.33
6.97
7.68
`1 (off)
0.1616
0.1880
0.1655
0.1703
0.11
0.13
0.18
0.21
0.23
FPR
0.48
0.24
0.06
0.01
0.52
0.35
0.16
0.07
0.02
TPR
0.99
0.99
0.99
0.97
1.00
0.99
0.99
0.94
0.86
Elem-GM
QUIC
K
0.05
0.1
0.5
1
2
2.5
3
3.5
Time(sec)
47.3
46.3
45.8
46.2
*
*
4.8 ? 104
2.7 ? 104
`F (off)
11.73
8.91
5.66
8.63
*
*
9.85
10.51
`1 (off)
0.1501
0.1479
0.1308
0.1111
*
*
0.1083
0.1111
FPR
0.13
0.03
0.0
0.0
*
*
0.06
0.04
TPR
1.00
1.00
1.00
0.99
*
*
1.00
0.99
Table 2: Performance of Elem-DMRF vs. the regularized MLE-based approach of [12] for structure
recovery of DRMFs.
Graph Type
Chain Graph
Grid Graph
# Parameters
128
2000
128
2000
Method
Elem-DMRF
Regularized MLE
Elem-DMRF
Regularized MLE
Elem-DMRF
Regularized MLE
Elem-DMRF
Regularized MLE
Time(sec)
0.17
7.30
21.67
4315.10
0.17
7.99
21.68
4454.44
TPR
0.87
0.81
0.79
0.75
0.97
0.84
0.80
0.77
FNR
0.01
0.01
0.12
0.21
0.01
0.02
0.12
0.18
?? is positive definite. Finally, we normalize ?? with maxpi=1 ??ii so that the maximum diagonal
entry is equal to 1. We control the number of non-zeros in U so that the number of non-zeros in the
final ?? is approximately 10p. We additionally set the number of samples n to half of the number
of variables p. Note that though the number of variables is p, the total number of entries in the
canonical parameter consisting of the covariance matrix is O(p2 ).
Table 1 summarizes the performance of our closed-form estimators in terms of computation
time,
p
b ?? k1,off and |||?
b ?? |||F,off . We fix the thresholding parameter ? = 2.5 log p/n for all
k?
p
settings, and vary the regularization parameter n = K log p/n to investigate how this regularizer
affects the final estimators. Baselines are `1 regularized MLE estimators in (4); we use QUIC
algorithms [24], which is one of the fastest way to solve (4). In the table, we show the results of the
QUIC algorithm run with a tolerance ? = 10 4 ; * indicates that the algorithm does not stop within
15 hours. In Appendix, we provide more extensive comparisons including receiver operator curves
(ROC) for these methods for settings in Table 1. As can be seen from the table and the figure, the
performance of Elem-GM estimators is both statistically competitive in terms of all types of errors
and support set recovery, while performing much better computationally than classical methods
based on `1 regularized MLE.
Discrete Graphical Models We consider two different classes of pairwise graphical models:
chain graphs and grids. For each case, the size of the alphabet is set to m = 3; the true parameter vector ?? is generated by sampling each non-zero entry from N (0, 1).
We compare Elem-DMRF with the group-sparse regularized MLE-based approach of Jalali et al.
[12], which uses group `1 /`2 regularization, where all the parameters of an edge form a group, so as
to encourage sparsity in terms of the edges, and which we solved using proximal gradient descent.
While our estimator in (11) used vanilla sparsity, we used a simple extension to the group-sparse
structured setting;
pplease see Appendix E for more details. For both methods, the tuning parameter
is set to n = c log p/n, where c is selected using cross-validation. We use 20 simulation runs
where for each run n = p/2 samples are drawn from the distribution specified by ?? .
We report true positive rates, false positive rates and timing for running each method. We note
that the timing is for running each method without counting the time spent in the cross-validation
process (Had we taken the cross-validation into account, the advantage of our method would be
even more pronounced, since the entire path of solutions can be computed via simple group-wise
thresholding operations.) The results in Table 2 show that Elem-DMRF is much faster than its
MLE-based counterpart, and yield competitive results in terms of structure recovery.
Acknowledgments E.Y and P.R. acknowledge the support of ARO via W911NF-12-1-0390 and
NSF via IIS-1149803, IIS-1320894, IIS-1447574, and DMS-1264033
8
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9
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4,770 | 5,319 | Structure learning of
antiferromagnetic Ising models
Guy Bresler1
David Gamarnik2
Devavrat Shah1
Laboratory for Information and Decision Systems
Department of EECS1 and Sloan School of Management2
Massachusetts Institute of Technology
{gbresler,gamarnik,devavrat}@mit.edu
Abstract
In this paper we investigate the computational complexity of learning the
graph structure underlying a discrete undirected graphical model from i.i.d.
samples. Our first result is an unconditional computational lower bound
of (pd/2 ) for learning general graphical models on p nodes of maximum
degree d, for the class of so-called statistical algorithms recently introduced
by Feldman et al. [1]. The construction is related to the notoriously difficult
learning parities with noise problem in computational learning theory. Our
? d+2 ) runtime required by Bresler, Mossel,
lower bound suggests that the O(p
and Sly?s [2] exhaustive-search algorithm cannot be significantly improved
without restricting the class of models.
Aside from structural assumptions on the graph such as it being a tree,
hypertree, tree-like, etc., many recent papers on structure learning assume
that the model has the correlation decay property. Indeed, focusing on ferromagnetic Ising models, Bento and Montanari [3] showed that all known
low-complexity algorithms fail to learn simple graphs when the interaction
strength exceeds a number related to the correlation decay threshold. Our
second set of results gives a class of repelling (antiferromagnetic) models
that have the opposite behavior: very strong interaction allows efficient
? 2 ). We provide an algorithm whose performance inlearning in time O(p
? 2 ) and O(p
? d+2 ) depending on the strength of the
terpolates between O(p
repulsion.
1
Introduction
Graphical models have had tremendous impact in a variety of application domains. For
unstructured high-dimensional distributions, such as in social networks, biology, and finance,
an important first step is to determine which graphical model to use. In this paper we
focus on the problem of structure learning: Given access to n independent and identically
distributed samples ? (1) , . . . ? (n) from an undirected graphical model representing a discrete
random vector ? = (?1 , . . . , ?p ), the goal is to find the graph G underlying the model. Two
basic questions are 1) How many samples are required? and 2) What is the computational
complexity?
In this paper we are mostly interested in the computational complexity of structure learning.
We first consider the problem of learning a general discrete undirected graphical model of
bounded degree.
1
1.1
Learning general graphical models
Several algorithms based on exhaustively searching over possible node neighborhoods have
appeared in the last decade [4, 2, 5]. Abbeel, Koller, and Ng [4] gave algorithms for learning
general graphical models close to the true distribution in Kullback-Leibler distance. Bresler,
Mossel, and Sly [2] presented algorithms guaranteed to learn the true underlying graph.
The algorithms in both [4] and [2] perform a search over candidate neighborhoods, and for
a graph of maximum degree d, the computational complexity for recovering a graph on p
? d+2 ) (where the O
? notation hides logarithmic factors).
nodes scales as O(p
While the algorithms in [2] are guaranteed to reconstruct general models under basic
nondegeneracy conditions using an optimal number of samples n = O(d log p) (sample
complexity lower bounds were proved by Santhanam and Wainwright [6] as well as [2]), the
? d+2 ) run-time is impractically high even for constant but large graph
exponent d in the O(p
degrees. This has motivated a great deal of work on structure learning for special classes of
graphical models. But before giving up on general models, we ask the following question:
Question 1: Is it possible to learn the structure of general graphical models on p
nodes with maximum degree d using substantially less computation than pd ?
Our first result suggests that the answer to Question 1 is negative. We show an uncond
ditional computational lower bound of p 2 for the class of statistical algorithms introduced
by Feldman et al. [1]. This class of algorithms was introduced in order to understand the
apparent difficulty of the Planted Clique problem, and is based on Kearns? statistical query
model [7]. Kearns showed in his landmark paper that statistical query algorithms require
exponential computation to learn parity functions subject to classification noise, and our
hardness construction is related to this problem. Most known algorithmic approaches (including Markov chain Monte Carlo, semidefinite programming, and many others) can be
implemented as statistical algorithms, so the lower bound is fairly convincing.
We give background and prove the following theorem in Section 4.
Theorem 1.1. Statistical algorithms require at least (p 2 ) computation steps in order to
learn the structure of a general graphical models of degree d.
d
If complexity pd is to be considered intractable, what shall we consider as tractable? Writing
algorithm complexity in the form c(d)pf (d) , for high-dimensional (large p) problems the
exponent f (d) is of primary importance, and we will think of tractable algorithms as having
an f (d) that is bounded by a constant independent of d. The factor c(d) is also important,
and we will use it to compare algorithms with the same exponent f (d).
In light of Theorem 1.1, reducing computation below p (d) requires restricting the class
of models. One can either restrict the graph structure or the nature of the interactions
between variables. The seminal paper of Chow and Liu [8] makes a model restriction of
the first type, assuming that the graph is a tree; generalizations include to polytrees [9],
hypertrees [10], and others. Among the many possible assumptions of the second type,
the correlation decay property is distinguished: to the best of our knowledge all existing
low-complexity algorithms require the correlation decay property [3].
1.2
Correlation decay property
Informally, a graphical model is said to have the correlation decay property (CDP) if any
two variables ?s and ?t are asymptotically independent as the graph distance between s and
t increases. Exponential decay of correlations holds when the distance from independence
decreases exponentially fast in graph distance, and we will mean this stronger form when
referring to correlation decay. Correlation decay is known to hold for a number of pairwise
graphical models in the so-called high-temperature regime, including Ising, hard-core lattice
gas, Potts (multinomial) model, and others (see, e.g., [11, 12, 13, 14, 15, 16]).
2
Bresler, Mossel, and Sly [2] observed that it is possible to efficiently learn models with (exponential) decay of correlations, under the additional assumption that neighboring variables
have correlation bounded away from zero (as is true, e.g., for the ferromagnetic Ising model
in the high temperature regime). The algorithm they proposed for this setting pruned the
candidate set of neighbors for each node to roughly size O(d) by retaining only those variables
with sufficiently high correlations, and then within this set performed the exhaustive search
? 2 ). The
over neighborhoods mentioned before, resulting in a computational cost of dO(d) O(p
greedy algorithms of Netrapali et al. [17] and Ray et al. [18] also require the correlation decay property and perform a similar pruning step by retaining only nodes with high pairwise
correlation; they then use a different method to select the true neighborhood.
A number of papers consider the problem of reconstructing Ising models on graphs with
few short cycles, beginning with Anandkumar et al. [19]. Their results apply to the case of
Ising models on sparsely connected graphs such as the Erd?os-Renyi random graph G(p, dp ).
They additionally require the interaction parameters to be either generic or ferromagnetic.
Ferromagnetic models have the benefit that neighbors always have a non-negligible correlation because the dependencies cannot cancel, but in either case the results still require the
CDP to hold. Wu et al. [20] remove the assumption of generic parameters in [19], but again
require the CDP.
Other algorithms for structure learning are based on convex optimization, such as Ravikumar et al.?s [21] approach using regularized node-wise logistic regression. While this
algorithm does not explicitly require the CDP, Bento and Montanari [3] found that the
logistic regression algorithm of [21] provably fails to learn certain ferromagnetic Ising model
on simple graphs without correlation decay. Other convex optimization-based algorithms
such as [22, 23, 24] require similar incoherence or restricted isometry-type conditions that
are difficult to verify, but likely also require correlation decay. Since all known algorithms
for structure learning require the CDP, we ask the following question (paraphrasing Bento
and Montanari):
Question 2: Is low-complexity structure learning possible for models which do not
exhibit the CDP, on general bounded degree graphs?
Our second main result answers this question affirmatively by showing that a broad class of
repelling models on general graphs can be learned using simple algorithms, even when the
underlying model does not exhibit the CDP.
1.3
Repelling models
The antiferromagnetic Ising model has a negative interaction parameter, whereby neighboring nodes prefer to be in opposite states. Other popular antiferromagnetic models include
the Potts or coloring model, and the hard-core model.
Antiferromagnetic models have the interesting property that correlations between neighbors
can be zero due to cancellations. Thus algorithms based on pruning neighborhoods using
pairwise correlations, such as the algorithm in [2] for models with correlation decay, does not
work for anti-ferromagnetic models. To our knowledge there are no previous results that
improve on the pd computational complexity for structure learning of antiferromagnetic
models on general graphs of maximum degree d.
Our first learning algorithm, described in Section 2, is for the hard-core model.
Theorem 1.2 (Informal). It is possible to learn strongly repelling models, such as the hard? 2 ).
core model, with run-time O(p
We extend this result to weakly repelling models (equivalent to the antiferromagnetic Ising
model parameterized in a nonstandard way, see Section 3). Here ? is a repelling strength
and h is an external field.
Theorem 1.3 (Informal). Suppose ? ? (d ? ?)(h + ln 2) for an integer 0 ? ? < d. Then
? 2+? ).
it is possible to learn a repelling model with interaction ?, with run-time O(p
3
? 2 ), achievable for
The computational complexity of the algorithm interpolates between O(p
d+2
?
strongly repelling models, and O(p ), achievable for general models using exhaustive
search. The complexity depends on the repelling strength of the model, rather than structural assumptions on the graph as in [19, 20].
We remark that the strongly repelling models exhibit long-range correlations, yet the algorithmic task of graph structure learning is possible using a local procedure.
The focus of this paper is on structure learning, but the problem of parameter estimation
is equally important. It turns out that the structure learning problem is strictly more
challenging for the models we consider: once the graph is known, it is not difficult to
estimate the parameters with low computational complexity (see, e.g., [4]).
2
Learning the graph of a hard-core model
We warm up by considering the hard-core model. The analysis in this section is straightforward, but serves as an example to highlight the fact that correlation decay is not a necessary
condition for structure learning.
Given a graph G = (V, E) on |V | = p nodes, denote by I(G) ? {0, 1}p the set of independent
set indicator vectors ?, for which at least one of ?i or ?j is zero for each edge {i, j} ? E(G).
The hardcore model with fugacity ? > 0 assigns nonzero probability only to vectors in I(G),
with
?|?|
P(?) =
, ? ? I(G) .
(2.1)
Z
q
|?|
Here |?| is the number of entries of ? equal to one and Z = ??I(G) ? is the normalizing
constant called the partition function. If ? > 1 then more mass is assigned to larger
independent sets. (We use indicator vectors to define the model in order to be consistent
with the antiferromagnetic Ising model in the next section.)
Our goal is to learn the graph G = (V, E) underlying the model (2.1) given access to independent samples ? (1) , . . . , ? (n) . The following simple algorithm reconstructs G efficiently.
Algorithm 1 simpleHC(? (1) , . . . , ? (n) )
1: FOR each i, j, k:
(k)
(k)
2: IF ?i = ?j = 1, THEN S = S ? {i, j}
? = Sc
3: OUTPUT E
The idea behind the algorithm is very simple. If {i, j} belongs to the edge set E(G), then
(k)
(k)
for every sample ? (k) either ?i = 0 or ?j = 0 (or both). Thus for every i, j and k such
(k)
(k)
that ?i = ?j = 1 we can safely declare {i, j} not to be an edge. To show correctness of
the algorithm it is therefore sufficient to argue that for every non-edge {i, j} there is a high
likelihood that such an independent set ? (k) will be sampled.
Before doing this, we observe that simpleHC actually computes the maximum-likelihood
(k)
(k)
estimate for the graph G. To see this, note that an edge e = {i, j} for which ?i = ?j = 1
? since P(? (k) |G+e)
?
? Thus the ML estimate contains
for some k cannot be in G,
= 0 for any G.
a subset of those edges e which have not been ruled out by ? (1) , . . . , ? (n) . But adding any
such edge e to the graph decreases the value of the partition function in (2.1) (the sum is
over fewer independent sets), thereby increasing the likelihood of each of the samples.
The sample complexity and computational complexity of simpleHC is as follows, with proof
in the Supplement.
Theorem 2.1. Consider the hard-core model (2.1) on a graph G = (V, E) on |V | = p nodes
and with maximum degree d. The sample complexity of simpleHC is
n = O((2?)2d?2 log p) ,
4
(2.2)
i.e. with this many samples the algorithm simpleHC correctly reconstructs the graph with
probability 1 ? o(1). The computational complexity is
O(np2 ) = O((2?)2d?2 p2 log p) .
(2.3)
We next show that the sample complexity bound in Theorem 2.1 is basically tight:
Theorem 2.2 (Sample complexity lower bound). Consider the hard-core model (2.1). There
is a family of graphs on p nodes with maximum degree d such that for the probability of
successful reconstruction to approach one, the number of samples must scale as
1
p2
n=
(2?)2d log
.
d
Lemma 2.3. Suppose edge e = (i, j) ?
/ G, and let I be an independent set chosen according
to the Gibbs distribution (2.1). Then P({i, j} ? I) ? (9 ? max{1, (2?)2d?2 })?1 , ? .
The Supplementary Material contains proofs for Theorem 2.2 and Lemma 2.3.
3
Learning anti-ferromagnetic Ising models
In this section we consider the anti-ferromagnetic Ising model on a graph G = (V, E). We
parametrize the model in such a way that each configuration has probability
)
*
1
P(?) = exp H(?) , ? ? {0, 1}p ,
(3.1)
Z
where
?
?
H(?) = ??
?i ?j +
hi ? i .
(3.2)
(i,j)?E
i?V
Here ? > 0 and {hi }i?V are real-valued parameters, and we assume that |hi | ? h for all i.
Working with configurations in {0, 1}p rather than the more typical {?1, +1}p amounts to
a reparametrization (which is without loss of generality as shown for example in Appendix 1
of [25]). Setting hi = h = ln ? for all i, we recover the hard-core model with fugacity ? in
the limit ? ? ?, so we think of (3.2) as a ?soft? independent set model.
3.1
Strongly antiferromagnetic models
We start by considering the situation in which the repelling strength ? is sufficiently large
that we can modify the approach used for the hard-core model. We require some notation
to work with conditional probabilities: for each vertex b ? V , let
(i)
and
Bb = {? (i) : ?b = 1} ,
? a = 1|?b = 1) := 1 |{i ? B : ? (i) = 1}| .
P(?
a
|B|
!
"
? a = 1|?b = 1) = P(?a = 1|?b = 1). The algorithm, described next,
Of course, E P(?
? to a threshold.
determines whether each edge {a, b} is present based on comparing P
Algorithm 2 StrongRepelling
?
Input: ?, h, d, and n samples ? (1) , . . . , ? (n) ? {0, 1}p . Output: edge set E.
d h(d?1) ?2
1: Let ? = (1 + 2 e
)
! "
2: FOR each possible edge {a, b} ? V2 :
?
3: IF P? (?a = 1|?b = 1) ? (1 + e??h )?1 + ? THEN add edge (a, b) to E
?
4: OUTPUT E
Algorithm StrongRepelling obtains the following performance. The proof of Proposition 3.1 is similar to that of Theorem 2.1, replacing Lemma 2.3 by Lemma 3.2 below.
5
Proposition 3.1. Consider the antiferromagnetic Ising model (3.2) on a graph G = (V, E)
on p nodes and with maximum degree d. If
? ? d(h + ln 2) ,
then algorithm StrongRepelling has sample complexity
1
2
n = O 22d e2h(d+1) log p ,
i.e. this many samples are sufficient to reconstruct the graph with probability 1 ? o(1). The
computational complexity of StrongRepelling is
1
2
O(np2 ) = O 22d e2h(d+1) p2 log p .
When the interaction parameter ? ? d(h+ln 2) it is possible to identify edges using pairwise
statistics. The next lemma, proved in the Supplement, shows the desired separation.
Lemma 3.2. We have the following estimates:
(i) If (a, b) ?
/ E(G), then P(?a = 1|?b = 1) ?
1
1+2deg(a) eh(deg(a)+1)
(ii) Conversely, if (a, b) ? E(G), then P(?a = 1|?b = 1) ?
(ii) For any b ? V , P(?b = 1) ?
3.2
1
1+2deg(b) eh(deg(b)+1)
.
1
1+e??h
.
.
Weakly antiferromagnetic models
In this section we focus on learning weakly repelling models and show a trade-off between
computational complexity and strength of the repulsion. Recall that for strongly repelling
models our algorithm has run-time O(p2 log p), the same as for the hard-core model (infinite
repulsion).
For a subset of nodes U ? V , let G\U denote the graph obtained from G by removing nodes
in U (as well as any edges incident to nodes in U ). The following corollary is immediate
from Lemma 3.2.
Corollary 3.3. We have the conditional probability estimates for deleting subsets of nodes:
(i) If (a, b) ?
/ E(G), then for any subset of nodes U ? V \ {a, b},
PG\U (?a = 1|?b = 1) ?
1
1+2
degG\U (a) h(degG\U (a)+1)
e
.
(ii) Conversely, if (a, b) ? E(G), then for any subset of nodes U ? V \ {a, b}
PG\U (?a = 1|?b = 1) ?
1
.
1 + e??h
We can effectively remove nodes from the graph by conditioning: The family of models (3.2)
has the property that conditioning on ?i = 0 amounts to removing node i from the graph.
Fact 3.4 (Self-reducibility). Let G = (V, E), and consider the model 3.2. Then for any
subset of nodes U ? V , the probability law PG (? ? ? |?U = 0) is equal to PG\U (?V \U ? ? ).
The final ingredient is to show that we can condition by restricting attention to a subset of
the observed data, ? (1) , . . . , ? (n) , without throwing away too many samples.
Lemma 3.5. Let U ? V be a subset of nodes and denote the subset of samples with variables
(i)
?U equal to zero by AU = {? (i) : ?u = 0 for all u ? U }. Then with probability at least
h 2|U |
1 ? exp(n/2(1 + e )
) the number |AU | of such samples is at least n2 ? (1 + eh )?|U | .
We now present the algorithm. Effectively, it reduces node degree by removing nodes (which
can be done by conditioning on value zero), and then applies the strong repelling algorithm
to the residual graph.
6
Algorithm 3 WeakRepelling
?
Input: ?, h, d, and n samples ? (1) , . . . , ? (n) ? {0, 1}p . Output: edge set E.
1: Let ? = (1 + 2d eh(d?1) )?2
! "
2: FOR each possible edge (a, b) ? V2 :
3: FOR each U ? V \ {a, b} of size |U | ? ?d ? ?/(h + ln 2)?
4:
Compute P?G\U (?a = 1|?b = 1)
?
5: IF minU :|U |= P?G\U (?a = 1|?b = 1) ? (1 + e??h ) + ? THEN add edge (a, b) to E
?
6: OUTPUT E
Theorem 3.6. Let ? be a nonnegative integer strictly smaller than d, and consider the
antiferromagnetic Ising model 3.2 with
? ? (d ? ?)(h + ln 2)
on a graph G. Algorithm WeakRepelling reconstructs the graph with probability 1 ? o(1)
as p ? ? using
1
2
n = O (1 + eh )? 22d e2h(d+1) log p
i.i.d. samples, with run-time
4
!
"
?h,d (p2+? ) .
O np2+? = O
Statistical algorithms and proof of Theorem 1.1
We start by describing the statistical algorithm framework introduced by [1]. In this section
it is convenient to work with variables taking values in {?1, +1} rather than {0, 1}.
4.1
Background on statistical algorithms
Let X = {?1, +1}p denote the space of configurations and let D be a set of distributions
over X . Let F be a set of solutions (in our case, graphs) and Z : D ? 2F be a map taking
each distribution D ? D to a subset of solutions Z(D) ? F that are defined to be valid
solutions for D. In our setting, since each graphical model is identifiable, there is a single
graph Z(D) corresponding to each distribution D. For n > 0, the distributional search
problem Z over D and F using n samples is to find a valid solution f ? Z(D) given access
to n random samples from an unknown D ? D.
The class of algorithms we are interested in are called unbiased statistical algorithms, defined
by access to an unbiased oracle. Other related classes of algorithms are defined in [1], and
similar lower bounds can be derived for those as well.
Definition 4.1 (Unbiased Oracle). Let D be the true distribution. The algorithm is given
access to an oracle, which when given any function h : X ? {0, 1}, takes an independent
random sample x from D and returns h(x).
These algorithms access the sampled data only through the oracle: unbiased statistical
algorithms outsource the computation. Because the data is accessed through the oracle, it
is possible to prove unconditional lower bounds using information-theoretic methods. As
noted in the introduction, many algorithmic approaches can be implemented as statistical
algorithms.
We now define a key quantity called average correlation. The average correlation of a subset
of distributions D? ? D relative to a distribution D is denoted ?(D? , D),
> ? -= D1
1
D2
?
?(D , D) := ? 2
(4.1)
- D ? 1, D ? 1 - ,
|D |
D1 ,D2 ?D ?
D
where ?f, g?D := Ex?D [f (x)g(x)] and the ratio D1 /D represents the ratio of probability
mass functions, so (D1 /D)(x) = D1 (x)/D(x).
We quote the definition of statistical dimension with average correlation from [1], and then
state a lower bound on the number of queries needed by any statistical algorithm.
7
Definition 4.2 (Statistical dimension). Fix ? > 0, ? > 0, and search problem Z over set
of solutions F and class of distributions D over X. We consider pairs (D, DD ) consisting
of a ?reference distribution? D over X and a finite set of distributions DD ? D with the
following property: for any solution f ? F, the set Df = DD \ Z ?1 (f ) has size at least
(1 ? ?) ? |DD |. Let ?(D, DD ) be the largest integer ? so that for any subset D? ? Df with
|D? | ? |Df |/?, the average correlation is |?(D? , D)| < ? (if there is no such ? one can take
? = 0). The statistical dimension with average correlation ? and solution set bound ? is
defined to be the largest ?(D, DD ) for valid pairs (D, DD ) as described, and is denoted by
SDA(Z, ?, ?).
Theorem 4.3 ([1]). Let X be a domain and Z a search problem over a set of solutions F
and a class of distributions D over X . For ? > 0 and ? ? (0, 1), let ? = SDA(Z, ?, ?). Any
(possibly randomized) unbiased statistical algorithm that solves Z with probability ? requires
at least m calls to the Unbiased Oracle for
;
<
?(? ? ?) (? ? ?)2
m = min
,
.
2(1 ? ?)
12?
In particular, if ? ? 1/6, then any algorithm with success probability at least 2/3 requires at
least min{?/4, 1/48?} samples from the Unbiased Oracle.
In order to show that a graphical model on p nodes of maximum degree d requires
computation p (d) in this computational model, we therefore would like to show that
SDA(Z, ?, ?) = p (d) with ? = p? (d) .
4.2
Soft parities
r
For any subset S ? [p] of cardinality |S| = d, let ?S (x) = i?S xi be the parity of variables
in S. Define a probability distribution by assigning mass to x ? {?1, +1}p according to
1
pS (x) = exp(c ? ?S (x)) .
(4.2)
Z
Here c is a constant, and the partition function is
?
Z=
exp(c ? ?S (x)) = 2p?1 (ec + e?c ) .
(4.3)
x
Our
!p" family of distributions D is given by these soft parities over subsets S ? [p], and |D| =
d . The following lemma, proved in the supplementary material, computes correlations
between distributions.
Lemma 4.4. Let U denote the uniform distribution on {?1, +1}p . For S ?= T , the correlation ? pUS ? 1, pUT ? 1? is exactly equal to zero for any value of c. If S = T , the correlation
? pUS ? 1, pUS ? 1? = 1 ? (ec +e4?c )2 ? 1.
Lemma 4.5. For any set D? ? D of size at least |D|/pd/2 , the average correlation satisfies
?(D? , U ) ? dd p?d/2 .
Proof. By the preceding lemma, the only contributions to the sum (4.1) comes from choosing
?
the same set S in the sum, of which there are a fraction 1/|D
such correlation is at
! " |. Each
?
d/2
d/2 p
most one !by" Lemma 4.4, so ? ? 1/|D | ? p /|D| = p / d ? dd /pd/2 . Here we used the
estimate nk ? ( nk )k .
Proof of Theorem 1.1. Let ? = 1/6 and ? = dd p?d/2 , and consider the set of distributions
D given by soft parities as defined above. With reference distribution D = U , the uniform
distribution, Lemma 4.5 implies that SDA(Z, ?, ?) of the structure learning problem over
distribution (4.2) is at least ? = pd/2 /dd . The result follows from Theorem 4.3.
Acknowledgments
This work was supported in part by NSF grants CMMI-1335155 and CNS-1161964, and by
Army Research Office MURI Award W911NF-11-1-0036.
8
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9
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4,771 | 532 | A Cortico-Cerebellar Model that Learns to
Generate Distributed Motor Commands to
Control a Kinematic Arm
N.E. Berthier S.P. Singh A.G. Barto
Department of Computer Science
University of Massachusetts
Amherst, MA 01002
.T.C. Honk
Department of Physiology
Northwestern University Medical School
Chicago, IL 60611
Abstract
A neurophysiologically-based model is presented that controls a simulated
kinematic arm during goal-directed reaches. The network generates a
quasi-feedforward motor command that is learned using training signals
generated by corrective movements. For each target, the network selects
and sets the output of a subset of pattern generators. During the movement, feedback from proprioceptors turns off the pattern generators. The
task facing individual pattern generators is to recognize when the arm
reaches the target and to turn off. A distributed representation of the motor command that resembles population vectors seen in vivo was produced
naturally by these simulations.
1
INTRODUCTION
We have recently begun to explore the properties of sensorimotor networks with
architectures inspired by the anatomy and physiology of the cerebellum and its interconnections with the red nucleus and the motor cortex (Houk 1989; Houk et al..
611
612
Berthier, Singh, Barto, and Houk
1990). It is widely accepted that these brain regions are important in the control
of limb movements (Kuypers, 1981; Ito, 1984), although relatively little attention
has been devoted to probing how the different regions might function together in
a cooperative manner. Starting from a foundation of known anatomical circuitry
and the results of microelectrode recordings from neurons in these circuits, we proposed the concept of rubrocerebellar and corticocerebellar information processing
modules that are arranged in parasagittal arrays and function as adjustable pattern
generators (APGs) capable of the storage, recall and execution of motor programs.
The aim of the present paper is to extend the APG Model to a multiple degreeof-freedom task and to investigate how the motor representation developed by the
model compares to the population vector representations seen by Georgopoulos
and coworkers (e.g., Georopoulos, 1988). A complete description of the model and
simulations reported here is contained in Berthier et al. (1991).
2
THE APG ARRAY MODEL
As shown in Figure 1 the model has three parts: a neural network that generates
control signals, a muscle model that controls joint angle, and a planar, kinematic
arm. The control network is an array of APGs that generate signals that are
fed to the limb musculature. Because here we are interested in the basic issue of
how a collection of APGs might cooperatively control multiple degree-of-freedom
movements, we use a very simplified model of the limb that ignores dynamics. The
muscles convert APG activity to changes in muscle length, which determine the
changes in the joint angles. Activation of an APG causes movement of the arm in
a direction in joint-angle space that is specific to that APG 1 , and the magnitude
of an APG's activity determines the velocity of that movement. The simultaneous
activation of selected APGs determines the arm trajectory as the superposition of
these movements. A learning rule, based on long-term depression (e.g., Ito, 1984),
adjusts the subsets of APGs that are selected as well as characteristics of their
activity in order to achieve desired movements .
Each APG consists of a positive feedback loop and a set of Purkinje cells (PCs).
The positive feedback loop is a highly simplified model of a component of a complex
cerebrocerebellar recurrent network. In the simplified model simulated here, each
APG has its own feedback loop, and the loops associated with different APGs do
not interact. When triggered by sufficiently strong activation, the neurons in these
loops fire repetitively in a self-sustaining manner. An APG's motor command is
generated through the action of its PCs which inhibit and modulate the buildup of
activity in the feedback loop. The activity of loop cells is conveyed to spinal motor
areas by rubrospinal fibers. PCs receive information that specifies and constrains
the desired movements via parallel fibers.
We hypothesize that the response of PCs to particular parallel fiber inputs is adaptively adjusted through the influence of climbing fibers that respond to corrective
movements (Houk & Barto, 1991). The APG array model assumes that climbing
fibers and PCs are aligned in a way that climbing fibers provide specialized inforITo simplify these initial simulations we ignore changes in muscle moment arms with
posture of the arm.
A Cortico-Cerebellar Model that Learns to Generate Distributed Motor Commands
Network
...........................,
APG Modules
?l?
1
Muscles
T
?
m
T
?
??
................................... .1
M
Figure 1: APG Control of Joint Angles. A collection of of APGs (adjustable pattern
generators) is connected to a simulated two degree-of-freedom, kinematic, planar
arm with antagonistic muscles at each joint. The task is to move the arm in the
plane from a central starting location to one of eight symmetrically placed targets.
Activation of an APG causes a movement of the arm that is specific to that APG,
and the magnitude of an APG's activity determines the velocity of that movement.
The simultaneous activation of selected APGs determines the arm trajectory as a
superposition of these movements.
mation to PCs. Gellman et al. (1985) showed that proprioceptive climbing fibers
are inhibited during planned movements, but the data of Gilbert and Thach (1977)
suggest that they fire during corrective movements. In the present simulations, we
assume that corrective movements are made when a movement fails to reach the
target. These corrective movements stimulate proprioceptive climbing fibers which
provides information to higher centers about the direction of the corrective movement. More detailed descriptions of APGs and relevant anatomy and physiology
can be found in Houk (1989), Houk et al. (1990), and Berthier et al. (1991).
The generation of motor commands occurs in three phases. In the first phase, we
assume that all positive feedback loops are off, and inputs provided by teleceptive
and proprioceptive parallel fibers and basket cells determine the outputs of the PCs.
We call this first phase selection. We assume that noise is present during the selection process so that individual PCs are turned off (Le., selected) probabilistic ally.
To begin the second phase, called the execution phase, loop activity is triggered by
cortical activity. Once triggered, loop activity is self-sustaining because the loop
cells have reciprocal positive connections. The triggering of loop activity causes the
motor command to be "read out." The states of the PCs in the selection phase
determine the speed and direction of the arm movement. As the movement is being performed, proprioceptive feedback and efference copy gradually depolarize the
PCs. When a large proportion of the PCs are depolarized, PC inhibition reaches a
critical value and terminates loop activity. In the third phase, the correction phase,
corrective movements trigger climbing fiber activity that alters parallel fiber-PC
connection weights.
613
614
Berthier, Singh, Barto, and Houk
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Figure 2: A. Movement Trajectories After Training. The starting point for each
movement is the center of the workspace, and the target location is the center of
the open square. The position of the arm at each time step is shown as a dot.
Three movements are shown to each target. B. APG selection. APG selection for
movements to a given target is illustrated by a vector plot at the position of the
target. An individual APG is represented by a vector, the direction of which is
equal to the direction of movement caused by that APG in Cartesian space. The
vector length is proportional to output of the Purkinje cells during the selection
phase. The arrow points in the direction of the vector sum.
3
SIMULATIONS
We trained the APG model to control a two degree-of-freedom, kinematic, planar
arm. The task was similar to Georgopoulos (1988) and required APGs to move the
arm from a central starting point to one of eight radially symmetric, equidistant
targets. Each simulated trial started by placing the endpoint of the arm in the central starting location. The selection, execution, and correction phases of operation
were then simulated. The task facing each of the selected APGs was to turn off at
the proper time so that the movement stopped at the target.
Simulations showed that the model could learn to control movements to the eight
targets. Training typically required about 700 trials per target until the arm endpoint was consistently moved to within 1 em of the target. Figure 2 shows sample
trajectories and population vectors of APG activity. Performance never resulted
in precise movements due to the probabilistic nature of selection. Movement trajectories tended to follow straight lines in joint-angle space and were thus slightly
curved lines in the workspace. About half of the APGs in the model were used
to move to an individual target with population vectors similar to those seen by
Georgopoulos (1988). The number of APGs used for each target was dependent
on the sharpness of the climbing fiber receptive fields, with cardioid shaped receptive fields in joint-angle space giving population vectors that most resembled those
experimentally observed.
A Carrico-Cerebellar Model that Learns to Generate Distributed Motor Commands
4
ANALYSIS
In order to understand how the model worked we undertook a theoretical analysis of
its simulated behavior. Analysis indicated that the expected trajectory of a movement was a straight line in joint-angle space from the starting position to the target.
This is a special case of a mathematical result by Mussa-Ivaldi (1988). Because selection is probabilistic in the APG Array Model, trajectories in the workspace varied
from the expected trajectory. In these cases, trajectories were piecewise linear because of the asynchronous termination of APG activity. Because of the Law of
Large Numbers, the more PCs in each APG, the more closely the movement will
resemble the expected movement.
The expected population of vectors of APG activity can be shown to be cosineshaped in joint-angle space. That is, the length of the vector representing the
activity of APG m is proportional to the cosine of the angle between the direction
of action of APG m and the direction of the target in joint-angle space. The shape
of the population vectors in Cartesian space is dependent on the Jacobian of the
arm, which is a function of the arm posture.
The manner in which the outputs of PCs were set during selection leads to scaling of
movement velocity with target distance. For any given movement direction, targets
that are farther from the starting location lead to more rapid movements than closer
targets.
Updating network weights based on the expected corrective movement will, in some
cases, result in changing the weights in a way that they converge to the correct
values. However, in other cases inappropriate changes are made. In the current
simulations, we could largely avoid this problem by selecting parameter and initial
weight values so that movements were initially small in amplitude. Random initialization of the weight values sometimes led to instances from which the learning rule
could not recover.
5
DISCUSSION
In general, the present implementation of the modelled to adequate control of the
kinematic arm and mimicked the general output of nervous system seen in actual
experiments. The network implemented a spatial to temporal transformation that
transformed a target location into a time varying motor command. The model
naturally generated population vectors that were similar to those seen in vivo. Further research is needed to improve the model's robustness and to extend it to more
realistic control of a dynamical limb.
In the APG array model, APGs control arm movement in parallel so that the activity of all the modules taken together forms a distributed representation. The APG
array executes a distributed motor program because it produces a spatiotemporal
pattern of activity in the cerebrocerebellar recurrent network that is transmitted to
the spinal cord to comprise a distributed motor command.
615
616
Berthier, Singh, Barto, and Houk
5.1
PARAMETRIZED MOTOR PROGRAMS
Certain features of the APG array model relate well to the ideas about parameterized motor programs discussed by Keele (1973), Schmidt (1988), and Adams (1971,
1977). The selection phase of the APG array model provides a feasible neuronal
mechanism for preparing a parameterized motor program in advance of movement.
The execution phase is also consistent with the open-loop ideas associated with
motor programming concepts, except that, like Adams (1977), we explain the termination of the execution phase as being a consequence of proprioceptive feedback
and efference copy.
In the APG array model, the counterpart of a generalized motor program is a
set of parallel fiber weights for proprioceptive, efference copy, and target inputs.
Given these weights, a particular constellation of parallel fiber inputs signifies that
the desired endpoint of a movement is about to be reached, causing PCs to become
depolarized. Once a set of parallel fiber weights corresponding to a desired endpoint
is learned, the neuronal architecture and neurodynamics of the cerebellar network
functions in a manner that parameterizes the motor program.
Movement velocity is parameterized in the selection phase of the model's operation.
The velocity that is selected is automatically scaled so that velocity increases as the
amplitude of the movement increases. While this type of scaling is often observed
in motor performance studies, velocity can also be varied in an independent manner where velocity scaling can be applied simultaneously to all elements of a motor
program to slow down or speed up the entire movement. Although we have not addressed this issue in the present report, simulation of velocity scaling under control
of a neuromodulator can naturally be accomplished in the APG array model.
Movements terminate when the endpoint is recognized by PCs so that movement
duration is dependent on the course of the movement instead of being determined by
some internal clock because. Movement amplitude is parameterized by the weights
of the target inputs, with smaller weights corresponding to larger amplitude movements.
5.2
CORRECTIVE MOVEMENTS
We assume that the training information conveyed to the APGs is the result of crude
corrective movements stimulating proprioceptive receptors. This sensory information is conveyed to the cerebellum by climbing fibers. Learning in the APG array
model therefore requires the existence of a low-level system capable of generating
movements to spatial targets with at least a ballpark level of accuracy. Lesion (Yu
et al., 1980) and developmental studies (von Hofsten, 1982) support the existence
of a low-level system. Other evidence indicates that when limb movements are not
proceeding accurately toward their intended targets, corrective components of the
movements are generated by an unconscious, automatic control system (Goodale et
aI., 1986).
We assume that collaterals from the corticospinal and rubrospinal system that convey the motor commands to the spinal cord gate off sensory transmission through
the proprioceptive climbing fiber pathway, thus preventing sensory responses to the
initial limb movement. As the initial movement proceeds, the low-level system re-
A Corrico-Cerebellar Model that Learns to Generate Distributed Motor Commands
ceives proprioceptive feedback from the limb and feedforward information about
target location from the gaze control system. The latter information is updated as
a consequence of corrective eye movements that typically occur after an initial gaze
shift toward a visual target. Updated gaze information causes the spinal processor to generate a corrective component that is superimposed on the original motor
command (Gielen & van Gisbergen, 1990; Flash & Henis, 1991). Since climbing
fiber pathways would not be gated off by this low-level corrective process, climbing
fibers should fire to indicate the direction of the corrective movement.
We assume that the network by which climbing fiber activity is generated is specifically wired to provide appropriate training information to the APGs (Houk &
Barto, 1991). The training signal provided by a climbing fiber is specialized for the
recipient APG in that it provides directional information in joint-angle space that is
relative to the direction in which that APG moves the arm. The fact that training
information is provided in terms of joint-angle space greatly simplifies the problem
of providing errors in the correct system of reference. For example, if the network
used visual error information, the error information would have to be transformed
to joint errors.
The specialized training signals provided by the climbing fibers are determined by
the structure of the ascending network conveying proprioceptive information. This
ascending network has the same structure-but works in the opposite direction-as
the network by which the APG array influences joint movement. This is reminiscent of the error backpropagation algorithm (e.g., Rumelhart et al., 1986, Parker,
1985) where the forward and backward passes through the network in the backpropagation algorithm are accomplished by the descending and ascending networks
of the APG Array Model. This use of the ascending network to transform errors in
the workspace to errors that are relative to a particular APG's direction of action
is closely related to the use of error backpropagation for "learning with a distal
teacher" as suggested by Jordan and Rumelhart (1991).
Houk and Barto (1991) suggested that the alignment of the ascending and descending networks might come about through trophic mechanisms stimulated by
use-dependent alterations in synaptic efficacy. In the context of the present model,
this hypothesis implies that the ascending network to the inferior olive, is established first, and that the descending network by which APGs influence motoneurons
changes. We have not yet simulated this mechanism to see if it could actually generate the kind of alignment we assume in the present model.
Acknowledgements
This research was supported by ONR N00014-88-K-0339, NIMH Center Grant P50
MH48185, and a grant from the McDonnell-Pew Foundation for Cognitive Neuroscience supported by the James S. McDonnell Foundation and the Pew Charitable
Trusts.
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4,772 | 5,320 | On Sparse Gaussian Chain Graph Models
Seyoung Kim
Lane Center for Computational Biology
Carnegie Mellon University
[email protected]
Calvin McCarter
Machine Learning Department
Carnegie Mellon University
[email protected]
Abstract
In this paper, we address the problem of learning the structure of Gaussian chain
graph models in a high-dimensional space. Chain graph models are generalizations of undirected and directed graphical models that contain a mixed set of directed and undirected edges. While the problem of sparse structure learning has
been studied extensively for Gaussian graphical models and more recently for
conditional Gaussian graphical models (CGGMs), there has been little previous
work on the structure recovery of Gaussian chain graph models. We consider linear regression models and a re-parameterization of the linear regression models
using CGGMs as building blocks of chain graph models. We argue that when the
goal is to recover model structures, there are many advantages of using CGGMs
as chain component models over linear regression models, including convexity of
the optimization problem, computational efficiency, recovery of structured sparsity, and ability to leverage the model structure for semi-supervised learning. We
demonstrate our approach on simulated and genomic datasets.
1
Introduction
Probabilistic graphical models have been extensively studied as a powerful tool for modeling a set
of conditional independencies in a probability distribution [12]. In this paper, we are concerned with
a class of graphical models, called chain graph models, that has been proposed as a generalization of
undirected graphical models and directed acyclic graphical models [4, 9, 14]. Chain graph models
are defined over chain graphs that contain a mixed set of directed and undirected edges but no
partially directed cycles.
In particular, we study the problem of learning the structure of Gaussian chain graph models in a
high-dimensional setting. While the problem of learning sparse structures from high-dimensional
data has been studied extensively for other related models such as Gaussian graphical models
(GGMs) [8] and more recently conditional Gaussian graphical models (CGGMs) [17, 20], to our
knowledge, there is little previous work that addresses this problem for Gaussian chain graph models. Even with a known chain graph structure, current methods for parameter estimation are hindered
by the presence of multiple locally optimal solutions [1, 7, 21].
Since the seminal work on conditional random fields (CRFs) [13], a general recipe for constructing
chain graph models [12] has been given as using CRFs as building blocks for the model. We employ
this construction for Gaussian chain graph models and propose to use the recently-introduced sparse
CGGMs [17, 20] as a Gaussian equivalent of general CRFs. When the goal is to learn the model
structure, we show that this construction is superior to the popular alternative approach of using
linear regression as component models. Some of the key advantages of our approach are due to the
fact that the sparse Gaussian chain graph models inherit the desirable properties of sparse CGGM
such as convexity of the optimization problem and structured output prediction. In fact, our work is
the first to introduce a joint estimation procedure for both the graph structure and parameters as a
convex optimization problem, given the groups of variables for chain components. Another advan1
xj
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xj
3
xj
2
xj
3 ..2
. xj
xj
3 - xj
2
...
..
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xj
1
xj
3
xj
2
xj
3 ..2
. xj
.....
..
.
.
.
. R1
xj
xj
xj
xj
xj
xj
1
1
4
1
4 . - xj
4 - xj
1
(a)
(b)
(c)
(d)
(e)
(f)
Figure 1: Illustration of chain graph models. (a) A chain graph with two components, {x1 , x2 } and
{x3 }. (b) The moralized graph of the chain graph in (a). (c) After inference in the chain graph
in (a), inferred indirect dependencies are shown as the dotted line. (d) A chain graph with three
components, {x1 , x2 }, {x3 }, and {x4 }. (e) The moralized graph of the chain graph in (d). (f) After
inference in the chain graph in (d), inferred indirect dependencies are shown as the dotted lines.
tage of our approach is the ability to model a functional mapping from multiple related variables to
other multiple related variables in a more natural way via moralization in chain graphs than other
approaches that rely on complex penalty functions for inducing structured sparsity [11, 15].
Our work on sparse Gaussian chain graphs is motivated by problems in integrative genomic data
analyses [6, 18]. While sparse GGMs have been extremely popular for learning networks from
datasets of single modality such as gene-expression levels [8], we propose that sparse Gaussian chain
graph models with CGGM components can be used to learn a cascade of networks by integrating
multiple types of genomic data in a single statistical analysis. We show that our approach can
reveal the module structures as well as the functional mapping between modules in different types
of genomic data effectively. Furthermore, as the cost of collecting each data type differs, we show
that semi-supervised learning can be used to make effective use of both fully-observed and partiallyobserved data.
2
Sparse Gaussian Chain Graph Models
We consider a chain graph model for a probability distribution over J random variables x =
{x1 , . . . , xJ }. The chain graph model assumes that the random variables are partitioned into C
chain components {x1 , . . . , xC }, the ? th component having size |? |. In addition, it assumes a partially directed graph structure, where edges between variables within each chain component are
undirected and edges across two chain components are directed. Given this chain graph structure,
the joint probability distribution factorizes as follows:
p(x) =
C
Y
p(x? |xpa(? ) ),
? =1
where xpa(? ) is the set of variables that are parents of one or more variables in x? . Each factor
p(x? |xpa(? ) ) models the conditional distribution of the chain component variables x? given xpa(? ) .
This model can also be viewed as being constructed with CRFs for p(x? |xpa(? ) )?s [13].
The conditional independence properties of undirected and directed graphical models have been
extended to chain graph models [9, 14]. This can be easily seen by first constructing a moralized
graph, where undirected edges are added between any pairs of nodes in xpa(? ) for each chain component ? and all the directed edges are converted into undirected edges (Figure 1). Then, subsets of
variables xa and xb are conditionally independent given xc , if xa and xb are separated by xc in the
moralized graph. This conditional independence criterion for a chain graph is called c-separation
and generalizes d-separation for Bayesian networks [12].
In this paper, we focus on Gaussian chain graph models, where both p(x) and p(x? |xpa(? ) )?s are
Gaussian distributed. Below, we review linear regression models and CGGMs as chain component
models, and introduce our approach for learning chain graph model structures.
2.1
Sparse Linear Regression as Chain Component Model
As the specific functional form of p(x? |xpa(? ) ) in Gaussian chain graphs models, a linear regression
model with multivariate responses has been widely considered [2, 3, 7]:
p(x? |xpa(? ) ) = N (B? xpa(? ) , ??1
? ),
|? |?|pa(? )|
(1)
where B? ? R
is the matrix of regression coefficients and ?? is the |? | ? |? | inverse
covariance matrix that models correlated noise. Then, the non-zero elements in B? indicate the
2
presence of directed edges from xpa(? ) to x? , and the non-zero elements in ?? correspond to the
undirected edges among the variables in x? . When the graph structure is known, an iterative procedure has been proposed to estimate the model parameters, but it converges only to one of many
locally-optimal solutions [7].
When the chain component model has the form of Eq. (1), in order to jointly estimate the sparse
graph structure and the parameters, we adopt sparse multivariate regression with covariance estimation (MRCE) [16] for each chain component and solve the following optimization problem:
min
C
X
tr((X? ?Xpa(? ) BT? )?? (X? ?Xpa(? ) BT? )T )?N log |?? | +?
? =1
C
X
||B? ||1 + ?
? =1
C
X
||?? ||1 ,
? =1
where X? ? RN ?|?| is a dataset for N samples, || ? ||1 is the sparsity-inducing L1 penalty, and ?
and ? are the regularization parameters that control the amount of sparsity in the parameters. As in
MRCE [16], the problem above is not convex, but only bi-convex.
2.2
Sparse Conditional Gaussian Graphical Model as Chain Component Model
As an alternative model for p(x? |xpa(? ) ) in Gaussian chain graph models, a re-parameterization of
the linear regression model in Eq. (1) with natural parameters has been considered [14]. This model
also has been called a CGGM [17] or Gaussian CRF [20] due to its equivalence to a CRF. A CGGM
for p(x? |xpa(? ) ) takes the standard form of undirected graphical models as a log-linear model:
1
p(x? |xpa(? ) ) = exp ? xT? ?? x? ? xT? ??,pa(? ) xpa(? ) /A(xpa(? ) ),
(2)
2
where ?? ? R|? |?|? | and ??,pa(? ) ? R|? |?|pa(? )| are the parameters for the feature weights between
pairs of variables within x? and between pairs of variables across x? and xpa(? ) , respectively, and
A(xpa(? ) ) is the normalization constant. The non-zero elements of ?? and ??,pa(? ) indicate edges
among the variables in x? and between x? and xpa(? ) , respectively.
The linear regression model in Eq. (1) can be viewed as the result of performing inference in the
probabilistic graphical model given by the CGGM in Eq. (2). This relationship between the two
models can be seen by re-writing Eq. (2) in the form of a Gaussian distribution:
?1
p(x? |xpa(? ) ) = N (???1
? ??,pa(? ) xpa(? ) , ?? ),
where marginalization in a CGGM involves computing B? xpa(? ) =
a linear regression model parameterized by B? .
???1
? ??,pa(? ) xpa(? )
(3)
to obtain
In order to estimate the graph structure and parameters for Gaussian chain graph models with CGGMs as chain component models, we adopt the procedure for learning a sparse CGGM [17, 20] and
minimize the negative log-likelihood of data along with sparsity-inducing L1 penalty:
min ?L(X; ?) + ?
C
X
||??,pa(? ) ||1 + ?
? =1
C
X
||?? ||1 ,
? =1
where ? = {?? , ??,pa(? ) , ? = 1, . . . , C} and L(X; ?) is the data log-likelihood for dataset X ?
RN ?J for N samples. Unlike MRCE, the optimization problem for a sparse CGGM is convex,
and efficient algorithms have been developed to find the globally-optimal solution with substantially
lower computation time than that for MRCE [17, 20].
While maximum likelihood estimation leads to the equivalent parameter estimates for CGGMs and
linear regression models via the transformation B? = ???1
? ??,pa(? ) , imposing a sparsity constraint on each model leads to different estimates for the sparsity pattern of the parameters and the
model structure [17]. The graph structure of a sparse CGGM directly encodes the probabilistic dependencies among the variables, whereas the sparsity pattern of B? = ???1
? ??,pa(? ) obtained after
marginalization can be interpreted as indirect influence of covariates xpa(? ) on responses x? . As illustrated in Figures 1(c) and 1(f), the CGGM parameters ??,pa(? ) (directed edges with solid line)
can be interpreted as direct dependencies between pairs of variables across x? and xpa(? ) , whereas
B? = ???1
? ??,pa(? ) obtained from inference can be viewed as indirect and inferred dependencies
(directed edges with dotted line).
3
We argue in this paper that when the goal is to learn the model structure, performing the estimation
with CGGMs for chain component models can lead to a more meaningful representation of the
underlying structure in data than imposing a sparsity constraint on linear regresssion models. Then
the corresponding linear regression model can be inferred via marginalization. This approach also
inherits many of the advantages of sparse CGGMs such as convexity of optimization problem.
2.3
Markov Properties and Chain Component Models
When a CGGM is used as the component model, the overall chain graph model is known to have
Lauritzen-Wermuth-Frydenberg (LWF) Markov properties [9]. The LWF Markov properties also
correspond to the standard probabilistic independencies in more general chain graphs constructed
by using CRFs as building blocks [12].
Many previous works have noted that LWF Markov properties do not hold for the chain graph models with linear regression models [2, 3]. The alternative Markov properties (AMP) were therefore
introduced as the set of probabilistic independencies associated with chain graph models with linear
regression component models [2, 3]. It has been shown that the LWF and AMP Markov properties are equivalent only for chain graph structures that do not contain the graph in Figure 1(a) as a
subgraph [2, 3]. For example, according to the LWF Markov property, in the chain graph model in
Figure 1(a), x1 ? x3 |x2 as x1 and x3 are separated by x2 in the moralized graph in Figure 1(b).
However, the corresponding AMP Markov property implies a different probabilistic independence
relationship, x1 ? x3 . In the model in Figure 1(d), according to the LWF Markov property, we have
x1 ? x3 |{x2 , x4 }, whereas the AMP Markov property gives x1 ? x3 |x4 .
We observe that when using sparse CGGMs as chain component models, we estimate a model with
the LWF Markov properties and perform marginalization in this model to obtain a model with linearregression chain components that can be interpreted with the AMP Markov properties.
3
Sparse Two-Layer Gaussian Chain Graph Models for Structured Sparsity
Another advantage of using CGGMs as chain component models instead of linear regression is
that the moralized graph, which is used to define the LWF Markov properties, can be leveraged to
discover the underlying structure in a correlated functional mapping from multiple inputs to multiple
outputs. In this section, we show that a sparse two-layer Gaussian chain graph model with CGGM
components can be used to learn structured sparsity. The key idea behind our approach is that
while inference in CGGMs within the chain graph model can reveal the shared sparsity patterns for
multiple related outputs, a moralization of the chain graph can reveal those for multiple inputs.
Statistical methods for learning models with structured sparsity were extensively studied in the literature of multi-task learning, where the goal is to find input features that influence multiple related
outputs simultaneously [5, 11, 15]. Most of the previous works assumed the output structure to be
known a priori. Then, they constructed complex penalty functions that leverage this known output structure, in order to induce structured sparsity pattern in the estimated parameters in linear
regression models. In contrast, a sparse CGGM was proposed as an approach for performing a joint
estimation of the output structure and structured sparsity for multi-task learning. As was discussed
in Section 2.2, once the CGGM structure is estimated, the inputs relevant for multiple related outputs
could be revealed via probabilistic inference in the graphical model.
While sparse CGGMs focused on leveraging the output structure for improved predictions, another
aspect of learning structured sparsity is to consider the input structure to discover multiple related
inputs jointly influencing an output. As CGGM is a discriminative model that does not model the
input distribution, it is unable to capture input relatedness directly, although discriminative models
in general are known to improve prediction accuracy. We address this limitation of CGGMs by
embedding CGGMs within a chain graph and examining the moralized graph.
We set up a two-layer Gaussian chain graph model for inputs x and outputs y as follows:
1
1
p(y, x) = p(y|x)p(x) = exp(? yT ?yy y ? xT ?xy y)/A1 (x)
exp(? xT ?xx x)/A2 ,
2
2
where a CGGM is used for p(y|x) and a GGM for p(x), and A1 (x) and A2 are normalization constants. As the full model factorizes into two factors p(y|x) and p(x) with distinct sets of parameters,
4
a sparse graph structure and parameters can be learned by using the optimization methods for sparse
CGGM [20] and sparse GGM [8, 10].
The estimated Gaussian chain graph model leads to a GGM over both the inputs and outputs, which
reveals the structure of the moralized graph:
?1 !
?yy
?Txy
p(y, x) = N 0,
.
T
?xy ?xx + ?xy ??1
yy ?xy
In the above GGM, we notice that the graph structure over inputs x consists of two components,
one for ?xx describing the conditional dependencies within the input variables and another for
T
?xy ??1
yy ?xy that reflects the results of moralization in the chain graph. If the graph ?yy contains
T
connected components, the operation ?xy ??1
yy ?xy for moralization induces edges among those
inputs influencing the outputs in each connected component.
Our approach is illustrated in Figure 2.
yl
yl
yl
yl
yl
yl
yl
yl
yl
yl
1
2
3
4
5
1
2
3
4
5
Given the model in Figure 2(a), Figure
.
.
. ... . . . . .1...
.
.. 3
Y
..
..
..3
.
.
.
.
. .A
. .
2(b) illustrates the inferred structured
I
K.A....
AKA
.. .
.... .
.. ..
.
sparsity for a functional mapping from
xl
xl
xl
xl
xl
xl
xl
xl
xl
xl
xl
xl
1
2
3
4
5
6
1
2
3
4
5
6
multiple inputs to multiple outputs. In
(a)
(b)
Figure 2(b), the dotted edges correspond
to inferred indirect dependencies intro- Figure 2: Illustration of sparse two-layer Gaussian chain
duced via marginalization in the CGGM graphs with CGGMs. (a) A two-layer Gaussian chain
p(y|x), which reveals how each input graph. (b) The results of performing inference and moralis influencing multiple related outputs. ization in (a). The dotted edges correspond to indirect deOn the other hand, the additional edges pendencies inferred by inference. The edges among xj ?s
among xj ?s have been introduced by represent the dependencies introduced by moralization.
T
moralization ?xy ??1
yy ?xy for multiple inputs jointly influencing each output. Combining the results of marginalization and moralization, the two connected components in Figure 2(b) represent
the functional mapping from {x1 , x2 } to {y1 , y2 } and from {x3 , x4 , x5 } to {y3 , y4 , y5 }, respectively.
4
Sparse Multi-layer Gaussian Chain Graph Models
In this section, we extend the two-layer Gaussian chain graph model from the previous section into
a multi-layer model to model data that are naturally organized into multiple layers. Our approach is
motivated by problems in integrative genomic data analysis. In order to study the genetic architecture of complex diseases, data are often collected for multiple data types, such as genotypes, gene
expressions, and phenotypes for a population of individuals [6, 18]. The primary goal of such studies
is to identify the genotype features that influence gene expressions, which in turn influence phenotypes. In such problems, data can be naturally organized into multiple layers, where the influence of
features in each layer propagates to the next layer in sequence. In addition, it is well-known that the
expressions of genes within the same functional module are correlated and influenced by the common genotype features and that the coordinated expressions of gene modules affect multiple related
phenotypes jointly. These underlying structures in the genomic data can be potentially revealed by
inference and moralization in sparse Gaussian chain graph models with CGGM components.
In addition, we explore the use of semi-supervised learning, where the top and bottom layer data
are fully observed but the middle-layer data are collected only for a subset of samples. In our
application, genotype data and phenotype data are relatively easy to collect from patients? blood
samples and from observations. However, gene-expression data collection is more challenging, as
invasive procedure such as surgery or biopsy is required to obtain tissue samples.
4.1
Models
Given variables, x = {x1 , . . . , xJ }, y = {y1 , . . . , yK }, and z = {z1 , . . . , zL }, at each of the three
layers, we set up a three-layer Gaussian chain graph model as follows:
p(z, y|x) = p(z|y)p(y|x)
1 T
1 T
T
T
= exp(? z ?zz z ? y ?yz z)/C2 (y) exp(? y ?yy y ? x ?xy y)/C1 (x) , (4)
2
2
5
where C1 (x) and C2 (y) are the normalization constants. In our application, x, y, and z correspond
to genotypes, gene-expression levels, and phenotypes, respectively. As the focus of such studies
lies on discovering how the genotypic variability influences gene expressions and phenotypes rather
than the structure in genotype features, we do not model p(x) directly.
Given the estimated sparse model for Eq. (4), structured sparsity pattern can be recovered via
T
?1 T
inference and moralization. Computing Bxy = ???1
yy ?xy and Byz = ??zz ?yz corresponds
to performing inference to reveal how multiple related yk ?s in ?yy (or zl ?s in ?zz ) are jointly
influenced by a common set of relevant xj ?s (or yk ?s). On the other hand, the effects of moralization
can be seen from the joint distribution p(z, y|x) derived from Eq. (4):
T
?1
p(z, y|x) = N (???1
(zz,yy) ?(yz,xy) x, ?(zz,yy) ),
?zz
?Tyz
where ?(yz,xy) = (0J?L , ?xy ) and ?(zz,yy) =
. ?(zz,yy) corT
?yz ?yy + ?yz ??1
zz ?yz
responds to the undirected graphical model over z and y conditional on x after moralization.
4.2
Semi-supervised Learning
Given a dataset D = {Do , Dh }, where Do = {Xo , Yo , Zo } for the fully-observed data and Dh =
{Xh , Zh } for the samples with missing gene-expression levels, for semi-supervised learning, we
adopt an EM algorithm that iteratively maximizes the expected log-likelihood of complete data:
L(Do ; ?) + E L(Dh , Yh ; ?) ,
combined with L1 -regularization, where L(Do ; ?) is the data log-likelihood with respect to the
model in Eq. (4) and the expectation is taken with respect to:
p(y|z, x) = N (?y|x,z , ?y|x,z ),
T ?1
.
?y|x,z = ??y|x,z (?yz z + ?Txy x) and ?y|x,z = (?yy + ?yz ??1
zz ?yz )
5
Results
In this section, we empirically demonstrate that CGGMs are more effective components for sparse
Gaussian chain graph models than linear regression for various tasks, using synthetic and real-world
genomic datasets. We used the sparse three-layer structure for p(z, y|x) in all our experiments.
5.1
Simulation Study
In simulation study, we considered two scenarios for true models, CGGM-based and linearregression-based Gaussian chain graph models. We evaluated the performance in terms of graph
structure recovery and prediction accuracy in both supervised and semi-supervised settings.
In order to simulate data, we assumed the problem size of J=500, K=100, and L=50 for x, y, and
z, respectively, and generated samples from known true models. Since we do not model p(x), we
used an arbitrary choice of multinomial distribution to generate samples for x. The true parameters
for CGGM-based simulation were set as follows. We set the graph structure in ?yy to a randomlygenerated scale-free network with a community structure [19] with six communities. The edge
weights were drawn randomly from a uniform distribution [0.8, 1.2]. We then set ?yy to the graph
Laplacian of this network plus small positive values along the diagonal so that ?yy is positive
definite. We generated ?zz using a similar strategy, assuming four communities. ?xy was set to
a sparse random matrix, where 0.4% of the elements have non-zero values drawn from a uniform
distribution [-1.2,-0.8]. ?yz was generated using a similar strategy, with a sparsity level of 0.5%. We
set the sparsity pattern of ?yz so that it roughly respects the functional mapping from communities
in y to communities in z. Specifically, after reordering the variables in y and z by performing
hierarchical clustering on each of the two networks ?yy and ?zz , the non-zero elements were
selected randomly around the diagonal of ?yz .
We set the true parameters for the linear-regression-based models using the same strategy as the
CGGM-based simulation above for ?yy and ?zz . We set Bxy so that 50% of the variables in x
have non-zero influence on five randomly chosen variables in y in one randomly chosen community
in ?yy . We set Byz in a similar manner, assuming 80% of the variables in y are relevant to eight
randomly-chosen variables in z from a randomly-chosen community in ?zz .
6
1
0.8
0.6
0.4
CG?semi
CG
LR?semi
LR
0.5
Recall
0.2
0
0
0.6
0.4
0.2
1
0.6
0.4
0.2
0
0
0.5
Recall
(a)
1
Precision
1
0.8
Precision
1
0.8
Precision
1
0.8
Precision
Precision
1
0.8
0.6
0.4
0.2
0
0
0.5
Recall
(b)
1
0.6
0.4
0.2
0
0
0.5
Recall
(c)
1
0
0
(d)
0.5
Recall
1
(e)
Figure 4: Precision/recall curves for graph structure recovery in CGGM-based simulation study. (a)
?yy , (b) ?zz , (c) Bxy , (d) Byz , and (e) ?xy . (CG: CGGM-based models with supervised learning,
CG-semi: CG with semi-supervised learning, LR: linear-regression-based models with supervised
learning, LR-semi: LR with semi-supervised learning.)
1.2
5
3
2
0.6
1
0.8
0.4
1
LR
0.8
0.6
0.4
0.2
CG?semi CG LR?semi
test err
0.6
test err
0.8
1.2
1
4
test err
test err
1
CG?semi CG LR?semi
CG?semi CG LR?semi
LR
0.4
CG?semi CG LR?semi
LR
LR
(a)
(b)
(c)
(d)
Figure 5: Prediction errors in CGGM-based simulation study. The same estimated models in Figure
4 were used to predict (a) y given x, z, (b) z given x, (c) y given x, and (d) z given y.
0.4
0.2
0
0
1
1
0.8
0.8
0.6
0.4
0.2
0.5
Recall
1
0
0
Precision
Precision
0.6
1
0.8
Precision
CG?semi
CG
LR?semi
LR
Precision
1
0.8
0.6
0.4
0.2
0.5
Recall
0
0
1
0.6
0.4
0.2
0.5
Recall
1
0
0
0.5
Recall
1
(a)
(b)
(c)
(d)
Figure 6: Performance for graph structure recovery in linear-regression-based simulation study.
Precision/recall curves are shown for (a) ?yy , (b) ?zz , (c) Bxy , and (d) Byz .
Each dataset consisted of 600 samples, of which 400
and 200 samples were used as training and test sets.
To select the regularization parameters, we estimated
a model using 300 samples, evaluated prediction errors on the other 100 samples in the training set, and
selected the values with the lowest prediction errors.
We used the optimization methods in [20] for CGGMbased models and the MRCE procedure [16] for linearregression-based models.
(c)
(a)
(b)
(d)
(e)
Figure 3 illustrates how the model with CGGM chain Figure 3: Illustration of the structured sparcomponents can be used to discover the structured sity recovered by the model with CGGM
sparsity via inference and moralization. In each panel, components, simulated dataset. (a) ?zz .
T
black and bright pixels correspond to zero and non- (b) Byz = ???1
zz ?yz shows the effects of
zero values, respectively. While Figure 3(a) shows marginalization (white vertical bars). The
how variables in z are related in ?zz , Figure 3(b) effects of moralization are shown in (c)
T
?1 T
shows Byz = ???1
zz ?yz obtained via marginaliza- ?yy + ?yz ?zz ?yz , and its decomposiT
tion within the CGGM p(z|y), where functional map- tion into (d) ?yy and (e) ?yz ??1
zz ?yz .
pings from variables in y to multiple related variables
in z can be seen as white vertical bars. In Figure 3(c), the effects of moralization ?yy +
T
?1 T
?yz ??1
zz ?yz are shown, which further decomposes into ?yy (Figure 3(d)) and ?yz ?zz ?yz
(Figure 3(e)). The additional edges among variables in y in Figure 3(e) correspond to the edges
introduced via moralization and show the groupings of the variables y as the block structure along
the diagonal. By examining Figures 3(b) and 3(e), we can infer a functional mapping from modules
in y to modules in z.
In order to systematically compare the performance of the two types of models, we examined the
average performance over 30 randomly-generated datasets. We considered both supervised and
semi-supervised settings. Assuming that 200 samples out of the total 400 training samples were
7
30
40
1
0.5
CG?semi CG LR?semi
30
20
10
LR
CG?semi CG LR?semi
test err
1.5
2
test err
test err
test err
2
1.5
1
20
10
0.5
LR
CG?semi CG LR?semi
LR
0
CG?semi CG LR?semi
LR
(a)
(b)
(c)
(d)
Figure 7: Prediction errors in linear-regression-based simulation study. The same estimated models
in Figure 6 were used to predict (a) y given x, z, (b) z given x, (c) y given x, and (d) z given y.
missing data for y, for supervised learning, we used only those samples with complete data; for
semi-supervised learning, we used all samples, including partially-observed cases.
The precision/recall curves for recovering the true graph structures are shown in Figure 4, using
datasets simulated from the true models with CGGM components. Each curve was obtained as an
average over 30 different datasets. We observe that in both supervised and semi-supervised settings, the models with CGGM components outperform the ones with linear regression components.
In addition, the performance of the CGGM-based models improves significantly, when using the
partially-observed data in addition to the fully-observed samples (the curve for CG-semi in Figure 4), compared to using only the fully-observed samples (the curve for CG in Figure 4). This
improvement from using partially-observed data is substantially smaller for the linear-regressionbased models. The average prediction errors from the same set of estimated models in Figure 4 are
shown in Figure 5. The CGGM-based models outperform in all prediction tasks, because they can
leverage the underlying structure in the data and estimate models more effectively.
For the simulation scenario using the linear-regression-based true models, we show the results for
precision/recall curves and prediction errors in Figures 6 and 7, respectively. We find that even
though the data were generated from chain graph models with linear regression components, the
CGGM-based methods perform as well as or better than the other models.
5.2
Integrative Genomic Data Analysis
We applied the two types of three-layer chain graph Table 1: Prediction errors, mouse diabetes data
models to single-nucleotide-polymorphism (SNP),
gene-expression, and phenotype data from the pancreTask CG-semi CG LR-semi LR
atic islets study for diabetic mice [18]. We selected
y | x, z 0.9070 0.9996 1.0958 0.9671
200 islet gene-expression traits after performing hierz | x 1.0661 1.0585 1.0505 1.0614
archical clustering to find several gene modules. Our
y | x 0.8989 0.9382 0.9332 0.9103
dataset also included 1000 SNPs and 100 pancreatic
z | y 1.0712 1.0861 1.1095 1.0765
islet cell phenotypes. Of the total 506 samples, we
used 406 as training set, of which 100 were held out as a validation set to select regularization
parameters, and used the remaining 100 samples as test set to evaluate prediction accuracies. We
considered both supervised and semi-supervised settings, assuming gene expressions are missing
for 150 mice. In supervised learning, only those samples without missing gene expressions were
used.
As can be seen from the prediction errors in Table 1, the models with CGGM chain components are
more accurate in various prediction tasks. In addition, the CGGM-based models can more effectively
leverage the samples with partially-observed data than linear-regression-based models.
6
Conclusions
In this paper, we addressed the problem of learning the structure of Gaussian chain graph models
in a high-dimensional space. We argued that when the goal is to recover the model structure, using
sparse CGGMs as chain component models has many advantages such as recovery of structured
sparsity, computational efficiency, globally-optimal solutions for parameter estimates, and superior
performance in semi-supervised learning.
Acknowledgements
This material is based upon work supported by an NSF CAREER Award No. MCB-1149885, Sloan
Research Fellowship, and Okawa Foundation Research Grant.
8
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4,773 | 5,321 | Provable Submodular Minimization using
Wolfe?s Algorithm
Deeparnab Chakrabarty?
Prateek Jain?
Pravesh Kothari?
Abstract
Owing to several applications in large scale learning and vision problems, fast
submodular function minimization (SFM) has become a critical problem. Theoretically, unconstrained SFM can be performed in polynomial time [10, 11]. However,
these algorithms are typically not practical. In 1976, Wolfe [21] proposed an
algorithm to find the minimum Euclidean norm point in a polytope, and in 1980,
Fujishige [3] showed how Wolfe?s algorithm can be used for SFM. For general
submodular functions, this Fujishige-Wolfe minimum norm algorithm seems to
have the best empirical performance.
Despite its good practical performance, very little is known about Wolfe?s minimum
norm algorithm theoretically. To our knowledge, the only result is an exponential
time analysis due to Wolfe [21] himself. In this paper we give a maiden convergence
analysis of Wolfe?s algorithm. We prove that in t iterations, Wolfe?s algorithm
returns an O(1/t)-approximate solution to the min-norm point on any polytope. We
also prove a robust version of Fujishige?s theorem which shows that an O(1/n2 )approximate solution to the min-norm point on the base polytope implies exact
submodular minimization. As a corollary, we get the first pseudo-polynomial time
guarantee for the Fujishige-Wolfe minimum norm algorithm for unconstrained
submodular function minimization.
1
Introduction
An integer-valued1 function f : 2X ? Z defined over subsets of some finite ground set X of n
elements is submodular if it satisfies the following diminishing marginal returns property: for every
S ? T ? X and i ? X \ T , f (S ? {i}) ? f (S) ? f (T ? {i}) ? f (T ). Submodularity arises
naturally in several applications such as image segmentation [17], sensor placement [18], etc. where
minimizing an arbitrary submodular function is an important primitive.
In submodular function minimization (SFM), we assume access to an evaluation oracle for f which
for any subset S ? X returns the value f (S). We denote the time taken by the oracle to answer a
single query as EO. The objective is to find a set T ? X satisfying f (T ) ? f (S) for every S ? X.
In 1981, Grotschel, Lovasz and Schrijver [8] demonstrated the first polynomial time algorithm for
SFM using the ellipsoid algorithm. This algorithm, however, is practically infeasible due to the
running time and the numerical issues in implementing the ellipsoid algorithm. In 2001, Schrijver [19]
and Iwata et al. [9] independently designed combinatorial polynomial time algorithms for SFM.
Currently, the best algorithm is by Iwata and Orlin [11] with a running time of O(n5 EO + n6 ).
However, from a practical stand point, none of the provably polynomial time algorithms exhibit good
performance on instances of SFM encountered in practice (see ?4). This, along with the widespread
applicability of SFM in machine learning, has inspired a large body of work on practically fast
procedures (see [1] for a survey). But most of these procedures focus either on special submodular
?
Microsoft Research, 9 Lavelle Road, Bangalore 560001.
University of Texas at Austin (Part of the work done while interning at Microsoft Research)
1
One can assume any function is integer valued after suitable scaling.
?
1
functions such as decomposable functions [16, 20] or on constrained SFM problems [13, 12, 15,
14].
Fujishige-Wolfe?s Algorithm for SFM: For any submodular function f , the base polytope Bf of f
is defined as follows:
Bf = {x ? Rn : x(A) ? f (A), ?A ? X, and x(X) = f (X)},
(1)
P
n
where x(A) := i?A xi and xi is the i-th coordinate of x ? R . Fujishige [3] showed that if one can
obtain the minimum norm point on the base polytope, then one can solve SFM. Finding the minimum
norm point, however, is a non-trivial problem; at present, to our knowledge, the only polynomial
time algorithm known is via the ellipsoid method. Wolfe [21] described an iterative procedure to find
minimum norm points in polytopes as long as linear functions could be (efficiently) minimized over
them. Although the base polytope has exponentially many constraints, a simple greedy algorithm
can minimize any linear function over it. Therefore using Wolfe?s procedure on the base polytope
coupled with Fujishige?s theorem becomes a natural approach to SFM. This was suggested as early
as 1984 in Fujishige [4] and is now called the Fujishige-Wolfe algorithm for SFM.
This approach towards SFM was revitalized in 2006 when Fujishige and Isotani [6, 7] announced
encouraging computational results regarding the minimum norm point algorithm. In particular, this
algorithm significantly out-performed all known provably polynomial time algorithms. Theoretically,
however, little is known regarding the convergence of Wolfe?s procedure except for the finite, but
exponential, running time Wolfe himself proved. Nor is the situation any better for its application
on the base polytope. Given the practical success, we believe this is an important, and intriguing,
theoretical challenge.
In this work, we make some progress towards analyzing the Fujishige-Wolfe method for SFM and, in
fact, Wolfe?s algorithm in general. In particular, we prove the following two results:
? We prove (in Theorem 4) that for any polytope B, Wolfe?s algorithm converges to an ?approximate solution, in O(1/?) steps. More precisely, in O(nQ2 /?) iterations, Wolfe?s
algorithm returns a point kxk22 ? kx? k22 + ?, where Q = maxp?B kpk2 .
? We prove (in Theorem 5) a robust version of a theorem by Fujishige [3] relating min-norm
points on the base polytope to SFM. In particular, we prove that an approximate min-norm
point solution provides an approximate solution to SFM as well. More precisely, if x
satisfies kxk22 ? z T x + ?2 for all z ? Bf , then, f (Sx ) ? minS f (S) + 2n?, where Sx can
be constructed efficiently using x.
Together, these two results gives us our main result which is a pseudopolynomial bound on the
running time of the Fujishige-Wolfe algorithm for submodular function minimization.
Theorem 1. (Main Result.) Fix a submodular function f : 2X ? Z. The FujishigeWolfe algorithm returns the minimizer of f in O((n5 EO + n7 )F 2 ) time where F :=
maxni=1 (|f ({i})|, |f ([n]) ? f ([n] \ i)|).
Our analysis suggests that the Fujishige-Wolfe?s algorithm is dependent on F and has worse dependence on n than the Iwata-Orlin [11] algorithm. To verify this, we conducted empirical study on
several standard SFM problems. However, for the considered benchmark functions, running time of
Fujishige-Wolfe?s algorithm seemed to be independent of F and exhibited better dependence on n
than the Iwata-Orlin algorithm. This is described in ?4.
2
Preliminaries: Submodular Functions and Wolfe?s Algorithm
2.1 Submodular Functions and SFM
Given a ground set X on n elements, without loss of generality we think of it as the first n integers
[n] := {1, 2, . . . , n}. f be a submodular function. Since submodularity is translation invariant,
we assume f (?) = 0. For a submodular function f , we write Bf ? Rn for the associated base
polyhedron of f defined in (1). Given x ? Rn , one can find the minimum value of q > x over
q ? Bf in O(n log n + nEO) time using the following greedy algorithm: Renumber indices such
that x1 ? ? ? ? ? xn . Set qi? = f ([i]) ? f ([i ? 1]). Then, it can be proved that q ? ? Bf and is the
minimizer of the x> q for q ? Bf .
The connection between the SFM problem and the base polytope was first established in the following
minimax theorem of Edmonds [2].
2
Theorem 2 (Edmonds [2]). Given any submodular function f with f (?) = 0, we have
!
X
min f (S) = max
xi
S?[n]
x?Bf
i:xi <0
The following theorem of Fujishige [3] shows the connection between finding the minimum norm
point in the base polytope Bf of a submodular function f and the problem of SFM on input f . This
forms the basis of Wolfe?s algorithm. In ?3.2, we prove a robust version of this theorem.
Theorem 3 (Fujishige?s Theorem [3]). Let f : 2[n] ? Z be a submodular function and let Bf be the
associated base polyhedron. Let x? be the optimal solution to minx?Bf ||x||. Define S = {i | x?i <
0}. Then, f (S) ? f (T ) for every T ? [n].
2.2
Wolfe?s Algorithm for Minimum Norm Point of a polytope.
We now present Wolfe?s algorithm for computing the minimum-norm point in an arbitrary polytope
B ? Rn . We assume a linear optimization oracle (LO) which takes input a vector x ? Rn and
outputs a vector q ? arg minp?B x> p.
We
some definitions. The affine hull of a finite set S ? Rn is aff(S) = {y | y =
P start by recalling
P
?
?
z,
?
z?S z
z?S z = 1}. The affine minimizer of S is defined as y = arg minz?aff(S) ||z||2 ,
and y satisfies the following affine minimizer property: for any v ? aff(S), v > y = ||y||2 . The
procedure AffineMinimizer(S) returns (y, ?) where y is the affine minimizer and ? = (?s )s?S is
the set of coefficients expressing y as an affine combination of points in S. This procedure can be
naively implemented in O(|S|3 + n|S|2 ) as follows. Let B be the n ? |S| matrix where each column
in a point in S. Then ? = (B > B)?1 1/1> (B > B)?1 1 and y = B?.
Algorithm 1 Wolfe?s Algorithm
P
1. Let q be an arbitrary vertex of B. Initialize x ? q. We always maintain x = i?S ?i qi as a
convex combination of a subset S of vertices of B. Initialize S = {q} and ?1 = 1.
2. WHILE(true): (MAJOR CYCLE)
(a) q := LO(x).
// Linear Optimization: q ? arg minp?B x> p.
(b) IF ||x||2 ? x> q + ?2 THEN break.
// Termination Condition. Output x.
(c) S := S ? {q}.
(d) WHILE(true): (MINOR CYCLE)
i. (y, ?) = AffineMinimizer(S).
//y = arg minz?aff(S) ||z||.
ii. IF ?i ? 0 for all i THEN break.
//If y ? conv(S), then end minor loop.
iii. ELSE
// If y ?
/ conv(S), then update x to the intersection of the boundary of conv(S) and the segment joining y and
previous x. Delete points from S which are not required to describe the new x as a convex combination.
P
? := mini:?i <0 ?i /(?i ? ?i )
// Recall, x =
i ?i qi .
Update x ? ?y + (1 ? ?)x.
// By definition of ?, the new x lies in conv(S).
Update ?i ? ??i + (1 ? ?)?i .
//This sets the coefficients of the new x
S = {i : ?i > 0}.
// Delete points which have ?i = 0. This deletes at least one point.
(e) Update x ? y.
// After the minor loop terminates, x is updated to be the affine minimizer of the current set S.
3. RETURN x.
When ? = 0, the algorithm on termination (if it terminates) returns the minimum norm point in B
since ||x||2 ? x> x? ? ||x|| ? ||x? ||. For completeness, we sketch Wolfe?s argument in [21] of finite
termination. Note that |S| ? n always; otherwise the affine minimizer is 0 which either terminates
the program or starts a minor cycle which decrements |S|. Thus, the number of minor cycles in a
major cycle ? n, and it suffices to bound the number of major cycles. Each major cycle is associated
with a set S whose affine minimizer, which is the current x, lies in the convex hull of S. Wolfe calls
such sets corrals. Next, we show that ||x|| strictly decreases across iterations (major or minor cycle)
of the algorithm, which proves that no corral repeats,
thus bounding the number of major cycles by
the number of corrals. The latter is at most N
,
where
N is the number of vertices of B.
n
Consider iteration j which starts with xj and ends with xj+1 . Let Sj be the set S at the beginning
of iteration j. If the iteration is a major cycle, then xj+1 is the affine minimizer of Sj ? {qj }
3
2
where qj = LO(xj ). Since x>
j qj < ||xj || (the algorithm doesn?t terminate in iteration j) and
>
2
xj+1 qj = ||xj+1 || (affine minimizer property), we get xj 6= xj+1 , and so ||xj+1 || < ||xj || (since
the affine minimizer is unique). If the iteration is a minor cycle, then xj+1 = ?xj + (1 ? ?)yj , where
yj is the affine minimizer of Sj and ? < 1. Since ||yj || < ||xj || (yj 6= xj since yj ?
/ conv(Sj )), we
get ||xj+1 || < ||xj ||.
3
Analysis
Our refined analysis of Wolfe?s algorithm is encapsulated in the following theorem.
Theorem 4. Let B be an arbitrary polytope such that the maximum Euclidean norm of any vertex of
B is at most Q. After O(nQ2 /?2 ) iterations, Wolfe?s algorithm returns a point x ? B which satisfies
||x||2 ? x> q + ?2 , for all points q ? B. In particular, this implies ||x||2 ? ||x? ||2 + 2?2 .
The above theorem shows that Wolfe?s algorithm converges to the minimum norm point at an 1/t-rate.
We stress that the above is for any polytope. To apply this to SFM, we prove the following robust
version of Fujishige?s theorem connecting the minimum norm point in the base polytope and the set
minimizing the submodular function value.
Theorem 5. Fix a submodular function f with base polytope Bf . Let x ? Bf be such that ||x||2 ?
x> q + ?2 for all q ? Bf . Renumber indices such that x1 ? ? ? ? ? xn . Let S = {1, 2, . . . , k},where
k is smallest index satisfying (C1) xk+1 ? 0 and (C2) xk+1 ? xk ? ?/n. Then, f (S) ? f (T ) + 2n?
1
for any subset T ? S. In particular, if ? = 4n
and f is integer-valued, then S is a minimizer.
Theorem 4 and Theorem 5 implies our main theorem.
Theorem 1. (Main Result.) Fix a submodular function f : 2X ? Z. The FujishigeWolfe algorithm returns the minimizer of f in O((n5 EO + n7 )F 2 ) time where F :=
maxni=1 (|f ({i})|, |f ([n]) ? f ([n] \ i)|).
Proof. The vertices of Bf are well understood: for every permutation ? of [n], we have a vertex
with x?(i) = f ({?(1), . . . , ?(i)}) ? f ({?(1), . . . , ?(i ? 1)}). By submodularity of f , we get
for all i, |xi | ? F . Therefore, for any point x ? Bf , ||x||2 ? nF 2 . Choose ? = 1/4n. From
Theorem 4 we know that if we run O(n4 F 2 ) iterations of Wolfe, we will get a point x ? Bf such
that ||x||2 ? x> q + ?2 for all q ? Bf . Theorem 5 implies this solves the SFM problem. The running
time for each iteration is dominated by the time for the subroutine to compute the affine minimizer of
S which is at most O(n3 ), and the linear optimization oracle. For Bf , LO(x) can be implemented in
O(n log n + nEO) time. This proves the theorem.
We prove Theorem 4 and Theorem 5 in ?3.1 and ?3.2, respectively.
3.1
Analysis of Wolfe?s Min-norm Point Algorithm
The stumbling block in the analysis of Wolfe?s algorithm is the interspersing of major and minor
cycles which oscillates the size of S preventing it from being a good measure of progress. Instead, in
our analysis, we use the norm of x as the measure of progress. Already we have seen that ||x|| strictly
decreases. It would be nice to quantify how much the decrease is, say, across one major cycle. This,
at present, is out of our reach even for major cycles which contain two or more minor cycles in them.
However, we can prove significant drop in norm in major cycles which have at most one minor cycle
in them. We call such major cycles good. The next easy, but very useful, observation is the following:
one cannot have too many bad major cycles without having too many good major cycles.
Lemma 1. In any consecutive 3n + 1 iterations, there exists at least one good major cycle.
Proof. Consider a run of r iterations where all major cycles are bad, and therefore contain ? 2
minor cycles. Say there are k major cycles and r ? k minor cycles, and so r ? k ? 2k implying
r ? 3k. Let SI be the set S at the start of these iterations and SF be the set at the end. We have
|SF | ? |SI | + k ? (r ? k) ? |SI | + 2k ? r ? n ? 3r . Therefore, r ? 3n, since |SF | ? 0.
Before proceeding, we introduce some notation.
Definition 1. Given a point x ? B, let us denote err(x) := ||x||2 ? ||x? ||2 . Given a point x and
q, let ?(x, q) := ||x||2 ? x> q and let ?(x) := maxq?B ?(x, q) = ||x||2 ? minq?B x> q. Observe
that ?(x) ? err(x)/2 since ?(x) ? ||x||2 ? x> x? ? (||x||2 ? ||x? ||2 )/2.
4
We now use t to index all good major cycles. Let xt be the point x at the beginning of the t-th
good major cycle. The next theorem shows that the norm significantly drops across good major
cycles.
Theorem 6. For t iterating over good major cycles, err(xt ) ? err(xt+1 ) ? ?2 (xt )/8Q2 .
We now complete the proof of Theorem 4 using Theorem 6.
Proof of Theorem 4. Using Theorem 6, we get that err(xt ) ? err(xt+1 ) ? err(xt )2 /32Q2 since
?(x) ? err(x)/2 for all x. We claim that in t? ? 64Q2 /?2 good major cycles, we reach xt with
err(xt? ) ? ?2 . To see this rewrite as follows:
err(xt )
, for all t.
err(xt+1 ) ? err(xt ) 1 ?
32Q2
Now let e0 := err(x0 ). Define t0 , t1 , . . . such that for all k ? 1 we have err(xt ) > e0 /2k for
t ? [tk?1 , tk ). That is, tk is thefirst time t at which err(xt ) ? e0 /2k . Note that for t ? [tk?1 , tk ),
we have err(xt+1 ) ? err(xt ) 1 ? 32Qe02 2k . This implies in 32Q2 2k /e0 time units after tk?1 , we
will have err(xt ) ? err(xtk?1 )/2; we have used the fact that (1 ? ?)1/? < 1/2 when ? < 1/32.
That is, tk ? tk?1 + 32Q2 2k /e0 . We are interested in t? = tK where 2K = e0 /?2 . We get
2
t? ? 32Q
1 + 2 + ? ? ? + 2K ? 64Q2 2K /e0 = 64Q2 /?2 .
e0
Next, we claim that in t?? < t? + t0 good major cycles, where t0 = 8Q2 /?2 , we obtain an xt??
with ?(xt?? ) ? ?2 . This is because, if not, then, using Theorem 6, in each of the good major
cycles t? + 1, t? + 2, . . . t? + t0 , err(x) falls additively by > ?4 /8Q2 and thus err(xt? +t0 ) <
err(xt? ) ? ?2 ? 0, which is a contradiction. Therefore, in O(Q2 /?2 ) good major cycles, the
algorithm obtains an x = xt?? with ?(x) ? ?2 , proving Theorem 4.
The rest of this subsection is dedicated to proving Theorem 6.
Proof of Theorem 6: We start off with a simple geometric lemma.
Lemma 2. Let S be a subset of Rn and suppose y is the minimum norm point of aff(S). Let x and
q be arbitrary points in aff(S). Then,
||x||2 ? ||y||2 ?
?(x, q)2
4Q2
(2)
where Q is an upper bound on ||x||, ||q||.
Proof. Since y is the minimum norm point in aff(S), we have x> y = q > y = ||y||2 . In particular,
||x ? y||2 = ||x||2 ? ||y||2 . Therefore,
?(x, q) = kxk2 ? xT q = kxk2 ? x> y + y > q ? xT q = (y ? x)T (q ? x) ? ky ? xk ? kq ? xk
? ky ? xk(kxk + kqk) ? 2Qky ? xk,
where the first inequality is Cauchy-Schwartz and the second is triangle inequality. Lemma now
follows by taking square of the above expression and by observing that ky ? xk2 = kxk2 ? kyk2 .
The above lemma takes case of major cycles with no minor cycles in them.
Lemma 3 (Progress in Major Cycle with no Minor Cycles). Let t be the index of a good major cycle
with no minor cycles. Then err(xt ) ? err(xt+1 ) ? ?2 (xt )/4Q2 .
Proof. Let St be the set S at start of the tth good major cycle, and let qt be the point minimizing x>
t q.
Let S = St ? qt and let y be the minimum norm point in aff(S). Since there are no minor cycles,
y ? conv(S). Abuse notation and let xt+1 = y be the iterate at the call of the next major cycle (and
not the next good major cycle). Since the norm monotonically decreases, it suffices to prove the
lemma statement for this xt+1 . Now apply Lemma 2 with x = xt and q = qt and S = St ? qt . We
have that err(xt ) ? err(xt+1 ) = ||xt ||2 ? ||y||2 ? ?(xt , qt )2 /4Q2 = ?(xt )2 /4Q2 .
Now we have to argue about major cycles with exactly one minor cycle. The next observation is a
useful structural result.
5
Lemma 4 (New Vertex Survives a Minor Cycle.). Consider any (not necessarily good) major
cycle. Let xt , St , qt be the parameters at the beginning of this cycle, and let xt+1 , St+1 , qt+1 be the
parameters at the beginning of the next major cycle. Then, qt ? St+1 .
Proof. Clearly St+1 ? St ? qt since qt is added and then maybe minor cycles remove some points
from S. Suppose qt ?
/ St+1 . Well, then St+1 ? St . But xt+1 is the affine minimizer of St+1 and xt
is the affine minimizer of St . Since St is the larger set, we get ||xt || ? ||xt+1 ||. This contradicts the
strict decrease in the norm.
Lemma 5 (Progress in an iteration with exactly one minor cyvle). Suppose the tth good major cycle
has exactly one minor cycle. Then, err(xt ) ? err(xt+1 ) ? ?(xt )2 /8Q2 .
Proof. Let xt , St , qt be the parameters at the beginning of the tth good major cycle. Let y be the
affine minimizer of St ?qt . Since there is one minor cycle, y ?
/ conv(St ?qt ). Let z = ?xt +(1??)y
be the intermediate x, that is, point in the line segment [xt , y] which lies in conv(St ? qt ). Let S 0 be
the set after the single minor cycle is run. Since there is just one minor cycle, we get xt+1 (abusing
notation once again since the next major cycle maynot be good) is the affine minimizer of S 0 .
Let A , ||xt ||2 ? ||y||2 . From Lemma 2, and using qt is the minimizer of x>
t q over all q, we have:
A = ||xt ||2 ? ||y||2 ? ?2 (xt )/4Q2
(3)
Recall, z = ?xt + (1 ? ?)y for some ? ? [0, 1]. Since y is the min-norm point of aff(St ? qt ), and
xt ? St , we get ||z||2 = ?2 ||xt ||2 + (1 ? ?2 )||y||2 . this yields:
||xt ||2 ? ||z||2 = (1 ? ?2 ) ||xt ||2 ? ||y||2 = (1 ? ?2 )A
(4)
Further, recall that S 0 is the set after the only minor cycle in the tth iteration is run and thus, from
Lemma 4, qt ? S 0 . z ? conv(S 0 ) by definition. And since there is only one minor cycle, xt+1 is the
affine minimizer of S 0 . We can apply Lemma 2 with z, qt and xt+1 , to get
?2 (z, qt )
(5)
||z||2 ? ||xt+1 ||2 ?
4Q2
Now we lower bound ?2 (z, qt ). By definition of z, we have:
>
>
2
z > qt = ?x>
t qt + (1 ? ?)y qt = ?xt qt + (1 ? ?)||y||
>
2
where the last equality follows since y qt = ||y|| (since qt ? St ? qt and y is affine minimizer of
St ? qt ). This gives
?(z, qt )
=
||z||2 ? z > qt
=
=
2
?2 ||xt ||2 + (1 ? ?2 )||y||2 ? ?x>
t qt + (1 ? ?)||y||
2
2
?(||xt ||2 ? x>
t qt ) ? ?(1 ? ?) ||xt || ? ||y||
=
? (?(xt ) ? (1 ? ?)A)
(6)
From (4),(5), and (6), we get
2
?2 (?(xt ) ? (1 ? ?)A)
(7)
4Q2
We need to show that the RHS is at least ?(xt )2 /8Q2 . Intuitively, if ? is small (close to 0), the
first term implies this using (3), and if ? is large (close to 1), then the second term implies this. The
following paragraph formalizes this intuition for any ?.
errt ? errt+1 ? (1 ? ?2 )A +
Now, if (1 ? ?2 )A > ?(xt )2 /8Q2 , we are done. Therefore, we assume (1 ? ?2 )A ? ?(xt )2 /8Q2 .
