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2,200 | 2,998 | Stochastic Relational Models
for Discriminative Link Prediction
Wei Chu
CCLS, Columbia University
New York, NY 10115
Kai Yu
NEC Laboratories America
Cupertino, CA 95014
Shipeng Yu, Volker Tresp, Zhao Xu
Siemens AG, Corporate Research & Technology,
81739 Munich, Germany
Abstract
We introduce a Gaussian process (GP) framework, stochastic relational models
(SRM), for learning social, physical, and other relational phenomena where interactions between entities are observed. The key idea is to model the stochastic
structure of entity relationships (i.e., links) via a tensor interaction of multiple
GPs, each defined on one type of entities. These models in fact define a set of
nonparametric priors on infinite dimensional tensor matrices, where each element
represents a relationship between a tuple of entities. By maximizing the marginalized likelihood, information is exchanged between the participating GPs through
the entire relational network, so that the dependency structure of links is messaged
to the dependency of entities, reflected by the adapted GP kernels. The framework
offers a discriminative approach to link prediction, namely, predicting the existences, strengths, or types of relationships based on the partially observed linkage
network as well as the attributes of entities (if given). We discuss properties and
variants of SRM and derive an efficient learning algorithm. Very encouraging experimental results are achieved on a toy problem and a user-movie preference link
prediction task. In the end we discuss extensions of SRM to general relational
learning tasks.
1
Introduction
Relational learning concerns the modeling of physical, social, or other phenomena, where rich types
of entities interact via complex relational structures. If compared to the traditional machine learning
settings, the entity relationships provide additional structural information.
A simple example of a relational setting is the user-movie rating database, which contains user entities with user attributes (e.g., age, gender, education), movie entities with movie attributes (e.g., year,
genre, director), and ratings (i.e., relations between users and movies). A typical relational learning
problem is entity classification, for example, segmenting users into groups. One may apply usual
clustering or classification methods based on a flat structure of data, where each user is associated
with a vector of user attributes. However a sound model should additionally explore the user-movie
relations as well as even the movie attributes, since like-minded users tend to give similar ratings on
the same movie, and may like (or dislike) movies with similar genres. Relational learning addresses
this and similar situation where it is not natural to transform the data into a flat structure.
Entity classification in a relational setting has gained considerable attentions, like webpage classification using both textual contents and hyperlinks. However, in other occasions relationships
themselves are often of central interest. For example, one may want to predict protein-protein in-
teractions, or in another application, user ratings for products. The family of these problems has
been called link prediction1 , which is the primary topic of this paper. In general, link prediction
includes link existence prediction (i.e., does a link exist?), link classification (i.e., what type of the
relationship?), and link regression (i.e., how does the user rate the item?).
In this paper we propose a family of stochastic relational models (SRM) for link prediction and
other relational learning tasks. The key idea of SRM is a stochastic link-wise process induced by
a tensor interplay of multiple entity-wise Gaussian processes (GP). These models in fact define a
set of nonparametric priors on an infinite dimensional tensor matrix, where each element represents
a relationship between a tuple of entities. By maximizing the marginalized likelihood, information is exchanged between the participating GPs through the entire relational network, so that the
dependency structure of links is messaged to the dependency of entities, reflected by the learned
entity-wise GP kernels (i.e., GP covariance functions). SRM is discriminative because training is
on a conditional model of links. We present various models of SRM and address the computational
issue, which is crucial in link prediction because the number of potential links grows exponentially
with the entity size. SRM has shown encouraging results in our experiments.
This paper is organized as follows. We introduce the stochastic relational models in Sec. 2, and
describe the algorithms for inference and parameter estimation in Sec. 3 and Sec. 4, followed by Sec.
5 on implementation details. Then we discuss the related work in Sec. 6 and report experimental
results in Sec. 7, followed by conclusions and extensions in Sec. 8.
2
Stochastic Relational Models
We first consider pairwise asymmetric links r between entities u ? U and v ? V. The two sets
U and V can be identical or different. We use u or v to represent the attribute vectors of entities
or their identity when entity attributes are unavailable. Note that ri,n ? r(ui , vn ) does not have
to be identical to rn,i when U = V, i.e. relationships can be asymmetrical. Extensions to involve
more than two entity sets, multi-way relations (i.e., links connecting more than two entities), and
symmetric links are straightforward and will be briefly discussed in Sec. 8.
We assume that the observable links r are derived as local measurements of a real-valued latent
relational function t : U ? V ? R, and each link ri,n is solely dependent on its latent value ti,n ,
modeled by the likelihood p(ri,n |ti,n ). The focus of this paper is a family of stochastic relational
processes acting on U ? V, the space of entity pairs, to generate the latent relational function t, via
a tensor interaction of two independent entity-specific GPs, one acting on U and the other on V.
We call them processes because U and V can both encompass an infinite number of entities. Let
the relational processes be characterized by hyperparameters ? = {?? , ?? }, ?? for the GP kernel
function on U and ?? for the GP kernel function on V, a SRM thus defines a Bayesian prior p(t|?)
for the latent variables t. Let I be the index set of entity pairs having observed links, the marginal
likelihood (also called evidence) under such a prior is
p(RI |?) =
p(ri,n |ti,n )p(t|?)dt, ? = {?? , ?? }
(1)
(i,n)?I
where RI = {ri,n }(i,n)?I . We estimate the hyperparameters ? by maximizing the evidence, which is
an empirical Bayesian approach to learning the relational structure of data. Once ? are determined,
we can predict the link for a new pair of entities via marginalization over the a posteriori p(t|RI , ?).
Choices for the Piror p(t|?)
2.1
In order to represent a rich class of link structures, p(t|?) should be sufficiently expressive. In the
following subsections, we will present three cases of p(t|?), from specific to general, by gradually
extending conventional GP models.
2.1.1
A Brief Introduction to Gaussian Processes
A GP defines a nonparametric prior distribution over functions in Bayesian inference. A random
real-valued function f : X ? R follows a GP prior, denoted by GP(?, ?), if for every finite set
1
We will use ?link? and ?relationship? interchangeably throughout this paper.
N
N
{xi }N
i=1 , f = {f (xi }i=1 follows a multivariate Gaussian distribution with mean ? = {?(xi )}i=1
N
N
and covariance (or kernel) ? = {?(xi , xj ; ?? )}i,j=1 with parameter ?? . Given D = {xi , yi }i=1 ,
by Gaussian noise, one can make predictions via
where yi is a measurement of f (xi ) corrupted
the marginal likelihood p(y|x, D, ?? ) = p(y|f, x)p(f |D, ?? )df . For non-Gaussian measurement
processes, as in classification models, the integral cannot be solved analytically, and approximation
for inference is required. A comprehensive coverage of GP models can be found in [9].
2.1.2
Hierarchical Gaussian Processes
By observing the relational data collectively, one may notice that two entities ui and uj in U demonstrate correlated relationships to entities in V. For example, two users often show opposite or close
opinions on movies, or two hub web pages are co-linked by a set of other authority web pages. In
this case, the dependency structure of links can be reduced to a dependency structure of entities
in U. A hierarchical GP (HGP) model [13], originally proposed for multi-task learning, can be
conveniently applied in such a situation. The model assumes that, for every v ? V, its relational
function t(?, v) : U ? R is an i.i.d. sample drawn from a common entity-wise GP with covariance
? : U ? U ? R. This provides a case of p(t|?) in a SRM, where ? = ?? determines the GP kernel
function ?. Optimizing the GP kernel ? via evidence maximization means to learn the dependency
of entities in U, summarized over all the entities v ? V.
There is a drawback if applying HGP to link prediction. The model only learns a one-side structure,
while ignoring the dependency in V. In particular, the attributes of entities v cannot be incorporated
even if their entity attributes are available. However, for the mentioned applications, it also makes
sense to explore the dependency between movies, or the dependency between authority web pages.
2.1.3
Tensor Gaussian Processes
To overcome the shortcoming of HGP, we consider a more complex SRM, which employs two
GP kernel functions ? : U ? U ? R and ? : V ? V ? R. The model explains the relational
dependency through the entity dependencies of both V and U. Let ? = {?? , ?? }, we describe a
stochastic relational process p(t|?) as the following:
Definition 2.1 (Tensor Gaussian Processes). Given two sets U and V, a collection of random variables {t(u, v)|t : U ? V ? R} follow a tensor Gaussian processes (TGP), if for every finite sets
{u1 , . . . , uN } and {v1 , . . . , vM }, random variables T = {t(ui , vn )}, organized into an N ? M
matrix, have a matrix-variate normal distribution
MN
M
N
1
NN ?M (T|B, ?, ?) = (2?)? 2 |?|? 2 |?|? 2 etr ? ??1 (T ? B)??1 (T ? B)
2
characterized by mean B = {b(ui , vn )} and positive definite covariance matrices ? =
{?(ui , uj ; ?? )} and ? = {?(vn , vm ; ?? )}. This random process is denoted as T GP(b, ?, ?).2
In the above theorem etr[?] is a shortcut for exp[trace(?)]. It is easy to see that the model reduces
to the HGP model if ? = I. As a key difference, the new model treats the relational function
t as a whole sample from a TGP, instead of being formed by i.i.d. functions in the HGP model.
Let vec(T ) = [t1,1 , t1,2 , . . . , t1,M , t2,1 , . . . , t2,M , . . . , tN,M ] . If T ? NN ?M (T|B, ?, ?), then
vec(T ) ? N (0, ?), where the covariance ? = ? ? ? is the Kronecker product of two covariance matrices [6]. In other words, TGP is in fact a GP for the relational function t, where the
kernel function ? : (U ? V) ? (U ? V) ? R is defined via a tensor product of two GP kernels
Cov(ti,n , tj,m ) = ?[(ui , vn ), (uj , vm )] = ?(ui , uj )?(vn , vm ). The model explains the dependence
structure of links by the dependence structure of participating entities.
It is well known that a linear predictive model has a GP interpretation if its linear weights follow a
Gaussian prior. A similar connection exists for TGP.
Theorem 2.2. Let U ? RP , V ? RQ , and W ? NP ?Q (0, IP , IQ ), where IP denotes a P ? P
identity matrix and ?, ? denotes the inner product, then the bilinear function t(u, v) = u Wv
follows T GP(0, ?, ?) with ?(ui , uj ) = ui , uj and ?(vn , vm ) = vn , vm .
2
Hereafter we always assume b(u, v) = 0 in TGP for simplicity.
The proof is straightforward through Cov[t(ui , vn ), t(uj , vm )] = ui , uj vn , vm and
E[t(ui , vn )] = 0, where E[?] means expectation. In practice, the linear model will help to provide an
efficient discriminative approach to link prediction.
It appears that link prediction using TGP is almost the same as a normal GP regression or classification, except that the hyperparameters ? now have two parts, ?? for ? and ?? for ?. Unfortunately
TGP inference suffers from a serious computational problem ? it does not scale well for even a small
size of entities. For example, if there is a fixed portion of missing data for pairwise relationships between N u-entities and M v-entities, the size of observations scales in O(N M ). Since GP inference
has the complexity cubic to the size of data, the complexity O(N 3 M 3 ) of TGP is computationally
prohibitive.
2.1.4
A Family of Stochastic Processes for Entity Relationships
To improve the scalability of SRM, and also describe the relational dependency in a way similar to
TGP, we propose a family of stochastic processes p(t|?) for entity relationships.
Definition 2.3 (Stochastic Relational Processes). A relational function t : U ? V ? R is said to
d
iid
follow a stochastic relational process (SRP), if t(u, v) = ?1d k=1 fk (u)gk (v), fk (u) ? GP(0, ?),
iid
gk (v) ? GP(0, ?). We denote t ? SRP d (0, ?, ?), where d is the degrees of freedom.
Interestingly, there exists an intimate connection between SRP and TGP:
Theorem 2.4. SRP d (0, ?, ?) converges to T GP(0, ?, ?) in the limit d ? ?.
Proof. Based on the central limit theory, for every (ui , vn ), ti,n ? t(ui , vn ) converges to a
Gaussian random variable. In the next steps, we prove E[ti,n ] = 0 and Cov(ti,n , tj,m ) =
d
d
E[ti,n tj,m ] = d1 { k=1 E[fk (ui )fk (uj )gk (vn )gk (vm )] + k=? E[fk (ui )f? (uj )gk (vn )g? (vm )]} =
d
1
k=1 E[fk (ui )fk (uj )gk (vn )gk (vm )] = ?(ui , uj )?(vn , vm ).
d
The theorem suggests that there is a constructive definition of TGP, where relationships are formed
via interactions between infinite samples from two GPs. Moreover, given a sufficiently large d, SRP
will provide a close approximation to TGP.
SRP is a general family of priors for modeling the relationships between entities, in which HGP and
TGP are special cases. The generalization offers several advantages: (1) SRP can model symmetric
links between the same set of entities. If we build a generative process where U = V, ? = ? and
fk = gk , then on every finite sets {ui }N
i=1 , T = {t(ui , uj )} is always a symmetric matrix; (2) Given
a fixed d, the inference with SRP obtains a much reduced complexity. In Sec. 3 we will introduce
an inference algorithm that scales in O[d(N 3 + M 3 )], which is much less than O(N 3 M 3 ).
2.2
Choices for the Likelihood p(ri,n |ti,n )
The likelihood term describes the dependency of observable relationships on the latent variables. It
should be tailored to the types of observations to be modeled. Here we list three possible situations:
? Binary Classification: Relationships may take categorical states, e.g., ?cue? or ?no cue? in
disease-treatment relationship prediction. It is popular to consider binary classification and employ the probit function to model the Bernoulli distribution over class labels, i.e. p(ri,n |ti,n ) =
? (ri,n ti,n ), where ?(?) is a cumulative normal function, and ri,n ? {?1, +1}.
? Regression: In this case we consider ri,n taking continuous values. For example, one may want to
predict the rating of user u for movie v. The corresponding likelihood function is essentially defined
by a noise model, e.g. a univariate Gaussian noise with variance ?2 and zero mean.
? One-class Problem: Sometimes one observed the presence of links between entities, for example,
the hyperlinks between web pages. Based on the open-world assumption, if a web page does not
link to another, it does not mean that they are unrelated. Therefore, we employ the likelihood
p(ri,n |ti,n ) = ?(ri,n ti,n ? b) for those observed links ri,n = 1, where b is an offset.
3
Inference with Laplacian Approximation
We have described the relational model under a prior of SRP, in which HGP and TGP are subcases.
Now we develop the inference algorithm to compute the sufficient statistics of the a posteriori distribution of latent variables.
Let F = {fi,k }, G = {gn,k }, f k = [f1,k , . . . , fN,k ] and gk = [g1,k , . . . , gM,k ] , where fi,k =
fk (ui ), gn,k = gk (vn ). Then the posterior distribution p(F, G|RI , ?) is proportional to the joint
distribution of the complete data:
d
d f g
1
?1
k=1 i,k n,k
?1
?
p RI , F, G|? ?
f k ? f k + gk ? gk
p ri,n
exp ?
2
d
k=1
(i,n)?I
An exact inference is intractable due to the coupling between fi,k and gn,k in the likelihood term. In
this paper we apply Laplacian approximation, which approximates p(F, G|RI , ?) by a multivariate
normal distribution q(F, G|?) with sufficient statistics ?. At the first step, we compute the means
by finding the mode in the posterior,
{F? , G? } = arg min J(F, G) = ? log p(RI , F, G|?)
(2)
{F,G}
We solve the minimization by the conjugate gradient method. The gradients are calculated by
1
?J(F, G)
= ? S F + ??1 G,
?G
d
?
d
N ?M
where S ? R
have elements si,n = ?[? log p(ri,n |ti,n )]/?ti,n , ti,n = k=1 fi,k gn,k / d,
if (i, n) ? I, otherwise si,n = 0. At the second step we calculate the covariance by C = H?1 ,
where H is the Hessian, i.e., the second-order derivatives of J(F, G) with respect to {F, G}.
However the inverse is prohibitive because H is a huge matrix. To reduce the complexity, we
assume that there exist disjoint groups of latent variables, each group is second-order independent
to any other at their equilibriums. We factorize the approximating distribution as q(F, G|?) =
d
?
?
?
?
k=1 q(f k |f k , ?k )q(gk |gk , ?k ), where f k and gk are the solution from Eq. (2), and ?k , ?k are
the covariances matrices. This follows
the
facts:
(1)
Each f k (or gk ) indirectly interacts with other
f ? (or g? ), ? = k, through the sum ?=k f ? g
,
indicating
that f k (or gk ) across different k are
?
only loosely dependent to each other, especially for a large d; (2) The dependency between fi,k and
gn,k is witnessed via at most only one observation in RI . Therefore we can compute the Hessian for
each group separately and obtain the covariances:
1
?J(F, G)
= ? SG + ??1 F,
?F
d
?k = (?(k) + ??1 )?1 ,
with
(k)
?i,i =
n:(i,n)?I
?k = (?(k) + ??1 )?1 ,
with
(k)
?n,n
=
i:(i,n)?I
2
?i,n gn,k
,
d
2
?i,n fi,k
,
d
(k)
?i,j = 0
(3)
(k)
?n,m
=0
(4)
where ?i,n = ? 2 [? log p(ri,n |ti,n )]/?t2i,n . Then we obtain the sufficient statistics F? , G? , {?k }
and {?k }. Finally we note that, the posterior distribution of {F, G} has many modes (Simply,
shuffling the order of latent dimensions or changing the signs of both f k and gk does not change the
probability.). However each mode is equally well in constructing the relational function t.
4
Structural Learning by Hyperparameter Estimation
We assign a hyper prior p(?|?) and estimate ? by maximizing a penalized marginal likelihood
?
log
p(RI , F, G|?)dFdG + log p(?|?)
(5)
? = arg max
?={?? ,?? }
G
F
So far the optimization (5) is quite general. In principal, it allows to learn some parametric forms
of kernel functions ?(ui , uj ; ?? ) and ?(vn , vm ; ?? ) that are generalizable to new entities. In this
paper we particularly consider an situation where entity attributes are not fully informative or even
absent. Therefore we introduce a direct parameterization ?? = ?, ?? = ?, and assign conjugate
inverse-Wishart priors ? ? IW N (?d, ?0 ) and ? ? IW M (?d, ?0 ), namely
?d ?1
?d
? ?0 ,
p(?|?d, ?0 ) ? det(?)? 2 etr ?
2
?d ?1
?d
p(?|?d, ?0 ) ? det(?)? 2 etr ?
? ?0 ,
2
where ? > 0 so that ?d denotes the degrees of freedom, ?0 and ?0 are the base kernels. Then
we apply an iterative expectation-maximization (EM) algorithm to solve the problem (5). In the
E-step, we follow Sec. 3 to compute q(F, G|?). In the M-step, we update the hyperparameters by
maximizing the expected log-likelihood of the complete data
max Eq [log p(RI , F, G|?, ?)] + log p(?|?d, ?0 ) + log p(?|?d, ?0 )
{?,?}
where Eq [?] is the expectation over q(F, G|?). Due to the conjugacy of the hyper prior, the maximization have an analytical solution,
d
d
??0 + d1 k=1 (g?k g?k + ?k )
??0 + d1 k=1 (f ?k f ?k + ?k )
, ?=
.
(6)
?=
?+1
?+1
5
Implementation Details
The parameters ?0 , ?0 , d and ? have to be pre-specified. We let the base kernels have the form
?0 (ui , uj ) = (1 ? a)?(ui , uj ) + a?i,j and ?0 (vn , vm ) = (1 ? ?)?(vn , vm ) + ??n,m , where 1 ?
a, ? ? 0, ? is a Dirac delta kernel (?i,j = 1 if i = j, otherwise ?i,j = 0), ?(?, ?) and ?(?, ?)
are some kernel functions defined on entity attributes, which reflect our prior notion of similarities
between entities. We use a and ? to penalize the effects of ?(?, ?) and ?(?, ?), respectively, when entity
attributes are deficient. If the attributes are unavailable, we set a = ? = 1. The dimensionality d
should be properly chosen, otherwise a too small d may deteriorate the modeling flexibility. We
determine d and ? based on the prediction performance on a validation set of links. The learning
algorithm iterates the E-step with Eq. (2), (3), (4), and the M-step with Eq. (6) until convergence.
In the experiments of this paper we use p(ri,n |t?i,n ) to make predictions, where t? is computed from
F? and G? . In a longer version the predictive uncertainty of ti,n will be considered.
6
Related Work
There is a history of probabilistic relational models (PRM) [8] in machine learning. Getoor et al.
[5] introduced link uncertainty and defined a generative model for both entity attributes and links.
Recently, [12] and [7] independently introduced an infinite (hidden) relational model to avoid the
difficulty of structural learning in PRM by explaining links via a potentially infinite number of
hidden states of entities. Since discriminatively trained models generally outperform generative
models in prediction tasks, Taskar et al. proposed relational Markov networks (RMNs) for link
prediction [11], by describing a conditional distribution of links given entity attributes and other
links. RMN has to define a class of potential functions on cliques of random variables based on
the observed relational structure. Compared to RMN, SRM is nonparametric because structural
information (e.g., cliques as well as the classes of potential functions) is not pre-defined but learned
from data. Very recently a GP model was developed to learn from undirected graphs [4], which
turns out to be a special rank-one case of SRM with d = 1, ? = ?, and fk = hk . In another work
[1] a SVM using a tensor kernel based on user and item attributes was used to predict user ratings
on items, which is similar to our TGP case and suffers a salability problem. When attributes are
deficient or unavailable, the model does not work well, while SRM can learn informative kernels
purely from only links (see Sec. 7). SRM is interestingly related to the recent fast maximum-margin
matrix factorization (MMMF) in [10]. If we fix ? and ? as uninformative Dirac kernels, the mode
of our Laplacian approximation is equivalent to the solution of Eq.(5) in [10] with the loss function
l(ri,n , ti,n ) = ? log p(ri,n |ti,n ). However SRM significantly differs from MMMF in two important
aspects: (1) SRM is a supervised predictive model because entity attributes enter the model by
forming informative priors (?, ?) and hyper priors (?0 , ?0 ); (2) More importantly, SRM deals with
(a)
(b)
(c)
(d)
(e)
10
10
10
10
10
20
20
20
20
20
30
5
10
15
20
30
5
10
(f)
15
20
5
(g)
5
10
15
20
30
5
10
(h)
15
20
20
5
10
15
20
30
10
20
30
20
30
5
10
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20
10
(j)
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(i)
5
10
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20
30
5
10
15
20
30
15
10
20
30
20
5
10
15
20
Figure 1: Link prediction on synthetic data: (a) training data, where black entry means positive links, white
means negative links, and gray means missing; (b) prediction of MMMF (classification rate 0.906); (c) prediction of SRM with noninformative prior (classification rate 0.942); (d) prediction of SRM with informative prior
(classification rate 0.965); (e-f) informative ?0 and ?0 ; (g-h) learned ? and ? with noninformative prior; (i-j)
learned ? and ? with informative prior.
structural learning by adapting the kernels and marginalizing out the latent relational function, while
MMMF only estimates the mode of the latent relational function with fixed Dirac kernels.
7
Experiments
30
Synthetic Data: We generated two sets of entities U = {ui }20
i=1 and V = {vn }n=1 on a real line
such that ui = 0.1i and vn = 0.1n. The positions of entities were used to compute two RBF kernels
that serve as informative ?0 and ?0 . Then we further made a deformation on the real line to form
2 clusters in U and 3 clusters in V. RBF function computed on the deformed scale gives two kernel
matrices ? and ? whose diagonal block structure reflects the clusters. Binary links between U
and V are obtained by taking the sign of a function, which is a sample from T GP(0, ?, ?). We
randomly withdrew 50% of links for training, and left the remaining for test (see Fig. 1-(a)). We
performed two variants of SRM, one with informative ?0 and ?0 (see Fig. 1-(e,f)) and the other
with noninformative Dirac kernels ?0 = ?0 = I, and compared with MMMF [10]. In all the cases
we set d = 20. The classification accuracy rates of two SRMs, 0.942 and 0.965, are both better than
0.906 of MMMF. As shown in Fig. 1, the block structures of learned kernels indicate that both SRMs
can learn the cluster structure of entities from links. The structural kernel optimization enables SRM
to outperform MMMF, even with a completely noninformative prior. Note that the informative prior
really helps SRM to achieve the best accuracy.
Eachmovie Data: We tested our algorithms on a data set from [3], which is a subset of EachMovie
data, containing 5000 users? ratings, i.e., 1, 2, 3, 4, 5, or 6, on 1623 movies. We selected the first 1000
users and organized the data into a 1000 ? 1623 table with 63, 592 observed ratings. We compared
SRM with MMMF in a regression task to predict the ?rating link? between users and movies. In
SRM we set ?0 = ?0 = I. For both methods the dimensionality was chosen as d = 20. In MMMF
we used the square error loss. We repeated the experiments for 10 times, where at each time we
randomly withdrew 70% ratings for training and left the remaining for test. Root-mean-square error
(RMSE) and mean-absolute error (MAE) were used to evaluate the accuracy. The results of all the
repeats, as well as the means and standard deviations, are shown in Table 1 and Table 2. Compared
to MMMF, SRM significantly reduces the prediction error by over 12% in terms of both RMSE and
MAE.
8
Conclusions and Future Extensions
In this paper we proposed a stochastic relational model (SRM) for learning relational data. Entity relationships are modeled by a tensor interaction of multiple Gaussian processes (GPs). We proposed
a family of relational processes and showed its convergence to a tensor Gaussian process if the degrees of freedom goes to infinity. The process imposes an effective prior on the entity relationships,
Table 1: User-movie rating prediction error measured by RMSE
Repeats
MMMF
SRM
1
1.366
1.195
2
1.367
1.199
3
1.372
1.192
4
1.377
1.200
5
1.363
1.198
6
1.368
1.209
7
1.356
1.204
8
1.380
1.208
9
1.358
1.189
10
1.373
1.209
mean ? std.
1.368 ? 0.008
1.200?0.007
Table 2: User-movie rating prediction error measured by MAE
Repeats
MMMF
SRM
1
1.067
0.924
2
1.066
0.928
3
1.074
0.924
4
1.076
0.923
5
1.066
0.924
6
1.073
0.934
7
1.060
0.931
8
1.074
0.932
9
1.062
0.918
10
1.072
0.933
mean ? std.
1.060?0.006
0.927? 0.005
and leads to a discriminative link prediction model. We demonstrated the excellent results of SRM
on a synthetic data set and a user-movie rating prediction problem.
Though the current work focused on the application of link prediction, the model can be used for
general relational learning purposes. There are several directions to extend the current model: (1)
SRM can describe a joint distribution of entity links and entity classes conditioned on entity-wise
GP kernels. Therefore entity classification can be solved in a relational context; (2) One can extend
SRM to model multi-way relations where more than two entities participate in a single relationship;
(3) SRM can also be applied to model pairwise relations between multiple entity sets, where kernel
updates amount to propagation of information through the entire relational network; (4) As discussed
in Sec. 2.1.2, SRM is a natural extension of hierarchical Bayesian multi-task models, by explicitly
modeling the dependency over tasks. In a recent work [2] a tensor GP for multi-task learning was
independently suggested; (5) Finally, it is extremely important to make the algorithm scalable to very
large relational data, like the Netflix problem, containing about 480,000 users and 17,000 movies.
Acknowledgement
The authors thank Andreas Krause, Chris Williams, Shenghuo Zhu, and Wei Xu for the fruitful
discussions.
References
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2,201 | 2,999 | Computation of Similarity Measures for
Sequential Data using Generalized Suffix Trees
Konrad Rieck
Fraunhofer FIRST.IDA
Kekul?estr. 7
12489 Berlin, Germany
[email protected]
Pavel Laskov
Fraunhofer FIRST.IDA
Kekul?estr. 7
12489 Berlin, Germany
[email protected]
S?oren Sonnenburg
Fraunhofer FIRST.IDA
Kekul?estr. 7
12489 Berlin, Germany
[email protected]
Abstract
We propose a generic algorithm for computation of similarity measures for sequential data. The algorithm uses generalized suffix trees for efficient calculation
of various kernel, distance and non-metric similarity functions. Its worst-case
run-time is linear in the length of sequences and independent of the underlying
embedding language, which can cover words, k-grams or all contained subsequences. Experiments with network intrusion detection, DNA analysis and text
processing applications demonstrate the utility of distances and similarity coefficients for sequences as alternatives to classical kernel functions.
1 Introduction
The ability to operate on sequential data is a vital prerequisite for application of machine learning
techniques in many challenging domains. Examples of such applications are natural language processing (text documents), bioinformatics (DNA and protein sequences) and computer security (byte
streams or system call traces). A key instrument for handling such data is the efficient computation
of pairwise similarity between sequences. Similarity measures can be seen as an abstraction between
particular structure of data and learning theory.
One of the most successful similarity measures thoroughly studied in recent years is the kernel
function [e.g. 1?3]. Various kernels have been developed for sequential data, starting from the
original ideas of Watkins [4] and Haussler [5] and extending to application-specific kernels such
as the ones for text and natural language processing [e.g. 6?8], bioinformatics [e.g. 9?14], spam
filtering [15] and computer security [e.g. 16; 17].
Although kernel-based learning has gained a major focus in machine learning research, a kernel
function is obviously only one of various possibilities for measuring similarity between objects.
The choice of a similarity measure is essentially determined by (a) understanding of a problem and
(b) properties of the learning algorithm to be applied. Some algorithms operate in vector spaces,
others in inner product, metric or even non-metric feature spaces. Investigation of techniques for
learning in spaces other than RKHS is currently one of the active research fields in machine learning
[e.g. 18?21].
The focus of this contribution lies on general similarity measures for sequential data, especially on
efficient algorithms for their computation. A large number of such similarity measures can be expressed in a generic form so that a simple linear-time algorithm can be applied for computation of a
wide class of similarity measures. This algorithm enables the investigation of alternative representations of problem domain knowledge other than kernel functions. As an example, two applications
are presented for which replacement of a kernel ? or equivalently, the Euclidean distance ? with a
different similarity measure yields a significant improvement of accuracy in an unsupervised learning scenario.
The rest of the paper is organized as follows. Section 2 provides a brief review of common similarity measures for sequential data and introduces a generic form in which a large variety of them
can be cast. The generalized suffix tree and a corresponding algorithm for linear-time computation
of similarity measures are presented in Section 3. Finally, the experiments in Section 4 demonstrate efficiency and utility of the proposed algorithm on real-world applications: network intrusion
detection, DNA sequence analysis and text processing.
2 Similarity measures for sequences
2.1 Embedding of sequences
A common way to define similarity measures for sequential data is via explicit embedding into a
high-dimensional feature space. A sequence x is defined as concatenation of symbols from a finite
alphabet ?. To model the content of a sequence, we consider a language L ? ?? comprising subsequences w ? L. We refer to these subsequences as words, even though they may not correspond to
a natural language. Typical examples for L are a ?bag of words? [e.g. 22], the set of all sequences of
fixed length (k-grams or k-mers) [e.g. 10; 23] or the set of all contained subsequences [e.g. 8; 24].
Given a language L, a sequence x can be mapped into an |L|-dimensional feature space by calculating an embedding function ?w (x) for every w ? L appearing in x. The funcion ?w is defined as
follows
?w : ?? ? R+ ? {0}, ?w (x) := ?(occ(w, x)) ? Ww
(1)
where occ(w, x) is the number of occurrences of w in x, ? a numerical transformation, e.g. a
conversion to frequencies, and W a weighting assigned to individual words, e.g. length-dependent
or position-dependent weights [cf. 3; 24]. By employing the feature space induced through L and
?, one can adapt many vectorial similarity measures to operate on sequences.
The feature space defined via explicit embedding is sparse, since the number of non-zero dimensions
for each feature vector is bounded by the sequence length. Thus the essential parameter for measuring complexity of computation is the sequence length, denoted hereinafter as n. Furthermore, the
length of a word |w| or in case of a set of words the maximum length is denoted by k.
2.2 Vectorial similarity measures
Several vectorial kernel and distance functions can be applied to the proposed embedding of sequential data. A list of common functions in terms of L and ? is given in Table 1.
k(x, y)
Kernel function
P
Linear
w?L
Polynomial
P
?w (x)?w (y) + ?
2
exp ?d(x,y)
?
w?L
RBF
P
w?L
|?w (x) ? ?w (y)|
|?w (x)??w (y)|
w?L ?w (x)+?w (y)
P
Canberra
Minkowski
qP
k
Hamming
P
Chebyshev
d
d(x, y)
Distance function
Manhattan
?w (x)?w (y)
w?L
w?L
|?w (x) ? ?w (y)|k
sgn |?w (x) ? ?w (y)|
maxw?L |?w (x) ? ?w (y)|
Table 1: Kernels and distances for sequential data
s(x, y)
Similarity coefficient
Simpson
a/ min(a + b, a + c)
Jaccard
a/(a + b + c)
a/ max(a + b, a + c)
Braun-Blanquet
Czekanowski, Sorensen-Dice
2a/(2a + b + c)
Sokal-Sneath, Anderberg
a/(a + 2(b + c))
a/(b + c)
Kulczynski (1st)
1
+
2 (a/(a
p
Kulczynski (2nd)
b) + a/(a + c))
(a + b)(a + c)
a/
Otsuka, Ochiai
Table 2: Similarity coefficients for sequential data
Beside kernel and distance functions, a set of rather exotic similarity coefficients is also suitable for
application to sequential data [25]. The coefficients are constructed using three summation variables
a, b and c, which in the case of binary vectors correspond to the number of matching component
pairs (1-1), left mismatching pairs (0-1) and right mismatching pairs (1-0) [cf. 26; 27] Common
similarity coefficients are given in Table 2. For application to non-binary data these summation
variables can be extended as proposed in [25]:
X
a=
min(?w (x), ?w (y))
w?L
b=
X
[?w (x) ? min(?w (x), ?w (y))]
X
[?w (y) ? min(?w (x), ?w (y))]
w?L
c=
w?L
2.3 A generic representation
One can easily see that the presented similarity measures can be cast in a generic form that consists
of an outer function ? and an inner function m:
s(x, y) =
M
m(?w (x), ?w (y))
(2)
w?L
Given this definition, the kernel and distance functions presented in Table 1 can be re-formulated
in terms of ? and m. Adaptation of similarity coefficients to the generic form (2) involves a reformulation of the summation variables a, b and c. The particular definitions of outer and inner
functions for the presented similarity measures are given in Table 3. The polynomial and RBF kernels are not shown since they can be expressed in terms of a linear kernel or a distance respectively.
Kernel function
?
m(x, y)
Linear
+
x?y
Similarity coef.
?
m(x, y)
Distance function
?
m(x, y)
Manhattan
+
|x ? y|
+
|x ? y|/(x + y)
Canberra
k
Variable a
+
min(x, y)
Minkowski
+
|x ? y|k
Variable b
+
x ? min(x, y)
Hamming
+
sgn |x ? y|
Variable c
+
y ? min(x, y)
Chebyshev
max
|x ? y|
Table 3: Generalized formulation of similarity measures
3 Generalized suffix trees for comparison of sequences
The key to efficient comparison of two sequences lies in considering only the minimum of words
necessary for computation of the generic form (2) of similarity measures. In the case of kernels
only the intersection of words in both sequences needs to be considered, while the union of words
is needed for calculating distances and non-metric similarity coefficients. A simple and well-known
approach for such comparison is representing the words of each sequence in a sorted list. For words
of maximum length k such a list can be constructed in O(kn log n) using general sorting or O(kn)
using radix-sort. If the length of words k is unbounded, sorted lists are no longer an option as the
sorting time becomes quadratic.
Thus, special data structures are needed for efficient comparison of sequences. Two data structures
previously used for computation of kernels are tries [28; 29] and suffix trees [30]. Both have been
applied for computation of a variety of kernel functions in O(kn) [3; 10] and also in O(n) run-time
using matching statistics [24]. In this contribution we will argue that a generalized suffix tree is
suitable for computation of all similarity measures of the form (2) in O(n) run-time.
A generalized suffix tree (GST) is a tree containing all suffixes of a set of strings x1 , . . . , xl [31]. The
simplest way to construct a generalized suffix tree is to extend each string xi with a delimiter $i and
to apply a suffix tree construction algorithm [e.g. 32] to the concatenation of strings x1 $1 . . . xl $l .
In the remaining part we will restrict ourselves to the case of two strings x and y delimited by
# and $, computation of an entire similarity matrix using a single GST for a set of strings being
a straightforward extension. An example of a generalized suffix tree for the strings ?aab#? and
?babab$? is shown in Fig. 1(a).
b
a
#...
#...
#...
m(1,3)
$
ab
b
$
b
ab#...
#...
$
ab
m(0,2)
(1,0)
(0,1)
x
ab$
$
#...
$
ab$
(1,0)
(a) Generalized suffix tree (GST)
ab$
(0,1)
(1,0)
$
(0,1)
(b) Traversal of a GST
Figure 1: Generalized suffix tree for ?aab#? and ?babab$? and a snapshot of its traveral
Once a generalized suffix tree is constructed, it remains to determine the number of occurences
occ(w, x) and occ(w, y) of each word w present in the sequences x and y. Unlike the case for
kernels for which only nodes corresponding to both sequences need to be considered [24], the contributions must be correctly computed for all nodes in the generalized suffix tree. The following
simple recursive algorithm computes a generic similarity measure between the sequence x and y in
one depth-first traversal of the generalized suffix tree (cf. Algorithm 1).
The algorithm exploits the fact that a leaf in a GST representing a suffix of x contributes exactly 1 to
occ(w, x) if w is the prefix of this suffix ? and similarly for y and occ(w, y). As the GST contains
all suffixes of x and y, every word w in x and y is represented by at least one leaf. Whether a leaf
contributes to x or y can be determined by considering the edge at the leaf. Due to the uniqueness
of the delimiter #, no branching nodes can occur below an edge containing #, thus a leaf node
at an edge starting before the index of # must contain a suffix of x; otherwise it contains a suffix
of y. The contributions of all leaves are aggregated in two variables x and y during a post-order
traversal. At each node the inner function m of (2) is calculated using ?(x) and ?(y) according to
the embedding ? in (1). A snapshot of the traversal procedure is illustrated in Fig. 1(b).
To account implicit nodes along the edges of the GST and to support weighted embeddings ?, the
weighting function W EIGHT introduced in [24] is employed. At a node v the function takes the
beginning (begin[v]) and the end (end[v]) of the incoming edge and the depth of node (depth[v]) as
arguments to determine how much the node and edge contribute to the similarity measure, e.g. for
k-gram models only nodes up to a path depth of k need to be considered.
Algorithm 1 Suffix tree comparison
1: function C OMPARE (x, y)
2:
S ? S UFFIX T REE(x # y $)
3:
(x, y, s) ? M ATCH(root[S])
4:
return s
5:
6: function M ATCH(v)
7:
if v is leaf then
8:
s?0
9:
if begin[v] ? index# then
10:
(x, y) ? (1, 0)
? Leaf of a suffix of x
11:
j ? index# ? 1
12:
else
13:
(x, y) ? (0, 1)
? Leaf of a suffix of y
14:
j ? index$ ? 1
15:
else
16:
(x, y, s) ? (0, 0, 0)
17:
for all c in children[v] do
18:
(?
x, y?, s?) ? M ATCH(c)
? Traverse GST
19:
(x, y, s) ? (x + x
?, y + y?, s ? s?)
20:
j ? end[v]
21:
W ? W EIGHT(begin[v], j, depth[v])
22:
s ? s ? m(?(x)W, ?(y)W)
? Cf. definitions in (1) and (2)
23:
return (x, y, s)
Similarly to the extension of string kernels proposed in [33], the GST traversal can be performed on
an enhanced suffix array [34] for further run-time and space reduction.
To prove correctness of our algorithm, a different approach must be taken than the one in [24]. We
cannot claim that the computed similarity value is equivalent to the one returned by the matching
statistic algorithm, since the latter is restricted to kernel functions. Instead we show that at each
recursive call to the M ATCH function correct numbers of occurences are maintained.
Theorem 1. A word w occurs occ(w, x) and occ(w, y) times in x and y if and only if M ATCH(w)
?
returns x = occ(w, x) and y = occ(w, y), where w
? is the node at the end of a path from the root
reassembling w in the generalized suffix tree of x and y .
Proof. If w occurs m times in x, there exist exactly m suffixes of x with w as prefix. Since w
corresponds to a path from the root of the GST to a node w
? all m suffixes must pass w.
? Due to
the unique delimiters # each suffix of x corresponds to one leaf node in the GST whose incoming
edge contains #. Hence m equals occ(w, x) and is exactly the aggregated quantity x returned by
M ATCH(w).
? Likewise, occ(w, y) is the number of suffixes beginning after # and having a prefix w,
which is computed by y.
4 Experimental Results
4.1 Run-time experiments
In order to illustrate the efficiency of the proposed algorithm, we conducted run-time experiments on
three benchmark data sets for sequential data: network connection payloads from the DARPA 1999
IDS evaluation [35], news articles from the Reuters-21578 data set [36] and DNA sequences from
the human genome [14]. Table 4 gives an overview of the data sets and their specific properties.
We compared the run-time of the generalized suffix tree algorithm with a recent trie-based method
supporting computation of distances. Tries yield better or equal run-time complexity for computation of similarity measures over k-grams than algorithms using indexed arrays and hash tables. A
detailed description of the trie-based approach is given in [25]. Note that in all of the following
experiments tries were generated in a pre-processing step and the reported run-time corresponds to
the comparison procedure only.
For each of the three data sets, we implemented the following experimental protocol: the Manhattan
distances were calculated for 1000 pairs of randomly selected sequences using k-grams as an em-
Name
DNA
NIDS
TEXT
Type
Human genome sequences
TCP connection payloads
Reuters Newswire articles
Alphabet
4
108
93
Min. length
2400
53
43
Max. length
2400
132753
10002
Table 4: Sequential data sets
bedding language. The procedure was repeated 10 times for various values of k, and the run-time
was averaged over all runs. Fig. 2 compares the run-time of sequence comparison algorithms using
the generalized suffix trees and tries. On all three data sets the trie-based comparison has a low
run-time for small values of n but grows linearly with k. The algorithm using a generalized suffix
tree is independent from complexity of the embedding language, although this comes at a price of
higher constants due to a more complex data structure. It is obvious that a generalized suffix tree is
the algorithm of choice for higher values of k.
8
6
4
2
0
5
10
15
k?gram length
(a) NIDS data set
Manhattan distance runtime (TEXT dataset)
20
5
Trie comparison
GST comparison
4
3
2
1
0
Manhattan distance runtime (DNA dataset)
mean runtime per 1000 cmp. (s)
10
Trie comparison
GST comparison
mean runtime per 1000 cmp. (s)
mean runtime per 1000 cmp. (s)
Manhattan distance runtime (NIDS dataset)
Trie comparison
GST comparison
12
10
8
6
4
2
5
10
15
k?gram length
(b) TEXT data set
20
0
5
10
15
k?gram length
20
(c) DNA data set
Figure 2: Run-time performance for varying k-gram lengths
4.2 Applications
As a second part of our evaluation, we show that the ability of our approach to compute diverse similarity measures pays off when it comes to real applications, especially in an unsupervised learning
scenario. The experiments were performed for (a) intrusion detection in real network traffic and (b)
transcription start site (TSS) recognition in DNA sequences.
For the first application, network data was generated by members of our laboratory using virtual
network servers. Recent attacks were injected by a penetration-testing expert. The distance-based
anomaly detection method Zeta [17] was applied to 5-grams extracted from byte sequences of TCP
connections using different similarity measures: the linear kernel, the Manhattan distance and the
Kulczynski coefficient. The results on network data from the HTTP protocol are shown in Fig. 3(a).
Application of the Kulczynski coefficient yields the highest detection accuracy. Over 78% of all
attacks are identified with no false-positives in an unsupervised setup. In comparison, the linear
kernel yields roughly 30% lower detection rates.
The second application focused on TSS recognition in DNA sequences. The data set comprises fixed
length DNA sequences that either cover the TSS of protein coding genes or have been extracted randomly from the interior of genes [14]. We evaluated three methods on this data: an unsupervised
k-nearest neighbor (kNN) classifier, a supervised and bagged kNN classifier and a Support Vector Machine (SVM). Each method was trained and tested using a linear kernel and the Manhattan
distance as a similarity measure over 4-grams. Fig. 3(b) shows the performance achieved by the
unsupervised and supervised versions of the kNN classifier1 . Even though the linear kernel and
the Manhattan distance yield similar accuracy in a supervised setup, their performance differs significantly in unsupervised application. In the absence of prior knowledge of labels the Manhattan
1
Results for the SVM are similar to the supervised kNN and have been omitted.
ROC for transcription site recognition
ROC for intrusion detection in HTTP
1
1
0.8
true positive rate
true positive rate
0.8
0.6
0.4
0.2
0
Linear kernel (unsup. knn)
Linear kernel (sup. knn)
Manhattan distance (unsup. knn)
Manhattan distance (sup. knn)
Kulczynski coefficient (unsup. knn)
Linear kernel (unsup. knn)
Manhattan distance (unsup. knn)
0
0.002
0.004
0.006
false positive rate
0.008
(a) Results for network application
0.01
0.6
0.4
0.2
0
0
0.02
0.04
0.06
false positive rate
0.08
0.1
(b) Results for DNA application
Figure 3: Comparison of similarity measures on the network and DNA data
distance expresses better discriminative properties for TSS recognition than the linear kernel. For the
supervised application the classication performance is bounded for both similarity measures, since
only some discriminative features for TSS recognition are encapsulated in n-gram models [14].
5 Conclusions
Kernel functions for sequences have recently gained strong attention in many applications of machine learning, especially in bioinformatics and natural language processing. In this contribution
we have shown that other similarity measures such as metric distances or non-metric similarity coefficients can be computed with the same run-time complexity as kernel functions. The proposed
algorithm is based on a post-order traversal of a generalized suffix tree of two or more sequences.
During the traversal, the counts of matching and mismatching words from an embedding language
are computed in time linear in sequence length ? regardless of the particular kind of chosen language: words, k-grams or even all consecutive subsequences. By using a generic representation of
the considered similarity measures based on an outer and inner function, the same algorithm can be
applied for various kernel, distance and similarity functions on sequential data.
Our experiments demonstrate that the use of general similarity measures can bring significant improvement to learning accuracy ? in our case observed for unsupervised learning ? and emphasize
importance of further investigation of distance- and similarity-based learning algorithms.
Acknowledgments
The authors gratefully acknowledge the funding from Bundesministerium f?ur Bildung und
Forschung under the project MIND (FKZ 01-SC40A) and would like to thank Klaus-Robert M?uller
and Mikio Braun for fruitful discussions and support.
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2,202 | 3 | 52
Supervised Learning of Probability Distributions
by Neural Networks
Eric B. Baum
Jet Propulsion Laboratory, Pasadena CA 91109
Frank Wilczek t
Department of Physics,Harvard University,Cambridge MA 02138
Abstract:
We propose that the back propagation algorithm for supervised learning can be generalized, put on a satisfactory conceptual
footing, and very likely made more efficient by defining the values of the output and input neurons as probabilities and varying
the synaptic weights in the gradient direction of the log likelihood,
rather than the 'error'.
In the past thirty years many researchers have studied the
question of supervised learning in 'neural'-like networks. Recently
a learning algorithm called 'back propagation H -
4
or the 'general-
ized delta-rule' has been applied to numerous problems including
the mapping of text to phonemes 5 , the diagnosis of illnesses 6 and
the classification of sonar targets 7 ? In these applications, it would
often be natural to consider imperfect, or probabilistic information. We believe that by considering supervised learning from this
slightly larger perspective, one can not only place back propagat Permanent address: Institute for Theoretical Physics, Univer-
sity of California, Santa Barbara CA 93106
? American Institute of Physics 1988
53
tion on a more rigorous and general basis, relating it to other well
studied pattern recognition algorithms, but very likely improve its
performance as well.
The problem of supervised learning is to model some mapping
between input vectors and output vectors presented to us by some
real world phenomena. To be specific, coqsider the question of
medical diagnosis. The input vector corresponds to the symptoms
of the patient; the i-th component is defined to be 1 if symptom i
is present and 0 if symptom i is absent. The output vector corresponds to the illnesses, so that its j-th component is 1 if the j-th
illness is present and 0 otherwise. Given a data base consisting
of a number of diagnosed cases, the goal is to construct (learn) a
mapping which accounts for these examples and can be applied to
diagnose new patients in a reliable way. One could hope, for instance, that such a learning algorithm might yield an expert system
to simulate the performance of doctors. Little expert advice would
be required for its design, which is advantageous both because experts' time is valuable and because experts often have extraodinary
difficulty in describing how they make decisions.
A feedforward neural network implements such a mapping between input vectors and output vectors. Such a network has a set
of input nodes, one or several layers of intermediate nodes, and a
layer of output nodes. The nodes are connected in a forward directed manner, so that the output of a node may be connected to
the inputs of nodes in subsequent layers, but closed loops do not
occur. See figure 1. The output of each node is assumed to be a
bounded semilinear function of its inputs. That is, if
the output of the j-th node and
Wij
Vj
denotes
denotes the weight associated
with the connection of the output of the j-th node to the input of
54
the i-th, then the i-th neuron takes value Vi = g(L,i Wi:jV:j), where
g is a bounded, differentiable function called the activation function. g(x)
= 1/(1 + e- X ),
called the logistic function, is frequently
used. Given a fixed set of weights {Wi:j}, we set the input node
values to equal some input vector, compute the value of the nodes
layer by layer until we compute the output nodes, and so generate
an output vector.
Figure 1: A 5 layer network. Note bottleneck at layer 3.
55
Such networks have been studied because of analogies to neurobiology, because it may be easy to fabricate them in hardware,
and because learning algorithms such as the Perceptron learning
algorithm 8 , Widrow- Hoff9, and backpropagation have been able
to choose weights
Wi,.
that solve interesting problems.
Given a set of input vectors
values
tj,
sr, together with associated target
back propagation attempts to adjust the weights so as
to minimize the error E in achieving these target values, defined as
E
= E EJL = E(tj - oj)2
JL
where
oj
input.
(1)
JL,i
is the output of the j-th node when sJL is presented as
Back propagation starts with randomly chosen
Wi,.
and
then varies in the gradient direction of E until a local minimum
is obtained. Although only a locally optimal set of weights is obtained, in a number of experiments the neural net so generated
has performed surprisingly well not only on the training set but on
subsequent data. 4 -
6
This performance is probably the main reason
for widespread interest in backpropagation.
It seems to us natural, in the context of the medical diagnosis
pro blem, the other real world problems to which backpropagation
has been applied, and indeed in any mapping problem where one
desires to generalize from a limited and noisy set of examples, to
interpret the output vector in probabilistic terms. Such an interpretation is standard in the literature on pattern classification. 1o
Indeed, the examples might even be probabilistic themselves. That
is to say it might not be certain whether symptom i was present
in case /L or not.
Let
sr represent the probability symptom i is present in case
/L, and let
tj
represent the probability disease j ocurred in case
56
fL.
Consider for the moment the case where the
tJ
are 1 or 0,
A
so that the cases are in fact fully diagnosed. Let
Ii (s, 0)
be our
prediction of the probability of disease i given input vector 5, where
{; is some set of parameters determined by our learning algorithm.
In the neural network case, the {; are the connection weights and
Ii (sl' , {Wi.i })
=
oJ.
Now lacking a priori knowledge of good
0, the best one can do
is to choose the parameters {; to maximize the likelihood that the
given set of examples should have occurred. 10 The formula for this
likelihood, p, is immediate:
or
The extension of equation (2), and thus equation (3) to the
case where the f are probabilities, taking values in [0,1]' is straight-
57
forward * 1 and yields
log(p) =
~ [tjlog(Jj (s", 0)) + (1 -
tj)log(1 - Ij (W, 0))]
(4)
p. ,3
Expressions of this sort often arise in physics and information theory and are generally interpreted as an entropy. 11
We may now vary the {O} in the gradient direction of the entropy. The back propagation algorithm generalizes immediately
from minimizing 'Error' or 'Energy' to maximizing entropy or log
likelihood, or indeed any other function of the outputs and the
inputs 12 . Of course it remains true that the gradient can be computed by back propagation with essentially the same number of
computations as are required to compute the output of the network.
A backpropagation algorithm based on log-likelihood is not
only more intuitively appealing than one based on an ad-hoc definition of error, but will make quite different and more accurate
predictions as well. Consider e.g. training the net on an example which it already understands fairly well.
/j(80)
=L
Now, from eqn(l) BE/B/j
Say
tj
= 2?, so using
= 0, and
'Error' as a
* 1 We may see this by constructing an equivalent larger set of
examples with the f taking only values 0 or 1 with the appropriate
frequency. Thus assume the
tj
are rational numbers with denomi-
nator dj and numerator nj and let p
= IIp.,j dj.
What we mean by
the set of examples {tp. : J-t = 1, ... , M} can be represented by con-
ij = 0
for p(J-t- 1) < v < pJ-t and 1 < vmod(dj) < (dj - nj), and ij = 1
sidering a set of N
= Mp examples {ij}
where for each J-t,
otherwise. N ow applying equation (3) gives equation (4), up to an
overall normalization.
58
criterion the net learns very little from this example, whereas, using eqn(3), Blog(p)/B!;j
= 1/(1 -
f), so the net continues to learn
and can in fact converge to predict probabilities near 1. Indeed
because back propagation using the standard 'Error' measure can
not converge to generate outputs of 1 or 0, it has been customary in the literature 4 to round the target values so that a target
of 1 would be presented in the learning algorithm as some ad hoc
number such as .8, whereas a target of 0 would be presented as .2.
In the context of our general discussion it is natural to ask
whether using a feedforward network and varying the weights is in
fact the most effective alternative. Anderson and Abrahams 13 have
discussed this issue from a Bayesian viewpoint. From this point of
view, fitting output to input using normal distributions and varying
the means and covariance matrix may seem to be more logical.
Feedforward networks do however have several advantages for
complex problems. Experience with neural networks has shown the
importance of including hidden units wherein the network can form
an internal representation of the world. If one simply uses normal
distributions, any hidden variables included will simply integrate
out in calculating an output. It will thus be necessary to include at
least third order correlations to implement useful hidden variables.
Unfortunately, the number of possible third order correlations is
very large, so that there may be practical obstacles to such an
approach. Indeed it is well known folklore in curve fitting and
pattern classification that the number of parameters must be small
compared to the size of the data set if any generalization to future
cases is expected. 10
In feedforward nets the question takes a different form. There
can be bottlenecks to information flow. Specifically, if the net is
59
constructed with an intermediate layer which is not bypassed by
any connections (i.e. there are no connections from layers preceding
to layers subsequent), and if furthermore the activation functions
are chosen so that the values of each of the intermediate nodes
tend towards either 1 or 0*2, then this layer serves as a bottleneck
to information flow. No matter how many input nodes, output
nodes, or free parameters there are in the net, the output will be
constrained to take on no more than 21 different patterns, where
I is the number of nodes in the bottleneck layer.
Thus if I is
small, some sort of 'generalization' must occur even if the number
of weights is large. One plausible reason for the success of back
propagation in adequately solving tasks, in spite of the fact that
it finds only local minima, is its ability to vary a large number of
parameters. This freedom may allow back propagation to escape
from many putative traps and to find an acceptable solution.
A good expert system, say for medical diagnosis, should not
only give a diagnosis based on the available information, but should
be able to suggest, in questionable cases, which lab tests might be
performed to clarify matters. Actually back propagation inherently has such a capability. Back propagation involves calculation
of 81og(p)/8wij. This information allows one to compute immediately 81og(p)/8s j . Those input nodes for which this partial derivative is large correspond to important experiments.
In conclusion, we propose that back propagation can be generalized, put on a satisfactory conceptual footing, and very likely
made more efficient, by defining the values of the output and in*2
Alternatively when necessary this can be enforced by adding
an energy term to the log-likelihood to constrain the parameter
variation so that the neuronal values are near either 1 or O.
60
put neurons as probabilities, and replacing the 'Error' by the loglikelihood.
Acknowledgement: E. B. Baum was supported in part by DARPA
through arrangement with NASA and by NSF grant DMB-840649,
802. F. Wilczek was supported in part by NSF grant PHY82-17853
References
(1)Werbos,P, "Beyond Regression: New Tools for Prediction and
Analysis in the Behavioral Sciences" , Harvard University Dissertation (1974)
(2)Parker D. B., "Learning Logic" ,MIT Tech Report TR-47, Center
for Computationl Research in Economics and Management Science,
MIT, 1985
(3)Le Cun, Y., Proceedings of Cognitiva 85,p599-604, Paris (1985)
(4)Rumelhart, D. E., Hinton, G. E., Williams, G. E., "Learning
Internal Representations by Error Propagation", in "Parallel Distributed Processing" , vol 1, eds. Rumelhart, D. E., McClelland, J.
L., MIT Press, Cambridge MA,( 1986)
(5)Sejnowski, T. J., Rosenberg, C. R., Complex Systems, v 1, pp
145-168 (1987)
(6)LeCun, Y., Address at 1987 Snowbird Conference on Neural
Networks
(7)Gorman, P., Sejnowski, T. J., "Learned Classification of Sonar
Targets Using a Massively Parallel Network", in "Workshop on
Neural Network Devices and Applications", JPLD-4406, (1987)
pp224-237
(8)Rosenblatt, F., "Principles of Neurodynamics: Perceptrons and
61
the theory of brain mechanisms", Spartan Books, Washington DC
(1962)
(9)Widrow, B., Hoff, M. E., 1960 IRE WESCON Cony. Record,
Part 4, 96-104 (1960)
(10)Duda, R. 0., Hart, P. E., "Pattern Classification and Scene
Analysis", John Wiley and Sons, N.Y., (1973)
(11)Guiasu, S., "Information Theory with Applications", McGraw
Hill, NY, (1977)
(12)Baum,E.B., "Generalizing Back Propagation to Computation" ,
in "Neural Networks for Computing", AlP Conf. Proc. 151, Snowbird UT (1986)pp47-53
(13)Anderson, C.H., Abrahams, E., "The Bayes Connection" , Proceedings of the IEEE International Conference on Neural N etwor ks,
San Diego,(1987)
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2,203 | 30 | 794
A 'Neural' Network that Learns to Play Backgammon
G. Tesauro
Center for Complex Systems Research, University of Illinois
at Urbana-Champaign, 508 S. Sixth St., Champaign, IL 61820
T. J. Sejnowski
Biophysics Dept., Johns Hopkins University, Baltimore, MD 21218
ABSTRACT
We describe a class of connectionist networks that have learned to play backgammon at an intermediate-to-advanced level. TIle networks were trained by a
supervised learning procedure on a large set of sample positions evaluated by a
human expert. In actual match play against humans and conventional computer
programs, the networks demonstrate substantial ability to generalize on the basis of
expert knowledge. Our study touches on some of the most important issues in network learning theory, including the development of efficient coding schemes and
training procedures, scaling, generalization, the use of real-valued inputs and outputs, and techniques for escaping from local minima. Practical applications in
games and other domains are also discussed.
INTRODUCTION
A potentially quite useful testing ground for studying issues of knowledge representation and
learning in networks can be found in the domain of game playing. Board games such as chess, go,
backgammon, and Othello entail considerable sophistication and complexity at the advanced level,
and mastery of expert concepts and strategies often takes years of intense study and practice for
humans. However, the complexities in board games are embedded in relatively "clean" structured
tasks with well-defined rules of play, and well-defined criteria for success and failure. This makes
them amenable to automated play, and in fact most of these games have been exten')ively studied
with conventional computer science techniques. Thus, direct comparisons of the results of network
learning can be made with more conventional approaches.
In this paper, we describe an application of network learning to the game of backgammon.
Backgammon is a difficult board game which appears to be well-suited to neural networks, because
the way in which moves are selected is primarily on the basis of pattern-recognition or "judgemental" reasoning, as opposed to explicit "look-ahead," or tree-search computations. This is due to
the probabilistic dice rolls in backgammon, which greatly expand the branching factor at each ply in
the search (to over 400 in typical positions).
Our learning procedure is a supervised one 1 that requires a database of positions and moves
that have been evaluated by an expert "teacher." In contrast, in an unsupervised procedure 2-4
learning would be based on the consequences of a given move (e.g., whether it led to a won or lost
position), and explicit teacher instructions would not be required. However, unsupervised learning
procedures thus far have been much less efficient at reaching high levels of performance than supervised learning procedures. In part, this advantage of supervised learning can be traced to the higher
? American Institute of Physics 1988
795
quantity and quality of information available from the teacher.
Studying a problem of the scale and complexity of backgammon leads one to confront important general issues in network learning. Amongst the most important are scaling and generalization.
Most of the problems that have been examined with connectionist learning algorithms are relatively
small scale and it is not known how well they will perform on much larger problems. Generalization
is a key issue in learning to play backgammon since it is estimated that there are 1020 possible board
positions, which is far in excess of the number of examples that can be provided during training. In
this respect our study is the most severe test of generalization in any connectionist network to date.
We have also identified in this study a novel set of special techniques for training the network
which were necessary to achieve good performance. A training set based on naturally occurring or
random examples was not sufficient to bring the network to an advanced level of performance.
Intelligent data-base design was necessary. Performance also improved when noise was added to
the training procedure under some circumstances. Perhaps the most important factor in the success
of the network was the method of encoding the input information. The best perfonnance was
achieved when the raw input infonnation was encoded in a conceptually significant way, and a certain number of pre-computed features were added to the raw infonnation. These lessons may also
be useful when connectionist learning algorithms are applied to other difficult large-scale problems.
NElWORK AND DATA BASE SET-UP
Our network is trained to select moves (i.e. to produce a real-valued score for any given
move), rather than to generate them. This avoids the difficulties of having to teach the network the
concept of move legality. Instead, we envision our network operating in tandem with a preprocessor which would take the board position and roll as input, and produce all legal moves as output. The network would be trained to score each move, and the system would choose the move with
the highest network score. Furthermore, the network is trained to produce relative scores for each
move, rather than an absolute evaluation of each final position. This approach would have greater
sensitivity in distinguishing between close alternatives, and corresponds more closely to the way
humans actually evaluate moves.
The current data base contains a totaJ of 3202 board positions, taken from various sources 5?
For each position there is a dice roll and a set of legal moves of that roll from that pOSition. The
moves receive commentary from a human expert in the form of a relative score in the range [100,+100), with +100 representing the best possible move and -100 representing the worst possible
move. One of us (G.T.) is a strong backgammon player, and played the role of human expert in
entering these scores. Most of the moves in the data base were not scored, because it is not feasible
for a human expert to comment on all possible moves. (The handling of these unscored lines of
data in the training procedure will be discussed in the following section.)
An important result of our study is that in order to achieve the best perfonnance, the data base
of examples must be intelligently designed, rather than haphazardly accumulated. If one simply
accumulates positions which occur in actual game play, for example, one will find that certain principles of play will appear over and over again in these positions, while other important principles
may be used only rarely. This causes problems for the network, as it tends to "overlearn" the commonly used principles, and not learn at aJl the rarely used principles. Hence it is necessary to have
both an intelligent selection mechanism to reduce the number of over-represented situations, and an
intelligent design mechanism to enhance the number of examples which illustrate under-represented
situations. This process is described in more detail elsewhere 5 .
We use a detenninistic, feed-forward network with an input layer, an output layer, and either
one or two layers of hidden units, with full connectivity between adjacent layers. (We have tried a
number of experiments with restricted receptive fields, and generally have not found them to be useful.) Since the desired output of the network is a single real value, only one output unit is required.
796
TIle coding of the input patterns is probably the most difficult and most important design
issue. In its current configuration the input layer contains 459 input units. A location-based
representation scheme is used, in which a certain number of input units are assigned to each of the
26 locations (24 basic plus White and Black bar) on the board. TIle input is inverted if necessary so
that the network always sees a problem in which White is to play.
An example of the coding scheme used until very recently is shown in Fig. I. This is essentially a unary encoding of the number of men at each board location, with a few exceptions as indicated in the diagram. This representation scheme worked fairly well, but had one peculiar problem
in that after training, the network tended to prefer piling large numbers of men on certain points, in
particular White's 5 point (the 20 point in the 1-24 numbering scheme). Fig. 2 illustrates an example
of this peculiar behavior. In this position White is to play 5-1. Most humans would play 4-5,4-9 in
this position; however, the network chose the move 4-9,19-20. This is actually a bad move, because
it reduces White's chances of making further points in his inner board. The fault lies not with the
data base used to train the network, but rather with the representation scheme used. In Fig. I a,
notice that unit 12 is turned on whenever the final position is a point, and the number of men is different from the initial position. For the 20 point in particular, this unit will develop strong excitatory
weights due to cases in which the initial position is not a point (i.e., the move makes the point). The
20 point is such a valuable point to make that the excitation produced by turning unit 12 on might
overwhelm the inhibition produced by the poor distribution of builders.
(0)
( b)
~-5
-4 -3
oI 02
~-5
~-2
-I
000
3
4
5
-4 -3 5-2 -I
oI DOD
0
234 5
I
~2
o6 ?7
I
~2
3
4
~5
0 0 0
8
9 10
3
4 >5
o6 7? 8DOn
910
o
I
oII 120
o
It
~2
3
4 ~5
? ? 0 0
13 14 15 16
I~ ~2t 2~
oII 12000
13 14
3
4
~5
0 .00
15 16 17 18
Figure 1-- Two schemes used to encode the raw position infonnation in the network's input.
illustrated in each case is the encoding of two White men p~sent befo~ the move, and three
White men p~Jent after the move. (a) An essentiaUy unary coding of the number of men at a
particular board location. Units 1-10 encode the initial position, units 11-16 encode the final
position if the~ has been a change from the initial position. Units are tumed on in the cases
indicated on top of each unit, e.g., unit 1 is turned on if the~ are 5 or more Black men p~sent,
etc.. (b) A superior coding scheme with more units u~ed to characterize the type of transition
from initial to final position. An up arrow indicates an increase in the number of men. a down
arrow indicates a decrease. Units 11-15 have conceptual interpretations: l1="dearing."
12="slotting," 13="b~aking," 14="making," 15="stripping" a point.
797
12 11 10 9
8
7
6
5
4
321
DO
13 14 15 16 17 18
19 20 21 22 23 24
Figure 2-- A sample position illustrating a defect of the coding scheme in Fig. 1a. White is to
play 5-1. With coding scheme (1a). the network prefers 4-9. 19-20. With coding scheme (lb).
the network prefers 4-9. 4-5. The graphic display was generated on a Sun Microsystems
workstation using the Garnmontool program.
In conceptual tenns, humans would say that unit 12 participates in the representation of two
different concepts: the concept of making a point, and the concept of changing the number of men
occupying a made point. These two concepts are unrelated, and there is no point in representing
them with a common input unit. A superior representation scheme in which these concepts are
separated is shown in Fig. 1b: In this representation unit 13 is turned on only for moves which
make the point. Other moves which change the number of men on an already-made point do not
activate unit 13, and thus do not receive any undeserved excitation. With this representation
scheme the network no longer tends to pile large numbers of men on certain points, and its overall
perfonnance is significantly better.
In addition to this representation of the raw board position, we also utilize a number of input
units to represent certain "pre-computed" features of the raw input. The principal goal of this
study has been to investigate network learning, rather than simply to obtain high perfonnance, and
thus we have resisted the temptation of including sophisticated hand-crafted features in the input
encoding. However, we have found that a few simple features are needed in practice to obtain
minimal standards of competent play. With only' 'raw" board infonnation, the order of the desired
computation (as defined by Minsky and Papert6 ) is probably quite high, and the number of examples
needed to learn such a difficult computation might be intractably large. By giving the network
"hints" in the fonn of pre-computed features, this reduces the order of the computation, and thus
might make more of the problem learnable in a tractable number of examples.
798
TRAINING AND TESTING PROCEDURES
To train the network, we have used the standard "back-propagation" learning algorithm 7-9
for modifying the connections in a multilayer feed-forward network. (A detailed discussion of
learning parameters, etc., is provided elsewhere s.) However, our procedure differs from the standard procedure due to the necessity of dealing with the large number of uncommented moves in the
data base. One solution would be simply to avoid presenting these moves to the network. However,
this would limit the variety of input patterns presented to the network in training, and certain types
of inputs probably would be eliminated completely. TIle alternative procedure which we have
adopted is to skip the uncommented moves most of the time (75% for ordinary rolls and 92% for
double rolls), and the remainder of the time present the pattern to the network and generate a random teacher signal with a slight negative bias. This makes sense, because if a move has not received
comment by the human expert, it is more likely to be a bad move than a good move. The random
teacher signal is chosen uniformly from the interval [-65,+35].
We have used the following four measures to assess the network's performance after it has
been trained: (i) performance on the training data, (ii) perfonnance on a set of test data (1000 positions) which was not used to train the network, (iii) perfonnance in actual game play against a conventional computer program (the program Gammontool of Sun Microsystems Inc.), and (iv) performance in game play against a human expert (G.T.). In the first two measures, we define the performance as the fraction of positions in which the network picks the correct move, i.e., those positions
for which the move scored highest by the network agrees with the choice of the human expert. In
the latter two measures, the perfonnance is defined simply as the fraction of games won, without
considering the complications of counting gammons or backgammons.
QUANTITATIVE RESULTS
A summary of our numerical results as measured by perfonnance on the training set and
against Gammontool is presented in Table 1. The best network that we have produced so far
appears to defeat Gammontool nearly 60% of the time. Using this as a benchmark, we find that the
most serious decrease in performance occurs by removing aU pre-computed features from the input
coding. This produces a network which wins at most about 41 % of the time. 'The next most important effect is the removal of noise from the training procedure; this results in a network which wins
45% of the time. Next in importance is the presence of hidden units; a network without hidden units
wins about 50% of the games against Gammontool. In contrast, effects such as varying the exact
number of hidden units, the number of layers, or the size of the training set, results in only a few
(1-3) percentage point decrease in the number of games won.
Also included in Table 1 is the result of an interesting experiment in which we removed our
usual set of pre-computed features and substituted instead the individual tenns of the Gammontool
evaluation function. We found that the resulting network, after being trained on our expert training
set, was able to defeat the Gammontool program by a small margin of 54 to 46 percent. 'The purpose
of this experiment was to provide evidence of the usefulness of network learning as an adjunct to
standard AI techniques for hand-crafting evaluation functions. Given a set of features to be used in
an evaluation function which have been designed, for example; by interviewing a human expert, the
problem remains as to how to "tune" these features, i.e., the relative weightings to associate to
each feature, and at an advanced level, the context in which each feature is relevant. Little is known
in general about how to approach this problem, and often the human programmer must resort to
painstaking trial-and-error tuning by hand. We claim that network learning is a powerful, generaJpurpose, automated method of approaching this problem, and has the potentiaJ to produce a tuning
which is superior to those produced by humans, given a data base of sufficiently high quality, and a
suitable scheme for encoding the features. The result of our experiment provides evidence to support this claim, although it is not firmly established since we do not have highly accurate statistics,
and we do not know how much human effort went into the tuning of the Gammontool evaluation
799
function. More conclusive evidence would be provided if the experiment were repeated with a more
sophisticated program such as Berliner's BKO lO, and similar results were obtained.
Network
sIZe
Training
cycles
Perf. on
test set
Perf. va.
Ga.mmontool
(a) 459-24-24-1
(b) 459-24-1
(c) 459-24-1
(d) 459-12-1
20
22
24
10
.540
.542
.518
.538
.59
.57
.58
.54
?
?
?
?
.03
.05
.05
.05
(e) 410-24-12-1
(f) 459-1
(g) 459-24-12-1
(h) 393-24-12-1
16
22
10
12
.493
.485
.499
.488
.54
.50
.45
.41
?
?
?
?
.03
.03
.03
.02
Comments
1600 posn. D.B .
Gammontool features
No hidden units
No training noise
No features
Table 1-- Summary of perfonnance statistics for various networks. (a) The best network we
have produced. containing two layers of hidden units. with 24 units in each layer. (b) A
network with only one layer of 24 hidden units. (c) A network with 24 hidden units in a single
layer, trained on a training set half the nonnal size. (d) A network with half the number of
hidden units as in (b). (e) A network with features from the Gammontool evaluation function
substituted for the nonnal features. (f) A network without hidden units. (g) A network trained
with no noise in the training procedure. (h) A network with only a raw board description as
input.
QUALITATIVE RESULTS
Analysis of the weights produced by training a network is an exceedingly difficult problem,
which we have only been able to approach qualitatively. In Fig. 3 we present a diagram showing the
connection strengths in a network with 651 input units and no hidden units. lbe figure shows the
weights from each input unit to the output unit. (For purposes of illustration, we have shown a coding scheme with more units than nonnal to explicitly represent the transition from initial to final
position.) Since the weights go directly to the output, the corresponding input units can be clearly
interpreted as having either an overall excitatory or inhibitory effect on the score produced by the
network.
A great deal of columnar structure is apparent in Fig. 3. This indicates that the network has
learned that a particular number of men at a given location, or a particular type of transition at a
given location, is either good or bad independent of the exact location on the board where it occurs.
Furthennore, we can see the importance of each of the pre-computed features in the input coding.
The most significant features seem to be the number of points made in the network's inner board,
and the total blot exposure.
800
features {
roll
{ ........
bar
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
ABC 0 E F G H I J K L M N 0 P Q R STU V W
Figure 3-- A Hinton diagram for a network with 651 input units and no hidden units. Small
squares indicate weights from a particular input unit to the output unit. White squares indicate
positive weights, and black squares indicate negative weights. Size of square indicates
24 rows from bottom up indicate raw board infonnation. Letting
magnitude of weight.
x=number of men before the move and y=number of men after the move, the interpretations of
columns are as follows: A: x<=-5; B: x=-4; C: x=-3; D: x<=-2; E: x=-I: F: x=l: G: x>=2; H:
x=3: I: x=4: J: x>=5: K: x<1 & y=l; L: x<2 & y>=2: M: x<3 & y=3: N: x<4 & y=4: 0: x<y &
y>=5: P: x=1 & y=O; Q: x>=2 & y=O; R: x>=2 & y=l: S: x>=3 & y=2: T: x>=4 & y=3: U:
x>=5 & y=4: V: x>y & y>=5: W: prob. of a White blot at thi~ location being hit (precomputed feature). The next row encodes number of men on White and Blnck bar~. The next
3 rows encode roll infonnation. Remaining rows encode various pre-computed feature~.
rust
Much insight into the basis for the network's judgement of various moves has been gained by
actually playing games against it. In fact, one of the most revealing tests of what the network has
and has not learned came from a 20-game match played by G.T. against one of the latest generation
of networks with 48 hidden units. (A detailed description of the match is given in Ref. II.) The
surprising result of this match was that the network actually won, 11 games to 9. However, a
801
detailed analysis of the moves played by the network during the match indicates that the network
was extremely lucky to have won so many games, and could not reasonably be expected to continue
to do so well over a large number of games. Out of the 20 games played, there were 11 in which
the network did not make any serious mistakes. The network won 6 out of these 11 games, a result
which is quite reasonable. However, in 9 of the 20 games, the network made one or more serious
(i.e. potentially fatal) "blunders." The seriousness of these mistakes would be equivalently to dropping a piece in chess. Such a mistake is nearly always fatal in chess against a good opponent; however in backgammon there are still chances due to the element of luck involved. In the 9 games in
which the network blundered, it did manage to survive and win 5 of the games due to the element of
luck. (We are assuming that the mistakes made by the human, if any, were only minor mistakes.) It
is highly unlikely that this sort of result would be repeated. A much more likely result would be that
the network would win only one or two of the games in which it made a serious error. This would
put the network's expected perfonnance against expert or near-expert humans at about the 35-40%
level. (This has also been confinned in play against other networks.)