In this case, using the fact that ?(xt ) ? ||xt ||2 + ||xt ||||qt || ? 2Q2 , we get that
?(xt )
(1 ? ?)A ? (1 ? ?2 )A ? ?(xt ) ?
? ?(xt )/4
8Q2
Substituting in (7), and using (3), we get
9?2 ?(xt )2
?(xt )2
(1 ? ?2 )?(xt )2
errt ? errt+1 ?
+
?
(8)
2
2
4Q
64Q
8Q2
This completes the proof of the lemma.
Lemma 3 and Lemma 5 complete the proof of Theorem 6.
6
3.2 A Robust version of Fujishige?s Theorem
In this section we prove Theorem 5 which we restate below.
Theorem 5. Fix a submodular function f with base polytope Bf . Let x ? Bf be such that ||x||2 ?
x> q + ?2 for all q ? Bf . Renumber indices such that x1 ? ? ? ? ? xn . Let S = {1, 2, . . . , k},where
k is smallest index satisfying (C1) xk+1 ? 0 and (C2) xk+1 ? xk ? ?/n. Then, f (S) ? f (T ) + 2n?
1
for any subset T ? S. In particular, if ? = 4n
and f is integer-valued, then S is a minimizer.
Before proving the theorem, note that setting ? = 0 gives Fujishige?s theorem Theorem 3.
Proof. We claim that the following inequality holds. Below, [i] := {1, . . . , i}.
n?1
X
(xi+1 ? xi ) ? (f ([i]) ? x([i])) ? ?2
(9)
i=1
P
We prove this shortly. Let S and k be as defined in the theorem statement. Note that i?S:xi ?0 xi ?
n?, since (C2) doesn?t hold for any index i < k with xi ? 0.PFurthermore, since xk+1 ? xk ? ?/n,
we get using (9), f (S) ? x(S) ? n?. Therefore, f (S) ? i?S:xi <0 xi + 2n? which implies the
theorem due to Theorem 2.
Now we prove (9). Let z ? Bf be the point which minimizes z > x. By the Greedy algorithm
described in Section 2.1, we know that zi = f ([i]) ? f ([i ? 1]). Next, we write x in a different basis
Pn?1
as follows: x = i=1 (xi ? xi+1 )1[i] + xn 1[n] . Here 1[i] is used as the shorthand for the vector
which has 1?s in the first i coordinates and 0s everywhere else. Taking dot product with (x ? z), we
get
||x||2 ? x> z = (x ? z)> x =
n?1
X
(xi ? xi+1 ) x> 1[i] ? z > 1[i] + xn x> 1[n] ? z > 1[n] (10)
i=1
Since zi = f ([i]) ? f ([i ? 1]), we get x> 1[i] ? z > 1[i] is x([i]) ? f ([i]). Therefore the RHS of (10)
is the LHS of (9). The LHS of (10), by the assumption of the theorem, is at most ?2 implying (9).
4
Discussion and Conclusions
(a)
(b)
(c)
Figure 1: Running time comparision of Iwata-Orlin?s (IO) method [11] vs Wolfe?s method. (a):
s-t mincut function, (b) Iwata?s 3 groups function [16]. (c): Total number of iterations required by
Wolfe?s method for solving s-t mincut with increasing F
We have shown that the Fujishige-Wolfe algorithm solves SFM in O((n5 EO + n7 )F 2 ) time, where
F is the maximum change in the value of the function on addition or deletion of an element. Although
this is the first pseudopolynomial time analysis of the algorithm, we believe there is room for
improvement and hope our work triggers more interest.
Note that our anlaysis of the Fujishige-Wolfe algorithm is weaker than the best known method in
terms of time complexity (IO method by [11]) on two counts: a) dependence on n, b) dependence
on F . In contrast, we found this algorithm significantly outperforming the IO algorithm empirically
? we show two plots here. In Figure 1 (a), we run both on Erdos-Renyi graphs with p = 0.8 and
randomly chosen s, t nodes. In Figure 1 (b), we run both on the Iwata group functions [16] with 3
groups. Perhaps more interestingly, in Figure 1 (c), we ran the Fujishige-Wolfe algorithm on the
simple path graph where s, t were the end points, and changed the capacities on the edges of the
graph which changed the parameter F . As can be seen, the number of iterations of the algorithm
remains constant even for exponentially increasing F .
7
References
[1] Francis Bach. Convex analysis and optimization with submodular functions: a tutorial. CoRR,
abs/1010.4207, 2010. 1
[2] Jack Edmonds. Matroids, submodular functions and certain polyhedra. Combinatorial Structures
and Their Applications, pages 69?87, 1970. 2, 3
[3] Satoru Fujishige. Lexicographieally optimal base of a polymatroid with respect to a weight
vector. Math. Oper. Res., 5:186?196, 1980. 1, 2, 3
[4] Satoru Fujishige. Submodular systems and related topics. Math. Programming Study, 1984. 2
[5] Satoru Fujishige. Submodular functions and optimization. Elsevier, 2005.
[6] Satoru Fujishige, Takumi Hayashi, and Shigueo Isotani. The minimum-norm-point algorithm
applied to submodular function minimization and linear programming. 2006. 2
[7] Satoru Fujishige and Shigueo Isotani. A submodular function minimization algorithm based on
the minimum-norm base. Pacific Journal of Optimization, 7:3, 2011. 2
[8] Martin Gr?otschel, L?aszl?o Lov?asz, and Alexander Schrijver. The ellipsoid method and its
consequences in combinatorial optimization. Combinatorica, 1(2):169?197, 1981. 1
[9] Satoru Iwata, Lisa Fleischer, and Satoru Fujishige. A combinatorial, strongly polynomial-time
algorithm for minimizing submodular functions. In STOC, pages 97?106, 2000. 1
[10] Satoru Iwata, Lisa Fleischer, and Satoru Fujishige. A combinatorial strongly polynomial
algorithm for minimizing submodular functions. J. ACM, 48(4):761?777, 2001. 1
[11] Satoru Iwata and James B. Orlin. A simple combinatorial algorithm for submodular function
minimization. In SODA, pages 1230?1237, 2009. 1, 2, 7
[12] Rishabh Iyer, Stefanie Jegelka, and Jeff Bilmes. Curvature and optimal algorithms for learning
and minimizing submodular functions. CoRR, abs/1311.2110, 2013. 1
[13] Rishabh Iyer, Stefanie Jegelka, and Jeff Bilmes. Fast semidifferential-based submodular function
optimization. In ICML (3), pages 855?863, 2013. 1
[14] Rishabh K. Iyer and Jeff A. Bilmes. Submodular optimization with submodular cover and
submodular knapsack constraints. In NIPS, pages 2436?2444, 2013. 1
[15] Stefanie Jegelka, Francis Bach, and Suvrit Sra. Reflection methods for user-friendly submodular
optimization. In NIPS, pages 1313?1321, 2013. 1
[16] Stefanie Jegelka, Hui Lin, and Jeff A. Bilmes. On fast approximate submodular minimization.
In NIPS, pages 460?468, 2011. 1, 7
[17] Pushmeet Kohli and Philip H. S. Torr. Dynamic graph cuts and their applications in computer
vision. In Computer Vision: Detection, Recognition and Reconstruction, pages 51?108. 2010. 1
[18] Andreas Krause, Ajit Paul Singh, and Carlos Guestrin. Near-optimal sensor placements in
gaussian processes: Theory, efficient algorithms and empirical studies. Journal of Machine
Learning Research, 9:235?284, 2008. 1
[19] Alexander Schrijver. A combinatorial algorithm minimizing submodular functions in strongly
polynomial time. J. Comb. Theory, Ser. B, 80(2):346?355, 2000. 1
[20] Peter Stobbe and Andreas Krause. Efficient minimization of decomposable submodular functions. In NIPS, pages 2208?2216, 2010. 1
[21] Phillip Wolfe. Finding the nearest point in a polytope. Math. Programming, 11:128 ? 149, 1976.
1, 2, 3
8
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4,774 | 5,322 | A Differential Equation for Modeling Nesterov?s
Accelerated Gradient Method: Theory and Insights
Weijie Su1
2
Stephen Boyd2
Emmanuel J. Cand`es1,3
1
Department of Statistics, Stanford University, Stanford, CA 94305
Department of Electrical Engineering, Stanford University, Stanford, CA 94305
3
Department of Mathematics, Stanford University, Stanford, CA 94305
{wjsu, boyd, candes}@stanford.edu
Abstract
We derive a second-order ordinary differential equation (ODE), which is the limit
of Nesterov?s accelerated gradient method. This ODE exhibits approximate equivalence to Nesterov?s scheme and thus can serve as a tool for analysis. We show that
the continuous time ODE allows for a better understanding of Nesterov?s scheme.
As a byproduct, we obtain a family of schemes with similar convergence rates.
The ODE interpretation also suggests restarting Nesterov?s scheme leading to an
algorithm, which can be rigorously proven to converge at a linear rate whenever
the objective is strongly convex.
1
Introduction
As data sets and problems are ever increasing in size, accelerating first-order methods is both of
practical and theoretical interest. Perhaps the earliest first-order method for minimizing a convex
function f is the gradient method, which dates back to Euler and Lagrange. Thirty years ago, in a
seminar paper [11] Nesterov proposed an accelerated gradient method, which may take the following
form: starting with x0 and y0 = x0 , inductively define
xk = yk?1 ? s?f (yk?1 )
k?1
yk = x k +
(xk ? xk?1 ).
k+2
(1.1)
For a fixed step size s = 1/L, where L is the Lipschitz constant of ?f , this scheme exhibits the
convergence rate
Lkx ? x? k2
0
.
f (xk ) ? f ? ? O
k2
?
?
?
Above, x is any minimizer of f and f = f (x ). It is well-known that this rate is optimal among
all methods having only information about the gradient of f at consecutive iterates [12]. This is in
contrast to vanilla gradient descent methods, which can only achieve a rate of O(1/k) [17]. This
improvement relies on the introduction of the momentum term xk ? xk?1 as well as the particularly
tuned coefficient (k ? 1)/(k + 2) ? 1 ? 3/k. Since the introduction of Nesterov?s scheme, there
has been much work on the development of first-order accelerated methods, see [12, 13, 14, 1, 2] for
example, and [19] for a unified analysis of these ideas.
In a different direction, there is a long history relating ordinary differential equations (ODE) to optimization, see [6, 4, 8, 18] for references. The connection between ODEs and numerical optimization
is often established via taking step sizes to be very small so that the trajectory or solution path converges to a curve modeled by an ODE. The conciseness and well-established theory of ODEs provide
deeper insights into optimization, which has led to many interesting findings [5, 7, 16].
1
In this work, we derive a second-order ordinary differential equation, which is the exact limit of
Nesterov?s scheme by taking small step sizes in (1.1). This ODE reads
? + 3 X? + ?f (X) = 0
(1.2)
X
t
?
for t > 0, with initial conditions X(0) = x0 , X(0)
= 0; here, x0 is the starting point in Nesterov?s
?
? = d2 X/dt2 denotes the
scheme, X denotes the time derivative or velocity dX/dt and similarly X
?
acceleration. The time parameter in this ODE is related to the step size in (1.1) via t ? k s. Case
studies are provided to demonstrate that the homogeneous and conceptually simpler ODE can serve
as a tool for analyzing and generalizing Nesterov?s scheme. To the best of our knowledge, this work
is the first to model Nesterov?s scheme or its variants by ODEs.
We denote by FL the class of convex functions f with L?Lipschitz continuous gradients defined on
Rn , i.e., f is convex, continuously differentiable, and obeys
k?f (x) ? ?f (y)k ? Lkx ? yk
for any x, y ? Rn , where k ? k is the standard Euclidean norm and L > 0 is the Lipschitz constant
throughout this paper. Next, S? denotes the class of ??strongly convex functions f on Rn with
continuous gradients, i.e., f is continuously differentiable and f (x) ? ?kxk2 /2 is convex. Last, we
set S?,L = FL ? S? .
2
Derivation of the ODE
Assume f ? FL for L > 0. Combining the two equations of (1.1) and applying a rescaling give
xk+1 ? xk
k ? 1 xk ? xk?1 ?
?
?
(2.1)
=
? s?f (yk ).
k+2
s
s
?
Introduce the ansatz xk ? X(k s) for some smooth curve X(t)
? defined for t ? 0. For fixed t,
as the step size s goes to zero, X(t) ? xt/?s = xk and X(t + s) ? x(t+?s)/?s = xk+1 with
?
k = t/ s. With these approximations, we get Taylor expansions:
?
?
1 ? ?
?
(xk+1 ? xk )/ s = X(t)
+ X(t)
s + o( s)
2
?
?
1 ? ?
?
s + o( s)
(xk ? xk?1 )/ s = X(t) ? X(t)
2
?
?
?
s?f (yk ) = s?f (X(t)) + o( s),
where in the last equality we use yk ? X(t) = o(1). Thus (2.1) can be written as
?
1 ? ?
?
s + o( s)
X(t)
+ X(t)
2
?
? ?
?
3 s ?
1 ? ?
= 1?
s + o( s) ? s?f (X(t)) + o( s).
X(t) ? X(t)
t
2
?
By comparing the coefficients of s in (2.2), we obtain
(2.2)
? + 3 X? + ?f (X) = 0
X
t
?
for t > 0. The first initial condition is X(0) = x0 . Taking k = 1 in (2.1) yields (x2 ? x1 )/ s =
?
?
= 0 (vanishing initial
? s?f (y1 ) = o(1). Hence, the second initial condition is simply X(0)
velocity). In the formulation of [1] (see also [20]), the momentum coefficient (k ? 1)/(k + 2) is
?1
? 1), where ?k are iteratively defined as
replaced by ?k (?k?1
p
?k4 + 4?k2 ? ?k2
?k+1 =
(2.3)
2
?1
starting from ?0 = 1. A bit of analysis reveals that ?k (?k?1
? 1) asymptotically equals 1 ? 3/k +
2
O(1/k ), thus leading to the same ODE as (1.1).
2
Classical results in ODE theory do not directly imply the existence or uniqueness of the solution to
this ODE because the coefficient 3/t is singular at t = 0. In addition, ?f is typically not analytic
at x0 , which leads to the inapplicability of the power series method for studying singular ODEs.
Nevertheless, the ODE is well posed: the strategy we employ for showing this constructs a series of
ODEs approximating (1.2) and then chooses a convergent subsequence by some compactness arguments such as the Arzel?a-Ascoli theorem. A proof of this theorem can be found in the supplementary
material for this paper.
Theorem 2.1. For any f ? F? , ?L>0 FL and any x0 ? Rn , the ODE (1.2) with initial conditions
?
X(0) = x0 , X(0)
= 0 has a unique global solution X ? C 2 ((0, ?); Rn ) ? C 1 ([0, ?); Rn ).
3
Equivalence between the ODE and Nesterov?s scheme
We study the stable step size allowed for numerically solving the ODE in the presence of accumulated errors. The finite difference approximation of (1.2) by the forward Euler method is
X(t + ?t) ? 2X(t) + X(t ? ?t) 3 X(t) ? X(t ? ?t)
+
+ ?f (X(t)) = 0,
(3.1)
?t2
t
?t
which is equivalent to
3?t
3?t
X(t) ? ?t2 ?f (X(t)) ? 1 ?
X(t ? ?t).
X(t + ?t) = 2 ?
t
t
Assuming that f is sufficiently smooth, for small perturbations ?x, ?f (x + ?x) ? ?f (x) +
?2 f (x)?x, where ?2 f (x) is the Hessian of f evaluated at x. Identifying k = t/?t, the characteristic equation of this finite difference scheme is approximately
3?t
3?t
det ?2 ? 2 ? ?t2 ?2 f ?
?+1?
= 0.
(3.2)
t
t
The numerical stability of (3.1) with respect to accumulated errors is equivalent to this: all the roots
2
2
of (3.2) lie in the unit circle
? [9]. When ? f LIn (i.e., LIn ? ? f is positive semidefinite), if
?t/t small and ?t? < 2/ L, we see that all the roots of (3.2) lie in the unit circle. On the other
hand, if ?t > 2/ L, (3.2) can possibly have a root ? outside the unit circle, causing numerical
instability. Under our identification s = ?t2 , a step
? size of s = 1/L in Nesterov?s scheme (1.1) is
approximately equivalent to a step size of ?t = 1/ L in the forward Euler method, which is stable
for numerically integrating (3.1).
As a comparison, note that the corresponding ODE for gradient descent with updates xk+1 = xk ?
s?f (xk ), is
?
X(t)
+ ?f (X(t)) = 0,
whose finite difference scheme has the characteristic equation det(? ? (1 ? ?t?2 f )) = 0. Thus,
to guarantee ?In 1 ? ?t?2 f In in worst case analysis,
? one can only choose ?t ? 2/L for a
fixed step size, which is much smaller than the step size 2/ L for (3.1) when ?f is very variable,
i.e., L is large.
Next, we exhibit approximate equivalence between the ODE and Nesterov?s scheme in terms of
convergence rates. We first recall the original result from [11].
Theorem 3.1 (Nesterov). For any f ? FL , the sequence {xk } in (1.1) with step size s ? 1/L obeys
f (xk ) ? f ? ?
2kx0 ? x? k2
.
s(k + 1)2
Our first result indicates that the trajectory of ODE (1.2) closely resembles the sequence {xk } in
terms of the convergence rate to a minimizer x? .
Theorem 3.2. For any f ? F? , let X(t) be the unique global solution to (1.2) with initial condi?
tions X(0) = x0 , X(0)
= 0. For any t > 0,
f (X(t)) ? f ? ?
3
2kx0 ? x? k2
.
t2
Proof of Theorem 3.2. Consider the energy functional defined as
t
E(t) , t2 (f (X(t)) ? f ? ) + 2kX + X? ? x? k2 ,
2
whose time derivative is
?
? + 4hX + t X? ? x? , 3 X? + t Xi.
(3.3)
E? = 2t(f (X) ? f ? ) + t2 h?f, Xi
2
2
2
? + tX/2
? with ?t?f (X)/2, (3.3) gives
Substituting 3X/2
t
E? = 2t(f (X) ? f ? ) + 4hX ? x? , ? ?f (X)i = 2t(f (X) ? f ? ) ? 2thX ? x? , ?f (X)i ? 0,
2
where the inequality follows from the convexity of f . Hence by monotonicity of E and non?
negativity of 2kX + tX/2
? x? k2 , the gap obeys f (X(t)) ? f ? ? E(t)/t2 ? E(0)/t2 =
? 2 2
2kx0 ? x k /t .
4
A family of generalized Nesterov?s schemes
In this section we show how to exploit the power of the ODE for deriving variants of Nesterov?s
scheme. One would be interested in studying the ODE (1.2) with the number 3 appearing in the
? replaced by a general constant r as in
coefficient of X/t
? + r X? + ?f (X) = 0, X(0) = x0 , X(0)
?
X
= 0.
(4.1)
t
Using arguments similar to those in the proof of Theorem 2.1, this new ODE is guaranteed to assume
a unique global solution for any f ? F? .
4.1
Continuous optimization
To begin with, we consider a modified energy functional defined as
2
2t2
t
? ? x?
.
E(t) =
X(t)
(f (X(t)) ? f ? ) + (r ? 1)
X(t)
+
r?1
r?1
? = ?t?f (X), the time derivative E? is equal to
Since rX? + tX
2t2
4t
? + 2hX + t X? ? x? , rX? + tXi
?
(f (X) ? f ? ) +
h?f, Xi
r?1
r?1
r?1
4t
=
(f (X) ? f ? ) ? 2thX ? x? , ?f (X)i. (4.2)
r?1
A consequence of (4.2) is this:
Theorem 4.1. Suppose r > 3 and let X be the unique solution to (4.1) for some f ? F? . Then X
obeys
(r ? 1)2 kx0 ? x? k2
f (X(t)) ? f ? ?
2t2
and
Z ?
(r ? 1)2 kx0 ? x? k2
.
t(f (X(t)) ? f ? )dt ?
2(r ? 3)
0
Proof of Theorem 4.1. By (4.2), the derivative dE/dt equals
2(r ? 3)t
2(r ? 3)t
(f (X) ? f ? ) ? ?
(f (X) ? f ? ), (4.3)
r?1
r?1
where the inequality follows from the convexity of f . Since f (X) ? f ? , (4.3) implies that E is
non-increasing. Hence
2t(f (X) ? f ? ) ? 2thX ? x? , ?f (X)i ?
2t2
(f (X(t)) ? f ? ) ? E(t) ? E(0) = (r ? 1)kx0 ? x? k2 ,
r?1
4
yielding the first inequality of the theorem as desired. To complete the proof, by (4.2) it follows that
Z ?
Z ?
dE
2(r ? 3)t
?
(f (X) ? f )dt ? ?
dt = E(0) ? E(?) ? (r ? 1)kx0 ? x? k2 ,
r?1
dt
0
0
as desired for establishing the second inequality.
We now demonstrate faster convergence rates under the assumption of strong convexity. Given a
strongly convex function f , consider a new energy functional defined as
2
2t ?
(2r ? 3)2 t
?
3
?
?
E(t) = t (f (X(t)) ? f ) +
X(t) + 2r ? 3 X(t) ? x
.
8
? gives
As in Theorem 4.1, a more refined study of the derivative of E(t)
Theorem 4.2. For any f ? S?,L (Rn ), the unique solution X to (4.1) with r ? 9/2 obeys
5
f (X(t)) ? f ? ?
Cr 2 kx0 ? x? k2
?
t3 ?
for any t > 0 and a universal constant C > 1/2.
The restriction r ? 9/2 is an artifact required in the proof. We believe that this theorem should be
valid as long as r ? 3. For example, the solution to (4.1) with f (x) = kxk2 /2 is
X(t) =
2
r?1
2
?((r + 1)/2)J(r?1)/2 (t)
x0 ,
(4.4)
r?1
t 2
where J(r?1)/2 (?) is the first kind Bessel function of order (r?1)/2. For large t, this Bessel function
p
obeys J(r?1)/2 (t) = 2/(?t)(cos(t ? (r ? 1)?/4 ? ?/4) + O(1/t)). Hence,
f (X(t)) ? f ? . kx0 ? x? k2 /tr ,
in which the inequality fails if 1/tr is replaced by any higher order rate. For general strongly convex
functions, such refinement, if possible, might require a construction of a more sophisticated energy
functional and careful analysis. We leave this problem for future research.
4.2
Composite optimization
Inspired by Theorem 4.2, it is tempting to obtain such analogies for the discrete Nesterov?s scheme
as well. Following the formulation of [1], we consider the composite minimization:
minimize
n
x?R
f (x) = g(x) + h(x),
where g ? FL for some L > 0 and h is convex on Rn with possible extended value ?. Define the
proximal subgradient
h
i
x ? argminz kz ? (x ? s?g(x))k2 /(2s) + h(z)
.
Gs (x) ,
s
Parametrizing by a constant r, we propose a generalized Nesterov?s scheme,
xk = yk?1 ? sGs (yk?1 )
(4.5)
k?1
(xk ? xk?1 ),
yk = x k +
k+r?1
starting from y0 = x0 . The discrete analog of Theorem 4.1 is below, whose proof is also deferred to
the supplementary materials as well.
Theorem 4.3. The sequence {xk } given by (4.5) with r > 3 and 0 < s ? 1/L obeys
f (xk ) ? f ? ?
and
?
X
(r ? 1)2 kx0 ? x? k2
2s(k + r ? 2)2
(k + r ? 1)(f (xk ) ? f ? ) ?
k=1
5
(r ? 1)2 kx0 ? x? k2
.
2s(r ? 3)
The idea behind the proof is the same as that employed for Theorem 4.1; here, however, the energy
functional is defined as
E(k) = 2s(k + r ? 2)2 (f (xk ) ? f ? )/(r ? 1) + k(k + r ? 1)yk ? kxk ? (r ? 1)x? k2 /(r ? 1).
The first inequality in Theorem 4.3 suggests that the generalized Nesterov?s scheme still achieves
O(1/k 2 ) convergence rate. However, if the error bound satisfies
c
f (xk? ) ? f ? ? ?2
k
for some c > 0 and a dense subsequence {k ? }, i.e., |{k ? } ? {1, . . . , m}| ? ?m for any positive
integer m and some ? > 0, then the second inequality of the theorem is violated. Hence, the second
inequality is not trivial because it implies the error bound is in some sense O(1/k 2 ) suboptimal.
In closing, we would like to point out this new scheme is equivalent to setting ?k = (r?1)/(k+r?1)
?1
and letting ?k (?k?1
? 1) replace the momentum coefficient (k ? 1)/(k + r ? 1). Then, the equal
sign ? = ? in (2.3) has to be replaced by ? ? ?. In examining the proof of Theorem 1(b) in [20], we
can get an alternative proof of Theorem 4.3 by allowing (2.3), which appears in Eq. (36) in [20], to
be an inequality.
5
Accelerating to linear convergence by restarting
Although an O(1/k 3 ) convergence rate is guaranteed for generalized Nesterov?s schemes (4.5), the
example (4.4) provides evidence that O(1/poly(k)) is the best rate achievable under strong convexity. In contrast, the vanilla gradient method achieves linear convergence
O((1 ? ?/L)k ) and
p
[12] proposed a first-order method with a convergence rate of O((1 ? ?/L)k ), which, however,
requires knowledge of the condition number ?/L. While it is relatively easy to bound the Lipschitz
constant L by the use of backtracking [3, 19], estimating the strong convexity parameter ?, if not
impossible, is very challenging. Among many approaches to gain acceleration via adaptively estimating ?/L, [15] proposes a restarting procedure for Nesterov?s scheme in which (1.1) is restarted
with x0 = y0 := xk whenever ?f (yk )T (xk+1 ? xk ) > 0. In the language of ODEs, this gradi? negative along the trajectory. Although it has been
ent based restarting essentially keeps h?f, Xi
empirically observed that this method significantly boosts convergence, there is no general theory
characterizing the convergence rate.
In this section, we propose a new restarting scheme we call the speed restarting scheme. The underlying motivation is to maintain a relatively high velocity X? along the trajectory. Throughout this
section we assume f ? S?,L for some 0 < ? ? L.
?
Definition 5.1. For ODE (1.2) with X(0) = x0 , X(0)
= 0, let
2
?
dkX(u)k
> 0}
T = T (f, x0 ) = sup{t > 0 : ?u ? (0, t),
du
be the speed restarting time.
? decreases. The definition itself does not imply that
In words, T is the first time the velocity kXk
0 < T < ?, which is proven in the supplementary materials. Indeed, f (X(t)) is a decreasing
function before time T ; for t ? T ,
? 2
df (X(t))
? = ? 3 kXk
? 2 ? 1 dkXk ? 0.
= h?f (X), Xi
dt
t
2 dt
The speed restarted ODE is thus
? + 3 X(t)
?
X(t)
+ ?f (X(t)) = 0,
(5.1)
tsr
? Xi
? = 0 and between two consecutive restarts, tsr grows just
where tsr is set to zero whenever hX,
as t. That is, tsr = t ? ? , where ? is the latest restart time. In particular, tsr = 0 at t = 0. The
theorem below guarantees linear convergence of the solution to (5.1). This is a new result in the
literature [15, 10].
Theorem 5.2. There exists positive constants c1 and c2 , which only depend on the condition number
L/?, such that for any f ? S?,L , we have
c1 Lkx0 ? x? k2 ?c2 t?L
.
e
f (X sr (t)) ? f (x? ) ?
2
6
5.1
Numerical examples
Below we present a discrete analog to the restarted scheme. There, kmin is introduced to avoid
having consecutive restarts that are too close. To compare the performance of the restarted scheme
with the original (1.1), we conduct four simulation studies, including both smooth and non-smooth
objective functions. Note that the computational costs of the restarted and non-restarted schemes are
the same.
Algorithm 1 Speed Restarting Nesterov?s Scheme
input: x0 ? Rn , y0 = x0 , x?1 = x0 , 0 < s ? 1/L, kmax ? N+ and kmin ? N+
j?1
for k = 1 to kmax do
1
xk ? argminx ( 2s
kx ? yk?1 + s?g(yk?1 )k2 + h(x))
j?1
yk ? xk + j+2 (xk ? xk?1 )
if kxk ? xk?1 k < kxk?1 ? xk?2 k and j ? kmin then
j?1
else
j ?j+1
end if
end for
Quadratic. f (x) = 21 xT Ax + bT x is a strongly convex function, in which A is a 500 ? 500 random
positive definite matrix and b a random vector. The eigenvalues of A are between 0.001 and 1. The
vector b is generated as i. i. d. Gaussian random variables with mean 0 and variance 25.
Log-sum-exp.
f (x) = ? log
m
hX
i=1
i
exp((aTi x ? bi )/?) ,
where n = 50, m = 200, ? = 20. The matrix A = {aij } is a random matrix with i. i. d. standard
Gaussian entries, and b = {bi } has i. i. d. Gaussian entries with mean 0 and variance 2. This function
is not strongly convex.
Matrix completion. f (X) = 21 kXobs ? Mobs k2F + ?kXk? , in which the ground truth M is a
rank-5 random matrix of size 300 ? 300. The regularization parameter is set to ? = 0.05. The 5
singular values of M are 1, . . . , 5. The observed set is independently sampled among the 300 ? 300
entries so that 10% of the entries are actually observed.
Lasso in ?1 ?constrained form with large sparse design. f = 12 kAx ? bk2 s.t. kxk1 ? ?, where
A is a 5000 ? 50000 random sparse matrix with nonzero probability 0.5% for each entry and b is
generated as b = Ax0 + z. The nonzero entries of A independently follow the Gaussian distribution
with mean 0 and variance 1/25. The signal x0 is a vector with 250 nonzeros and z is i. i. d. standard
Gaussian noise. The parameter ? is set to kx0 k1 .
In these examples, kmin is set to be 10 and the step sizes are fixed to be 1/L. If the objective is in
composite form, the Lipschitz bound applies to the smooth part. Figures 1(a), 1(b), 1(c) and 1(d)
present the performance of the speed restarting scheme, the gradient restarting scheme proposed in
[15], the original Nesterov?s scheme and the proximal gradient method. The objective functions
include strongly convex, non-strongly convex and non-smooth functions, violating the assumptions
in Theorem 5.2. Among all the examples, it is interesting to note that both speed restarting scheme
empirically exhibit linear convergence by significantly reducing bumps in the objective values. This
leaves us an open problem of whether there exists provable linear convergence rate for the gradient
restarting scheme as in Theorem 5.2. It is also worth pointing that compared with gradient restarting,
the speed restarting scheme empirically exhibits more stable linear convergence rate.
6
Discussion
This paper introduces a second-order ODE and accompanying tools for characterizing Nesterov?s
accelerated gradient method. This ODE is applied to study variants of Nesterov?s scheme. Our
7
8
2
10
10
srN
grN
oN
PG
6
10
srN
grN
oN
PG
0
10
4
10
?2
10
2
10
?4
f ? f*
f ? f*
10
0
10
?6
10
?2
10
?8
10
?4
10
?10
10
?6
10
?12
0
200
400
600
800
1000
1200
10
1400
0
500
1000
iterations
1500
iterations
(a) min 12 xT Ax + bx
(b) min ? log(
2
Pm
i=1
(aT
i x?bi )/?
e
)
5
10
10
srN
grN
oN
PG
0
10
srN
grN
oN
PG
?2
10
0
10
?4
f ? f*
f?f
*
10
?6
10
?8
?5
10
10
?10
10
?12
10
?10
0
20
40
(c) min
60
80
1
kXobs
2
?
100
iterations
120
Mobs k2F
140
160
180
10
200
0
200
400
600
800
1000
1200
1400
iterations
+ ?kXk?
(d) min
1
kAx
2
? bk
2
s.t. kxk1 ? C
Figure 1: Numerical performance of speed restarting (srN), gradient restarting (grN) proposed in
[15], the original Nesterov?s scheme (oN) and the proximal gradient (PG)
approach suggests (1) a large family of generalized Nesterov?s schemes that are all guaranteed to
converge at the rate 1/k 2 , and (2) a restarted scheme provably achieving a linear convergence rate
whenever f is strongly convex.
In this paper, we often utilize ideas from continuous-time ODEs, and then apply these ideas to
discrete schemes. The translation, however, involves parameter tuning and tedious calculations.
This is the reason why a general theory mapping properties of ODEs into corresponding properties
for discrete updates would be a welcome advance. Indeed, this would allow researchers to only
study the simpler and more user-friendly ODEs.
7
Acknowledgements
We would like to thank Carlos Sing-Long and Zhou Fan for helpful discussions about parts of this
paper, and anonymous reviewers for their insightful comments and suggestions.
8
References
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4,775 | 5,323 | Learning Distributed Representations for Structured
Output Prediction
Vivek Srikumar?
University of Utah
[email protected]
Christopher D. Manning
Stanford University
[email protected]
Abstract
In recent years, distributed representations of inputs have led to performance gains
in many applications by allowing statistical information to be shared across inputs. However, the predicted outputs (labels, and more generally structures) are
still treated as discrete objects even though outputs are often not discrete units
of meaning. In this paper, we present a new formulation for structured prediction where we represent individual labels in a structure as dense vectors and allow
semantically similar labels to share parameters. We extend this representation
to larger structures by defining compositionality using tensor products to give a
natural generalization of standard structured prediction approaches. We define a
learning objective for jointly learning the model parameters and the label vectors
and propose an alternating minimization algorithm for learning. We show that
our formulation outperforms structural SVM baselines in two tasks: multiclass
document classification and part-of-speech tagging.
1
Introduction
In recent years, many computer vision and natural language processing (NLP) tasks have benefited
from the use of dense representations of inputs by allowing superficially different inputs to be related
to one another [26, 9, 7, 4]. For example, even though words are not discrete units of meaning, traditional NLP models use indicator features for words. This forces learning algorithms to learn separate
parameters for orthographically distinct but conceptually similar words. In contrast, dense vector
representations allow sharing of statistical signal across words, leading to better generalization.
Many NLP and vision problems are structured prediction problems. The output may be an atomic
label (tasks like document classification) or a composition of atomic labels to form combinatorial
objects like sequences (e.g. part-of-speech tagging), labeled trees (e.g. parsing) or more complex
graphs (e.g. image segmentation). Despite both the successes of distributed representations for
inputs and the clear similarities over the output space, it is still usual to handle outputs as discrete
objects. But are structures, and the labels that constitute them, really discrete units of meaning?
Consider, for example, the popular 20 Newsgroups dataset [13] which presents the multiclass
classification problem of identifying a newsgroup label given the text of a posting. Labels
include comp.os.mswindows.misc, sci.electronics, comp.sys.mac.hardware,
rec.autos and rec.motorcycles. The usual strategy is to train a classifier that uses separate
weights for each label. However, the labels themselves have meaning that is independent of the training data. From the label, we can see that comp.os.mswindows.misc, sci.electronics
and comp.sys.mac.hardware are semantically closer to each other than the other two. A similar argument can be made for not just atomic labels but their compositions too. For example, a
part-of-speech tagging system trained as a sequence model might have to learn separate parameters
?
This work was done when the author was at Stanford University.
1
for the JJ?NNS and JJR?NN transitions even though both encode a transition from an adjective to
a noun. Here, the similarity of the transitions can be inferred from the similarity of its components.
In this paper, we propose a new formulation for structured output learning called D ISTRO (DIStributed STRuctred Output), which accounts for the fact that labels are not atomic units of meaning.
We model label meaning by representing individual labels as real valued vectors. Doing so allows us
to capture similarities between labels. To allow for arbitrary structures, we define compositionality
of labels as tensor products of the label vectors corresponding to its sub-structures. We show that
doing so gives us a natural extension of standard structured output learning approaches, which can
be seen as special cases with one-hot label vectors.
We define a learning objective that seeks to jointly learn the model parameters along with the label
representations and propose an alternating algorithm for minimizing the objective for structured
hinge loss. We evaluate our approach on two tasks which have semantically rich labels: multiclass
classification on the newsgroup data and part-of-speech tagging for English and Basque. In all cases,
we show that D ISTRO outperforms the structural SVM baselines.
1.1
Related Work
This paper considers the problem of using distributed representations for arbitrary structures and is
related to recent work in deep learning and structured learning. Recent unsupervised representation
learning research has focused on the problem of embedding inputs in vector spaces [26, 9, 16, 7].
There has been some work [22] on modeling semantic compositionality in NLP, but the models do
not easily generalize to arbitrary structures. In particular, it is not easy to extend these approaches
to use advances in knowledge-driven learning and inference that standard structured learning and
prediction algorithms enable.
Standard learning approaches for structured output allow for modeling arbitrarily complex structures
(subject to inference difficulties) and structural SVMs [25] or conditional random fields [12] are
commonly used. However, the output itself is treated as a discrete object and similarities between
outputs are not modeled. For multiclass classification, the idea of classifying to a label set that
follow a known hierarchy has been explored [6], but such a taxonomy is not always available.
The idea of distributed representations for outputs has been discussed in the connectionist literature
since the eighties [11, 21, 20]. In recent years, we have seen several lines of research that address the
problem in the context of multiclass classification by framing feature learning as matrix factorization
or sparse encoding [23, 1, 3]. As in this paper, the goal has often explicitly been to discover shared
characteristics between the classes [2]. Indeed, the inference formulation we propose is very similar
to inference in these lines of work. Also related is recent research in the NLP community that
explores the use of tensor decompositions for higher order feature combinations [14]. The primary
novelty in this paper is that in addition to representing atomic labels in a distributed manner, we
model their compositions in a natural fashion to generalize standard structured prediction.
2
Preliminaries and Notation
In this section, we give a very brief overview of structured prediction with the goal of introducing
notation and terminology for the next sections. We represent inputs to the structured prediction
problem (such as, sentences, documents or images) by x ? X and output structures (such as labels
or trees) by y ? Y. We define the feature function ? : X ? Y ? <n that captures the relationship
between the input x and the structure y as an n dimensional vector. A linear model scores the
structure y with a weight vector w ? <n as wT ?(x, y). We predict the output for an input x as
arg maxy wT ?(x, y). This problem of inference is a combinatorial optimization problem.
We will use the structures in Figure 1 as running examples. In the case of multiclass classification,
the output y is one of a finite set of labels (Figure 1, left). For more complex structures, the feature
vector is decomposed over the parts of the structure. For example, the usual representation of a
first-order linear sequence model (Figure 1, middle) decomposes the sequence into emissions and
transitions and the features decompose over these [8]. In this case, each emission is associated with
one label and a transition is associated with an ordered pair of labels.
2
Compositional part
Label yp = (y0 , y1 )
y
y0
y1
y2
y0
y1
y2
Atomic part
Label yp = (y)
x
Atomic part
Label yp = (y0 )
Multiclass classification
x
Compositional part
Label yp = (y0 , y1 , y2 )
Sequence labeling. The emissions are
atomic and the transitions are compositional.
x
A purely compositional part
Figure 1: Three examples of structures. In all cases, x represents the input and the y?s denote the outputs to
be predicted. Here, each square represents a part as defined in the text and circles represent random variables
for inputs and outputs (as in factor graphs). The left figure shows multiclass classification, which has an atomic
part associated with exactly one label. The middle figure shows a first-order sequence labeling task that has both
atomic parts (emissions) and compositional ones (transitions). The right figure shows a purely compositional
part where all outputs interact. The feature functions for these structures are shown at the end of Section 3.1.
In the general case, we denote the parts (or equivalently, factors in a factor graph) in the structure
for input x by ?x . Each part p ? ?x is associated with a list of discrete labels, denoted by yp =
(yp0 , yp1 , ? ? ? ). Note that the size of the list yp is a modeling choice; for example, transition parts in
the first-order Markov model correspond to two consecutive labels, as shown in Figure 1.
We denote the set of labels in the problem as L = {l1 , l2 , ? ? ? , lM } (e.g. the set of part-of-speech
tags). All the elements of the part labels yp are members of this set. For notational convenience,
we denote the first element of the list yp by yp (without boldface) and the rest by yp1: . In the rest of
the paper, we will refer to a part associated with a single label as atomic and all other parts where
yp has more than one element as compositional. In Figure 1, we see examples of a purely atomic
structure (multiclass classification), a purely compositional structure (right) and a structure that is a
mix of the two (first order sequence, middle).
The decomposition of the structure decomposes the feature function over the parts as
X
?(x, y) =
?p (x, yp ) .
(1)
p??x
The scoring function wT ?(x, y) also decomposes along this sum. Standard definitions of structured prediction models leave the definition of the part-specific feature function ?p to be problem
dependent. We will focus on this aspect in Section 3 to define our model.
With definitions of a scoring function and inference, we can state the learning objective. Given a
collection of N training examples of the form (xi , yi ), training is the following regularized risk
minimization problem:
?
1 X
minn wT w +
L(xi , yi ; w).
(2)
w?< 2
N i
Here, L represents a loss function such as the hinge loss (for structural SVMs) or the log loss (for
conditional random fields) and penalizes model errors.The hyper-parameter ? trades off between
generalization and accuracy.
3
Distributed Representations for Structured Output
As mentioned in Section 2, the choice of the feature function ?p for a part p is left to be problem
specific. The objective is to capture the correlations between the relevant attributes of the input x
and the output labels yp . Typically, this is done by conjoining the labels yp with a user-defined
feature vector ?p (x) that is dependent only on the input.
3
When applied to atomic parts (e.g. multiclass classification), conjoining the label with the input features effectively allocates a different portion of the weight vector for each label. For compositional
parts (e.g. transitions in sequence models), this ensures that each combination of labels is associated
with a different portion of the weight vector. The implicit assumption in this design is that labels and
label combinations are distinct units of meaning and hence do not share any parameters across them.
In this paper, we posit that in most naturally occurring problems and their associated labels, this
assumption is not true. In fact, labels often encode rich semantic information with varying degrees
of similarities to each other. Because structures are composed of atomic labels, the same applies to
structures too.
From Section 2, we see that for the purpose of inference, structures are completely defined by
their feature vectors, which are decomposed along the atomic and compositional parts that form the
structure. Thus, our goal is to develop a feature representation for labeled parts that exploits label
similarity. More explicitly, our desiderata are:
1. First, we need to be able to represent labeled atomic parts using a feature representation
that accounts for relatedness of labels in such a way that statistical strength (i.e. weights)
can be shared across different labels.
2. Second, we need an operator that can construct compositional parts to build larger structures so that the above property can be extended to arbitrary structured output.
3.1
The D ISTRO model
In order to assign a notion of relatedness between labels, we associate a d dimensional unit vector
al to each label l ? L. We will refer to the d ? M matrix comprising of all the M label vectors as
A, the label matrix.
We can define the feature vectors for parts, and thus entire structures, using these label vectors. To
do so, we define the notion of a feature tensor function for a part p that has been labeled with a list
of m labels yp . The feature tensor function is a function ?p that maps the input x and the label list
yp associated with the part to a tensor of order m + 1. The tensor captures the relationships between
the input and all the m labels associated with it. We recursively define the feature tensor function
using the label vectors as:
alyp ? ?p (x),
p is atomic,
(3)
?p (x, yp , A) =
1:
alyp ? ?p x, yp , A , p is compositional.
Here, the symbol ? denotes the tensor product operation. Unrolling the recursion in this definition
shows that the feature tensor function for a part is the tensor product of the vectors for all the labels
associated with that part and the feature vector associated with the input for the part. For an input x
and a structure y, we use the feature tensor function to define its feature representation as
X
?A (x, y) =
vec (?p (x, yp , A))
(4)
p??x
Here, vec(?) denotes the vectorization operator that converts a tensor into a vector by stacking its
elements. Figure 2 shows an example of the process of building the feature vector for a part that is
labeled with two labels. With this definition of the feature vector, we can use the standard approach
to score structures using a weight vector as wT ?A (x, y).
In our running examples from Figure 1, we have the following definitions of feature functions for
each of the cases:
1. Purely atomic part, multiclass classification (left): Denote the feature vector associated
with x as ?. For an atomic part, the definition of the feature tensor function in Equation (3)
T
effectively produces a d ?
|?| matrix aly ? . Thus the feature vector for the structure y is
T
?A (x, y) = vec aly ? . For this case, the score for an input x being assigned a label y
can be explicitly be written as the following summation:
wT ?A (x, y) =
|?|
d X
X
i=0 j=0
4
wdj+i aly ,i ?j
vec (
?
?
)?
vec (
al1 ? <d
al2 ? <d
?p (x) ? <N
)
d?d?N
Feature tensor
?
Feature vector ? <d
2
N
Figure 2: This figure summarizes feature vector generation for a compositional part labeled with two labels
l1 and l2 . Each label is associated with a d dimensional label vector and the feature vector for the input is N
dimensional. Vectorizing the feature tensor produces a final feature vector that is a d2 N -dimensional vector.
2. Purely compositional part (right): For a compositional part, the feature tensor function
produces a tensor whose elements effectively enumerate every possible combination of
elements of input vector ?p (x) and the associated label vectors.
So, the feature vector for
the structure is ?A (x, y) = vec aly0 ? aly1 ? aly2 ? ?p (x) .
3. First order sequence (middle): This structure presents a combination of atomic and compositional parts. Suppose we denote the input emission features by ?E,i for the ith label
and the input features corresponding to the transition1 from yi to yi+1 by ?T,i . With this
notation, we can define the feature vector for the structure as
X
X
?A (x, y) =
vec alyi ? ?E,i +
vec alyi ? alyi+1 ? ?T,i .
i
3.2
i
Discussion
Connection to standard structured prediction For a part p, a traditional structured model conjoins all its associated labels to the input feature vector to get the feature vector for that assignment
of the labels. According to the definition of Equation (3), we propose that these label conjunctions
should be replaced with a tensor product, which generalizes the standard method. Indeed, if the
labels are represented via one-hot vectors, then we would recover standard structured prediction
where each label (or group of labels) is associated with a separate section of the weight vector. For
example, for multiclass classification, if each label is associated with a separate one-hot vector, then
the feature tensor for a given label will be a matrix where exactly one column is the input feature
vector ?p (x) and all other entries are zero. This argument also extends to compositional parts.
Dimensionality of label vectors If labels are represented by one-hot vectors, the dimensionality
of the label vectors will be M , the number of labels in the problem. However, in D ISTRO, in addition
to letting the label vectors be any unit vector, we can also allow them to exist in a lower dimensional
space. This presents us with a decision with regard to the dimensionality d.
The choice of d is important for two reasons. First, it determines the number of parameters in the
model. If a part is associated with m labels, recall that the feature tensor function produces a m + 1
order tensor formed by taking the tensor product of the m label vectors and the input features. That
is, the feature vector for the part is a dm |?p (x)| dimensional vector. (See 2 for an illustration.)
Smaller d thus leads to smaller weight vectors. Second, if the dimensionality of the label vectors
is lower, it encourages more weights to be shared across labels. Indeed, for purely atomic and
compositional parts if the labels are represented by M dimensional vectors, we can show that for
any weight vector that scores these labels via the feature representation defined in Equation (4), there
is another weight vector that assigns the same scores using one-hot weight vectors.
4
Learning Weights and Label Vectors
In this section, we will address the problem of learning the weight vectors w and the label vectors
A from data. We are given a training set with N examples of the form (xi , yi ). The goal of learning
1
In a linear sequence model defined as a CRF or a structural SVM, these transition input features can simply
be an indicator that selects a specific portion of the weight vector.
5
is to minimize regularized risk over the training set. This leads to a training objective similar to
that of structural SVMs or conditional random fields (Equation (2)). However, there are two key
differences. First, the feature vectors for structures are not fixed as in structural SVMs or CRFs but
are functions of the label vectors. Second, the minimization is over not just the weight vectors, but
also over the label vectors that require regularization.
In order to encourage the labels to share weights, we propose to impose a rank penalty over the
label matrix A in the learning objective. Since the rank minimization problem is known to be
computationally intractable in general [27], we use the well known nuclear norm surrogate to replace
the rank [10]. This gives us the learning objective defined as f below:
?1 T
1 X
f (w, A) =
w w + ?2 ||A||? +
L(xi , yi ; w, A)
(5)
2
N i
Here, the ||A||? is the nuclear norm of A, defined as the sum of the singular values of the matrix.
Compared to the objective in Equation (2), the loss function L is also dependent of the label matrix
via the new definition of the features. In this paper, we instantiate the loss using the structured hinge
loss [25]. That is, we define L to be
L(xi , yi ; w, A) = max wT ?A (xi , y) + ?(y, yi ) ? wT ?A (xi , yi )
(6)
y
Here, ? is the Hamming distance. This defines the D ISTRO extension of the structural SVM.
The goal of learning is to minimize the objective function f in terms of both its parameters w and
A, where each column of A is restricted to be a unit vector by definition. However, the objective
is not longer jointly convex in both w and A because of the product terms in the definition of the
feature tensor.
We use an alternating minimization algorithm for solving the optimization problem (Algorithm 1).
If the label matrix A is fixed, then so are the feature representations of structures (from Equation
(4)). Thus, for a fixed A (lines 2 and 5), the problem of minimizing f (w, A) with respect to only w
is identical to the learning problem of structural SVMs. Since gradient computation and inference
do not change from the usual setting, we can solve this minimization over w using stochastic subgradient descent (SGD). For fixed weight vectors (line 4), we implemented stochastic sub-gradient
descent using the proximal gradient method [18] for solving for A. The supplementary material
gives further details about the steps of the algorithm.
Algorithm 1 Learning algorithm by alternating minimization.
The goal is to solve
minw,A f (w, A). The input to the problem is a training set of examples consisting of pairs of
labeled inputs (xi , yi ) and T , the number of iterations.
1:
2:
3:
4:
5:
6:
7:
Initialize A0 randomly
Initialize w0 = minw f (w, A0 )
for t = 1, ? ? ? , T do
At ? minA f (wt?1 , A)
wt+1 ? minw f (w, At )
end for
return (wT +1 , AT )
Even though the objective function is not jointly convex in w and A, in our experiments (Section
5), we found that in all but one trial, the non-convexity of the objective did not affect performance.
Because the feature functions are multilinear in w and A, multiple equivalent solutions can exist
(from the perspective of the score assigned to structures) and the eventual point of convergence is
dependent on the initialization.
For regularizing the label matrix, we also experimented with the Frobenius norm and found that not
only does the nuclear norm have an intuitive explanation (rank minimization) but also performed
better. Furthermore, the proximal method itself does not add significantly to the training time because the label matrix is small. In practice, training time is affected by the density of the label
vectors and sparser vectors correspond to faster training because the sparsity can be used to speed
up dot product computation. Prediction is as fast as inference in standard models, however, because
the only change is in feature computation via the vectorization operator, which can be performed
efficiently.
6
5
Experiments
We demonstrate the effectiveness of D ISTRO on two tasks ? document classification (purely atomic
structures) and part-of-speech (POS) tagging (both atomic and compositional structures). In both
cases, we compare to structural SVMs ? i.e. the case of one-hot label vectors ? as the baseline.
We selected the hyper-parameters for all experiments by cross validation. We ran the alternating
algorithm for 5 epochs for all cases with 5 epochs of SGD for both the weight and label vectors.
We allowed the baseline to run for 25 epochs over the data. For the proposed method, we ran all
the experiments five times with different random initializations for the label vectors and report the
average accuracy. Even though the objective is not convex, we found that the learning algorithm
converged quickly in almost all trials. When it did not, the objective value on the training set at the
end of each alternating SGD step in the algorithm was a good indicator for ill-behaved initializations.
This allowed us to discard bad initializations during training.
5.1
Atomic structures: Multiclass Classification
Our first application is the problem of document classification with the 20 Newsgroups Dataset [13].
This dataset is collection of about 20,000 newsgroup posts partitioned roughly evenly among 20
newsgroups. The task is to predict the newsgroup label given the post. As observed in Section 1,
some newsgroups are more closely related to each other than others.
We used the ?bydate? version of the data with tokens as features. Table 1 reports the performance of
the baseline and variants of D ISTRO for newsgroup classification. The top part of the table compares
the baseline to our method and we see that modeling the label semantics gives us a 2.6% increase in
accuracy. In a second experiment (Table 1, bottom), we studied the effect of explicitly reducing the
label vector dimensionality. We see that even with 15 dimensional vectors, we can outperform the
baseline and the performance of the baseline is almost matched with 10 dimensional vectors. Recall
that the size of the weight vector increases with increasing label vector dimensionality (see Figure
2). This motivates a preference for smaller label vectors.
Algorithm
Structured SVM
D ISTRO
D ISTRO
D ISTRO
Label Matrix Rank Average accuracy (%)
20
81.4
19
84.0
Reduced dimensionality setting
15
83.1
10
80.9
Table 1: Results on 20 newsgroup classification. The top part of the table compares the baseline against the
full D ISTRO model. The bottom part shows the performance of two versions of D ISTRO where the dimensionality of the label vectors is fixed. Even with 10-dimensional vectors, we can almost match the baseline.
5.2
Compositional Structures: Sequence classification
We evaluated D ISTRO for English and Basque POS tagging using first-order sequence models.
English POS tagging has been long studied using the Penn Treebank data [15]. We used the standard
train-test split [8, 24] ? we trained on sections 0-18 of the Treebank and report performance on
sections 22-24. The data is labeled with 45 POS labels. Some labels are semantically close to each
other because they express variations of a base part-of-speech tag. For example, the labels NN, NNS,
NNP and NNPS indicate singular and plural versions of common and proper nouns
We used the Basque data from the CoNLL 2007 shared task [17] for training the Basque POS tagger.
This data comes from the 3LB Treebank. There are 64 fine grained parts of speech. Interestingly,
the labels themselves have a structure. For example, the labels IZE and ADJ indicate a noun and
an adjective respectively. However, Basque can take internal noun ellipsis inside noun-forms, which
are represented with tags like IZE IZEELI and ADJ IZEELI to indicate nouns and adjectives
with internal ellipses.
In both languages, many labels and transitions between labels are semantically close to each other.
This observation has led, for example, to the development of the universal part-of-speech tag set
7
[19]. Clearly, the labels should not be treated as independent units of meaning and the model should
be allowed to take advantage of the dependencies between labels.
Language
English
Basque
Algorithm
Structured SVM
D ISTRO
D ISTRO
Structured SVM
D ISTRO
Label Matrix Rank
45
5
20
64
58
Average accuracy (%)
96.2
95.1
96.7
91.5
92.4
Table 2: Results on part-of-speech tagging. The top part of the table shows results on English, where we see
a 0.5% gain in accuracy. The bottom part shows Basque results where we see a nearly 1% improvement.
For both languages, we extracted the following emission features: indicators for the words, their
prefixes and suffixes of length 3, the previous and next words and the word shape according to the
Stanford NLP pipeline2,3 . Table 2 presents the results for the two languages. We evaluate using the
average accuracy over all tags. In the English case, we found that the performance plateaued for any
label matrix with rank greater than 20 and we see an improvement of 0.5% accuracy. For Basque,
we see an improvement of 0.9% over the baseline.
Note that unlike the atomic case, the learning objective for the first order Markov model is not even
bilinear in the weights and the label vectors. However, in practice, we found that this did not cause
any problems. In all but one run, the test performance remained consistently higher than the baseline.
Moreover, the outlier converged to a much higher objective value; it could easily be identified. As
an analysis experiment, we initialized the model with one-hot vectors (i.e. the baseline) and found
that this gives us similar improvements as reported in the table.
6
Conclusion
We have presented a new model for structured output prediction called Distributed Structured Output
(D ISTRO). Our model is motivated by two observations. First, distributed representations for inputs
have led to performance gains by uncovering shared characteristics across inputs. Second, often,
structures are composed of semantically rich labels and sub-structures. Just like inputs, similarities
between components of structures can be exploited for better performance. To take advantage of
similarities among structures, we have proposed to represent labels by real-valued vectors and model
compositionality using tensor products between the label vectors. This not only lets semantically
similar labels share parameters, but also allows construction of complex structured output that can
take advantage of similarities across its component parts.
We have defined the objective function for learning with D ISTRO and presented a learning algorithm
that jointly learns the label vectors along with the weights using alternating minimization. We
presented an evaluation of our approach for two tasks ? document classification, which is an instance
of multiclass classification, and part-of-speech tagging for English and Basque, modeled as firstorder sequence models. Our experiments show that allowing the labels to be represented by realvalued vectors improves performance over the corresponding structural SVM baselines.
Acknowledgments
We thank the anonymous reviewers for their valuable comments. Stanford University gratefully acknowledges the support of the Defense Advanced Research Projects Agency (DARPA) Deep Exploration and Filtering of Text (DEFT) Program under Air Force Research Laboratory (AFRL) contract
no. FA8750-13-2-0040. Any opinions, findings, and conclusion or recommendations expressed in
this material are those of the authors and do not necessarily reflect the view of the DARPA, AFRL,
or the US government.
2
http://nlp.stanford.edu/software/corenlp.shtml
Note that our POS systems are not state-of-the-art implementations, which typically use second order
Markov models with additional features and specialized handling of unknown words. However, surprisingly,
for Basque, even the baseline gives better accuracy than the second order TnT tagger[5, 19].
3
8
References
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9
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4,776 | 5,324 | Predictive Entropy Search for Efficient Global
Optimization of Black-box Functions
Jos?e Miguel Hern?andez-Lobato
[email protected]
University of Cambridge
Matthew W. Hoffman
[email protected]
University of Cambridge
Zoubin Ghahramani
[email protected]
University of Cambridge
Abstract
We propose a novel information-theoretic approach for Bayesian optimization
called Predictive Entropy Search (PES). At each iteration, PES selects the next
evaluation point that maximizes the expected information gained with respect to
the global maximum. PES codifies this intractable acquisition function in terms
of the expected reduction in the differential entropy of the predictive distribution. This reformulation allows PES to obtain approximations that are both more
accurate and efficient than other alternatives such as Entropy Search (ES). Furthermore, PES can easily perform a fully Bayesian treatment of the model hyperparameters while ES cannot. We evaluate PES in both synthetic and realworld applications, including optimization problems in machine learning, finance,
biotechnology, and robotics. We show that the increased accuracy of PES leads to
significant gains in optimization performance.
1
Introduction
Bayesian optimization techniques form a successful approach for optimizing black-box functions
[5]. The goal of these methods is to find the global maximizer of a nonlinear and generally nonconvex function f whose derivatives are unavailable. Furthermore, the evaluations of f are usually
corrupted by noise and the process that queries f can be computationally or economically very
expensive. To address these challenges, Bayesian optimization devotes additional effort to modeling the unknown function f and its behavior. These additional computations aim to minimize the
number of evaluations that are needed to find the global optima.
Optimization problems are widespread in science and engineering and as a result so are Bayesian
approaches to this problem. Bayesian optimization has successfully been used in robotics to adjust
the parameters of a robot?s controller to maximize gait speed and smoothness [16] as well as parameter tuning for computer graphics [6]. Another example application in drug discovery is to find the
chemical derivative of a particular molecule that best treats a given disease [20]. Finally, Bayesian
optimization can also be used to find optimal hyper-parameter values for statistical [29] and machine
learning techniques [24].
As described above, we are interested in finding the global maximizer x? = arg maxx?X f (x) of
a function f over some bounded domain, typically X ? Rd . We assume that f (x) can only be
evaluated via queries to a black-box that provides noisy outputs of the form yi ? N (f (xi ), ? 2 ).
We note, however, that our framework can be extended to other non-Gaussian likelihoods. In this
setting, we describe a sequential search algorithm that, after n iterations, proposes to evaluate f at
some location xn+1 . To make this decision the algorithm conditions on all previous observations
eN
Dn = {(x1 , y1 ), . . . , (xn , yn )}. After N iterations the algorithm makes a final recommendation x
for the global maximizer of the latent function f .
We take a Bayesian approach to the problem described above and use a probabilistic model for the
eN . In this work we use a zero-mean Gaussian
latent function f to guide the search and to select x
1
Algorithm 2 PES acquisition function
Input: a black-box with unknown mean f
1: for n = 1, . . . , N do
2:
select xn = arg maxx?X ?n?1 (x)
3:
query the black-box at xn to obtain yn
4:
augment data Dn = Dn?1 ? {(xn , yn )}
5: end for
eN = arg maxx?X ?N (x)
6: return x
Input: a candidate x; data Dn
1: sample M hyperparameter values {? (i) }
2: for i = 1, . . . , M do
3:
sample f (i) ? p(f |Dn , ?, ? (i) )
(i)
4:
set x? ? arg maxx?X f (i) (x)
(i)
(i)
e (i) , v
e (i)
5:
compute m0 , V0 and m
(i)
(i)
(i)
6:
compute vn (x) and vn (x|x? )
7: end for
8: return ?n (x) as in (10)
precomputed
Algorithm 1 Generic Bayesian optimization
process (GP) prior for f [22]. This prior is specified by a positive-definite kernel function k(x, x0 ).
Given any finite collection of points {x1 , . . . , xn }, the values of f at these points are jointly zeromean Gaussian with covariance matrix Kn , where [Kn ]ij = k(xi , xj ). For the Gaussian likelihood
described above, the vector of concatenated observations yn is also jointly Gaussian with zero-mean.
Therefore, at any location x, the latent function f (x) conditioned on past observations Dn is then
Gaussian with marginal mean ?n (x) and variance vn (x) given by
?n (x) = kn (x)T (Kn + ? 2 I)?1 yn ,
vn (x) = k(x, x) ? kn (x)T (Kn + ? 2 I)?1 kn (x) ,
(1)
where kn (x) is a vector of cross-covariance terms between x and {x1 , . . . , xn }.
Bayesian optimization techniques use the above predictive distribution p(f (x)|Dn ) to guide the
search for the global maximizer x? . In particular, p(f (x)|Dn ) is used during the computation of an
acquisition function ?n (x) that is optimized at each iteration to determine the next evaluation location xn+1 . This process is shown in Algorithm 1. Intuitively, the acquisition function ?n (x) should
be high in areas where the maxima is most likely to lie given the current data. However, ?n (x)
eN is a
should also encourage exploration of the search space to guarantee that the recommendation x
global optimum of f , not just a global optimum of the posterior mean. Several acquisition functions
have been proposed in the literature. Some examples are the probability of improvement (PI) [14],
the expected improvement (EI) [19, 13] or upper confidence bounds (UCB) [26]. Alternatively, one
can combine several of these acquisition functions [10].
The acquisition functions described above are based on probabilistic measures of improvement (PI
an EI) or on optimistic estimates of the latent function (UCB) which implicitly trade off between
exploiting the posterior mean and exploring based on the uncertainty. An alternate approach, introduced by [28], proposes maximizing the expected posterior information gain about the global
maximizer x? evaluated over a grid in the input space. A similar strategy was later employed by [9]
which although it requires no such grid, instead relies on a difficult-to-evaluate approximation. In
Section 2 we derive a rearrangement of this information-based acquisition function which leads to a
more straightforward approximation that we call Predictive Entropy Search (PES). In Section 3 we
show empirically that our approximation is more accurate than that of [9]. We evaluate this claim on
both synthetic and real-world problems and further show that this leads to real gains in performance.
2
Predictive entropy search
We propose to follow the information-theoretic method for active data collection described in [17].
We are interested in maximizing information about the location x? of the global maximum, whose
posterior distribution is p(x? |Dn ). Our current information about x? can be measured in terms
of the negative differential entropy of p(x? |Dn ). Therefore, our strategy is to select xn+1 which
maximizes the expected reduction in this quantity. The corresponding acquisition function is
?n (x) = H[p(x? |Dn )] ? Ep(y|Dn ,x) [H[p(x? |Dn ? {(x, y)})]] ,
(2)
R
where H[p(x)] = ? p(x) log p(x)dx represents the differential entropy of its argument and the
expectation above is taken with respect to the posterior predictive distribution of y given x. The
exact evaluation of (2) is infeasible in practice. The main difficulties are i) p(x? |Dn ? {(x, y)})
must be computed for many different values of x and y during the optimization of (2) and ii) the
entropy computations themselves are not analytical. In practice, a direct evaluation of (2) is only
2
possible after performing many approximations [9]. To avoid this, we follow the approach described
in [11] by noting that (2) can be equivalently written as the mutual information between x? and y
given Dn . Since the mutual information is a symmetric function, ?n (x) can be rewritten as
?n (x) = H[p(y|Dn , x)] ? Ep(x? |Dn ) [H[p(y|Dn , x, x? )]] ,
(3)
where p(y|Dn , x, x? ) is the posterior predictive distribution for y given the observed data Dn and
the location of the global maximizer of f . Intuitively, conditioning on the location x? pushes the
posterior predictions up in locations around x? and down in regions away from x? . Note that, unlike
the previous formulation, this objective is based on the entropies of predictive distributions, which
are analytic or can be easily approximated, rather than on the entropies of distributions on x? whose
approximation is more challenging.
The first term in (3) can be computed analytically using the posterior marginals for f (x) in (1), that
is, H[p(y|Dn , x)] = 0.5 log[2?e (vn (x) + ? 2 )], where we add ? 2 to vn (x) because y is obtained
by adding Gaussian noise with variance ? 2 to f (x). The second term, on the other hand, must be
(i)
approximated. We first approximate the expectation in (3) by averaging over samples x? drawn
approximately from p(x? |Dn ). For each of these samples, we then approximate the corresponding
(i)
entropy function H[p(y|Dn , x, x? )] using expectation propagation [18]. The code for all these
operations is publicly available at http://jmhl.org.
2.1
Sampling from the posterior over global maxima
In this section we show how to approximately sample from the conditional distribution of the global
maximizer x? given the observed data Dn , that is,
p(x? |Dn ) = p f (x? ) = max f (x) Dn .
(4)
x?X
If the domain X is restricted to some finite set of m points, the latent function f takes the form
of an m-dimensional
vector f . The probability that the ith element of f is optimal can then be
R
Q
written as p(f |Dn ) j?m I[fi ? fj ] df . This suggests the following generative process: i) draw
a sample from the posterior distribution p(f |Dn ) and ii) return the index of the maximum element
in the sampled vector. This process is known as Thompson sampling or probability matching when
used as an arm-selection strategy in multi-armed bandits [8]. This same approach could be used for
sampling the maximizer over a continuous domain X . At first glance this would require constructing
an infinite-dimensional object representing the function f . To avoid this, one could sequentially
construct f while it is being optimized. However, evaluating such an f would ultimately have cost
O(m3 ) where m is the number of function evaluations necessary to find the optimum. Instead,
we propose to sample and optimize an analytic approximation to f . We will briefly derive this
approximation below, but more detail is given in Appendix A.
Given a shift-invariant kernel k, Bochner?s theorem [4] asserts the existence of its Fourier dual s(w),
which is equal to the spectral density of k. Letting p(w) = s(w)/? be the associated normalized
density, we can write the kernel as the expectation
T
0
k(x, x0 ) = ? Ep(w) [e?iw (x?x ) ] = 2? Ep(w,b) [cos(wT x + b) cos(wT x0 + b)] ,
(5)
p
where b ? U[0, 2?]. Let ?(x) = 2?/m cos(Wx + b) denote an m-dimensional feature mapping
where W and b consist of m stacked samples from p(w, b). The kernel k can then be approximated
by the inner product of these features, k(x, x0 ) ? ?(x)T ?(x0 ). This approach was used by [21]
as an approximation method in the context of kernel methods. The feature mapping ?(x) allows
us to approximate the Gaussian process prior for f with a linear model f (x) = ?(x)T ? where
? ? N (0, I) is a standard Gaussian. By conditioning on Dn , the posterior for ? is also multivariate
Gaussian, ?|Dn ? N (A?1 ?T yn , ? 2 A?1 ) where A = ?T ? + ? 2 I and ?T = [?(x1 ) . . . ?(xn )].
Let ?(i) and ? (i) be a random set of features and the corresponding posterior weights sampled both
according to the generative process given above. They can then be used to construct the function
f (i) (x) = ?(i) (x)T ? (i) , which is an approximate posterior sample of f ?albeit one with a finite
(i)
parameterization. We can then maximize this function to obtain x? = arg maxx?X f (i) (x), which
is approximately distributed according to p(x? |Dn ). Note that for early iterations when n < m, we
can efficiently sample ? (i) with cost O(n2 m) using the method described in Appendix B.2 of [23].
This allows us to use a large number of features in ?(i) (x).
3
2.2
Approximating the predictive entropy
We now show how to approximate H[p(y|D
R n , x, x? )] in (3). Note that we can write the argument to
H in this expression as p(y|Dn , x, x? ) = p(y|f (x))p(f (x)|Dn , x? ) df (x). Here p(f (x)|Dn , x? )
is the posterior distribution on f (x) given Dn and the location x? of the global maximizer of f .
When the likelihood p(y|f (x)) is Gaussian, we have that p(f (x)|Dn ) is analytically tractable since it
is the predictive distribution of a Gaussian process. However, by further conditioning on the location
x? of the global maximizer we are introducing additional constraints, namely that f (z) ? f (x? )
for all z ? X . These constraints make p(f (x)|Dn , x? ) intractable. To circumvent this difficulty, we
instead use the following simplified constraints:
C1. x? is a local maximum. This is achieved by letting ?f (x? ) = 0 and ensuring that
?2 f (x? ) is negative definite. We further assume that the non-diagonal elements of
?2 f (x? ), denoted by upper[?2 f (x? )], are known, for example they could all be zero.
This simplifies the negative-definite constraint. We denote by C1.1 the constraint given by
?f (x? ) = 0 and upper[?2 f (x? )] = 0. We denote by C1.2 the constraint that forces the
elements of diag[?2 f (x? )] to be negative.
C2. f (x? ) is larger than past observations. We also assume that f (x? ) ? f (xi ) for all
i ? n. However, we only observe f (xi ) noisily via yi . To avoid making inference on these
latent function values, we approximate the above hard constraints with the soft constraint
f (x? ) > ymax + , where ? N (0, ? 2 ) and ymax is the largest yi seen so far.
C3. f (x) is smaller than f (x? ). This simplified constraint only conditions on the given x
rather than requiring f (x? ) ? f (z) for all z ? X .
We incorporate these simplified constraints into p(f (x)|Dn ) to approximate p(f (x)|Dn , x? ). This
is achieved by multiplying p(f (x)|Dn ) with specific factors that encode the above constraints. In
what follows we briefly show how to construct these factors; more detail is given in Appendix B.
Consider the latent variable z = [f (x? ); diag[?2 f (x? )]]. To incorporate constraint C1.1 we
can condition on the data and on the ?observations? given by the constraints ?f (x? ) = 0 and
upper[?2 f (x? )] = 0. Since f is distributed according to a GP, the joint distribution between z and
these observations is multivariate Gaussian. The covariance between the noisy observations yn and
the extra noise-free derivative observations can be easily computed [25]. The resulting conditional
distribution is also multivariate Gaussian with mean m0 and covariance V0 . These computations
are similar to those performed in (1). Constraints C1.2 and C2 can then be incorporated by writing
hQ
i
d
2
p(z|Dn , C1, C2) ? ??2 (f (x? ) ? ymax )
(6)
i=1 I [? f (x? )]ii ? 0 N (z|m0 , V0 ) ,
where ??2 is the cdf of a zero-mean Gaussian distribution with variance ? 2 . The first new factor in
this expression guarantees that f (x? ) > ymax + , where we have marginalized out, and the second
set of factors guarantees that the entries in diag[?2 f (x? )] are negative.
Later integrals that make use of p(z|Dn , C1, C2), however, will not admit a closed-form expression. As a result we compute a Gaussian approximation q(z) to this distribution using Expectation
Propagation (EP) [18]. The resulting algorithm is similar to the implementation of EP for binary
classification with Gaussian processes [22]. EP approximates each non-Gaussian factor in (6) with
a Gaussian factor whose mean and variance are m
e i and vei , respectively. The EP approximation can
Qd+1
e i , vei )]N (z|m0 , V0 ). Note that these computations have so
then be written as q(z) ? [ i=1 N (zi |m
e v
e } once and store them for later use, where
far not depended on x, so we can compute {m0 , V0 , m,
e = (m
e = (?
m
? 1, . . . , m
? d+1 ) and v
v1 , . . . , v?d+1 ).
We will now describe how to compute the predictive variance of some latent function value f (x)
given these constraints. Let f = [f (x); f (x? )] be a vector given by the concatenation of the values
of the latent function at x and x? . The joint distribution between f , z, the evaluations yn collected
so far and the derivative ?observations? ?f (x? ) = 0 and upper[?2 f (x? )] = 0 is multivariate
Gaussian. Using q(z), we then obtain the following approximation:
R
p(f |Dn , C1, C2) ? p(f |z, Dn , C1.1) q(z) dz = N (f |mf , Vf ) .
(7)
Implicitly we are assuming above that f depends on our observations and constraint C1.1, but is
independent of C1.2 and C2 given z. The computations necessary to obtain mf and Vf are similar
4
to those used above and in (1). The required quantities are similar to the ones used by EP to make
predictions in the Gaussian process binary classifier [22]. We can then incorporate C3 by multiplying
N (f |mf , Vf ) with a factor that guarantees f (x) < f (x? ). The predictive distribution for f (x) given
Dn and all the constraints can be approximated as
R
p(f (x)|Dn , C1, C2, C3) ? Z ?1 I(f1 < f2 ) N (f |mf , Vf ) df2 ,
(8)
where Z is a normalization constant. The variance of the right hand size of (8) is given by
vn (x|x? ) = [Vf ]1,1 ? v ?1 ?(? + ?){[Vf ]1,1 ? [Vf ]1,2 }2 ,
(9)
?
T
T
where v = [?1, 1] Vf [?1, 1], ? = m/ v, m = [?1, 1] mf , ? = ?(?)/?(?), and ?(?) and
?(?) are the standard Gaussian density function and cdf, respectively. By further approximating (8) by a Gaussian distribution with the same mean and variance we can write the entropy as
H[p(y|Dn , x, x? )] ? 0.5 log[2?e(vn (x|x? ) + ? 2 )].
The computation of (9) can be numerically unstable when s is very close to zero. This occurs when
[Vf ]1,1 is very similar to [Vf ]1,2 . To avoid these numerical problems, we multiply [Vf ]1,2 by the
largest 0 ? ? ? 1 that guarantees that s > 10?10 . This can be understood as slightly reducing
the amount of dependence between f (x) and f (x? ) when x is very close to x? . Finally, fixing
upper[?2 f (x? )] to be zero can also produce poor predictions when the actual f does not satisfy
this constraint. To avoid this, we instead fix this quantity to upper[?2 f (i) (x? )], where f (i) is the
(i)
ith sample function optimized in Section 2.1 to sample x? .
2.3
Hyperparameter learning and the PES acquisition function
We now show how the previous approximations are integrated to compute the acquisition function
used by predictive entropy search (PES). This acquisition function performs a formal treatment of the
hyperparameters. Let ? denote a vector of hyperparameters which includes any kernel parameters
as well as the noise variance ? 2 . Let p(?|Dn ) ? p(?) p(Dn |?) denote the posterior distribution
over these parameters where p(?) is a hyperprior and p(Dn |?) is the GP marginal likelihood. For a
fully Bayesian treatment of ? we must marginalize the acquisition function (3) with respect to this
posterior. The corresponding integral has no analytic expression and must be approximated using
Monte Carlo. This approach is also taken in [24].
(i)
We draw M samples {? (i) } from p(?|Dn ) using slice sampling [27]. Let x? denote a sampled
(i)
global maximizer drawn from p(x? |Dn , ? (i) ) as described in Section 2.1. Furthermore, let vn (x)
(i)
(i)
and vn (x|x? ) denote the predictive variances computed as described in Section 2.2 when the
model hyperparameters are fixed to ? (i) . We then write the marginalized acquisition function as
o
PM n
(i)
(i)
(i)
1
2
2
?n (x) = M
(10)
i=1 0.5 log[vn (x) + ? ] ? 0.5 log[vn (x|x? ) + ? ] .
Note that PES is effectively marginalizing the original acquisition function (2) over p(?|Dn ). This
is a significant advantage with respect to other methods that optimize the same information-theoretic
acquisition function but do not marginalize over the hyper-parameters. For example, the approach
of [9] approximates (2) only for fixed ?. The resulting approximation is computationally very expensive and recomputing it to average over multiple samples from p(?|Dn ) is infeasible in practice.
Algorithm 2 shows pseudo-code for computing the PES acquisition function. Note that most of the
computations necessary for evaluating (10) can be done independently of the input x, as noted in the
pseudo-code. This initial cost is dominated by a matrix inversion necessary to pre-compute V for
each hyperparameter sample. The resulting complexity is O[M (n+d+d(d?1)/2)3 ]. This cost can
be reduced to O[M (n + d)3 ] by ignoring the derivative observations imposed on upper[?2 f (x? )]
by constraint C1.1. Nevertheless, in the problems that we consider d is very small (less than 20).
After these precomputations are done, the evaluation of (10) is O[M (n + d + d(d ? 1)/2)].
3
Experiments
In our experiments, wePuse Gaussian process priors for f with squared-exponential kernels
k(x, x0 ) = ? 2 exp{?0.5 i (xi ? x0i )2 /`2i }. The corresponding spectral density is zero-mean Gaus2
sian with covariance given by diag([`?2
i ]) and normalizing constant ? = ? . The model hyperpa2
rameters are {?, `1 , . . . , `d , ? }. We use broad, uninformative Gamma hyperpriors.
5
1
0.0
0.35
x
x
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x
0.05
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x
0.01
0.03
0.00
0.00
Figure 1: Comparison of different estimates of the objective function ?n (x) given by (2). Left, ground truth
obtained by the rejection sampling method RS. Middle, approximation produced by the ES method. Right,
approximation produced by the proposed PES method. These plots show that the PES objective is much more
similar to the RS ground truth than the ES objective.
First, we analyze the accuracy of PES in the task of approximating the differential entropy (2).
We compare the PES approximation (10), with the approximation used by the entropy search (ES)
method [9]. We also compare with the ground truth for (2) obtained using a rejection sampling (RS)
algorithm based on (3). For this experiment we generate the data Dn using an objective function f
sampled from the Gaussian process prior as in [9]. The domain X of f is fixed to be [0, 1]2 and data
are generated using ? 2 = 1, ? 2 = 10?6 , and `2i = 0.1. To compute (10) we avoid sampling the
hyperparameters and use the known values directly. We further fix M = 200 and m = 1000.
The ground truth rejection sampling scheme works as follows. First, X is discretized using a uniform
grid. The expectation with respect to p(x? |Dn ) in (3) is then approximated using sampling. For this,
we sample x? by evaluating a random sample from p(f |Dn ) on each grid cell and then selecting the
cell with highest value. Given x? , we then approximate H[p(y|Dn , x, x? )] by rejection sampling.
We draw samples from p(f |Dn ) and reject those whose corresponding grid cell with highest value is
not x? . Finally, we approximate H[p(y|Dn , x, x? )] by first, adding zero-mean Gaussian noise with
variance ? 2 to the the evaluations at x of the functions not rejected during the previous step and
second, we estimate the differential entropy of the resulting samples using kernels [1].
Figure 1 shows the objective functions produced by RS, ES and PES for a particular Dn with 10
measurements whose locations are selected uniformly at random in [0, 1]2 . The locations of the
collected measurements are displayed with an ?x? in the plots. The particular objective function
used to generate the measurements in Dn is displayed in the left part of Figure 2. The plots in
Figure 1 show that the PES approximation to (2) is more similar to the ground truth given by RS
than the approximation produced by ES. In this figure we also see a discrepancy between RS and
PES at locations near x = (0.572, 0.687). This difference is an artifact of the discretization used in
RS. By zooming in and drawing many more samples we would see the same behavior in both plots.
We now evaluate the performance of PES in the task of finding the optimum of synthetic black-box
objective functions. For this, we reproduce the within-model comparison experiment described in
[9]. In this experiment we optimize objective functions defined in the 2-dimensional unit domain
X = [0, 1]2 . Each objective function is generated by first sampling 1024 function values from the
GP prior assumed by PES, using the same ? 2 , `i and ? 2 as in the previous experiment. The objective
function is then given by the resulting GP posterior mean. We generated a total of 1000 objective
functions by following this procedure. The left plot in Figure 2 shows an example function.
In these experiments we compared the performance of PES with that of ES [9] and expected improvement (EI) [13], a widely used acquisition function in the Bayesian optimization literature. We
again assume that the optimal hyper-parameter values are known to all methods. Predictive performance is then measured in terms of the immediate regret (IR) |f (e
xn ) ? f (x? )|, where x? is the
en is the recommendation of each algorithm had we
known location of the global maximum and x
stopped at step n?for all methods this is given by the maximizer of the posterior mean. The right
plot in Figure 2 shows the decimal logarithm of the median of the IR obtained by each method
across the 1000 different objective functions. Confidence bands equal to one standard deviation are
obtained using the bootstrap method. Note that while averaging these results is also interesting,
corresponding to the expected performance averaged over the prior, here we report the median IR
6
1.5
0
0.5
2
? ?
?
1.5
2
?0.5
0.5
1
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1
x
x
Log10 Median IR
x
Results on Synthetic Cost Functions
1
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x
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10
20
30
40
Number of Function Evaluations
50
Figure 2: Left, example of objective functions f . Right, median of the immediate regret (IR) for the methods
PES, ES and EI in the experiments with synthetic objective functions.
Results on Cosines Cost Function
Results on Branin Cost Function
?
?
Results on Hartmann Cost Function
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?
40
50
Number of Function Evaluations
Figure 3: Median of the immediate regret (IR) for the methods EI, ES, PES and PES-NB in the experiments
with well-known synthetic benchmark functions.
because the empirical distribution of IR values is very heavy-tailed. In this case, the median is more
representative of the exact location of the bulk of the data. These results indicate that the best method
in this setting is PES, which significantly outperforms ES and EI. The plot also shows that in this
case ES is significantly better than EI.
We perform another series of experiments in which we optimize well-known synthetic benchmark
functions including a mixture of cosines [2] and Branin-Hoo (both functions defined in [0, 1]2 ) as
well as the Hartmann-6 (defined in [0, 1]6 ) [15]. In all instances, we fix the measurement noise
to ? 2 = 10?3 . For both PES and EI we marginalize the hyperparameters ? using the approach
described in Section 2.3. ES, by contrast, cannot average its approximation of (2) over the posterior
on ?. Instead, ES works by fixing ? to an estimate of its posterior mean (obtained using slice
sampling) [27]. To evaluate the gains produced by the fully Bayesian treatment of ? in PES, we also
compare with a version of PES (PES-NB) which performs the same non-Bayesian (NB) treatment
of ? as ES. In PES-NB we use a single fixed hyperparameter as in previous sections with value
given by the posterior mean of ?. All the methods are initialized with three random measurements
collected using latin hypercube sampling [5].
The plots in Figure 3 show the median IR obtained by each method on each function across 250
random initializations. Overall, PES is better than PES-NB and ES. Furthermore, PES-NB is also
significantly better than ES in most of the cases. These results show that the fully Bayesian treatment
of ? in PES is advantageous and that PES can produce better approximations than ES. Note that
PES performs better than EI in the Branin and cosines functions, while EI is significantly better on
the Hartmann problem. This appears to be due to the fact that entropy-based strategies explore more
aggressively which in higher-dimensional spaces takes more iterations. The Hartmann problem,
however, is a relatively simple problem and as a result the comparatively more greedy behavior of
EI does not result in significant adverse consequences. Note that the synthetic functions optimized
in the previous experiment were much more multimodal that the ones considered here.
3.1
Experiments with real-world functions
We finally optimize different real-world cost functions. The first one (NNet) returns the predictive
accuracy of a neural network on a random train/test partition of the Boston Housing dataset [3].
7
Hydrogen
NNet Cost
Portfolio
Methods
? ?
0.6
EI
Log10 Median IR
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Figure 4: Median of the immediate regret (IR) for the methods PES, PES-NB, ES and EI in the experiments
with non-analytic real-world cost functions.
The variables to optimize are the weight-decay parameter and the number of training iterations for
the neural network. The second function (Hydrogen) returns the amount of hydrogen production of
a particular bacteria in terms of the PH and Nitrogen levels of the growth medium [7]. The third
one (Portfolio) returns the ratio of the mean and the standard deviation (the Sharpe ratio) of the
1-year ahead returns generated by simulations from a multivariate time-series model that is adjusted
to the daily returns of stocks AXP, BA and HD. The time-series model is formed by univariate
GARCH models connected with a Student?s t copula [12]. These three functions (NNet, Hydrogen
and Portfolio) have as domain [0, 1]2 . Furthermore, in these examples, the ground truth function
that we want to optimize is unknown and is only available through noisy measurements. To obtain
a ground truth, we approximate each cost function as the predictive distribution of a GP that is
adjusted to data sampled from the original function (1000 uniform samples for NNet and Portfolio
and all the available data for Hydrogen [7]). Finally, we also consider another real-world function
that returns the walking speed of a bipedal robot [30]. This function is defined in [0, 1]8 and its
inputs are the parameters of the robot?s controller. In this case the ground truth function is noiseless
and can be exactly evaluated through expensive numerical simulation. We consider two versions of
this problem (Walker A) with zero-mean, additive noise of ? = 0.01 and (Walker B) with ? = 0.1.
Figure 4 shows the median IR values obtained by each method on each function across 250 random
initializations, except in Hydrogen where we used 500 due to its higher level of noise. Overall, PES,
ES and PES-NB perform similarly in NNet, Hydrogen and Portfolio. EI performs rather poorly in
these first three functions. This method seems to make excessively greedy decisions and fails to
explore the search space enough. This strategy seems to be advantageous in Walker A, where EI
obtains the best results. By contrast, PES, ES and PES-NB tend to explore more in this latter dataset.
This leads to worse results than those of EI. Nevertheless, PES is significantly better than PES-NB
and ES in both Walker datasets and better than EI in the noisier Walker B. In this case, the fully
Bayesian treatment of hyper-parameters performed by PES produces improvements in performance.
4
Conclusions
We have proposed a novel information-theoretic approach for Bayesian optimization. Our method,
predictive entropy search (PES), greedily maximizes the amount of one-step information on the location x? of the global maximum using its posterior differential entropy. Since this objective function
is intractable, PES approximates the original objective using a reparameterization that measures
entropy in the posterior predictive distribution of the function evaluations. PES produces more accurate approximations than Entropy Search (ES), a method based on the original, non-transformed
acquisition function. Furthermore, PES can easily marginalize its approximation with respect to
the posterior distribution of its hyper-parameters, while ES cannot. Experiments with synthetic and
real-world functions show that PES often outperforms ES in terms of immediate regret. In these experiments, we also observe that PES often produces better results than expected improvement (EI),
a popular heuristic for Bayesian optimization. EI often seems to make excessively greedy decisions, while PES tends to explore more. As a result, EI seems to perform better for simple objective
functions while often getting stuck with noisier objectives or for functions with many modes.
Acknowledgements J.M.H.L acknowledges support from the Rafael del Pino Foundation.
8
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9
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4,777 | 5,325 | Submodular meets Structured: Finding Diverse
Subsets in Exponentially-Large Structured Item Sets
Adarsh Prasad
UT Austin
[email protected]
Stefanie Jegelka
UC Berkeley
[email protected]
Dhruv Batra
Virginia Tech
[email protected]
Abstract
To cope with the high level of ambiguity faced in domains such as Computer
Vision or Natural Language processing, robust prediction methods often search
for a diverse set of high-quality candidate solutions or proposals. In structured
prediction problems, this becomes a daunting task, as the solution space (image
labelings, sentence parses, etc.) is exponentially large. We study greedy algorithms for finding a diverse subset of solutions in structured-output spaces by
drawing new connections between submodular functions over combinatorial item
sets and High-Order Potentials (HOPs) studied for graphical models. Specifically,
we show via examples that when marginal gains of submodular diversity functions
allow structured representations, this enables efficient (sub-linear time) approximate maximization by reducing the greedy augmentation step to inference in a
factor graph with appropriately constructed HOPs. We discuss benefits, tradeoffs, and show that our constructions lead to significantly better proposals.
1
Introduction
Many problems in Computer Vision, Natural Language Processing and Computational Biology involve mappings from an input space X to an exponentially large space Y of structured outputs.
For instance, Y may be the space of all segmentations of an image with n pixels, each of which
may take L labels, so |Y| = Ln . Formulations such as Conditional Random Fields (CRFs) [24],
Max-Margin Markov Networks (M3 N) [31], and Structured Support Vector Machines (SSVMs) [32]
have successfully provided principled ways of scoring all solutions y ? Y and predicting the single
highest scoring or maximum a posteriori (MAP) configuration, by exploiting the factorization of a
structured output into its constituent ?parts?.
In a number of scenarios, the posterior P(y|x) has several modes due to ambiguities, and we seek
not only a single best prediction but a set of good predictions:
(1) Interactive Machine Learning. Systems like Google Translate (for machine translation) or
Photoshop (for interactive image segmentation) solve structured prediction problems that are often
ambiguous ("what did the user really mean?"). Generating a small set of relevant candidate solutions
for the user to select from can greatly improve the results.
(2) M-Best hypotheses in cascades. Machine learning algorithms are often cascaded, with the
output of one model being fed into another [33]. Hence, at the initial stages it is not necessary to
make a single perfect prediction. We rather seek a set of plausible predictions that are subsequently
re-ranked, combined or processed by a more sophisticated mechanism.
In both scenarios, we ideally want a small set of M plausible (i.e., high scoring) but non-redundant
(i.e., diverse) structured-outputs to hedge our bets.
Submodular Maximization and Diversity. The task of searching for a diverse high-quality subset of items from a ground set V has been well-studied in information retrieval [5], sensor placement [22], document summarization [26], viral marketing [17], and robotics [10]. Across these
domains, submodularity has emerged as an a fundamental and practical concept ? a property of
functions for measuring diversity of a subset of items. Specifically, a set function F : 2V ? R is
submodular if its marginal gains, F (a|S) ? F (S ?a)?F (S) are decreasing, i.e. F (a|S) ? F (a|T )
1
argmax F (a | S) ?
a?V
(a) Image
(b) All segmentations: |V | = Ln
r(y)
+
d(y | S)
(c) Structured Representation.
Figure 1: (a) input image; (b) space of all possible object segmentations / labelings (each item is a segmentation); (c) we convert the problem of finding the item with the highest marginal gain F (a|S) to a MAP inference
problem in a factor graph over base variables y with an appropriately defined HOP.
for all S ? T and a ?
/ T . In addition, if F is monotone, i.e., F (S) ? F (T ), ?S ? T , then a simple
greedy algorithm (that in each iteration t adds to the current set S t the item with the largest marginal
gain F (a|S t )) achieves an approximation factor of (1 ? 1e ) [27]. This result has had significant
practical impact [21]. Unfortunately, if the number of items |V | is exponentially large, then even a
single linear scan for greedy augmentation is infeasible.
In this work, we study conditions under which it is feasible to greedily maximize a submodular
function over an exponentially large ground set V = {v1 , . . . , vN } whose elements are combinatorial objects, i.e., labelings of a base set of n variables y = {y1 , y2 , . . . , yn }. For instance, in image
segmentation, the base variables yi are pixel labels, and each item a ? V is a particular labeling of
the pixels. Or, if each base variable ye indicates the presence or absence of an edge e in a graph,
then each item may represent a spanning tree or a maximal matching. Our goal is to find a set of
M plausible and diverse configurations efficiently, i.e. in time sub-linear in |V | (ideally scaling as
a low-order polynomial in log |V |). We will assume F (?) to be monotone submodular, nonnegative
and normalized (F (?) = 0), and base our study on the greedy algorithm. As a running example, we
focus on pixel labeling, where each base variable takes values in a set [L] = {1, . . . , L} of labels.
Contributions. Our principal contribution is a conceptual one. We observe that marginal gains of a
number of submodular functions allow structured representations, and this enables efficient greedy
maximization over exponentially large ground sets ? by reducing the greedy augmentation step to
a MAP inference query in a discrete factor graph augmented with a suitably constructed HighOrder Potential (HOP). Thus, our work draws new connections between two seemingly disparate
but highly related areas in machine learning ? submodular maximization and inference in graphical
models with structured HOPs. As specific examples, we construct submodular functions for three
different, task-dependent definitions of diversity, and provide reductions to three different HOPs for
which efficient inference techniques have already been developed. Moreover, we present a generic
recipe for constructing such submodular functions, which may be ?plugged? with efficient HOPs
discovered in future work. Our empirical contribution is an efficient algorithm for producing a set of
image segmentations with significantly higher oracle accuracy1 than previous works. The algorithm
is general enough to transfer to other applications. Fig. 1 shows an overview of our approach.
Related work: generating multiple solutions. Determinental Point Processesare an elegant probabilistic model over sets of items with a preference for diversity. Its generalization to a structured
setting [23] assumes a tree-structured model, an assumption that we do not make. Guzman-Rivera et
al. [14, 15] learn a set of M models, each producing one solution, to form the set of solutions. Their
approach requires access to the learning sub-routine and repeated re-training of the models, which
is not always possible, as it may be expensive or proprietary. We assume to be given a single (pretrained) model from which we must generate multiple diverse, good solutions. Perhaps the closest
to our setting are recent techniques for finding diverse M -best solutions [2, 28] or modes [7, 8]
in graphical models. While [7] and [8] are inapplicable since they are restricted to chain and tree
graphs, we compare to other baselines in Section 3.2 and 4.
1.1 Preliminaries and Notation
We select from a ground set V of N items. Each item is a labeling y = {y1 , y2 , . . . , yn }
of n base variables. For clarity, we use non-bold letters a ? V for items, and boldface letters y for base set configurations. Uppercase letters refer to functions over the ground set items
F (a|A), R(a|A), D(a|A), and lowercase letters to functions over base variables f (y), r(y), d(y).
1
The accuracy of the most accurate segmentation in the set.
2
Formally, there is a bijection ? : V 7? [L]m that maps items a ? V to their representation as base
variable labelings y = ?(a). For notational simplicity, we often use y ? S to mean ??1 (y) ? S,
i.e. the item corresponding to the labeling y is present in the set S ? V . We write ` ? y if the label
` is used in y, i.e. ?j s.t. yj = `. For a set c ? [n], we use yc to denote the tuple {yi | i ? c}.
Our goal to find an ordered set or list of items S ? V that maximizes a scoring function F . Lists
generalize the notation of sets, and allow for reasoning of item order and repetitions. More details
about list vs set prediction can be found in [29, 10].
Scoring Function. We trade off the relevance and diversity of list S ? V via a scoring function
F : 2V ? R of the form
F (S) = R(S) + ?D(S),
(1)
P
where R(S) = a?S R(a) is a modular nonnegative relevance function that aggregates the quality
of all items in the list; D(S) is a monotone normalized submodular function that measure the diversity of items in S; and ? ? 0 is a trade-off parameter. Similar objective functions were used e.g. in
[26]. They are reminiscent of the general paradigm in machine learning of combining a loss function that measures quality (e.g. training error) and a regularization term that encourages desirable
properties (e.g. smoothness, sparsity, or ?diversity?).
Submodular Maximization. We aim to find a list S that maximizes F (S) subject to a cardinality
constraint |S| ? M . For monotone submodular F , this may be done via a greedy algorithm that
starts out with S 0 = ?, and iteratively adds the next best item:
S t = S t?1 ? at ,
at ? argmaxa?V F (a | S t?1 ).
(2)
The final solution S M is within a factor of (1 ? 1e ) of the optimal solution S ? : F (S M ) ? (1 ?
1
?
e )F (S ) [27]. The computational bottleneck is that in each iteration, we must find the item with
the largest marginal gain. Clearly, if |V | has exponential size, we cannot touch each item even once.
Instead, we propose ?augmentation sub-routines? that exploit the structure of V and maximize the
marginal gain by solving an optimization problem over the base variables.