We find that the network does act as if it has picked up many of the global concepts and strategies of advanced play. The network has also learned many important tactical elements of play at
the advanced level. As for the specific kinds of mistakes made by the network, we find that they are
not at all random, senseless mistakes, but instead fall into clear, well-defined conceptual categories,
and furthennore, one can understand the reasons why these categories of mistakes are made. We do
not have space here to describe these in detail, and refer the reader instead to Ref. 5.
To summarize, qualitative analysis of the network's play indicates that it has learned many
important strategies and tactics of advanced backgammon. This gives the network very good overall
perfonnance in typical positions. However, the network's worst case perfonnance leaves a great
deal to be desired. The network is capable of making both serious, obvious, "blunders," as well
more subtle mistakes, in many different types of positions. Worst case perfonnance is important,
because the network must make long sequences of moves throughout the course of a game without
any serious mistakes in order to have a reasonable chance of winning against a skilled opponent.
The prospects for improving the network's worst case perfonnance appear to be mixed. It seems
quite likely that many of the current "blunders" can be fixed with a reasonable number of handcrafted examples added to the training set. However, many of the subtle mistakes are due to a lack
of very sophisticated knowledge, such as the notion of timing. It is difficult to imagine that this kind
of knowledge could be imparted to the network in only a few examples. Probably what is required is
either an intractably large number of examples, or a major overhaul in either the pre-computed
features or the training paradigm.
DISCUSSION
We have seen from both quantitative and qualitative measures that the network has learned a
great deal about the general principles of backgammon play, and has not simply memorized the
individual positions in the training set. Quantitatively, the measure of game perfonnace provides a
clear indication of the network's ability to generalize, because apart from the first couple of moves
at the start of each game, the network must operate entirely on generalization. Qualitatively, one can
see after playing several games against the network that there are certain characteristic kinds of
positions in which it does well, and other kinds of positions in which it systematically makes welldefined types of mistakes. Due to the network's frequent "blunders," its overall level of play is
only intennediate level, although it probably is somewhat better than the average intennediate-Ievel
player. Against the intennediate-level program Gammontool, our best network wins almost 60% of
the games. However, against a human expert the network would only win about 35-40% of the time.
Thus while the network does not play at expert level, it is sufficiently good to give an expert a hard
time, and with luck in its favor can actually win a match to a small number of games.
Our simple supervised learning approach leaves out some very important sources of
802
infonnation which are readily available to humans. 1be network is never told that the underlying
topological structure of its input space really corresponds to a one-dimensional spatial structure; all
it knows is that the inputs form a 459-dimensional hypercube. It has no idea of the object of the
game, nor of the sense of temporal causality, i.e. the notion that its actions have consequences, and
how those consequences lead to the achievement of the objective. The teacher signal only says
whether a given move is good or bad, without giving any indication as to what the teacher's reasons
are for making such a judgement. Finally, the network is only capable of scoring single moves in
isolation, without any idea of what other moves are available. 1bese sources of knowledge are
essential to the ability of humans to play backgammon well, and it seems likely that some way of
incorporating them into the network learning paradigm will be necessary in order to achieve further
substantial improvements in performance.
111ere are a number of ways in which these additional sources of knowledge might be incorporated, and we shall be exploring some of them in future work. For example, knowledge of alternative moves could be introduced by defining a more sophisticated error signal which takes into
account not only the network and teacher scores for the current move, but also the network and
teacher scores for other moves from the same position. However, the more immediate plans involve
a continuation of the existing strategies of hand-crafting examples and coding scheme modifications
to eliminate the most serious errors in the network's play. If these errors can be eliminated, and we
are confident that this can be achieved, then the network would become substantially better than any
commercially available program, and would be a serious challenge for human experts. We would
expect 65% performance against Gammontool, and 45% performance against human experts.
Some of the results of our study have implications beyond backgammon to more general
classes of difficult problems. One of the limitations we have found is that substantial human effort
is required both in the design of the coding scheme and in the design of the training set. It is not
sufficient to use a simple coding scheme and random training patterns, and let the automated network learning procedure take care of everything else. We expect this to be generally true when
connectionist learning is applied to difficult problem domains.
On the positive side, we foresee a potential for combining connectionist learning techniques
with conventional AI techniques for hand-crafting knowledge to make significant progress in the
development of intelligent systems. From the practical point of view, network learning can be
viewed as an "enhancer" of traditional techniques, which might produce systems with superior perfonnance. For this particular application, the obvious way to combine the two approaches is in the
use of pre-computed features in the input encoding. Any set of hand-crafted features used in a conventional evaluation function could be encoded as discrete or continuous activity levels of input
units which represent the current board state along with the units representing the raw information.
Given a suitable encoding scheme for these features, and a training set of sufficient size and quality
(i.e., the scores in the training set should be better than those of the original evaluation function), it
seems possible that the resulting network could outperform the original evaluation function, as evidenced by our experiment with the Gammontool features.
Networlc learning might also hold promise as a means of achieving the long-sought goal of
automated feature discovery2. Our network certainly appears to have learned a great deal of
knowledge from the training set which goes far beyond the amount of knowledge that was explicitJy
encoded in the input features. Some of this knowledge (primarily the lowest level components) is
apparent from the weight diagram when there are no hidden units (Fig. 3). However, much of the
network's knowledge remains inaccessible. What is needed now is a mean'! of disentangling the
novel features discovered by the network from either the patterns of activity in the hidden units, or
from the massive number of connection strengths which characterize the network. This is one our
top priorities for future research, although techniques for such "reverse engineering" of parallel
networlcs are only beginning to be developed l2 .
803
ACKNOWLEDGEMrnNTS
lhis work was inspired by a conference on "Evolution, Games and Learning" held at Los
Alamos National Laboratory, May 20-24, 1985. We thank Sun Microsystems Inc. for providing the
source code for their Gammontool program, Hans Berliner for providing some of the po!>itions used
in the data base, Subutai Ahmad for writing the weight display graphics package, Bill Bogstad for
assistance in programming the back-propagation simulator, and Bartlett Mel, Peter Frey, and Scott
Kirkpatrick for critical reviews of the manuscript. G.T. was supponed in part by the National
Center for Supercomputing Applications. TJ.S. was supponed by a NSF Presidential Young Investigator Award, and by grants from the Seaver Institute and the Lounsbury Foundation.
REFERENCES
1. D. E. Rumelart and J. L. McClelland, eds., Parallel Distributed Processing: Explorations in the
Microstructure o/Cognition, Vols. 1 and 2 (Cambridge: MIT Press, 1986).
2. A. L. Samuel, "Some studies in machine learning using the game of checkers." IBM J. Res.
Dev. 3, 210--229 (1959).
3. J. H. Holland, "Escaping brittleness: the possibilities of general-purpose learning algorithms
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artificial ;ntelligence approach. Vol. II (Los Altos CA: Morgan-Kaufman, 1986).
4. R. S. Sutton, "Learning to predict by the methods of temporal differences," GTE Labs Tech.
Repon TR87-509.1 (1987).
5. G. Tesauro and T. J. Sejnowski, "A parallel network that learns to play backgammon." Univ. of
Illinois at Urbana-Champaign, Center for Complex Systems Research Technical Repon (1987).
6. M. Minsky and S. Papen, Perceptrons (Cambridge: MIT Press, 1969).
7. D. E. Rumelhart, G. E. Hinton, and R. J. Williams, "Learning representations by backpropagating errors." Nature 323,533--536 (1986).
8. Y. Le Cun, "A learning procedure for asymmetric network." Proceedings o/Cognitiva (Par;s)
85,599--604 (1985).
9. D. B. Parker, "Learning-logic." MIT Center for Computational Research in Economics and
Management Science Tech. Repon TR-47 (1985).
10. H. Berliner, "Backgammon computer program beats world champion." Artificial Intelligence
14,205--220 (1980).
11. G. Tesauro, "Neural network defeats creator in backgammon match." Univ. of llIinois at
Urbana-Champaign, Center for Complex Systems Research Technical Repon (1987).
12. C. R. Rosenberg, "Revealing the structure of NETtalk's internal representations." Proceedings
of the Ninth Annual Conference of the Cognitive Science Society (Hillsdale, NJ: Lawrence Erlbaum
Associates, 1987).
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2,204 | 300 | Multi-Layer Perceptrons
with B-SpIine Receptive Field Functions
Stephen H. Lane, Marshall G. Flax, David A. Handelman and JackJ. Gelfand
Human Information Processing Group
Department of Psychology
Princeton University
Princeton, New Jersey 08544
ABSTRACT
Multi-layer perceptrons are often slow to learn nonlinear functions
with complex local structure due to the global nature of their function
approximations. It is shown that standard multi-layer perceptrons are
actually a special case of a more general network formulation that
incorporates B-splines into the node computations. This allows novel
spline network architectures to be developed that can combine the
generalization capabilities and scaling properties of global multi-layer
feedforward networks with the computational efficiency and learning
speed of local computational paradigms. Simulation results are
presented for the well known spiral problem of Weiland and of Lang
and Witbrock to show the effectiveness of the Spline Net approach.
1.
INTRODUCTION
Recently, it has been shown that multi-layer feedforward neural networks, such as
Multi-Layer Perceptrons (MLPs) , are theoretically capable of representing arbitrary
mappings, provided that a sufficient number of units are included in the hidden layers
(Hornik et aI., 1989). Since all network weights are updated with each training
exemplar, these networks construct global approximations to multi-input/multi-output
function data in a manner analogous to fitting a low-order polynomial through a set of
684
Multi-Layer Perceptrons with B-Spline Receptive Field Functions
data points. This is illustrated by the cubic polynomial "Global Fit" of the data points
in Fig. 1.
~LocalFit
I
~GlobaiFit
Figure 1. Global vs. Local Function Approximation
Consequently, multi-layer perceptrons are capable of generalizing (extrapolating!
interpolating) their response to regions of the input space where little or no training data
is present, using a quantity of connection weights that typically scales quadratically
with the number of hidden nodes. The global nature of the weight updating, however,
tends to blur the details of local structures, slows the rate of learning, and makes the
accuracy of the resulting function approximation sensitive to the order of presentation
of the training data.
It is well known that many sensorimotor structures in the brain are organized using
neurons that possess locally-tuned overlapping receptive fields (Hubel and Wiesel,
1962). Several neural network computational paradigms such as CMACs (Cerebel1ar
Model Articulation Controllers) (Albus, 1973) and Radial Basis Functions (RBFs)
(Moody and Darken, 1988) have been quite successful representing complex nonlinear
functions using this same organizing principle. These networks construct local
approximations to multi-input/multi-output function data that are analogous to fitting a
least-squares spline through a set of data points using piecewise polynomials or other
basis functions. This is illustrated as the cubic spline "Local Fit" in Fig. 1. The main
benefits of using local approximation techniques to represent complex nonlinear
functions include fast learning and reduced sensitivity to the order of presentation of
training data. In many cases, however, in order to represent the function to the desired
degree of smoothness, the number of basis functions required to adequately span the
input space can scale exponentially with the number of inputs (Lane et aI., 1991a,b).
The work presented in this paper is part of a larger effort (Lane et aI, 1991a) to develop
a general neural network formulation that can combine the generalization capabilities
and scaling properties of global multi-layer feed forward networks with the
computational efficiency and learning speed of local network paradigms. It is shown in
the sequel that this can be accomplished by incorporating B-Spline receptive fields into
the node connection functions of Multi-Layer Perceptrons.
685
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Lane, Flax, Handelman, and Gelfand
2.
MULTI?LAYER PERCEPTRONS
WITH B?SPLINE RECEPTIVE FIELD FUNCTIONS
Standard Multi-Layer Perceptrons (MLPs) can be represented using node equations of
the form,
(1)
where llL is the number of nodes in layer L and the
cf;
are linear connection functions
between nodes in layers Land (L-1) such that,
(2)
0'(-) is the standard sigmoidal nonlinearity, yf-l is the output of a node in layer L-1,
wf; are adjustable network weights.
y~-l = 1, and the
functions are shown in Fig. 2.
Some typical linear connection
cfo corresponds to a threshold input.
~o 1 - - - - -
~2
y' L-1
2
Figure 2. Typical MLP Node Connection Functions
Incorporating B-Spline receptive field functions (Lane et aI., 1991a) into the node
computations of eq. (1) allows more general connection functions (e.g. piecewise linear,
quadratic, cubic, etc.) to be formulated. The corresponding B-Spline MLP (Spline Net)
is derived by redefining the connection functions of eq. (2) such that,
\' W L BG
( L-l)
cijL(YjL-l) = ~
ijk nk Yj
(3)
This enables the construction of a more general neural network architecture that has
node equations of the form,
L
Yi
=
(4)
Multi-Layer Perceptrons with B-Spline Receptive Field Functions
The B~(Yf-1) are B-spline receptive field functions (Lane et al? 1989,19913) of order
n and support G? while the
'wtk
are the spline network weights. The order, n?
corresponds to the number of coefficients in the polynomial pieces. For example, linear
splines are of order n=2? whereas cubic splines are of order n=4. The advantage of the
more general B-Spline connection functions of eq. (3) is that it allows varying degrees
of "locality" to be added to the network computations since network weights are now
yf-1.
wtk
The
are modified by backpropagating the
activated based on the value of
output error only to the G weights in each connection function associated with active
(i.e. nonzero) receptive field functions. The Dh-Iayer weights are updated using the
method of steepest descent learning such that,
L
Wijk
Eo-
L + f3 ejLYiL(I -
W ijk
L)BnkG(YjL-1)
Yi
(5)
where ef is the output error back-propagated to the ith node in layer L and ~ is the
learning rate (Lane et aI., 19913). In the more general Spline Net formulation of eqs.
(3-5), each node input has P+G-1 receptive fields and P+G-1 weights associated with it,
but only G are active at anyone time. P determines the number of partitions in the
input space of the connection functions. Standard MLP networks are a degenerate case
of the Spline Net architecture? as they can be realized with B-Spline receptive field
functions of order n=2, with P=1 and G=2. Due to the connectivity of the B-Spline
receptive field functions, for the case when P> 1, the resulting network architecture
corresponds to multiply-connected MLPs? where any given MLP is active within only
one hypercube in the input space, but has weights that are shared with MLPs on the
neighboring hypercubes. The amount of computation required in each layer of a Spline
Net during both learning and function approximation is proportional to G, and
independent of P.
Formulating the connection functions of eq. (3) with linear (n=2) B-Splines allows
connection functions such as those shown in Fig. 3 to be learned.
Figure 3. Spline Net Connection Functions Using Linear B-Splines (n=2)
The connection functions shown in Fig. 3 have P=4 partitions (5 knots) on the interval
yj-1 E[O,1].
The number of input partitions, P? determines the degree of locality of
687
688
Lane, Flax, Handelman, and Gelfand
the resulting function approximation since the local shape of the connection function is
determined from the current node input activation interval.
Networks constructed using the Spline Net formulation are reminiscent of the form and
function of Kolmogorov-Lorenz networks (Baron and Baron, 1988). A neurobiological
interpretation of a Spline Net is that it is composed of neurons that have dendritic
branches with synapses that operate as a function of the level of activation at a given
node or network input. This is shown in the network architecture of Fig. 4b where the
standard three-layer MLP network of Fig. 4a has been redrawn using B-Spline receptive
field functions with n=2, P=4 and G=2.
~- ,
5
Figure 4. Three-Layer Spline Net Architecture, n=2,P=4,G=2
The horizontal arrows projecting from the right of each network node in Fig. 4b
represent the node outputs. The overlapping triangles on the node output represent the
receptive field functions of neurons in the next layer. These receptive field functions
are summed with weighted connections in the dendritic branches to form the inputs to
the next network layer. In the architecture shown in Fig. 4b, only two receptive fields
are active for any given value of a node output. Therefore for this single hidden-layer
network architecture, given any value for the inputs (x1,xV, at most N w = 30 weights
will be active where,
(6)
s is the number of network inputs and 11 is the number of nodes in the hidden layer,
which for this case is 2s+ 1 =5.
Multi-Layer Perceptrons with B-Spline Receptive Field Functions
3.
SIMULATION RESULTS
In order to evaluate the impact of local computation on MLP performance, the well
known spiral problem of Weiland and of Lang and Witbrock (1988) was chosen as a
benchmark. Simulations were conducted using a Spline Net architecture having one
hidden layer with 5 hidden nodes and linear B-Splines with support, G=2 (Fig. 4). All
trials used the "vanilla" back-prop learning rule of eq. (5) with ~ = l/{2P). The
connection function weights were initialized in each node such that the resulting
connection functions were continuous linear functions with arbitrary slope. From
previous experience (Lane et aI., 1989), it was known that the number of receptive field
partitions can drastically affect network learning and performance. Therefore, the
connection function partitions were bifurcated during training to see the effect on
network generalization capability and learning speed. The bifurcation consisted of
splitting every receptive field in half after increments of lOOK (100,(00) training
points, each time doubling the number of connection function partitions and weights in
the network nodes. A more adaptive approach would monitor the slope of the learning
curve to determine when to split the partitions. New weights were initializing such that
the connection functions before and after the bifurcation retained the same shape. All
simulation results presented in Figs. 5-12 were generated using 800K training points.
The left-most column of Fig. 5 represents the two learned connection functions that
lead to each hidden node depicted in Fig. 4. The elements in the second column are the
hidden node response to excitation over the unit square, while the plots in the third
column are the connection functions from the hidden layer to the output node. The
fourth column shows the hidden node outputs after being passed through their
respective connection functions. The network output shown in the fifth column is the
algebraic sum of the hidden node responses shown in the fourth column. The Spline
Net was initialized as a standard MLP with P=1. Figure 6 shows the evolution of the
two connection functions to the third hidden node in Fig. 4 after every lOOK training
points. Around 400K (P=8) the connection functions start to take on a characteristic
shape. For 1'>8, the creation of additional partitions has little effect on the shape of the
connection functions. Figure 7 shows the associated learning curve, while Fig. 8 is an
enlarged version of the network output. These results indicate that the bifurcation
schedule introduces additional degrees of freedom (weights) to the network in such a
way as to carve out coarse global features first, then incrementally capture finer and
finer localized details later. This is in contrast to the results shown in Figs. 9 and 10
where the training (using the same 800K points as in Figs. 7 and 8) was begun on a
network having P=l28 initial partitions. Figure 11 shows the Spline Net output after
800K training iterations using 112 discrete points located on the two spirals. Lang and
Witbrock (1988) state that similar spiral results could only be obtained using a MLP
network with 3 hidden layers (including jump connections) and 50,000,000 training
iterations. The use of a Spline Net with a bifurcation schedule enabled the learning to
be sped up by almost two orders of magnitude, indicating there is a significant
performance advantage in trading-off number of hidden layers for node complexity.
689
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Lane, Flax, Handelman, and Gelfand
Hidden Node
Connection
Functions
Hidden Node
Response
Output Node
Connection
Functions
Hidden Node
Outputs After
Connection
Functions
Output Node
Response
Figure 5. Spiral Learning with Bifurcation Schedule
P=l
P=2
P=16
P=32
P=4
P=8
P=128
Figure 6. Evolution of Connection Functions to Third Hidden Node
Multi-Layer Perceptrons with B-Spline Receptive Field Functions
3000
2500
~
-
:
~
\.
2000
CI)
~
\
1500
\....
1000
,
i'--
500
o
o
200
400
600
800
1000
Training Iteration
Figure 8. Output Node Response
with Bifurcation
Figure 7. Learning Curve with Bifurcation Schedule
Mean Square Error vs. Training Iteration
,. ,.
3000
2500
2000
(P=128)
~ "'\.
'" ~ -
1500
-
1000
~
.
500
o
o
200
400
600
800
1000
Figure 9. Learning Curve without Bifurcation Schedule
Mean Square Error vs. Training Iteration
3000
2500
without Bifurcation
~
~
\
2000
\
1500
1000
500
Figure 10. Output Node Response
~
~
;
-"
,e
o
o
:i
'" ~
200
400
~
600
800
1000
Figure 11. Learning Curve with Bifurcation Schedule
Mean Square Error vs. Training Iteration
(112 Discrete Points)
Figure 12. Output Node Response
with Bifurcation
(112 Discrete Points)
691
692
Lane, Flax, Handelman, and Gelfand
4.
CONCLUSIONS
It was shown that the introduction of B-Splines into the node connection functions of
Multi-Layer Perceptrons allows more general neural network architectures to be
developed. The resulting Spline Net architecture combines the fast learning and
computational efficiency of strictly local neural network approaches with the scaling
and generalization properties of the more established global MLP approach. Similarity
to Kolmogorov-Lorenz networks can be used to suggest an initial number of hidden
layer nodes. The number of node connection function partitions chosen affects both
network generalization capability and learning performance. It was shown that use of a
bifurcation schedule to determine the number of node input partitions speeds learning
and improves network generalization. Results indicate that Spline Nets solve difficult
learning problems by trading-off number of hidden layers for node complexity.
Acknowledgements
Stephen H. Lane and David A. Handelman are also employed by Robicon Systems Inc.,
Princeton, NJ. This research has been supported through a grant from the James S.
McDonnell Foundation and a contract from the DARPA Neural Network Program.
References
Albus, J. (1975) "A New Approach to Manipulator Control: The Cerebellar Model
Articulation Controller (CMAC)," 1. Dyn. Sys. Meas. Control, vol. 97, pp. 270-277.
Barron, A.R. and Barron, R.L. (1988) "Statistical Learning Networks: A Unifying
View," Proc. 20th Symp. on the Interface - Computing and Statistics, pp. 192-203.
Hornik, K. Stinchcombe, M. and White, H. (1989) "Multi-layer Feedforward Networks
are Universal Approximators," Neural Networks, vol. 2, pp. 359-366.
Hubel, D. and Wiesel, T.N. (1962) "Receptive Fields, Binocular Interaction and
Functional Architecture in Cat's Visual Cortex," 1. Physiology, vol. 160, no. 106.
Lane, S.H., Handelman, D.A. and Gelfand, JJ. (1989) "Development of Adaptive BSplines Using CMAC Neural Networks", 1989 I1CNN, Wash. DC., June 1989.
Lane, S.H., Flax, M.B., Handelman, D.A. and Gelfand, JJ. (1991a) "Function
Approximation in Multi-Layer Neural Networks with B-Spline Receptive Field
Functions," Princeton University Cognitive Science Lab Report No. 42, in prep for 1. of
Int'l Neural Network Society.
Lane, S.H., Handelman, D.A. and Gelfand, JJ. (1991b) "Higher-Order CMAC Neural
Networks-Theory and Practice," to appear Amer. Contr. Conf., Boston, MA, June,1991.
Lang, K.J. and Witbrock, MJ. (1988) "Learning to Tell Two Spirals Apart," Proc. 1988
Connectionist Model Summer School, D. Touretzky, G. Hinton, and T. Sejnowski, Eds.
Moody, J. and Darken, C. (1988) "Learning with Localized Receptive Fields," Proc.
1988 Connectionist Model Summer School, D. Touretzky, G. Hinton, T.Sejnowski, Eds.
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2,205 | 3,000 | On the Relation Between Low Density Separation, Spectral
Clustering and Graph Cuts
Hariharan Narayanan
Department of Computer Science
University of Chicago
Chicago IL 60637
[email protected]
Mikhail Belkin
Department of Computer Science and Engineering
The Ohio State University
Columbus, OH 43210
[email protected]
Partha Niyogi
Department of Computer Science
University of Chicago
Chicago IL 60637
[email protected]
Abstract
One of the intuitions underlying many graph-based methods for clustering and semi-supervised
learning, is that class or cluster boundaries pass through areas of low probability density. In this
paper we provide some formal analysis of that notion for a probability distribution. We introduce
a notion of weighted boundary volume, which measures the length of the class/cluster boundary
weighted by the density of the underlying probability distribution. We show that sizes of the cuts of
certain commonly used data adjacency graphs converge to this continuous weighted volume of the
boundary.
keywords: Clustering, Semi-Supervised Learning
1
Introduction
Consider the probability distribution with density p(x) depicted in Fig. 1, where darker color denotes higher probability
density. Asked to cluster this probability distribution, we would probably separate it into two roughly Gaussian bumps
as shown in the left panel.
Same intuition applies to semi-supervised learning. Asked to point out more likely groups of data of the same type,
we would be inclined to believe that these two bumps contain data points with the same labels. On the other hand,
the class boundary shown in the right panel seems rather less likely. One way to state this basic intuition is the Low
Density Separation assumption [5], saying that the class/cluster boundary tends to pass through regions of low density.
In this paper we propose a formal measure on the complexity of the boundary, which intuitively corresponds to the
Low Density Separation assumption. We will show that given a class boundary, this measure can be computed from
a finite sample from the probability distribution. Moreover, we show this is done by computing the size of a cut for a
partition of a certain standard adjacency graph, defined on that sample, and point out some interesting connections to
spectral clustering.
To fix our intuitions, let us consider the question of what makes the cut in the left panel more intuitively acceptable
than the cut in the right. Two features of the left cut make it more pleasing: the cut is shorter in length and the cut
Figure 1: A likely cut and a less likely cut.
passes through a low-density area. Note that a very jagged cut through a low-density area or a short cut through the
middle of a high-density bump would be unsatisfactory. It will therefore appear reasonable to take the length of the
cut as a measure of its complexity but weight it depending on the density of the probability distribution p through
which it passes. In other
words, we propose the weighted length of the boundary, represented by the contour integral
R
along the boundary cut p(s)ds to measure the complexity of a cut. It is clear that the boundary in the left panel has a
considerably lower weighted length than the boundary in the right panel of our Fig. 1.
To formalize this notion further consider a (marginal) probability distribution with density p(x) supported on some
domain or manifold M . This domain is partitioned in two disjoint clusters/parts.
Assuming that the boundary S is a
R
smooth hypersurface we define the weighted volume of the cut to be S p(s)ds. Note that just as in the example above,
the integral is taken over the surface of the boundary.
We will show how this quantity can be approximated given empirical data and establish connections with some popular
graph-based methods.
2
2.1
Connections and related work
Spectral Clustering
Over the last two decades there has been considerable interest in various spectral clustering techniques (see, e.g., [6]
for an overview). The idea of spectral clustering can be expressed very simply. Given a graph, we would often like
to construct a balanced partitioning of the vertex set, i.e. a partitioning such which minimizes the number (or total
weight) of edges across the cut. This is generally an NP-hard optimization problem. It turns out, however, that a
simple real-valued relaxation can be used to reduce it to standard linear algebra, typically to finding eigenvectors of a
certain graph Laplacian. We note that the quality of partition is usually measured in terms of the corresponding cut
size.
A critical question, when this notion is applied to general purpose clustering in the context of machine learning is how
to construct the graph given data points. A typical choice here is the Gaussian weights (e.g., [14]). To summarize, a
graph is obtained from a point cloud, using Gaussian or other weights, and partitioned using spectral clustering or a
different algorithm, which attempts to approximate the smallest (balanced) cut.
We note that while the intuition is that spectral clustering is an approximation to the minimum cut, and is closely
related to random walks and diffusions on graphs and the underlying probability distributions ([13, 12]), existing
results on convergence of spectral clustering ([11]) do not provide a formal interpretation of the limiting partition or
connect it to the size of the resulting cut.
2.2
Graph-based semi-supervised learning
Similarly to spectral clustering, graph-based semi-supervised learning constructs a graph from the data. In contrast
to clustering, however, some of the data is labeled. The problem is typically to either label the unlabeled points
(transduction) or, more generally, to build a classifier defined on the whole space. This may be done trying to find the
2
Figure 2: Curves of small and high condition number respectively
minimum cut, which respects the labels of the data directly ([3]), or, using graph Laplacian as a penalty functional
(e.g.,[15, 1]).
One of the important intuitions of semi-supervised learning is the cluster assumption(e.g., [4]) or, more specifically,
the low density separation assumption suggested in [5], which states that the class boundary passes through a low
density region. We argue that this intuition needs to slightly modified by suggesting that cutting through a high density
region may be acceptable as long as the length of the cut is very short. For example imagine two high-density round
clusters connected by a very thin high-density thread. Cutting the thread is appropriate as long as the width of the
thread is much smaller than the radii of the clusters.
2.3
Convergence of Manifold Laplacians
Another closely related line of research is the connections between point-cloud graph Laplacians and Laplace-Beltrami
operators on manifolds, which have been explored recently in [9, 2, 10]. A typical result in that setting shows that for
a fixed function f and points sampled from a probability distribution on a manifold or a domain, the graph Laplacian
applied to f converges to the manifold Laplace-Beltrami operator ?M f . We note that the results of those papers
cannot be directly applied in our situation as for us f is the indicator function of a subset and is not differentiable.
Even more importantly, this paper establishes an explicit connection between the point-cloud Laplacian applied to
such characteristic functions (weighted graph cuts) and a geometric quantity, which is the weighted volume of the cut
boundary. This geometric connection does not easily follow from results of those papers and the techniques used in
the proof of our Theorem 3 are significantly different.
3
Summary of the Main Results
Let p be a probability density function on a domain M ? Rd .
Let S be a smooth hypersurface that separates M into two parts, S1 and S2 . The smoothness of S will be quantified
by a condition number 1/? , where ? is the radius of the largest ball that can be placed tangent to the manifold at any
point, intersecting the manifold at only one point. It bounds the curvature of the manifold.
Definition 1 Let Kt (x, y) be the heat kernel in Rd given by
Kt (x, y) :=
Let Mt := Kt (x, x) =
2
1
e?kx?yk /4t .
(4?t)d/2
1
.
(4?t)d/2
Let X := {x1 , . . . , xN } be a set of N points chosen independently at random from p. Consider the complete graph
whose vertices are associated with the points in X, and where the weight of the edge between xi and xj , i 6= j is given
3
by
Wij = Kt (xi , xj )
Let W be the weight matrix. Let X1 = X ? S1 and X2 = X ? S2 be the data point which land in S1 and S2
respectively. Let D be the diagonal matrix whose entries are row sums of W (degrees of the corresponding vertices)
X
Dii =
Wij
j
The normalized Laplacian associated to the data X (and parameter t) is the matrix L(t, X) := I ? D?1/2 W D?1/2 .
Let f = (f1 , . . . , fN ) be the indicator vector for X1 :
?
1
fi =
0
if xi ? X1
otherwise
There are two quantities of interest:
R
1. S p(s)ds, which measures the quality of the partition S in accordance with the weighted volume of the
boundary.
2. f T Lf , which measures the quality of the empirical partition in terms of its cut size.
Our main Theorem shows that after an appropriate scaling, the empirical cut size converges to the volume of the
boundary.
Theorem 1 Let the number of points, |X| = N tend to infinity and {tN }?
0 , be a sequence of values of t that tend to
zero such that tN > 11 . Then with probability 1,
N 2d+2
lim
?
Z
?
? f T L(tN , X)f =
p(s)ds
N t
S
Further, for any ? ? (0, 1) and any ? ? (0, 1/2), there exists a positive constant C and an integer N0 (depending on ?,
? and certain generic invariants of p and S) such that with probability 1 ? ?, (?N > N0 ),
? ?
?
Z
? ? T
?
? ? f L(tN , X)f ?
? < Ct?N .
p(s)ds
?N t
?
S
?
This theorem is proved by first relating the empirical quantity N ??t f T L(tN , X)f to a heat flow across the relevant cut
(on the continuous domain), and then relating the heat flow to the measure of the cut. In order to state these results,
we need the following notation.
Definition 2 Let
p(x)
?t (x) = qR
M
Let
.
Kt (x, z)p(z)dz
?
?
?(t, X) := ? f T L(t, X)f
N t
and
r
?(t) :=
?
t
Z
Z
Kt (x, y)?t (x)?t (y)dxdy.
S1
S2
Where t and X are clear from context, we shall abbreviate ?(t, X) to ? and ?(t) to ?. In theorem 2 we show that for
a fixed t, as the number of points |X| = N tends to infinity, with probability 1, ?(t, X) tends to ?(t).
In theorem 3 we show that ?(t) can be made arbitrarily close to the weighted volume of the boundary by making t
tend to 0.
4
S
S2
S1
t1/2
Figure 3: Heat flow ? tends to
R
S
p(s)ds
?
Theorem 2 Let 0 < ? < 1. Let u := 1/ t2d+1 N 1?? . Then, there exist positive constants C1 , C2 depending only on
p and S such that with probability greater than 1 ? exp (?C1 N ? )
?
?
d+1
(1)
|?(t, X) ? ?(t)| < C2 1 + t 2 u?(t).
e
Theorem 3 For any ? ? (0, 21 ), there exists a constant C such that for all t such that 0 < t < ? (2d)? e?1 ,
?r Z Z
?
Z
? ?
?
?
Kt (x, y)?t (x)?t (y)dxdy ?
p(s)ds?? < C t? .
? t
S 2 S1
S
(2)
By letting N ? ? and tN ? 0 at suitable rates and putting together theorems 2 and 3, we obtain the following
theorem:
?
Theorem 4 Let the number of random data points N ? ?, and tN ? 0, at rates so that u := 1/ t2d+1 N 1?? ? 0.
Then, for any ? ? (0, 1/2), there exist positive constants C1 , C2 depending only on p and S , such that for any N > 1
with probability greater than 1 ? exp (?C1 (N ? )),
?
?
Z
?
?
?? (tN , X) ?
p(s)ds?? < C2 (t? + u)
(3)
?
S
4
Outline of Proofs
1?2?
. N0 is chosen to
Theorem 1 is a corollary of Theorem 4, obtained by setting u to be t? , and setting ? to 2d+2
bePa large enough integer so that an application of the union bound on all N > N0 , still gives us a probability
exp (?C1 (N ? )) < ?, of the rate of convergence being worse than stated in Theorem 1.
N >N0
Theorem 4 is a direct consequence of Theorem 2 and Theorem 3.
Theorem 2:
We prove theorem 2 using a generalization of McDiarmid?s inequality from [7, 8]. McDiarmid?s inequality asserts
that a function of a large number of independent random variables, that is not very influenced by the value of any one
of these, takes a value close to its mean. In the generalization that we use, it is permitted that over a bad set that has
a small probability mass, the function is highly influenced by some of the random variables. In our setting, it can be
shown that our measure of a cut, f T Lf is such a function of the independent random points in X, and so the result is
applicable. There is another step involved, since the mean ?
of f T Lf is not ?, the quantity to which we wish to prove
convergence. Therefore we need to prove that the mean E[ N ??t f T L(t, X)f ] tends to ?(t) as N tends to infinity. Now,
?
X X
p
?
Kt (x, y)
P
P
? f T L(t, X)f = 1/N ?/t
.
{( z6=x Kt (x, z))( z6=y Kt (y, z))}1/2
N t
x?X y?X
1
2
5
If, instead, we had in the denominator of the right side
vZ
Z
u
u
t p(z)Kt (x, z)dz p(z)Kt (y, z)dz,
M
M
using the linearity of Expectation,
?
?
?
X X
p
?
1
s?
E?
? N (N ? 1) ?/t
R
?
x?X1 y?X2
?
?
Kt (x, y)
?
??
? ? = ?(t).
R
?
p(z)Kt (x, z)dz
p(z)Kt (y, z)dz
M
M
Using Chernoff bounds, we can show that with high probability, for all x ? X,
P
Z
z6=x Kt (x, z)
? p(z)Kt (x, z)dz.
N ?1
M
Putting the last two facts together and using the Generalization of McDiarmid?s inequality from [7, 8], the result
follows. Since the exact details require fairly technical calculations, we leave them to the Journal version.
Theorem 3:
The quantity
r Z Z
?
Kt (x, y)?t (x)?t (y)dxdy
? :=
t
S1 S2
is similar to the heat that would flow in time t from one part to another if the first were heated proportional to p.
Intuitively, the heat that would flow from one part to the otherRin a small interval ought to be related to the volume of
the boundary between these two parts, which in our setting is S p(s)ds. To prove this relationship, we bound ? both
above and below in terms of the weighted volume and condition number of the boundary. These bounds are obtained
by making comparisons with the ?worst case?, given condition number ?1 , which is when S is a sphere of radius ? . In
order to obtain a lower bound on ?, we observe that if B2 is the nearest ball of radius ? contained in S1 to a point P
in S2 that is within ? of S1 ,
Z
Z
Kt (x, P )?t (x)?t (P )dx ? Kt (x, P )?t (x)?t (P )dx,
S1
B2
as in Figure 4.
Similarly, to obtain an upper bound on ?, we observe that if B1 is a ball or radius ? in S2 , tangent to B2 at the point
of S nearest to P ,
Z
Z
Kt (x, P )?t (x)?t (P )dx ?
Kt (x, P )?t (x)?t (P )dx.
B1c
S1
We now indicate how a lower bound is obtained for
Z
Kt (x, P )?t (x)?t (P )dx.
B2
p
A key observation is that for R = 2dt ln(1/t),
R
Kt (x, P )dx ? 1. For this reason, only the portions of B2
kx?P k>R
near P contribute to the the integral
Z
Kt (x, P )?t (x)?t (P )dx.
B2
It turns out that a good lower bound can be obtained by considering the integral over H2 instead, where H2 is a
2
halfspace whose boundary is at a distance ? ? R
2? from the center as in figure 4.
6
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B
B
P
Figure 4: The density of heat diffusing to point P from S1 in the left panel is less or equal to the density of heat
diffusing to P from B2 in the right panel.
H2
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P
P
2?
Figure 5: The density of heat received by point P from B2 in the left panel can be approximated by the density of heat
received by P from the halfspace H2 in the right panel.
7
An upper bound for
Z
Kt (x, P )?t (x)?t (P )dx
B1c
is obtained along similar lines.
The details of this proof will be presented in the Journal version.
5
Conclusion
In this paper we take a step towards a probabilistic analysis of graph based methods for clustering. The nodes of the
graph are identified with data points drawn at random from an underlying probability distribution on a continuous
domain. For a fixed partition we show that the cut-size of the graph partition converges to the weighted volume of
the boundary separating the two regions of the domain. The rates of this convergence are analyzed. If one is able
to generalize our result uniformly over all partitions, this allows us to relate ideas around graph based partitioning to
ideas surrounding Low Density Separation. The most important future direction would be to achieve similar results
uniformly over balanced partitions.
References
[1] M. Belkin and P. Niyogi (2004).?Semi-supervised Learning on Riemannian Manifolds.? In Machine Learning
56, Special Issue on Clustering, 209-239.
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
M. Belkin and P. Niyogi. ?Toward a theoretical foundation for Laplacian-based manifold methods.? COLT 2005.
A. Blum and S. Chawla, ?Learning from labeled and unlabeled data using graph mincuts?, ICML 2001.
O.Chapelle, J. Weston, B. Scholkopf, ?Cluster kernels for semi-supervised learning?, NIPS 2002.
O. Chapelle and A. Zien, ?Semi-supervised Classification by Low Density Separation?, AISTATS 2005.
Chris Ding, Spectral Clustering, ICML 2004 Tutorial.
Samuel Kutin, Partha Niyogi, ?Almost-everywhere Algorithmic Stability and Generalization Error.?, UAI 2002,
275-282
S. Kutin, TR-2002-04, ?Extensions to McDiarmid?s inequality when differences are bounded with high probability.? Technical report TR-2002-04 at the Department of Computer Science, University of Chicago.
S. Lafon, Diffusion Maps and Geodesic Harmonics, Ph. D. Thesis, Yale University, 2004.
M. Hein, J.-Y. Audibert, U. von Luxburg, From Graphs to Manifolds ? Weak and Strong Pointwise Consistency
of Graph Laplacians, COLT 2005.
U. von Luxburg, M. Belkin, O. Bousquet, Consistency of Spectral Clustering, Max Planck Institute for Biological
Cybernetics Technical Report TR 134, 2004.
M. Meila and J. Shi. A Random Walks View of Spectral Segmentation, NIPS 2001.
[13] B. Nadler, S. Lafon, R. R. Coifman,and I. G. Kevrekidis. Diffusion Maps, Spectral Clustering and Eigenfunctions
of Fokker-Planck Operators, NIPS 2006.
[14] J. Shi and J. Malik. ?Normalized cuts and image segmentation.?
[15] X. Zhu, J. Lafferty and Z. Ghahramani, Semi-supervised learning using Gaussian fields and harmonic functions,
ICML 2003.
8
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2,206 | 3,001 | Learning to be Bayesian without Supervision
Martin Raphan
Courant Inst. of Mathematical Sciences
New York University
[email protected]
Eero P. Simoncelli
Center for Neural Science, and
Courant Inst. of Mathematical Sciences
New York University
[email protected]
Bayesian estimators are defined in terms of the posterior distribution. Typically,
this is written as the product of the likelihood function and a prior probability
density, both of which are assumed to be known. But in many situations, the prior
density is not known, and is difficult to learn from data since one does not have
access to uncorrupted samples of the variable being estimated. We show that for a
wide variety of observation models, the Bayes least squares (BLS) estimator may
be formulated without explicit reference to the prior. Specifically, we derive a
direct expression for the estimator, and a related expression for the mean squared
estimation error, both in terms of the density of the observed measurements. Each
of these prior-free formulations allows us to approximate the estimator given a
sufficient amount of observed data. We use the first form to develop practical
nonparametric approximations of BLS estimators for several different observation
processes, and the second form to develop a parametric family of estimators for
use in the additive Gaussian noise case. We examine the empirical performance
of these estimators as a function of the amount of observed data.
1
Introduction
Bayesian methods are widely used throughout engineering for estimating quantities from corrupted
measurements. Those that minimize the mean squared error (known as Bayes least squares, or BLS)
are particularly widespread. These estimators are usually derived assuming explicit knowledge of
the observation process (expressed as the conditional density of the observation given the quantity
to be estimated), and the prior density over that quantity. Despite its appeal, this approach is often
criticized for the reliance on knowledge of the prior distribution, since the true prior is usually
not known, and in many cases one does not have data drawn from this distribution with which to
approximate it. In this case, it must be learned from the same observed measurements that are
available in the estimation problem. In general, learning the prior distribution from the observed
data presents a difficult, if not impossible task, even when the observation process is known. In the
commonly used ?empirical Bayesian? approach [1], one assumes a parametric family of densities,
whose parameters are obtained by fitting the data. This prior is then used to derive an estimator
that may be applied to the data. If the true prior is not a member of the assumed parametric family,
however, such estimators can perform quite poorly.
An estimator may also be obtained in a supervised setting, in which one is provided with many
pairs containing a corrupted observation along with the true value of the quantity to be estimated. In
this case, selecting an estimator is a classic regression problem: find a function that best maps the
observations to the correct values, in a least squares sense. Given a large enough number of training
samples, this function will approach the BLS estimate, and should perform well on new samples
drawn from the same distribution as the training samples. In many real-world situations, however,
one does not have access to such training data.
In this paper, we examine the BLS estimation problem in a setting that lies between the two cases
described above. Specifically, we assume the observation process (but not the prior) is known, and
we assume unsupervised training data, consisting only of corrupted observations (without the correct
values). We show that for many observation processes, the BLS estimator may be written directly in
terms of the observation density. We also show a dual formulation, in which the BLS estimator may
be obtained by minimizing an expression for the mean squared error that is written only in terms of
the observation density. A few special cases of the first formulation appear in the empirical Bayes
literature [2], and of the second formulation in another branch of the statistical literature concerned
with improvement of estimators [3, 4, 5]. Our work serves to unify these prior-free methods within
a linear algebraic framework, and to generalize them to a wider range of cases. We develop practical
nonparametric approximations of estimators for several different observation processes, demonstrating empirically that they converge to the BLS estimator as the amount of observed data increases.
We also develop a parametric family of estimators for use in the additive Gaussian case, and examine
their empirical convergence properties. We expect such BLS estimators, constructed from corrupted
observations without explicit knowledge of, assumptions about, or samples from the prior, to prove
useful in a variety real-world estimation problems faced by machine or biological systems that must
learn from examples.
2
Bayes least squares estimation
Suppose we make an observation, Y , that depends on a hidden variable X, where X and Y may
be scalars or vectors. Given this observation, the BLS estimate of X is simply the conditional
expectation of the posterior density, E{X|Y = y}. If the prior distribution on X is PX , and the
likelihood function is PY |X then this can be written using Bayes? rule as
E{X|Y = y} =
x PX|Y (x|y) dx
(1)
=
x PY |X (y|x) PX (x) dx PY (y) ,
where the denominator is the distribution of the observed data:
PY (y) = PY |X (y|x) PX (x) dx .
(2)
If we know PX and PY |X , we can calculate this explicitly.
Alternatively, if we do not know PX or PY |X , but are given independent identically distributed
(i.i.d.) samples (Xn , Yn ) drawn from the joint distribution of (X, Y ), then we can solve for the
estimator f (y) = E{X|Y = y} nonparametrically, or we could choose a parametric family of
estimators {f? }, and choose ? to minimize the empirical squared error:
N
1
|f? (Yn ) ? Xn |2 .
?? = arg min
? N
n=1
However, in many situations, one does not have access to PX , or to samples drawn from PX .
2.1
Prior-free reformulation of the BLS estimator
In many cases, the BLS estimate may be written without explicit reference to the prior distribution.
We begin by noting that in Eq. (1), the prior appears only in the numerator
N (y) = PY |X (y|x) x PX (x) dx.
This equation may be viewed as a composition of linear transformations of the function PX (x)
N (y) = (A ? X){PX }(y),
where
X{f }(x) = xf (x),
and the operator A computes an inner product with the likelihood function
A{f }(y) = PY |X (y|x) f (x) dx.
Similarly, Eq. (2) may be viewed as the linear transformation A applied to PX (x). If the linear
transformation A is 1-1, and we restrict PY to lie in the range of A, then we can then write the
numerator as a linear transformation on PY alone, without explicit reference to PX :
N (y)
=
=
(A ? X ? A?1 ){PY }(y)
L{PY }(y).
(3)
In the discrete case, PY (y) and N (y) are each vectors, A is a matrix containing PY |X , X is a
diagonal matrix containing values of x, and ? is matrix multiplication.
This allows us to write the BLS estimator as
E{X|Y = y} =
L{PY }(y)
.
PY (y)
(4)
Note that if we wished to calculate E{X n |Y }, then Eq. (3) would be replaced by (A ? Xn ?
A?1 ){PY } = (A ? X ? A?1 )n {PY } = Ln {PY } . By linearity of the conditional expectation,
we may extend this to any polynomial function (and thus to any function that can be approximated
with a polynomial):
M
M
ck Lk {PY }(y)
E
.
ck X k |Y = y = k=?N
PY (y)
k=?N
In the definition of the operator L, A?1 effectively inverts the observation process, recovering PX
from PY . In many situations, this operation will not be well-behaved. For example, in the case
of additive Gaussian noise, A?1 is a deconvolution operation which is inherently unstable for high
frequencies. The usefulness of Eq. (4) comes from the fact that in many cases, the composite operation L may be written explicitly, even when the inverse operation is poorly defined or unstable. In
section 3, we develop examples of operators L for a variety of observation processes.
2.2
Prior-free reformulation of the mean squared error
In some cases, developing a stable nonparametric approximation of the ratio in Eq. (4) may be difficult. However, the linear operator formulation of the BLS estimator also leads to a dual expression
for the mean squared error that does not depend explicitly on the prior, and this may be used to select
an optimal estimator from a parametric family of estimators. Specifically, for any estimator f? (Y )
parameterized by ?, the mean squared error may be decomposed into two orthogonal terms:
2
2
2
E |f? (Y ) ? X| = E |f? (Y ) ? E(X|Y )| + E |E(X|Y ) ? X| .
The second term is the minimum possible MSE, obtained when using the optimal estimator. Since
it does not depend on f? , it is irrelevant for optimizing ?. The first term may be expanded as
2
2
2
E |f? (Y ) ? E(X|Y )| = E |f? (Y )| ? 2f? (Y )E(X|Y ) + E |E(X|Y )| .
Again, the second expectation does not depend on f? . Using the prior-free formulation of the previous section, the second component of the first expectation may be written as
L{PY }(Y )
E {f? (Y )E(X|Y )} = E f? (Y )
PY (Y )
L{PY }(y)
PY (y)dy
=
f? (y)
PY (y)
=
f? (y) L{PY }(y)dy
=
L? {f? }(y)PY (y)dy
= E {L? {f? }(Y )} ,
where L? is the dual operator of L (in the discrete case, L? is the matrix transpose of L). Combining
all of the above, we have:
2
2
arg min E |f? (Y ) ? X| = arg min E |f? (Y )| ? 2L? {f? }(Y ) .
(5)
?
?
where the expectation on the right is over the observation variable, Y . In practice, we can solve for
an optimal ? by minimizing the sample mean of this quantity:
N
1
2
|f? (Yn )| ? 2L? {f? }(Yn ) .
(6)
?? = arg min
? N
n=1
where {Yn } is a set of observed data. Again this does not require any knowledge of, or samples
drawn from, the prior PX .
3
Example estimators
In general, it can be difficult to obtain the operator L directly from the definition in Eq. (3), because
inversion of the operator A could be unstable or undefined. Instead, a solution may often be obtained
by noting that the definition implies that
L ? A = A ? X,
or, equivalently
L{PY |X (y|x)} = xPY |X (y|x).
This is an eigenfunction equation: for each value of x, the conditional density PY |X (y|x) must be
an eigenfunction (eigenvector, for discrete variables) of eoperator L, with associated eigenvalue x.
Consider a standard example, in which the variable of interest is corrupted by independent additive
noise: Y = X + W . The conditional density is
PY |X (y|x) = PW (y ? x).
We wish to find an operator which when applied to this conditional density (viewed as a function of
y) will give
L{PW (y ? x)} = x PW (y ? x)
(7)
for all x. Subtracting y PW (y ? x) from both sides gives
M {PW (y ? x)} = ?(y ? x) PW (y ? x).
(8)
where
M {f } (y) = L{f }(y) ? y f (y)
is a linear shift-invariant operator (acting in y).
Taking Fourier transforms and using the convolution and differentiation properties gives:
P
M(?)
W (?) = ?(yPW )(?)
?i?? P
W (?),
(9)
?? P
W (?)
M(?)
= ?i
PW (?)
= ?i?? ln P
W (?) .
(10)
L{f }(y) = y f (y) ? F ?1 i?? ln P
W (?) f (?) (y),
(11)
=
so that
This gives us the linear operator
where F ?1 denotes the inverse Fourier transform. Note that throughout this discussion X and
W played symmetric roles. Thus, in cases with known prior density and unknown additive noise
density, one can formulate the estimator entirely in terms of the prior.
Our prior-free estimator methodology is quite general, and can often be applied to more complicated
observation processes. In order to give some sense of the diversity of forms that can arise, Table 1
provides additional examples. References for the specific cases that we have found in the statistics
literature are provided in table.
Obs. process
Obs. density: PY |X (y|x)
Numerator: N (y) = L{PY }(y)
A
(A ? X ? A?1 )PY (y)
yPY ? F ?1 i?? ln P
W (?) PY (?)
Discrete
PW (y ? x)
Gen. add.
Add.
[6]/[4]*
Gaussian
Add. Poisson
exp{
Add. uniform
Add. random # of
components
?
|2??|
?k e??
Add. Laplacian
Add. Cauchy
}
?1
T ?1
(y?x??)
2 (y?x??) ?
k! ?(y ? x ? ks)
1 ?|(y?x)/?|
2? e
1
?
? ( (?(y?x))2 +1 )
1
2a ,
0,
|y ? x| ? a
|y ? x| > a
PW (y ? x), where:
K
W ? k=0 Wk ,
Wk i.i.d. (Pc ),
K ? Poiss(?)
(y ? ?)PY (y) + ??y PY (y)
yPY (y) ? ?sPY (y ? s)
yPY (y) + 2?2 {PW
PY }(y)
1
yPY (y) ? { 2??y
PY }(y)
yPY (y)+ a sgn(k)PY (y ? ak)
? 12 PY (?
y )sgn(y ? y?)d?
y
yPY (y) ? ?{(yPc ) PY }(y)
Disc. exp. [2]/[5]*
h(x)g(n)xn
g(n)
g(n+1) PY
(n + 1)
Disc. inv. exp. [5]*
h(x)g(n)x?n
g(n)
g(n?1) PY
(n ? 1)
Cnt. exp. [2]/[3]*
h(x)g(y)eT (y)x
Cnt. inv. exp. [3]*
h(x)g(y)eT (y)/x
xn e?x
n!
Poisson [7]/[5]*
Lapl. scale mixture
?1 e
2?x
y
1 ?x
; x, y
xe
g(y)
y
y)
T (?
y ) PY
?? g(?
(?
y )d?
y
(n + 1)PY (n + 1)
2
? y2x
Gauss. scale mixture
g(y) d PY (y)
T (y) dy { g(y) }
>0
?EY {Y ; Y < y}
PY {Y > y}
Table 1: Prior-free estimation formulas. Functions written with hats or in terms of ? represent
multiplication in the Fourier Domain. n replaces y for discrete distributions. Bracketed numbers are
references for operators L, with * denoting references for the parametric (dual) operator, L? .
4
4.1
Simulations
Non-parametric examples
Since each of the prior-free estimators discussed above relies on approximating values from the
observed data, the behavior of such estimators should approach the BLS estimator as the number of
data samples grows. In Fig. 1, we examine the behavior of three non-parametric prior-free estimators
based on Eq. (4). The first case corresponds to data drawn independently from a binary source,
which are observed through a process in which bits are switched with probability 14 . The estimator
does not know the binary distribution of the source (which was a ?fair coin? for our simulation),
but does know the bit-switching probability. For this estimator we approximate PY using a simple
histogram, and then use the matrix version of the linear operator in Eq. (3). We characterize the
behavior of this estimator as a function of the number of data points, N , by running many Monte
Carlo simulations for each N and indicating the mean improvement in MSE (compared with the
ML estimator, which is the identity function), the mean improvement using the conventional BLS
estimation function, E{X|Y = y} assuming the prior density is known, and the standard deviations
of the improvements taken over our simulations.
Figure 1b shows similar results for additive Gaussian noise, with SNR replacing MSE. Signal density is a generalized Gaussian with exponent 0.5. In this case, we compute Eq. (4) using a more
3
0.2
4
0
?0.1
?0.2
SNR improvement
SNR improvement
0.1
MSE improvement
5
2.5
2
1.5
1
0.5
?0.3 0
10
1
10
2
10
# samples
3
10
(a)
4
10
0 2
10
3
2
1
0
?1
?2
3
4
10
10
5
10
?3 2
10
3
10
4
# samples
10
# samples
(b)
(c)
5
10
6
10
Fig. 1: Empirical convergence of prior-free estimator to optimal BLS solution, as a function number
of observed samples of Y . For each number of observations, each estimator is simulated many times.
Black dashed lines show the improvement of the prior-free estimator, averaged over simulations,
relative to the ML estimator. White line shows the mean improvement using the conventional BLS
solution, E{X|Y = y}, assuming the prior density is known. Gray regions denote ? one standard
deviation. (a) Binary noise (10,000 simulations for each number of observations); (b) additive
Gaussian noise (1,000 simulations); (c) Poisson noise (1,000 simulations).
sophisticated approximation method, as described in [8]. We fit a local exponential model similar
to that used in [9] to the data in bins, with binwidth adaptively selected so that the product of the
number of points in the bin and the squared binwidth is constant. This binwidth selection procedure, analogous to adaptive binning procedures developed in the density estimation literature [10],
provides a reasonable tradeoff between bias and variance, and converges to the correct answer for
any well-behaved density [8]. Note that in this case, convergence is substantially slower than for the
binary case, as might be expected given that we are dealing with a continuous density rather than a
single scalar probability. But the variance of the estimates is very low.
Figure 1c shows the case of estimating a randomly varying rate parameter that governs an inhomogeneous Poisson process. The prior on the rate (unknown to the estimator) is exponential. The
observed values Y are the (integer) values drawn from the Poisson process. In this case the histogram of observed data was used to obtain a naive approximation of PY (n). It should be noted
that improved performance for this estimator is expected if we were to use a more sophisticated
approximation of the ratio of densities.
4.2
Parametric examples
In this section we discuss the empirical behavior of the parametric approach applied to the additive
Gaussian case. From the derivation in section 3, and restricting to the scalar case, we have
L? = y ? ? 2
d
.
dy
In this particular case,, it is easier to represent the estimator as
f (y) = y + g(y).
Substituting into Eq. (5) gives
E{|f (Y ) ? X|2 } = E{|g(Y )|2 + ? 2 g (Y )} + const,
where the constant does not depend on g. Therefore, if we have a parametric family {g? } of such g,
and are given data {Yn } we can try and minimize
N
1
{|g? (Yn )|2 + ? 2 g? (Yn )}.
N n=1
(12)
This expression, known as Stein?s unbiased risk estimator (SURE) [4], favors estimators g? that have
small amplitude, and highly negative derivatives at the data values. This is intuitively sensible: the
resulting estimators will ?shrink? the data toward regions of high probability.
Recently, an expression similar to Eq. (12) was used as a criterion for density estimation in cases
where the normalizing constant, or partition function, is difficult to obtain [11]. The prior-free
3
3
2.5
2.5
2.5
2
1.5
1
0.5
0 2
10
SNR improvement
3
SNR improvement
SNR improvement
Fig. 2: Example bump functions, used for linear parameterization of estimators in Figs. 3(a) and
3(b).
2
1.5
1
0.5
3
4
10
10
5
10
2
1.5
1
0.5
0 2
10
3
4
10
10
5
10
0 2
10
3
4
10
10
# samples
# samples
# samples
(a)
(b)
(c)
5
10
Fig. 3: Empirical convergence of parametric prior-free method to optimal BLS solution, as a function
number of data observations, for three different parameterized estimators. (a)3 bump; (b)15 bumps;
(c) Soft thresholding. All cases use a generalized Gaussian prior (exponent 0.5), and assume additive
Gaussian noise.
approach we are discussing provides an interpretation for this procedure: the optimal density is the
one which, when converted into an estimator using the formula in Table 1 for the additive Gaussian
case, gives the best MSE. This may be extended to any of the linear operators in Table 1.
As an example, we parametrize g as a linear combination of nonlinear ?bump? functions
g? (y) =
?k gk (y)
(13)
k
where the functions gk are of the form
1
k?
sgn(y) log2 (|y|/? + 1) ?
,
gk (y) = y cos
?
2
as illustrated in Fig. 2. Recently, linear parameterizations have been used in conjunction with
Eq. (12) for image denoising in the wavelet domain [12].
2
We can substitute Eq. (13) into Eq. (12) and pick coefficients {?k } to minimize this criteria, which
is a quadratic function of the coefficients. For our simulations, we used a generalized Gaussian
prior, with exponent 0.5. Figure 3 shows the empirical behavior of these ?SURE-bump? estimators
when using three bumps ( Fig. 3a) and fifteen bumps (Fig. 3b), illustrating the bias-variance tradeoff inherent in the fixed parameterization. Three bumps behaves fairly well, though the asymptotic
behavior for large amounts of data is biased and thus falls short of ideal. Fifteen bumps asymptotes correctly but has very large variance for small amounts of data (overfitting). For comparison
purposes, we have included the behavior of SURE thresholding [13], in which Eq. (4.2) is used to
choose an optimal threshold, ?, for the function
f? (y) = sgn(y)(|y| ? ?)+ .
As can be seen, SURE thresholding shows significant asymptotic bias although the variance behavior
is nearly ideal.
5
Discussion
We have reformulated the Bayes least squares estimation problem for a setting in which one knows
the observation process, and has access to many observations. We do not assume the prior density
is known, nor do we assume access to samples from the prior. Our formulation thus acts as a bridge
between a conventional Bayesian setting in which one derives the optimal estimator from known
prior and likelihood functions, and a data-oriented regression setting in which one learns the optimal
estimator from samples of the prior paired with corrupted observations of those samples. In many
cases, the prior-free estimator can be written explicitly, and we have shown a number of examples
to illustrate the diversity of estimators that can arise under different observation processes. For three
simple cases, we developed implementations and demonstrated that these converge to optimal BLS
estimators as the amount of data grows. We also have derived a prior-free formulation of the MSE,
which allows selection of an estimator from a parametric family. We have shown simulations for a
linear family of estimators in the additive Gaussian case.
These simulations serve to demonstrate the potential of this approach, which holds particular appeal for real-world systems (machine or biological) that must learn the priors from environmental
observations. Both methods can be enhanced by using data-adaptive parameterizations or fitting
procedures in order to properly trade off bias and variance (see, for example [8]). It is of particular interest to develop incremental implementations, which would update the estimator based on
incoming observations. This would further enhance the applicability of this approach for systems
that must learn to do optimal estimation from corrupted observations.
Acknowledgments
This work was partially funded by the Howard Hughes Medical Institute, and by New York University through a McCracken Fellowship to MR.
References
[1] G. Casella, ?An introduction to empirical Bayes data analysis,? Amer. Statist., vol. 39, pp. 83?
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[2] J. S. Maritz and T. Lwin, Empirical Bayes Methods. Chapman & Hall, 2nd ed., 1989.
[3] J. Berger, ?Improving on inadmissible estimators in continuous exponential families with applications to simultaneous estimation of gamma scale parameters,? The Annals of Staistics,
vol. 8, pp. 545?571, 1980.
[4] C. M. Stein, ?Estimation of the mean of a multivariate normal distribution,? Annals of Statistics,
vol. 9, pp. 1135?1151, November 1981.
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1982.
[6] K. Miyasawa, ?An empirical bayes estimator of the mean of a normal population,? Bull. Inst.
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[7] H. Robbins, ?An empirical bayes approach to statistics,? Proc. Third Berkley Symposium on
Mathematcal Statistics, vol. 1, pp. 157?163, 1956.
[8] M. Raphan and E. P. Simoncelli, ?Empirical Bayes least squares estimation without an explicit
prior.? NYU Courant Inst. Tech. Report, 2007.
[9] C. R. Loader, ?Local likelihood density estimation,? Annals of Statistics, vol. 24, no. 4,
pp. 1602?1618, 1996.
[10] D. W. Scott, Multivariate Density Estimation: Theory, Practice, and Visualization. John Wiley,
1992.
[11] A. Hyvarinen, ?Estimation of non-normalized statistical models by score matching,? Journal
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[12] F. Luisier, T. Blu, and M. Unser, ?SURE-based wavelet thresholding integrating inter-scale
dependencies,? in Proc IEEE Int?l Conf on Image Proc, (Atlanta GA, USA), pp. 1457?1460,
October 2006.
[13] D. Donoho and I. Johnstone, ?Adapting to unknown smoothness via wavelet shrinkage,? J
American Stat Assoc, vol. 90, December 1995.
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2,207 | 3,002 | Linearly-solvable Markov decision problems
Emanuel Todorov
Department of Cognitive Science
University of California San Diego
[email protected]
Abstract
We introduce a class of MPDs which greatly simplify Reinforcement Learning.
They have discrete state spaces and continuous control spaces. The controls have
the effect of rescaling the transition probabilities of an underlying Markov chain.
A control cost penalizing KL divergence between controlled and uncontrolled
transition probabilities makes the minimization problem convex, and allows analytical computation of the optimal controls given the optimal value function. An
exponential transformation of the optimal value function makes the minimized
Bellman equation linear. Apart from their theoretical signi cance, the new MDPs
enable ef cient approximations to traditional MDPs. Shortest path problems are
approximated to arbitrary precision with largest eigenvalue problems, yielding
an O (n) algorithm. Accurate approximations to generic MDPs are obtained via
continuous embedding reminiscent of LP relaxation in integer programming. Offpolicy learning of the optimal value function is possible without need for stateaction values; the new algorithm (Z-learning) outperforms Q-learning.
This work was supported by NSF grant ECS?0524761.
1
Introduction
In recent years many hard problems have been transformed into easier problems that can be solved
ef ciently via linear methods [1] or convex optimization [2]. One area where these trends have not
yet had a signi cant impact is Reinforcement Learning. Indeed the discrete and unstructured nature
of traditional MDPs seems incompatible with simplifying features such as linearity and convexity.
This motivates the search for more tractable problem formulations. Here we construct the rst MDP
family where the minimization over the control space is convex and analytically tractable, and where
the Bellman equation can be exactly transformed into a linear equation. The new formalism enables
ef cient numerical methods which could not previously be applied in Reinforcement Learning. It
also yields accurate approximations to traditional MDPs.
Before introducing our new family of MDPs, we recall the standard formalism. Throughout the
paper S is a nite set of states, U (i) is a set of admissible controls at state i 2 S, ` (i; u) 0 is a cost
for being in state i and choosing control u 2 U (i), and P (u) is a stochastic matrix whose element
pij (u) is the transition probability from state i to state j under control u. We focus on problems
where a non-empty subset A
S of states are absorbing and incur zero cost: pij (u) = ji and
` (i; u) = 0 whenever i 2 A. Results for other formulations will be summarized later. If A can be
reached with non-zero probability in a nite number of steps from any state, then the undiscounted
in nite-horizon optimal value function is nite and is the unique solution [3] to the Bellman equation
n
o
X
pij (u) v (j)
(1)
v (i) = min ` (i; u) +
u2U (i)
j
For generic MDPs this equation is about as far as one can get analytically.
2
A class of more tractable MDPs
In our new class of MDPs the control u 2 RjSj is a real-valued vector with dimensionality equal to
the number of discrete states. The elements uj of u have the effect of directly modifying the transition probabilities of an uncontrolled Markov chain. In particular, given an uncontrolled transition
probability matrix P with elements pij , we de ne the controlled transition probabilities as
(2)
pij (u) = pij exp (uj )
Note that P (0) = P . In some sense this is the most general notion of "control" one can imagine ?
we are allowing the controller to rescale the underlying transition probabilities in any way it wishes.
However there are two constraints implicit in (2). First, pij = 0 implies pij (u) = 0. In this case uj
has no effect and so we set it to 0 for concreteness. Second, P (u) must have row-sums equal to 1.
Thus the admissible controls are
n
o
X
U (i) = u 2 RjSj ;
pij exp (uj ) = 1; pij = 0 =) uj = 0
(3)
j
Real-valued controls make it possible to de ne a natural control cost. Since the control vector
acts directly on the transition probabilities, it makes sense to measure its magnitude in terms of
the difference between the controlled and uncontrolled transition probabilities. Differences between
probability distributions are most naturally measured using KL divergence, suggesting the following
de nition. Let pi (u) denote the i-th row-vector of the matrix P (u), that is, the vector of transition
probabilities from state i to all other states under control u. The control cost is de ned as
r (i; u) = KL (pi (u) jjpi (0)) =
X
j:pij 6=0
pij (u) log
From the properties of KL divergence it follows that r (i; u)
Substituting (2) in (4) and simplifying, the control cost becomes
X
r (i; u) =
pij (u) uj
pij (u)
pij (0)
(4)
0, and r (i; u) = 0 iff u = 0.
(5)
j
This has an interesting interpretation. The Markov chain likes to behave according to P but can
be paid to behave according to P (u). Before each transition the controller speci es the price uj it
is willing to pay (or collect, if uj < 0) for every possible next state j. When the actual transition
occurs, say to state k, the controller pays the price uk it promised. Then r (i; u) is the price the
controller expects to pay before observing the transition.
Coming back to the MDP construction, we allow an arbitrary state cost q (i)
above control cost:
` (i; u) = q (i) + r (i; u)
0 in addition to the
(6)
We require q (i) = 0 for absorbing states i 2 A so that the process can continue inde nitely without
incurring extra costs. Substituting (5, 6) in (1), the Bellman equation for our MDP is
n
o
X
v (i) = min q (i) +
pij exp (uj ) (uj + v (j))
(7)
j
u2U (i)
We can now exploit the bene ts of this unusual construction. The minimization in (7) subject to the
constraint (3) can be performed in closed form using Lagrange multipliers, as follows. For each i
de ne the Lagrangian
X
X
L (u; i ) =
pij exp (uj ) (uj + v (j)) + i
pij exp (uj ) 1
(8)
j
j
The necessary condition for an extremum with respect to uj is
0=
@L
= pij exp (uj ) (uj + v (j) +
@uj
i
+ 1)
(9)
When pij 6= 0 the only solution is
uj (i) =
v (j)
i
1
(10)
Taking another derivative yields
@2L
@uj @uj
= pij exp uj (i) > 0
(11)
uj =uj (i)
and therefore (10) is a minimum. The Lagrange multiplier i can be found by applying the constraint
(3) to the optimal control (10). The result is
X
pij exp ( v (j))
1
(12)
i = log
j
and therefore the optimal control law is
uj (i) =
v (j)
log
X
k
pik exp ( v (k))
(13)
Thus we have expressed the optimal control law in closed form given the optimal value function.
Note that the only in uence of the current state i is through the second term, which serves to normalize the transition probability distribution pi (u ) and is identical for all next states j. Thus the
optimal controller is a high-level controller: it tells the Markov chain to go to good states without
specifying how to get there. The details of the trajectory emerge from the interaction of this controller and the uncontrolled stochastic dynamics. In particular, the optimally-controlled transition
probabilities are
pij exp ( v (j))
pij (u (i)) = P
(14)
k pik exp ( v (k))
These probabilities are proportional to the product of two terms: the uncontrolled transition probabilities pij which do not depend on the costs or values, and the (exponentiated) next-state values
v (j) which do not depend on the current state. In the special case pij = consti the transition probabilities (14) correspond to a Gibbs distribution where the optimal value function plays the role of
an energy function.
Substituting the optimal control (13) in the Bellman equation (7) and dropping the min operator,
X
v (i) = q (i) +
pij (u (i)) uj (i) + v (j)
(15)
j
X
= q (i) +
pij (u (i)) ( i 1)
j
= q (i)
i
= q (i)
log
1
X
j
pij exp ( v (j))
Rearranging terms and exponentiating both sides of (15) yields
X
exp ( v (i)) = exp ( q (i))
pij exp ( v (j))
j
(16)
We now introduce the exponential transformation
z (i) = exp ( v (i))
(17)
which makes the minimized Bellman equation linear:
z (i) = exp ( q (i))
X
j
pij z (j)
(18)
De ning the vector z with elements z (i), and the diagonal matrix G with elements exp ( q (i))
along its main diagonal, (18) becomes
z = GP z
Thus our class of optimal control problems has been reduced to a linear eigenvalue problem.
(19)
2.1
Iterative solution and convergence analysis
From (19) it follows that z is an eigenvector of GP with eigenvalue 1. Furthermore z (i) > 0 for
all i 2 S and z (i) = 1 for i 2 A. Is there a vector z with these properties and is it unique? The
answer to both questions is af rmative, because the Bellman equation has a unique solution and v
is a solution to the Bellman equation iff z = exp ( v) is an admissible solution to (19). The only
remaining question then is how to nd the unique solution z. The obvious iterative method is
zk+1 = GP zk ;
(20)
z0 = 1
This iteration always converges to the unique solution, for the following reasons. A stochastic matrix
P has spectral radius 1. Multiplication by G scales down some of the rows of P , therefore GP has
spectral radius at most 1. But we are guaranteed than an eigenvector z with eigenvalue 1 exists,
therefore GP has spectral radius 1 and z is a largest eigenvector. Iteration (20) is equivalent to
the power method (without the rescaling which is unnecessary here) so it converges to a largest
eigenvector. The additional constraints on z are clearly satis ed at all stages of the iteration. In
particular, for i 2 A the i-th row of GP has elements ji , and so the i-th element of zk remains equal
to 1 for all k.
We now analyze the rate of convergence. Let m = jAj and n = jSj. The states can be permuted so
that GP is in canonical form:
T1 T 2
GP =
(21)
0
I
where the absorbing states are last, T1 is (n m) by (n m), and T2 is (n m) by m. The reason
we have the identity matrix in the lower-right corner, despite multiplication by G, is that q (i) = 0
for i 2 A. From (21) we have
GP
k
=
T1k
0
T1k
1
+
+ T1 + I T2
I
=
T1k
0
I
T1k (I T1 )
I
1
T2
(22)
A stochastic matrix P with m absorbing states has m eigenvalues 1, and all other eigenvalues are
smaller than 1 in absolute value. Since the diagonal elements of G are no greater than 1, all eigenvalues of T1 are smaller than 1 and so limk!1 T1k = 0. Therefore iteration (20) converges exponentially as k where < 1 is the largest eigenvalue of T1 . Faster convergence is obtained for
smaller . The factors that can make small are: (i) large state costs q (i) resulting in small terms
exp ( q (i)) along the diagonal of G; (ii) small transition probabilities among non-absorbing states
(and large transition probabilities from non-absorbing to absorbing states). Convergence is independent of problem size because has no reason to increase as the dimensionality of T1 increases.
Indeed numerical simulations on randomly generated MDPs have shown that problem size does not
systematically affect the number of iterations needed to reach a given convergence criterion. Thus
the average running time scales linearly with the number of non-zero elements in P .
2.2
Alternative problem formulations
While the focus of this paper is on in nite-horizon total-cost problems with absorbing states, we
have obtained similar results for all other problem formulations commonly used in Reinforcement
Learning. Here we summarize these results. In nite-horizon problems equation (19) becomes
z (t) = G (t) P (t) z (t + 1)
(23)
where z (t nal ) is initialized from a given nal cost function. In in nite-horizon average-cost-perstage problems equation (19) becomes
z = GP z
(24)
where is the largest eigenvalue of GP , z is a differential value function, and the average cost-perstage turns out to be log ( ). In in nite-horizon discounted-cost problems equation (19) becomes
z = GP z
(25)
where < 1 is the discount factor and z is de ned element-wise. Even though the latter equation
is nonlinear, we have observed that the analog of iteration (20) still converges rapidly.
Fig 1A
3
Fig 1B
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6
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10
9
31
10
10
10
11
12
13
14
31
11
11
11
11
12
13
14
Shortest paths as an eigenvalue problem
Suppose the state space S of our MDP corresponds to the vertex set of a directed graph, and let D be
the graph adjacency matrix whose element dij indicates the presence (dij = 1) or absence (dij = 0)
of a directed edge from vertex i to vertex j. Let A S be a non-empty set of destination vertices.
Our goal is to nd the length s (i) of the shortest path from every i 2 S to some vertex in A. For
i 2 A we have s (i) = 0 and dij = ji .
We now show how the shortest path lengths s (i) can be obtained from our MDP. De ne the elements
of the stochastic matrix P as
dij
pij = P
(26)
k dik
corresponding to a random walk on the graph. Next choose > 0 and de ne the state costs
q (i) = when i 2
= A,
q (i) = 0 when i 2 A
(27)
This cost model means that we pay a price whenever the current state is not in A. Let v (i) denote
the optimal value function for the MDP de ned by (26, 27). If the control costs were 0 then the
shortest paths would simply be s (i) = 1 v (i). Here the control costs are not 0, however they are
bounded. This can be shown using
(28)
pij (u) = pij exp (uj ) 1
which implies that for pij 6= 0 we have uj
log pij . Since r (i; u) is a convex combination of
the elements of u, the following bound holds:
r (i; u)
maxj (uj )
log minj:pij 6=0 pij
(29)
The control costs are bounded and we are free to choose arbitrarily large, so we can make the state
costs dominate the optimal value function. This yields the following result:
v (i)
s (i) = lim
(30)
!1
Thus we have reduced the shortest path problem to an eigenvalue problem. In spectral graph theory
many problems have previously been related to eigenvalues of the graph Laplacian [4], but the
shortest path problem was not among them until now. Currently the most widely used algorithm is
Dijkstra's algorithm. In sparse graphs its running time is O (n log (n)). In contrast, algorithms for
nding largest eigenpairs have running time O (n) for sparse matrices.
Of course (30) involves a limit and so we cannot obtain the exact shortest paths by solving a single
eigenvalue problem. However we can obtain a good approximation by setting large enough ? but
not too large because exp ( ) may become numerically indistinguishable from 0. Fig 1 illustrates
the solution obtained from (30) and rounded down to the nearest integer, for = 1 in 1A and = 50
in 1B. Transitions are allowed to all neighbors. The result in 1B matches the exact shortest paths.
Although the solution for = 1 is numerically larger, it is basically a scaled-up version of the
correct solution. Indeed the R2 between the two solutions before rounding was 0:997.
4
Approximating discrete MDPs via continuous embedding
In the previous section we replaced the shortest path problem with a continuous MDP and obtained
an excellent approximation. Here we obtain approximations of similar quality in more general settings, using an approach reminiscent of LP-relaxation in integer programming. As in LP-relaxation,
theoretical results are hard to derive but empirically the method works well.
We construct an embedding which associates the controls in the discrete MDP with speci c control
vectors of a continuous MDP, making sure that for these control vectors the continuous MDP has the
same costs and transition probabilities as the discrete MDP. This turns out to be possible under mild
and reasonable assumptions, as follows. Consider a discrete MDP with transition probabilities and
e De ne the matrix B (i) of all controlled transition probabilities from state i.
costs denoted pe and `.
This matrix has elements
baj (i) = peij (a) ;
a 2 U (i)
(31)
We need two assumptions to guarantee the existence of an exact embedding: for all i 2 S the matrix
B (i) must have full row-rank, and if any element of B (i) is 0 then the entire column must be 0. If
the latter assumption does not hold, we can replace the problematic 0 elements of B (i) with a small
and renormalize. Let N (i) denote the set of possible next states, i.e. states j for which peij (a) > 0
for any/all a 2 U (i). Remove the zero-columns of B (i) and restrict j 2 N (i).
The rst step in the construction is to compute the real-valued control vectors ua corresponding to
the discrete actions a. This is accomplished by matching the transition probabilities of the discrete
and continuous MDPs:
pij exp uaj = peij (a) ;
8 i 2 S; j 2 N (i) ; a 2 U (i)
(32)
These constraints are satis ed iff the elements of the vector u are
a
uaj = log (e
pij (a))
(33)
log pij
The second step is to compute the uncontrolled transition probabilities pij and state costs q (i) in the
continuous MDP so as to match the costs in the discrete MDP. This yields the set of constraints
q (i) + r (i; ua ) = `e(i; a) ;
8 i 2 S; a 2 U (i)
For the control vector given by (33) the KL-divergence cost is
X
X
r (i; ua ) =
pij exp uaj uaj = h (i; a)
peij (a) log pij
j
j
where h (i; a) is the entropy of the transition probability distribution in the discrete MDP:
X
h (i; a) =
peij (a) log (e
pij (a))
j
The constraints (34) are then equivalent to
X
q (i)
baj (i) log pij = `e(i; a)
j
h (i; a)
(34)
(35)
(36)
(37)
De ne the vector y (i) with elements `e(i; a) h (i; a), and the vector x (i) with elements log pij .
The dimensionality of y (i) is jU (i)j while the dimensionality of x (i) is jN (i)j jU (i)j. The latter
inequality follows from the assumption that B (i) has full row-rank. Suppressing the dependence on
the current state i, the constraints (34) can be written in matrix notation as
q1
(38)
Bx = y
Since the probabilities pij must sum up to 1, the vector x must satisfy the additional constraint
X
exp (xj ) = 1
(39)
j
b be any vector such that
We are given B; y and need to compute q; x satisfying (38, 39). Let x
b = B y y where y denotes the Moore-Penrose pseudoinverse. Since B is
Bb
x = y, for example x
a stochastic matrix we have B1 = 1, and so
q1
B (b
x + q1) =
Bb
x=y
(40)
Fig 2A
Fig 2B
Fig 2C
40
*
*
30
20
R 2 = 0.986
10
*
0
*
0
20
40
60
value in discrete MDP
b + q1 satis es (38) for all q, and we can adjust q to also satisfy (39), namely
Therefore x = x
X
q = log
exp (b
xj )
(41)
j
This completes the embedding. If the above q turns out to be negative, we can either choose another
b by adding an element from the null-space of B, or scale all costs `e(i; a) by a positive constant.
x
Such scaling does not affect the optimal control law for the discrete MDP, but it makes the elements
of B y y more negative and thus q becomes more positive.
We now illustrate this construction with the example in Fig 2. The grid world has a number of
obstacles (black squares) and two absorbing states (white stars). The possible next states are the
immediate neighbors including the current state. Thus jN (i)j is at most 9. The discrete MDP has
jN (i)j 1 actions corresponding to stochastic transitions to each of the neighbors. For each action,
the transition probability to the "desired" state is 0:8 and the remaining 0:2 is equally distributed
among the other states. The costs `e(i; a) are random numbers between 1 and 10 ? which is why
the optimal value function shown in grayscale appears irregular. Fig 2A shows the optimal value
function for the discrete MDP. Fig 2B shows the optimal value function for the corresponding continuous MDP. The scatterplot in Fig 2C shows the optimal values in the discrete and continuous
MDP (each dot is a state). The values in the continuous MDP are numerically smaller ? which is
to be expected since the control space is larger. Nevertheless, the correlation between the optimal
values in the discrete and continuous MDPs is excellent. We have observed similar performance in
a number of randomly-generated problems.
5
Z-learning
So far we assumed that a model of the continuous MDP is available. We now turn to stochastic
approximations of the optimal value function which can be used when a model is not available. All
we have access to are samples (ik ; jk ; qk ) where ik is the current state, jk is the next state, qk is the
state cost incurred at ik , and k is the sample number. Equation (18) can be rewritten as
X
z (i) = exp ( q (i))
pij z (j) = exp ( q (i)) EP [z (j)]
(42)
j
This suggests an obvious stochastic approximation zb to the function z, namely
zb (ik )
b (ik )
k) z
(1
+
k
exp ( qk ) zb (jk )
(43)
where the sequence of learning rates k is appropriately decreased as k increases. The approximation to v (i) is simply log (b
z (i)). We will call this algorithm Z-learning.
Let us now compare (43) to the Q-learning algorithm applicable to discrete MDPs. Here we have
samples (ik ; jk ; `k ; uk ). The difference is that `k is now a total cost rather than a state cost, and we
have a control uk generated by some control policy. The update equation for Q-learning is
b (ik ; uk )
Q
(1
b
k ) Q (ik ; uk )
+
k
min
u0 2U (jk )
b (jk ; u0 )
`k + Q
(44)
Fig 3A
Fig 3B
1
1
Z-learning
*
Q-learning
Q
*
*
Z
0
0
0
1
2
3 x 10 4
0
5
10
15
Number of state transitions
To compare the two algorithms, we rst constructed continuous MDPs with q (i) = 1 and transitions
to the immediate neighbors in the grid worlds shown in Fig 3. For each state we found the optimal
transition probabilities (14). We then constructed a discrete MDP which had one action (per state)
that caused the same transition probabilities, and the corresponding cost was the same as in the
continuous MDP. We then added jN (i)j 1 other actions by permuting the transition probabilities.
Thus the discrete and continuous MDPs were guaranteed to have identical optimal value functions.
Note that the goal here is no longer to approximate discrete with continuous MDPs, but to construct
pairs of problems with identical solutions allowing fair comparison of Z-learning and Q-learning.
We run both algorithms with the same random policy. The learning rates decayed as k =
c= (c + t (k)) where the constant c was optimized separately for each algorithm and t (k) is the
run to which sample k belongs. When the MDP reaches an absorbing state a new run is started from
a random initial state. The approximation error plotted in Fig 3 is de ned as
maxi jv (i) vb (i)j
maxi v (i)
(45)
and is computed at the end of each run. For small problems (Fig 3A) the two algorithms had identical
convergence, however for larger problems (Fig 3B) the new Z-learning algorithm was clearly faster.
This is not surprising: even though Z-learning is as model-free as Q-learning, it bene ts from the
analytical developments in this paper and in particular it does not need a maximization operator or
state-action values. The performance of Q-learning can be improved by using a non-random (say greedy) policy. If we combine Z-learning with importance sampling, the performance of Z-learning
can also be improved by using such a policy.
6
Summary
We introduced a new class of MDPs which have a number of remarkable properties, can be solved
ef ciently, and yield accurate approximations to traditional MDPs. In general, no single approach is
likely to be a magic wand which simpli es all optimal control problems. Nevertheless the results so
far are very encouraging. While the limitations remain to be clari ed, our approach appears to have
great potential and should be thoroughly investigated.
References
[1]
[2]
[3]
[4]
B. Scholkopf and A. Smola, Learning with kernels. MIT Press (2002)
S. Boyd and L. Vandenberghe, Convex optimization. Cambridge University Press (2004)
D. Bertsekas, Dynamic programming and optimal control (2nd ed). Athena Scienti c (2000)
F. Chung, Spectral graph theory. CMBS Regional Conference Series in Mathematics (1997)
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2,208 | 3,003 | Efficient Structure Learning of Markov Networks
using L1-Regularization
Su-In Lee Varun Ganapathi Daphne Koller
Department of Computer Science
Stanford University
Stanford, CA 94305-9010
{silee,varung,koller}@cs.stanford.edu
Abstract
Markov networks are commonly used in a wide variety of applications, ranging
from computer vision, to natural language, to computational biology. In most
current applications, even those that rely heavily on learned models, the structure
of the Markov network is constructed by hand, due to the lack of effective algorithms for learning Markov network structure from data. In this paper, we provide
a computationally efficient method for learning Markov network structure from
data. Our method is based on the use of L1 regularization on the weights of
the log-linear model, which has the effect of biasing the model towards solutions
where many of the parameters are zero. This formulation converts the Markov network learning problem into a convex optimization problem in a continuous space,
which can be solved using efficient gradient methods. A key issue in this setting
is the (unavoidable) use of approximate inference, which can lead to errors in the
gradient computation when the network structure is dense. Thus, we explore the
use of different feature introduction schemes and compare their performance. We
provide results for our method on synthetic data, and on two real world data sets:
pixel values in the MNIST data, and genetic sequence variations in the human
HapMap data. We show that our L1 -based method achieves considerably higher
generalization performance than the more standard L2 -based method (a Gaussian
parameter prior) or pure maximum-likelihood learning. We also show that we can
learn MRF network structure at a computational cost that is not much greater than
learning parameters alone, demonstrating the existence of a feasible method for
this important problem.
1 Introduction
Undirected graphical models, such as Markov networks or log-linear models, have been used in an
ever-growing variety of applications, including computer vision, natural language, computational
biology, and more. However, as this modeling framework is used in increasingly more complex
and less well-understood domains, the problem of selecting from the exponentially large space of
possible network structures becomes of great importance. Including all of the possibly relevant
interactions in the model generally leads to overfitting, and can also lead to difficulties in running
inference over the network. Moreover, learning a ?good? structure can be an important task in its
own right, as it can provide insight about the underlying structure in the domain.
Unfortunately, the problem of learning Markov networks remains a challenge. The key difficulty is that the maximum likelihood (ML) parameters of these networks have no analytic closed
form; finding these parameters requires an iterative procedure (such as conjugate gradient [15] or
BFGS [5]), where each iteration runs inference over the current model. This type of procedure is
computationally expensive even for models where inference is tractable. The problem of structure
learning is considerably harder. The dominant type of solution to this problem uses greedy local
heuristic search, which incrementally modifies the model by adding and possibly deleting features.
One approach [6, 14] adds features so as to greedily improve the model likelihood; once a feature
is added, it is never removed. As the feature addition step is heuristic and greedy, this can lead
to the inclusion of unnecessary features, and thereby to overly complex structures and overfitting.
An alternative approach [1, 7] explicitly searches over the space of low-treewidth models, but the
utility of such models in practice is unclear; indeed, hand-designed models for real-world problems
generally do not have low tree-width. Moreover, in all of the greedy heuristic search methods, the
learned network is (at best) a local optimum of a penalized likelihood score.
In this paper, we propose a different approach for learning the structure of a log-linear graphical model (or a Markov network). Rather than viewing it as a combinatorial search problem, we
embed the structure selection step within the problem of parameter estimation: features that have
weight zero (in log-space) are degenerate, and do not induce dependencies between the variables
they involve. To appropriately bias the model towards sparsity, we use a technique that has become
increasingly popular in the context of supervised learning, in problems that involve a large number of
features, many of which may be irrelevant. It has been long known [21] that using L 1 -regularization
over the model parameters ? optimizing a joint objective that trades off fit to data with the sum of
the absolute values of the parameters ? tends to lead to sparse models, where many weights have
value 0. More recently, Dudik et al. (2004) showed that density estimation in log-linear models
using L1 -regularized likelihood has sample complexity that grows only logarithmically in the number of features of the log-linear model; Ng (2004) shows a similar result for L 1 -regularized logistic
regression. These results show that this approach is useful for selecting the relevant features from a
large number of irrelevant ones. Other recent work proposes effective algorithms for L 1 -regularized
generalized linear models (e.g., [18, 10, 9]), support vector machines (e.g., [25]), and feature selection in log-linear models encoding natural language grammars [19].
Surprisingly, the use of L1 -regularization has not been proposed for the purpose of structure
learning in general Markov networks. In this paper, we explore this approach, and discuss issues
that are important to its effective application to large problems. A key point is that, for a given L 1 based model score and given candidate feature set F, we have a fixed convex optimization problem
that admits a unique optimal solution. Due to the properties of the L 1 score, in this solution, many
features will have weight 0, generally leading to a sparse network structure. However, it is generally
impractical to simply initialize the model to include all possible features: exact inference in such
a model is almost invariably intractable, and approximate inference methods such as loopy belief
propagation [17] are likely to give highly inaccurate estimates of the gradient, leading to poorly
learned models. Thus, we propose an algorithm schema that gradually introduces features into
the model, and lets the L1 -regularization scheme eliminate them via the optimization process. We
explore the use of different approaches for feature introduction, one based on the gain-based method
of Della Pietra, Della Pietra and Lafferty [6] and one on the grafting method of Perkins, Lacker and
Theiler [18]. We provide a sound termination condition for the algorithm based on the criterion
proposed by Perkins et al. [18]; given correct estimates of the gradient, this algorithm is guaranteed
to terminate only at the unique global optimum, for any reasonable feature introduction method.
We test our method on synthetic data generated from known MRFs and on two real-world tasks:
modeling the joint distribution of pixel values in the MNIST data [12], and modeling the joint
distribution of genetic sequence variations ? single-nucleotide polymorphisms (SNPs) ? in the
human HapMap data [3]. Our results show that L1 -regularization out-performs other approaches,
and provides an effective method for learning MRF structure even in large, complex domains.
2 Preliminaries
We focus our presentation on the framework of log-linear models, which forms a convenient basis
for a discussion of learning. Let X = {X1 , . . . , Xn } be a set of discrete-valued random variables. A
log-linear model is a compact representation of a probability distribution over assignments to X . The
log-linear model is defined in terms of a set of feature functions fk (X k ), each of which is a function
that defines a numerical value for each assignment xk to some subset X k ? X . Given a set of
feature functions F = {fk }, the parameters of the log-linear model
P are weights ? = {?k : fk ? F }.
1
The overall distribution is then defined as: P? (x) = Z(?)
exp( fk ?F ?k fk (xk )), where xk is the
assignment to X k within x, and Z(?) is the partition function that ensures that the distribution is
normalized (so that all entries sum to 1). Note that this definition of features encompasses both
?standard? features that relate unobserved network variables (e.g., the part of speech of a word
in a sentence) to observed elements in the data (e.g., the word itself), and structural features that
encode the interaction between hidden variables in the model. A log-linear model induces a Markov
network over X , where there is an edge between every pair of variables X i , Xj that appear together
in some feature fk (X k ) (Xi , Xj ? X k ). The clique potentials are constructed from the log-linear
features in the obvious way. Conversely, every Markov network can be encoded as a log-linear
model by defining a feature which is an indicator function for every assignment of variables x c to a
clique X c . The mapping to Markov networks is useful, as most inference algorithms, such as belief
propagation [17, 16], operate on the graph structure of the Markov network.
The standard learning problem for MRFs is formulated as follows. We are given a set of IID
training instances D = {x[1], . . . , x[M ]}, each consisting of a full assignment to the variables in
X . Our goal is to output a log-linear model M over X , which consists of a set of features F and a
parameter vector ? that specifies a weight for each fk ? F .
The log-likelihood function log P (D | M) has the following form:
X
`(M : D) =
?k fk (D) ? M log Z(?) = ? > f (D) ? M log Z(?),
(1)
fk ?F
where fk (D) = m=1 fk (xk [m]) is the sum of the feature values over the entire data set, f (D)
is the vector where all of these aggregate features have been arranged in the same order as the
parameter vector, and ? > f (D) is a vector dot-product operation. There is no closed-form solution
for the parameters that maximize Eq. (1), but the objective is concave, and can therefore be optimized
using numerical optimization procedures such as conjugate gradient [15] or BFGS [5]. The gradient
of the log-likelihood is:
?`(M : D)
(2)
= fk (D) ? M IEx?P? [fk (x)].
??k
PM
This expression has a particularly intuitive form: the gradient attempts to make the feature counts in
the empirical data equal to their expected counts relative to the learned model. Note that, to compute
the expected feature counts, we must perform inference relative to the current model. This inference
step must be performed at every iteration of the gradient process.
3 L1 -Regularized Structure Learning
We formulate our structure learning problem as follows. We assume that there is a (possibly very
large) set of features F, from which we wish to select a subset F ? F for inclusion in the model.
This problem is generally solved using a heuristic search over the combinatorial space of possible
feature subsets. Our approach addresses it as a search over the possible parameter vectors ? ? IR |F | .
Specifically, rather than optimizing the log-likelihood itself, we introduce a Laplacian parameter
Q prior for each feature fk takes the form P (?k ) = ?k /2 ? exp(??k |?k |). We define P (?) =
k P (?k ). We now optimize the posterior probability P (D, ?) = P (D | ?)P (?) rather than the
likelihood. Taking the logarithm and eliminating constant terms, we obtain the following objective:
X
max[?> f (D) ? M log Z(?) ?
?k |?k |].
(3)
?
k
In most cases, the same prior is used for all features, so we have ? k = ? for all k. This objective
is convex, and can be optimized efficiently using methods such as conjugate gradient or BFGS,
although care needs to be taken with the discontinuity of the derivative at 0. Thus, in principle, we
can simply optimize this objective to obtain its globally optimal parameter assignment.
The objective of Eq. (3) should be contrasted with the one obtained for the more standard parameter prior used for log-linear models: the mean-zero Gaussian prior P (? k ) ? exp(??k2 /2? 2 ). The
gaussian prior induces a regularization term that is quadratic in ?k , which penalizes large features
much more than smaller ones. Conversely, L1 -regularization still penalizes small terms strongly,
thereby forcing parameters to 0. Overall, it is known that, empirically, optimizing an L 1 -regularized
objective leads to a sparse representation with a relative small number of non-zero parameters.
Aside from this intuitive argument, recent theoretical results also provide a formal justification
for the use of L1 -regularization over other approaches: The analysis of Dudik et al. (2004) and Ng
(2004) suggests that this form of regularization is effective at identifying relevant features even with
a relatively small number of samples. Building directly on the results of Dudik et al. (2004), we can
show the following result:
Corollary 3.1: Let X = {X1 , . . . , Xn } be a set of variables each of domain size d, and P ? (X )
be a distribution. Let F be the set of indicator features over all subsets of variables X ? X of
cardinality at most c, and ?, , B > 0. Let be the parameterization over F that optimizes
h
i
??B = arg max IE??P ? `(? : ???B ) .
? : k?k?B
? D be the assignment that optimizes Eq. (3), for regularization parameter
For a data p
set D, let ?
?k = ? = c ln(2nd/?)/(2m) for all k. Then with probability at least 1 ? ?, for a data set D of
IID instances from P ? of size
1
2nd
m ? 2cB 2 2 ln
.
?
we have that:
h
i
? D ) ? IE??P ? [`(? : ? ? )] ? .
IE??P ? `(? : ?
B
In words, using the L1 -regularized log-likelihood objective, we can learn a Markov network with
a maximal clique size c, whose expected log-likelihood relative to the true underlying distribution is
at most worse than the log-likelihood of the optimal Markov network in this class whose L 1 -norm
is at most B. The number of samples required grows logarithmically in the number of nodes in
the network, and polynomially in B. The dependence on B is quite natural, indicating that more
samples are required to learn networks containing more ?strong? interactions. Note, however, that
if we bound the magnitude of each potential in the Markov network, then B = O((nd) c ), so that a
polynomial number of data instances suffices.
4 Incremental Feature Introduction
The above discussion implicitly assumed that we can find the global optimum of Eq. (3) by simple
convex optimization. However, we cannot simply include all of the features in the model in advance,
and use only parameter optimization to prune away the irrelevant ones. Recall that computing the
gradient requires performing inference in the resulting model. If we have too many features, the
model may be too densely connected to allow effective inference. Even approximate inference
algorithms, such as belief propagation, tend to degrade as the density of the network increases; for
example, BP algorithms are less likely to converge, and the answers they return are typically much
less accurate. Thus, our approach also contains a feature introduction component, which gradually
selects features to add into the model, allowing the optimization process to search for the optimal
values for their parameters. More precisely, our algorithm maintains a set of active features F ? F.
An inactive feature fk has its parameter ?k set to 0; the parameters of active features are free to be
changed when optimizing the objective Eq. (3).
In addition to various simple baseline methods, we explore two feature introduction methods,
both of which are greedy and myopic, in that they compute some heuristic estimate of the likely
benefit to be gained from introducing a single feature into the active set.
The grafting procedure of Perkins et al. [18], which was developed for feature selection in standard classification tasks, selects features based on the gradient of these parameters: We first optimize
the objective relative to the current active features F and their weights, so that, at convergence, the
gradient relative to these features is zero. Then, for each inactive feature f , we compute the partial
derivative of the objective Eq. (3) relative to ?f , and select the one whose gradient is largest.
A more conservative estimate is obtained from the gain-based method of Della Pietra et al. [6].
This method was designed for the log-likelihood objective. It begins by optimizing the parameters
relative to the current active set F . Then, for each inactive feature f , it computes the log-likelihood
gain of adding that feature, assuming that we could optimize its feature weight arbitrarily, but that
the weights of all other features are held constant. It then introduces the feature with the greatest
gain. Della Pietra et al. show that the gain is a concave objective that can be computed efficiently
using a one-dimensional line search. For the restricted case of binary-valued features, they provide a
closed-form solution for the gain. Our task is to compute not the optimal gain in log-likelihood, but
rather the optimal gain of Eq. (3). It is not difficult to see that the gain in this objective, which differs
from the log-likelihood in only a linear term, is also a concave function that can be optimized using
line search. Moreover, for the case of binary-valued features, we can also provide a closed-form
solution for the gain. The change in the objective function for introducing a feature f k is:
?L1 = ?k fk (D) ? ?k?k k ? M log[exp(?k )P? (fk ) + P? (?fk )],
where M is the number of training instances. If we take the derivative of the above function and set
it to zero, we also get a closed form solution:
(fk (D) ? ?sign(?k ))P? (?fk )
?k = log
.
(M ? fk (D) + ?sign(?k ))P? (fk )
Both methods are heuristic, in that they consider only the potential gain of adding a single feature
in isolation, assuming all other weights are held constant. However, the grafting method is more
optimistic, in that it estimates the value of adding a single feature via the slope of adding it, whereas
the gain-based approach also considers, intuitively, how far one can go in that direction before the
gain ?peaks out?. The gain based heuristic is, in fact, a lower bound on the actual gain obtained from
adding this feature, while allowing the other features to also adapt. Overall, the gain-based heuristic
provides a better estimate of the value of adding the feature, albeit at slightly greater computational
cost (except in the limited cases where a closed-form solution can be found).
As observed by Perkins et al. [18], the use of the L1 -regularized objective also provides us with
a sound stopping criterion for any incremental feature-induction algorithm. If we have that, for
every inactive fk 6? F , the gradient of the log-likelihood | ?`(M:D)
| ? ?, then the gradient of the
??k
objective in any direction is non-positive, and the objective is at a stationary point. Importantly,
as the overall objective is a concave function, it has a unique global maximum. Hence, once the
termination condition is achieved, we are guaranteed that we are at the local maximum, regardless
of the feature introduction method used. Thus, assuming the algorithm is run until the convergence
criterion is satisfied, there is no impact of the feature introduction heuristic on the final outcome, but
only on the computational complexity.
Finally, constantly evaluating all of the non-active candidate features can be computationally
prohibitive when many features are possible. Even in pairwise Markov networks, when the number
of nodes is large, a quadratic number of candidate edges can become unmanageable. In this case,
we must generally pre-select a smaller set of candidate features, and ignore the others entirely. One
very natural method for pre-selecting edges is to train an L1 -regularized logistic regression classifier
for each variable given all of the others, and then use only the edges that are used in these individual
classifiers. This approach is similar to the work of Wainwright et al. [22] (done in parallel with
our work), who proposed the use of L1 -regularized pseudo-likelihood for asymptotically learning a
Markov network structure.
5 The Use of Approximate Inference
All of the steps in the above algorithm rely on the use of inference for computing key quantities: The
gradient is needed for the parameter optimization, for the grafting method, and for the termination
condition, and the expression for the gradient requires the computation of marginal probabilities
relative to our current model. Similarly, the computation of the gain also requires inference. As
we discussed above, in most of the networks that are useful models for real applications, exact
inference is intractable. Therefore, we must resort to approximate inference, which results in errors
in the gradient. While many approximate inference methods have been proposed, one of the most
commonly used is the general class of loopy belief propagation (BP) algorithms [17, 16, 24]. The
use of an approximate inference algorithm such as BP raises several important points.
One important question issue relates to the computation of the gradient or the gain for features that
are currently inactive. The belief propagation algorithm, when executed on a particular network with
a set of active features F , creates a cluster for every subset of variables X k that appear as the scope of
a feature fk (X k ). The output of the BP inference process is a set of marginal probabilities over all of
the clusters; thus, it returns the necessary information for computing the expected sufficient statistics
in the gradient of the objective (see Eq. (2)). However, for features f k (X k ) that are currently
inactive, there is no corresponding cluster in the induced Markov network, and hence, in most cases,
the necessary marginal probabilities over X k will not be computed by BP. We can approximate this
marginal probability by extracting a subtree of the calibrated loopy graph that contains all of the
variables in X k . At convergence of the BP algorithm, every subtree of the loopy graph is calibrated,
in that all of the belief potentials must agree [23]. Thus, we can view the subtree as a calibrated
clique tree, and use standard dynamic programming methods over the tree (see, e.g.. [4]) to extract
an approximate joint distribution over X k . We note that this computation is exact for tree-structured
cluster graphs, but approximate otherwise, and that the choice of tree is not obvious, and affects the
accuracy of the answers.
A second key issue is that the performance of BP algorithms generally degrades significantly as
the density of the network increases: they are less likely to converge, and the answers they return are
typically much less accurate. Moreover, non-convergence of the inference is more common when the
network parameters are allowed to take larger, more extreme values; see, for example, [20, 11, 13]
for some theoretical results supporting this empirical phenomenon. Thus, it is important to keep the
model amenable to approximate inference, and thereby continue to improve, for as long as possible. This observation has two important consequences. First, while different feature introduction
schemes achieve the same results when using exact inference, their outcomes can vary greatly when
using approximate inference, due to differences in the structure of the networks arising during the
learning process. Thus, as we shall see, better feature introduction methods, which introduce the
more relevant features first, work much better in practice. Second, in order to keep the inference
feasible for as long as possible, we utilize an annealing schedule for the regularization parameter ?,
beginning with large values of ?, leading to greater sparsification of the structure, and then gradually
reducing ?, allowing additional (weaker) features to be introduced. This method allows a greater
part of the learning to be executed with a more robust model.
6 Results
In our experiments, we focus on binary pairwise Markov networks, where each feature function
is an indicator function for a certain assignment to a pair of nodes. As computing the exact loglikelihood is intractable in most networks, we use the conditional marginal log-likelihood (CMLL) as
our evaluation metric on the learned network. To calculate CMLL, we first divide the variables into
two groups: Xhidden and Xobserved . Then, for any test instance X[m], we compute CM LL(X[m]) =
P
Xh ?Xhidden [m] log P (Xh |Xobserved [m]). In practice, we divide the variables into four groups and
calculate the average CMLL when observing only one group and hiding the rest. Note that the
CMLL is defined only with respect to the marginals (but not the global partition function Z(?)),
which are empirically thought to be more accurate.
We considered three feature induction schemes: (a) Gain: based on the estimated change of gain,
(b) Grad: using grafting and (c) Simple: based on pairwise similarity. Under the Simple scheme,
the score of a pairwise feature between Xi and Xj is the mutual information between Xi and Xj .
For each scheme, we varied the regularization method: (a) None: no regularization, (b) L1: L 1
regularization and (c) L2: L2 regularization. We note that Gain and Grad performed similarly for
L1 and None. Moreover, we used only Grad for L2, because L 2 regularization does not admit a
closed form solution for the approximate gain.
Experiments on Synthetically Generated Data. We generated synthetic data through Gibbs
sampling on a synthetic network. A network structure with N nodes was generated by treating
each possible edge as a Bernoulli random variable and sampling the edges. We chose the parameter
of Bernoulli distribution so that each node had K neighbors on average. In order to analyze the
dependence of the performance on the size and connectivity of a network, we varied N and K.
Figure 1: Results from the experiments on the synthetic data (See text for details.)
We compare our algorithm using L1 regularization against no regularization and L2 regularization
in three different ways. Figure 1 summarizes our results on this data sets, and includes information
about the synthetic networks used for each experiment. The method labeled ?True? simply learns
the parameters given the true model. In Figure 1(a), we measure performance using CMLL and
reconstruction error as the number of training examples increases. As expected, L1 produces the
biggest improvement when the number of training instances is small, whereas L2 and None are more
prone to overfitting. This effect is much more pronounced when measuring the Hamming distance,
the number of disagreeing edges between the learned structure and the true structure. The figure
shows that L2 and None learn many spurious edges. Not surprisingly, L1 shows sparser distribution
on the weights, thereby it has smaller number of edges with non-negligible weights; the structures
from None and L2 tend to have many edges with small values. In Figure 1(b), we plot performance
as a function of the density of the synthetic network. As the synthetic network gets denser, L1
increasingly outperforms the other algorithms. This may be because as the graph gets more dense,
each node is indirectly correlated with more other nodes. Therefore, the feature induction algorithm
is more likely to introduce an spurious edge, which L1 may later remove, whereas None and L2 do
not. In Figure 1(c), we measure the wall-clock time as a function of the size of the synthetic network.
Figure 1(c) shows that the computational cost of learning the structure of the network using Gain-L1
not much more than that of learning the parameters alone. Moreover, L1 increasingly outperforms
other regularization methods as the number of nodes increases.
Experiments on MNIST Digit Dataset. Moving to real data, we applied our algorithm to handwritten digits. The MNIST training set consists of 32 ? 32 binary images of handwritten digits. In
order to speed up inference and learning, we resized the image to 16 ? 16. We trained the model
where each pixel is a variable for each digit separately, using a training set consisting of 189?195
images per digit. For each digit, we used 50 images as training instances and the remainder as test
instances.
Figure 2(a) compares CMLL of the different methods. To save space, we show the digits on which
the relative difference in performance of L1 compared to the next best competitor is the lowest (digit
5) and highest (digit 0), as well as the average performance. As mentioned earlier, the performance
Figure 2: Results from the experiments on MNIST dataset
of the regularized algorithm should be insensitive to the feature induction method, assuming inference is exact. However, in practice, because inference is approximate, an induction algorithm that
introduces spurious features will affect the quality of inference, and therefore the performance of
the algorithm. This effect is substantiated by the poor performance of the Simple-L1 and SimpleL2 methods that introduce features based on mutual information rather than gradient (Grad-) or
approximate gain (Gain-). Nevertheless L1 still outperforms None and L2, regardless the feature
induction algorithm with which it is paired. Figure 2(b) shows a visualization of the MRF learned
when modeling digits 4 and 7. Of course, one would expect many short-range interactions, such as
the associativity between neighboring pixels, and the algorithm does indeed capture these relationships. (They are not shown in the graph to simplify the analysis of the relationships.) Interestingly,
the algorithm picks up long-range interactions, which presumably allow the algorithm to model the
variations in the size and shape of hand-written digits.
Experiments on Human Genetic Variation Data. The Human HapMap data set 1 represents
the genetic variation over human individuals. Six data sets contain the genotype values over 6141,052 genetic markers (SNPs) from 120 individuals. For each data set, we learned the structure of
the Markov network whose nodes are binary valued SNPs such that it captures the structure of the
human genetic variation. Figure 2(c) compares CMLLs among three methods for these data sets.
For all data sets, L1 shows better performance than L2 and None.
7 Discussion and Future Work
We have presented a simple and effective method for learning the structure of Markov networks. We
view the structure learning problem as an L1 -regularized parameter estimation task, allowing it to be
solved using convex optimization techniques. We show that the computational cost of our method is
not considerably greater than pure parameter estimation for a fixed structure, suggesting that MRF
structure learning is a feasible option for many applications.
1
The Human HapMap data are available at: http://www.hapmap.org.
There are some important directions in which our work can be extended. Currently, our method
handles each feature in the log-linear model independently, with no attempt to bias the learning
towards sparsity in the structure of the induced Markov network. We can extend our approach to
introduce such a bias by using a variant of L1 regularization that penalizes blocks of parameters
together, such as the block-L1-norm of [2].
From a theoretical perspective, it would be interesting to show that, at the large sample limit,
redundant features are eventually eliminated, so that the learning eventually converges to a minimal
structure consistent with the underlying distribution. Similar results were shown by Donoho [8], and
can perhaps be adapted to this case.
A key limiting factor in MRF learning, and in our approach, is the fact that it requires inference
over the model. While our experiments suggest that approximate inference is a viable solution, as
the network structure becomes dense, its performance does degrade, especially as the approximate
gradient does not always move the parameters to 0, diminishing the sparsifying effect of the L 1
regularization, and rendering the inference even less precise. It would be interesting to explore
inference methods whose goal is correctly estimating the direction (even if not the magnitude) of
the gradient.
Finally, it would be interesting to explore the viability of the learned network structures in realworld applications, both for density estimation and for knowledge discovery, for example, in the
context of the HapMap data.
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2,209 | 3,004 | Towards a general independent subspace analysis
Fabian J. Theis
Max Planck Institute for Dynamics and Self-Organisation &
Bernstein Center for Computational Neuroscience
Bunsenstr. 10, 37073 G?ottingen, Germany
[email protected]
Abstract
The increasingly popular independent component analysis (ICA) may only be applied to data following the generative ICA model in order to guarantee algorithmindependent and theoretically valid results. Subspace ICA models generalize the
assumption of component independence to independence between groups of components. They are attractive candidates for dimensionality reduction methods,
however are currently limited by the assumption of equal group sizes or less general semi-parametric models. By introducing the concept of irreducible independent subspaces or components, we present a generalization to a parameter-free
mixture model. Moreover, we relieve the condition of at-most-one-Gaussian by
including previous results on non-Gaussian component analysis. After introducing this general model, we discuss joint block diagonalization with unknown block
sizes, on which we base a simple extension of JADE to algorithmically perform
the subspace analysis. Simulations confirm the feasibility of the algorithm.
1 Independent subspace analysis
A random vector Y is called an independent component of the random vector X, if there exists
an invertible matrix A and a decomposition X = A(Y, Z) such that Y and Z are stochastically
independent. The goal of a general independent subspace analysis (ISA) or multidimensional independent component analysis is the decomposition of an arbitrary random vector X into independent
components. If X is to be decomposed into one-dimensional components, this coincides with ordinary independent component analysis (ICA). Similarly, if the independent components are required
to be of the same dimension k, then this is denoted by multidimensional ICA of fixed group size k
or simply k-ISA. So 1-ISA is equivalent to ICA.
1.1 Why extend ICA?
An important structural aspect in the search for decompositions is the knowledge of the number of
solutions i.e. the indeterminacies of the problem. Without it, the result of any ICA or ISA algorithm
cannot be compared with other solutions, so for instance blind source separation (BSS) would be
impossible. Clearly, given an ISA solution, invertible transforms in each component (scaling matrices L) as well as permutations of components of the same dimension (permutation matrices P) give
again an ISA of X. And indeed, in the special case of ICA, scaling and permutation are already all
indeterminacies given that at most one Gaussian is contained in X [6]. This is one of the key theoretical results in ICA, allowing the usage of ICA for solving BSS problems and hence stimulating
many applications. It has been shown that also for k-ISA, scalings and permutations as above are
the only indeterminacies [11], given some additional rather weak restrictions to the model.
However, a serious drawback of k-ISA (and hence of ICA) lies in the fact that the requirement
fixed group-size k does not allow us to apply this analysis to an arbitrary random vector. Indeed,
crosstalking error
4
3
2
1
0
FastICA
JADE
Extended Infomax
Figure 1: Applying ICA to a random vector X = AS that does not fulfill the ICA model; here S
is chosen to consist of a two-dimensional and a one-dimensional irreducible component. Shown are
the statistics over 100 runs of the Amari error of the random original and the reconstructed mixing
matrix using the three ICA-algorithms FastICA, JADE and Extended Infomax. Clearly, the original
mixing matrix could not be reconstructed in any of the experiments. However, interestingly, the
latter two algorithms do indeed find an ISA up to permutation, which will be explained in section 3.
theoretically speaking, it may only be applied to random vectors following the k-ISA blind source
separation model, which means that they have to be mixtures of a random vector that consists of
independent groups of size k. If this is the case, uniqueness up to permutation and scaling holds as
noted above; however if k-ISA is applied to any random vector, a decomposition into groups that are
only ?as independent as possible? cannot be unique and depends on the contrast and the algorithm.
In the literature, ICA is often applied to find representations fulfilling the independence condition as
well as possible, however care has to be taken; the strong uniqueness result is not valid any more,
and the results may depend on the algorithm as illustrated in figure 1.
This work aims at finding an ISA model that allows applicability to any random vector. After reviewing previous approaches, we will provide such a model together with a corresponding uniqueness
result and a preliminary algorithm.
1.2 Previous approaches to ISA for dependent component analysis
Generalizations of the ICA model that are to include dependencies of multiple one-dimensional
components have been studied for quite some time. ISA in the terminology of multidimensional ICA
has first been introduced by Cardoso [4] using geometrical motivations. His model as well as the
related but independently proposed factorization of multivariate function classes [9] is quite general,
however no identifiability results were presented, and applicability to an arbitrary random vector
was unclear; later, in the special case of equal group sizes (k-ISA) uniqueness results have been
extended from the ICA theory [11]. Algorithmic enhancements in this setting have been recently
studied by [10]. Moreover, if the observation contain additional structures such as spatial or temporal
structures, these may be used for the multidimensional separation [13].
Hyv?arinen and Hoyer presented a special case of k-ISA by combining it with invariant feature subspace analysis [7]. They model the dependence within a k-tuple explicitly and are therefore able
to propose more efficient algorithms without having to resort to the problematic multidimensional
density estimation. A related relaxation of the ICA assumption is given by topographic ICA [8],
where dependencies between all components are assumed and modelled along a topographic structure (e.g. a 2-dimensional grid). Bach and Jordan [2] formulate ISA as a component clustering
problem, which necessitates a model for inter-cluster independence and intra-cluster dependence.
For the latter, they propose to use a tree-structure as employed by their tree dependepent component
analysis. Together with inter-cluster independence, this implies a search for a transformation of the
mixtures into a forest i.e. a set of disjoint trees. However, the above models are all semi-parametric
and hence not fully blind. In the following, no additional structures are necessary for the separation.
1.3 General ISA
Definition 1.1. A random vector S is said to be irreducible if it contains no lower-dimensional
independent component. An invertible matrix W is called a (general) independent subspace analysis
of X if WX = (S1 , . . . , Sk ) with pairwise independent, irreducible random vectors Si .
Note that in this case, the Si are independent components of X. The idea behind this definition is
that in contrast to ICA and k-ISA, we do not fix the size of the groups Si in advance. Of course,
some restriction is necessary, otherwise no decomposition would be enforced at all. This restriction
is realized by allowing only irreducible components. The advantage of this formulation now is that
it can clearly be applied to any random vector, although of course a trivial decomposition might be
the result in the case of an irreducible random vector. Obvious indeterminacies of an ISA of X are,
as mentioned above, scalings i.e. invertible transformations within each Si and permutation of Si
of the same dimension1. These are already all indeterminacies as shown by the following theorem,
which extends previous results in the case of ICA [6] and k-ISA [11], where also the additional
slight assumptions on square-integrability i.e. on existing covariance have been made.
Theorem 1.2. Given a random vector X with existing covariance and no Gaussian independent
component, then an ISA of X exists and is unique except for scaling and permutation.
Existence holds trivially but uniqueness is not obvious. Due to the limited space, we only give
a short sketch of the proof in the following. The uniqueness result can easily be formulated as a
subspace extraction problem, and theorem 1.2 follows readily from
Lemma 1.3. Let S = (S1 , . . . , Sk ) be a square-integrable decomposition of S into irreducible
independent components Si . If X is an irreducible component of S, then X ? Si for some i.
Here the equivalence relation ? denotes equality except for an invertible transformation. The following two lemmata each give a simplification of lemma 1.3 by ordering the components Si according
to their dimensions. Some care has to be taken when showing that lemma 1.5 implies lemma 1.4.
Lemma 1.4. Let S and X be defined as in lemma 1.3. In addition assume that dim Si = dim X for
i ? l and dim Si < dim X for i > l. Then X ? Si for some i ? l.
Lemma 1.5. Let S and X be defined as in lemma 1.4, and let l = 1 and k = 2. Then X ? S1 .
In order to prove lemma 1.5 (and hence the theorem), it is sufficient to show the following lemma:
Lemma 1.6. Let S = (S1 , S2 ) with S1 irreducible and m := dim S1 > dim S2 =: n. If X = AS
is again irreducible for some m ? (m + n)-matrix A, then (i) the left m ? m-submatrix of A is
invertible, and (ii) if X is an independent component of S, the right m ? n-submatrix of A vanishes.
(i) follows after some linear algebra, and is necessary to show the more difficult part (ii). For this,
we follow the ideas presented in [12] using factorization of the joint characteristic function of S.
1.4 Dealing with Gaussians
In the previous section, Gaussians had to be excluded (or at most one was allowed) in order to
avoid additional indeterminacies. Indeed, any orthogonal transformation of two decorrelated hence
independent Gaussians is again independent, so clearly such a strong identification result would not
be possible.
Recently, a general decomposition model dealing with Gaussians was proposed in the form of the socalled non-Gaussian subspace analysis (NGSA) [3]. It tries to detect a whole non-Gaussian subspace
within the data, and no assumption of independence within the subspace is made. More precisely,
given a random vector X, a factorization X = AS with an invertible matrix A, S = (SN , SG )
and SN a square-integrable m-dimensional random vector is called an m-decomposition of X if
SN and SG are stochastically independent and SG is Gaussian. In this case, X is said to be mdecomposable. X is denoted to be minimally n-decomposable if X is not (n ? 1)-decomposable.
According to our previous notation, SN and SG are independent components of X. It has been
shown that the subspaces of such decompositions are unique [12]:
1
Note that scaling here implies a basis change in the component Si , so for example in the case of a twodimensional source component, this might be rotation and sheering. In the example later in figure 3, these
indeterminacies can easily be seen by comparing true and estimated sources.
Theorem 1.7 (Uniqueness of NGSA). The mixing matrix A of a minimal decomposition is unique
except for transformations in each of the two subspaces.
Moreover, explicit algorithms can be constructed for identifying the subspaces [3]. This result enables us to generalize theorem 1.2and to get a general decomposition theorem, which characterizes
solutions of ISA.
Theorem 1.8 (Existence and Uniqueness of ISA). Given a random vector X with existing covariance, an ISA of X exists and is unique except for permutation of components of the same dimension
and invertible transformations within each independent component and within the Gaussian part.
Proof. Existence is obvious. Uniqueness follows after first applying theorem 1.7 to X and then
theorem 1.2 to the non-Gaussian part.
2 Joint block diagonalization with unknown block-sizes
Joint diagonalization has become an important tool in ICA-based BSS (used for example in JADE)
or in BSS relying on second-order temporal decorrelation. The task of (real) joint diagonalization
(JD) of a set of symmetric real n?n matrices M := {M1 , . . . , MK } is to find an orthogonal matrix
? := PK kE
? ? Mk E
??
E such that E? Mk E is diagonal for all k = 1, . . . , K i.e. to minimizef (E)
k=1
?
2
?
?
?
diagM(E Mk E)kF with respect to the orthogonal matrix E, where diagM(M) produces a matrix
where all off-diagonal elements of M have been set to zero, and kMk2F := tr(MM? ) denotes the
squared Frobenius norm. The Frobenius norm is invariant under conjugation by an orthogonal ma? := PK k diag(E
? ? Mk E)k
? 2 , where now
trix, so minimizing f is equivalent to maximizing g(E)
k=1
diag(M) := (mii )i denotes the diagonal of M. For the actual minimization of f respectively maximization of g, we will use the common approach of Jacobi-like optimization by iterative applications
of Givens rotation in two coordinates [5].
2.1 Generalization to blocks
In the following we will use a generalization of JD in order to solve ISA problems. Instead of fully
diagonalizing all n ? n matrices Mk ? M, in joint block diagonalization (JBD) of M we want to
determine E such that E? Mk E is block-diagonal. Depending on the application, we fix the blockstructure in advance or try to determine it from M. We are not interested in the order of the blocks,
so the block-structure is uniquely specified by fixing a partition of n i.e. a way of writing n as a sum
of positive integers, where the order of the addends is not significant. So let2 n = m1 + . . . + mr
with m1 ? m2 ? . . . ? mr and set m := (m1 , . . . , mr ) ? Nr . An n ? n matrix is said to be
m-block diagonal if it is of the form
?
?
D1 ? ? ?
0
? ..
.. ?
..
? .
.
. ?
0 ? ? ? Dr
with arbitrary mi ? mi matrices Di .
As generalization of JD in the case of known the block structure, we can formulate the joint m? := PK kE
? ? Mk E
? ?
block diagonalization (m-JBD) problem as the minimization of f m (E)
k=1
m ??
m
2
?
?
diagM (E Mk E)kF with respect to the orthogonal matrix E, where diagM (M) produces a
m-block diagonal matrix by setting all other elements of M to zero. In practice due to estimation
errors, such E will not exist, so we speak of approximate JBD and imply minimizing some errormeasure on non-block-diagonality. Indeterminacies of any m-JBD are m-scaling i.e. multiplication
by an m-block diagonal matrix from the right, and m-permutation defined by a permutation matrix
that only swaps blocks of the same size.
Finally, we speak of general JBD if we search for a JBD but no block structure is given; instead it is to be determined from the matrix set. For this it is necessary to require a block
2
We do not use the convention from Ferrers graphs of specifying partitions in decreasing order, as a visualization of increasing block-sizes seems to be preferable in our setting.
structure of maximal length, otherwise trivial solutions or ?in-between? solutions could exist (and
obviously contain high indeterminacies). Formally, E is said to be a (general) JBD of M if
(E, m) = argmaxm | ?E:f m (E)=0 |m|. In practice due to errors, a true JBD would always result
in the trivial decomposition m = (n), so we define an approximate general JBD by requiring
f m (E) < ? for some fixed constant ? > 0 instead of f m (E) = 0.
2.2 JBD by JD
A few algorithms to actually perform JBD have been proposed, see [1] and references therein. In
the following we will simply perform joint diagonalization and then permute the columns of E to
achieve block-diagonality ? in experiments this turns out to be an efficient solution to JBD [1].
This idea has been formulated in a conjecture [1] essentially claiming that a minimum of the JD cost
function f already is a JBD i.e. a minimum of the function f m up to a permutation matrix. Indeed,
in the conjecture it is required to use the Jacobi-update algorithm from [5], but this is not necessary,
and we can prove the conjecture partially:
We want to show that JD implies JBD up to permutation, i.e. if E is a minimum of f , then there
exists a permutation P such that f m (EP) = 0 (given existence of a JBD of M). But of course
f (EP) = f (E), so we will show why (certain) JBD solutions are minima of f . However, JD might
have additional minima. First note that clearly not any JBD minimizes f , only those such that in
each block of size mk , f (E) when restricted to the block is maximal over E ? O(mk ). We will call
such a JBD block-optimal in the following.
Theorem 2.1. Any block-optimal JBD of M (zero of f m ) is a local minimum of f .
Proof. Let E ? O(n) be block-optimal with f m (E) = 0. We have to show that E is a local
minimum of f or equivalently a local maximum of the squared diagonal sum g. After substituting
each Mk by E? Mk E, we may already assume that Mk is m-block diagonal, so we have to show
that E = I is a local maximum of g.
Consider the elementary Givens rotation Gij (?) defined for i < j and ??? (?1, 1) as the orthogonal
matrix, where all diagonal elements are 1 except for the two elements 1 ? ?2 in rows i and j and
with all off-diagonal elements equal to 0 except for the two elements ? and ?? at (i, j) and (j, i),
respectively. It can be used to construct local coordinates
Q of the d := n(n ? 1)/2-dimensional
manifold O(n) at I, simply by ?(?12 , ?13 , . . . , ?n?1,n ) := i<j Gij (?ij ) This is an embedding, and
?(0) = I, so we only have to show that h(??) := g(?(??)) has a local maximum at ? = 0. We do this
by considering h partially in each coordinate. Let i < j. If i, j are in the same block of m, then h is
locally maximal i.e. negative semi-definite at 0 in the direction ?ij because of block-optimality.
Now assume i and j are from different blocks. After possible permutation, we may assume that j =
i + 1 so that each matrix Mk ? M has (Mk )ij = (Mk )ji = 0, and ak := (Mk )ii , bk := (Mk )jj .
Then Gij (?)? Mk Gij (?) can be easily calculated at coordinates (i, i) to (j, j), and indeed entries
on the diagonal other than at indices (i, i) and (j, j) are not changed, so
k diag(Gij (?)? Mk Gij (?))k2 ? k diag(Mk )k2 =
= ?2ak (ak ? bk )?2 + 2bk (ak ? bk )?2 + 2(ak ? bk )2 ?4
= ?2(a2k + b2k )?2 + 2(ak ? bk )2 ?4 .
P
2
2
Hence h(0, . . . , 0, ?ij , 0, . . . , 0) ? h(0) = ?c?2ij + d?4ij with c = 2 K
k=1 (ak + bk ) and d =
PK
2 k=1 (ak ? bk )2 . Now either c = 0, then also d = 0 and h is constant zero in the direction
?ij . Or, more interestingly, c 6= 0, then c > 0 and therefore h is negative definite in the direction
?ij . Altogether we get a negative definite h at 0 except for ?trivial directions?, and hence a local
maximum at 0.
2.3 Recovering the permutation
In order to perform JBD, we therefore only have to find a JD E of M. What is left according to the
above theorem is to find a permutation matrix P such that EP block-diagonalizes M. In the case of
known block-order m, we can employ similar techniques as used in [1, 10], which essentially find
P by some combinatorial optimization.
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??
(a) (unknown) block diagonal M1 (b) E E w/o recovered permutation
5
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??
(c) E E
Figure 2: Performance of the proposed general JBD algorithm in the case of the (unknown) blockpartition 40 = 1 + 2 + 2 + 3 + 3 + 5 + 6 + 6 + 6 + 6 in the presence of noise with SNR of 5dB. The
? ? E of the inverse of the estimated block diagonalizer and the original one is an m-block
product E
diagonal matrix except for permutation within groups of the same sizes as claimed in section 2.2.
In the case of unknown block-size, we propose to use the following simple permutation-recovery
PK
algorithm: consider the mean diagonalized matrix D := K ?1 k=1 E? Mk E. Due to the assumption that M is m-block-diagonalizable (with unknown m), each E? Mk E and hence also D must
be m-block-diagonal except for a permutation P, so it must have the corresponding number of zeros
in each column and row. In the approximate JBD case, thresholding with a threshold ? is necessary,
whose choice is non-trivial.
We propose using algorithm 1 to recover the permutation; we denote its resulting permuted matrix by
P(D) when applied to the input D. P(D) is constructed from possibly thresholded D by iteratively
permuting columns and rows in order to guarantee that all non-zeros of D are clustered along the
diagonal as closely as possible. This recovers the permutation as well as the partition m of n.
Algorithm 1: Block-diagonality permutation finder
Input: (n ? n)-matrix D
Output: block-diagonal matrix P(D) := D? such that D? = PDPT for a permutation matrix P
D? ? D
for i ? 1 to n do
repeat
if (j0 ? min{j|j ? i and d?ij = 0 and d?ji = 0}) exists then
if (k0 ? min{k|k > j0 and (d?ik 6= 0 or d?ki 6= 0)}) exists then
swap column j0 of D? with column k0
swap row j0 of D? with row k0
until no swap has occurred ;
We illustrate the performance of the proposed JBD algorithm as follows: we generate a set of K =
100 m-block-diagonal matrices Dk of dimension 40 ? 40 with m = (1, 2, 2, 3, 3, 5, 6, 6, 6, 6).
They have been generated in blocks of size m with coefficients chosen randomly uniform from
[?1, 1], and symmetrized by Dk ? (Dk + D?
k )/2. After that, they have been mixed by a random
orthogonal mixing matrix E ? O(40), i.e. Mk := EDk E? + N, where N is a noise matrix with
independent Gaussian entries such that the resulting signal-to-noise ratio is 5dB. Application of
the JBD algorithm from above to {M1 , . . . , MK } with threshold ? = 0.1 correctly recovers the
? equals E up to m-scaling and permutation, as
block sizes, and the estimated block diagonalizer E
illustrated in figure 2.
3 SJADE ? a simple algorithm for general ISA
As usual by preprocessing of the observations X by whitening we may assume that Cov(X) = I.
The indeterminacies allow scaling transformations in the sources, so without loss of generality let
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(f) (S?1 , S?2 )
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(g) histogram of S?3
0
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?1.5
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(i) S5
?4
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?2.5
1
?5
0
(j) S6
Figure 3: Example application of general ISA for unknown sizes m = (1, 2, 2, 2, 3). Shown are the
? ?1 A.
scatter plots i.e. densities of the source components and the mixing-separating map A
also Cov(S) = I. Then I = Cov(X) = A Cov(S)A? = AA? so A is orthogonal. Due to the ISA
assumptions, the fourth-order cross cumulants of the sources have to be trivial between different
groups, and within the Gaussians. In order to find transformations of the mixtures fulfilling this
property, we follow the idea of the JADE algorithmbut now in the ISA setting. We perform JBD of
the (whitened) contracted quadricovariance matrices defined by Cij (X) := E X? Eij XXX? ?
Eij ? E?
ij ? tr(Eij )I. Here RX := Cov(X) and Eij is a set of eigen-matrices of Cij , 1 ? i, j ? n.
One simple choice is to use n2 matrices Eij with zeros everywhere except 1 at index (i, j). More
elaborate choices of eigen-matrices (with only n(n + 1)/2 or even n entries) are possible. The
resulting algorithm, subspace-JADE (SJADE) not only performs NGCA by grouping Gaussians as
one-dimensional components with trivial Cii ?s, but also automatically finds the subspace partition
m using the general JBD algorithm from section 2.3.
4 Experimental results
In a first example, we consider a general ISA problem in dimension n = 10 with the unknown
partition m = (1, 2, 2, 2, 3). In order to generate 2- and 3-dimensional irreducible random vectors,
we decided to follow the nice visual ideas from [10] and to draw samples from a density following
a known shape ? in our case 2d-letters or 3d-geometrical shapes. The chosen sources densities are
shown in figure 3(a-d). Another 1-dimensional source following a uniform distribution was constructed. Altogether 104 samples were used. The sources S were mixed by a mixing matrix A with
coefficients uniformly randomly sampled from [?1, 1] to give mixtures X = AS. The recovered
? was then estimated using the above block-JADE algorithm with unknown block
mixing matrix A
size; we observed that the method is quite sensitive to the choice of the threshold (here ? = 0.015).
? ?1 A; clearly the matrices are equal
Figure 3(e) shows the composed mixing-separating system A
except for block permutation and scaling, which experimentally confirms theorem 1.8. The algo? = (1, 1, 1, 2, 2, 3), so one 2d-source was misinterpreted as two 1d-sources,
rithm found a partition m
but by using previous knowledge combination of the correct two 1d-sources yields the original 2d? := A
? ?1 X, figures 3(f-j), then equal the original sources
source. The resulting recovered sources S
except for permutation and scaling within the sources ? which in the higher-dimensional cases
implies transformations such as rotation of the underlying images or shapes. When applying ICA
(1-ISA) to the above mixtures, we cannot expect to recover the original sources as explained in
figure 1; however, some algorithms might recover the sources up to permutation. Indeed, SJADE
equals JADE with additional permutation recovery because the joint block diagonalization is per-
formed using joint diagonalization. This explains why JADE retrieves meaningful components even
in this non-ICA setting as observed in [4].
In a second example, we illustrate how the algorithm deals with Gaussian sources i.e. how the
subspace JADE also includes NGCA. For this we consider the case n = 5, m = (1, 1, 1, 2) and
sources with two Gaussians, one uniform and a 2-dimensional irreducible component as before;
105 samples were drawn. We perform 100 Monte-Carlo simulations with random mixing matrix
? is compared with A by
A, and apply SJADE with ? = 0.01. The recovered mixing matrix A
P3 P2
2
2
?1
?
taking the ad-hoc measure ?(P) := i=1 j=1 (pij + pji ) for P := A A. Indeed, we get nearly
perfect recovery in 99 out of 100 runs, the median of ?(P) is very low with 0.0083. A single run
diverges with ?(P ) = 3.48. In order to show that the algorithm really separates the Gaussian part
from the other components, we compare the recovered source kurtoses. The median kurtoses are
?0.0006 ? 0.02, ?0.003 ? 0.3, ?1.2 ? 0.3, ?1.2 ? 0.2 and ?1.6 ? 0.2. The first two components
have kurtoses close to zero, so they are the two Gaussians, whereas the third component has kurtosis
of around ?1.2, which equals the kurtosis of a uniform density. This confirms the applicability of
the algorithm in the general, noisy ISA setting.
5 Conclusion
Previous approaches for independent subspace analysis were restricted either to fixed group sizes
or semi-parametric models. In neither case, general applicability to any kind of mixture data set
was guaranteed, so blind source separation might fail. In the present contribution we introduce the
concept of irreducible independent components and give an identifiability result for this general,
parameter-free model together with a novel arbitrary-subspace-size algorithm based on joint block
diagonalization. As in ICA, the main uniqueness theorem is an asymptotic result (but includes
noisy case via NGCA). However in practice in the finite sample case, due to estimation errors the
general joint block diagonality only approximately holds. Our simple solution in this contribution
was to choose appropriate thresholds. But this choice is non-trivial, and adaptive methods are to be
developed in future works.
References
[1] K. Abed-Meraim and A. Belouchrani. Algorithms for joint block diagonalization. In Proc. EUSIPCO
2004, pages 209?212, Vienna, Austria, 2004.
[2] F.R. Bach and M.I. Jordan. Finding clusters in independent component analysis. In Proc. ICA 2003, pages
891?896, 2003.
[3] G. Blanchard, M. Kawanabe, M. Sugiyama, V. Spokoiny, and K.-R. M?uller. In search of non-gaussian
components of a high-dimensional distribution. JMLR, 7:247?282, 2006.
[4] J.F. Cardoso. Multidimensional independent component analysis. In Proc. of ICASSP ?98, Seattle, 1998.
[5] J.F. Cardoso and A. Souloumiac. Jacobi angles for simultaneous diagonalization. SIAM J. Mat. Anal.
Appl., 17(1):161?164, January 1995.
[6] P. Comon. Independent component analysis - a new concept? Signal Processing, 36:287?314, 1994.
[7] A. Hyv?arinen and P.O. Hoyer. Emergence of phase and shift invariant features by decomposition of
natural images into independent feature subspaces. Neural Computation, 12(7):1705?1720, 2000.
[8] A. Hyv?arinen, P.O. Hoyer, and M. Inki. Topographic independent component analysis. Neural Computation, 13(7):1525?1558, 2001.
[9] J.K. Lin. Factorizing multivariate function classes. In Advances in Neural Information Processing Systems, volume 10, pages 563?569, 1998.
[10] B. Poczos and A. L?orincz. Independent subspace analysis using k-nearest neighborhood distances. In
Proc. ICANN 2005, volume 3696 of LNCS, pages 163?168, Warsaw, Poland, 2005. Springer.
[11] F.J. Theis. Uniqueness of complex and multidimensional independent component analysis. Signal Processing, 84(5):951?956, 2004.
[12] F.J. Theis and M. Kawanabe. Uniqueness of non-gaussian subspace analysis. In Proc. ICA 2006, pages
917?925, Charleston, USA, 2006.
[13] R. Vollgraf and K. Obermayer. Multi-dimensional ICA to separate correlated sources. In Proc. NIPS
2001, pages 993?1000, 2001.
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2,210 | 3,005 | Large-Scale Sparsified Manifold Regularization
Ivor W. Tsang
James T. Kwok
Department of Computer Science and Engineering
The Hong Kong University of Science and Technology
Clear Water Bay, Kowloon, Hong Kong
{ivor,jamesk}@cse.ust.hk
Abstract
Semi-supervised learning is more powerful than supervised learning by using both
labeled and unlabeled data. In particular, the manifold regularization framework,
together with kernel methods, leads to the Laplacian SVM (LapSVM) that has
demonstrated state-of-the-art performance. However, the LapSVM solution typically involves kernel expansions of all the labeled and unlabeled examples, and
is slow on testing. Moreover, existing semi-supervised learning methods, including the LapSVM, can only handle a small number of unlabeled examples. In this
paper, we integrate manifold regularization with the core vector machine, which
has been used for large-scale supervised and unsupervised learning. By using a
sparsified manifold regularizer and formulating as a center-constrained minimum
enclosing ball problem, the proposed method produces sparse solutions with low
time and space complexities. Experimental results show that it is much faster than
the LapSVM, and can handle a million unlabeled examples on a standard PC;
while the LapSVM can only handle several thousand patterns.
1
Introduction
In many real-world applications, collection of labeled data is both time-consuming and expensive.
On the other hand, a large amount of unlabeled data are often readily available. While traditional
supervised learning methods can only learn from the limited amount of labeled data, semi-supervised
learning [2] aims at improving the generalization performance by utilizing both the labeled and
unlabeled data. The label dependencies among patterns are captured by exploiting the intrinsic
geometric structure of the data. The underlying smoothness assumption is that two nearby patterns
in a high-density region should share similar labels [2]. When the data lie on a manifold, it is
common to approximate this manifold by a weighted graph, leading to graph-based semi-supervised
learning methods. However, many of these are designed for transductive learning, and thus cannot
be easily extended to out-of-sample patterns.
Recently, attention is drawn to the development of inductive methods, such as harmonic mixtures
[15] and Nystr?om-based methods [3]. In this paper, we focus on the manifold regularization framework proposed in [1]. By defining a data-dependent reproducing kernel Hilbert space (RKHS),
manifold regularization incorporates an additional regularizer to ensure that the learned function is
smooth on the manifold. Kernel methods, which have been highly successful in supervised learning, can then be integrated with this RKHS. The resultant Laplacian SVM (LapSVM) demonstrates
state-of-the-art semi-supervised learning performance [10]. However, a deficiency of the LapSVM
is that its solution, unlike that of the SVM, is not sparse and so is much slower on testing.
Moreover, while the original motivation of semi-supervised learning is to utilize the large amount
of unlabeled data available, existing algorithms are only capable of handling a small to moderate
amount of unlabeled data. Recently, attempts have been made to scale up these methods. Sindhwani
et al. [9] speeded up manifold regularization by restraining to linear models, which, however, may
not be flexible enough for complicated target functions. Garcke and Griebel [5] proposed to use
discretization with a sparse grid. Though it scales linearly with the sample size, its time complexity
grows exponentially with data dimensionality. As reported in a recent survey [14], most semisupervised learning methods can only handle 100 ? 10,000 unlabeled examples. More recently,
G?artner et al. [6] presented a solution in the more restrictive transductive setting. The largest graph
they worked with involve 75,888 labeled and unlabeled examples. Thus, no one has ever been
experimented on massive data sets with, say, one million unlabeled examples.
On the other hand, the Core Vector Machine (CVM) is recently proposed for scaling up kernel methods in both supervised (including classification [12] and regression [13]) and unsupervised learning
(e.g., novelty detection). Its main idea is to formulate the learning problem as a minimum enclosing
ball (MEB) problem in computational geometry, and then use an (1 + )-approximation algorithm
to obtain a close-to-optimal solution efficiently. Given m samples, the CVM has an asymptotic time
complexity that is only linear in m and a space complexity that is even independent of m for a fixed
. Experimental results on real world data sets with millions of patterns demonstrated that the CVM
is much faster than existing SVM implementations and can handle much larger data sets.
In this paper, we extend the CVM to semi-supervised learning. To restore sparsity of the LapSVM
solution, we first introduce a sparsified manifold regularizer based on the -insensitive loss. Then, we
incorporate manifold regularization into the CVM. It turns out that the resultant QP can be casted as
a center-constrained MEB problem introduced in [13]. The rest of this paper is organized as follows.
In Section 2, we first give a brief review on manifold regularization. Section 3 then describes the
proposed algorithm for semi-supervised classification and regression. Experimental results on very
large data sets are presented in Section 4, and the last section gives some concluding remarks.
2
Manifold Regularization
Given a training set {(xi , yi )}m
i=1 with input xi ? X and output yi ? R. The regularized risk
functional is the sum of the empirical risk (corresponding to a loss function `) and a regularizer ?.
Given a kernel k and its RKHS Hk , we minimize the regularized risk over function f in Hk :
m
min
f ?Hk
1 X
`(xi , yi , f (xi )) + ??(kf kHk ).
m i=1
(1)
Here, k ? kHk denotes the RKHS norm and ? > 0 is a regularization
Pm parameter. By the representer
theorem, the minimizer f admits the representation f (x) =
i=1 ?i k(xi , x), where ?i ? R.
Therefore, the problem is reduced to the optimization over the finite-dimensional space of ? i ?s.
In semi-supervised learning, we have both labeled examples {(xi , yi )}m
i=1 and unlabeled examples
m+n
{xi }i=m+1 . Manifold regularization uses an additional regularizer kf k2I to ensure that the function
f is smooth on the intrinsic structure of the input. The objective function in (1) is then modified to:
m
1 X
`(xi , yi , f (xi )) + ??(kf kHk ) + ?I kf k2I ,
m i=1
(2)
where
It can be shown that the minimizer f is of the form f (x) =
Pm ?I is anotherRtradeoff0 parameter.
0
0
i=1 ?i k(xi , x) + M ?(x )k(x, x )dPX (x ), where M is the support of the marginal distribution
PX of X [1].
In practice, we do not have access
to PX . Now, assume that the support M of PX is a compact subR
manifold, and take kf k2I = M h?f, ?f i where ? is the gradient of f . It is common to approximate
this manifold by a weighted graph defined on all the labeled and unlabeled data, as G = (V, E) with
V and EP
being the sets of vertices and edges respectively. Denote the weight function w and degree
d(u) = v?u w(u, v). Here, v ? u means that u, v are adjacent. Then, kf k2I is approximated as1
2
X p
f (xue ) f (xve )
2
(3)
kf kI =
w(ue , ve ) s(ue ) ? s(ve ) ,
e?E
1
When the set of labeled and unlabeled data is small, a function that is smooth on this small set may not be
interesting. However, this is not an issue here as our focus is on massive data sets.
p
where ue and ve are vertices of the edge e, and s(u) = d(u) when the normalized graph Laplacian
is used, and s(u) = 1 with the unnormalized one. As shown in [1], the minimizer of (2) becomes
Pm+n
f (x) = i=1 ?i k(xi , x), which depends on both labeled and unlabeled examples.
2.1
Laplacian SVM
Using the hinge loss `(xi , yi , f (xi )) = max(0, 1 ? yi f (xi )) in (2), we obtain the Laplacian
SVM (LapSVM) [1]. Its training involves two steps. First, solve the quadratic program (QP)
1
1 to obtain the ? ? . Here, ? = [?1 , . . . , ?m ]0 , 1 =
max ? 0 1 ? 12 ? 0 Q? : ? 0 y = 0, 0 ? ? ? m
0
?1 0
[1, . . . , 1] , Qm?m = YJK(2?I + 2?I LK) J Y, Ym?m is the diagonal matrix with Yii = yi ,
K(m+n)?(m+n) is the kernel matrix over both the labeled and unlabeled data, L(m+n)?(m+n) is the
graph Laplacian, and Jm?(m+n) with Jij = 1 if i = j and xi is a labeled example, and Jij = 0
otherwise. The optimal ? = [?1 , . . . , ?m+n ]0 solution is then obtained by solving the linear system:
?? = (2?I + 2?I LK)?1 J0 Y? ? . Note that the matrix 2?I + 2?I LK is of size (m + n) ? (m + n),
and so its inversion can be very expensive when n is large. Moreover, unlike the standard SVM, the
?? obtained is not sparse and so evaluation of f (x) is slow.
3
Proposed Algorithm
3.1
Sparsified Manifold Regularizer
To restore sparsity of the LapSVM solution, we replace the square function in the manifold regularizer (3) by the -insensitive loss function2 , as
2
X p
w(ue , ve ) f (xue ) ? f (xve ) ,
kf k2I =
(4)
s(ue )
s(ve ) ??
e?E
where |z|?? = 0 if |z| ? ??; and |z| ? ?? otherwise. Obviously, it reduces to (3) when ?? = 0. As will
be shown in Section 3.3, the ? solution obtained will be sparse. Substituting (4) into (2), we have:
( m
2 )
X p
f (xue ) f (xve )
1X
`(xi , yi , f (xi ))+?I
min
w(ue , ve ) s(ue ) ? s(ve ) +??(kf kHk ).
f ?Hk m
??
i=1
e?E
By treating the terms inside the braces as the ?loss function?, this can be regarded as regularized risk
minimization and, using the standard representer theorem, the minimizer f then admits the form
Pm+n
f (x) = i=1 ?i k(xi , x), same as that of the original manifold regularization.
P
2
0
Moreover, putting f(x) = w0 ?(x) + b into (4), we obtain kfk2I = e?E |w
? e + b?e |??, where
p
p
?(x )
?(x )
? e = w(ue , ve ) s(uuee) ? s(vvee) , and ?e = w(ue , ve ) s(u1 e ) ? s(v1e ) . The primal of the
LapSVM can then be formulated as:
m
min
C X 2
C? X 2
kwk + b +
?i + 2C ?? +
(?e + ?e? 2 )
m? i=1
|E|?
2
2
(5)
e?E
s.t.
yi (w0 ?(xi ) + b) ? 1 ? ?? ? ?i , i = 1, . . . , m,
?(w0 ? e + b?e ) ? ?? + ?e , w0 ? e + b?e ? ?? + ?e? ,
e ? E.
(6)
(7)
Here, |E| is the number of edges in the graph, ?i is the slack variable for the error, ?e , ?e? are slack
variables for edge e, and C, ?, ? are user-defined parameters. As in previous CVM formulations
2
?2
[12, 13], the bias b is penalized and the two-norm errors (?i2 , ?ij
and ?ij
) are used. Moreover, the
?
constraints ?i , ?ij , ?ij , ?? ? 0 are automatically satisfied. When ?? = 0, (5) reduces to the original
LapSVM (using two-norm errors). When ? is also zero, it becomes the Lagrangian SVM.
The dual can be easily obtained as the following QP:
max [? 0 ? 0 ? ? 0 ][
2
2 0 0 00
? 0 ? 0 ? ? 0 ]0 : [? 0 ? 0 ? ? 0 ]1 = 1, ?, ?, ? ? ? 0, (8)
1 0 0 ] ? [? 0 ? 0 ? ? 0 ]K[?
C
To avoid confusion with the in the (1 + )-approximation, we add a bar to the ? here.
? 0
? =
where ? = [?1 , . . . , ?m ]0 , ? = [?1 , . . . , ?|E| ]0 , ? ? = [?1? , . . . , ?|E|
] are the dual variables, and K
?
?
?m
0
0
?
(K` + 11 + C I) yy
V0
?V0
V
I
U + |E|?
C?
?U
?V
? is the transformed ?kernel matrix?. Here,
?U
|E|?
U + C? I
K` is the kernel matrix defined using kernel k on the m labeled examples, U|E|?|E| = [? 0e ? f +
?e ?f ], and Vm?|E| = [yi ?(xi )0 ? e + ?e ]. Note that while each entry of the matrix Q in LapSVM
? here takes only
(Section 2.1) requires O((m + n)2 ) kernel k(xi , xj ) evaluations, each entry in K
O(1) kernel evaluations. This is particularly favorable to decomposition methods such as SMO as
most of the CPU computations are typically dominated by kernel evaluations.
Moreover, it can be shown that ? is a parameter that controls the size of ??, analogous to the
? parameter in ?-SVR. Hence, only ?, but not ??, appears in (8). Moreover, the primal variables can P
be easily recoveredPfrom the dual variables
by the P
KKT conditions.
In particular,
P
m
m
?
?
w = C
i=1 ?i yi ?(xi ) +
e?E (?e ? ?e )? e and b = C
i=1 ?i yi +
e?E (?e ? ?e )?e .
Subsequently, the decision function f (x) = w 0 ?(x) + b is a linear combination of k(xi , x)?s defined on both the labeled and unlabeled examples, as in standard manifold regularization.
3.2
Transforming to a MEB Problem
We now show that CVM can be used for solving the possibly very large QP in (8). In particular, we
will transform this QP to the dual of a center-constrained MEB problem [13], which is of the form:
max ?0 (diag(K) + ? ? ?1) ? ?0 K? : ? ? 0, ?0 1 = 1,
(9)
0
for some 0 ? ? ? Rm and ? ? R. From the variables in (8), define ?
? = ? 0 ? 0 ? ? 0 and
? + ?1 + 2 [10 00 0]0 s.t. ? ? 0 for some sufficiently large ?. (8) can then be written
? = ?diag(K)
C
? + ? ? ?1) ? ?
??
as max ?
? 0 (diag(K)
? 0K
? : ?
? ? 0, ?
? 0 1 = 1, which is of the form in (9).
The above formulation can be easily extended to the regression case, with the pattern output changed
from ?1 to yi ? R, and the hinge loss replaced by the -insensitive loss. Converting the resultant
QP to the form in (9) is also straightforward.