2
Marginal Gains in Configuration Space
To solve the greedy augmentation step via optimization over y, we transfer the marginal gain from
the world of items to the world of base variables and derive functions on y from F :
(3)
F (??1 (y) | S) = R(??1 (y)) +? D(??1 (y) | S) .
{z
} | {z }
|
{z
}
|
f (y|S)
r(y)
d(y|S)
Maximizing F (a|S) now means maximizing f (y|S) for y = ?(a). This can be a hard combinatorial
optimization problem in general. However, as we will see, there is a broad class of useful functions
F for which f inherits exploitable structure, and argmaxy f (y|S) can be solved efficiently, exactly
or at least approximately.
Relevance Function. We use a structured relevance function R(a) that is the score of a factor graph
defined over the base variables y. Let G = (V, E) be a graph defined over {y1 , y2 , . . . , yn }, i.e.
V = [n], E ? V2 . Let C = {C | C ? V} be a set of cliques in the graph, and let ?C : [L]|C| 7? R
be the log-potential functions (or factors)
for these cliques. The quality of an item a = ??1 (y)
P
is then given by R(a) = r(y)
P = C?C ?C (y
PC ). For instance, with only node and edge factors,
this quality becomes r(y) = p?V ?p (yp ) + (p,q)?E ?pq (yp , yq ). In this model, finding the single
highest quality item corresponds to maximum a posteriori (MAP) inference in the factor graph.
Although we refer to terms with probabilistic interpretations such as ?MAP?, we treat our relevance
function P
as output of an energy-based model [25] such as a Structured SVM [32]. For instance,
r(y) = C?C ?C (yC ) = w| ?(y) for parameters w and feature vector ?(y). Moreover, we assume that the relevance function r(y) is nonnegative2 . This assumption ensures that F (?) is monotone. If F is non-monotone, algorithms other than the greedy are needed [4, 12]. We leave this
generalization for future work. In most application domains the relevance function is learned from
data and thus our positivity assumption is not restrictive ? one can simply learn a positive relevance
function. For instance, in SSVMs, the relevance weights are learnt to maximize the margin between
the correct labeling and all incorrect ones. We show in the supplement that SSVM parameters that
assign nonnegative scores to all labelings achieve exactly the same hinge loss (and thus the same
generalization error) as without the nonnegativity constraint.
2
Strictly speaking, this condition is sufficient but not necessary. We only need nonnegative marginal gains.
3
Figure 2: Diversity via groups:
(a) groups defined by the presence of labels (i.e. #groups
= L); (b) groups defined by
Hamming balls around each
item/labeling (i.e. #groups =
Ln ). In each case, diversity is
measured by how many groups
are covered by a new item. See
text for details.
(a) Label Groups
3
(b) Hamming Ball Groups
Structured Diversity Functions
We next discuss a general recipe for constructing monotone submodular diversity functions D(S),
and for reducing their marginal gains to structured representations over the base variables
d(y|S).
S
Our scheme relies on constructing groups Gi that cover the ground set, i.e. V = i Gi . These
groups will be defined by task-dependent characteristics ? for instance, in image segmentation, G`
can be the set of all segmentations that contain label `. The groups can be overlapping. For instance,
if a segmentation y contains pixels labeled ?grass? and ?cow?, then y ? Ggrass and y ? Gcow .
Group Coverage: Count Diversity. Given V and a set of groups {Gi }, we measure the diversity
of a list S in terms of its group coverage, i.e., the number of groups covered jointly by items in S:
D(S) = i | Gi ? S 6= ? ,
(4)
where we define Gi ? S as the intersection of Gi with the set of unique items in S. It is easy to show
that this function is monotone submodular. If G` is the group of all segmentations that contain label
`, then the diversity measure of a list of segmentations S is the number of object labels that appear
in any a ? S. The marginal gain is the number of new groups covered by a:
D(a | S) = i | a ? Gi and S ? Gi = ? .
(5)
Thus, the greedy algorithm will try to find an item/segmentation that belongs to as many as yet
unused groups as possible.
Group Coverage: General Diversity. More generally, instead of simply counting the number of
groups covered by S, we can use a more refined decay
X
D(S) =
h Gi ? S .
(6)
i
where h is any nonnegative nondecreasing concave scalar function. This is a sum of submodular
functions and hence submodular.
Eqn. (4) is a special case of Eqn. (6) with h(y) = min{1, y}.
?
Other possibilities are ?, or log(1 + ?). For this general definition of diversity, the marginal gain is
X
D(a | S) =
h 1 + Gi ? S ? h Gi ? S .
(7)
i:Gi 3a
Since h is concave, the gain h 1 + Gi ? S ? h Gi ? S decreases as S becomes larger. Thus,
the marginal gain of an item a is proportional to how rare each group Gi 3 a is in the list S.
In each step of the greedy algorithm, we maximize r(y) + ?d(y|S). We already established a
structured representation of r(y) via a factor graph on y. In the next few subsections, we specify
three example definitions of groups Gi that instantiate three diversity functions D(S). For each
D(S), we show how the marginal gains D(a|S) can be expressed as a specific High-Order Potential
(HOP) d(y|S) in the factor graph over y. These HOPs are known to be efficiently optimizable, and
hence we can solve the augmentation step efficiently. Table 1 summarizes these connections.
Diversity and Parsimony. If the groups Gi are overlapping, some y can belong to many groups
simultaneously. While such a y may offer an immediate large gain in diversity, in many applications
it is more natural to seek a small list of complementary labelings rather than having all labels occur
in the same y. For instance, in image segmentation with groups defined by label presence (Sec. 3.1),
natural scenes are unlikely to contain many labels at the same time. Instead, the labels should
be spread across the selected labelings y ? S. Hence, we include a parsimony factor p(y) that
biases towards simpler labelings y. This term is a modular function and does not affect the diversity
functions directly. We next outline some example instantiations of the functions (4) and (6).
4
Groups (Gi )
Higher Order Potentials
Section 3.1
Labels
Label Cost
Supplement Label Transitions Co-operative Cuts
Section 3.2
Hamming Balls
Cardinality Potentials
Table 1: Different diversity functions and corresponding HOPs.
3.1 Diversity of Labels
For the first example, let G` be the set of all labelings y containing the label `, i.e. y ? G` if and only
if yj = ` for some j ? [n]. Such a diversity function arises in multi-class image segmentation ? if the
highest scoring segmentation contains ?sky? and ?grass?, then we would like to add complementary
segmentations that contain an unused class label, say ?sheep? or ?cow?.
Structured Representation of Marginal Gains. The marginal gain for this diversity function turns
out to be a HOP called label cost [9]. It penalizes each label that occurs in a previous segmentation.
Let lcountS (`) be the number of segmentations in S that contain label `. In the simplest case of
coverage diversity (4), the marginal gain provides a constant reward for every as yet unseen label `:
X
d(y | S) = ` | y ? G` , S ? G` = ? =
1.
(8)
`?y,lcountS (`)=0
For the general group coverage diversity (6), the gain becomes
d(y|S) =
X
X
h 1 + G` ? S ? h G` ? S =
h 1 + lcountS (`) ? h lcountS (`) .
`:G` 3y
`?y
Thus, d(y|S) rewards the presence of a label ` in y by an amount proportional to howP
rare ` is in
the segmentations already chosen in S. The parsimony factor in this setting is p(y) = `?y c(`).
In the simplest case, c(`) = ?1, i.e. we are charged a constant for every label used in y.
With this type of diversity (and parsimony terms), the greedy augmentation step is equivalent to
performing MAP inference in a factor graph augmented with label reward HOPs: argmaxy r(y) +
?(d(y | S) + p(y)). Delong et al. [9] show how to perform approximate MAP inference with such
label costs via an extension to the standard ?-expansion [3] algorithm.
Label Transitions. Label Diversity can be extended to reward not just the presence of previously
unseen labels, but also the presence of previously unseen label transitions (e.g., a person in front
of a car or a person in front of a house). Formally, we define one group G`,`0 per label pair `, `0 ,
and y ? G`,`0 if it contains two adjacent variables yi , yj with labels yi = `, yj = `0 . This diversity
function rewards the presence of a label pair (`, `0 ) by an amount proportional to how rare this pair
is in the segmentations that are part of S. For such functions, the marginal gain d(y|S) becomes
a HOP called cooperative cuts [16]. The inference algorithm in [19] gives a fully polynomial-time
approximation scheme for any nondecreasing, nonnegative h, and the exact gain maximizer for the
count function h(y) = min{1, y}. Further details may be found in the supplement.
3.2 Diversity via Hamming Balls
The label diversity function simply rewarded the presence of a label `, irrespective of which or how
many variables yi were assigned
that label. The next diversity function rewards a large Hamming disPn
tance Ham(y1 , y2 ) = i=1 [[yi1 6= yi2 ]] between configurations (where [[?]] is the Iverson bracket.)
Let Bk (y) denote the k-radius Hamming ball centered at y, i.e. B(y) = {y0 | Ham(y0 , y) ? k}.
The previous section constructed one group per label `. Now, we construct one group Gy for each
configuration y, which is the k-radius Hamming ball centered at y, i.e. Gy = Bk (y).
Structured Representation of Marginal Gains. For this diversity, the marginal gain d(y|S) becomes a HOP called cardinality potential [30]. For count group coverage, this becomes
d(y|S) = y0 | Gy0 ? (S ? y) 6= ? ? y0 | Gy0 ? S 6= ?
(9a)
[
[
h [
i
=
Bk (y0 ) ?
Bk (y0 ) = Bk (y) ? Bk (y) ?
Bk (y0 ) ,
(9b)
y0 ?S?y
y0 ?S
0
y0 ?S
i.e., the marginal gain of adding y is the number of new configurations y covered by the Hamming
ball centered at y. Since the size of the intersection of Bk (y) with a union of Hamming balls does
not have a straightforward structured representation, we maximize a lower bound on d(y|S) instead:
X
Bk (y) ? Bk (y0 )
(10)
d(y | S) ? dlb (y | S) ? Bk (y) ?
y0 ?S
5
This lower bound dlb (y|S) overcounts the intersection in Eqn. (9b) by summing the intersections
with each Bk (y0 ) separately. We can also interpret this lower bound as clipping the series arising
from the inclusion-exclusion principle to the first-order terms. Importantly, (10) depends on y only
via its Hamming distance to y0 . This is a cardinality potential that depends only on the number of
variables yi assigned to a particular label. Specifically, ignoring constant terms, the lower bound can
be
factors (one for each previous solution y0 ? S): dlb (y|S) =
P written as a summation of cardinality
b
0
0
0
y0 ?S ?y (y), where ?y (y) = |S| ? Iy (y), b is a constant (size of a k-radius Hamming ball), and
Iy0 (y) is the number of points in the intersection of k-radius Hamming balls centered at y0 and y.
With this approximation, the greedy step means performing MAP inference in a factor graph augmented with cardinality potentials: argmaxy r(y) + ?dlb (y|S). This may be solved via messagepassing, and all outgoing messages from cardinality factors can be computed in O(n log n) time
[30]. While this algorithm does not offer any approximation guarantees, it performs well in practice. A subtle point to note is that dlb (y|S) is always decreasing w.r.t. |S| but may become negative
due to over-counting. We can fix this by clamping dlb (y|S) to be greater than 0, but in our experiments this was unnecessary ? the greedy algorithm never chose a set where dlb (y|S) was negative.
Comparison to DivMBest. The greedy algorithm for Hamming diversity is similar in spirit to the
recent work of Batra et al. [2], who also proposed a greedy algorithm (DivMBest) for finding diverse
MAP solutions in graphical models. They did not provide any justification for greedy, and our
formulation sheds some light on their work. Similar to our approach, at
Peach greedy step, DivMBest
involves maximizing a diversity-augmented score: argmaxy r(y)+? y0 ?S ?y0 (y). However, their
Pn
diversity function grows linearly with the Hamming distance, ?y0 (y) = Ham(y0 , y) = i=1 [[yi0 6=
yi ]]. Linear diversity rewards are not robust, and tend to over-reward diversity. Our formulation uses
b
a robust diversity function ?y0 (y) = |S|
? Iy0 (y) that saturates as y moves far away from y0 .
In our experiments, we make the saturation behavior smoothly tunable via a parameter ?: Iy0 (y) =
0
e?? Ham(y ,y) . A larger ? corresponds to Hamming balls of smaller radius, and can be set to optimize
performance on validation data. We found this to work better than directly tuning the radius k.
4
Experiments
We apply our greedy maximization algorithms to two image segmentation problems: (1) interactive
binary segmentation (object cutout) (Section 4.1); (2) category-level object segmentation on the
PASCAL VOC 2012 dataset [11] (Section 4.2). We compare all methods by their respective oracle
accuracies, i.e. the accuracy of the most accurate segmentation in the set of M diverse segmentations
returned by that method. For a small value of M ? 5 to 10, a high oracle accuracy indicates that
the algorithm has achieved high recall and has identified a good pool of candidate solutions for
further processing in a cascaded pipeline. In both experiments, the label ?background? is typically
expected to appear somewhere in the image, and thus does not play a role in the label cost/transition
diversity functions. Furthermore, in binary segmentation there is only one non-background label.
Thus, we report results with Hamming diversity only (label cost and label transition diversities are
not applicable). For the multi-class segmentation experiments, we report experiments with all three.
Baselines. We compare our proposed methods against DivMBest [2], which greedily produces
diverse segmentation by explicitly adding a linear Hamming distance term to the factor graph. Each
Hamming term isP
decomposable along the variables yi and simply modifies the node potentials
0
? i ) = ?(yi )+?
?(y
y0 ?S [[yi 6= yi ]]. DivMBest has been shown to outperform techniques such as MBest-MAP [34, 1], which produce high scoring solutions without a focus on diversity, and samplingbased techniques, which produce diverse solutions without a focus on the relevance term [2]. Hence,
we do not include those methods here. We also report results for combining different diversity
functions via two operators: (?), where we generate the top M
k solutions for each of k diversity
functions and then concatenate these lists; and (?), where we linearly combine diversity functions
(with coefficients chosen by k-D grid search) and generate M solutions using the combined diversity.
4.1 Interactive segmentation
In interactive foreground-background segmentation, the user provides partial labels via scribbles.
One way to minimize interactions is for the system to provide a set of candidate segmentations for
the user to choose from. We replicate the experimental setup of [2], who curated 100 images from
the PASCAL VOC 2012 dataset, and manually provided scribbles on objects contained in them.
For each image, the relevance model r(y) is a 2-label pairwise CRF, with a node term for each
6
Label Cost (LC)
min{1,
?}
p
(?)
log(1 + ?)
MAP
M=5
M=15
42.35
42.35
42.35
45.43
45.72
46.28
45.58
50.01
50.39
Hamming Ball (HB)
DivMBest
HB
MAP
M=5
M=15
43.43
43.43
51.21
51.71
52.90
55.32
? Combined Diversity
HB ? LC ? LT
DivMBest ? HB ? LC ? LT
M=15
M=16
56.97
-
57.39
Label Transition (LT)
min{1,
?}
p
(?)
log(1 + ?)
MAP
M=5
M=15
42.35
42.35
42.35
44.26
45.43
45.92
44.78
46.21
46.89
? Combined Diversity
M=15
DivMBest ? HB
DivMBest ? LC ? LT
55.89
53.47
Table 2: PASCAL VOC 2012 val oracle accuracies for different diversity functions.
superpixel in the image and an edge term for each adjacent pair of superpixels. At each superpixel,
we extract colour and texture features. We train a Transductive SVM from the partial supervision
provided by the user scribbles. The node potentials are derived from the scores of these TSVMs. The
edge potentials are contrast-sensitive Potts. Fifty of the images were used for tuning the diversity
parameters ?, ?, and the other 50 for reporting oracle accuracies. The 2-label contrast-sensitive Potts
model results in a supermodular relevance function r(y), which can be efficiently maximized via
graph cuts [20]. The Hamming ball diversity dlb (y|S) is a collection of cardinality factors, which
we optimize with the Cyborg implementation [30].
Results. For each of the 50 test images in our dataset we generated the single best y1 and 5 additional solutions {y2 , . . . , y6 } using each method. Table 3 shows the average oracle accuracies for
DivMBest, Hamming ball diversity, and their two combinations. We can see that the combinations
slightly outperform both approaches.
DivMBest
Hamming Ball
DivMBest?Hamming Ball
DivMBest?Hamming Ball
MAP
M=2
M=6
91.57
91.57
-
93.16
93.95
-
95.02
94.86
95.16
95.14
Table 3: Interactive segmentation: oracle pixel accuracies averaged over 50 test images
4.2
Category level Segmentation
In category-level object segmentation, we label each pixel with one of 20 object categories or background. We construct a multi-label pairwise CRF on superpixels. Our node potentials are outputs of
category-specific regressors trained by [6], and our edge potentials are multi-label Potts. Inference
in the presence of diversity terms is performed with the implementations of Delong et al. [9] for
label costs, Tarlow et al. [30] for Hamming ball diversity, and Boykov et al. [3] for label transitions.
Figure 3: Qualitative Results:
each row shows the original image, ground-truth segmentation
(GT) from PASCAL, the singlebest segmentation y1 , and oracle
segmentation from the M = 15
segmentations (excluding y1 ) for
different definitions of diversity.
Hamming typically performs the
best. In certain situations (row3),
label transitions help since the
single-best segmentation y1 included a rare pair of labels (dogcat boundary).
Results. We evaluate all methods on the PASCAL VOC 2012 data [11], consisting of train, val
and test partitions with about 1450 images each. We train the regressors of [6] on train, and
report oracle accuracies of different methods on val (we cannot report oracle results on test since
those annotations are not publicly available). Diversity parameters (?, ?) are chosen by performing cross-val on val. The standard PASCAL accuracy is the corpus-level intersection-over-union
measure, averaged over all categories. For both label cost and transition, we try 3 different concave
7
p
functions h(?) = min{1, ?}, (?) and log(1 + ?). Table 2 shows the results.3 Hamming ball diversity performs the best, followed by DivMBest, and label cost/transitions are worse here. We found
that while worst on average, label transition diversity helps in an interesting scenario ? when the
first best segmentation y1 includes a pair of rare or mutually confusing labels (say dog-cat). Fig. 3
shows an example, and more illustrations are provided in the supplement. In these cases, searching
for a different label transition produces a better segmentation. Finally, we note that lists produced
with combined diversity significantly outperform any single method (including DivMBest).
5
Discussion and Conclusion
In this paper, we study greedy algorithms for maximizing scoring functions that promote diverse
sets of combinatorial configurations. This problem arises naturally in domains such as Computer
Vision, Natural Language Processing, or Computational Biology, where we want to search for a set
of diverse high-quality solutions in a structured output space.
The diversity functions we propose are monotone submodular functions by construction. Thus, if
r(y) + p(y) ? 0 for all y, then the entire scoring function F is monotone submodular. We showed
that r(y) can simply be learned to be positive. The greedy algorithm for maximizing monotone
submodular functions has proved useful in moderately-sized unstructured spaces. To the best of our
knowledge, this is the first generalization to exponentially large structured output spaces. In particular, our contribution lies in reducing the greedy augmentation step to inference with structured,
efficiently solvable HOPs. This insight makes new connections between submodular optimization
and work on inference in graphical models. We now address some questions.
Can we sample? One question that may be posed is how random sampling would perform for large
ground sets V . Unfortunately, the expected value of a random sample of M elements can be much
worse than the optimal value F (S ? ), especially if N is large. Lemma 1 is proved in the supplement.
Lemma 1. Let S ? V be a sample of size M taken uniformly at random. There exist monotone
submodular functions where E[F (S)] ? M
N max|S|=M F (S).
Guarantees? If F is nonnegative, monotone submodular, then using an exact HOP inference algorithm will clearly result in an approximation factor of 1 ? 1/e. But many HOP inference procedures
are approximate. Lemma 2 formalizes how approximate inference affects the approximation bounds.
Lemma 2. Let F ? 0 be monotone submodular. If each step of the greedy algorithm uses an
approximate marginal gain maximizer bt+1 with F (bt+1 | S t ) ? ? maxa?V F (a | S t ) ? t+1 , then
PM
F (S M ) ? (1 ? e1? ) max|S|?M F (S) ? i=1 t .
Parts of Lemma 2 have been observed in previous work [13, 29]; we show the combination in the
supplement. If F is monotone but not nonnegative,
then Lemma 2 can be extended to a relative error
P
F (S M )?Fmin
1
i i
bound F (S ? )?Fmin ? (1 ? e? ) ? F (S ? )?Fmin that refers to Fmin = minS F (S) and the optimal
solution S ? . While stating these results, we add that further additive approximation losses occur if
the approximation bound for inference is computed on a shifted or reflected function (positive scores
vs positive energies). We pose theoretical improvements as an open question for future work. That
said, our experiments convincingly show that the algorithms perform very well in practice, even
when there are no guarantees (as with Hamming Ball diversity).
Generalization. In addition to the three specific examples in Section 3, our constructions generalize
to the broad HOP class of upper-envelope potentials [18]. The details are provided in the supplement.
Acknowledgements. We thank Xiao Lin for his help. The majority of this work was done while AP was an
intern at Virginia Tech. AP and DB were partially supported by the National Science Foundation under Grant
No. IIS-1353694 and IIS-1350553, the Army Research Office YIP Award W911NF-14-1-0180, and the Office
of Naval Research Award N00014-14-1-0679, awarded to DB. SJ was supported by gifts from Amazon Web
Services, Google, SAP, The Thomas and Stacey Siebel Foundation, Apple, C3Energy, Cisco, Cloudera, EMC,
Ericsson, Facebook, GameOnTalis, Guavus, HP, Huawei, Intel, Microsoft, NetApp, Pivotal, Splunk, Virdata,
VMware, WANdisco, and Yahoo!.
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4,778 | 5,326 | The Noisy Power Method:
A Meta Algorithm with Applications
Moritz Hardt?
IBM Research Almaden
Eric Price?
IBM Research Almaden
Abstract
We provide a new robust convergence analysis of the well-known power method for
computing the dominant singular vectors of a matrix that we call the noisy power
method. Our result characterizes the convergence behavior of the algorithm when
a significant amount noise is introduced after each matrix-vector multiplication.
The noisy power method can be seen as a meta-algorithm that has recently found a
number of important applications in a broad range of machine learning problems
including alternating minimization for matrix completion, streaming principal
component analysis (PCA), and privacy-preserving spectral analysis. Our general
analysis subsumes several existing ad-hoc convergence bounds and resolves a
number of open problems in multiple applications:
Streaming PCA. A recent work of Mitliagkas et al. (NIPS 2013) gives a spaceefficient algorithm for PCA in a streaming model where samples are drawn from a
gaussian spiked covariance model. We give a simpler and more general analysis that
applies to arbitrary distributions confirming experimental evidence of Mitliagkas
et al. Moreover, even in the spiked covariance model our result gives quantitative
improvements in a natural parameter regime. It is also notably simpler and follows
easily from our general convergence analysis of the noisy power method together
with a matrix Chernoff bound.
Private PCA. We provide the first nearly-linear time algorithm for the problem
of differentially private principal component analysis that achieves nearly tight
worst-case error bounds. Complementing our worst-case bounds, we show that the
error dependence of our algorithm on the matrix dimension can be replaced by an
essentially tight dependence on the coherence of the matrix. This result resolves the
main problem left open by Hardt and Roth (STOC 2013). The coherence is always
bounded by the matrix dimension but often substantially smaller thus leading to
strong average-case improvements over the optimal worst-case bound.
1
Introduction
Computing the dominant singular vectors of a matrix is one of the most important algorithmic
tasks underlying many applications including low-rank approximation, PCA, spectral clustering,
dimensionality reduction, matrix completion and topic modeling. The classical problem is wellunderstood, but many recent applications in machine learning face the fundamental problem of
approximately finding singular vectors in the presence of noise. Noise can enter the computation
through a variety of sources including sampling error, missing entries, adversarial corruptions and
privacy constraints. It is desirable to have one robust method for handling a variety of cases without
the need for ad-hoc analyses. In this paper we consider the noisy power method, a fast general purpose
method for computing the dominant singular vectors of a matrix when the target matrix can only be
accessed through inaccurate matrix-vector products.
?
?
Email: [email protected]
Email: [email protected]
1
Figure 1 describes the method when the target matrix A is a symmetric d ? d matrix?a generalization
to asymmetric matrices is straightforward. The algorithm starts from an initial matrix X0 ? Rd?p
and iteratively attempts to perform the update rule X` ? AX` . However, each such matrix product is
followed by a possibly adversarially and adaptively chosen perturbation G` leading to the update rule
X` ? AX` + G` . It will be convenient though not necessary to maintain that X` has orthonormal
columns which can be achieved through a QR-factorization after each update.
Input: Symmetric matrix A ? Rd?d , number of iterations L, dimension p
1. Choose X0 ? Rd?p .
2. For ` = 1 to L:
(a) Y` ? AX`?1 + G` where G` ? Rd?p is some perturbation
(b) Let Y` = X` R` be a QR-factorization of Y`
Output: Matrix XL
Figure 1: Noisy Power Method (NPM)
The noisy power method is a meta algorithm that when instantiated with different settings of G`
and X0 adapts to a variety of applications. In fact, there have been a number of recent surprising
applications of the noisy power method:
1. Jain et al. [JNS13, Har14] observe that the update rule of the well-known alternating least
squares heuristic for matrix completion can be considered as an instance of NPM. This lead
to the first provable convergence bounds for this important heuristic.
2. Mitgliakas et al. [MCJ13] observe that NPM applies to a streaming model of principal
component analysis (PCA) where it leads to a space-efficient and practical algorithm for
PCA in settings where the covariance matrix is too large to process directly.
3. Hardt and Roth [HR13] consider the power method in the context of privacy-preserving
PCA where noise is added to achieve differential privacy.
In each setting there has so far only been an ad-hoc analysis of the noisy power method. In the first
setting, only local convergence is argued, that is, X0 has to be cleverly chosen. In the second setting,
the analysis only holds for the spiked covariance model of PCA. In the third application, only the
case p = 1 was considered.
In this work we give a completely general analysis of the noisy power method that overcomes
limitations of previous analyses. Our result characterizes the global convergence properties of the
algorithm in terms of the noise G` and the initial subspace X0 . We then consider the important
case where X0 is a randomly chosen orthonormal basis. This case is rather delicate since the initial
correlation between a random matrix X0 and the target subspace is vanishing in the dimension d for
small p. Another important feature of the analysis is that it shows how X` converges towards the first
k 6 p singular vectors. Choosing p to be larger than the target dimension leads to a quantitatively
stronger result. Theorem 2.3 formally states our convergence bound. Here we highlight one useful
corollary to illustrate our more general result.
Corollary 1.1. Let k 6 p. Let U ? Rd?k represent the top k singular vectors of A and let
?1 > ? ? ? > ?n > 0 denote its singular values. Suppose X0 is an orthonormal basis of a random
p-dimensional subspace. Further suppose that at every step of NPM we have
5kG` k 6 ?(?k ? ?k+1 ) and
5kU > G` k 6 (?k ? ?k+1 )
? ?
p? k?1
?
? d
??(d)
for some fixed parameter ? and ? < 1/2. Then with all but ? ??(p+1?k) +
probability,
e
there
?k
exists an L = O( ?k ??k+1 log(d? /?)) so that after L steps we have that (I ? XL XL> )U
6 ?.
The corollary shows that the algorithm converges in the strong sense that the entire spectral norm
of U up to an ? error is contained in the space spanned by XL . To achieve this the result places two
assumptions on the magnitude of the noise. The total spectral norm of G` must be bounded by ?
times the separation between ?k and ?k+1 . This dependence on the singular value separation arises
even in the classical perturbation theory of Davis-Kahan [DK70]. The second condition is specific to
the power method and requires that the noise term is proportionally smaller when projected onto the
space spanned by the top k singular vectors. This condition ensures that the correlation between X`
2
and U that is initially very small is not destroyed by the noise addition step. If the noise term has
some sphericalpproperties (e.g. a Gaussian matrix), we expect the projection onto U to be smaller
by a factor of k/d, since the space U is k-dimensional. In the case where p = k + ?(k) this is
precisely what the condition requires. When p = k the requirement is stronger by a factor of k. This
phenomenon stems from the fact that the smallest singular value of a random p ? k gaussian matrix
behaves differently in the square and the rectangular case.
We demonstrate the usefulness of our convergence bound with several novel results in some of the
aforementioned applications.
1.1
Application to memory-efficient streaming PCA
In the streaming PCA setting we receive a stream of samples z1 , z2 , . . . zn ? Rd drawn i.i.d. from
an unknown distribution D over Rd . Our goal is to compute the dominant k eigenvectors of the
covariance matrix A = Ez?D zz > . The challenge is to do this in space linear in the output size,
namely O(kd). Recently, Mitgliakas et al. [MCJ13] gave an algorithm for this problem based on the
noisy power method. We analyze the same algorithm, which we restate here and call SPM:
Input: Stream of samples z1 , z2 , . . . , zn ? Rd , iterations L, dimension p
1. Let X0 ? Rd?p be a random orthonormal basis. Let T = bm/Lc
2. For ` = 1 to L:
P`T
(a) Compute Y` = A` X`?1 where A` = i=(`?1)T +1 zi zi>
(b) Let Y` = X` R` be a QR-factorization of Y`
Output: Matrix XL
Figure 2: Streaming Power Method (SPM)
The algorithm can be executed in space O(pd) since the update step can compute the d ? p matrix
A` X`?1 incrementally without explicitly computing A` . The algorithm maps to our setting by
defining G` = (A` ? A)X`?1 . With this notation Y` = AX`?1 + G` . We can apply Corollary 1.1
directly once we have suitable bounds on kG` k and kU > G` k.
The result of [MCJ13] is specific to the spiked covariance model. The spiked covariance model
is defined by an orthonormal basis U ? Rd?k and a diagonal matrix ? ? Rk?k with diagonal
entries ?1 > ?2 > ? ? ? > ?k > 0. The distribution D(U, ?) is defined as the normal distribution
N(0, (U ?2 U > + ? 2 Idd?d )). Without loss of generality we can scale the examples
such that ?1
= 1.
One corollary of our result shows that the algorithm outputs XL such that
(I ? XL XL> )U
6 ?
with probability 9/10 provided p = k + ?(k) and the number of samples satisfies
6
? +1
n=?
kd .
?2 ?6k
Previously, the same bound1 was known with a quadratic dependence on k in the case where p = k.
Here we can strengthen the bound by increasing p slightly.
While we can get some improvements even in the spiked covariance model, our result is substantially
more general and applies to any distribution. The sample complexity bound we get varies according
to a technical parameter of the distribution. Roughly speaking, we get a near linear sample complexity
if the distribution is either ?round? (as in the spiked covariance setting) or is very well approximated
by a k dimensional subspace. To illustrate the latter condition, we have the following result without
making any assumptions other than scaling the distribution:
Corollary 1.2. Let D be any distribution scaled so that Pr {kzk > t} 6 exp(?t) for every t > 1.
Let U represent the top k eigenvectors of the covariance matrix E zz > and ?1 > ? ? ? > ?d > 0 its
eigenvalues. Then,
SPM invoked
with p = k + ?(k) outputs a matrix XL such with probability
(I ? XL X > )U
6 ? provided SPM receives n samples where n satisfies n =
9/10
we
have
L
?k
? 2
O
?
d
.
3
? k(?k ??k+1 )
1
That the bound stated in [MCJ13] has a ? 6 dependence is not completely obvious. There is a O(? 4 ) in the
numerator and log((? 2 + 0.75?2k )/(? 2 + 0.5?2k )) in the denominator which simplifies to O(1/? 2 ) for constant
?k and ? 2 > 1.
3
The corollary establishes a sample complexity that?s linear in d provided that the spectrum decays
quickly, as is common in applications. For example, if the spectrum follows a power law so that
? 2c+2 d/?2 ).
?j ? j ?c for a constant c > 1/2, the bound becomes n = O(k
1.2
Application to privacy-preserving spectral analysis
Many applications of singular vector computation are plagued by the fact that the underlying matrix
contains sensitive information about individuals. A successful paradigm in privacy-preserving data
analysis rests on the notion of differential privacy which requires all access to the data set to be
randomized in such a way that the presence or absence of a single data item is hidden. The notion of
data item varies and could either refer to a single entry, a single row, or a rank-1 matrix of bounded
norm. More formally, Differential Privacy requires that the output distribution of the algorithm
changes only slightly with the addition or deletion of a single data item. This requirement often
necessitates the introduction of significant levels of noise that make the computation of various
objectives challenging. Differentially private singular vector computation has been studied actively
since the work of Blum et al. [BDMN05]. There are two main objectives. The first is computational
efficiency. The second objective is to minimize the amount of error that the algorithm introduces.
In this work, we give a fast algorithm for differentially private singular vector computation based
on the noisy power method that leads to nearly optimal bounds in a number of settings that were
considered in previous work. The algorithm is described in Figure 3. It?s a simple instance of NPM
in which each noise matrix G` is a gaussian random matrix scaled so that the algorithm achieves
(?, ?)-differential privacy (as formally defined in Definition E.1). It is easy to see that the algorithm
can be implemented in time nearly linear in the number of nonzero entries of the input matrix (input
sparsity). This will later lead to strong improvements in running time compared with several previous
works.
Input: Symmetric A ? Rd?d , L, p, privacy parameters ?, ? > 0
1. Let pX0 be a random orthonormal basis and put ?
??1 4pL log(1/?)
2. For ` = 1 to L:
(a) Y` ? AX`?1 + G` where G` ? N(0, kX`?1 k2? ? 2 )d?p .
(b) Compute the QR-factorization Y` = X` R`
Output: Matrix XL
=
Figure 3: Private Power Method (PPM). Here kXk? = maxij |Xij |.
We first state a general purpose analysis of PPM that follows from Corollary 1.1.
Theorem 1.3. Let k 6 p. Let U ? Rd?k represent the top k singular vectors of A and let
?1 > ? ? ? > ?d > 0 denote its singular values. Then, PPM satisfies (?, ?)-differential privacy and
?k
log(d)) iterations we have with probability 9/10 that
after L = O( ?k ??
k+1
(I ? XL XL> )U
6 O
!
?
?
p
? max kX` k? d log L
?
??
.
?k ? ?k+1
p? k?1
When p = k + ?(k) the trailing factor becomes a constant. If p = k it creates a factor k overhead.
In the worst-case we can always bound kX` k? by 1 since X` is an orthonormal basis. However, in
principle we could hope that a much better bound holds provided that the target subspace U has small
coordinates. Hardt and Roth [HR12, HR13] suggested a way to accomplish a stronger bound by
considering a notion of coherence of A, denoted as ?(A). Informally, the coherence is a well-studied
parameter that varies between 1 and n, but is often observed to be small. Intuitively, the coherence
measures the correlation between the singular vectors of the matrix with the standard basis. Low
coherence means that the singular vectors have small coordinates in the standard basis. Many results
on matrix completion and robust PCA crucially rely on the assumption that the underlying matrix
has low coherence [CR09, CT10, CLMW11] (though the notion of coherence here will be somewhat
different).
4
Theorem 1.4. Under the assumptions of Theorem 1.3, we have the conclusion
!
p
?
p
(I ? XL XL> )U
6 O ? ?(A) log d log L ? ?
?
.
?k ? ?k+1
p? k?1
Hardt
? and Roth proved this result for the case where p = 1. The extension to p > 1 lost a factor
of d in general and therefore gave no improvement over Theorem 1.3. Our result resolves the
main problem left open in their work. The strength of Theorem 1.4 is that the bound is essentially
dimension-free under a natural assumption on the matrix and never worse than our worst-case result.
It is also known that in general the dependence on d achieved in Theorem 1.3 is best possible in the
worst case (see discussion in [HR13]) so that further progress requires making stronger assumptions.
Coherence is a natural such assumption.
The proof of Theorem 1.4 proceeds by showing that each
p
iterate X` satisfies kX` k? 6 O( ?(A) log(d)/d) and applying Theorem 1.3. To do this we exploit
a non-trivial symmetry of the algorithm that we discuss in Section E.3.
Other variants of differential privacy. Our discussion above applied to (?, ?)-differential privacy
under changing a single entry of the matrix. Several works consider other variants of differential
privacy. It is generally easy to adapt the power method to these settings by changing the noise
distribution or its scaling. To illustrate this aspect, we consider the problem of privacy-preserving
principal component analysis as recently studied by [CSS12, KT13]. Both works consider an
algorithm called exponential mechanism. The first work gives a heuristic implementation that may
not converge, while the second work gives a provably polynomial time algorithm though the running
time is more than cubic. Our algorithm gives strong improvements in running time
? while giving
? k) factor. We
nearly optimal accuracy guarantees as it matches a lower bound of [KT13] up to a O(
also improve the error dependence on k by
?polynomial factors compared to previous work. Moreover,
we get an accuracy improvement of O( d) for the case of (?, ?)-differential privacy, while these
previous works only apply to (?, 0)-differential privacy. Section E.2 provides formal statements.
1.3
Related Work
Numerical Analysis. One might expect that a suitable analysis of the noisy power method would
have appeared in the numerical analysis literature. However, we are not aware of a reference and
there are a number of points to consider. First, our noise model is adaptive thus setting it apart from
the classical perturbation theory of the singular vector decomposition [DK70]. Second, we think
of the perturbation at each step as large making it conceptually different from floating point errors.
Third, research in numerical analysis over the past decades has largely focused on faster Krylov
subspace methods. There is some theory of inexact Krylov methods, e.g., [SS07] that captures the
effect of noisy matrix-vector products in this context. Related to our work are also results on the
perturbation stability of the QR-factorization since those could be used to obtain convergence bounds
for subspace iteration. Such bounds, however, must depend on the condition number of the matrix
that the QR-factorization is applied to. See Chapter 19.9 in [Hig02] and the references therein for
background. Our proof strategy avoids this particular dependence on the condition number.
Streaming PCA. PCA in the streaming model is related to a host of well-studied problems that we
cannot survey completely here. We refer to [ACLS12, MCJ13] for a thorough discussion of prior
work. Not mentioned therein is a recent work on incremental PCA [BDF13] that leads to space
efficient algorithms computing the top singular vector; however, it?s not clear how to extend their
results to computing multiple singular vectors.
Privacy. There has been much work on differentially private spectral analysis starting with Blum
et al. [BDMN05] who used an algorithm known as Randomized Response which adds a single
noise matrix N either to the input matrix A or the covariance matrix AA> . This approach appears
in a number of papers, e.g. [MM09]. While often easy to analyze it has the disadvantage that it
converts sparse matrices to dense matrices and is often impractical on large data sets. Chaudhuri
et al. [CSS12] and Kapralov-Talwar [KT13] use the so-called exponential mechanism to sample
approximate eigenvectors of the matrix. The sampling is done using a heuristic approach without
convergence polynomial time convergence guarantees in the first case and using a polynomial time
algorithm in the second. Both papers achieve a tight dependence on the matrix dimension d (though
5
the dependence on k is suboptimal in general). Most closely related to our work are the results of
Hardt and Roth [HR13, HR12] that introduced matrix coherence as a way to circumvent existing
worst-case lower bounds on the error. They also analyzed a natural noisy variant of power iteration
for the case of computing the dominant eigenvector of A. When multiple eigenvectors are needed,
their algorithm uses the well-known deflation technique. However, this step loses
p control of the
coherence of the original matrix and hence results in suboptimal bounds. In fact, a rank(A) factor
is lost.
1.4
Open Questions
We believe Corollary 1.1 to be a fairly precise characterization of the convergence of the noisy power
method to the top k singular vectors when p = k. The main flaw is that the noise tolerance depends
on the eigengap ?k ? ?k+1 , which could be very small. We have some conjectures for results that do
not depend on this eigengap.
First, when p > k, we think that Corollary 1.1 might hold using the gap ?k ? ?p+1 instead of
?k ? ?k+1 . Unfortunately, our proof technique relies on the principal angle decreasing at each step,
which does not necessarily hold with the larger level of noise. Nevertheless we expect the principal
angle to decrease fairly fast on average, so that XL will contain a subspace very close to U . We are
actually unaware of this sort of result even in the noiseless setting.
Conjecture 1.5. Let X0 be a random p-dimensional basis for p > k. Suppose at every step we have
?
?
p? k?1
?
100kG` k 6 ?(?k ? ?p+1 ) and 100kU T G` k 6
d
?k
Then with high probability, after L = O( ?k ??
log(d/?)) iterations we have
p+1
k(I ? XL XL> )U k 6 ?.
The second way of dealing with a small eigengap would be to relax our goal. Corollary 1.1 is quite
stringent in that it requires XL to approximate the top k singular vectors U , which gets harder when
the eigengap approaches zero and the kth through p + 1st singular vectors are nearly indistinguishable.
A relaxed goal would be for XL to spectrally approximate A, that is
k(I ? XL XL> )Ak 6 ?k+1 + ?.
(1)
This weaker goal is known to be achievable in the noiseless setting without any eigengap at all.
In particular, [?] shows that (1) happens after L = O( ?k+1
? log n) steps in the noiseless setting. A
plausible extension to the noisy setting would be:
Conjecture 1.6. Let X0 be a random 2k-dimensional basis. Suppose at every step we have
p
kG` k 6 ? and kU T G` k 6 ? k/d
Then with high probability, after L = O( ?k+1
? log d) iterations we have that
k(I ? XL XL> )Ak 6 ?k+1 + O(?).
1.5
Organization
All proofs can be found in the supplementary material. In the remaining space, we limit ourselves to
a more detailed discussion of our convergence analysis and the application to streaming PCA. The
entire section on privacy is in the supplementary materials in Section E.
2
Convergence of the noisy power method
Figure 1 presents our basic algorithm that we analyze in this section. An important tool in our analysis
are principal angles, which are useful in analyzing the convergence behavior of numerical eigenvalue
methods. Roughly speaking, we will show that the tangent of the k-th principal angle between X and
the top k eigenvectors of A decreases as ?k+1 /?k in each iteration of the noisy power method.
6
Definition 2.1 (Principal angles). Let X and Y be subspaces of Rd of dimension at least k. The
principal angles 0 6 ?1 6 ? ? ? 6 ?k between X and Y and associated principal vectors x1 , . . . , xk
and y1 , . . . , yk are defined recursively via
hx, yi
?i (X , Y) = min arccos
: x ? X , y ? Y, x ? xj , y ? yj for all j < i
kxk2 kyk2
and xi , yi are the x and y that give this value. For matrices X and Y , we use ?k (X, Y ) to denote the
kth principal angle between their ranges.
2.1
Convergence argument
Fix parameters 1 6 k 6 p 6 d. In this section we consider a symmetric d ? d matrix A with singular
values ?1 > ?2 > ? ? ? > ?d . We let U ? Rd?k contain the first k eigenvectors of A. Our main
lemma shows that tan ?k (U, X) decreases multiplicatively in each step.
Lemma 2.2. Let U contain the largest k eigenvectors of a symmetric matrix A ? Rd?d , and let
X ? Rd?p for p > k. Let G ? Rd?p satisfy
4kU > Gk 6 (?k ? ?k+1 ) cos ?k (U, X)
4kGk 6 (?k ? ?k+1 )?.
for some ? < 1. Then
tan ?k (U, AX + G) 6 max ?, max ?,
?k+1
?k
1/4 !
!
tan ?k (U, X) .
We can inductively apply the previous lemma to get the following general convergence result.
Theorem 2.3. Let U represent the top k eigenvectors of the matrix A and ? = 1 ? ?k+1 /?k . Suppose
that the initial subspace X0 and noise G` is such that
5kU > G` k 6 (?k ? ?k+1 ) cos ?k (U, X0 )
5kG` k 6 ?(?k ? ?k+1 )
at every stage `, for some ? < 1/2. Then there exists an L . ?1 log tan ?k ?(U,X0 ) such that for all
` > L we have tan ?(U, XL ) 6 ?.
2.2
Random initialization
The next lemma essentially follows from bounds on the smallest singular value of gaussian random
matrices [RV09].
Lemma 2.4. For an arbitrary orthonormal U and random subspace X, we have
?
d
?
tan ?k (U, X) 6 ? ?
p? k?1
with all but ? ??(p+1?k) + e??(d) probability.
With this lemma we can prove the corollary that we stated in the introduction.
Proof of Corollary 1.1. By Lemma 2.4, with the desired probability ?
we have tan ?k (U, X0 ) 6
?
?
? ?d
.
p? k?1
?
p? k?1
?
.
2?? d
>
XL XL )U k
Hence cos ?k (U, X0 ) > 1/(1 + tan ?k (U, X0 )) >
ply Theorem 2.3 to get that tan ?k (U, XL ) 6 ?. Then k(I ?
tan ?k (U, XL ) 6 ?.
7
Rescale ? and ap= sin ?k (U, XL ) 6
3
Memory efficient streaming PCA
In the streaming PCA setting we receive a stream of samples z1 , z2 , ? ? ? ? Rd . Each sample is drawn
i.i.d. from an unknown distribution D over Rd . Our goal is to compute the dominant k eigenvectors
of the covariance matrix A = Ez?D zz > . The challenge is to do this with small space, so we cannot
store the d2 entries of the sample covariance matrix. We would like to use O(dk) space, which is
necessary even to store the output.
The streaming power method (Figure 2, introduced by [MCJ13]) is a natural algorithm that performs
streaming PCA with O(dk) space. The question that arises is how many samples it requires to
achieve a given level of accuracy, for various distributions D. Using our general analysis of the noisy
power method, we show that the streaming power method requires fewer samples and applies to more
distributions than was previously known. We analyze a broad class of distributions:
Definition 3.1. A distribution D over Rd is (B, p)-round if for
n every p-dimensional
oprojection P and
p
all t > 1 we have Prz?D {kzk > t} 6 exp(?t) and Prz?D kP zk > t ? Bp/d 6 exp(?t) .
The first condition just corresponds to a normalization of the samples drawn from D. Assuming the
first condition holds, the second condition always holds with B = d/p. For this reason our analysis
in principle applies to any distribution, but the sample complexity will depend quadratically on B.
Let us illustrate this definition through the example of the spiked covariance model studied
by [MCJ13]. The spiked covariance model is defined by an orthonormal basis U ? Rd?k and a
diagonal matrix ? ? Rk?k with diagonal entries ?1 > ?2 > ? ? ? > ?k > 0. The distribution D(U,
P ?)
is defined as the normal distribution N(0, (U ?2 U > + ? 2 Idd?d )/D) where D = ?(d? 2 + i ?2i )
is a normalization factor chosen so that the distribution satisfies the norm bound. Note that the the
i-th eigenvalue of the covariance matrix is ?i = (?2i + ? 2 )/D for 1 6 i 6 k and ?i = ? 2 /D for
i > k. We show in Lemma D.2 that the spiked covariance model D(U, ?) is indeed (B, p)-round for
?21 +? 2
B = O( tr(?)/d+?
2 ), which is constant for ? & ?1 . We have the following main theorem.
Theorem 3.2. Let D be a (B, p)-round distribution over Rd with covariance matrix A whose
eigenvalues are ?1 > ?2 > ? ? ? > ?d > 0. Let U ? Rd?k be an orthonormal basis for the
eigenvectors corresponding to the first k eigenvalues of A. Then, the streaming power method SPM
returns an orthonormal basis X ? Rd?p such that tan ?(U, X) 6 ? with probability 9/10 provided
that SPM receives n samples from D for some n satisfying
B 2 ?k k log2 d
?
n6O
?2 (?k ? ?k+1 )3 d
if p = k + ?(k). More generally, for all p > k one can get the slightly stronger result
!
?
?
2
2
2
Bp?
max{1/?
,
Bp/(
p
?
k
?
1)
}
log
d
k
?
n6O
.
(?k ? ?k+1 )3 d
Instantiating with the spiked covariance model gives the following:
Corollary 3.3. In the spiked covariance model D(U, ?) the conclusion of Theorem 3.2 holds for
p = 2k with
2
(?1 + ? 2 )2 (?2k + ? 2 )
?
n=O
dk .
?2 ?6k
? ?6 +1
When ?1 = O(1) and ?k = ?(1) this becomes n = O
?
dk
.
?2
We can apply Theorem 3.2 to all distributions that have exponentially concentrated norm by setting
B = d/p. This gives the following result.
Corollary 3.4. Let D be any distribution scaled such that Prz?D [kzk > t] 6 exp(?t) for all t > 1.
Then the conclusion of Theorem 3.2 holds for p = 2k with
?k
?
n=O
?d .
?2 k(?k ? ?k+1 )3
If the eigenvalues follow a power law, ?j ? j ?c for a constant c > 1/2, this gives an n =
? 2c+2 d/?2 ) bound on the sample complexity.
O(k
8
References
[ACLS12]
Raman Arora, Andrew Cotter, Karen Livescu, and Nathan Srebro. Stochastic optimization for pca and pls. In Communication, Control, and Computing (Allerton), 2012 50th
Annual Allerton Conference on, pages 861?868. IEEE, 2012.
[BDF13]
Akshay Balsubramani, Sanjoy Dasgupta, and Yoav Freund. The fast convergence of
incremental PCA. In Proc. 27th Neural Information Processing Systems (NIPS), pages
3174?3182, 2013.
[BDMN05] Avrim Blum, Cynthia Dwork, Frank McSherry, and Kobbi Nissim. Practical privacy:
the SuLQ framework. In Proc. 24th PODS, pages 128?138. ACM, 2005.
[CLMW11] Emmanuel J. Cand?s, Xiaodong Li, Yi Ma, and John Wright. Robust principal component analysis? J. ACM, 58(3):11, 2011.
[CR09]
Emmanuel J. Cand?s and Benjamin Recht. Exact matrix completion via convex optimization. Foundations of Computional Mathematics, 9:717?772, December 2009.
[CSS12]
Kamalika Chaudhuri, Anand Sarwate, and Kaushik Sinha. Near-optimal differentially
private principal components. In Proc. 26th Neural Information Processing Systems
(NIPS), 2012.
Emmanuel J. Cand?s and Terence Tao. The power of convex relaxation: near-optimal
matrix completion. IEEE Transactions on Information Theory, 56(5):2053?2080, 2010.
[CT10]
[DK70]
Chandler Davis and W. M. Kahan. The rotation of eigenvectors by a perturbation. iii.
SIAM J. Numer. Anal., 7:1?46, 1970.
[Har14]
Moritz Hardt. Understanding alternating minimization for matrix completion. In Proc.
55th Foundations of Computer Science (FOCS). IEEE, 2014.
[Hig02]
Nicholas J. Higham. Accuracy and Stability of Numerical Algorithms. Society for
Industrial and Applied Mathematics, 2002.
[HR12]
Moritz Hardt and Aaron Roth. Beating randomized response on incoherent matrices.
In Proc. 44th Symposium on Theory of Computing (STOC), pages 1255?1268. ACM,
2012.
[HR13]
Moritz Hardt and Aaron Roth. Beyond worst-case analysis in private singular vector
computation. In Proc. 45th Symposium on Theory of Computing (STOC). ACM, 2013.
[JNS13]
Prateek Jain, Praneeth Netrapalli, and Sujay Sanghavi. Low-rank matrix completion
using alternating minimization. In Proc. 45th Symposium on Theory of Computing
(STOC), pages 665?674. ACM, 2013.
[KT13]
Michael Kapralov and Kunal Talwar. On differentially private low rank approximation.
In Proc. 24rd Symposium on Discrete Algorithms (SODA). ACM-SIAM, 2013.
[MCJ13]
Ioannis Mitliagkas, Constantine Caramanis, and Prateek Jain. Memory limited, streaming PCA. In Proc. 27th Neural Information Processing Systems (NIPS), pages 2886?
2894, 2013.
[MM09]
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[RV09]
Mark Rudelson and Roman Vershynin. Smallest singular value of a random rectangular
matrix. Communications on Pure and Applied Mathematics, 62(12):1707?1739, 2009.
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9
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4,779 | 5,327 | Two-Layer Feature Reduction for Sparse-Group
Lasso via Decomposition of Convex Sets
Jie Wang, Jieping Ye
Computer Science and Engineering
Arizona State University, Tempe, AZ 85287
{jie.wang.ustc, jieping.ye}@asu.edu
Abstract
Sparse-Group Lasso (SGL) has been shown to be a powerful regression technique for simultaneously discovering group and within-group sparse patterns by
using a combination of the `1 and `2 norms. However, in large-scale applications,
the complexity of the regularizers entails great computational challenges. In this
paper, we propose a novel two-layer feature reduction method (TLFre) for SGL
via a decomposition of its dual feasible set. The two-layer reduction is able to
quickly identify the inactive groups and the inactive features, respectively, which
are guaranteed to be absent from the sparse representation and can be removed
from the optimization. Existing feature reduction methods are only applicable
for sparse models with one sparsity-inducing regularizer. To our best knowledge,
TLFre is the first one that is capable of dealing with multiple sparsity-inducing
regularizers. Moreover, TLFre has a very low computational cost and can be integrated with any existing solvers. Experiments on both synthetic and real data sets
show that TLFre improves the efficiency of SGL by orders of magnitude.
1
Introduction
Sparse-Group Lasso (SGL) [5, 16] is a powerful regression technique in identifying important
groups and features simultaneously. To yield sparsity at both group and individual feature levels,
SGL combines the Lasso [18] and group Lasso [28] penalties. In recent years, SGL has found great
success in a wide range of applications, including but not limited to machine learning [20, 27], signal processing [17], bioinformatics [14] etc. Many research efforts have been devoted to developing
efficient solvers for SGL [5, 16, 10, 21]. However, when the feature dimension is extremely high,
the complexity of the SGL regularizers imposes great computational challenges. Therefore, there
is an increasingly urgent need for nontraditional techniques to address the challenges posed by the
massive volume of the data sources.
Recently, El Ghaoui et al. [4] proposed a promising feature reduction method, called SAFE screening, to screen out the so-called inactive features, which have zero coefficients in the solution, from
the optimization. Thus, the size of the data matrix needed for the training phase can be significantly
reduced, which may lead to substantial improvement in the efficiency of solving sparse models.
Inspired by SAFE, various exact and heuristic feature screening methods have been proposed for
many sparse models such as Lasso [25, 11, 19, 26], group Lasso [25, 22, 19], etc. It is worthwhile
to mention that the discarded features by exact feature screening methods such as SAFE [4], DOME
[26] and EDPP [25] are guaranteed to have zero coefficients in the solution. However, heuristic feature screening methods like Strong Rule [19] may mistakenly discard features which have nonzero
coefficients in the solution. More recently, the idea of exact feature screening has been extended
to exact sample screening, which screens out the nonsupport vectors in SVM [13, 23] and LAD
[23]. As a promising data reduction tool, exact feature/sample screening would be of great practical
importance because they can effectively reduce the data size without sacrificing the optimality [12].
1
However, all of the existing feature/sample screening methods are only applicable for the sparse
models with one sparsity-inducing regularizer. In this paper, we propose an exact two-layer feature
screening method, called TLFre, for the SGL problem. The two-layer reduction is able to quickly
identify the inactive groups and the inactive features, respectively, which are guaranteed to have zero
coefficients in the solution. To the best of our knowledge, TLFre is the first screening method which
is capable of dealing with multiple sparsity-inducing regularizers.
We note that most of the existing exact feature screening methods involve an estimation of the dual
optimal solution. The difficulty in developing screening methods for sparse models with multiple
sparsity-inducing regularizers like SGL is that the dual feasible set is the sum of simple convex
sets. Thus, to determine the feasibility of a given point, we need to know if it is decomposable with
respect to the summands, which is itself a nontrivial problem (see Section 2). One of our major
contributions is that we derive an elegant decomposition method of any dual feasible solutions of
SGL via the framework of Fenchel?s duality (see Section 3). Based on the Fenchel?s dual problem
of SGL, we motivate TLFre by an in-depth exploration of its geometric properties and the optimality
conditions. We derive the set of the regularization parameter values corresponding to zero solutions.
To develop TLFre, we need to estimate the upper bounds involving the dual optimal solution. To this
end, we first give an accurate estimation of the dual optimal solution via the normal cones. Then,
we formulate the estimation of the upper bounds via nonconvex optimization problems. We show
that these nonconvex problems admit closed form solutions. Experiments on both synthetic and real
data sets demonstrate that the speedup gained by TLFre in solving SGL can be orders of magnitude.
All proofs are provided in the long version of this paper [24].
Notation: Let k ? k1 , k ? k and k ? k? be the `1 , `2 and `? norms, respectively. Denote by B1n , B n , and
n
the unit `1 , `2 , and `? norm balls in Rn (we omit the superscript if it is clear from the context).
B?
For a set C, let int C be its interior. If C is closed and convex, we define the projection operator as
PC (w) := argminu?C kw ? uk. We denote by IC (?) the indicator function of C, which is 0 on C and
? elsewhere. Let ?0 (Rn ) be the class of proper closed convex functions on Rn . For f ? ?0 (Rn ),
let ?f be its subdifferential. The domain of f is the set dom f := {w : f (w) < ?}. For w ? Rn ,
let [w]i be its ith component. For ? ? R, let sgn(?) = sign(?) if ? 6= 0, and sgn(0) = 0. We define
sign([w]i ), if [w]i 6= 0;
SGN(w) = s ? Rn : [s]i ?
[?1, 1],
if [w]i = 0.
We denote by ?+ = max(?, 0). Then, the shrinkage operator S? (w) : Rn ? Rn with ? ? 0 is
[S? (w)]i = (|[w]i | ? ?)+ sgn([w]i ), i = 1, . . . , n.
2
(1)
Basics and Motivation
In this section, we briefly review some basics of SGL. Let y ? RN be the response vector and
X ? RN ?p be the matrix of features. With the group information available, the SGL problem [5] is
2
XG
XG ?
1
minp
y
?
X
?
ng k?g k + ?2 k?k1 ,
(2)
g g
+ ?1
g=1
g=1
??R 2
where ng is the number of features in the g th group, Xg ? RN ?ng denotes the predictors in that
group with the corresponding coefficient vector ?g , and ?1 , ?2 are positive regularization parameters. Without loss of generality, let ?1 = ?? and ?2 = ? with ? > 0. Then, problem (2) becomes:
2
X
XG
G ?
1
min
y?
Xg ? g
+ ? ?
ng k?g k + k?k1 .
(3)
g=1
g=1
??Rp 2
By the Lagrangian multipliers method [24], the dual problem of SGL is
n
o
2
?
sup 12 kyk2 ? 12
y? ? ?
: XTg ? ? Dg? := ? ng B + B? , g = 1, . . . , G .
(4)
?
It is well-known that the dual feasible set of Lasso is the intersection of closed half spaces (thus a
polytope); for group Lasso, the dual feasible set is the intersection of ellipsoids. Surprisingly, the
geometric properties of these dual feasible sets play fundamentally important roles in most of the
existing screening methods for sparse models with one sparsity-inducing regularizer [23, 11, 25, 4].
When we incorporate multiple sparse-inducing regularizers to the sparse models, problem (4) indicates that the dual feasible set can be much more complicated. Although (4) provides a geometric
2
description of the dual feasible set of SGL, it is not suitable for further analysis. Notice that, even
the feasibility of a given point ? is not easy to determine, since it is nontrivial to tell if XTg ? can
?
be decomposed into b1 + b2 with b1 ? ? ng B and b2 ? B? . Therefore, to develop screening
methods for SGL, it is desirable to gain deeper understanding of the sum of simple convex sets.
In the next section, we analyze the dual feasible set of SGL in depth via the Fenchel?s Duality
Theorem. We show that for each XTg ? ? Dg? , Fenchel?s duality naturally leads to an explicit decom?
position XTg ? = b1 + b2 , with one belonging to ? ng B and the other one belonging to B? . This
lays the foundation of the proposed screening method for SGL.
3
The Fenchel?s Dual Problem of SGL
In Section 3.1, we derive the Fenchel?s dual of SGL via Fenchel?s Duality Theorem. We then
motivate TLFre and sketch our approach in Section 3.2. In Section 3.3, we discuss the geometric
properties of the Fenchel?s dual of SGL and derive the set of (?, ?) leading to zero solutions.
3.1 The Fenchel?s Dual of SGL via Fenchel?s Duality Theorem
To derive the Fenchel?s dual problem of SGL, we need the Fenchel?s Duality Theorem as stated in
Theorem 1. The conjugate of f ? ?0 (Rn ) is the function f ? ? ?0 (Rn ) defined by
f ? (z) = supw hw, zi ? f (w).
Theorem 1. [Fenchel?s Duality Theorem] Let f ? ?0 (RN ), ? ? ?0 (Rp ), and T (?) = y ? X?
be an affine mapping from Rp to RN . Let p? , d? ? [??, ?] be primal and dual values defined,
respectively, by the Fenchel problems:
p? = inf ??Rp f (y ? X?) + ??(?); d? = sup??RN ?f ? (??) ? ??? (XT ?) + ?hy, ?i.
One has p? ? d? . If, furthermore, f and ? satisfy the condition 0 ? int (dom f ? y + Xdom ?),
then the equality holds, i.e., p? = d? , and the supreme is attained in the dual problem if finite.
We omit the proof of Theorem 1 since it is a slight modification of Theorem 3.3.5 in [2].
Let f (w) = 21 kwk2 , and ??(?) be the second term in (3). Then, SGL can be written as
min? f (y ? X?) + ??(?).
To derive the Fenchel?s dual problem of SGL, Theorem 1 implies that we need to find f ? and ?? . It
is well-known that f ? (z) = 21 kzk2 . Therefore, we only need to find ?? , where the concept infimal
convolution is needed. Let h, g ? ?0 (Rn ). The infimal convolution of h and g is defined by
(hg)(?) = inf ? h(?) + g(? ? ?),
and it is exact at a point ? if there exists a ? ? (?) such that (hg)(?) = h(? ? (?)) + g(? ? ? ? (?)).
hg is exact if it is exact at every point of its domain, in which case it is denoted by h g.
PG ?
?
Lemma 2. Let ??
1 (?) = ?
g=1 ng k?g k, ?2 (?) = k?k1 and ?(?) = ?1 (?) + ?2 (?). More?
over, let Cg? = ? ng B ? Rng , g = 1, . . . , G. Then, the following hold:
PG
PG
?
?
(i): (??
(?2 )? (?) = g=1 IB? (?g ),
1 ) (?) =
g=1 ICg (?g ) ,
PG
? ?P
(? )
?
(ii): ?? (?) = ((??
(?2 )? ) (?) = g=1 IB g ??Bn?g g ,
1)
where ?g ? Rng is the sub-vector of ? corresponding to the g th group.
Note that PB? (?g ) admits a closed form solution, i.e., [PB? (?g )]i = sgn ([?g ]i ) min (|[?g ]i | , 1).
Combining Theorem 1 and Lemma 2, the Fenchel?s dual of SGL can be derived as follows.
Theorem 3. For the SGL problem in (3), the following hold:
(i): The Fenchel?s dual of SGL is given by:
?
inf 12 k y? ? ?k2 ? 12 kyk2 :
XTg ? ? PB? (XTg ?)
? ? ng , g = 1, . . . , G .
?
(5)
(ii): Let ? ? (?, ?) and ?? (?, ?) be the optimal solutions of problems (3) and (5), respectively. Then,
??? (?, ?) =y ? X? ? (?, ?),
?
XTg ?? (?, ?) ?? ng ?k?g? (?, ?)k + ?k?g? (?, ?)k1 , g = 1, . . . , G.
3
(6)
(7)
Remark 1. We note that the shrinkage operator can also be expressed by
S? (w) = w ? P?B? (w), ? ? 0.
(8)
Therefore, problem (5) can be written more compactly as
?
inf 21 k y? ? ?k2 ? 12 kyk2 :
S1 (XTg ?)
? ? ng , g = 1, . . . , G .
(9)
?
Remark 2. Eq. (6) and Eq. (7) can be obtained by the Fenchel-Young inequality [2, 24]. They
are the so-called KKT conditions [3] and can also be obtained by the Lagrangian multiplier method
[24]. Moreover, for the SGL problem, its Lagrangian dual in (4) and Fenchel?s dual in (5) are indeed
equivalent to each other [24].
Remark 3. An appealing advantage of the Fenchel?s dual in (5) is that we have a natural decomposition of all points ?g ? Dg? : ?g = PB? (?g ) + S1 (?g )) with PB? (?g ) ? B? and S1 (?g ) ? Cg? . As a
result, this leads to a convenient way to determine the feasibility of any dual variable ? by checking
if S1 (XTg ?) ? Cg? , g = 1, . . . , G.
3.2 Motivation of the Two-Layer Screening Rules
We motive the two-layer screening rules via the KKT condition in Eq. (7). As implied by the name,
there are two layers in our method. The first layer aims to identify the inactive groups, and the
second layer is designed to detect the inactive features for the remaining groups.
by Eq. (7), we have the following cases by noting ?kwk1 = SGN(w) and
o
(n
w
,
if w 6= 0,
kwk
?kwk =
{u : kuk ? 1}, if w = 0.
Case 1. If ?g? (?, ?) 6= 0, we have
( ? [? ? (?,?)]
i
? ng k?g? (?,?)k + sign([?g? (?, ?)]i ), if [?g? (?, ?)]i 6= 0,
T ?
g
[Xg ? (?, ?)]i ?
[?1, 1],
if [?g? (?, ?)]i = 0.
(10)
In view of Eq. (10), we can see that
?
?
? ? (?1 ,?2 )
(a): S1 (XTg ?? (?, ?)) = ? ng k?g? (?1 ,?2 )k and kS1 (XTg ?? (?, ?))k = ? ng ,
g
(b): If [XTg ?? (?, ?]i ? 1 then [?g? (?, ?)]i = 0.
Case 2. If
?g? (?, ?)
(11)
(12)
= 0, we have
?
[XTg ?? (?, ?)]i ? ? ng [ug ]i + [?1, 1], kug k ? 1.
(13)
The first layer (group-level) of TLFre From (11) in Case 1, we have
S1 (XTg ?? (?, ?))
< ??ng ? ?g? (?, ?) = 0.
(R1)
Clearly, (R1) can be used to identify the inactive groups and thus a group-level screening rule.
The second layer (feature-level) of TLFre Let xgi be the ith column of Xg .
[XTg ?? (?, ?)]i = xTgi ?? (?, ?). In view of (12) and (13), we can see that
T ?
xg ? (?, ?) ? 1 ? [?g? (?, ?)]i = 0.
i
We have
(R2)
Different from (R1), (R2) detects the inactive features and thus it is a feature-level screening rule.
However, we cannot directly apply (R1) and (R2) to identify the inactive groups/features because
both need to know ?? (?, ?). Inspired by the SAFE rules [4], we can first estimate a region ?
containing ?? (?, ?). Let XTg ? = {XTg ? : ? ? ?}. Then, (R1) and (R2) can be relaxed as follows:
?
(R1? )
sup?g kS1 (?g )k : ?g ? ?g ? XTg ? < ? ng ? ?g? (?, ?) = 0,
T
sup? xg ? : ? ? ? ? 1 ? [?g? (?, ?)]i = 0.
(R2? )
i
Inspired by (R1? ) and (R2? ), we develop TLFre via the following three steps:
Step 1. Given ? and ?, we estimate a region ? that contains ?? (?, ?).
Step 2. We solve for the supreme values in (R1? ) and (R2? ).
Step 3. By plugging in the supreme values from Step 2, (R1? ) and (R2? ) result in the desired
two-layer screening rules for SGL.
4
3.3
The Set of Parameter Values Leading to Zero Solution
?
For notational convenience, let Fg? = {? : kS1 (XTg ?)k ? ? ng }, g = 1, . . . , G; and thus the
?
feasible set of the Fenchel?s dual of SGL is F = ?g=1,...,G Fg? . In view of problem (5) [or (9)],
we can see that ?? (?, ?) is the projection of y/? on F ? , i.e., ?? (?, ?) = PF ? (y/?). Thus, if
y/? ? F ? , we have ?? (?, ?) = y/?. Moreover, by (R1), we can see that ? ? (?, ?) = 0 if y/? is an
interior point of F ? . Indeed, we have the following stronger result.
?
T
Theorem 4. For the SGL problem, let ??
max = maxg {?g : S1 (Xg y/?g ) = ? ng }. Then,
y
y
?
?
?
?
? ? F ? ? (?, ?) = ? ? ? (?, ?) = 0 ? ? ? ?max .
?g in the definition of ??
max admits a closed form solution [24]. Theorem 4 implies that the optimal
solution ? ? (?, ?) is 0 as long as y/? ? F ? . This geometric property also leads to an explicit
characterization of the set of (?1 , ?2 ) such that the corresponding solution of problem (2) is 0. We
denote by ??? (?1 , ?2 ) the optimal solution of problem (2).
Corollary 5. For the SGL problem in (2), let ?max
(?2 ) = maxg ?1ng kS?2 (XTg y)k. Then,
1
(i): ??? (?1 , ?2 ) = 0 ? ?1 ? ?max (?2 ).
1
(ii): If ?1 ? ?max
:= maxg
1
4
?1 kXT
g yk
ng
or ?2 ? ?max
:= kXT yk? , then ??? (?1 , ?2 ) = 0.
2
The Two-Layer Screening Rules for SGL
We follow the three steps in Section 3.2 to develop TLFre. In Section 4.1, we give an accurate
estimation of ?? (?, ?) via normal cones [15]. Then, we compute the supreme values in (R1? ) and
(R2? ) by solving nonconvex problems in Section 4.2. We present the TLFre rules in Section 4.3.
4.1 Estimation of the Dual Optimal Solution
Because of the geometric property of the dual problem in (5), i.e., ?? (?, ?) = PF ? (y/?), we have
a very useful characterization of the dual optimal solution via the so-called normal cones [15].
Definition 1. [15] For a closed convex set C ? Rn and a point w ? C, the normal cone to C at w is
NC (w) = {v : hv, w0 ? wi ? 0, ?w0 ? C}.
(14)
? ?).
? ?) is known if ?
? = ?? . Thus, we can estimate ?? (?, ?) in terms of ?? (?,
By Theorem 4, ?? (?,
max
?
for
?
(?,
?)
to
be
estimated.
Due to the same reason, we only consider the cases with ? < ??
max
Remark 4. In many applications, the parameter values that perform the best are usually unknown.
To determine appropriate parameter values, commonly used approaches such as cross validation
and stability selection involve solving SGL many times over a grip of parameter values. Thus, given
{?(i) }Ii=1 and ?(1) ? ? ? ? ? ?(J ) , we can fix the value of ? each time and solve SGL by varying the
value of ?. We repeat the process until we solve SGL for all of the parameter values.
? ?) is known with ?
? ? ?? . Let ?g ,
Theorem 6. For the SGL problem in (3), suppose that ?? (?,
max
?
g = 1, . . . , G, be defined by Theorem 4. For any ? ? (0, ?), we define
? ?
?
? < ?? ,
if ?
max
? = y/? ? ? (?, ?),
where X? = argmaxXg ?g ,
n? (?)
T
?
?
X? S1 (X? y/?max ), if ? = ??
max ,
? = y ? ?? (?,
? ?),
v? (?, ?)
?
? ? = v? (?, ?)
? ?
v? (?, ?)
?
?
hv? (?,?),n
? (?)i
?
n? (?).
? 2
kn? (?)k
Then, the following hold:
? ? NF ? (?? (?,
? ?)),
(i): n? (?)
?
? ?) + 1 v? (?, ?))k
?
?
(ii): k?? (?, ?) ? (?? (?,
? 21 kv?
(?, ?)k.
2 ?
? Theorem 6 shows that ?? (?, ?)
? = ?? (?,
? ?) + 1 v? (?, ?).
For notational convenience, let o? (?, ?)
2 ?
?
? centered at o? (?, ?).
?
lies inside the ball of radius 21 kv?
(?, ?)k
4.2 Solving for the supreme values via Nonconvex Optimization
We solve the optimization problems in (R1? ) and (R2? ). To simplify notations, let
? ? 1 kv? (?, ?)k},
?
? = {? : k? ? o? (?, ?)k
?
2
? ? 1 kv? (?, ?)kkX
?
?g = ?g : k?g ? XT o? (?, ?)k
g k2 , g = 1, . . . , G.
g
2
5
?
(15)
(16)
Theorem 6 indicates that ?? (?, ?) ? ?. Moreover, we can see that XTg ? ? ?g , g = 1, . . . , G. To
develop the TLFre rule by (R1? ) and (R2? ), we need to solve the following optimization problems:
? ?) = sup {kS1 (?g )k : ?g ? ?g }, g = 1, . . . , G,
s?g (?, ?;
(17)
?g
?
T
?
t (?, ?; ?) = sup {|x ?| : ? ? ?}, i = 1, . . . , ng , g = 1, . . . , G.
(18)
?
gi
gi
Solving problem (17) We consider the following equivalent problem of (17):
1
1
? ?) 2 = sup
kS1 (?g )k2 : ?g ? ?g .
s? (?, ?;
(19)
We can see that the objective function of problem (19) is continuously differentiable and the feasible
set is a ball. Thus, (19) is a nonconvex problem because we need to maximize a convex function
subject to a convex set. We derive the closed form solutions of problems (17) and (19) as follows.
?
? r = 1 kv? (?, ?)kkX
?
Theorem 7. For problems (17) and (19), let c = XTg o? (?, ?),
g k2 and ?g be
?
2
the set of the optimal solutions.
(i) Suppose that c ?
/ B? , i.e., kck? > 1. Let u = rS1 (c)/kS1 (c)k. Then,
? ?) = kS1 (c)k + r and ?? = {c + u}.
s?g (?, ?;
(20)
g
(ii) Suppose that c is a boundary point of B? , i.e., kck? = 1. Then,
? ?) = r and ?? = {c + u : u ? NB (c), kuk = r} .
s?g (?, ?;
(21)
g
?
?
?
(iii) Suppose that c ? int B? , i.e., kck? < 1. Let i ? I = {i : |[c]i | = kck? }. Then,
? ?) = (kck? + r ? 1) ,
s?g (?, ?;
(22)
+
?
if ?g ? B? ,
??g ,
?
?
?
?
?
?g = {c + r ? sgn([c]i )ei : i ? I } , if ?g 6? B? and c 6= 0,
?
{r ? ei? , ?r ? ei? : i? ? I ? } ,
if ?g 6? B? and c = 0,
2
g
?g
2
where ei is the ith standard basis vector.
Solving problem in (18) Problem (18) can be solved directly via the Cauchy-Schwarz inequality.
? + 1 kv? (?, ?)kkx
? ?) = |xT o? (?, ?)|
?
Theorem 8. For problem (18), we have t?gi (?, ?;
gi k.
gi
?
2
4.3 The Proposed Two-Layer Screening Rules
To develop the two-layer screening rules for SGL, we only need to plug the supreme values
? 2 ; ?1 ) in (R1? ) and (R2? ). We present the TLFre rule as follows.
? 2 ; ?1 ) and t? (?2 , ?
s?g (?2 , ?
gi
Theorem 9. For the SGL problem in (3), suppose that we are given ? and a sequence of parameter
(0)
values ??
> ?(1) > . . . > ?(J ) . For each integer 0 ? j < J , we assume that ? ? (?(j) , ?)
max = ?
? (j)
? (j+1)
(?
, ?(j) ) and s?g (?(j+1) , ?(j) ; ?) be given by Eq. (6), Theorems 6
is known. Let ? (? , ?), v?
and 7, respectively. Then, for g = 1, . . . , G, the following holds
?
s?g (?(j+1) , ?(j) ; ?) < ? ng ? ?g? (?(j+1) , ?) = 0.
(L1 )
For the g?th group that does not pass the rule in (L1 ), we have [?g?? (?(j+1) , ?)]i = 0 if
T y?X? ? (?(j) ,?) 1 ? (j+1) (j) 1 ? (j+1) (j)
+
v
(?
,
?
)
, ? )kkxg?i k ? 1.
xg?i
+ 2 kv? (?
(j)
?
2
?
(L2 )
(L1 ) and (L2 ) are the first layer and second layer screening rules of TLFre, respectively.
5
Experiments
We evaluate TLFre on both synthetic and real data sets. To measure the performance of TLFre, we
compute the rejection ratios of (L1 ) and (L2 ), respectively. Specifically, let m be the number of
features that have 0 coefficients in the solution, G be the index set of groups that are discarded by
(L1 ) and p be the number of inactive
features that are detected by (L2 ). The rejection ratios of (L1 )
P
ng
and (L2 ) are defined by r1 = g?G
and r2 = |p|
m
m , respectively. Moreover, we report the speedup
gained by TLFre, i.e., the ratio of the running time of solver without screening to the running time
of solver with TLFre. The solver used in this paper is from SLEP [9].
To determine appropriate values of ? and ? by cross validation or stability selection, we can run
TLFre with as many parameter values as we need. Given a data set, for illustrative purposes only,
we select seven values of ? from {tan(?) : ? = 5? , 15? , 30? , 45? , 60? , 75? , 85? }. Then, for each
value of ?, we run TLFre along a sequence of 100 values of ? equally spaced on the logarithmic
scale of ?/??
max from 1 to 0.01. Thus, 700 pairs of parameter values of (?, ?) are sampled in total.
6
0.3
0.1
200
400
?2
600
0.01 0.02 0.04
800
Rejection Ratio
Rejection Ratio
0.1 0.2 0.4
?/??max
1
0.7
0.5
0.3
0.1
0.1 0.2 0.4
?/??max
1
0.7
0.5
0.3
0.1 0.2 0.4
?/??max
1
?
(e) ? = tan(45 )
1
0.5
0.3
0.01 0.02 0.04
0.7
0.5
0.3
0.1 0.2 0.4
?/??max
1
1
0.9
0.7
0.5
0.3
0.1
0.1 0.2 0.4
?/??max
1
0.01 0.02 0.04
?
(f) ? = tan(60 )
0.7
(d) ? = tan(30? )
1
0.9
0.01 0.02 0.04
1
0.9
0.1
0.1 0.2 0.4
?/??max
0.1
0.01 0.02 0.04
?
0.3
(c) ? = tan(15? )
1
0.9
0.1
0.01 0.02 0.04
0.5
0.01 0.02 0.04
(b) ? = tan(5? )
(a)
1
0.9
0.7
0.1
Rejection Ratio
0
0
0.5
Rejection Ratio
100
0.7
1
0.9
Rejection Ratio
200
1
0.9
Rejection Ratio
?1
300
Rejection Ratio
?max
1 (?2 )
? = tan(5? )
? = tan(15? )
? = tan(30? )
? = tan(45? )
? = tan(60? )
? = tan(75? )
? = tan(85? )
400
0.1 0.2 0.4
?/??max
1
?
(g) ? = tan(75 )
(h) ? = tan(85 )
Figure 1: Rejection ratios of TLFre on the Synthetic 1 data set.
0.5
0.3
0.1
500
?2
0.01 0.02 0.04
1000
Rejection Ratio
Rejection Ratio
0.7
0.5
0.3
0.1
0.01 0.02 0.04
0.1 0.2 0.4
?/??max
1
?
(e) ? = tan(45 )
1
0.3
0.7
0.5
0.3
1
0.1 0.2 0.4
?/??max
1
0.7
0.5
0.3
0.01 0.02 0.04
?
0.5
0.3
0.01 0.02 0.04
0.1 0.2 0.4
?/??max
1
1
0.9
0.7
0.5
0.3
0.1
0.1 0.2 0.4
?/??max
?
(f) ? = tan(60 )
0.7
(d) ? = tan(30? )
1
0.9
0.1
0.01 0.02 0.04
1
0.9
0.1
0.1 0.2 0.4
?/??max
(c) ? = tan(15? )
1
0.9
0.1
0.1 0.2 0.4
?/??max
0.5
0.01 0.02 0.04
(b) ? = tan(5? )
(a)
1
0.9
0.7
0.1
Rejection Ratio
0
0
Rejection Ratio
0.7
1
0.9
Rejection Ratio
200
1
0.9
Rejection Ratio
?1
400
Rejection Ratio
?max
1 (?2 )
? = tan(5? )
? = tan(15? )
? = tan(30? )
? = tan(45? )
? = tan(60? )
? = tan(75? )
? = tan(85? )
600
(g) ? = tan(75 )
1
0.01 0.02 0.04
0.1 0.2 0.4
?/??max
1
?
(h) ? = tan(85 )
Figure 2: Rejection ratios of TLFre on the Synthetic 2 data set.
5.1 Simulation Studies
We perform experiments on two synthetic data sets that are commonly used in the literature [19, 29].
The true model is y = X? ? + 0.01, ? N (0, 1). We generate two data sets with 250 ? 10000
entries: Synthetic 1 and Synthetic 2. We randomly break the 10000 features into 1000 groups. For
Synthetic 1, the entries of the data matrix X are i.i.d. standard Gaussian with pairwise correlation
zero, i.e., corr(xi , xi ) = 0. For Synthetic 2, the entries of the data matrix X are drawn from i.i.d.
standard Gaussian with pairwise correlation 0.5|i?j| , i.e., corr(xi , xj ) = 0.5|i?j| . To construct ? ? ,
we first randomly select ?1 percent of groups. Then, for each selected group, we randomly select ?2
percent of features. The selected components of ? ? are populated from a standard Gaussian and the
remaining ones are set to 0. We set ?1 = ?2 = 10 for Synthetic 1 and ?1 = ?2 = 20 for Synthetic 2.
The figures in the upper left corner of Fig. 1 and Fig. 2 show the plots of ?max
(?2 ) (see Corollary
1
5) and the sampled parameter values of ? and ? (recall that ?1 = ?? and ?2 = ?). For the other
figures, the blue and red regions represent the rejection ratios of (L1 ) and (L2 ), respectively. We
can see that TLFre is very effective in discarding inactive groups/features; that is, more than 90%
of inactive features can be detected. Moreover, we can observe that the first layer screening (L1 )
becomes more effective with a larger ?. Intuitively, this is because the group Lasso penalty plays a
more important role in enforcing the sparsity with a larger value of ? (recall that ?1 = ??). The top
and middle parts of Table 1 indicate that the speedup gained by TLFre is very significant (up to 30
times) and TLFre is very efficient. Compared to the running time of the solver without screening,
the running time of TLFre is negligible. The running time of TLFre includes that of computing
kXg k2 , g = 1, . . . , G, which can be efficiently computed by the power method [6]. Indeed, this can
be shared for TLFre with different parameter values.
5.2 Experiments on Real Data Set
We perform experiments on the Alzheimer?s Disease Neuroimaging Initiative (ADNI) data set
(http://adni.loni.usc.edu/). The data matrix consists of 747 samples with 426040 single
7
Table 1: Running time (in seconds) for solving SGL along a sequence of 100 tuning parameter
values of ? equally spaced on the logarithmic scale of ?/??
max from 1.0 to 0.01 by (a): the solver [9]
without screening; (b): the solver combined with TLFre. The top and middle parts report the results
of TLFre on Synthetic 1 and Synthetic 2. The bottom part reports the results of TLFre on the ADNI
data set with the GMV data as response.
tan(5? ) tan(15? ) tan(30? ) tan(45? ) tan(60? ) tan(75? ) tan(85? )
?
solver
TLFre
Synthetic 1
TLFre+solver
speedup
298.36
0.77
10.26
29.09
301.74
0.78
12.47
24.19
308.69
0.79
15.73
19.63
307.71
0.79
17.69
17.40
311.33
0.81
19.71
15.79
307.53
0.79
21.95
14.01
291.24
0.77
22.53
12.93
solver
TLFre
Synthetic 2
TLFre+solver
speedup
294.64
0.79
11.05
26.66
294.92
0.80
12.89
22.88
297.29
0.80
16.08
18.49
297.50
0.81
18.90
15.74
297.59
0.81
20.45
14.55
295.51
0.81
21.58
13.69
292.24
0.82
22.80
12.82
30838.29
64.96
386.80
79.73
31096.10
65.00
402.72
77.22
30850.78
64.89
391.63
78.78
30728.27
65.17
385.98
79.61
30572.35
65.05
382.62
79.90
0.5
0.3
0.1
50
100
?2
150
1
0.7
0.5
0.3
0.1
0.1 0.2 0.4
?/??max
(e) ? = tan(45? )
1
0.3
1
0.7
0.5
0.3
0.1 0.2 0.4
?/??max
1
(f) ? = tan(60? )
0.3
0.1 0.2 0.4
?/??max
1
(d) ? = tan(30 )
1
0.9
0.7
0.5
0.3
0.01 0.02 0.04
0.5
?
0.1
0.01 0.02 0.04
0.7
0.01 0.02 0.04
(c) ? = tan(15 )
1
0.9
1
0.9
0.1
0.1 0.2 0.4
?/??max
?
0.1
0.01 0.02 0.04
0.5
0.01 0.02 0.04
(b) ? = tan(5 )
Rejection Ratio
Rejection Ratio
0.1 0.2 0.4
?/??max
?
(a)
0.7
0.1
0.01 0.02 0.04
1
0.9
1
0.9
Rejection Ratio
0.7
Rejection Ratio
0
0
1
0.9
Rejection Ratio
50
Rejection Ratio
?max
1 (?2 )
? = tan(5? )
? = tan(15? )
? = tan(30? )
? = tan(45? )
? = tan(60? )
? = tan(75? )
? = tan(85? )
100
?1
solver
30652.56 30755.63
TLFre
64.08
64.56
TLFre+solver 372.04
383.17
speedup
82.39
80.27
Rejection Ratio
ADNI+GMV
1
0.9
0.7
0.5
0.3
0.1
0.1 0.2 0.4
?/??max
(g) ? = tan(75? )
1
0.01 0.02 0.04
0.1 0.2 0.4
?/??max
1
(h) ? = tan(85? )
Figure 3: Rejection ratios of TLFre on the ADNI data set with grey matter volume as response.
nucleotide polymorphisms (SNPs), which are divided into 94765 groups. The response vector is the
grey matter volume (GMV).
The figure in the upper left corner of Fig. 3 shows the plots of ?max
(?2 ) (see Corollary 5) and the
1
sampled parameter values of ? and ?. The other figures present the rejection ratios of (L1 ) and (L2 )
by blue and red regions, respectively. We can see that almost all of the inactive groups/features are
discarded by TLFre. The rejection ratios of r1 + r2 are very close to 1 in all cases. The bottom part
of Table 1 shows that TLFre leads to a very significant speedup (about 80 times). In other words, the
solver without screening needs about eight and a half hours to solve the 100 SGL problems for each
value of ?. However, combined with TLFre, the solver needs only six to eight minutes. Moreover,
we can observe that the computational cost of TLFre is negligible compared to that of the solver
without screening. This demonstrates the efficiency of TLFre.
6
Conclusion
In this paper, we propose a novel feature reduction method for SGL via decomposition of convex
sets. We also derive the set of parameter values that lead to zero solutions of SGL. To the best
of our knowledge, TLFre is the first method which is applicable to sparse models with multiple
sparsity-inducing regularizers. More importantly, the proposed approach provides novel framework
for developing screening methods for complex sparse models with multiple sparsity-inducing regularizers, e.g., `1 SVM that performs both sample and feature selection, fused Lasso and tree Lasso
with more than two regularizers. Experiments on both synthetic and real data sets demonstrate the
effectiveness and efficiency of TLFre. We plan to generalize the idea of TLFre to `1 SVM, fused
Lasso and tree Lasso, which are expected to consist of multiple layers of screening.
8
References
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9
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4,780 | 5,328 | Median Selection Subset Aggregation for Parallel
Inference
Xiangyu Wang
Dept. of Statistical Science
Duke University
[email protected]
Peichao Peng
Statistics Department
University of Pennsylvania
[email protected]
David B. Dunson
Dept. of Statistical Science
Duke University
[email protected]
Abstract
For massive data sets, efficient computation commonly relies on distributed algorithms that store and process subsets of the data on different machines, minimizing
communication costs. Our focus is on regression and classification problems involving many features. A variety of distributed algorithms have been proposed
in this context, but challenges arise in defining an algorithm with low communication, theoretical guarantees and excellent practical performance in general settings. We propose a MEdian Selection Subset AGgregation Estimator (message)
algorithm, which attempts to solve these problems. The algorithm applies feature
selection in parallel for each subset using Lasso or another method, calculates the
?median? feature inclusion index, estimates coefficients for the selected features in
parallel for each subset, and then averages these estimates. The algorithm is simple, involves very minimal communication, scales efficiently in both sample and
feature size, and has theoretical guarantees. In particular, we show model selection
consistency and coefficient estimation efficiency. Extensive experiments show excellent performance in variable selection, estimation, prediction, and computation
time relative to usual competitors.
1
Introduction
The explosion in both size and velocity of data has brought new challenges to the design of statistical
algorithms. Parallel inference is a promising approach for solving large scale problems. The typical
procedure for parallelization partitions the full data into multiple subsets, stores subsets on different
machines, and then processes subsets simultaneously. Processing on subsets in parallel can lead to
two types of computational gains. The first reduces time for calculations within each iteration of
optimization or sampling algorithms via faster operations; for example, in conducting linear algebra
involved in calculating likelihoods or gradients [1?7]. Although such approaches can lead to substantial reductions in computational bottlenecks for big data, the amount of gain is limited by the
need to communicate across computers at each iteration. It is well known that communication costs
are a major factor driving the efficiency of distributed algorithms, so that it is of critical importance
to limit communication. This motivates the second type of approach, which conducts computations
completely independently on the different subsets, and then combines the results to obtain the final
output. This limits communication to the final combining step, and may lead to simpler and much
faster algorithms. However, a major issue is how to design algorithms that are close to communication free, which can preserve or even improve the statistical accuracy relative to (much slower)
algorithms applied to the entire data set simultaneously. We focus on addressing this challenge in
this article.
There is a recent flurry of research in both Bayesian and frequentist settings focusing on the second
approach [8?14]. Particularly relevant to our approach is the literature on methods for combining
point estimators obtained in parallel for different subsets [8, 9, 13]. Mann et al. [9] suggest using
1
averaging for combining subset estimators, and Zhang et al. [8] prove that such estimators will
achieve the same error rate as the ones obtained from the full set if the number of subsets m is well
chosen. Minsker [13] utilizes the geometric median to combine the estimators, showing robustness
and sharp concentration inequalities. These methods function well in certain scenarios, but might
not be broadly useful. In practice, inference for regression and classification typically contains two
important components: One is variable or feature selection and the other is parameter estimation.
Current combining methods are not designed to produce good results for both tasks.
To obtain a simple and computationally efficient parallel algorithm for feature selection and coefficient estimation, we propose a new combining method, referred to as message. The detailed
algorithm will be fully described in the next section. There are related methods, which were proposed with the very different goal of combining results from different imputed data sets in missing
data contexts [15]. However, these methods are primarily motivated for imputation aggregation, do
not improve computational time, and lack theoretical guarantees. Another related approach is the
bootstrap Lasso (Bolasso) [16], which runs Lasso independently for multiple bootstrap samples, and
then intersects the results to obtain the final model. Asymptotic properties are provided under fixed
number of features (p fixed) and the computational burden is not improved over applying Lasso to
the full data set. Our message algorithm has strong justification in leading to excellent convergence
properties in both feature selection and prediction, while being simple to implement and computationally highly efficient.
The article is organized as follows. In section 2, we describe message in detail. In section 3, we
provide theoretical justifications and show that message can produce better results than full data inferences under certain scenarios. Section 4 evaluates the performance of message via extensive numerical experiments. Section 5 contains a discussion of possible generalizations of the new method
to broader families of models and online learning. All proofs are provided in the supplementary
materials.
2
Parallelized framework
Consider the linear model which has n observations and p features,
Y = X? + ?,
where Y is an n ? 1 response vector, X is an n ? p matrix of features and ? is the observation error,
which is assumed to have mean zero and variance ? 2 . The fundamental idea for communication
efficient parallel inference is to partition the data set into m subsets, each of which contains a small
portion of the data n/m. Separate analysis on each subset will then be carried out and the result will
be aggregated to produce the final output.
As mentioned in the previous section, regression problems usually consist of two stages: feature
selection and parameter estimation. For linear models, there is a rich literature on feature selection
and we only consider two approaches. The risk inflation criterion (RIC), or more generally, the generalized information criterion (GIC) is an l0 -based feature selection technique for high dimensional
data [17?20]. GIC attempts to solve the following optimization problem,
? ? = arg
M
min
M ?{1,2,??? ,p}
kY ? XM ?M k22 + ?|M |? 2
(1)
for some well chosen ?. For ? = 2(log p + log log p) it corresponds to RIC [18], for ? =
(2 log p + log n) it corresponds to extended BIC [19] and ? = log n reduces to the usual BIC.
Konishi and Kitagawa [18] prove the consistency of GIC for high dimensional data under some
regularity conditions.
Lasso [21] is an l1 based feature selection technique, which solves the following problem
1
?? = arg min kY ? X?k22 + ?k?k1
? n
(2)
for some well chosen ?. Lasso transfers the original NP hard l0 -based optimization to a problem
that can be solved in polynomial time. Zhao and Yu [22] prove the selection consistency of Lasso
under the Irrepresentable condition. Based on the model selected by either GIC or Lasso, we could
then apply the ordinary least square (OLS) estimator to find the coefficients.
2
As briefly discussed in the introduction, averaging and median aggregation approaches possess different advantages but also suffer from certain drawbacks. To carefully adapt these features to regression and classification, we propose the median selection subset aggregation (message) algorithm,
which is motivated as follows.
Averaging of sparse regression models leads to an inflated number of features having non-zero coefficients, and hence is not appropriate for model aggregation when feature selection is of interest.
When conducting Bayesian variable selection, the median probability model has been recommended
as selecting the single model that produces the best approximation to model-averaged predictions
under some simplifying assumptions [23]. The median probability model includes those features
having inclusion probabilities greater than 1/2. We can apply this notion to subset-based inference
by including features that are included in a majority of the subset-specific analyses, leading to select(i)
(i)
ing the ?median model?. Let ? (i) = (?1 , ? ? ? , ?p ) denote a vector of feature inclusion indicators
(i)
for the ith subset, with ?j = 1 if feature j is included so that the coefficient ?j on this feature is
(i)
non-zero, with ?j = 0 otherwise. The inclusion indicator vector for the median model M? can be
obtained by
? = arg min
??{0,1}p
m
X
i=1
k? ? ? (i) k1 ,
or equivalently,
(i)
?j = median{?j , i = 1, 2, ? ? ? , m} for j = 1, 2, ? ? ? , p.
If we apply Lasso or GIC to the full data set, in the presence of heavy-tailed observation errors, the
estimated feature inclusion indicator vector will converge to the true inclusion vector at a polynomial
rate. It is shown in the next section that the convergence rate of the inclusion vector for the median
model can be improved to be exponential, leading to substantial gains in not only computational
time but also feature selection performance. The intuition for this gain is that in the heavy-tailed
case, a proportion of the subsets will contain outliers having a sizable influence on feature selection.
By taking the median, we obtain a central model that is not so influenced by these outliers, and hence
can concentrate more rapidly. As large data sets typically contain outliers and data contamination,
this is a substantial practical advantage in terms of performance even putting aside the computational gain. After feature selection, we obtain estimates of the coefficients for each selected feature
by averaging the coefficient estimates from each subset, following the spirit of [8]. The message
algorithm (described in Algorithm 1) only requires each machine to pass the feature indicators to
a central computer, which (essentially instantaneously) calculates the median model, passes back
the corresponding indicator vector to the individual computers, which then pass back coefficient
estimates for averaging. The communication costs are negligible.
3
Theory
In this section, we provide theoretical justification for the message algorithm in the linear model
case. The theory is easily generalized to a much wider range of models and estimation techniques,
as will be discussed in the last section.