3.3
Sparsity
In Section 3.3.1, we first explain why a sparse solution can be obtained by using the KKT conditions.
Alternatively, by building on [7], we show in Section 3.3.2 that the -insensitive loss achieves a
similar effect as the `1 penalty in LASSO [11], which is known to produce sparse approximation.
3.3.1
KKT Perspective
Basically, this follows from the standard argument as for sparse solutions with the -insensitive loss
in SVR. From the KKT condition associated with (6): ?i (yi (w0 ?(xi ) + b) ? 1 + ?? + ?i ) = 0. As
for the SVM, most patterns are expected to lie outside the margin (i.e. yi (w0 ?(xi ) + b) > 1 ? ??)
and so most ?i ?s are zero. Similarly, manifold regularization finds a f that is locally smooth. Hence,
from the definition of ? e and ?e , many values of (w0 ? e + b?e )?s will be inside the ??-tube. Using the
KKT conditions associated with (7), the corresponding ?e ?s and ?e? ?s are zero. As f (x) is a linear
combination of the k(xi , x)?s weighted by ?i and ?e ? ?e? (Section 3.1), f is thus sparse.
3.3.2
LASSO Perspective
Our exposition will be along the line pioneered by Girosi [7], who established a connection
between
the -insensitive loss in SVR and sparse approximation. Given a predictor f (x) =
Pm
0
i=1 ?i k(xi , x) = K?, we consider minimizing the error between f = [f (x 1 ), . . . f (xm )] and
y = [y1 , . . . , ym ]0 . While sparse approximation techniques such as basis pursuit typically use the
L2 norm for the error, Girosi argued that the norm of the RKHS Hk is a better measure of smoothness. However, the RKHS norm operates on functions, while here we have vectors f and y w.r.t.
x1 , . . . , xm . Hence, we will use the kernel PCA map with ky ? f k2K ? (y ? f )K?1 (y ? f ).
First, consider the simpler case where the manifold regularizer is replaced by a simple regularizer
k?k22 . As in LASSO, we also add a `1 penalty on ?. The optimization problem is formulated as:
?m 0
? ? : k?k1 = C,
(10)
min ky ? f k2K +
C
where C and ? are constants. As in [7], we decompose ? as ??? ? , where ?, ? ? ? 0 and ?i ?i? = 0.
Then, (10) can be rewritten as:
0
0 ?
0
0 ?0 0
max [? 0 ? ? ][2y0 ? 2y0 ]0 ? [? 0 ? ? ]K[?
? ] : ?, ? ? ? 0, ? 0 1 + ? ? 1 = C,
? =
where3 K
I
K + ?m
C
?K
using the -insensitive loss:
?K
K + ?m
I
C
(11)
. On the other hand, consider the following variant of SVR
m
min kwk2 +
C X 2
(? + ?i?2 ) + 2C ?? : yi ? w0 ?(xi ) ? ?? + ?i , w0 ?(xi ) ? yi ? ?? + ?i? . (12)
m? i=1 i
It can be shown that its dual is identical to (11), with ?, ? ? as dual variables. Moreover, the LASSO
penalty (i.e., the equality constraint in (11)) is induced from the ?? in (12). Hence, the -insensitive
loss in SVR achieves a similar effect as using the error ky ? f k2K and the LASSO penalty.
We now add back the manifold regularizer. The derivation is similar, though more involved, and so
details are skipped. As above, the key steps are on replacing the `2 norm by the kernel PCA map,
and adding a `1 penalty on the variables. It can then be shown that sparsified manifold regularizer
(based on the -insensitive loss) can again be recovered by using the LASSO penalty.
3.4
Complexities
As the proposed algorithm is an extension of the CVM, its properties are analogous to those in [12].
For example, its approximation ratio is (1 + )2 , and so the approximate solution obtained is very
close to the exact optimal solution. As for the computational complexities, it can be shown that the
SLapCVM only takes O(1/8 ) time and O(1/2 ) space when probabilistic speedup is used. (Here,
we ignore O(m+|E|) space required for storing the m training patterns and 2|E| edge constraints, as
these may be stored outside the core memory.) They are thus independent of the numbers of labeled
and unlabeled examples for a fixed . In contrary, LapSVM involves an expensive matrix inversion
for K(m+n)?(m+n) and requires O((m + n)3 ) time and O((m + n)2 ) space.
3.5
Remarks
The reduced SVM [8] has been used to scale up the standard SVM. Hence, another natural alternative
is to extend it for the LapSVM. This ?reduced LapSVM? solves a smaller optimization problem that
involves a random r?(m+n) rectangular subset of the kernel matrix, where the r patterns are chosen
from both the labeled and unlabeled data. It can be easily shown that it requires O((m + n) 2 r) time
and O((m + n)r) space. Experimental comparisons based on this will be made in Section 4.
Note that the CVM [12] is in many aspects similar to the column generation technique [4] commonly
used in large-scale linear or integer programs. Both start with only a small number of nonzero
variables, and the restricted master problem in column generation corresponds to the inner QP that
is solved at each CVM iteration. Moreover, both can be regarded as primal methods that maintain
primal4 feasibility and work towards dual feasibility. Also, as is typical in column generation, the
dual variable whose KKT condition is most violated is added at each iteration. The key difference 5 ,
3
For simplicity, here we have only considered the case where f does not have a bias. In the presence of a
? has to be replaced by K + 110 .
bias, it can be easily shown that K (in the expression of K)
4
By convention, column generation takes the optimization problem to be solved as the primal. Hence, in
this section, we also regard the QP to be solved as CVM?s primal, and the MEB problem as its dual. Note that
each dual variable then corresponds to a training pattern.
5
Another difference is that an entire column is added at each iteration of column generation. However, in
CVM, the dual variable added is just a pattern and the extra space required for the QP is much smaller. Besides,
there are other implementation tricks (such as probabilistic speedup) that further improves the speed of CVM.
however, is that CVM exploits the ?approximateness? as in other approximation algorithms. Instead
of requiring the dual solution to be strictly feasible, CVM only requires it to be feasible within a
factor of (1 + ). This, together with the fact that its dual is a MEB problem, allows its number of
iterations for convergence to be bounded and thus the total time and space complexities guaranteed.
On the other hand, we are not aware of any similar results for column generation.
By regarding the CVM as the approximation algorithm counterpart of column generation, this suggests that the CVM can also be used in the same way as column generation in speeding up other
optimization problems. For example, the CVM can also be used for SVM training with other loss
functions (e.g. 1-norm error). However, as the dual may no longer be a MEB problem, the downside
is that its convergence bound and complexity results in Section 3.4 may no longer be available.
4
Experiments
In this section, we perform experiments on some massive data sets 6 (Table 1). The graph (for the
manifold) is constructed by using the 6 nearest neighbors of each
w(u e , ve )
P pattern, and the weight
1
2
kx
?
x
k
.
For
simplicity,
in (3) is defined as exp(?kxue ? xve k2 /?g ), where ?g = |E|
e
e
u
v
e?E
we use the unnormalized Laplacian and so all s(?)?s in (3) are 1. The value of m? in (5) is always
fixed at 1, and the other parameters are tuned by a small validation
specified,
Pmset. Unless otherwise
1
? k2 . For comparison,
we use the Gaussian kernel exp(?kx ? zk2 /?), with ? = m
i=1 kxi ? x
we also run the LapSVM7 and another LapSVM implementation based on the reduced SVM [8]
(Section 3.5). All the experiments are performed on a 3.2GHz Pentium?4 PC with 1GB RAM.
Table 1: A summary of the data sets used.
data set
4.1
two-moons
#attrib
2
extended USPS
676
extended MIT face
361
class
+
?
+
?
+
?
#training patns
labeled unlabeled
1
500,000
1
500,000
1
144,473
1
121,604
5
408,067
5
481,909
#test patns
2,500
2,500
43,439
31,944
472
23,573
Two-Moons Data Set
We first perform experiments on the popular two-moons data set, and use one labeled example for
each class (Figure 1(a)). To better illustrate the scaling behavior, we vary the number of unlabeled
patterns used for training (from 1, 000 up to a maximum of 1 million). Following [1], the width of
the Gaussian kernel is set to ? = 0.25. For the reduced LapSVM implementation, we fix r = 200.
4
3
SLapCVM
LapSVM
Reduced LapSVM
2
10
number of kernel expansions
CPU time (in seconds)
10
1
10
0
10
?1
10
(a) Data distribution.
(b) Typical decision boundary
obtained by SLapCVM.
3
10
4
5
10
10
number of unlabeled points
(c) CPU time.
6
10
10
SLapCVM
core?set Size
LapSVM
Reduced LapSVM
3
10
2
10 3
10
4
5
10
10
number of unlabeled points
(d) #kernel
sions.
6
10
expan-
Figure 1: Results on the two-moons data set (some abscissas and ordinates are in log scale). The
two labeled examples are labeled in red in Figure 1(a).
Results are shown in Figure 1. Both the LapSVM and SLapCVM always attain 100% accuracy on
the test set, even with only two labeled examples (Figure 1(b)). However, SLapCVM is faster than
LapSVM (Figure 1(c)). Moreover, as mentioned in Section 2.1, the LapSVM solution is non-sparse
6
7
Both the USPS and MIT face data sets are downloaded from http://www.cs.ust.hk/?ivor/cvm.html.
http://manifold.cs.uchicago.edu/manifold regularization/.
and all the labeled and unlabeled examples are involved in the solution (Figure 1(d))). On the other
hand, SLapCVM uses only a small fraction of the examples. As can be seen from Figures 1(c)
and 1(d), both the time and space required by the SLapCVM are almost constant, even when the
unlabeled data set gets very large. The reduced LapSVM, though also fast, is slightly inferior to
both the SLapCVM and LapSVM. Moreover, note that both the standard and reduced LapSVMs
cannot be run on the full data set on our PC because of their large memory requirements.
4.2
Extended USPS Data Set
The second experiment is performed on the USPS data from [12]. One labeled example is randomly
sampled from each class for training. To achieve comparable accuracy, we use r = 2, 000 for the
reduced LapSVM. For comparison, we also train a standard SVM with the two labeled examples.
Results are shown in Figure 2. As can be seen, the SLapCVM is again faster (Figures 2(a)) and
produces a sparser solution than LapSVM (Figure 2(b)). For the SLapCVM, both the time required
and number of kernel expansions involved grow only sublinearly with the number of unlabeled
examples. Figure 2(c) demonstrates that semi-supervised learning (using either the LapSVMs or
SLapCVM) can have much better generalization performance than supervised learning using the
labeled examples only. Note that although the use of the 2-norm error in SLapCVM could in theory
be less robust than the use of the 1-norm error in LapSVM, the SLapCVM solution is indeed always
more accurate than that of LapSVM. On the other hand, the reduced LapSVM has comparable speed
with the SLapCVM, but its performance is inferior and cannot handle large data sets.
5
5
CPU time (in seconds)
number of kernel expansions
4
10
80
10
SLapCVM
LapSVM
Reduced LapSVM
3
10
2
10
1
10
SLapCVM
core?set Size
LapSVM
Reduced LapSVM
SLapCVM
LapSVM
Reduced LapSVM
SVM (#labeled = 2)
70
60
4
10
error rate (in %)
10
3
10
50
40
30
20
10
0
10 3
10
2
4
5
10
10
number of unlabeled points
6
10
(a) CPU time.
10 3
10
4
5
10
10
number of unlabeled points
(b) #kernel expansions.
6
10
0 3
10
4
5
10
10
number of unlabeled points
6
10
(c) Test error.
Figure 2: Results on the extended USPS data set (some abscissas and ordinates are in log scale).
4.3
Extended MIT Face Data Set
In this section, we perform face detection using the extended MIT face database in [12]. Five labeled
example are randomly sampled from each class and used in training. Because of the imbalanced
nature of the test set (Table 1), the classification error is inappropriate for performance evaluation
here. Instead, we will use the area under the ROC curve (AUC) and the balanced loss 1 ? (TP +
TN)/2, where TP and TN are the true positive and negative rates respectively. Here, faces are treated
as positives while non-faces as negatives. For the reduced LapSVM, we again use r = 2, 000. For
comparison, we also train two SVMs: one uses the 10 labeled examples only while the other uses
all the labeled examples (a total of 889,986) in the original training set of [12].
Figure 3 shows the results. Again, the SLapCVM is faster and produces a sparser solution than
LapSVM. Note that the SLapCVM, using only 10 labeled examples, can attain comparable AUC and
even better balanced loss than the SVM trained on the original, massive training set (Figures 3(c) and
3(d)). This clearly demonstrates the usefulness of semi-supervised learning when a large amount of
unlabeled data can be utilized. On the other hand, note that the LapSVM again cannot be run with
more than 3,000 unlabeled examples on our PC because of its high space requirement. The reduced
LapSVM performs very poorly here, possibly because this data set is highly imbalanced.
5
Conclusion
In this paper, we addressed two issues associated with the Laplacian SVM: 1) How to obtain a sparse
solution for fast testing? 2) How to handle data sets with millions of unlabeled examples? For the
number of kernel expansions
CPU time (in seconds)
10
3
10
2
10
1
10
0
10 3
10
50
SLapCVM
core?set Size
LapSVM
Reduced LapSVM
45
4
10
3
10
2
4
10
number of unlabeled points
(a) CPU time.
5
10
10 3
10
4
10
number of unlabeled points
(b) #kernel expansions.
5
10
40
35
1.1
SLapCVM
LapSVM
Reduced LapSVM
SVM (#labeled = 10)
CVM (w/ all training labels)
1.05
1
0.95
AUC
5
10
SLapCVM
LapSVM
Reduced LapSVM
4
balanced loss (in %)
5
10
30
25
0.9
0.85
20
0.8
15
0.75
10 3
10
SLapCVM
LapSVM
Reduced LapSVM
SVM (#labeled = 10)
CVM (w/ all training labels)
4
10
number of unlabeled points
(c) Balanced loss.
5
10
0.7 3
10
4
10
number of unlabeled points
5
10
(d) AUC.
Figure 3: Results on the extended MIT face data (some abscissas and ordinates are in log scale).
first issue, we introduce a sparsified manifold regularizer based on the -insensitive loss. For the
second issue, we integrate manifold regularization with the CVM. The resultant algorithm has low
time and space complexities. Moreover, by avoiding the underlying matrix inversion in the original LapSVM, a sparse solution can also be recovered. Experiments on a number of massive data
sets show that the SLapCVM is much faster than the LapSVM. Moreover, while the LapSVM can
only handle several thousand unlabeled examples, the SLapCVM can handle one million unlabeled
examples on the same machine. On one data set, this produces comparable or even better performance than the (supervised) CVM trained on 900K labeled examples. This clearly demonstrates the
usefulness of semi-supervised learning when a large amount of unlabeled data can be utilized.
References
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2,211 | 3,006 | Optimal Change-Detection and Spiking Neurons
Angela J. Yu
CSBMB, Princeton University
Princeton, NJ 08540
[email protected]
Abstract
Survival in a non-stationary, potentially adversarial environment requires animals
to detect sensory changes rapidly yet accurately, two oft competing desiderata.
Neurons subserving such detections are faced with the corresponding challenge
to discern ?real? changes in inputs as quickly as possible, while ignoring noisy
fluctuations. Mathematically, this is an example of a change-detection problem
that is actively researched in the controlled stochastic processes community. In
this paper, we utilize sophisticated tools developed in that community to formalize an instantiation of the problem faced by the nervous system, and characterize the Bayes-optimal decision policy under certain assumptions. We will derive
from this optimal strategy an information accumulation and decision process that
remarkably resembles the dynamics of a leaky integrate-and-fire neuron. This correspondence suggests that neurons are optimized for tracking input changes, and
sheds new light on the computational import of intracellular properties such as
resting membrane potential, voltage-dependent conductance, and post-spike reset
voltage. We also explore the influence that factors such as timing, uncertainty,
neuromodulation, and reward should and do have on neuronal dynamics and sensitivity, as the optimal decision strategy depends critically on these factors.
1
Introduction
Animals interacting with a changeable, potentially adversarial environment need to excel in the
detection of changes in its sensory inputs. This detection, however, is riddled by the inherently competing goals of accuracy and speed. Due to the noisy and incomplete nature of sensory inputs, the
animal can generally achieve more accurate detection by waiting for more sensory inputs. However,
gathering this extra information incurs an opportunity cost, as the extra time can be used to gather
more food, attract a mate, or escape a predator. Neurons subserving the detection process face a
similar speed-accuracy trade-off. In this work, we aim to understand the computations performed
by a neuron at the time-scale of single spikes. How sensitive a neuron is to each input spike should
depend on the relative probabilities of the input representing noise and useful information, and the
relative costs of mis-interpretation. We formulate the problem as an example of change-detection,
and characterize the optimal decision policy in this context. The formal tools we utilize to formalize the change-detection problem are built upon work in the area of controlled stochastic processes.
Controlled stochastic processes refer to decision-making in environments plagued not only by inferential uncertainty about the state of the world, but also uncertainty associated with the consequences
of an action or decision on the world itself. Finding optimal decision policies for such processes is
an actively researched problem in financial mathematics and operations research. As we will discuss
below, neuronal change-detection is a prime example of such a problem.
In Sec. 2, we introduce the general framework of change-detection. In Sec. 3, we apply the framework to a specific scenario similar to that faced by the neuron, and characterize the optimal solution.
In Sec. 3, we demonstrate that the optimal information accumulation and decision process has dynamics remarkably resembling that of a spiking neuron. We examine the computational import of
certain intracellular properties, characterize the input-output firing rate relationship, and extend the
framework of multi-source detection. In Sec. 4, we explore the behavioral consequences of opti-
mal change-detection and examine issues such as the speed-accuracy trade-off, temporal and spatial
cueing, and neuromodulation.
2
A Bayesian Formulation of the Change-Detection Problem
The Generative Model
Suppose we have sequential inputs x1 , x2 , . . ., which are generated iid by a distribution f0 (x) before
time ? ? {0, 1, . . .}, and by a distribution f1 (x) afterwards, where the random variable (r.v.) ?
denotes the sudden, hidden change time. ? has an initial probability P (? = 0) = q0 , and a geometric
distribution thereafter: P (? = t) = (1 ? q0 )(1 ? q)t?1 q, t > 0. The change-detection problem is
concerned with finding the optimal decision policy for reporting the change from f0 to f1 as early as
possible while minimizing false-alarms [1]. A decision policy ? is a mapping, possibly stochastic,
from all observations made so far to the control (or action) set, ?(xt , {x1 , . . . , xt }) 7? {a1 , a2 }.
The action a1 terminates the observation process and reports ? ? t, and a2 continues the observation
for another time step. Every unique decision policy is identified by a corresponding r.v. of stopping
times ? ? {0, 1, . . .}. In the following, we will use ? and ? interchangeably to refer to a policy.
The Loss Function
Following convention [2], we assume a loss function linear in false alarms and detection delay:
l? (?, ? ) = 1{? <?;?} + 1{? ??;?} c(? ? ?)
(1)
where 1 is the indicator function, and c > 0 is a constant that specifies the relative importance of
speed and accuracy. The total loss is the expectation of this loss function over ? and ? :
!
?X
=? X
??1
?
X
L? , hl? (?, ? ); ?i =
P (?, ? ) +
c(? ??)P (?, ? ) = P (? < ?) + ch(? ??)+ i (2)
?=0
? =0
? =?
?
An optimal policy ? minimizes L? . Due to the linear loss in detection delay, the expected loss
blows up for all policies that do not stop almost surely (a.s.; probability=1) in finite time; therefore, we restrict the optimization problem in the following to the class of almost-surely finite-time
policies. Using the notation Pt , P (? ? t|xt ), we have the following:
?
? Z
X
X
P (? > ? ) =
P (? = t, ? > ? ) =
P (? > ? |x? )P (? = t|xt )p(xt )dxt = h1?P? i?,x?
t=0
h(???)+ i =
?
X
t=0
h1{? >t} ?1{??t} i?,? =
t=0
? ?
?1
X
X
P (? )P (? ? t) =
? =0 t=0
?
X
P (? )
? =0
t?1
X
?
?1
X
hPt i?,xt = h
t=0
Pt i?,xt ,?
t=0
The cumulative posterior probability P? at the detection time ? , therefore, is the critical factor in
loss evaluation and policy optimization:
?1
L? = hc??k=0
Pk + (1 ? P ? )i?,Pk ,? ;? .
(3)
Bayes Rule gives us the iterative update rule for the cumulative posterior Pt , P (? ? t|xt ),
Pt+1 =
(Pt + (1 ? Pt )q)f1 (xt+1 )
,
(Pt + (1 ? Pt )q)f1 (xt+1 ) + (1 ? Pt )(1 ? q)f0 (xt+1 )
P0 = q0 .
(4)
Pt+1 is a deterministic function of Pt and xt+1 , but appears to take a stochastic trajectory since
xt+1 is an i.i.d.-distributed r.v. The expectation of hPt+1 |xt i is Pt +(1?Pt )q. We also define the
Pt
monotonically related posterior ratio ?t = 1?P
, which has the update rule
t
?t+1 =
f1 (xt+1 )(?t +q)
,
f0 (xt+1 )(1 ? q)
?0 =
q
.
1?q
(5)
Optimal Policy: Threshold Crossing
In order to optimize over the space of all possible stopping rules (policies), we define the following:
(1) the conditional termination cost, Ct , associated with stopping at time t after observing xt : Ct ,
Pt?1
c i=0 Pi + (1 ? Pt ); (2) the minimal conditional cost, ?t , to be expected after observation xt :
?t , ess inf? hC? |xt i, where ? ranges over all stopping rules that terminate no earlier than t, and
the expectation is taken over all future observations (which can be a function of the decision taken
at every time step); (3) ess inf, the largest (a.s.) r.v. less than (a.s.) every r.v. Xn , n ? N .
As an example of Bellman?s Equation, ?t satisfies the dynamic programming equation ?t =
min{Ct , h?t+1 |xt i}, and that the stationary, deterministic stopping rule ? ? = min{t ? 1|?t = Ct }
achieves optimality (Eq. 2). This implies that the optimal policy consists of a stopping region S ? [0, 1] and a continuation region C = [0, 1] \ S, such that ?(Pt : Pt ? S) = a1 and
?(Pt : Pt ? C) = a2 . We will state and prove a useful theorem below, which will imply that C
and S neatly fall into two contiguous blocks, such that the optimal policy requires the termination
action as soon as Pt exceeds some fixed threshold B ? ? ie the optimal policy is a first-passage
process in Pt !
Before we present the theorem, we first introduce the method of truncation. The difficulty of solving
the dynamic equation for ?t lies in its infinite recursiveness. If we can impose a finite horizon
T
T on ? , then the finitely recursive relation ?tT = min {Ct , h?t+1
|xt i} has a corresponding finite?
T
horizon optimal policy ?T , where ?T = CT . Taking the infinite limit ?t? , limT ?? ?tT , it has been
shown [2] that when the expected loss is finite (which is the case here, since the expression in Eq. 2
is finite for all decision policies that stop a.s. in finite time), ?t = ?t? , and ?T? converges to the
infinite-horizon optimal policy ? ? . We also note the following self-evident lemma.
P
Lemma.
functions in t, and h(t) = i gi (t)wi (t),
P Suppose {gi (t)}i?I is a family of decreasing
where i wi (t) = 1 ?t. If gi (t) ? gj (t) implies wi0 (t) ? wj0 (t), then h(t) decreases with t.
T
Theorem. Ct ?h?t+1
|xt i is a decreasing function of Pt .
Proof. CT ?1 ?h?TT |xt?1 i decreases with PT ?1 . Assume that the theorem holds for t+1, and note:
T
Ct ? h?t+1
|xt i = ?(c + q)Pt + q +
X
gi wi
i
T
where gi , max(0, li ), li , Ct+1 ? h?t+2
|xt , xt+1 = ii, and wi , P (xt+1 = i|x). gi decreases
with Pt for each i, since li decreases with Pt+1 by the inductive hypothesis, and Pt+1 increases
with Pt by Eq. 4. Suppose i, j are such that f1 (i)?f0 (i) > f1 (j)?f0 (j), then ?t+1 (i) > ?t+1 (j),
and Pt+1 (i) > Pt+1 (j), for any given xt . The inductive hypothesis implies gi ? gj . Also note
T
dwk /dPt = (f1 (k)?f0 (k))(1?q), so dwi /dPt ? dwj /dPt . Thus, Ct ?h?t+1
|xt i decreases with Pt .
This theorem states that the cost of stopping at time t relative to continuing gets smaller when it is
more certain that ? ? t. This is true for any finite stopping time T and therefore also for the infinitehorizon limit. If Ct ?h?t+1 |xt i is negative for some value of Pt , then the optimal policy is to select
action a1 ; this is also true for any larger values of Pt . Define B ? ? [0, 1] as the lower bound of all
such Pt , then the stopping and continuation regions have the form [B ? , 1] and [0, B ? ), respectively.
Ideally, we would like to have an exact solution for the optimal policy as a function of the generative
and cost parameters of the change-detection problem as defined in Sec. 1. While the explicit form
of B ? is not known, the theorem allows us to find the optimal policy numerically by evaluating and
minimizing the empirical loss as a function of the decision threshold B ? [0, 1].
3
Neuronal change-detection
In the following, we focus on the specific case where f0 and f1 are Bernoulli processes with respective rate parameters ?0 and ?1 . This case resembles the problem faced by neurons, which receive
sequential binary inputs (spike=1, no spike=0) with approximately Poisson statistics. The Bernoulli
process is a discrete-time analog of the Poisson process, and obviates the problematic assumption
(made by the Poisson model) that spikes could be fired infinitely close to one another. For now, we
assume that the generative parameters ?1 , ?0 , q0 , q and the cost parameter c are known. We also
assume, without loss of generality, that ?1 > ?0 (rate increases), since otherwise we can just redefine
the inputs (0 or 1). When the parameters satisfy c ? (?1 ? ?0 ? q(1 ? ?0 ))/(1 ? ?1 ), we have the
explicit solution B ? = q/(q = c), or ? ? q/c (proof omitted). This corresponds to the one-step
look-ahead policy, and is optimal when the cost of detection is large or when the probability of the
change taking place is very high. This turns out not to be a very interesting case as the detection
process is driven to cross the threshold even in the absence of any input spikes.
Although we do not have an explicit solution for the optimal detection threshold B ? in general, we
can numerically compare different values of B for any specific problem. Fig. 1(a) shows the empirical cost averaged over 1000 trials for different threshold values. For these particular parameters, the
minimum is around B = 0.65, although the cost function is quite shallow for a large range of values
of B around the optimum, implying that performance is not particularly sensitive to relatively large
perturbation around the optimal value.
Repeated Change-Detection and Firing Rate
From the problem formulation in sec. 2, it might seem like the framework only applies to detecting
a single change, or multiple unrelated changes. However, the same policy formulation can apply
to the case of repeated detection of changes, one after another, in a temporally contiguous fashion.
As long as each detection event is generated from the same model parameters (q, q0 , f1 , f0 ), and
the cost parameter (c) remains constant, the threshold-crossing policy is still optimal in minimizing
the empirical expected loss over these repeated events. The only generative parameter affected by
the repetition is q0 , which represents the probability of the inputs already being generated from
f1 before the current observation process began. In this repeated detection scenario, q0 should in
general be high if the detection threshold B ? is high, and low if B ? is low. However, the strength
of this coupling is tempered by (i) whether each detection termination resets the generative process,
as happens when visual detection leads to saccades and thus the resetting of input statistics, and (ii)
the amount of time elapsed during the refractory period after a detection spike. Fortunately, while
q0 is influenced by the detection policy, the optimization of the policy is not influenced by q0 , since
it consists of comparing Ct and h?t+1 |xt i at every time step. This comparison does not depend on
q0 , which simply adds a linear factor to both terms.
In this repeated firing scenario, where the number of spikes is relatively high relative to the frequency
of changes, the loss function of Eq. 2 can be rewritten as L? = p0 r0 + c/r1 , where ri is the
mean firing rate when the inputs are generated from fi , and p0 is the fraction of time when f0 is
applicable (as opposed to f1 ). In other words, if the rate of change is slow compared to neuronal
firing rates, then optimal processing amounts to minimizing the ?spontaneous? firing rate during f0
and maximizing the ?stimulus-evoked? firing rate during f1 .
Optimality and Dynamics of Leaky Integrate-and-Fire
Fig. 1(b) illustrates this concept of repeated firing. The top panel shows an example tracing of the
dynamical variable ?t in the repeated optimal change-detection process. Whenever ?t reaches the
threshold 0.65/(1?0.65) (or equivalently when Pt reaches 0.65, the optimal threshold as determined
in the last section), a change is reported and the whole process resets to ?0 . The dynamics of ?t
is remarkably similar to a leaky integrate-and-fire neuron. The bottom panel shows a raster plot of
input and output spikes over 25 trials, and again the resemblance to spiking neurons is remarkable.
Closer inspection indicates that the update rule for the posterior ratio in Eq. 5 indeed approximates
f1 (xt )
the dynamics of a leaky integrate-and-fire neuron [3]. Let a , (1?q)f
, we can rewrite Eq. 5 as
0 (xt )
?t = a(?t?1 + q)
(6)
?1
(1?q)?0
When xt = 1, a =
> 1, ?t increases, and the rate of increase is larger when ?t itself
is larger. This is reminiscent of the near-threshold dynamics of the Hodgkin-Huxley model, in
which the voltage-dependent activation of sodium conductance drives the neuron to fire [4]. When
xt = 0, ?t converges to ?0? = f1 q/(f0 (1 ? q) ? f1 ) (by Eq. 5), which is greater than 0 when
f0 (0)/f1 (0) ? 1?q. We can think of ?0? as the resting membrane potential. Since ?0? increases
with q, it implies that the resting potential should be higher and closer to the firing threshold, making
the neuron more sensitive to synaptic inputs, when there is a stronger expectation that a change is
imminent.
Relationship Between Input and Output Firing Rates
We can also look at the input-output relationship at the firing-rate level. The state-dependent rate
parameter a has the expected values:
1
1 ?21 +?0 ?2?0 ?1
a0 , ha|f0 i =
a1 , ha|f1 i =
.
1?q
1?q
?0 ??20
Given Eqs. 5 and 6, we can write down an approximate, explicit expression for h?t |fi i:
t?1
X
ai q(1?ati )
q
t
k
t
t
h?t |fi i ? ai (h?t?1 i+q) = ai h?0 i + ai q
ai = ai h?0 i +
? ai ?0 +
. (7)
1?ai
ai ?1
k=0
(b)
0.5
0
0
100
200
300
400
(c)
Distribution of input spikes
0.7
0.1
0
500
Cost as a function of thresholds
0.8
0.2
0.6
?200
Input and output spikes
?100
0
100
200
Cost
Dynamics of ?
1
Frequency
(a)
Distribution of output spikes
Frequency
0
10
20
0
100
200
300
Time (samples)
400
500
0.04
0.4
0.3
0.02
0
0.5
0.2
?200
?100
0
Time
100
0.1
0.6
200
0.65
0.7
0.75
0.8
Threshold
0.85
0.9
Figure 1: Optimal change-detection and dynamics. (a) The empirical average cost (over 1000 trials)
has a single shallow minimum at B = 0.65. ?0 = 0.13, ?1 = 0.17, q = 0.0125, q0 = 0.05,
c = 0.0005; these parameters apply for the remainder of the paper unless otherwise specified. (b)
Top panel: a typical example of the dynamics of ?t over time. Superimposed on ?t are the spikes,
which are arbitrarily set to a fixed high value. Black bars near the bottom indicate input spikes.
Green line indicates time of actual change. In this example, a chance flurry of input spikes near
the start causes the optimal change-detector to fire; after the change, the increased input firing rate
induces the change-detector fire much more frequently. Note that ?t decreases whenever there
is a lull in input spikes. Bottom panel: Raster plot of input (blue) and output (red) spikes; both
more frequent after the the change indicated by the green line. (c) Output spikes (bottom) increase
frequency quickly after the increase in input spikes (top).
Given the decision threshold B, h?t0 |f0 i = h?t1 |f0 i = B, where ti is the average number of time
steps it takes to reach the threshold for for xt = fi , and can be assumed to be 1 (it takes many
time steps of input integration to reach the threshold). We therefore have
1
q
q
q/(a0 ?1) + ?0 t1
t /t
t /t
at00 ?0 +
= at11 ?1 +
=? a1 = a00 1
? a00 1 . (8)
a0 ?1
a1 ?1
q/(a1 ?1) + ?0
And therefore the ratio of the output firing rates, ri , 1/ti for i = 1, 2, is
log
r1
t0
log a1
=
=
=
r0
t1
log a0
1
1?q
?21+?0?2?0 ?1
?0??20
1
1?q
+ log
log
=1+
log
?21+?0?2?0 ?1
?0??20
1
log 1?q
.
(9)
Since the arguments of log in both the denominator and numerator are greater than 1, r1 /r0 > 1.
Therefore, when the input rates are such that ?1 > ?0 , then the respective output rates are also such
that r1 > r0 . To see exactly how the output firing rate ratio changes as a function of the input rates,
?2+? ?2? ?1
we define the function g(?0 , ?1 ) , 1 ?00??20 , and take its partial derivatives with respect to ?0
0
and ?1 . Then we see that the output firing ratio Eq. 9 also increases with ?1 and decreases with
?0 , consistent with intuitions. Fig. 1(c) shows the average detection/firing rate over time: the rise in
output firing rate closely follows that in the input, despite the small change in the input firing rates.
Multi-source change-detection
So far, we have only considered the case of the Bernoulli inputs uniformly changing from one
rate to another. However, sometimes the problem at hand is one of multi-source change-detection.
For instance, a visual neuron detecting the onset of a stimulus might get inputs from up-stream
neurons sensitive to stimuli with different properties (different colors, orientations, depth of view,
etc.). Here, we extend our framework to the case of two independent sources of inputs, using an
approach similar to that taken in [5]. The source f i , i ? {1, 2} emits observations xi1 , xi2 , . . . from a
Bernoulli process that changes from rate ?i0 to ?i1 at an unknown time ?i , where ?i is generated by
a geometric distribution with parameter q i , and the prior probability P (?i = 0) is q0i . The objective
is to detect ? , min(?1 , ?2 ) with the cost function specified as before (Eqs. 1-2).
Defining the individual posteriors Pti , P (?i ? t|xit ), where xit , xi1 , . . . , xit , we have the following
Pt , P (min(?1 , ?2 ) ? t|x1t , x2t ) = 1 ? (1 ? Pt1 )(1 ? Pt2 ) = Pt1 + Pt2 ? Pt1 Pt2 .
(10)
We can also define the corresponding overall posterior ratio
?t , Pt /(1 ? Pt ) = ?1t + ?2t + ?1t ?2t
(11)
(a)
(b)
(c)
Figure 2: Effect of cueing on change-detection. (a) Distribution of first spikes for the optimal
stopping policy; spikes aligned to time 0 when the actual change takes place. (b) This distribution is
significantly tightened with mean brought closer to the actual change, when there is extra temporal
information about an imminent change (q = .02). (c) The distribution of spikes is also slightly
tightened and brought closer to the actual change time, when there is stronger prior probability of
a stimulus appearing (q0 = .1), as during special cueing. The effect is smaller because the higher
prior leads to false alarms as well as reducing detection delay.
as a function of the individual posterior ratios ?it , Pti /(1 ? Pti ). Following reasoning very close
to that of Sec. 2, we can show that if the generative and cost parameters are such that ?t is lowerbounded by ?0? for t 1, then the optimal stopping/detection policy is to terminate at the smallest
t, such that ?t = ?1t +?2t +?1t ?2t ? (q 1 +q 2 ?q 1 q 2 )/c. Despite the generative independence of
the two Bernoulli processes, we note that the optimal policy is different from the na??ve strategy of
running two single-source change-detectors, and report a change as soon as one of them reports a
change. To see this, consider the case when ?1t = q 1 /c, but ?2t ? 0, so that ?t ? ?1t = q 1 /c <
(q 1 +q 2 (1 ? q 1 ))/c. Therefore, the individual detector for process 1 would have reported a change,
but the overall detector would not.
4
Optimal Change-Detection and Neuromodulation
A sizeable body of behavioral studies suggest that stimulus processing is influenced by cognitive
factors such as knowledge about the timing of stimulus onset, or whether or not a stimulus would
appear in a particular location. There is evidence that the neuromodulators norepinephrine [6], and
acetylcholine [7] are respectively involved in those two aspects of stimulus processing. Separately,
there is a rich literature on the effects of these various neuromodulators at the single-cell level [8].
Since we have here an explicit model of neuronal dynamics as a function of the statistical properties
associated with the stimulus, we are ideally positioned to examine how these properties should
affect the cellular properties, and whether the known behavioral consequences of neuromodulation
are consistent with their observed effects at the cellular level.
If the system has some prior knowledge about the onset time of a stimulus, we can model the information accumulation process as starting shortly before the mean change-time, with a tight distribution over the random variable ?. Making q larger achieves both effects in our model. Fig. 2A shows
the distribution of first spikes for 1000 trials; Fig. 2B shows that this distribution is more tightly clustered immediately after the actual change time ? for larger q. Experimentally, it has been observed
that norepinephrine makes sensory neurons fire more vigorously to bottom-up sensory inputs [8]. It
is also known from behavioral studies that a temporal cue improves detection performance, and that
noradrenergic depletion diminishes this advantage [6].
If there is some prior knowledge about the stimulus being in a particular location, we can model this
with a higher prior probability q0 of the stimulus being present. This also has the effect of increasing
the responsiveness of the change-detection process to input spikes (Fig. 2C), as well as making the
detection (spiking) process more sensitive. It has been shown experimentally that a (correct) spatial
cue improves stimulus detection, and that acetylcholine is implicated in this process [7], and that
acetylcholine potentiates neurons and increases their responsiveness to sensory inputs [8].
5
Discussion
Responding accurately and rapidly to changes in the environment is a problem confronted by the
brain at every level, from single neurons to behavior. In this work, we have presented a formal treatment of the change-detection problem and obtained important properties of the optimal policy ? for a
broad class of problems, the optimal detection algorithm is a threshold-crossing process based on the
posterior probability of the change having taken place, which can be iteratively updated using Bayes?
Rule. Applying these ideas to the case of neurons that must rapidly and accurately detect changes
in input spike statistics, we saw that the optimal algorithm yields dynamics remarkably similar to
the intracellular dynamics of spiking neurons. This suggests that neurons are optimized for tracking
discrete, abrupt changes in the inputs. The model yields insight into the computational import of
cellular properties such as resting membrane potential, post-spike reset potential, voltage-dependent
conductances, and the input-output spiking relationship. The basic framework was extended to examine the case of multi-source change-detection, a problem faced by a neuron tasked with detecting
a stimulus when it could be one of two possible sub-categories. We also explored the computational
consequences of spatial and temporal cueing on stimulus detection, and saw that the behavioral
and biophysical effects of neuromodulation (eg by acetylcholine and norepinephrine) are consistent
within the framework.
This novel framework for modeling single-neuron computations is attractive, as it suggests explicit
design principles underlying neuronal dynamics, and not merely provides a descriptive model. Since
the computational objects are well-specified at the outset, it provides a natural theoretica link between cellular properties and behavioral constraints. It is also appealing as a self-consistent and
elegantly simple model of the computations taking place in single neurons. Every neuron in this
scheme simply detects changes in its synaptic inputs, on a spike-to-spike time scale, and propagates
its knowledge according to its own speed-accuracy trade-off. All that a down-stream neuron needs
from this neuron for its own change-detection computations are this neuron?s average firing rate in
different states, the rate of change among these states, and the prior probability of of this neuron
being in one of those states ? all of these quantities can be learned over a longer time-scale. In
particular, the down-stream neuron does not need to know about this neuron?s inputs, its internal dynamics, its decision policy, its objective function, its model of the world, etc. In this scheme, more
sophisticated computations can be achieved by pooling together the outputs of different neurons in
various configurations ? we explored this briefly with the example of multi-source change-detection.
Another advantage of this framework is that it eliminates the boundary between inference and decision. In this scheme, neurons make inferences about their inputs and make decisions at every level
of processing. It therefore obviates the problem of where in a hierarchical nervous system does the
nature of the computation change from input-processing to decision-making.
While the incorporation of formal tools from controlled stochastic processes into the modeling of
single-cell computations is a novel approach, this work is related to several other theoretical works.
The idea of neurons processing and representing probabilistic information has received much attention in recent years, with most work focusing on the level of neuronal populations [9?12]. Theoretical work on the representation and processing of probabilistic information in single neurons are
comparatively more rare. It has been suggested [13] that certain decision-making neurons may accumulate probabilistic information and spike when the evidence exceeds a certain threshold. However,
it was typically assumed that the neurons already receive continuously-valued inputs that represent
probabilistic information. Moreover, the tasks considered in these earlier works involved stationary
discrimination, such that there was no explicit non-stationarity in the state of the world/inputs. We
note that our framework is a generalization of the commonly studied 2AFC task, which is equivalent to setting the change probability q to 0 in our model. Consistent with this characterization, our
optimal policy is a generalization of the SPRT algorithm which is known to be optimal for stationary
2AFC discrimination [14].
One closely related piece of work proposed that single neurons track the log posterior ratio of the
state of an underlying binary variable, and spike when the new inputs imply a value for this log
posterior ratio that is sufficiently different from the neuron?s current estimate based on previous
inputs [15]. The key difference at the conceptual level is that this previous work focused on the
explicit propagation of probabilistic information across neurons, thus introducing complications into
processing and learning that are necessary to make this probabilistic knowledge consistent across
neurons. Also, there was no explicit analysis of the optimality of the output spike generation process:
how much of a discrepancy merits a spike, and how this depends on the relevant statistical and cost
parameters. At the mechanistic level, having the membrane potential represent the log posterior
ratio, as opposed to the posterior ratio, requires the dynamical update rule to involve exponentiation.
While it was shown in that work that the dynamics is approximately leaky integrate-and-fire during
steady state, it does not help the most interesting case, when the world is rapidly changing and the
linear approximation is most detrimental. We showed in this work that there are good reasons for
neurons not to integrate inputs linearly. The amount of new evidence provided by each input (spike
or not spike) at every time step is state-dependent, and should be so according to optimal information
integration. This work suggests that the particular types of nonlinearity we see in neuronal dynamics
are desirable from a computational point of view.
One important assumption we made in our model is that the cost of detection delay is linear in
time, parameterized by the constant c. Without this assumption, the controlled dynamic process
framework would not apply, as the decision policy would not only depend on a state variable, but on
time in an explicit way. However, in general, there might not be a fixed c that relates the trade-off
between false alarms and detection delay. Intuitively, c should be related to how much reward could
be obtained per unit of time if the system were not engaged in prolonging the current observation
process. In particular, if a new ?trial? begins as soon as the current ?trial? terminates, regardless of
detection accuracy, then c should be set to P (? ? ? )/h? i, which also places the two cost terms in the
same dimension. If we had analytical expressions for P (? ? ? ) and h? i as a function of the decision
threshold B, then we could solve the optimization problem through the self-consistency constraint
placed on the optimal threshold B ? through its dependence on c. Unfortunately, there is no known
analytical expressions for P (? ? ? ; B) and h? ; Bi. Alternatively, one might still numerically obtain
a value for a fixed detection threshold that incurs the lowest cost among all thresholds. There is no
guarantee, however, that the optimal policy lives in this parameterized family of policies. It may be
that the best fixed threshold policy is still far from optimal detection.
There are several important and exciting directions in which we plan to extend the current work.
One is the consideration of more complex state transitions. In this work, we assumed that the state
transition is always from f0 to f1 . But in more general scenarios, the inputs are likely to revert back
to f0 before another transition into f1 , and so on. Thus, we need at least two populations of detectors,
one that detects the onset (f0 to f1 ), and one that detects the offset (f1 to f0 ). Intuitively, there ought
to be recurrent connections between them, to propagate and aggregate the total information about
what states the inputs are in. A related problem is when the inputs can be in multiple (> 2) possible
states, or even a continuous range of states, with complex transitions among these states. Another
interesting question is what happens when we have a different or more complex distribution for the
change variable ?. We know, for instance, that animals are capable of utilizing independent temporal
information about the mean and variance of the stimulus onset. In the geometric model we assumed,
these two variables are coupled. Finally, we note that the formal framework we presented, that of
optimal detection of changes in input statistics, is not only applicable to the level of single neuron,
but also to systems and cognitive level problems. For example, certain problems in reinforcement
learning, such as reversal learning and exploration versus exploitation in general, are also amenable
to analysis by a similar approach. We intend to explore some of these problems in the future using
similar formal tools from controlled dynamic processes.
Acknowledgments
We thank Bill Bialek, Peter Dayan, Savas Dayanik, and Sophie Deneve for helpful discussions.
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2,212 | 3,007 | Attribute-efficient learning
of decision lists and linear threshold functions
under unconcentrated distributions
Philip M. Long
Google
Mountain View, CA
[email protected]
Rocco A. Servedio
Department of Computer Science
Columbia University
New York, NY
[email protected]
Abstract
We consider the well-studied problem of learning decision lists using few examples when many irrelevant features are present. We show that smooth boosting algorithms such as MadaBoost can efficiently learn decision lists of length k over n
boolean variables using poly(k, log n) many examples provided that the marginal
distribution over the relevant variables is ?not too concentrated? in an L 2 -norm
sense. Using a recent result of H?astad, we extend the analysis to obtain a similar
(though quantitatively weaker) result for learning arbitrary linear threshold functions with k nonzero coefficients. Experimental results indicate that the use of
a smooth boosting algorithm, which plays a crucial role in our analysis, has an
impact on the actual performance of the algorithm.
1 Introduction
A decision list is a Boolean function defined over n Boolean inputs of the following form:
if `1 then b1 else if `2 then b2 ... else if `k then bk else bk+1 .
Here `1 , ..., `k are literals defined over the n Boolean variables and b1 , . . . , bk+1 are Boolean values.
Since the work of Rivest [24] decision lists have been widely studied in learning theory and machine
learning.
A question that has received much attention is whether it is possible to attribute-efficiently learn
decision lists, i.e. to learn decision lists of length k over n variables using only poly(k, log n) many
examples. This question was first asked by Blum in 1990 [3] and has since been re-posed numerous
times [4, 5, 6, 29]; as we now briefly describe, a range of partial results have been obtained along
different lines.
Several authors [4, 29] have noted that Littlestone?s Winnow algorithm [17] can learn decision lists
of length k using 2O(k) log n examples in time 2O(k) n log n. Valiant [29] and Nevo and El-Yaniv
[21] sharpened the analysis of Winnow in the special case where the decision list has only a bounded
number of alternations in the sequence of output bits b1 , . . . , bk+1 . It is well known that the ?halving
algorithm? (see [1, 2, 19]) can learn length-k decision lists using only O(k log n) examples, but the
running time of the algorithm is nk . Klivans and Servedio [16] used polynomial threshold functions together with Winnow to obtain a tradeoff between running time and the number of examples
? 1/3
? 1/3
required, by giving an algorithm that runs in time nO(k ) and uses 2O(k ) log n examples.
In this work we take a different approach by relaxing the requirement that the algorithm work under
any distribution on examples or in the mistake-bound model. This relaxation in fact allows us to handle not just decision lists, but arbitrary linear threshold functions with k nonzero coefficients. (Recall
Pn
that a linear threshold function f : {?1, 1}n ? {?1, 1}n is a function f (x) = sgn( i=1 wi xi ? ?)
where wi , ? are real numbers and the sgn function outputs the ?1 numerical sign of its argument.)
The approach and results. We will analyze a smooth boosting algorithm (see Section 2) together
with a weak learner that exhaustively considers all 2n possible literals x i , ?xi as weak hypotheses.
The algorithm, which we call Algorithm A, is described in more detail in Section 6.
The algorithm?s performance can be bounded in terms of the L 2 -norm of the distribution
over examP
ples. Recall that the L2 -norm of a distribution D over a finite set X is kDk2 := ( x?X D(x)2 )1/2 .
The L2 norm can be used to evaluate the ?spread? of a probability distribution: if the probability is
concentrated on a constant number of elements of the domain then the L 2 norm is constant, ?
whereas
if the probability mass is spread uniformly over a domain of size N then the L 2 norm is 1/ N .
Our main results are as follows. Let D be a distribution over {?1, 1} n. Suppose the target function
f has k relevant variables. Let D rel denote the marginal distribution over {?1, 1}k induced by the
relevant variables to f (i.e. if the relevant variables are xi1 , . . . , xik , then the value that D rel puts
on an input (z1 , . . . , zk ) is Prx?D [xi1 . . . xik = z1 . . . zk ]. Let Uk be the uniform distribution over
{?1, 1}k and suppose that ||D rel ||2 /||Uk ||2 = ? . (Note that for any D we have ? ? 1, since Uk has
minimal L2 -norm among all distributions over {?1, 1}k .) Then we have:
Theorem 1 Suppose the target function is an arbitrary decision list in the setting described above.
Then given poly(log n, 1 , ?, log ?1 ) examples, Algorithm A runs in poly(n, ?, 1 , log ?1 ) time and with
probability 1 ? ? constructs a hypothesis h that is -accurate with respect to D.
Theorem 2 Suppose the target function is an arbitrary linear threshold function in the setting de2
?
scribed above. Then given poly(k, log n, 2O((? /) ) , log 1? ) examples, Algorithm A runs in poly(n,
2
?
2O((? /) ) , log 1? ) time and with probability 1 ? ? constructs a hypothesis h that is -accurate with
respect to D.
Relation to Previous Work. Jackson and Craven [14] considered a similar approach of using
Boolean literals as weak hypotheses for a boosting algorithm (in their case, AdaBoost). Jackson and Craven proved that for any distribution over examples, the resulting algorithm requires
poly(K, log n)P
examples to learn any weight-K linear threshold function, i.e. any function of
n
the
form
sgn(
i=1 wi xi ? ?) over Boolean variables where all weights w i are integers and
Pn
|w
|
?
K
(this
clearly implies that there are at most K relevant variables). It is well known
i
i=1
[12, 18] that general decision lists of length k can only be expressed by linear threshold functions
of weight 2?(k) , and thus the result of [14] does not give an attribute efficient learning algorithm for
decision lists.
More recently Servedio [27] considered essentially the same algorithm we analyze in this work
by specifically studying smooth boosting algorithms with the ?best-single-variable? weak learner.
He considered a general linear threshold learning problem (with no assumption that there are few
relevant variables) and showed that if the distribution satisfies a margin condition then the algorithm
has some level of resilience to malicious noise. The analysis of this paper is different from that of
[27]; to the best of our knowledge ours is the first analysis in which the smoothness property of
boosting is exploited for attribute efficient learning.
2 Boosting and Smooth Boosting
Fix a target function f : {?1, 1}n ? {?1, 1} and a distribution D over {?1, 1}n. A hypothesis
function h : {?1, 1}n ? {?1, 1} is a ?-weak hypothesis for f with respect to D if ED [f h] ? ?.
We sometimes refer to ED [f h] as the advantage of h with respect to f.
We remind the reader that a boosting algorithm is an algorithm which operates in a sequence of
stages and at each stage t maintains a distribution Dt over {?1, 1}n. At stage t the boosting algorithm is given a weak hypothesis ht for f with respect to D; the boosting algorithm then uses this
to construct the next distribution Dt+1 over {?1, 1}n. After T such stages the boosting algorithm
constructs a final hypothesis h based on the weak hypotheses h 1 , . . . , hT that is guaranteed to have
high accuracy with respect to the initial distribution D. See [25] for more details.
Let D1 , D2 be two distributions. For ? ? 1 we say that D1 is ?-smooth with respect to D2 if
for all x ? {?1, 1}n,
D1 (x)/D2 (x) ? ?.
Following [15], we say that a boosting algorithm B is ?(, ?)-smooth if for any initial distribution D
and any distribution Dt that is generated starting from D when B is used to boost to -accuracy with
?-weak hypotheses at each stage, Dt is ?(, ?)-smooth w.r.t. D. It is known that there are algorithms
that are ?-smooth for ? = ?( 1 ) with no dependence on ?, see e.g. [8]. For the rest of the paper B
will denote such a smooth boosting algorithm.
It is easy to see that every distribution D which is 1 -smooth w.r.t. the uniform distribution U satisfies
p
kDk2 /kUk2 ?
1/. On the other hand, there are distributions D that are highly non-smooth
relative to U but which still have kDk2 /kUk2 small. For instance, the distribution D over {?1, 1}k
1
which puts weight 2k/2
on a single point and distributes the remaining weight uniformly on the other
k
2 ? 1 points is only 2k/2 -smooth (i.e. very non-smooth) but satisfies kDk2 /kUk k2 = ?(1). Thus
the L2 -norm condition we consider in this paper is a weaker condition than smoothness with respect
to the uniform distribution.
3 Total variation distance and L2 -norm of distributions
The total variation distance between twoP
probability distributions D 1 , D2 over a finite set X is
dT V := maxS?X D1 (S) ? D2 (S) = 21 x?X |D1 (x) ? D2 (x)| . It is easy to see that the total
variation distance between any two distributions is at most 1, and equals 1 if and only if the supports
of the distributions are disjoint. The following is immediate:
Lemma
P 1 For any two distributions D1 and D2 over a finite domain X, we have dT V (D1 , D2 ) =
1 ? x?X min{D1 (x), D2 (x)}.
We can bound the total variation distance between a distribution D and the uniform distribution in
terms of the ratio kDk2 /kUk2 of the L2 -norms as follows:
Lemma 2 For any distribution D over a finite domain X, if U is the uniform distribution over X,
||U ||2
we have dT V (D, U) ? 1 ? 4||D||22 .
2
2
2
2
2
Proof: Let M = ||D||
||U ||2 . Since ||D||2 = Ex?D [D(x)], we have Ex?D [D(x)] = M ||U||2 =
By Markov?s inequality,
Pr [D(x) ? 2M 2 U(x)] = Pr [D(x) ?
x?D
x?D
By Lemma 1, we have
1 ? dT V (D, U)
=
X
x
?
min{D(x), U(x)} ?
X
x:D(x)?2M 2 U (x)
2M 2
] ? 1/2.
|X|
X
x:D(x)?2M 2 U (x)
M2
|X| .
(1)
min{D(x), U(x)}
D(x)
1
?
,
2
2M
4M 2
where the second inequality uses the fact that M ? 1 (so D(x)/2M 2 < D(x)) and the third
inequality uses (1). Using the definition of M and solving for dT V (D, U) completes the proof.
4 Weak hypotheses for decision lists
Let f be any decision list that depends on k variables:
if `1 then output b1 else ? ? ? else if `k then output bk else output bk+1
where each `i is either ?(xi = 1)? or ?(xi = ?1).?
(2)
The following folklore lemma can be proved by an easy induction (see e.g. [12, 26] for proofs of
essentially equivalent claims):
Lemma 3 The decision list f can be represented by a linear threshold function of the form f (x) =
sgn(c1 x1 + ? ? ? + ck xk ? ?) where each ci = ?2k?i and ? is an even integer in the range [?2k , 2k ].
It is easy to see that for any fixed c1 , . . . , ck as in the lemma, as x = (x1 , . . . , xk ) varies over
{?1, 1}k the linear form c1 x1 +? ? ?+ck xk will assume each odd integer value in the range [?2k , 2k ]
exactly once. Now we can prove:
Lemma 4 Let f be any decision list of length k over the n Boolean variables x 1 , . . . , xn . Let D be
any distribution over {?1, 1}n, and let Drel denote the marginal distribution over {?1, 1}k induced
by the k relevant variables of f. Suppose that dT V (Drel , Uk ) ? 1 ? ?. Then there is some weak
2
hypothesis h ? {x1 , ?x1 , . . . , xn , ?xn , 1, ?1} which satisfies EDrel [f h] ? ?16 .
Proof: We first observe that by Lemma 3 and the well-known ?discriminator lemma? of [23, 11],
under any distribution D some weak hypothesis h from {x1 , ?x1 , . . . , xn , ?xn , 1, ?1} must have
4
ED [f h] ? 21k . This immediately establishes the lemma for all ? ? 2k/2
, and thus we may suppose
4
.
w.l.o.g. that ? > 2k/2
We may assume w.l.o.g. that f is the decision list (2), that is, that the first literal concerns x 1 , the
second concerns x2 , and so on. Let L(x) denote the linear form c1 x1 +? ? ?+ck xk ?? from Lemma 3,
so f (x) = sgn(L(x)). If x is drawn uniformly from {?1, 1}k , then L(x) is distributed uniformly
over the 2k odd integers in the interval [?2k ? ?, 2k ? ?], as c1 x1 is uniform over ?2k , c2 x2 over
?2k?1 , and so on.
Let S denote the set of those x ? {?1, 1}k that satisfy |L(x)| ? ?4 2k . Note that there are at most
? k
4 2 + 1 elements in S, corresponding to L(x) = ?1, ?3, . . . , ?(2j ? 1), where j is the greatest
4
integer such that 2j ? 1 ? ?4 2k . Since ? > 2k/2
, certainly |S| ? 1 + ?4 2k ? ?2 2k . We thus have
PrUk [|L(x)| > ?4 2k ] ? 1 ? ?/2. It follows that PrDrel [|L(x)| > ?4 2k ] ? ?2 (for otherwise we would
have dT V (Drel , Uk ) > 1 ? ?), and consequently we have EDrel [|L(x)|] ?
?2 k
8 2 .
Now we follow the simple argument used to prove the ?discriminator lemma? [23, 11]. We have
EDrel [|L(x)|] = EDrel [f (x)L(x)] = c1 E[f (x)x1 ] + ? ? ? + ck E[f (x)xk ] ? ?E[f (x)] ?
?2 k
2 . (3)
8
Recalling that each |ci | = 2k?i , it follows that some h ? {x1 , ?x1 , . . . , xn , ?xn , 1, ?1} must
2
2
satisfy EDrel [f h] ? ( ?8 2k )/(2k?1 + ? ? ? + 20 + |?|). Since |?| ? 2k this is at least ?16 , and the proof
is complete.
5 Weak hypotheses for linear threshold functions
Now we consider the more general setting of arbitrary linear threshold functions. Though there are
additional technical complications the basic idea is as in the previous section.
We will use the following fact due to H?astad:
Fact 3 (H?astad) (see [28], Theorem 9) Let f : {?1, 1}k ? {?1, 1} be any linear threshold funcPk
tion that depends on all k variables x1 , . . . , xk . There is a representation sgn( i=1 wi xi ? ?)
for f which is such that (assuming the weights w1 , . . . , wk are ordered by decreasing magnitude
1
1 = |w1 | ? |w2 | ? ? ? ? ? |wk | > 0) we have |wi | ? i!(k+1)
for all i = 2, . . . , k.
The main result of this section is the following lemma. The proof uses ideas from the proof of
Theorem 2 in [28].
Lemma 5 Let f : {?1, 1}n ? {?1, 1} be any linear threshold function that depends on k variables. Let D be any distribution over {?1, 1}n, and let Drel denote the marginal distribution over
{?1, 1}k induced by the k relevant variables of f. Suppose that dT V (Drel , Uk ) ? 1 ? ?. Then
there is some weak hypothesis h ? {x1 , ?x1 , . . . , xn , ?xn , 1, ?1} which satisfies EDrel [f h] ?
2
?
1/(k 2 2O(1/? ) ).
Proof sketch: We may assume that f (x) = sgn(L(x)) where L(x) = w1 x1 + ? ? ? + wk xk ? ? with
w1 , . . . , wk as described in Fact 3.
2
?
Let ` := O(1/?
) = O((1/? 2 )poly(log(1/?))). (We will specify ` in more detail later.)
Suppose first that ` ? k. By a well-known result of Muroga et al. [20], every linear threshold
function f that depends on k variables can be represented using integer weights each of magnitude 2O(k log k) . Now the discriminator lemma [11] implies that for any distribution P, for
some h ? {x1 , ?x1 , . . . , xn , ?xn , 1, ?1} we have EP [f h] ? 1/2O(k log k) . If ` ? k and ` =
2
?
2
?
O((1/? 2 )poly(log(1/?))), we have k log k = O(1/?
). Thus, in this case, EP [f h] ? 1/2O(1/? ) ,
so the lemma holds if ` ? k.
Thus we henceforth assume that ` < k. It remains only to show that
?
(4)
2
EDrel [|L(x)|] ? 1/(k2O(1/? ) );
once we have this, following (3) we get
?
2
EDrel [|L(x)|] = EDrel [f L] = w1 E[f (x)x1 ] + ? ? ? + wk E[f (x)xk ] ? ?E[f (x)] ? 1/(k2O(1/? ) ),
and now since each |wi | ? 1 (and w.l.o.g. |?| ? k) this implies that some h satisfies EDrel [f h] ?
2
?
1/(k 2 2O(1/? ) ) as desired.
Similar to [28] we consider two cases (which are slightly different from the cases in [28]).
Pk
Case I: For all 1 ? i ? ` we have wi2 /( j=i wj2 ) > ? 2 /576.
r
Pk
2 ln(8/?). Recall the following version of Hoeffding?s bound: for any
Let ? := 2
w
j=`+1 j
0 6= w ? Rk and any ? > 0, we have Prx?{?1,1}k [|w ? x| ? ?kwk] ? 2e?? /2 (where we write
q
Pk
2
kwk to denote
i=1 wi ). This bound directly gives us that
?
Pr [|w`+1 x`+1 + ? ? ? + wk xk | ? ?] ? 2e?2 ln(8/?)/2 = .
(5)
x?Uk
4
2
Moreover, the argument in [28] that establishes equation (4) of [28] also yields
?
(6)
Pr [|w1 x1 + ? ? ? + w` x` ? ?| ? 2?] ?
x?Uk
4
in our current setting. (The only change that needs to be made to the argument of [28] is adjusting
various constant factors in the definition of `). Equations (5) and (6) together yield Prx?Uk [|w1 x1 +
? ? ? + wk xk ? ?| ? ?] ? 1 ? ?2 . Now as before, taken together with the dT V bound this yields
PrDrel [|L(x)| ? ?] ? ?2 and hence we have EDrel [|L(x)|] ? ??/2. Since ? > w`+1 and w`+1 ?
1/((k + 1)(` + 1)!) by Fact 3, we have established (4) in Case I.
Pk
Case II: For some value J ? ` we have wJ2 /( i=J wi2 ) ? ? 2 /576. Let us fix any setting z ?
{?1, 1}J?1 of the variables x1 , . . . , xJ?1 . By an inequality due to Petrov [22] (see [28], Theorem
4) we have
6wJ
6?
?
Pr
[|w1 z1 +? ? ?+wJ?1 zJ?1 +wJ xJ +? ? ?+wk xk ??| ? wJ ] ? qP
?
= .
xJ ,...,xk ?Uk?J+1
24
4
k
2
i=J wi
Thus for each z ? {?1, 1}J?1 we have Prx?Uk [|L(x)| ? wJ | x1 . . . xJ?1 = z1 . . . zJ?1 ] ? ?4 .
This immediately yields Prx?Uk [|L(x)| > wJ ] ? 1 ? ?4 , which in turn gives Prx?Drel [|L(x)| >
3?wJ
wJ ] ? 3?
by our usual arguments. Now (4) follows using Fact 3
4 and hence ED rel [|L(x)|] ?
4
and J ? `.
6 Putting it all together
Algorithm A works by running a ?( 1 )-smooth boosting-by-filtering algorithm; for concreteness
we use the MadaBoost algorithm of Domingo and Watanabe [8]. At the t-th stage of boosting,
when MadaBoost simulates the distribution Dt , the weak learning algorithm works as follows:
0
)
O( log n+log(1/?
) many examples are drawn from the simulated distribution Dt , and these examples
?2
are used to obtain an empirical estimate of EDt [f h] for each h ? {x1 , ?x1 , . . . , xn , ?xn , ?1, 1}.
(Here ? is an upper bound on the advantage EDt [f h] of the weak hypotheses used at each stage;
we discuss this more below.) The weak hypothesis used at this stage is the one with the highest
observed empirical estimate. The algorithm is run for T = O( ?12 ) stages of boosting.
Consider any fixed stage t of the algorithm?s execution. As shown in [8], at most O( 1 ) draws from
the original distribution D are required for MadaBoost to simulate a draw from the distribution D t .
(This is a direct consequence of the fact that MadaBoost is O( 1 )-smooth; the distribution Dt is
simulated using rejection sampling from D.) Standard tail bounds show that if the best hypothesis
h has E[f h] ? ? then with probability 1 ? ? 0 the hypothesis selected will have E[f h] ? ?/2. In
[8] it is shown that if MadaBoost always has an ?(?)-accurate weak hypothesis at each stage, then
after at most T = O( ?12 ) stages the algorithm will construct a hypothesis which has error at most
. Thus it suffices to take ? 0 = O(?2 ?). The overall number of examples used by Algorithm A is
0
)
O( log n+log(1/?
).
2 ? 4
Thus to establish Theorems 1 and 2, it remains only to show that for any initial distribution D with
kDrel k2 /kUk k2 = ? , the distributions Dt that arise in the course of boosting are always such that
the best weak hypothesis h ? {x1 , ?x1 , . . . , xn , ?xn , ?1, 1} has sufficiently large advantage.
Suppose f is a target function that depends on some set of k (out of n) variables. Consider
what happens if we run a 1 -smooth boosting algorithm, where the initial distribution D satisfies
kDrel k/kUk k = ?. At each stage we will have Dtrel (x) ? 1 ? Drel (x) for all x ? {?1, 1}k , and
consequently we will have
||Dtrel ||22 =
X
x?{?1,1}k
Dtrel (x)2 ?
1
2
X
x?{?1,1}k
Drel (x)2 ?
?2
2
X
x?{?1,1}k
Uk (x)2 .
Thus, by Lemma 2 each distribution Dt will satisfy dT V (Dtrel , Uk ) ? 1?2 /(4? 2 ). Now Lemmas 4
and 5 imply that in both cases (decision lists and LTFs) the best weak hypothesis h does indeed have
the required advantage.
7 Experiments
The smoothness property enabled the analysis of this paper. Is smoothness really helpful for learning
decision lists with respect to diffuse distributions? Is it critical?
This section is aimed at addressing these questions experimentally. We compared the accuracy of
the classifiers output by a number of smooth boosters from the literature with AdaBoost (which is
known to not be a smooth booster in general, see e.g. Section 4.2 of [7]) on synthetic data in which
the examples were distributed uniformly, and the class designations were determined by applying a
randomly generated decision list. The number of relevant variables was fixed at 10. The decision
list was determined by picking `1 , ..., `10 and b1 , ..., b11 from (2) independently uniformly at random
from among the possibilities.
We evaluated the following algorithms: (a) AdaBoost [9], (b) MadaBoost [8], (c) SmoothBoost [27],
and (d) a smooth booster proposed by Gavinsky [10]. Due to space constraints, we cannot describe
each of these in detail.1
Each booster was used to reweight the training data, and in each round, the literal which minimized
the weighted training error was chosen. Some of the algorithms choose the number of rounds of
1
Very roughly speaking, AdaBoost reweights the data to assign more weight to examples that previously
chosen base classifiers have often classified incorrectly; it then outputs a weighted vote over the outputs of the
base classifiers, where each voting weight is determined as a function of how well its base classifier performed.
MadaBoost modifies AdaBoost to place a cap on the weight, prior to normalization. SmoothBoost [27] caps
the weight more aggressively as learning progresses, but also reweights the data and weighs the base classifiers
in a manner that does not depend on how well they performed. The form of the manner in which Gavinsky?s
booster updates weights is significantly different from AdaBoost, and reminiscent of [13, 15].
m
100
200
500
1000
100
200
500
1000
n
100
100
100
100
1000
1000
1000
1000
Ada
0.086
0.052
0.022
0.016
0.123
0.079
0.045
0.033
Mada
0.077
0.045
0.018
0.014
0.119
0.072
0.039
0.026
Gavinsky
0.088
0.050
0.024
0.024
0.116
0.083
0.045
0.035
SB(0.05)
0.071
0.067
0.056
0.063
0.093
0.071
0.050
0.048
SB(0.1)
0.067
0.047
0.031
0.036
0.101
0.064
0.040
0.038
SB(0.2)
0.077
0.047
0.025
0.028
0.117
0.072
0.040
0.032
SB(0.4)
0.089
0.051
0.031
0.033
0.128
0.081
0.044
0.036
SB(0.2)
7.5
9.4
11.5
12.1
6.1
9.5
10.9
12.1
SB(0.4)
9.1
9.9
12.2
13.0
7.4
11.7
11.5
13.3
Table 1: Average test set error rate
m
100
200
500
1000
100
200
500
1000
n
100
100
100
100
1000
1000
1000
1000
Ada
13.6
19.8
32.2
37.2
13.3
19.8
28.1
36.7
Mada
8.8
13.1
20.7
19.2
7.7
11.5
16.7
20.1
Gavinsky
11.7
12.5
15.2
15.3
26.8
19.4
16.2
14.7
SB(0.05)
3.9
4.1
5.0
7.1
3.7
4.4
4.9
7.2
SB(0.1)
6.0
6.9
9.1
10.7
5.3
7.4
8.6
11.0
Table 2: Average smoothness
boosting as a function of the desired accuracy; instead, we ran all algorithms for 100 rounds. All
boosters reweighted the data by normalizing some function that assigns weight to examples based
on how well previously chosen based classifiers are doing at classifying them correctly. The booster
proposed by Gavinsky might set all of these weights to zero: in such cases, it was terminated.
For each choice of the number of examples m and the number of features n, we repeated the following steps: (a) generate a random target, (b) generate m random examples, (c) split them into a
training set with 2/3 of the examples and a test set with the remaining 1/3, (d) apply all the algorithms on the training set, and (e) apply all the resulting classifiers on the test set. We repeated the
steps enough times so that the total size of the test sets was at least 10000; that is, we repeated them
d30000/me times. The average test-set error is reported.
SmoothBoost [27] has two parameters, ? and ?. In his analysis, ? = ?/(2 + ?), so we used the same
setting. We tried his algorithm with ? set to each of 0.05, 0.1, 0.2 and 0.4.
The test set error rates are tabulated in Table 1. MadaBoost always improved on the accuracy of AdaBoost. The results are consistent with the possibility that AdaBoost learns decision lists attributeefficiently with respect to the uniform distribution; this motivates theoretical study of whether this
is true. One possible route is to prove that, for sources like this, AdaBoost is, with high probability,
a smooth boosting algorithm. The average smoothnesses are given in Table 2.
SmoothBoost [27] was seen to be fairly robust to the choice of ?; with a good choice it sometimes
performed the best. This motivates research into adaptive boosters along the lines of SmoothBoost.
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2,213 | 3,008 | Clustering Under Prior Knowledge with Application
to Image Segmentation
M?ario A. T. Figueiredo
Instituto de Telecomunicac?o? es
Instituto Superior T?ecnico
Technical University of Lisbon
Portugal
Dong Seon Cheng, Vittorio Murino
Vision, Image Processing, and Sound Laboratory
Dipartimento di Informatica
University of Verona
Italy
[email protected]
[email protected], [email protected]
Abstract
This paper proposes a new approach to model-based clustering under prior knowledge. The proposed formulation can be interpreted from two different angles: as
penalized logistic regression, where the class labels are only indirectly observed
(via the probability density of each class); as finite mixture learning under a grouping prior. To estimate the parameters of the proposed model, we derive a (generalized) EM algorithm with a closed-form E-step, in contrast with other recent
approaches to semi-supervised probabilistic clustering which require Gibbs sampling or suboptimal shortcuts. We show that our approach is ideally suited for
image segmentation: it avoids the combinatorial nature Markov random field priors, and opens the door to more sophisticated spatial priors (e.g., wavelet-based)
in a simple and computationally efficient way. Finally, we extend our formulation
to work in unsupervised, semi-supervised, or discriminative modes.
1
Introduction
Most approaches to semi-supervised learning (SSL) see the problem from one of two (dual) perspectives: supervised classification with additional unlabelled data (see [20] for a recent survey);
clustering with prior information or constraints (e.g., [4, 10, 11, 15, 17]). The second perspective,
usually termed semi-supervised clustering (SSC), is usually adopted when labels are totaly absent,
but there are (usually pair-wise) relations that one wishes to enforce or encourage.
Most SSC techniques work by incorporating the constrains (or prior) into classical algorithms such
as K-means or EM for mixtures. The semi-supervision may be hard (i.e., grouping constraints
[15, 17]), or have the form of a prior under which probabilistic clustering is performed [4, 11]. The
later is clearly the most natural formulation for cases where one wishes to encourage, not enforce,
certain relations; an obvious example is image segmentation, seen as clustering under a spatial
prior, where neighboring sites should be encouraged, but not constrained, to belong to the same
cluster/segment. However, the previous EM-type algorithms for this class of methods have a major
drawback: the presence of the prior makes the E-step non-trivial, forcing the use of expensive Gibbs
sampling [11] or suboptimal methods such as the iterated conditional modes algorithm [4].
In this paper, we introduce a new approach to mixture-based SSC, leading to a simple, fully deterministic, generalized EM (GEM) algorithm. The keystone is the formulation of SSC as a penalized
logistic regression problem, where the labels are only indirectly observed. The linearity of the
resulting complete log-likelihood, w.r.t. the missing group labels, underlies the simplicity of the
resulting GEM algorithm. When applied to image segmentation, our method allows using spatial
priors which are typical of image estimation problems (e.g., restoration/denoising), such as Gaussian
fields or wavelet-based priors. Under these priors, the M-step of our GEM algorithm reduces to a
simple image denoising procedure, for which there are several extremely efficient algorithms.
2
Formulation
We start from the standard formulation of finite mixture models: X = {x1 , ..., xn } is an observed
data set, where each xi ? IRd was generated (independently) according to one of a set of K probability (density or mass) functions {p(?|?(1) ), ..., p(?|?(K) )}. In image segmentation, each xi is a
pixel value (gray scale, d = 1; color, d = 3) or a vector of local (e.g., texture) features. Associated
(1)
(K)
with X , there is a hidden label set Y = {y1 , ..., yn }, where yi = [yi , ..., yi ]T ? {0, 1}K , with
(k)
yi = 1 if and only if xi was generated by source k (the so-called ?1-of-K? binary encoding). Thus,
K
n
K h
iyi(k)
Y
Y
Y
Y
(k)
(k)
p (X |Y, ? ) =
p(xi |? ) =
p(xi |? )
,
(1)
k=1
(k)
i: yi
i=1
=1
k=1
where ? = (?(1) , ..., ?(K) ) is the set of parameters of the generative models of classes.
In standard mixture models, all the yi are assumed to be independent and identically distributed samples following a multinomial distribution with probabilities {? (1) , ..., ? (K) }, i.e., P (Y) =
Q Q (k) y(k)
) i . This is the part of standard mixture models that has to be modified in order to
i
k (?
insert grouping constraints [15] or a grouping prior p(Y) [4, 11]. However, this prior destroys the
simplicity of the standard E-step for finite mixtures, which is critically based on the independence
assumption. We follow a different route to avoid that roadblock.
Let the hidden labels Y = {y1 , ..., yn } depend on a new set of variables Z = {z1 , ..., zn }, where
(1)
(K)
each zi = [zi , ..., zi ]T ? IRK, following a multinomial logistic model [5]:
(k)
n Y
K
yi(k)
Y
ezi
(k)
(k)
P (Y|Z) =
P [yi = 1|zi ]
,
where
P [yi = 1|zi ] = PK
.
(2)
(l)
zi
i=1 k=1
l=1 e
(K)
Due to the normalization, we can set (w.l.o.g.) zi = 0, for i = 1, ..., n [5]. We?re thus left with
(k)
(k)
n (K ? 1) real variables, i.e., Z = {z(1) , ..., z(K?1) }, where z(k) = [z1 , ..., zn ]T ; of course, Z
can be seen as an n ? (K ? 1) matrix, where z(k) is the k-th column and zi is the i-th row.
With this formulation, certain grouping preferences may be expressed by a prior p(Z). For example,
preferred pair-wise relations can be easily embodied in a Gaussian prior
?
?
K?1
n X
n
K?1
Y
X
Y
1
1
(k)
(k)
Ai,j (zi ? zj )2 ? =
exp ? (z(k) )T ? z(k) , (3)
p(Z) ?
exp ??
4 i=1 j=1
2
k=1
k=1
where A is a matrix (with a null diagonal) encoding pair-wise preferences (Ai,j > 0 expresses
preference, with strength proportional to Ai,j , for having points i and j in the same cluster) and ?
is the well-known graph-Laplacian matrix [20],
nP
o
Pn
n
? = diag
A
,
...,
A
? A.
(4)
1,j
n,j
j=1
j=1
For image segmentation, each z(k) is an image with real-valued elements and a natural choice for
A is to have Ai,j = ?, if i and j are neighbors, and zero otherwise. Assuming periodic boundary
conditions for the neighborhood system, ? is a block-circulant matrix with circulant blocks [2].
However, as shown below, other more sophisticated priors (such as wavelet-based priors) can also
be used at no additional computational cost [1].
3
3.1
Model Estimation
Marginal Maximum A Posteriori and the GEM Algorithm
Based on the formulation presented in the previous section, SSC is performed by estimating Z
and ?, seeing Y as missing data. The marginal maximum a posteriori estimate is obtained by
marginalizing out the hidden labels (over all the possible label configurations),
X
X
b = arg max
b ?
Z,
p(X , Y, Z|?) = arg max
p(X |Y, ?) P (Y|Z) p(Z),
Z,? Y
Z,? Y
(5)
where we?re assuming a flat prior for ?. One of the key advantages of this approach is that (5) is
a continuous (not combinatorial) optimization problem. This is in contrast Markov random field
approaches to image segmentation, which lead to hard combinatorial problems, since they perform
optimization directly with respect to the (discrete) label variables Y. Finally, notice that once in
b one may compute P (Y|Z)
b which gives the probability that each data
possession of an estimate Z,
(k)
point belongs to each class. By finding arg maxk P [yi = 1|zi ], for every i, one may obtain a hard
clustering/segmentation.
We handle (5) with a generalized EM (GEM) algorithm [13], i.e., by applying the following iterative
procedure (until some convergence criterion is satisfied):
E-step: Compute the conditional expectation of the complete log-posterior, given the current estib and the observations X :
b ?)
mates (Z,
h
i
b = EY log p(Y, Z, ?|X ) Z,
b X .
b ?)
b ?,
Q(Z, ?|Z,
(6)
b ? (Zbnew , ?
b ), with new values such that
b ?)
M-step: Update the estimate: (Z,
new
b |Z,
b ? Q(Z,
b Z,
b
b ?)
b ?|
b ?).
Q(Zbnew , ?
new
(7)
Under mild conditions, it is well known that GEM algorithms converge to a local maximum of the
marginal log-posterior [18].
3.2
E-step
The complete log-posterior is
.
log p(Y, Z, ?|X ) = log p(X |Y, ?) + log P (Y|Z) + log p(Z)
"K
#
n K
n
K
X
X (k) (k)
X
(k)
. X X (k)
(k)
=
yi log p(xi |? ) +
yi zi ? log
ezi
+ log p(Z)
i=1 k=1
i=1
k=1
(8)
k=1
.
where = stands for ?equal up to an additive constant?. The key observation is that this function is lin(k)
ear w.r.t. the hidden variables yi . Consequently, the E-step reduces to computing their conditional
expectations, which are then plugged into (8).
(k)
(k)
As in standard mixtures, each missing yi is binary, thus its expectation (denoted ybi ) equals its
posterior probability of being equal to one, easily obtained via Bayes law:
(k)
ybi
?
(k) b b
E[yi |Z,
?, X ]
=
(k)
P [yi
b (k) ) P [y (k) = 1|b
p(xi |?
zi ]
i
b
= 1|b
zi , ?, xi ] = P
.
(j)
(j)
K
b
= 1|b
zi ]
j=1 p(xi |? ) P [yi
(9)
Notice that this is the same as the E-step for a standard finite mixture, where the probabilities
(k)
P [yi = 1|b
zi ] (given by (2)) play the role of the probabilities of the classes/components. Finally,
(k)
the Q function is obtained by plugging the expectations ybi into (8).
3.3
M-Step
It?s clear from (8) that the maximization w.r.t. ? can be performed separately w.r.t. to each ?(k) ,
n
X
(k)
b (k) = arg max
?
ybi log p(xi |?(k) ).
(10)
new
(k)
? i=1
This is the familiar weighted maximum likelihood criterion, exactly as it appears in EM for standard
mixtures. The explicit form of this update depends on the choice of p(?|?(k) ); e.g., this step can be
easily applied to any finite mixture of exponential family densities [3].
In supervised image segmentation, these parameters are known (e.g., previously estimated from
training data) and thus it?s not necessary to estimate them; the M-step reduces to the estimation of
Z. In unsupervised image segmentation, ? is unknown and (10) will have to be applied.
To update the estimate of Z, we need to maximize (or at least improve, see (7))
"K
#
n
K
X
X (k) (k)
X
(k)
z
b ?
L(Z|Y)
yb z ? log
ei
+ log p(Z).
i
i=1
k=1
i
(11)
k=1
Without the prior, this would be a simple logistic regression (LR) problem, with an identity design
(k)
(k)
matrix [5]; however, instead of the usual hard labels yi ? {0, 1}, we have ?soft? labels ybi ? [0, 1].
Arguably, the two standard approaches to maximum likelihood LR are the Newton-Raphson algorithm (a.k.a. iteratively reweighted least squares ? IRLS [7]) and the minorize-maximize (MM)
approach (formerly known as bound optimization) [5, 9]. We will show below that the MM approach
can be easily modified to accommodate the presence of a prior.
Let?s briefly review the MM approach for maximizing a twice differentiable concave function E(?)
with bounded Hessian [5, 9]. Let the Hessian H(?) of E(?) be bounded below by ?B (that is,
H(?) ?B, in the matrix sense, meaning that H(?) + B is positive definite), where B is a positive
b has a minimum at ? = ?,
b where
definite matrix. It?s trivial to show that E(?) ? R(?, ?)
T
b = ?1 ? ? ?
b ? B?1 g(?)
b B ???
b ? B?1 g(?)
b ,
R(?, ?)
(12)
2
b denoting the gradient of E(?) at ?.
b Thus, the iteration
with g(?)
bnew = arg max R(?, ?)
b =?
b + B?1 g(?)
b
?
?
bnew ) ? E(?).
b
is guaranteed to monotonically improve E(?), i.e., E(?
(13)
It was shown in [5] that the gradient and the Hessian of the logistic log-likelihood function, i.e., (11)
without the log-prior, verify (with Ia denoting an a ? a identity matrix and 1a a vector of a ones)
!
1K?1 1TK?1
1
b ? ?(z)
IK?1 ?
? In ? ?B,
(14)
g(z) = y
and
H(z) ?
2
K
(1)
(1)
(2)
(K?1)
b denotes
]T denotes the lexicographic vectorization of Z, y
where z = [z1 , ..., zn , z1 , ..., zn
(1)
(1) (2)
(K?1) T
b
the corresponding lexicographic vectorization of Y, and ?(z) = [p1 , ..., pn , p1 , ..., pn
]
(k)
(k)
with pi = P [yi = 1|zi ].
Defining v = b
z + B?1 (b
y ? ?(b
z)), the MM update equation for solving (11) is thus
1
T
b
znew (v) = arg min
(z ? v) B (z ? v) ? log p(z) ,
z
2
(15)
where p(z) is equivalent to p(Z), because z is simply the lexicographic vectorization of Z.
We now summarize our GEM algorithm:
b , using (9), for all i = 1, ..., n and k = 1, ..., K ? 1.
E-step: compute y
b fixed, that is, loop through
(Generalized) M-step: Apply one or more iterations (15), keeping y
the following two steps: v ? b
z + B?1 (b
y ? ?(b
z)) and b
z?b
znew (v).
3.4
Speeding Up the Algorithm
In image segmentation, the MM update equation (15) is formally equivalent to the MAP estimation
of an image with n pixels in IRK?1 , under prior p(z), where v plays the role of observed image, and
B is the inverse covariance matrix of the noise. Due to the structure of B, even if the prior models
the several z(k) as independent, i.e., if log p(z) = log p(z(1) ) + ? ? ? + log p(z(K?1) ), (15) can not be
decoupled into the several components {z(1) , ..., z(K?1) }. We sidestep this difficulty, at the cost of
using a less tight bound in (14), based the following lemma:
Lemma 1 Let ?K = 1/2, if K > 2, and ?K = 1/4, if K = 2. Then, B ?K In(K?1) .
Proof: Inserting K = 2 in (14) yields B = I/4, which proves the case K = 2. For K > 2, the
inequality I/2 B is equivalent to ?min (I/2 ? B) ? 0, which is equivalent to ?max (B) ? (1/2).
Since the eigenvalues of the Kronecker product are the products of the eigenvalues of the matrices,
?max (B) = ?max (I ? (1/K) 1 1T )/2. Since 1 1T is a rank-1 matrix with eigenvalues {0, ..., 0, K ?
1}, the eigenvalues of (I ? (1/K) 1 1T ) are {1, ..., 1, 1/K}, thus ?max (I ? (1/K) 1 1T ) = 1, and
?max (B) = 1/2.
This lemma allows replacing B with ?K In(K?1) in (15) which (assuming independent priors, as is
the case of (3)) becomes decoupled, leading to
2
?K
(k)
(k)
(k)
(k)
b
p(z
)
,
for k = 1, ..., K ? 1, (16)
?
log
z(k)
(v
)
=
arg
min
?
v
z
new
2
z(k)
where v(k) = b
z(k) + (1/?K )(b
y(k) ? ? (k) (b
z(k) )). Moreover, the ?noise? in each of these ?denoising?
problems is white and Gaussian, of variance 1/?K .
3.5
Stationary Gaussian Field Priors
Consider a Gaussian prior of form (3), where Ai,j only depends on the relative position of i and j and
the neighborhood system defined by A has periodic boundary conditions. In this case, both A and ?
are block-circulant matrices, with circulant blocks [2], thus diagonalizable by a 2D discrete Fourier
transform (2D-DFT). Formally, ? = UH DU, where D is diagonal, U is the orthogonal matrix
representing the 2D-DFT, and (?)H denotes conjugate transpose. The log-prior is then expressed in
.
the DFT domain, log p(z(k) ) = 21 (Uz(k) )H D(Uz(k) ), and the solution of (16) is
?1
(k)
b
z(k)
) = ?K UH [?K In + D] U v(k) ,
new (v
for
k = 1, ..., K ? 1.
(17)
(k)
Observe that (17) corresponds to filtering each image v , in the DFT domain, with a fixed filter
?1
with frequency response [?K In + D] ; this inversion can be computed off-line and is trivial because ?K In + D is diagonal. Finally, it?s worth stressing that the matrix-vector products by U and
UH are not carried out explicitly but more efficiently via the FFT algorithm, with cost O(n log n).
3.6
Wavelet-Based Priors for Segmentation
It?s known that piece-wise smooth images have sparse wavelet-based representations (see [12]
and the many references therein); this fact underlies the state-of-the-art denoising performance of
wavelet-based methods. Piece-wise smoothness of the z(k) translates into segmentations in which
pixels in each class tend to form connected regions. Consider a wavelet expansion of each z(k)
z(k) = W? (k) ,
k = 1, ..., K ? 1,
(18)
(k)
where the ?
are sets of coefficients and W is the matrix representation of an inverse wavelet
transform; W may be orthogonal or have more columns than lines (over-complete representations)
[12]. A wavelet-based prior for z(k) is induced by placing a prior on the coefficients ? (k) . A classical
choice for p(? (k) ) is a generalized Gaussian [14]. Without going into details, under this class of
priors (and others), (16) becomes a non-linear wavelet-based denoising step, which has been widely
studied in the image processing literature. For several choices of p(? (k) ) and W, this denoising
step has a very simple closed form, which essentially corresponds to computing a wavelet transform
of the observations, applying a coefficient-wise non-linear shrinkage/thresholding operation, and
applying the inverse transform to the processed coefficients. This is computationally very efficient,
due to the existence of fast algorithms for computing direct and inverse wavelet transforms; e.g.,
O(n) for an orthogonal wavelet transform or O(n log n) for a shift-invariant redundant transform.
4
4.1
Extensions
Semi-Supervised Segmentation
For semi-supervised image segmentation, the user defines regions in the image for which the true
label is known. Our GEM algorithm is trivially modified to handle this case: if at location i the label
(k)
(j)
is known to be (say) k, we freeze ybi = 1, and ybi = 0, for j 6= k. The E-step is only applied to
those locations for which the label is unknown. The M-step remains unchanged.
4.2
Discriminative Features
Our formulation (as most probabilistic segmentation methods) adopts a generative perspective,
where each p(?|?(k) ) models the data generation mechanism in the corresponding class. However,
discriminative methods (such as support vector machines) are seen as the current state-of-the-art in
classification [7]. We will now show how a pre-trained discriminative classifier can be used in our
GEM algorithm instead of the generative likelihoods.
The E-step (see (9)) obtains the posterior probability that xi was generated by the k-th model, by
b (k) ) with the local prior probability
combining (via Bayes law) the corresponding likelihood p(xi |?
(k)
P [yi = 1|b
zi ]. Consider that, instead of likelihoods derived from generative models, we have a
discriminative classifier, i.e., one that directly provides estimates of the posterior class probabilities
(k)
P [yi = 1|xi ]. To use these values in our segmentation algorithm, we need a way to bias these
(k)
estimates according to the local prior probabilities P [yi = 1|b
zi ], which are responsible for encouraging spatial coherence. Let us assume that we know that the discriminative classifier was trained
using mk samples from the k-th class. It can thus be assumed that these posterior class probabilities
(k)
(k)
verify P [yi = 1|xi ] ? mk p(xi |yi = 1). It is then possible to ?bias? these classifiers, with the
(k)
local prior probabilities P [yi = 1|b
zi ], simply by computing
?
??1
K
(k)
(k)
(j)
(j)
X
P
[y
=
1|x
]
P
[y
=
1|b
z
]
P
[y
=
1|x
]
P
[y
=
1|b
z
]
i
i ?
i
i ?
(k)
i
i
i
i
P [yi = 1|xi ] =
.
mk
m
j
j=1
5
Experiments
In this section we will show experimental results of image segmentation in supervised, unsupervised,
semi-supervised, and discriminative modes. Assessing the performance of a segmentation method
is not a trivial task. Moreover, the performance of segmentation algorithms depends more critically
on the adopted features (which is not the focus of this paper) than on the spatial coherence prior. For
these reasons, we will not present any careful comparative study, but simply a set of experimental
examples testifying for the promising behavior of the proposed approach.
5.1
Supervised and Unsupervised Image Segmentation
The first experiment, reported in Fig. 1, illustrates the algorithm on a synthetic gray scale image
with four Gaussian classes of means 1, 2, 3, and 4, and standard deviation 0.6. For this image, both
supervised and unsupervised segmentation lead to almost visually indistinguishable results, so we
only show the supervised segmentation results. In the Gaussian prior, matrix A corresponds to a
first order neighborhood, that is, Ai,j = ? if and only if j is one of the four nearest neighbors of
i. For wavelet-based segmentation, we have used undecimated Haar wavelets and the Bayes-shrink
denoising procedure [6].
Figure 1: From left to right: observed image, maximum likelihood segmentation, GEM result with
Gaussian prior, GEM result with wavelet-based prior.
5.2
Semi-supervised Image Segmentation
We illustrate the semi-supervised mode of our approach on two real RGB images, shown in Fig. 2.
Each region is modelled by a single multivariate Gaussian density in RGB space. In the example
in the first row, the goal is to segment the image into skin, cloth, and background regions; in the
second example, the goal is to segment the horses from the background. These examples show how
the semi-supervised mode of our algorithm is able to segment the image into regions which ?look
like? the seed regions provided by the user.
Figure 2: From left to right (in each row): observed image with regions indicated by the user as
belonging to each class, segmentation result, region boundaries.
5.3
Discriminative Texture Segmentation
Finally, we illustrate the behavior of the algorithm when used with discriminative classifiers by
applying it to texture segmentation. We build on the work in [8], where SVM classifiers are used
for texture classification (see [8] for complete details about the kernels and texture features used).
Fig. 3 shows two experiments; one with a two-texture 256?512 image and the other with a 5-texture
256 ? 256 image. In the two-class case, one binary SVM was trained on 1000 random patterns from
each class. For the 5-class case, 5 binary SVMs were trained in the ?1-vs-all? mode, with 500
samples from each class. In the 2-class and 5-class cases, the error rates of the SVM classifier are
12.69% and 13.92%, respectively. Our GEM algorithm achieves 0.51% and 2.22%, respectively.
These examples show that our method is able to take class predictions produced by a classifier
lacking any spatial prior and produce segmentations with a high degree of spatial coherence.
6
Conclusions
We have introduced an approach to probabilistic semi-supervised clustering which is particularly
suited for image segmentation. The formulation allows supervised, unsupervised, semi-supervised,
and discriminative modes, and can be used with classical standard image priors (such as Gaussian
fields, or wavelet-based priors). Unlike the usual Markov random field approaches, which involve
combinatorial optimization, our segmentation algorithm consists of a simple generalized EM algorithm. Several experimental examples illustrated the promising behavior of our method. Ongoing
work includes a thorough experimental comparison with state-of-the-art segmentation algorithms,
namely, spectral methods [16] and techniques based on ?graph-cuts? [19].
Acknowledgement: This work was partially supported by the (Portuguese) Fundac?a? o para a
Ci?encia e Tecnologia (FCT), grant POSC/EEA-SRI/61924/2004.
Figure 3: From left to right (in each row): observed image, direct SVM segmentation, segmentation
produced by our algorithm.
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2,214 | 3,009 | Emergence of conjunctive visual features by
quadratic independent component analysis
J.T. Lindgren
Department of Computer Science
University of Helsinki
Finland
[email protected]
Aapo Hyv?arinen
HIIT Basic Research Unit
University of Helsinki
Finland
[email protected]
Abstract
In previous studies, quadratic modelling of natural images has resulted in cell
models that react strongly to edges and bars. Here we apply quadratic Independent Component Analysis to natural image patches, and show that up to a small
approximation error, the estimated components are computing conjunctions of
two linear features. These conjunctive features appear to represent not only edges
and bars, but also inherently two-dimensional stimuli, such as corners. In addition, we show that for many of the components, the underlying linear features
have essentially V1 simple cell receptive field characteristics. Our results indicate
that the development of the V2 cells preferring angles and corners may be partly
explainable by the principle of unsupervised sparse coding of natural images.
1
Introduction
Sparse coding of natural images has led to models that resemble the receptive fields in the primate
primary visual cortex area V1 (see e.g. [1, 2, 3]). An ongoing research effort is in trying to understand and model the computational principles in visual areas following V1, commonly thought to
provide representations for more complicated stimuli. For example, it has recently been shown that
in the Macaque monkey, the V2 area following V1 contains neurons responding favourably to angles
and corners, but not necessarily to their constituent edges if presented alone [4, 5]. This behaviour
can not be easily attained with linear models [6].
In this paper we estimate quadratic models for natural images using Independent Component Analysis (ICA). The used quadratic functions are a natural extension to linear functions (i.e. lT x), and
give the value of a single feature or component as
s = xT Hx + lT x,
(1)
where the matrix H specifies weights for second-order interactions between the input variables in
stimulus x. This class of functions is equivalent to second-order polynomials of the input, and
can compute linear combinations of squared responses of linear models (see e.g. [7]). Another wellknown interpretation of components in a quadratic model is as outputs of two-layer neural networks,
which is based on an eigenvalue decomposition and will be discussed below.
Estimating a quadratic model for natural images with ICA, we report here the emergence of receptive
field models that respond strongly only if the stimulus contains two features that are in a correct
spatial arrangement. With a heavy dimensionality reduction, the conjuncted features are mostly
collinear (i.e. prefer edges or bars), but with a smaller reduction, additional components emerge
that appear to prefer more complex stimuli such as angles or corners. We show that in both cases,
the emerging components approximately operate by computing products between the outputs of two
linear submodels that have V1 simple cell characteristics.
The rest of this paper is organized as follows. In section 2 we describe the quadratic ICA in detail.
Section 3 outlines the dataset and the preprocessing we used, and section 4 describes the results.
Finally, section 5 concludes with discussion and future work.
2
Quadratic ICA
Let x ? Rn be a vectorized grayscale input image patch. A basic form of linear ICA assumes that
each data point is generated as
x = As,
(2)
where A is a linear mixing matrix and s the vector of unknown source signals or independent
components. The dimension of s is assumed to be equal to the dimension of x, possibly after
the x have been reduced by PCA to a smaller dimension. ICA estimation tries to recover s and
the parameter matrix W = A?1 . If the independent components are sparse, this is equivalent to
performing sparse coding (for an account of ICA, see e.g. [8]).
It has been proposed that ICA for quadratic models can be performed by first making a quadratic
basis expansion on each x and then applying standard linear ICA [9]. Let the new data vectors
z ? Rn(n+1)/2+n in quadratic space be
z = ?([x1 , x2 , ..., xn ]) = [x21 , x1 x2 , ..., x22 , x2 x3 , ..., x2n , x1 , x2 , ..., xn ],
(3)
that is, ?(x) generates all the monomials for a second-order polynomial of x, except for the constant
term. Such a dimension expansion is also implicit in kernel methods, where a second-order polynomial kernel would be used instead of ?. Here we work with the more traditional input transformation
for simplicity.
From now on, assume that ICA has been performed on the transformed data z. Then the columns
wi of WT make up the quadratic components (cell models, polynomial filters) of the model. As the
coefficients in wi are weights for a second-order polynomial, it is straightforward to decompose the
response si of each quadratic component to x as
si = wiT z = xT Hi x + lTi x,
(4)
where Hi is a symmetric square matrix corresponding to the weights given to all the cross-terms
and li weights the first- order monomials. It is well known that the Hi can be represented in another
form by eigenvalue decomposition, leading to the expression
si =
n
X
?j (vjT x)2 + lTi x,
(5)
j=1
where ?j are the decreasingly sorted eigenvalues of Hi and vj the corresponding eigenvectors. In
some cases the representation of eq. 5 can help to understand the model, since the individual eigenvectors can be interpreted as linear receptive fields. A quadratic function in this form is illustrated
on the left in figure 1.
However, for our model estimated with quadratic ICA, many of the eigenvectors vj did not resemble
V1 simple cell receptive fields. Here we propose another decomposition which leads to a simple
network computation based on linear receptive fields similar to those of V1 simple cells. Assume
that the two eigenvalues of Hi which are largest in absolute value have opposite signs; we will refer
to these as the dominant eigenvalues and denote them by ?1 and ?n . This assumption will turn out
to hold empirically for our estimated models. Now, including just the two corresponding dominant
eigenvectors and ignoring the linear term, we denote
p
p
p
p
v+ = ( |?1 |v1 + |?n |vn ),
v? = ( |?1 |v1 ? |?n |vn ),
(6)
x
x
x
XYZ[
/ _^]\
(v1T x)2
DD
DD ?
DD1
... ...
DD
DD
! P
?n
XYZ[
HIJK
/ ONML
/ _^]\
(vnT x)2
<
zz
z
z
1 z
z
zz
z
z
z
PQRS
/ WVUT
lT x
x
/s
x
T
_^]\
/ XYZ[
(v+
x)
BB
BB
BB1
BB
BB
! Q
1
T
_^]\
HIJK
/ XYZ[
/ ONML
(v?
x)
/ s?
Figure 1: Quadratic components as networks. Using the eigenvalue decomposition, quadratic forms
can be interpreted as networks. Left, the computation of a single component w, where the vi are the
eigenvectors and the ?i the eigenvalues of the matrix H. Right, its product approximation, which is
possible if the variance is concentrated on just two eigenvectors with eigenvalues of opposite signs.
This turns out to be the case for natural images.
and by using simple arithmetic we obtain a product approximation for eq. 5 as
T
T
s?i = ?1 (v1T x)2 + ?n (vnT x)2 = (v+
x)(v?
x).
(7)
This approximation is shown as a network on the right in figure 1, and will be justified later by
its relatively small empirical error for our models. Providing that the approximation is good, the
intuition is that the component is essentially computing the product of the responses of two linear
filters, analogous to a logical AND operation, or a conjunction. We will empirically show that the
vectors v+ and v? have resemblance to V1 simple cell receptive fields for our model even if the
respective two dominant eigenvectors have more complicated shapes.
3
Materials and methods
In our experiments we used the natural image dataset provided by van Hateren and van der Schaaf
[2]. This dataset contains over 4000 grayscale images representing natural scenes, each image having a resolution of 1024?1536. The intensity distribution over this image set has a very long right tail
related to variation in the overall image contrast and intensity [2, 10]. In addition, it is known that the
high frequencies in natural images contain sampling artifacts due to rectangular sampling, and that
the spectral characteristics are not uniform across frequencies, causing difficulties for gradient-based
estimation methods [1]. To alleviate these problems for ICA, we adopt a two-phase preprocessing
for the raw images, following [1, 2]. This processing can be considered a very simple model of the
physiological pathway containing the retina and the lateral geniculate nucleus (LGN).
First, we address the problem of heavy contrast variation and the long-tailed intensity distribution
by taking a natural logarithm of the input images, effectively compressing their dynamic range. This
preprocessing is similar to what happens in the first stages of natural visual systems, and has been
previously suggested for the current dataset [2]. Next, to correct for the spectral imbalances in the
data, we use the whitening filter proposed by Olshausen and Field [1]. This whitening filter cuts the
highest frequencies, and balances the frequency distribution otherwise by dampening the dominant
low frequencies. We use the filter with the same parameters as in [1]. The whitening filter has bandpass characteristics and hence resembles the center-surround behaviour of LGN cells. In practice,
the filtering approximately decorrelates the data.
After preprocessing each image as a whole, we sampled 300, 000 small image patches from the
images, each patch having a resolution of 9 ? 9. Then we subtracted the local DC-component (mean
intensity) from each patch. These patches then formed the data we used to estimate the quadratic
ICA model. The model fitting was done by transforming the data to the quadratic space using eq.
3, followed by linear ICA. For ICA, we used the FastICA algorithm [11] with tanh nonlinearity
and symmetric estimation of the components. The input dimension in the quadratic space was
Figure 2: The quadratic ICA components when the model size is very small (81 components).
Each quadruple displays the two dominant eigenvectors v1 and vn (top row), and the corresponding
vectors v+ and v? (bottom row). Light and dark areas correspond to positive and negative weights,
respectively. The components have been sorted by collinearity of the conjuncted features.
81 ? (81 + 1)/2 + 81 = 3402. We used PCA to drop the dimension by selecting the 400 most
dominant principal axes, covering approximately 50% of the summed eigenvalues. This resulted in
estimation of 400 independent components (or second-order polynomial filters). We also performed
additional experiments with 81 and 1024 dominant principal axes, corresponding to 18% and 80%
coverage. Due to space constraints, we are unable to discuss the 1024 component model, other than
to briefly mention that it conformed to the main results presented in this paper.
To ensure replicable research, the source codes performing the experiments described in this paper
have been made publicly available1 .
4
Results
In general, interpreting quadratic models can be difficult, and several strategies have been proposed
in the literature (see e.g. [12]). However, in the current work the estimated components turned out to
be fairly simple (up to a small approximation error, as shown later), and as discussed in section 2, it
will be illustrative to display the estimated components in terms of their two dominant eigenvectors
v1 and vn of H, and the respective vectors v+ and v? (see eq. 6). Since either pair of the two
vectors can be used to compute the approximate component response to any stimuli using eq. 7, the
analysis of the components can be based on the vectors v+ and v? if preferred.
Figure 2 shows the quadratic ICA components when a small model was estimated with only 81
components. If we ignore the linear term as in eq. 7, the dominant eigenvectors shown at the top
row of each quadruple are equal to the two unit-norm stimuli that the component reacts most highly
to (e.g. [12]). Note that the reaction to the eigenvector vn (top right) will be highly negative. On the
1
http://www.cs.helsinki.fi/u/jtlindgr/stuff/
Figure 3: Quadratic ICA components picked from 10 bootstrap iterations with 400 components
estimated on each run. All 4000 components were ordered by collinearity of the conjuncted features,
and a small sample of each tail is shown. The presentation is the same as in Figure 2. Top, some
components that prefer conjunctions of two collinear features. Bottom, components that conjunct
two highly orthogonal features. The latter components become more apparent if the model size is
large. Clear Gabor-like V1 characteristics can be seen in both cases in the vectors v+ and v? , even
if the corresponding eigenvectors are more complex.
other hand, both vectors v+ and v? must respond to a stimuli with a non-zero value if the component
is to respond strongly. In the case of this small model size, many of the conjuncted features v+ and
v? are collinear, and respond strongly to edge- or bar-like stimuli. The feature conjunctions that
are not collinear remain more unstructured and appear to react highly to blob-like stimuli. However,
both component types are quite different from ordinary linear detectors for edges, bars and blobs,
since their conjunctive nature makes them much more selective.
In the following, we will limit the discussion to larger models consisting of 400 components (unless
mentioned otherwise). With the higher dimensionality allowed, the diversity of the emerging components increased. Figure 3 shows quadratic ICA components picked from 10 experiments repeated
with different subsets of the input patches and different random seeds. Now, in addition to collinear
conjunctions (on the top in the image), we also get components that conjunct more orthogonal stimuli (on the bottom). The latter components appear to respond favourably to intuitive visual concepts
such as angles and corners. In this case, the benefits of the decomposition to vectors v+ and v?
becomes more apparent, as many of the receptive field models retain their resemblance to Gabor-like
filters (as in e.g. [1, 2, 8]) even if the corresponding eigenvectors become more complicated.
Next we will validate the above characterization by showing that the approximation of eq. 7 holds
up to a generally small error.
First, it turns out that the eigenvalue distributions decay fast for the quadratic forms Hi of the
estimated components. This is illustrated on the left in figure 4, which shows the mean sorted
eigenvalues for the 400 components (for a model of 81 components, the figure was similar). Since
all the eigenvectors have equal norms, the eigenvalues imply the magnitude of the contribution of
its respective eigenvector to the component output value. Due to the fast decay of the eigenvalues,
the two dominant eigenvectors are largely responsible for the component output, providing that the
linear term l is insignificant (for some discussion on the linear term in quadratic models, see [12]).
Figure 4: The conjunctive nature of the components is due to the eigenvalues of the quadratic forms
Hi typically emerging as heavily dominated by just two eigenvectors with opposite-sign eigenvalues. This conjunctiveness is further confirmed by the relatively small approximation error caused
by ignoring the non-dominant eigenvectors and the linear term. Left, sorted eigenvalues of Hi averaged over all 400 components for both quadratic ICA and quadratic PCA. It can be seen that the
ICA-based eigenvalue distributions tend to decay faster. Right, the relative mean square error of the
product approximation for the 400 quadratic ICA components. The components have been sorted
by the error of approximation when Gabor functions have been used to model v+ and v? .
Here the quadratic part tended to dominate the component responses, which may be because the
(co)variances were much larger for the quadratic dimensions of the space than for the linear ones.
The above reasoning is supported by analysis of the prediction error of the product approximation.
We examined this by sampling 100, 000 new image patches (not used in the training), and computing
the mean square error of the approximation divided by the variance of the component response, i.e.
err(?
s) = E[(s??
s)2 ]/V ar(s). This error is shown on the right in figure 4 for all the 400 components.
On average, this relative error was 12% of the respective component variance, ranging from 2% to
57%. Hence, the product approximation appears to capture the behaviour of the components rather
well. The plot also shows the effect of approximating the vectors v+ and v? with Gabor functions,
which are commonly used to model V1 receptive fields. Using Gabor functions, the approximation
error increased, ranging from 8% to 93%, with mean of 34%.
To better understand the obtained error rates, we also fitted linear models to approximate the estimated quadratic cells using least-squares regression. This revealed the quadratic components to
have highly nonlinear behaviour. For all components, the error of the linear approximator was over
90%, coming close to the baseline 100% error attained if the empirical mean is used as a (constant)
predictor.
Since the product approximation only covers the two dominant eigenvectors, it is possible that the
rest of the eigenvectors might code for interesting phenomena through further excitatory and inhibitory effects. However, the quick decay of the eigenvalues in our estimated model should make
any such effects rather minor. Following the ideas and methods of [12], we explored the possibility
that the nondominant eigenvectors coded for invariances of the component. The only strong invariance we found was insensitivity to (possibly local) input sign changes, which is at least partly a
structural property of the model, originating from taking squares in eq. 5. In particular, we observed
no shift-invariance, consistent with some recent findings in the V2 area of the Macaque monkey [5].
We leave more in-depth exploration of the role of the nondominant eigenvectors as future work.
Finally, we performed some experiments to examine to what extent the method of quadratic ICA
on the one hand, and the natural image input data on the other, are responsible for the reported
results. For example, it could be argued that the quadratic ICA components might be very similar
to the quadratic PCA components. Figure 5 illustrates that this is not trivially so by showing 16
PCA components with large eigenvalues. It can be seen that the PCA components quickly lose
resemblance to Gabor-like filters as the eigenvalues decrease. Also, the conjunctive nature of the
estimated features is not as clear for the PCA-based components. This is shown on the left in
Figure 5: Left, the two top rows show the vectors v+ and v? for the first 8 quadratic PCA components. The two bottom rows display the PCA components 41 ? 48. It can be seen that the PCA
components quickly lose any resemblance to Gabor-like receptive fields of V1 simple cells. Right,
some typical quadratic ICA components in terms of the vectors v+ and v? when the model was
estimated on white noise. The circular shapes are likely artifacts due to the whitening filter.
figure 4, demonstrating that on the average, the eigenvalues of the quadratic forms decay slower for
the PCA components. If the whole set of PCA components is studied (not shown), it can be seen
that the components appear to change from low-pass filters to high-pass filters as the eigenvalues
decrease. Comparing figure 5 to figure 3, both outputs seem characteristic to the method applied,
the differences resembling those observed when linear ICA and linear PCA are used to code natural
image data.
To verify that the emerging component structures are not artifacts of the modelling methodology,
we generated a dataset of artificial images of white noise, having the luminance distribution of the
original dataset, but with no spatial or spectral structure. By repeating the model estimation (including preprocessing) on this new dataset, the resulting components did not respond favourably to the
same stimuli as before, and they were no longer clearly conjunctive: the eigenvalue distributions
decayed fast, but tended to have only one dominant eigenvector. Based on these vectors, the components could be roughly categorized to two classes. The first class responded to spatial forms of
center-surround filters, possibly caused by the use of the whitening filter. The second class preferred
apparently random configurations of inhibitory and excitatory effects. Some of the components
estimated on random data are displayed on the right in figure 5 in terms of vectors v+ and v? .
5
Discussion
In this paper, we specified a quadratic model for natural images and estimated its parameters with
independent component analysis. We reported the emergence of cell models exhibiting strongly nonlinear behaviour. In particular, we demonstrated that the estimated cells were essentially computing
products between outputs of two linear filters that had V1 simple cell characteristics. Many of these
feature conjunctions preferred two collinear features, and yet others corresponded to combinations
of more orthogonal stimuli, reacting strongly to angles and corners. Our results indicate that sparse
coding of natural images might partially explain the development of angle- or corner-preferring cells
in V2.
There has been some previous work describing quadratic models of natural image data (i.e. [13, 7,
9]). Of these, the ICA-based approaches [13, 9] resemble ours the most. Bartsch and Obermayer
[13] report curvature detecting cells, but the patch size used and the number of components estimated
were very small, making the results inconclusive. Hashimoto [7] sought to replicate complex cell
properties with an algorithm based on Kullback-Leibler divergences, and does not report conjunctive
features or cells with preferences for angles or corners. Instead, most of the estimated quadratic
forms on static image data had only one dominant eigenvector. Our work extends the previous
research by reporting the emergence of conjunctive components that combine responses of V1-like
linear filters.
The differences of our work to the previous research can be due to various reasons. The number of
estimated components (i.e., number of principal components retained) was seen to affect the feature
diversity, and with only 81 components, the conjuncted features were mostly collinear, producing
highly selective edge or bar detectors. Even larger differences to previous work are likely due to
different input preprocessing: it is known that unprocessed image data can cause difficulties to
statistical estimation of linear models [1, 10] and that both the preprocessing and the size of the
used image patches can affect the estimated features [10]. In quadratic modelling, taking products
between the dimensions of the input data can cause additional problems for any methods relying
on non-robust estimation (such as covariance-based PCA) since the quadratic transform has the
strongest boosting effect on outliers and tails of the marginal distributions.
It is worthwhile to note that despite differences to previous work [13, 7, 9], invariances resembling
complex cell behaviour did not emerge with our method either, although the class of quadratic models contains the classic energy-detector models of complex cells (fitted in e.g. [3]). It could be that
static images and optimization of sparsity alone may not work towards the emergence of invariances,
or equivalently, behaviour resembling logical OR operation, unless the model is further constrained
(for example as in [3]). Optimizing model likelihood can also be preferable to optimizing output
sparseness, but for quadratic ICA it is not clear how to construct a proper probabilistic model.
Finally, an important open question regarding the current work is to what extent the obtained conjunctive features reflect real structures present in the image data. At the time of writing, we have not
been able to either prove or disprove the possibility that the pairings are an algorithmic artifact in
the following sense: it could be that after the effects of quadratic-space PCA have been accounted
for, the quadratic ICA components are only combinations of two rather randomly selected sparse
T
T
x) which are as independent as possible. We are currently investigating
x and v?
components (v+
this issue.
Acknowledgments
The authors wish to thank Jarmo Hurri and the anonymous reviewers for helpful comments. This
work was supported in part by the IST Programme of the European Community, under the PASCAL
Network of Excellence, IST-2002-506778. This publication only reflects the authors? views.
References
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2,215 | 301 | Constructing Hidden Units
using Examples and Queries
Eric B. Baum
Kevin J. Lang
NEC Research Institute
4 Independence Way
Princeton, NJ 08540
ABSTRACT
While the network loading problem for 2-layer threshold nets is
NP-hard when learning from examples alone (as with backpropagation), (Baum, 91) has now proved that a learner can employ queries
to evade the hidden unit credit assignment problem and PAC-load
nets with up to four hidden units in polynomial time. Empirical
tests show that the method can also learn far more complicated
functions such as randomly generated networks with 200 hidden
units. The algorithm easily approximates Wieland's 2-spirals function using a single layer of 50 hidden units, and requires only 30
minutes of CPU time to learn 200-bit parity to 99.7% accuracy.
1
Introd uction
Recent theoretical results (Baum & Haussler, 89) promise good generalization from
multi-layer feedforward nets that are consistent with sufficiently large training sets.
Unfortunately, the problem of finding such a net has been proved intractable due
to the hidden unit credit assignment problem - even for nets containing only 2
hidden units (Blum & Rivest, 88). While back-propagation works well enough on
simple problems, its luck runs out on tasks requiring more than a handful of hidden
units. Consider, for example, Alexis Wielands "2-spirals" mapping from ~2 to
{O, I}. There are many sets of weights that would cause a 2-50-1 network to be
consistent with the training set of figure 3a, but backpropagation seems unable to
find any of them starting from random initial weights. Instead, the procedure drives
the net into a suboptimal configuration like the one pictured in figure 2b.
904
Constructing Hidden Units Using Examples and Queries
Figure 1: The geometry of query learning.
In 1984, Valiant proposed a query learning model in which the learner can ask an
oracle for the output values associated with arbitrary points in the input space.
In the next section we shall see how this additional source of information can be
exploited to locate and pin down a network's hidden units one at a time, thus
avoiding the combinatorial explosion of possible hidden unit configurations which
can arise when one attempts to learn from examples alone.
2
How to find a hidden unit using queries
For now, assume that our task is to build a 2-layer network of binary threshold units
which computes the same function as an existing "target" network. Our first step
will be to draw a positive example x+ and a negative example x_ from our training
set. Because the target net maps these points to different output values, its hidden
layer representations for the points must also be different, so the hyperplane through
input space corresponding to one of the net's hidden units must intersect the line
segment bounded by the two points (see figure 1). We can reduce our uncertainty
about the location of this intersection point by a factor of 2 by asking the oracle
for the target net's output at m, the line segment's midpoint. If, for example, m
is mapped to the same output as x+, then we know that the hidden plane must
intersect the line segment between x_ and m, and we can then further reduce our
uncertainty by querying the midpoint of this segment. By performing b of queries
of this sort, we can determine to within b bits of accuracy the location of a point
Po that lies on the hidden plane. Assuming that our input space has n dimensions,
after finding n - 1 more points on this hyperplane we can solve n equations in n
unknowns to find the weights of the corresponding hidden unit. 1
1 The additional points Pi are obtained by perturbing PO with various small vectors 1r i and then
diving back to the plane via a search that is slightly more complicated than the bisection method
by which we found PO. (Baum, 91) describes this search procedure in detail, &8 well &8 a technique
for verifying that all the points Pi lie on the .ame hidden plane.
90S
906
Baum and Lang
Figure 2: A backprop net before and after being trained on the 2-spirals task.
In these plots over input space, the net's hidden units are shown by lines while its
output is indicated by grey-level shading.
3
Can we find all of a network's hidden units?
Here is the crucial question: now that we have a procedure for finding one hidden unit whose hyperplane passes between a given pair of positive and negative
examples,2 can we discover all of the net's hidden units by invoking this procedure
on a sequence of such example pairs? If the answer is yes, then we have got a viable
learning method because the net's output weights can be efficiently computed via
the linear programming problem that arises from forward-propagating the training
set through the net's first layer of weights. (Baum, 91) proves that for target nets
with four or fewer hidden units we can always find enough of them to compute the
required function. This result is a direct counterpoint to the theorem of (Blum &
Rivest, 88): by using queries, we can PAC learn in polynomial time small threshold
nets that would be NP-hard to learn from examples alone.
However, it is possible for an adversary to construct a larger target net and an
input distribution such that we may not find enough hidden units to compute the
target function even by searching between every pair of examples in our training
set. The problem is that more than one hidden plane can pass between a given pair
of points, so we could repeatedly encounter some of the hidden units while never
seeing others.
2This "positive" and ''negative'' terminology suggests that the target net possesses a single
output unit, but the method is not actually restricted to this case.
Constructing Hidden Units Using Examples and Queries
Figure 3: 2-spirals oracle, and net built by query learning.
Fortunately, the experiments described in the next section suggest that one can
find most of a net's hidden units in the average case. In fact, we may not even
need to find all of a network's hidden units in order to achieve good generalization.
Suppose that one of a network's hidden units is hard to find due to the rarity of
nearby training points. As long as our test set is drawn from the same distribution
as the training set, examples that would be misclassified due to the absence of
this plane will also be rare. Our experiment on learning 200-bit parity illustrates
this point: only 1/4 of the possible hidden units were needed to achieve 99.7%
generalization.
4
Learning random target nets
Although query learning might fail to discover hidden units in the worse case, the
following empirical study suggests that the method has good behavior in the average
case. In each of these learning experiments the target function was computed by a
2-layer threshold net whose k hidden units were each chosen by passing a hyperplane
through a set of n points selected from the uniform distribution on the unit n-sphere.
The output weights of each target net corresponded to a random hyperplane through
the origin of the unit k-sphere. Our training examples were drawn from the uniform
distribution on the corners of the unit n-cube and then classified according to the
target net.
To establish a performance baseline, we attempted to learn several of these functions
using backpropagation. For (n = 20, k = 20) we succeeded in training a net to
97% accuracy in less than a day, but when we increased the size of the problem
to (n
100, k = 50) or (n
200, k
30), 150 hours of CPU time dumped our
backprop nets into local minima that accounted for only 90% of the training data.
=
=
=
907
908
Baum and Lang
In contrast, query learning required only 1.5 hours to learn either of the latter
two functions to 99% accuracy. The method continued to function well when we
increased the problem size to (n = 200, k = 200). In each of five trials at this
scale, a check of 104 training pairs revealed 197 or more hidden planes. Because the
networks were missing a couple of hidden units, their hidden-to-output mappings
were not quite linearly separable. Nevertheless, by running the percept ron algorithm
on 100 x k random examples, in each trial we obtained approximate output weights
whose generalization was 98% or better.
5
Learning 200-bit parity
Because the learning method described above needs to make real-valued queries in
order to localize a hidden plane, it cannot be used to learn a function that is only
defined on boolean inputs. Thus, we defined the parity of a real-valued vector to be
the function computed by the 2-layer parity net of (Rumelhart, Hinton & Williams,
1986), which has input weights of 1, hidden unit thresholds of ~, ~, ... , n - ~, and
output weights alternating between 1 and -1. The n parallel hidden planes of this
net carve the input space into n + 1 diagonal slabs, each of which contains all of the
binary patterns with a particular number of 1'so
After adopting this definition of parity (which agrees with the standard definition on
boolean inputs), we applied the query learning algorithm to 200-dimensionalinput
patterns. A search of 30,000 pairs of examples drawn randomly and uniformly
from the corners of the unit cube revealed 46 of the 200 decision planes of the
target function. Using approximate output weights computed by the perceptron
algorithm, we found the nets generalization rate to be 99.7%. If it seems surprising
that the net could perform so well while lacking so many hidden planes, consider the
following. The target planes that we did find were the middle ones with thresholds
near 100, and these are the relevant ones for classifying inputs that contain about
the same number of 1's and O's. Because vectors of uniform random bits are unlikely
to contain many more 1's than O's or vice versa, we had little chance of stumbling
across hidden planes with high or low thresholds while learning, but we were also
unlikely to need them for classifying any given test case.
6
Function approximation using queries
Suppose now that our goal in building a threshold net is to approximate an arbitrary
function rather than to duplicate an existing threshold net. Earlier, we were worried
about whether we could locate all of a target net's hidden units, but at least we
knew how many of them there were, and we knew that we had made real progress
when we found one of them. Now, the hidden units constructed by our algorithm
are merely tangents to the true decision boundaries of the target fuction, and we do
not know ahead of time how many such units will be required to construct a decent
approximation to the function.
While one could keep adding hidden units to a net until the hidden layer representation of the training set becomes linearly separable, the fact that there are
Constructing Hidden Units Using Examples and Queries
learning
algorithm
additional
heuristics
none
querIes
reject redundant units
two-stage construction
conjugate gradient backprop
hidden units
min
max
90
160
80
65
49
59
60
train
errors
0
0
0
avg=9
test errors
min max
70
136
47
72
15
45
80
125
Table 1: 2-spirals performance summary.
infinitely many of tangents to a given curve can result in the creation of an oversized net which generalizes poorly. This problem can be addressed heuristically by
rejecting new hidden units that are too similar to existing ones. For example, the
top two rows of the above table summarize the results of 10 learning trials on the
two-spirals problem with and without such a heuristic. 3 By imposing a floor on the
difference between two hidden units,4 we reduced the size of our nets and the rate
of generalization errors by 40%.
The following two-stage heuristic training method resulted in even better networks.
During the first stage of learning we attempted to create a minimally necessary
set of hidden units by searching only between training examples that were not
yet divided by an existing hidden unit. During the second stage of learning we
tried to increase the separability of our hidden codes by repeatedly computing an
approximate set of output weights and then searching for hidden units between
misclassified examples and nearby counterexamples. This heuristic was motivated
by the observation that examples tend to be misclassified when a nearby piece of
the target function's decision boundary has not been discovered. Ten trials of this
method resulted in networks containing an average of just 54 hidden units, and the
nets committed an average of only 29 mistakes on the test set. An example of a
network generated by this method is shown in figure 3b.
For comparison, we made 10 attempts to train a 60-hidden-unit backprop net on
the 2-spirals problem starting from uniform random weights and using conjugate
gradient code provided by Steve Nowlan. While these nets had more than enough
hidden units to compute the required function, not one ofthem succeeded in learning
the complete training set. s
3To employ query learning, we defined the oracle function indicated by shading in figure 3a.
The 194 training points are shown by dots in the figure. Our 576-element test set consisted of 3
points between each pair of adjacent same-class training points.
? Specifically, we required a minimum euclidean distance of 0.3 between the weights of two
hidden units (after first normalizing the weight vectors so that the length of the non-threshold
part of each vector was 1.
5lnterestingly, a 2-50-1 backprop net whose initial weights were drawn from a handcrafted
distribution (hidden units with uniform random positions together with the appropriate output
weights) came much closer to success than 2-50-1 nets with uniform random initial weights (compare figures 4 and 2). We can sometimes address tough problems with backprop when our prior
knowledge gives us a head start.
909
910
Baum and Lang
Figure 4: Backprop works better when started near a solution.
These results illustrate the main point of this paper: the currently prevalent training
methodology (local optimization of random initial weights) is too weak to solve the
NP-hard problem of hidden unit deployment. We believe that methods such as
query learning which avoid the credit assignment problem are essential to the future
of connectionism.
References
E. Baum & D. Haussler. (1989) What size net gives valid generalization? Neural
Computation 1(1): 151-160.
E. Baum. (1991) Neural Net Algorithms that Learn in Polynomial Time from Examples and Queries. IEEE Transactions on Neural Networks 2(1), January, 1991.
A. Blum & R. L. Rivest. (1988) Training a 3-node neural network is NP-complete.
In D. S. Touretzky (ed.), Advances in Neural Information Processing Systems 1,
494-501. San Mateo, CA: Morgan Kaufmann.
K. Lang & M. Witbrock. (1988) Learning to Tell Two Spirals Apart. Proceedings
of the 1988 Connectionist Models Summer School, Morgan Kaufmann.
D. Rumelhart, G. Hinton, & R. Williams. (1986) Learning internal representations
by error propagation. In D. Rumelhart & J. McClelland (eds.) Parallel Distributed
Processing, MIT Press.
L. G. Valiant. (1984) A theory of the learnable. Comm. ACM 27(11): 1134-1142.
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2,216 | 3,010 | Robotic Grasping of Novel Objects
Ashutosh Saxena, Justin Driemeyer, Justin Kearns, Andrew Y. Ng
Computer Science Department
Stanford University, Stanford, CA 94305
{asaxena,jdriemeyer,jkearns,ang}@cs.stanford.edu
Abstract
We consider the problem of grasping novel objects, specifically ones that are being seen for the first time through vision. We present a learning algorithm that
neither requires, nor tries to build, a 3-d model of the object. Instead it predicts,
directly as a function of the images, a point at which to grasp the object. Our algorithm is trained via supervised learning, using synthetic images for the training
set. We demonstrate on a robotic manipulation platform that this approach successfully grasps a wide variety of objects, such as wine glasses, duct tape, markers, a translucent box, jugs, knife-cutters, cellphones, keys, screwdrivers, staplers,
toothbrushes, a thick coil of wire, a strangely shaped power horn, and others, none
of which were seen in the training set.
1
Introduction
In this paper, we address the problem of grasping novel objects that a robot is perceiving for the first
time through vision.
Modern-day robots can be carefully hand-programmed or ?scripted? to carry out many complex manipulation tasks, ranging from using tools to assemble complex machinery, to balancing a spinning
top on the edge of a sword. [15] However, autonomously grasping a previously unknown object still
remains a challenging problem. If the object was previously known, or if we are able to obtain a full
3-d model of it, then various approaches, for example ones based on friction cones, [5] form- and
force-closure, [1] pre-stored primitives, [7] or other methods can be applied. However, in practical
scenarios it is often very difficult to obtain a full and accurate 3-d reconstruction of an object seen
for the first time through vision. This is particularly true if we have only a single camera; for stereo
systems, 3-d reconstruction is difficult for objects without texture, and even when stereopsis works
well, it would typically reconstruct only the visible portions of the object. Finally, even if more
specialized sensors such as laser scanners (or active stereo) are used to estimate the object?s shape,
we would still have only a 3-d reconstruction of the front face of the object.
In contrast to these approaches, we propose a learning algorithm that neither requires, nor tries to
build, a 3-d model of the object. Instead it predicts, directly as a function of the images, a point at
which to grasp the object. Informally, the algorithm takes two or more pictures of the object, and
then tries to identify a point within each 2-d image that corresponds to a good point at which to grasp
the object. (For example, if trying to grasp a coffee mug, it might try to identify the mid-point of the
handle.) Given these 2-d points in each image, we use triangulation to obtain a 3-d position at which
to actually attempt the grasp. Thus, rather than trying to triangulate every single point within each
image in order to estimate depths?as in dense stereo?we only attempt to triangulate one (or at
most a small number of) points corresponding to the 3-d point where we will grasp the object. This
allows us to grasp an object without ever needing to obtain its full 3-d shape, and applies even to
textureless, translucent or reflective objects on which standard stereo 3-d reconstruction fares poorly.
To the best of our knowledge, our work represents the first algorithm capable of grasping novel
objects (ones where a 3-d model is not available), including ones from novel object classes, that we
are perceiving for the first time using vision.
Figure 1: Examples of objects on which the grasping algorithm was tested.
In prior work, a few others have also applied learning to robotic grasping. [1] For example, Jebara et
al. [8] used a supervised learning algorithm to learn grasps, for settings where a full 3-d model of the
object is known. Hsiao and Lozano-Perez [4] also apply learning to grasping, but again assuming a
fully known 3-d model of the object. Piater?s algorithm [9] learned to position single fingers given
a top-down view of an object, but considered only very simple objects (specifically, square, triangle
and round ?blocks?). Platt et al. [10] learned to sequence together manipulation gaits, but again
assumed a specific, known, object. There is also extensive literature on recognition of known object
classes (such as cups, mugs, etc.) [14], but this seems unlikely to apply directly to grasping objects
from novel object classes.
To pick up an object, we need to identify the grasping point?more formally, a position for the
robot?s end-effector. This paper focuses on the task of grasp identification, and thus we will consider
only objects that can be picked up without performing complex manipulation.1 We will attempt to
grasp a number of common office and household objects such as toothbrushes, pens, books, mugs,
martini glasses, jugs, keys, duct tape, and markers. (See Fig. 1.)
The remainder of this paper is structured as follows. In Section 2, we describe our learning approach,
as well as our probabilistic model for inferring the grasping point. In Section 3, we describe the
motion planning/trajectory planning (on our 5 degree of freedom arm) for moving the manipulator
to the grasping point. In Section 4, we report the results of extensive experiments performed to
evaluate our algorithm, and Section 5 concludes.
2
Learning the Grasping Point
Because even very different objects can have similar sub-parts, there are certain visual features that
indicate good grasps, and that remain consistent across many different objects. For example, jugs,
cups, and coffee mugs all have handles; and pens, white-board markers, toothbrushes, screw-drivers,
etc. are all long objects that can be grasped roughly at their mid-point. We propose a learning
approach that uses visual features to predict good grasping points across a large range of objects.
Given two (or more) images of an object taken from different camera positions, we will predict the
3-d position of a grasping point. An image is a projection of the three-dimensional world into an
image plane, and does not have depth information. In our approach, we will predict the 2-d location
of the grasp in each image; more formally, we will try to identify the projection of a good grasping
point onto the image plane. If each of these points can be perfectly identified in each image, we
can then easily ?triangulate? from these images to obtain the 3-d grasping point. (See Fig. 4a.) In
practice it is difficult to identify the projection of a grasping point into the image plane (and, if there
are multiple grasping points, then the correspondence problem?i.e., deciding which grasping point
in one image corresponds to which point in another image?must also be solved). On our robotic
platform, this problem is further exacerbated by uncertainty in the position of the camera when the
1
For example, picking up a heavy book lying flat on table might require a sequence of complex manipulations, such as to first slide the book slightly past the edge of the table so that the manipulator can place its
fingers around the book.
Figure 2: Examples of different edge and texture filters used to calculate the features.
Figure 3: Synthetic images of the objects used for training. The classes of objects used for training
were martini glasses, mugs, teacups, pencils, whiteboard erasers, and books.
images were taken. To address all of these issues, we develop a probabilistic model over possible
grasping points, and apply it to infer a good position at which to grasp an object. 2
2.1
Features
In our approach, we begin by dividing the image into small rectangular patches, and for each patch
predict if it is a projection of a grasping point onto the image plane. For this prediction problem,
we chose features that represent three types of local cues: edges, textures, and color. [11, 13] We
compute features representing edges by convolving the intensity channel3 with 6 oriented edge filters
(Fig. 2). Texture information is mostly contained within the image intensity channel, so we apply
9 Laws masks to this channel to compute the texture energy. For the color channels, low frequency
information is most useful to identify grasps; our color features are computed by applying a local
averaging filter (the first Laws mask) to the 2 color channels. We then compute the sum-squared
energy of each of these filter outputs. This gives us an initial feature vector of dimension 17.
To predict if a patch contains a grasping point, local image features centered on the patch are insufficient, and one has to use more global properties of the object. We attempt to capture this information
by using image features extracted at multiple spatial scales (3 in our experiments) for the patch. Objects exhibit different behaviors across different scales, and using multi-scale features allows us to
capture these variations. In detail, we compute the 17 features described above from that patch as
well as the 24 neighboring patches (in a 5x5 window centered around the patch of interest). This
gives us a feature vector x of dimension 1 ? 17 ? 3 + 24 ? 17 = 459.
2.2
Synthetic Data for Training
We apply supervised learning to learn to identify patches that contain grasping points. To do so,
we require a labeled training set, i.e., a set of images of objects labeled with the 2-d location of the
grasping point in each image. Collecting real-world data of this sort is cumbersome, and manual
labeling is prone to errors. Thus, we instead chose to generate, and learn from, synthetic data that is
automatically labeled with the correct grasps.
In detail, we generate synthetic images along with correct grasp (Fig. 3) using a computer graphics ray tracer,4 as this produces more realistic images than other simpler rendering methods. 5 The
advantages of using synthetic images are multi-fold. First, once a synthetic model of an object has
been created, a large number of training examples can be automatically generated by rendering the
object under different (randomly chosen) lighting conditions, camera positions and orientations, etc.
2
An earlier version of this work without the probabilistic model and using simpler learning/inference was
described in [12].
3
We use YCbCr color space, where Y is the intensity channel, and Cb and Cr are color channels.
4
Ray tracing [3] is a standard image rendering method from computer graphics. It handles many real-world
optical phenomenon such as multiple specular reflections, textures, soft shadows, smooth curves, and caustics.
We used PovRay, an open source ray tracer.
5
There is a relation between the quality of the synthetically generated images and the accuracy of the
algorithm. The better the quality of the synthetically generated images and graphical realism, the better the
accuracy of the algorithm. Therefore, we use a ray tracer instead of faster, but cruder, openGL style graphics.
Michels, Saxena and Ng [6] used synthetic openGL images to learn distances in natural scenes. However,
because openGL style graphics have less realism, the learning performance sometimes decreased with added
complexity in the scenes.
In addition, to increase the diversity of the training data generated, we randomized different properties of the objects such as color, scale, and text (e.g., on the face of a book). The time-consuming
part of synthetic data generation is the creation of the mesh models of the objects. However, there
are many objects for which models are available on the internet, and can be used with only minor
modifications. We generated 2500 examples from synthetic data, comprising objects from six object
classes (see Figure 3). Using synthetic data also allows us to generate perfect labels for the training
set with the exact location of a good grasp for each object. In contrast, collecting and manually
labeling a comparably sized set of real images would have been extremely time-consuming.
2.3
Probabilistic Model
On our manipulation platform, we have a camera mounted on the robotic arm. (See Fig. 6) When
asked to grasp an object, we command the arm to move the camera to two or more positions, so as
to acquire images of the object from these positions. However, there are inaccuracies in the physical
positioning of the arm, and hence some slight uncertainty in the position of the camera when the
images are acquired. We now describe how we model these position errors. Formally, let C be the
image that would have been taken if the robot had moved exactly to the commanded position and
orientation. However, due to robot positioning error, instead an image C? is taken from a slightly
different location. Let (u, v) be a 2-d position in image C, and let (?
u, v?) be the corresponding image
? Thus C(u, v) = C(?
? u, v?), where C(u, v) is the pixel value at (u, v) in image C. The
position in C.
errors in camera position/pose should usually be small,6 and we model the difference between (u, v)
and (?
u, v?) using an additive Gaussian model: u
? = u + u , v? = v + v , where u , v ? N (0, ? 2 ).
For each location (u, v) in an image C, we define the class label to be z(u, v) =
1{(u, v) is the projection of a grasping point into image plane}. (Here, 1{?} is the indicator func? we similarly
tion; 1{True} = 1, 1{False} = 0.) For a corresponding location (?
u, v?) in image C,
? Since,
define z?(?
u, v?) to indicate whether position (?
u, v?) represents a grasping point in the image C.
?
(u, v) and (?
u, v?) are corresponding pixels in C and C, we assume z?(?
u, v?) = z(u, v). Thus:
?
P (z(u, v) = 1|C) = P (?
z (?
u, v?) = 1|C)
R R
? u dv
= u v P (u , v )P (?
z (u + u , v + v ) = 1|C)d
(1)
(2)
Here, P (u , v ) is the (Gaussian) density over u and v . Further, we use logistic regression to model
the probability of a 2-d position (u + u , v + v ) in C? being a good grasping point:
T
? = P (?
P (?
z (u + u , v + v ) = 1|C)
z (u + u , v + v ) = 1|x; w) = 1/(1 + e?w x )
(3)
where x ? R459 are the features for the rectangular patch centered at (u + u , v + v ) in image
C? (described in Section 2.1). The parameter of this model wQ? R459 is learned using standard
maximum likelihood for logistic regression: w = arg maxw0 i P (zi |xi ; w0 ), where (xi , zi ) are
the synthetic training examples (image patches and labels), as described in Section 2.2. Fig. 5 shows
the result of applying the learned logistic regression model to some real (non-synthetic) images.
Given two or more images of a new object from different camera positions, we want to infer the
3-d position of the grasping point. (See Fig. 4.) Because logistic regression may have predicted
multiple grasping points per image, there is usually ambiguity in the correspondence problem (i.e.,
which grasping point in one image corresponds to which graping point in another). To address this
while also taking into account the uncertainty in camera position, we propose a probabilistic model
over possible grasping points in 3-d space. In detail, we discretize the 3-d work-space of the robotic
arm into a regular 3-d grid G ? R3 , and associate with each grid element j a random variable
yj = 1{grid cell j is a grasping point}.
From each camera location i = 1, ..., N , one image is taken. In image Ci , let the ray passing through
(u, v) be denoted Ri (u, v). Let Gi (u, v) ? G be the set of grid-cells through which the ray Ri (u, v)
passes. Let r1 , ...rK ? Gi (u, v) be the indices of the grid-cells lying on the ray Ri (u, v) .
6
The robot position/orientation error is typically small (position is usually accurate to within 1mm), but it
is still important to model this error. From our experiments (see Section 4), if we set ? 2 = 0, the triangulation
is highly inaccurate, with average error in predicting grasping point being 15.40 cm, as compared to 1.85 cm
when appropriate ? 2 is chosen.
Figure 4: (a) Diagram illustrating rays from two images C1 and C2 intersecting at a grasping point
(shown in dark blue). (b) Actual plot in 3-d showing multiple rays from 4 images intersecting at
the grasping point. All grid-cells with at least one ray passing nearby are colored using a light
blue-green-dark blue colormap, where dark blue represents those grid-cells which have many rays
passing near them. (Best viewed in color.)
We know that if any of the grid-cells rj along the ray represent a grasping point, then its projection
is a grasp point. More formally, zi (u, v) = 1 if and only if yr1 = 1 or yr2 = 1 or . . . or yrK = 1.
For simplicity, we use a (arguably unrealistic) naive Bayes-like assumption of independence, and
model the relation between P (zi (u, v) = 1|Ci ) and P (yr1 = 1 or . . . or yrK = 1|Ci ) as
QK
P (zi (u, v) = 0|Ci ) = P (yr1 = 0, ..., yrK = 0|Ci ) = j=1 P (yrj = 0|Ci )
(4)
Assuming that any grid-cell along a ray is equally likely to be a grasping point, this therefore gives
P (yrj = 1|Ci ) = 1 ? (1 ? P (zi (u, v) = 1|Ci ))1/K
(5)
Next, using another naive Bayes-like independence assumption, we estimate the probability of a
particular grid-cell yj ? G being a grasping point as:
QN
P (yj =1)
P (yj =1)P (C1 ,...,CN |yj =1)
= P (C1 ,...,C
(6)
P (yj = 1|C1 , ..., CN ) =
i=1 P (Ci |yj = 1)
P (C1 ,...,CN )
N)
Q
Q
P (yj =1)
N P (yj =1|Ci )P (Ci )
N
= P (C1 ,...,C
? i=1 P (yj = 1|Ci ) (7)
i=1
P (yj =1)
N)
where P (yj = 1) is the prior probability of a grid-cell being a grasping point (set to a constant
value in our experiments). Using Equations 2, 3, 5, and 7, we can now compute (up to a constant
of proportionality that does not depend on the grid-cell) the probability of any grid-cell y j being a
valid grasping point, given the images.
2.4
MAP Inference
We infer the best grasping point by choosing the 3-d position (grid-cell) that is most likely to be a
valid grasping point. More formally, using Eq. 5 and 7, we will choose:
QN
arg maxj log P (yj = 1|C1 , ..., CN ) = arg maxj log i=1 P (yj = 1|Ci )
(8)
PN
1/K
= arg maxj
(9)
i=1 log 1 ? (1 ? P (zi (u, v) = 1|Ci ))
where P (zi (u, v) = 1|Ci ) is given by Eq. 2 and 3. A straightforward implementation that explicitly computes the sum above for every single grid-cell would give good grasping performance, but
be extremely inefficient (over 110 seconds). For real-time manipulation, we therefore used a more
efficient search algorithm in which we explicitly consider only grid-cells yj that at least one ray
Ri (u, v) intersects. Further, the counting operation in Eq. 9 is implemented using an efficient counting algorithm that accumulates the sums over all grid-cells by iterating over all the images N and
rays Ri (u, v).7 This results in an algorithm that identifies a grasping position in 1.2 sec.
7
Since there are only a few places in an image where P (zi (u, v) = 1|Ci ) > 0, the counting algorithm is
computationally much less expensive than enumerating over all grid-cells. In practice, we found that restricting
attention to areas where P (zi (u, v) = 1|Ci ) > 0.1 allows us to further reduce the number of rays to be
considered, with no noticeable degradation in performance.
Figure 5: Grasping point classification. The red points in each image show the most likely locations,
predicted to be candidate grasping points by our logistic regression model. (Best viewed in color.)
Figure 6: The robotic arm picking up various objects: box, screwdriver, duct-tape, wine glass, a
solder tool holder, powerhorn, cellphone, and martini glass and cereal bowl from dishwasher.
3
Control
Having identified a grasping point, we have to move the end-effector of the robotic arm to it, and
pick up the object. In detail, our algorithm plans a trajectory in joint angle space [5] to take the endeffector to an approach position,8 and then moves the end-effecter in a straight line forward towards
the grasping point. Our robotic arm uses two classes of grasps: downward grasps and outward
grasps. These arise as a direct consequence of the shape of the workspace of our 5 dof robotic arm
(Fig. 6). A ?downward? grasp is used for objects that are close to the base of the arm, which the
arm will grasp by reaching in a downward direction. An ?outward? grasp is for objects further away
from the base, for which the arm is unable to reach in a downward direction. The class is determined
based on the position of the object and grasping point.
4
Experiments
4.1
Hardware Setup
Our experiments used a mobile robotic platform called STAIR (STanford AI Robot) on which are
mounted a robotic arm, as well as other equipment such as our web-camera, microphones, etc.
STAIR was built as part of a project whose long-term goal is to create a robot that can navigate home
and office environments, pick up and interact with objects and tools, and intelligently converse with
and help people in these environments. Our algorithms for grasping novel objects represent a first
step towards achieving some of these goals. The robotic arm we used is the Harmonic Arm made
by Neuronics. This is a 4 kg, 5-dof arm equipped with a parallel plate gripper, and has a positioning
accuracy of ?1 mm. Our vision system used a low-quality webcam mounted near the end-effector.
8
The approach position is set to be a fixed distance away from the predicted grasp point.
Table 1: Average absolute error in locating the grasp point for different objects, as well as grasp
success rate for picking up the different objects using our robotic arm. (Although training was done
on synthetic images, testing was done on the real robotic arm and real objects.)
O BJECTS S IMILAR TO O NES T RAINED ON
T ESTED ON
M EAN
G RASP E RROR ( CM )
RATE
M UGS
P ENS
W INE G LASS
B OOKS
E RASER /
C ELLPHONE
2.4
0.9
1.2
2.9
75%
100%
100%
75%
1.6
100%
OVERALL
1.80
90%
4.2
N OVEL O BJECTS
M EAN
E RROR ( CM )
D UCT TAPE
1.8
K EYS
1.0
M ARKERS /S CREWDRIVER
1.1
T OOTHBRUSH /C UTTER
1.1
J UG
1.7
T RANSLUCENT B OX
3.1
P OWERHORN
3.6
C OILED W IRE
1.4
OVERALL
1.85
T ESTED ON
G RASP RATE
100%
100%
100%
100%
75%
75%
50%
100%
87.5%
Results and Discussion
We first evaluated the predictive accuracy of the algorithm on synthetic images (not contained in
the training set). (See Fig. 5.) The average accuracy for classifying whether a 2-d image patch is a
projection of a grasping point was 94.2% (evaluated on a balanced test set), although the accuracy
in predicting 3-d grasping points was higher because the probabilistic model for inferring a 3-d
grasping point automatically aggregates data from multiple images, and therefore ?fixes? some of
the errors from individual classifiers.
We then tested the algorithm on the physical robotic arm. Here, the task was to use input from a
web-camera, mounted on the robot, to pick up an object placed in front of the robot. Recall that the
parameters of the vision algorithm were trained from synthetic images of a small set of six object
classes, namely books, martini glasses, white-board erasers, coffee mugs, tea cups and pencils. We
performed experiments on coffee mugs, wine glasses (partially filled with water), pencils, books,
and erasers?but all of different dimensions and appearance than the ones in the training set?
as well as a large set of objects from novel object classes, such as rolls of duct tape, markers, a
translucent box, jugs, knife-cutters, cellphones, pens, keys, screwdrivers, staplers, toothbrushes,
a thick coil of wire, a strangely shaped power horn, etc. (See Fig. 1.) We note that many of these
objects are translucent, textureless, and/or reflective, making 3-d reconstruction difficult for standard
stereo systems. (Indeed, a carefully-calibrated Point Gray stereo system, the Bumblebee BB-COL20, ?with higher quality cameras than our web-camera?fails to accurately reconstruct the visible
portions of 9 out of 12 objects.)
In extensive experiments, the algorithm for predicting grasps in images appeared to generalize very
well. Despite being tested on images of real (rather than synthetic) objects, including many very
different from ones in the training set, it was usually able to identify correct grasp points. We note
that test set error (in terms of average absolute error in the predicted position of the grasp point) on
the real images was only somewhat higher than the error on synthetic images; this shows that the
algorithm trained on synthetic images transfers well to real images. (Over all 5 object types used
in the synthetic data, average absolute error was 0.81cm in the synthetic images; and over all the
13 real test objects, average error was 1.83cm.) For comparison, neonate humans can grasp simple
objects with an average accuracy of 1.5cm. [2]
Table 1 shows the errors in the predicted grasping points on the test set. The table presents results
separately for objects which were similar to those we trained on (e.g., coffee mugs) and those which
were very dissimilar to the training objects (e.g., duct tape). In addition to reporting errors in grasp
positions, we also report the grasp success rate, i.e., the fraction of times the robotic arm was able
to physically pick up the object (out of 4 trials). On average, the robot picked up the novel objects
87.5% of the time.
For simple objects such as cellphones, wine glasses, keys, toothbrushes, etc., the algorithm performed perfectly (100% grasp success rate). However, grasping objects such as mugs or jugs (by
the handle) allows only a narrow trajectory of approach?where one ?finger? is inserted into the
handle?so that even a small error in the grasping point identification causes the arm to hit and
move the object, resulting in a failed grasp attempt. Although it may be possible to improve the
algorithm?s accuracy, we believe that these problems can best be solved by using a more advanced
robotic arm that is capable of haptic (touch) feedback.
In many instances, the algorithm was able to pick up completely novel objects (a strangely shaped
power-horn, duct-tape, solder tool holder, etc.; see Fig. 1 and 6). Perceiving a transparent wine glass
is a difficult problem for standard vision (e.g., stereopsis) algorithms because of reflections, etc.
However, as shown in Table 1, our algorithm successfully picked it up 100% of the time. The same
rate of success holds even if the glass is 2/3 filled with water. Videos showing the robot grasping
the objects, are available at
http://ai.stanford.edu/?asaxena/learninggrasp/
We also applied our learning algorithm to the task of unloading items from a dishwasher. 9 Fig. 5
demonstrates that the algorithm correctly identifies the grasp on multiple objects even in the presence
of clutter and occlusion. Fig. 6 shows our robot unloading some items from a dishwasher.
5
Conclusions
We proposed an algorithm to enable a robot to grasp an object that it has never seen before. Our
learning algorithm neither tries to build, nor requires, a 3-d model of the object. Instead it predicts,
directly as a function of the images, a point at which to grasp the object. In our experiments,
the algorithm generalizes very well to novel objects and environments, and our robot successfully
grasped a wide variety of objects, such as wine glasses, duct tape, markers, a translucent box, jugs,
knife-cutters, cellphones, keys, screwdrivers, staplers, toothbrushes, a thick coil of wire, a strangely
shaped power horn, and others, none of which were seen in the training set.
Acknowledgment
We give warm thanks to Anya Petrovskaya, Morgan Quigley, and Jimmy Zhang for help with the robotic arm
control driver software. This work was supported by the DARPA transfer learning program under contract
number FA8750-05-2-0249.
References
[1] A. Bicchi and V. Kumar. Robotic grasping and contact: a review. In ICRA, 2000.
[2] T. G. R. Bower, J. M. Broughton, and M. K. Moore. Demonstration of intention in the reaching behaviour
of neonate humans. Nature, 228:679?681, 1970.
[3] A. S. Glassner. An Introduction to Ray Tracing. Morgan Kaufmann Publishers, Inc., San Francisco, 1989.
[4] K. Hsiao and T. Lozano-Perez. Imitation learning of whole-body grasps. In IEEE/RJS International
Conference on Intelligent Robots and Systems (IROS), 2006.
[5] M. T. Mason and J. K. Salisbury. Manipulator grasping and pushing operations. In Robot Hands and the
Mechanics of Manipulation. The MIT Press, Cambridge, MA, 1985.