Throughout the paper we will assume X = (x1 , ? ? ? , xp ) is an n ? p feature matrix, s = |S| is the
number of non-zero coefficients and ?(A) is the eigenvalue for matrix A. Before we proceed to the
theorems, we enumerate several conditions that are required for establishing the theory. We assume
there exist constants V1 , V2 > 0 such that
A.1 Consistency condition for estimation.
?
?
1 T
n xi xi ? V1 for
?min ( n1 XST XS )
i = 1, 2, ? ? ? , p
? V2
A.2 Conditions on ?, |S| and ?
? E(?2k ) < ? for some k > 0
? s = |S| ? c1 n? for some 0 ? ? < 1
3
Algorithm 1 Message algorithm
Initialization:
1: Input (Y, X), n, p, and m
2:
# n is the sample size, p is the number of features and m is the number of subsets
3: Randomly partition (Y, X) into m subsets (Y (i) , X (i) ) and distribute them on m machines.
Iteration:
4: for i = 1 to m do
5:
? (i) = minM? loss(Y (i) , X (i) ) # ? (i) is the estimated model via Lasso or GIC
6: # Gather all subset models ? (i) to obtain the median model M?
7: for j = 1 to p do
(i)
8:
?j = median{?j , i = 1, 2, ? ? ? , m}
9: # Redistribute the estimated model M? to all subsets
10: for i = 1 to m do
(i)T
(i)
(i)T (i)
# Estimate the coefficients
11:
? (i) = (X? X? )?1 X? Y?
(i)
12: # Gather
Pm all subset estimations ?
13: ?? = i=1 ? (i) /m
14:
? ?
15: return ?,
1??
? mini?S |?i | ? c2 n? 2 for some 0 < ? ? 1
A.3 (Lasso) The strong irrepresentable condition.
? Assuming XS and XS c are the features having non-zero and zero coefficients, respectively, there exists some positive constant vector ? such that
|XSTc XS (XST XS )?1 sign(?S )| < 1 ? ?
A.4 (Generalized information criterion, GIC) The sparse Riesz condition.
? There exist constants ? ? 0 and c > 0 such that ? > cn?? , where
? = inf ?min (X?T X? /n)
|?|?|S|
A.1 is the usual consistency condition for regression. A.2 restricts the behaviors of the three key
terms and is crucial for model selection. These are both usual assumptions. See [19,20,22]. A.3 and
A.4 are specific conditions for model selection consistency for Lasso/GIC. As noted in [22], A.3 is
almost sufficient and necessary for sign consistency. A.4 could be relaxed slightly as shown in [19],
but for simplicity we rely on this version. To ameliorate possible concerns on how realistic these
conditions are, we provide further justifications via Theorem 3 and 4 in the supplementary material.
Theorem 1. (GIC) Assume each subset satisfies A.1, A.2 and A.4, and p ? n? for some ? < k(? ?
?), where ? = max{?/k, 2?}. If ? < ? , 2? < ? and ? in (1) are chosen so that ? = c0 /? 2 (n/m)? ??
?1
for some c0 < cc2 /2, then there exists some constant C0 such that for n ? (2C0 p)(k? ?k?) and
?1
?1
m = ?(4C0 )?(k? ?k?) ? n/p(k? ?k?) ?, the selected model M? follows,
n1??/(k? ?k?)
P (M? = MS ) ? 1 ? exp ?
,
24(4C0 )(k? ?k?)
(i)T
and defining C0? = mini ?min (X?
(i)
X? /ni ), the mean square error follows,
?1
n1??/(k? ?k?)
? 2 V2 s
??1
??1 2
2
+ exp ?
.
(1
+
2C
sV
)k?k
+
C
?
Ek?? ? ?k22 ?
1
2
0
0
n
24(4C0 )(k? ?k?)
Theorem 2. (Lasso) Assume each subset satisfies A.1, A.2 and A.3, and p ? n? for some ? <
??? +1
k(? ? ?). If ? < ? and ? in (2) are chosen so that ? = c0 (n/m) 2 for some c0 < c1 V2 /c2 ,
?1
?1
then there exists some constant C0 such that for n ? (2C0 p)(k? ?k?) and m = ?(4C0 )(k? ?k?) ?
?1
n/p(k? ?k?) ?, the selected model M? follows
n1??/(k? ?k?)
P (M? = MS ) ? 1 ? exp ?
,
24(4C0 )(k? ?k?)
4
and with the same C0? defined in Theorem 1, we have
? 2 V2?1 s
n1??/(k? ?k?)
??1
??1 2
2
2
?
(1 + 2C0 sV1 )k?k2 + C0 ? .
Ek? ? ?k2 ?
+ exp ?
n
24(4C0 )(k? ?k?)
The above two theorems boost the model consistency property from the original polynomial rate
[20, 22] to an exponential rate for heavy-tailed errors. In addition, the mean square error, as shown
in the above equation, preserves almost the same convergence rate as if the full data is employed
and the true model is known. Therefore, we expect a similar or better performance of message with
a significantly lower computation load. Detailed comparisons are demonstrated in Section 4.
4
Experiments
This section assesses the performance of the message algorithm via extensive examples, comparing
the results to
? Full data inference. (denoted as ?full data?)
? Subset averaging. Partition and average the estimates obtained on all subsets. (denoted as
?averaging?)
? Subset median. Partition and take the marginal median of the estimates obtained on all
subsets (denoted as ?median?)
? Bolasso. Run Lasso on multiple bootstrap samples and intersect to select model. Then
estimate the coefficients based on the selected model. (denoted as ?Bolasso?)
The Lasso part of all algorithms will be implemented by the ?glmnet? package [24]. (We did not
use ADMM [25] for Lasso as its actual performance might suffer from certain drawbacks [6] and is
reported to be slower than ?glmnet? [26])
4.1
Synthetic data sets
We use the linear model and the logistic model for (p; s) = (1000; 3) or (10,000; 3) with different
sample size n and different partition number m to evaluate the performance. The feature vector is
drawn from a multivariate normal distribution with correlation ? = 0 or 0.5. Coefficients ? are
chosen as,
?
?i ? (?1)ber(0.4) (8 log n/ n + |N (0, 1)|), i ? S
Since GIC is intractable to implement (NP hard), we combine it with Lasso for variable selection:
Implement Lasso for a set of different ??s and determine the optimal one via GIC. The concrete
setup of models are as follows,
Case 1 Linear model with ? ? N (0, 22 ).
Case 2 Linear model with ? ? t(0, df = 3).
Case 3 Logistic model.
For p = 1, 000, we simulate 200 data sets for each case, and vary the sample size from 2000 to
10,000. For each case, the subset size is fixed to 400, so the number of subsets will be changing
? probability of selecting the
from 5 to 25. In the experiment, we record the mean square error for ?,
true model and computational time, and plot them in Fig 1 - 6. For p = 10,000, we simulate 50
data sets for each case, and let the sample size range from 20,000 to 50,000 with subset size fixed to
2000. Results for p = 10,000 are provided in supplementary materials.
It is clear that message had excellent performance in all of the simulation cases, with low MSE,
high probability of selecting the true model, and low computational time. The other subset-based
methods we considered had similar computational times and also had computational burdens that
effectively did not increase with sample size, while the full data analysis and bootstrap Lasso approach both were substantially slower than the subset methods, with the gap increasing linearly in
sample size. In terms of MSE, the averaging and median approaches both had dramatically worse
5
2000
4000
6000
8000
10000
0.5
1.0
1.5
median
fullset
average
message
bolasso
0.0
0.0
0.0
0.2
seconds
2.0
1.0
0.8
0.6
prob
median
fullset
average
message
bolasso
0.4
0.2
0.3
median
fullset
average
message
bolasso
0.1
value
Computational time
2.5
Probability to select the true model
0.4
Mean square error
2000
4000
Sample size n
6000
8000
10000
2000
4000
Sample size n
6000
8000
10000
Sample size n
Figure 1: Results for case 1 with ? = 0.
Probability to select the true model
2000
4000
6000
8000
10000
3.0
2.0
1.0
median
fullset
average
message
bolasso
0.0
0.2
0.0
0.0
seconds
0.8
0.6
prob
median
fullset
average
message
bolasso
0.4
0.4
0.6
median
fullset
average
message
bolasso
0.2
value
Computational time
1.0
0.8
Mean square error
2000
4000
Sample size n
6000
8000
10000
2000
4000
Sample size n
6000
8000
10000
Sample size n
Figure 2: Results for case 1 with ? = 0.5.
4000
6000
8000
10000
2.5
2.0
1.5
seconds
0.5
0.0
0.0
0.00
2000
median
fullset
average
message
bolasso
1.0
1.0
0.6
0.8
Computational time
0.2
0.04
median
fullset
average
message
bolasso
0.4
prob
0.10
0.06
0.08
Probability to select the true model
median
fullset
average
message
bolasso
0.02
value
Mean square error
2000
4000
Sample size n
6000
8000
10000
2000
4000
Sample size n
6000
8000
10000
Sample size n
Figure 3: Results for case 2 with ? = 0.
2000
4000
6000
Sample size n
8000
10000
2.5
2.0
0.5
1.0
1.5
median
fullset
average
message
bolasso
0.0
seconds
1.0
0.6
0.0
prob
0.4
median
fullset
average
message
bolasso
0.2
0.20
0.00
0.10
value
median
fullset
average
message
bolasso
Computational time
0.8
Probability to select the true model
0.30
Mean square error
2000
4000
6000
8000
10000
Sample size n
Figure 4: Results for case 2 with ? = 0.5.
6
2000
4000
6000
Sample size n
8000
10000
Probability to select the true model
6
median
fullset
average
message
bolasso
4
seconds
0.6
median
fullset
average
message
bolasso
0.4
prob
0.8
8
6
3
4
5
median
fullset
average
message
bolasso
2000
4000
6000
8000
10000
0
0
0.0
1
0.2
2
2
value
Computational time
1.0
Mean square error
2000
4000
Sample size n
6000
8000
10000
2000
4000
Sample size n
6000
8000
10000
Sample size n
Figure 5: Results for case 3 with ? = 0.
Probability to select the true model
Computational time
12
10
8
median
fullset
average
message
bolasso
6
seconds
0.6
prob
median
fullset
average
message
bolasso
2000
4000
6000
8000
Sample size n
10000
2
0
0
0.0
2
0.2
4
4
value
6
median
fullset
average
message
bolasso
0.4
8
0.8
10
1.0
Mean square error
2000
4000
6000
8000
10000
Sample size n
2000
4000
6000
8000
10000
Sample size n
Figure 6: Results for case 3 with ? = 0.5.
performance than message in every case, while bootstrap Lasso was competitive (MSEs were same
order of magnitude with message ranging from effectively identical to having a small but significant
advantage), with both message and bootstrap Lasso clearly outperforming the full data approach. In
terms of feature selection performance, averaging had by far the worst performance, followed by the
full data approach, which was substantially worse than bootstrap Lasso, median and message, with
no clear winner among these three methods. Overall message clearly had by far the best combination
of low MSE, accurate model selection and fast computation.
4.2
Individual household electric power consumption
This data set contains measurements of electric power consumption for every household with a
one-minute sampling rate [27]. The data have been collected over a period of almost 4 years and
contain 2,075,259 measurements. There are 8 predictors, which are converted to 74 predictors due
to re-coding of the categorical variables (date and time). We use the first 2,000,000 samples as the
training set and the remaining 75,259 for testing the prediction accuracy. The data are partitioned
into 200 subsets for parallel inference. We plot the prediction accuracy (mean square error for test
samples) against time for full data, message, averaging and median method in Fig 7. Bolasso is
excluded as it did not produce meaningful results within the time span.
To illustrate details of the performance, we split the time line into two parts: the early stage shows
how all algorithms adapt to a low prediction error and a later stage captures more subtle performance
of faster algorithms (full set inference excluded due to the scale). It can be seen that message
dominates other algorithms in both speed and accuracy.
4.3
HIGGS classification
The HIGGS data have been produced using Monte Carlo simulations from a particle physics model
[28]. They contain 27 predictors that are of interest to physicists wanting to distinguish between two
classes of particles. The sample size is 11,000,000. We use the first 10,000,000 samples for training
a logistic model and the rest to test the classification accuracy. The training set is partitioned into
1,000 subsets for parallel inference. The classification accuracy (probability of correctly predicting
the class of test samples) against computational time is plotted in Fig 8 (Bolasso excluded for the
same reason as above).
7
Mean prediction error (later stage)
0.0
0.0016
value
message
median
average
0.0020
0.4
message
median
average
fullset
0.2
value
0.6
0.0024
0.8
Mean prediction error (earlier stage)
0.060
0.065
0.070
0.075
0.080
0.084
0.086
Time (sec)
0.088
0.090
0.092
0.094
Time (sec)
Figure 7: Results for power consumption data.
0.55
0.60
message
median
average
fullset
0.50
value
0.65
Mean prediction accuracy
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time (sec)
Figure 8: Results for HIGGS classification.
Message adapts to the prediction bound quickly. Although the classification results are not as good
as the benchmarks listed in [28] (due to the choice of a simple parametric logistic model), our new
algorithm achieves the best performance subject to the constraints of the model class.
5
Discussion and conclusion
In this paper, we proposed a flexible and efficient message algorithm for regression and classification with feature selection. Message essentially eliminates the computational burden attributable to
communication among machines, and is as efficient as other simple subset aggregation methods. By
selecting the median model, message can achieve better accuracy even than feature selection on the
full data, resulting in an improvement also in MSE performance. Extensive simulation experiments
show outstanding performance relative to competitors in terms of computation, feature selection and
prediction.
Although the theory described in Section 3 is mainly concerned with linear models, the algorithm
is applicable in fairly wide situations. Geometric median is a topological concept, which allows the
median model to be obtained in any normed model space. The properties of the median model result
from independence of the subsets and weak consistency on each subset. Once these two conditions
are satisfied, the property shown in Section 3 can be transferred to essentially any model space. The
follow-up averaging step has been proven to be consistent for all M estimators with a proper choice
of the partition number [8].
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4,781 | 5,329 | Asymmetric LSH (ALSH) for Sublinear Time
Maximum Inner Product Search (MIPS)
Ping Li
Department of Statistics and Biostatistics
Department of Computer Science
Rutgers University
Piscataway, NJ 08854, USA
[email protected]
Anshumali Shrivastava
Department of Computer Science
Computing and Information Science
Cornell University
Ithaca, NY 14853, USA
[email protected]
Abstract
We present the first provably sublinear time hashing algorithm for approximate
Maximum Inner Product Search (MIPS). Searching with (un-normalized) inner
product as the underlying similarity measure is a known difficult problem and
finding hashing schemes for MIPS was considered hard. While the existing Locality Sensitive Hashing (LSH) framework is insufficient for solving MIPS, in this
paper we extend the LSH framework to allow asymmetric hashing schemes. Our
proposal is based on a key observation that the problem of finding maximum inner products, after independent asymmetric transformations, can be converted into
the problem of approximate near neighbor search in classical settings. This key
observation makes efficient sublinear hashing scheme for MIPS possible. Under
the extended asymmetric LSH (ALSH) framework, this paper provides an example of explicit construction of provably fast hashing scheme for MIPS. Our proposed algorithm is simple and easy to implement. The proposed hashing scheme
leads to significant computational savings over the two popular conventional LSH
schemes: (i) Sign Random Projection (SRP) and (ii) hashing based on p-stable
distributions for L2 norm (L2LSH), in the collaborative filtering task of item recommendations on Netflix and Movielens (10M) datasets.
1
Introduction and Motivation
The focus of this paper is on the problem of Maximum Inner Product Search (MIPS). In this problem,
we are given a giant data vector collection S of size N , where S ? RD , and a given query point
q ? RD . We are interested in searching for p ? S which maximizes (or approximately maximizes)
the inner product q T p. Formally, we are interested in efficiently computing
p = arg max q T x
x?S
(1)
The MIPS problem is related to near neighbor search (NNS), which instead requires computing
p = arg min ??q ? x??22 = arg min(??x??22 ? 2q T x)
x?S
x?S
(2)
These two problems are equivalent if the norm of every element x ? S is constant. Note that the
value of the norm ??q??2 has no effect as it is a constant and does not change the identity of arg max
or arg min. There are many scenarios in which MIPS arises naturally at places where the norms of
the elements in S have significant variations [13] and cannot be controlled, e.g., (i) recommender
system, (ii) large-scale object detection with DPM, and (iii) multi-class label prediction.
Recommender systems: Recommender systems are often based on collaborative filtering which
relies on past behavior of users, e.g., past purchases and ratings. Latent factor modeling based on
matrix factorization [14] is a popular approach for solving collaborative filtering. In a typical matrix
factorization model, a user i is associated with a latent user characteristic vector ui , and similarly,
an item j is associated with a latent item characteristic vector vj . The rating ri,j of item j by user i
is modeled as the inner product between the corresponding characteristic vectors.
1
In this setting, given a user i and the corresponding learned latent vector ui finding the right item j,
to recommend to this user, involves computing
j = arg max
ri,j ? = arg max
uTi vj ?
?
?
j
j
(3)
which is an instance of the standard MIPS problem. It should be noted that we do not have control
over the norm of the learned vector, i.e., ?vj ?2 , which often has a wide range in practice [13].
If there are N items to recommend, solving (3) requires computing N inner products. Recommendation systems are typically deployed in on-line application over web where the number N is huge.
A brute force linear scan over all items, for computing arg max, would be prohibitively expensive.
Large-scale object detection with DPM: Deformable Part Model (DPM) based representation of
images is the state-of-the-art in object detection tasks [8]. In DPM model, firstly a set of part filters
are learned from the training dataset. During detection, these learned filter activations over various
patches of the test image are used to score the test image. The activation of a filter on an image patch
is an inner product between them. Typically, the number of possible filters are large (e.g., millions)
and so scoring the test image is costly. Recently, it was shown that scoring based only on filters with
high activations performs well in practice [7]. Identifying those filters having high activations on a
given image patch requires computing top inner products. Consequently, an efficient solution to the
MIPS problem will benefit large scale object detections based on DPM.
Multi-class (and/or multi-label) prediction: The models for multi-class SVM (or logistic regression) learn a weight vector wi for each of the class label i. After the weights are learned, given a
new test data vector xtest , predicting its class label is basically an MIPS problem:
ytest = arg max xTtest wi
i?L
(4)
where L is the set of possible class labels. Note that the norms of the vectors ?wi ?2 are not constant.
The size, ?L?, of the set of class labels differs in applications. Classifying with large number of possible class labels is common in multi-label learning and fine grained object classification, for instance,
prediction task with ?L? = 100, 000 [7]. Computing such high-dimensional vector multiplications for
predicting the class label of a single instance can be expensive in, e.g., user-facing applications.
1.1
The Need for Hashing Inner Products
Solving the MIPS problem can have significant practical impact. [19, 13] proposed solutions based
on tree data structure combined with branch and bound space partitioning technique similar to k-d
trees [9]. Later, the same method was generalized for general max kernel search [5], where the runtime guarantees, like other space partitioning methods, are heavily dependent on the dimensionality
and the expansion constants. In fact, it is well-known that techniques based on space partitioning
(such as k-d trees) suffer from the curse of dimensionality. For example, [24] showed that techniques
based on space partitioning degrade to linear search, even for dimensions as small as 10 or 20.
Locality Sensitive Hashing (LSH) [12] based randomized techniques are common and successful
in industrial practice for efficiently solving NNS (near neighbor search). Unlike space partitioning
techniques, both the running time as well as the accuracy guarantee of LSH based NNS are in a way
independent of the dimensionality of the data. This makes LSH suitable for large scale processing
system dealing with ultra-high dimensional datasets which are common in modern applications.
Furthermore, LSH based schemes are massively parallelizable, which makes them ideal for modern
?Big? datasets. The prime focus of this paper will be on efficient hashing based algorithms for
MIPS, which do not suffer from the curse of dimensionality.
1.2
Our Contributions
We develop Asymmetric LSH (ALSH), an extended LSH scheme for efficiently solving the approximate MIPS problem. Finding hashing based algorithms for MIPS was considered hard [19, 13]. We
formally show that, under the current framework of LSH, there cannot exist any LSH for solving
MIPS. Despite this negative result, we show that it is possible to relax the current LSH framework to
allow asymmetric hash functions which can efficiently solve MIPS. This generalization comes with
no extra cost and the ALSH framework inherits all the theoretical guarantees of LSH.
Our construction of asymmetric LSH is based on an interesting fact that the original MIPS problem,
after asymmetric transformations, reduces to the problem of approximate near neighbor search in
2
classical settings. Based on this key observation, we provide an example of explicit construction of
asymmetric hash function, leading to the first provably sublinear query time hashing algorithm for
approximate similarity search with (un-normalized) inner product as the similarity. The new ALSH
framework is of independent theoretical interest. We report other explicit constructions in [22, 21].
We also provide experimental evaluations on the task of recommending top-ranked items with collaborative filtering, on Netflix and Movielens (10M) datasets. The evaluations not only support our
theoretical findings but also quantify the obtained benefit of the proposed scheme, in a useful task.
2
2.1
Background
Locality Sensitive Hashing (LSH)
A commonly adopted formalism for approximate near-neighbor search is the following:
Definition: (c-Approximate Near Neighbor or c-NN) Given a set of points in a D-dimensional space
RD , and parameters S0 > 0, ? > 0, construct a data structure which, given any query point q, does
the following with probability 1 ? ?: if there exists an S0 -near neighbor of q in P , it reports some
cS0 -near neighbor of q in P .
In the definition, the S0 -near neighbor of point q is a point p with Sim(q, p) ? S0 , where Sim is the
similarity of interest. Popular techniques for c-NN are often based on Locality Sensitive Hashing
(LSH) [12], which is a family of functions with the nice property that more similar objects in the
domain of these functions have a higher probability of colliding in the range space than less similar
ones. In formal terms, consider H a family of hash functions mapping RD to a set I.
Definition: (Locality Sensitive Hashing (LSH)) A family H is called (S0 , cS0 , p1 , p2 )-sensitive if,
for any two point x, y ? RD , h chosen uniformly from H satisfies the following:
? if Sim(x, y) ? S0 then P rH (h(x) = h(y)) ? p1
? if Sim(x, y) ? cS0 then P rH (h(x) = h(y)) ? p2
For efficient approximate nearest neighbor search, p1 > p2 and c < 1 is needed.
Fact 1 [12]: Given a family of (S0 , cS0 , p1 , p2 ) -sensitive hash functions, one can construct a data
log p1
structure for c-NN with O(n? log n) query time and space O(n1+? ), where ? = log
< 1.
p2
2.2
LSH for L2 Distance (L2LSH)
[6] presented a novel LSH family for all Lp (p ? (0, 2]) distances. In particular, when p = 2, this
scheme provides an LSH family for L2 distances. Formally, given a fixed (real) number r, we choose
a random vector a with each component generated from i.i.d. normal, i.e., ai ? N (0, 1), and a scalar
b generated uniformly at random from [0, r]. The hash function is defined as:
hL2
a,b (x) = ?
aT x + b
?
r
(5)
where ?? is the floor operation. The collision probability under this scheme can be shown to be
2
2
L2
P r(hL2
(1 ? e?(r/d) /2 ) (6)
Fr (d) = 1 ? 2?(?r/d) ? ?
a,b (x) = ha,b (y)) = Fr (d);
2?(r/d)
where ?(x) = ??? ?12? e? 2 dx is the cumulative density function (cdf) of standard normal distribution and d = ??x ? y??2 is the Euclidean distance between the vectors x and y. This collision
probability Fr (d) is a monotonically decreasing function of the distance d and hence hL2
a,b is an LSH
for L2 distances. This ?
scheme is also the part of LSH package [1]. Here r is a parameter. As argued
previously, ??x ? y??2 = (??x??22 + ??y??22 ? 2xT y) is not monotonic in the inner product xT y unless the
given data has a constant norm. Hence, hL2
a,b is not suitable for MIPS.
x
x2
The recent work on coding for random projections [16] showed that L2LSH can be improved when
the data are normalized for building large-scale linear classifiers as well as near neighbor search [17].
In particular, [17] showed that 1-bit coding (i.e., sign random projections (SRP) [10, 3]) or 2-bit
coding are often better compared to using more bits. It is known that SRP is designed for retrieving
T
y
. Again, ordering under this similarity can be very
with cosine similarity: Sim(x, y) = ??x??x2 ??y??
2
different from the ordering of inner product and hence SRP is also unsuitable for solving MIPS.
3
3
3.1
Hashing for MIPS
A Negative Result
We first show that, under the current LSH framework, it is impossible to obtain a locality sensitive
hashing scheme for MIPS. In [19, 13], the authors also argued that finding locality sensitive hashing
for inner products could be hard, but to the best of our knowledge we have not seen a formal proof.
Theorem 1 There cannot exist any LSH family for MIPS.
Proof: Suppose there exists such hash function h. For un-normalized inner products the self similarity of a point x with itself is Sim(x, x) = xT x = ??x??22 and there may exist another points y, such that
Sim(x, y) = y T x > ??x??22 + C, for any constant C. Under any single randomized hash function h,
the collision probability of the event {h(x) = h(x)} is always 1. So if h is an LSH for inner product
then the event {h(x) = h(y)} should have higher probability compared to the event {h(x) = h(x)},
since we can always choose y with Sim(x, y) = S0 + ? > S0 and cS0 > Sim(x, x) ?S0 and c < 1.
This is not possible because the probability cannot be greater than 1. This completes the proof. ?
3.2
Our Proposal: Asymmetric LSH (ALSH)
The basic idea of LSH is probabilistic bucketing and it is more general than the requirement of
having a single hash function h. The classical LSH algorithms use the same hash function h for both
the preprocessing step and the query step. One assigns buckets in the hash table to all the candidates
x ? S using h, then uses the same h on the query q to identify relevant buckets. The only requirement
for the proof of Fact 1, to work is that the collision probability of the event {h(q) = h(x)} increases
with the similarity Sim(q, x). The theory [11] behind LSH still works if we use hash function h1
for preprocessing x ? S and a different hash function h2 for querying, as long as the probability of
the event {h2 (q) = h1 (x)} increases with Sim(q, x), and there exist p1 and p2 with the required
property. The traditional LSH definition does not allow this asymmetry but it is not a required
condition in the proof. For this reason, we can relax the definition of c-NN without losing runtime
guarantees. [20] used a related (asymmetric) idea for solving 3-way similarity search.
We first define a modified locality sensitive hashing in a form which will be useful later.
Definition: (Asymmetric Locality Sensitive Hashing (ALSH)) A family H, along with the two
?
?
vector functions Q ? RD ? RD (Query Transformation) and P ? RD ? RD (Preprocessing
Transformation), is called (S0 , cS0 , p1 , p2 )-sensitive if, for a given c-NN instance with query q and
any x in the collection S, the hash function h chosen uniformly from H satisfies the following:
? if Sim(q, x) ? S0 then P rH (h(Q(q))) = h(P (x))) ? p1
? if Sim(q, x) ? cS0 then P rH (h(Q(q)) = h(P (x))) ? p2
When Q(x) = P (x) = x, we recover the vanilla LSH definition with h(.) as the required hash
function. Coming back to the problem of MIPS, if Q and P are different, the event {h(Q(x)) =
h(P (x))} will not have probability equal to 1 in general. Thus, Q ? P can counter the fact that self
similarity is not highest with inner products. We just need the probability of the new collision event
{h(Q(q)) = h(P (y))} to satisfy the conditions in the definition of c-NN for Sim(q, y) = q T y. Note
that the query transformation Q is only applied on the query and the pre-processing transformation
P is applied to x ? S while creating hash tables. It is this asymmetry which will allow us to solve
MIPS efficiently. In Section 3.3, we explicitly show a construction (and hence the existence) of
asymmetric locality sensitive hash function for solving MIPS. The source of randomization h for
both q and x ? S is the same. Formally, it is not difficult to show a result analogous to Fact 1.
Theorem 2 Given a family of hash function H and the associated query and preprocessing transformations P and Q, which is (S0 , cS0 , p1 , p2 ) -sensitive, one can construct a data structure for
log p1
c-NN with O(n? log n) query time and space O(n1+? ), where ? = log
.
p2
3.3
From MIPS to Near Neighbor Search (NNS)
Without loss of any generality, let U < 1 be a number such that ??xi ??2 ? U < 1, ?xi ? S. If this is
not the case then define a scaling transformation,
U
? x;
M = maxxi ?S ??xi ??2 ;
(7)
S(x) =
M
4
Note that we are allowed one time preprocessing and asymmetry, S is the part of asymmetric transformation. For simplicity of arguments, let us assume that ??q??2 = 1, the arg max is anyway independent of the norm of the query. Later we show in Section 3.6 that it can be easily removed.
We are now ready to describe the key step in our algorithm. First, we define two vector transformations P ? RD ? RD+m and Q ? RD ? RD+m as follows:
m
P (x) = [x; ??x??22 ; ??x??42 ; ....; ??x??22 ];
Q(x) = [x; 1/2; 1/2; ....; 1/2],
(8)
i
where [;] is the concatenation. P (x) appends m scalers of the form ??x??22 at the end of the vector x,
while Q(x) simply appends m ?1/2? to the end of the vector x. By observing that
m
m+1
1
Q(q)T P (xi ) = q T xi + (??xi ??22 + ??xi ??42 + ... + ??xi ??22 ); ??P (xi )??22 = ??xi ??22 + ??xi ??42 + ... + ??xi ??22
2
we obtain the following key equality:
??Q(q) ? P (xi )??22 = (1 + m/4) ? 2q T xi + ??xi ??22
m+1
(9)
m+1
Since ??xi ??2 ? U < 1, ??xi ??2
? 0, at the tower rate (exponential to exponential). The term
(1 + m/4) is a fixed constant. As long as m is not too small (e.g., m ? 3 would suffice), we have
arg max q T x ? arg min ??Q(q) ? P (x)??2
x?S
(10)
x?S
This gives us the connection between solving un-normalized MIPS and approximate near neighbor
search. Transformations P and Q, when norms are less than 1, provide correction to the L2 distance
??Q(q) ? P (xi )??2 making it rank correlate with the (un-normalized) inner product. This works only
m+1
after shrinking the norms, as norms greater than 1 will instead blow the term ??xi ??22 .
3.4
Fast Algorithms for MIPS
Eq. (10) shows that MIPS reduces to the standard approximate near neighbor search problem which
m+1
m+1
can be efficiently solved. As the error term ??xi ??22
< U2
goes to zero at a tower rate, it quickly
becomes negligible for any practical purposes. In fact, from theoretical perspective, since we are
interested in guarantees for c-approximate solutions, this additional error can be absorbed in the
approximation parameter c. Formally, we can state the following theorem.
Theorem 3 Given a c-approximate instance of MIPS, i.e., Sim(q, x) = q T x, and a query q such
that ??q??2 = 1 along with a collection S having ??x??2 ? U < 1 ?x ? S. Let P and Q be the vector
transformations defined in (8). We have the following two conditions for hash function hL2
a,b (5)
?
m+1
T
L2
L2
2
)
1) if q x ? S0 then P r[ha,b (Q(q)) = ha,b (P (x))] ? Fr ( 1 + m/4 ? 2S0 + U
?
L2
(
)
2) if q T x ? cS0 then P r[hL2
(Q(q))
=
h
(P
(x))]
?
F
1
+
m/4
?
2cS
r
0
a,b
a,b
where the function Fr is defined in (6).
?
?
Thus, we have obtained p1 = Fr ( (1 + m/4) ? 2S0 + U 2m+1 ) and p2 = Fr ( (1 + m/4) ? 2cS0 ).
Applying Theorem 2, we can construct data structures with worst case O(n? log n) query time
guarantees for c-approximate MIPS, where
?
log Fr ( 1 + m/4 ? 2S0 + U 2m+1 )
(11)
?=
?
log Fr ( 1 + m/4 ? 2cS0 )
We need p1 > p2 in order for ? < 1. This requires us to have ?2S0 + U 2
m+1
U2
2S0
m+1
< ?2cS0 , which boils
m+1
U2
2S0
down to the condition c < 1 ?
. Note that
can be made arbitrarily close to zero with the
appropriate value of m. For any given c < 1, there always exist U < 1 and m such that ? < 1. This
way, we obtain a sublinear query time algorithm for MIPS.
We also have one more parameter r for the hash function ha,b . Recall the definition of Fr in Eq. (6):
2
2
Fr (d) = 1 ? 2?(?r/d) ? ?2?(r/d)
(1 ? e?(r/d) /2 ). Thus, given a c-approximate MIPS instance, ?
5
1
1
0.9
0.9
S0 = 0.5U
0.6
0.7
0.7
0.6
0.6
0.7
0.7
0.6
0.8
0.5
0.8
0.5
0.4
0.3
1
S0 = 0.5U
0.8
?
?*
0.8
m=3,U=0.83, r=2.5
0.4
S0 = 0.9U
0.8
0.6
0.4
0.2
0.3
1
0
c
S0 = 0.9U
0.8
0.6
0.4
0.2
0
c
?
Figure 1: Left panel: Optimal values of ? with respect to approximation ratio c for different S0 .
The optimization of Eq. (14) was conducted by a grid search over parameters r, U and m, given S0
and c. Right Panel: ? values (dashed curves) for m = 3, U = 0.83 and r = 2.5. The solid curves are
?? values. See more details about parameter recommendations in arXiv:1405.5869.
is a function of 3 parameters: U , m, r. The algorithm with the best query time chooses U , m and r,
which minimizes the value of ?. For convenience, we define
?
m+1
log Fr ( 1 + m/4 ? 2S0 + U 2m+1 )
U2
?
?
? = min
s.t.
< 1 ? c, m ? N+ , 0 < U < 1. (12)
U,m,r
2S
0
log Fr ( 1 + m/4 ? 2cS0 )
See Figure 1 for the plots of ?? . With this best value of ?, we can state our main result in Theorem 4.
Theorem 4 (Approximate MIPS is Efficient) For the problem of c-approximate MIPS with ??q??2 =
?
?
1, one can construct a data structure having O(n? log n) query time and space O(n1+? ), where
?
? < 1 is the solution to constraint optimization (14).
3.5
Practical Recommendation of Parameters
Just like in the typical LSH framework, the value of ?? in Theorem 4 depends on the c-approximate
instance we aim to solve, which requires knowing the similarity threshold S0 and the approximation
ratio c. Since, ??q??2 = 1 and ??x??2 ? U < 1, ?x ? S, we have q t x ? U . A reasonable choice of the
threshold S0 is to choose a high fraction of U, for example, S0 = 0.9U or S0 = 0.8U .
The computation of ?? and the optimal values of corresponding parameters can be conducted via a
grid search over the possible values of U , m and r. We compute ?? in Figure 1 (Left Panel). For
convenience, we recommend m = 3, U = 0.83, and r = 2.5. With this choice of the parameters,
Figure 1 (Right Panel) shows that the ? values using these parameters are very close to ?? values.
3.6
Removing the Condition ??q??2 = 1
Changing norms of the query does not affect the arg maxx?C q T x. Thus in practice for retrieving topranked items, normalizing the query should not affect the performance. But for theoretical purposes,
we want the runtime guarantee to be independent of ??q??2 . We are interested in the c-approximate
instance which being a threshold based approximation changes if the query is normalized.
Previously, transformations P and Q were precisely meant to remove the dependency on the norms
of x. Realizing the fact that we are allowed asymmetry, we can use the same idea to get rid of the
norm of q. Let M be the upper bound on all the norms or the radius of the space as defined in
Eq (7). Let the transformation S ? RD ? RD be the ones defined in Eq (7). Define asymmetric
transformations P ? ? RD ? RD+2m and Q? ? RD ? RD+2m as
P ? (x) = [x; ??x??22 ; ??x??42 ; ....; ??x??22 ; 1/2; ...1/2]; Q? (x) = [x; 1/2; ....; 1/2; ??x??22 ; ??x??42 ; ....; ??x??22 ],
m
m
Given the query q and data point x, our new asymmetric transformations are Q? (S(q)) and
P ? (S(x)) respectively. We observe that
??Q? (S(q)) ? P ? (S(x))??22 =
m+1
Both ??S(x)??22
m+1
, ??S(q)??22
m+1
? U2
m+1
m+1
U2
m
+ ??S(x)??22
+ ??S(q)??22
? 2q t x ? ( 2 )
2
M
(13)
? 0. Using exactly same arguments as before, we obtain
6
Theorem 5 (Unconditional Approximate MIPS is Efficient) For the problem of c-approximate
?
MIPS in a bounded space, one can construct a data structure having O(n?u log n) query time and
?
space O(n1+?u ), where ??u < 1 is the solution to constraint optimization (14).
?
m+1
U2
2m+1 )
(
m/2 ? 2S0 ( M
log
F
r
2 ) + 2U
U (2 ?2) M 2
?
< 1 ? c,
(14)
s.t.
?u =
min
?
2
0<U <1,m?N,r
S0
log F ( m/2 ? 2cS ( U ))
r
0
M2
Again, for any c-approximate MIPS instance, with S0 and c, we can always choose m big enough
such that ??u < 1. The theoretical guarantee only depends on the radius of the space M .
3.7
A Generic Recipe for Constructing Asymmetric LSHs
We are allowed any asymmetric transformation on x and q. This gives us a lot of flexibility to construct ALSH for new similarities S that we are interested in. The generic idea is to take a particular
similarity Sim(x, q) for which we know an existing LSH or ALSH. Then we construct transformations P and Q such Sim(P (x), Q(q)) is monotonic in the similarity S that we are interested in.
The other observation that makes it easier to construct P and Q is that LSH based guarantees are
independent of dimensions, thus we can expand the dimensions like we did for P and Q.
This paper focuses on using L2LSH to convert near neighbor search of L2 distance into an ALSH
(i.e., L2-ALSH) for MIPS. We can devise new ALSHs for MIPS using other similarities and hash
functions. For instance, utilizing sign random projections (SRP), the known LSH for correlations,
we can construct different P and Q leading to a better ALSH (i.e., Sign-ALSH) for MIPS [22]. We
are aware another work [18] which performs very similarly to Sign-ALSH. Utilizing minwise hashing [2, 15], which is the LSH for resemblance and is known to outperform SRP in sparse data [23],
we can construct an even better ALSH (i.e., MinHash-ALSH) for MIPS over binary data [21].
4
Evaluations
Datasets. We evaluate the proposed ALSH scheme for the MIPS problem on two popular collaborative filtering datasets on the task of item recommendations: (i) Movielens(10M), and (ii) Netflix.
Each dataset forms a sparse user-item matrix R, where the value of R(i, j) indicates the rating
of user i for movie j. Given the user-item ratings matrix R, we follow the standard PureSVD procedure [4] to generate user and item latent vectors. This procedure generates latent vectors ui for
each user i and vector vj for each item j, in some chosen fixed dimension f . The PureSVD method
returns top-ranked items based on the inner products uTi vj , ?j. Despite its simplicity, PureSVD
outperforms other popular recommendation algorithms [4]. Following [4], we use the same choices
for the latent dimension f , i.e., f = 150 for Movielens and f = 300 for Netflix.
4.1
Ranking Experiment for Hash Code Quality Evaluations
We are interested in knowing, how the two hash functions correlate with the top-10 inner products.
For this task, given a user i and its corresponding user vector ui , we compute the top-10 gold
standard items based on the actual inner products uTi vj , ?j. We then compute K different hash
codes of the vector ui and all the item vectors vj s. For every item vj , we compute the number of
times its hash values matches (or collides) with the hash values of query which is user ui , i.e., we
compute M atchesj = ?K
t=1 1(ht (ui ) = ht (vj )), based on which we rank all the items.
Figure 2 reports the precision-recall curves in our ranking experiments for top-10 items, for comparing our proposed method with two baseline methods: the original L2LSH and the original sign
random projections (SRP). These results confirm the substantial advantage of our proposed method.
4.2 LSH Bucketing Experiment
We implemented the standard (K, L)-parameterized (where L is number of hash tables) bucketing
algorithm [1] for retrieving top-50 items based on PureSVD procedure using the proposed ALSH
hash function and the two baselines: SRP and L2LSH. We plot the recall vs the mean ratio of inner
product required to achieve that recall. The ratio being computed relative to the number of inner
products required in a brute force linear scan. In order to remove the effect of algorithm parameters
(K, L) on the evaluations, we report the result from the best performing K and L chosen from
K ? {5, 6, ..., 30} and L ? {1, 2, ..., 200} for each query. We use m = 3, U = 0.83, and r = 2.5 for
7
Top 10, K = 16
5
0
0
20
40
60
Recall (%)
10
NetFlix
6
4
Top 10, K = 16
2
0
0
20
40
60
Recall (%)
80
100
Top 10, K = 64
10
0
0
100
Proposed
L2LSH
SRP
8
Precision (%)
80
20
20
40
60
Recall (%)
20
80
15
10
Top 10, K = 64
5
0
0
Movielens
Proposed
L2LSH
SRP
40
Top 10, K = 256
20
0
0
100
Proposed
L2LSH
SRP
NetFlix
60
Precision (%)
10
Proposed
L2LSH
SRP
Movielens
20
40
60
Recall (%)
80
50
40
Precision (%)
Precision (%)
Movielens
30
Precision (%)
Proposed
L2LSH
SRP
Precision (%)
15
100
Proposed
L2LSH
SRP
NetFlix
30
Top 10, K = 256
20
10
20
40
60
Recall (%)
80
100
0
0
20
40
60
Recall (%)
80
100
Figure 2: Ranking. Precision-Recall curves (higher is better), of retrieving top-10 items, with the
number of hashes K ? {16, 64, 256}. The proposed algorithm (solid, red if color is available) significantly outperforms L2LSH. We fix the parameters m = 3, U = 0.83, and r = 2.5 for our proposed
method and we present the results of L2LSH for all r values in {1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5}.
0.8
1
Proposed
Top 50
SRP
Movielens
L2LSH
Fraction Multiplications
Fraction Multiplications
1
0.6
0.4
0.2
0
0
0.2
0.4
0.6
Recall
0.8
0.8
Top 50
Netflix
0.6
0.4
0.2
0
0
1
Proposed
SRP
L2LSH
0.2
0.4
0.6
Recall
0.8
1
Figure 3: Bucketing. Mean number of inner products per query, relative to a linear scan, evaluated by different hashing schemes at different recall levels, for generating top-50 recommendations
(Lower is better). The results corresponding to the best performing K and L (for a wide range of K
and L) at a given recall value, separately for all the three hashing schemes, are shown.
our hashing scheme. For L2LSH, we observe that using r = 4 usually performs well and so we show
results for r = 4. The results are summarized in Figure 3, confirming that the proposed ALSH leads
to significant savings compared to baseline hash functions.
5
Conclusion
MIPS (maximum inner product search) naturally arises in numerous practical scenarios, e.g., collaborative filtering. This problem is challenging and, prior to our work, there existed no provably
sublinear time hashing algorithms for MIPS. Also, the existing framework of classical LSH (locality
sensitive hashing) is not sufficient for solving MIPS. In this study, we develop ALSH (asymmetric
LSH), which generalizes the existing LSH framework by applying (appropriately chosen) asymmetric transformations to the input query vector and the data vectors in the repository. We present an
implementation of ALSH by proposing a novel transformation which converts the original inner
products into L2 distances in the transformed space. We demonstrate, both theoretically and empirically, that this implementation of ALSH provides provably efficient as well as practical solution
to MIPS. Other explicit constructions of ALSH, for example, ALSH through cosine similarity, or
ALSH through resemblance (for binary data), will be presented in followup technical reports.
Acknowledgments
The research is partially supported by NSF-DMS-1444124, NSF-III-1360971, NSF-Bigdata1419210, ONR-N00014-13-1-0764, and AFOSR-FA9550-13-1-0137. We appreciate the constructive comments from the program committees of KDD 2014 and NIPS 2014. Shrivastava would also
like to thank Thorsten Joachims and the Class of CS6784 (Spring 2014) for valuable feedbacks.
8
References
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931?939, 2012.
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9
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4,782 | 533 | Adaptive Elastic Models for Hand-Printed
Character Recognition
Geoffrey E. Hinton, Christopher K. I. Williams and Michael D. Revow
Department of Computer Science, University of Toronto
Toronto, Ontario, Canada M5S lA4
Abstract
Hand-printed digits can be modeled as splines that are governed by about
8 control points . For each known digit. the control points have preferred
., home" locations, and deformations of the digit are generated by moving
the control points away from their home locations. Images of digits can be
produced by placing Gaussian ink generators uniformly along the spline.
Real images can be recognized by finding the digit model most likely to
have generated the data. For each digit model we use an elastic matching
algorithm to minimize an energy function that includes both the deformation energy of the digit model and the log probability that the model
would generate the inked pixels in the image. The model with the lowest
total energy wins. If a uniform noise process is included in the model of
image generation, some of the inked pixels can be rejected as noise as a
digit model is fitting a poorly segmented image. The digit models learn
by modifying the home locations of the control points.
1
Introduction
Given good bottom-up segmentation and normalization, feedforward neural networks are an efficient way to recognize digits in zip codes. (Ie Cun et al., 1990).
However. in some cases. it is not possible to correctly segment and normalize the
digits without using knowledge of their shapes, so to achieve close to human performance on images of whole zip codes it will be necessary to use models of shapes
to influence the segmentation and normalization of the digits. One way of doing
this is to use a large cooperative network that simultaneously segments, normalizes
and recognizes all of the digit.s in a zip code. A first step in this direct.ion is to take
a poorly segmented image of a single digit and to explain the image properly in
terms of an appropriately normalized, deformed digit model plus noise. The ability
of t.he model to reject some parts of the image as noise is the first step towards
model-driven segmentation.
512
Adaptive Elastic Models for Hand)Printed Character Recognition
2
Elastic models
One technique for recognizing a digit is to perform an elastic match with many
different exemplars of each known digit-class and to pick the class of the nearest
neighbor. Unfortunately this requires a large number of elastic matches, each of
which is expensive . By using one elastic model to capture all the variations of a given
digit we greatly reduce the number of elastic matches required . Burr (1981a, 1981b)
has investigated several types of elastic model and elastic matching procedure. We
describe a different kind of elastic model that is based on splines. Each elastic
model contains parameters that define an ideal shape and also define a deformation
energy for departures from this ideal. These parameters are initially set by hand
but can be improved by learning. They are an efficient way to represent the many
possible instances of a given digit .
Each digit is modelled by a deformable spline whose shape is determined by the
positions of 8 control points . Every point on the spline is a weighted average of
four control points, with the weighting coefficients changing smoothly as we move
along the spline. 1 To generate an ideal example of a digit we put the 8 control
points at their home locations for that model. To deform the digit we move the
control points away from their home locations. Currently we assume that, for each
model, the control points have independent, radial Gaussian distributions about
their home locations. So the negative log probability of a deformation (its energy)
is proportional to the sum of the squares of the departures of the control points
from their home locations.
The deformation energy function only penalizes shape deformations . Translation,
rotation, dilation , elongation, and shear do not change the shape of an object so we
want the deformation energy to be invariant under these affine transformations. We
achieve this by giving each model its own "object-based frame". Its deformation
energy is computed relative to this frame, not in image coordinates. When we fit
the model to data, we repeatedly recompute the best affine transformation between
the object-based frame and the image (see section 4). The repeated recomputation
of the affine transform during the model fit means that the shape of the digit is
influencing the normalization.
Although we will use our digit models for recognizing images, it helps to start by
considering how we would use them for generating images . The generative model is
an elaboration of the probabilistic interpretation of the elastic net given by Durbin,
Szeliski & Yuille (1989) . Given a particular spline, we space a number of "beads"
uniformly along the spline. Each bead defines the center of a Gaussian ink generator.
The number of beads on the spline and the variance of the ink generators can easily
be changed without changing the spline itself.
To generate a noisy image of a particular digit class, run the following procedure:
? Pick an affine transformation from the model's intrinsic reference frame to the
image frame (i .e. pick a size, position, orientation, slant and elongation for the
digit) .
1 In computing the weighting coefficients we use a cubic B-spline and we treat the first
and last control points as if they were doubled.
513
514
Hinton, Williams, and Revow
? Pick a defo~mation of the mo.d~l (i.e. ~~ve the control !)Qi~ts awa1 from their
home locatIOns). The probabIlIty of pIckmg a deformatIOn IS ~ e- de.Jornl
? Repeat many times:
Either (with probability 1I"noi.H) add a randomly positioned noise pixel
Or pick a bead at random and generate a pixel from the Gaussian
distribution defined by the bead.
3
Recognizing isolated digits
We recognize an image by finding which model is most likely to have generated it.
Each possible model is fitted to the image and the one that has the lowest cost fit is
the winner. The cost of a fit is the negative log probability of generating the image
gi ven the model.
- log
J
P(I) P( image
I 1)
dI
(1 )
IE model
instances
We can approximate this by just considering the best fitting model instance 2 and
ignoring the fact that the model should not generate ink where there is no ink in
the image: 3
E
= A EdeJorm
-
L
logP(pixel
I best
model instance)
(2)
inked
pixels
The probability of an inked pixel is the sum of the probabilities of all the possible
ways of generating it from the mixture of Gaussian beads or the uniform noise field.
P(i)
= 1I"noi.H + 1I"model
N
B
(3)
where N is the total number of pixels, B is the number of beads, 11" is a mlxmg
proportion', and Pb( i) is the probability density of pixel i under Gaussian bead b.
4
The search procedure for fitting a model to an image
Every Gaussian bead in a model has the same variance. When fitting data, we start
with a big variance and gradually reduce it as in the elastic net algorithm of Durbin
and Willshaw (1987) . Each iteration of the elastic matching algorithm involves
three steps:
21n effect, we are assuming that the integral in equation 1 can be approximated by the
height of the highest peak, and so we are ignoring variations between models in the width
of the peak or the number of peaks.
3If the inked pixels are rare, poor models sin mainly by not inking those pixels that
should be inked rather than by inking those pixels that should not be inked.
Adaptive Elastic Models for Hand) Printed Character Recognition
? Given the current locations of the Gaussians, compute the responsibility that
each Gaussian has for each inked pixel. This is just the probability of generating
the pixel from that Gaussian, normalized by the total probability of generating
the pixel.
? Assuming that the responsibilities remain fixed, as in the EM algorithm of
Dempster, Laird and Rubin (1977), we invert a 16 x 16 matrix to find the
image locations for the 8 control points at which the forces pulling the control
points towards their home locations are balanced by the forces exerted on the
control points by the inked pixels. These forces come via the forces that the
inked pixels exert on the Gaussian beads.
? Given the new image locations of the control points, we recompute the affine
transformation from the object-based frame to the image frame. We choose
the affine transformation that minimizes the sum of the squared distances, in
object-based coordinates, between the control points and their home locations.
The residual squared differences determine the deformation energy.
Some stages in the fitting of a model to data are shown in Fig. 1. This search
technique avoids nearly all local minima when fitting models to isolated digits. But
if we get a high deformation energy in the best fitting model, we can try alternative
starting configurations for the models.
5
Learning the digit models
We can do discriminative learning by adjusting the home positions and variances
of the control points to minimize the objective function
c =-
L
e-Ecorrect
10gp(cor7'ect digit),
p(correct digit) =
training
cases
=-----~"""
Lall digits e-Ed'Y'1
(4)
For a model parameter such as the x coordinate of the home location of one of the
control points we need oC /
in order to do gradient descent learning. Equation
4 allows us to express oC /
in terms of oE /
but there is a subtle problem:
Changing a parameter of an elastic model causes a simple change in the energy
of the configuration that the model previously settled to, but the model no longer
settles to that configuration. So it appears that we need to consider how the energy
is affected by the change in the configuration. Fortunately, derivatives are simple at
an energy minimum because small changes in the configuration make no change in
the energy (to first order). Thus the inner loop settling leads to simple derivatives
for the outer loop learning, as in the Boltzmann machine (Hinton, 1989).
ax
ax
6
ax
Results on the hand-filtered dataset
We are trying out the scheme out on a relatively simple task - we have a model of
a two and a model of a three, and we want the two model to win on "two" images,
and the three model to win on "three" images.
We have tried many variations of the character models, the preprocessing, the initial
affine transformations of the models, the annealing schedule for the variances, the
515
516
Hinton, Williams, and Revow
c:=
(b)
(a)
,,--,.
,
"",ue:;~,. .~~4P
(c)
(d)
Figure 1: The sequellce> (n) 1.0 (d) shows SOIIlC stages or rHf.illg a model :~ 1.0 SOllie
daf.1\.. The grey circles I?e>presellf. the heads Oil the splille, alld t.he> m,dius or t.he rircl(~
represents t.he standard deviation or t.he Gaussian. (a.) shows the illitia.1 conliglll'atioll, with eight beads equally spaced along the spline. 111 (b) and (c) the va.riallce
is 11I'ogl:es~ively decrca.~ed and t.he Humber or heads is incrf~ased. The ri lIal ra \lsi IIg
GO beads is showlI in (d). We use about. three iterat.ions al. each or nve variallces
on our "annealing schedule". III this example, we used 1Tnoiu = 0.3 which lIIa.kes it.
cheaper to explain the extrft,nCOliS 1I0ise pixels and the flourishes 011 t.he cllds or t.11!~
:~ as noise rather lhall deformillg t.he llIodel to briug Gallssiall heads cI()s(~ t.o t.hese
pixels.
Adaptive Elastic Models for Hand)Printed Character Recognition
mixing proportion of the noise, and the relative importance of deformation energy
versus data-fit energy.
Our current best performance is 10 errors (1.6%) on a test set of 304 two's and 304
three's. We reject cases if the best-fitting model is highly deformed, but on this
test set the deformation energy never reached the rejection criterion. The training
set has 418 cases, and we have a validation set of 200 cases to tell us when to stop
learning. Figure 2 shows the effect of learning on the models. The initial affine
transform is defined by the minimal vertical rectangle around the data.
(BEFORE)
[0
lTI
IAFTER
[i]
I
lTI
Figure 2: The two and three models before and after learning. The control points
are labelled 1 through 8. We used maximum likelihood learning in which each digit
model is trained only on instances of that digit. After each pass through all those
instances, the home location of each control point (in the object-based frame) is
redefined to be the average location of the control point in the final fits of the model
of the digit to the instances of the digit. Most of the improvement in performance
occurred after the fist pass, and after five updates of the home locations of the
control points, performance on the validation set started to decrease. Similar results
were obtained with discriminative training. We could also update the variance of
each control point to be its variance in the final fits, though we did not adapt the
variances in this simulation.
517
518
Hinton, Williams, and Revow
The images are preprocessed to eliminate variations due to stroke-width and paper
and ink intensities. First, we use a standard local thresholding algorithm to make
a binary decision for each pixel. Then we pick out the five largest connected components (hopefully digits). We put a box around each component , then thin all the
data in the box . If we ourselves cannot recognize the resulting image we eliminate
it from the data set. The training, validation and test data is all from the training
portion of the United States Postal Service Handwritten ZIP Code Database (1987)
which was made available by the USPS Office of Advanced Technology.
7
Discussion
Before we tried using splines to model digits, we used models that consisted of a
fixed number of Gaussian beads with elastic energy constraints operating between
neighboring beads. To constrain the curvature we used energy terms that involved
triples of beads. With this type of energy function, we had great difficulty using
a single model to capture topologically different instances of a digit. For example,
when the central loop of a 3 changes to a cusp and then to an open bend, the sign
of the curvature reverses. With a spline model it is easy to model these topological
variants by small changes in the relative vertical locations of the central two control
points (see figure 2). This advantage of spline models is pointed out by (Edelman,
Ullman and Flash, 1990) who use a different kind of spline that they fit to character
data by directly locating candidate knot points in the image.
Spline models also make it easy to increase the number of Gaussian beads as their
variance is decreased. This coarse-to-fine strategy is much more efficient than using
a large number of beads at all variances, but it is much harder to implement if the
deformation energy explicitly depends on particular bead locations, since changiug
the number of beads then requires a new function for the deformation energy.
In determining where on the spline to place the Gaussian beads, we initially used
a fixed set of blending coefficients for each bead . These coefficients are the weight:s
used to specify the bead location as a weighted center of gravity of the loca.l-iollS of
4 control points. Unfortunately this yields too few beads in portions of a digit such
as a long tail of a 2 which are governed by just a few control points. Performance
was much improved by spacing the beads uniformly along the curve.
By using spline models, we build in a lot of prior knowledge about wha.t characters
look like, so we can describe the shape of a character using only a small number
of parameters (16 coordinates and 8 variances). This means that the learning is
exploring a much smaller space than a conventional feed-forward network. Also,
because the parameters are easy to interpret, we can start with fairly good initial
models of the characters. So learning only requires a few updates of the parameters.
Obvious extensions of the deformation energy function include using elliptical Gaussians for the distributions of the control points, or using full covariance matrices for
neighboring pairs of control points. Another obvious modification is to use elliptical rather than circular Gaussians for the beads . If strokes curve gently relative to
their thickness, the distribution of ink can be modelled much better using elliptical
Gaussians. However, an ellipse takes about twice as many operations to fit and is
not helpful in regions of sharp curvature. Our simulations suggest that, on average,
two circular beads are more flexible than one elliptical bead.
Adaptive Elastic Models for Hand) Printed Character Recognition
Currently we do not impose any penalty on extremely sheared or elongated affine
transformations, though this would probably improve performance. Having an explicit representation of the affine transformation of each digit should prove very
helpful for recognizing multiple digits, since it will allow us to impose a penalty on
differences in the affine transformations of neighboring digits.
Presegmented images of single digits contain many different kinds of noise that
cannot be eliminated by simple bottom-up operations. These include descenders,
underlines, and bits of other digits; corrections; dirt in recycled paper; smudges and
misplaced postal franks. To really understand the image we probably need to model
a wide variety of structured noise. We are currently experimenting with one simple
way of incorporating noise models. After each digit model has been used to segment
a noisy image into one digit instance plus noise, we try to fit more complicated noise
models to the residual noise. A good fit greatly decreases the cost of that noise and
hence improves this interpretation of the image. We intend to handle flourishes on
the ends of characters in this way rather than using more elaborate digit models
that include optional flourishes.
One of our main motivations in developing elastic models is the belief that a strong
prior model should make learning easier, should reduce confident errors, and should
allow top-down segmentation . Although we have shown that elastic spline models can be quite effective, we have not yet demonstrated that they are superior to
feedforward nets and there is a serious weakness of our approach: Elastic matching is slow. Fitting the models to the data takes much more computation than a
feedforward net. So in the same number of cycles, a feedforward net can try many
alternative bottom-up segmentations and normalizations and select the overall segmentation that leads to the most recognizable digit string.
Acknowledgements
This research was funded by Apple and by the Ontario Information Technology Research
Centre. We thank Allan Jepson and Richard Durbin for suggesting spline models.
References
Burr, D. J. (1981a) . A dynamic model for image registration. Comput. Gmphics image
Process., 15:102-112.
Burr, D. J. (1981b). Elastic matching of line drawings. IEEE Trans. Pattern Analysis and
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Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood from
incomplete data via the EM algorithm . Proc. Roy. Stat. Soc., B-39:1-38 .
Durbin, R., Szeliski, R., and Yuille, A. L. (1989). An analysis of the elastic net approach
to the travelling salesman problem. Neural Computation, 1:348-358.
Durbin, R. and Willshaw, D. (1987). An analogue approach to the travelling salesman
problem. Nature, 326:689-691.
Edelman, S., Ullman, S., and Flash, T. (1990). Reading cursive handwriting by alignment
of letter prototypes. Internat. Journal of Comput. Vision, 5(3):303-33l.
Hinton, G. E. (1989). Deterministic Boltzmann learning performs steepest descent in
weight-space. Neural Computation, 1:143-150.
Ie Cun, Y., Boser, B., Denker, J., Henderson, D., Howard, R., Hubbard, W., and Jackel, L.
(1990). Handwritten digit recognition with a back-propagation network. In Advances
in Neural Information Processing Systems 2, pages 396-404. Morgan Kaufmann.
519
PART IX
CONTROL
AND PLANNING
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4,783 | 5,330 | A Latent Source Model for
Online Collaborative Filtering
Guy Bresler
George H. Chen
Devavrat Shah
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
Cambridge, MA 02139
{gbresler,georgehc,devavrat}@mit.edu
Abstract
Despite the prevalence of collaborative filtering in recommendation systems, there
has been little theoretical development on why and how well it works, especially
in the ?online? setting, where items are recommended to users over time. We
address this theoretical gap by introducing a model for online recommendation
systems, cast item recommendation under the model as a learning problem, and
analyze the performance of a cosine-similarity collaborative filtering method. In
our model, each of n users either likes or dislikes each of m items. We assume
there to be k types of users, and all the users of a given type share a common
string of probabilities determining the chance of liking each item. At each time
step, we recommend an item to each user, where a key distinction from related
bandit literature is that once a user consumes an item (e.g., watches a movie),
then that item cannot be recommended to the same user again. The goal is to
maximize the number of likable items recommended to users over time. Our main
result establishes that after nearly log(km) initial learning time steps, a simple
collaborative filtering algorithm achieves essentially optimal performance without
knowing k. The algorithm has an exploitation step that uses cosine similarity and
two types of exploration steps, one to explore the space of items (standard in the
literature) and the other to explore similarity between users (novel to this work).
1
Introduction
Recommendation systems have become ubiquitous in our lives, helping us filter the vast expanse of
information we encounter into small selections tailored to our personal tastes. Prominent examples
include Amazon recommending items to buy, Netflix recommending movies, and LinkedIn recommending jobs. In practice, recommendations are often made via collaborative filtering, which boils
down to recommending an item to a user by considering items that other similar or ?nearby? users
liked. Collaborative filtering has been used extensively for decades now including in the GroupLens
news recommendation system [20], Amazon?s item recommendation system [17], the Netflix Prize
winning algorithm by BellKor?s Pragmatic Chaos [16, 24, 19], and a recent song recommendation
system [1] that won the Million Song Dataset Challenge [6].
Most such systems operate in the ?online? setting, where items are constantly recommended to
users over time. In many scenarios, it does not make sense to recommend an item that is already
consumed. For example, once Alice watches a movie, there?s little point to recommending the same
movie to her again, at least not immediately, and one could argue that recommending unwatched
movies and already watched movies could be handled as separate cases. Finally, what matters is
whether a likable item is recommended to a user rather than an unlikable one. In short, a good
online recommendation system should recommend different likable items continually over time.
1
Despite the success of collaborative filtering, there has been little theoretical development to justify
its effectiveness in the online setting. We address this theoretical gap with our two main contributions
in this paper. First, we frame online recommendation as a learning problem that fuses the lines of
work on sleeping bandits and clustered bandits. We impose the constraint that once an item is
consumed by a user, the system can?t recommend the item to the same user again. Our second
main contribution is to analyze a cosine-similarity collaborative filtering algorithm. The key insight
is our inclusion of two types of exploration in the algorithm: (1) the standard random exploration
for probing the space of items, and (2) a novel ?joint? exploration for finding different user types.
Under our learning problem setup, after nearly log(km) initial time steps, the proposed algorithm
achieves near-optimal performance relative to an oracle algorithm that recommends all likable items
first. The nearly logarithmic dependence is a result of using the two different exploration types. We
note that the algorithm does not know k.
Outline. We present our model and learning problem for online recommendation systems in Section
2, provide a collaborative filtering algorithm and its performance guarantee in Section 3, and give
the proof idea for the performance guarantee in Section 4. An overview of experimental results is
given in Section 5. We discuss our work in the context of prior work in Section 6.
2
A Model and Learning Problem for Online Recommendations
We consider a system with n users and m items. At each time step, each user is recommended an
item that she or he hasn?t consumed yet, upon which, for simplicity, we assume that the user immediately consumes the item and rates it +1 (like) or ?1 (dislike).1 The reward earned by the recommendation system up to any time step is the total number of liked items that have been recommended
so far across all users. Formally, index time by t ? {1, 2, . . . }, and users by u ? [n] , {1, . . . , n}.
(t)
Let ?ut ? [m] , {1, . . . , m} be the item recommended to user u at time t. Let Yui ? {?1, 0, +1}
be the rating provided by user u for item i up to and including time t, where 0 indicates that no rating
has been given yet. A reasonable objective is to maximize the expected reward r(T ) up to time T :
r(T ) ,
T X
n
X
(T )
E[Yu?
]=
ut
t=1 u=1
m X
n
X
(T )
E[Yui ].
i=1 u=1
The ratings are noisy: the latent item preferences for user u are represented by a length-m vector
pu ? [0, 1]m , where user u likes item i with probability pui , independently across items. For a user
u, we say that item i is likable if pui > 1/2 and unlikable if pui < 1/2. To maximize the expected
reward r(T ) , clearly likable items for the user should be recommended before unlikable ones.
In this paper, we focus on recommending likable items. Thus, instead of maximizing the expected
reward r(T ) , we aim to maximize the expected number of likable items recommended up to time T :
(T )
r+ ,
T X
n
X
E[Xut ] ,
(1)
t=1 u=1
where Xut is the indicator random variable for whether the item recommended to user u at time t is
(T )
likable, i.e., Xut = +1 if pu?ut > 1/2 and Xut = 0 otherwise. Maximizing r(T ) and r+ differ
since the former asks that we prioritize items according to their probability of being liked.
Recommending likable items for a user in an arbitrary order is sufficient for many real recommendation systems such as for movies and music. For example, we suspect that users wouldn?t actually
prefer to listen to music starting from the songs that their user type would like with highest probability to the ones their user type would like with lowest probability; instead, each user would listen to
songs that she or he finds likable, ordered such that there is sufficient diversity in the playlist to keep
the user experience interesting. We target the modest goal of merely recommending likable items,
in any order. Of course, if all likable items have the same probability of being liked and similarly
(T )
for all unlikable items, then maximizing r(T ) and r+ are equivalent.
1
In practice, a user could ignore the recommendation. To keep our exposition simple, however, we stick to
this setting that resembles song recommendation systems like Pandora that per user continually recommends
a single item at a time. For example, if a user rates a song as ?thumbs down? then we assign a rating of ?1
(dislike), and any other action corresponds to +1 (like).
2
The fundamental challenge is that to learn about a user?s preference for an item, we need the user to
rate (and thus consume) the item. But then we cannot recommend that item to the user again! Thus,
the only way to learn about a user?s preferences is through collaboration, or inferring from other
users? ratings. Broadly, such inference is possible if the users preferences are somehow related.
In this paper, we assume a simple structure for shared user preferences. We posit that there are
k < n different types of users, where users of the same type have identical item preference vectors.
The number of types k represents the heterogeneity in the population. For ease of exposition, in this
paper we assume that a user belongs to each user type with probability 1/k. We refer to this model
as a latent source model, where each user type corresponds to a latent source of users. We remark
that there is evidence suggesting real movie recommendation data to be well modeled by clustering
of both users and items [21]. Our model only assumes clustering over users.
Our problem setup relates to some versions of the multi-armed bandit problem. A fundamental
difference between our setup and that of the standard stochastic multi-armed bandit problem [23, 8]
is that the latter allows each item to be recommended an infinite number of times. Thus, the solution
concept for the stochastic multi-armed bandit problem is to determine the best item (arm) and keep
choosing it [3]. This observation applies also to ?clustered bandits? [9], which like our work seeks to
capture collaboration between users. On the other hand, sleeping bandits [15] allow for the available
items at each time step to vary, but the analysis is worst-case in terms of which items are available
over time. In our setup, the sequence of items that are available is not adversarial. Our model
combines the collaborative aspect of clustered bandits with dynamic item availability from sleeping
bandits, where we impose a strict structure on how items become unavailable.
3
A Collaborative Filtering Algorithm and Its Performance Guarantee
This section presents our algorithm C OLLABORATIVE -G REEDY and its accompanying theoretical
performance guarantee. The algorithm is syntactically similar to the ?-greedy algorithm for multiarmed bandits [22], which explores items with probability ? and otherwise greedily chooses the best
item seen so far based on a plurality vote. In our algorithm, the greedy choice, or exploitation, uses
the standard cosine-similarity measure. The exploration, on the other hand, is split into two types, a
standard item exploration in which a user is recommended an item that she or he hasn?t consumed
yet uniformly at random, and a joint exploration in which all users are asked to provide a rating for
the next item in a shared, randomly chosen sequence of items. Let?s fill in the details.
Algorithm. At each time step t, either all the users are asked to explore, or an item is recommended
to each user by choosing the item with the highest score for that user. The pseudocode is described
in Algorithm 1. There are two types of exploration: random exploration, which is for exploring the
space of items, and joint exploration, which helps to learn about similarity between users. For a
pre-specified rate ? ? (0, 4/7], we set the probability of random exploration to be ?R (n) = 1/n?
Algorithm 1: C OLLABORATIVE -G REEDY
Input: Parameters ? ? [0, 1], ? ? (0, 4/7].
Select a random ordering ? of the items [m]. Define
?R (n) =
1
,
n?
and
?J (t) =
1
.
t?
for time step t = 1, 2, . . . , T do
With prob. ?R (n): (random exploration) for each user, recommend a random item that the
user has not rated.
With prob. ?J (t): (joint exploration) for each user, recommend the first item in ? that the user
has not rated.
With prob. 1 ? ?J (t) ? ?R (n): (exploitation) for each user u, recommend an item j that the
(t)
user has not rated and that maximizes score peuj , which depends on threshold ?.
end
3
(decaying with the number of users), and the probability of joint exploration to be ?J (t) = 1/t?
(decaying with time).2
(t)
(t)
Next, we define user u?s score peui for item i at time t. Recall that we observe Yui = {?1, 0, +1}
as user u?s rating for item i up to time t, where 0 indicates that no rating has been given yet. We
define
?P
(t)
?
(t)
? v?Neu(t) 1{Yvi = +1} if P
P
(t)
eu(t) 1{Yvi 6= 0} > 0,
(t)
v?N
peui ,
eu(t) 1{Yvi 6= 0}
v?N
?
?
1/2
otherwise,
where the neighborhood of user u is given by
eu(t) , {v ? [n] : hYeu(t) , Yev(t) i ? ?|supp(Yeu(t) ) ? supp(Yev(t) )|},
N
(t)
and Yeu consists of the revealed ratings of user u restricted to items that have been jointly explored.
In other words,
(
(t)
Yui if item i is jointly explored by time t,
(t)
e
Yui =
0
otherwise.
The neighborhoods are defined precisely by cosine similarity with respect to jointed explored items.
(t)
(t)
(t)
(t)
To see this, for users u and v with revealed ratings Yeu and Yev , let ?uv , supp(Yeu )?supp(Yev )
(t)
(t)
be the support overlap of Yeu and Yev , and let h?, ?i?uv be the dot product restricted to entries in
?uv . Then
(t)
(t)
(t)
(t)
hYeu , Yev i?uv
hYeu , Yev i
q
,
=q
|?uv |
(t)
(t)
(t)
(t)
hYeu , Yeu i?uv hYev , Yev i?uv
(t)
(t)
which is the cosine similarity of revealed rating vectors Yeu and Yev restricted to the overlap of
their supports. Thus, users u and v are neighbors if and only if their cosine similarity is at least ?.
Theoretical performance guarantee. We now state our main result on the proposed collaborative
filtering algorithm?s performance with respect to the objective stated in equation (1). We begin with
two reasonable, and seemingly necessary, conditions under which our the results will be established.
A1 No ?-ambiguous items. There exists some constant ? > 0 such that
|pui ? 1/2| ? ?
for all users u and items i. (Smaller ? corresponds to more noise.)
A2 ?-incoherence. There exist a constant ? ? [0, 1) such that if users u and v are of different
types, then their item preference vectors pu and pv satisfy
1
h2pu ? 1, 2pv ? 1i ? 4??2 ,
m
where 1 is the all ones vector. Note that a different way to write the left-hand side is
1
E[ m
hYu? , Yv? i], where Yu? and Yv? are fully-revealed rating vectors of users u and v, and
the expectation is over the random ratings of items.
The first condition is a low noise condition to ensure that with a finite number of samples, we can
correctly classify each item as either likable or unlikable. The incoherence condition asks that the
different user types are well-separated so that cosine similarity can tease apart the users of different
types over time. We provide some examples after the statement of the main theorem that suggest the
incoherence condition to be reasonable, allowing E[hYu? , Yv? i] to scale as ?(m) rather than o(m).
We assume that the number of users satisfies n = O(mC ) for some constant C > 1. This is without
loss of generality since otherwise, we can randomly divide the n users into separate population
2
For ease of presentation, we set the two explorations to have the same decay rate ?, but our proof easily
extends to encompass different decay rates for the two exploration types. Furthermore, the constant 4/7 ? ?
is not special. It could be different and only affects another constant in our proof.
4
pools, each of size O(mC ) and run the recommendation algorithm independently for each pool to
achieve the same overall performance guarantee.
Finally, we define ?, the minimum proportion of likable items for any user (and thus any user type):
Pm
1{pui > 1/2}
? , min i=1
.
m
u?[n]
Theorem 1. Let ? ? (0, 1) be some pre-specified tolerance. Take as input to C OLLABORATIVE G REEDY ? = 2?2 (1 + ?) where ? ? [0, 1) is as defined in A2, and ? ? (0, 4/7]. Under the latent
source model and assumptions A1 and A2, if the number of users n = O(mC ) satisfies
1 4 1/?
n = ? km log +
,
?
?
then for any Tlearn ? T ? ?m, the expected proportion of likable items recommended by
C OLLABORATIVE -G REEDY up until time T satisfies
(T )
r+
Tlearn
? 1?
(1 ? ?),
Tn
T
where
Tlearn = ?
log km
??
?4 (1 ? ?)2
1/(1??)
+
4 1/?
?
.
Theorem 1 says that there are Tlearn initial time steps for which the algorithm may be giving poor
recommendations. Afterward, for Tlearn < T < ?m, the algorithm becomes near-optimal, recommending a fraction of likable items 1?? close to what an optimal oracle algorithm (that recommends
all likable items first) would achieve. Then for time horizon T > ?m, we can no longer guarantee
that there are likable items left to recommend. Indeed, if the user types each have the same fraction
of likable items, then even an oracle recommender would use up the ?m likable items by this time.
Meanwhile, to give a sense of how long the learning period Tlearn is, note that when ? = 1/2, we
have Tlearn scaling as log2 (km), and if we choose ? close to 0, then Tlearn becomes nearly log(km).
In summary, after Tlearn initial time steps, the simple algorithm proposed is essentially optimal.
To gain intuition for incoherence condition A2, we calculate the parameter ? for three examples.
Example 1. Consider when there is no noise, i.e., ? = 21 . Then users? ratings are deterministic
given their user type. Produce k vectors of probabilities by drawing m independent Bernoulli( 12 )
random variables (0 or 1 with probability 21 each) for each user type. For any item i and pair of
?
users u and v of different types, Yui
? Yvi? is a Rademacher random variable (?1 with probability 12
each), and thus the inner product of two user rating vectors is equal to the sum of m Rademacher
q
log m
random variables. Standard concentration inequalities show that one may take ? = ?
to
m
satisfy ?-incoherence with probability 1 ? 1/poly(m).
Example 2. We expand on the previous example by choosing an arbitrary ? > 0 and making all
latent source probability vectors have entries equal to 21 ? ? with probability 12 each. As before let
?
user u and v are from different type. Now E[Yui
? Yvi? ] = ( 12 + ?)2 + ( 12 ? ?)2 ? 2( 41 ? ?2 ) = 4?2
1
?
?
2
if pui = pvi and E[Yui ? Yvi ] = 2( 4 ? ? ) ? ( 12 + ?)2 ? ( 21 ? ?)2 = ?4?2 if pui = 1 ? pvi .
The value of the inner product E[hYu? , Yv? i] is again equal to the sum of m Rademacher
random
q
log m
2
suffices to
variables, but this time scaled by 4? . For similar reasons as before, ? = ?
m
satisfy ?-incoherence with probability 1 ? 1/poly(m).
Example 3. Continuing with the previous example, now suppose each entry is 12 +? with probability
? ? (0, 1/2) and 12 ? ? with probability 1 ? ?. Then for two users u and v of different types,
?
2
2
pui = pvi with probability ?2 + (1 ? ?)2 . This implies that E[hYu? , Yq
v i] = 4m? (1 ? 2?) .
log m
Again, using standard concentration, this shows that ? = (1 ? 2?)2 + ?
suffices to satisfy
m
?-incoherence with probability 1 ? 1/poly(m).
5
4
Proof of Theorem 1
Recall that Xut is the indicator random variable for whether the item ?ut recommended to user u
at time t is likable, i.e., pu?ut > 1/2. Given assumption A1, this is equivalent to the event that
pu?ut ? 12 + ?. The expected proportion of likable items is
(T )
T
T
n
n
r+
1 XX
1 XX
=
E[Xut ] =
P(Xut = 1).
Tn
T n t=1 u=1
T n t=1 u=1
Our proof focuses on lower-bounding P(Xut = 1). The key idea is to condition on what we call the
?good neighborhood? event Egood (u, t):
n
n
neighbors from the same user type (?good neighbors?),
Egood (u, t) = at time t, user u has ?
5k
o
?tn1??
and ?
neighbors from other user types (?bad neighbors?) .
10km
This good neighborhood event will enable us to argue that after an initial learning time, with high
probability there are at most ? as many ratings from bad neighbors as there are from good neighbors.
The proof of Theorem 1 consists of two parts. The first part uses joint exploration to show that after
a sufficient amount of time, the good neighborhood event Egood (u, t) holds with high probability.
Lemma 1. For user u, after
1/(1??)
2 log(10kmn? /?)
t?
?4 (1 ? ?)2
time steps,
?4 (1 ? ?)2 t1??
n
? 12 exp ?
.
P(Egood (u, t)) ? 1 ? exp ?
8k
20
In the above lower bound, the first exponentially decaying term could be thought of as the penalty
for not having enough users in the system from the k user types, and the second decaying term could
be thought of as the penalty for not yet clustering the users correctly.
The second part of our proof to Theorem 1 shows that, with high probability, the good neighborhoods
have, through random exploration, accurately estimated the probability of liking each item. Thus,
we correctly classify each item as likable or not with high probability, which leads to a lower bound
on P(Xut = 1).
Lemma 2. For user u at time t, if the good neighborhood event Egood (u, t) holds and t ? ?m, then
?2 tn1??
1
1
P(Xut = 1) ? 1 ? 2m exp ?
? ?? ?.
40km
t
n
Here, the first exponentially decaying term could be thought of as the cost of not classifying items
correctly as likable or unlikable, and the last two decaying terms together could be thought of as the
cost of exploration (we explore with probability ?J (t) + ?R (n) = 1/t? + 1/n? ).
We defer the proofs of Lemmas 1 and 2 to the supplementary material. Combining these lemmas
and choosing appropriate constraints on the numbers of users and items, we produce the following
lemma.
Lemma 3. Let ? ? (0, 1) be some pre-specified tolerance. If the number of users n and items m
satisfy
n
4 4 1/? o
,
n ? max 8k log ,
? ?
1/(1??)
1/(1??)
2 log(10kmn? /?)
20 log(96/?)
4 1/?
,
,
?m ? t ? max
,
4
2
4
2
? (1 ? ?)
? (1 ? ?)
?
16m
40km
log
nt1?? ?
,
?2
?
then P(Xut = 1) ? 1 ? ?.
6
Proof. With the above conditions on n and t satisfied, we combine Lemmas 1 and 2 to obtain
?4 (1 ? ?)2 t1??
?2 tn1??
n
P(Xut = 1) ? 1 ? exp ?
? 12 exp ?
? 2m exp ?
8k
20
40km
1
?
1
?
?
?
?
? ? ? ? ? 1 ? ? ? ? ? = 1 ? ?.
t
n
4 8 8 4 4
Theorem 1 follows as a corollary to Lemma 3. As previously mentioned, without loss of generality,
we take n = O(mC ). Then with number of users n satisfying
1 4 1/?
,
O(mC ) = n = ? km log +
?
?
and for any time step t satisfying
?m ? t ? ?
log km
??
?4 (1 ? ?)2
1/(1??)
+
4 1/?
?
, Tlearn ,
we simultaneously meet all of the conditions of Lemma 3. Note that the upper bound on number of
users n appears since without it, Tlearn would depend on n (observe that in Lemma 3, we ask that t
be greater than a quantity that depends on n). Provided that the time horizon satisfies T ? ?m, then
n
n
T
T
(T )
r+
1 X X
(T ? Tlearn )(1 ? ?)
1 X X
P(Xut = 1) ?
(1 ? ?) =
?
,
Tn
Tn
Tn
T
u=1
u=1
t=Tlearn
t=Tlearn
yielding the theorem statement.
5
Experimental Results
We provide only a summary of our experimental results here, deferring full details to the supplementary material. We simulate an online recommendation system based on movie ratings from the
Movielens10m and Netflix datasets, each of which provides a sparsely filled user-by-movie rating
matrix with ratings out of 5 stars. Unfortunately, existing collaborative filtering datasets such as the
two we consider don?t offer the interactivity of a real online recommendation system, nor do they
allow us to reveal the rating for an item that a user didn?t actually rate. For simulating an online system, the former issue can be dealt with by simply revealing entries in the user-by-item rating matrix
over time. We address the latter issue by only considering a dense ?top users vs. top items? subset
of each dataset. In particular, we consider only the ?top? users who have rated the most number of
items, and the ?top? items that have received the most number of ratings. While this dense part of the
dataset is unrepresentative of the rest of the dataset, it does allow us to use actual ratings provided
by users without synthesizing any ratings. A rigorous validation would require an implementation
of an actual interactive online recommendation system, which is beyond the scope of our paper.
First, we validate that our latent source model is reasonable for the dense parts of the two datasets
we consider by looking for clustering behavior across users. We find that the dense top users vs. top
movies matrices do in fact exhibit clustering behavior of users and also movies, as shown in Figure
1(a). The clustering was found via Bayesian clustered tensor factorization, which was previously
shown to model real movie ratings data well [21].
Next, we demonstrate our algorithm C OLLABORATIVE -G REEDY on the two simulated online movie
recommendation systems, showing that it outperforms two existing recommendation algorithms
Popularity Amongst Friends (PAF) [4] and a method by Deshpande and Montanari (DM) [12]. Following the experimental setup of [4], we quantize a rating of 4 stars or more to be +1 (likable), and
a rating less than 4 stars to be ?1 (unlikable). While we look at a dense subset of each dataset, there
are still missing entries. If a user u hasn?t rated item j in the dataset, then we set the corresponding
true rating to 0, meaning that in our simulation, upon recommending item j to user u, we receive
0 reward, but we still mark that user u has consumed item j; thus, item j can no longer be recommended to user u. For both Movielens10m and Netflix datasets, we consider the top n = 200 users
and the top m = 500 movies. For Movielens10m, the resulting user-by-rating matrix has 80.7%
nonzero entries. For Netflix, the resulting matrix has 86.0% nonzero entries. For an algorithm that
7
Average cumulative reward
0
50
100
150
200
0
100
200
300
400
60
50
40
30
20
Collaborative-Greedy
Popularity Amongst Friends
Deshpande-Montanari
10
0
?10
0
100
200
300
400
500
Time
500
(a)
(b)
Figure 1: Movielens10m dataset: (a) Top users by top movies matrix with rows and columns reordered to show clustering of users and items. (b) Average cumulative rewards over time.
recommends item ?ut to user u at time t, we measure the algorithm?s average cumulative reward up
PT Pn
(T )
to time T as n1 t=1 u=1 Yu?ut , where we average over users. For all four methods, we recommend items until we reach time T = 500, i.e., we make movie recommendations until each user has
seen all m = 500 movies. We disallow the matrix completion step for DM to see the users that we
actually test on, but we allow it to see the the same items as what is in the simulated online recommendation system in order to compute these items? feature vectors (using the rest of the users in the
dataset). Furthermore, when a rating is revealed, we provide DM both the thresholded rating and the
non-thresholded rating, the latter of which DM uses to estimate user feature vectors over time. We
discuss choice of algorithm parameters in the supplementary material. In short, parameters ? and
? of our algorithm are chosen based on training data, whereas we allow the other algorithms to use
whichever parameters give the best results on the test data. Despite giving the two competing algorithms this advantage, C OLLABORATIVE -G REEDY outperforms the two, as shown in Figure 1(b).
Results on the Netflix dataset are similar.
6
Discussion and Related Work
This paper proposes a model for online recommendation systems under which we can analyze the
performance of recommendation algorithms. We theoretical justify when a cosine-similarity collaborative filtering method works well, with a key insight of using two exploration types.
The closest related work is by Biau et al. [7], who study the asymptotic consistency of a cosinesimilarity nearest-neighbor collaborative filtering method. Their goal is to predict the rating of the
next unseen item. Barman and Dabeer [4] study the performance of an algorithm called Popularity Amongst Friends, examining its ability to predict binary ratings in an asymptotic informationtheoretic setting. In contrast, we seek to understand the finite-time performance of such systems.
Dabeer [11] uses a model similar to ours and studies online collaborative filtering with a moving
horizon cost in the limit of small noise using an algorithm that knows the numbers of user types and
item types. We do not model different item types, our algorithm is oblivious to the number of user
types, and our performance metric is different. Another related work is by Deshpande and Montanari [12], who study online recommendations as a linear bandit problem; their method, however,
does not actually use any collaboration beyond a pre-processing step in which offline collaborative
filtering (specifically matrix completion) is solved to compute feature vectors for items.
Our work also relates to the problem of learning mixture distributions (c.f., [10, 18, 5, 2]), where
one observes samples from a mixture distribution and the goal is to learn the mixture components
and weights. Existing results assume that one has access to the entire high-dimensional sample
or that the samples are produced in an exogenous manner (not chosen by the algorithm). Neither
assumption holds in our setting, as we only see each user?s revealed ratings thus far and not the user?s
entire preference vector, and the recommendation algorithm affects which samples are observed (by
choosing which item ratings are revealed for each user). These two aspects make our setting more
challenging than the standard setting for learning mixture distributions. However, our goal is more
modest. Rather than learning the k item preference vectors, we settle for classifying them as likable
or unlikable. Despite this, we suspect having two types of exploration to be useful in general for
efficiently learning mixture distributions in the active learning setting.
Acknowledgements. This work was supported in part by NSF grant CNS-1161964 and by Army
Research Office MURI Award W911NF-11-1-0036. GHC was supported by an NDSEG fellowship.
8
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4,784 | 5,331 | Clustering from Labels and Time-Varying Graphs
Shiau Hong Lim
National University of Singapore
[email protected]
Yudong Chen
EECS, University of California, Berkeley
[email protected]
Huan Xu
National University of Singapore
[email protected]
Abstract
We present a general framework for graph clustering where a label is observed to
each pair of nodes. This allows a very rich encoding of various types of pairwise
interactions between nodes. We propose a new tractable approach to this problem
based on maximum likelihood estimator and convex optimization. We analyze our
algorithm under a general generative model, and provide both necessary and sufficient conditions for successful recovery of the underlying clusters. Our theoretical
results cover and subsume a wide range of existing graph clustering results including planted partition, weighted clustering and partially observed graphs. Furthermore, the result is applicable to novel settings including time-varying graphs such
that new insights can be gained on solving these problems. Our theoretical findings are further supported by empirical results on both synthetic and real data.
1
Introduction
In the standard formulation of graph clustering, we are given an unweighted graph and seek a partitioning of the nodes into disjoint groups such that members of the same group are more densely
connected than those in different groups. Here, the presence of an edge represents some sort of
affinity or similarity between the nodes, and the absence of an edge represents the lack thereof.
In many applications, from chemical interactions to social networks, the interactions between nodes
are much richer than a simple ?edge? or ?non-edge?. Such extra information may be used to improve
the clustering quality. We may represent each type of interaction by a label. One simple setting of
this type is weighted graphs, where instead of a 0-1 graph, we have edge weights representing
the strength of the pairwise interaction. In this case the observed label between each pair is a
real number. In a more general setting, the label need not be a number. For example, on social
networks like Facebook, the label between two persons may be ?they are friends?, ?they went to
different schools?, ?they liked 21 common pages?, or the concatenation of these. In such cases
different labels carry different information about the underlying community structure. Standard
approaches convert these pairwise interactions into a simple edge/non-edge, and then apply standard
clustering algorithms, which might lose much of the information. Even in the case of a standard
weighted/unweighted graph, it is not immediately clear how the graph should be used. For example,
should the absence of an edge be interpreted as a neutral observation carrying no information, or as
a negative observation which indicates dissimilarity between the two nodes?
We emphasize that the forms of labels can be very general. In particular, a label can take the form
of a time series, i.e., the record of time varying interaction such as ?A and B messaged each other
on June 1st, 4th, 15th and 21st?, or ?they used to be friends, but they stop seeing each other since
2012?. Thus, the labeled graph model is an immediate tool for analyzing time-varying graphs.
1
In this paper, we present a new and principled approach for graph clustering that is directly based on
pairwise labels. We assume that between each pair of nodes i and j, a label Lij is observed which
is an element of a label set L. The set L may be discrete or continuous, and need not have any
structure. The standard graph model corresponds to a binary label set L = {edge, non-edge}, and
a weighted graph corresponds to L = R. Given the observed labels L = (Lij ) ? Ln?n , the goal
is to partition the n nodes into disjoint clusters. Our approach is based on finding a partition that
optimizes a weighted objective appropriately constructed from the observed labels. This leads to a
combinatorial optimization problem, and our algorithm uses its convex relaxation.
To systematically evaluate clustering performance, we consider a generalization of the stochastic
block model [1] and the planted partition model [2]. Our model assumes that the observed labels
are generated based on an underlying set of ground truth clusters, where pairs from the same cluster generate labels using a distribution ? over L and pairs from different clusters use a different
distribution ?. The standard stochastic block model corresponds to the case where ? and ? are twopoint distributions with ?(edge) = p and ?(edge) = q. We provide theoretical guarantees for our
algorithm under this generalized model.
Our results cover a wide range of existing clustering settings?with equal or stronger theoretical
guarantees?including the standard stochastic block model, partially observed graphs and weighted
graphs. Perhaps surprisingly, our framework allows us to handle new classes of problems that are not
a priori obvious to be a special case of our model, including the clustering of time-varying graphs.
1.1
Related work
The planted partition model/stochastic block model [1, 2] are standard models for studying graph
clustering. Variants of the models cover partially observed graphs [3, 4] and weighted graphs [5, 6].
All these models are special cases of ours. Various algorithms have been proposed and analyzed
under these models, such as spectral clustering [7, 8, 1], convex optimization approaches [9, 10, 11]
and tensor decomposition methods [12]. Ours is based on convex optimization; we build upon and
extend the approach in [13], which is designed for clustering unweighted graphs whose edges have
different levels of uncertainty, a special case of our problem (cf. Section 4.2 for details).
Most related to our setting is the labelled stochastic block model proposed in [14] and [15]. A
main difference in their model is that they assume each observation is a two-step process: first
an edge/non-edge is observed; if it is an edge then a label is associated with it. In our model
all observations are in the form of labels; in particular, an edge or no-edge is also a label. This
covers their setting as a special case. Our model is therefore more general and natural?as a result
our theory covers a broad class of subproblems including time-varying graphs. Moreover, their
analysis is mainly restricted to the two-cluster setting with edge probabilities on the order of ?(1/n),
while we allow for an arbitrary number of clusters and a wide range of edge/label distributions.
In addition, we consider the setting where the distributions of the labels are not precisely known.
Algorithmically, they use belief propagation [14] and spectral methods [15].
Clustering time-varying graphs has been studied in various context; see [16, 17, 18, 19, 20] and
the references therein. Most existing algorithms use heuristics and lack theoretical analysis. Our
approach is based on a generative model and has provable performance guarantees.
2
Problem setup and algorithms
We assume n nodes are partitioned into r disjoint clusters of size at least K, which are unknown and
considered as the ground truth. For each pair (i, j) of nodes, a label Lij ? L is observed, where L is
the set of all possible labels.1 These labels are generated independently across pairs according to the
distributions ? and ?. In particular, the probability of observing the label Lij is ?(Lij ) if i and j are
in the same cluster, and ?(Lij ) otherwise. The goal is to recover the ground truth clusters given the
labels. Let L = (Lij ) ? Ln?n be the matrix of observed labels. We represent the true clusters by
an n ? n cluster matrix Y ? , where Yij? = 1 if nodes i and j belong to the same cluster and Yij? = 0
otherwise (we use the convention Yii? = 1 for all i). The problem is therefore to find Y ? given L.
1
Note that L does not have to be finite. Although some of the results are presented for finite L, they can be
easily adapted to the other cases, for instance, by replacing summation with integration.
2
We take an optimization approach to this problem. To motivate our algorithm, first consider the
case of clustering a weighted graph, where all labels are real numbers. Positive weights indicate
in-cluster interaction while negative weights indicate cross-cluster interaction. A natural approach
is to cluster the nodes in a way that maximizes the total weight inside the clusters (this is equivalent
to correlation clustering
P [21]). Mathematically, this is to find a clustering, represented by a cluster
matrix Y , such that i,j Lij Yij is maximized. For the case of general labels, we pick a weight
function w : L 7? R, which assigns a number Wij = w(Lij ) to each label, and then solve the
following max-weight problem:
max hW, Y i s.t. Y is a cluster matrix;
Y
(1)
P
here hW, Y i := ij Wij Yij is the standard trace inner product. Note that this effectively converts
the problem of clustering from labels into a weighted clustering problem.
The program (1) is non-convex due to the constraint. Our algorithm is based on a convex relaxation
of (1), using the now well-known fact that a cluster matrix is a block-diagonal 0-1 matrix and thus
has nuclear norm2 equal to n [22, 3, 23]. This leads to the following convex optimization problem:
max
Y
s.t.
hW, Y i
(2)
kY k? ? n; 0 ? Yij ? 1, ?(i, j).
We say that this program succeeds if it has a unique optimal solution equal to the true cluster matrix
Y ? . We note that a related approach is considered in [13], which is discussed in section 4.
One has the freedom of choosing the weight function w. Intuitively, w should assign w(Lij ) > 0
to a label Lij with ?(Lij ) > ?(Lij ), so the program (2) is encouraged to place i and j in the same
cluster, the more likely possibility; similarly we should have w(Lij ) < 0 if ?(Lij ) < ?(Lij ). A
good weight function should further reflect the information in ? and ?. Our theoretical results in
section 3 characterize the performance of the program (2) for any given weight function; building
on this, we further derive the optimal choice for the weight function.
3
Theoretical results
In this section, we provide theoretical analysis for the performance of the convex program (2) under
the probabilistic model described in section 2. The proofs are given in the supplementary materials.
Our main result is a general theorem that gives sufficient conditions for (2) to recover the true cluster
matrix Y ? . The conditions are stated in terms of the label distribution ? and ?,Pthe minimum size
of the true clusters K, and any given weight function w. Define E? w :=
l?L w(l)?(l) and
P
Var? w := l?L [w(l) ? E? w]2 ?(l); E? w and Var? w are defined similarly.
Theorem 1 (Main). Suppose b is any number that satisfies |w(l)| ? b, ?l ? L almost surely. There
exists a universal constant c > 0 such that if
?
?
b log n + K log n Var? w
?E? w ? c
,
(3)
K p
?
b log n + n log n max(Var? w, Var? w)
E? w ? c
,
(4)
K
then Y ? is the unique solution to (2) with probability at least 1 ? n?10 . 3
The theorem holds for any given weight function w. In the next two subsections, we show how to
choose w optimally, and then address the case where w deviates from the optimal choice.
3.1
Optimal weights
A good candidate for the weight function w can be derived from the maximum likelihood estimator (MLE) of Y ? . Given the observed labels L, the log-likelihood of the true cluster matrix taking
2
The nuclear norm of a matrix is defined as the sum of its singular values. A cluster matrix is positive
semidefinite so its nuclear norm is equal to its trace.
3
In all our results, the choice n?10 is arbitrary. In particular, the constant c scales linearly with the exponent.
3
the value Y is
log Pr(L|Y ? = Y ) =
X
log ?(Lij )Yij ?(Lij )1?Yij = hW, Y i + c
i,j
where c is independent of Y and W is given by the weight function w(l) = wMLE (l) := log ?(l)
?(l) .
MLE
The MLE thus corresponds to using the log-likelihood ratio w
(?) as the weight function. The
following theorem is a consequence of Theorem 1 and characterizes the performance of using the
MLE weights. In the sequel, we use D(?k?) to denote the KL divergence between two distributions.
MLE
Theorem 2 (MLE). Suppose
w is used, and b and ? are any numbers which satisfy with
?(l)
D(?||?) ? ?D(?||?) and log ?(l) ? b, ?l ? L. There exists a universal constant c > 0 such
that Y ? is the unique solution to (2) with probability at least 1 ? n?10 if
log n
D(?||?) ? c(b + 2)
,
K
n log n
.
D(?||?) ? c(? + 1)(b + 2)
K2
Moreover, we always have D(?||?) ? (2b + 3)D(?||?), so we can take ? = 2b + 3.
(5)
(6)
Note that the theorem has the intuitive interpretation that the in/cross-cluster label distributions ? and
? should be sufficiently different, measured by their KL divergence. Using a classical result in information theory [24], we may replace the KL divergences with a quantity that is often easier to work
with, as summarized below. The LHS of (7) is sometimes called the triangle discrimination [24].
Corollary 1 (MLE 2). Suppose wMLE is used, and b, ? are defined as in Theorem 2. There exists a
universal constant c such that Y ? is the unique solution to (2) with probability at least 1 ? n?10 if
X (?(l) ? ?(l))2
n log n
? c(? + 1)(b + 2)
.
(7)
?(l) + ?(l)
K2
l?L
We may take ? = 2b + 3.
The MLE weight wMLE turns out to be near-optimal, at least in the two-cluster case, in the sense that
no other weight function (in fact, no other algorithm) has significantly better performance. This is
shown by establishing a necessary condition for any algorithm to recover Y ? . Here, an algorithm is
a measurable function Y? that maps the data L to a clustering (represented by a cluster matrix).
Theorem 3 (Converse). The following holds for some universal constants c, c0 > 0. Suppose K =
n
0
2 , and b defined in Theorem 2 satisfies b ? c . If
X (?(l) ? ?(l))2
c log n
?
,
(8)
?(l) + ?(l)
n
l?L
then inf Y? supY ? P(Y? =
6 Y ? ) ? 12 , where the supremum is over all possible cluster matrices.
Under the assumption of Theorem 3, the conditions (7) and (8) match up to a constant factor.
Remark. The MLE weight |wMLE (l)| becomes large if ?(l) = o(?(l)) or ?(l) = o(?(l)), i.e., when
the in-cluster probability is negligible compared to the cross-cluster one (or the other way around).
It can be shown that in this case the MLE weight is actually order-wise better than a bounded weight
function. We give this result in the supplementary material due to space constraints.
3.2
Monotonicity
We sometimes do not know the exact true distributions ? and ? to compute wMLE . Instead, we might
compute the weight using the log likelihood ratios of some ?incorrect? distribution ?
? and ??. Our
algorithm has a nice monotonicity property: as long as the divergence of the true ? and ? is larger
than that of ?
? and ?? (hence an ?easier? problem), then the problem should still have the same, if not
better probability of success, even though the wrong weights are used.
We say that (?, ?) is more divergent then (?
?, ??) if, for each l ? L, we have that either
?(l)
?(l)
?
?(l)
?(l)
?(l)
??(l)
?
?
? 1 or
?
?
? 1.
?(l)
??(l)
??(l)
?(l)
?
?(l)
?
?(l)
4
(l)
Theorem 4 (Monotonicity). Suppose we use the weight function w(l) = log ????(l)
, ?l, while the
actual label distributions are ? and ?. If the conditions in Theorem 2 or Corollary 1 hold with ?, ?
replaced by ?
?, ??, and (?, ?) is more divergent than (?
?, ??), then with probability at least 1 ? n?10
Y ? is the unique solution to (2).
This result suggests that one way to choose the weight function is by using the log-likelihood ratio
based on a ?conservative? estimate (i.e., a less divergent one) of the true label distribution pair.
3.3
Using inaccurate weights
In the previous subsection we consider using a conservative log-likelihood ratio as the weight. We
now consider a more general weight function w which need not be conservative, but is only required
to be not too far from the true log-likelihood ratio wMLE . Let
?(l)
?(l) := w(l) ? wMLE (l) = w(l) ? log
?(l)
P
P
be the error for each label l ? L. Accordingly, let ?? := l?L ?(l)?(l) and ?? := l?L ?(l)?(l)
be the average errors with respect to ? and ?. Note that ?? and ?? can be either positive or negative.
The following characterizes the performance of using such a w.
Theorem 5 (Inaccurate Weights). Let b and ? be defined as in Theorem 2. If the weight w satisfies
?(l)
, ?l ? L, |?? | ? ?D(?||?), |?? | ? ?D(?||?)
|w(l)| ? ? log
?(l)
for some ? < 1 and ? > 0. Then Y ? is unique solution to (2) with probability at least 1 ? n?10 if
?2
?2
log n
n log n
D(?||?) ? c
and D(?||?) ? c
(b + 2)
(? + 1)(b + 2)
.
(1 ? ?)2
K
(1 ? ?)2
K2
Therefore, as long as the errors ?? and ?? in w are not too large, the condition for recovery will be
order-wise similar to that in Theorem 2 for using the MLE weight. The numbers ? and ? measure
the amount of inaccuracy in w w.r.t. wMLE . The last two conditions in Theorem 5 thus quantify the
relation between the inaccuracy in w and the price we need to pay for using such a weight.
4
Consequences and applications
We apply the general results in the last section to different special cases. In sections 4.1 and 4.2, we
consider two simple settings and show that two immediate corollaries of our main theorems recover,
and in fact improve upon, existing results. In sections 4.3 and 4.4, we turn to the more complicated
setting of clustering time-varying graphs and derive several novel results.
4.1
Clustering a Gaussian matrix with partial observations
Analogous to the planted partition model for unweighted graphs, the bi-clustering [5] or submatrixlocalization [6, 23] problem concerns with weighted graph whose adjacency matrix has Gaussian
entries. We consider a generalization of this problem where some of the entries are unobserved.
n?n
Specifically, we observe a matrix L ? (R ? {?})
, which has r submatrices of size K ? K with
disjoint row and column support, such that Lij =? (meaning unobserved) with probability 1 ? s and
otherwise Lij ? N (uij , 1). Here the means of the Gaussians satisfy: uij = u
? if (i, j) is inside the
? > u ? 0. Clustering is equivalent to locating these
submatrices and uij = u if outside, where u
submatrices with elevated mean, given the large Gaussian matrix L with partial observations.4
This is a special case of our labeled framework with L = R ? {?}. Computing the log-likelihood
MLE
ratios for two Gaussians, we obtain wMLE (Lij ) = 0 if Lij =?,
(Lij ) ? Lij ? (?
u + u)/2
? and w
otherwise. This problem is interesting only when u
? ? u . log n (otherwise simple element-wise
thresholding [5, 6] finds the submatrices), which we assume to hold. Clearly D (?k?) = D (?k?) =
1
u ? u)2 . The following can be proved using our main theorems (proof in the appendix).
4 s(?
4
Here for simplicity we consider the clustering setting instead of bi-clustering. The latter setting corresponds
to rectangular L and submatrices. Extending our results to this setting is relatively straightforward.
5
Corollary 2 (Gaussian Graphs). Under the above setting, Y ? is the unique solution to (2) with
weights w = wMLE with probability at least 1 ? 2n?10 provided
2
s (?
u ? u) ? c
n log3 n
.
K2
In the fully observed case, this recovers the results in [23, 5, 6] up to log factors. Our results are
more general as we allow for partial observations, which is not considered in previous work.
4.2
Planted Partition with non-uniform edge densities
The work in [13] considers a variant of the planted partition model with non-uniform edge densities,
where each pair (i, j) has an edge with probability 1 ? uij > 1/2 if they are in the same cluster, and
with probability uij < 1/2 otherwise. The number uij can be considered as a measure of the level of
uncertainty in the observation between i and j, and is known or can be estimated in applications like
cloud-clustering. They show that using the knowledge of {uij } improves clustering performance,
and such a setting covers clustering of partially observed graphs that is considered in [11, 3, 4].
Here we consider a more general setting that does not require the in/cross-cluster edge density to be
symmetric around 12 . Suppose each pair (i, j) is associated with two numbers pij and qij , such that
if i and j are in the same cluster (different clusters, resp.), then there is an edge with probability pij
(qij , resp.); we know pij and qij but not which of them is the probability that generates the edge.
The values of pij and qij are generated i.i.d. randomly as (pij , qij ) ? D by some distribution D on
[0, 1] ? [0, 1]. The goal is to find the clusters given the graph adjacency matrix A, (pij ) and (qij ).
This model is a special case of our labeled framework. The labels have the form Lij =
(Aij , pij , qij ) ? L = {0, 1} ? [0, 1] ? [0, 1], generated by the distributions
pD(p, q),
l = (1, p, q)
qD(p, q),
l = (1, p, q)
?(l) =
?(l) =
(1 ? p)D(p, q), l = (0, p, q)
(1 ? q)D(p, q), l = (0, p, q).
p
1?p
ij
ij
+(1?Aij ) log 1?qij
. It turns out it is more
The MLE weight has the form wMLE (Lij ) = Aij log qij
convenient to use a conservative weight in which we replace pij and qij with p?ij = 34 pij + 14 qij and
q?ij = 14 pij + 34 qij . Applying Theorem 4 and Corollary 1, we immediately obtain the following.
Corollary 3 (Non-uniform Density). Program (2) recovers Y ? with probability at least 1 ? n?10 if
n log n
(pij ? qij )2
, ?(i.j).
?c
ED
pij (1 ? qij )
K2
Here ED is the expectation w.r.t. the distribution D, and LHS above is in fact independent of (i, j).
Corollary 3 improves upon existing results for several settings.
? Clustering partially observed graphs. Suppose D is such that pij = p and qij = q with probability s, and pij = qij otherwise, where p > q. This extends the standard planted partition model:
each pair is unobserved with probability 1 ? s. For this setting we require
s(p ? q)2
n log n
&
.
p(1 ? q)
K2
When s = 1. this matches the best existing bounds for standard planted partition [9, 12] up to a
log factor. For the partial observation setting with s ? 1, the work in [4] gives a similar bound
under the additional assumption p > 0.5 > q, which is not required by our result. For general
p and q, the best existing bound is given in [3, 9], which replaces unobserved entries with 0 and
s(p?q)2
log n
requires the condition p(1?sq)
& nK
. Our result is tighter when p and q are close to 1.
2
? Planted partition with non-uniformity. The model and algorithm in [13] isa special case
of ours
with symmetric densities pij ? 1 ? qij , for which we recover their result ED (1?2qij )2 & nlogn
K2 .
Corollary 3 is more general as it removes the symmetry assumption.
6
4.3
Clustering time-varying multiple-snapshot graphs
Standard graph clustering concerns with clustering on a single, static graph. We now consider a
setting where the graph can be time-varying. Specifically, we assume that for each time interval
t = 1, 2, . . . , T , we observed a snapshot of the graph L(t) ? Ln?n . We assume each snapshot is
generated by the distributions ? and ?, independent of other snapshots.
We can map this problem into our original labeled framework, by considering the whole time se? ij := (L(1) , . . . , L(T ) ) observed at the pair (i, j) as a single label. In this case the label
quence of L
ij
ij
set is thus the set of all possible sequences, i.e., L? = (L)T , and the label distributions are (with a
? ij ) = ?(L(1) ) . . . ?(L(T ) ), with ?(?) given similarly. The MLE weight
slight abuse of notation) ?(L
ij
ij
(normalized by T ) is thus the average log-likelihood ratio:
w
MLE
(1)
(T )
(t)
T
?(Lij ) . . . ?(Lij )
?(Lij )
1
1X
?
(Lij ) = log
log
=
.
(1)
(T )
(t)
T
T t=1
?(Lij ) . . . ?(Lij )
?(Lij )
? ij ) is the average of T independent random variables, its variance scales with
Since wMLE (L
Applying Theorem 1, with almost identical proof as in Theorem 2 we obtain the following:
1
T
.
Corollary 4 (Independent Snapshots). Suppose | log ?(l)
?(l) | ? b, ?l ? L and D(?||?) ? ?D(?||?).
?
The program (2) with MLE weights given recovers Y with probability at least 1 ? n?10 provided
log n
,
K
n log n
n log n o
D(?||?) ? c(b + 2) max
, (? + 1)
.
K
T K2
D(?||?) ? c(b + 2)
(9)
(10)
Setting T = 1 recovers Theorem 2. When the second term in (10) dominates, the corollary says that
the problem becomes easier if we observe more snapshots, with the tradeoff quantified precisely.
4.4
Markov sequence of snapshots
We now consider the more general and useful setting where the snapshots form a Markov chain. For
simplicity we assume that the Markov chain is time-invariant and has a unique stationary distribution
(t)
which is also the initial distribution. Therefore, the observations Lij at each (i, j) are generated by
first drawing a label from the stationary distribution ?
? (or ??) at t = 1, then applying a one-step
transition to obtain the label at each subsequent t. In particular, given the previously observed label
l, let the intra-cluster and inter-cluster conditional distributions be ?(?|l) and ?(?|l). We assume that
the Markov chains with respect to both ? and ? are geometrically ergodic such that for any ? ? 1,
and label-pair L(1) , L(? +1) ,
| Pr? (L(? +1) |L(1) ) ? ?
?(L(? +1) )| ? ?? ?
and
| Pr? (L(? +1) |L(1) ) ? ??(L(? +1) )| ? ?? ?
for some constants ? ? 1 and ? < 1 that only depend on ? and ?. Let Dl (?||?)
Pbe the KL-divergence
between ?(?|l) and ?(?|l); Dl (?||?) is similarly defined. Let E?? Dl (?||?) = l?L ?
?(l)Dl (?||?) and
similarly for E?? Dl (?||?). As in the previous subsection, we use the average log-likelihood ratio as
the weight. Define ? = (1??) minl?{??(l),?? (l)} . Applying Theorem 1 gives the following corollary.
See sections H?I in the supplementary material for the proof and additional discussion.
Corollary
Snapshots).
Under the above setting, suppose for each label-pair (l, l0 ),
5 (Markov
?
? (l)
?(l0 |l)
? ||?
?) ? ?D(?
?||?
? ) and E?? Dl (?||?) ? ?E?? Dl (?||?). The
log ??(l) ? b, log ?(l0 |l) ? b, D(?
program (2) with MLE weights recovers Y ? with probability at least 1 ? n?10 provided
1
1
log n
D(?
? ||?
?) + 1 ?
E?? Dl (?||?) ? c(b + 2)
T
T
K
n log n
1
1
n log n o
D(?
?||?
?) + 1 ?
E?? Dl (?||?) ? c(b + 2) max
, (? + 1)?
.
T
T
K
T K2
7
(11)
(12)
As an illuminating example, consider the case where ?
? ? ??, i.e., the marginal distributions for
individual snapshots are identical or very close. It means that the information is contained in the
change of labels, but not in the individual labels, as made evident in the LHSs of (11) and (12).
In this case, it is necessary to use the temporal information in order to perform clustering. Such
information would be lost if we disregard the ordering of the snapshots, for example, by aggregating
or averaging the snapshots then apply a single-snapshot clustering algorithm. This highlights an
essential difference between clustering time-varying graphs and static graphs.
5
Empirical results
To solve the convex program (2), we follow [13, 9] and adapt the ADMM algorithm by [25]. We
perform 100 trials for each experiment, and report the success rate, i.e., the fraction of trials where
the ground-truth clustering is fully recovered. Error bars show 95% confidence interval. Additional
empirical results are provided in the supplementary material.
We first test the planted partition model with partial observations under the challenging sparse (p
and q close to 0) and dense settings (p and q close to 1); cf. section 4.2. Figures 1 and 2 show the
results for n = 1000 with 4 equal-size clusters. In both cases, each pair is observed with probability
0.5. For comparison, we include results for the MLE weights as well as the linear weights (based on
linear approximation of the log-likelihood ratio), uniform weights and a imputation scheme where
all unobserved entries are assumed to be ?no-edge?.
q = 0.02, s = 0.5
14 weeks
p = 0.98, s = 0.5
1
1
0.9
0.6
0.4
0.2
MLE
linear
uniform
no partial
0.6
0.4
0.2
0
0.1
0.2
0.3
p?q
0.4
Figure 1: Sparse graphs
0.5
accuracy
0.8
MLE
linear
uniform
no partial
success rate
success rate
0.8
0.85
0.8
0.75
0
0.1
0.2
0.3
p?q
0.4
Figure 2: Dense graphs
0.5
0.7
0
Markov
independent
aggregate
0.005
0.01
0.015
0.02
fraction of data used in estimation
0.025
Figure 3: Reality Mining dataset
2
Corollary 3 predicts more success as the ratio s(p?q)
p(1?q) gets larger. All else being the same, distributions with small ? (sparse) are ?easier? to solve. Both predictions are consistent with the empirical
results in Figs. 1 and 2. The results also show that the MLE weights outperform the other weights.
For real data, we use the Reality Mining dataset [26], which contains individuals from two main
groups, the MIT Media Lab and the Sloan Business School, which we use as the ground-truth
clusters. The dataset records when two individuals interact, i.e., become proximal of each other or
make a phone call, over a 9-month period. We choose a window of 14 weeks (the Fall semester)
where most individuals have non-empty interaction data. These consist of 85 individuals with 25 of
them from Sloan. We represent the data as a time-varying graph with 14 snapshots (one per week)
and two labels?an ?edge? if a pair of individuals interact within the week, and ?no-edge? otherwise.
We compare three models: Markov sequence, independent snapshots, and the aggregate (union)
graphs. In each trial, the in/cross-cluster distributions are estimated from a fraction of randomly
selected pairwise interaction data. The vertical axis in Figure 3 shows the fraction of pairs whose
cluster relationship are correctly identified. From the results, we infer that the interactions between
individuals are likely not independent across time, and are better captured by the Markov model.
Acknowledgments
S.H. Lim and H. Xu were supported by the Ministry of Education of Singapore through AcRF Tier
Two grant R-265-000-443-112. Y. Chen was supported by NSF grant CIF-31712-23800 and ONR
MURI grant N00014-11-1-0688.
8
References
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9
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4,785 | 5,332 | Discrete Graph Hashing
?
Wei Liu? Cun Mu? Sanjiv Kumar Shih-Fu Chang?
IBM T. J. Watson Research Center ? Columbia University Google Research
[email protected] [email protected]
[email protected] [email protected]
Abstract
Hashing has emerged as a popular technique for fast nearest neighbor search in gigantic databases. In particular, learning based hashing has received considerable
attention due to its appealing storage and search ef?ciency. However, the performance of most unsupervised learning based hashing methods deteriorates rapidly
as the hash code length increases. We argue that the degraded performance is due
to inferior optimization procedures used to achieve discrete binary codes. This
paper presents a graph-based unsupervised hashing model to preserve the neighborhood structure of massive data in a discrete code space. We cast the graph
hashing problem into a discrete optimization framework which directly learns the
binary codes. A tractable alternating maximization algorithm is then proposed to
explicitly deal with the discrete constraints, yielding high-quality codes to well
capture the local neighborhoods. Extensive experiments performed on four large
datasets with up to one million samples show that our discrete optimization based
graph hashing method obtains superior search accuracy over state-of-the-art unsupervised hashing methods, especially for longer codes.
1
Introduction
During the past few years, hashing has become a popular tool for tackling a variety of large-scale
computer vision and machine learning problems including object detection [6], object recognition
[35], image retrieval [22], linear classi?er training [19], active learning [24], kernel matrix approximation [34], multi-task learning [36], etc. In these problems, hashing is exploited to map similar
data points to adjacent binary hash codes, thereby accelerating similarity search via highly ef?cient
Hamming distances in the code space. In practice, hashing with short codes, say about one hundred
bits per sample, can lead to signi?cant gains in both storage and computation. This scenario is called
Compact Hashing in the literature, which is the focus of this paper.
Early endeavors in hashing concentrated on using random permutations or projections to construct
randomized hash functions. The well-known representatives include Min-wise Hashing (MinHash)
[3] and Locality-Sensitive Hashing (LSH) [2]. MinHash estimates the Jaccard set similarity and is
improved by b-bit MinHash [18]. LSH can accommodate a variety of distance or similarity metrics
such as p distances for p ? (0, 2], cosine similarity [4], and kernel similarity [17]. Due to randomized hashing, one needs more bits per hash table to achieve high precision. This typically reduces
recall, and multiple hash tables are thus required to achieve satisfactory accuracy of retrieved nearest
neighbors. The overall number of hash bits used in an application can easily run into thousands.
Beyond the data-independent randomized hashing schemes, a recent trend in machine learning is to
develop data-dependent hashing techniques that learn a set of compact hash codes using a training
set. Binary codes have been popular in this scenario for their simplicity and ef?ciency in computation. The compact hashing scheme can accomplish almost constant-time nearest neighbor search,
after encoding the whole dataset to short binary codes and then aggregating them into a hash table.
Additionally, compact hashing is particularly bene?cial to storing massive-scale data. For example, saving one hundred million samples each with 100 binary bits costs less than 1.5 GB, which
1
can easily ?t in memory. To create effective compact codes, several methods have been proposed.
These include the unsupervised methods, e.g., Iterative Quantization [9], Isotropic Hashing [14],
Spectral Hashing [38, 37], and Anchor Graph Hashing [23], the semi-supervised methods, e.g.,
Weakly-Supervised Hashing [25], and the supervised methods, e.g., Semantic Hashing [30], Binary
Reconstruction Embeddings [16], Minimal Loss Hashing [27], Kernel-based Supervised Hashing
[22], Hamming Distance Metric Learning [28], and Column Generation Hashing [20].
This paper focuses on the problem of unsupervised learning of compact hash codes. Here we argue
that most unsupervised hashing methods suffer from inadequate search performance, particularly
low recall, when applied to learn relatively longer codes (say around 100 bits) in order to achieve
higher precision. The main reason is that the discrete (binary) constraints which should be imposed
on the codes during learning itself have not been treated adequately. Most existing methods either
neglect the discrete constraints like PCA Hashing and Isotropic Hashing, or discard the constraints
to solve the relaxed optimizations and afterwards round the continuous solutions to obtain the binary codes like Spectral Hashing and Anchor Graph Hashing. Crucially, we ?nd that the hashing
performance of the codes obtained by such relaxation + rounding schemes deteriorates rapidly when
the code length increases (see Fig. 2). Till now, very few approaches work directly in the discrete
code space. Parameter-Sensitive Hashing [31] and Binary Reconstruction Embeddings (BRE) learn
the parameters of prede?ned hash functions by progressively tuning the codes generated by such
functions; Iterative Quantization (ITQ) iteratively learns the codes by explicitly imposing the binary
constraints. While ITQ and BRE work in the discrete space to generate the hash codes, they do not
capture the local neighborhoods of raw data in the code space well. ITQ targets at minimizing the
quantization error between the codes and the PCA-reduced data. BRE trains the Hamming distances
to mimic the 2 distances among a limited number of sampled data points, but could not incorporate
the entire dataset into training due to its expensive optimization procedure.
In this paper, we leverage the concept of Anchor Graphs [21] to capture the neighborhood structure
inherent in a given massive dataset, and then formulate a graph-based hashing model over the whole
dataset. This model hinges on a novel discrete optimization procedure to achieve nearly balanced
and uncorrelated hash bits, where the binary constraints are explicitly imposed and handled. To
tackle the discrete optimization in a computationally tractable manner, we propose an alternating
maximization algorithm which consists of solving two interesting subproblems. For brevity, we call
the proposed discrete optimization based graph hashing method as Discrete Graph Hashing (DGH).
Through extensive experiments carried out on four benchmark datasets with size up to one million,
we show that DGH consistently obtains higher search accuracy than state-of-the-art unsupervised
hashing methods, especially when relatively longer codes are learned.
2
Discrete Graph Hashing
First we de?ne a few main notations used throughout this paper: sgn(x) denotes the sign function
which returns 1 for x > 0 and ?1 otherwise; In denotes the n?n identity matrix; 1 denotes a vector
with all 1 elements; 0 denotes a vector or matrix of all 0 elements; diag(c) represents a diagonal
matrix with elements of vector c being its diagonal entries; tr(?), ? F , ? 1 , and ?, ? express
matrix trace norm, matrix Frobenius norm, 1 norm, and inner-product operator, respectively.
Anchor Graphs. In the discrete graph hashing model, we need to choose a neighborhood graph that
can easily scale to massive data points. For simplicity and ef?ciency, we choose Anchor Graphs [21],
which involve no special indexing scheme but still have linear construction time in the number of
data points. An anchor graph uses a small set of m points (called anchors), U = {uj ? Rd }m
j=1 , to
d n
approximate the neighborhood structure underlying the input dataset X = {xi ? R }i=1 . Af?nities
(or similarities) of all n data points are computed with respect to these m anchors in linear time
O(dmn) where m n. The true af?nity matrix Ao ? Rn?n is then approximated by using these
af?nities.
Speci?cally, an anchor graph leverages a nonlinear data-to-anchor mapping (Rd ? Rm ) z(x) =
2
2
1)
m)
?1 exp(? D (x,u
), ? ? ? , ?m exp(? D (x,u
) /M , where ?j ? {1, 0} and ?j = 1 if and only
t
t
if anchor uj is one of s m closest anchors of x in U according to some distance function
m
D 2 (x,uj )
)
D() (e.g., 2 distance), t > 0 is the bandwidth parameter, and M =
j=1 ?j exp(?
t
leading to z(x)1 = 1. Then, the anchor graph builds a data-to-anchor af?nity matrix Z =
2
z(x1 ), ? ? ? , z(xn ) ? Rn?m that is highly sparse. Finally, the anchor graph gives a data-to-data
af?nity matrix as A = Z??1 Z ? Rn?n where ? = diag(Z 1) ? Rm?m . Such an af?nity
matrix empirically approximates the true af?nity matrix Ao , and has two nice characteristics: 1)
A is a low-rank positive semide?nite (PSD) matrix with rank at most m, so the anchor graph does
not need to compute it explicitly but instead keeps its low-rank form and only saves Z and ? in
memory; 2) A has unit row and column sums, so the resulting graph Laplacian is L = In ? A. The
two characteristics permit convenient and ef?cient matrix manipulations upon A, as shown later on.
We also de?ne an anchor graph af?nity function as A(x, x ) = z (x)??1 z(x ) in which (x, x )
is any pair of points in Rd .
Learning Model. The purpose of unsupervised hashing is to learn to map each data point xi to an
r-bit binary hash code b(xi ) ? {1, ?1}r given a training dataset X = {xi }ni=1 . For simplicity, let
us denote b(xi ) as bi , and the corresponding code matrix as B = [b1 , ? ? ? , bn ] ? {1, ?1}n?r . The
standard graph-based hashing framework, proposed by [38], aims to learn the hash codes such that
the neighbors in the input space have small Hamming distances in the code space. This is formulated
as:
n
1
min
bi ? bj 2 Aoij = tr B Lo B , s.t. B ? {?1}n?r , 1 B = 0, B B = nIr , (1)
B
2 i,j=1
where Lo is the graph Laplacian based on the true af?nity matrix Ao1 . The constraint 1 B = 0
is imposed to maximize the information from each hash bit, which occurs when each bit leads to
a balanced partitioning of the dataset X . Another constraint B B = nIr makes r bits mutually
uncorrelated to minimize the redundancy among these bits. Problem (1) is NP-hard, and Weiss et al.
[38] therefore solved a relaxed problem by dropping the discrete (binary) constraint B ? {?1}n?r
and making a simplifying assumption of data being distributed uniformly.
We leverage the anchor graph to replace Lo by the anchor graph Laplacian L = In ? A. Hence, the
objective in Eq. (1) can be rewritten as a maximization problem:
max tr B AB , s.t. B ? {1, ?1}n?r , 1 B = 0, B B = nIr .
(2)
B
In [23], the solution to this problem is obtained via spectral relaxation [33] in which B is relaxed
to be a matrix of reals followed by a thresholding step (threshold is 0) that brings the ?nal discrete
B. Unfortunately, this procedure may result in poor codes due to ampli?cation of the error caused
by the relaxation as the code length r increases. To this end, we propose to directly solve the binary
codes B without resorting to such error-prone relaxations.
Let us de?ne a set ? = Y ? Rn?r |1 Y = 0, Y Y = nIr }. Then we formulate a more general
graph hashing framework which softens the last two hard constraints in Eq. (2) as:
?
(3)
max tr B AB ? dist2 (B, ?), s.t. B ? {1, ?1}n?r ,
B
2
where dist(B, ?) = minY?? B ? YF measures the distance from any matrix B to the set ?, and
? ? 0 is a tuning parameter. If problem (2) is feasible, we can enforce dist(B, ?) = 0 in Eq. (3) by
imposing a very large ?, thereby turning problem (3) into problem (2). However, in Eq. (3) we allow
a certaindiscrepancy between B and ? (controlled by ?), which makes problem (3) more ?exible.
Since tr B B) = tr Y Y) = nr, problem (3) can be equivalently transformed to the following
problem:
max Q(B, Y) := tr B AB + ?tr B Y ,
B,Y
(4)
s.t. B ? {1, ?1}n?r , Y ? Rn?r , 1 Y = 0, Y Y = nIr .
We call the code learning model formulated in Eq. (4) as Discrete Graph Hashing (DGH). Because
concurrently imposing B ? {?1}n?r and B ? ? will make graph hashing computationally intractable, DGH does not pursue the latter constraint but penalizes the distance from the target code
matrix B to ?. Different from the previous graph hashing methods which discard the discrete constraint B ? {?1}n?r to obtain continuously relaxed B, our DGH model enforces this constraint to
directly achieve discrete B. As a result, DGH yields nearly balanced and uncorrelated binary bits. In
Section 3, we will propose a computationally tractable optimization algorithm to solve this discrete
programming problem in Eq. (4).
1
The spectral hashing method in [38] did not compute the true af?nity matrix Ao because of the scalability
issue, but instead used a complete graph built over 1D PCA embeddings.
3
Algorithm 1 Signed Gradient Method (SGM) for B-Subproblem
Input: B(0) ? {1, ?1}n?r and Y ? ?.
j := 0; repeat B(j+1) := sgn C 2AB(j) + ?Y, B(j) , j := j + 1, until B(j) converges.
Output: B = B(j) .
Out-of-Sample Hashing. Since a hashing scheme should be able to generate the hash code for any
data point q ? Rd beyond the points in the training set X , here we address the out-of-sample extension of the DGH model. Similar to the objective in Eq. (1), we minimize the Hamming distances
between a novel data point q and its neighbors (revealed by the af?nity function A) in X as
n
1
b(q) ? b?i 2 A(q, xi ) = arg max
b(q) ? arg min r
b(q), (B? ) Z??1 z(q) ,
r
b(q)?{?1} 2
b(q)?{?1}
i=1
where B? = [b?1 , ? ? ? , b?n ] is the solution of problem (4). After pre-computing a matrix
W =
?1
?
r?m
?
(B ) Z? ? R
in the training phase, one can compute the hash code b (q) = sgn Wz(q)
for any novel data point q very ef?ciently.
3
Alternating Maximization
The graph hashing problem in Eq. (4) is essentially a nonlinear mixed-integer program involving
both discrete variables in B and continuous variables in Y. It turns out that problem (4) is generally
NP-hard and also dif?cult to approximate. In speci?c, since the Max-Cut problem is a special case
of problem (4) when ? = 0 and r = 1, there exists no polynomial-time algorithm which can achieve
the global optimum, or even an approximate solution with its objective value beyond 16/17 of the
global maximum unless P = NP [11]. To this end, we propose a tractable alternating maximization
algorithm to optimize problem (4), leading to good hash codes which are demonstrated to exhibit
superior search performance through extensive experiments conducted in Section 5.
The proposed algorithm proceeds by alternately solving the B-subproblem
max
f (B) := tr B AB + ?tr Y B
B?{?1}n?r
and the Y-subproblem
max
Y?Rn?r
tr B Y ,
s.t. 1 Y = 0, Y Y = nIr .
(5)
(6)
In what follows, we propose an iterative ascent procedure called Signed Gradient Method for subproblem (5) and derive a closed-form optimal solution to subproblem (6). As we can show, our
alternating algorithm is provably convergent. Schemes for choosing good initializations are also
discussed. Due to the space limit, all the proofs of lemmas, theorems and propositions presented in
this section are placed in the supplemental material.
3.1
B-Subproblem
We tackle subproblem (5) with a simple iterative ascent procedure described in Algorithm 1. In the
j-th iteration, we de?ne a local function f?j (B) that linearizes f (B) at the point B(j) , and employ
f?j (B) as a surrogate of f (B) for discrete optimization. Given B(j) , the next discrete point is
derived as B(j+1) ? arg maxB?{?1}n?r f?j (B) := f B(j) + ?f B(j) , B ? B(j) . Note that
(j)
(j+1)
since ?f B
may include zero entries, multiple
could exist. To avoid this
solutions for B
x, x = 0
ambiguity, we introduce the function C(x, y) =
to specify the following update:
y, x = 0
B(j+1) := sgn C ?f B(j) , B(j) = sgn C 2AB(j) + ?Y, B(j) ,
(7)
in which
C is applied in an element-wise manner, and no update is carried out to the entries where
?f B(j) vanishes.
Due to the PSD property of the matrix A, f is a convex function and thus f (B) ? f?j (B) for any B.
Taking advantage of the fact f B(j+1) ? f?j B(j+1) ? f?j B(j) ? f B(j) , Lemma 1 ensures
that both the sequence of cost values f (B(j) ) and the sequence of iterates B(j) converge.
4
Algorithm 2 Discrete Graph Hashing (DGH)
Input: B0 ? {1, ?1}n?r and Y0 ? ?.
k := 0;
repeat Bk+1 := SGM(Bk , Yk ), Yk+1 ? ?(JBk+1 ), k := k + 1, until Q(Bk , Yk ) converges.
Output: B? = Bk , Y? = Yk .
Lemma 1. If B(j) is the sequence of iterates produced by Algorithm 1, then f B(j+1) ?
(j)
f B
holds for any integer j ? 0, and both f (B(j) ) and B(j) converge.
Our idea of optimizing a proxy function f?j (B) can be considered as a special case of majorization
methodology exploited in the ?eld of optimization. The majorization method typically deals with a
generic constrained optimization problem: min g(x), s.t. x ? F, where g : Rn ? R is a continuous function and F ? Rn is a compact set. The majorization method starts with a feasible point
x0 ? F, and then proceeds by setting xj+1 as a minimizer of g?j (x) over F, where g?j satisfying
g?j (xj ) = g(xj ) and g?j (x) ? g(x) ?x ? F is called a majorization function of g at xj . In speci?c,
in our scenario, problem (5) is equivalent to minB?{?1}n?r ?f (B), and the linear surrogate ?f?j
is a majorization function of ?f at point B(j) . The majorization method was ?rst systematically
introduced by [5] to deal with multidimensional scaling problems, although the EM algorithm [7],
proposed at the same time, also falls into the framework of majorization methodology. Since then,
the majorization method has played an important role in various statistics problems such as multidimensional data analysis [12], hyperparameter learning [8], conditional random ?elds and latent
likelihoods [13], and so on.
Y-Subproblem
3.2
An analytical solution to subproblem (6) can be obtained with the aid of a centering matrix J = In ?
r
1
k=1 ?k uk vk ,
n 11 . Write the singular value decomposition (SVD) of JB as JB = U?V =
where r ? r is the rank of JB, ?1 , ? ? ? , ?r are the positive singular values, and U = [u1 , ? ? ? , ur ]
and V = [v1 , ? ? ? , vr ] contain the left- and right-singular vectors, respectively. Then, by employing
? ? Rn?(r?r ) and V
? ? Rr?(r?r )
a Gram-Schmidt process, one can easily construct matrices U
?
2
?
?
?
?
?
such that U U = Ir?r , [U 1] U = 0, and V V = Ir?r , V V = 0 . Now we are ready to
characterize a closed-form solution of the Y-subproblem by Lemma 2.
?
?
? is an optimal solution to the Y-subproblem in Eq. (6).
Lemma 2. Y = n[U U][V
V]
?
?
?
V]
For notational convenience, we de?ne the set of all matrices in the form of n[U U][V
as ?(JB). Lemma 2 reveals that any matrix in ?(JB) is an optimal solution to subproblem (6).
In practice, to compute such an optimal Y , we perform the
eigendecomposition over the small
2
?
0
?
? ?, and
? , which gives V, V,
r ? r matrix B JB to have B JB = [V V]
[V V]
0 0
? is initially set to a random matrix followed
immediately leads to U = JBV??1 . The matrix U
by the aforementioned Gram-Schmidt orthogonalization. It can be seen that Y is uniquely optimal
when r = r (i.e., JB is full column rank).
3.3
DGH Algorithm
The proposed alternating maximization algorithm, also referred to as Discrete Graph Hashing
(DGH), for solving the raw problem in Eq. (4) is summarized in Algorithm 2, in which we introduce
SGM(?, ?) to represent the functionality of Algorithm 1. The convergence of Algorithm 2 is guaranteed by Theorem 1, whose proof is based on the nature of the proposed alternating maximization
procedure that always generates a monotonically non-decreasing and bounded sequence.
Theorem 1. If (Bk , Yk ) is the sequence generated
by Algorithm
2, then Q(Bk+1 , Yk+1 ) ?
Q(Bk , Yk ) holds for any integer k ? 0, and Q(Bk , Yk ) converges starting with any feasible
initial point (B0 , Y0 ).
Initialization. Since the DGH algorithm deals with discrete and non-convex optimization, a good
choice of an initial point (B0 , Y0 ) is vital. Here we suggest two different initial points which are
both feasible to problem (4).
2
? and V
? are nothing but 0.
Note that when r = r, U
5
m
Let us perform the eigendecomposition over A to obtain A = P?P = k=1 ?k pk p
k , where
?1 , ? ? ? , ?m are the eigenvalues arranged in a non-increasing order, and p1 , ? ? ? , pm are the corresponding normalized eigenvectors.
We write ? = diag(?1 , ? ? ? , ?m )?
and P = [p1 , ? ? ? , pm
]. Note
?
that ?1 = 1 and p1 = 1/ n. The ?rst initialization used is Y0 = nH, B0 = sgn(H) , where
H = [p2 , ? ? ? , pr+1 ] ? Rn?r . The initial codes B0 were used as the ?nal codes by [23].
Alternatively,
? Y0 can be allowed to consist of orthonormal columns within the column space of H,
i.e., Y0 = nHR subject to some orthogonal matrix R ? Rr?r . We can obtain R along with B0
by solving a new discrete optimization problem:
(8)
max tr R H AB0 , s.t. R ? Rr?r , RR = Ir , B0 ? {1, ?1}n?r ,
R,B0
which is motivated by the proposition below.
Proposition 1. For any orthogonal matrix R ? Rr?r and any binary matrix B ? {1, ?1}n?r , we
1
have tr B AB ? tr2 R H AB .
r
Proposition 1 implies that the optimization in Eq. (8) can be interpreted as to maximize a lower
bound of tr B AB which is the ?rst term of the objective Q(B, Y) in the original problem
(4). We still exploit an alternating maximization procedure to solve problem (8).
AH =
Noticing
? where ?
? = diag(?2 , ? ? ? , ?r+1 ), the objective in Eq. (8) is equal to tr R ?H
? B0 ). The
H?
? j ,
alternating procedure starts with R0 = Ir , and then makes the simple updates Bj0 := sgn H?R
? jV
? j, V
? j ? Rr?r stem from the full SVD U
? j?
? for j = 0, 1, 2, ? ? ? , where U
? jV
? of
Rj+1 := U
j
j
j
? B . When convergence is reached, we obtain the optimized rotation R that yields
the matrix ?H
0
?
?
the second initialization Y0 = nHR, B0 = sgn(H?R)
.
Empirically, we ?nd that the second initialization typically gives a better objective value Q(B0 , Y0 )
at the start than the ?rst one, as it aims to maximize the lower bound of the ?rst term in the objective
Q. We also observe that the second initialization often results in a higher objective value Q(B? , Y? )
at convergence (Figs. 1-2 in the supplemental material show convergence curves of Q starting from
the two initial points). We call DGH using the ?rst and second initializations as DGH-I and DGH-R,
respectively. Regarding the convergence property, we would like to point out that since the DGH algorithm (Algorithm 2) works on a mixed-integer objective, it is hard to quantify the convergence to a
local optimum of the objective function Q. Nevertheless, this does not affect the performance of our
algorithm in practice. In our experiments in Section 5, we consistently ?nd a convergent sequence
{(Bk , Yk )} arriving at a good objective value when started with the suggested initializations.
4
Discussions
Here we analyze space and time complexities of DGH-I/DGH-R. The space complexity is O (d +
s + r)n in the training stage and O(rn) for storing hash codes in the test stage for DGH-I/DGH-R.
and the whole DGH
Let TB and TG be the budget iteration numbers of optimizing the B-subproblem
problem, respectively.
Then,
the
training
time
complexity
of
DGH-I
is
O
dmn
+ m2 n + (mTB +
2
sTB + r)rTG n , and the training time complexity of DGH-R is O dmn + m n + (mTB + sTB +
r)rTG n + r2 TR n , where TR is the budget iteration number for seeking the initial point via Eq. (8).
Note that the time for ?nding anchors and building the anchor graph is O(dmn) which is included
in the above training time. Their test time (referring to encoding a query to an r-bit code) is both
O(dm + sr). In our experiments, we ?x m, s, TB , TG , TR to constants independent of the dataset
size n, and make r ? 128. Thus, DGH-I/DGH-R enjoy linear training time and constant test time.
It is worth mentioning again that the low-rank PSD property of the anchor graph af?nity matrix A is
advantageous for training DGH, permitting ef?cient matrix computations in O(n) time, such as the
eigendecomposition of A (encountered in initializations) and multiplying A with B (encountered
in solving the B-subproblem with Algorithm 1).
It is interesting to point out that DGH falls into the asymmetric hashing category [26] in the sense
that hash codes are generated differently for samples within the dataset and queries outside the
dataset. Unlike most existing hashing techniques, DGH directly solves the hash codes B? of the
training samples via the proposed discrete optimization in Eq. (4) without relying on any explicit or
prede?ned hash functions. On the other hand, the hash code
for any
query q is induced from the
solved codes B? , leading to a hash function b? (q) = sgn Wz(q) parameterized by the matrix
6
(b) Hash lookup success rate @ SUN397
Success rate
(a) Hash lookup success rate @ CIFAR?10
(c) Hash lookup success rate @ YouTube Faces
1
1
1
0.9
0.9
0.9
0.9
0.8
0.8
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.4
0.3
0.2
0.1
0
LSH
KLSH
ITQ
IsoH
SH
IMH
1?AGH
2?AGH
BRE
DGH?I
DGH?R
8 1216
24
LSH
KLSH
ITQ
IsoH
SH
IMH
1?AGH
2?AGH
BRE
DGH?I
DGH?R
0.4
0.3
0.2
0.1
48
# bits
64
96
0
8 1216
24
0.6
0.5
0.4
0.3
0.2
32
48
# bits
64
96
0.1
(d) Hash lookup success rate @ Tiny?1M
LSH
KLSH
ITQ
IsoH
SH
IMH
1?AGH
2?AGH
BRE
DGH?I
DGH?R
0.7
0.7
0.5
32
1
0.6
LSH
KLSH
ITQ
IsoH
SH
IMH
1?AGH
2?AGH
BRE
DGH?I
DGH?R
1216 24 32
0.5
0.4
0.3
0.2
0.1
48
64
# bits
96
128
0
1216 24 32
48
64
# bits
96
128
Figure 1: Hash lookup success rates for different hashing techniques. DGH tends to achieve nearly
100% success rates even for longer code lengths.
F?measure within Hamming radius 2
0.3
(a) Hash lookup F?measure @ CIFAR?10
0.2
LSH
KLSH
ITQ
IsoH
SH
IMH
1?AGH
2?AGH
BRE
DGH?I
DGH?R
0.25
0.2
0.15
(c) Hash lookup F?measure @ YouTube Faces
(b) Hash lookup F?measure @ SUN397
0.8
0.18
0.16
(d) Hash lookup F?measure @ Tiny?1M
0.2
0.6
0.14
0.5
0.12
0.1
0.15
0.4
0.08
0.1
0.1
0.3
0.06
0.2
0.04
0.05
8 1216
24
32
48
# bits
64
96
0
0.05
0.1
0.02
0
0.25
0.7
8 1216
24
32
48
# bits
64
96
0
1216 24 32
48
64
# bits
96
128
0
1216 24 32
48
64
# bits
96
128
Figure 2: Mean F-measures of hash lookup within Hamming radius 2 for different techniques. DGH
tends to retain good recall even for longer codes, leading to much higher F-measures than the others.
W which was computed using B? . While the hashing mechanisms for producing B? and b? (q) are
distinct, they are tightly coupled and prone to be adaptive to speci?c datasets. The ?exibility of the
asymmetric hashing nature of DGH is validated through the experiments shown in the next section.
5
Experiments
We conduct large-scale similarity search experiments on four benchmark datasets: CIFAR-10 [15],
SUN397 [40], YouTube Faces [39], and Tiny-1M. CIFAR-10 is a labeled subset of the 80 Million
Tiny Images dataset [35], which consists of 60K images from ten object categories with each image
represented by a 512-dimensional GIST feature vector [29]. SUN397 contains about 108K images
from 397 scene categories, where each image is represented by a 1,600-dimensional feature vector
extracted by PCA from 12,288-dimensional Deep Convolutional Activation Features [10]. The raw
YouTube Faces dataset contains 1,595 different people, from which we choose 340 people such that
each one has at least 500 images to form a subset of 370,319 face images, and represent each face
image as a 1,770-dimensional LBP feature vector [1]. Tiny-1M is one million subset of the 80M
tiny images, where each image is represented by a 384-dimensional GIST vector. In CIFAR-10, 100
images are sampled uniformly randomly from each object category to form a separate test (query)
set of 1K images; in SUN397, 100 images are sampled uniformly randomly from each of the 18
largest scene categories to form a test set of 1.8K images; in YouTube Faces, the test set includes
3.8K face images which are evenly sampled from the 38 people each containing more than 2K faces;
in Tiny-1M, a separate subset of 5K images randomly sampled from the 80M images is used as the
test set. In the ?rst three datasets, groundtruth neighbors are de?ned based on whether two samples
share the same class label; in Tiny-1M which does not have full annotations, we de?ne groundtruth
neighbors for a given query as the samples among the top 2% 2 distances from the query in the 1M
training set, so each query has 20K groundtruth neighbors.
We evaluate twelve unsupervised hashing methods including: two randomized methods LSH [2] and
Kernelized LSH (KLSH) [17], two linear projection based methods Iterative Quantization (ITQ) [9]
and Isotropic Hashing (IsoH) [14], two spectral methods Spectral Hashing (SH) [38] and its weighted version MDSH [37], one manifold based method Inductive Manifold Hashing (IMH) [32], two
existing graph-based methods One-Layer Anchor Graph Hashing (1-AGH) and Two-Layer Anchor
Graph Hashing (2-AGH) [23], one distance preservation method Binary Reconstruction Embeddings (BRE) [16] (unsupervised version), and our proposed discrete optimization based methods
DGH-I and DGH-R. We use the publicly available codes of the competing methods, and follow the
conventional parameter settings therein. In particular, we use the Gaussian kernel and 300 randomly
sampled exemplars (anchors) to run KLSH; IMH, 1-AGH, 2-AGH, DGH-I and DGH-R also use
m = 300 anchors (obtained by K-means clustering with 5 iterations) for fair comparison. This
choice of m gives a good trade-off between hashing speed and performance. For 1-AGH, 2-AGH,
DGH-I and DGH-R that all use anchor graphs, we adopt the same construction parameters s, t on
each dataset (s = 3 and t is tuned following AGH), and 2 distance as D(?). For BRE, we uniformly
7
Table 1: Hamming ranking performance on YouTube Faces and Tiny-1M. r denotes the number of
hash bits used in the hashing methods. All training and test times are in seconds.
Method
2 Scan
YouTube Faces
Mean Precision / Top-2K
TrainTime
TestTime
r = 48
r = 96
r = 128
r = 128
r = 128
0.7591
?
LSH
KLSH
ITQ
IsoH
SH
MDSH
IMH
1-AGH
2-AGH
BRE
DGH-I
DGH-R
0.0830
0.3982
0.7017
0.6093
0.5897
0.6110
0.3150
0.7138
0.6727
0.5564
0.7086
0.7245
0.1005
0.5210
0.7493
0.6962
0.6655
0.6752
0.3641
0.7571
0.7377
0.6238
0.7644
0.7672
0.1061
0.5871
0.7562
0.7058
0.6736
0.6795
0.3889
0.7646
0.7521
0.6483
0.7750
0.7805
6.4
16.1
169.0
73.6
108.9
118.8
92.1
84.1
94.7
10372.0
402.6
408.9
1.8?10?5
4.8?10?5
1.8?10?5
1.8?10?5
2.0?10?4
4.9?10?5
2.3?10?5
2.1?10?5
3.5?10?5
9.0?10?5
2.1?10?5
2.1?10?5
Tiny-1M
Mean Precision / Top-20K
TrainTime
TestTime
r = 48
r = 96
r = 128
r = 128
r = 128
1
?
0.1155
0.3054
0.3925
0.3896
0.1857
0.3312
0.2257
0.4061
0.3925
0.3943
0.4045
0.4208
0.1324
0.4105
0.4726
0.4816
0.1923
0.3878
0.2497
0.4117
0.4099
0.4836
0.4865
0.5006
0.1766
0.4705
0.5052
0.5161
0.2079
0.3955
0.2557
0.4107
0.4152
0.5218
0.5178
0.5358
6.1
20.7
297.3
13.5
61.4
193.6
139.3
141.4
272.5
8419.0
1769.4
2793.4
1.0?10?5
4.6?10?5
1.0?10?5
1.0?10?5
1.6?10?4
2.8?10?5
2.7?10?5
3.4?10?5
4.7?10?5
8.8?10?5
3.3?10?5
3.3?10?5
randomly sample 1K, and 2K training samples to train the distance preservations on CIFAR-10 &
SUN397, and YouTube Faces & Tiny-1M, respectively. For DGH-I and DGH-R, we set the penalty
parameter ? to the same value in [0.1, 5] on each dataset, and ?x TR = 100, TB = 300, TG = 20.
We employ two widely used search procedures hash lookup and Hamming ranking with 8 to 128
hash bits for evaluations. The Hamming ranking procedure ranks the dataset samples according to
their Hamming distances to a given query, while the hash lookup procedure ?nds all the points within
a certain Hamming radius away from the query. Since hash lookup can be achieved in constant time
by using a single hash table, it is the main focus of this work. We carry out hash lookup within a
Hamming ball of radius 2 centered on each query, and report the search recall and F-measure which
are averaged over all queries for each dataset. Note that if table lookup fails to ?nd any neighbors
within a given radius for a query, we call it a failed query and assign it zero recall and F-measure. To
quantify the failed queries, we report the hash lookup success rate which gives the proportion of the
queries for which at least one neighbor is retrieved. For Hamming ranking, mean average precision
(MAP) and mean precision of top-retrieved samples are computed.
The hash lookup results are shown in Figs. 1-2. DGH-I/DGH-R achieve the highest (close to 100%)
hash lookup success rates, and DGH-I is slightly better than DGH-R. The reason is that the asymmetric hashing scheme exploited by DGH-I/DGH-R poses a tight linkage to connect queries and
database samples, providing a more adaptive out-of-sample extension than the traditional symmetric hashing schemes used by the competing methods. Also, DGH-R achieves the highest F-measure
except on CIFAR-10, where DGH-I is highest while DGH-R is the second. The F-measures of
KLSH, IsoH, SH and BRE deteriorate quickly and are with very poor values (< 0.05) when r ? 48
due to poor recall3 . Although IMH achieves nice hash lookup succuss rates, its F-measures are
much lower than DGH-I/DGH-R due to lower precision. MDSH produces the same hash bits as SH,
so is not included in the hash lookup experiments. DGH-I/DGH-R employ the proposed discrete
optimization to yield high-quality codes that preserve the local neighborhood of each data point
within a small Hamming ball, so obtain much higher search accuracy in F-measure and recall than
SH, 1-AGH and 2-AGH which rely on relaxed optimizations and degrade drastically when r ? 48.
Finally, we report the Hamming ranking results in Table 1 and the table in the sup-material, which
clearly show the superiority of DGH-R over the competing methods in MAP and mean precision;
on the ?rst three datasets, DGH-R even outperforms exhaustive 2 scan. The training time of DGHI/DGH-R is acceptable and faster than BRE, and their test time (i.e., coding time since hash lookup
time is small enough to be ignored) is comparable with 1-AGH.
6 Conclusion
This paper investigated a pervasive problem of not enforcing the discrete constraints in optimization pertaining to most existing hashing methods. Instead of resorting to error-prone continuous
relaxations, we introduced a novel discrete optimization technique that learns the binary hash codes
directly. To achieve this, we proposed a tractable alternating maximization algorithm which solves
two interesting subproblems and provably converges. When working with a neighborhood graph,
the proposed method yields high-quality codes to well preserve the neighborhood structure inherent
in the data. Extensive experimental results on four large datasets up to one million showed that our
discrete optimization based graph hashing technique is highly competitive.
3
The recall results are shown in Fig. 3 of the supplemental material, which indicate that DGH-I achieves the
highest recall except on YouTube Faces, where DGH-R is highest while DGH-I is the second.
8
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9
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4,786 | 5,333 | Large-scale L-BFGS using MapReduce
Weizhu Chen, Zhenghao Wang, Jingren Zhou
Microsoft
{wzchen,zhwang,jrzhou}@microsoft.com
Abstract
L-BFGS has been applied as an effective parameter estimation method for various
machine learning algorithms since 1980s. With an increasing demand to deal
with massive instances and variables, it is important to scale up and parallelize
L-BFGS effectively in a distributed system. In this paper, we study the problem
of parallelizing the L-BFGS algorithm in large clusters of tens of thousands of
shared-nothing commodity machines. First, we show that a naive implementation
of L-BFGS using Map-Reduce requires either a significant amount of memory or a
large number of map-reduce steps with negative performance impact. Second, we
propose a new L-BFGS algorithm, called Vector-free L-BFGS, which avoids the
expensive dot product operations in the two loop recursion and greatly improves
computation efficiency with a great degree of parallelism. The algorithm scales
very well and enables a variety of machine learning algorithms to handle a massive
number of variables over large datasets. We prove the mathematical equivalence
of the new Vector-free L-BFGS and demonstrate its excellent performance and
scalability using real-world machine learning problems with billions of variables
in production clusters.
1
Introduction
In the big data era, many applications require solving optimization problems with billions of variables on a huge amount of training data. Problems of this scale are more common nowadays, such
as Ads CTR prediction[1] and deep neural network[2]. The other trend is the wide adoption of mapreduce [3] environments built with commodity hardware. Those large-scale optimization problems
are often expected to be solved in a map-reduce environment where big data are stored.
When a problem is with huge number of variables, it can be solved efficiently only if the storage and
computation cost are maintained effectively. Among a diverse collection of large-scale optimization
methods, Limited-memory BFGS (L-BFGS)[4] is one of the frequently used optimization methods
in practice[5]. In this paper, we study the L-BFGS implementation for billion-variable scale problems in a map-reduce environment. The original L-BFGS algorithm and its update procedure were
proposed in 1980s. A lot of popular optimization software packages implement it as a fundamental building block. Approaches to apply it in a problem with up to millions of variables are well
studied and implemented in various optimization packages [6]. However, studies about how to scale
L-BFGS into billions of variables are still in their very early stages. For such a massive scale, the
parameters, their gradients, and the associated L-BFGS historical states are not only too large to
be stored in the memory of a single computation node, but also create too huge computation complexity for a processor or multicores to conquer it within reasonable time. Therefore, it is critical
to explore an effective decomposition over both examples and models via distributed learning. Yet,
to our knowledge, there is still very limited work to explore billion-variable scale L-BFGS. This
directly leads to the consequence that very little work can scale various machine learning algorithms
up to billion-variable scale using L-BFGS on map-reduce.
1
In this paper, we start by carefully studying the implementation of L-BFGS in map-reduce environment. We examine two typical L-BFGS implementations in map-reduce and present their scaling
obstacles. Particularly, given a problem with d variables and m historical states to approximate Hessian [5], traditional implementation[6][5], either need to store 2md variables in memory or need to
perform 2m map-reduce steps per iteration. This clearly creates huge overhead for the problem with
billions of variables and prevents a scalable implementation in map-reduce.
To conquer these limitations, we reexamine the original L-BFGS algorithm and propose a new LBFGS update procedure, called Vector-free L-BFGS (VL-BFGS), which is specifically devised for
distributed learning with huge number of variables. In particular, we replace the original L-BFGS
update procedure depending on vector operations, as known as two-loop recursion, by a new procedure only relying on scalar operations. The new two-loop recursion in VL-BFGS is mathematically
equivalent to the original algorithm but independent on the number of variable. Meanwhile, it reduces the memory requirement from O(md) to O(m2 ) where d could be billion-scale but m is often
less than 10. Alternatively, it only require 3 map-reduce steps compared to 2m map-reduce steps in
another naive implementation.
This new algorithm enables the implementation of a collection of machine learning algorithms to
scale to billion variables in a map-reduce environment. We demonstrate its scalability and advantage
over other approaches designed for large scale problems with billions of variables, and share our
experience after deploying it into an industrial cluster with tens of thousands of machines.
2
Related Work
L-BFGS [4][7] is a quasi-newton method based on the BFGS [8][9] update procedure, while maintaining a compact approximation of Hessian with modest storage requirement. Traditional implementation of L-BFGS follows [6] or [5] using the compact two-loop recursion update procedure.
Although it has been applied in the industry to solve various optimization problems for decades,
recent work, such as [10][11], continue to demonstrate its reliability and effectiveness over other
optimization methods. In contrast to our work, theirs implemented L-BFGS on a single machine
while we focus on the L-BFGS implementation in a distributed environment.
In the context of distributed learning, there recently have been extensive research break-through.
GraphLab [12] built a parallel distributed framework for graph computation. [13] introduced a
framework to parallelize various machine learning algorithms in a multi-core environment. [14] applied the ADMM technique into distributed learning. [15] proposed a delayed version of distributed
online learning. General distributed learning techniques closer to our work are the approaches based
on parallel gradient calculation followed by a centralized algorithm ([7][16][17]). Different from our
work, theirs built on fully connected environment such as MPI while we focus on the map-reduce
environment with loose connection. Their centralized algorithm is often the bottleneck of the whole
procedure and limits the scalability of the algorithm. For example, [17] clearly stated that it is impractical for their L-BFGS algorithm to run their large dataset due to huge memory consumption in
the centralized algorithm although L-BFGS has been shown to be an excellent candidate for their
problem. Moreover, the closest to our work lies in applying L-BFGS in the map-reduce-like environment, such as [18][2]. They are solving large-scale problems in a map-reduce adapted environment
using L-BFGS. [18] run L-BFGS on a map-reduce plus AllReduce environment to demonstrate the
power of large-scale learning with map-reduce. Although it has been shown to scale up to billion
of data instances with trillion entries in their data matrix, the number of variables in their problem
is only about 16 million due to the constraints in centralized computation of L-BFGS direction. [2]
used L-BFGS to solve the deep learning problem. It introduced the parameter servers to split a
global model into multiple partitions and store each partition separately. Despite their successes,
from the algorithmic point of view, their two-loop recursion update procedure is still highly dependent on the number of variable. Compared with these work, our proposed two-loop recursion
updating procedure is independent on the number of variables and with much better parallelism.
Furthermore, the proposed algorithm can run on pure map-reduce environment while previous work
[2] and [18] require special components such as AllReduce or parameter servers. In addition, it is
straightforward for previous work, such as [2][18][17], to leverage our proposal to scale up their
problem into another order of magnitude in terms of number of variables.
2
3
L-BFGS Algorithm
Given an optimization problem with d variables, BFGS requires to store a dense d by d matrix to
approximate the inverse Hessian, where L-BFGS only need to store a few vectors of length d to
approximate the Hessian implicitly. Let us denote f as the objective function, g as the gradient and
? as the dot product between two vectors. L-BFGS maintains the historical states of previous m
(generally m = 10) updates of current position x and its gradient g = ?f (x).
In L-BFGS algorithm, the historical states are represented as the last m updates of form sk =
xk+1 ? xk and yk = gk+1 ? gk where sk represents the position difference and yk represents the
gradient difference in iteration k. Each of them is a vector of length d. All of these 2m vector
with the original gradient gk will be used to calculate a new direction in line 3 of Algorithm 1.
Algorithm 1: L-BFGS Algorithm Outline
Input: starting point x0 , integer history size m > 0, k=1;
Output: the position x with a minimal objective function
1 while no converge do
2
Calculate gradient ?f (xk ) at position xk ;
3
Compute direction pk using Algorithm 2 ;
4
Compute xk+1 = xk + ?k pk where ?k is chosen to satisfy Wolfe conditions;
5
if k > m then
6
Discard vector pair sk?m , yk?m from memory storage;;
7
end
8
Update sk = xk+1 ? xk , yk = ?f (xk+1 ) ? ?f (xk ), k = k + 1 ;
9 end
Algorithm 2: L-BFGS two-loop recursion
Input: ?f (xk ), si , yi where i = k ? m, ..., k ? 1
Output: new direction p
1 p = ??f (xk ) ;
2 for i ? k ? 1 to k ? m do
3
?i ? ssii?y?pi ;
4
p = p ? ?i ? yi ;
5 end
s
?y
6 p = ( yk?1 ?yk?1 )p
k?1 k?1
7 for i ? k ? m to k ? 1 do
8
? = syii?y?pi ;
9
p = p + (?i ? ?) ? si ;
10 end
The core update procedure in Algorithm 1 is the line 3 to calculate a new direction pk using s and
y with current gradient ?f (xk ). The most common approach for this calculation is the two-loop
recursion in Algorithm 2[5][6]. It initializes the direction p with gradient and continues to update it
using historical states y and s. More information about two-loop recursion could be found from [5].
4
A Map-Reduce Implementation
The main procedure in Algorithm 1 lies in Line 2, 3 and 4. The calculation of gradient in Line 2
can be straightforwardly parallelized by dividing the data into multiple partitions. In the map-reduce
environment, we can use one map step to calculate the partial gradient for partial data and one reduce
to aggregate them into a global gradient vector. The verification of the Wolfe condition only depends
on the calculation of the objective function following the line search procedure[5]. So thus Line 4
can also be easily parallelized following the same approach as Line 2. Therefore, the challenge in
the L-BFGS algorithm is Line 3. In other words,the difficulties come from the calculation of the
two-loop recursion, as shown in Algorithm 2.
3
4.1
Centralized Update
The simplest implementation for Algorithm 2 may be to run it in a single processor. We can easily
perform this in a singleton reduce. However, the challenge is that Algorithm 2 requires 2m + 1
vectors and each of them has a length of d. This could be feasible when d is in million scale. Nevertheless, when d is in billion scale, either the storage or the computation cost becomes a significant
challenge and makes it impractical to implement it in map-reduce. Given the Ads CTR prediction
task [1] as an example, there are more than 1 billion of features. If we set m = 10 in a linear model,
it will produce 21 ? 1 = 21 billion variables. Even if we compactly use a single-precision floating
point to represent a variable, it requires 84 GB memory to store the historical states and gradient.
For a map-reduce cluster built from commodity hardware and shared with other applications, this is
generally unfeasible nowadays. For example, for the cluster into which we deployed the L-BFGS,
its maximal memory limitation for a map-reduce step is 6 GB.
4.2
Distributed Update
Due to the storage limitation in centralized update, an alternative is to store s and y into multiple
partitions without overlap and use a map-reduce step to calculate every dot product, such as si ?p and
si ? yi in Line 3 of Algorithm 2. Yet, if each dot product within the for-loop in Algorithm 2 requires a
map-reduce step to perform the calculation, this will result in at least 2m map-reduce steps in a twoloop recursion. If we call Algorithm 2 for N times(iterations) in Algorithm 1, it will lead to 2mN
map-reduce steps. For example, if m = 10 and N = 100, this will produce 2000 map-reduce steps
in a map-reduce job. Unfortunately, each map-reduce step will bring significant overhead due to the
scheduling cost and application launching cost. For a job with thousands of map-reduce steps, both
these cost often dominate the overall running time and make the useful computational time spent
in algorithmic vector operations negligible. Moreover, given our current production cluster as an
example, a job with such a huge number of map-reduce step is too large for execution. It will trigger
a compilation timeout error before becoming too complicated for an execution engine to execute it.
5
Vector-free L-BFGS
For the reasons mentioned, a feasible two-loop recursion procedure has to limit both the memory
consumption and the number of map-reduce steps per iteration. To strictly limit the memory consumption in Algorithm 2, we can not store the 2m + 1 vectors with length d in memory unless d is
only up to million scale. To comply with the allowable map-reduce steps per iteration, it is neither
practical to perform map-reduce steps within the for-loop in Algorithm 2. Both of these assumptions
motivate us to carefully re-examine Algorithm 2 and lead to the proposed algorithm in this section.
5.1
Basic Idea
Before illustrating the new procedure, let us describe following three observations in Algorithm 2
that guide the design of the new procedure in Algorithm 3:
1. All inputs are invariable during Algorithm 2.
2. All operations applied on p are linear with respect to the inputs. In other words, p could be
formalized as a linear combination of the inputs although its coefficients are unknown.
3. The core numeric operation is the dot product between two vectors.
Observation 1 and 2 motivate us to formalize the inputs as (2m + 1) invariable base vectors.
b1 = sk?m , b2 = sk?m+1 , ..., bm = sk?1
(1)
bm+1 = yk?m , bm+2 = yk?m+1 , ..., b2m = yk?1
(2)
b2m+1 = ?f (xi )
(3)
So thus we can represent p as a linear combination of bi . Assume ? as the scalar coefficients in this
linear combination, we can write p as:
2m+1
X
p=
?k bk
(4)
k=1
4
Since bk are the inputs and invariants during the two-loop recursion, if we can calculate the coefficients ?k , we can proceed to calculate the direction p.
Following observation 3 with an re-examination of Algorithm 2, we classify the dot product operations into two categories in terms of whether p is involved in the calculation. For the first category
only involving the dot product between the inputs (si , yi ), a straightforward intuition is to precompute their dot products to produce a scalar, so as to replace each dot product with a scalar in
the two-loop recursion. However, the second category of dot products involving p can not follow
this same procedure. Because the direction p is ever-changing during the for loop, any dot products
involving p can not be settled or pre-computed. Fortunately, thanks to the linear decomposition of
p in observation 2 and Eqn.4, we can decompose any dot product involving p into a summation of
dot products with its based vectors and corresponding coefficients. This new elegant mathematical
procedure only happens after we formalize p as the linear combination of the base vectors.
5.2
The VL-BFGS Algorithm
We present the algorithmic procedure in Algorithm 3. Let us denote the results of dot products
between every two base vectors as a scalar matrix of (2m + 1) ? (2m + 1) scalars. The proposed
VL-BFGS algorithm only takes it as the input. Similar as the original L-BFGS algorithm, it has a
two-loop recursion, but all the operations are only dependent on scalar operations. In Line 1-2, it
assigns the initial values for ?i . This is equivalent to Line 1 in Algorithm 2 to use opposite direction
of gradient as the initial direction. The original calculation of ?i in Line 6 relies on the direction
vector p. It is worth noting that p is variable within the first loop in which ? is updated. So thus we
can not pre-compute any dot product involving p. However, as mentioned earlier and according to
observation 2 and Eqn.4, we can formalize bj ? p as a summation from a list of dot products between
base vectors and corresponding coefficients, as shown in Line 6 of Algorithm 3. Meanwhile, since all
base vectors are invariable, their dot products can be pre-computed and replaced with scalars,which
then multiply the ever-changing ?l . But these are only scalar operations and they are extremely
efficient. Line 7 continues to update scalar coefficient ?m+j , which is equivalent to update the
direction p with respect to the base vector bm+j or corresponding yj . This whole procedure is the
same when we apply it to Line 14 and 15. With the new formalization of p in Eqn.4 and the
Algorithm 3: Vector-free L-BFGS two-loop recursion
Input: (2m + 1) ? (2m + 1) dot product matrix between bi
Output: The coefficients ?i where i = 1, 2, ...2m + 1
1 for i ? 1 to 2m + 1 do
2
?i = i ? 2m ? 0 : ?1
3 end
4 for i = k ? 1 to k ? m do
5
j = i ? (k ? m) + 1 ;
6
7
8
9
10
11
12
13
b ?p
?i ? ssii?y?pi = bj ?bjm+j =
?m+j = ?m+j ? ?i ;
end
for i ? 1 to 2m + 1 do
m ?b2m
?i = ( bb2m
?b2m )?i
end
for i ? k ? m to k ? 1 do
j = i ? (k ? m) + 1 ;
b
?p
P2m+1
P2m+1
? b
?l bl ?bj
bj ?bm+j
l=1
l m+j
l=1
? = bjm+j
?bm+j =
bj ?bm+j
15
?j = ?j + (?i ? ?)
16 end
14
?bl
;
;
invariability of yi and si during Algorithm 2, Line 4 in Algorithm 2 updating with yi (equivalent to
bm+j ) is mathematically equivalent to Line 7 in Algorithm 3, so as Line 9 in Algorithm 2 and Line
15 in Algorithm 3. For other lines between these two algorithms, it is easy to infer their equivalence
with the consideration of Eqn.1-4. Thus, Algorithm 3 is mathematically equivalent to Algorithm 2.
5
5.3
Complexity Analysis and Comparison
Using the dot product matrix of scalars as the input, the calculation in Algorithm 3 is substantially
efficient, since all the calculation is based on scalars. Altogether, it only requires 8m2 multiplications
between scalars in the two for-loops. This is tiny compared to any vector operation involving billionscale of variables. Thus, it is not necessary to parallelize Algorithm 3 in implementation.
To integrate Algorithm 3 as the core step in Algorithm 1, there are two extra steps we need to
perform before and after it. One is to calculate the dot product matrix between the (2m + 1) base
vectors. Because all base vectors have the same dimension d, we can partition them using the
same way and use one map-reduce step to calculate the dot product matrix. This computation is
greatly parallelizable and intrinsically suitable for map-reduce. Even without the consideration of
parallization, a first glance tells us it may require about 4m2 dot products. However, since all the
si and yi except the first ones are unchanged in a new iteration, we can save the tiny dot product
matrix and reuse most entries across iterations. With the consideration of the commutative law of
multiplication since si ? yj ? yj ? si , each new iteration only need to calculate 6m new dot products
which involve new sk , yk and gk . Thus, the complexity is only 6md and this calculation is fully
parallel in map-reduce, with each partition only calculating a small portion of 6md multiplications.
The other and the final step is to calculate the new direction p based on ?i and the base vectors. The
complexity is another 2md multiplications, which means the overall complexity of the algorithm
is 8md multiplications. Since the overall ? is just a tiny vector with 2m + 1 dimensions, we can
join it with all the other base vectors, and then use the same approach as dot product calculation to
produce the final direction p using Eqn.4. A single map-reduce step is sufficient for this final step.
Altogether, without considering the gradient calculation which is same to all algorithms, VL-BFGS
only require 3 map-reduce steps for one iteration in the update.
For the centralized update approach in section 4.1, it also requires 6md multiplications in each
two loop recursion. In addition to being a centralized approach, as we analyzed above, it requires
(2m + 1) ? d memory storage. This clearly limits its applications to large-scale problems. On the
other hand, VL-BFGS in Algorithm 3 only requires (2m+1)2 memory storage and is independent on
d. For the distributed approach in section 4.2, it requires at least 2m map-reduce step in a two-loop
recursion. Given the number of iteration as N (generally N > 100), the total number of map-reduce
steps is 2mN . Fortunately, the VL-BFGS only requires 3N map-reduce steps. In summary, VLBFGS algorithm enjoys a similar overall complexity but it is born with massive degree of parallelism.
For problem with billion scale of variables, it is the only map-reduce friendly implementation of the
three different approaches.
6
Experiment and Discussion
As demonstrated above, it is clear that VL-BFGS has a better scalability property than original LBFGS. Although it is always desirable to invent an exact algorithm that could be mathematically
proved to obtain a better scalability property, it is beneficial to demonstrate the value of larger
number of variables with an industrial application. On the other hand, for a problem with billions
of variables, there are existing practical approaches to reduce it into a smaller number of variables
and then solve it with traditional approaches designed for centralized algorithm. In this section, we
justify the value of learning large scale variables and simultaneously compare it with the hashing
approach, and finally demonstrate the scalability advantage of VL-BFGS.
6.1
Dataset and Experimental Setting
The dataset we used is from an Ads Click-through Rate (CTR) prediction problem [1] collected from
an industrial search engine. The click event (click or not) is used as the label for each instance. The
features include the terms from a query and an Ad keyword along with the contextual information
such as Ad position, session-related information and time. We collect 30 days of data and split them
into training and test set chronologically. The data from the first 20 days are used as the training
set and rest 10 days are used as test set. The total training data have about 12 billions instances and
another 6 billion in testing data. There are 1,038,934,683 features the number of non-zero features
per instance is about 100 on average. Altogether it has about 2 trillion entries in the data matrix.
6
Table 1: Relative AUC Performance over different number of variables
K
Relative AUC Performance
Baseline(K=1,038,934,683) 0.0%
K=250 millions
-0.1007388%
K=100 millions
-0.1902843%
K= 10 millions
-0.3134094%
K= 1 millions
-0.5701142%
Table 2: Relative AUC Performance over different number of Hash bits
K
Relative AUC Performance
Baseline(K=1,038,934,683) 0.0%
K=64 millions(26 bits)
-0.1063033%
K=16 millions(24 bits)
-0.2323647%
K= 4 millions(22 bits)
-0.3300788%
K= 1 millions(20 bits)
-0.5080904%
We run logistic regression training, so thus each feature corresponds to a variable. The model is
evaluated based on the testing data using Area Under ROC Curve [19], denoted as AUC. We set
the historical state length m = 10 and enforce L1[20] regularizer to avoid overfitting and achieve
sparsity. The regularizer parameter is tuned following the approach in [18].
We run the experiment in a shared cluster with tens of thousands of machines. Each machine has up
to 12 concurrent vertices. A vertex is generally a map or reduce step with an allocation of 2 cores
and 6G memory. There are more than 1000 different jobs running simultaneously but this number
also varies significantly. We split the training data into 400 partitions and allocate 400 tokens for this
job, which means this job can use up to 400 vertices at the same time. When we partition vectors to
calculate their dot products, our strategy is to allocate up to 5 million entries in a partial vector. For
example, 1 billion variables will be split into 200 partitions evenly.
We use the model trained with original 1 billion features as the baseline. All the other experiments
are compared with it. Since we are not allowed to exhibit the exact AUC number due to privacy
consideration, we report the relative change compared with the baseline. The scale of the dataset
makes any relative AUC change over 0.001% produce a p-value less than 0.01.
6.2
Value of Large Number of Variables
To reduce the number of variables in the original problem, we sort the features based on their frequency in the training data. If we plan to reduce the problem to K variables, we keep the top K
frequent features. The baseline without filtering is equivalent to K = 1, 038, 934, 683. We choose
different K values and report the relative AUC number in Table 1.
The table shows that while we reduce the number of variables, the results consistently decline significantly. When the number of variables is 1 million, the drop is more than 0.5% . This is considerably
significant for the problem. Even when we increase the number of variable up to 250 million, the
decline is still obvious and significant. This demonstrates that the large number of variables is really
needed to learn a good model and the value of learning with billion-scale of variables.
6.3
Comparison with Hashing
We follow the approach in [21][18] to calculate a new hash value for each original feature value
based on a hash function in [18]. The number of hash bits ranges from 20 to 26. Experimental
results compared with the baseline in terms of relative AUC performance are presented in Table 2
Consistently with previous results, all the hashing experiments result in degradation. For the experiment with 20 bits, the degradation is 0.5%. This is a substantial decline for this problem. When we
increase the number of bits till 26, the gap becomes smaller but still noticeable. All of these consis7
tently demonstrate that the hashing approach will sacrifice noticeable performance. It is beneficial
to train with large-scale number of raw features.
6.4
Training Time Comparison
We compare the L-BFGS in section 4.1 with the proposed VL-BFGS. To enable a larger number of
variable support for L-BFGS, we reduce the m parameter into 3. We conduct the experiments with
varying number of feature number and report their corresponding running time. We use the original
data after hashing into 1M features as the baseline and compare all the other experiments with it and
report the relative training time for same number of iterations. We run each experiment 5 times and
report their mean to cope with the variance in each run. The results with respect to different hash bits
range from 20 to 29 and the original 1B features are shown in figure 1. When the number of features
is less than 10M, the original L-BFGS has a small advantage over VL-BFGS. However, when we
continue to increase the feature number, the running time of L-BFGS grows quickly while that of
VL-BFGS increases slowly. On the other hand, when we increase the feature number to 512M, the
L-BFGS fails with an out-of-memory exception, while VL-BFGS can easily scale to 1B features.All
of these clearly show the scalability advantage of VL-BFGS over traditional L-BFGS.
Figure 1: Training time over feature number.
7
Conclusion
We have presented a new vector-free exact L-BFGS updating procedure called VL-BFGS. As opposed to original L-BFGS algorithm in map-reduce, the core two-loop recursion in VL-BFGS is
independent on the number of variables. This enables it to be easily parallelized in map-reduce
and scale up to billions of variables. We present its mathematical equivalence to original L-BFGS,
show its scalability advantage over traditional L-BFGS in map-reduce with a great degree of parallelism, and perform experiments to demonstrate the value of large-scale learning with billions of
variables using VL-BFGS. Although we emphasis the implementation on map-reduce in this paper,
VL-BFGS can be straightforwardly utilized by other distributed frameworks to avoid their centralized problem and scale up their algorithms. In short, VL-BFGS is highly beneficial for machine
learning algorithms relying on L-BFGS to scale up to another order of magnitude.
8
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9
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4,787 | 5,334 | Recovery of Coherent Data via Low-Rank
Dictionary Pursuit
Ping Li
Department of Statistics and Biostatistics
Department of Computer Science
Rutgers University
Piscataway, NJ 08854, USA
[email protected]
Guangcan Liu
Department of Statistics and Biostatistics
Department of Computer Science
Rutgers University
Piscataway, NJ 08854, USA
[email protected]
Abstract
The recently established RPCA [4] method provides a convenient way to restore
low-rank matrices from grossly corrupted observations. While elegant in theory
and powerful in reality, RPCA is not an ultimate solution to the low-rank matrix recovery problem. Indeed, its performance may not be perfect even when
data are strictly low-rank. This is because RPCA ignores clustering structures of
the data which are ubiquitous in applications. As the number of cluster grows,
the coherence of data keeps increasing, and accordingly, the recovery performance of RPCA degrades. We show that the challenges raised by coherent data
(i.e., data with high coherence) could be alleviated by Low-Rank Representation
(LRR) [13], provided that the dictionary in LRR is configured appropriately. More
precisely, we mathematically prove that if the dictionary itself is low-rank then
LRR is immune to the coherence parameter which increases with the underlying
cluster number. This provides an elementary principle for dealing with coherent
data and naturally leads to a practical algorithm for obtaining proper dictionaries
in unsupervised environments. Experiments on randomly generated matrices and
real motion sequences verify our claims. See the full paper at arXiv:1404.4032.
1 Introduction
Nowadays our data are often high-dimensional, massive and full of gross errors (e.g., corruptions,
outliers and missing measurements). In the presence of gross errors, the classical Principal Component Analysis (PCA) method, which is probably the most widely used tool for data analysis and
dimensionality reduction, becomes brittle ? A single gross error could render the estimate produced
by PCA arbitrarily far from the desired estimate. As a consequence, it is crucial to develop new statistical tools for robustifying PCA. A variety of methods have been proposed and explored in the
literature over several decades, e.g., [2, 3, 4, 8, 9, 10, 11, 12, 24, 13, 16, 19, 25]. One of the most exciting methods is probably the so-called RPCA (Robust Principal Component Analysis) method [4],
which was built upon the exploration of the following low-rank matrix recovery problem:
Problem 1 (Low-Rank Matrix Recovery) Suppose we have a data matrix X ? Rm?n and we
know it can be decomposed as
X = L 0 + S0 ,
m?n
(1.1)
where L0 ? R
is a low-rank matrix each column of which is a data point drawn from some
low-dimensional subspace, and S0 ? Rm?n is a sparse matrix supported on ? ? {1, ? ? ? , m} ?
{1, ? ? ? , n}. Except these mild restrictions, both components are arbitrary. The rank of L0 is unknown, the support set ? (i.e., the locations of the nonzero entries of S0 ) and its cardinality (i.e.,
the amount of the nonzero entries of S0 ) are unknown either. In particular, the magnitudes of the
nonzero entries in S0 may be arbitrarily large. Given X, can we recover both L0 and S0 , in a
scalable and exact fashion?
1
cluster 1
cluster 2
Figure 1: Exemplifying the extra structures of low-rank data. Each entry of the data matrix is a grade
that a user assigns to a movie. It is often the case that such data are low-rank, as there exist wide
correlations among the grades that different users assign to the same movie. Also, such data could
own some clustering structure, since the preferences of the same type of users are more similar to
each other than to those with different gender, personality, culture and education background. In
summary, such data (1) are often low-rank and (2) exhibit some clustering structure.
The theory of RPCA tells us that, very generally, when the low-rank matrix L0 is meanwhile incoherent (i.e., with low coherence), both the low-rank and the sparse matrices can be exactly recovered
by using the following convex, potentially scalable program:
min kLk? + ?kSk1 , s.t.
L,S
X = L + S,
(1.2)
where k ? k? is the nuclear norm [7] of a matrix, k ? k1 denotes the ?1 norm of a matrix seen as
a long vector, and ? > 0 is a parameter. Besides of its elegance in theory, RPCA also has good
empirical performance in many practical areas, e.g., image processing [26], computer vision [18],
radar imaging [1], magnetic resonance imaging [17], etc.
While complete in theory and powerful in reality, RPCA cannot be an ultimate solution to the lowrank matrix recovery Problem 1. Indeed, the method might not produce perfect recovery even when
L0 is strictly low-rank. This is because RPCA captures only the low-rankness property, which
however is not the only property of our data, but essentially ignores the extra structures (beyond
low-rankness) widely existing in data: Given the low-rankness constraint that the data points (i.e.,
columns vectors of L0 ) locate on a low-dimensional subspace, it is unnecessary for the data points
to locate on the subspace uniformly at random and it is quite normal that the data may have some
extra structures, which specify in more detail how the data points locate on the subspace. Figure 1
demonstrates a typical example of extra structures; that is, the clustering structures which are ubiquitous in modern applications. Whenever the data are exhibiting some clustering structures, RPCA
is no longer a method of perfection. Because, as will be shown in this paper, while the rank of L0 is
fixed and the underlying cluster number goes large, the coherence of L0 keeps heightening and thus,
arguably, the performance of RPCA drops.
To better handle coherent data (i.e., the cases where L0 has large coherence parameters), a seemingly straightforward idea is to avoid the coherence parameters of L0 . However, as explained in [4],
the coherence parameters are indeed necessary (if there is no additional condition assumed on the
data). This paper shall further indicate that the coherence parameters are related in nature to some
extra structures intrinsically existing in L0 and therefore cannot be discarded simply. Interestingly,
we show that it is possible to avoid the coherence parameters by using some additional conditions,
which are easy to obey in supervised environment and can also be approximately achieved in unsupervised environment. Our study is based on the following convex program termed Low-Rank
Representation (LRR) [13]:
min kZk? + ?kSk1 , s.t.
Z,S
X = AZ + S,
(1.3)
where A ? Rm?d is a size-d dictionary matrix constructed in advance1, and ? > 0 is a parameter. In
order for LRR to avoid the coherence parameters which increase with the cluster number underlying
1
It is not crucial to determine the exact value of d. Suppose Z ? is the optimal solution with respect to Z.
Then LRR uses AZ ? to restore L0 . LRR falls back to RPCA when A = I (identity matrix). Furthermore, it can
be proved that the recovery produced by LRR is the same as RPCA whenever the dictionary A is orthogonal.
2
L0 , we prove that it is sufficient to construct in advance a dictionary A which is low-rank by itself.
This gives a generic prescription to defend the possible infections raised by coherent data, providing
an elementary criteria for learning the dictionary matrix A. Subsequently, we propose a simple and
effective algorithm that utilizes the output of RPCA to construct the dictionary in LRR. Our extensive experiments demonstrated on randomly generated matrices and motion data show promising
results. In summary, the contributions of this paper include the following:
? For the first time, this paper studies the problem of recovering low-rank, and coherent (or
less incoherent as equal) matrices from their corrupted versions. We investigate the physical
regime where coherent data arise. For example, the widely existing clustering structures
may lead to coherent data. We prove some basic theories for resolving the problem, and
also establish a practical algorithm that outperforms RPCA in our experimental study.
? Our studies help reveal the physical meaning of coherence, which is now standard and
widely used in various literatures, e.g., [2, 3, 4, 25, 15]. We show that the coherence
parameters are not ?assumptions? for a proof, but rather some excellent quantities that
relate in nature to the extra structures (beyond low-rankness) intrinsically existing in L0 .
? This paper provides insights regarding the LRR model proposed by [13]. While the special
case of A = X has been extensively studied, the LRR model (1.3) with general dictionaries
is not fully understood yet. We show that LRR (1.3) equipped with proper dictionaries
could well handle coherent data.
? The idea of replacing L with AZ is essentially related to the spirit of matrix factorization
which has been explored for long, e.g., [20, 23]. In that sense, the explorations of this paper
help to understand why factorization techniques are useful.
2 Summary of Main Notations
Capital letters such as M are used to represent matrices, and accordingly, [M ]ij denotes its (i, j)th
entry. Letters U , V , ? and their variants (complements, subscripts, etc.) are reserved for left singular
vectors, right singular vectors and support set, respectively. We shall abuse the notation U (resp. V )
to denote the linear space spanned by the columns of U (resp. V ), i.e., the column space (resp. row
space). The projection onto the column space U , is denoted by PU and given by PU (M ) = U U T M ,
and similarly for the row space PV (M ) = M V V T . We shall also abuse the notation ? to denote
the linear space of matrices supported on ?. Then P? and P?? respectively denote the projections
onto ? and ?c such that P? + P?? = I, where I is the identity operator. The symbol (?)+ denotes
T
the Moore-Penrose pseudoinverse of a matrix: M + = VM ??1
M UM for a matrix M with Singular
2
T
Value Decomposition (SVD) UM ?M VM .
Six different matrix norms are used in this paper. The first three norms are functions of the singular
values: 1) The operator norm (i.e., the largest singular value) denoted by kM k, 2) the Frobenius
norm (i.e., square root of the sum of squared singular values) denoted by kM kF , and 3) the nuclear
norm (i.e., the sum of singular values) denoted byP
kM k? . The other three are the ?1 , ?? (i.e.,
sup-norm) and ?2,? norms of a matrix: kM k1 =
i,j |[M ]ij |, kM k? = maxi,j {|[M ]ij |} and
qP
2
kM k2,? = maxj {
i [M ]ij }, respectively.
The Greek letter ? and its variants (e.g., subscripts and superscripts) are reserved for the coherence
parameters of a matrix. We shall also reserve two lower case letters, m and n, to respectively denote
the data dimension and the number of data points, and we use the following two symbols throughout
this paper:
n1 = max(m, n)
and n2 = min(m, n).
3 On the Recovery of Coherent Data
In this section, we shall firstly investigate the physical regime that raises coherent (or less incoherent) data, and then discuss the problem of recovering coherent data from corrupted observations,
providing some basic principles and an algorithm for resolving the problem.
2
In this paper, SVD always refers to skinny SVD. For a rank-r matrix M ? Rm?n , its SVD is of the form
T
UM ?M VM
, with UM ? Rm?r , ?M ? Rr?r and VM ? Rn?r .
3
3.1 Coherence Parameters and Their Properties
As the rank function cannot fully capture all characteristics of L0 , it is necessary to define some
quantities to measure the effects of various extra structures (beyond low-rankness) such as the clustering structure as demonstrated in Figure 1. The coherence parameters defined in [3, 4] are excellent
exemplars of such quantities.
3.1.1 Coherence Parameters: ?1 , ?2 , ?3
For an m ? n matrix L0 with rank r0 and SVD L0 = U0 ?0 V0T , some important properties can
be characterized by three coherence parameters, denoted as ?1 , ?2 and ?3 , respectively. The first
coherence parameter, 1 ? ?1 (L0 ) ? m, which characterizes the column space identified by U0 , is
defined as
m
?1 (L0 ) =
max kU T ei k22 ,
(3.4)
r0 1?i?m 0
where ei denotes the ith standard basis. The second coherence parameter, 1 ? ?2 (L0 ) ? n, which
characterizes the row space identified by V0 , is defined as
?2 (L0 ) =
n
max kV T ej k22 .
r0 1?j?n 0
(3.5)
The third coherence parameter, 1 ? ?3 (L0 ) ? mn, which characterizes the joint space identified
by U0 V0T , is defined as
?3 (L0 ) =
mn
mn
(kU0 V0T k? )2 =
max(|hU0T ei , V0T ej i|)2 .
r0
r0 i,j
(3.6)
The analysis in RPCA [4] merges the above three parameters into a single one: ?(L0 ) =
max{?1 (L0 ), ?2 (L0 ), ?3 (L0 )}. As will be seen later, the behaviors of those three coherence parameters are different from each other, and hence it is more adequate to consider them individually.
3.1.2 ?2 -phenomenon
According to the analysis in [4], the success condition (regarding L0 ) of RPCA is
rank (L0 ) ?
cr n 2
,
?(L0 )(log n1 )2
(3.7)
where ?(L0 ) = max{?1 (L0 ), ?2 (L0 ), ?3 (L0 )} and cr > 0 is some numerical constant. Thus,
RPCA will be less successful when the coherence parameters are considerably larger. In this subsection, we shall show that the widely existing clustering structure can enlarge the coherence parameters
and, accordingly, downgrades the performance of RPCA.
Given the restriction that rank (L0 ) = r0 , the data points (i.e., column vectors of L0 ) are unnecessarily sampled from a r0 -dimensional subspace uniformly at random. A more realistic interpretation
is to consider the data points as samples from the union of k number of subspaces (i.e., clusters),
and the sum of those multiple subspaces together has a dimension r0 . That is to say, there are
multiple ?small? subspaces inside one r0 -dimensional ?large? subspace, as exemplified in Figure 1.
Whenever the low-rank matrix L0 is meanwhile exhibiting such clustering behaviors, the second
coherence parameter ?2 (L0 ) (and so ?3 (L0 )) will increase with the number of clusters underlying
L0 , as shown in Figure 2. When the coherence is heightening, (3.7) suggests that the performance
of RPCA will drop, as verified in Figure 2(d). Note here that the variation of ?3 is mainly due
to the variation of the row space, which is characterized by ?2 . We call the phenomena shown in
Figure 2(b)?(d) as the ??2 -phenomenon?. Readers can also refer to the full paper to see why the
second coherence parameter increases with the cluster number underlying L0 .
Interestingly, one may have noticed that ?1 is invariant to the variation of the clustering number, as
can be seen from Figure 2(a). This is because the clustering behavior of the data points can only
affect the row space, while ?1 is defined on the column space. Yet, if the row vectors of L0 also
own some clustering structure, ?1 could be large as well. Such kind of data can exist widely in text
documents and we leave this as future work.
4
(a)
?3
?
2
?1
40
2
20
1
1
2
5 10 20 50
#cluster
0
0.3
recover error
3
0.5
(d)
60
4
1
0
(c)
(b)
1.5
1
2
0
5 10 20 50
1
2
5 10 20 50
#cluster
#cluster
0.2
0.1
0
1
2
5 10 20 50
#cluster
Figure 2: Exploring the influence of the cluster number, using randomly generated matrices. The
size and rank of L0 are fixed to be 500 ? 500 and 100, respectively. The underlying cluster number
varies from 1 to 50. For the recovery experiments, S0 is fixed as a sparse matrix with 13% nonzero
entries. (a) The first coherence parameter ?1 (L0 ) vs cluster number. (b) ?2 (L0 ) vs cluster number.
(c) ?3 (L0 ) vs cluster number. (d) Recover error (produced by RPCA) vs cluster number. The
numbers shown in these figure are averaged from 100 random trials. The recover error is computed
? 0 ? L0 kF /kL0 kF , where L
? 0 denotes an estimate of L0 .
as kL
3.2 Avoiding ?2 by LRR
The ?2 -phenomenon implies that the second coherence parameter ?2 is related in nature to some
intrinsic structures of L0 and thus cannot be eschewed without using additional conditions. In the
following, we shall figure out under what conditions the second coherence parameter ?2 (and ?3 )
can be avoided such that LRR could well handle coherent data.
Main Result: We show that, when the dictionary A itself is low-rank, LRR is able to avoid ?2 .
Namely, the following theorem is proved without using ?2 . See the full paper for a detailed proof.
Theorem 1 (Noiseless) Let A ? Rm?d with SVD A = UA ?A VAT be a column-wisely unit-normed
(i.e., kAei k2 = 1, ?i) dictionary matrix which satisfies PUA (U0 ) = U0 (i.e., U0 is a subspace of
UA ). For any 0 < ? < 0.5 and some numerical constant ca > 1, if
rank (L0 ) ? rank (A) ?
? 2 n2
ca ?1 (A) log n1
and |?| ? (0.5 ? ?)mn,
(3.8)
?10
then
? with probability at least 1 ? n1 , the optimal solution to the LRR problem (1.3) with ? =
1/ n1 is unique and exact, in a sense that
Z ? = A+ L0
and S ? = S0 ,
where (Z ? , S ? ) is the optimal solution to (1.3).
It is worth noting that the restriction rank (L0 ) ? O(n2 / log n1 ) is looser than that of PRCA3 , which
requires rank (L0 ) ? O(n2 /(log n1 )2 ). The requirement of column-wisely unit-normed
? dictionary
(i.e., kAei k2 = 1, ?i) is purely for complying the parameter estimate of ? = 1/ n1 , which is
consistent with RPCA. The condition PUA (U0 ) = U0 , i.e., U0 is a subspace of UA , is indispensable
if we ask for exact recovery, because PUA (U0 ) = U0 is implied by the equality AZ ? = L0 . This
necessary condition, together with the low-rankness condition, provides an elementary criterion for
learning the dictionary matrix A in LRR. Figure 3 presents an example, which further confirms our
main result; that is, LRR is able to avoid ?2 as long as U0 ? UA and A is low-rank. It is also
worth noting that it is unnecessary for A to satisfy UA = U0 , and that LRR is actually tolerant to the
?errors? possibly existing in the dictionary.
The program (1.3) is designed for the case where the uncorrupted observations are noiseless. In
reality this assumption is often not true and all entries of X can be contaminated by a small amount
of noises, i.e., X = L0 + S0 + N , where N is a matrix of dense Gaussian noises. In this case, the
formula of LRR (1.3) need be modified to
min kZk? + ?kSk1 , s.t.