[6] J. Michels, A. Saxena, and A. Y. Ng. High speed obstacle avoidance using monocular vision and reinforcement learning. In ICML, 2005.
[7] Miller and et. al. Automatic grasp planning using shape primitives. In ICRA, 2003.
[8] R. Pelossof and et. al. An svm learning approach to robotic grasping. In ICRA, 2004.
[9] J. H. Piater. Learning visual features to predict hand orientations. In ICML Workshop on Machine Learning of Spatial Knowledge, 2002.
[10] R. Platt, A. H. Fagg, and R. Grupen. Reusing schematic grasping policies. In IEEE-RAS International
Conference on Humanoid Robots, Tsukuba, Japan, 2005.
[11] A. Saxena, S. H. Chung, and A. Y. Ng. Learning depth from single monocular images. In NIPS 18, 2005.
[12] A. Saxena, J. Driemeyer, J. Kearns, C. Osondu, and A. Y. Ng. Learning to grasp novel objects using
vision. In 10th International Symposium of Experimental Robotics (ISER), 2006.
[13] A. Saxena, J. Schulte, and A. Y. Ng. Depth estimation using monocular and stereo cues. In 20th International Joint Conference on Artificial Intelligence (IJCAI), 2007.
[14] H. Schneiderman and T. Kanade. Probabilistic modeling of local appearance and spatial relationships for
object recognition. In CVPR, 1998.
[15] T. Shin-ichi and M. Satoshi. Living and working with robots. Nipponia, 2000.
9
To improve performance, we also used depth-based features. More formally, we applied our texture based
features to the depth image obtained from a stereo camera, and appended them to the feature vector used in
classification. We also appended some hand-labeled real examples of dishwasher images to the training set to
prevent the algorithm from identifying grasping points on background clutter, such as dishwasher prongs.
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2,217 | 3,011 | Neurophysiological Evidence of Cooperative
Mechanisms for Stereo Computation
Jason M. Samonds
Brian R. Potetz
Tai Sing Lee
Center for the Neural Basis
CNBC and Computer
CNBC and Computer
of Cognition (CNBC)
Science Department
Science Department
Carnegie Mellon University Carnegie Mellon University Carnegie Mellon University
Pittsburgh, PA 15213
Pittsburgh, PA 15213
Pittsburgh, PA 15213
[email protected]
[email protected]
[email protected]
Abstract
Although there has been substantial progress in understanding the neurophysiological mechanisms of stereopsis, how neurons interact in a network
during stereo computation remains unclear. Computational models on
stereopsis suggest local competition and long-range cooperation are important for resolving ambiguity during stereo matching. To test these predictions, we simultaneously recorded from multiple neurons in V1 of awake,
behaving macaques while presenting surfaces of different depths rendered
in dynamic random dot stereograms. We found that the interaction between
pairs of neurons was a function of similarity in receptive fields, as well as
of the input stimulus. Neurons coding the same depth experienced common
inhibition early in their responses for stimuli presented at their nonpreferred disparities. They experienced mutual facilitation later in their responses for stimulation at their preferred disparity. These findings are consistent with a local competition mechanism that first removes gross mismatches, and a global cooperative mechanism that further refines depth estimates.
1
In trod u ction
The human visual system is able to extract three-dimensional (3D) structures in random
noise stereograms even when such images evoke no perceptible patterns when viewed
monocularly [1]. Bela Julesz proposed that this is accomplished by a stereopsis mechanism
that detects correlated shifts in 2D noise patterns between the two eyes. He also suggested
that this mechanism likely involves cooperative neural processing early in the visual system.
Marr and Poggio formalized the computational constraints for solving stereo matching (Fig.
1a) and devised an algorithm that can discover the underlying 3D structures in a variety of
random dot stereogram patterns [2]. Their algorithm was based on two rules: (1) each element or feature is unique (i.e., can be assigned only one disparity) and (2) surfaces of objects
are cohesive (i.e., depth changes gradually across space). To describe their algorithm in neurophysiological terms, we can consider neurons in primary visual cortex as simple element
or feature detectors. The first rule is implemented by introducing competitive interactions
(mutual inhibition) among neurons of different disparity tuning at each location (Fig. 1b,
blue solid horizontal or vertical lines), allowing only one disparity to be detected at each
location. The second rule is implemented by introducing cooperative interactions (mutual
facilitation) among neurons tuned to the same depth (image disparity) across different spatial
locations (Fig. 1b, along the red dashed diagonal lines). In other words, a disparity estimate
at one location is more likely to be correct if neighboring locations have similar disparity
estimates. A dynamic system under such constraints can relax to a stable global disparity
map. Here, we present neurophysiological evidence of interactions between disparity-tuned
neurons in the primary visual cortex that is consistent with this general approach. We sampled from a variety of spatially distributed disparity tuned neurons (see electrodes Fig. 1b)
while displaying DRDS stimuli defined at various disparities (see stimulus Fig.1b). We then
measured the dynamics of interactions by assessing the temporal evolution of correlation in
neural responses.
a
Left Image
b
Right Image
Electrodes
Disparity
Left Image
?
Stimulus
Right Image
Figure 1: (a) Left and right images of random dot stereogram (right image has been shifted
to the right). (b) 1D graphical depiction of competition (blue solid lines) and cooperation
(red dashed lines) among disparity-tuned neurons with respect to space as defined by Marr
and Poggio?s stereo algorithm [2].
2
2.1
Methods
Recording and stimulation
a
Posterior - Anterior
Recordings were made in V1 of two awake, behaving macaques. We simultaneously recorded from 4-8 electrodes providing data from up to 10 neurons in a single recording session (some electrodes recorded from as many as 3 neurons). We collected data from 112
neurons that provided 224 pairs for cross-correlation analysis. For stimuli, we used 12 Hz
dynamic random dot stereograms (DRDS; 25% density black and white pixels on a mean
luminance background) presented in a 3.5-degree aperture. Liquid crystal shutter goggles
were used to present random dot patterns to each eye separately. Eleven horizontal disparities between the two eyes, ranging from ?0.9 degrees, were tested. Seventy-four neurons
(66%) had significant disparity tuning and 99 pairs (44%) were comprised of neurons that
both had significant disparity tuning (1-way ANOVA, p<0.05).
b
5mm
Medial - Lateral
100?V
0.2ms
1?
Figure 2: (a) Example recording session from five electrodes in V1. (b) Receptive field
(white box?arrow represents direction preference) and random dot stereogram locations for
same recording session (small red square is the fixation spot).
2.2
Data analysis
Interaction between neurons was described as "effective connectivity" defined by crosscorrelation methods [3]. First, the probability of all joint spikes (x and y) between the two
neurons was calculated for all times from stimulus onset (t1 and t2) including all possible lag
times (t1 - t2) between the two neurons (2D joint peristimulus time histogram?JPSTH).
Next, the cross-product of each neuron?s PSTH (joint probabilities expected from chance)
was subtracted from the JPSTH; this difference is referred to as the cross-covariance histogram. Finally, the cross-covariance histogram was normalized by the geometric mean of the
auto-covariance histograms:
C x ,y ( t1 ,t 2 ) =
x( t1 )y ( t 2 ) ? x( t1 ) y ( t 2 )
( x( t
1 )x( t1 ) ? x( t1 ) x( t1 )
)( y ( t
2 )y ( t 2 ) ? y ( t 2 ) y ( t 2 )
)
(1)
This normalized cross-covariance histogram is a 2D matrix of Pearson?s correlation coefficients between the two neurons where the axes represent time from stimulus onset (Figure
3). The principal diagonal also represents time from stimulus onset for correlation and the
opposite diagonal represents lag time between the two neurons. We derived three measurements from this matrix to describe the ?effective connectivity? between neuron pairs. Using
bootstrapped samples of stimulus trials, we estimated 95% confidence intervals for these
three measurements [4]. We first integrated along the principal diagonal to produce correlation versus lag time (i.e., the traditional cross-correlation histogram?CCH). We used CCHs
to find significant correlation at or near 0 ms lag times (suggesting synaptic connectivity
between the neurons). Second, we integrated under the half-height full bandwidth of significant correlation peaks to quantify effective connectivity. Figure 4 shows the population average of normalized CCHs (n = 27) and 95% confidence intervals. Finally, we repeated this
integration along the principal diagonal to obtain the temporal evolution of effective connectivity (computed with a running 100 ms window).
CCH
y(t 2 )
t2
JPSTH
x(t 1 )
t1
Figure 3: Example normalized cross-covariance histogram.
In computing effective connectivity with Equation 1, we assume trial-to-trial stationarity. If
this is not true (e.g., due to difference in attentional effort in different trials), correlation
peaks can emerge that are not due to effective connectivity [5]. We applied a correction to
equation 1 [5,6] based on the average firing rate for each trial. However, no significant difference in correlation peaks was observed. In addition, changes in DRDS properties other
than disparity did not cause significant changes to correlation peak properties. Finally, alternative cross-correlation methods (CCG) [7] using responses to the same exact random dot
pattern to predict correlation expected from chance, again, lead to no significant difference
in correlation peak properties. These observations justify our assumption that the effective
connectivity computed in our case does not arise due to trial-to-trial non-stationarity.
0.008
b
0.006
a
Correlation
0.004
0.002
0.004
0.17 ? 0.02
0.000
-0.002
-50
0.000
-25
25
50
0
25
Lag Time (ms)
50
0
Half-Height
Half Bandwidth
-0.004
-300
Peak Lag Time
-100
100
Lag Time (ms)
300
Figure 4: (a) Population average CCH for 27 neuron pairs with a significant correlation
peak. (b) Same as (a), but zoomed into ?50 ms lag times with statistics of peak properties
(mean ? s.e.m.).
3
Interaction depends on tuning properties
The primary indicator of whether or not a neuron pair had a significant correlation peak at or
near a 0 ms lag time, for this class of stimuli, was similarity in disparity tuning between the
two neurons. Neuron pairs with significant correlation peaks (n = 27; 27%) tended to have
more similar disparity peaks, bandwidths, and frequencies (determined from fitted Gabor
functions) than neuron pairs that did not have significant correlation peaks. We quantified
similarity in tuning using the similarity index (SI), which is Pearson?s product-moment corn
relation [8]:
SI =
?( x
i =1
?
?
?
?
n
?( x
i =1
i
i
? x )( y i ? y )
??
? x )2 ??
??
??
n
?( y
i =1
i
?
? y )2 ?
?
?
(2)
0.4
0.3
0.2
0.1
0
a
b
0.4
Correlation
Percentage Pairs ( n=27 )
where i is each point on the disparity tuning curve, x and y are the firing rates at each point
for each neuron, and x and y are the mean firing rates across the tuning curve.
Figure 5a and 5b clearly show that both the probability of correlation and strength in correlation increase with greater SI (n = 27 pairs). This relationship is limited to long-range interactions among neurons because our electrodes were all at least 1 mm apart. This suggests
they are likely mediated by the well known long-range intracortical connections in V1 that
link neurons of similar orientation across space [9]. Our results suggest that these connections might also be shared to link similar disparity neurons together. Because connectivity
also depended on orientation (Figure 5c), V1 connectivity among neurons appears to depend
on similarity across multiple cue dimensions.
r = 0.49
p = 0.01
0.3
c
r = -0.40
p = 0.04
0.2
-0.9
-0.3
0.3
0.9
Similarity Index (SI)
0.1
0.0
-1.0
-0.5
0.0
SI
0.5
1.0
0
20
40
60
80
? Orientation Preference
Figure 5: (a) Likelihood of significant correlation peak with respect to similar disparity tuning. (b) Strength of correlation increases with similarity. (c) Correlation is also more likely
if orientation preference is similar.
From the 12 pairs of neurons recorded on a single electrode, correlation was observed among
neuron pairs with very similar disparity tuning as well as among neurons with nearly opposite disparity tuning (see also [8]). This suggests that antagonistic disparity-tuned neurons
tend to spatially coexist, and their interactions are likely competitive.
4
Interaction is stimulus-dependent
The interaction between pairs of neurons was not simply a function of the similarity between
their receptive field properties but was also a function of the input stimuli (or stimulus disparity in our case). The effective connectivity was significantly modulated (1-way ANOVA,
p<0.05) by the stimulus disparity for 25 out of the 27 pairs. We are not suggesting synaptic
connections physically change, but rather that the effectiveness of those connections can
change depending on the spiking activity and therefore the stimulus input. For neuron pairs
with similar disparity tuning, the strongest correlation was observed at their shared preferred
disparity, i.e. the peak of the disparity tuning curves based on firing rate (as shown in Figure
6). This suggests facilitation is strongest when a frontal parallel plane activated these neurons simultaneously at their preferred depth. As the stimulus plane moved away from this
depth, the effective connectivity between the neurons became weaker. This was observed in
10 pairs (e.g., Figure 6c). For the other 17 pairs (e.g., Figure 6d), the correlation or effective
connectivity was again strongest at the neuron pair's shared preferred disparity. However,
these pairs in addition exhibited secondary correlation peaks for disparity stimuli that produced the lowest firing rates (even below the baseline for DRDSs).
50 sps
80 sps
b
40 sps
20 sps
Firing Rate
a
-0.9
0.30
Correlation
c
0.0
0.9
-0.9
0.15
0.20
0.10
0.10
0.05
0.0
0.9
d
0.00
0.00
-0.9
0.0
0.9
-0.9
0.0
0.9
Horizontal Disparity (?)
Horizontal Disparity (?)
Figure 6: Top row are disparity tuning curves based on firing rates (mean ? s.e.m.). Bottom
row are disparity tuning curves based on correlation for the corresponding pairs of neurons
in the top row. Error bars are 95% confidence intervals and dashed lines represent 95% confidence of the mean correlation.
Cross-correlation peaks are interpreted as a result of effective circuits that may represent any
combination of a variety of synaptic connections that may have a bias in direction (one neuron drives the other) or may not have a bias in direction (zero lag time; both neurons receive
a common drive) [10]. As correlation peaks become broader, as in our case (mean = 42 ms),
this interpretation becomes more ambiguous (more possible circuits). The broader positive
correlation peaks can even be caused by common inhibitory circuitry. One way to potentially disambiguate our interpretations is to consider firing rate behavior. The positive correlation measured at the preferred disparity suggests that the interaction was likely facilitatory
in nature based on the increased firing of the neurons. The positive correlation measured at
the disparity where both neurons' firing rates were depressed, i.e. at the valley of the firing-
rate based disparity tuning curves, suggests that the correlation likely arose from common
inhibition (presumably from neurons that preferred that disparity).
5
Temporal dynamics of interaction
We can compare the temporal dynamics of the correlation with the temporal dynamics of the
firing rate of the neurons to gain more insight into the possible underlying circuitry. We
computed the correlation every 1 msec over a 100 ms running window, and found that the
correlation peak at the preferred disparity (based on firing rates) occurred at a later time
(250-350 ms post-stimulus onset) than the correlation peaks at the non-preferred disparity
(100-200 ms). Figure 7 illustrates the temporal dynamics of correlation for the example neuron pair shown in Figure 6b and 6d. The distinct interval in which correlation emerged at
the preferred and the non-preferred disparities was consistently observed for all 27 pairs of
neurons. Even for the example shown in Figure 6c, there were peaks in correlation in the
early part of the response at the most non-preferred disparities. The timing of these two
phases of correlation was also rather consistent over the population of pairs.
Correlation
400
100
0.4
0.0
0.2
Correlation
0.2
0.1
0
200
Correlation
Time (ms)
300
0.3
-0.9
0
Horizontal Disparity (?)
0.9
-0.1
0.3
0.2
0.1
0
-0.9
0
0.9
-0.1 Horizontal Disparity (?)
Figure 7: Temporal dynamics of correlation for example neuron pair shown in Figure 6,
right. From left to right: Correlation versus time for preferred (red) and non-preferred (blue)
disparities. Contour map of correlation versus time and disparity. Disparity tuning based on
correlation for the early (blue) and late (red) portion of the response (95% confidence intervals). Correlation was calculated every 1 ms over 100 ms windows.
By examining the interplay between firing rate and correlation, we were able to gain even
greater insight about the interactions among neuron pairs. To summarize this interplay across
our population, we compared the temporal evolution of the correlation at three distinct disparities with the temporal evolution of the firing rates at the same disparities (also smoothed
with 100 ms time windows). The first disparity, the preferred disparity A, is where we
measured the strongest correlation and was at or near the highest firing rate measured in individual neurons (see Figure 8, left). The second important disparity, the most non-preferred
disparity C, was where we measured secondary correlation peaks and coincided with the
lowest firing rates observed in individual neurons. Lastly, we looked at a disparity B that
was in between disparities A and C.
Figure 8 shows that neurons responded better to their preferred disparity over other disparities very early, resulting in immediate moderate firing rate-based disparity tuning. Then
shortly after (100 ms), a correlation peak emerges at the least preferred disparity C. This
coincides with suppression of firing rate for all disparities (Figure 8, blue dashed line).
However, the suppression in firing rate is much stronger for C where the firing rate diverges
downward from the firing rates for A and B sharpening the disparity tuning (Figure 8, blue
arrow; see also [11]). The strong correlation coupled with the decrease in firing suggests
strong common inhibition.
0.25
Correlation
A
B
C
B
A
Correlation
0.20
C
n = 27 pairs
0.15
0.10
0.05
0.00
Firing Rate
B
C
A
Firing Rate (sps)
-100
-0.05
30
0
100
A
B
C
25
20
200
300
400
300
400
n = 32 cells
15
10
5
0
-100
0
100
200
Time (ms)
Figure 8: Population average of normalized correlation versus time (top) for three disparities
shown on the left. Population average of normalized PSTHs for same three disparities (bottom). Both correlation and firing rates were calculated every 1 ms over 100 ms windows.
Once the correlation peak at C subsided (200 ms), the correlation increased for A (red
dashed line). When the correlation for A peaked, the correlation decreased for B and C, leading to very sharp correlation-based disparity tuning (see also Figure 7). This correlationbased tuning can facilitate depth estimates by changing how effectively these signals are
integrated downstream as a function of disparity [12].
Our interpretation is that the initial firing rate bias leads to antagonistic disparity-tuned neurons generating common inhibition that suppresses firing at non-preferred disparities, removing potential mismatches. The immediacy suggests that mutual inhibition was local,
which is consistent with our observation that many opposing disparity-tuned neurons spatially coexisted. The slower correlation peak at the preferred disparity A is indicative of mutual facilitation that occurred when the depth estimates of spatially distinct neurons matched.
This facilitation leads to a more precise estimate of depth.
6
D i s c u s s i on a n d c o n c l u s i o n s
The findings from this study provide support to Julesz?s proposal that cooperative and competitive mechanisms in primary visual cortex are utilized for estimating global depth in random dot stereograms [1], which was later described formally by Marr and Poggio [2]. More
recent cooperative stereo computation models allow excitatory interaction between neurons
of different disparities separated by long distance. This is used to accommodate the computation of slanted surfaces [13,14]. In this experiment, we only tested frontal parallel planes,
thus, we cannot answer whether or not effective connections and facilitation exist between
neurons with larger disparity differences over long distances. This will require further experiments using planes with disparity gradients.
The observation that initial correlation peaks occurred at disparities that evoked the lowest
firing rates in neurons, suggests that correlation peaks emerged from common inhibition for
non-preferred disparities. The observation that later correlation occurred at disparities that
evoked the highest firing rates suggests that neurons were mutually exciting each other at
their preferred disparity. Our neurophysiological data reveal interesting dynamics between
network-based (effective connectivity) and firing rate-based encoding of depth estimates.
The observation that inhibition precedes facilitation suggests that competition is local (re-
calling neurons at the same electrode tend to have opposite disparity tuning) and cooperation
is more global (mediated through long-range connectivity). Local competition between neurons encoding different depths is consistent with the uniqueness principle of Marr and Poggio's algorithm [2]. In addition, cooperation among neurons encoding the same depth across
space was predicted by the second rule of their algorithm: matter is cohesive. These two
interactions are robust at removing potential ambiguity during stereo matching and depth
inference.
Previous neurophysiological data had suggested that intracortical connectivity in primary
visual cortex underlies competitive [15] and cooperative [16] mechanisms for improving
estimates of orientation. Our data suggests similar circuitry might play a role also in stereo
matching [17]. However, this study is distinct in that it provides detailed empirical support
for computational algorithms for solving stereo matching. It thus highlights the importance
of computational algorithms in generating hypotheses to guide future neurophysiological
studies.
Acknowledgments
We thank George Gerstein and Jeff Keating for JPSTH software. Supported by NIMH IBSC
MH64445 and NSF CISE IIS-0413211 grants.
References
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[2] Marr, D. & Poggio, T. (1976) Cooperative computation of stereo disparity.
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2,218 | 3,012 | Handling Advertisements of Unknown Quality
in Search Advertising
Sandeep Pandey
Carnegie Mellon University
[email protected]
Christopher Olston
Yahoo! Research
[email protected]
Abstract
We consider how a search engine should select advertisements to display
with search results, in order to maximize its revenue. Under the standard
?pay-per-click? arrangement, revenue depends on how well the displayed
advertisements appeal to users. The main difficulty stems from new advertisements whose degree of appeal has yet to be determined. Often the
only reliable way of determining appeal is exploration via display to users,
which detracts from exploitation of other advertisements known to have
high appeal. Budget constraints and finite advertisement lifetimes make it
necessary to explore as well as exploit.
In this paper we study the tradeoff between exploration and exploitation,
modeling advertisement placement as a multi-armed bandit problem. We
extend traditional bandit formulations to account for budget constraints
that occur in search engine advertising markets, and derive theoretical
bounds on the performance of a family of algorithms. We measure empirical performance via extensive experiments over real-world data.
1
Introduction
Search engines are invaluable tools for society. Their operation is supported in large part
through advertising revenue. Under the standard ?pay-per-click? arrangement, search engines earn revenue by displaying appealing advertisements that attract user clicks. Users
benefit as well from this arrangement, especially when searching for commercial goods or
services.
Successful advertisement placement relies on knowing the appeal or ?clickability? of advertisements. The main difficulty is that the appeal of new advertisements that have not yet
been ?vetted? by users can be difficult to estimate. In this paper we study the problem of
placing advertisements to maximize a search engine?s revenue, in the presence of uncertainty
about appeal.
1.1
The Advertisement Problem
Consider the following advertisement problem [8], illustrated in Figure 1. There are m
advertisers A1 , A2 . . . Am who wish to advertise on a search engine. The search engine runs
a large auction where each advertiser submits its bids to the search engine for the query
phrases in which it is interested. Advertiser Ai submits advertisement ai,j to target query
phrase Qj , and promises to pay bi,j amount of money for each click on this advertisement,
where bi,j is Ai ?s bid for advertisement ai,j . Advertiser Ai can also specify a daily budget (di )
that is the total amount of money it is willing to pay for the clicks on its advertisements
in a day. Given a user search query on phrase Qj , the search engine selects a constant
number C ? 1 of advertisements from the candidate set of advertisements {a?,j }, targeted
to Qj . The objective in selecting advertisements is to maximize the search engine?s total
daily revenue. The arrival sequence of user queries is not known in advance. For now we
assume that each day a new set of advertisements is given to the search engine and the set
remains fixed through out the day; we drop both of these assumptions later in Section 4.
Budgets Advertisers
d1
A1
Ads
a 1,1
Query
phrases
CTR = 1
for all ads
Q
A2
d3
A3
d4
A4
d5
A5
general CTR,
CTR unknown
V
this paper
1
a 2,1
d2
general CTR,
CTR known
a 1,3
Q
2
a 3,2
a 3,4
a 4,4
a 5,5
no budget
constraints
Q
I
III
GREEDY
GREEDY
ratio=1
ratio=1
3
Q
4
a 5,4
budget
constraints
Q
5
Figure 1: Advertiser and query model.
II
IV
MSVV
GREEDY
ratio=1?1/e
ratio=1/2
VI
this paper
Figure 2: Problem variants.
High revenue is achieved by displaying advertisements that have high bids as well as high
likelihood of being clicked on by users. Formally, the click-through rate (CTR) ci,j of
advertisement ai,j is the probability of a user to click on advertisement ai,j given that the
advertisement was displayed to the user for query phrase Qj . In the absence of budget
constraints, revenue is maximized by displaying advertisements with the highest ci,j ? bi,j
value. The work of [8] showed how to maximize revenue in the presence of budget constraints,
but under the assumption that all CTRs are known in advance. In this paper we tackle the
more difficult but realistic problem of maximizing advertisement revenue when CTRs are
not necessarily known at the outset, and must be learned on the fly.
We show the space of problem variants (along with the best known advertisement policies)
in Figure 2. GREEDY refers to selection of advertisements according to expected revenue
(i.e., ci,j ? bi,j ). In Cells I and III GREEDY performs as well as the optimal policy, where the
optimal policy also knows the arrival sequence of queries in advance. We write ?ratio=1?
in Figure 2 to indicate that GREEDY has the competitive ratio of 1. For Cells II and IV
the greedy policy is not optimal, but is nevertheless 1/2 competitive. An alternative policy
for Cell II was given in [8], which we refer to as MSVV; it achieves a competitive ratio
of 1 ? 1/e. In this paper we give the first policies for Cells V and VI, where we must
choose which advertisements to display while simultaneously estimating click-through rates
of advertisements.
1.2
Exploration/Exploitation Tradeoff
The main issue we face while addressing Cells V and VI is to balance the exploration/exploitation tradeoff. To maximize short-term revenue, the search engine should
exploit its current, imperfect CTR estimates by displaying advertisements whose estimated
CTRs are large. On the other hand, to maximize long-term revenue, the search engine
needs to explore, i.e., identify which advertisements have the largest CTRs. This kind of
exploration entails displaying advertisements whose current CTR estimates are of low confidence, which inevitably leads to displaying some low-CTR ads in the short-term. This kind
of tradeoff between exploration and exploitation shows up often in practice, e.g., in clinical
trials, and has been extensively studied in the context of the multi-armed bandit problem
[4].
In this paper we draw upon and extend the existing bandit literature to solve the advertisement problem in the case of unknown CTR. In particular, first in Section 3 we show that the
unbudgeted variant of the problem (Cell V in Figure 2) is an instance of the multi-armed
bandit problem. Then, in Section 4 we introduce a new kind of bandit problem that we
termed the budgeted multi-armed multi-bandit problem (BMMP), and show that the budgeted unknown-CTR advertisement problem (Cell VI) is an instance of BMMP. We propose
policies for BMMP and give performance bounds. We evaluate our policies empirically over
real-world data in Section 5. In the extended technical version of the paper [9] we show how
to extend our policies to address various practical considerations, e.g., exploiting any prior
information available about the CTRs of ads, permitting advertisers to submit and revoke
advertisements at any time, not just at day boundaries.
2
Related Work
We have already discussed the work of [8], which addresses the advertisement problem under
the assumption that CTRs are known. There has not been much published work on estimating CTRs. Reference [7] discusses how contextual information such as user demographic
or ad topic can be used to estimate CTRs, and makes connections to the recommender and
bandit problems, but stops short of presenting technical solutions. Some methods for estimating CTRs are proposed in [5] with the focus of thwarting click fraud.
Reference [1] studies how to maximize user clicks on banner ads. The key problem addressed in [1] is to satisfy the contracts made with the advertisers in terms of the minimum
guaranteed number of impressions (as opposed to the budget constraints in our problem).
Reference [10] looks at the advertisement problem from an advertiser?s point of view, and
gives an algorithm for identifying the most profitable set of keywords for the advertiser.
3
Unbudgeted Unknown-CTR Advertisement Problem
In this section we address Cell V of Figure 2, where click-through rates are initially unknown
and budget constraints are absent (i.e., di = ? for all advertisers Ai ). Our unbudgeted
problem is an instance of the multi-armed bandit problem [4], which is the following: we
have K arms where each arm has an associated reward and payoff probability. The payoff
probability is not known to us while the reward may or may not be known (both versions of
the bandit problem exist). With each invocation we activate exactly C ? K arms. 1 Each
activated arm then yields the associated reward with its payoff probability and nothing with
the remaining probability. The objective is to determine a policy for activating the arms so
as to maximize the total reward over some number of invocations.
To solve the unbudgeted unknown-CTR advertisement problem, we create a multi-armed
bandit problem instance for each query phrase Q, where ads targeted for the query phrase
are the arms, bid values are the rewards and CTRs are the payoff probabilities of the
bandit instance. Since there are no budget constraints, we can treat each query phrase
independently and solve each bandit instance in isolation. 2 The number of invocations for
a bandit instance is not known in advance because the number of queries of phrase Q in a
given day is not known in advance.
A variety of policies have been proposed for the bandit problem, e.g., [2, 3, 6], any of which
can be applied to our unbudgeted advertisement problem. The policies proposed in [3] are
particularly attractive because they have a known performance bound for any number of
invocations not known in advance (in our context the number of queries is not known a
priori). In the case of C = 1, the policies of [3] make O(ln n) number of mistakes, on
expectation, in n invocations (which is also the asymptotic lower bound on the number of
mistakes [6]). A mistake occurs when a suboptimal arm is chosen by a policy (the optimal
arm is the one with the highest expected reward).
We consider a specific policy from [3] called UCB and apply it to our problem (other policies
from [3] can also be used). UCB is proposed under a slightly different reward model; we adapt
it to our context to produce the following policy that we call MIX (for mixing exploration
with exploitation). We prove a performance bound of O(ln n) mistakes for MIX for any
C ? 1 in [9].
Policy MIX :
Each time a query for phrase Qj arrives:
1. Display the C ads targeted for Qj that have the highest priority. The priority Pi,j
of ad ai,j is a function of its current CTR estimate (?
ci,j ), its bid value (bi,j ), the
number of times it has been displayed so far (ni,j ), and the number of times phrase
Qj has been queried so far in the day (nj ). Formally, priority Pi,j is defined as:
(
q
2 ln nj
? bi,j if ni,j > 0
c?i,j +
ni,j
Pi,j =
?
otherwise
1
The conventional multi-armed bandit problem is defined for C = 1. We generalize it to any
C ? 1 in this paper.
2
We assume CTRs to be independent of one another.
2. Monitor the clicks made by users and update the CTR estimates c?i,j accordingly.
c?i,j is the average click-through rate observed so far, i.e., the number of times ad
ai,j has been clicked on divided by the total number of times it has been displayed.
Policy MIX manages the exploration/exploitation q
tradeoff in the following way. The priority
2 ln nj
function has two factors: an exploration factor
that diminishes with time, and
ni,j
an exploitation factor (?
ci,j ). Since c?i,j can be estimated only when ni,j ? 1, the priority
value is set to ? for an ad which has never been displayed before.
Importantly, the MIX policy is practical to implement because it can be evaluated efficiently
using a single pass over the ads targeted for a query phrase. Furthermore, it incurs minimal
storage overhead because it keeps only three numbers (?
ci,j , ni,j and bi,j ) with each ad and
one number (nj ) with each query phrase.
4
Budgeted Unknown-CTR Advertisement Problem
We now turn to the more challenging case in which advertisers can specify daily budgets
(Cell VI of Figure 2). Recall from Section 3 that in the absence of budget constraints,
we were able to treat the bandit instance created for a query phrase independent of the
other bandit instances. However, budget constraints create dependencies between query
phrases targeted by an advertiser. To model this situation, we introduce a new kind of
bandit problem that we call Budgeted Multi-armed Multi-bandit Problem (BMMP), in
which multiple bandit instances are run in parallel under overarching budget constraints.
We derive generic policies for BMMP and give performance bounds.
4.1 Budgeted Multi-armed Multi-bandit Problem
BMMP consists of a finite set of multi-armed bandit instances, B = {B1 , B2 . . . B|B| }. Each
bandit instance Bi has a finite number of arms and associated rewards and payoff probabilities as described in Section 3. In BMMP each arm also has an associated type. Each type
Ti ? T has budget di ? [0, ?] which specifies the maximum amount of reward that can be
generated by activating all the arms of that type. Once the specified budget is reached for
a type, the corresponding arms can still be activated but no further reward is earned.
With each invocation of the bandit system, one bandit instance from B is invoked; the policy
has no control over which bandit instance is invoked. Then the policy activates C arms of
the invoked bandit instance, and the activated arms generate some (possibly zero) total
reward.
It is easy to see that the budgeted unknown-CTR advertisement problem is an instance
of BMMP. Each query phrase acts as a bandit instance and the ads targeted for it act as
bandit arms, as described in Section 3. Each advertiser defines a unique type of arms and
gives a budget constraint for that type; all ads submitted by an advertiser belong to the
type defined by it. When a query is submitted by a user, the corresponding bandit instance
is invoked.
We now show how to derive a policy for BMMP given as input a policy POL for the regular
multi-armed bandit problem such as one of the policies from [3]. The derived policy, denoted
by BPOL (Budget-aware POL), is as follows:
? Run |B| instances of POL in parallel, denoted POL1 , POL2 , . . . POL|B| .
? Whenever bandit instance Bi is invoked:
1. Discard any arm(s) of Bi whose type?s budget is newly depleted, i.e., has
become depleted since the last invocation of Bi .
2. If one or more arms of Bi was discarded during step 1, restart POLi .
3. Let POLi decide which of the remaining arms of Bi to activate.
Observe that in the second step of BPOL, when POL is restarted, POL loses any state
it has built up, including any knowledge gained about the payoff probabilities of bandit
arms. Surprisingly, despite this seemingly imprudent behavior, we can still derive a good
performance bound for BPOL, provided that POL has certain properties, as we discuss in
the next section. In practice, since most bandit policies can take prior information about
the payoff probabilities as input, when restarting POL we can supply the previous payoff
probability estimates as the prior (as done in our experiments).
4.2 Performance Bound for BMMP Policies
Let S denote the sequence of bandit instances that are invoked, i.e., S =
{S(1), S(2) . . . S(N )} where S(n) denotes the index of the bandit instance invoked at the nth
invocation. We compare the performance of BPOL with that of the optimal policy, denoted
by OPT, where OPT has advance knowledge of S and the exact payoff probabilities of all
bandit instances.
We claim that bpol (N ) ? opt(N )/2?O(f (N )) for any N , where bpol (N ) and opt(N ) denote
the total expected reward obtained after N invocations by BPOL and OPT, respectively, and
f (n) denotes the expected number of mistakes made by POL after n invocations of the the
regular multi-armed bandit problem (for UCB, f (n) is O(ln n) [3]). Our complete proof is
rather involved. Here we give a high-level outline of the proof (the complete proof is given
in [9]). For simplicity we focus on the C = 1 case; C ? 1 is a simple extension thereof.
Since bandit arms generate rewards stochastically, it is not clear how we should compare
BPOL and OPT. For example, even if BPOL and OPT behave in exactly the same way
(activate the same arm on each bandit invocation), we cannot guarantee that both will have
the same total reward in the end. To enable meaningful comparison, we define a payoff
instance, denoted by I, such that I(i, n) denotes the reward generated by arm i of bandit
instance S(n) for invocation n in payoff instance I. The outcome of running BPOL or
OPT on a given payoff instance is deterministic because the rewards are fixed in the payoff
instance. Hence, we can compare BPOL and OPT on per payoff instance basis. Since each
payoff instance arises with a certain probability, denoted as P(I), by taking expectation
over all possible payoff instances of execution we can compare the expected performance of
BPOL and OPT.
Let us consider invocation n in payoff instance I. Let B (I, n) and O(I, n) denote the arms
of bandit instance S(n) activated under BPOL and OPT respectively. Based on the different
possibilities that can arise, we classify invocation n into one of three categories:
? Category 1: The arm activated by OPT, O(I, n), is of smaller or equal expected
reward in comparison to the arm activated by BPOL, B (I, n). The expected reward
of an arm is the product of its payoff probability and reward.
? Category 2: Arm O(I, n) is of greater expected reward than B (I, n), but O(I, n) is
not available for BPOL to activate at invocation n due to budget restrictions.
? Category 3: Arm O(I, n) is of greater expected reward than B (I, n) and both arms
O(I, n) and B (I, n) are available for BPOL to activate, but BPOL prefers to activate
B (I, n) over O(I, n).
Let us denote the invocations of category k (1, 2 or 3) by N k (I) for payoff instance I.
Let bpol k (N ) and opt k (N ) denote the expected reward obtained during the invocations of
category k (1, 2 or 3) by BPOL and OPT respectively. In [9] we show that
X
X
I(B (I, n), n)
bpol k (N ) =
P(I) ?
I?I
n?N k (I)
Similarly,
opt k (N ) =
X
I?I
P(I) ?
X
I(O(I, n), n)
n?N k (I)
Then for each k we bound opt k (N ) in terms of bpol (N ). In [9] we provide proof of each of
the following bounds:
Lemma 1 opt 1 (N ) ? bpol 1 (N ).
Lemma 2 opt 2 (N ) ? bpol (N ) + (|T | ? rmax ), where |T | denotes the number of arm types
and rmax denotes the maximum reward.
Lemma 3 opt 3 (N ) = O(f (N )).
From the above bounds we obtain our overall claim:
Theorem 1 bpol (N ) ? opt(N )/2 ? O(f (N )), where bpol (N ) and opt(N ) denote the total
expected reward obtained under BPOL and OPT respectively.
Proof:
opt(N )
= opt 1 (N ) + opt 2 (N ) + opt 3 (N )
? bpol 1 (N ) + bpol (N ) + |T | ? rmax + O(f (N ))
? 2 ? bpol (N ) + O(f (N ))
Hence, bpol (N ) ? opt(N )/2 ? O(f (N )).
If we supply MIX (Section 3) as input to our generic BPOL framework, we obtain BMIX,
a policy for the budgeted unknown-CTR advertisement problem. Due to the way MIX
structures and maintains its internal state, it is not necessary to restart a MIX instance when
an advertiser?s budget is depleted in BMIX, as specified in the generic BPOL framework (the
exact steps of BMIX are given in [9]).
So far, for modeling purposes, we have assumed the search engine receives an entirely new
batch of advertisements each day. In reality, ads may persist over multiple days. With
BMIX, we can carry forward an ad?s CTR estimate (?
ci,j ) and display count (ni,j ) from day
to day until an ad is revoked, to avoid having to re-learn CTR?s from scratch each day. Of
course the daily budgets reset daily, regardless of how long each ad persists. In fact, with a
little care we can permit ads to be submitted and revoked at arbitrary times (not just at day
boundaries). We describe this extension, as well as how we can incorporate and leverage
prior beliefs about CTR?s, in [9].
5
Experiments
From our general result of Section 4, we have a theoretical performance guarantee for BMIX.
In this section we study BMIX empirically. In particular, we compare it with the greedy policy proposed for the known-CTR advertisement problem (Cells 1-IV in Figure
2). GREEDY
displays the C ads targeted for a query phrase that have the highest c?i,j ?bi,j values among
the ads whose advertisers have enough remaining budgets; to induce a minimal amount of
exploration, for an ad which has never been displayed before, GREEDY treats c?i,j as ?
(our policies do this as well). GREEDY is geared exclusively toward exploitation. Hence, by
comparing GREEDY with our policies, we can gauge the importance of exploration.
We also propose and evaluate the following variants of BMIX that we expect to perform well
in practice:
1. Varying the Exploration Factor. Internally, BMIX runs instances of MIX to select
which ads to display.qAs mentioned in Section 4, the priority function of MIX consists of an
2 ln nj
and an exploitation factor (ci,j ). In [3] it was shown empiriexploration factor
ni,j
cally that the following heuristical exploitation factor performs well, despite the absence of
a known performance guarantee:
s
s
n1
o
ln nj
2 ln nj
? min
, Vi,j (ni,j , nj )
where Vi,j (ni,j , nj ) = c?i,j ? (1 ? c?i,j ) +
ni,j
4
ni,j
q
2 ln nj
Substituting this expression in place of
in the priority function of BMIX gives us
ni,j
a new (heuristical) policy we call BMIX-E.
2. Budget Throttling. It is shown in [8] that in the presence of budget constraints, it is
beneficial to display the ads of an advertiser less often as the advertiser?s remaining budget
decreases. In particular, they propose to multiply bids from advertiser Ai by the following
discount factor :
0
?(d0i ) = 1 ? e?di /di
where d0i is the current remaining budget of advertiser Ai for the day and di is its total daily
budget. Following this idea we can replace bi,j by ?(d0i ) ? bi,j in the priority function of
BMIX, yielding a variant we call BMIX-T. Policy BMIX-ET refers to use of heuristics 1 and
2 together.
5.1
Experiment Setup
We evaluate advertisement policies by conducting simulations over real-world data. Our
data set consists of a sample of 85,000 query phrases selected at random from the Yahoo!
query log for the date of February 12, 2006. Since we have the frequency counts of these
query phrases but not the actual order, we ran the simulations multiple times with random
orderings of the query instances and report the average revenue in all our experiment results.
The total number of query instances is 2 million. For each query phrase we have the list
of advertisers interested in it and the ads submitted by them to Yahoo!. We also have the
budget constraints of the advertisers. Roughly 60% of the advertisers in our data set impose
daily budget constraints.
In our simulation, when an ad is displayed, we decide whether a click occurs by flipping a
coin weighted by the true CTR of the ad. Since true CTRs are not known to us (this is the
problem we are trying to solve!), we took the following approach to assign CTRs to ads:
from a larger set of Yahoo! ads we selected those ads that have been displayed more than
thousand times, and therefore we have highly accurate CTR estimates. We regarded the
distribution of these CTR estimates as the true CTR distribution. Then for each ad ai,j in
the dataset we sampled a random value from this distribution and assigned it as CTR ci,j
of the ad. (Although this method may introduce some skew compared with the (unknown)
true distribution, it is the best we could do short of serving live ads just for the purpose of
measuring CTRs).
We are now ready to present our results. Due to lack of space we consider a simple setting
here where the set of ads is fixed and no prior information about CTR is available. We
study the more general setting in [9].
5.2
Exploration/Exploitation Tradeoff
We ran each of the policies for a time horizon of ten days; each policy carries over its CTR
estimates from one day to the next. Budget constraints are renewed each day. For now
we fix the number of displayed ads (C) to 1. Figure 3 plots the revenue generated by each
policy after a given number of days (for confidentiality reasons we have changed the unit
of revenue). All policies (including GREEDY) estimate CTRs based on past observations,
so as time passes by their estimates become more reliable and their performance improves.
Note that the exploration factor of BMIX-E causes it to perform substantially better than
that of BMIX. The budget throttling heuristic (BMIX-T and BMIX-ET) did not make much
difference in our experiments.
All of our proposed policies perform significantly better than GREEDY, which underscores
the importance of balancing exploration and exploitation. GREEDY is geared exclusively
toward exploitation, so one might expect that early on it would outperform the other policies.
However, that does not happen because GREEDY immediately fixates on ads that are not
very profitable (i.e., low ci,j ? bi,j ).
Next we vary the number of ads displayed for each query (C). Figure 4 plots total revenue
over ten days on the y-axis, and C on the x-axis. Each policy earns more revenue when
more ads are displayed (larger C). Our policies outperform GREEDY consistently across
different values of C. In fact, GREEDY must display almost twice as many ads as BMIX-E
to generate the same amount of revenue.
12
5
Total revenue
4
3
Total revenue
GREEDY
BMIX
BMIX-T
BMIX-E
BMIX-ET
2
8
GREEDY
BMIX
BMIX-T
BMIX-E
BMIX-ET
4
1
0
1
4
7
10
Time horizon (days)
Figure 3: Revenue generated by different advertisement policies (C=1).
6
0
1
4
7
10
Ads per query (C)
Figure 4: Effect of C (number of ads
displayed per query).
Summary and Future Work
In this paper we studied how a search engine should select which ads to display in order
to maximize revenue, when click-through rates are not initially known. We dealt with the
underlying exploration/exploitation tradeoff using multi-armed bandit theory. In the process
we contributed to bandit theory by proposing a new variant of the bandit problem that we
call budgeted multi-armed multi-bandit problem (BMMP). We proposed a policy for solving
BMMP and derived a performance guarantee. Practical extensions of our advertisement
policies are given in the extended version of the paper. Extensive experiments over real
ad data demonstrate substantial revenue gains compared to a greedy strategy that has no
provision for exploration.
Several useful extensions of this problem can be conceived. One such extension would be
to exploit similarity in ad attributes while inferring CTRs, as suggested in [7], instead
of estimating the CTR of each ad independently. Also, an adversarial formulation of this
problem merits study, perhaps leading to general consideration of how to manage exploration
versus exploitation in game-theoretic scenarios.
References
[1] N. Abe and A. Nakamura. Learning to Optimally Schedule Internet Banner Advertisements. In ICML, 1999.
[2] R. Agrawal. Sample Mean Based Index Policies with O(log n) Regret for the MultiArmed Bandit Problem. Advances in Applied Probability, 27:1054?1078, 1995.
[3] P. Auer, N. Cesa-Bianchi, and P. Fischer. Finite-time Analysis of the Multi-Armed
Bandit Problem. Machine Learning, 47:235?256, 2002.
[4] D. A. Berry and B. Fristedt. Bandit Problems: Sequential Allocation of Experiments.
Chapman and Hall, London, 1985.
[5] N. Immorlica, K. Jain, M. Mahdian, and K. Talwar. Click Fraud Resistant Methods
for Learning Click-Through Rates. In WINE, 2005.
[6] T. Lai and H. Robbins. Asymptotically Efficient Adaptive Allocation Rules. Advances
in Applied Mathematics, 6:4?22, 1985.
[7] O. Madani and D. Decoste. Contextual Recommender Problems. In Proceedings of the
1st International Workshop on Utility-based Data Mining, 2005.
[8] A. Mehta, A. Saberi, U. Vazirani, and V. Vazirani. AdWords and Generalized On-line
Matching. In FOCS, 2005.
[9] S. Pandey and C. Olston. Handling advertisements of unknown quality in search advertising, October, 2006. Technical report, available via http://www.cs.cmu.edu/
?spandey/publications/ctrEstimation.pdf.
[10] P. Rusmevichientong and D. Williamson. An Adaptive Algorithm for Selecting Profitable Keywords for Search-Based Advertising Services. In EC, 2006.
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2,219 | 3,013 | Near-Uniform Sampling of Combinatorial Spaces
Using XOR Constraints
Carla P. Gomes
Ashish Sabharwal
Bart Selman
Department of Computer Science
Cornell University, Ithaca NY 14853-7501, USA
{gomes,sabhar,selman}@cs.cornell.edu ?
Abstract
We propose a new technique for sampling the solutions of combinatorial problems in a near-uniform manner. We focus on problems specified as a Boolean formula, i.e., on SAT instances. Sampling for SAT problems has been shown to have
interesting connections with probabilistic reasoning, making practical sampling
algorithms for SAT highly desirable. The best current approaches are based on
Markov Chain Monte Carlo methods, which have some practical limitations. Our
approach exploits combinatorial properties of random parity (XOR) constraints to
prune away solutions near-uniformly. The final sample is identified amongst the
remaining ones using a state-of-the-art SAT solver. The resulting sampling distribution is provably arbitrarily close to uniform. Our experiments show that our
technique achieves a significantly better sampling quality than the best alternative.
1
Introduction
We present a new method, XORSample, for uniformly sampling from the solutions of hard combinatorial problems. Although our method is quite general, we focus on problems expressed in the
Boolean Satisfiability (SAT) framework. Our work is motivated by the fact that efficient sampling
for SAT can open up a range of interesting applications in probabilistic reasoning [6, 7, 8, 9, 10, 11].
There has also been a growing interest in combining logical and probabilistic constraints as in the
work of Koller, Russell, Domingos, Bacchus, Halpern, Darwiche, and many others (see e.g. statistical relational learning and Markov logic networks [1]), and a recently proposed Markov logic
system for this task uses efficient SAT sampling as its core reasoning mechanism [2].
Typical approaches for sampling from combinatorial spaces are based on Markov Chain Monte
Carlo (MCMC) methods, such as the Metropolis algorithm and simulated annealing [3, 4, 5]. These
methods construct a Markov chain with a predefined stationary distribution. One can draw samples
from the stationary distribution by running the Markov chain for a sufficiently long time. Unfortunately, on many combinatorial problems, the time taken by the Markov chain to reach its stationary
distribution scales exponentially with the problem size.
MCMC methods can also be used to find (globally optimal) solutions to combinatorial problems. For
example, simulated annealing (SA) uses the Boltzmann distribution as the stationary distribution.
By lowering the temperature parameter to near zero, the distribution becomes highly concentrated
around the minimum energy states, which correspond to the solutions of the combinatorial problem
under consideration. SA has been successfully applied to a number of combinatorial search problems. However, many combinatorial problems, especially those with intricate constraint structure,
are beyond the reach of SA and related MCMC methods. Not only does problem structure make
reaching the stationary distribution prohibitively long, even reaching a single (optimal) solution is
often infeasible. Alternative combinatorial search techniques have been developed that are much
more effective at finding solutions. These methods generally exploit clever search space pruning
? This
work was supported by the Intelligent Information Systems Institute (IISI) at Cornell University
(AFOSR grant F49620-01-1-0076) and DARPA (REAL grant FA8750-04-2-0216).
techniques, which quickly focus the search on small, but promising, parts of the overall combinatorial space. As a consequence, these techniques tend to be highly biased, and sample the set of
solutions in an extremely non-uniform way. (Many are in fact deterministic and will only return one
particular solution.)
In this paper, we introduce a general probabilistic technique for obtaining near-uniform samples
from the set of all (globally optimal) solutions of combinatorial problems. Our method can use any
state-of-the-art specialized combinatorial solver as a subroutine, without requiring any modifications
to the solver. The solver can even be deterministic. Most importantly, the quality of our sampling
method is not affected by the possible bias of the underlying specialized solver ? all we need is
a solver that is good at finding some solution or proving that none exists. We provide theoretical
guarantees for the sampling quality of our approach. We also demonstrate the practical feasibility
of our approach by sampling near-uniformly from instances of hard combinatorial problems.
As mentioned earlier, to make our discussion more concrete, we will discuss our method in the context of SAT. In the SAT problem, we have a set of logical constraints on a set of Boolean (True/False)
variables. The challenge is to find a setting of the variables such that all logical constraints are satisfied. SAT is the prototypical NP-complete problem, and quite likely the most widely studied
combinatorial problem in computer science. There have been dramatic advances in recent years in
the state-of-the-art of SAT solvers [e.g. 12, 13, 14]. Current solvers are able to solve problems with
millions of variables and constraints. Many practical combinatorial problems can be effectively
translated into SAT. As a consequence, one of the current most successful approaches to solving
hard computational problems, arising in, e.g., hardware and software verification and planning and
scheduling, is to first translate the problem into SAT, and then use a state-of-the-art SAT solver to
find a solution (or show that it does not exist). As stated above, these specialized solvers derive much
of their power from quickly focusing their search on a very small part of the combinatorial space.
Many SAT solvers are deterministic, but even when the solvers incorporate some randomization,
solutions will be sampled in a highly non-uniform manner.
The central idea behind our approach can be summarized as follows. Assume for simplicity that
our original SAT instance on n Boolean variables has 2s solutions or satisfying assignments. How
can we sample uniformly at random from the set of solutions? We add special randomly generated
logical constraints to our SAT problem. Each random constraint is constructed in such a way that
it rules out any given truth assignment exactly with probability 1/2 . Therefore, in expectation, after
adding s such constraints, we will have a SAT instance with exactly one solution. 1 We then use a
SAT solver to find the remaining satisfying assignment and output this as our first sample. We can
repeat this process with a new set of s randomly generated constraints and in this way obtain another
random solution. Note that to output each sample, we can use whatever off-the-shelf SAT solver is
available, because all it needs to do is find the single remaining assignment.2 The randomization in
the added constraints will guarantee that the assignment is selected uniformly at random.
How do we implement this approach? For our added constraints, we use randomly generated parity or ?exclusive-or? (XOR) constraints. In recent work, we introduced XOR constraints for the
problem of counting the number of solutions using MBound [15]. Although the building blocks of
MBound and XORSample are the same, this work relies much more heavily on the properties of XOR
constraints, namely, pairwise and even 3-wise independence. As we will discuss below, an XOR constraint eliminates any given truth assignment with probability 1/2 , and therefore, in expectation, cuts
the set of satisfying assignments in half. For this expected behavior to happen often, the elimination of each assignment should ideally be fully independent of the elimination of other assignments.
Unfortunately, as far as is known, there are no compact (polynomial size) logical constraints that
can achieve such complete independence. However, XOR constraints guarantee at least pairwise
independence, i.e., if we know that an XOR constraint C eliminates assignment ? 1 , this provides no
information as to whether C will remove any another assignment ?2 . Remarkably, as we will see,
such pairwise independence already leads to near-uniform sampling.
Our sampling approach is inspired by earlier work in computational complexity theory by Valiant
and Vazirani [16], who considered the question whether having one or more assignments affects
1 Of course, we don?t know the true value of s. In practice, we use a binary style search to obtain a rough
estimate. As we will see, our algorithms work correctly even with over- and under-estimates for s.
2 The practical feasibility of our approach exploits the fact that current SAT solvers are very effective in
finding such truth assignments in many real-world domains.
the hardness of combinatorial problems. They showed that, in essence, the number of solutions
should not affect the hardness of the problem instances in the worst case [16]. This was received as
a negative result because it shows that finding a solution to a Unique SAT problem (a SAT instance
that is guaranteed to have at most one solution) is not any easier than finding a solution to an arbitrary
SAT instance. Our sampling strategy turns this line of research into a positive direction by showing
how a standard SAT solver, tailored to finding just one solution of a SAT problem, can now be used
to sample near-uniformly from the set of solutions of an arbitrary SAT problem.
In addition to introducing XORSample and deriving theoretical guarantees on the quality of the samples it generates, we also provide an empirical validation of our approach. One question that arises
is whether the state-of-the-art SAT solvers will perform well on problem instances with added XOR
(or parity) constraints. Fortunately, as our experiments show, a careful addition of such constraints
does generally not degrade the performance of the solvers. In fact, the addition of XOR constraints
can be beneficial since the constraints lead to additional propagation that can be exploited by the
solvers.3 Our experiments show that we can effectively sample near-uniformly from hard practical
combinatorial problems. In comparison with the best current alternative method on such instances,
our sampling quality is substantially better.
2
Preliminaries
For the rest of this paper, fix the set of propositional variables in all formulas to be V , |V | = n. A
variable assignment ? : V ? {0, 1} is a function that assigns a value in {0, 1} to each variable in V .
We may think of the value 0 as FALSE and the value 1 as TRUE. We will often abuse notation and
write ? (i) for valuations of entities i 6? V when the intended meaning is either already defined or is
clear from the context. In particular, ? (1) = 1 and ? (0) = 0. When ? (i) = 1, we say that ? satisfies
i. For x ? V , ?x denotes the corresponding negated variable; ? (?x) = 1 ? ? (x). Let F be a formula
over variables V . ? (F) denotes the valuation of F under ? . If ? satisfies F, i.e., ? (F) = 1, then ?
is a model, solution, or satisfying assignment for F. Our goal in this paper is to sample uniformly
from the set of all solutions of a given formula F.
An XOR constraint D over variables V is the logical ?xor? or parity of a subset of V ? {1}; ? satisfies
D if it satisfies an odd number of elements in D. The value 1 allows us to express even parity. For
instance, D = {a, b, c, 1} represents the xor constraint a ? b ? c ? 1, which is TRUE when an even
number of a, b, c are TRUE. Note that it suffices to use only positive variables. E.g., ?a ? b ? ?c
and ?a ? b are equivalent to D = {a, b, c} and D = {a, b, 1}, respectively. Our focus will be on
formulas which are a logical conjunction of a formula in Conjunctive Normal Form (CNF) and some
XOR constraints. In all our experiments, XOR constraints are translated into CNF using additional
variables so that the full formula can be fed directly to standard (CNF-based) SAT solvers.
We will need basic concepts from linear algebra. Let F2 denote the field of two elements, 0 and 1,
and Fn2 the vector space of dimension n over F. An assignment ? can be thought of as an element of
Fn2 . Similarly, an XOR constraint D can be seen as a linear constraint a1 x1 + a2 x2 + . . .+ an xn + b = 1,
where ai , b ? {0, 1}, + denotes addition modulo 2 for F2 , ai = 1 iff D has variable i, and b = 1 iff
D has the parity constant 1. In this setting, we can talk about linear transformations of F n2 as well as
linear independence of ? , ? 0 ? Fn2 (see standard texts for details). We will use two properties: every
linear transformation maps the all-zeros vector to itself, and there exists a linear transformation that
maps any k linearly independent vectors to any other k linearly independent vectors.
Consider the set X of all XOR constraints over V . Since an XOR constraint is a subset of V ? {1},
|X| = 2n+1 . Our method requires choosing XOR constraints from X at random. Let X(n, q) denote
the probability distribution over X defined as follows: select each v ? V independently at random
with probability q and include the constant 1 independently with probability 1/2 . This produces XORs
of average length nq. In particular, note that every two complementary XOR constraints involving
the same subset of V (e.g., c ? d and c ? d ? 1) are chosen with the same probability irrespective of
q. Such complementary XOR constraints have the simple but useful property that any assignment ?
satisfies exactly one of them. Finally, when the distribution X(n,1/2 ) is used, every XOR constraint
in X is chosen with probability 2?(n+1) .
3
Note that there are certain classes of structured instances based on parity constraints that are designed to
be hard for SAT solvers [17]. Our augmented problem instances appear to behave quite differently from these
specially constructed instances because of the interaction between the constraints in the original instance and
the added random parity constraints.
We will be interested in the random variables which are the sum of indicator random variables:
Y = ?? Y? . Linearity of expectation says that E [Y ] = ?? E [Y? ]. When various Y? are pairwise
independent, i.e., knowing Y?2 tells us nothing about Y?1 , even variance behaves linearly: Var [Y ] =
?? Var [Y? ]. We will also need conditional probabilities. Here, for a random event X, linearity of
conditional expectation says that E [Y | X] = ?? E [Y? | X]. Let X = Y?0 . When various Y? are 3-wise
independent, i.e., knowing
Y?2 and
Y?3 tells
us nothing about Y?1 , even conditional variance behaves
linearly: Var Y | Y?0 = ?? Var Y? | Y?0 . This will be key to the analysis of our second algorithm.
3
Sampling using XOR constraints
In this section, we describe and analyze two randomized algorithms, XORSample and XORSample?,
for sampling solutions of a given Boolean formula F near-uniformly using streamlining with random
XOR constraints. Both algorithms are parameterized by two quantities: a positive integer s and a
real number q ? (0, 1), where s is the number of XORs added to F and X(n, q) is the distribution
from which they are drawn. These parameters determine the degree of uniformity achieved by the
algorithms, which we formalize as Theorems 1 and 2. The first algorithm, XORSample, uses a SAT
solver as a subroutine on the randomly streamlined formula. It repeatedly performs the streamlining
process until the resulting formula has a unique solution. When s is chosen appropriately, it takes
XORSample a small number of iterations (on average) to successfully produce a sample. The second
algorithm, XORSample?, is non-iterative. Here s is chosen to be relatively small so that a moderate
number of solutions survive. XORSample? then uses stronger subroutines, namely a SAT model
counter and a model selector, to output one of the surviving solutions uniformly at random.
3.1
XOR -based
sampling using SAT solvers: XORSample
Let F be a formula over n variables, and q and s be the parameters of XORSample. The algorithm
works by adding to F, in each iteration, s random XOR constraints Qs drawn independently from the
distribution X(n, q). This generates a streamlined formula Fsq whose solutions (called the surviving
solutions) are a subset of the solutions of F. If there is a unique surviving solution ? , XORSample
outputs ? and stops. Otherwise, it discards Qs and Fsq , and iterates the process (rejection sampling).
The check for uniqueness of ? is done by adding the negation of ? as a constraint to Fsq and testing
whether the resulting formula is still satisfiable. See Algorithm 1 for a full description.
Params: q ? (0, 1), a positive integer s
Input : A CNF formula F
Output : A solution of F
begin
iterationSuccess f ul ? FALSE
while iterationSuccess f ul = FALSE do
Qs ? {s random constraints independently drawn from X(n, q)}
q
Fs ? F ? Qs
// Add s random X O R constraints to F
q
result ? SATSolve(Fs )
// Solve using a SAT solver
if result = TRUE then
? ? solution returned by SATSolve (Fsq )
q
F 0 ? Fs ? {?? }
// Remove ? from the solution set
0
result ? SATSolve(F 0 )
if result 0 = FALSE then
iterationSuccess f ul = TRUE
q
return ?
// Output ? ; it is the unique solution of Fs
end
Algorithm 1: XORSample, sampling solutions with XORs using a SAT solver
We now analyze how uniform the samples produced by XORSample are. For the rest of this section,
?
fix q = 1/2 . Let F be satisfiable and have exactly 2s solutions; s? ? [0, n]. Ideally, we would like
?
each solution ? of F to be sampled with probability 2?s . Let pone,s (? ) be the probability that
?
XORSample outputs ? in one iteration. This is typically much lower than 2?s , which is accounted
for by rejection sampling. Nonetheless, we will show that when s is larger than s ? , the variation in
pone,s (? ) over different ? is small. Let ps (? ) be the overall probability that XORSample outputs ? .
?
This, we will show, is very close to 2?s , where ?closeness? is formalized as being within a factor
of c(? ) which approaches 1 very fast. The proof closely follows the argument used by Valiant and
Vazirani [16] in their complexity theory work on unique satisfiability. However, we give a different,
non-combinatorial argument for the pairwise independence property of XORs needed in the proof,
relying on linear algebra. This approach is insightful and will come handy in Section 3.2. We
describe the main idea below, deferring details to the full version of the paper.
Lemma 1. Let ? > 0, c(? ) = 1 ? 2?? , and s = s? + ? . Then c(? )2?s < pone,s (? ) ? 2?s .
Proof sketch. We first prove the upper bound on pone,s (? ). Recall that for any two complementary
XORs (e.g. c ? d and c ? d ? 1), ? satisfies exactly one XOR. Hence, the probability that ? satisfies
an XOR chosen randomly from the distribution X(n, q) is 1/2 . By independence of the s XORs in Qs
in XORSample, ? survives with probability exactly 2?s , giving the desired upper bound on pone,s (? ).
For the lower bound, we resort to pairwise independence. Let ? 6= ? 0 be two solutions of F. Let
D be an XOR chosen randomly from X(n,1/2 ). We use linear algebra arguments to show that the
probability that ? (D) = 1 (i.e., ? satisfies D) is independent of the probability that ? 0 (D) = 1.
Recall the interpretation of variable assignments and XOR constraints in the vector space F n2 (cf.
Section 2). First suppose that ? and ? 0 are linearly dependent. In Fn2 , this can happen only if exactly
one of ? and ? 0 is the all-zeros vector. Suppose ? = (0, 0, . . . , 0) and ? 0 is non-zero. Perform a linear
transformation on Fn2 so that ? 0 = (1, 0, . . . , 0). Let D be the constraint a1 x1 + a2 x2 + . . . + an xn + b =
1. Then, ? 0 (D) = a1 + b and ? (D) = b. Since a1 is chosen uniformly from {0, 1} when D is drawn
from X(n,1/2 ), knowing a1 + b gives us no information about b, proving independence. A similar
argument works when ? is non-zero and ? 0 = (0, 0, . . . , 0), and also when ? and ? 0 are linearly
independent to begin with. We skip the details.
This proves that ? (D) and ? 0 (D) are independent when D is drawn from X(n,1/2 ). In particular,
Pr [? 0 (D) = 1 | ? (D) = 1] = 1/2 . This reasoning easily extends to s XORs in Qs and we have that
Pr [? 0 (Qs ) = 1 | ? (Qs ) = 1] = 2?s . Now,
pone,s (? ) = Pr ? (Qs ) = 1 and for all other solutions ? 0 of F, ? 0 (Qs ) = 0
= Pr [? (Qs ) = 1] ? 1 ? Pr for some solution ? 0 6= ? , ? 0 (Qs ) = 1 | ? (Qs ) = 1 .
Evaluating this using the union bound and pairwise independence shows p one,s (? ) > c(? ) 2?s .
Theorem 1. Let F be a formula with 2s solutions. Let ? > 0, c(? ) = 1 ? 2?? , and s = s? + ? . For
any solution ? of F, the probability ps (? ) with which XORSample with parameters q = 1/2 and s
outputs ? satisfies
?
?
1
c(? ) 2?s < ps (? ) <
2?s
and
min {ps (? )} > c(? ) max {ps (? )} .
?
?
c(? )
Further, the number of iterations needed to produce one sample has a geometric distribution with
expectation between 2? and 2? /c(? ).
Proof. Let p? denote the probability that XORSample finds some unique solution in any single iteration. pone,s (? ), as before, is the probability that ? is the unique surviving solution. ps (? ), the
overall probability of sampling ? , is given by the infinite geometric series
?
ps (? ) = pone,s (? ) + (1 ? p)p
? one,s (? ) + (1 ? p)
? 2 pone,s (? ) + . . .
which sums to pone,s (? )/ p.
? In particular, ps (? ) is proportional to pone,s (? ).
Lemma 1 says that for any two solutions ?1 and ?2 of F, pone,s (?1 ) and pone,s (?2 ) are strictly within
a factor of c(? ) of each other. By the above discussion, ps (?1 ) and ps (?2 ) must also be strictly
within a factor of c(? ) of each other, already proving the min vs. max part of the result. Further,
?? ps (? ) = 1 because of rejection sampling.
For the first part of the result, suppose for the sake of contradiction that ps (?0 ) ? c(? )2?s for some
?0 , violating the claimed lower bound. By the above argument, ps (? ) is within a factor of c(? ) of
?
ps (?0 ) for every ? , and would therefore be at most 2?s . This would make ?? ps (? ) strictly less
than one, a contradiction. A similar argument proves the upper bound on ps (? ).
?
Finally, the number of iterations needed to find a unique solution (thereby successfully producing a
sample) is a geometric random variable with success parameter p? = ?? pone,s (? ), and has expected
value 1/ p.
? Using the bounds on pone,s (? ) from Lemma 1 and the fact that the unique survival of each
?
?
?
of the 2s solutions ? are disjoint events, we have p? ? 2s 2?s = 2?? and p? > 2s c(? )2?s = c(? )2?? .
This proves the claimed bounds on the expected number of iterations, 1/ p.
?
3.2
XOR -based
sampling using model counters and selectors: XORSample?
We now discuss our second parameterized algorithm, XORSample?, which also works by adding to
F s random XORs Qs chosen independently from X(n, q). However, now the resulting streamlined
formula Fsq is fed to an exact model counting subroutine to compute the number of surviving solutions, mc. If mc > 0, XORSample? succeeds and outputs the ith surviving solution using a model
selector on Fsq , where i is chosen uniformly from {1, 2, . . . , mc}. Note that XORSample?, in contrast
to XORSample, is non-iterative. Also, the model counting and selecting subroutines it uses are more
complex than SAT solvers; these work well in practice only because Fsq is highly streamlined.
Params: q ? (0, 1), a positive integer s
Input : A CNF formula F
Output : A solution of F, or Failure
begin
Qs ? {s constraints randomly drawn from X(n, p)}
q
Fs ? F ? Qs
// Add s random X O R constraints to F
q
q
mc ? SATModelCount(Fs )
// Compute the exact model count of Fs
if mc 6= 0 then
i ? a random number chosen uniformly from {1, 2, . . . , mc}
? ? SATFindSolution(Fsq , i)
// Compute the ith solution
return ?
// Sampled successfully!
else return Failure
end
Algorithm 2: XORSample?, sampling with XORs using a model counter and selector
The sample-quality analysis of XORSample? requires somewhat more complex ideas than that of
?
XORSample. Let F have 2s solutions as before. We again fix q = 1/2 and prove that if the parameter
s is sufficiently smaller than s? , the sample-quality is provably good. The proof relies on the fact that
XORs chosen randomly from X(n,1/2 ) act 3-wise independently on different solutions, i.e., knowing
the value of an XOR constraint on two variable assignments does not tell us anything about its value
on a third assignment. We state this as the following lemma, which can be proved by extending the
linear algebra arguments we used in the proof of Lemma 1 (see the full version for details).
Lemma 2 (3-wise independence). Let ?1 , ?2 , and ?3 be three distinct assignments to n Boolean
variables. Let D be an XOR constraint chosen at random from X(n,1/2 ). Then for i ? {0, 1},
Pr [?1 (D) = i | ?2 (D), ?3 (D)] = Pr [?1 (D) = i].
Recall the discussion of expectation, variance, pairwise independence, and 3-wise independence in
Section 2. In particular, when a number of random variables are 3-wise independent, the conditional
variance of their sum (conditioned on one of these variables) equals the sum of their individual
conditional variances. We use this to compute bounds on the sampling probability of XORSample?.
The idea is to show that the number of solutions surviving, given that any fixed solution ? survives,
is independent of ? in expectation and is highly likely to be very close to the expected value. As
a result, the probability with which ? is output, which is inversely proportional to the number of
solutions surviving along with ? , will be very close to the uniform probability. Here ?closeness? is
one-sided and is measured as being within a factor of c0 (? ) which approaches 1 very quickly.
Theorem 2. Let F be a formula with 2s solutions. Let ? > 0 and s = s? ? ? . For any solution ? of
F, the probability p0s (? ) with which XORSample? with parameters q = 1/2 and s outputs ? satisfies
?
?
p0s (? ) > c0 (? ) 2?s ,
where c0 (? ) =
1 ? 2?? /3
.
(1 + 2?? )(1 + 2?? /3 )
Further, XORSample? succeeds with probability larger than c0 (? ).
Proof sketch. See the full version for a detailed proof. We begin by setting up a framework for
analyzing the number of surviving solutions after s XORs Qs drawn from X(n,1/2 ) are added to F. Let
Y? 0 be the indicator random variable which is 1 iff ? 0 (Qs ) = 1, i.e., ? 0 survives Qs . E [Y? 0 ] = 2?s and
Var [Y? 0 ] ? E [Y? 0 ] = 2?s . Further, a straightforward generalization of Lemma 2 from a single XOR
constraint D to s independent XORs Qs implies that the random variables Y? 0 are 3-wise independent.
The variable mc (see Algorithm 2), which is the number of surviving solutions, equals ?? 0 Y? 0 .
Consider the distribution of mc conditioned on the fact that ? survives. Using pairwise independence, the corresponding conditional expectation can be shows to satisfy: ? = E [mc | ? (Q s ) = 1] =
1 + (2s ? 1)2?s . More interesting, using 3-wise independence, the corresponding conditional variation can also be bounded: Var [mc | ? (Qs ) = 1] < E [mc | ? (Qs ) = 1].
?
Since s = s? ? ? , 2? < ? < 1 + 2? . We show that mc conditioned on ? (Qs ) = 1 indeed lies very
close to ? . Let ? ? 0 be a parameter whose value we will fix later. By Chebychev?s inequality,
h
i 22? Var [mc | ? (Q ) = 1]
?
22?
22?
s
<
=
Pr |mc ? ? | ? ? | ? (Qs ) = 1 ?
(E [mc | ? (Qs ) = 1])2
E [mc | ? (Qs ) = 1]
?
2
Therefore, conditioned on ? (Qs ) = 1, with probability more than 1 ? 22? /? , mc lies between (1 ?
2?? )? and (1 + 2?? )? . Recall that p0s (? ) is the probability that XORSample? outputs ? .
p0s (? ) = Pr [? (Qs ) = 1]
n
i=1
? 2?s Pr mc ? (1 + 2?? )? | ? (Qs ) = 1
h
1
? Pr [mc = i | ? (Qs ) = 1] i
i
2?
1
?s 1 ? 2 / ?
?
2
(1 + 2?? )?
(1 + 2?? )?
Simplifying this expression and optimizing it by setting ? = ? /3 gives the desired bound on p 0s (? ).
Lastly, the success probability of XORSample? is ?? p0s (? ) > c0 (? ).
Remark 1. Theorems 1 and 2 show that both XORSample and XORSample? can be used to sample
arbitrarily close to the uniform distribution when q = 1/2 . For example, as the number of XORs
used in XORSample is increased, ? increases, the deviation c(? ) from the truly uniform sampling
probability p? approaches 0 exponentially fast, and we get progressively smaller error bands around
p? . However, for any fixed ? , these algorithms, somewhat counter-intuitively, do not always sample
truly uniformly (see the full version). As a result, we expect to see a fluctuation around p ? , which,
as we proved above, will be exponentially small in ? .
4
Empirical validation
To validate our XOR-sampling technique, we consider two kinds of formulas: a random 3-SAT instance generated near the SAT phase transition [18] and a structured instance derived from a logistics
planning domain (data and code available from the authors). We used a complete model counter,
Relsat [12], to find all solutions of our problem instances. Our random instance with 75 variables
has a total of 48 satisfying assignments, and our logistics formula with 352 variables has 512 satisfying assignments. (We used formulas with a relatively small number of assignments in order to
evaluate the quality of our sampling. Note that we need to draw many samples for each assignment.)
We used XORSample with MiniSat [14] as the underlying SAT solver to generate samples from the
set of solutions of each formula. Each sample took a fraction of a second to generate on a 4GHz processor. For comparison, we also ran the best alternative method for sampling from SAT problems,
SampleSAT [19, 2], allowing it roughly the same cumulative runtime as XORSample.
Figure 1 depicts our results. In the left panel, we consider the random SAT instance, generating
200,000 samples total. In pure uniform sampling, in expectation we have 200, 000/48 ? 4, 167
samples for each solution. This level is indicated with the solid horizontal line. We see that the
samples produced by XORSample all lie in a narrow band centered around this line. Contrast this
with the results for SampleSAT: SampleSAT does sample quite uniformly from solutions that lie
near each other in Hamming distance but different solution clusters are sampled with different frequencies. This SAT instance has two solution clusters: the first 32 solutions are sampled around
2, 900 times each, i.e., not frequently enough, whereas the remaining 16 solutions are sampled too
frequently, around 6,700 times each. (Although SampleSAT greatly improves on other sampling
strategies for SAT, the split into disjoint sampling bands appears inherent in the approach.) The
Kullback-Leibler (KL) divergence between the XORSample data and the uniform distribution is
0.002. For SampleSAT the KL-divergence from uniform is 0.085. It is clear that the XORSample
approach leads to much more uniform sampling.
The right panel in Figure 1 gives the results for our structured logistics planning instance. (To improve the readability of the figure, we plot the sample frequency only for every fifth assignment.) In
this case, the difference between XORSample and SampleSAT is even more dramatic. SampleSAT
in fact only found 256 of the 512 solutions in a total of 100,000 samples. We also see that one of
these solutions is sampled nearly 60,000 times, whereas many other solutions are sampled less than
Absolute Frequency (log scale)
Absolute Frequency (log scale)
10000
1000
100000
XORsample
SampleSat
uniform
XORSample
SampleSat
uniform
10000
1000
100
10
1
0
10
20
Solution #
30
40
50
0
100
200
300
Solution #
400
500
Figure 1: Results of XORSample and SampleSAT on a random 3-SAT instance, the left panel, and a logistics
planning problem, the right panel. (See color figures in PDF.)
five times. The KL divergence from uniform is 4.16. (Technically the KL divergence is infinite, but
we assigned a count of one to the non-sampled solutions.) The expected number of samples for each
assignment is 100, 000/512 ? 195. The figure also shows that the sample counts from XORSample
all lie around this value; their KL divergence from uniform is 0.013.