Z,S
3
kX ? AZ ? SkF ? ?,
(3.9)
In terms of exact recovery, O(n2 / log n1 ) is probably the ?finest? bound one could accomplish in theory.
5
AZ*
*
S
0.2
recover error
X
0.1
0
1 5
10 15 20
rank(A)
Figure 3: Exemplifying that LRR can void ?2 . In this experiment, L0 is a 200 ? 200 rank-1 matrix
with one column being 1 (i.e., a vector of all ones) and everything else being zero. Thus, ?1 (L0 ) = 1
and ?2 (L0 ) = 200. The dictionary is set as A = [1, W ], where W is a 200 ? p random Gaussian
matrix (with varying p). As long as rank (A) = p + 1 ? 10, LRR with ? = 0.08 can exactly recover
L0 from a grossly corrupted observation matrix X.
where ? is a parameter that measures the noise level of data. In the experiments of this paper,
we consistently set ? = 10?6 kXkF . In the presence of dense noises, the latent matrices, L0 and
S0 , cannot be exactly restored. Yet we have the following theorem to guarantee the near recovery
property of the solution produced by the program (3.9):
Theorem 2 (Noisy) Suppose kX ? L0 ? S0 kF ? ?. Let A ? Rm?d with SVD A = UA ?A VAT be a
column-wisely unit-normed dictionary matrix which satisfies PUA (U0 ) = U0 (i.e., U0 is a subspace
of UA ). For any 0 < ? < 0.35 and some numerical constant ca > 1, if
rank (L0 ) ? rank (A) ?
? 2 n2
ca ?1 (A) log n1
and |?| ? (0.35 ? ?)mn,
(3.10)
?
then with probability at least 1 ? n?10
, any solution (Z ??
, S ? ) to (3.9) with ? = 1/ n1?gives a near
1
recovery to (L0 , S0 ), in a sense that kAZ ? ? L0 kF ? 8 mn? and kS ? ? S0 kF ? (8 mn + 2)?.
3.3 An Unsupervised Algorithm for Matrix Recovery
To handle coherent (equivalently, less incoherent) data, Theorem 1 suggests that the dictionary matrix A should be low-rank and satisfy U0 ? UA . In certain supervised environment, this might not be
difficult as one could potentially use clear, well processed training data to construct the dictionary. In
an unsupervised environment, however, it will be challenging to identify a low-rank dictionary that
obeys U0 ? UA . Note that U0 ? UA can be viewed as supervision information (if A is low-rank).
In this paper, we will introduce a heuristic algorithm that can work distinctly better than RPCA in
an unsupervised environment. As can be seen from (3.7), RPCA is actually not brittle with respect
to coherent data (although its performance is depressed). Based on this observation, we propose
a simple algorithm, as summarized in Algorithm 1, to achieve a solid improvement over RPCA.
Our idea is straightforward: We first obtain an estimate of L0 by using RPCA and then utilize the
estimate to construct the dictionary matrix A in LRR. The post-processing steps (Step 2 and Step 3)
that slightly modify the solution of RPCA is to encourage well-conditioned dictionary, which is the
circumstance favoring LRR.
Whenever the recovery produced by RPCA is already exact, the claim in Theorem 1 gives that the
recovery produced by our Algorithm 1 is exact as well. That is to say, in terms of exactly recovering
L0 from a given X, the success probability of our Algorithm 1 is greater than or equal to that of
RPCA. From the computational perspective, Algorithm 1 does not really double the work of RPCA,
although there are two convex programs in our algorithm. In fact, according to our simulations,
usually the computational time of Algorithm 1 is merely about 1.2 times as much as RPCA. The
reason is that, as has been explored by [13], the complexity of solving the LRR problem (1.3) is
O(n2 rA ) (assuming m = n), which is much lower than that of RPCA (which requires O(n3 ))
provided that the obtained dictionary matrix A is fairly low-rank (i.e., rA is small).
One may have noticed that the procedure of Algorithm 1 could be made iterative, i.e., one can
? ? as a new estimate of L0 and use it to further update the dictionary matrix A, and so
consider AZ
on. Nevertheless, we empirically find that such an iterative procedure often converges within two
iterations. Hence, for the sake of simplicity, we do not consider iterative strategies in this paper.
6
Algorithm 1 Matrix Recovery
input: Observed data matrix X ? Rm?n .
adjustable parameter: ?.
? 0 by optimizing the RPCA problem (1.2) with ? = 1/?n1 .
1. Solve for L
? 0 by
2. Estimate the rank of L
r?0 = #{i : ?i > 10?3 ?1 },
? 0.
where ?1 , ?2 , ? ? ? , ?n2 are the singular values of L
? 0 by using the rank-?
? 0 . That is,
3. Form L
r0 approximation of L
? 0 = arg min kL ? L
? 0 k2 , s.t. rank (L) ? r?0 ,
L
F
L
which is solved by SVD.
? 0 by normalizing the column vectors of L
? 0:
4. Construct a dictionary A? from L
? :,i =
[A]
? 0 ]:,i
[L
, i = 1, ? ? ? , n,
? 0 ]:,i k2
k[L
where [?]:,i denotes the ith column of a matrix.
?
5. Solve for Z ? by optimizing the LRR problem (1.3) with A = A? and ? = 1/ n1 .
? ?.
output: AZ
4 Experiments
4.1 Results on Randomly Generated Matrices
We first verify the effectiveness of our Algorithm 1 on randomly generated matrices. We generate
a collection of 200 ? 1000 data matrices according to the model of X = P?? (L0 ) + P? (S0 ):
? is a support set chosen at random; L0 is created by sampling 200 data points from each of 5
randomly generated subspaces; S0 consists of random values from Bernoulli ?1. The dimension of
each subspace varies from 1 to 20 with step size 1, and thus the rank of L0 varies from 5 to 100 with
step size 5. The fraction |?|/(mn) varies from 2.5% to 50% with step size 2.5%. For each pair of
rank and support size (r0 , |?|), we run 10 trials, resulting in a total of 4000 (20 ? 20 ? 10) trials.
RPCA
Algorithm 1
32
22
12
2
0.1 0.2 0.3 0.4 0.5
rank(L0)/n2
42
corruption (%)
corruption (%)
corruption (%)
50
42
32
22
12
2
0.1 0.2 0.3 0.4 0.5
rank(L0)/n2
40
RPCA
Algorithm 1
30
20
10
0.1 0.2 0.3 0.4 0.5
rank(L0)/n2
Figure 4:?Algorithm 1 vs RPCA for the task of recovering randomly generated matrices, both using
? = 1/ n1 . A curve shown in the third subfigure is the boundary for a method to be successful
? The recovery is successful for any pair (r0 /n2 , |?|/(mn)) that locates below the curve. Here, a
? 0 ? L0 kF < 0.05kL0kF , where L
? 0 denotes an estimate of L0 .
success means kL
?
Figure 4 compares our Algorithm 1 to RPCA, both using ? = 1/ n1 . It can be seen that, using the
learned dictionary matrix, Algorithm 1 works distinctly better than RPCA. In fact, the success area
(i.e., the area of the white region) of our algorithm is 47% wider than that of RPCA! We should also
mention that it is possible for RPCA to be exactly successful on coherent (or less incoherent) data,
provided that the rank of L0 is low enough and/or S0 is sparse enough. Our algorithm in general
improves RPCA when L0 is moderately low-rank and/or S0 is moderately sparse.
7
4.2 Results on Corrupted Motion Sequences
We now present our experiment with 11 additional sequences attached to the Hopkins155 [21]
database. In those sequences, about 10% of the entries in the data matrix of trajectories are unobserved (i.e., missed) due to vision occlusion. We replace each missed entry with a number from
Bernoulli ?1, resulting in a collection of corrupted trajectory matrices for evaluating the effectiveness of matrix recovery algorithms. We perform subspace clustering on both the corrupted trajectory
matrices and the recovered versions, and use the clustering error rates produced by existing subspace
clustering methods as the evaluation metrics. We consider three state-of-the-art subspace clustering
methods: Shape Interaction Matrix (SIM) [5], Low-Rank Representation with A = X [14] (which
is referred to as ?LRRx?) and Sparse Subspace Clustering (SSC) [6].
Table 1: Clustering error rates (%) on 11 corrupted motion sequences.
SIM
RPCA + SIM
Algorithm 1 + SIM
LRRx
RPCA + LRRx
Algorithm 1 + LRRx
SSC
RPCA + SSC
Algorithm 1 + SSC
Mean
29.19
14.82
8.74
21.38
10.70
7.09
22.81
9.50
5.74
Median
27.77
8.38
3.09
22.00
3.05
3.06
20.78
2.13
1.85
Maximum
45.82
45.78
42.61
56.96
46.25
32.33
58.24
50.32
27.84
Minimum
12.45
0.97
0.23
0.58
0.20
0.22
1.55
0.61
0.20
Std.
11.74
16.23
12.95
17.10
15.63
10.59
18.46
16.17
8.52
Time (sec.)
0.07
9.96
11.64
1.80
10.75
12.11
3.18
12.51
13.11
Table 1 shows the error rates of various algorithms. Without the preprocessing of matrix recovery,
all the subspace clustering methods fail to accurately categorize the trajectories of motion objects,
producing error rates higher than 20%. This illustrates that it is important for motion segmentation
to correct
?the gross corruptions possibly existing in the data matrix of trajectories. By using RPCA
(? = 1/ n1 ) to correct the corruptions, the clustering performances of all considered methods are
improved dramatically. For example, the error rate of ?
SSC is reduced from 22.81% to 9.50%. By
choosing an appropriate dictionary for LRR (? = 1/ n1 ), the error rates can be reduced again,
from 9.50% to 5.74%, which is a 40% relative improvement. These results verify the effectiveness
of our dictionary learning strategy in realistic environments.
5 Conclusion and Future Work
We have studied the problem of disentangling the low-rank and sparse components in a given data
matrix. Whenever the low-rank component exhibits clustering structures, the state-of-the-art RPCA
method could be less successful. This is because RPCA prefers incoherent data, which however may
be inconsistent with data in the real world. When the number of clusters becomes large, the second
and third coherence parameters enlarge and hence the performance of RPCA could be depressed. We
have showed that the challenges arising from coherent (equivalently, less incoherent) data could be
effectively alleviated by learning a suitable dictionary under the LRR framework. Namely, when the
dictionary matrix is low-rank and contains information about the ground truth matrix, LRR can be
immune to the coherence parameters that increase with the underlying cluster number. Furthermore,
we have established a practical algorithm that outperforms RPCA in our extensive experiments.
The problem of recovering coherent data essentially concerns the robustness issues of the Generalized PCA (GPCA) [22] problem. Although the classic GPCA problem has been explored for several
decades, robust GPCA is new and has not been well studied. The approach proposed in this paper is in a sense preliminary, and it is possible to develop other effective methods for learning the
dictionary matrix in LRR and for handling coherent data. We leave these as future work.
Acknowledgement
Guangcan Liu was a Postdoctoral Researcher supported by NSF-DMS0808864, NSF-SES1131848,
NSF-EAGER1249316, AFOSR-FA9550-13-1-0137, and ONR-N00014-13-1-0764. Ping Li is also
partially supported by NSF-III1360971 and NSF-BIGDATA1419210.
8
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9
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4,788 | 5,335 | Scalable Methods for Nonnegative Matrix
Factorizations of Near-separable Tall-and-skinny
Matrices
Jason D. Lee
ICME
Stanford University
Stanford, CA
[email protected]
Austin R. Benson
ICME
Stanford University
Stanford, CA
[email protected]
Bartek Rajwa
Bindley Biosciences Center
Purdue University
West Lafeyette, IN
[email protected]
David F. Gleich
Computer Science Department
Purdue University
West Lafeyette, IN
[email protected]
Abstract
Numerous algorithms are used for nonnegative matrix factorization under the assumption that the matrix is nearly separable. In this paper, we show how to
make these algorithms scalable for data matrices that have many more rows than
columns, so-called ?tall-and-skinny matrices.? One key component to these improved methods is an orthogonal matrix transformation that preserves the separability of the NMF problem. Our final methods need to read the data matrix only
once and are suitable for streaming, multi-core, and MapReduce architectures.
We demonstrate the efficacy of these algorithms on terabyte-sized matrices from
scientific computing and bioinformatics.
1
Nonnegative matrix factorizations at scale
A nonnegative matrix factorization (NMF) for an m ? n matrix X with real-valued, nonnegative
entries is
X = WH
(1)
where W is m ? r, H is r ? n, r < min(m, n), and both factors have nonnegative entries. While
there are already standard dimension reduction techniques for general matrices such as the singular
value decomposition, the advantage of NMF is in interpretability of the data. A common example is
facial image decomposition [17]. If the columns of X are pixels of a facial image, the columns of W
may be facial features such as eyes or ears, and the coefficients in H represent the intensity of these
features. For this reason, among a host of other reasons, NMF is used in a broad range of applications
including graph clustering [21], protein sequence motif discovery [20], and hyperspectral unmixing
[18].
An important property of matrices in these applications and other massive scientific data sets is that
they have many more rows than columns (m n). For example, this matrix structure is common
in big data applications with hundreds of millions of samples and a small set of features?see,
e.g., Section 4.2 for a bioinformatics application where the data matrix has 1.6 billion rows and 25
columns. We call matrices with many more rows than columns tall-and-skinny. The number of
columns of these matrices is small, so there is no problem storing or manipulating them. Our use
1
of NMF is then to uncover the hidden structure in the data rather than for dimension reduction or
compression.
In this paper, we present scalable and computationally efficient NMF algorithms for tall-and-skinny
matrices as prior work has not taken advantage of this structure for large-scale factorizations. The
advantages of our method are: we preserve the geometry of the problem, we only read the data
matrix once, and we can test several different nonnegative ranks (r) with negligible cost. Furthermore, we show that these methods can be implemented in parallel (Section 3) to handle large data
sets. In Section 2.3, we present a new dimension reduction technique using orthogonal transformations. These transformations are particularly effective for tall-and-skinny matrices and lead to
algorithms that only need to read the data matrix once. We compare this method with a Gaussian
projection technique from the hyperspectral unmixing community [5, 7]. We test our algorithms on
data sets from two scientific applications, heat transfer simulations and flow cytometry, in Section 4.
Our new dimension reduction technique outperforms Gaussian projections on these data sets. In
the remainder of the introduction, we review the state of the art for computing non-negative matrix
factorizations.
1.1
Separable NMF
We first turn to the issue of how to practically compute the factorization in Equation (1). Unfortunately, for a fixed non-negative rank r, finding the factors W and H for which the residual kX ? WHk
is minimized is NP-complete [26]. To make the problem tractable, we make assumptions about the
data. In particular, we require a separability condition on the matrix. A nonnegative matrix X is
separable if
X = X(:, K)H,
where K is an index set with |K| = r and X(:, K) is Matlab notation for the matrix X restricted to
the columns indexed by K. Since the coefficients of H are nonnegative, all columns of X live in
the conical hull of the ?extreme? columns indexed by K. The idea of separability was developed
by Donoho and Stodden [15], and recent work has produced tractable NMF algorithms by assuming
that X almost satisfies a separability condition [3, 6].
A matrix X is noisy r-separable or near-separable if X = X(:, K)H + N, where N is a noise matrix
whose entries are small. Near-separability means that all data points approximately live in the
conical hull of the extreme columns. The algorithms for near-separable NMF are typically based on
convex geometry (see Section 2.1) and can be described by the same two-step approach:
1. Determine the extreme columns, indexed by K, and let W = X(:, K).
2. Solve H = arg minY?Rr?n
kX ? WYk.
+
The bulk of the literature is focused on the first step. In Section 3, we show how to implement both
steps in a single pass over the data and provide the details of a MapReduce implementation. We
note that separability (or near-separability) is a severe and restrictive assumption. The tradeoff is
that our algorithms are extremely scalable and provably correct under this assumption. In big data
applications, scalability is at a premium, and this provides some justification for using separability
as a tool for exploratory data analysis. Furthermore, our experiments on real scientific data sets in
Section 4 under the separability assumption lead to new insights.
1.2
Alternative NMF algorithms and related work
There are several approaches to solving Equation (1) that do not assume the separability condition. These algorithms typically employ block coordinate descent, optimizing over W and H while
keeping one factor fixed. Examples include the seminal work by Lee and Seung [23], alternating
least squares [10], and fast projection-based least squares [19]. Some of these methods are used in
MapReduce architectures at scale [24].
Alternating methods require updating the entire factor W or H after each optimization step. When
one of the factors is large, repeated updates can be prohibitively expensive. The problem is exacerbated in Hadoop MapReduce, where intermediate results are written to disk. In addition, alternating
methods can take an intolerable number of iterations to converge. Regardless of the approach or
computing platform, the algorithms are too slow when the matrices cannot fit in main memory In
2
contrast, we show in Sections 2 and 3 that the separability assumption leads to algorithms that do
not require updates to large matrices. This approach is scalable for large tall-and-skinny matrices in
big data problems.
2
Algorithms and dimension reduction for near-separable NMF
There are several popular algorithms for near-separable NMF, and they are motivated by convex
geometry. The goal of this section is to show that when X is tall-and-skinny, we can apply dimension
reduction techniques so that established algorithms can execute on n ? n matrices, rather than the
original m ? n. Our new dimension reduction technique in Section 2.3 is also motivated by convex
geometry. In Section 3, we leverage the dimension reduction into scalable algorithms.
2.1
Geometric algorithms
There are two geometric strategies typically employed for near-separable
P NMF. The first deals with
conical hulls. A cone C ? Rm is a non-empty convex set with C = { i ?i xi | ?i ? R+ , xi ? Rm }.
The xi are generating vectors. In separable NMF,
X = X(:, K)H
implies that all columns of X lie in the cone generated by the columns indexed by K. For any k ? K,
{?X(:, k) | ? ? R+ } is an extreme ray of this cone, In other words, the set of columns indexed by
K are the set of extreme rays of the cone. The goal of the XRAY algorithm [22] is to find these
extreme rays (i.e., to find K). In particular, the greedy variant of XRAY selects the maximum
column norm arg max j kRT X(:, j)k2 /kX(:, j)k2 , where R is a residual matrix that gets updated with
each new extreme column.
The second approach deals with convex hulls, where the columns of X are `1 -normalized. If D is a
diagonal matrix with Dii = kX(:, i)k1 and X is separable, then
?
XD?1 = X(:, K)D(K, K)?1 D(K, K)HD?1 = (XD?1 )(:, K)H.
Thus, XD?1 is also separable (in fact, this holds for any nonsingular diagonal matrix D). Since the
columns are `1 -normalized, the columns of H? have non-negative entries and sum to one. In other
words, all columns of XD?1 are in the convex hull of the columns indexed by K. The problem of
determining K is reduced to finding the extreme points of a convex hull. Popular approaches in the
context of NMF include the Successive Projection Algorithm (SPA, [2]) and its generalization [16].
Another alternative, based on linear programming, is Hott Topixx [6]. Other geometric approaches
had good heuristic performance [9, 25] before the more recent theoretical work. As an example of
the particulars of one such method, SPA, which we will use in Section 4, finds extreme points by
computing arg max j kR(:, j)k22 , where R is a residual matrix related to the data matrix X.
In any algorithm, we call the columns indexed by K extreme columns. The next two subsections are
devoted to dimension reduction techniques for finding the extreme columns in the case when X is
tall-and-skinny.
2.2
Gaussian projection
A common dimension reduction technique is random Gaussian projections, and the idea has been
used in hyperspectral unmixing problems [5]. In the hyperspectral unmixing literature, the separability is referred to as the pure-pixel assumption, and the random projections are motivated by
convex geometry [7]. In particular, given a matrix G ? Rm?k with Gaussian i.i.d. entries, the extreme
columns of X are taken as the extreme columns of GT X, which is of dimension k ? n. Recent work
shows that when X is nearly r-separable and k = O(r log r), then all of the extreme columns are
found with high probability [13].
2.3
Orthogonal transformations
Our new alternative dimension reduction technique is also motivated by convex geometry. Consider
a cone C ? Rm and a nonsingular m ? m matrix M. It is easily shown that x is an extreme ray of C
3
if and only if Mx is an extreme ray of MC = {Mz | z ? C}. Similarly, for any convex set, invertible
transformations preserve extreme points.
We take advantage of these facts by applying specific orthogonal transformations as the nonsingular
? T be the full QR factorization and singular value decomposition
matrix M. Let X = QR? and X = U ?V
(SVD) of X, so that Q and U are m ? m orthogonal (and hence nonsingular) matrices. Then
!
!
R
?V T
T
T
, U X=
Q X=
,
0
0
where R and ? are the top n ? n blocks of R? and ?? and 0 is an (m ? n) ? n matrix of zeroes. The zero
rows provide no information on which columns of QT X or U T X are extreme rays or extreme points.
Thus, we can restrict ourselves to finding the extreme columns of R and ?V T . These matrices are
n ? n, and we have significantly reduced the dimension of the problem. In fact, if X = X(:, K)H is a
separable representation, we immediately have separated representations for R and ?V T :
R = R(:, K)H,
?V T = ?V T (:, K)H.
We note that, although any invertible transformation preserves extreme columns, many transformations will destroy the geometric structure of the data. However, orthogonal transformations are
either rotations or reflections, and they preserve the data?s geometry. Also, although QT and U T are
m ? m, we will only apply them implicitly (see Section 3.1), i.e., these matrices are never formed or
computed.
This dimension reduction technique is exact when X is r-separable, and the results will be the same
for orthogonal transformations QT and U T . This is a consequence of the transformed data having
the same separability as the original data. The SPA and XRAY algorithms briefly described in
Section 2.1 only depend on computing column 2-norms, which are preserved under orthogonal
transformations. For these algorithms, applying QT or U T preserves the column 2-norms of the data,
and the selected extreme columns are the same. However, other NMF algorithms do not possess this
invariance. For this reason, we present both of the orthogonal transformations.
Finally, we highlight an important benefit of this dimension reduction technique. In many applications, the data is noisy and the separation rank (r in Equation (1)) is not known a priori. In
Section 2.4, we show that the H factor can be computed in the small dimension. Thus, it is viable to
try several different values of the separation rank and pick the best one. This idea is extremely useful
for the applications presented in Section 4, where we do not have a good estimate of the separability
of the data.
2.4
Computing H
Selecting the extreme columns indexed by K completes one half of the NMF factorization in Equation (1). How do we compute H? We want H = arg minY?Rr?n
kX ? X(:, K)Yk2 for some norm.
+
Choosing the Frobenius norm results in a set of n nonnegative least squares (NNLS) problems:
H(:, i) = arg minr kX(:, K)y ? X(:, i)k22 ,
y?R+
i = 1, . . . , n.
? Then H(:, i) is computed by finding y ? Rr+ that
Let X = QR? with R the upper n ? n block of R.
minimizes
kX(:, K)y ? X(:, i)k22 = kQT (X(:, K)y ? X(:, i)) k22 = kR(:, K)y ? R(:, i)k22
Thus, we can solve the NNLS problem with matrices of size n ? n. After computing just the small
R factor from the QR factorization, we can compute the entire nonnegative matrix factorization by
working with matrices of size n ? n. Analogous results hold for the SVD, where we replace Q by
U, the left singular vectors. In Section 3, we show that these computations are simple and scalable.
Since m n, computations on O(n2 ) data are fast, even in serial. Finally, note that we can also
compute the residual in this reduced space, i.e.:
min kX(:, K)y ? X(:, i)k22 = minn kR(:, K)y ? R(:, i)k22 .
y?Rn+
y?R+
This simple fact is significant in practice. When there are several candidate sets of extreme columns
K, the residual error for each set can be computed quickly. In Section 4, we compute many residual
errors for different sets K in order to choose an optimal separation rank.
4
We have now shown how to use dimension reduction techniques for tall-and-skinny matrix data in
near-separable NMF algorithms. Following the same strategy as many NMF algorithms, we first
compute extreme columns and then solve for the coefficient matrix H. Fortunately, once the upfront
cost of the orthogonal transformation is complete, both steps can be computed using O(n2 ) data.
3
Implementation
Remarkably, when the matrix is tall-and-skinny, we only need to read the data matrix once. The
reads can be performed in parallel, and computing platforms such as MapReduce, Spark, distributed
memory MPI, and GPUs can all achieve optimal parallel communication. For our implementation,
we use Hadoop MapReduce for convenience.1 While all of the algorithms use sophisticated computation, these routines are only ever invoked with matrices of size n ? n. Furthermore, the local
memory requirements of these algorithms are only O(n2 ). Thus, we get extremely scalable implementations. We note that, using MapReduce, computing GT X for the Gaussian projection technique
is a simple variation of standard methods to compute X T X [4].
3.1
TSQR and R-SVD
The thin QR factorization of an m ? n real-valued matrix X with m > n is X = QR where Q is an
m ? n orthogonal matrix and R is an n ? n upper triangular matrix. This is precisely the factorization
we need in Section 2. For our purposes, QT is applied implicitly, and we only need to compute
R. When m n, communication-optimal algorithms for computing the factorization are referred
to as TSQR [14]. Implementations and specializations of the TSQR ideas are available in several
environments, including MapReduce [4, 11], distributed memory MPI [14], and GPUs [1]. All of
these methods avoid computing X T X and hence are numerically stable.
The thin SVD used in Section 2.3 is a small extension of the thin QR factorization. The thin SVD is
X = U?V T , where U is m ? n and orthogonal, ? is diagonal with decreasing, nonnegative diagonal
entries, and V is n?n and orthogonal. Let X = QR be the thin QR factorization of X and R = UR ?V T
be the SVD of R. Then X = (QUR )?V T = U?V T . The matrix U = QUR is m ? n and orthogonal, so
this is the thin SVD of X. The dimension of R is n ? n, so computing its SVD takes O(n3 ) floating
point operations (flops), a trivial cost when n is small. When m n, this method for computing the
SVD is called the R-SVD [8]. Both TSQR and R-SVD require O(mn2 ) flops. However, the dominant
cost is data I/O, and TSQR only reads the data matrix once.
3.2
Column normalization
The convex hull algorithms from Section 2.1 and the Gaussian projection algorithm from Section 2.2
require the columns of the data matrix X to be normalized. A naive implementation of the column
normalization in a MapReduce environment is: (1) read X and compute the column norms; (2) read
X, normalize the columns, and write the normalized data to disk; (3) use TSQR on the normalized
matrix. This requires reading the data matrix twice and writing O(mn) data to disk once just to
normalize the columns. The better approach is a single step: use TSQR on the unnormalized data X
and simultaneously compute the column norms. If D is the diagonal matrix of column norms, then
X = QR ? XD?1 = Q(RD?1 ).
The matrix R? = RD?1 is upper triangular, so QR? is the thin QR factorization of the columnnormalized data. This approach reads the data once and only writes O(n2 ) data. The same idea
applies to Gaussian projection since GT (XD?1 ) = (GT X)D?1 . Thus, our algorithms only need to
read the data matrix once in all cases. (We refer to the algorithm output as selecting the columns
and computing the matrix H, which is typically what is used in practice. Retrieving the entries from
the columns of A from K does require a subsequent pass.)
4
Applications
In this section, we test our dimension reduction technique on massive scientific data sets. The data
are nonnegative, but we do not know a priori that the data is separable. Experiments on synthetic
1
The code is available at https://github.com/arbenson/mrnmf.
5
data sets are provided in an online version of this paper and show that our algorithms are effective
and correct on near-separable data sets.2
All experiments were conducted on a 10-node, 40-core MapReduce cluster. Each node has 6 2-TB
disks, 24 GB of RAM, and a single Intel Core i7-960 3.2 GHz processor. They are connected via
Gigabit ethernet. We test the following three algorithms: (1) dimension reduction with the SVD
followed by SPA; (2) Dimension reduction with the SVD followed by the greedy variant of the
XRAY algorithm; (3) Gaussian projection (GP) as described in Section 2.2. We note that the greedy
variant of XRAY is not exact in the separable case but works well in practice [22].
Using our dimension reduction technique, all three algorithms require reading the data only once.
The algorithms were selected to be a representative set of the approaches in the literature, and we
will refer to the three algorithms as SPA, XRAY, and GP. As discussed in Section 2.3, the choice of
QR or SVD does not matter for these algorithms (although it may matter for other NMF algorithms).
Thus, we only consider the SVD transformation in the subsequent numerical experiments.
4.1
Heat transfer simulation
The heat transfer simulation data contains the simulated heat in a high-conductivity stainless steel
block with a low-conductivity foam bubble inserted in the block [12].3 Each column of the matrix
corresponds to simulation results for a foam bubble of a different radius. Several simulations for random foam bubble locations are included in a column. Each row corresponds to a three-dimensional
spatial coordinate, a time step, and a bubble location. An entry of the matrix is the temperature of
the block at a single spatial location, time step, bubble location, and bubble radius. The matrix is
constructed such that columns near 64 have far more variability in the data ? this is then responsible
for additional ?rank-like? structure. Thus, we would intuitively expect the NMF algorithms to select
additional columns closer to the end of the matrix. (And indeed, this is what we will see shortly.) In
total, the matrix has approximately 4.9 billion rows and 64 columns and occupies a little more than
2 TB on the Hadoop Distributed File System (HDFS).
The left plot of Figure 1 shows the relative error for varying separation ranks. The relative error is
defined as kX ? X(:, K)Hk2F /kXk2F . Even a small separation rank (r = 4) results in a small residual.
SPA has the smallest residuals, and XRAY and GP are comparable. An advantage of our projection
method is that we can quickly test many values of r. For the heat transfer simulation data, we choose
r = 10 for further experiments. This value is near an ?elbow? in the residual plot for the GP curve.
We note that the original SPA and XRAY algorithms would achieve the same reconstruction error
if applied to the entire data set. Our dimension reduction technique allows us to accelerate these
established methods for this large problem.
The middle plot of Figure 1 shows the columns selected by each algorithm. Columns 5 through
30 are not extreme in any algorithm. Both SPA and GP select at least one column in indices one
through four. Columns 41 through 64 have the highest density of extreme columns for all algorithms.
Although the extreme columns are different for the algorithms, the coefficient matrix H exhibits
remarkably similar characteristics in all cases. Figure 2 visualizes the matrix H for each algorithm.
Each non-extreme column is expressed as a conic combination of only two extreme columns. In
general, the two extreme columns corresponding to column i are j1 = arg max{ j ? K | j < i} and
arg min{ j ? K | j > i}. In other words, a non-extreme column is a conic combination of the two
extreme columns that ?sandwich? it in the data matrix. Furthermore, when the index i is closer to
j1 , the coefficient for j1 is larger and the coefficient for j2 is smaller. This phenomenon is illustrated
in the right plot of Figure 1.
4.2
Flow cytometry
The flow cytometry (FC) data represent abundances of fluorescent molecules labeling antibodies
that bind to specific targets on the surface of blood cells.4 The phenotype and function of individual
cells can be identified by decoding these label combinations. The analyzed data set contains measurements of 40,000 single cells. The measurement fluorescence intensity conveying the abundance
2
http://arxiv.org/abs/1402.6964.
The heat transfer simulation data is available at https://www.opensciencedatacloud.org.
4
The FC data is available at https://github.com/arbenson/mrnmf/tree/master/data.
3
6
Figure 1: (Left) Relative error in the separable factorization as a function of separation rank (r)
for the heat transfer simulation data. Our dimension reduction technique lets us test all values of
r quickly. (Middle) The first 10 extreme columns selected by SPA, XRAY, and GP. We choose 10
columns as there is an ?elbow? in the GP curve there (left plot). The columns with larger indices
are more extreme, but the algorithms still select different columns. (Right) Values of H(K ?1 (1), j)
and H(K ?1 (34), j) computed by SPA for j = 2, . . . , 33, where K ?1 (1) and K ?1 (34) are the indices
of the extreme columns 1 and 34 in W (X = WH). Columns 2 through 33 of X are roughly convex
combinations of columns 1 and 34, and are not selected as extreme columns by SPA. As j increases,
H(K ?1 (1), j) decreases and H(K ?1 (34), j) increases.
Figure 2: Coefficient matrix H for SPA, XRAY, and GP for the heat transfer simulation data when
r = 10. In all cases, the non-extreme columns are conic combinations of two of the selected columns,
i.e., each column in H has at most two non-zero values. Specifically, the non-extreme columns are
conic combinations of the two extreme columns that ?sandwich? them in the matrix. See the right
plot of Figure 1 for a closer look at the coefficients.
information were collected at five different bands corresponding to the FITC, PE, ECD, PC5, and
PC7 fluorescent labels tagging antibodies against CD4, CD8, CD19, CD45, and CD3 epitopes.
The measurements are represented as the data matrix A of size 40, 000 ? 5. Our interest in the presented analysis was to study pairwise interactions in the data (cell vs. cell, and marker vs. marker).
Thus, we are interested in the matrix X = A ? A, the Kronecker product of A with itself. Each row
of X corresponds to a pair of cells and each column to a pair of marker abundance values. X has
dimension 40, 0002 ? 52 and occupies 345 GB on HDFS.
The left plot of Figure 3 shows the residuals for the three algorithms applied to the FC data for
varying values of the separation rank. In contrast to the heat transfer simulation data, the relative
errors are quite large for small r. In fact, SPA has large relative error until nearly all columns are
selected (r = 22). XRAY has the smallest residual for any value of r. The right plot of Figure 3
shows the columns selected when r = 16. XRAY and GP only disagree on one column. SPA
chooses different columns, which is not surprising given the relative residual error. Interestingly, the
columns involving the second marker defining the phenotype (columns 2, 6, 7, 8, 9, 10, 12, 17, 22)
are underrepresented in all the choices. This suggests that the information provided by the second
marker may be redundant. In biological terms, it may indicate that the phenotypes of the individual
cells can be inferred from a smaller number of markers. Consequently, this opens a possibility that in
modified experimental conditions, the FC researchers may omit this particular label, and still be able
to recover the complete phenotypic information. Owing to the preliminary nature of these studies,
a more in-depth analysis involving multiple similar blood samples would be desirable in order to
confirm this hypothesis.
7
Figure 3: (Left) Relative error in the separable factorization as a function of nonnegative rank (r)
for the flow cytometry data. (Right) The first 16 extreme columns selected by SPA, XRAY, and GP.
We choose 16 columns since the XRAY and GP curve levels for larger r (left plot).
Figure 4: Coefficient matrix H for SPA, XRAY, and GP for the flow cytometry data when r =
16. The coefficients tend to be clustered near the diagonal. This is remarkably different to the
coefficients for the heat transfer simulation data in Figure 2.
Finally, Figure 4 shows the coefficient matrix H. The coefficients are larger on the diagonal, which
means that the non-extreme columns are composed of nearby extreme columns in the matrix.
5
Discussion
We have shown how to compute nonnegative matrix factorizations at scale for near-separable talland-skinny matrices. Our main tool was TSQR, and our algorithms only needed to read the data
matrix once. By reducing the dimension of the problem, we can easily compute the efficacy of
factorizations for several values of the separation rank r. With these tools, we have computed the
largest separable nonnegative matrix factorizations to date. Furthermore, our algorithms provide
new insights into massive scientific data sets. The coefficient matrix H exposed structure in the
results of heat transfer simulations. Extreme column selection in flow cytometry showed that one
of the labels used in measurements may be redundant. In future work, we would like to analyze
additional large-scale scientific data sets. We also plan to test additional NMF algorithms.
The practical limits of our algorithm are imposed by the tall-and-skinny requirement where we
assume that it is easy to manipulate n ? n matrices. The synthetic examples we explored used up
to 200 columns, and regimes up to 5000 columns have been explored in prior work [11]. A rough
rule of thumb is that our implementations should be possible as long as an n ? n matrix fits into
main memory. This means that implementations based on our work will scale up to 30, 000 columns
on machines with more than 8 GB of memory; although at this point communication begins to
dominate. Solving these problems with more columns is a challenging opportunity for the future.
Acknowledgments
ARB and JDL are supported by an Office of Technology Licensing Stanford Graduate Fellowship.
JDL is also supported by a NSF Graduate Research Fellowship. DFG is supported by NSF CAREER
award CCF-1149756. BR is supported by NIH grant 1R21EB015707-01.
8
References
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9
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4,789 | 5,336 | Analog Memories in a Balanced Rate-Based
Network of E-I Neurons
Dylan Festa
[email protected]
Guillaume Hennequin
[email protected]
M?at?e Lengyel
[email protected]
Computational & Biological Learning Lab, Department of Engineering
University of Cambridge, UK
Abstract
The persistent and graded activity often observed in cortical circuits is sometimes seen as a signature of autoassociative retrieval of memories stored earlier
in synaptic efficacies. However, despite decades of theoretical work on the subject, the mechanisms that support the storage and retrieval of memories remain
unclear. Previous proposals concerning the dynamics of memory networks have
fallen short of incorporating some key physiological constraints in a unified way.
Specifically, some models violate Dale?s law (i.e. allow neurons to be both excitatory and inhibitory), while some others restrict the representation of memories to
a binary format, or induce recall states in which some neurons fire at rates close
to saturation. We propose a novel control-theoretic framework to build functioning attractor networks that satisfy a set of relevant physiological constraints. We
directly optimize networks of excitatory and inhibitory neurons to force sets of
arbitrary analog patterns to become stable fixed points of the dynamics. The resulting networks operate in the balanced regime, are robust to corruptions of the
memory cue as well as to ongoing noise, and incidentally explain the reduction
of trial-to-trial variability following stimulus onset that is ubiquitously observed
in sensory and motor cortices. Our results constitute a step forward in our understanding of the neural substrate of memory.
1
Introduction
Memories are thought to be encoded in the joint, persistent activity of groups of neurons. According
to this view, memories are embedded via long-lasting modifications of the synaptic connections
between neurons (storage) such that partial or noisy initialization of the network activity drives
the collective dynamics of the neurons into the corresponding memory state (recall) [1]. Models of
memory circuits following these principles abound in the theoretical neuroscience literature, but few
respect some of the most fundamental properties of brain networks, including: i) the separation of
neurons into distinct classes of excitatory (E) and inhibitory (I) cells ? known as Dale?s law ?, ii) the
presence of recurrent and sparse synaptic connections, iii) the possibility for each neuron to sustain
graded levels of activity in different memories, iv) the firing of action potentials at reasonably low
rates, and v) a dynamic balance of E and I inputs.
In the original Hopfield network [1], connectivity must be symmetrical, which violates Dale?s law.
Moreover, just as in much of the work following it up, memories are encoded in binary neuronal
responses and so converge towards effectively binary recall states even if the recall dynamics formally uses graded activities [2]. Subsequent work considered non-binary pattern distributions [3, 4],
and derived high theoretical capacity limits for them, but those capacities proved difficult ? if not
impossible ? to realise in practice [5, 6], and the network dynamics therein did not explicitly model
inhibitory neurons thus implicitly assuming instantaneous inhibitory feedback. More recent work
1
a
c
b
exc. (prescribed distribution)
20 Hz
memories
5 Hz
inh. (optimized distribution)
0
exc. neurons
inh. neurons
10
20
30
firing rate [Hz]
Figure 1: (a) Examples of analog patterns of excitatory neuronal activities, drawn from a log-normal
distribution. In all our training experiments, network parameters were optimized to stabilize a set
of such analog patterns and the baseline, uniform activity state (top row). For ease of visualization,
only 30 of the 100 excitatory neurons are shown. (b) Optimized values of the inhibitory (auxiliary)
neuronal firing rates for 5 of 30 learned memories (corresponding to those in panel a). Only 30 of
the 50 auxiliary neurons are shown. (c) Empirical distributions of firing rates across neurons and
memory patterns, for each population.
incorporated Dale?s law, and described neurons using the more realistic, leaky integrate-and-fire
(LIF) neuron model [7]. However, the stability of the recall states still relied critically on the saturating behavior of the LIF input-output transfer function at high rates. Although it was later shown
that dynamic feedback inhibition can stabilize relatively low firing rates in subpopulations of more
tightly connected neurons [8, 9], inhibitory feedback in these models is global, and calibrated for a
single stereotypical level of excitation for all memories, implying effectively binary memories again.
Finally, spatially connected networks are able to sustain graded activity patterns (spatial ?bumps?),
but make strong assumptions about the spatial structure of both the connectivity and the memory
patterns, and are sensitive to ongoing noise (e.g. [10, 11]). Ref. [12] provides a rare example of
spike timing-based graded memory network, but it again did not contain inhibitory units.
Here we propose a general control-theoretic framework that overcomes all of the above limitations
with minimal additional assumptions. We formalize memory storage as implying two conditions:
that the desired activity states be fixed points of the dynamics, and that the dynamics be stable
around those fixed points. We directly optimize the network parameters, including the synaptic
connectivity, to satisfy both conditions for a collection of arbitrary, graded memory patterns (Fig. 1).
The fixed point condition is achieved by minimizing the time derivative of the neural activity, such
that ideally it reaches zero, at each of the desired attractor states. Stability, however, is more difficult
to achieve because the fixed-point constraints tend to create strong positive feedback loops in the
recurrent circuitry, and direct measures of dynamical stability (eg. the spectral abscissa) do not admit
efficient, gradient-based optimization. Thus, we use recently developed methods from robust control
theory, namely the minimization of the Smoothed Spectral Abscissa (SSA, [13, 14]) to perform
robust stability optimization. To satisfy biological constraints, we parametrize the networks that we
optimize such that they have realistic firing rate dynamics and their connectivities obey Dale?s law.
We show that despite these constraints the resulting networks perform memory recall that is robust
to noise in both the recall cue and the ongoing dynamics, and is stabilized through a tight dynamic
balance of excitation and inhibition. This novel way of constructing structurally realistic memory
networks should open new routes to the understanding of memory and its neural substrate.
2
Methods
We study a network of n = nE (excitatory) +nI (inhibitory) neurons. The activity of neuron i is
represented by a single scalar potential vi , which is converted into a firing rate ri via a thresholdquadratic gain function (e.g. [15]):
2
?vi
if vi > 0
:=
ri = g(vi )
(1)
0
otherwise
2
We set ? to 0.04, such that g(vi ) spans a few tens of Hz when vi spans a few tens of mV, as
experimentally observed in cortical areas (e.g. cat?s V1 [16]). The instantaneous state of the system
can be expressed as a vector v(t) := (v1 (t), . . . , vn (t)). We denote the activity of the excitatory or
inhibitory subpopulation by vexc and vinh , respectively. The recurrent interactions between neurons
are governed by a synaptic weight matrix W, in which the sign of each element Wij depends on
the nature (excitatory or inhibitory) of the presynaptic neuron j. We enforce Dale?s law via a reparameterization of the synaptic weights:
+1
if j ? nE
(2)
Wij = sj log(1 + exp ?ij ) with sj =
?1
otherwise
where the ?ij ?s are free, unconstrained parameters. (We do not allow for autapses, i.e. we fix Wii =
0). The network dynamics are thus given by:
?i
n
X
dvi
= ?vi +
Wij g(vj ) + hi ,
dt
j=1
(3)
where ?i is the membrane time constant, and hi is a constant external input, independent of the
memory we wish to recall.
It is worth noting that, since the gain function g(vi ) defined in Eq (1) has no upper saturation,
recurrent interactions can easily result in runaway excitation and firing rates growing unbounded.
However, our optimization algorithm will naturally seek stable solutions, in which firing rates are
kept within a limited range due to a fine dynamic balance of excitation and inhibition [14].
Optimizing network parameters to embed attractor memories
We are going to build and study networks that have a desired set of analog activity patterns as stable
?
}?=1,...,m be a set of m target analog patterns (Fig. 1),
fixed points of their dynamics. Let {vexc
defined in the space of excitatory neuronal activity (potentials). For a given pattern ?, the inhibitory
?
neurons will be free to adjust their steady state firing rates vinh
to whatever pattern proves to be
optimal to maintain stability. In other words, we think of the activity of inhibitory neurons as
?auxiliary? variables.
>
>
? >
?
A given activity pattern v? ? (vexc
, vinh
) is a stable fixed point of the network dynamics if, and
only if, it satisfies the following two conditions:
dv
=0
and
? (J? ) < 0
(4)
dt
?
v=v
?
:= Wij g 0 (vj? ) ? ?ij (Kronecker?s
where J? is the Jacobian matrix of the dynamics in Eq. 3, i.e. Jij
?
delta), and ?(J ) denotes the spectral abscissa (SA), defined as the largest real part in the eigenvalue
spectrum of J? . The first condition makes v? a fixed point of the dynamics, while the second
condition makes that fixed point asymptotically stable with respect to small local perturbations.
Note that the width of the basin of attraction is not captured by the SA.
The two conditions in Eq. 4 depend on a set of network parameters that we will allow ourselves
to optimize. These are all the synaptic weight parameters (?ij , i 6= j), as well as the values of the
?
inhibitory neurons? firing rates in each attractor (vinh
, ? = 1, . . . , m). Thus, we may adjust a total
of n(n ? 1) + nI m parameters.
Pn
Using Eq. 3, the first condition in Eq. 4 can be rewritten as vi? ? j=1 Wij g(vj? ) ? hi = 0.
Despite this equation being linear in the synaptic weights, the re-parameterization of Eq. 2 makes
?
it nonlinear in ?, and it is in any case nonlinear in vinh
. We will therefore seek to satisfy this
2
condition by minimizing k dv/dt|v=v? k , which quantifies how fast the potentials drift away when
initialized in the desired attractor state v? . When it is zero, v? is a fixed point of the dynamics. Our
optimization procedure (see below) may not be able to set this term to exactly zero, especially as we
try to store a large number of memories, but in practice we find it becomes small enough that the
Jacobian-based stability criterion remains valid.
Meeting the stability condition (second condition in Eq. 4) turns out to be more involved. The SA
is, in general, a non-smooth function of the matrix elements and is therefore difficult to minimize.
3
A more suitable stability measure has been introduced recently in the context of robust control
theory [13, 14], called the Smoothed Spectral Abscissa (SSA), which we will use here and denote
by ?
? ? (J? ). The SSA, defined for some smoothness parameter ? > 0, is a differentiable relaxation of
the SA, with the properties ?(J? ) < ?
? ? (J? ) and lim??0 ?
? ? (J? ) = ?(J? ). Therefore, the criterion
?
? ? (J? ) ? 0 implies ?(J? ) < 0, and can therefore be used as an indication of local stability.
Both the SSA and its gradient are straightforward to evaluate numerically, making it amenable to
minimization through gradient descent. Note that the SSA depends on the Jacobian matrix elements
?
?
{Jij
}, which in turn depend both on the connectivity parameters {?ij } and on vinh
. Note also that
the parameter ? > 0 controls how tightly the SSA hugs the SA. Small values make it a tight upper
bound, with increasingly ill-behaved gradients. Large values imply more smoothness, but may no
longer guarantee that the SSA has a negative minimum even though the SA might have one. In our
system of n = 150 neurons we found ? = 0.01 to yield a good compromise. In the general case the
distance between SA and SSA grows linearly with the number of dimensions.To keep it invariant, ?
should be scaled accordingly. We therefore used the following heuristic rule ? = 0.01 ? 150/n.
We summarize the above objective into a global cost function by lumping together the fixed point
and stability conditions, summing over the entire set of m target memory patterns, and adding an L2
penalty term on the synaptic weights to regularize:
!
2
m
1 X 1
?F
dv
2
?
?
? ({?ij }, {vinh }) :=
+ ?s ?
? ? (J ) + 2 kWkF .
(5)
m ?=1 n
dt
v=v?
n
where kWk2F is the squared Frobenius norm of W, i.e. the sum of its squared elements, and the
parameters ?s and ?F control the relative importance of each component of the objective function.
We set them heuristically (Table 1). We used a variant of the low-storage BFGS algorithm included
in the open source library NLopt [17] to minimize ?.
Choice of initial parameters and attractors
The synaptic weights are initially drawn randomly from a Gamma distribution with a shape factor of
2 and a mean that depends only on the type of pre- and post-synaptic population. The mean synaptic
weights of the four synapse types were computed using a mean-field reduction of the full network
?=1
in which all
to meet the condition that the network initially exhibits a stable baseline state vexc
excitatory firing rates equal rbaseline = 5 Hz (Table 1, and Supplementary Material). This baseline state was included in every set of m target attractors that we used and was thus stable from
?
}?=2,...,m were generated
the beginning, by construction. For the remaining target patterns, {vexc
?1
by inverting (using g ) firing rates that were sampled from a log-normal distribution with a mean
matching the baseline firing rate, rbaseline (Fig. 1a) and a variance of 5 Hz. This log-normal distribution was chosen to roughly capture the skewed and heavy-tailed nature of firing rate distributions
?
observed in vivo (see e.g. for a review [18]). The inhibitory potentials in the memory states, {vinh
},
?1
were initialized to the baseline, g (5 Hz), and were subsequently used as free parameters by the
learning algorithm (cf. above; see also Fig. 1b).
3
Results
Example of successful storage
Figure 2 shows an example of stability optimization: in this specific run we used 150 neurons to embed 30 graded attractors (examples of which where shown in Fig. 1), yielding a storage capacity of
0.2. Other parameters are listed in Table 1. Gradient descent gradually reduces each of the attractorspecific sub-objectives in Eq. 5, namely the SSA, the SA, and the potential velocities kdv/dtk2 in
each target state (Fig. 2). After convergence, the SSA has become negative for all desired states,
indicating stability. Note, however, that kdv/dtk after convergence is small but non-zero in each
of the target memories. Thus, strictly speaking, the target patterns haven?t become fixed points of
the dynamics, but only slow points from which the system will eventually drift away. In practice
though, we found that stability was robust enough that an exact, stable fixed point had in fact been
created very near each target pattern. This is detailed below.
4
? ? (J? )i?
h?
m = 50
b
?
0
?0.5
?1
m = 30
10?2
D
E
v? (?)
2
SA / SSA
a
h? (J? )i?
10?4
0
0
20
40
60
time (hours)
20
40
60
time (hours)
Figure 2: (a) Decrease of the SA (solid line) and of the SSA (dotted line) during learning in systems
with 30 (purple) and 50 attractors (orange). Thick lines show averages across attractors, flanking
lines show the corresponding standard deviations. The x-axis marks the actual duration of the run of
the learning algorithm. (b) Euclidean norm of the velocity at the fixed point during learning. Lines
and colors as in a. Note the logarithmic y-axis.
Table 1: Parameter settings
nE
nI
m
100
50
30
?E
?I
rbaseline
20 ms
10 ms
5 Hz
?s
?F
0.02
0.001
Memory recall performance and robustness
For recall, we initialize neuronal activities at a noisy version of one of the target patterns, and study
the subsequent evolution of the network state. The network performs well if its dynamics clean up
the noise and home in on the target pattern (autoassociative behavior) and if it achieves this robustly
even in the face of large amounts of noise.
Initial cues are chosen to be linear combinations of the form r(t = 0) = ? ?r + (1 ? ?) r? , where r?
is the memory we intend to recall and ?r is an independent random vector with the same lognormal
statistics used to generate the memory patterns themselves. The parameter ? regulates the noise
level: ? = 0 sets the network activity directly in the desired attractor, while ? = 1 initializes it with
completely random values.
The deviation of the momentary network state r(t) ? g(v(t)) from the target pattern r? ? g(v? )
is measured in terms of the squared Euclidean distance, further normalized by the expected squared
distance between r? and a random pattern drawn from the same distribution (log-normal in our
case). Formally:
krexc (t) ? r?exc k2
d? (t) :=
.
(6)
hk?rexc ? r?exc k2 i?r
Figure 3a shows the temporal evolution of d? (t) on a few sample recall trials, for two different noise
levels ?. For ? = 0.5, recalls are always successful, as the network state converges to the right target
pattern on each trial. For ? = 0.75, the network activity occasionally settles in another, well distinct
attractor.
We used the convention that a trial is deemed successful if the distance d? (t) falls below 0.001. (A
? 3 Hz deviation from the target in only one of the 100 exc. neurons, with all other 99 neurons
behaving perfectly, would be sufficient to cross this threshold and fail the test.) We further measure
performance as the probability of successful recall, which we estimated from many independent
trials with different realizations of the noise ?r in the initial condition (Figure 3b). The network
performance is also compared to an ?ideal observer? [6] that has direct access to all the stored
memories (rather than just their reflection in the synaptic weights) and simply returns that pattern
in the training set {r? } to which the initial cue is closest (Fig. 3b). Thus, as an upper bound on
performance, the ideal observer only produces a wrong recall when the added noise brings the
initial state closer to an attractor that is different from the target. Remarkably, our network dynamics
5
2
? = 0.50
? = 0.75
d? (t )
1.5
1
0.5
0
0
0.1
t (s)
b
probability of success
(a)
a
0.2
network
ideal
memories
baseline
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4 0.6
d? (t0 )
0.8
1
Figure 3: (a) Example recall trials for a single memory r? , which is presented to the network at
time t = 0 in a corrupted version that is different on every trial, for two different values of the
noise level ? (colors). Shown here is the temporal evolution of the momentary distance between the
vector of excitatory firing rates rexc (t) and the memory pattern r?exc . Different lines correspond to
different trials. (b) Fraction of trials that converged onto the correct attractor (final distance d? (t =
?) < 0.001, cf. text) as a function of the normalized distance between the initial condition and the
desired attractor, d? (t = 0). Thick lines show medians across attractors, flanking thin lines show
the 25th and 75th percentiles. The performance of the baseline state is shown separately (orange).
The dashed lines show the performance of an ?ideal observer?, always selecting the memory closest
to the initial condition, for the same trials.
(continuous lines) and the ideal observer (dashed lines) have comparable performances. When trying
to recall the uniform pattern of baseline activity, the performance appears much better (orange line)
both for the ideal observer and the network. This is simply because the random vectors used to
perturb the system have a high probability of lying closer to the mean of the log normal distribution
(that is, the baseline state) than to any other memory pattern. Moreover, the network was initialized
prior to learning with the baseline as the single global attractor, and this might account for the
additional tendency of the network (solid orange line) to fall on such state, as compared to the ideal
observer (dotted orange line).
Only a few strong synaptic weights contribute to memory recall
Synaptic weights after learning (Fig. 4a) are sparse: their distribution shows the characteristic peak
near zero and the long tail observed in real cortical circuits [19, 20] (Fig. 4b). This sparseness cannot
be accounted for by the L2 norm regularizer in the cost function (Eq. 5) as it does not promote
sparsity as an L1 term would. Thus, the observed sparsity in the trained network must be a genuine
consequence of having optimized the connectivity for robust stability.
If we assume that weights |Wij | ? 0.01 correspond to functionally silent synapses, then the trained
network contains 52% of silent excitatory synapses and 46% of silent inhibitory ones (Fig. 4c). We
wondered if those weak, ?silent? synapses are necessary for stability of memory recall, or could be
removed altogether without affecting performance. To test that, we clipped those synapses {|Wij | <
0.01} to zero, and computed recall performance again (Fig. 4d). This clipping turns out to slightly
shift the position of the attractors in state space, so we increased the distance threshold that defines
a successful recall trial to 0.08. The test reveals that one of the attractors loses stability, reducing
the average performance. However the remaining 29 attractors are robust to this removal of weak
synapses and show near-equal recall performance as above. This demonstrates that small weights,
though numerous, are not necessary for competent recall performance.
Balanced state
As a result of the connection weight distributions and robust stability, the trained network produces
a regime in which excitation and inhibition balance each other, precisely tuning each neuron to
its
attractor.
as hexc
i (t) =
Pntarget frequency in each
PnExcitatory and inhibitory inputs are defined
inh
exc
bW
c
r
(t)
and
h
(t)
=
b?
W
c
r
(t)
so
that
the
difference
h
(t)
?
hinh
ij
+
j
ij
+
j
i
i
i (t)
j=1
j=1
corresponds to the total recurrent input, i.e. the second term on the r.h.s. of Eq. 3.
6
c
1
a
exc.
150
inh.
b
15
5
0
exc.
inh.
1
postsynaptic
0.5
10?4 10?2 100
weight
0.1
0
-1
-5
-15
1
success rate
-0.1
1
?10
Wij
150
presynaptic
?5
d
1
0.75
0.5
0.25
0
0
5
weight
10
clipped
full
0
0.5
1
starting distance from attr.
Figure 4: (a) Synaptic weight matrix after learning. Note the logarithmic color scale. (b) Distribution of the excitatory (red) and inhibitory (blue) weights. (c) Cumulative weight distribution of
absolute weight values. Gray line marks the 0.01 threshold we use to defined ?silent? synapses. (d)
Performance of the network after clipping the weights below 0.01 to zero (black, median with 25th
and 75th percentiles), compared to the performance of the unperturbed network redrawn from Fig. 3
(purple).
hkexc (t ), hkinh (t )
40
20
0
k =3
k = 72
k = 101
b 60
hkinh (t? )
k =3
k = 72
k = 101
a
40
k = 15
c
20
0
0
20
40 60
t (ms)
0
20
40
hkexc (t? )
60
0
0.5
1
correlation
Figure 5: (a) Dynamics of the excitatory and inhibitory inputs during a memory recall trial, for
three sample neurons. (b) Scatter plot of steady-state excitatory versus inhibitory inputs. Each dot
corresponds to a different memory pattern, and several neurons are shown in different colors. (c)
Histogram of E and I input correlations across all memories for each neuron (for example, one value
binned in this histogram would be the correlation between all green dots in b).
inh
Figure 5a shows the evolution of hexc
i (t) and hi (t) during a recall trial for one of the stored random
attractors, for 3 different neurons. Neuron 3 has rate target of 9Hz, well above average, therefore its
excitation is much higher than inhibition. Neuron 72 has a steady state firing rate of 2 Hz, below
average: its inhibitory input is greater than the excitatory one, and firing is driven by the external
current. Finally, neuron 101 is inhibitory and has a target rate 0, and indeed its inhibitory input
is large enough to overwhelm the combined effects of the external and recurrent excitatory inputs.
Notably, in all these cases, both E and I input currents are fairly large but cancel each other to leave
something smaller, either positive or negative.
Figure 5b shows the E vs. I inputs at steady-state across all the embedded attractors, for various
neurons plotted in different colors. These E and I inputs tend to be correlated across attractors for
every single neuron (dots in Fig. 5 tend to hug the identity line), with relative differences fine-tuned
to yield the desired firing rates. These across-attractors E/I correlations are summarized in Fig. 5c
as a histogram over neurons.
Robustness to ongoing noise and reduction of across-trial variability following recall onset
Finally, to probe the system under more realistic dynamics, we added time-varying, Gaussian white
noise such that, in an excitatory neuron free from network interactions, the potential would fluctuate
7
nearest
a
others
b
hstd [vi (t )]ii
d? (t )
2
1
3
2
1
0
0
0
0.2
0.4 0.6
t (s)
0.8
0
1
0.2
0.4 0.6
t (s)
0.8
1
Figure 6: (a) Normalized distance calculated according to Eq. 6 between the network activity and
each of the attractors (targeted attractor: green line; others: orange lines) during a noisy recall
episode. (b) Trial-to-trial variability, expressed as the standard deviation of a neuron?s activity across
multiple repetitions with random initial conditions. At time t = 0.5 s the network receives a pulse
in the direction of one target attractor (? = 2). Gray lines are for single neurons; the black line is
an average over the population.
with standard deviation 0.33. Figure 6a shows the momentary distance d? (t) of the network state
from the attractor closest to the initial cue (green), and for all other attractors (orange), during a
recall trial. It is clear that the system revolves around the desired attractor, performing successful
recall despite the ongoing noise. In a second experiment, we ran many trials in which the initialization at time t = 0 was random, while the same spatially patterned stimulation ? aligned onto a
chosen attractor ? is given to the network in each trial at time t = 0.5 sec. Figure 6b shows the standard deviation of the internal state of a neuron across trials, averaged across the neural population.
Following stimulus onset, neurons are always pushed towards the target attractor, and this greatly
reduces trial-by-trial variability, compared to the initial spontaneous regime in which the neurons
would fluctuate around any of the activity levels corresponding to its assigned attractors. Interestingly, such stimulus-induced variability reduction has been observed very broadly across sensory
and motor cortical areas [21]. This extends previous work, e.g. [22] and [23], showing variability
reduction in a multiple-attractor scenario with effectively binary patterns, to the case of patterns with
graded activities.
4
Discussion
We have provided a proof of concept that a model cortical networks of E and I neurons can embed
multiple analog memories as stable fixed-points of their dynamics. Memories are stable in the face
of ongoing noise and corruption of the recall cues. Neuronal activities do not saturate, and indeed,
our single-neuron model did not explicitly incorporate an upper saturation mechanism: dynamic
feedback inhibition, precisely matched to the level of excitation incurred by each attractor, ensures
that each neuron can fire at a relatively low rate during recall. As a result, excitation and inhibition
are tightly balanced.
We have used a rate-based formulation of the circuit dynamics, which raises the question of the
applicability of our method to understanding spiking memory networks. Once the connectivity
in the rate model is generated and optimized, it could still be used in a spiking model, provided
the gain function we have used here matches that of the single spiking neurons. In this respect,
the gain function we have used here is likely an appropriate choice: in physiological conditions,
cortical neurons have input-output gain functions that are well approximated by a rectified powerlaw function over their entire dynamic range [24, 25, 26].
An important question for future research is how local synaptic learning rules can achieve the stabilization objective that we have approached here from an optimal, algorithmic viewpoint. Inhibitory
synaptic plasticity is a promising candidate, as it has already been shown to enable self-regulation of
the spontaneous, baseline activity regime, and also to promote the stable storage of binary memory
patterns [27]. More work is required in this direction.
Acknowledgements. This work was supported by the Wellcome Trust (GH, ML), the European
Union Seventh Framework Programme (FP7/20072013) under grant agreement no. 269921 (BrainScaleS) (DF, ML), and the Swiss National Science Foundation (GH).
8
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4,790 | 5,337 | Feedforward Learning of Mixture Models
Matthew Lawlor?
Applied Math
Yale University
New Haven, CT 06520
[email protected]
Steven W. Zucker
Computer Science
Yale University
New Haven, CT 06520
[email protected]
Abstract
We develop a biologically-plausible learning rule that provably converges to the
class means of general mixture models. This rule generalizes the classical BCM
neural rule within a tensor framework, substantially increasing the generality of
the learning problem it solves. It achieves this by incorporating triplets of samples
from the mixtures, which provides a novel information processing interpretation
to spike-timing-dependent plasticity. We provide both proofs of convergence, and
a close fit to experimental data on STDP.
1
Introduction
Spectral tensor methods and tensor decomposition are emerging themes in machine learning, but
they remain global rather than ?online.? While incremental (online) learning can be useful for applications, it is essential for neurobiology. Error back propagation does operate incrementally, but
its neurobiological relevance remains a question for debate. We introduce a triplet learning rule
for mixture distributions based on a tensor formulation of the BCM biological learning rule. It is
implemented in a feedforward fashion, removing the need for backpropagation of error signals.
The triplet requirement is natural biologically. Informally imagine your eyes microsaccading during
a fixation, so that a tiny image fragment is ?sampled? repeatedly until the next fixation. Viewed
from visual cortex, edge selective neurons will fire repeatedly. Importantly, they exhibit strong
statistical dependencies due to the geometry of objects and their relationships in the world. ?Hidden?
information such as edge curvatures, the presence of textures, and lighting discontinuities all affect
the probability distribution of firing rates among orientation selective neurons, leading to complex
statistical interdependencies between neurons.
Latent variable models are powerful tools in this context. They formalize the idea that highly coupled
random variables can be simply explained by a small number of hidden causes. Conditioned on these
causes, the input distribution should be simple. For example, while the joint distribution of edges in
a small patch of a scene might be quite complex, the distribution conditioned on the presence of a
curved object at a particular location might be comparatively simple [14]. The specific question is
whether brains can learn these mixture models, and how.
Example: Imagine a stimulus space of K inputs. These could be images of edges at particular
orientations, or audio tones at K frequencies. These stimuli are fed into a network of n LinearNonlinear Poisson (LNP) spiking neurons. Let rij denote the firing rate of neuron i to stimulus j.
Assuming the stimuli are drawn independently with probability ?k , then the number of spikes d in
an interval where a single stimulus is shown is distributed according to a mixture model.
X
P (d) =
?k Pk (d)
k
1
Now at Google Inc.
1
where Pk (d) is a vector of independent Poisson distributions, and the rate parameter of the ith
component is rik . We seek a filter that responds (in expectation) to one and only one stimulus. To
do this, we must learn a set of weights that are orthogonal to all but one of the vectors of rates r ?j .
Each rate vector corresponds to the mean of one of the mixtures. Our problem is thus to learn the
means of mixtures. We will demonstrate that this can be done non-parametrically over a broad class
of firing patterns, not just Poisson spiking neurons.
Although fitting mixture models can be exponentially hard, under a certain multiview assumption,
non-parametric estimation of mixture means can be done by tensor decomposition [2][1]. This
multiview assumption requires access to at least 3 independent copies of the samples; i.e., multiple
samples drawn from the same mixture component. For the LNP example above, this multiview
assumption requires only that we have access to the number of spikes in three disjoint intervals,
while the stimulus remains constant. After these intervals, the stimulus is free to change ? in vision,
say, after a saccade ? after which point another sample triple is taken.
Our main result is that, with a slight modification of classical Bienenstock-Cooper-Munro [5] synaptic update rule a neuron can perform a tensor decomposition of the input data. By incorporating the
interactions between input triplets, our online learning rule can provably learn the mixture means
under an extremely broad class of mixture distributions and noise models. (The classical BCM
learning rule will not converge properly in the presence of noise.) Specifically we show how the
classical BCM neuron performs gradient ascent in a tensor objective function, when the data consists of discrete input vectors, and how our modified rule converges when the data are drawn from a
general mixture model.
The multiview requirement has an intriguing implication for neuroscience. Since spikes arrive in
waves, and spike trains matter for learning [9], our model suggests that the waves of spikes arriving during adjacent epochs in time provide multiple samples of a given stimulus. This provides an
unusual information processing interpretation for the functional role of spike trains. To realize it
fully, we point out that classical BCM can be implemented via spike timing dependent plasticity
[17][10][6][18]. However, most of these approaches require much stronger distributional assumptions on the input data (generally Poisson), or learn a much simpler decomposition of the data than
our algorithm. Other, Bayesian methods [16], require the computation of a posterior distribution
which requires an implausible normalization step. Our learning rule successfully avoids these issues, and has provable guarantees of convergence to the true mixture means. At the end of this
paper we show how our rule predicts pair and triple spike timing dependent plasticity data.
2
Tensor Notation
Let ? denote the tensor product. We denote application of a k-tensor to k vectors by T (w1 , ..., wk ),
so in the simple case where T = v 1 ? ... ? v k ,
Y
T (w1 , ..., wk ) =
hv j , wj i
j
We further denote the application of a k-tensor to k matrices by T (M1 , ..., Mk ) where
X
T (M1 , ..., Mk )i1 ,...,ik =
Tj1 ,...,jk [M1 ]j1 ,i1 ...[Mk ]jk ,ik
j1 ,...,jk
Thus if T is a symmetric 2-tensor, T (M1 , M2 ) = M1T T M2 with ordinary matrix multiplication.
Similarly, T (v 1 , v 2 ) = v T1 T v 2 .
We say that T has an orthogonal tensor decomposition if
X
T =
?k v k ? v k ? ... ? v k and hv i , v j i = ?ij
k
3
Connection Between BCM Neuron and Tensor Decompositions
The BCM learning rule was introduced in 1982 in part to correct failings of the classical Hebbian
learning rule [5]. The Hebbian learning rule [11] is one of the simplest and oldest learning rules. It
2
posits that the selectivity of a neuron to input i, mt (i) is increased in proportion to the post-synaptic
activity of that neuron ct = hmt?1 , dt i, where m is a vector of synaptic weights.
mt ? mt?1 = ?t ct dt
This learning rule will become increasingly correlated with its input. As formulated this rule does
not converge for most input, as kmk ? ?. In addition, in the presence of multiple inputs Hebbian
learning rule will always converge to an ?average? of the inputs, rather than becoming selective to
one particular input. It is possible to choose a normalization of m such that m will converge to
the first eigenvector of the input data. The BCM rule tries to correct for the lack of selectivity, and
for the stabilization problems. Like the Hebbian learning rule, it always updates its weights in the
direction of the input, however it also has a sliding threshold that controls the magnitude and sign of
this weight update.
The original formulation of the BCM rule is as follows: Let c be the post-synaptic firing rate, d ? RN
be the vector of presynaptic firing rates, and m be the vector of synaptic weights. Then the BCM
synaptic modification rule is
c = hm, di
? = ?(c, ?)d
m
? is a non-linear function of the firing rate, and ? is a sliding threshold that increases as a superlinear
function of the average firing rate.
There are many different formulations of the BCM rule. The primary features that are required are
?(c, ?) is convex in c, ?(0, ?) = 0, ?(?, ?) = 0, and ? is a super-linear function of E[c].
These properties guarantee that the BCM learning rule will not grow without bound. There have
been many variants of this rule. One of the most theoretically well analyzed variants is the Intrator
and Cooper model [12], which has the following form for ? and ?.
?(c, ?) = c(c ? ?) with ? = E[c2 ]
Definition 3.1 (BCM Update Rule). With the Intrator and Cooper definition, the BCM rule is defined
as
mt = mt?1 + ?t ct (ct ? ?t?1 )dt
(1)
2
where
ct = hmt?1
P
P, dt i2 and ? = E[c ]. ?t is a sequence of positive step sizes with the property that
?
?
?
and
<
?
?
t t
t t
The traditional application of this rule is a system where the input d is drawn from linearly independent vectors {d1 , ..., dk } with probabilities ?1 , ..., ?K , with K = N , the dimension of the
space.
These choices are quite convenient because they lead to the following objective function formulation
of the synaptic update rule.
R(m) =
i 1 h
i2
1 h
3
2
E hm, di ? E hm, di
3
4
Thus,
h
i
2
2
?R = E hm, di d ? E[hm, di ] hm, di d
= E[c(c ? ?)d]
= E[?(c, ?)d]
So in expectation, the BCM rule performs a gradient ascent in R(m). For random, discrete input
this rule would then be a form of stochastic gradient ascent.
With this model, we observe that the objective function can be rewritten in tensor notation. Note
that this input model can be seen as a kind of degenerate mixture model.
3
This objective function can be written as a tensor objective function, by noting the following:
X
T =
?k dk ? dk ? dk
k
M=
X
k
R(m) =
?k dk ? dk
1
1
T (m, m, m) ? M (m, m)2
3
4
(2)
For completeness, we present a proof that the stable points of the expected BCM update are selective
for only one of the data vectors.
? = 0. Let ck = hm, dk i and ?k =
The stable points of the expected update occur when E[m]
?(ck , ?). Let c = [c1 , . . . , cK ]T and ? = [?1 , . . . , ?K ]T .
DT = [d1 | ? ? ? |dK ]
P = diag(?)
Theorem 3.2. (Intrator 1992) Let K = N , let each dk be linearly independent, and let ?k > 0 and
? = ?R occur when
distinct. Then stable points (in the sense of Lyapunov) of the expected update m
c = ?k?1 ek or m = ?k?1 D?1 ek . ek is the unit basis vector, so there is activity for only one stimuli.
P
? = DT P ? which is 0 only when ? = 0. Note ? = k ?k c2k . ?k = 0 if ck = 0 or
Proof. E[m]
ck = ?. Let S+ = {k : ck 6= 0}, and S? = {k : ck = 0}. Then for all k ? S+ , ck = ?S+
?
??1
X
X
?S+ ? ?S2 +
?i = 0
?S+ = ?
?i ?
k?S+
k?S+
Therefore the solutions of the BCM learning rule are c = 1S+ ?S+ , for all subsets S+ ? {1, . . . , K}.
We now need to check which solutions are stable. The stable points (in the sense of Lyapunov) are
points where the matrix
?
?? ?c
??
?E[m]
T
T
=D P
=D P
D
H=
?m
?c ?m
?c
is negative semidefinite.
Let S be an index set S ? {1, . . . , n}. We will use the following notation for the diagonal matrix
IS :
1 i?S
(IS )ii =
(3)
0 i?
/S
So IS + IS c = I, and ei eTi = I{i}
a quick calculation shows
??i
= ?S+ IS+ ? ?S+ IS? ? 2?S2 + diag(?) 1S+ 1TS+
?cj
This is negative semidefinite iff A = IS+ ? 2?S+ diag(?) 1S+ 1TS+ is negative semidefinite.
Assuming a non-degeneracy of the probabilities ?, and assume |S+ | > 1. Let j = arg mink?S+ ?k .
Then ?S+ ?j < 12 so A is not negative semi-definite. However, if |S+ | = 1 then A = ?IS+ so the
stable points occur when c = ?1i ei
The triplet version of BCM can be viewed as a modification of the classical BCM rule which allows
it to converge in the presence of zero-mean noise. This indicates that the stable solutions of this
learning rule are selective for only one data vector, dk .
Building off of the work of [2] we will use this characterization of the objective function to build a
triplet BCM update rule which will converge for general mixtures, not just discrete data points.
4
hm1,di
Noise sensitivity of m after 10e6 steps
14
12
10
Triplet Rule
BCM
3
8
6
d1
4
2
0
?10
?5
0
5
10
15
20
km ? m0 k
2
m1
25
hm2,di
m2
22
1
20
18
d2
16
14
12
10
0
8
6
10?2
4
2
0
?10
?5
0
5
10
15
20
100
101
Noise ?
(a) Geometry of stable solutions. Each stable
solution is selective in expectation for a single
mixture. Note that the classical BCM rule will
not converge to these values in the presence of
noise.
4
10?1
25
(b) Noise response of triplet BCM update
rule vs BCM update. Input data was a mixture of Gaussians with standard deviation ?.
The selectivity of the triplet BCM rule remains unchanged in the presence of noise.
Triplet BCM Rule
We now show that by modifying the update rule to incorporate information from triplets of input
vectors, the generality of the input data can be dramatically increased. Our new BCM rule will learn
selectivity for arbitrary mixture distributions, and learn weights which in expectation are selective
for only one mixture component. Assume that
X
P (d) =
?k Pk (d)
k
where EPk [d] = dk . For example, the data could be a mixture of axis-aligned Gaussians, a mixture
of independent Poisson variables, or mixtures of independent Bernoulli random variables to name a
few. We also require EPk [kdk2 ] < ?. We emphasize that we do not require our data to come from
any parametric distribution.
We interpret k to be a latent variable that signals the hidden cause of the underlying input distribution, with distribution Pk . Critically, we assume that the hidden variable k changes slowly compared
to the inter-spike period of the neuron. In particular, we need at least 3 samples from each Pk . This
corresponds to the multi-view assumption of [2]. A particularly relevant model meeting this assumption is that of spike counts in disjoint intervals under a Poisson process, with a discrete, time
varying rate parameter. For the purpose of this paper, we assume the number of mixed distributions,
k, is equal to the number of dimensions, n, however it is possible to relax this to k < n.
Let {d1 , d2 , d3 } be a triplet of independent copies from some Pk (d), i.e. each are drawn from
the same mixture. It is critical to note that if {d1 , d2 , d3 } are not drawn from the same class, this
update will not converge to the global maximum. Numerical experiments show this assumption can
be violated somewhat without severe changes to the fixed points of the algorithm. Our sample
i
is
i
then a sequence of triplets, each triplet drawn from the same latent distribution. Let c = d , m .
With these independent triples, we note that the tensors T and M from equation (2) can be written
as moments of the independent triplets
T = E[d1 ? d2 ? d3 ]
M = E[d1 ? d2 ]
1
1
R(m) = T (m, m, m) ? M (m, m)2
3
4
This is precisely the same objective function optimized by the classical BCM update, with the conditional means of the mixture distributions taking the place of discrete data points. With access to
independent triplets, selectivity for significantly richer input distributions can be learned.
5
As with classical BCM, we can perform gradient ascent in this objective function which leads to the
expected update
E[?R] = E[c1 c2 d3 + (c1 d2 + c2 d1 )(c3 ? 2?)]
where ? = E[c1 c2 ]. This update is rather complicated, and couples pre and post synaptic firing rates
across multiple time intervals. Since each ci and di are identically distributed, this expectation is
equal to
E[c2 (c3 ? ?)d1 ]
which suggests a much simpler update. This ordering was chosen to match the spike timing dependency of synaptic modification. This update depends on the presynaptic input, and the postsynaptic
excitation in two disjoint time periods.
Definition 4.1 (Full-rank Triplet BCM). The full-rank Triplet BCM update rule is:
mt = ?(mt?1 + ?t ?(c2 , c3 , ?t?1 )d1 )
(4)
P
P 2
2 3
2 3
where ?(c , c , ?) = c (c ? ?), the step size ?t obeys t ?t ? ?, and t ?t < ?. ? is a
projection into an arbitrary large compact ball, which is needed for technical reasons to guarantee
convergence.
5
Stochastic Approximation
Having found the stable points of the expected update for BCM and triplet BCM, we now turn to
a proof of convergence for the stochastic update generated by mixture models. For this, we turn to
results from the theory of stochastic approximation.
We will decompose our update into two parts, the expected update, and the (random) deviation.
This deviation will be a L2 bounded martingale, while the expected update will be a ODE with the
previously calculated stable points. Since the expected update is the gradient of a objective function
R, the Lyapunov functions required for the stability analysis are simply this objective function.
The decomposition of the triplet BCM stochastic process is as follows:
mt ? mt?1 = ?t ?(c2t , c3t , ?t?1 )d1
= ?n E[?(c2 , c3 , ?t?1 )d1 ] + ?n ?(c2 , c3 , ?t?1 )d1 ? E[?(c2 , c3 , ?t?1 )d1 ]
= ?t h(mt ) ? ?t ?t
Here, h(mt ) is the deterministic expected update, and ?t is a martingale. All our expectations are
taken with respect to triplets of input data. The decomposition for classical BCM is similar.
This is the Doob decomposition [8] of the sequence. Using a theorem of Delyon [7], we will show
that our triplet BCM algorithm will converge with probability 1 to the stable points of the expected
update. As was shown previously, these stable points are selective for one and only one mixture
component in expectation.
Theorem 5.1. For the full rank case, the projected update converges w.p. 1 to the zeros of ??
Proof. See supplementary material, or an extended discussion in a forthcoming arXiv preprint [13].
6
Triplet BCM Explains STDP Up to Spike Triplets
Biophysically synaptic efficiency in the brain is more closely modeled by spike timing dependent
plasticity (STDP). It depends precisely on the interval between pre- and post-synaptic spikes. Initial
research on spike pairs [15, 3] showed that a presynaptic spike followed in close succession by
a postsynaptic spike tended to strengthen a synapse, while the reverse timing weakened it. Later
work on natural spike chains [9], triplets of spikes [4, 19], and quadruplets have shown interaction
effects beyond pairs. Most closely to ours, recent work by Pfister and Gerstner [17] suggested that
a synaptic modification function depending only on spike triplets is sufficient to explain all current
experimental data. Furthermore, their rule resembles a BCM learning rule when the pre- and postsynaptic firing distributions are independent Poisson.
6
We now demonstrate that our learning rule can model both the pairwise and triplet results from
Pfister and Gerstner using a smaller number of free parameters and without the introduction of
hidden leaky timing variables. Instead, we work directly with the pre- and post-synaptic voltages,
and model the natural voltage decay during the falling phase of an action potential. Our (four)
free variables are the voltage decay, which we set within reasonable biological limits; a bin width,
controlling the distance between spiking triplet periods; ?, our sliding voltage threshold; and an
overall multiplicative constant. We emphasize that our model was not designed to fit these data; it
was designed to learn selectivity for the multi-view mixture task. Spike timing dependence falls out
as a natural consequence of our multi-view assumption.
Change in EPSC Amplitude (%)
100
50
0
?50
?100
?80
?60
?40
?20
0
20
Spike Timing (ms)
40
60
80
100
Figure 2: Fit of triplet BCM learning rule to synaptic strength STDP curve from [3]. Data points
were recreated from [3] . Spike timing measures the time between post synaptic and presynaptic
spikes, tpost ? tpre . A positive time means the presynaptic spike was followed by a postsynaptic
spike.
We first model hippocampus data from Mu-ming Poo [3], who applied repeated electrical stimulation to the pre- and post-synaptic neurons in a pairing protocol within which the relative timing of
the two spike chains was varied. After repeated stimulation at a fixed timing offset, the change in
synaptic strength (postsynaptic current) was measured.
We take the average voltage in triplet intervals to be the measure of pre- and post-synaptic activity,
and consider a one-dimensional version of our synaptic update:
?m = Ac2 (c3 ? ?)d1
(5)
2
3
where c and c are the postsynaptic voltage averaged over the second and third time bins, and d1
is the presynaptic voltage averaged over the first time bin. We assume our pre and post synaptic
voltages are governed by the differential equation:
dV
= ?? V
(6)
dt
such that, if t = sk where sk is the kth spike, V (t) ? 1. That is, the voltage is set to 1 at each spike
time before decaying again.
Let Vpre be the presynaptic voltage trace, and Vpost be the postsynaptic voltage trace. They are
determined by the timing of pre- and post-synaptic spikes, which occur at r1 , r2 , . . . , rn for the
presynaptic spikes, and o1 , o2 , . . . om for the postsynaptic spikes.
To model the pairwise experiments, we let ri = r0 + iT where T = 1000ms, a large time constant.
Then oi = ri + ?t where ?t is the spike timing. Let ?b be the size of the bins. That is to say,
Z t+ ?2b
Z t+ ?2b
1
0
0
2
d (t) =
Vpre (t + ?b )dt
c (t) =
Vpost (t0 )dt0
?b
2
?
t+ 2b
t?
c3 (t) =
Z
t?
?b
2
t?
Vpost (t0 ? ?b )dt0
?b
2
Vpost (t) = Vpre (t ? ?t )
Then the overall synaptic modification is given by
Z
Ac2 (t)(c3 (t) ? ?)d1 (t)dt
t
7
We fit A, ? , ?, and the bin size of integration. Recall that the sliding threshold, ? is a function of the
expected firing rate of the neuron. Therefore we would not expect it to be a fixed constant. Instead,
it should vary slowly over a time period much longer than the data sampling period. For the purpose
of these experiments it would be at an unknown level that depends on the history of neural activity.
See figure 2 for the fit for Mu-ming Poo?s synaptic modification data.
Froemke and Dan also investigated higher order spike chains, and found that two spikes in short
succession did not simply multiply in their effects. This would be the expected result if the spike
timing dependence treated each pair in a triplet as an independent event. Instead, they found that a
presynaptic spike followed by two postsynaptic spikes resulted in significantly less excitation than
expected if the two pairs were treated as independent events. They posited that repeated spikes
interacted suppressively, and fit a model based on that hypothesis. They performed two triplet experiments with pre- pre-post triplets, and pre-post-post triplets. Results of their experiment along
with the predictions based on our model are presented in figure 3.
Figure 3: Measured excitation and inhibition for spike triplets from Froemke and Dan are demarcated in circles and triangles. A red circle or triangle indicates excitation, while a blue circle or
triangle indicates inhibition. The predicted results from our model are indicated by the background
color. Numerical results for our model, with boundaries for the Froemke and Dan model are reproduced.
Left figure is pairs of presynaptic spikes, and a single post-synaptic spike. The right figure is pairs of
postsynaptic spikes, and a presynaptic spike. For each figure, t1 measures the time between the first
paired spike with the singleton spike, with the convention that each t is positive if the presynaptic
spike happens before the post synaptic spike. See paired STDP experiments for our spiking model.
For the top figure, ? = .65, our bin width was 2ms, and our spike voltage decay rate ? = 8ms. For
the right figure ? = .45. Red is excitatory, blue is inhibitory, white is no modification. A positive t
indicates a presynaptic spike occurred before a postsynaptic spike.
7
Conclusion
We introduced a modified formulation of the classical BCM neural update rule. This update rule
drives the synaptic weights toward the components of a tensor decomposition of the input data.
By further modifying the update to incorporate information from triplets of input data, this tensor decomposition can learn the mixture means for a broad class of mixture distributions. Unlike
other methods to fit mixture models, we incorporate a multiview assumption that allows us to learn
asymptotically exact mixture means, rather than local maxima of a similarity measure. This is in
stark contrast to EM and other gradient ascent based methods, which have limited guarantees about
the quality of their results. Conceptually our model suggests a different view of spike waves during
adjacent time epochs: they provide multiple independent samples of the presynaptic ?image.?
Due to size constraints, this abstract has has skipped some details, particularly in the experimental
sections. More detailed explanations will be provided in future publications.
Research supported by NSF, NIH, The Paul Allen Foundation, and The Simons Foundation.
8
References
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[4] Guo-Qiang Bi and Huai-Xing Wang. Temporal asymmetry in spike timing-dependent synaptic
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9
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4,791 | 5,338 | A Bayesian model for identifying hierarchically
organised states in neural population activity
Patrick Putzky1,2,3 , Florian Franzen1,2,3 , Giacomo Bassetto1,3 , Jakob H. Macke1,3
1
Max Planck Institute for Biological Cybernetics, T?
ubingen
2
Graduate Training Centre of Neuroscience, University of T?
ubingen
3
Bernstein Center for Computational Neuroscience, T?
ubingen
[email protected], [email protected]
[email protected], [email protected]
Abstract
Neural population activity in cortical circuits is not solely driven by external inputs, but is also modulated by endogenous states which vary on
multiple time-scales. To understand information processing in cortical circuits, we need to understand the statistical structure of internal states
and their interaction with sensory inputs. Here, we present a statistical
model for extracting hierarchically organised neural population states from
multi-channel recordings of neural spiking activity. Population states are
modelled using a hidden Markov decision tree with state-dependent tuning
parameters and a generalised linear observation model. We present a variational Bayesian inference algorithm for estimating the posterior distribution
over parameters from neural population recordings. On simulated data, we
show that we can identify the underlying sequence of population states and
reconstruct the ground truth parameters. Using population recordings from
visual cortex, we find that a model with two levels of population states outperforms both a one-state and a two-state generalised linear model. Finally,
we find that modelling of state-dependence also improves the accuracy with
which sensory stimuli can be decoded from the population response.
1
Introduction
It has long been recognised that the firing properties of cortical neurons are not constant
over time, but that neural systems can exhibit multiple distinct firing regimes. For example,
cortical circuits can be in a ?synchronised? state during slow-wave sleep, exhibiting synchronised fluctuations of neural excitability [1] or in a ?desynchronised? state in which firing is
irregular. Neural activity in anaesthetised animals exhibits distinct states which lead to
widespread modulations of neural firing rates and contribute to cross-neural correlations
[2]. Changes in network state can be brought about through the influence of inter-area
interactions [3] and affect communication between cortical and subcortical structures [4].
Given the strong impact of cortical states on neural firing [3, 5, 4], an understanding of the
interplay between internal states and external stimuli is essential for understanding how populations of cortical neurons collectively process information. Multi-cell recording techniques
allow to record neural activity from dozens or even hundreds of neurons simultaneously,
making it possible to identify the signatures of underlying states by fitting appropriate
statistical models to neural population activity.
It is thought that the state-dependence of neocortical circuits is not well described using a
global bi-modal state. Instead, the structure of cortical states is more accurately described
1
Figure 1: Illustration of the model. A) Generative model. At time t, the cortical state st
is determined using a Hidden Markov Decision Tree (HMDT) and depends on the previous
state st?1 , population activity yt?1 and on the current stimulus xt . In our simulations,
we assumed that the first split of the tree determined whether to transition into an up
or down-state. Up-states contained transient periods of high firing across the population
(up-high) as well as sustained periods of irregular firing (up-low). Each cortical state is
then associated with different spike-generation dynamics, modelling state-dependence of
firing properties such as ?burstiness?. B) State-transition probabilities depend on the treestructure. Transition matrices are depicted as Hinton diagrams where each block represents
a probability and each column sums to 1. Each row corresponds to the possible future state
st (see colour), and each column to the current state.
(1) A model in which transition-probabilities in the first level of the tree (up/down) are
biased towards the up-state (green squares are bigger than gray ones), and weakly depend on
the previous state st?1 . In this example, both high/low phases are equally likely within upstates (second level of tree, depicted in second column) and do not depend on the previous
state (all orange/red squares have same size). The resulting 3 ? 3 matrix of transition
probabilities across all states can be calculated from the transition-probabilities in the tree.
(2) Changing the properties of the second-level node only leads to a local change in the
transition matrix: It affects the proportion between the orange/red states, but leaves the
green state unchanged.
using multiple states which vary both between and within brain regions [6]. In addition,
the ?state? of a neural population can vary across multiple time scales from milliseconds to
seconds or more [6]: For example, cortical recordings can switch between up- and downphases. During an up-phase cortical activity can exhibit ?volleys? of synchronised activity
[7]?sometimes referred to as population bursts?which can be modelled as transient states.
These observations suggest that the structure of cortical states could be captured by a
hierarchical organisation in which each state can give rise to multiple temporally nested
?sub-states?. This structure naturally yields a binary tree: States can be divided into subclasses, with states further down the tree operating at faster time-scales determined by
their parent node. We hypothesise that other cortical states also exhibit similar hierarchical
structure. Our goal here is to provide a statistical model which can identify cortical states
and their hierarchical organisation from recordings of population activity. As a running
example of such a hierarchical organisation we use a model in which the population exhibits
synchronised population bursts during up-states, but not during down-states. This system
is modelled using a first level of state (up/down), and for which the up-state is further
divided into two states (transient high-firing events and normal firing, see 1A).
We present an inhomogeneous hidden Markov model (HMM) [8] to model the temporal
dynamics of state-transitions [9, 10]. Our approach is most closely related to [10], who
developed a state-dependent generalised linear model [11] in which both the tuning prop2
erties and state-transitions can be modelled to depend on external covariates. However,
our formulation also allows for hierarchically organised state-structures. In addition, previous population models based on discrete latent states [10, 12] used point-estimation for
parameter learning. In contrast, we present algorithms for full Bayesian inference over the
parameters of our model, making it possible to identify states in smaller or noisier data
[13]. This is important for neural population recordings which are typically characterised
by short recording times relative to the dimensionality of the data and by high variability.
In addition, estimates of posterior distributions are important for visualising uncertainty
and for optimising experimental paradigms with active-learning methods [14, 15].
2
Methods
We use a hidden Markov decision tree (HMDT) [16] to model hierarchically organised states
with binary splits and a generalised linear observation model (GLM). An HMDT combines
the properties of a hidden Markov model (to model temporal structure) with a hierarchical
mixture of experts (HME, to model a hierarchy of latent states) [17]. In general the hierarchical approach can represent richer dependence of states on external covariates, analogous
to the difference between multi-class logistic regression and multi-class binary decision trees.
For example, a two-level binary tree can separate four point clouds situated at the corners
of a square whereas a 4-class multinomial regression cannot. We use Bayesian logistic regression [18] to model transition gates and emissions. In the following, we describe the model
structure and propose a variational algorithm [8, 19] for inferring its parameters.
2.1
Hierarchical hidden Markov model for multivariate binary data
We consider discrete time-series data of multivariate binary1 neural spiking events yt ?
{0, 1}C where C is the number of cells. We assume that neural spiking can be influenced
by (observed) covariates xt ? RD . The covariates xt could represent external stimuli,
spiking history of neurons or other measures such as the total population spike count. In
our analyses below, we assume that correlations across neurons arise only from the joint
coupling to the population state, and we do not include couplings between neurons as is
sometimes done with GLMs [11]. Dependence of neural firing on internal states is modelled
by including a 1-of-K latent state vector st , where K is the number of latent states. The
emission probabilities for the observable vector yt (i.e. the probability of spiking for each
neuron) are thus given by
C
K Y
s(i)
Y
(c) (c)
(c) t
,
(1)
p yt |xt , ?i
p (yt |xt , st , ?) =
i=1 c=1
where ? is a set of model parameters. We allow the external covariate xt to be different for
each neuron c.
To model temporal dynamics over st , we use a hidden Markov model (HMM) [10], where
the state transitions take the form
(j)
K Y
K
s(i)
Y
t st?1
(i) (j)
p (st |st?1 , xt , ?) =
p st |st?1 , xt , ?
,
(2)
i=1 j=1
where ? is a set of parameters of the transition model. The model allows state-transitions
to be dependent on an external input xt ? this can e.g. be used to model state-transitions
caused by stimulation of subcortical structures involved in controlling cortical states [20].
Moving beyond this standard input output HMM formulation [21], we introduce hierarchically organised auxiliary latent variables zt which represent the current state st through a
binary tree. Using HME terminology, we refer to the nodes representing zt as ?gates?. Each
of the K leaves of the tree (or, equivalently, each path through the tree) corresponds to one
of the K entries of st and we can thus represent st in the form
L
A(l,k)
A(l,k)
Y
L
R
(k)
(l)
(l)
st =
zt
1 ? zt
,
(3)
l=1
1
All derivations below can be generalised to model the emission probabilities by any kind of
generalised linear model.
3
where AL and AR are adjacency matrices which indicate whether state k is in the left or
right branch of gate l, respectively (see [19]). Using this representation, st is deterministic
given zt which significantly simplifies the inference process. The auxiliary latent variables
(l)
zt are Bernoulli random variables and we chose their conditional probability distribution
to be
(l)
(l)
(l)
p(zt = 1|xt , st?1 , vl ) = ? v>
u
.
(4)
t
l
Here, ?(?) is the logistic sigmoid, vl are the parameters of the l-th gate and ut represents
a concatenation of the previous state st?1 , the input xt (which could for example represent
population firing rate, time in trial or an external stimulus) and a constant term of unit
(l)
value to model the prior probability of z0 = 1. This parametrisation significantly reduces
the number of parameters used for the transition probabilities as compared to [10]. To
enforce stronger temporal locality and less jumping between states we could also reduce
this probability to be conditioned only on previous activations of a sub-tree of the HMDT
instead of all population states.
2.2
Learning & Inference
For posterior inference over the model parameters we would need to infer the joint distribution over all stochastic variables conditioned on X,
p (Y, S, ?, ?, ?, ?|X) =p (Y|S, X, ?) p (S|X, ?) p (?|?) p (?) p (?|?) p (?)
(5)
where Y is the set of yt ?s, ? and ? are the sets of parameters for the emission and gating
distributions, respectively, and ? and ? are the hyperparameters for the parameter priors.
Since there is no closed form solution for this distribution, we use a variational approximation
[8]. We assume that the posterior factorises as
q (S, ?, ?, ?, ?) =q (S) q (?) q (?) q (?) q (?)
(6)
L
K Y
C
Y
Y
(c)
(c)
q (? l ) q (? l ) ,
(7)
=q (S)
q ? k q ?k
k=1 c=1
l=1
and find the variational approximation to the posterior over parameters, q (S, ?, ?, ?, ?),
by optimising the variational lower bound L(q) to the evidence
X ZZZZ
p (Y, S, ?, ?, ?, ?|X)
L(q) :=
q (S, ?, ?, ?, ?) ln
d?d?d?d?
(8)
q (S, ?, ?, ?, ?)
S
X ZZZZ
? ln
p (Y, S, ?, ?, ?, ?|X) d?d?d?d? = ln p (Y|X) .
(9)
S
We use variational Expectation-Maximisation (VBEM) to perform alternating updates on
the posterior over latent state variables and the posterior over model parameters. To infer
the posterior over latent variables (i.e. responsibilities), we use a modified forward-backward
algorithm as proposed in [22] (see also [8]). In order to perform the forward and backward
steps, they propose the use of subnormalised probabilities of the form
i
h
(i) (j)
(i) (j)
(10)
p? st |st?1 , xt , ? := exp E? ln p st |st?1 , xt , ?
p? (yt |xt , ?i ) := exp (E?i [ln p (yt |xt , ?i )])
(11)
for the state-transition probabilities and emission probabilities. Since all relevant probabilities in our model are over discrete variables, it would be straightforward to normalise those
probabilities, but we found that normalisation did not noticeably change results.
With the approximations from above, the forward probability can thus be written as
K
X
1
(i)
(i)
(j)
(i) (j)
? st
=
p? yt |st , xt , ?
? st?1 p? st |st?1 , xt , ? ,
(12)
C?t
j=1
(i)
(i)
where ?(st ) is the probability-mass of state st given previous time steps and C?t is a
normalisation constant. Similar to the forward step, the backward recursion takes the form
4
K
1 X (j)
(i)
(j)
(j)
(i)
? st
=
?t st+1 p? yt+1 |st+1 , xt+1 , ? p? st+1 |st , xt , ? .
C?t
(13)
j=1
Using the forward and backward equation steps we can infer the state posteriors [8]. Given
the state posteriors, the logarithm of the approximate parameter posterior for each of the
nodes takes the form
T
X
(n)
(n) (n)
ln q ? (? n ) =
?t ln p ?t |xt , ? n , (. . . ) + E? n [ln p (? n |? n )] + const.
(14)
t=1
where ? n are the parameters of the n-th node and p (? n |? n ) is the prior over the param(n)
eters. Here, ?t is the posterior responsibility or estimated influence of node n on the tth
(n)
observation and ?t denotes the expected output (known for state nodes) of node n (see
supplement for details). This equation also holds for a tree structure with multinomial gates
and for non-binary emission models such as Poisson and linear models. The above equations
are valid for maximum likelihood inference, except that all parameter priors are removed,
and the expectations of log-likelihoods reduce to log-likelihoods We use logistic regression
for all emission probabilities and gates, and a local variational approximation to the logistic
sigmoid as presented in [18].
As parameter priors we use anisotropic Gaussians with individual Gamma priors on each
diagonal entry of the precision matrix. With this prior structure we can perform automatic
relevance determination [23]. We chose shape parameter a0 =1 ? 10?2 and rate parameter
b0 = 1 ? 10?4 , leading to a broad Gamma hyperprior [19]. In many applications, it will
be reasonable to assume that neurons in close-by states of the tree show similar response
characteristics (similar parameters). The hierarchical organisation of the model yields a
natural structure for hierarchical priors which can encourage parameter similarity2 .
2.3
Details of simulated and neurophysiological data
To assess and illustrate our model, we simulated a population recording with trials of 3 s
length (20 neurons, 10 ms time bins). As illustrated in Fig. 1 A, we modelled one low-firingrate down state (down, base firing rate 0.5 Hz) and two up states (up-low and up-high, with
base firing rates of 5, and 50 Hz respectively). The root node switched between up and
down states, whereas a second node controlled transitions between the two types of upstates. Up-high states only occurred transiently, modelling synchronised bouts of activity.
In the down state, neurons have a 10 ms refractory period, during up states they exhibit
bursting activity. Transitions from down to up go mainly via up-high to up-low, while downtransitions go from up-low to down; stimulation increases the probability of being in one of
the up states. A pulse-stimulus occurred at time 1 s of each trial. Each model was fit on a
set of 20 trials and evaluated on a different test set of 20 trials. For each training set, 24
random parameter initialisations were drawn and the one with highest evidence was chosen
for evaluation. State predictions were evaluated using the Viterbi algorithm [24, Ch. 13].
We analysed a recording from visual cortex (V1) of an anaesthetised macaque [2]. The
data-set consisted of 1600 presentations of drifting gratings (16 directions, 100 trials each),
each lasting 2 s. Experimental details are described in [2]. For each trial, we kept a segment
of 500 ms before and after a stimulus presentation, resulting in trials of length 3 s each. We
binned and binarised spike trains in 50 ms bins. Additional spikes (present in (5.45 ? 1.56) %
of bins) were discarded by the binarisation procedure. We chose the representation of
the stimulus to be the outer product of the two vectors [1, sin(?), cos(?)], where ? is
the phase of the grating, and [1, sin(?), cos(?), sin(2?), cos(2?)] for the direction ? of the
grating. This resulted in a 15 dimensional stimulus-parametrisation, and made it possible to
represent tuning-curves with orientation and direction selectivity, as well as modulation of
firing rates by stimulus phase. The only gate input was chosen to be an indicator function
with unit value during stimulus presentation and zero value otherwise. Post-spike filters
were parametrised using five cubic b-splines for the last 10 bins with a bin width of 50 ms.
2
See supplement for an example of how this could be implemented with Gaussian priors.
5
Figure 2: Performance of the model on simulated data. A) Example rasters sampled using ground truth (GT) parameters, colors indicate sequence of underlying population
states. B) For the sample from (A), the state-sequence decoded with our variational Bayes
(VB) method matches the decoded sequence using GT parameters. C) Comparison of statedecoding performance using GT parameters, VB and maximum likelihood (ML) learning
(Wilcoxon ranksum, * p < 0.05; *** p 0.001). D) Model performance quantified using
per-data-point log-likelihood difference between estimated and GT-model on test-set. Our
VB method outperforms ML (Wilcoxon ranksum, *** p 0.001), and both models considerably outperform a 1-state GLM (not shown). E) Estimated post-spike filters match
the GT values well (depicted are the filters from one of the cross-validated models). F)
Comparison of the autocorrelation of the ground truth data and samples drawn from the
VB fit as in (E). G) GT (top) and VB estimated (bottom) transition matrices in absence
(left) or presence (right) of a stimulus.
3
3.1
Results
Results on simulated data
To illustrate our model and to evaluate the estimation procedure on data with known ground
truth, we used a simulated population recording of 20 neurons by sampling from our model
(details in Methods, see Fig. 2 A). In this simulation, the up-state had much higher firing
rates than the down-state. It was therefore possible to decode the underlying states from
the population spike trains with high accuracy (Fig. 2 B). For the VB method, we used
the posterior mean over parameters for state-inference. In addition, we compared both of
these approaches to state-decoding based on a model estimated using maximum likelihood
learning. All three models showed similar performance, but the decoding advantage of the
3-state VB model was statistically significant (using pairwise comparisons, Fig. 2 C).
We also directly evaluated performance of the VB and ML methods for parameter estimation
by calculating the log-likelihood of the data on held-out test-data, and found that our VB
method performed significantly better than the ML method (Fig. 2 D). Finally, we also
compared the estimated post-spike filters (Fig. 2 E), auto-correlation functions (Fig. 2 F)
and state-transition matrices (Fig. 2 G) and found an excellent agreement between the GT
parameters and the estimates returned by VB.
To test whether the VB method is able to determine the correct model complexity, we
fit an over-parameterised model with 3 layers and potentially 8 states to the simulation
data. The best model fit from 200 random restarts (lower bound of ?2.24 ? 104 , no crossvalidation, results not shown) only used 3 out of the 8 possible states (the other 5 states
had a probability of less than 0.5 %). Therefore, in this example, the best lower bound is
achieved by a model with correct, and low, complexity.
6
C 50
*** *** *** *** ***
0
-1
40
30
20
10
-2
90
180 270
direction (deg)
360
0
0
360
0
0
1000
2000
time (ms)
1.5
1
0.5
0
0
G
10
10
10
250
500
empirical ITIs (ms)
750
population rate (%)
F
sampeled ITIs
events per trial
E
90
180 270
direction (deg)
0.2
0
-2
-4
-2
10
empirical ITIs
10
0
1S
1S
2S
3S
1
0.5
0
0
1000
2000
time (ms)
50
250
time (ms)
500
H 50
40
30
20
10
0
PR
d
le
co
up
5
0
L
3S
1S
PR
le
d
10
v
1.5
modulation
5
0.4
iv
15
spikes (hz)
10
iii
0.6
population rate (%)
0
co
up
M
0.2
ii
15
spikes (hz)
p?(spike)
0.4
0
1S-GLM
empirical
i
pt(spike|?=67.5?)
sampled
D0.6
1S
0
2S
2000
1S
1000
3S
0
L
-500
M
?loglikelihood
B
1S
down
up low-rate
accuracy (%)
up high-rate
A
0
5
10
number of spikes (per bin)
40
30
20
10
0
0
5
10
number of spikes (per bin)
Figure 3: Results for population recordings from V1. A) Raster plot of population
response to a drifting grating with orientation 67.5? . Arrows indicate stimulus onset and
offset, colours show the most likely state sequence inferred with the 3-state variational Bayes
(3S-VB) model. B) Cross-validated log-likelihoods per trial, relative to the 3S-VB model.
C) Stimulus decoding performance, in percentage of correctly decoded stimuli (16 discrete
stimuli, chance level 6.25 %), using maximum-likelihood decoding.
D) Tuning properties of an example neuron. i) Orientation tuning calculated from the
tuning-parameters of 3S-VB (red, orange, green) or 1-state GLM (purple). iii) Temporal
component of tuning parameters. ii) Orientation tuning measured from sampled data of the
estimated model, each line representing one state. Note that the firing rate also depends
on state-transitions and post-spike filters. iv) Peri-stimulus time-histograms (PSTHs) estimated from samples of the estimated models. v) Post-spike filters for each state, and
comparison with 1-state GLM (purple). E) Distributions of times spent in each state, i.e.
inter-transition intervals (ITIs), estimated from the empirical data using 3S-VB. F) Comparison between distribution of ITIs in samples from model 3S-VB and in the Viterbi-decoded
path (from E).
G) Histogram of population rates (i.e. number of synchronous spikes across the population in each 50 ms bin) for 3S-VB (blue), 1S (purple), and data (gray). H) Histograms of
population rate for each state.
3.2
Results on neurophysiological recordings
We analysed a neural population recording from V1 to determine whether we could successfully identify cortical states by decoding the activity of the neural population, and whether
accounting for state-dependence resulted in a more accurate statistical model of neural firing.
While neurons generally responded robustly to the stimulus (3 D), firing rates were strongly
modulated by internal states [2] (Fig. 3 A). We fit different models to data, and found that
our 3-state model estimated with VB resulted in better cross-validation performance than
either the 3-state model estimated with ML, the 2-state model or a 1-state GLM (i.e. a
GLM without cross-neural couplings, Fig. 3 B). In addition we fit a fully coupled GLM
(with cross-history terms as in [11, 13]), as well as one in which the total population count
was used as a history feature using VB. These models were intermediate between the 1-state
GLM and the 2-state model, i.e. both worse than the 3-state one. A ?flat? 3-states model
with a single multinomial gate estimated with ML performed similarly to the hierarchical
3S-ML model. This is to be expected, as any differences in expressive power between the
two models will only become substantial for a different choice of xt or larger models.
7
We also evaluated the ability of different models to decode the stimulus, (i.e. the direction
of the presented grating) from population spike trains. We evaluated the likelihood of each
population spike train for each of the 16 stimulus directions, and decoded the stimulus which
yielded the highest likelihood. The 3-state VB model shows best decoding performance
among all tested models (3 C), and all models with state-dependence (3-state VB, 3-state
ML, 2-state) outperformed the 1-state GLM. We sampled from the estimated 3S-VB model
to evaluate to what extent the model captures the tuning properties of neurons (Fig. 3 D(ii
& iv)). The example neuron shows strong modulation of base firing rate dependent on the
population state, but not a qualitative change of the tuning properties (Fig. 3 D i-iv). The
down-state post-spike filter (Fig. 3 D v) exhibits a small oscillatory component which is not
present in the post-spike filters of the other states or the 1-state GLM.
Investigation of inter-transition-interval (ITI) distributions from the data (after Viterbidecoding) shows heavy tails (Fig. 3 E). Comparison of ITI-distribution estimated from the
empirical data and from sampled data (3S-VB) show good agreement, apart from small
deficiencies of the model to capture the heavy tails of the empirical ITI distribution (Fig.
3 F). Finally, population rates (i.e. total number of spikes across the population) are often
used as a summary-measure for characterizing cortical states [6]. We found that the distribution of population rates in the data was well matched by the distribution estimated
from our model (Fig. 3 G) with the three states having markedly different population rate
distributions (Fig. 3 H). Although a 1-state GLM also captured the tuning-properties of
this neuron (Fig. 3 D) it failed to recover the distribution of population rates (Fig. 3 G).
4
Discussion
We presented a statistical method for extracting cortical states from multi-cell recordings of
spiking activity. Our model is based on a ?state-dependent? GLM [10] in which the states are
organised hierarchically and evolve over time according to a hidden Markov model. Whether,
and in which situations, the best descriptions of cortical states are multi-dimensional, discrete or continuous [25, 2] is an open question [6], and models like the one presented here
will help shed light on these questions. We showed that the use of variational inference
methods makes it possible to estimate the posterior over parameters. Bayesian inference
provides better model performance on limited data [13], uncertainty information, and is
also an important building block for active learning approaches [14]. Finally, it can be used
to determine the best model complexity: For example, one could start inference with a
model containing only one state and iteratively add states (as in divisive clustering) until
the variational bound stops increasing.
Cortical states can have a substantial impact on the firing and coding properties of cortical
neurons [6] and interact with inter-area communication [4, 3]. Therefore, a better understanding of the interplay between cortical states and sensory information, and the role of
cortical states in gating information in local cortical circuits will be indispensable for our
understanding of how populations of neurons collectively process information. Advances in
experimental technology enable us to record neural activity in large populations of neurons
distributed across brain areas. This makes it possible to empirically study how cortical
states vary across the brain, to identify pathways which influence state, and ultimately to
understand their role in neural coding and computation. The combination of such data with
statistical methods for identifying the organisation of cortical states holds great promise for
making progress on understanding state-dependent information processing in the brain.
Acknowledgements
We are grateful to the authors of [2] for sharing their data (toliaslab.org/publications/eckeret-al-2014/) and to Alexander Ecker, William McGhee, Marcel Nonnenmacher and David
Janssen for comments on the manuscript. This work was funded by the German Federal Ministry of Education and Research (BMBF; FKZ: 01GQ1002, Bernstein Center T?
ubingen) and
the Max Planck Society. Supplementary details and code are available at www.mackelab.org.
8
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9
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4,792 | 5,339 | Clustered factor analysis of multineuronal spike data
Lars Buesing1 , Timothy A. Machado1,2 , John P. Cunningham1 and Liam Paninski1
1
Department of Statistics, Center for Theoretical Neuroscience
& Grossman Center for the Statistics of Mind
2
Howard Hughes Medical Institute & Department of Neuroscience
Columbia University, New York, NY
{lars,cunningham,liam}@stat.columbia.edu
Abstract
High-dimensional, simultaneous recordings of neural spiking activity are often
explored, analyzed and visualized with the help of latent variable or factor models. Such models are however ill-equipped to extract structure beyond shared,
distributed aspects of firing activity across multiple cells. Here, we extend unstructured factor models by proposing a model that discovers subpopulations or
groups of cells from the pool of recorded neurons. The model combines aspects
of mixture of factor analyzer models for capturing clustering structure, and aspects of latent dynamical system models for capturing temporal dependencies. In
the resulting model, we infer the subpopulations and the latent factors from data
using variational inference and model parameters are estimated by Expectation
Maximization (EM). We also address the crucial problem of initializing parameters for EM by extending a sparse subspace clustering algorithm to integer-valued
spike count observations. We illustrate the merits of the proposed model by applying it to calcium-imaging data from spinal cord neurons, and we show that it
uncovers meaningful clustering structure in the data.
1
Introduction
Recent progress in large-scale techniques for recording neural activity has made it possible to study
the joint firing statistics of 102 up to 105 cells at single-neuron resolution. Such data sets grant
unprecedented insight into the temporal and spatial structure of neural activity and will hopefully
lead to an improved understanding of neural coding and computation.
These recording techniques have spurred the development of statistical analysis tools which help to
make accessible the information contained in simultaneously recorded activity time-series. Amongst
these tools, latent variable models prove to be particularly useful for analyzing such data sets [1,
2, 3, 4]. They aim to capture shared structure in activity across different neurons and therefore
provide valuable summary statistics of high-dimensional data that can be used for exploratory data
analysis as well as for visualization purposes. The majority of latent variable models, however,
being relatively general purpose tools, are not designed to extract additional structure from the data.
This leads to latent variables that can be hard to interpret biologically. Furthermore, additional
information from other sources, such as spatial structure or genetic cell type information, cannot be
readily integrated into these models.
An approach to leveraging simultaneous activity recordings that is complementary to applying unstructured factor models, is to infer detailed circuit properties from the data. By modelling the
detailed interactions between neurons in a local micro-circuit, multiple tools aim at inferring the
existence, type, and strength of synaptic connections between neurons [5, 6]. In spite of algorithmic
progress [7], the feasibility of this approach has only been demonstrated in circuits of up to three
1
neurons [8], as large scale data with ground truth connectivity is currently only rarely available.
This lack of validation data sets also makes it difficult to asses the impact of model mismatch and
unobserved, highly-correlated noise sources (?common input?).
Here, we propose a statistical tool for analyzing multi-cell recordings that offers a middle ground
between unstructured latent variable models and models for inferring detailed network connectivity.
The basic goal of the model is to cluster neurons into groups based on their joint activity statistics.
Clustering is a ubiquitous and valuable tool in statistics and machine learning as it often yields
interpretable structure (a partition of the data), and is of particular relevance in neuroscience because
neurons often can be categorized into distinct groups based on their morphology, physiology, genetic
identity or stimulus-response properties. In many experimental setups, side-information allowing for
a reliable supervised partitioning of the recorded neurons is not available. Hence, the main goal of
the paper is to develop a method for clustering neurons based on their activity recordings.
We model the firing time-series of a cluster of neurons using latent factors, assuming that different
clusters are described by disjoint sets of factors. The resulting model is similar to a mixture of factor
analyzers [9, 10] with Poisson observations, where each mixture component describes a subpopulation of neurons. In contrast to a mixture of factor analyzers model which assumes independent
factors, we put a Markovian prior over the factors, capturing temporal dependencies of neural activity as well as interactions between different clusters over time. The resulting model, which we
call mixture of Poisson linear dynamical systems (mixPLDS) model, is able to capture more structure using the cluster assignments compared to latent variable models previously applied to neural
recordings, while at the same time still providing low-dimensional latent trajectories for each cluster
for exploratory data analysis and visualization. In contrast to the lack of connectivity ground truth
for neurons from large-scale recordings, there are indeed large-scale activity recordings available
that exhibit rich and biologically interpretable clustering structure, allowing for a validation of the
mixPLDS model in practice.
2
2.1
Mixture of Poisson linear dynamical systems for modelling neural
subpopulations
Model definition
Let ykt denote the observed spike count of neuron k = 1, . . . , K in time-bin t = 1, . . . , T . For
the mixture of Poisson linear dynamical systems (mixPLDS) model, we assume that each neuron k
belongs to exactly one of M groups (subpopulations, clusters), indicated by the discrete (categorical)
variable sk ? {1, . . . , M }. The sk are modelled as i.i.d.:
p(s)
=
K
Y
p(sk ) =
K
Y
Disc(sk |?0 ),
(1)
k=1
k=1
where ?0 := (?10 , . . . , ?M
0 ) are the natural parameters of the categorical distribution. In the remainder of the paper we use the convention that the group-index m = 1, . . . , M is written as superscript.
dm
The activity of each subpopulation m at time t is modeled by a latent variable xm
. We
t ? R
assume that these latent variables (we will also call them factors) are jointly normal and we model
interactions between different groups by a linear dynamical system (LDS) prior:
? 1?
? 11
?? 1 ?
xt?1
xt
A
? ? ? A1M
? .. ?
?
?
? ..
.
.. ? ? ... ?
xt = ? . ? = Axt?1 + ?t = ? .
(2)
? + ?t ,
xM
t
AM 1
m
???
AM M
xM
t?1
l
where the block-matrices Aml ? Rd ?d capture the interactions between groups m and l. The
innovations ?t are i.i.d. from N (0, Q) and the starting distribution is given by x1 ? N (?1 , Q1 ). If
neuron k belongs to group m, i.e. sk = m, we model its activity ykt at time t as Poisson distributed
spike count with a log-rate given by an affine combination of the factors of group m:
zkt | sk = m
ykt | zkt , sk
m m
= Ck:
xt
? Poisson(exp(zkt + bk )),
2
(3)
(4)
m
where b ? RK captures the baseline of the firing rates. We denote with C m ? RK?d the
m
group loading matrix with rows Ck:
for neurons k in group m and fill in the remaining rows
with 0s for all neurons not in group m. We concatenate these into the total loading matrix
PM
C := (C 1 ? ? ? C M ) ? RK?d , where d := m=1 dm is the total latent dimension. If the neurons are sorted with respect to their group membership, then the total loading C has block-diagonal
structure. Further, we denote with yk: := (yk,1 ? ? ? yk,T ) the activity time series of neuron k and
m
m
1?T
use an analogous notation for xm
for n = 1, . . . , dm . The model
n := (xn,1 ? ? ? xn,T ) ? R
parameters are ? := (A, Q, Q1 , ?1 , C, b); we consider the hyper-parameters ?0 to be given and
fixed.
For known clusters s, the mixPLDS model can be regarded as a special case of the Poisson linear
dynamical system (PLDS) model [3], where the loading C is block-diagonal. For unknown group
memberships s, the mixPLDS model defined above is similar to a mixture of factor analyzers (e.g.
see [9, 10]) with Poisson observations over neurons k = 1, . . . , K. In the mixPLDS model however,
we do not restrict the factors of the mixture components to be independent but allow for interactions
over time which are modeled by a LDS.
2.2
Variational inference and parameter estimation for the mixPLDS model
When applying the mixPLDS model to data y, we are interested in inferring the group memberships
s and the latent trajectories x as well as estimating the parameters ?. For known parameters ?,
the posterior p(x, s|y, ?) (even in the special case of a single mixture component M = 1) is not
available in closed form and needs approximating. Here we propose to approximate the posterior
using variational inference with the following factorization assumption:
p(x, s|y, ?) ? q(x)q(s).
(5)
We further restrict q(x) to be a normal distribution q(x) = N (x|m, V ) with
Q mean m and covariance
V . Under the assumption (5), q(s) further factorizes into the product k q(sk ) where q(sk ) is a
categorical distribution with natural parameters ?k = (?1k , . . . , ?M
k ). The variational parameters
m, V and ? = (?1 , . . . , ?K ) are obtained by maximizing the variational lower bound of the log
marginal likelihood log p(y|?):
L(m, V, ?, ?)
=
K
X
1
log |V | ? tr[??1 V ] ? (m ? ?)> ??1 (m ? ?) +
DKL [q(sk )kp(sk )]
2
k=1
+
M X
K X
T
X
m
m
?km (ykt hm
kt ? exp(hkt + ?kt /2)) + const
(6)
m=1 k=1 t=1
m
hm
t := C mt + b,
m
m>
?m
),
t := diag(C Vt C
?km ? exp(?m
k ),
where Vt = Covq(x) [xt ] and ? ? RdT , ? ? RdT ?dT are the mean and covariance of the LDS
prior over x. The first two terms in (6) are the Kullback-Leibler divergence between the prior
p(x, s) = p(x)p(s) and its approximation q(x)q(s), penalizing a variational posterior that is far
away from the prior. The third term in (6) is given by the expected log-likelihood of the data,
promoting a posterior approximation that explains the observed data well. We optimize L in a
coordinate ascent manner, i.e. we hold ? fixed and optimize jointly over m, V and vice versa. A
naive implementation of the optimization of L over {m, V } is prohibitively costly for data sets with
large T , as the posterior covariance V has O((dT )2 ) elements and has to be optimized over the set
of semi-definite matrices. Instead of solving this large program, we apply a method proposed in
[11], where the authors show that Gaussian variational inference for latent Gaussian models with
Poisson observations can be solved more efficiently using the dual problem. We generalize their
approach to the mixture of Poisson observation model (3) considered here, and we also leverage the
Markovian structure of the LDS prior to speed up computations (see below). In the supplementary
material, we derive this approach to inference in the mixPLDS model in detail. The optimization
over ? is available in closed form and is also given in the supplementary material. We iterate updates
over m, V and ?. In practice, this method converges very quickly, often requiring only two or three
iterations to reach a reasonable convergence criterion.
The most computationally intensive part of the proposed variational inference method is the update
of m, V . Using properties of the LDS prior (i.e. the prior precision ??1 is block-tri-diagonal),
3
we can show that evaluation of L, its dual and the gradient of the latter all cost O(KT d + T d3 ),
which is the same complexity as Kalman smoothing in a LDS with Gaussian observations or a
single iteration of Laplace inference over x. While having the same cost as Laplace approximation,
variational inference has the advantage of a non-deceasing variational lower bound L, which can be
used for monitoring convergence as well as for model comparison.
We can also get estimates for the model parameters by maximizing the lower bound L over ?. To
this end, we interleave updates of ? and m, V with maximizations over ?. The latter corresponds to
standard parameter updates in a LDS model with Poisson observations and are discussed e.g. in [3].
This procedure implements variational Expectation Maximization (VEM) in the mixPLDS model.
2.3
Initialization by Poisson subspace clustering
In principle, for a given number of groups M with given dimensions d1 , . . . , dM one can estimate
the parameters of the mixPLDS using VEM as described above. In practice we find however that
this yields poor results without having reasonable initial membership assignments s, i.e. reasonable
initial values for the variational parameters ?. Furthermore, VEM requires the a priori specification
of the latent dimensions d1 , . . . , dM . Here we show that a simple extension to an existing subspace
clustering algorithm provides, given the number of groups M , a sufficiently accurate initializer for
? and allows for an informed choice for the dimensions d1 , . . . , dM .
We first illustrate the connection of the mixPLDS model to the subspace clustering problem (for
a review of the latter see e.g. [12]). Assume that we observe the log-rates zkt defined in equation
(3) directly; we denote the corresponding data matrix as Z ? RK?T . For unknown loading C, the
m
row Zk: lies on a dm -dimensional subspace spanned by the ?basis-trajectories? xm
1,: , . . . , xdm ,: , if
neuron k is in group m. If s and x are unobserved, we only know that the rows of Z lie on a union
of M subspaces of dimensions d1 , . . . , dm in an ambient space of dimension T . Reconstructing the
subspaces and the subspace assignments is known as a subspace clustering problem and connections
to mixtures of factor analyzers have been pointed out in [13]. The authors of [13] propose to solve
the subspace clustering problem by the means of the following sparse regression problem:
1
min
kZ ? W Zk2F + ?kW k1
(7)
2
W ?RK?K
s.t.
diag(W ) = 0.
This optimization can be interpreted as trying to reconstruct each row Zk: by the remaining rows
Z\k: using sparse reconstruction weights W . Intuitively, a point on a subspace can be reconstructed
using the fewest reconstruction weights by points on the same subspace, i.e. Wkl = 0 if k and l lie
on different subspaces. The symmetrized, sign-less weights |W | + |W |> are then interpreted as the
adjacency matrix of a graph and spectral clustering, with a user defined number of clusters M , is
applied to obtain a subspace clustering solution. In the noise-free case (and taking ? ? 0 in eqn 7),
under linear independence assumptions on the subspaces, [13] shows that this procedure recovers
the correct subspace assignments.
If the matrix Z is not observed directly but only through the observation model (3), the subspace
clustering approach does not directly apply. The observed data Y generated from the model (3)
is corrupted by Poisson noise and furthermore the non-linear link function transforms the union of
subspaces into a union of manifolds. We can circumvent these problems using the simple observation that not only Z but also the rows Ck: of the loading matrix C lie on a union of subspaces of
dimensions d1 , . . . , dm (where the ambient space has dimension d). This can be easily seen from the
block-diagonal structure of C (if the neurons are sorted by their true cluster assignments) mentioned
in section 2.1. Hence we can use an estimate C? of the loading C as input to the subspace clustering
optimization (7). In order to get an initial estimate C? we can use a variety of dimensionality reduction methods with exp-Poisson observations, e.g. exponential family PCA [14], a nuclear norm
based method [15], subspace identification methods [16] and EM-based PLDS learning [16]; here
we use the nuclear norm based method [15] for reasons that will become obvious below. Because of
the non-identifiability of latent factor models, these methods only yield an estimate of C ? D with an
unknown, invertible transformation D ? Rd?d . Nevertheless, the rows of C ?D still lie on a union of
subspaces (which are however not axis-aligned anymore as is the case for C), and therefore the cluster assignments can still be recovered. Given these cluster assignments, we can get initial estimates
of the non-zero rows of C m by applying nuclear norm minimization to the individual clusters. This
4
method also returns a singular value spectrum associated with each subspace, which can be used to
determine the dimension dm . One can specify e.g. a threshold ?min , and determine the dimension
dm as the number of singular values > ?min .
2.4
The full parameter estimation algorithm
We briefly summarize the proposed parameter estimation algorithm for the mixPLDS model. The
procedure requires the user to define the number of groups M . This choice can either be informed
by biological prior knowledge or one can use standard model selection methods, such as crossvalidation on the variational approximation of the marginal likelihood. We first get an initial estimate C? of the total loading matrix by nuclear-norm-penalized Poisson dimensionality reduction.
Then, subspace clustering on C? yields initial group assignments. Based on these assignments, for
each cluster we estimate the group dimension dm and the group loading C? m . Keeping the cluster
assignments fixed, we do a few VEM steps in the mixPLDS model with an initial estimation for the
loading matrix given by (C? 1 , . . . , C? M ). This last step provides reasonable initial parameters for the
parameters A, Q, Q1 , ?1 of the dynamical system prior. Finally, we do full VEM iterations in the
mixPLDS model to refine the initial parameters. We monitor the increase of the variational lower
bound L and use its increments in a termination criterion for the VEM iterations.
2.5
Non-negativity constraints on the loading C
Each component m of the mixPLDS model, representing a subpopulation of neurons, can be a
very flexible model by itself (depending on the latent dimension dm ). This flexibility can in some
situations lead to counter-intuitive clustering results. Consider the following example. Let half of
the recorded neurons oscillate in phase and the remaining neurons oscillate with a phase shift of
? relative to the first half. Depending on the context, we might be interested in clustering the first
and second half of the neurons into separate groups reflecting oscillation phase. The mixPLDS
model could however end up putting all neurons into a single cluster, by modelling them with one
oscillating latent factor that has positive loadings on the first half of neurons and negative on the
second half (or vice versa). We can prevent this behavior, by imposing element-wise non-negativity
constraints on the loading matrix C, denoted as C ? 0 (and by simultaneously constraining the
latent dimensions of each group). The constraints guarantee that the influence of each factor on its
group has the same sign across all neurons. The suitability of these constraints strongly depends on
the biological context. In the application of the mixPLDS model in section 3.2, we found them to
be essential for obtaining meaningful results.
We modify the subspace clustering initialization to respect the constraints C ? 0 in the following way. Instead of solving the unconstrained reconstruction problem (7) with respect to W , we
add non-negativity constraints W ? 0. These sign constraints restrict the points that can be reconstructed from a given set of points to the convex cone of these points (instead of the subspace
containing these points). Hence, under these assumptions, all data points in a cluster can be approximately reconstructed by a (non-negative) convex combination of some ?time-series basis?. We
empirically observed that this yields initial loading matrix estimates with only very few negative
elements (after possible row-wise sign inversions). For the full mixPLDS model we enforce C ? 0
by the reparametrization C = exp(?) and doing VEM updates on ?.
3
3.1
Experiments
Artificial data
Here we validate the parameter estimation procedure for the mixPLDS model on artificial data. We
generate 35 random ground truth mixPLDS models with M = 3, d1 = d2 = d3 = 2 and 20 observed
neurons per cluster. We sampled from each ground truth model a data set consisting of 4 i.i.d. trials
with T = 250 times steps each. Ground truth parameters were generated such that the resulting data
was sparse (12% of the bins non-empty). We compared the ability of different clustering methods
to recover the 3 clusters from each data set. We report the results in fig. 1A in terms of the fraction
of misclassified neurons (class labels were determined by majority vote in each cluster). We applied
K-Means with careful initialization of the cluster centers [17] to the data. For K-Means, we pre5
B
0.5
0.5
freq. of misclassification
freq. of misclassification
A
0
Kmeans
specCl
subCl
PsubCl
mixPLDS
0
0
0.1 0.2 0.3 0.4
assignment uncertainty
Figure 1: Finding clusters of neurons in artificial data. A: Performance of different clustering
algorithms, reported in terms of frequency of misclassified neurons, on artificial data sampled from
ground truth mixPLDs models. Red bars indicate medians and blue boxes the 25% and 75% percentiles. Standard clustering methods (data plotted in black) such as K-Means, spectral clustering
(?specCl?), and subspace clustering (?subCl?) are substantially outperformed by the two methods
proposed here (data plotted in red). Poisson subspace clustering (?PsubCl?) yielded accurate initial
cluster estimates that were significantly improved by application of the full mixPLDs model. B:
Misclassification rate as a function of the cluster assignment uncertainty for the mixPLDS model.
This shows that the posterior over cluster assignments returned by the mixPLDS model is well calibrated, as neurons with low assignment uncertainty as rarely misclassified.
processed the data in a standard way by smoothing (Gaussian kernel, standard deviation 10 timesteps), mean-centering and scaling (such that each dimension k = 1, . . . , K has variance 1). We
found K-Means yielded reasonable clusters when all populations are one-dimensional (i.e. ?m dm =
1, data not shown) but it fails when clustering multi-dimensional groups of neurons. An alternative
approach is to cluster the cross-correlation matrix of neurons (computed from pre-processed data as
above) with standard spectral clustering [18]. We found that this approach works well when all the
factors have small variances, as in this case the link function of the observation model is only mildly
non-linear. However, with growing variances of the factors (larger dynamic ranges of neurons)
spectral clustering performance quickly degrades. Standard sparse subspace clustering [13] on the
spike trains (pre-processed as above) yielded very similar results to spectral clustering. We found
our novel Poisson subspace clustering algorithm proposed in section 2.3 to robustly outperform the
other approaches, as long as reasonable amounts of data were available (roughly T > 100 for the
above system). The mixPLDS model initialized with the Poisson subspace clustering consistently
yielded the best results, as it is able to integrate information over time and denoise the observations.
One advantage of the mixPLDS model is that it not only returns cluster assignments for neurons
but also provides a measure of uncertainty over these assignments. However, variational inference
tends to return over-confident posteriors in general and the factorization approximation (5) might
yield posterior uncertainty that is uninformative. To show that the variational posterior uncertainty
is well-calibrated we computed the entropy of the posterior cluster assignment q(sk ) for all neurons
as a measure for assignment uncertainty. We binned the neurons according to their assignment
uncertainty and report the misclassification rate for each bin in fig. 1B. 89% of the neurons have low
posterior uncertainty and reside in the first bin having a low misclassification rate of ? 0.1, whereas
few neurons (5%) have an assignment uncertainty larger than 0.3 nats and they are misclassified
with a rate of ? 0.4.
3.2
Calcium imaging of spinal cord neurons
We tested the mixPLDS model on calcium imaging data obtained from an in vitro, neonatal mouse
spinal cord that expressed the calcium indicator GCaMP3 in all motor neurons. When an isolated
spinal cord is tonically excited by a cocktail of rhythmogenic drugs (5 ?M NMDA, 10 ?M 5-HT,
50 ?M DA), motor neurons begin to fire rhythmically. In this network state, spatially clustered ensembles of motor neurons fire in phase with each other [19]. Since multiple ensembles that have
distinct phase tunings can be visualized in a single imaging field, this data represents a convenient
6
B
1
1
sorted neuron #
unsorted
neuron #
A
1
70
sorted
neuron #
factors
cluster 1
70
1
4
latent dim
C
factors
cluster 2
70
sorted
neuron #
1
70
1
500
frames
Figure 2: Application of the mixPLDS model to recordings from spinal cord neurons. A, top
panel: 500 frames of input data to the mixPLDS model. Middle panel: Same data as in upper panel,
but rows are sorted by mixPLDS clusters and factor loading. Inferred latent factors (red: cluster 1,
blue: cluster 2, solid: factor 1, dashed: factor 2) are also shown. Bottom panel: Inferred (smoothed)
firing rates. B: Loading matrix C of the mixPLDS model showing how factors 1,2 of cluster 1 and
factors 3,4 of cluster 2 influence the neurons. C: Preferred phases shown as a function of (sorted)
neuron index and colored by posterior probability of belonging to cluster 1. Clearly visible are two
clusters as well as an (approximately) increasing ordering within a cluster.
setting for testing our algorithm. The data (90 second long movies) were acquired at 15 Hz from
a custom two-photon microscope equipped with a resonant scanner (downsampled from 60 Hz to
boost SNR). The frequency of the rhythmic activity was typically 0.2 Hz. In addition, aggregate motor neuron activity was simultaneously acquired with each movie using a suction electrode attached
to a ventral root. This electrophysiology recording (referred to here as ephys-trace) was used as an
external phase reference point to compute phase tuning curves for imaged neurons, which we used
to validate our mixPLDS results.
A deconvolution algorithm [20] was applied to the recorded calcium time-series to estimate the
spiking activity of 70 motor neurons. The output of the deconvolution, a 70 ? 1140 (neurons ?
frames) matrix of posterior expected number of spikes, was used as input to the mixPLDS model.
The non-empty bins of the the first 500 out of the 1140 frames of input data (thresholded at 0.1)
are shown in fig. 2A (upper panel). We used a mixPLDS model with M = 2 groups with two
latent dimensions each, i.e. d1 = d2 = 2. We imposed the non-negativity constraints C ? 0 on the
loading matrix; these were found to be crucial for finding a meaningful clustering of the neurons,
as discussed above. The mixPLDS clustering reveals two groups with strongly periodic but phaseshifted population activities, as can be seen from the inferred latent factors shown in fig. 2A (middle
panel, factors of cluster 1 shown in red, factors of cluster 2 in blue). For each cluster, the model
learned a stronger (higher variance) latent factor (solid line) and a weaker one (dashed line); we
interpret the former as capturing the main activity structure in a cluster and the latter as describing
deviations. Based on the estimated mixPLDS model, we sorted the neurons for visualization into
two clusters according to their most likely cluster assignment argmaxsk =1,2 q(sk ). Within each
cluster, we sorted the neurons according to the ratio of the loading coefficient onto the stronger
factor over the loading onto the weaker factor. Re-plotting the spike-raster with this sorting in fig.
2A (middle panel) reveals interesting structure. First, it shows that the initial choice of two clusters
was well justified for this data set. Second, the sorting reveals that the majority of neurons tend to
7
fire at a preferred phase relative to the oscillation cycle, and the mixPLDS-based sorting corresponds
to an increasing ordering of preferred phases. Fig. 2B shows the loading matrix C of the mixPLDS,
which is found to be approximately block-diagonal.
On this data set we also have the opportunity to validate the unsupervised clustering by taking into
account the simultaneously recorded ephys-trace. We computed for each neuron a phase tuning
curve based on the ephys-trace history of the last 80 times steps (estimated via L2 regularized generalized linear model estimation, with an exp-Poisson observation model). For each neuron, we
extracted the peak location of this phase tuning curve, which we call the preferred phase. Fig. 2C
shows these preferred phases as a function of (sorted) neuron index, revealing that the two clusters
found by the mixPLDS model coincide well with the two modes of the bi-model distribution of preferred phases. Furthermore, within each cluster, the preferred phases are (approximately) increasing,
showing that the mixPLDS-sorting of neurons reflects the phase-relation of the neurons to the global,
oscillatory ephys-trace. We emphasize that the latter was not used for fitting the mixPLDS; i.e., this
constitutes an independent validation of our results.
We conclude that the mixPLDS model successfully uncovered clustering structure from the recordings that can be validated using the side information from electrophysiological tuning, and furthermore allowed for a meaningful sorting within each cluster capturing neural response properties. In
addition, the mixPLDS model leverages the temporal structure in recordings, automatically optimizing for the temporal smoothness level and revealing the main time-constants in the data (in the above
data set 1.8 and 6.5 sec) as well as main oscillation frequencies (0.2 and 0.45Hz). Furthermore, either the latent trajectories or the inferred firing rates shown in fig. 2A can be used as smoothed
proxies for their corresponding population activities for subsequent analyses.
4
Discussion
One can generalize the mixPLDS model in several ways. Here we assumed that, given the latent factors, all neurons fire independently. This is presumably a good assumption if the recorded neurons
are spatially distant, but it might break down if neurons are densely sampled from a local population
and have strong, monosynaptic connections. This more general case can be accounted for by incorporating direct interaction terms between neurons into the observation model in the spirit of coupled
GLMs (see [21]); inference and parameter learning are still tractable in this model using VEM. Furthermore, in addition to the activity recordings, one might have access to other covariates that are
informative about the clustering structure of the population, such as cell location, genetic markers,
or cell morphology. We can add such data as additional observations into the mixPLDS model to
facilitate clustering of the cells. An especially relevant example are stimulus-response properties of
cells. We can add a mixture model over receptive-field parameters using the cluster assignments s.
This extension would provide a clustering of neurons based on their joint activity statistics (such as
shared trial-to-trial variability) as well as on their receptive field properties.
We presented three technical contributions, that we expect to be useful outside the context of the
mixPLDS model. First, we proposed a simple extension of the sparse subspace clustering algorithm
to Poisson observations. We showed that if the dimension of the union of subspaces is much smaller
than the ambient dimension, our method substantially outperforms other approaches. Second, we
introduced a version of subspace clustering with non-negativity constraints on the reconstruction
weights, which therefore clusters points into convex cones. We expect this variant to be particularly
useful when clustering activity traces of cells, allowing for separating anti-phasic oscillations. Third,
we applied the dual variational inference approach of [11] to a model with a Markovian prior and
with mixtures of Poisson observations. The resulting inference method proved itself numerically
robust, and we expect it to be a valuable tool for analyzing time-series of sparse count variables.
Acknowledgements This work was supported by Simons Foundation (SCGB#325171 and
SCGB#325233), Grossman Center at Columbia University, and Gatsby Charitable Trust as well
as grants MURI W911NF-12-1-0594 from the ARO, vN00014-14-1-0243 from the ONR, W91NF14-1-0269 from DARPA and an NSF CAREER award (L.P.).
8
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4,793 | 534 | Learning to Make Coherent Predictions in
Domains with Discontinuities
Suzanna Becker and Geoffrey E. Hinton
Department of Computer Science, University of Toronto
Toronto, Ontario, Canada M5S 1A4
Abstract
We have previously described an unsupervised learning procedure that
discovers spatially coherent propertit>_<; of the world by maximizing the information that parameters extracted from different parts of the sensory
input convey about some common underlying cause. When given random
dot stereograms of curved surfaces, this procedure learns to extract surface depth because that is the property that is coherent across space. It
also learns how to interpolate the depth at one location from the depths
at nearby locations (Becker and Hint.oll. 1992). 1n this paper, we propose two new models which handle surfaces with discontinuities. The first
model attempts to detect cases of discontinuities and reject them. The
second model develops a mixture of expert interpolators. It learns to detect the locations of discontinuities and to invoke specialized, asymmetric
interpolators that do not cross the discontinuities .
1
Introd uction
Standard backpropagation is implausible as a model of perceptual learning because
it requires an external teacher to specify the desired output of the network. We
have shown (Becker and Hinton, 1992) how the external teacher can be replaced
by internally derived teaching signals. These signals are generated by using the
assumption that different parts of the perceptual input have common causes in
the external world. Small modules that look at separate but related parts of the
perceptual input discover these common causes by striving to produce outputs that
agree with each other (see Figure 1 a). The modules may look at different modalities
(e.g. vision and touch), or the same modality at different times (e.g. the consecutive
2-D views of a rotating 3-D object), or even spatially adjacent parts of the same
image. In previous work, we showed that when our learning procedure is applied
372
Learning to Make Coherent Predictions in Domains with Discontinuities
to adjacent patches of 2-dimensional images, it allows a neural network that has no
prior knowledge of the third dimension to discover depth in random dot stereograms
of curved surfaces. A more general version of the method allows the network to
discover the best way of interpolating the depth at one location from the depths
at nearby locations. We first summarize this earlier work, and then introduce
two new models which allow coherent predictions to be made in the presence of
discontinuities.
a)
left
rightm~m~
patch A
patch B
Figure 1: a) Two modules that receive input from corresponding parts of stereo
images. The first module receives input from stereo patch A, consisting of a horizontal strip from the left image (striped) and a corresponding strip from the right
image (hatched). The second module receives input from an adjacent stereo patch
B . The modules try to make their outputs, d a and db, convey as much information as possible about some underlying signal (i. e., the depth) which is common to
both patches. b) The architecture of the interpolating network, consisting of multiple
copies of modules like those in a) plus a layer of interpolating units. The network
tries to maximize the information that the locally extracted parameter de and the
contextually predicted parameter de convey about some common underlying signal.
We actually used 10 modules and the central 6 modules tried to maximize agreement
between their outputs and contextually predicted values. We used weight averaging
to constrain the interpolating function to be identical for all modules.
2
Learning spatially coherent features in images
The simplest way to get the outputs of two modules to agree is to use the squared
difference between the outputs as a cost function, and to adjust the weights in each
module so as to minimize this cost. Unfortunately, this usually causes each module
to produce the same constant output that is unaffected by the input to the module
and therefore conveys no information about it. What we want is for the outputs
of two modules to agree closely (i.e. to have a small expected squared difference)
relative to how much they both vary as the input is varied. When this happens, the
two modules must be responding to something that is common to their two inputs.
In the special case when the outputs, d a , db, of the two modules are scalars, a good
373
374
Becker and Hinton
measure of agreement is:
(1)
where V is the variance over the training cases. If d a and db are both versions
of the same underlying Gaussian signal that have been corrupted by independent
Gaussian noise, it can be shown that I is the mutual information between the
underlying signal and the average of d a and db. By maximizing I we force the two
modules to extract as pure a version as possible of the underlying common signal.
2.1
The basic stereo net
We have shown how this principle can be applied to a multi-layer network that learns
to extract depth from random dot stereograms (Becker and Hinton, 1992). Each
network module received input from a patch of a left image and a corresponding
patch of a right image, as shown in Figure 1 a). Adjacent modules received input
from adjacent stereo image patches, and learned to extract depth by trying to
maximize agreement between their outputs. The real-valued depth (relative to the
plane of fixation) of each patch of the surface gives rise to a disparity between
features in the left and right images; since that disparity is the only property that
is coherent across each stereo image, the output units of modules were able to learn
to accurately detect relative depth.
2.2
The interpolating net
The basic stereo net uses a very simple model of coherence in which an underlying
parameter at one location is assumed to be approximately equal to the parameter at
a neighbouring location. This model is fine for the depth of fronto-parallel surfaces
but it is far from the best model of slanted or curved surfaces. Fortunately, we can
use a far more general model of coherence in which the parameter at one location
is assumed to be an unknown linear function of the parameters at nearby locations.
The particular linear function that is appropriate can be learned by the network.
We used a network of the type shown in Figure 1 b). The depth computed locally
by a module, dc, was compared with the depth predicted by a linear combination de
of the outputs of nearby modules, and the network tried to maximize the agreement
between de and de.
The contextual prediction, dc, was produced by computing a weighted sum of
the outputs of two adjacent modules on either side. The interpolating weights
used in this sum, and all other weights in the network, were adjusted so as to
maximize agreement between locally computed and contextually predicted depths.
To speed the learning, we first trained the lower layers of the network as before, so that agreement was maximized between neighbouring locally computed
outputs. This made it easier to learn good interpolating weights. When the
network was trained on stereograms of cubic surfaces, it learned interpolating
weights of -0.147,0.675,0.656, -0.131 (Becker and Hinton, 1992). Given noise
free estimates of local depth, the optimal linear interpolator for a cubic surfa.ce
is -0.167,0.667,0.667, -0.167.
Learning to Make Coherent Predictions in Domains with Discontinuities
3
Throwing out discontinuities
If the surface is continuous, the depth at one patch can be accurately predicted from
the depths of two patches on either side. If, however, the training data contains cases
in which there are depth discontinuities (see figure 2) the interpolator will also try
to model these cases and this will contribute considerable noise to the interpolating
weights and to the depth estimates. One way of reducing this noise is to treat the
discontinuity cases as outliers and to throw them out. Rather than making a hard
decision about whether a case is an outlier, we make a soft decision by using a
mixture model. For each training case, the network compares the locally extracted
depth, dc, with the depth predicted from the nearby context, de. It assumes that
de - de is drawn from a zero-mean Gaussian if it is a continuity case and from a
uniform distribution if it is a discontinuity case. It can then estimate the probability
of a continuity case:
--------
Spline
curve
Left
Image
I 1
l
Right
Image
"I
I
II I II \
I I
I
II I
III
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til
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Figure 2: Top: A curved surface strip with a discontinuity created by fitting 2
cubic splines through randomly chosen control points, 25 pixels apart, separated by
a depth discontinuity. Feature points are randomly scattered on each spline with an
average of 0.22 features per pixel. Bottom: A stereo pair of "intensity" images
of the surface strip formed by taking two different projections of the feature points,
filtering them through a gaussian, and sampling the filtered projections at evenly
spaced sample points. The sample values in corresponding patches of the two images
are used as the inputs to a module. The depth of the surface for a particular zmage
region is directly related to the disparity between corresponding features in the left
and right patch. Disparity ranges continuously from -1 to + 1 image pixels. Each
stereo image was 120 pixels wide and divided into 10 receptive fields 10 pixels wide
and separated by 2 pixel gaps, as input for the networks shown in figure 1. The
receptive field of an interpolating unit spanned 58 image pixels, and discontinuities
were randomly located a minimum of 40 pixels apart, so only rarely would more
than one discontinuity lie within an interpolator's receptive field.
375
376
Becker and Hinton
(2)
where N is a gaussian, and kdi3eont is a constant representing a uniform density.
1
We can now optimize the average information de and de transmit about their common cause. We assume that no information is transmitted in discontinuity cases,
so the average information depends on the probability of continuity and on the
variance of de + de and de - de measured only in the continuity cases.
(3)
We tried several variations of this mixture approach. The network is quite good at
rejecting the discontinuity cases, but this leads to only a modest improvement in
the performance of the interpolator. In cases where there is a depth discontinuity
between d a and db or between dd and de the interpolator works moderately well
because the weights on d a or de are small. Because of the term Peont in equation
3 there is pressure to include these cases as continuity cases, so they probably
contribute noise to the interpolating weights. In the next section we show how to
avoid making a forced choice between rejecting these cases or treating them just
like all the other continuity cases.
4
Learning a mixture of expert interpolators
The presence of a depth discontinuity somewhere within a strip of five adjacent
patches does not entirely eliminate the coherence of depth across these patches. It
just restricts the range over which this coherence operates. So instead of throwing
out cases that contain a discontinuity, the network could try to develop a number
of different, specialized interpolators each of which captures the particular type of
coherence that remains in the presence of a discontinuity at a particular location.
If, for example, there is a depth discontinuity between de and de, an extrapolator
with weights of -1.0, +2.0,0, would be an appropriate predictor of de .
?
Figure 3 shows the system of five expert interpolators that we used for predicting
de from the neighboring depths. To allow the system to invoke the appropriate
interpolator, each expert has its own "controller" which must learn to detect the
presence of a discontinuity at a particular location (or the absence of a discontinuity in the case of the interpolator for pure continuity cases). The outputs of the
controllers are normalized, as shown in figure 3, so that they form a probability distribution. We can think of these normalized outputs as the probability with which
the system selects a particular expert. The controllers get to see all five local depth
estimates and most of them learn to detect particular depth discontinuities by using
large weights of opposite sign on the local depth estimates of neighboring patches.
lWe empirically select a good (fixed) value of kdiseont, and we choose a starting value
of Veont{de - de) (some proportion of the initial variance of de - de), and gradually shrink
it during learning.
Learning to Make Coherent Predictions in Domains with Discontinuities
expert 1
expert 2
expert 3
de , I
PI
de ,2
P2
de ,3
P3
Xl
Normalization
Pi
=
J
expert 4
expert 5
X2
controller 2
X3
controller 3
e x ,2
I: e
controller 1
x J? 2
de ,4
P4
X4
de ,5
P5
X5
controller 4
controller 5
Figure 3: The architecture of the mixture of interpolators and discontinuzty detec.
tors . We actually used a larger modular network and equality constraints between
modules, as described in figure 1 b), with 6 copies of the architecture shown here .
Each copy received input from different but overlapping parts of the input.
Figure 4 shows the weights learned by the experts and by their controllers. As
expected, there is one interpolator (the top one) that is appropriate for continuity
cases and four other interpolators that are appropriate for the four different locations of a discontinuity. In interpreting the weights of the controllers it is important
to remember that a controller which produces a small X value for a particular case
may nevertheless assign high probability to its expert if all the other controllers
produce even smaller x values.
4.1
The learning procedure
In the example presented here, we first trained the network shown in figure 1b)
on images with discontinuities. We then used the outputs of the depth extracting
layer, d a , ... ,de as the inputs to the expert interpolators and their controllers. The
system learned a set of expert interpolators without backpropagating derivatives all
the way down to the weights of the local depth extracting modules. So the local
depth estimates d a , ... ,de did not change as the interpolators were learned .
To train the system we used an unsupervised version of the competing experts
algorithm described by Jacobs, Jordan, Nowlan and Hinton (1991) . The output of
the ith expert, de,i, is treated as the mean of a Gaussian distribution with variance 0- 2
and the normalized output of each controller, Pi , is treated as the mixing proportion
of that Gaussian. So, for each training case, the outputs of the experts and their
controllers define a probability distribution that is a mixture of Gaussians . The aim
377
378
Becker and Hinton
,a)
b)
Mean output vs. distance
to nearest discontinuity
Interpolator Discontinuity
weights
detector weights
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Figure 4: a) Typical weights lear~ed by the five competing interpolators and corresponding five discontinuity detectors. Positive weights are shown in white, and
negative weights in black. b) The mean probabilities computed by each discontinuity
detector are plotted against the the distance from the center of the units' receptive
field to the nearest discontinuity. The probabilistic outputs are averaged over an
ensemble of 1000 test cases. If the nearest discontinuity is beyond ? thirty pixels,
it is outside the units' receptive field and the case is therefore a continuity example.
of the learning is to maximize the log probability density of the desired output, de,
under this mixture of Gaussians distribution. For a particular training case this log
probability is given by :
'"
log P( de) = log L.,; Pi
.
I
1
to=
v2~u
exp
((d
-
e
ei )2)
-d
2 2 '
u
(4)
By taking derivatives of this objective function we can simultaneously learn the
weights in the experts and in the controllers. For the results shown here, the
nework was trained for 30 conjugate gradient iterations on a set of 1000 random
dot stereograms with discontinuities.
The rationale for the use of a variance ratio in equation 1 is to prevent the variances
of d a and db collapsing to zero. Because the local estimates d 1 , ... , d s did not change
as the system learned the expert interpolators, it was possible to use (de - dc ,d 2 in
the objective function without worrying about the possibility that the variance of
de across cases would collapse to zero during the learning . Ideally we would like to
Learning (0 Make Coherent Predictions in Domains with Discontinuities
refine the weights of the local depth estimators to maximize their agreement with
the contextually predicted depths produced by the mixture of expert interpolators.
One way to do this would be to generalize equation 3 to handle a mixture of expert
interpolators:
(5)
Alternatively we could modify equation 4 by normalizing the difference (de - de.i )2
by the actual variance of dc, though this makes the derivatives considerably more
complicated.
5
Discussion
The competing controllers in figure 3 explicitly represent which regularity applies in
a particular region. The outputs of the controllers for nearby regions may themselves
exhibit coherence at a larger spatial scale, so the same learning technique could be
applied recursively. In 2-D images this should allow the continuity of depth edges
to be discovered.
The approach presented here should be applicable to other domains which contain
a mixture of alternative local regularities aCl?OSS space or time. For example, a l?igiJ
shape causes a linear constraint between the locations of its parts in an image, so if
there are many possible shapes, there are many alternative local regularities (Zemel
and Hinton, 1991).
Our learning procedure differs from methods that try to capture as much information as possible about the input (Linsker, 1988; Atick and Redlich, 1990) because
we ignore information in the input that is not coherent across space.
Acknowledgements
This research was funded by grants from NSERC and the Ontario Information Technology Research Centre. Hinton is Noranda fellow of the Canadian Institute for Advanced
Research. Thanks to John Bridle and Steve Nowlan for helpful discussions.
References
Atick, J. J. and Redlich, A. N. (1990). Towards a theory of early visual processing.
Technical Report IASSNS-HEP-90j10, Institute for Advanced Study, Princeton.
Becker, S. and Hinton, G. E. (1992). A self-organizing neural network that discovers
surfaces in random-dot stereograms. January 1992 Nature.
Jacobs, R. A., Jordan, M. 1., Nowlan, S. J., and Hinton, G. E. (1991). Adaptive mixtures
of local experts. Neural Computation, 3(1).
Linsker, R. (1988). Self-organization in a perceptual network. IEEE Computer, March,
21:105-117.
Zemel, R. S. and Hinton, G. E. (1991). Discovering viewpoint-invariant relationships that
characterize objects. In Advances In Neural Information Processing Systems 3, pages
299-305. Morgan Kaufmann Publishers.
379
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4,794 | 5,340 | Design Principles of the Hippocampal Cognitive Map
Kimberly L. Stachenfeld1 , Matthew M. Botvinick1 , and Samuel J. Gershman2
Princeton Neuroscience Institute and Department of Psychology, Princeton University
2
Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology
[email protected], [email protected], [email protected]
1
Abstract
Hippocampal place fields have been shown to reflect behaviorally relevant aspects
of space. For instance, place fields tend to be skewed along commonly traveled
directions, they cluster around rewarded locations, and they are constrained by the
geometric structure of the environment. We hypothesize a set of design principles
for the hippocampal cognitive map that explain how place fields represent space
in a way that facilitates navigation and reinforcement learning. In particular, we
suggest that place fields encode not just information about the current location,
but also predictions about future locations under the current transition distribution. Under this model, a variety of place field phenomena arise naturally from
the structure of rewards, barriers, and directional biases as reflected in the transition policy. Furthermore, we demonstrate that this representation of space can
support efficient reinforcement learning. We also propose that grid cells compute
the eigendecomposition of place fields in part because is useful for segmenting an
enclosure along natural boundaries. When applied recursively, this segmentation
can be used to discover a hierarchical decomposition of space. Thus, grid cells
might be involved in computing subgoals for hierarchical reinforcement learning.
1
Introduction
A cognitive map, as originally conceived by Tolman [46], is a geometric representation of the environment that can support sophisticated navigational behavior. Tolman was led to this hypothesis
by the observation that rats can acquire knowledge about the spatial structure of a maze even in the
absence of direct reinforcement (latent learning; [46]). Subsequent work has sought to formalize the
representational content of the cognitive map [13], the algorithms that operate on it [33, 35], and its
neural implementation [34, 27]. Much of this work was galvanized by the discovery of place cells
in the hippocampus [34], which selectively respond when an animal is in a particular location, thus
supporting the notion that the brain contains an explicit map of space. The later discovery of grid
cells in the entorhinal cortex [16], which respond periodically over the entire environment, indicated
a possible neural substrate for encoding metric information about space.
Metric information is very useful if one considers the problem of spatial navigation to be computing the shortest path from a starting point to a goal. A mechanism that accumulates a record of
displacements can easily compute the shortest path back to the origin, a technique known as path
integration. Considerable empirical evidence supports the idea that animals use this technique for
navigation [13]. Many authors have proposed theories of how grid cells and place cells can be used
to carry out the necessary computations [27].
However, the navigational problems faced by humans and animals are inextricably tied to the more
general problem of reward maximization, which cannot be reduced to the problem of finding the
shortest path between two points. This raises the question: does the brain employ the same machinery for spatial navigation and reinforcement learning (RL)? A number of authors have suggested
how RL mechanisms can support spatial learning, where spatial representations (e.g., place cells or
1
grid cells), serve as the input to the learning system [11, 15]. In contrast to the view that spatial representation is extrinsic to the RL system, we pursue the idea that the brain?s spatial representations
are designed to support RL. In particular, we show how spatial representations resembling place
cells and grid cells emerge as the solution to the problem of optimizing spatial representation in the
service of RL.
We first review the formal definition of the RL problem, along with several algorithmic solutions.
Special attention is paid to the successor representation (SR) [6], which enables a computationally
convenient decomposition of value functions. We then show how the successor representation naturally comes to represent place cells when applied to spatial domains. The eigendecomposition of
the successor representation reveals properties of an environment?s spectral graph structure, which
is particularly useful for discovering hierarchical decompositions of space. We demonstrate that the
eigenvectors resemble grid cells, and suggest that one function of the entorhinal cortex may be to
encode a compressed representation of space that aids hierarchical RL [3].
2
The reinforcement learning problem
Here we consider the problem of RL in a Markov decision process, which consists of the following
elements: a set of states S, a set of actions A, a transition distribution P (s0 |s, a) specifying the
probability of transitioning to state s0 ? S from state s ? S after taking action a ? A, a reward
function R(s) specifying the expected reward in state s, and a discount factor ? ? [0, 1]. An agent
chooses actions according to a policy ?(a|s) and collects rewards as it moves through the state space.
The standard RL problem
P? is to choose a policy that maximizes the value (expected discounted future
return), V (s) = E? [ t=0 ? t R(st ) | s0 = s]. Our focus here is on policy evaluation (computing V ).
In our simulations we feed the agent the optimal policy; in the Supplementary Materials we discuss
algorithms for policy improvement. To simplify notation,
P we assume implicit dependence on ? and
define the state transition matrix T , where T (s, s0 ) = a ?(a|s)P (s0 |s, a).
Most work on RL has focused on two classes of algorithms for policy evaluation: ?model-free?
algorithms that estimate V directly from sample paths, and ?model-based? algorithms that estimate
T and R from sample paths and then compute V by some form of dynamic programming or tree
search [44, 5]. However, there exists a third class that has received less attention. As shown by
Dayan [6], the value function can be decomposed into the inner product of the reward function with
the SR, denoted by M :
P
V (s) = s0 M (s, s0 )R(s0 ),
M = (I ? ?T )?1
(1)
where I denotes the identity matrix. The SR encodes the expected discounted future occupancy of
state s0 along a trajectory initiated in state s:
P?
M (s, s0 ) = E [ t=0 ? t I{st = s0 } | s0 = s] ,
(2)
where I{?} = 1 if its argument is true, and 0 otherwise.
The SR obeys a recursion analogous to the Bellman equation for value functions:
P
M (s, j) = I{s = j} + ? s0 T (s, s0 )M (s0 , j).
(3)
This recursion can be harnessed to derive a temporal difference learning algorithm for incrementally
? of the SR [6, 14]. After observing a transition s ? s0 , the estimate is
updating an estimate M
updated according to:
h
i
? (s, j) ? M
? (s, j) + ? I{s = j} + ? M
? (s0 , j) ? M
? (s, j) ,
M
(4)
where ? is a learning rate (unless specified otherwise, ? = 0.1 in our simulations). The SR combines
some of the advantages of model-free and model-based algorithms: like model-free algorithms,
policy evaluation is computationally efficient, but at the same time the SR provides some of the same
flexibility as model-based algorithms. As we illustrate later, an agent using the SR will be sensitive
to distal changes in reward, whereas a model-free agent will be insensitive to these changes.
3
The successor representation and place cells
In this section, we explore the neural implications of using the SR for policy evaluation: if the brain
encoded the SR, what would the receptive fields of the encoding population look like, and what
2
Empty Room
1.8
1.8
Single Barrier
c
e
1.8
1.8
2.1
1.2
5.6
1.9
1.8
1.8
1.2
1.3
1.2
1.4
1.8
f
1.6
1.3
1.3
Reward (+)
d
5.6
1.2
1.9
1.8
1.8
b
Multiple Rooms
No Reward
a
1.4
Discounted expected visiations (SR)
Figure 1: SR place fields. Top two rows show place fields without reward, bottom two show
retrospective place fields with reward (marked by +). Maximum firing rate (a.u.) indicated for each
plot. (a, b) Empty room. (c, d) Single barrier. (e, f) Multiple rooms.
Direction Selectivity
4
0.2
0.4
0.6
0.8
1
3
Figure 2: Direction selectivity along a
track. Direction selectivity arises in SR
place fields when the probability p?
of transitioning in the preferred left-toright direction along a linear track is
greater than the probability p? of transitioning in the non-preferred direction.
The legend shows the ratio of p? to p?
for each simulation.
2
1
0
100
150
200
250
300
Distance along Track
350
400
would the population look like at any point in time? This question is most easily addressed in spatial
domains, where states index spatial locations (see Supplementary Materials for simulation details).
For an open field with uniformly distributed rewards we assume a random walk policy, and the
resulting SR for a particular location is an approximately symmetric, gradually decaying halo around
that location (Fig. 1a)?the canonical description of a hippocampal place cell. In order for the
population to encode the expected visitations to each state in the domain from the current starting
state (i.e. a row of M ), each receptive field corresponds to a column of the SR matrix. This allows
the current state?s value to be computed by taking the dot product of its population vector with the
reward vector. The receptive field (i.e. column of M ) will encode the discounted expected number
of times that state was visited for each starting state, and will therefore skew in the direction of the
states that likely preceded the current state.
More interesting predictions can be made when we examine the effects of obstacles and direction
preference that shape the transition structure. For instance, when barriers are inserted into the environment, the probability of transitioning across these obstacles will go to zero. SR place fields
are therefore constrained by environmental geometry, and the receptive field will be discontinuous
across barriers (Fig. 1c,e). Consistent with this idea, experiments have shown that place fields become distorted around barriers [32, 40]. When an animal has been trained to travel in a preferred
direction along a linear track, we expect the response of place fields to become skewed opposite the
direction of travel (Fig. 2), a result that has been observed experimentally [28, 29].
Another way to alter the transition policy is by introducing a goal, which induces a tendency to move
in the direction that maximizes reward. Under these conditions, we expect firing fields centered near
rewarded locations to expand to include the surrounding locations and to increase their firing rate,
as has been observed experimentally [10, 21]. Meanwhile, we expect the majority of place fields
3
Percentage of Neurons Firing
f
a4
2
0
0.4
Depth
b
0.2
0
Distance around annular track
Figure 3: Reward clustering in annular maze. (a) Histogram of number of cells firing above
baseline at each displacement around an annular track. (b) Heat map of number of firing cells at
each location on unwrapped annular maze. Reward is centered on track. Baseline firing rate set to
10% maximum.
late detour
early detour
no detour
a
b
c
d
1.25
1.15
1.49
1.60
1.15
1.49
2.36
1.08
1.49
Firing Fields
Value
Figure 4: Tolman detour task. The starting location is at the bottom of the maze where the
three paths meet, and the reward is at the top. Barriers are shown as black horizontal lines. Three
conditions are shown: No detour, early detour, and late detour. (a, b, c) SR place fields centered near
and far from detours. Maximum firing rate (a.u.) indicated by each plot. (d) Value function.
that encode non-rewarded states to skew slightly away from the reward. Under certain settings
for what firing rate constitutes baseline (see Supplementary Materials), the spread of the rewarded
locations? fields compensates for the skew of surrounding fields away from the reward, and we
observe ?clustering? around rewarded locations (Fig. 3), as has been observed experimentally in the
annular water maze task [18]. This parameterization sensitivity may explain why goal-related firing
is not observed in all tasks [25, 24, 41].
As another illustration of the model?s response to barriers, we simulated place fields in a version
of the Tolman detour task [46], as described in [1]. Rats are trained to move from the start to the
rewarded location. At some point, an ?early? or a ?late? transparent barrier is placed in the maze
so that the rat must take a detour. For the early barrier, a short detour is available, and for the later
barrier, the only detour is a longer one. Place fields near the detour are more strongly affected than
places far away from the detour (Fig. 4a,b,c), consistent with experimental findings [1]. Fig. 4d
shows the value function in each of these detour conditions.
4
Behavioral predictions: distance estimation and latent learning
In this section, we examine some of the behavioral consequences of using the SR for RL. We first
show that the SR anticipates biases in distance estimation induced by semi-permeable boundaries.
We then explore the ability of the SR to support latent learning in contextual fear conditioning.
4
b
75
4
50
3
2
25
1
0
0
a 18
16
14
0.5
0
1
Permeability
SR Distance
b
Lesion
Control
12
10
6
4
2
0
1
2
Preexposure Duration (steps)
3
x 10
5
Figure 6: Context preexposure facilitation
effect. (a) Simulated conditioned response
(CR) to the context following one-trial contextual fear conditioning, shown as a function of
preexposure duration. The CR was approximated as the negative value summed over the
environment. The ?Lesion? corresponds to
agents with hippocampal damage, simulated by
setting the SR learning rate to 0.01. The ?Control? group has a learning rate of 0.1. (b) value
for a single location after preexposure in a control agent. (c) same as (b) in a lesioned agent.
0
?0.2
?0.4
?0.6
?0.8
c
8
0
Control
0
?0.1
Value
Conditioned Response
Figure 5: Distance estimates. (a) Increase in
the perceived distance between two points on
opposite sides of a semipermeable boundary
(marked with + and ? in 5b) as a function of
barrier permeability. (b) Perceived distance between destination (market with +) and all other
locations in the space (barrier permeability =
0.05).
5
Value
Distance (% Increase)
a
?0.2
Lesion
?0.3
Stevens and Coupe [43] reported that people overestimated the distance between two locations when
they were separated by a boundary (e.g., a state or country line). This bias was hypothesized to arise
from a hierarchical organization of space (see also [17]). We showp(Fig. 5) how distance estimates
(using the Euclidean distance between SR state representations, (M (s0 ) ? M (s))2 , as a proxy
for the perceived distance between s and s0 ) between points in different regions of the environment
are altered when an enclosure is divided by a soft (semi-permeable) boundary. We see that as the
permeability of the barrier decreases (making the boundary harder), the percent increase in perceived
distance between locations increases without bound. This gives rise to a discontinuity in perceived
travel time at the soft boundary. Interestingly, the hippocampus is directly involved in distance
estimation [31], suggesting the hippocampal cognitive map as a neural substrate for distance biases
(although a direct link has yet to be established).
The context preexposure facilitation effect refers to the finding that placing an animal inside a conditioning chamber prior to shocking it facilitates the acquisition of contextual fear [9]. In essence, this
is a form of latent learning [46]. The facilitation effect is thought to arise from the development of a
conjunctive representation of the context in the hippocampus, though areas outside the hippocampus
may also develop a conjunctive representation in the absence of the hippocampus, albeit less efficiently [48]. The SR provides a somewhat different interpretation: over the course of preexposure,
the hippocampus develops a predictive representation of the context, such that subsequent learning
is rapidly propagated across space. Fig. 6 shows a simulation of this process and how it accounts
for the facilitation effect. We simulated hippocampal lesions by reducing the SR learning rate from
0.1 to 0.01, resulting in a more punctate SR following preexposure and a reduced facilitation effect.
5
Eigendecomposition of the successor representation: hierarchical
decomposition and grid cells
Reinforcement learning and navigation can often be made more efficient by decomposing the environment hierarchically. For example, the options framework [45] utilizes a set of subgoals to divide
and conquer a complex learning environment. Recent experimental work suggests that the brain may
exploit a similar strategy [3, 36, 8]. A key problem, however, is discovering useful subgoals; while
progress has been made on this problem in machine learning, we still know very little about how the
brain solves it (but see [37]). In this section, we show how the eigendecomposition of the SR can
be used to discover subgoals. The resulting eigenvectors strikingly resemble grid cells observed in
entorhinal cortex.
5
a
Open Room
b
Single Barrier
c
Multiple Rooms
Figure 7: Eigendecomposition of the SR. Each panel shows the same 20 eigenvectors randomly
sampled from the top 100 (excluding the constant first eigenvector) for the environmental geometries
shown in Fig. 1 (no reward). (a) Empty room. (b) Single barrier. (c) Multiple rooms.
Eigendecomposition
Figure 8: Eigendecomposition of the SR in a
hairpin maze. Since the walls of the maze effectively elongate a dimension of travel (the track
of the maze), the low frequency eigenvectors resemble one-dimensional sinusoids that have been
folded to match the space. Meanwhile, the low
frequency eigenvectors exhibit the compartmentalization shown by [7].
A number of authors have used graph partitioning techniques to discover subgoals [30, 39]. These
approaches cluster states according to their community membership (a community is defined as a
highly interconnected set of nodes with relatively few outgoing edges). Transition points between
communities (bottleneck states) are then used as subgoals. One important graph partitioning technique, used by [39] to find subgoals, is the normalized cuts algorithm [38], which recursively thresholds the second smallest eigenvector (the Fiedler vector) of the normalized graph Laplacian to obtain
a graph partition. Given an undirected graph with symmetric weight matrix W , the graph Laplacian
is given by L = D ? W . The normalized graph Laplacian
is given by L = I ? D?1/2 W D?1/2 ,
P
where D is a diagonal degree matrix with D(s, s) = s0 W (s, s0 ). When states are projected onto
the second eigenvector, they are pulled along orthogonal dimensions according to their community
membership. Locations in distinct regions but close in Euclidean distance ? for instance, nearby
points on opposite sides of a boundary ? will be represented as distant in the eigenspace.
The normalized graph Laplacian is closely related to the SR [26]. Under a random walk policy,
the transition matrix is given by T = D?1 W . If ? is an eigenvector of the random walk?s graph
Laplacian I?T , then D1/2 ? is an eigenvector of the normalized graph Laplacian. The corresponding
eigenvector for the discounted Laplacian, I ? ?T , is ??. Since the matrix inverse preserves the
eigenvectors, the normalized graph Laplacian has the same eigenvectors as the SR, M = (I??T )?1 ,
scaled by ?D?1/2 . These spectral eigenvectors can be approximated by slow feature analysis [42].
Applying hierarchical slow feature analysis to streams of simulated visual inputs produces feature
representations that resemble hippocampal receptive fields [12].
A number of representative SR eigenvectors are shown in Fig. 7, for three different room topologies.
The higher frequency eigenvectors display the latticing characteristic of grid cells [16]. The eigendecomposition is often discontinuous at barriers, and in many cases different rooms are represented
by independent sinusoids. Fig. 8 shows the eigendecomposition for a hairpin maze. The eigenvectors resemble folded up one-dimensional sinusoids, and high frequency eigenvectors appear as
repeating phase-locked ?submaps? with firing selective to a subset of hallways, much like the grid
cells observed by Derdikman and Moser [7].
In the multiple rooms environment, visual inspection reveals that the SR eigenvector with the second
smallest eigenvalue (the Fiedler vector) divides the enclosure along the vertical barrier: the left half
is almost entirely blue and the right half almost entirely red, with a smooth but steep transition
at the doorway (Fig. 9a). As discussed above, this second eigenvector can therefore be used to
segment the enclosure along the vertical boundary. Applying this segmentation recursively, as in
the normalized cuts algorithm, produces a hierarchical decomposition of the environment (Figure
6
Segmentation
b
Figure 9: Segmentation using normalized cuts.
(a) The results of segmentation by thresholding
the second eigenvector of the multiple rooms environment in Fig. 1. Dotted lines indicate the
segment boundaries. (b, c) Eigenvector segmentation applied recursively to fully parse the enclosure into the four rooms.
a
c
First Level
Second Level
9b,c). By identifying useful subgoals from the environmental topology, this decomposition can be
exploited by hierarchical learning algorithms [3, 37].
One might reasonably question why the brain should represent high frequency eigenvectors (like
grid cells) if only the low frequency eigenvectors are useful for hierarchical decomposition. Spectral
features also serve as generally useful representations [26, 22], and high frequency components are
important for representing detail in the value function. The progressive increase in grid cell spacing
along the dorsal-ventral axis of the entorhinal cortex may function as a multi-scale representation
that supports both fine and coarse detail [2].
6
Discussion
We have shown how many empirically observed properties of spatial representation in the brain,
such as changes in place fields induced by manipulations of environmental geometry and reward,
can be explained by a predictive representation of the environment. This predictive representation
is intimately tied to the problem of RL: in a certain sense, it is the optimal representation of space
for the purpose of computing value functions, since it reduces value computation to a simple matrix
multiplication [6]. Moreover, this optimality principle is closely connected to ideas from manifold
learning and spectral graph theory [26]. Our work thus sheds new computational light on Tolman?s
cognitive map [46].
Our work is connected to several lines of previous work. Most relevant is Gustafson and Daw
[15], who showed how topologically-sensitive spatial representations recapitulate many aspects of
place cells and grid cells that are difficult to reconcile with a purely Euclidean representation of
space. They also showed how encoding topological structure greatly aids reinforcement learning in
complex spatial environments. Earlier work by Foster and colleagues [11] also used place cells as
features for RL, although the spatial representation did not explicitly encode topological structure.
While these theoretical precedents highlight the importance of spatial representation, they leave
open the deeper question of why particular representations are better than others. We showed that
the SR naturally encodes topological structure in a format that enables efficient RL.
Spectral graph theory provides insight into the topological structure encoded by the SR. In particular,
we showed that eigenvectors of the SR can be used to discover a hierarchical decomposition of the
environment for use in hierarchical RL. These eigenvectors may also be useful as a representational
basis for RL, encoding multi-scale spatial structure in the value function. Spectral analysis has
frequently been invoked as a computational motivation for entorhinal grid cells (e.g., [23]). The
fact that any function can be reconstructed by sums of sinusoids suggested that the entorhinal cortex
implements a kind of Fourier transform of space, and that place cells are the result of reconstructing
spatial signals from their spectral decomposition. Two problems face this interpretation. Fist, recent
evidence suggests that the emergence of place cells does not depend on grid cell input [4, 47].
Second, and more importantly for our purposes, Fourier analysis is not the right mathematical tool
when dealing with spatial representation in a topologically structured environment, since we do not
expect functions to be smooth over boundaries in the environment. This is precisely the purpose of
spectral graph theory: the eigenvectors of the graph Laplacian encode the smoothest approximation
of a function that respects the graph topology [26].
Recent work has elucidated connections between models of episodic memory and the SR. Specifically, in [14] it was shown that the SR is closely related to the Temporal Context Model (TCM)
of episodic memory [20]. The core idea of TCM is that items are bound to their temporal context
(a running average of recently experienced items), and the currently active temporal context is used
7
to cue retrieval of other items, which in turn cause their temporal context to be retrieved. The SR
can be seen as encoding a set of item-context associations. The connection to episodic memory is
especially interesting given the crucial mnemonic role played by the hippocampus and entorhinal
cortex in episodic memory. Howard and colleagues [19] have laid out a detailed mapping between
TCM and the medial temporal lobe (including entorhinal and hippocampal regions).
An important question for future work concerns how biologically plausible mechanisms can implement the computations posited in our paper. We described a simple error-driven updating rule for
learning the SR, and in the Supplementary Materials we derive a stochastic gradient learning rule
that also uses a simple error-driven update. Considerable attention has been devoted to the implementation of error-driven learning rules in the brain, so we expect that these learning rules can be
implemented in a biologically plausible manner.
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4,795 | 5,341 | Scalable Inference for Neuronal Connectivity from
Calcium Imaging
Alyson K. Fletcher
Sundeep Rangan
Abstract
Fluorescent calcium imaging provides a potentially powerful tool for inferring
connectivity in neural circuits with up to thousands of neurons. However, a key
challenge in using calcium imaging for connectivity detection is that current systems often have a temporal response and frame rate that can be orders of magnitude slower than the underlying neural spiking process. Bayesian inference methods based on expectation-maximization (EM) have been proposed to overcome
these limitations, but are often computationally demanding since the E-step in the
EM procedure typically involves state estimation for a high-dimensional nonlinear dynamical system. In this work, we propose a computationally fast method
for the state estimation based on a hybrid of loopy belief propagation and approximate message passing (AMP). The key insight is that a neural system as viewed
through calcium imaging can be factorized into simple scalar dynamical systems
for each neuron with linear interconnections between the neurons. Using the structure, the updates in the proposed hybrid AMP methodology can be computed by a
set of one-dimensional state estimation procedures and linear transforms with the
connectivity matrix. This yields a computationally scalable method for inferring
connectivity of large neural circuits. Simulations of the method on realistic neural
networks demonstrate good accuracy with computation times that are potentially
significantly faster than current approaches based on Markov Chain Monte Carlo
methods.
1
Introduction
Determining connectivity in populations of neurons is fundamental to understanding neural computation and function. In recent years, calcium imaging has emerged as a promising technique for
measuring synaptic activity and mapping neural micro-circuits [1?4]. Fluorescent calcium-sensitive
dyes and genetically-encoded calcium indicators can be loaded into neurons, which can then be imaged for spiking activity either in vivo or in vitro. Current methods enable imaging populations of
hundreds to thousands of neurons with very high spatial resolution. Using two-photon microscopy,
imaging can also be localized to specific depths and cortical layers [5]. Calcium imaging also has
the potential to be combined with optogenetic stimulation techniques such as in [6].
However, inferring neural connectivity from calcium imaging remains a mathematically and computationally challenging problem. Unlike anatomical methods, calcium imaging does not directly
measure connections. Instead, connections must be inferred indirectly from statistical relationships
between spike activities of different neurons. In addition, the measurements of the spikes from calcium imaging are indirect and noisy. Most importantly, the imaging introduces significant temporal
blurring of the spike times: the typical time constants for the decay of the fluorescent calcium concentration, [Ca2+ ], can be on the order of a second ? orders of magnitude slower than the spike rates
and inter-neuron dynamics. Moreover, the calcium imaging frame rate remains relatively slow ?
often less than 100 Hz. Hence, determining connectivity typically requires super-resolution of spike
times within the frame period.
1
To overcome these challenges, the recent work [7] proposed a Bayesian inference method to estimate functional connectivity from calcium imaging in a systematic manner. Unlike ?model-free?
approaches such as in [8], the method in [7] assumed a detailed functional model of the neural dynamics with unknown parameters including a connectivity weight matrix W. The model parameters
including the connectivity matrix can then be estimated via a standard EM procedure [9]. While the
method is general, one of the challenges in implementing it is the computational complexity. As
we discuss below, the E-step in the EM procedure essentially requires estimating the distributions
of hidden states in a nonlinear dynamical system whose state dimension grows linearly with the
number of neurons. Since exact computation of these densities grows exponentially in the state dimension, [7] uses an approximate method based on blockwise Gibbs sampling where each block of
variables consists of the hidden states associated with one neuron. Since the variables within a block
are described as a low-dimensional dynamical system, the updates of the densities for the Gibbs
sampling can be computed efficiently via a standard particle filter [10, 11]. However, simulations of
the method show that the mixing between blocks can still take considerable time to converge.
This paper provides a novel method that can potentially significantly improve the computation time
of the state estimation. The key insight is to recognize that a high-dimensional neural system can be
?factorized? into simple, scalar dynamical systems for each neuron with linear interactions between
the neurons. As described below, we assume a standard leaky integrate-and-fire model for each
neuron [12] and a first-order AR process for the calcium imaging [13]. Under this model, the
dynamics of N neurons can be described by 2N systems, each with a scalar (i.e. one-dimensional)
state. The coupling between the systems will be linear as described by the connectivity matrix
W. Using this factorization, approximate state estimation can then be efficiently performed via
approximations of loopy belief propagation (BP) [14]. Specifically, we show that the loopy BP
updates at each of the factor nodes associated with the integrate-and-fire and calcium imaging can
be performed via a scalar standard forward?backward filter. For the updates associated with the
linear transform W, we use recently-developed approximate message passing (AMP) methods.
AMP was originally proposed in [15] for problems in compressed sensing. Similar to expectation
propagation [16], AMP methods use Gaussian and quadratic approximations of loopy BP but with
further simplifications that leverage the linear interactions. AMP was used for neural mapping from
multi-neuron excitation and neural receptive field estimation in [17, 18]. Here, we use a so-called
hybrid AMP technique proposed in [19] that combines AMP updates across the linear coupling
terms with standard loopy BP updates on the remainder of the system. When applied to the neural
system, we show that the estimation updates become remarkably simple: For a system with N
neurons, each iteration involves running 2N forward?backward scalar state estimation algorithms,
along with multiplications by W and WT at each time step. The practical complexity scales as
O(N T ) where T is the number of time steps. We demonstrate that the method can be significantly
faster than the blockwise Gibbs sampling proposed in [7], with similar accuracy.
2
System Model
We consider a recurrent network of N spontaneously firing neurons. All dynamics are approximated
in discrete time with some time step ?, with a typical value ? = 1 ms. Importantly, this time step
is typically smaller than the calcium imaging period, so the model captures the dynamics between
observations. Time bins are indexed by k = 0, . . . , T ?1, where T is the number of time bins so that
T ? is the total observation time in seconds. Each neuron i generates a sequence of spikes (action
potentials) indicated by random variables ski taking values 0 or 1 to represent whether there was a
spike in time bin k or not. It is assumed that the discretization step ? is sufficiently small such that
there is at most one action potential from a neuron in any one time bin. The spikes are generated
via a standard leaky integrate-and-fire (LIF) model [12] where the (single compartment) membrane
voltage vik of each neuron i and its corresponding spike output sequence ski evolve as
v?ik+1 = (1 ? ?IF )vik + qik + dkvi ,
qik =
N
X
Wij sk??
+ bIF,i ,
j
dkvi ? N (0, ?IF ),
(1)
j=1
and
(vik+1 , sk+1
i )
=
(?
vik , 0)
(0, 1)
2
if vik < ?,
if v?ik ? ?,
(2)
where ?IF is a time constant for the integration leakage; ? is the threshold potential at which the
neurons spikes; bIF,i is a constant bias term; qik is the increase in the membrane potential from the
pre-synaptic spikes from other neurons and dkvi is a noise term including both thermal noise and
currents from other neurons that are outside the observation window. The voltage has been scaled
so that the reset voltage is zero. The parameter ? is the integer delay (in units of the time step
?) between the spike in one neuron and the increase in the membrane voltage in the post-synaptic
neuron. An implicit assumption in this model is the post-synaptic current arrives in a single time bin
with a fixed delay.
To determine functional connectivity, the key parameter to estimate will be the matrix W of the
weighting terms Wij in (1). Each parameter Wij represents the increase in the membrane voltage in
neuron i due to the current triggered from a spike in neuron j. The connectivity weight Wij will be
zero whenever neuron j has no connection to neuron i. Thus, determining W will determine which
neurons are connected to one another and the strengths of those connections.
For the calcium imaging, we use a standard model [7], where the concentration of fluorescent Calcium has a fast initial rise upon an action potential followed by a slow exponential decay. Specifically, we let zik = [Ca2+ ]k be the concentration of fluorescent Calcium in neuron i in time bin k and
assume it evolves as first-order auto-regressive AR(1) model,
zik+1 = (1 ? ?CA,i )zik + ski ,
(3)
where ?CA is the Calcium time constant. The observed net fluorescence level is then given by a
noisy version of zik ,
yik = aCA,i zik + bCA,i + dkyi , dkyi ? N (0, ?y ),
(4)
where aCA,i and bCA,i are constants and dyi is white Gaussian noise with variance ?y . Nonlinearities
such as saturation described in [13] can also be modeled.
As mentioned in the Introduction, a key challenge in calcium imaging is the relatively slow frame
rate which has the effect of subsampling of the fluorescence. To model the subsampling, we
let IF denote the set of time indices k on which we observe Fik . We will assume that fluorescence values are observed once every TF time steps for some integer period TF so that
IF = {0, TF , 2TF , . . . , KTF } where K is the number of Calcium image frames.
3
3.1
Parameter Estimation via Message Passing
Problem Formulation
Let ? be set of all the unknown parameters,
? = {W, ?IF , ?CA , ?IF , bIF,i , ?CA , aCA,i , bCA,i , i = 1, . . . , N },
(5)
which includes the connectivity matrix, time constants and various variances and bias terms. Estimating the parameter set ? will provide an estimate of the connectivity matrix W, which is our main
goal.
To estimate ?, we consider a regularized maximum likelihood (ML) estimate
?b = arg max L(y|?) + ?(?),
L(y|?) = ? log p(y|?),
(6)
?
where y is the set of observed values; L(y|?) is the negative log likelihood of y given the parameters
? and ?(?) is some regularization function. For the calcium imaging problem, the observations y
are the observed fluorescence values across all the neurons,
y = {y1 , . . . , yN } , yi = yik , k ? IF ,
(7)
where yi is the set of fluorescence values from neuron i, and, as mentioned above, IF is the set of
time indices k on which the fluorescence is sampled.
The regularization function ?(?) can be used to impose constraints or priors on the parameters. In
this work, we will assume a simple regularizer that only constrains the connectivity matrix W,
X
?(?) = ?kWk1 , kWk1 :=
|Wij |,
(8)
ij
3
where ? is a positive constant. The `1 regularizer is a standard convex function used to encourage
sparsity [20], which we know in this case must be valid since most neurons are not connected to one
another.
3.2
EM Estimation
Exact computation of ?b in (6) is generally intractable, since the observed fluorescence values y
depend on the unknown parameters ? through a large set of hidden variables. Similar to [7], we thus
use a standard EM procedure [9]. To apply the EM procedure to the calcium imaging problem, let
x be the set of hidden variables,
x = {v, z, q, s} ,
(9)
where v are the membrane voltages of the neurons, z the calcium concentrations, s the spike outputs
and q the linearly combined spike inputs. For any of these variables, we will use the subscript i (e.g.
vi ) to denote the values of the variables of a particular neuron i across all time steps and superscript
k (e.g. vk ) to denote the values across all neurons at a particular time step k. Thus, for the membrane
voltage
k
v = vik , vk = v1k , . . . , vN
, vi = vi0 , . . . , viT?1 .
The EM procedure alternately estimates distributions on the hidden variables x given the current
parameter estimate for ? (the E-step); and then updates the estimates for parameter vector ? given
the current distribution on the hidden variables x (the M-step).
? E-Step: Given parameter estimates ?b` , estimate
P (x|y, ?b` ),
(10)
which is the posterior distribution of the hidden variables x given the observations y and
current parameter estimate ?b` .
? M-step Update the parameter estimate via the minimization,
h
i
?b`+1 = arg min E L(x, y|?)|?b` + ?(?),
(11)
?
where L(x, y|?) is the joint negative log likelihood,
L(x, y|?) = ? log p(x, y|?).
(12)
In (11) the expectation is with respect to the distribution found in (10) and ?(?) is the
parameter regularization function.
The next two sections will describe how we approximately perform each of these steps.
3.3
E-Step estimation via Approximate Message Passing
For the calcium imaging problem, the challenging step of the EM procedure is the E-step, since
the hidden variables x to be estimated are the states and outputs of a high-dimensional nonlinear
dynamical system. Under the model in Section 2, a system with N neurons will require N states
for the membrane voltages vik and N states for the bound Ca concentration levels zik , resulting in
a total state dimension of 2N . The E-step for this system is essentially a state estimation problem,
and exact inference of the states of a general nonlinear dynamical system grows exponentially in the
state dimension. Hence, exact computation of the posterior distribution (10) for the system will be
intractable even for a moderately sized network.
As described in the Introduction, we thus use an approximate messaging passing method that exploits the separable structure of the system. For the remainder of this section, we will assume the
parameters ? in (5) are fixed to the current parameter estimate ?b` . Then, under the assumptions of
Section 2, the joint probability distribution function of the variables can be written in a factorized
form,
P (x, y) = P (q, v, s, z, y) =
N
T?1
Y
1 Y
1{qk =Wsk } ?iIF (qi , vi , si )?iCA (si , zi , yi ),
Z
i=1
k=0
4
(13)
input currents
qi
membrane voltage
Ca2+ concentration
vi
zi
?iIF (qi , vi , si )
Integrate-and-fire
dynamics
spike outputs
si
?iCA (si , zi , yi )
Ca imaging
dynamics
observed
fluorescence
yi
Neuron i, i = 1, . . . , N
qk = Wsk
Connectivity
between neurons
Time step k, k = 0, . . . , T ?1
Figure 1: Factor graph plate representation of the system where the spike dynamics are described
by the factor node ?iIF (qi , vi , si ) and the calcium image dynamics are represented via the factor
node ?iCA (si , zi , yi ). The high-dimensional dynamical system is described as 2N scalar dynamical
systems (2 for each neuron) with linear interconnections, qk = Wsk between the neurons. A
computational efficient approximation of loopy BP [19] is applied to this graph for approximate
Bayesian inference required in the E-step of the EM algorithm.
where Z is a normalization constant; ?iIF (qi , vi , si ) is the potential function relating the summed
spike inputs qi to the membrane voltages vi and spike outputs si ; ?iCA (si , zi , yi ) relates the spike
outputs si to the bound calcium concentrations zi and observed fluorescence values yi ; and the term
1{qk =Wsk } indicates that the distribution is to be restricted to the set satisfying the linear constraints
qk = Wsk across all time steps k.
As in standard loopy BP [14], we represent the distribution (13) in a factor graph as shown in
Fig. 1. Now, for the E-step, we need to compute the marginals of the posterior distribution p(x|y)
from the joint distribution (13). Using the factor graph representation, loopy BP iteratively updates
estimates of these marginal posterior distributions using a message passing procedure, where the
estimates of the distributions (called beliefs) are passed between the variable and factor nodes in
the graph. In general, the computationally challenging component of loopy BP is the updates on
the belief messages at the factor nodes. However, using the factorized structure in Fig. 1 along
with approximate message passing (AMP) simplifications as described in [19], these updates can be
computed easily.
Details are given in the full paper [21], but the basic procedure for the factor node updates and the
reasons why these computations are simple can be summarized as follows. At a high level, the factor
graph structure in Fig. 1 partitions the 2N -dimensional nonlinear dynamical system into N scalar
systems associated with each membrane voltage vik and an additional N scalar systems associated
with each calcium concentration level zik . The only coupling between these systems is through the
linear relationships qk = Wsk . As shown in Appendix ??, on each of the scalar systems, the factor
node updates required by loopy BP essentially reduces to a state estimation problem for this system.
Since the state space of this system is scalar (i.e. one-dimensional), we can discretize the state space
well with a small number of points ? in the experiments below we use L = 20 points per dimension.
Once discretized, the state estimation can be performed via a standard forward?backward algorithm.
If there are T time steps, the algorithm will have a computational cost of O(T L2 ) per scalar system.
Hence, all the factor node updates across all the 2N scalar systems has total complexity O(N T L2 ).
For the factor nodes associated with the linear constraints qk = Wsk , we use the AMP approximations [19]. In this approximation, the messages for the transform outputs qik are approximated as
Gaussians which is, at least heuristically, justified since the they are outputs of a linear transform of
a large number of variables, ski . In the AMP algorithm, the belief updates for the variables qk and
sk can then be computed simply by linear transformations of W and WT . Since W represents a
connectivity matrix, it is generally sparse. If each row of W has d non-zero values, multiplication
5
by W and WT will be O(N d). Performing the multiplications across all time steps results in a total
complexity of O(N T d).
Thus, the total complexity of the proposed E-step estimation method is O(N T L2 + N T d) per loopy
BP iteration. We typically use a small number of loopy BP iterations per EM update (in fact, in the
experiments below, we found reasonable performance with one loopy BP update per EM update).
In summary, we see that while the overall neural system is high-dimensional, it has a linear + scalar
structure. Under the assumption of the bounded connectivity d, this structure enables an approximate
inference strategy that scales linearly with the number of neurons N and time steps T . Moreover,
the updates in different scalar systems can be computed separately allowing a readily parallelizable
implementation.