These experiments show that XORSample is a promising practical technique (with theoretical guarantees) for obtaining near-uniform samples from intricate combinatorial spaces.
References
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2,220 | 3,014 | Generalized Regularized Least-Squares Learning
with Predefined Features in a Hilbert Space
Wenye Li, Kin-Hong Lee, Kwong-Sak Leung
Department of Computer Science and Engineering
The Chinese University of Hong Kong
Shatin, Hong Kong, China
{wyli, khlee, ksleung}@cse.cuhk.edu.hk
Abstract
Kernel-based regularized learning seeks a model in a hypothesis space by minimizing the empirical error and the model?s complexity. Based on the representer
theorem, the solution consists of a linear combination of translates of a kernel.
This paper investigates a generalized form of representer theorem for kernel-based
learning. After mapping predefined features and translates of a kernel simultaneously onto a hypothesis space by a specific way of constructing kernels, we proposed a new algorithm by utilizing a generalized regularizer which leaves part of
the space unregularized. Using a squared-loss function in calculating the empirical error, a simple convex solution is obtained which combines predefined features
with translates of the kernel. Empirical evaluations have confirmed the effectiveness of the algorithm for supervised learning tasks.
1 Introduction
Supervised learning, or learning from examples, refers to the task of training a system by a set of
examples which are specified by input-output pairs. The system is used to predict the output value
for any valid input object after training. Examples of such tasks include regression which produces
continuous values, and classification which predicts a class label for an input object.
Vapnik?s seminal work[1] shows that the key to effectively solving this problem is by controlling
the solution?s complexity, which leads to the techniques known as regularized kernel methods[1]
[2][3] and regularization networks[4]. The work championed by Poggio and other researchers[5][6]
implicitly treats learning as an approximation problem and gives a general scheme with ideas going
back to modern regularization theory[7][8][9]. For both frameworks, a solution is sought by simultaneously minimizing the empirical error and the complexity. More precisely, given a training set
m
D = (xi ; yi )i=1 , an estimator f : X ? Y, where X is a closed subset of Rd and Y ? R, is given
by
m
1 X
2
min
V (yi , f (xi )) + ? kf kK
(1)
f ?HK m
i=1
where V is a convex loss function, kf kK is the norm of f in a reproducing kernel Hilbert space
(RKHS) HK induced by a positive definite function (a kernel) Kx (x0 ) = K (x, x0 ), and ? is a
regularization parameter that makes a trade-off between the empirical error and the complexity.
? kf k2K is also called a regularizer.
According to representer theorem [10][11] [12], the minimizer of (1) admits a simple solution as a
linear combination of translates of the kernel K by the training data
m
X
f? =
ci Kxi , ci ? R, 1 ? i ? m
i=1
for a variety of loss functions. Different loss functions lead to different learning algorithms. For
2
example, when used for classification, a squared-loss (y ? f (x)) brings about the regularized
least-squares classification (RLSC) algorithm[13][14][15]; while a hinge loss (1 ? yf (x)) + ?
max (1 ? yf (x) , 0) corresponds to the classical support vector machines(SVM).
Using this model, data are implicitly projected onto the hypothesis space H K via a transformation
?K : x ? K x
and a linear functional is sought by finding its representer in HK , which generally has infinite
dimensions. It is generally believed that learning problems associated with infinite dimensions are
ill-posed and need regularization. However, finite dimensional problems are often associated with
well-posedness and do not need regularization. Motivated by this, we unified these two views in
this paper. Using an existing trick in designing kernels, an RKHS is constructed which contains
a subspace spanned by some predefined features and this subspace is left unregularized during the
learning process. Empirical results have shown the embedding of these features often has the effect
of stabilizing the algorithms?s performance for different choices of kernels and prevents the results
from deteriorating for inappropriate kernels.
The paper is organized as follows. First, a generalized regularized learning model and its associated
representer theorem are studied. Then, we introduce an existing trick with which we constructed a
hypothesis space which has a subspace of the predefined features. Next, a generic learning algorithm
is proposed based on the model and especially evaluated for classification problems. Empirical
results have confirmed the benefits brought by the algorithm.
A note on notation. Throughout the paper, vectors and matrices are represented in bold notation and
scalars in normal script, e.g. x1 , ? ? ? , xm ? Rd , K ? Rm?m , and y1 , ? ? ? , ym ? R. I and O are
used to denote an identity matrix and a zero matrix of appropriate sizes, respectively. For clarity, the
size of a matrix is sometimes added as a subscript, such as Om?` .
2 Generalized regularized least-squares learning model
Suppose the space HK decomposes into the direct sum: HK = H0 ? H1 , where H0 is spanned
by ` (? m) linearly independent features: H0 = span (?1 , ? ? ? , ?` ). We propose the generalized
regularized least-squares (G-RLS) learning model as
m
min L (f ) =
f ?HK
1 X
(yi ? f (xi ))2 + ? kf ? P f k2K ,
m i=1
(2)
where P f is the orthogonal projection of f onto H0 .
Suppose f ? is the minimizer of (2). For any f ? HK , let f = f ? + ?g where ? ? R and g ? HK .
Now take derivative w.r.t. ? and notice that ?L
?? |?=0 = 0 . Then
m
?
2 X
(yi ? f ? (xi )) g (xi ) + 2? hf ? ? P f ? , giK = 0,
m i=1
(3)
where h?, ?iK denotes the inner product in HK . This equation holds for any g ? HK . In particular,
setting g = Kx gives
Pm
(yi ? f ? (xi )) Kxi
.
(4)
f ? ? P f ? = i=1
m?
P f ? is the orthogonal projection of f ? onto H0 and hence,
P f? =
`
X
?p ?p , ?p ? R, 1 ? p ? `.
(5)
p=1
So (4) is simplified to
f? =
`
X
p=1
?p ? p +
m
X
i=1
c i Kxi ,
(6)
where
yi ? f ? (xi )
, 1 ? i ? m.
(7)
m?
The coefficients ?1 , ? ? ? , ?` , c1 , ? ? ? , cm are uniquely specified by m + ` linear equations. The first
m equations are obtained by substituting (6) into (7). The rest ` equations are derived from the
orthogonality constraint between P f ? and f ? P f ? , which can be written as
*
+
m
X
?p ,
c i Kxi
= 0, 1 ? p ? `,
(8)
ci =
i=1
K
or equivalently due to the property of reproducing kernels,
m
X
(9)
ci ?p (xi ) = 0, 1 ? p ? `.
i=1
m
The solution (6) derived from (2) satisfies the reproduction property. Suppose (x i ; yi )i=1 comes
purely from a model which is perfectly linearly related to ?1 , ? ? ? , ?` , it is desirable to get back a
solution that is independent of the other features. As an evident result of (2), the property is satisfied.
The parameters c1 , ? ? ? , cm in the resulting estimator (6) are all zero, which makes the regularizer in
(2) equal to zero.
3 Kernel construction
By decomposing a hypothesis space HK and studying a generalized regularizer, we have proposed
the G-RLS model and derived a solution which consists of predefined features as well as translates
of a kernel function. In this section, starting with predefined features ? 1 , ? ? ? , ?` and a kernel ?, we
will construct a hypothesis space which contains the features and translates of the kernel by using
an existing trick.
3.1 A kernel construction trick
Let?s consider the following reproducing kernel
K (x, x0 ) = H (x, x0 ) +
`
X
?0p (x) ?0p (x0 )
(10)
p=1
where
H (x, x0 )
=
? (x, x0 ) ?
`
X
?0p (x) ? (xp , x0 ) ?
p=1
+
` X
`
X
`
X
?0q (x0 ) ? (x, xq )
(11)
q=1
?0p (x) ?0q (x0 ) ? (xp , xq ) ,
p=1 q=1
? is any strictly positive definite function, and ?01 , ? ? ? , ?0` defines a linear transformation of
?1 , ? ? ? , ?` w.r.t. x1 , ? ? ? , x` ,
" 0
# "
#?1 "
#
?1 (x)
?1 (x1 ) ? ? ? ?1 (x` )
?1 (x)
???
???
???
???
=
(12)
?0` (x)
?` (x1 ) ? ? ? ?` (x` )
?` (x)
which satisfies
?0q
(xp ) =
1
0
1?p=q?`
.
1 ? p 6= q ? `
(13)
This trick is studied in [16] to provide an alternative basis for radial basis functions and first used in
a fast RBF interpolation algorithm[17]. A sketch of properties which are peripheral to our concerns
in this paper are given below.
Kxp = ?0p , 1 ? p ? `
(14)
?0p , ?0q
=
K
1
0
1?p=q?`
1 ? p 6= q ? `
(15)
(16)
Hxp = H (xp , ?) = 0, 1 ? p ? `
= H (xi , ?) , ?0p K = 0, ` + 1 ? i ? m, 1 ? p ? `
Hxi , Hxj K = H (xi , xj ) , ` + 1 ? i, j ? m
(17)
Hxi , ?0p K
Another property is that the matrix H = (H
be used in the computations below.
m
(xi , xj ))i,j=`+1
(18)
is strictly positive definite, which will
By constructing a kernel K using this trick, predefined features ?1 , ? ? ? , ?` are explicitly mapped
onto HK which has a subspace H0 = span (?01 , ? ? ? , ?0` ) = span (?1 , ? ? ? , ?` ). By property (15),
we can see that ?01 , ? ? ? , ?0` also forms an orthonormal basis of H0 .
3.2 Computation
After projecting the features ?1 , ? ? ? , ?` onto an RKHS HK , let?s study the regularized minimization problem in (2). As shown in (6), the minimizer has a form of a linear combination of predefined
features and translates of a kernel. By the properties of K in (14)-(17), the minimizer can be rewritten as:
f?
=
`
X
?p ? p +
p=1
=
`
X
=
?0p ?0p
+
`
X
`
X
ci ?0i
+
i=1
?0p + cp +
p=1
=
c i Kxi
i=1
p=1
`
X
m
X
m
X
p=1
m
X
Hx i +
m
X
?0p
(xi ) ?0p
p=1
i=`+1
!
ci ?0p (xi ) ?0p +
i=`+1
? p ?0 +
?
p
ci
`
X
m
X
!!
c i Hx i
i=`+1
(19)
c?i Hxi
i=`+1
?1 , ? ? ? , ?
?` , c?`+1 , ? ? ? , c?m are m parameters to be determined. Furthermore, from the orthogwhere ?
onal property between ?0p and Hxi in (17), we have
f? ? P f? =
m
X
(20)
c?i Hxi .
i=`+1
T
T
?1 , ? ? ? , ?
?`
and ?
c = (?
c`+1 , ? ? ? , c?m ) , we need
To determine the values of ?
?= ?
2
kf ? ? P f ? kK =
m
X
c?i c?j H (xi , xj ) =
i,j=`+1
? =
where H
O`?`
O(m?`)?`
O`?(m?`)
H
?
?
?
c
T
?
H
?
?
?
c
(21)
. Substituting (21) into (2), we have
T
?
?
1
?
?
?
?
y?K
y?K
+?
L=
?
c
?
c
m
`,m
I`?` O`?(m?`)
?
where K =
and E = ?0p (xi ) p=1,i=`+1 .
ET
H
and set the derivative to zero, and we get
?
?
?
?2 ?
?
?
K
+ ?mH
= Ky.
?
c
?
c
?
?
?
c
T
?
H
?
?
?
c
Take derivative w.r.t.
(22)
?
?
?
c
(23)
? ?1 =
Since K
we have
i.e.
I`?`
?H?1 ET
O(m?`)?`
H?1
? ?1 H
? =?
and K
I=
O`?`
O`?(m?`)
O(m?`)?`
I(m?`)?(m?`)
?
?
? + ?m?
K
I
= y,
?
c
?
O`?(m?`)
y1
?
=
,
y2
H + ?mI
?
c
I`?`
ET
,
(24)
(25)
T
T
? by
where y1 = (y1 , ? ? ? , y` ) and y2 = (y`+1 , ? ? ? , ym ) . Equation (25) uniquely specifies ?
?
? = y1 ,
(26)
and ?
c by
?.
(27)
(H + ?mI) ?
c = y 2 ? ET ?
H+?mI is a strictly positive definite matrix. The equation can be efficientlysolved either
by
conju
3
gate gradient or by Cholesky factorization. The worst case complexity is O (m ? `) ? O m3 .
It is also possible to investigate iterative methods for solving linear systems coupled with recent advances in fast matrix-vector multiplication methods (e.g. fast multipole method), and the complexity
reduces to nearly O (m log m), which provides the potential to solve large scale problems.
4 A generic learning algorithm
Based on the discussions above, a generic learning algorithm (G-RLS algorithm) is summarized
below.
m
1. Start with data (xi ; yi )i=1 .
2. For ` (? m) predefined linearly independent features ?1 , ? ? ? , ?` of the data, define
?01 , ? ? ? , ?0` according to equation (12).
3. Choose a symmetric, strictly positive definite function ?x (x0 ) = ? (x, x0 ) which is continuous on X ? X . Define H according to equation (11).
4. The estimator f : X ? Y is given by
f (x) =
`
X
?p ?0 (x) +
?
p
p=1
m
X
c?i Hxi (x)
(28)
i=`+1
?1 , ? ? ? , ?
? ` , c?`+1 , ? ? ? , c?m are obtained by solving equations (26) and (27).
where ?
The algorithm can be applied to a number of applications including regression and binary classification. As a simple example for regression, noisy points were randomly generated via a function y = |5 ? x|, and we fitted the data by a curve. Polynomial features up to the second degree
(?1 = 1, ?2 = x, ?3 = x2 ) were used for G-RLS algorithm along with a Gaussian RBF kernel
kx??k2
?x (?) = e? ?2 .
We selected ridge regression with the Gaussian RBF kernel for a comparison, which can be regarded
as an implementation of standard regularized least-squares model for regression tasks. For both
algorithms, three trials were made in which the parameter ? was set to a large value, to a small
value, and by cross validation respectively. For each ?, the parameter ? was set by cross validation.
Comparing with ridge regression in figure 1(b), the existence of polynomial features in G-RLS has
the effect of stabilizing the results, as shown in figure 1(a). Varying ?, different fitting results were
obtained by ridge regression. However, for G-RLS algorithm, the difference was not evident.
In the case of generalized regularized least-squares classification (G-RLSC), each y i of the training
set takes the values {?1, 1}. The predicted label of any x depends on the sign of (28)
1,
f (x) > 0
y=
.
?1 otherwise
G-RLSC uses the ?classical? squared-loss as a classification loss criterion. The effectiveness of this
criterion has been reported by the empirical results[13][14][15].
5
5
data
data
2
2
? =cv
? =cv
2
? =1000
4
? =0.001
3
3
2
2
1
1
0
0
?1
?5
?4
?3
?2
?1
0
1
2
3
4
2
? =1000
4
2
5
(a) G-RLS Regression
?1
?5
2
? =0.001
?4
?3
?2
?1
0
1
2
3
4
5
(b) Ridge Regression
Figure 1: A Regression Example. The existence of polynomial features in G-RLS helped to improve
the stability of the algorithm.
5 Experiments
To evaluate the performance of G-RLS algorithm, empirical results are reported on text categorization tasks using the three datasets from CMU text mining group1. The 7-sectors dataset has 4, 573
web pages belonging to seven economic sectors, with each sector containing pages varying from
300 to 1, 099. The 4-universities dataset consists of 8, 282 webpages collected mainly from four
universities, in which the pages belong to seven classes and each class has 137 to 3, 764 pages.
The 20-newsgroups dataset collects UseNet postings into twenty newsgroups and each group has
about 1, 000 messages. We experimented with its four major subsets. The first subset has 5 groups
(comp.*), the second 4 groups (rec.*), the third 4 groups (sci.*) and the last 4 groups (talk.*).
For each dataset, we removed all but the 2, 000 words with highest mutual information with the
class variable by rainbow package[18]. The document was represented as bag-of-words with linear
normalization into [?1, 1]. Probabilistic latent semantic analysis[19] (pLSA) was used to get ten
latent features ?1 , ? ? ? , ?10 out of the data. Experiments were carried out with different number
(100?3, 200) of data for training and the rest for testing. Each experiment consisted of ten runs
and the average accuracy is reported. In each run, the data were separated by the xval-prep utility
accompanied in C4.5 package2.
Figure 2 compares the performance of G-RLSC, RLSC and SVM. It is shown that G-RLSC reports
improved results on most of the datasets except on 4-universities. Moreover, an insightful observation may find that although SVM excels on the dataset when the number of training data increases,
G-RLSC shows better performance than standard RLSC. A possible reason is that the hinge loss
used by SVM is more appropriate than the squared-loss used by RLSC and G-RLSC on this dataset;
while the embedding of pLSA features still improves the accuracy.
6 Conclusion
In this paper, we first proposed a generic G-RLS learning model. Unlike the standard kernel-based
methods which only consider the translates of a kernel for model learning, the new model takes
predefined features into special consideration. A generalized regularizer is studied which leaves
part of the hypothesis space unregularized. Similar ideas were explored in spline smoothing[9] in
which low degree polynomials are not regularized. Another example is semi-parametric SVM[2],
which considers the addition of some features to the kernel expansion for SVM. However, to our
knowledge, few learning algorithms and applications have been studied along this line from a unified
RKHS regularization point of view, or investigated for empirical evaluations.
The second part of our work presented a practical computation method based on the model. An
RKHS that contains the combined solutions is explicitly constructed based on a special trick in
designing kernels. (The idea of a conditionally positive definite function[20] is lurking in the back1
2
http://www.cs.cmu.edu/?TextLearning/datasets.html
http://www.rulequest.com/Personal/c4.5r8.tar.gz.
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Figure 2: Classification accuracies on CMU text datasets with different number of training samples.
Ten pLSA features along with a linear kernel ? were used for G-RLSC. Both bag-of-words (BoW)
and pLSA representations of documents were experimented for RLSC and SVM with a linear kernel.
The parameter ? was selected via cross validation. For multi-classification, G-RLSC and RLSC used
one-versus-all strategy. SVM used one-versus-one strategy.
ground of this trick, which goes beyond the discussion of this paper.) With the construction of the
RKHS, the computation is further optimized and the theoretical analysis of such algorithms is also
potentially facilitated.
We evaluated G-RLS learning algorithm in text categorization. The empirical results from real-world
applications have confirmed the effectiveness of the algorithm.
Acknowledgments
The authors thank Dr. Haixuan Yang for useful discussions. This research was partially supported
by RGC Earmarked Grant #4173/04E and #4132/05E of Hong Kong SAR and RGC Research Grant
Direct Allocation of the Chinese University of Hong Kong.
References
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[2] B. Sch?olkopf and A.J. Smola. Learning with Kernels. The MIT Press, 2002.
[3] J.S. Taylor and N. Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, 2004.
[4] T. Evgeniou, M. Pontil, and T. Poggio. Regularization networks and support vector machines. Adv.
Comput. Math., 13:1?50, 2000.
[5] T. Poggio and F. Girosi. Regularization algorithms for learning that are equivalent to multilayer networks.
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[6] T. Poggio and S. Smale. The mathematics of learning: Dealing with data. Not. Am. Math. Soc, 50:537?
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[7] A.N. Tikhonov and V.Y. Arsenin. Solutions of Ill-Posed Problems. Winston and Sons, 1977.
[8] V.A. Morozov. Methods for Solving Incorrectly Posed Problems. Springer-Verlag, 1984.
[9] G. Wahba. Spline Models for Observational Data. SIAM, 1990.
[10] G. Kimeldorf and G. Wahba. Some results on Tchebycheffian spline functions. J. Math. Anal. Appl.,
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[11] F. Girosi, M.J. Jones, and T. Poggio. Regularization theory and neural networks architectures. Neural
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[12] B. Sch?olkopf, R. Herbrich, and A.J. Smola. A generalized representer theorem. In COLT?2001 and
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[13] R.M. Rifkin. Everything Old is New Again: A Fresh Look at Historical Approaches in Machine Learning.
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[14] G. Fung and O.L. Mangasarian. Proximal support vector machine classifiers. In KDD?01, 2001.
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[16] W. Light and H. Wayne. Spaces of distributions, interpolation by translates of a basis function and error
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[17] R.K. Beatson, W.A. Light, and S. Billings. Fast solution of the radial basis function interpolation equations: Domain decomposition methods. SIAM J. Sci. Comput., 22:1717?1740, 2000.
[18] A.K. McCallum. Bow: A toolkit for statistical language modeling, text retrieval, classification and clustering. http://www.cs.cmu.edu/?mccallum/bow, 1996.
[19] T. Hofmann. Probabilistic latent semantic analysis. In UAI?99, 1999.
[20] C.A. Micchelli. Interpolation of scattered data: Distances, matrices, and conditionally positive definite
functions. Constr. Approx., 2:11?22, 1986.
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2,221 | 3,015 | Stratification Learning: Detecting Mixed Density and
Dimensionality in High Dimensional Point Clouds
Gloria Haro, Gregory Randall, and Guillermo Sapiro
IMA and Electrical and Computer Engineering
University of Minnesota, Minneapolis, MN 55455
[email protected],[email protected],[email protected]
Abstract
The study of point cloud data sampled from a stratification, a collection of manifolds with possible different dimensions, is pursued in this paper. We present a
technique for simultaneously soft clustering and estimating the mixed dimensionality and density of such structures. The framework is based on a maximum likelihood estimation of a Poisson mixture model. The presentation of the approach
is completed with artificial and real examples demonstrating the importance of
extending manifold learning to stratification learning.
1
Introduction
Data in high dimensions is becoming ubiquitous, from image analysis and finances to computational
biology and neuroscience. This data is often given or represented as samples embedded in a high
dimensional Euclidean space, point cloud data, though it is assumed to belong to lower dimensional
manifolds. Thus, in recent years, there have been significant efforts in the development of methods to
analyze these point clouds and their underlying manifolds. These include numerous techniques for
the estimation of the intrinsic dimension of the data and also its projection onto lower dimensional
representations. These disciplines are often called manifold learning and dimensionality reduction.
A few examples include [2, 3, 4, 9, 10, 11, 12, 16].
The vast majority of the manifold learning and dimensionality reduction techniques developed in
the literature assume, either explicitly or implicitly, that the given point cloud are samples of a
unique manifold. It is very easy to realize that a significant part of the interesting data has mixed
dimensionality and complexity. The work here presented deals with this more general case, where
there are different dimensionalities/complexities present in the point cloud data. That is, we have
samples not of a manifold but of a stratification. The main aim is to cluster the data according to the
complexity (dimensionality) of the underlying possible multiple manifolds. Such clustering can be
used both to better understand the varying dimensionality and complexity of the data, e.g., states in
neural recordings or different human activities for video analysis, or as a pre-processing step for the
above mentioned manifold learning and dimensionality reduction techniques.
This clustering-by-dimensionality task has been recently explored in a handful of works. Barbar?a
and Chen, [1], proposed a hard clustering technique based on the fractal dimension (box-counting).
Starting from an initial clustering, they incrementally add points into the cluster for which the change
in the fractal dimension after adding the point is the lowest. They also find the number of clusters
and the intrinsic dimension of the underlying manifolds. Gionis et al., [7], use local growth curves
to estimate the local correlation dimension and density for each point. The new two-dimensional
representation of the data is clustered using standard techniques. Souvenir and Pless, [14], use an
Expectation Maximization (EM) type of technique, combined with weighted geodesic multidimensional scaling. The weights measure how well each point fits the underlying manifold defined by
the current set of points in the cluster. After clustering, each cluster dimensionality is estimated
following [10]. Huang et al., [8], cluster linear subspaces with an algebraic geometric method based
on polynomial differentiation and a Generalized PCA. They search for the best combination of linear subspaces that explains the data, and find the number of linear subspaces and their intrinsic
dimension. The work of Mordohai and Medioni, [11], estimates the local dimension using tensor
voting.
These recent works have clearly shown the necessity to go beyond manifold learning, into ?stratification learning.? In our work, we do not assume linear subspaces, and we simultaneously estimate
the soft clustering and the intrinsic dimension and density of the clusters. This collection of attributes is not shared by any of the pioneering works just described. Our approach is an extension
of the Levina and Bickel?s local dimension estimator [10]. They proposed to compute the intrinsic
dimension at each point using a Maximum Likelihood (ML) estimator based on a Poisson distribution. The local estimators are then averaged, under the assumption of a single uniform manifold. We
propose to compute a ML on the whole point cloud data at the same time (and not one for each point
independently), and use a Poisson mixture model, which permits to have different classes, each one
with their own dimension and sampling density. This technique automatically gives a soft clustering
according to dimensionality and density, with an estimation of both quantities for each class. Our
approach assumes that the number of classes is given, but we are discovering the actual number
of underlying manifolds. If we search for a larger than needed number of classes, we obtain some
classes with the same dimensionality and density or some classes with very few representatives, as
shown in the examples later presented.
The remainder of this paper is organized as follows: In Section 2 we review the method proposed by
Levina and Bickel, [10], which gives a local estimation of the intrinsic dimension and has inspired
our work. In Section 3 we present our core contribution of simultaneous soft clustering and dimensionality and density estimation. We present experiments with synthetic and real data in Section 4,
and finally, some conclusions are presented in Section 5.
2
Local intrinsic dimension estimation
Levina and Bickel (LB), [10], proposed a geometric and probabilistic method which estimates the
local dimension (and density) of a point cloud data1 . This is the approach we here extend, which
is based on the idea that if we sample an m-dimensional manifold with T points, the proportion
of points that fall into a ball around a point xt is Tk ? f (xt )V (m)Rk (xt )m , where the given point
cloud, embedded in high dimensions D, is X = {xt ? RD ; t = 1, . . . , T }, k is the number of points
inside the ball, f (xt ) is the local sampling density at point xt , V (m) is the volume of the unit sphere
in Rm , and Rk (xt ) is the Euclidean distance from xt to its k-th nearest neighbor (kNN). Then, they
consider the inhomogeneous process N (R, xt ), which counts the number of points falling into a
small D-dimensional sphere B(R, xt ) of radius R centered at xt . This is a binomial process, and
some assumptions need to be done to proceed. First, if T ? ?, k ? ?, and k/T ? 0, then we
can approximate the binomial process by a Poisson process. Second, the density f (xt ) is constant
inside the sphere, a valid assumption for small R. With these assumptions, the rate ? of the counting
process N (R, xt ) can be written as ?(R, xt ) = f (xt )V (m)mRm?1 . The log-likelihood of the
process N (R, xt ) is then given by
Z R
Z R
L(m(xt ), ?(xt )) =
log ?(r, xt )dN (r, xt ) ?
?(r, xt )dr,
0
0
where ?(xt ) := log f (xt ) is the density parameter and the first integral is a Riemann-Stieltjes
integral [13]. The maximum likelihood estimators satisfy ?L/?? = 0 and ?L/?m = 0, leading to
a computation for the local dimension at point xt , m(xt ), depending on all the neighbors within a
distance R from xt [10]. In practice, it is more convenient to compute a fixed amount k of nearest
i?1
h
Pk?1
Rk (xt )
1
log
. This
neighbors. Thus, the local dimension at point xt is m(xt ) = k?2
j=1
Rj (xt )
estimator is asymptotically unbiased (see [10] for more details). If the data points belong to the
same manifold, we can average over all m(xt ) in order to obtain a more robust estimator. However,
if there are two or more manifolds with different dimensions, the average does not make sense,
unless we first cluster according to dimensionality and then we estimate the dimensionality for each
1
M. Hein pointed us out in NIPS that this dimension estimator is equivalent to the one proposed in [15].
cluster. We briefly toy with this idea now, as a warm up to our simultaneous soft clustering and
estimation technique described in Section 3.
2.1
A two step clustering approach
As a first simple approach to detect and cluster mixed dimensionality (and/or densities), we can
combine a local dimensionality estimator such as the one just described and a clustering technique.
For the second step we use the Information Bottleneck (IB) [17], which is an elegant framework to
eventually combine several local dimension estimators and other possible features such as density
[6]. The IB is a technique that allows to cluster (compress) a variable according to another related
variable. Let X be the set of variables to be clustered and S the relevance variable that gives some
information about X. An example is the information that different words provide about documents
? the clustered version of X. The optimal X
? is the one that minimizes
of different topics. We call X
? X) ? ?I(X;
? S), where I(?; ?) denotes mutual information and
the functional L(p(x?t |xt )) = I(X;
p(?) the probability density function. There is a trade-off, controlled by ?, between compressing the
representation and preserving the meaningful information. In our context, we want to cluster the
data according to the intrinsic dimensionality (and/or density). Then, our relevant variable S will
be the set of (quantized) estimated local intrinsic dimensions. For the joint distribution p(xt , si ),
si ? S, we use the histogram of local dimensions inside a ball of radius R0 around xt ,2 computed
by the LB technique.
Examples of this technique will be presented in the experimental Section. Instead of a two-steps
algorithm, with local dimensionality and/or density estimation followed by clustering, we now propose a maximum likelihood technique that combines these steps.
3
Poisson mixture model
The core approach that we propose to study stratifications (mixed manifolds) is based on extending
the LB technique [10]. Instead of modelling each point and its local ball of radius R as a Poisson
process and computing the ML for each ball separately, we consider all the possible balls at the
same time in the same ML function. As the probability density function for all the point cloud
we consider a mixture of Poisson distributions with different parameters (dimension and density).
Thus, we allow the presence of different intrinsic dimensions and densities in the dataset. These are
automatically computed while being used for soft clustering.
Let us denote by J the number of different Poisson distributions considered in the mixture, each
one with a (possibly) different dimension m and density parameter ?. We consider the vector set of
parameters ? = {? j = (? j , ?j , mj ); j = 1, . . . , J}, where ? j is the mixture coefficient for class j
j
(the proportion of distribution j in the dataset), ?j is its density parameter (f j = e? ), and mj is its
dimension. We denote by p(?) the probability density function and by P (?) the probability.
As in the LB approach, the observable event will be yt = N (R, xt ), the number of points inside
the ball B(R, xt ) of radius R centered at point xt . The total number of observations is T 0 and
Y = {yt ; t = 1, . . . , T 0 } is the observation sequence. If we consider every possible ball in the
dataset then, T 0 coincides with the total number of points T in the point cloud. From now on, we
will consider this case and T 0 ? T . The density function of the Poisson mixture model is given by
!
!
Z R
Z R
J
J
X
X
j
j
j
j
j
j
p(yt |?) =
? p(yt |? , m ) =
? exp
log ? (r) dN (r, xt ) exp ?
? (r)dr ,
j=1
j=1
j
0
j
0
where ?j (r) = e? V (mj )mj rm ?1 . Usually, problems involving a mixture of experts are solved by
the Expectation Maximization (EM) algorithm [5]. In our context, there are two kinds of unknown
parameters: The membership function of an expert (class), ? j , and the parameters of each expert,
mj and ?j . The membership information is originally unknown, thereby making the parameter
estimation for each class difficult. The EM algorithm computes its expected value (E-step) and then
this value is used for the parameter estimation procedure (M-step). These two steps are iterated.
2
The value of R0 determines the amount of regularity in the classification.
If Y contains T statistically independent variables, then the incomplete data log-likelihood is:
L(Y |?) = log p(Y |?) = log
T
Y
p(yt |?) = Q(?) + R(?),
t=1
Q(?) :=
X
P (Z|Y, ?) log p(Z, Y |?), R(?) := ?
Z
X
P (Z|Y, ?) log P (Z|Y, ?),
Z
where Z = {zt ? C; t = 1, . . . , T } is the missing data (hidden-state information), and the set
of class labels is C = {C 1 , C 2 , . . . C J }. Here, zt = C j means that the j-th mixture generates
yt . We call Q the expectation of log p(Z, Y |?) with respect to Z. The EM algorithm is based on
maximizing Q, since while improving (maximizing) the function Q at each iteration, the likelihood
function L is also improved. The probability density that appears in the function Q can be written
QT
as p(Z, Y |?) = t=1 p(zt , yt |?), and the complete-data log-likelihood becomes
log p(Z, Y |?) =
T X
J
X
?tj log p(yt |zt = C j , ? j )? j ,
(1)
t=1 j=1
where a set of indicator variables ?tj is used in order to indicate the status of the hidden variables:
1 if yt generated by mixture C j ,
j
j
?t ? ?(zt , C ) =
0 else.
Considering the expectation, with respect to Z, EZ (?) of (1) and setting ? to a fixed known value
?n (the value at step n of the algorithm), everywhere except for the log function, we get a function
Q of ?. We denote it by Q(?|?n ), and it has the following form
Q(?|?n ) =
J
T X
X
h
i
hjn (yt ) log p(yt |?tj = 1, ? j )? j ,
t=1 j=1
where
p(yt |?tj = 1, ?nj )?nj
(2)
hjn (yt ) = EZ [?tj |yt , ?n ] = P (?tj = 1|yt , ?n ) = PJ
l
l
l
l=1 p(yt |?t = 1, ?n )?n
is the probability that observation t belongs to mixture j. Finally, the probability density in (2) is
!
!
Z R
Z R
p(yt |?tj = 1, ?nj ) = exp
log ?jn (r)dN (r, xt ) exp ?
?jn (r)dr ,
(3)
0
j
?n
0
mjn ?1
. As mentioned above, the EM algorithm consists of two main
where ?jn (r) = e V (mjn )mjn r
steps. In the E-step, the function Q(?|?n ) is computed, for that, we determine the best guess of
the membership function, i.e., the probabilities hjn (yt ). Once we know these probabilities, Q(?|?n )
can be considered as a function of the only unknown, ?, and it is maximized in order to compute
the values of ?n+1 , i.e., the maximum likelihood parameters ? at step n + 1; this is called the Mstep. The EM suffers from local maxima, hitting a local maximum can be prevented running the
algorithm several times with different initializations. Different random subset of points, from the
point cloud, may be used in each run. We have experimented with both approaches and the results
are always similar if we initialize all the probabilities equally. The Algorithm PMM describes the
j
j
are obtained by
, mjn+1 , and ?n+1
main components of this proposed approach. The estimators ?n+1
PJ
j
l
computing ?n+1 = arg max?j Q(?|?n ) + ?( l=1 ? ? 1) in the M-step, where ? is the Lagrange
PJ
multiplier that allows to introduce the constraint l=1 ? l = 1. This gives equations (4)-(5), where
R? j
j
mj
mj
V (mjn ) = (2? mn /2 )/(mjn ?( 2n )), and ?( 2n ) = 0 tmn /2?1 e?t dt. In order to compute mjn+1
we have used the same approach as in [10], by means of a k nearest neighbor graph.
4
Experimental results
We now present a number of experimental results for the technique proposed in Section 3. We often
compare it with the two-steps algorithm described in Section 2, and denote this algorithm by LD+IB.
Algorithm PMM Poisson Mixture Model
Require: The point cloud data, J (number of desired classes) and k (scale of observation).
Ensure: Soft clustering according to dimensionality and density.
PJ
j
j
j
j
1: Initialization of ?0 = {?0 , m0 , ?0 } to any set of values which ensures that j=1 ?0 = 1.
2: EM iterations on n,
For all j = 1, . . . J, compute:
? E-step: Compute hjn (yt ) by (2).
? M-step: Compute
?P
??1
Pk?1
T
Rk (yt )
j
T
X
h
(y
)
log
t
1
t=1 n
j=1
Rj (yt )
j
?
=
?n+1
hj (yt ) ; mjn+1 = ?
PT
j
T t=1 n
t=1 hn (yt )(k ? 1)
!
T
T
X
X
j
j
j
j
mjn
?n+1 = log
hn (yt )(k ? 1) ? log V (mn )
hn (yt )Rk (yt )
t=1
(4)
(5)
t=1
Until convergence of ?n , that is, when ||?n+1 ? ?n ||2 < , for a certain small value .
In all the experiments we use the initialization ?0j = 1/J, ?0j = 0, and mj0 = j, for all j = 1, . . . , J.
The distances are normalized so that the maximum distance is 1. The embedding dimension in all
the experiments on synthetic data is 3, although the results were found to be consistent when we
increased the embedding dimension.
The first experiment consists of a mixture of a Swiss roll manifold (700 points) and a line (700
points) embedded in a three dimensional space. The algorithm (with J = 2 and k = 10) is able
to separate both manifolds. The estimated parameters are collected in Table 1. For each table,
we display the estimated dimension m, density ?, and mixture coefficient ? for each one of the
classes. We also show the percentage of points of each manifold that are classified in each class
(after thresholding the soft assignment). Figure 1(a) displays both manifolds ? each point is colored
according to the probability of belonging to each one of the two possible classes. Tables 1(a) and
1(c) contain the results for both PMM and LD+IB using J = 2. Table 1(b) shows the results for
the PMM algorithm with k = 10 and J = 3. Note how the parameters of the first two classes are
quite similar to the ones obtained with J = 2, and the third class is marginal (very small ?). Figure
1(b) shows the PMM classification when J = 3. Note that all the points of the line belong to the
class of dimension 1. The points of the Swiss roll are mainly concentrated in the other class with
dimension 2. A slight amount of Swiss roll points belong to a third class with roughly the same
dimension as the second class. Actually, these points are located in the point cloud boundaries,
where the underlying assumptions are not always valid.
If we estimate the dimension of the mixture using the LB technique with k = 10, we obtain 1.70 with
a standard deviation of 5.31. If we use the method proposed by Costa and Hero [4], the estimated
dimension is 2. In both cases, the estimated intrinsic dimension is the largest one present in the
mixture, ignoring that the data actually lives in two manifolds of different intrinsic dimension.
The same Table and Figure, second rows, show results for noisy data. We add to the point coordinates Gaussian noise with ? = 0.6. The results obtained with k = 10 are displayed in Tables 1(d),
1(e) and 1(f), and in Figures 1(d), 1(e) and 1(f). Note how the classification still separates the two
different manifolds, although the line is much more affected by the noise and it does not look like a
one dimensional manifold anymore. This is reflected also by the estimated dimension which now is
bigger. This phenomena is related to the scale of observation and to the level of noise. If the level
of noise is large ? e.g., compared to the mean distance to the k nearest neighbors for a small k ?
intuitively the estimated intrinsic dimension will be closer to the embedding dimension (this behavior was experimentally verified). We can again compare the results with the ones obtained with the
LB estimator alone: Estimated dimension 2.71 and standard deviation 1.12. Using Costa and Hero
[4], the estimated dimension varies between 2 and 3 (depending on the number of bootstrap loops).
Both techniques do not consider the possibility of mixed dimensionality.
The experiment in Figure 2 illustrates how the soft clustering is done according to both dimensionality and density. The data consists of 2500 points on the Swiss roll, 100 on a line with high density
Estimated parameters
m
1.00
2.01
?
5.70
2.48
?
0.5000 0.5000
% points in each class
Line
100
0
SR
0
100
Estimated parameters
1.00
2.01
2.16
5.70
2.55
1.52
0.5000 0.4792 0.0208
% points in each class
Line
100
0
0
SR
0
96.57
3.43
Estimated dimension
m
1.67
2.00
% points in each class
Line
100
0
Swiss roll
3.45 96.55
(a) PMM (J = 2).
(b) PMM (J = 3).
(c) LD+IB (J = 2).
Estimated parameters
m
3.02
2.38
?
7.69
2.73
?
0.4951 0.5049
% points in each class
Line
98.14
1.86
SR
0.86
99.14
Estimated parameters
3.01
2.40
2.26
7.70
2.88
1.72
0.4910 0.4766 0.0325
% points in each class
Line
97.71
2.29
0
SR
0.71
93.00
6.29
Estimated dimension
m
3.09
2.30
% points in each class
Line
79.71 20.29
Swiss roll
24.71 75.29
(d) PMM (J = 2).
(e) PMM (J = 3).
(f) LD+IB (J = 2).
m
?
?
m
?
?
Table 1: Clustering results for the Swiss roll (SR) and a line (k = 10), without noise (first row) and
with noise (second row).
(a) PMM (J = 2)
(b) PMM (J = 3)
(c) LD+IB (J = 2)
(d) PMM (J = 2)
(e) PMM (J = 3)
(f) LD+IB (J = 2)
Figure 1: Clustering of a line and a Swiss roll (k = 10). First row without noise, second row with
Gaussian noise (? = 0.6). Points colored according to the probability of belonging to each class.
and 50 on another less dense line. We have set J = 4 and the algorithm gives an ?empty class,?
thus discovering that three classes, with correct dimensionality and density, is enough for a good
representation. The only errors are in the borders, as expected.
m
?
?
Line
Line (dense)
Swiss Roll
Estimated parameters
1.94
1.04
0.98
7.12
3.82
2.66
0.9330 0.0498 0.0167
% points in each class
0.0
15.69
84.31
0.0
99.00
1.00
98.92
1.08
0.0
1.93
2.57
0.0004
0.0
0.0
0.0
Figure 2: Clustering with mixed dimensions and density (k = 20, J = 4).
In order to test the algorithm with real data, we first work with the MNIST database of handwritten
digits,3 which has a test set of 10.000 examples. Each digit is an image of 28 ? 28 pixels and we
treat the data as 784-dimensional vectors.
3
http://yann.lecun.com/exdb/mnist/
We study the mixture of digits one and two and apply PMM and LD+IB with J = 2 and k = 10.
The results are shown in Figure 3. Note how the digits are well separated.4 The LB estimator alone
gives dimensions 9.13 for digits one, 13.02 for digits two, and 11.26 for the mixture of both digits.
The Costa and Hero?s method, [4], gives 8, 11 and 9 respectively. Both methods assume a single
intrinsic dimension and give an average of the dimensions of the underlying manifolds.
Estimated parameters
m
8.50
12.82
?
11.20
6.80
?
0.4901 0.5099
% points in each class
Ones
93.48
6.52
Twos
0
100
Estimated dimension
m
9.17
13.74
% points in each class
Ones
94.71
5.29
Twos
9.08
90.02
(a) PMM
(b) LD+IB
(c) Some image examples.
Figure 3: Results for digits 1 and 2 (k = 10, J = 2).
Next, we experiment with 9-dimensional vectors formed of image patches of 3 ? 3 pixels. If we
impose J = 3 and use PMM, we obtain the results in Figure 4. Notice how roughly one class
corresponds to patches in homogeneous zones (approximately constant gray value), a second class
corresponds to textured zones and a third class to patches containing edges. The estimated dimensions in each region are in accordance to the estimated dimensions using Isomap or Costa and Hero?s
technique in each region after separation. This experiment is just a proof of concept, in the future
we will study how to adapt this clustering approach to image segmentation.
Figure 4: Clustering of image patches of 3 ? 3 pixels with PMM, colors indicating the different
classes (complexity) (J = 3, k = 30). Left: original and segmented images of a house. Right:
original and segmented images of a portion of biological tissue. Adding spatial regularization is the
subject of current research.
Finally, as an additional proof of the validity of our approach and its potential applications, we
use the PMM framework to separate activities in video, Figure 5 (see also [14]). Each original
frame is 480 ? 640, sub-sampled to 48 ? 64 pixels, with 1673 frames. Four classes are present:
standing, walking, jumping, and arms waving. The whole run took 361 seconds in Matlab, while
the classification time (PMM) can be neglected compared to the kNN component.
Samples in each cluster
C1
C2
C3
Standing
416
0
95
Walking
0
429
69
Waving
0
5
423
Jumping
0
18
0
C4
0
25
4
189
Figure 5: Classifying human activities in video (k = 10, J = 4). Four sample frames are shown
followed by the classification results (confusion matrix). Visual analysis of the wrongly classified
frames show that these are indeed very similar to the misclassified class members. Adding features,
e.g., optical flow, will improve the results.
4
Since the clustering is done according to dimensionality and density, digits which share these characteristics won?t be separated into different classes.
5
Conclusions
In this paper we discussed the concept of ?stratification learning,? where the point cloud data is not
assumed to belong to a single manifold, as commonly done in manifold learning and dimensionality
reduction. We extended the work in [10] in the sense that the maximum likelihood is computed
once for the whole dataset, and the probability density function is a mixture of Poisson laws, each
one modeling different intrinsic dimensions and densities. The soft clustering and the estimation
are simultaneously computed. This framework has been contrasted with a more standard two-steps
approach, a combination of the local estimator introduced in [10] with the Information Bottleneck
clustering technique [17]. In both methods we need to compute a kNN-graph which is precisely the
computationally more expensive part. The mixture of Poisson estimators is faster than the two-steps
approach one, it uses an EM algorithm, linear in the number of classes and observations, which
converges in a few iterations.
The mixture of Poisson model is not only clustering according to dimensionality, but to density as
well. The introduction of additional observations and estimates can also help to separate points that
although have the same dimensionality and density, belong to different manifolds. We would also
like to study the use of ellipsoids instead of balls in the counting process in order to better follow
the geometry of the intrinsic manifolds. Another aspect to study is the use of metrics more adapted
to the nature of the data instead of the Euclidean distance. At the theoretical level, the bias of the
PMM model needs to be studied. Results in these directions will be reported elsewhere.
Acknowledgments: This work has been supported by ONR, DARPA, NSF, NGA, and the McKnight Foundation. We thank Prof. Persi Diaconis and Prof. Ren?e Vidal for important feedback and comments. We also
thank Pablo Arias and J?er?emie Jakubowicz for their help. GR was on sabbatical from the Universidad de la
Republica, Uruguay, while performing this work.
References
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2,222 | 3,016 | Dynamic Foreground/Background Extraction from
Images and Videos using Random Patches
Le Lu?
Integrated Data Systems Department
Siemens Corporate Research
Princeton, NJ 08540
[email protected]
Gregory Hager
Department of Computer Science
Johns Hopkins University
Baltimore, MD 21218
[email protected]
Abstract
In this paper, we propose a novel exemplar-based approach to extract dynamic
foreground regions from a changing background within a collection of images
or a video sequence. By using image segmentation as a pre-processing step, we
convert this traditional pixel-wise labeling problem into a lower-dimensional supervised, binary labeling procedure on image segments. Our approach consists of
three steps. First, a set of random image patches are spatially and adaptively sampled within each segment. Second, these sets of extracted samples are formed into
two ?bags of patches? to model the foreground/background appearance, respectively. We perform a novel bidirectional consistency check between new patches
from incoming frames and current ?bags of patches? to reject outliers, control
model rigidity and make the model adaptive to new observations. Within each
bag, image patches are further partitioned and resampled to create an evolving
appearance model. Finally, the foreground/background decision over segments
in an image is formulated using an aggregation function defined on the similarity measurements of sampled patches relative to the foreground and background
models. The essence of the algorithm is conceptually simple and can be easily implemented within a few hundred lines of Matlab code. We evaluate and validate
the proposed approach by extensive real examples of the object-level image mapping and tracking within a variety of challenging environments. We also show that
it is straightforward to apply our problem formulation on non-rigid object tracking
with difficult surveillance videos.
1 Introduction
In this paper, we study the problem of object-level figure/ground segmentation in images and video
sequences. The core problem can be defined as follows: Given an image X with known figure/ground
labels L, infer the figure/ground labels L? of a new image X? closely related to X. For example, we
may want to extract a walking person in an image using the figure/ground mask of the same person
in another image of the same sequence. Our approach is based on training a classifier from the
appearance of a pixel and its surrounding context (i.e., an image patch centered at the pixel) to
recognize other similar pixels across images. To apply this process to a video sequence, we also
evolve the appearance model over time.
A key element of our approach is the use of a prior segmentation to reduce the complexity of the
segmentation process. As argued in [22], image segments are a more natural primitive for image
modeling than pixels. More specifically, an image segmentation provides a natural dimensional
reduction from the spatial resolution of the image to a much smaller set of spatially compact and
relatively homogeneous regions. We can then focus on representing the appearance characteristics
?
The work has been done while the first author was a graduate student in Johns Hopkins University.
of these regions. Borrowing a term from [22], we can think of each region as a ?superpixel? which
represents a complex connected spatial region of the image using a rich set of derived image features. We can then consider how to classify each superpixel (i.e. image segment) as foreground or
background, and then project this classification back into the original image to create the pixel-level
foreground-background segmentation we are interested in.
The original superpixel representation in [22, 19, 18] is a feature vector created from the image segment?s color histogram [19], filter bank responses [22], oriented energy [18] and contourness [18].
These features are effective for image segmentation [18], or finding perceptually important boundaries from segmentation by supervised training [22]. However, as shown in [17], those parameters
do not work well for matching different classes of image regions from different images. Instead,
we propose using a set of spatially randomly sampled image patches as a non-parametric, statistical
superpixel representation. This non-parametric ?bag of patches? model1 can be easily and robustly
evolved with the spatial-temporal appearance information from video, while maintaining the model
size (the number of image patches per bag) using adaptive sampling. Foreground/background classification is then posed as the problem of matching sets of random patches from the image with
these models. Our major contributions are demonstrating the effectiveness and computational simplicity of a nonparametric random patch representation for semantically labelling superpixels and
a novel bidirectional consistency check and resampling strategy for robust foreground/background
appearance adaptation over time.
(a)
(b)
(c)
(d)
Figure 1: (a) An example indoor image, (b) the segmentation result using [6] coded in random colors, (c)
the boundary pixels between segments shown in red, the image segments associated with the foreground, a
walking person here, shown in blue, (d) the associated foreground/background mask. Notice that the color
in (a) is not very saturated. This is a common fact in our indoor experiments without any specific lighting
controls.
We organize the paper as follows. We first address several image patch based representations and
the associated matching methods are described. In section 3, the algorithm used in our approach is
presented with details. We demonstrate the validity of the proposed approach using experiments on
real examples of the object-level figure/ground image mapping and non-rigid object tracking under
dynamic conditions from videos of different resolutions in section 4. Finally, we summarize the
contributions of the paper and discuss possible extensions and improvements.
2 Image Patch Representation and Matching
Building stable appearance representations of images patches is fundamental to our approach. There
are many derived features that can be used to represent the appearance of an image patch. In this
paper, we evaluate our algorithm based on: 1) an image patch?s raw RGB intensity vector, 2) mean
color vector, 3) color + texture descriptor (filter bank response or Haralick feature [17]), and 4) PCA,
LDA and NDA (Nonparametric Discriminant Analysis) features [7, 3] on the raw RGB vectors. For
completeness, we give a brief description of each of these techniques.
Texture descriptors: To compute texture descriptions, we first apply the Leung-Malik (LM) filter
bank [13] which consists of 48 isotropic and anisotropic filters with 6 directions, 3 scales and 2
phases. Thus each image patch is represented by a 48 component feature vector. The Haralick
texture descriptor [10] was used for image classification in [17]. Haralick features are derived from
the Gray Level Co-occurrence Matrix, which is a tabulation of how often different combinations
of pixel brightness values (grey levels) occur in an image region. We selected 5 out of 14 texture
1
Highly distinctive local features [16] are not the adequate substitutes for image patches. Their spatial
sparseness nature limits their representativity within each individual image segment, especially for the nonrigid,
nonstructural and flexible foreground/background appearance.
descriptors [10] including dissimilarity, Angular Second Moment (ASM), mean, standard deviation
(STD) and correlation. For details, refer to [10, 17].
Dimension reduction representations: The Principal Component Analysis (PCA) algorithm is
used to reduce the dimensionality of the raw color intensity vectors of image patches. PCA makes
no prior assumptions about the labels of data. However, recall that we construct the ?bag of patches?
appearance model from sets of labelled image patches. This supervised information can be used to
project the bags of patches into a subspace where they are best separated using Linear discriminant Analysis (LDA) or Nonparametric Discriminant Analysis (NDA) algorithm [7, 3] by assuming
Gaussian or Non-Gaussian class-specific distributions.
Patch matching: After image patches are represented using one of the above methods, we must
match them against the foreground/background models. There are 2 methods investigated in this
paper: the nearest neighbor matching using Euclidean distance and KDE (Kernel Density Estimation) [12] in PCA/NDA subspaces. For nearest-neighbor matching, we find, for each patch p, its
B
F
F
nearest neighbors pF
n , pn in foreground/background bags, and then compute dp =k p ? pn k,
B
B
F
B
dp =k p ? pn k. On the other hand, an image patch?s matching scores mp and mp are evaluated
as probability density values from the KDE functions KDE(p, ?F ) and KDE(p, ?B ) where ?F |B
are bags of patch models. Then the segmentation-level classification is performed as section 3.2.
3 Algorithms
We briefly summarize our labeling algorithm as follows. We assume that each image of interest
has been segmented into spatial regions. A set of random image patches are spatially and adaptively sampled within each segment. These sets of extracted samples are formed into two ?bags of
patches? to model the foreground/background appearance respectively. The foreground/background
decision for any segment in a new image is computed using one of two aggregation functions on
the appearance similarities from its inside image patches to the foreground and background models.
Finally, for videos, within each bag, new patches from new frames are integrated through a robust
bidirectional consistency check process and all image patches are then partitioned and resampled to
create an evolving appearance model. As described below, this process prune classification inaccuracies in the nonparametric image patch representations and adapts them towards current changes in
foreground/background appearances for videos. We describe each of these steps for video tracking
of foreground/background segments in more detail below, and for image matching, which we treat
as a special case by simply omitting step 3 and 4 in Figure 2.
Non-parametric Patch Appearance Modelling-Matching Algorithm
inputs: Pre-segmented Images Xt , t = 1, 2, ..., T ; Label L1
F |B
outputs: Labels Lt , t = 2, ..., T ; 2 ?bags of patches? appearance model for foreground/background ?T
1. Sample segmentation-adaptive random image patches {P1 } from image X1 .
F |B
2. Construct 2 new bags of patches ?1 for foreground/background using patches {P1 } and label
L1 ; set t = 1.
3. t = t + 1; Sample segmentation-adaptive random image patches {Pt } from image Xt ; match
F |B
{Pt } with ?t?1 and classify segments of Xt to generate label Lt by aggregation.
4. Classify and reject ambiguous patch samples, probable outliers and redundant appearance patch
F |B
samples from new extracted image patches {Pt } against ?t?1 ; Then integrate the filtered {Pt }
F |B
F |B
into ?t?1 and evaluate the probability of survival ps for each patch inside ?t?1 against the
original unprocessed {Pt } (Bidirectional Consistency Check).
5. Perform the random partition and resampling process according to the normalized product of
F |B
F |B
probability of survival ps and partition-wise sampling rate ? ? inside ?t?1 to generate ?t .
F |B
6. If t = T , output Lt , t = 2, ..., T and ?T
; exit. If t < T , go to (3).
Figure 2: Non-parametric Patch Appearance Modelling-Matching Algorithm
Figure 3: Left: Segment adaptive random patch sampling from an image with known figure/ground labels.
Green dots are samples for background; dark brown dots are samples for foreground. Right: Segment adaptive
random patch sampling from a new image for figure/ground classification, shown as blue dots.
3.1 Sample Random Image Patches
We first employ an image segmentation algorithm2 [6] to pre-segment all the images or video frames
in our experiments. A typical segmentation result is shown in Figure 1. We use Xt , t = 1, 2, ..., T
to represent a sequence of video frames. Given an image segment, we formulate its representation
as a distribution on the appearance variation over all possible extracted image patches inside the
segment. To keep this representation to a manageable size, we approximate this distribution by
sampling a random subset of patches.
We denote an image segment as Si with SiF for a foreground segment, and SiB for a background
segment, where i is the index of the (foreground/background)image segment within an image. Accordingly, Pi , PiF and PiB represent a set of random image patches sampled from Si , SiF and SiB
respectively. The cardinality Ni of an image segment Si generated by [6] typically ranges from 50
to thousands. However small or large superpixels are expected to have roughly the same amount
of uniformity. Therefore the sampling rate ?i of Si , defined as ?i = size(Pi )/Ni , should decrease
with increasing Ni . For simplicity, we keep ?i as a constant for all superpixels, unless Ni is above a
predefined threshold ? , (typically 2500 ? 3000), above which size(Pi ) is held fixed. This sampling
adaptivity is illustrated in Figure 3. Notice that large image segments have much more sparsely
sampled patches than small image segments. From our experiments, this adaptive spatial sampling
strategy is sufficient to represent image segments of different sizes.
3.2 Label Segments by Aggregating Over Random Patches
For an image segment Si from a new frame to be classified, we again first sample a set of random
patches Pi as its representative set of appearance samples. For each patch p ? Pi , we calculate its
B
B
F
distances dF
p , dp or matching scores mp , mp towards the foreground and background appearance
models respectively as described in Section 2.
The decision of assigning Si to foreground or background, is an aggregating process over all
B
B
F
{dF
p , dp } or {mp ; mp } where p ? Pi . Since Pi is considered as a set of i.i.d. samples of the
B
B
F
appearance distribution of Si , we use the average of {dF
p , dp } or {mp ; mp } (ie. first-order statisB
F
B
F
tics) as its distances DPi , DPi or fitness values MPi , MPi with the foreground/background model.
B
F
B
B
F
In terms of distances {dF
p , dp }, DPi = meanp?Pi (dp ) and DPi = meanp?Pi (dp ). Then the
F
segment?s foreground/background fitness is set as the inverse of the distances: MPFi = 1/DP
i
F
B
F
F
B
B
and MPi = 1/DPi . In terms of KDE matching scores {mp ; mp }, MPi = meanp?Pi (mp ) and
B
F
MPBi = meanp?Pi (mB
p ). Finally, Si is classified as foreground if MPi > MPi , and vice versa. The
Median robust operator can also be employed in our experiments, without noticeable difference in
F
performance. Another choice is to classify each p ? Pi from mB
p and mp , then vote the majority
foreground/background decision for Si . The performance is similar with mean and median.
2
Because we are not focused on image segmentation algorithms, we choose Felzenszwalb?s segmentation
code which generates good results and is publicly available at http://people.cs.uchicago.edu/?pff/segment/.
3.3 Construct a Robust Online Nonparametric Foreground/Background Appearance Model
with Temporal Adaptation
From sets of random image patches extracted from superpixels with known figure/ground labels, 2
foreground/background ?bags of patches? are composed. The bags are the non-parametric form of
the foreground/background appearance distributions. When we intend to ?track? the figure/ground
model sequentially though a sequence, these models need to be updated by integrating new image
patches extracted from new video frames. However the size (the number of patches) of the bag will
be unacceptably large if we do not also remove the some redundant information over time. More
importantly, imperfect segmentation results from [6] can cause inaccurate segmentation level figure/ground labels. For robust image patch level appearance modeling of ?t , we propose a novel
bidirectional consistency check and resampling strategy to tackle various noise and labelling uncertainties.
More precisely, we classify new extracted image patches {Pt } as {PtF } or {PtB } according to
F |B
F |B
B
?t?1 ; and reject ambiguous patch samples whose distances dF
p , dp towards respective ?t?1 have
B
no good contrast (simply, the ratio between dF
p and dp falls into the range of 0.8 to 1/0.8). We
further sort the distance list of the newly classified foreground patches {PtF } to ?F
t?1 , filter out
image patches on the top of the list which have too large distances and are probably to be outliers,
and ones from the bottom of the list which have too small distances and contain probably redundant
3
B
B
appearances compared with ?F
t?1 . We perform the same process with {Pt } according to ?t?1 .
F |B
F ? |B ?
Then the filtered {Pt } are integrated into ?t?1 to form ?t?1 , and we evaluate the probability of
F ? |B ?
survival ps for each patch inside ?t?1
against the original unprocessed {Pt } with their labels4 .
F ? |B ?
Next, we cluster all image patches of ?t?1 into k partitions [8], and randomly resample image
patches within each partition. This is roughly equivalent to finding the modes of an arbitrary distribution and sampling for each mode. Ideally, the resampling rate ? ? should decrease with increasing
partition size, similar to the segment-wise sampling rate ?. For simplicity, we define ? ? as a constant
value for all partitions, unless setting a threshold ? ? to be the minimal required size5 of partitions after resampling. If we perform resampling directly over patches without partitioning, some modes of
the appearance distribution may be mistakenly removed. This strategy represents all partitions with
sufficient number of image patches, regardless of their different sizes. In all, we resample image
F |B
patches of ?t?1 , according to the normalized product of probability of survival ps and partitionF |B
wise sampling rate ? ? , to generate ?t . By approximately fixing the expected bag model size, the
number of image patches extracted from a certain frame Xt in the bag decays exponentially in time.
The problem of partitioning image patches in the bag can be formulated as the NP-hard k-center
problem. The definition of k-center is as follows: given a data set of n points and a predefined
cluster number k, find a partition of the points into k subgroups P1 , P2 , ..., Pk and the data centers c1 , c2 , ..., ck , to minimize the maximum radius of clusters maxi maxp?Pi k p ? ci k, where i
is the index of clusters. Gonzalez [8] proposed an efficient greedy algorithm, farthest-point clustering, which proved to give an approximation factor of 2 of the optimum. The algorithm operates as follows: pick a random point p1 as the first cluster center and add it to the center set C;
for iterations i = 2, ..., k, find the point pi with the farthest distance to the current center set C:
di (pi , C) = minc?C k pi ? c k and add pi to set C; finally assign data points to its nearest center and recompute the means of clusters in C. Compared with the popular k-means algorithm, this
algorithm is computationally efficient and theoretically bounded6. In this paper, we employ the Eu3
Simply, we reject patches with distances dF
that are larger than mean(dF
) + ? ? std(dF
) or smaller
pF
pF
pF
t
t
t
F |B
than mean(dF
) ? ? ? std(dF
) where ? controls the range of accepting patch samples of ?t?1 , called model
pF
pF
t
t
rigidity.
?
4
F
For example, we compute the distance of each patch in ?F
t?1 to {Pt }, and covert them as surviving
probabilities using a exponential function over negative covariance normalized distances. Patches with smaller
distances have higher survival chances during resampling; and vice versa. We perform the same process with
?
B
?B
t?1 according to {Pt }.
5
All image patches from partitions that are already smaller than ? ? are kept during resampling.
6
The random initialization of all k centers and the local iterative smoothing process in k-means, which is
time-consuming in high dimensional space and possibly converges to undesirable local minimum, are avoided.
clidean distance between an image patch and a cluster center, using the raw RGB intensity vector or
the feature representations discussed in section 2.
4 Experiments
We have evaluated the image patch representations described in Section 2 for figure/ground mapping
between pairs of image on video sequences taken with both static and moving cameras. Here we
summarize our results.
4.1 Evaluation on Object-level Figure/Ground Image Mapping
We first evaluate our algorithm on object-level figure/ground mapping between pairs of images under
eight configurations of different image patch representations and matching criteria. They are listed
as follows: the nearest neighbor distance matching on the image patch?s mean color vector (MCV);
raw color intensity vector of regular patch scanning (RCV) or segment-adaptive patch sampling over
image (SCV); color + filter bank response (CFB); color + Haralick texture descriptor (CHA); PCA
feature vector (PCA); NDA feature vector (NDA) and kernel density evaluation on PCA features
(KDE). In general, 8000 ? 12000 random patches are sampled per image. There is no apparent
difference on classification accuracy for the patch size ranging from 9 to 15 pixels and the sample
rate from 0.02 to 0.10. The PCA/NDA feature vector has 20 dimensions, and KDE is evaluated on
the first 3 ? 6 PCA features.
Because the foreground figure has fewer of pixels than background, we conservatively measure the
classification accuracy from the foreground?s detection precision and recall on pixels. Precision
is the ratio of the number of correctly detected foreground pixels to the total number of detected
foreground pixels; recall is is the ratio of the number of correctly detected foreground pixels to the
total number of foreground pixels in the image. The patch size is 11 by 11 pixels, and the segmentwise patch sampling rate ? is fixed as 0.06, unless stated otherwise. Using 40 pairs of (720 ?
480) images with the labelled figure/ground segmentation, we compare their average classification
accuracies in Tables 1.
MCV
0.46
0.28
RCV
0.81
0.89
SCV
0.97
0.95
CFB
0.92
0.85
CHA
0.89
0.81
PCA
0.93
0.85
NDA
0.96
0.87
KDE
0.69
0.98
Table 1: Evaluation on classification accuracy (ratio). The first row is precision; the second row is
recall.
For figure/ground extraction accuracy, SCV has the best classification ratio using the raw color intensity vector without any dimension reduction. MCV has the worst accuracy, which shows that pixelcolor leads to poor separability between figure and ground in our data set. Four feature based representations, CFB, CHA, PCA, NDA with reduced dimensions, have similar performance, whereas
NDA is slightly better than the others. KDE tends to be more biased towards the foreground class
because background usually has a wider, flatter density distribution. The superiority of SCV over
RCV proves that our segment-wise random patch sampling strategy is more effective at classifying
image segments than regularly scanning the image, even with more samples. As shown in Figure 4
(b), some small or irregularly-shaped image segments do not have enough patch samples to produce
stable classifications.
(a) MCV
(b) RCV
(c) SCV
(d) CFB
(e) CHA
(f) PCA
(g) NDA
(h) KDE
Figure 4: An example of evaluation on object-level figure/ground image mapping. The labeled figure image
segments are coded in blue.
4.2 Figure/Ground Segmentation Tracking with a Moving Camera
From Figure 4 (h), we see KDE tends to produce some false positives for the foreground. However
the problem can be effectively tackled by multiplying the appearance KDE by the spatial prior
which is also formulated as a KDE function of image patch coordinates. By considering videos with
complex appearance-changing figure/ground, imperfect segmentation results [6] are not completely
avoidable which can cause superpixel based figure/ground labelling errors. However our robust
bidirectional consistency check and resampling strategy, as shown below, enables to successfully
track the dynamic figure/ground segmentations in challenging scenarios with outlier rejection, model
rigidity control and temporal adaptation (as described in section 3.3).
Karsten.avi shows a person walking in an uncontrolled indoor environment while tracked with a
handheld camera. After we manually label the frame 1, the foreground/background appearance
model starts to develop, classify new frames and get updated online. Eight Example tracking frames
are shown in Figure 5. Notice that the significant non-rigid deformations and large scale changes of
the walking person, while the original background is completely substituted after the subject turned
his way. In frame 258, we manually eliminate some false positives of the figure. The reason for
this failure is that some image regions which were behind the subject begin to appear when the
person is walking from left to the center of image (starting from frame 220). Compared to the online
foreground/background appearance models by then, these newly appearing image regions have quite
different appearance from both the foreground and the background. Therefore the foreground?s
spatial prior dominates the classification. We leave this issue for future work.
(a) 12#
(b) 91#
(c) 155#
(d) 180#
(e) 221#
(f) 257#
(g) 308#
(h) 329#
Figure 5: Eight example frames (720 by 480 pixels) from the video sequence Karsten.avi of 330 frames. The
video is captured using a handheld Panasonic PV-GS120 in standard NTSC format. Notice that the significant non-rigid deformations and large scale changes of the walking person, while the original background is
completely substituted after the subject turned his way. The red pixels are on the boundary of segments; the
tracked image segments associated with the foreground walking person is coded in blue.
4.3 Non-rigid Object Tracking from Surveillance Videos
We can also apply our nonparametric treatment of dynamic random patches in Figure 2 into tracking non-rigid interested objects from surveillance videos. The difficult is that surveillance cameras
normally capture small non-rigid figures, such as a walking person or running car, in low contrast
and low resolution format. Thus to adapt our method to solve this problem, we make the following modifications. Because our task changes to localizing figure object automatically overtime, we
can simply model figure/ground regions using rectangles and therefore no pre-segmentation [6] is
needed. Random figure/ground patches are then extracted from the image regions within these two
rectangles. Using two sets of random image patches, we train an online classifier for figure/ground
classes at each time step, generate a figure appearance confidence map of classification for the next
frame and, similarly to [1], apply mean shift [4] to find the next object location by mode seeking.
In our problem solution, the temporal evolution of dynamic image patch appearance models are
executed by the bidirectional consistency check and resampling described in section 3.3. Whereas
[1] uses boosting for both temporal appearance model updating and classification, our online binary classification training can employ any off-the-shelf classifiers, such as k-Nearest Neighbors
(KNN), support vector machine (SVM). Our results are favorably competitive to the state-of-the-art
algorithms [1, 9], even under more challenging scenario.
5 Conclusion and Discussion
Although quite simple both conceptually and computationally, our algorithm of performing dynamic foreground-background extraction in images and videos using non-parametric appearance
models produces very promising and reliable results in a wide variety of circumstances. For tracking figure/ground segments, to our best knowledge, it is the first attempt to solve this difficult ?video
matting? problem [15, 25] by robust and automatic learning. For surveillance video tracking, our
results are very competitive with the state-of-art [1, 9] under even more challenging conditions.
Our approach does not depend on an image segmentation algorithm that totally respects the boundaries of the foreground object. Our novel bidirectional consistency check and resampling process
has been demonstrated to be effectively robust and adaptive. We leave the explorations on supervised dimension reduction and density modeling techniques on image patch sets, optimal random
patch sampling strategy, and self-tuned optimal image patch size searching as our future work.
In this paper, we extract foreground/background by classifying on individual image segments. It
might improve the figure/ground segmentation accuracy by modeling their spatial pairwise relationships as well. This problem can be further solved using generative or discriminative random field
(MRF/DRF) model or the boosting method on logistic classifiers [11]. In this paper, we focus on
learning binary dynamic appearance models by assuming figure/ground are somewhat distributionwise separatable. Other cues, as object shape regularization and motion dynamics for tracking, can
be combined to improve performance.
References
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[2] Y. Boykov and M. Jolly, Interactive Graph Cuts for Optimal boundary and Region Segmentation of Objects
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[3] M. Bressan and J. Vitri`
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Color, and Texture Cues, IEEE Trans. on PAMI, 26(5):530-549, May 2004.
[20] A. Mittal and N. Paragios, Motion-based Background Substraction using Adaptive Kernel Density Estimation, CVPR, 2004.
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ECCV, 2006.
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2,223 | 3,017 | Bayesian Detection of Infrequent Differences in
Sets of Time Series with Shared Structure
Jennifer Listgarten? , Radford M. Neal? , Sam T. Roweis? Rachel Puckrin? and Sean Cutler?
?
Department of Computer Science, ? Department of Botany,
University of Toronto, Toronto, Ontario, M5S 3G4
{jenn,radford,roweis}@cs.toronto.edu
rachel [email protected], [email protected]
Abstract
We present a hierarchical Bayesian model for sets of related, but different, classes
of time series data. Our model performs alignment simultaneously across all
classes, while detecting and characterizing class-specific differences. During inference the model produces, for each class, a distribution over a canonical representation of the class. These class-specific canonical representations are automatically aligned to one another ? preserving common sub-structures, and highlighting differences. We apply our model to compare and contrast solenoid valve
current data, and also, liquid-chromatography-ultraviolet-diode array data from a
study of the plant Arabidopsis thaliana.
1
Aligning Time Series From Different Classes
Many practical problems over a wide range of domains require synthesizing information from several noisy examples of one or more categories in order to build a model which captures common structure and also learns the patterns of variability between categories. In time series analysis, these modeling goals manifest themselves in the tasks of alignment and difference detection.
These tasks have diverse applicability, spanning speech & music processing, equipment & industrial
plant diagnosis/monitoring, and analysis of biological time series such as microarray & liquid/gas
chromatography-based laboratory data (including mass spectrometry and ultraviolet diode arrays).
Although alignment and difference detection have been extensively studied as separate problems in
the signal processing and statistical pattern recognition communities, to our knowledge, no existing
model performs both tasks in a unified way. Single class alignment algorithms attempt to align a
set of time series all together, assuming that variability across different time series is attributable
purely to noise. In many real-world situations, however, we have time series from multiple classes
(categories) and our prior belief is that there is both substantial shared structure between the class
distributions and, simultaneously, systematic (although often rare) differences between them. While
in some circumstances (if differences are small and infrequent), single class alignment can be applied
to multi-class data, it is much more desirable to have a model which performs true multi-class
alignment in a principled way, allowing for more refined and accurate modeling of the data.
In this paper, we introduce a novel hierarchical Bayesian model which simultaneously solves the
multi-class alignment and difference detection tasks in a unified manner, as illustrated in Figure
1. The single-class alignment shown in this figure coerces the feature in region A for class 1 to
be inappropriately collapsed in time, and the overall width of the main broad peak in class 2 to
be inappropriately narrowed. In contrast, our multi-class model handles these features correctly.
Furthermore, because our algorithm does inference for a fully probabilistic model, we are able to
obtain quantitative measures of the posterior uncertainty in our results, which, unlike the point estimates produced by most current approaches, allow us to assess our relative confidence in differences
learned by the model. Our basic setup for multi-class alignment assumes the class labels are known
for each time series, as is the case for most difference detection problems. However, as we discuss
at the end of the paper, our model can be extended to the completely unsupervised case.
normal
abnormal
3
common structure
3
1
1
20
0
3
?20
class?specific differences
20
1
0
?20
3
class?specific models
3
A
1
1
0
50
100
150
200
250
0
50
100
150
200
Figure 1: Nine time series from the NASA valve solenoid current data set [4]. Four belong to a
?normal? class, and five to an ?abnormal? class. On all figures, the horizontal axis is time, or latent
time for figures of latent traces and observed time series aligned to latent traces. The vertical axis
is current amplitude. Top left: The raw, unaligned data. Middle left: Average of the unaligned
data within each class in thick line, with the thin lines showing one standard deviation on either
side. Bottom left: Average of the aligned data (over MCMC samples) within each class, using the
single-class alignment version of the model (no child traces), again with one standard deviation lines
shown in the thinner style line. Right: Mean and one standard deviation over MCMC samples using
the HB-CPM. Top right: Parent trace. Middle right: Class-specific energy impulses with the topmost showing the class impulses for the less smooth class. Bottom right: Child traces superimposed.
Note that if one generates more HB-CPM MCMC samples, the parent cycles between the two classes
since the model has no preference for which class is seen as a modification of the other; the child
classes remain stable however.
2
A Hierarchical Bayesian Continuous Profile Model
Building on our previous Continuous Profile Model (CPM) [7], we propose a Hierarchical Bayesian
Continuous Profile Model (HB-CPM) to address the problems of multi-class alignment and difference detection, together, for sets of sibling time series data ? that is, replicate time series from
several distinct, but related classes. The HB-CPM is a generative model that allows simultaneous
alignment of time series and also provides aligned canonical representations of each class along with
measures of uncertainty on these representations. Inference in the model can be used, for example, to
detect and quantify similarities and differences in class composition. The HB-CPM extends the basic
CPM in two significant ways: i) it addresses the multi-class rather than the single-class alignment
problem, and ii) it uses a fully Bayesian framework rather than a maximum likelihood approach,
allowing us to estimate uncertainty in both the alignments and the canonical representations.
Our model, depicted in Figure 2, assumes that each observed time series is generated as a noisy
transformation of a single, class-specific latent trace. Each latent trace is an underlying, noiseless
representation of the set of replicated, observable time series belonging to a single class. An observed time series is generated from this latent trace exactly as in the original CPM, by moving
through a sequence of hidden states in a Markovian manner and emitting an observable value at
each step, as with an HMM. Each hidden state corresponds to a ?latent time? in the latent trace. Thus
different choices of hidden state sequences result in different nonlinear transformations of the underlying trace. The HB-CPM uses a separate latent trace for each class, which we call child traces.