3.4
Approximate M-step Optimization
The M-step (11) is computationally relatively simple. All the parameters in ? in (5) have a linear
relationship between the components of the variables in the vector x in (9). For example, the parameters aCA,i and bCA,i appear in the fluorescence output equation (4). Since the noise dkyi in this
equation is Gaussian, the negative log likelihood (12) is given by
L(x, y|?) =
1 X k
T
(yi ? aCA,i zik ? bCA,i )2 + log(?yi ) + other terms,
2?yi
2
k?IF
where ?other terms? depend on parameters other than aCA,i and bCA,i . The expectation
E(L(x, y|?)|?b` ) will then depend only on the mean and variance of the variables yik and zik , which
are provided by the E-step estimation. Thus, the M-step optimization in (11) can be computed via a
simple least-squares problem. Using the linear relation (1), a similar method can be used for ?IF,i
and bIF,i , and the linear relation (3) can be used to estimate the calcium time constant ?CA .
To estimate the connectivity matrix W, let rk = qk ? Wsk so that the constraints in (13) is equivalent to the condition that rk = 0. Thus, the term containing W in the expectation of the negative log
likelihood E(L(x, y|?)|?b` ) is given by the negative log probability density of rk evaluated at zero.
In general, this density will be a complex function of W and difficult to minimize. So, we approxb and b
imate the density as follows: Let q
s be the expectation of the variables q and s given by the
bk ? Wb
E-step. Hence, the expectation of rk is q
sk . As a simple approximation, we will then assume
that the variables rik are Gaussian, independent and having some constant variance ? 2 . Under this
simplifying assumption, the M-step optimization of W with the `1 regularizer (8) reduces to
c = arg min 1
W
2
W
T?1
X
kb
qk ? Wb
sk k2 + ? 2 ?kWk1 ,
(14)
k=0
For a given value of ? 2 ?, the optimization (14) is a standard LASSO optimization [22] which can be
evaluated efficiently via a number of convex programming methods. In this work, in each M-step,
we adjust the regularization parameter ? 2 ? to obtain a desired fixed sparsity level in the solution W.
3.5
Initial Estimation via Sparse Regression
Since the EM algorithm cannot be guaranteed to converge a global maxima, it is important to pick
the initial parameter estimates carefully. The time constants and noise levels for the calcium image
can be extracted from the second-order statistics of fluorescence values and simple thresholding can
provide a coarse estimate of the spike rate.
The key challenge is to obtain a good estimate for the connectivity matrix W. For each neuron i, we
first make an initial estimate of the spike probabilities P (ski = 1|yi ) from the observed fluorescence
values yi , assuming some i.i.d. prior of the form P (sti ) = ??, where ? is the estimated average spike
rate per second. This estimation can be solved with the filtering method in [13] and is also equivalent
to the method we use for the factor node updates. We can then threshold these probabilities to make
a hard initial decision on each spike: ski = 0 or 1. We then propose to estimate W from the spikes
as follows. Fix a neuron i and let wi be the vector of weights Wij , j = 1, . . . , N . Under the
assumption that the initial spike sequence ski is exactly correct, it is shown in the full paper [21], that
6
Parameter
Number of neurons, N
Connection sparsity
Value
100
10% with random connections. All connections are excitatory
with the non-zero weights Wij being exponentially distributed.
10 Hz
1 ms
10 sec (10,000 time steps)
20 ms
2 time steps = 2 ms
Produced from two unobserved neurons.
500 ms
Set to 20 dB SNR
100 Hz
Mean firing rate per neuron
Simulation time step, ?
Total simulation time, T ?
Integration time constant, ?IF
Conduction delay, ?
Integration noise, dkvi
Ca time constant, ?CA
Fluorescence noise, ?CA
Ca frame rate , 1/TF
Table 1: Parameters for the Ca image simulation.
Figure 2: Typical network simulation
trace. Top panel: Spike traces for
the 100 neuron simulated network.
Bottom panel: Calcium image fluorescence levels. Due to the random network topology, neurons often
fire together, significantly complicating connectivity detection. Also, as
seen in the lower panel, the slow decay of the fluorescent calcium blurs
the spikes in the calcium image.
a regularized maximum likelihood estimate of wi and bias term bIF,i is given by
b i , bbIF,i ) = arg min
(w
wi ,bIF,i
T?1
X
Lik (uTk wi + cik bIF,i ? ?, ski ) + ?
N
X
|Wij |,
(15)
j=1
k=0
where Lik is a probit loss function and the vector uk and scalar cik can be determined from the
spike estimates. The optimization (15) is precisely a standard probit regression used in sparse linear
classification [23]. This form arises due to the nature of the leaky integrate-and-fire model (1) and
(2). Thus, assuming the initial spike sequences are estimated reasonably accurately, one can obtain
good initial estimates for the weights Wij and bias terms bIF,i by solving a standard classification
problem.
4
Numerical Example
The method was tested using realistic network parameters, as shown in Table 1, similar to those
found in neurons networks within a cortical column [24]. Similar parameters are used in [7]. The
network consisted of 100 neurons with each neuron randomly connected to 10% of the other neurons. The non-zero weights Wij were drawn from an exponential distribution. As a simplification,
all weights were positive (i.e. the neurons were excitatory ? there were no inhibitory neurons in the
simulation). A typical random matrix W generated in this manner would not in general result in a
stable system. To stabilize the system, we followed the procedure in [8] where the system is simulated multiple times. After each simulation, the rows of the matrix W were adjusted up or down to
increase or decrease the spike rate until all neurons spiked at a desired target rate. In this case, we
assumed a desired average spike rate of 10 Hz.
7
Figure 3: Weight estimation accuracy. Left: Normalized mean-squared error as a function of the
iteration number. Right: Scatter plot of the true and estimated weights.
From the parameters in Table 1, we can immediately see the challenges in the estimation. Most
importantly, the calcium imaging time constant ?CA is set for 500 ms. Since the average neurons
spike rate is assumed to be 10 Hz, several spikes will typically appear within a single time constant.
Moreover, both the integration time constant and inter-neuron conduction time are much smaller
than the
A typical simulation of the network after the stabilization is shown in Fig. 2. Observe that due to
the random connectivity, spiking in one neuron can rapidly cause the entire network to fire. This
appears as the vertical bright stripes in the lower panel of Fig. 2. This synchronization makes the
connectivity detection difficult to detect under temporal blurring of Ca imaging since it is hard to
determine which neuron is causing which neuron to fire. Thus, the random matrix is a particularly
challenging test case.
The results of the estimation are shown in Fig. 3. The left panel shows the relative mean squared
error defined as
P
cij |2
min? ij |Wij ? ?W
P
relative MSE =
,
(16)
2
ij |Wij |
cij is the estimate for the weight Wij . The minimization over all ? is performed since the
where W
method can only estimate the weights up to a constant scaling. The relative MSE is plotted as a
function of the EM iteration, where we have performed only a single loopy BP iteration for each
EM iteration. We see that after only 30 iterations we obtain a relative MSE of 7% ? a number at
least comparable to earlier results in [7], but with significantly less computation. The right panel
cij against the true weights Wij .
shows a scatter plot of the estimated weights W
5
Conclusions
We have presented a scalable method for inferring connectivity in neural systems from calcium
imaging. The method is based on factorizing the systems into scalar dynamical systems with linear
connections. Once in this form, state estimation ? the key computationally challenging component
of the EM estimation ? is tractable via approximating message passing methods. The key next step
in the work is to test the methods on real data and also provide more comprehensive computational
comparisons against current techniques such as [7].
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9
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4,796 | 5,342 | Sparse space-time deconvolution
for Calcium image analysis
Ferran Diego
Fred A. Hamprecht
Heidelberg Collaboratory for Image Processing (HCI)
Interdisciplinary Center for Scientific Computing (IWR)
University of Heidelberg, Heidelberg 69115, Germany
{ferran.diego,fred.hamprecht}@iwr.uni-heidelberg.de
Abstract
We describe a unified formulation and algorithm to find an extremely sparse representation for Calcium image sequences in terms of cell locations, cell shapes,
spike timings and impulse responses. Solution of a single optimization problem
yields cell segmentations and activity estimates that are on par with the state of
the art, without the need for heuristic pre- or postprocessing. Experiments on real
and synthetic data demonstrate the viability of the proposed method.
1
Introduction
A detailed understanding of brain function is a still-elusive grand challenge. Experimental evidence
is collected mainly by electrophysiology and ?Calcium imaging?. In the former, multi-electrode
array recordings allow the detailed study of hundreds neurons, while field potentials reveal the collective action of dozens or hundreds of neurons. The more recent Calcium imaging, on the other
hand, is a fluorescent microscopy technique that allows the concurrent monitoring of the individual actions of thousands of neurons at the same time. While its temporal resolution is limited by
the chemistry of the employed fluorescent markers, its great information content makes Calcium
imaging an experimental technique of first importance in the study of neural processing, both in
vitro [16, 6] and in vivo [5, 7]. However, the acquired image sequences are large, and in laboratory
practice the analysis remains a semi-manual, tedious and subjective task.
Calcium image sequences reveal the activity of neural tissue over time. Whenever a neuron fires,
its fluorescence signal first increases and then decays in a characteristic time course. Evolutionary
and energetic constraints on the brain guarantee that, in most cases, neural activity is sparse in
both space (only a fraction of neurons fire at a given instant) and time (most neurons fire only
intermittently). The problem setting can be formalized as follows: given an image sequence as
input, the desired output is (i) a set of cells1 and (ii) a set of time points at which these cells were
triggered. We here propose an unsupervised learning formulation and algorithm that leverages the
known structure of the data to produce the sparsest representations published to date, and allow for
meaningful automated analysis.
1.1
Prior Art
Standard laboratory practice is to delineate each cell manually by a polygon, and then integrate their
fluorescence response over the polygon, for each point in time. The result is a set of time series, one
per cell.
1
Optical sectioning by techniques such as confocal or two-photon microscopy implies that we see only parts
of a neuron, such as a slice through its cell body or a dendrite, in an image plane. For brevity, we simply refer
to these as ?cells? in the following.
1
a) Matrix factorization [13, 15, 4, 3, 12]
b) Convolutional sparse coding [8, 25, 20, 17, 14]
Figure 1: Sketch of selected previous work. Left: Decomposition of an image sequence into a sum
of K components. Each component is given by the Cartesian product of a spatial component or
basis image Dk and its temporal evolution uk . In this article, we represent such Cartesian products
by the convolution of multidimensional arrays. Right: Description of a single image in terms of a
sum of latent feature maps Dk convolved with filters Hk
Given that the fluorescence signal impulse response to a stimulus is stereotypic, these time series
can then be deconvolved to obtain a sparse temporal representation for each cell using nonnegative
sparse deconvolution [24, 5, 10].
The problem of automatically identifying the cells has received much less attention, possibly due to
the following difficulties [16, 23]: i) low signal-to-noise ratio (SNR); ii) large variation in luminance
and contrast; iii) heterogeneous background; iv) partial occlusion and v) pulsations due to heartbeat
or breathing in live animals. Existing work either hard-codes prior knowledge on the appearance of
specific cell types [16, 22] or uses supervised learning (pixel and object level classification, [23]) or
unsupervised learning (convolutional sparse block coding, [14]).
Closest in spirit to our work are attempts to simultaneously segment the cells and estimate their time
courses. This is accomplished by matrix factorization techniques such as independent component
analysis [13], nonnegative matrix factorization [12] and (hierarchical) dictionary learning [4, 3].
However, none of the above give results that are truly sparse in time; and all of the above have to go
to some lengths to obtain reasonable cell segmentations: [13, 4] recur to heuristic post-processing,
while [3] invokes structured sparsity inducing norms and [15] use an additional clustering step as
initialization. All these extra steps are necessary to assure that each spatial component represents
exactly one cell.
In terms of mathematical modeling, we build on recent advances and experiments in convolutional
sparse coding such as [8, 25, 20, 17, 14]. Ref. [21] already applies convolutional sparse coding to
video, but achieves sparsity only in space and not in time (where only pairs of frames are used to
learn latent representations). Refs. [19, 18] consider time series which they deconvolve, without
however using (or indeed needing, given their data) a sparse spatial representation.
1.2
Contributions
Summarizing prior work, we see three strands: i) Fully automated methods that require an extrinsic cell segmentation, but can find a truly2 sparse representation of the temporal activity. ii) Fully
automated methods that can detect and segment cells, but do not estimate time courses in the same
framework. iii) Techniques that both segment cells and estimate their time courses. Unfortunately,
existing techniques either produce temporal representations that are not truly sparse [12, 4, 3] or do
not offer a unified formulation of segmentation and activity detection that succeeds without extraneous clustering steps [15].
In response, we offer the first unified formulation in terms of a single optimization problem: its
solution simultaneously yields all cells along with their actions over time. The representation of
activity is truly sparse, ideally in terms of a single nonzero coefficient for each distinct action of a
cell. This is accomplished by sparse space-time deconvolution. Given a motion-corrected sequence
of Calcium images, it estimates i) locations of cells and ii) their activity along with iii) typical cell
shapes and iv) typical impulse responses. Taken together, these ingredients afford the sparsest, and
thus hopefully most interpretable, representation of the raw data. In addition, our joint formulation
2
We distinguish a sparse representation, in which the estimated time course of a cell has many zeros; and a
?truly sparse? representation in which a single action of a cell is ideally represented in terms of a single nonzero
coefficient.
2
Figure 2: Summary of sparse space-time deconvolution. Top: Unified formulation in terms of a
single optimization problem. Bottom: Illustration on tiny subset of data. Left: raw data. The
fluorescence level to be estimated is heavily degraded by Poisson shot noise that is unavoidable
at the requisite short exposure times. Middle: smoothed raw data. Right: approximation of the
data in terms of a Cartesian product of estimated cell shapes and temporal activities. Each temporal
activity is further decomposed as a convolution of estimated impulse responses and very few nonzero
coefficients.
allows to estimate a nonuniform and temporally variable background. Experiments on difficult
artificial and real-world data show the viability of the proposed formulation.
Notation Boldface symbols describe multidimensional arrays. We define A ? B as a convolution of
multidimensional arrays A and mirror(B), with the result trimmed to the dimensions of A. Here,
the ?mirror? operation flips a multidimensional array along every dimension. A ~ B is the full
convolution result of multidimensional arrays A and mirror(B). These definitions are analogous to
the ?convn? command in matlab with shape arguments ?same? and ?full?, respectively. k ? k0 counts
the number of non-zero coeficients, and k ? kF is the Frobenius norm.
2
2.1
Sparse space-time deconvolution (SSTD)
No background subtraction
An illustration of the proposed formulation is given in Fig. 2 and our notation is summarized in
Table. 1. We seek to explain image sequence X in terms of up to K cells and their activity over time.
In so doing, all cells are assumed to have exactly one (Eq. 1.1) of J << K possible appearances,
and to reside at a unique location (Eq. 1.1). These cell locations are encoded in K latent binary
feature maps. The activity of each cell is further decomposed in terms of a convolution of impulses
(giving the precise onset of each burst) with exactly one of L << K types of impulse responses.
A single cell may ?use? different impulse responses at different times, but just one type at any one
time ((Eq. 1.2).
All of the above is achieved by solving the following optimization problem:
?
?
K
J
X
X
?
min X ?
Dk,j ? Hj ? ~
D,H,f ,s
j=1
k=1
3
! 2
sk,l ? fl
l=1
L
X
F
(1)
Constraint
P
Pj kDk,j k0 ? 1, ?k
such that
l kst,k,l k0 ? 1, ?k, t
kHj k2F ? 1, ?j
kfl k22 ? 1, ?l
(1.1)
(1.2)
(1.3)
(1.4)
Interpretation
at most one location and appearance per component
only one type of activation at each time per cell
prevent cell appearance from becoming large
prevent impulse filter from becoming large
Here, the optimization is with respect to the cell detection maps D, cell appearances H, activity
patterns or impulse responses f as well as ?truly sparse? activity indicator vectors s. This optimization is subject to the two constraints mentioned earlier plus upper bounds on the norm of the learned
filters.
The user needs to select the following parameters: an upper bound K on the number of cells as
well as the size in pixels H of their matched filters / convolution kernels H; upper bounds J on
the number of different appearances and L on the number of different activity patterns that cells
may have; as well as the number of coefficients F that the learned impulse responses may have.
Considering that we propose a method for both cell detection and sparse time course estimation,
the number of six user-adjustable parameters compares favourably to previous work. Methods that
decouple these steps typically need more parameters altogether, and the heuristics that prior work
on joint optimization uses also have a large number of (implicit) parameters.
PK
PK PJ
While many other approximations such as k=1 Dk ~ sk ? fk or k=1 j=1 Dk,j ? Hj ~ sk,j ? fj
are conceivable and may make sense in other applications areas, the proposed formulation is the
most parsimonious of its kind. Indeed, it uses a small pool of J shapes and L firing patterns, which
can be combined freely, to represent all cells and their activities. It is owing to this fact that we dub
the method sparse spatio-temporal deconvolution (SSTD).
2.2
SSTD with background subtraction
In actual experiments, the observed fluorescence level is a sum of the signal of interest plus a nuisance background signal. This background is typically nonuniform in the spatial domain and, while
it can be modeled as constant over time [15, 24], is often also observed to vary over time, prompting
robust local normalization as a preprocessing step [7, 4].
Here, we generalize the formulation from (1) to correct for a background that is assumed to be
spatially smooth and time-varying. In more detail, we model the background in terms of the direct
?N ?1
1?1?T
product Bs ~ bt of a spatial component Bs ? RM
and a time series bt ? R+
. Insights
+
into the physics and biology of Calcium imaging suggest that (except for saturation regimes characterized by high neuron firing rates), it is reasonable to assume that the normalized quantity (observed
fluorescence minus background) divided by background, typically dubbed ?F/F0 , is linearly related
to the intracellular Calcium concentration [24, 10]. In keeping with this notion, we now propose our
final model, viz.
?
2
?
?
?
!
K
J
L
X
X
X
s
t?
s
t
?
?
?
min
X?
Dk,j ? Hj ~
sk,l ? fl ? B ~ b B ~ b
D,H,f ,s,Bs ,bt
j=1
k=1
l=1
F
+ ?kBs kT V such that (1.1) ? (1.4), Bs > 0, bt > 0
(2)
with ? ? denoting an elementwise division. Note that the optimization now also runs over the
spatial and temporal components of the background, with the total variation (TV) regularization
term3 enforcing spatial smoothness of the spatial background component [2].
In addition to the previously defined parameters, the user also needs to select parameter ? which
determines the smoothness of the background estimate.
2.3
Optimization
The optimization problem in (2) is convex in either the spatial or the temporal filters H, f alone when
keeping all other unknowns fixed; but it is nonconvex in general. In our experiments, we use a block
coordinate descent strategy [1, Section 2.7] that iteratively optimizes one group of variables while
3
TV measures the sum of the absolute values of the spatial gradient.
4
Symbol
?N ?T
X ? RM
+
K ? N+
J ? N+
Hj ? RH?H?1
+
Dk,j ? {0, 1}M ?N ?1
L ? N+
fl ? R1?1?F
+
sk,l ? R1?1?T
+
Definition
image sequence of length T , each image is M ? N
number of cells
number of distinct cell appearances
j th cell appearance / spatial filter / matched filter of size H ? H
indicator matrix of the k th cell for the j th cell appearance
number of distinct impulse responses / activity patterns
lth impulse response of length F
indicator vector of the k th spike train for the lth impulse response
Table 1: Notation
fixing all others (see supplementary material for details). The nonconvex l0 -norm constraints require
that cell centroids D and spike trains s are estimated by techniques such as convolutional matching
pursuit [20]; while the spatio-temporal filters can be learned using simpler gradient descent [25],
K-SVD [20] or simple algebraic expressions.
All unknowns are initialized with standard Gaussian noise truncated to nonnegative values. The
limiting number of cells K can be set to a generous upper bound of the expected true number
because spatial components without activity are automatically set to zero during optimization.
3
Experimental Setup
This section describes the data and algorithms used for experiments and benchmarks.
3.1
Inferring Spike Trains
The following methods assume that cell segmentation has already been performed by some means,
and that the fluorescence signal of individual pixels has been summed up for each cell and every time
step. They can hence concentrate exclusively on the estimation of a ?truly sparse? representation of
the respective activities in terms of a ?spike train?.
Data We follow [24, 5] in generating 1100 sequences consisting of one-sided exponential decays
with a constant amplitude of 1 and decay rate ? = 1/2s, sampled at 30f ps with firing rates ranging
uniformly from 1 to 10Hz and different Gaussian noise levels ? ? [0.1, 0.6].
Fast non-negative deconvolution (FAST) [24] uses a one-sided exponential decay as parametric
model for the impulse response by invoking a first-order autoregressive process. Given that our
artificial data is free of a nuisance background signal, we disregard its ability to also model such
background. The sole remaining parameter, the rate of the exponential decay, can be fit using maximum likelihood estimation or a method-of-moments approach [15].
Peeling [5] finds spikes by means of a greedy approach that iteratively removes one impulse response
at a time from the residual fluorescence signal. Importantly, this stereotypical transient must be
manually defined a priori.
Sparse temporal deconvolution (STD) with a single impulse response is a special case of this work
for given nonoverlapping cell segmentations and L = 1; and it is also a special case of [14]. The
impulse response can be specified beforehand (amounting to sparse coding), or learned from the
data (that is, performing dictionary learning on time-series data).
3.2
Segmenting Cells and Estimating Activities
Data Following the procedure described in [4, 12, 13], we have created 80 synthetic sequences
with a duration of 15s each at a frame rate of 30f ps with image sizes M = N = 512 pixels.
The cells are randomly selected from 36 cell shapes extracted from real data, and are randomly
located in different locations with a maximum spatial overlap of 30%. Each cell fires according to
a dependent Poisson process, and its activation pattern follows a one-sided exponential decay with
5
a scale selected uniform randomly between 500 and 800ms. The average number of active cells
per frame varies from 1 to 10. Finally, the data has been distorted by additive white Gaussian noise
with a relative amplitude (max. intensity ? mean intensity)/?noise ? {3, 5, 7, 10, 12, 15, 17, 20}.
By construction, the identity, location and activity patterns of all cells are known. The supplemental
material shows an example with its corresponding inferred neural activity.
Real-world data comes from two-photon microscopy of mouse motor cortex recorded in vivo [7]
which has been motion-corrected. These sequences allow us to conduct qualitative experiments.
ADINA [4] relies on dictionary learning [11] to find both spatial components and their time courses.
Both have many zero coefficients, but are not ?truly sparse? in the sense of this paper. The method
comes with a heuristic post-processing to separate coactivated cells into distinct spatial components.
NMF+ADINA uses non-negative matrix factorization to infer both the spatial and temporal primitives of an image sequence as in [12, 15]. In contrast to [15] which uses a k-means clustering of
highly confident spike vectors to provide a good initialization in the search for spatial components,
we couple NMF with the postprocessing of ADINA.
CSBC+SC combines convolutional sparse block coding [14] based on a single still image (obtained
from the temporal mean or median image, or a maximum intensity projection across time) with
temporal sparse coding.
CSBC+STD combines convolutional sparse block coding [14] based on a single still image (obtained from the temporal mean or median image, or a maximum intensity projection across time)
with the proposed sparse temporal deconvolution in Sect. 3.1.
SSTD is the method described here. We used J = L = 2, K = 200, F = 200 and H = 31, 15 for
the artificial and real data, respectively.
4
4.1
Results
Inferring spike trains
To quantify the accuracy of activity detection, we first threshold the estimated activities and then
compute, by summing over each step in every time series, the number of true and false negatives
and positives. For a fair comparison, the thresholds were adjusted separately for each method to give
optimal accuracy. Sensitivity, precision and accuracy computed from the above implicitly measure
both the quality of the segmentation and the quality of the activity estimation. An additional measure, SPIKE distance [9], emphasizes any temporal deviations between the true and estimated spike
location in a truly sparse representation.
Fig. 3 shows that, unsurprisingly, best results are obtained when methods use the true impulse response rather than learning it from the data. This finding does not carry over to real data, where a
?true? impulse response is typically not known. Given the true impulse response, both FAST and
STD fare better than Peeling, showing that a greedy algorithm is faster but gives somewhat worse
results. Even when learning the impulse response, FAST and STD are no worse than Peeling. When
learning the parameters, FAST has an advantage over STD on this artificial data because FAST already uses the correct parametric form of the impulse response that was used to generate the data
and only needs to learn a single parameter; while STD learns a more general but nonparametric
activity model with many degrees of freedom.
The great spread of all quality measures results from the wide range of noise levels used, and the
overall deficiencies in accuracy attest to the difficulty of these simulated data sets.
4.2
Segmenting Cells and Inferring spike trains
Fig. 4 shows that all the methods from Sect. 3.2 reach respectable and comparable performance in
the task of identifying neural activity from non-trivial synthetic image sequences.
CSBC+SC reaches the highest sensitivity while SSTD has the greatest precision. SSTD apparently
achieves comparable performance to the other methods without the need for a heuristic pre- or
postprocessing. Multiple random initializations lead to similar learned filters (results not shown),
6
FAST (fixed param.)
FAST (learned param.)
Peeling (fixed param.)
STD (fixed param.)
STD (learned param.)
00
20 20 40
60
40
8060
Sensitivity
(%)
Sensitivity
(%)
100
80
1000
20
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60
Precision (%)
100 0
80
20
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60
Accuracy (%)
80
100 0
0.1
0.2
SPIKE distance
0.3
0.4
Figure 3: Sensitivity, precision, accuracy (higher is better) and SPIKE distance (lower is better) of
different methods for spike train estimation. Methods that need to learn the activation pattern perform worse than those using the true (but generally unknown) activation pattern and its parameters.
FAST is at an advantage here because it happens to use the very impulse response that was used in
generating the data.
so the optimization problem seems to be well-posed. The price to pay for the elegance of a unified
formulation is a much higher computational cost of this more involved optimization. Again, the
spread of sensitivities, precisions and accuracies results from the range of noise levels used in the
simulations. The plots suggest that SSTD may have fewer ?catastrophic failure? cases, but an even
larger set of sequences will be required to verify this tendency.
ADINA
NNMF+ADINA
CSBC+SC
CSBC+STD
SSTD
50
50
60 60
70
70 80
8090
Sensitivity
(%) (%)
Sensitivity
90
100
100 50
60
70
80
Precision (%)
90
100 50
60
70
80
Accuracy (%)
90
100
Figure 4: Quality of cell detection and and the estimation of their activities. SSTD does as well as
the competing methods that rely on heuristic pre- or post-processing.
Real Sequences: Qualitative results are shown in Fig. 5. SSTD is able to distinguish both cells with
spatial overlap and with high temporal correlation. It compensates large variations in luminance
and contrast, and can discriminate between different types of cells. Exploiting truly sparse but
independent representations in both the spatial and the temporal domain allows to infer plausible
neural activity and, at the same time, reduce the noise in the underlying Calcium image sequence.
5
Discussion
The proposed SSTD combines the decomposition of the data into low-rank components with the
finding of a convolutional sparse representation for each of those components. The formalism allows
exploiting sparseness and the repetitive motifs that are so characteristic of biological data. Users
need to choose the number and size of filters that indirectly determine the number of cell types
found and their activation patterns.
As shown in Fig. 5, the approach gives credible interpretations of raw data in terms of an extremely
sparse and hence parsimonious representation.
The decomposition of a spacetime volume into a Cartesian product of spatial shapes and their time
courses is only possible when cells do not move over time. This assumption holds for in vitro
experiments, and can often be satisfied by good fixation in in vivo experiments, but is not universally
valid. Correcting for motions in a generalized unified framework is an interesting direction for future
work. The experiments in section 4.1 suggest that it may also be worthwhile to investigate the use
of more parametric forms for the impulse response instead of the completely unbiased variant used
here.
7
70
48
64
56
62
2
65
29
135
142
150
88
54
66
69
41
137
18
15
37
144
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76
75
59 20
38
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141
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121
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96 71
125
61 44 43
106
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74 136
1
58
17
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Frames
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25 36
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69
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3510
89
27
95
148
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32
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47
12
100 65
5
81
63
47
0.05
16
31
98
23
76
96
93
0.25
75
38
28
1567
54
21
14
44
55
34
9
134
56
11
4
33
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78
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filter 1
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Figure 5: Qualitative results on two real data sets. The data on the left column shows mostly cell
bodies, while the data on the right shows both cell bodies (large) and dendrites (small). For each
data set, the top left shows an average projection of the relative fluorescence change across time with
cell centroids D (black dots) and contours of segmented cells, and the top right shows the learned
impulse responses. In the middle, the fluorescence levels integrated over the segmented cells are
shown in random colors. The bottom shows by means of small disks the location, type and strength
of the impulses that summarize all the data shown in the middle. Together with the cell shapes, the
impulses from part of the ?truly sparse? representation that we propose. When convolving these
spikes with the impulse responses from the top right insets, we obtain the time courses shown in
random colors.
Such advances will further help making Calcium imaging an enabling tool for the neurosciences.
8
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[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
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4,797 | 5,343 | Spatio-temporal Representations of Uncertainty in
Spiking Neural Networks
Sophie Deneve
Group for Neural Theory, ENS Paris
Rue d?Ulm, 29, Paris, France
[email protected]
Cristina Savin
IST Austria
Klosterneuburg, A-3400, Austria
[email protected]
Abstract
It has been long argued that, because of inherent ambiguity and noise, the brain
needs to represent uncertainty in the form of probability distributions. The neural encoding of such distributions remains however highly controversial. Here we
present a novel circuit model for representing multidimensional real-valued distributions using a spike based spatio-temporal code. Our model combines the computational advantages of the currently competing models for probabilistic codes
and exhibits realistic neural responses along a variety of classic measures. Furthermore, the model highlights the challenges associated with interpreting neural
activity in relation to behavioral uncertainty and points to alternative populationlevel approaches for the experimental validation of distributed representations.
Core brain computations, such as sensory perception, have been successfully characterized as probabilistic inference, whereby sensory stimuli are interpreted in terms of the objects or features that
gave rise to them [1, 2]. The tenet of this Bayesian framework is the idea that the brain represents uncertainty about the world in the form of probability distributions. While this notion seems
supported by behavioural evidence, the neural underpinnings of probabilistic computation remain
highly debated [1, 2]. Different proposals offer different trade-offs between flexibility, i.e. the class
of distributions they can represent, and speed, i.e. how fast can the uncertainty be read out from the
neural activity. Given these two dimensions, we can divide existing models in two main classes.
The first set, which we will refer to as spatial codes, distributes information about the distribution
across neurons; the activity of different neurons reflects different values of an underlying random
variable (alternatively, it can be viewed as encoding parameters of the underlying distribution [1,
2]). Linear probabilistic population codes (PPCs) are a popular instance of this class, whereby
the log-probability of a random variable can be linearly decoded from the responses of neurons
tuned to different values of that variable [3]. This encoding scheme has the advantage of speed, as
uncertainty can be decoded in a neurally plausible way from the quasi-instantaneous neural activity,
and reproduces aspects of the experimental data. However, these benefits come at the price of
flexibility: the class of distributions that the network can represent needs to be highly restricted,
otherwise the network size scales exponentially with the number of variables [1].
This limitation has lead to a second class of models, which we will refer to as temporal codes.These
use stochastic network dynamics to sample from the target distribution [4, 1]. Existing models
from this class assume that the activity of each neuron encodes a different random variable; the
network explores the state space such that the time spent in any particular state is proportional to its
probability under the distribution [4]. This representation is exact in the limit of infinite samples.
It has several important computational advantages (e.g. easy marginalization, parameter learning,
linear scaling of network size with the number of dimensions) and further accounts for trial-totrial variability in neural responses [1]. These benefits come at the cost of sampling time: a fair
representation of the underlying distribution requires pooling over several samples, i.e. integrating
neural activity over time. Some have argued that this feature makes sampling unfeasibly slow [2].
1
Here we show that it is possible to construct spatio-temporal codes that combine the best of both
worlds. The core idea is that the network activity evolves through recurrent dynamics such that
samples from the posterior distribution can be linearly decoded from the (quasi-)instantaneous neural responses. This distributed representation allows several independent samples to be encoded
simultaneously, thus enabling a fast representation of uncertainty that improves over time. Computationally, our model inherits all the benefits of a sampling-based representation, while overcoming potential shortcomings of classic temporal codes. We explored the general implications of the
new coding scheme for a simple inference problem and found that the network reproduces many
properties of biological neurons, such as tuning, variability, co-variability and their modulation by
uncertainty. Nonetheless, these single or pairwise measures provided limited information about the
underlying distribution represented by the circuit. In the context of our model, these results argue for
using decoding as tool for validating distributed probabilistic codes, an approach which we illustrate
with a simple example.
1
A distributed spatio-temporal representation of uncertainty
The main idea of the representation is simple: we want to approximate a real-valued D-dimensional
distribution P(x) by samples generated by K independent chains implementing Markov Chain
Monte Carlo (MCMC) sampling [5], y(t) = {yk (t)}k=1...K , with yk ? P(x) (Fig. 1). To this aim,
we encode the stochastic trajectory of the chains in a population of N spiking neurons (N > KD),
such that y(t) is linearly decodable from the neural responses. In particular, we adapt a recently
proposed coding scheme for representing time-varying signals [6] and construct stochastic neural
dynamics such that samples from the target distribution can be obtained by a linear mapping of the
spikes convolved with an epsp-like exponential kernel (Fig. 1a):
y
?(t) = ? ? r(t)
(1)
? (t) denotes the decoded state of the K MCMC chains at time t (of size D ? K), ? is the
where y
decoding matrix1 and r is the low-pass version of the spikes o, ?V r?i = ?ri + oi .
To facilitate the presentation of the model, we start by constructing recurrent dynamics for sampling
a single MCMC chain, which we then generalise to the multi-chain scenario. Based on these network
dynamics, we implement probabilistic inference in a linear Gaussian mixture, which we use in
Section 2 to investigate the neural implications of the code.
Distributed MCMC sampling
As a starting point, consider the computational task of representing an arbitrary temporal trajectory
(the gray line in Fig. 1b) as the linear combination of the responses of a set of neurons (one can think
of this as an analog-to-digital conversion of sorts). If the decoding weights of each neuron points in
a different direction (colour coded), then the trajectory could be efficiently reconstructed by adding
the proper weight vectors (the local derivative of the trajectory) at just the right moment. Indeed,
recent work has shown how to construct network dynamics enabling the network to track a trajectory
as closely as possible [6]. To achieve this, neurons use a greedy strategy: each neuron monitors
the current prediction error (the difference between the trajectory and its linear decoding from the
spikes) and spikes only when its weight vector points in the right direction. When the decoding
weights of several neurons point the same way (as in Fig. 1a), they compete to represent the signal
via recurrent inhibition:2 from the perspective of the decoder, it does not matter which of these
neurons spikes next, so the actual population responses depend on the previous spike history, initial
conditions and intrinsic neural noise.3 As a result, spikes are highly irregular and look ?random?
(with Poisson-like statistics), even when representing a constant signal. While competition is an
important driving force for the network, neurons can also act cooperatively ? when the change in the
signal is larger than the contribution of a single decoding vector, then several neurons need to spike
together to represent the signal (e.g. response to the step in Fig. 1a).
1
The decoding matrix can be arbitrary.
This competition makes spike correlations extremely weak in general [7].
3
When N D there is a strong degeneracy in the map between neural responses and the signal, such that
several different spike sequences yield the same decoded signal. In absence of internal noise, the encoding is
nonetheless deterministic despite apparent variability.
2
2
Figure 1: Overview of the model. a. We assume a linear decoder, where the estimated signal y?
is obtained as a weighted sum of neural responses (exponential kernel, blue). b. When the signal
is multidimensional, different neurons are responsible for encoding different directions along the
target trajectory (gray). c. Alternative network architectures: in the externally-driven version the
target trajectory is given as an external input, whereas in the self-generated case it is computed via
slow recurrent connections (green arrow); the input s is used during inference, when sampling from
P(x|s). d. Encoding an example MCMC trajectory in the externally-driven mode. Light colours
show ground truth; dark colours the decoded signal. e. Single-chain samples from a multivariate
distribution (shown as colormap) decoded from a spiking network; trajectory subsampled by a factor
of 10 for visibility. e. Decoded samples using 5 chains (colors) and a fifth of the time in e.
2
Formally, the network dynamics minimise the squared reconstruction error, (y ? y
?) , under certain
constraints on mean firing rate which ensure the representation is distributed (see Suppl. Info.).
The resulting network consists of spiking neurons with simple leaky-integrate-and-fire dynamics,
? denotes the temporal derivative of V, the binary vector o denotes
? = ? 1 V ? Wo + I, where V
V
?v
the spikes, oi (t) = ? iff Vi (t) > ?i , ?v is the membrane time constant (same as that of the decoder),
P
the neural threshold is ?i = j ?2ij + ? and the recurrent connections, W = ?T ? + ? ? I, can
be learned by STDP [8], where ? is a free parameter controlling neural sparseness. The membrane
potential of each neuron tracks the component of the reconstruction error along the direction of its
decoding weights. As a consequence, the network is balanced (because the dynamics aim to bring
the reconstruction error to zero) and membrane potentials are correlated, particularly in pairs of
neurons with similar decoding weights [7] (see Fig. 2c).
In the traditional form, which we refer to as the ?externally-driven? network (Fig. 1c), information
? In
about the target trajectory is provided as an external input to the neurons: I = ?T ? (1/?v y + y).
our particular case, this input implements a particular kind of MCMC sampling (Langevin). Briefly,
the sampler involves stochastic dynamics driven by the gradient of log P (y), with additive Gaussian
noise [5] (see Suppl.Info. for implementation details). Hence, the external input is stochastic I =
?T ? (1/?v y + F (y) + ), where F (y) = ? log P(y), and is D-dimensional white independent
Gaussian noise. Using our network dynamics, we can encode the MCMC trajectory with high
precision (Fig. 1d). Importantly, because of the distributed representation, the integration window
of the decoder does not restrict the frequency content of the signal. The network can represent
signals that change faster than the membrane time constant (Fig. 1a, d).
To construct a viable biological implementation of this network, we need to embed the sampling
dynamics within the circuit (?self-generated? architecture in Fig. 1c). We achieved this by approximating the current I using the decoded signal y
? instead of y. This results in a second recurrent input
? + F (?
to the neurons, ?I = ?T ? (1/?v y
y) + ). While this is an approximation, we found it does not
affect sampling quality in the parameter regime when the encoding scheme itself works well (see
example dynamics in Fig. 1e).
3
Such dynamics can be derived for any distribution from the broad class of product-of-(exponentialfamily) experts [9], with no restrictions on D; for simplicity and to ease visualisation, here we focus
on the multivariate Gaussian case and restrict the simulations to bivariate distributions (D = 2). For
a Gaussian distribution with mean ? and covariance ?, the resulting membrane potential dynamics
are linear:4
?V
1
= ? V ? Wfast o + Wslow r + D + ?T
(2)
?t
?v
where o denotes the spikes, r is a low-passed version of the spikes. The connections Wfast
correspond to the recurrent
dynamics derived above, while the slow5 connections, Wslow =
T
?1
1
1
?T ??1 ? correspond to
I??
? (e.g. NMDA currents) and the drift term D = ?slow
?slow ? ?
the deterministic component of the MCMC dynamics6 and is white independent Gaussian noise
(implemented for instance by a small chaotic subnetwork appropriately connected to the principal
neurons). In summary, relatively simple leaky integrate-and-fire neurons with appropriate recurrent
connectivity are sufficient for implementing Langevin sampling from a Gaussian distribution in a
distributed code. More complex distributions will likely involve nonlinearities in the slow connections (possibly computed in the dendrites) [10].
Multi-chain encoding: instantaneous representation of uncertainty
The earliest proposal for sampling-based neural representations of uncertainty suggested distributing
samples either across neurons or across time [4]. Nonetheless, all realisations of neural sampling use
the second solution. The reason is simple: when equating the activity of individual neurons (either
voltage or firing rate) to individual random variables, it is relatively straightforward to construct neural dynamics implementing MCMC sampling. It is less clear what kind of neural dynamics would
generate samples in several neurons at a time. One naive solution would be to construct several networks that each sample from the same distribution in parallel. This however seems to unavoidably
entail a ?copy-pasting? of all recurrent connections across different circuits, which is biologically
unrealistic. Our distributed representation, in which neurons jointly encode the sampling trajectory,
provides a potential solution to this problem. In particular, it allows several chains to be embedded
in a single network.
To extend the dynamics to a multi-chain scenario, we imagine an auxiliary probability distribution
over K random variables. We want each to correspond to one chain, so we take them to be independent and identically distributed according to P(x). Since the sampling dynamics derived above do
not restrict the dimensionality of the underlying distribution, we can use them to sample from this
D ? K-dimensional distribution instead. For the example
of a multivariate normal, for instance, we
would now sample from another Gaussian, P x?K , with mean ??K (K repetitions of ?) and covariance ??K , a block-diagonal matrix, obtained by K repetitions of ?. In general, the multi-chain
trajectory can be viewed as just another instance of MCMC sampling, where the encoding scheme
guarantees that the signals across different chains remain independent. What may change, however,
is the interpretability of neural responses in relation to the underlying encoded variable. We show
that under mild assumptions on the decoding matrix ?, the main features of single and pairwise
responses are preserved (see below and Suppl.Info. Sec.4).
Fig. 1f shows an example run for multi-chain sampling from a bivariate Gaussian. In a fifth of the
time used in the single-chain scenario (Fig. 1e), the network dynamics achieves a similar spread
across the state space, allowing for a quick estimation of uncertainty (see also Suppl.Info. 2). For a
certain precision of encoding (determined by the size of the decoding weights ?) and neural sparseness level, N scales linearly with the dimensionality of the state space D and the number of simultaneously encoded chains K. Thus, our representation provides a convenient trade-off between the
network size and the speed of the underlying computation. When N is fixed, faster sampling requires either a penalty on precision, or increased firing rates (N D). Overall, the coding scheme
allows for a linear trade-off between speed and resources (either neurons or spikes).
Since F (x) = ??1 (x ? ?), this results in a stochastic generalisation of the dynamics in [7].
?Slow? marks the fact that the term depends on the low-passed neural output r, rather than o.
6
Learning the connections goes beyond the scope of this paper; it seems parameter learning can be achieved
using the plasticity rules derived for the temporal code, if these are local (not shown).
4
5
4
2
Neural implications
To investigate the experimental implications of our coding scheme, we assumed the posterior distribution is centred around a stimulus-specific mean (a set of S = 12 values, equidistantly distributed
on a circle of radius 1 around the origin, see black dots in Fig. 3a), with a stimulus independent
covariance parametrizing the uncertainty about x. This kind of posterior arises e.g. as a result of
inference in a linear Gaussian mixture (since the focus here is not on a specific probabilistic model
of the circuit function, we keep the computation very basic, see Suppl. Info. for details). It allows us
quantify the general properties of distributed sampling in terms of classic measures (tuning curves,
Fano factors, FF, cross-correlogram, CCG, and spike count correlations, rsc ) and how these change
with uncertainty.
Since we found that, under mild assumptions for the decoding matrix ?, the results are qualitatively
similar in a single vs. a multi-chain scenario (see Suppl. Info.), and to facilitate the explanation, the
results reported in the main text used K = 1.
Figure 2: Our model recapitulates several known features of cortical responses. a. Mean firing rates
as a function of stimulus, for all neurons (N = 37); color reflects the phase of ?i (right). b. The
network is in an asynchronous state. Left: example spike raster. Right: Fano factor distribution. c.
Within-trial correlations in membrane potential for pairs of neurons as a function of the similarity
of their decoding weights. d. Spike count correlations (averaged across stimuli) as a function of
the neurons? tuning similarity. Right: distribution of rsc , with mean in magenta. e We use crosscorrelograms (CCG) to asses spike synchrony. Left: CCG for an example neuron. Middle: Area
under the peak ?10ms (between the dashed vertical bars) for all neuron pairs for 3 example stimuli;
neurons ordered by ?i phase. Right: the area under CCG peak as a function of tuning similarity.
a. The neural dynamics are consistent with a wide range of experimental observations
First, we measured the mean firing rate of the neurons for each stimulus (averaged across 50 trials, each 1s long). We found that individual neurons show selectivity to stimulus orientations, with
bell-shaped tuning curves, reminiscent of e.g. the orientation-tuning of V1 neurons (Fig. 2a). The inhomogeneity in the scale of the responses across the population is a reflection of the inhomogeneities
in the decoding matrix ?.7
7
The phase of the decoding weights was sampled uniformly around the circle, with an amplitude drawn
uniformly from the interval [0.005; 0.025].
5
Neural responses were asynchronous, with irregular firing (Fig. 2b), consistent with experimental
observations [11, 12]. To quantify neural variability, we estimated the Fano factors, measured as the
ratio between the variance and the mean of the spike counts in different trials, F Fi = ?f2i /?fi . We
found that the Fano factor distribution was centered around 1, a signature of Poisson variability. This
observation suggests that the sampling dynamics preserve the main features of the distributed code
described in Ref. [6]. Unlike the basic model, however, here neural variability arises both because
of indeterminacies, due to distributed coding, and because of ?true? stochasticity, owed to sampling.
The contribution of the latter, which is characteristic of our version, will depend on the underlying
distribution represented: when the distribution is highly peaked, the deterministic component of the
MCMC dynamics dominates, while the noise plays an increasingly important role the broader the
distribution.
At the level of the membrane potential, both sources of variability introduce correlations between
neurons with similar tuning (Fig. 2c), as seen experimentally [13]: the first because the reconstruction error acts as a shared latent cause, the second because the stochastic component ?which was
independent in the y space? is mapped through ?T in a distributed representation (see Eq. 2). While
the membrane correlations introduced by the first disappear at the level of the spikes [7], the addition
of the stochastic component turns out to have important consequences for the spike correlations both
on the fast time scale, measured by CCG, and for the across-trial spike count covariability, measured
by the noise correlations, rsc .
Fig. 2e shows the CCG of an example pair of neurons, with similar tuning; their activity synchronizes on the time scale of few milliseconds. In more detail, our CCG measure was normalised by
first computing the raw cross-correlogram (averaged across trials) and then subtracting a baseline
obtained as the CCG of shuffled data, where the responses of each neuron come from a different
trial. The raw cross-correlogram for a time delay, ? , CCG(? ) was computed as the Pearsons correlation of the neural responses, shifted in time time by ? .8 At the level of the population, the amount
of synchrony (measured as the area under the CCG peak ?10ms) was strongly modulated by the
input (Fig. 2e, middle), with synchrony most prominent in pairs of neurons that aligned with the
stimulus (not shown). This is consistent with the idea that synchrony is stimulus-specific [14, 15].
We also measured spike count correlation (the Pearsons correlation coefficient of spike counts
recorded in different trials for the same stimulus) and found they depend on the selectivity of the
neurons, with positive correlations for pairs of neurons with similar tuning (Fig. 2d), as seen in experiments [16]. The overall distribution was broad, with a small positive mean (Fig. 2d), as in recent
reports [11, 12]. Taken together, these results suggest that our model qualitatively recapitulates the
basic features of cortical neural responses.
b. Uncertainty modulates neural variability and covariability
We have seen that sampling introduces spike correlations, not seen when encoding a deterministic
dynamical system [7]. Since stochasticity seems to be key for these effects, this suggests uncertainty should significantly modulate pairwise correlations. To confirm this prediction, we varied the
covariance structure of the underlying distribution for the same circuit (Fig. 3a; the low variance condition corresponds to baseline measures reported above) and repeated all previous measurements.
We found that changes in uncertainty leave neuronal tuning invariant (Fig. 3b, not surprisingly since
the mean firing rates reflect the posterior mean). Nonetheless, increasing uncertainty had significant
effects on neural variability and co-variability.
Fano factors increased for broader distributions (Fig. 3b), congruent with the common observation
of the stimulus quenching response variability in experiments [17]. Second, we found a slower
component in the CCG, which increased with uncertainty (Fig. 3e), as in the data [15]. Lastly, the
dependence of different spike correlation measures on neural co-tuning increased with uncertainty
(Fig. 3c, d). In particular, neurons with similar stimulus preferences increased their synchrony
and spike-count correlations with increasing uncertainty, consistent with the stimulus quenching
response co-variability in neural data and increases in correlations at low contrast [17, 16].
Although we see a significant modulation of (co-)variability with changes in uncentainty, these measures provide limited information about the underlying distribution represented in the network. They
can be used to detect changes in the overall spread of the distribution, i.e. the high vs. low-variance
8
While this is not the most common expression for the CCG; we found it reliably detects synchronous firing
across neurons; spikes discretised in 2ms bins.
6
Figure 3: The effects of uncertainty on neural responses. a. Overview of different experimental conditions, posterior mean centred on different stimuli (black dots) with stimulus independent
covariance shown for 4 conditions. b. Left: Tuning curves for an example neuron, for different conditions. Right: firing rate in the low variance vs. all other conditions, summary across all neurons;
dots correspond to different neuron-stimulus pairs. c. Fano factor distribution for high-variance
condition (compare Fig.2b). d. Area under CCG peak ?10ms as a function of the tuning similarity
of the neurons, for different uncertainty conditions (colours as in b). e. Complete CCG, averaged
across 10 neurons with similar tuning while sampling from independent bivariate Gaussians with
different s.d. (0.1 for ?high variance?). f. Spike count correlations (averaged across stimuli) as a
function of the tuning similarity of the neurons, for different uncertainty conditions.
condition look different at the level of pairwise neural responses. However, they cannot discriminate
between distributions with similar spread, but very different dependency structure, e.g. between the
correlated and anti-correlated condition (Fig. 3d, f; also true for FF and the slow component of the
CCG, not shown). For this, we need to look at the population level.
a experimental setup
b
same condition (lowVar)
c
across condition(highVar)
across condition (Corr)
stimuli
neuron
S stimuli
(repeated trials)
1
2
3
4
5
estimate
true trajectory
estimate
estimate
Figure 4: A decoding approach to study the encoding of uncertainty. a. In a low-variability condition
we record neural responses for several repetitions of different stimuli (black dots); We estimated the
decoding matrix by linear regression and used it to project the activity of the population in individual
trials. b. The decoder captures well the underlying dynamics in a trial; ground-truth in black. c.
? can be used to visualise the structure of the underlying distribution in other
The same decoder ?
conditions. Note the method is robust to a misalignment in initial conditions (red trace).
c. Decoding can be used to assess neural representations of uncertainty
Since in a distributed representation single-neuron or pairwise measures tell us little about the dependency structure of the represented random variables, alternative methods need to be devised for
investigating the underlying computation performed by the circuit. The representational framework
proposed here suggests that linear decoding may be used for this purpose. In particular, we can
record neural responses for a variety of stimuli and reverse-engineer the map between spikes and
the relevant latent variables (or, if the assumed generative model is linear as here, the stimuli themselves). We can use the low-variance condition to get a reasonable estimate of the decoding matrix,
? (since the underlying sampling dynamics are close to the posterior mean) and then use the de?
coder for visualising the trajectory of the network while varying uncertainty. As an illustration, we
7
use simple linear regression of the stimuli s as a function of the neuron firing rates, scaled by ?v .9
Although the recovered decoding weights are imperfect and the initial conditions unknown, the pro? captures the main features of the underlying
jections of the neural responses in single trials along ?
sampler, both in the low-variance and in other conditions (Fig. 4b, c).
3
Discussion
How populations of neurons encode probability distributions in a central question for Bayesian approaches to understanding neural computation. While previous work has shown that spiking neural
networks could represent a probability over single real-valued variables [18], or the joint probability
of many binary random variables [19], the representation of complex multi-dimensional real-valued
distributions10 remains less clear [1, 2]. Here we have proposed a new spatio-temporal code for
representing such distributions quickly and flexibly. Our model relies on network dynamics which
approximate the target distribution by several MCMC chains, encoded in the spiking neural activity
such that the samples can be linearly decoded from the quasi-instantaneous neural responses. Unlike previous sampling-based codes [19], our model does not require a one-to-one correspondence
between random variables and neurons. This separation between computation and representation is
critical for the increased speed, as it allows multiple chains to be realistically embedded in the same
circuit, while preserving all the computational benefits of sampling. Furthermore, it makes the encoding robust to neural damage, which seems important when representing behaviourally-relevant
variables, e.g. in higher cortical areas. These benefits come at the cost of a linear increase in the
number of neurons with K, providing a convenient trade-off between speed and neural resources.
The speedup due to increases in network size is orthogonal to potential improvements in sampling
efficiency achieved by more sophisticated MCMC dynamics, e.g. relying on oscillations [21] or nonnormal stochastic dynamics [22], suggesting that distributed sampling could be made even faster by
combining the two approaches.
The distributed coding scheme has important consequences for interpreting neural responses: since
knowledge about the underlying distribution is spread across the population, the activity of single
cells does not reflect the underlying computation in any obvious way. In particular, although the
network did reproduce various properties of single neuron and pairs of neuron responses seen experimentally, we found that their modulation with uncertainty provides relatively limited information
about the underlying probabilistic computation. Changes in the overall spread (entropy) of the posterior are reflected in changes in variability (Fano factors) and covariability (synchrony on the ms
timescale and spike-count correlations across trials) of neural responses across the population, as
seen in the data. Since these features arise due to the interaction between sampling and distributed
coding, the model further predicts that the degree of correlations between a pair of neurons should
depend on their functional similarity, and that the degree of this modulation should be affected by
uncertainty. Nonetheless, the distributed representation occludes the structure of the underlying
distribution (e.g. correlations between random variables), something which would have been immediately apparent in a one-to-one sampling code.
Our results reinforce the idea that population, rather than single-cell, responses are key to understanding cortical computation, and points to linear decoding as a potential analysis tool for investigating probabilistic computation in a distributed code. In particular, we have shown that we can
train a linear decoder on spiking data and use it to reveal the underlying sampling dynamics in different conditions. While ours is a simple toy example, where we assume that we can record from
all the neurons in the population, the fact that the signal is low-dimensional relative to the number
of neurons gives hope that it should be possible to adapt more sophisticated machine learning techniques [23] for decoding the underlying trajectory traced by a neural circuit in realistic settings. If
this could be done reliability on data, then the analysis of probabilistic neural computation would
no longer be restricted to regions for which we have good ideas about the mathematical form of the
underlying distribution, but could be applied to any cortical circuit of interest.11 Thus, our coding
scheme opens exciting avenues for multiunit data analysis.
9
This requires knowledge of ?v and, in a multi-chain scenario, a grouping of neural responses by chain
preference. Proxies for which neurons should be decoded together are discussed in Suppl.Info. Sec.4.
10
Such distribution arise in many models of probabilistic inference in the brain, e.g. [20].
11
The critical requirement is to know (some of) the variables represented in the circuit, up to a linear map.
8
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9
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4,798 | 5,344 | Conditional Random Field Autoencoders
for Unsupervised Structured Prediction
Waleed Ammar
Chris Dyer
Noah A. Smith
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213, USA
{wammar,cdyer,nasmith}@cs.cmu.edu
Abstract
We introduce a framework for unsupervised learning of structured predictors with
overlapping, global features. Each input?s latent representation is predicted conditional on the observed data using a feature-rich conditional random field (CRF).
Then a reconstruction of the input is (re)generated, conditional on the latent structure, using a generative model which factorizes similarly to the CRF. The autoencoder formulation enables efficient exact inference without resorting to unrealistic
independence assumptions or restricting the kinds of features that can be used.
We illustrate connections to traditional autoencoders, posterior regularization, and
multi-view learning. We then show competitive results with instantiations of the
framework for two canonical tasks in natural language processing: part-of-speech
induction and bitext word alignment, and show that training the proposed model
can be substantially more efficient than a comparable feature-rich baseline.
1
Introduction
Conditional random fields [24] are used to model structure in numerous problem domains, including natural language processing (NLP), computational biology, and computer vision. They enable
efficient inference while incorporating rich features that capture useful domain-specific insights. Despite their ubiquity in supervised settings, CRFs?and, crucially, the insights about effective feature
sets obtained by developing them?play less of a role in unsupervised structure learning, a problem which traditionally requires jointly modeling observations and the latent structures of interest.
For unsupervised structured prediction problems, less powerful models with stronger independence
assumptions are standard.1 This state of affairs is suboptimal in at least three ways: (i) adhering
to inconvenient independence assumptions when designing features is limiting?we contend that
effective feature engineering is a crucial mechanism for incorporating inductive bias in unsupervised learning problems; (ii) features and their weights have different semantics in joint and conditional models (see ?3.1); and (iii) modeling the generation of high-dimensional observable data with
feature-rich models is computationally challenging, requiring expensive marginal inference in the
inner loop of iterative parameter estimation algorithms (see ?3.1).
Our approach leverages the power and flexibility of CRFs in unsupervised learning without sacrificing their attractive computational properties or changing the semantics of well-understood feature
sets. Our approach replaces the standard joint model of observed data and latent structure with a twolayer conditional random field autoencoder that first generates latent structure with a CRF (conditional on the observed data) and then (re)generates the observations conditional on just the predicted
structure. For the reconstruction model, we use distributions which offer closed-form maximum
1
For example, a first-order hidden Markov model requires that yi ? xi+1 | yi+1 for a latent sequence
y = hy1 , y2 , . . .i generating x = hx1 , x2 , . . .i, while a first-order CRF allows yi to directly depend on xi+1 .
1
Extension: partial reconstruction. In our running POS example, the reconstruction model
p? (?
xi | yi ) defines a distribution over words given tags. Because word distributions are heavytailed, estimating such a distribution reliably is quite challenging. Our solution is to define a function ? : X ? X? such that |X? | ? |X |, and let x
?i = ?(xi ) be a deterministic transformation of the
original structured observation. We can add indirect supervision by defining ? such that it represents
observed information relevant to the latent structure of interest. For example, we found reconstructing Brown clusters [5] of tokens instead of their surface forms to improve POS induction. Other
possible reconstructions include word embeddings, morphological and spelling features of words.
More general graphs. We presented the CRF autoencoder in terms of sequential Markovian assumptions for ease of exposition; however, this framework can be used to model arbitrary hidden
structures. For example, instantiations of this model can be used for unsupervised learning of parse
trees [21], semantic role labels [42], and coreference resolution [35] (in NLP), motif structures [1]
in computational biology, and object recognition [46] in computer vision. The requirements for
applying the CRF autoencoder model are:
? An encoding discriminative model defining p? (y | x, ?). The encoder may be any model family
where supervised learning from hx, yi pairs is efficient.
? A reconstruction model that defines p? (?
x | y, ?) such that inference over y given hx, x
?i is
efficient.
? The independencies among y | x, x
? are not strictly weaker than those among y | x.
2.1
Learning & Inference
Model parameters are selected to maximize the regularized conditional log likelihood of reconstructed observations x
? given the structured observation x:
P
P
x | y)
(2)
??(?, ?) = R1 (?) + R2 (?) + (x,?x)?T log y p? (y | x) ? p? (?
We apply block coordinate descent, alternating between maximizing with respect to the CRF parameters (?-step) and the reconstruction parameters (?-step). Each ?-step applies one or two iterations
of a gradient-based convex optimizer.5 The ?-step applies one or two iterations of EM [10], with a
closed-form solution in the M-step in each EM iteration. The independence assumptions among y
make the marginal inference required in both steps straightforward; we omit details for space.
In the experiments below, we apply a squared L2 regularizer for the CRF parameters ?, and a
symmetric Dirichlet prior for categorical parameters ?.
The asymptotic runtime complexity of each block coordinate descent iteration, assuming the firstorder Markov dependencies in Fig. 2 (right), is:
(3)
O |?| + |?| + |T | ? |x|max ? |Y|max ? (|Y|max ? |Fyi?1 ,yi | + |Fx,yi |)
where Fyi?1 ,yi are the active ?label bigram? features used in hyi?1 , yi i factors, Fx,yi are the active
emission-like features used in hx, yi i factors. |x|max is the maximum length of an observation
sequence. |Y|max is the maximum cardinality6 of the set of possible assignments of yi .
After learning the ? and ? parameters of the CRF autoencoder, test-time predictions are made using maximum a posteriori estimation, conditioning on both observations and reconstructions, i.e.,
? MAP = arg maxy p?,? (y | x, x
y
?).
3
Connections To Previous Work
This work relates to several strands of work in unsupervised learning. Two broad types of models
have been explored that support unsupervised learning with flexible feature representations. Both are
5
We experimented with AdaGrad [12] and L-BFGS. When using AdaGrad, we accummulate the gradient
vectors across block coordinate ascent iterations.
6
In POS induction, |Y| is a constant, the number of syntactic classes which we configure to 12 in our experiments. In word alignment, |Y| is the size of the source sentence plus one, therefore |Y|max is the maximum
length of a source sentence in the bitext corpus.
4
fully generative models that define joint distributions over x and y. We discuss these ?undirected?
and ?directed? alternatives next, then turn to less closely related methods.
3.1
Existing Alternatives for Unsupervised Learning with Features
Undirected models. A Markov random field (MRF) encodes the joint distribution through local
potential functions parameterized using features. Such models ?normalize globally,? requiring during training the calculation of a partition function summing over all possible inputs and outputs. In
our notation:
X X
exp ?? g
?(x, y)
(4)
Z(?) =
x?X ? y?Y |x|
where g
? collects all the local factorization by cliques of the graph, for clarity. The key difficulty
is in the summation over all possible observations. Approximations have been proposed, including
contrastive estimation, which sums over subsets of X ? [38, 43] (applied variously to POS learning
by Haghighi and Klein [18] and word alignment by Dyer et al. [14]) and noise contrastive estimation
[30].
Directed models. The directed alternative avoids the global partition function by factorizing the
joint distribution in terms of locally normalized conditional probabilities, which are parameterized
in terms of features. For unsupervised sequence labeling, the model was called a ?feature HMM?
by Berg-Kirkpatrick et al. [3]. The local emission probabilities p(xi | yi ) in a first-order HMM for
POS tagging are reparameterized as follows (again, using notation close to ours):
p? (xi | yi ) = P
exp ?? g(xi , yi )
?
x?X exp ? g(x, yi )
(5)
The features relating hidden to observed variables must be local within the factors implied by the
directed graph. We show below that this locality restriction excludes features that are useful (?A.1).
Put in these terms, the proposed autoencoding model is a hybrid directed-undirected model.
Asymptotic Runtime Complexity of Inference. The models just described cannot condition on
arbitrary amounts of x without increasing inference costs. Despite the strong independence assumptions of those models, the computational complexity of inference required for learning with CRF
autoencoders is better (?2.1).
Consider learning the parameters of an undirected model by maximizing likelihood of the observed
data. Computing the gradient for a training instance x requires time
O |?| + |T | ? |x| ? |Y| ? (|Y| ? |Fyi?1 ,yi |+|X | ? |Fxi ,yi |) ,
where Fxi ?yi are the emission-like features used in an arbitrary assignment of xi and yi . When the
multiplicative factor |X | is large, inference is slow compared to CRF autoencoders.
Inference in directed models is faster than in undirected models, but still slower than CRF autoencoder models. In directed models [3], each iteration requires time
O |?| + |T | ? |x| ? |Y| ? (|Y| ? |Fyi?1 ,yi | + |Fxi ,yi |)+|? ? | ? max(|Fyi?1 ,yi |, |FX ,yi |) ,
where Fxi ,yi are the active emission features used in an arbitrary assignment of xi and yi , FX ,yi
is the union of all emission features used with an arbitrary assignment of yi , and ? ? are the local
emission and transition probabilities. When |X | is large, the last term |? ? |?max(|Fyi?1 ,yi |, |FX ,yi |)
can be prohibitively large.
3.2
Other Related Work
The proposed CRF autoencoder is more distantly related to several important ideas in less-thansupervised learning.
5
Autoencoders and other ?predict self? methods. Our framework borrows its general structure,
Fig. 2 (left), as well as its name, from neural network autoencoders. The goal of neural autoencoders
has been to learn feature representations that improve generalization in otherwise supervised learning problems [44, 8, 39]. In contrast, the goal of CRF autoencoders is to learn specific interpretable
regularities of interest.7 It is not clear how neural autoencoders could be used to learn the latent
structures that CRF autoencoders learn, without providing supervised training examples. Stoyanov
et al. [40] presented a related approach for discriminative graphical model learning, including features and latent variables, based on backpropagation, which could be used to instantiate the CRF
autoencoder.
Daum?e III [9] introduced a reduction of an unsupervised problem instance to a series of singlevariable supervised classifications. The first series of these construct a latent structure y given the
entire x, then the second series reconstruct the input. The approach can make use of any supervised
learner; if feature-based probabilistic models were used, a |X | summation (akin to Eq. 5) would
be required. On unsupervised POS induction, this approach performed on par with the undirected
model of Smith and Eisner [38].
Minka [29] proposed cascading a generative model and a discriminative model, where class labels
(to be predicted at test time) are marginalized out in the generative part first, and then (re)generated
in the discriminative part. In CRF autoencoders, observations (available at test time) are conditioned
on in the discriminative part first, and then (re)generated in the generative part.
Posterior regularization. Introduced by Ganchev et al. [16], posterior regularization is an effective method for specifying constraint on the posterior distributions of the latent variables of interest;
a similar idea was proposed independently by Bellare et al. [2]. For example, in POS induction,
every sentence might be expected to contain at least one verb. This is imposed as a soft constraint,
i.e., a feature whose expected value under the model?s posterior is constrained. Such expectation
constraints are specified directly by the domain-aware model designer.8 The approach was applied
to unsupervised POS induction, word alignment, and parsing. Although posterior regularization was
applied to directed feature-less generative models, the idea is orthogonal to the model family and
can be used to add more inductive bias for training CRF autoencoder models.
4
Evaluation
We evaluate the effectiveness of CRF autoencoders for learning from unlabeled examples in POS
induction and word alignment. We defer the detailed experimental setup to Appendix A.
Part-of-Speech Induction Results. Fig. 3 compares predictions of the CRF autoencoder model
in seven languages to those of a featurized first-order HMM model [3] and a standard (feature-less)
first-order HMM, using V-measure [37] (higher is better). First, note the large gap between both
feature-rich models on the one hand, and the feature-less HMM model on the other hand. Second,
note that CRF autoencoders outperform featurized HMMs in all languages, except Italian, with an
average relative improvement of 12%.
These results provide empirical evidence that feature engineering is an important source of inductive
bias for unsupervised structured prediction problems. In particular, we found that using Brown
cluster reconstructions and specifying features which span multiple words significantly improve the
performance. Refer to Appendix A for more analysis.
Bitext Word Alignment Results. First, we consider an intrinsic evaluation on a Czech-English
dataset of manual alignments, measuring the alignment error rate (AER; [32]). We also perform an
7
This is possible in CRF autoencoders due to the interdependencies among variables in the hidden structure
and the manually specified feature templates which capture the relationship between observations and their
hidden structures.
8
In a semi-supervised setting, when some labeled examples of the hidden structure are available, Druck
and McCallum [11] used labeled examples to estimate desirable expected values. We leave semi-supervised
applications of CRF autoencoders to future work; see also Suzuki and Isozaki [41].
6
0.6
0.4
0.3
0.0
0.1
0.2
V?measure
0.5
Standard HMM
Featurized HMM
CRF autoencoder
Arabic
Basque
Danish
Greek
Hungarian
Italian
Turkish
Average
Figure 3: V-measure [37] of induced parts of speech in seven languages. The CRF autoencoder with
features spanning multiple words and with Brown cluster reconstructions achieves the best results in
all languages but Italian, closely followed by the feature-rich HMM of Berg-Kirkpatrick et al. [3].
The standard multinomial HMM consistently ranks last.
direction
forward
reverse
symmetric
fast align
model 4
auto
27.7
25.9
25.2
31.5
24.1
22.2
27.5
21.1
19.5
pair
cs-en
ur-en
zh-en
fast align
model 4
auto
15.2?0.3
20.0?0.6
56.9?1.6
15.3?0.1
20.1?0.6
56.7?1.6
15.5?0.1
20.8?0.5
56.1?1.7
Table 1: Left: AER results (%) for Czech-English word alignment. Lower values are better. . Right:
Bleu translation quality scores (%) for Czech-English, Urdu-English and Chinese-English. Higher
values are better. .
extrinsic evaluation of translation quality in three language pairs, using case-insensitive Bleu [33] of
a machine translation system (cdec9 [13]) built using the word alignment predictions of each model.
AER for variants of each model (forward, reverse, and symmetrized) are shown in Table 1 (left).
Our model significantly outperforms both baselines. Bleu scores on the three language pairs are
shown in Table 1; alignments obtained with our CRF autoencoder model improve translation quality
of the Czech-English and Urdu-English translation systems, but not of Chinese-English. This is unsurprising, given that Chinese orthography does not use letters, so that source-language spelling and
morphology features our model incorporates introduce only noise here. Better feature engineering,
or more data, is called for.
We have argued that the feature-rich CRF autoencoder will scale better than its feature-rich alternatives. Fig. 5 (in Appendix A.2) shows the average per-sentence inference runtime for the CRF
autoencoder compared to exact inference in an MRF [14] with a similar feature set, as a function of
the number of sentences in the corpus. For CRF autoencoders, the average inference runtime grows
slightly due to the increased number of parameters, while it grows substantially with vocabulary size
in MRF models [14].10
5
Conclusion
We have presented a general and scalable framework to learn from unlabeled examples for structured
prediction. The technique allows features with global scope in observed variables with favorable
asymptotic inference runtime. We achieve this by embedding a CRF as the encoding model in the
9
http://www.cdec-decoder.org/
We only compare runtime, instead of alignment quality, because retraining the MRF model with exact
inference was too expensive.
10
7
input layer of an autoencoder, and reconstructing a transformation of the input at the output layer
using simple categorical distributions. The key advantages of the proposed model are scalability and
modeling flexibility. We applied the model to POS induction and bitext word alignment, obtaining
results that are competitive with the state of the art on both tasks.
Acknowledgments
We thank Brendan O?Connor, Dani Yogatama, Jeffrey Flanigan, Manaal Faruqui, Nathan Schneider,
Phil Blunsom and the anonymous reviewers for helpful suggestions. We also thank Taylor BergKirkpatrick for providing his implementation of the POS induction baseline, and Phil Blunsom for
sharing POS induction evaluation scripts. This work was sponsored by the U.S. Army Research
Laboratory and the U.S. Army Research Office under contract/grant number W911NF-10-1-0533.
The statements made herein are solely the responsibility of the authors.
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9
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4,799 | 5,345 | Learning Generative Models with Visual Attention
Yichuan Tang, Nitish Srivastava, Ruslan Salakhutdinov
Department of Computer Science
University of Toronto
Toronto, Ontario, Canada
{tang,nitish,rsalakhu}@cs.toronto.edu
Abstract
Attention has long been proposed by psychologists to be important for efficiently
dealing with the massive amounts of sensory stimulus in the neocortex. Inspired
by the attention models in visual neuroscience and the need for object-centered
data for generative models, we propose a deep-learning based generative framework using attention. The attentional mechanism propagates signals from the
region of interest in a scene to an aligned canonical representation for generative modeling. By ignoring scene background clutter, the generative model can
concentrate its resources on the object of interest. A convolutional neural net is
employed to provide good initializations during posterior inference which uses
Hamiltonian Monte Carlo. Upon learning images of faces, our model can robustly
attend to the face region of novel test subjects. More importantly, our model can
learn generative models of new faces from a novel dataset of large images where
the face locations are not known.
1
Introduction
Building rich generative models that are capable of extracting useful, high-level latent representations from high-dimensional sensory input lies at the core of solving many AI-related tasks, including object recognition, speech perception and language understanding. These models capture
underlying structure in data by defining flexible probability distributions over high-dimensional data
as part of a complex, partially observed system. Some of the successful generative models that
are able to discover meaningful high-level latent representations include the Boltzmann Machine
family of models: Restricted Boltzmann Machines, Deep Belief Nets [1], and Deep Boltzmann Machines [2]. Mixture models, such as Mixtures of Factor Analyzers [3] and Mixtures of Gaussians,
have also been used for modeling natural image patches [4]. More recently, denoising auto-encoders
have been proposed as a way to model the transition operator that has the same invariant distribution
as the data generating distribution [5].
Generative models have an advantage over discriminative models when part of the images are occluded or missing. Occlusions are very common in realistic settings and have been largely ignored
in recent literature on deep learning. In addition, prior knowledge can be easily incorporated in
generative models in the forms of structured latent variables, such as lighting and deformable parts.
However, the enormous amount of content in high-resolution images makes generative learning difficult [6, 7]. Therefore, generative models have found most success in learning to model small
patches of natural images and objects: Zoran and Weiss [4] learned a mixture of Gaussians model
over 8?8 image patches; Salakhutdinov and Hinton [2] used 64?64 centered and uncluttered stereo
images of toy objects on a clear background; Tang et al. [8] used 24?24 images of centered and
cropped faces. The fact that these models require curated training data limits their applicability on
using the (virtually) unlimited unlabeled data.
In this paper, we propose a framework to infer the region of interest in a big image for generative modeling. This will allow us to learn a generative model of faces on a very large dataset of
(unlabeled) images containing faces. Our framework is able to dynamically route the relevant information to the generative model and can ignore the background clutter. The need to dynamically and
selectively route information is also present in the biological brain. Plethora of evidence points to
1
the presence of attention in the visual cortex [9, 10]. Recently, in visual neuroscience, attention has
been shown to exist not only in extrastriate areas, but also all the way down to V1 [11].
Attention as a form of routing was originally proposed by Anderson and Van Essen [12] and then
extended by Olshausen et al. [13]. Dynamic routing has been hypothesized as providing a way for
achieving shift and size invariance in the visual cortex [14, 15]. Tsotsos et al. [16] proposed a model
combining search and attention called the Selective Tuning model. Larochelle and Hinton [17] proposed a way of using third-order Boltzmann Machines to combine information gathered from many
foveal glimpses. Their model chooses where to look next to find locations that are most informative
of the object class. Reichert et al. [18] proposed a hierarchical model to show that certain aspects of
covert object-based attention can be modeled by Deep Boltzmann Machines. Several other related
models attempt to learn where to look for objects [19, 20] and for video based tracking [21]. Inspired
by Olshausen et al. [13], we use 2D similarity transformations to implement the scaling, rotation,
and shift operation required for routing. Our main motivation is to enable the learning of generative
models in big images where the location of the object of interest is unknown a-priori.
2
Gaussian Restricted Boltzmann Machines
Before we describe our model, we briefly review the Gaussian Restricted Boltzmann Machine
(GRBM) [22], as it will serve as the building block for our attention-based model. GRBMs are
a type of Markov Random Field model that has a bipartite structure with real-valued visible variables v ? RD connected to binary stochastic hidden variables h ? {0, 1}H . The energy of the joint
configuration {v, h} of the Gaussian RBM is defined as follows:
X
1 X (vi ? bi )2 X
?
c
h
?
Wij vi hj ,
(1)
EGRBM (v, h; ?) =
j
j
2 i
?i2
j
ij
where ? = {W, b, c, ?}
P are the model parameters. The marginal distribution over the visible vector
1
v is P (v; ?) = Z(?)
h exp (?E(v, h; ?)) and the corresponding conditional distributions take
the following form:
X
p(hj = 1|v) = 1/ 1 + exp(?
Wij vi ? cj ) ,
(2)
i
p(vi |h)
= N (vi ; ?i , ?i2 ), where ?i = bi + ?i2
X
Wij hj .
(3)
j
Observe that conditioned on the states of the hidden variables (Eq. 3), each visible unit is modeled
by a Gaussian distribution, whose mean is shifted by the weighted combination of the hidden unit
activations. Unlike directed models, an RBM?s conditional distribution over hidden nodes is factorial
and can be easily computed.
We can also add a binary RBM on top of the learned GRBM by treating the inferred h as the
?visible? layer together with a second hidden layer h2 . This results in a 2-layer Gaussian Deep
Belief Network (GDBN) [1] that is a more powerful model of v.
Specifically, in a GDBN model, p(h1 , h2 ) is modeled by the energy function of the 2nd-layer RBM,
while p(v1 |h1 ) is given by Eq. 3. Efficient inference can be performed using the greedy approach
of [1] by treating each DBN layer as a separate RBM model. GDBNs have been applied to various
tasks, including image classification, video action and speech recognition [6, 23, 24, 25].
3
The Model
Let I be a high resolution image of a scene, e.g. a 256?256 image. We want to use attention to
propagate regions of interest from I up to a canonical representation. For example, in order to learn
a model of faces, the canonical representation could be a 24?24 aligned and cropped frontal face
image. Let v ? RD represent this low resolution canonical image. In this work, we focus on a Deep
Belief Network1 to model v.
This is illustrated in the diagrams of Fig. 1. The left panel displays the model of Olshausen et.al. [13],
whereas the right panel shows a graphical diagram of our proposed generative model with an attentional mechanism. Here, h1 and h2 represent the latent hidden variables of the DBN model, and
1
Other generative models can also be used with our attention framework.
2
2d similarity
transformation
Our model
Olshausen et al. 93
Figure 1: Left: The Shifter Circuit, a well-known neuroscience model for visual attention [13]; Right: The
proposed model uses 2D similarity transformations from geometry and a Gaussian DBN to model canonical
face images. Associative memory corresponds to the DBN, object-centered frame correspond to the visible
layer and the attentional mechanism is modeled by 2D similarity transformations.
4x, 4y, 4?, 4s (position, rotation, and scale) are the parameters of the 2D similarity transformation.
The 2D similarity transformation is used to rotate, scale, and translate the canonical image v onto the
canvas that we denote by I. Let p = [x y]T be a pixel coordinate (e.g. [0, 0] or [0, 1]) of the canonical
image v. Let {p} be the set of all coordinates of v. For example, if v is 24?24, then {p} ranges
from [0, 0] to [23, 23]. Let the ?gaze? variables u ? R4 ? [4x, 4y, 4?, 4s] be the parameter
of the Similarity transformation. In order to simplify derivations and to make transformations be
linear w.r.t. the transformation parameters, we can equivalently redefine u = [a, b, 4x, 4y],
where a = s sin(?) ? 1 and b = s cos(?) (see [26] for details). We further define a function
w := w(p, u) ? p0 as the transformation function to warp points p to p0 :
h 0 i h
ih
i h
i
1+a
?b
x
4x
x
p0 ,
=
+
.
(4)
0
y
b
1+a
y
4y
We use the notation I({p}) to denote the bilinear interpolation of I at coordinates {p} with antialiasing. Let x(u) be the extracted low-resolution image at warped locations p0 :
x(u) , I(w({p}, u)).
(5)
Intuitively, x(u) is a patch extracted from I according to the shift, rotation and scale parameters
of u, as shown in Fig. 1, right panel. It is this patch of data that we seek to model generatively. Note
that the dimensionality of x(u) is equal to the cardinality of {p}, where {p} denotes the set of pixel
coordinates of the canonical image v. Unlike standard generative learning tasks, the data x(u) is
not static but changes with the latent variables u. Given v and u, we model the top-down generative
process over2 x with a Gaussian distribution having a diagonal covariance matrix ? 2 I:
1 X (xi (u) ? vi )2
p(x|v, u, I) ? exp ?
.
(6)
2 i
?i2
The fact that we do not seek to model the rest of the regions/pixels of I is by design. By using 2D
similarity transformation to mimic attention, we can discard the complex background of the scene
and let the generative model focus on the object of interest. The proposed generative model takes
the following form:
p(x, v, u|I) = p(x|v, u, I)p(v)p(u),
(7)
where for p(u) we use a flat prior that is constant for all u, and p(v) is defined by a 2-layer Gaussian
Deep Belief Network. The conditional p(x|v, u, I) is given by a Gaussian distribution as in Eq. 6.
To simplify the inference procedure, p(x|v, u, I) and the GDBN model of v, p(v), will share the
same noise parameters ?i .
2
We will often omit dependence of x on u for clarity of presentation.
3
4
Inference
While the generative equations in the last section are straightforward and intuitive, inference in these
models is typically intractable due to the complicated energy landscape of the posterior. During
inference, we wish to compute the distribution over the gaze variables u and canonical object v given
the big image I. Unlike in standard RBMs and DBNs, there are no simplifying factorial assumptions
about the conditional distribution of the latent variable u. Having a 2D similarity transformation is
reminiscent of third-order Boltzmann machines with u performing top-down multiplicative gating
of the connections between v and I. It is well known that inference in these higher-order models is
rather complicated.
One way to perform inference in our model is to resort to Gibbs sampling by computing the set of
alternating conditional posteriors: The conditional distribution over the canonical image v takes the
following form:
? + x(u)
(8)
; ?2 ,
p(v|u, h1 , I) = N
2
P
where ?i = bi + ?i2 j Wij h1j is the top-down influence of the DBN. Note that if we know the
gaze variable u and the first layer of hidden variables h1 , then v is simply defined by a Gaussian
distribution, where the mean is given by the average of the top-down influence and bottom-up information from x. The conditional distributions over h1 and h2 given v are given by the standard
DBN inference equations [1]. The conditional posterior over the gaze variables u is given by:
p(x|u, v)p(u)
p(u|x, v) =
,
p(x|v)
1 X (xi (u) ? vi )2
log p(u|x, v) ? log p(x|u, v) + log p(u) =
+ const.
(9)
2 i
?i2
Using Bayes? rule, the unnormalized log probability of p(u|x, v) is defined in Eq. 9. We stress that
this equation is atypical in that the random variable of interest u actually affects the conditioning
variable x (see Eq. 5) We can explore the gaze variables using Hamiltonian Monte Carlo (HMC)
algorithm [27, 28]. Intuitively, conditioned on the canonical object v that our model has in ?mind?,
HMC searches over the entire image I to find a region x with a good match to v.
If the goal is only to find the MAP estimate of p(u|x, v), then we may want to use second-order
methods for optimizing u. This would be equivalent to the Lucas-Kanade framework in computer
vision, developed for image alignment [29]. However, HMC has the advantage of being a proper
MCMC sampler that satisfies detailed balance and fits nicely with our probabilistic framework.
The HMC algorithm first specifies the Hamiltonian over the position variables u and auxiliary
momentum variables r: H(u, r) = U (u) + K(r), where the potential function is defined by
2
P
P
i)
U (u) = 21 i (xi (u)?v
and the kinetic energy function is given by K(r) = 12 i ri2 . The dy?i2
namics of the system is defined by:
?u
?r
?H
= r,
=?
(10)
?t
?t
?u
?H
(x(u) ? v) ?x(u)
=
,
(11)
?u
?2
?u
?x
?x
?w({p}, u) X
?xi
?w(pi , u)
=
=
.
(12)
?u
?w({p}, u)
?u
?w(p
,
u)
?u
i
i
Observe that Eq. 12 decomposes into sums over single coordinate positions pi = [x y]T . Let us
denote p0 i = w(pi , u) to be the coordinate pi warped by u. For the first term on the RHS of Eq. 12,
?xi
= ?I(p0 i ), (dimension 1 by 2 )
(13)
?w(pi , u)
where ?I(p0 i ) denotes the sampling of the gradient images of I at the warped location pi . For the
second term on the RHS of Eq. 12, we note that we can re-write Eq. 4 as:
" a #
h 0 i h
i
h
i
x
x ?y 1 0
b
x
=
+
,
(14)
0
y
y x 0 1
4x
y
4y
4
giving us
?w(pi , u) h x
=
y
?u
?y
x
1
0
i
0
.
1
(15)
HMC simulates the discretized system by performing leap-frog updates of u and r using Eq. 10.
Additional hyperparameters that need to be specified include the step size , number of leap-frog
steps, and the mass of the variables (see [28] for details).
4.1 Approximate Inference
HMC essentially performs gradient descent with momentum,
therefore it is prone to getting stuck at local optimums. This
is especially a problem for our task of finding the best transformation parameters. While the posterior over u should be
unimodal near the optimum, many local minima exist away
from the global optimum. For example, in Fig. 2(a), the big
image I is enclosed by the blue box, and the canonical image
v is enclosed by the green box. The current setting of u aligns
together the wrong eyes. However, it is hard to move the green
box to the left due to the local optima created by the dark intensities of the eye. Resampling the momentum variable every
iteration in HMC does not help significantly because we are
modeling real-valued images using a Gaussian distribution as
the residual, leading to quadratic costs in the difference between x(u) and v (see Eq. 9). This makes the energy barriers
between modes extremely high.
(a)
Average
A
B
To alleviate this problem we need to find good initializations
of u. We use a Convolutional Network (ConvNet) to perform efficient approximate inference, resulting in good initial
(b)
guesses. Specifically, given v, u and I, we predict the change Figure 2: (a) HMC can easily get
in u that will lead to the maximum log p(u|x, v). In other stuck at local optima. (b) Importance
words, instead of using the gradient field for updating u, we of modeling p(u|v, I). Best in color.
learn a ConvNet to output a better vector field in the space
of u. We used a fairly standard ConvNet architecture and the standard stochastic gradient descent
learning procedure.
We note that standard feedforward face detectors seek to model p(u|I), while completely ignoring
the canonical face v. In contrast, here we take v into account as well. The ConvNet is used to initialize u for the HMC algorithm. This is important in a proper generative model because conditioning
on v is appealing when multiple faces are present in the scene. Fig. 2(b) is a hypothesized Euclidean
space of v, where the black manifold represents canonical faces and the blue manifold represents
cropped faces x(u). The blue manifold has a low intrinsic dimensionality of 4, spanned by u. At A
and B, the blue comes close to black manifold. This means that there are at least two modes in the
posterior over u. By conditioning on v, we can narrow the posterior to a single mode, depending on
whom we want to focus our attention. We demonstrate this exact capability in Sec. 6.3.
Fig. 3 demonstrates the iterative process of how approximate inference works in our model. Specifically, based on u, the ConvNet takes a window patch around x(u) (72?72) and v (24?24) as input,
and predicts the output [4x, 4y, 4?, 4s]. In step 2, u is updated accordingly, followed by step 3
of alternating Gibbs updates of v and h, as discussed in Sec. 4. The process is repeated. For the
details of the ConvNet see the supplementary materials.
5
Learning
While inference in our framework localizes objects of interest and is akin to object detection, it is not
the main objective. Our motivation is not to compete with state-of-the-art object detectors but rather
propose a probabilistic generative framework capable of generative modeling of objects which are
at unknown locations in big images. This is because labels are expensive to obtain and are often not
available for images in an unconstrained environment.
To learn generatively without labels we propose a simple Monte Carlo based ExpectationMaximization algorithm. This algorithm is an unbiased estimator of the maximum likelihood objec5
1 Gibbs step
ConvNet
Step 1
ConvNet
Step 4
Step 3
Step 2
Figure 3: Inference process: u in step 1 is randomly initialized. The average v and the extracted x(u) form
the input to a ConvNet for approximate inference, giving a new u. The new u is used to sample p(v|I, u, h).
In step 3, one step of Gibbs sampling of the GDBN is performed. Step 4 repeats the approximate inference
using the updated v and x(u).
V
X
1
2
3
Inference steps
4
5
6
HMC
Figure 4: Example of an inference step. v is 24?24, x is 72?72. Approximate inference quickly finds a
good initialization for u, while HMC provides further adjustments. Intermediate inference steps on the right
are subsampled from 10 actual iterations.
tive. During the E-step, we use the Gibbs sampling algorithm developed in Sec. 4 to draw samples
from the posterior over the latent gaze variables u, the canonical variables v, and the hidden variables h1 , h2 of a Gaussian DBN model. During the M-step, we can update the weights of the
Gaussian DBN by using the posterior samples as its training data. In addition, we can update the
parameters of the ConvNet that performs approximate inference. Due to the fact that the first E-step
requires a good inference algorithm, we need to pretrain the ConvNet using labeled gaze data as
part of a bootstrap process. Obtaining training data for this initial phase is not a problem as we can
jitter/rotate/scale to create data. In Sec. 6.2, we demonstrate the ability to learn a good generative
model of face images from the CMU Multi-PIE dataset.
6
Experiments
We used two face datasets in our experiments. The first dataset is a frontal face dataset, called
the Caltech Faces from 1999, collected by Markus Weber. In this dataset, there are 450 faces of 27
unique individuals under different lighting conditions, expressions, and backgrounds. We downsampled the images from their native 896 by 692 by a factor of 2. The dataset also contains manually
labeled eyes and mouth coordinates, which will serve as the gaze labels. We also used the CMU
Multi-PIE dataset [30], which contains 337 subjects, captured under 15 viewpoints and 19 illumination conditions in four recording sessions for a total of more than 750,000 images. We demonstrate our model?s ability to perform approximate inference, to learn without labels, and to perform
identity-based attention given an image with two people.
6.1 Approximate inference
We first investigate the critical inference algorithm of p(u|v, I) on the Caltech Faces dataset. We
run 4 steps of approximate inference detailed in Sec. 4.1 and diagrammed in Fig. 3, followed by
three iterations of 20 leap-frog steps of HMC. Since we do not initially know the correct v, we
initialize v to be the average face across all subjects.
Fig. 4 shows the image of v and x during inference for a test subject. The initial gaze box is colored
yellow on the left. Subsequent gaze updates progress from yellow to blue. Once ConvNet-based
approximate inference gives a good initialization, starting from step 5, five iterations of 20 leap-frog
steps of HMC are used to sample from the the posterior.
Fig. 5 shows the quantitative results of Intersection over Union (IOU) of the ground truth face box
and the inferred face box. The results show that inference is very robust to initialization and requires
6
Accuracy of Approximate Inference
Accuracy of Approximate Inference
Accuracy Improvements
1.1
0.3
Average IOU Improvements
1
0.2
0.9
0.8
0.7
0.6
0.5
0
Trials with IOU > 0.5
Average IOU
20
40
60
0.8
Accuracy
Accuracy
Accuracy
1
0.6
0.4
Trials with IOU > 0.5
Average IOU
0.2
80
100
0
0
Initial Pixel Offset
5
10
0.1
0
?0.1
15
# of Inference Steps
?0.2
0
20
40
60
80
100
Initial Pixel Offset
(a)
(b)
(c)
Figure 5: (a) Accuracy as a function of gaze initialization (pixel offset). Blue curve is the percentage success
of at least 50% IOU. Red curve is the average IOU. (b) Accuracy as a function of the number of approximate
inference steps when initializing 50 pixels away. (c) Accuracy improvements of HMC as a function of gaze
initializations.
(a) DBN trained on Caltech
(b) DBN updated with Multi-PIE
Figure 6: Left: Samples from a 2-layer DBN trained on Caltech. Right: samples from an updated DBN after
training on CMU Multi-PIE without labels. Samples highlighted in green are similar to faces from CMU.
only a few steps of approximate inference to converge. HMC clearly improves model performance,
resulting in an IOU increase of about 5% for localization. This is impressive given that none of
the test subjects were part of the training and the background is different from backgrounds in the
training set.
We also compared our inference algorithm to
the template matching in the task of face deOur method OpenCV NCC template
tection. We took the first 5 subjects as test
IOU > 0.5
97%
97%
93%
78%
subjects and the rest as training. We can lo# evaluations
O(c)
O(whs) O(whs) O(whs)
Table 1: Face localization accuracy. w: image width; calize with 97% accuracy 3(IOU > 0.5) ush: image height; s: image scales; c: number of inference ing our inference algorithm . In comparison,
a near state-of-the-art face detection system
steps used.
from OpenCV 2.4.9 obtains the same 97% accuracy. It uses Haar Cascades, which is a form of AdaBoost4 . Normalized Cross Correlation [31]
obtained 93% accuracy, while Euclidean distance template matching achieved an accuracy of only
78%. However, note that our algorithm looks at a constant number of windows while the other
baselines are all based on scanning windows.
6.2
Generative learning without labels
The main advantage of our
model is that it can learn on
large images of faces without localization label information (no
manual cropping required). To
demonstrate, we use both the
Caltech and the CMU faces
Table 2: Variational lower-bound estimates on the log-density of the dataset. For the CMU faces, a
Gaussian DBNs (higher is better).
subset of 2526 frontal faces with
ground truth labels are used. We split the Caltech dataset into a training and a validation set. For
the CMU faces, we first took 10% of the images as training cases for the ConvNet for approximate
inference. This is needed due to the completely different backgrounds of the Caltech and CMU
datasets. The remaining 90% of the CMU faces are split into a training and validation set. We first
trained a GDBN with 1024 h1 and 256 h2 hidden units on the Caltech training set. We also trained
nats
No CMU training CMU w/o labels CMU w/ labels
Caltech Train
617?0.4
627?0.5
569?0.6
Caltech Valid
512?1.1
503?1.8
494?1.7
CMU Train
96?0.8
499?0.1
594?0.5
CMU Valid
85?0.5
387?0.3
503?0.7
log Z?
454.6
687.8
694.2
3
u is randomly initialized at ? 30 pixels, scale range from 0.5 to 1.5.
OpenCV detection uses pretrained model from haarcascade_frontalface_default.xml, scaleFactor=1.1,
minNeighbors=3 and minSize=30.
4
7
Figure 7: Left: Conditioned on different v will result in a different 4u. Note that the initial u is exactly the
same for two trials. Right: Additional examples. The only difference between the top and bottom panels is the
conditioned v. Best viewed in color.
a ConvNet for approximate inference using the Caltech training set and 10% of the CMU training
images.
Table 2 shows the estimates of the variational lower-bounds on the average log-density (higher is
better) that the GDBN models assign to the ground-truth cropped face images from the training/test
sets under different scenarios. In the left column, the model is only trained on Caltech faces. Thus it
gives very low probabilities to the CMU faces. Indeed, GDBNs achieve a variational lower-bound of
only 85 nats per test image. In the middle column, we use our approximate inference to estimate the
location of the CMU training faces and further trained the GDBN on the newly localized faces. This
gives a dramatic increase of the model performance on the CMU Validation set5 , achieving a lowerbound of 387 nats per test image. The right column gives the best possible results if we can train
with the CMU manual localization labels. In this case, GDBNs achieve a lower-bound of 503 nats.
We used Annealed Importance Sampling (AIS) to estimate the partition function for the top-layer
RBM. Details on estimating the variational lower bound are in the supplementary materials.
Fig. 6(a) further shows samples drawn from the Caltech trained DBN, whereas Fig. 6(b) shows
samples after training with the CMU dataset using estimated u. Observe that samples in Fig. 6(b)
show a more diverse set of faces. We trained GDBNs using a greedy, layer-wise algorithm of [1].
For the top layer we use Fast Persistent Contrastive Divergence [32], which substantially improved
generative performance of GDBNs (see supplementary material for more details).
6.3 Inference with ambiguity
Our attentional mechanism can also be useful when multiple objects/faces are present in the scene.
Indeed, the posterior p(u|x, v) is conditioned on v, which means that where to attend is a function of the canonical object v the model has in ?mind? (see Fig. 2(b)). To explore this, we first
synthetically generate a dataset by concatenating together two faces from the Caltech dataset. We
then train approximate inference ConvNet as in Sec. 4.1 and test on the held-out subjects. Indeed,
as predicted, Fig. 7 shows that depending on which canonical image is conditioned, the same exact
gaze initialization leads to two very different gaze shifts. Note that this phenomenon is observed
across different scales and location of the initial gaze. For example, in Fig. 7, right-bottom panel,
the initialized yellow box is mostly on the female?s face to the left, but because the conditioned
canonical face v is that of the right male, attention is shifted to the right.
7
Conclusion
In this paper we have proposed a probabilistic graphical model framework for learning generative
models using attention. Experiments on face modeling have shown that ConvNet based approximate
inference combined with HMC sampling is sufficient to explore the complicated posterior distribution. More importantly, we can generatively learn objects of interest from novel big images. Future
work will include experimenting with faces as well as other objects in a large scene. Currently the
ConvNet approximate inference is trained in a supervised manner, but reinforcement learning could
also be used instead.
Acknowledgements
The authors gratefully acknowledge the support and generosity from Samsung, Google, and ONR
grant N00014-14-1-0232.
5
We note that we still made use of labels coming from the 10% of CMU Multi-PIE training set in order to
pretrain our ConvNet. "w/o labels" here means that no labels for the CMU Train/Valid images are given.
8
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