Crucially, each of these child traces is generated from a single parent trace (also unobserved), which
250
captures the common structure among all of the classes. The joint prior distribution for the child
traces in the HB-CPM model can be realized by first sampling a parent trace, and then, for each
class, sampling a sparse ?difference vector? which dictates how and where each child trace should
differ from the common parent.
z
Figure 2: Core
elements
of
the
HB-CPM,
illustrated with
two-class data
(hidden
and
observed) drawn
from the model?s
prior.
parent
r1
r2
z1
impulse
z2
child
trace
x1
x2
class 1 observed time series
2.1
x3
impulse
child
trace
x4
x5
x6
class 2 observed time series
The Prior on Latent Traces
Let the vector xk = (xk1 , xk2 , ..., xkN ) represent the k th observed scalar time series, and w k ? 1..C
be the class label of this time series. Also, let z = (z1 , z2 , ..., zM ) be the parent trace, and
c
z c = (z1c , z2c , ..., zM
) be the child trace for the cth class. During inference, posterior samples of
c
z form a canonical representation of the observed times series in class c, and z contains their common sub-structure. Ideally, the length of the latent traces, M , would be very large relative to N so
that any experimental data could be mapped precisely to the correct underlying trace point. Aside
from the computational impracticalities this would pose, great care to avoid overfitting would have
to be taken. Thus in practice, we have used M = (2 + )N (double the resolution, plus some slack
on each end) in our experiments, and found this to be sufficient with < 0.2. Because the resolution
of the latent traces is higher than that of the observed time series, experimental time can be made to
effectively speed up or slow down by advancing along the latent trace in larger or smaller jumps.
As mentioned previously, the child traces in the HB-CPM inherit most of their structure from a
common parent trace. The differences between child and parent are encoded in a difference vector
for each class, dc = (dc1 , dc2 , ..., dcM ); normally, most elements of dc are close to zero. Child traces
are obtained by adding this difference vector to the parent trace: z c = z + dc .
We model both the parent trace and class-specific difference vectors with what we call an energy
impulse chain, which is an undirected Markov chain in which neighbouring nodes are encouraged to
be similar (i.e., smooth), and where this smoothness is perturbed by a set of marginally independent
energy impulse nodes, with one energy impulse node attached to each node in the chain. For the difc
ference vector of the cth class, the corresponding energy impulses are denoted r c = (r1c , r2c , ..., rM
),
and for the parent trace the energy impulses are denoted r = (r1 , r2 , ..., rM ). Conditioned on the
energy impulses, the probability of a difference vector is
"
!!#
M ?1
M
1
1 X (dci ? dci+1 )2 X (dci ? ric )2
c c
c c
p(d |r , ? , ? ) =
exp ?
+
.
(1)
Zr c
2 i=1
?c
?c
i=1
Here, Zrc is the normalizing constant for this probability density, ?c controls the smoothness of the
chain, and ?c controls the influence of the energy impulses. Together, ?c and ?c also control the
overall tightness of the distribution for dc . Presently, we set all ?c = ?0 , and similarly ?c = ?0 ?
that is, these do not differ between classes. Similarly, the conditional probability of the parent trace
is
"
1
1
p(z|r, ?, ?) =
exp ?
Zr
2
M
?1
X
i=1
M
(zi ? zi+1 )2 X (zi ? ri )2
+
?
?
i=1
!!#
.
(2)
These probability densities are each multivariate Gaussian with tridiagonal precision matrixes (corresponding to the Markov nature of the interactions).
Each component of each energy impulse for the parent, rj , is drawn independently from a single
univariate Gaussian, N (ri |?par , spar ), whose mean and variance are in turn drawn from a Gaussian
and inverse-gamma, respectively. The class-specific difference vector impulses, however, are drawn
from a mixture of two zero-mean Gaussians ? one ?no difference? (inlier) Gaussian, and one ?classdifference? (outlier) Gaussian. The means are zero so as to encourage difference vectors to be near
zero (and thus child traces to be similar to the parent trace). Letting ?ic denote the binary latent
mixture component indicator variables for each rjc ,
c
c
p(?jc ) = Multinomial(?jc |mcin , mcout ) = (mcin )?j (mcout )1??j
N (rjc |0, s2in ), if ?jc = 1
c c
p(rj |?j ) =
.
N (rjc |0, s2out ), if ?jc = 0
(3)
(4)
Each Gaussian mixture variance has an Inverse-Gamma prior, which for the ?no difference? variance,
s2in , is set to have very low mean (and not overly dispersed) so that ?no difference? regions truly have
little difference from the parent class, while for the ?class-difference? variance, s 2out , the prior is
set to have a larger mean, so as to model our belief that substantial class-specific differences do
occasionally exist. The priors for ?c , ?c , ?, ? are each log-normal (inverse-gamma priors would
not be conjugate in this model, so we use log-normals which are easier to specify). Additionally,
the mixing proportions, mcin , mcout , have a Dirichlet prior, which typically encodes our belief that the
proportion that are ?class differences? is likely to be small.
2.2
The HMM Portion of the Model
Each observed xk is modeled as being generated by an HMM conditioned on the appropriate child
k
trace, z w . The probability of an observed time series conditioned on a path of hidden time states,
QN
k
k
? k , and the child trace, is given by p(xk |z w , ? k ) = i=1 N (xki |z?wk uk , ? k ), where ? k is the
i
emission variance for time series k, and the scale factor, uk , allows for constant, global, multiplicak
tive rescaling. The HMM transition probabilities T k (?i?1
? ?ik ) are multinomial within a limited
k
k
range, with p (?i = a|?i?1 = b) = ?(a?b) for (a ? b) ? [1, J? ] and pk (?i = a|?i?1 = b) = 0
for (a ? b) < 1 or (a ? b) > J? where J? is the maximum allowable number of consecutive time
P J? k
states that can be advanced in a single transition. (Of course, i=1
?i = 1.) This multinomial distribution, in turn, has a Dirichlet prior. The HMM emission variances, ? k , have an inverse-gamma
prior. Additionally, the prior over the first hidden time state is a uniform distribution over a constant
number of states, 1..Q, where Q defines how large a shift can exist between any two observed time
series. The prior over each global scaling parameter, uk , is a log-normal with fixed variance and
mean of zero, which encourages the scaling factors to remain near unity.
3
Posterior Inference of Alignments and Parameters by MCMC
Given a set of observed time series (and their associated class labels), the main computational operation to be performed in the HB-CPM is inference of the latent traces, alignment state paths and
other model parameters. Exact inference is analytically intractable, but we are able to use Markov
Chain Monte Carlo (MCMC) methods to create an iterative algorithm which, if run for sufficiently
long, produces samples from the correct posterior distribution. This posterior provides simultaneous
alignments of all observed time series in all classes, and also, crucially, aligned canonical representations of each class, along with error bars on these representations, allowing for a principled
approach to difference detection in time series data from different classes.
We may also wish to obtain a posterior estimate of some of our parameters conditioned on the data,
and marginalized over the other parameters. In particular, we might be interested in obtaining the
posterior over hidden time state vectors for each time series, ? k , which together provide a simultaneous, multi-class alignment of our data. We may, in addition, or, alternatively, be interested in
the posterior of the child traces, z c , which together characterize how the classes agree and disagree.
The former may be more of interest for visualizing aligned observed time series, or in expanding out
aligned scalar time series to a related vector time series, while the latter would be more of interest
when looking to characterize differences in multi-class, scalar time series data.
We group our parameters into blocks, and sample these blocks conditioned on the values of the
other parameters (as in Gibbs sampling) ? however, when certain conditional distributions are not
amenable to direct sampling, we use slice sampling [8]. The scalar conditional distributions for each
of ?par , spar , mcin , mcout , ?jc , ?ki are known distributions, amenable to direct sampling. The conditional
distributions for the scalars ?c , ?c , ?, ? and uk are not tractable, and for each of these we use slice
sampling (doubling out and shrinking).
The conditional distribution for each of r and r c is multivariate Gaussian, and we sample directly
from each using a Cholesky decomposition of the covariance matrix.
1
p(r|z, ?, ?) = p(z|r, ?, ?)p(r) = N (r|c, C)
(5)
Z
1
p(r c |dc , ?c , ?c ) = p(dc |r, ?c , ?c )p(r) = N (r c |b, B),
(6)
Z
where, using I to denote the identity matrix,
?1
?par
z
S
?1
+
Is
+
I
(7)
,
c
=
C
C=
par
?2
?
spar
!?1
S?
dc
?1
B=
,
b=B c.
+ vc
(8)
2
?
(?c )
?1
The diagonal matrix v c consists of mixture component variances (s2in or s2out ). S ?1 [or S ? ] is the
tridiagonal precision matrix of the multivariate normal distribution p(z|r, ?, ?) [or p(d c |r c , ?c , ?c )],
?1
?1
and has entries Sj,j
= ?2 + ?1 for j = 2..(M ? 1), Sj,j
= ?1 + ?1 for j = 1, M , and
?1
?1
?1
Sj,j+1
= Sj+1,j
= ? ?1 [or analogously for S ? ]. The computation of C and B can be made
more efficient by using the Sherman-Morrison-Woodbury matrix inversion lemma. For example,
?1
?1
?1
?1
?1
B = (?1c )2 (S ? ? S ? (v c + S ? )?1 S ? ), and we have S ?1 [or S ? ] almost for free, and
no longer need to invert S [or S ? ] to obtain it.
The conditional distributions of each of z, z c are also multivariate Gaussians. However, because
of the underlying Markov dependencies, their precision matrixes are tridiagonal, and hence we can
use belief propagation, in the style of Kalman filtering, followed by a stochastic traceback to sample
from them efficiently. Thus each can be sampled in time proportional to M rather than M 3 , as
required for a general multivariate Gaussian.
Lastly, to sample from the conditional distribution of the hidden time vectors for each sample, ? k ,
we run belief propagation (analogous to the HMM forward-backward algorithm) followed by a
stochastic traceback.
In our experiments, the parent trace was initialized by averaging one smoothed example from each
class. The child traces were initialized to the initial parent trace. The HMM states were initialized
by a Viterbi decoding with respect to the initial values of the other parameters. The scaling factors
were initialized to unity, and the child energy impulses to zero. MCMC was run for 5000 iterations,
with convergence generally realized in less than 1000 iterations.
4
Experiments and Results
We demonstrate use of the HB-CPM on two data sets. The first data set is the part of the NASA
shuttle valve data [4], which measures valve solenoid current against time for some ?normal? runs
and some ?abnormal? runs. Measurements were taken at a rate of 1ms per sample, with 1000 samples per time series. We subsampled the data by a factor of 7 in time since it was extremely dense.
The results of performing posterior inference in our model on this two-class data set are shown in
Figure 1. They nicely match our intuition of what makes a good solution. In our experiments, we
also compared our model to a simple ?single-class? version of the HB-CPM in which we simply
remove the child trace level of the model, letting all observed data in both classes depend directly
on one single parent trace. The single-class alignment, while doing a reasonable job, does so by coercing the two classes to look more similar than they should. This is evident in one particular region
labeled on the graph and discussed in the legend. Essentially a single class alignment causes us to
lose class-specific fine detail ? the precise information we seek to retain for difference detection.
The second data set is from a botany study which uses reverse-phase HPLC (high performance liquid chromatography) as a high-throughput screening method to identify genes involved in xenobiotic
uptake and metabolism in the model plant Arabidopsis thaliana. Liquid-chromatography (LC) techniques are currently being developed and refined with the aim of providing a robust platform with
which to detect differences in biological organisms ? be they plants, animals or humans. Detected
differences can reveal new fundamental biological insight, or can be applied in more clinical settings. LC-mass spectrometry technology has recently undergone explosive growth in tackling the
problem of biomarker discovery ? for example, detecting biological markers that can predict treatment outcome or severity of disease, thereby providing the potential for improved health care and
better understanding of the mechanisms of drug and disease. In botany, LC-UV data is used to help
understand the uptake and metabolism of compounds in plants by looking for differences across
experimental conditions, and it is this type of data that we use here.
LC separates mixtures of analytes on the basis of some chemical property ? hydrophobicity, for
reverse-phase LC, used to generate our data. Components of the analyte in our data set were detected
as they came off the LC column with a Diode Array Detector (DAD), yielding UV-visible spectra
collected at 540 time points (we used the 280 nm band, which is informative for these experiments).
We performed background subtraction [2] and then subsampled this data by a factor of four. This is a
three-class data set, where the first class is untreated plant extract, followed by two classes consisting
of this same plant treated with compounds that were identified as possessing robust uptake in vivo,
and, hence, when metabolized, provide a differential LC-UV signal of interest.
Figure 3 gives an overview of the LC-UV results, while Figure 4 zooms in on a particular area
of interest to highlight how subtle differences can be detected by the HB-CPM, but not by a singleclass alignment scheme. As with the NASA data set, a single-class alignment coerces features across
classes that are in fact different to look the same, thereby preventing us from detecting them. Recall
that this data set consists of a ?no treatment? plant extract, and two ?treatments? of this same plant.
Though our model was not informed of these special relationships, it nevertheless elegantly captures
this structure by giving almost no energy impulses to the ?no treatment? class, meaning that this class
is essentially the parent trace, and allowing the ?treatment? classes to diverge from it, thereby nicely
matching the reality of the situation.
All averaging over MCMC runs shown is over 4000 samples, after a 1000 burn in period, which
took around 3 hours for the NASA data, and 5 hours for the LC data set, on machines with dual 3
GHz Pentium 4 processors.
5
Related Work
While much work has been done on time series alignment, and on comparison/clustering of time
series, none of this work, to our knowledge, directly addresses the problem presented in this paper
? simultaneously aligning and comparing sets of related time series in order to characterize how
they differ from one another.
The classical algorithm for aligning time series is Dynamic Time Warping (DTW) [10]. DTW works
on pairs of time series, aligning one time series to a specified reference time, in a non-probabilistic
way, without explicit allowance for differences in related time series. More recently, Gaffney et al [5]
jointly clustered and aligned time series data from different classes. However, their model does not
attempt to put time series from different classes into correspondence with one another ? only time
series within a class are aligned to one another. Ziv Bar-Joseph et al [1] use a similar approach to
cluster and align microarray time series data. Ramsay et al [9] have introduced a curve clustering
model, in which a time warping function, h(t), for each time series is learned by way of learning
its relative curvature, parameterized with order one B-spline coefficients. This model accounts for
5
5
3
3
9
0
5
?9
9
0
?9
9
3
0
?9
5
5
3
3
0
50
100
150
200
250
300
0
50
100
150
200
250
Figure 3: Seven time series from each of three classes of LC-UV data. On all figures, the horizontal
axis is time, or latent time for figures of latent traces and observed time series aligned to latent
traces. The vertical axis is log of UV absorbance. Top left: The raw, unaligned data. Middle
left: Average of the unaligned data within each class in thick line, with the thin lines showing one
standard deviation on either side. Bottom left: Average of the aligned data within each class, using
the single-class alignment version of the model (no child traces), again with one standard deviation
lines shown in the thinner style line. Right: Mean and one standard deviation over MCMC samples
using the HB-CPM model. Top right: Parent trace. Middle right: Class-specific energy impulses,
with the top-most showing the class impulses for the ?no treatment? class. Bottom right: Child traces
superimposed. See Figure 4 for a zoom-in in around the arrow.
systematic changes in the range and domain of time series in a way that aligns curves with the same
fundamental shape. However, their method does not allow for class-specific differences between
shapes to be taken into account. The anomaly detection (AD) literature deals with related, yet
distinct problems. For example, Chan et al [3] build a model of one class of time series data (they use
the same NASA valve data as in this paper), and then match test data, possibly belonging to another
class (e.g. ?abnormal? shuttle valve data) to this model to obtain an anomaly score. Emphasis in the
AD community is on detecting abnormal events relative to a normal baseline, in an on-line manner,
rather than comparing and contrasting two or more classes from a dataset containing examples of
all classes. The problem of ?elastic curve matching? is addressed in [6], where a target time series
that best matches a query series is found, by mapping the problem of finding the best matching
subsequence to the problem of finding the cheapest path in a DAG (directed acyclic graph).
6
Discussion and Conclusion
We have introduced a hierarchical, Bayesian model to perform detection of rare differences between
sets of related time series, a problem which arises across a wide range of domains. By training our
model, we obtain the posterior distribution over a set of class-specific canonical representations of
each class, which are aligned in a way that preserves their common sub-structures, yet retains and
highlights important differences.
This model can be extended in several interesting and useful ways. One small modification could be
useful for the LC-UV data set presented in this paper, in which one of the classes was ?no treatment?,
while the other two were each a different ?treatment?. We might model the ?no treatment? as the
parent trace, and each of the treatments as a child trace, so that the direct comparison of interest
would be made more explicit. Another direction would be to apply the HB-CPM in a completely
300
4
5
0
3
?5
4
5
0
3
?5
4
5
0
3
?5
100
105
110
115
120
125
130
135
140
145
150
155
100
105
110
115
120
125
130
135
140
145
150
155
Figure 4: Left: A zoom in of data displayed in Figure 3, from the region of time 100-150 (labeled in
that figure in latent time, not observed time). Top left: mean and standard deviation of the unaligned
data. Middle left: mean and standard deviation of the single-class alignment. Bottom left: mean
and standard deviation of the child traces from the HB-CPM. A case in point of a difference that
could be detected with the HB-CPM and not in the raw or single-class aligned data, is the difference
occurring at time point 127. Right: The mean and standard deviation of the child energy impulses,
with dashed lines showing correspondences with the child traces in the bottom left panel.
unsupervised setting where we learn not only the canonical class representations, but also obtain the
posterior over the class labels by introducing a latent class indicator variable. Lastly, one could use a
model with cyclical latent traces to model cyclic data such as electrocardiogram (ECG) and climate
data. In such a model, an observed trace being generated by the model would be allowed to cycle
back to the start of the latent trace, and the smoothness constraints on the trace would be extended
to apply to beginning and end of the traces, coercing these to be similar. Such a model would allow
one to do anomaly detection in cyclic data, as well as segmentation.
Acknowledgments: Thanks to David Ross and Roland Memisevic for useful discussions, and Ben Marlin for
his Matlab slice sampling code.
References
[1] Z. Bar-Joseph, G. Gerber, D. K. Gifford, T. Jaakkola, and I. Simon. A new approach to analyzing gene
expression time series data. In RECOMB, pages 39?48, 2002.
[2] H. Boelens, R. Dijkstra, P. Eilers, F. Fitzpatrick, and J. Westerhuis. New background correction method
for liquid chromatography with diode array detection, infrared spectroscopic detection and raman spectroscopic detection. Journal of Chromatography A, 1057:21?30, 2004.
[3] P. K. Chan and M. V. Mahoney. Modeling multiple time series for anomaly detection. In ICDM, 2005.
[4] B. Ferrell and S. Santuro. NASA shuttle valve data.
http://www.cs.fit.edu/?pkc/nasa/data/,
2005.
[5] S. J. Gaffney and P. Smyth. Joint probabilistic curve clustering and alignment. In Advances in Neural
Information Processing Systems 17, 2005.
[6] L. Latecki, V. Megalooikonomou, Q. Wang, R. Lakaemper, C. Ratanamahatana, and E. Keogh. Elastic
partial matching of time series, 2005.
[7] J. Listgarten, R. M. Neal, S. T. Roweis, and A. Emili. Multiple alignment of continuous time series. In
Advances in Neural Information Processing Systems 17, 2005.
[8] R. M. Neal. Slice sampling. Annals of Statistics, 31:705?767, 2003.
[9] J. Ramsay and X. Li. Curve registration. Journal of the Royal Statistical Society(B), 60, 1998.
[10] H. Sakoe and S. Chiba. Dynamic programming algorithm for spoken word recognition. Readings in
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2,224 | 3,018 | Recursive ICA
Honghao Shan, Lingyun Zhang, Garrison W. Cottrell
Department of Computer Science and Engineering
University of California, San Diego
La Jolla, CA 92093-0404
{hshan,lingyun,gary}@cs.ucsd.edu
Abstract
Independent Component Analysis (ICA) is a popular method for extracting independent features from visual data. However, as a fundamentally linear technique,
there is always nonlinear residual redundancy that is not captured by ICA. Hence
there have been many attempts to try to create a hierarchical version of ICA, but so
far none of the approaches have a natural way to apply them more than once. Here
we show that there is a relatively simple technique that transforms the absolute values of the outputs of a previous application of ICA into a normal distribution, to
which ICA maybe applied again. This results in a recursive ICA algorithm that
may be applied any number of times in order to extract higher order structure from
previous layers.
1
Introduction
Linear implementations of Barlow?s efficient encoding hypothesis1 , such as ICA [1] and sparse coding [2], have been used to explain the very first layers of auditory and visual information processing
in the cerebral cortex [1, 2, 3]. Nevertheless, many interesting structures are nonlinear functions of
the stimulus inputs, which are unlikely to be captured by a linear model. For example, for natural
images, it has been observed that there is still significant statistical dependency between the variance
of the filter outputs [4]. Several extensions of the linear ICA algorithm [5, 6, 7, 8] have been proposed to reduce such residual nonlinear redundancy, with an explicit or implicit aim of explaining
higher perceptual layers, such as complex cells in V1. However, none of these extensions are obviously recursive, so it is unclear how to generalize them to multi-layer models in order to account for
even higher perceptual layers.
In this paper, we propose a hierarchical redundancy reduction model in which the problem of modeling the residual nonlinear dependency is transformed into another LEE problem, as illustrated in
Figure 1. There are at least two reasons why we want to do this. First, this transforms a new and
hard problem into an easier and previously solved problem. Second, different parts of the brain share
similar anatomical structures and it is likely that they are also working under similar computational
principles. For example, fMRI studies have shown that removal of one sensory modality leads to
neural reorganization of the remaining modalities [9], suggesting that the same principles must be
at work across modalities. Since the LEE model has been so successful in explaining the very first
layer of perceptual information processing in the cerebral cortex, it seems reasonable to hypothesize
that higher layers might also be explained by a LEE model.
The problem at hand is then how to transform the problem of modeling the residual nonlinear dependency into a LEE problem. To achieve this goal, we need to first make clear what the input
constraints are that are imposed by the LEE model. This is done in Section 2. After that, we will
derive the transformation function that ?prepares? the output of ICA for its recursive application,
and then test this model on natural images.
1
We refer to such algorithms as linear efficient encoding (LEE) algorithms throughout this paper.
Figure 1: The RICA (Recursive ICA) model. After the first layer of linear efficient encoding,
sensory inputs X are now represented by S. The signs of S are discarded. Then coordinate-wise
nonlinear activation functions gi are applied to each dimension of S, so that the input of the next
layer X 0 = g(|S|) satisfies the input constraints imposed by the LEE model. The statistical structure
among dimensions of X 0 are then extracted by the next layer of linear efficient encoding.
2
Bayesian Explanation of Linear Efficient Encoding
It has long been hypothesized that the functional role of perception is to capture the statistical structure of the sensory stimuli so that appropriate action decisions can be made to maximize the chance
of survival (see [10] for a brief review). Barlow provided the insight that the statistical structure
is measured by the redundancy of the stimuli and that completely independent stimuli cannot be
distinguished from random noise [11]. He also hypothesized that one way for the neural system to
capture the statistical structure is to remove the redundancy in the sensory outputs. This so-called
redundancy reduction principle forms the foundation of ICA algorithms.
Algorithms following the sparse coding principle are also able to find interesting structures when
applied to natural image patches [2]. Later it was realized that although ICA and sparse coding
algorithms started out from different principles and goals, their implementations can be summarized
in the same Bayesian framework [12]. In this framework, the observed data X is assumed to be
generated by some underlying signal sources S:
X = AS +
where A is a linear mixing matrix and is additive Gaussian noise. Also, it is assumed that the
features Sj are independent from each other, and that the marginal distribution of Sj is sparse. For
the sparse coding algorithm described in [2], although it started from the goal of finding sparse features, the algorithm?s implementation implicitly assumes the independence of Sj ?s. For the infomax
ICA algorithm [1], although it aimed at finding independent features, the algorithm?s implementation assumes a sparse marginal prior (p(Sj ) ? sech(Sj )). The energy-based ICA algorithm using a
student-t prior [13] can also be placed in this framework for complete representations.
The moral here, though, is that in practice, the samples available are always insufficient to allow
any efficient inference without making some assumptions about the data distribution. A sparseness
and independence assumption about the data distribution is appropriate because: (1) independence
allows the system to capture the statistical structure of the stimuli, as described above, and (2)
sparse distribution of the sensory outputs is energy-economic. This is important for the survival of
the biological system, considering the fact that human brain consists 2% of the body weight but
accounts for 20% of its resting metabolism [14]. The linear efficient encoding model captures the
important characteristics of sensory coding: capturing the statistical structure (independence) of
sensory stimuli with minimum cost (sparseness).
This generative model describes our assumption about the data. How well the algorithms perform
depends on how well this assumption matches the real data. Hence, it is very important to check
what kind of data the model generates. If the input data strongly deviate from what can be generated
by the model (in other words, the observed data strongly deviate from our assumption), the results
could be errant no matter how much effort we put into the model parameter estimation. As to the
LEE model, there is a clear constraint on the marginal distribution of Xi .
Here we limit our study on those ICA algorithms that produce basis functions resembling the simple
cells? receptive fields when applied on natural image patches. Such algorithms [1, 13, 15] typically
adopt a symmetric 2 and sparse marginal prior for Sj ?s that can be well approximated by a generalized Gaussian distribution. In fact, if we apply linear filters resembling the receptive fields of
simple cells on natural images, the distribution of the filter responses can be well approximated by
a generalized Gaussian distribution.
Here we show that such a prior suggests that the Xi ?s should also be symmetric. A random variable
X is symmetric if and only if its characteristic function is real valued. In the above Bayesian
framework, we assume that Sj ?s are independent and the marginal distribution of Sj is symmetric
about zero. The characteristic function is then given by:
?
P
?
X
E[e ?1tXi ] = E[e ?1t j Ai,j Sj ] (Xi =
Ai,j Sj )
(1)
j
Y ?
= E[ e ?1tAi,j Sj ]
(2)
j
=
Y
?
E[e
?1tAi,j Sj
] (Sj ?s are independent from each other)
(3)
j
Since Ai,j Sj is symmetric, it is easy to see that Xi must also be symmetric.
A surprising fact about our perceptual system is that there does exist such a process that regularizes the marginal distribution of the sensory inputs. In the visual system, for example, the data is
whitened in the retina and the LGN before transmission to V1. The functional role of this process is
generally described as removing pairwise redundancy, as natural images (as well as natural sounds)
obey the 1/f power law in the frequency domain [16]. However, as shown in Figure 2, it also
regulates the marginal distribution of the input to follow a generalized-gaussian-like distribution3 .
This phenomenon has long been observed. We believe that besides the functional role of removing
second-order redundancy, whitening might also serve as a role of formatting the sensory input for
the cortex. For example, it has been observed [1] that without pre-whitening the images, the learned
basis functions by ICA do not cover a broad range of spatial frequencies.
Figure 2: The distribution of pixel values of whiten images follows a generalized Gaussian distribution (see Section 2). The shape parameter of the distribution is about 1.094, which means that the
marginal distribution of the inputs to the LEE model is already very sparse.
2
p(X) is symmetric if X and ?X have the same distribution.
For all the image patches we tried, the distribution of pixel values on whitened image patches can be well
fitted by a generalized Gaussian distribution. This is true even for small image patches. The only exception we
have discovered occurs when the original image contains only binomial noise.
3
In this work, we will make the assumption that the marginal distribution of the inputs to the LEE
model is a generalized gaussian distribution, as this enables the LEE model to work more efficiently.
Also, as just discussed, at least for sound and image processing, there is an effective way to achieve
this neurally.
3
Reducing Residual Redundancy
For the filter outputs S of a layer of LEE, we will first discard information that provides no interesting structure (i.e., redundancy), and find an activation function such that the marginal distribution
obeys the input requirements of the next layer.
3.1
Discarding the Signs
It has been argued that the signs of the filter outputs do not carry any redundancy [5]. The models
proposed in [6, 7, 8] also implicitly or explicitly discard the signs. We have observed the usefulness
of this process in a study of natural image statistics. We applied the FastICA algorithm [15] to 20x20
natural image patches, and studied the joint distribution of the filter outputs. As shown in the left
plot of Figure 3, p(si |sj ) = p(si | ? sj ), i.e. the conditional probability of si on sj only depends on
the absolute value of sj . In other words, the signs of S do not provide any dependency among the
dimensions. By removing the sign and applying our transformation (described in the next section),
the nonlinear dependency between the si ?s is exposed (see Figure 3, right).
Figure 3: Left: s1 and s2 are ICA filter responses on natural image patches. The red dashed lines
plot the linear regression between them. Right: After the coordinate-wise nonlinear transformation,
the two features are no longer uncorrelated.
3.2
Nonlinear Activation Function
The only problem left is to find the coordinate-wise activation function gi for each dimension of S
such that Xi0 = gi (|Si |) follows a generalized Gaussian distribution, as required by the next layer of
LEE. In this work, we make the transformed features have a normal distribution. By doing so, we
force the LEE model of the higher layer to set more A0j,i to nonzero values (so that the Central Limit
Theorem takes effect to make Xi0 a Gaussian distribution), which leads to more global structures at
the higher layer. We used two methods to find this activation function in our experiments.
Parametric Activation Function
Assume s approximately follows a generalized Gaussian distribution(GGD). The probability density
function of a GGD is given by:
s ?
?
exp{? }
(4)
f (s; ?, ?) =
2??(1/?)
?
where ? > 0 is a scale parameter and ? > 0 is a shape parameter and ? denotes the gamma function.
These two parameters can be estimated efficiently by an iterative algorithm developed by [17].
s is then transformed into a normally distributed N (0, 1) random variable by the function g:
?
u = g(|s|) = F
?1
?( |s|? , 1 )
( ?1 ? )
?( ? )
(5)
where F denotes the cumulative density function (cdf) of standard normal distribution and ? denotes
the incomplete gamma function. This transformation can be seen as three consecutive steps:
? Discard the sign: u ? |s|, now u bears pdf g(u; ?, ?) =
and cdf
?(
|s|? 1
, )
?? ?
?( ?1 )
?
??( ?1 )
exp{? ?u ? }, 0 ? u < ?
, 0 ? u < ?.
? Transform to a uniform distribution U [0, 1] by applying its own cdf: u ?
?(
|s|? 1
, )
?? ?
?( ?1 )
.
? Transform to a Gaussian distribution by applying the inverse cdf of N (0, 1): u ? F ?1 (u).
Nonparametric Activation Function
When the number of samples N is sufficiently large, a non-parametric activation function works
more efficiently. In this approach, all the samples |Si | are sorted in ascending order. For each sample
c (|s|))
s, cdf (|s|) is approximated by the ratio of its ranking in the list with N . Then u = F ?1 (cdf
will approximately follow the standard normal distribution. Note that since ui depends only on the
rank order of |si |, the results would be the same if the signs are discarded by taking s2i .
4
Experiments on Natural Images
To test the behavior of our model, we applied it to small patches taken from digitized natural images.
The image dataset is available on the World Wide Web from Bruno Olshausen 4 . It contains ten
512x512 pre-whitened images. We took 151,290 evenly distributed 20x20 image patches. We ran
the FastICA algorithm [15] and obtained 397 basis functions. As reported in other models, the basis
functions are Gabor-like filters (Figure 4).
The nonparametric method was used to transform the marginal distribution of the outputs? absolute
values to a standard normal distribution. Then the FastICA algorithm was applied again to retrieve
100 basis functions5 . We adopted the visualization method employed by [12] to investigate what
kind of structures the second layer units are fond of. The basis functions are fitted to Gabor filter
functions using a gradient descent algorithm [12]. The connection weights from a layer-2 unit to
layer-1 units are shown in Figure 5, arranged by either the center or frequency/orientation of the
fitted Gabor filters. The layer-2 units are qualitatively similar to those found in [18]. Some units
welcome strong activation of layer-1 units within a certain orientation range but have no preference
for locations, while others have a location preference but welcome activation of layer-1 units of all
frequencies and orientations, and some develop a picky appetite for both.
Again, the nonparametric method was used to transform the marginal distribution of the absolute
values of the outputs from the second layer to a standard normal distribution, and FastICA was
applied to retrieve 20 basis functions. We had no initial guess of what kind of statistical structure
these third layer units might capture. The activation map of a couple of these units, however, seemed
to suggest that they might be tuned to respond to complicated textures. In particular, one unit
seems more activated by seemingly blank background, while another seems to like textures of leaves
(Figure 6). We think that probably a larger database than merely 10 images, and larger image patches
would be helpful for producing cleaner high level units.
The same procedure can be repeated for multiple layers. However, at this point, until we develop
better methods for analyzing the representation developed by these deeply embedded units, we will
leave this for future work.
4
http://redwood.berkeley.edu/bruno/sparsenet/
This reduction in the number of units follows the example of [18]. In general, there appears to be less
information in later layers (as assessed by eigenvalue analysis), most likely due to the discarding of the sign.
5
Figure 4: A subset of the 397 ICA image basis functions. Each basis function is 20x20 pixels. They
are 2D Gabor like filters.
Figure 5: Sample units from the second layer. The upper panel arranges the connection weights
from layer-2 units to layer-1 units by the centers of the fitted Gabor filters. Every point corresponds
to one basis function of the first layer, located at the center of the fitted Gabor filter. Warm colors
represent strong positive connections; cold colors represent negative connections. For example, the
leftmost unit prefers strong activation of layer-1 units located on the right and weak activation of
layer-1 units on the left. The lower panel arranges the connection weights by the frequencies and the
orientations of the fitted Gabor filters. Now every point corresponds to the Gabor filter?s frequency
and orientation (in polar coordinates). The third leftmost unit welcomes strong activation of Gabor
filters whose orientations are around 34 ? but prefers no/little activation from those whose orientations
are around 14 ?.
5
Discussion
The key idea of our model is to transform the high-order residual redundancy to linear dependency
that can be easily exploited again by the LEE model. By using activation functions that are dependent on the marginal distribution of the outputs, a normal Gaussian interface is provided at every
layer. This procedure can then repeat itself and a hierarchical model with same structure at every
level can thus be constructed. As the redundancy is reduced progressively along the layers, statistical
structures are also captured to progressively higher orders.
Our simulation of a three layer Recursive ICA shows the effectiveness of our model. The first layer,
not surprisingly, produces the Gabor like basis functions as linear ICA always does. The second
layer, however, produces basis functions that qualitatively resemble those produced by a previous
hierarchical generative model [7]. This is remarkable given that our model is essentially a filtering
model with no assumptions of underlying independent variables, but merely targeting redundancy
reduction. The advantage of our model is the theoretical simplicity of generalization to a third layer
or more. For the Karklin and Lewicki model, the assumption that the ultimate independent causal
variables are two layers away from the images has to be reworked for a three layer system. It is not
clear how the variables at every layer should affect the next when an extra layer is added. Osindero
et al. [8] employed an energy based model. The energy function used at the first layer made it
essentially a linear ICA algorithm, thus it also produces Gabor like filters. The first layer outputs
are squared to discard the signs and then fed to the next layer. The inputs for the second layer are
thus all positive and bear a very different marginal distribution from those for the first layer. The
energy function is changed accordingly and the second layer is essentially doing nonnegative ICA.
The output of this layer, however, will all be positive, which makes discarding the signs no longer an
effective way of exposing higher-order dependence. Thus, to extend to another layer, new activation
functions and new energy function must be derived.
The third layer of our model produces some interesting results in that some units seem to have
preferences for complicated textures (Figure 6). However, as the statistical structure represented
here must be of very high order, we are still looking for an effective visualization method. Also, as
Figure 6: Activation maps on two images (upper and lower panel respectively) for two units per
layer. The leftmost two images are the raw images. The second left column to the rightmost column
are activation maps of two units from the first layer to the third respectively. The first layer units
respond to small local edges, the second layer units respond to larger borders, and the third layer
units seem to respond to large area of textures.
units at the second layer have larger receptive field than those at the first layer, it is reasonable to
expect the third layer to bear even larger ones. We believe that a wider range of visual structure will
be picked up by the third layer units with a larger patch size on a larger training set.
Acknowledgments
We thank Eric Wiewiora, Lei Zhang and members from GURU for helpful discussions. This work
was supported by NIH grant MH57075 to GWC.
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2,225 | 3,019 | Mixture Regression for Covariate Shift
Amos J Storkey
Institute of Adaptive and Neural Computation
School of Informatics, University of Edinburgh
[email protected]
Masashi Sugiyama
Department of Computer Science
Tokyo Institute of Technology
[email protected]
Abstract
In supervised learning there is a typical presumption that the training and test
points are taken from the same distribution. In practice this assumption is commonly violated. The situations where the training and test data are from different
distributions is called covariate shift. Recent work has examined techniques for
dealing with covariate shift in terms of minimisation of generalisation error. As
yet the literature lacks a Bayesian generative perspective on this problem. This
paper tackles this issue for regression models. Recent work on covariate shift can
be understood in terms of mixture regression. Using this view, we obtain a general
approach to regression under covariate shift, which reproduces previous work as
a special case. The main advantages of this new formulation over previous models for covariate shift are that we no longer need to presume the test and training
densities are known, the regression and density estimation are combined into a
single procedure, and previous methods are reproduced as special cases of this
procedure, shedding light on the implicit assumptions the methods are making.
1
Introduction
There is a common presumption in developing supervised methods that the distribution of training
points used for learning supervised models will match the distribution of points seen in a new test
scenario. The expectation that the training and test points follow the same distribution is explicitly
stated in [2, p. 10], is an assumption of empirical risk minimisation, [see e.g. 9, p. 25], and is
implicit in the common practice of randomized splitting given data into a ?training set? and a ?test
set?, where the latter is used in assessing performance [5, p. 482-495].
This paper, then, is concerned with the following issue. A set of real valued training data pairs of
the form (x, y) is provided to train a model for a supervised learning problem. In addition data of
the form x is provided from one (or more) test environments where the model will be used. The
question to be addressed is ?How should we predict a value of y given a value x from within that
particular test environment??
Cases where test scenarios truly match the training data are probably rare. The problem of mismatch
has been grappled within literature from a number of fields, and has become known as covariate
shift [14]. Specific examples of covariate shift include situations in reinforcement learning [c.f. 13]
and bio-informatics [c.f. 1]. The common issue of sample selection bias [7] is a particular case of
covariate shift.
Much of the recent analysis of covariate shift has been made in the context of assessing the asymptotic bias of various estimators [15]. In general it has been noted that in the case of mismatched
models (i.e. where the model from which the training data is generated is not included in the training
model class), some typical estimators, such as least squares approaches, produce biased asymptotic
estimators [14]. It might appear that the presumption of matched models in Bayesian analysis means
covariate shift is not an issue: failure or otherwise under situations of covariate shift is solved by
valid choice for the prior distribution over conditional models. The difficulty with this dismissal
of the subject is that modelling conditional distributions alone is not always valid. In fact we can
categorise at least three different types of covariate shift:
1. Independent covariate shift: Ptrain (y|x) = Ptest (y|x), but Ptrain (x) 6= Ptest (x).
2. Dependent prior probability change: Ptrain (x|y) = Ptest (x|y), but Ptrain (y) 6= Ptest (y).
3. Latent prior probability change: Ptrain (x, y|r) = Ptest (x, y|r) for all values of some
latent variable r, but Ptrain (r) 6= Ptest (r).
Let us presume that we are only interested in the quality of the conditional model Ptest (y|x). Then
Case 1 is the only one of the above where covariate shift will have no effect on modelling. Case
2 is the well known situation of class prior probability change and, for example, is considered in
comparing the benefits of a naive Bayes model, which allows for class prior probability change, and
discriminant models, which typically do not.
Case 3 involves a more general assumption, and arguably can be used to cover most situations of
covariate shift, by incorporating any known structural characteristics of the problem into some latent
variable r. Change in the distribution of x points implicitly informs us about variation in the targets
y via the shift in the latent variable r, which is the causal factor for the change. The purpose of this
paper is to provide a generative framework for analysis of covariate shift. The main advantages of
this new formulation over previous approaches are
? It provides an explicit model for the changes occurring in situations of covariate shift, and
hence the predictions that result from it.
? There is no need to presume the training and test distributions are known. Furthermore the
test covariates are also used as part of the model estimation procedure, resulting in better
predictions.
? Previous results, such as Importance Weighted Least Squares, are special cases of this
method with explicit presumptions that can be relaxed to gain more general models. Hence
this paper is a natural extension to the existing work.
? Utilising the test covariate distribution gives performance benefits over using the same
model for training data alone.
? All the usual machinery for mixture of experts are available, and so this approach allows
model selection and many natural extensions.
Outline. In Section 2, related work is discussed, before the problem is formally specified and a
general model is derived in Section 3. A specific form of mixture regression model is formulated
and an Expectation Maximisation solution is given in Section 3.1. The specific relationship to Importance Weighted Least Squares is discussed in Section 3.1.2. Test examples are given in section
4. The results and methods are discussed in Section 5.
2
Prior work
Covariate shift will be interpreted, in the context of this work, using mixture of regressor models,
where the regression model is dependent on a latent class variable. Clustered regression models have
been discussed widely [4, 18, 8, 16]. The benefits of the mixture of regressor approach for heterogeneous data was discussed in [17], but not formulated specifically for the problem of covariate shift.
This paper establishes for the first time the relationship between the mixture of regressor model and
the typical statistical results in the literature on covariate shift. The main differences of our approach
from a standard mixture of regressor formalism is that we utilise the training and test distributions
as part of the model and do not use only a conditional model, and we allow coupling of regressors
across different mixture components. The main significance with regard to the literature on covariate shift is that we establish covariate shift within a general probabilistic modelling paradigm and
hence extend the standard techniques to establish more general methods, which are also applicable
when the training and test distributions are not explicitly given. The mixture of regressors form for
(x, y) used in this paper is a specific from of mixture of experts [10]. Hence hierarchical extensions
are also possible in the form of [11].
The problem of sample selection bias is related to covariate shift. Sample selection bias has been
discussed in [19], where they estimate the distribution determining the bias for a classification problem. The problem of sample selection bias differs from the case in this paper as here there is no
fundamental requirement of distribution overlap between the training and test sets. First, each can
have zero density in regions the other is non-zero. Second, the presumption is different: rather than
there being a sample rejection process that characterised the difference between training and test
sets, there is a sample production process that differs.
3
Framework for Covariate Shift
This paper follows most others in considering the restricted case of a single training and single test
set. Each datum x is assumed to have been generated from one of a number of data sources using
a mixture distribution corresponding to the source. The proportions of each of the sources varies
across the training and test datasets. Hence, in the context of this paper, we understand covariate
shift to be effected by a change in the contribution of different sources to the data.
The motivation of the framework in this paper is that there is a latent feature set upon which each
dataset is dependent, and the the variations between the two datasets are dependent upon variation
of the proportions, but not the form, of those latent features. This is characterised by presuming
each data source is a member of one of two different sets. Each of the two sets of sources is also
associated with a regression model. The two sets of sources have the following characteristics:
? Source set 1 corresponds to sources that may occur in the test data, and potentially also in
the training data, and are associated with regression model P1 (y|x).
? Source set 2 corresponds to sources that occur only in the training data, and are associated
with regression model P2 (y|x).
By taking this approach we note that we will be able to separate out effects that we expect to be only
characteristics of the training data from effects that are common across training and test sets.
The full generative model for the observed data consists of the model for the training data D and
model for the test data T . The test data is just used to determine the nature of the covariate shift, and
consists of only of the covariates x, and not any targets y. We emphasise that we do not presume to
have seen the test data we wish to predict. Rather a prior model is built for the training and test data,
and this is then conditioned on the information from the training data and the known covariates for
the test data but not the unknown targets.
3.1
Mixture Regression for Covariate Shift
In this section the full model is introduced. This significantly extends the previous work on covariate
shift, in that the model allows for unknown training and test distributions, and utilises a mixture
model approach for representing the relationship between the two. In Section 3.1.2, we will show
how the previous results on covariate shift are special cases of the general model. We will develop
this formalism for any parametric form for the regressors P (y|x). In fact this restriction is mainly
for ease of explanation, and the method can be used with non-parametric models too, and will be
tested in the case of Gaussian process models1 .
The model takes the following form
? The distribution of the training data and test data are denoted PD and PT respectively, and
are unknown in general.
? Source set 1 consists of M mixture distributions, where mixture t is denoted P1t (x). Each
of the components is associated2 with regression model P1 (y|x).
? Source set 2 consists of M2 mixture distributions, where mixture t is denoted P2t (x). Each
of the components is associated with the regression model P2 (y|x).
1
The primary restriction is than we need to be able to compute standard EM responsibilites for a given
regressor, and hence for Gaussian processes a variational approximation is needed to do this.
2
If a component i is associated with a regression model j, this means that any datum x generated from
the mixture component i, will also have a corresponding y generated from the associated regression model
Pj (y|x).
? The training and test data distributions take the following form:
X
D
D
T
PD (x) =
?1 ?1t
P1t (x) + ?2 ?2t
P2t (x) and PT (x) = ?1t
P1t (x)
(1)
t
D
Hence ?1 and ?2 are parameters for the proportions of the two source sets in the training data, ?1t
are
D
the relative proportions of each mixture from source set 1 in the training data, and ?2t
are the relative
T
proportions of each mixture from source set 2 in the training data. Finally ?1t
are the proportions
of each mixture from source set 1 in the test data. All these parameters are presumed unknown. At
some points in the paper it will be presumed the mixtures are Gaussian, when the form N (x; m, K)
will be used to denote the Gaussian distribution function of x, with mean m and covariance K.
For a parametric model, with the collection of mixture parameters denoted by ?, the collection
of regression parameters denoted by ?, and the mixing proportions, ? and ? we have the full
probabilistic model
P ({i? , y? , x? |? ? D}, {i? , x? |? ? T }|?, ?, ?) =
Y
Y
P (s? |?)P (t? |?, s? )Ps? t? (x? |?t? )Ps? (y? |x? , ?)
P (t? |?)P1t? (x? |?). (2)
??D
??T
?
where s denotes the source set used to generate the data point ?, and t? denotes the particular
mixture from that source set used to generate the data point ?. In words, this says that the model for
the training dataset involves sampling the particular source set s? , then the mixture component t?
from that particular source set. Given these we then sample an x? from the relevant mixture and a
y? conditionally on x? from the relevant regressor. The same procedure is followed for the test set,
except now there is only one source set to consider.
3.1.1
EM algorithm
A maximum likelihood solution for the parameters (?, ?, ?, ?) can be obtained for this model
(given the training data and test covariate) using Expectation Maximisation (EM) [3]. The derivations are standard EM calculations (see e.g. [2]), and hence are not reiterated here. Denote the
responsibility of mixture i for data point ? by ?i? . Then the application of EM involves maximisation of
log P ({y? , x? |? ? D}, {x? |? ? T }|?, ?, ?, ?)
(3)
with respect to the parameters through iteration of E and M steps. The E-step update uses current
parameter values to compute the responsibility (denoted by ?s) of each mixture 1t and 2t for each
data point ? in the training set and each data point ? in the test set using
D
? T P1t (x? |?)
?s ?st
Pst (x? |?)Ps (y? |x? , ?)
?
and ?1t
= P 1t T
.
D
?
?
?
?
s,t ?s ?st Pst (x |?)Ps (y |x , ?)
t ?1t P1t (x |?)
?
?st
=P
(4)
?
We set ?2t
= 0 for ? ? T , as none of these mixtures are represented in the test set. The parameters
of the mixture model distributions are then updated with the usual M steps for the relevant mixture component, and the regression parameters are updated using maximum responsibility-weighted
likelihood. When each mixture component is a Gaussian of the form N (x; mst , Kst ), when we
have a Gaussian regression error term, and denoting the (vector of) regression functions by fs for
each source set s, these update rules are:
P
P
? ?
?
?
?
T
??(D,T ) ?st x
??(D,T ) ?st (x ? mst )(x ? mst )
P
P
mst =
, Kst =
(5)
?
?
??D,T ?st
??D,T ?st
?
1 X ?
1 X ?
1 X ?st
T
D
, ?1t
=
?1t
(6)
?s =
?st , ?st
=
|D|
|D|
?s
|T |
??D,t
??D
??T
"
#
X ?
?
? T
?
?
fs = arg min
?st (fs (x ) ? y ) (fs (x ) ? y )
(7)
fs
?,t
Given the learnt model, inference is straightforward. The test data is associated with a single regression model P1 (y|x), and so the predictive distribution for the test set is the learnt predictor P1 (y|xi )
for each point xi in the test set.
3.1.2 Importance Weighted Least Squares
Previous results in modelling covariate shift can be obtained as special cases of the general approach
taken in this paper. Suppose we make the assumptions that PD and PT are known, and that the
source set 1 contains just the one component, which must be PT by definition. Suppose also that the
two regressors have a large and identical variance ?. In this simple case, we do not need to know
the actual test points (in this framework these are only used to infer the test distribution, which is
assumed given here). The M step update only involves update to the regressor. For the E step we
use the approximation P (y? |x? , ?1 ) ? P (y? |x? , ?2 ), which becomes asymptotically true in the
case of infinite variance ?. The resulting E and M steps are
"
#
X
PT (x? )?1
?
?
?
? T
?
?
and f1 = arg min
? ?
? (f1 (x ) ? y ) (f1 (x ) ? y )
(8)
f1
PD (x? )
?
where we note that ?1 is a common constant and can be dropped from the calculations. Hence
we never need to learn ?1 or the parameters associated with mixture 2 in this procedure. Also no
iterative EM procedure is needed as the E step is independent of the M step results. Hence this
is a one shot process. This is the Importance Weighted Least Squares estimator for covariate shift
[14]. A simple extension of this model will allow the large variance assumption to be relaxed, so the
model can use the regressor information for computing responsibilities.
4
4.1
Examples
Generated Test Data
We demonstrate the mixture of regressors approach to covariate shift (MRCS) on generated test
data: a one dimensional regression problem with two sources each corresponding to different linear
regressors. Regression performance for MRCS with Gaussian mixtures and linear regressors is
compared with three other cases. The first is an importance weighted least squares estimator (IWLS)
given the best mixture model fit for the data, corresponding to the current standard for modelling
covariate shift. The second uses a mixture of regressors model that ignores the form of the test data,
but chooses the regressor corresponding to the mixtures which best match the test data distribution
using a KL divergence measure (MRKL). This corresponds to recognising that covariate shift can
happen, but ignoring the nature of the test distribution in the modelling process, and trying to choose
the best of the two regressors. The third case is where the mixture of regressors is used simply as a
standard regression model, ignoring the possibility of covariate shift (MRREG).
The generative procedure for each of the 100 test datasets involves generating random parameter
values for between 1 and 3 mixtures for each of two linear regressors. Test and training datasets
of 200 data points each are generated from these mixtures and regression models, using different
mixing proportions in each case. The various approaches were run 8 times with different random
starting parameters for all methods. 80 iterations of EM were used. A fixed number of iterations
was chosen to allow reasonable comparison. Analysis was done for fixed model sizes and for model
choice using a Bayesian Information Criterion (BIC). Even though the regularity conditions for BIC
do not hold for mixture models, it has been shown that BIC is consistent in this case [12]. It has also
been shown to be a good choice on practical grounds [6].
The results of these tests show the significant benefits of explicit recognition of covariate shift over
straight regression even compared with the use of the same mixture of regressors model, but without
reference to the test distribution. It also shows benefits of the approach of this paper over the current
state of the art for modelling covariate shift. Table 1 gives the result of these approaches for various
fixed choices of numbers of mixtures associated with each regressor. Independent of the use of any
model order choice, the Mixture of Regressors for Covariate Shift (MRCS) performs better than the
other approaches. Table 1 also gives the results when the Bayesian Information Criterion is used
for selecting the number of mixtures. Again MRCS performs best, and consistently gives better
performance on the test data for more than 70 percent of the test cases.
To illustrate the difference between the methods, Figure 1 plots the results of training a MRCS
model on some one dimensional data using a regularised cubic regressor. The fit to the test data is
also shown. Once again this is compared with IWLS and MRKL. It can be seen that both IWLS and
MRKL fail to untangle the regressors associated with the overlapping central clusters in the training
data and hence perform badly in that region of the test data.
5
6
4
5
4
3
3
2
2
1
1
0
0
?1
?1
?2
?3
?3
?2
?2
?1
0
1
2
3
4
?3
?3
?2
?1
(a)
0
1
2
0
1
2
0
1
2
(b)
5
6
4
5
4
3
3
2
2
1
1
0
0
?1
?1
?2
?3
?3
?2
?2
?1
0
1
2
3
4
?3
?3
?2
?1
(c)
(d)
5
6
4
5
4
3
3
2
2
1
1
0
0
?1
?1
?2
?3
?3
?2
?2
?1
0
1
(e)
2
3
4
?3
?3
?2
?1
(f)
Figure 1: Nonlinear regression using covariate shift. (a),(c),(e) Training set fit and (b),(d),(f) test data with
predictions for MRCS (top), IWLS (middle) and MRKL (bottom) respectively. In (a),(c),(e), the ?.? data labels
the points for which the test regressor has greater responsibility, and the ?+? data labels points for which the
training only regressor has greater responsibility.
Table 1: Average mean square error over all 100 datasets for each choice of fixed model mixture size. The
actual number of mixtures in the data varies. MRCS - mixture of regressors for covariate shift, IWLS - Importance weighted least squares, MRKL - Mixture of regressors, evaluated on regressor with best fit to test
distribution. MRREG - Mixture of regressors as a standard regression model, ignoring covariate shift. The
sixth row gives the average mean square error over all 100 datasets, with number of mixtures chosen using a
Bayesian information criterion for each case, and the last row gives the proportion of times MRCS performs
better than the other cases for a BIC choice of model. PValues: If two of the approaches were equivalent
performers, empirically better performance for 70/100 or more cases would only occur on less than 1 ? 10?4
such trials.
1 Mixture
2 Mixtures
3 Mixtures
4 Mixtures
5 Mixtures
BIC Choice
MCRS better
4.2
MRCS
0.588
0.536
0.601
0.623
0.612
0.6100
-
IWLS
0.797
0.804
0.831
0.817
0.837
0.7990
77/100
MRKL
3.274
2.673
3.390
2.823
2.817
2.8638
72/100
MRREG
0.890
0.881
0.887
0.894
0.898
0.8813
84/100
Auto-Mpg Test
It is useful to see that the approach does indeed make a noticeable difference on data that takes
the appropriate prior form, but that says nothing about how appropriate that prior is for real problems. Here we demonstrate the method on the auto-mpg problem from the UCI dataset. This
provides a natural scenario for demonstrating covariate shift. The auto-mpg data can be found
at http://www.ics.uci.edu/?mlearn/MLSummary.html and involves the problem of
predicting the city cycle fuel consumption of cars. One of the attributes is a class label dictating the
origin of a particular car. To demonstrate covariate shift we can consider the prediction task trained
on cars from one place of origin and tested on cars from another place of origin. Here we consider
predicting the fuel consumption (attribute 1) using the four continuous attributes. We train the model
using data on cars from origin 1, and test on cars from origin 2 and origin 3. We use the same test
algorithms as the previous section, but now using a Gaussian process regressors for each regression
function. The results of running this are in Table 2. The Gaussian process hyper-parameters were
optimised separately for each case. These are results obtained using a Bayesian Information Criterion for selecting the number of mixtures between 1 and 14 for each of the cases. We obtain similar
results if we compare methods with various fixed numbers of mixtures. Critically, we note that all
covariate shift methods performed better than a straight Gaussian Process predictor in this situation.
The mixture of Gaussian processes did not perform as well as the methods which explicitly recognised the covariate shift, although interestingly did perform better than a straight Gaussian process
predictor. Again the MRCS performed better overall.
5
Discussion
This paper establishes that explicit generative modelling of covariate shift can bring improvements
over conditional regression models, or over standard covariate shift methods that ignore the dependent data in the modelling process. The method is also better than using an identical mixture of
regressors model for the training data alone, as it utilises the positions of the independent test points
to help refine the mixture locations and the separation of regressors.
We expect significant improvements can be made with a fully Bayesian treatment of the parameters.
This framework is currently being extended to the case of multiple training and test datasets using a
fully Bayesian scheme, and will be the subject of future work. In this setting we have a Topic model,
Table 2: Tests of methods on the auto-mpg dataset. These are the (standardised) mean squared
errors for each approach. GP denotes the use of Gaussian Process regression for prediciton. Orgin
2, and Origin 3 denote the two different car origins used to test the model.
Origin 2
Origin 3
GP
1.192
0.898
MRCS
0.600
0.568
IWLS
0.700
0.691
MRKL
1.2243
1.3862
MRREG
0.7397
0.706
similar to Latent Dirichlet Allocation, where each dataset is built from a number of contributing
regression components, where each component is expressed in different proportions in each dataset.
The model and tests of this paper show that this multiple dataset extension could well be fruitful.
6
Conclusion
In this paper a novel approach to the problem of covariate shift has been developed that is demonstrably better than state of the art regression approaches, and better than the current standard for
covariate shift. These have been tested on both generated data, and on a real problem of covariate
shift, derived from a standard UCI dataset. Importance Weighted Least Squares is shown to be a
special case. Specifically we provide explicit modelling of the covariate shift process by assuming a
shift in the proportions of a number of latent components. A mixture of regressors model is used for
this purpose, but it differs from standard mixture of regressors by allowing sharing of the regression
functions between mixture components and explicitly including a model for the test set as part of
the process.
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2,226 | 302 | Phase-coupling in Two-Dimensional
Networks of Interacting Oscillators
Ernst Niebur, Daniel M. Kammen, Christof Koch,
Daniel Ruderman! & Heinz G. Schuster2
Computation and Neural Systems
Caltech 216-76
Pasadena, CA 91125
ABSTRACT
Coherent oscillatory activity in large networks of biological or artificial neural units may be a useful mechanism for coding information
pertaining to a single perceptual object or for detailing regularities
within a data set. We consider the dynamics of a large array of
simple coupled oscillators under a variety of connection schemes.
Of particular interest is the rapid and robust phase-locking that
results from a "sparse" scheme where each oscillator is strongly
coupled to a tiny, randomly selected, subset of its neighbors.
1
INTRODUCTION
Networks of interacting oscillators provide an excellent model for numerous physical
processes ranging from the behavior of magnetic materials to models of atmospheric
dynamics to the activity of populations of neurons in a variety of cortical locations.
Particularly prominent in the neurophysiological data are the 40-60 Hz oscillations
that have long been reported in the rat and rabbit olfactory bulb and cortex on the
basis of single-and multi-unit recordings as well as EEG activity (Freeman, 1978).
In addition, periodicities in eye movement reaction times (Poppel and Logothetis,
1986), as well as oscillations in the auditory evoked potential in response to single
click or a series of clicks (Madler and Poppel, 1987) all support a 30 - 50 Hz
framework for aspects of cortical activity. Two groups (Eckhorn et al., 1988, Gray
1 Permanent
2 Permanent
address: Department of Physics, University of California, Berkeley, CA 94720
address: Institut fiir Theoretische Physik, Universitat Kiel, 2300 Kiell, Germany.
123
124
Niebur, Kammen, Koch, Ruderman, and Schuster
and Singer, 1989; Gray et al., 1989) have recently reported highly synchronized,
stimulus specific oscillations in the 35 - 85 Hz range in areas 17, 18 and PMLS of
anesthetized as well as awake cats. Neurons with similar orientation tuning up to
7 mm apart show phase-locked oscillations with a phase shift of less than 1 msec
that have been proposed to play a role in the coding of visual information (Crick
and Koch, 1990, Niebur et al. 1991).
The complexity of networks of even relatively simple neuronal units - let alone
"real" cortical cells - warrants a systematic investigation of the behavior of two
dimensional systems. To address this question we begin with a network of mathematically simple limit-cycle oscillators. While the dynamics of pairs of oscillators are
well understood (Sakaguchi, et al. 1988, Schuster and Wagner, 1990a,b), this is not
the case for large networks with nontrivial connection schemes. Of general interest
is the phase-coupling that results in networks of oscillators with different coupling
schemes. We will summarize some generic features of simple nearest-neighbor coupled models, models where each oscillator receives input from a large neighborhood,
and of "sparse" connection geometries where each cell is connected to only a tiny
fraction of the units in its neighborhood, but with large coupling strength. The
numerical work was performed on a CM-2 Connection Machine and involved 16,384
oscillators in a 128 by 128 square grid.
2
The Model
The basic unit in our networks is an oscillator whose phase (Jij is 21r periodic and
which has the intrinsic frequency Wij. The dynamics of an isolated oscillator are
described by:
d;)
d(J? .
=
(1)
Wij.
The influence of the network can be expressed as an additional interaction term,
(2)
The coupling function, lij we used is expressed as the sum of terms, each one
consisting of the product of a coupling strength and the sine of a phase difference
(see below, eq. 3). The sinusoidal form of the interaction is, of course, linear for
small differences.
This system, and numerous variants, has received a considerable amount of attention
from solid state physicists (see, e.g. Kosterlitz and Thouless 1973, and Sakaguchi et
al. 1988), although primarily in the limit of t - 00. With an interest in the possible
role of networks of oscillators in the parsing or segregating of incident signals in
nervous systems, we will concentrate on short time, non-equilibrium, properties.
We shall confine ourselves to two generic network configurations described by
L
' -d(Jij
J .. 1:lszn((J??
dt = w .. + a
&)
&),
1:1
&)
(J1:I) ,
(3)
Phase-coupling in Two-Dimensional Networks of Interacting Oscillators
where 0' designates the global strength of the interaction, and the geometry of the
interactions is incorporated in Jij ,kl'
The networks are all defined on a square grid and they are characterized as follows:
1: Gaussian Connections. The cells are connected to every oscillator within a
specified neighborhood with Gaussian weighted connections. Hence,
J
ij ,kl
1
( (i = 271'0'
exp
k) 2
+ (j -
20'2
I) 2 )
.
(4)
We truncate this function at 20', i.e. Jij,kl = 0 if (i - k)2 + (j _1)2 > (20')2. While
the connectivity in the nearest neighbor case is 4, the connectivity is significantly
higher for the Gaussian connection schemes: Already 0'
2 yields 28 connections
per cell, and the largest network we studied, with 0' 6, results in 372 connections
per cell.
=
=
2: Sparse Gaussian Connections. In this scheme we no longer require symmetric connections, or that the connection pattern is identical from unit to unit. A
given cell is connected to a fixed number, n, of neighboring cells, with the probability
of a given connection determined by
'"
rij,kl
1
= 271'0' exp
( (i -
k?20'2
+ (j - I? ) .
(5)
Jij,kl is unity with probability Pij,kl and zero otherwise . This connection scheme
is constructed by drawing for each lattice site n coordinate pairs from a Gaussian
distribution, and use these as the indices of the cells that are connected with the
oscillator at location (i,j). Therefore, the probability of making a connection decreases with distance. If a connection is made, however, the weight is the same as
for all other connections. We typically used n = 5, and in all cases 2 ~ n < 10.
For all networks, the sum of the weights of all connections with a given oscillator i, j
was conserved and chosen as 0' 2::kl Jij,kl = 10 * w, where w is the average frequency
of all N oscillators in the system, w =
2::ij Wij. By this procedure, the total
impact of the interaction term is identical in all cases.
k
3
RESULTS
Perhaps the most basic, and most revealing, comparison of the behavior of the
models introduced above is the two-point correlation function of phase-coupling,
which is defined as
(6)
where R is defined as the separation between a pair of cells, R = hj - nIl. We
compute and then average C(R, t) over 10,000 pairs of oscillators separated by R in
the array. In all cases, the frequencies Wij are chosen randomly, with a Gaussian distribution with mean 0.5 and variance 1. In Figure 1 we plot C(R, t) for separations
125
126
Niebur, Kammen, Koch, Ruderman, and Schuster
of R = 20, 30, 40, 50, 6, and 70 oscillators. Time is measured in oscillation periods of the mean oscillator frequency, w. At t 0, phases are distributed randomly
between 0 and 27r with a uniform distribution. The case of Gaussian connectivity
with u 6 and hence 372 connection per cell is seen in Figure l(a), and the sparse
connectivity scheme with u
6 and n
5 is presented in Figure l(b). The most
striking difference is that correlation levels of over 0.9 are rapidly achieved in the
sparse scheme for all cases, even for separations of 70 oscillators (plotted as asterisks, *), while there are clear separation-dependent differences in the phase-locking
behavior of the Gaussian model. In fact, even after t
10 there is no significant
locking over the longer distances of R = 50,60, or 70 units. For local connectivity
schemes, like Gaussian connectivity with u
2 or nearest neighbors connections,
no long-range order evolves even at larger times (data not shown).
=
=
=
=
=
=
=
Data in Fig. 1 were computed with a uniform phase distribution for t
o. An interesting and robust feature of the dynamics emerges when the influence of different
types of initial phase distributions are examined. In Figure 2 we plot the probability distribution of phases at different early times. In Figure 2( a) the distribution
of phases is plotted at t
0 (diamonds), t
0.2 ("plus signs, +) and at t
0.4
(squares) for the sparse scheme with a uniform initial distribution. In Figure 2(b),
the evolution of a Gaussian initial distribution centered at () = 7r of the phases
is plotted. Note the slight curve in the distribution at t = 0, indicating that the
Gaussian initial seeding is rather slight (variance u = 27r) . Remarkably, however,
this has a dramatic impact on the phase-locking as after two-tenth of an average
cycle time (" plus" signs) there is already a pronounced peak in the distribution . At
t 0.4 (squares) the system that started with the uniform distribution begins to
only exhibit a slight increase in the phase-correlation while the system with Gaussian distributed initial phases is strongly peaked with virtually no probability of
encountering phase values that differ significantly from the mean.
=
=
=
=
4
DISCUSSION
The power of the sparse connection scheme to rapidly generate phase-locking throughout the network that is equivalent, or superior, to that of the massively interconnected Gaussian scheme highlights a trade-off in network dynamics: massive averaging versus strong, long-range, connections. With n = 5, the sparse scheme
effectively "tiles" a two-dimensional lattice and tightly phase-locks oscillators even
at opposite corners of the array. Similar results are obtained even with n = 2 (data
not shown).
In many ways the Gaussian and sparse geometries reperent opposing avenues to
achieve global coherence: exhaustive local coupling or distributed, but powerful
long-range coupling. The amount of wiring necessary to implement these schemes
is, however, radically different.
Phase-coupling in Two-Dimensional Networks of Interacting Oscillators
Acknowledgement
EN is supported by the Swiss National Science Foundation through Grant No. 822025941. DMK is a recipient of a Weizman Postdoctoral Fellowship. CK acknowledges
support from the Air Force Office of Scientific Research, a NSF Presidential Young
Investigator Award and from the James S. McDonnell Foundation. HGS is supported by the Volkswagen Foundation.
References
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visual cortex? Bioi. Cybern., 60, 121 - 130.
Freeman, W. J. 1978. Spatial properties of an EEG event in the olfactory bulb and
cortex. Elect. Clin. Neurophys., 44, 586 - 605.
Gray, C. M., Konig, P., Engel, A. K. and Singer, W. 1989. Oscillatory responses
in cat visual cortex exhibit inter-columnar synchronization which reflects global
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Kosterlitz, J. M. and Thouless, D. J. 1973. Ordering, metastability and phase
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Madler, C. and Poppel, E. 1987. Auditory evoked potentials indicate the loss of
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Cybernetics, 64, 83.
127
128
Niebur, Kammen, Koch, Ruderman, and Schuster
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Figure 1: Tw~point correlation functions, C(R, t), for various separations, R, in
(a) the (T = 6 Gaussian scheme with 372 connections per cell and (b) the spar~e
connection scheme with (T
6 and n
5 connections per cell. Separations of
R = 20 (diamonds), R = 30 ("Plus" signs, +), R = 40 (squares), R = 50 (crosses,
x), R = 60 (triangles), and R = 70 (asterisks, *) are shown. Note the rapid locking
for all lengths in the sparse scheme (b) while the Gaussian scheme (a) appears far
more "diffusive," with progressively poorer and slower locking as R increases.
=
=
Phase-coupling in Two-Dimensional Networks of Interacting Oscillators
14~----~----~----~-----r----~--~--~----~----~----~
(A I
10
PI' 1
8
r.
Figure 2: Snapshots of the distribution of phases in the sparse scheme
(n
5,0' = 6) when the system begins from (a) uniform and (b) a Gaussian
"biased" initial distribution. The figures show the probability P(O) to find a phase
between 0 and 0 + dO (bin size 'KIlO). At t 0, the distribution is fiat (a) or very
slightly curved (b); see text. The difference in the time evolution can clearly be
seen in the state of the system after t = 0.2 ("plus" signs, +) and t = 0.4 (squares).
=
=
129
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2,227 | 3,020 | Multi-dynamic Bayesian Networks
Karim Filali and Jeff A. Bilmes
Departments of Computer Science & Engineering and Electrical Engineering
University of Washington
Seattle, WA 98195
{karim@cs,bilmes@ee}.washington.edu
Abstract
We present a generalization of dynamic Bayesian networks to concisely describe
complex probability distributions such as in problems with multiple interacting
variable-length streams of random variables. Our framework incorporates recent
graphical model constructs to account for existence uncertainty, value-specific
independence, aggregation relationships, and local and global constraints, while
still retaining a Bayesian network interpretation and efficient inference and learning techniques. We introduce one such general technique, which is an extension
of Value Elimination, a backtracking search inference algorithm. Multi-dynamic
Bayesian networks are motivated by our work on Statistical Machine Translation (MT). We present results on MT word alignment in support of our claim that
MDBNs are a promising framework for the rapid prototyping of new MT systems.
1
INTRODUCTION
The description of factorization properties of families of probabilities using graphs (i.e., graphical
models, or GMs), has proven very useful in modeling a wide variety of statistical and machine
learning domains such as expert systems, medical diagnosis, decision making, speech recognition,
and natural language processing. There are many different types of graphical model, each with its
own properties and benefits, including Bayesian networks, undirected Markov random fields, and
factor graphs. Moreover, for different types of scientific modeling, different types of graphs are
more or less appropriate. For example, static Bayesian networks are quite useful when the size of
set of random variables in the domain does not grow or shrink for all data instances and queries of
interest.
Hidden Markov models (HMMs), on the other hand, are such that the number of underlying random
variables changes depending on the desired length (which can be a random variable), and HMMs
are applicable even without knowing this length as they can be extended indefinitely using online
inference. HMMs have been generalized to dynamic Bayesian networks (DBNs) and temporal conditional random fields (CRFs), where an underlying set of variables gets repeated as needed to fill
any finite but unbounded length. Probabilistic relational models (PRMs) [5] allow for a more complex template that can be expanded in multiple dimensions simultaneously. An attribute common
to all of the above cases is that the specification of rules for expanding any particular instance of a
model is finite. In other words, these forms of GM allow the specification of models with an unlimited number of random variables (RVs) using a finite description. This is achieved using parameter
tying, so while the number of RVs increases without bound, the number of parameters does not.
In this paper, we introduce a new class of model we call multi-dynamic Bayesian networks. MDBNs
are motivated by our research into the application of graphical models to the domain of statistical
machine translation (MT) and they have two key attributes from the graphical modeling perspective.
First, an MDBN generalizes a DBN in that there are multiple ?streams? of variables that can get
unrolled, but where each stream may be unrolled by a differing amount. In the most general case,
connecting these different streams together would require the specification of conditional probabil-
ity tables with a varying and potentially unlimited number of parents. To avoid this problem and
retain the template?s finite description length, we utilize a switching parent functionality (also called
value-specific independence). Second, in order to capture the notion of fertility in MT-systems
(defined later in the text), we employ a form of existence uncertainty [7] (that we call switching
existence), whereby the existence of a given random variable might depend on the value of other
random variables in the network.
Being fully propositional, MDBNs lie between DBNs and PRMs in terms of expressiveness. While
PRMs are capable of describing any MDBN, there are, in general, advantages to restricting ourselves
to a more specific class of model. For example, in the DBN case, it is possible to provide a bound on
inference costs just by looking at attributes of the DBN template only (e.g., the left or right interfaces
[12, 2]). Restricting the model can also make it simpler to use in practice. MDBNs are still relatively
simple, while at the same time making possible the easy expression of MT systems, and opening
doors to novel forms of probabilistic inference as we show below.
In section 2, we introduce MDBNs, and describe their application to machine translation showing
how it is possible to represent even complex MT systems. In section 3, we describe MDBN learning
and decoding algorithms. In section 4, we present experimental results in the area of statistical
machine translation, and future work is discussed in section 5.
2
MDBNs
A standard DBN [4] template consists of a directed acyclic graph G = (V, E) = (V1 ? V2 , E1 ?
E2 ? E2? ) with node set V and edge set E. For t ? {1, 2}, the sets Vt are the nodes at slice t, Et
are the intra-slice edges between nodes in Vt , and Et? are the inter-slice edges between nodes in V1
and V2 . To unroll a DBN to length T , the nodes V2 along with the edges adjacent to any node in V2
are cloned T ? 1 times (where parameters of cloned variables are constrained to be the same as the
template) and re-connected at the corresponding places.
An MDBN with K streams consists of the union of K DBN templates along with a template structure specifying rules to connect the various streams together. An MDBN template is a directed
graph
[
[
(k)
G = (V, E) = ( V (k) , E (k) ? Ell )
k
(k)
(k)
th
k
(k)
Ell
where (V , E ) is the k DBN, and the edges
are rules specifying how to connect stream
k to the other streams. These rules are general in that they specify the set of edges for all values of
Tk . There can be arbitrary nesting of the streams such as, for example, it is possible to specify a
model that can grow along several dimensions simultaneously.
An MDBN also utilizes ?switching existence?, meaning some subset of the variables in V bestow
existence onto other variables in the network. We call these variables existence bestowing (or ebnodes). The idea of bestowing existence is well defined over a discrete space, and is not dissimilar
to a variable length DBN. For example, we may have a joint distribution over lengths as follows:
p(X1 , . . . , XN , N ) = p(X1 , . . . , Xn |N = n)p(N = n)
where here N is an eb-node that determines the number of other random variables in the DGM.
Our notion of eb-nodes allows us to model certain characteristics found within machine translation
systems, such as ?fertility? [3], where a given English word is cloned a random number of times
in the generative process that explains a translation from French into English. This random cloning
might happen simultaneously at all points along a given MDBN stream. This means that even for a
given fixed stream length Ti = ti , each stream could have a randomly varying number of random
variables. Our graphical notation for eb-nodes consists of the eb-node as a square box containing
variables whose existence is determined by the eb-node.
We start by providing a simple example of an expanded MDBN for three well known MT systems,
namely the IBM models 1 and 2 [3], and the ?HMM? model [15].1 We adopt the convention in [3]
that our goal is to translate from a string of French words F = f of length M = m into a string
of English words E = e of length L = l ? of course these can be any two languages. The basic
generative (noisy channel) approach when translating from French to English is to represent the joint
1
We will refer to it as M-HMM to avoid confusion with regular HMMs.
distribution P (f , e) = P (f |e)P (e). P (e) is a language model specifying the prior over the word
string e. The key goal is to produce a finite-description length representation for P (f |e) where f
and e are of arbitrary length. A hidden P
alignment string, a, specifies how the English words align to
the French word, leading to P (f |e) = a P (f , a|e).
Figure 1(a) is a 2-stream MDBN expanded representation of the three models, in this case ? = 4
and m = 3. As shown, it appears that the fan-in to node fi will be ? and thus will grow without
bound. However, a switching mechanism whereby P (fi |e, ai ) = P (fi |eai ) limits the number of
parameters regardless of L. This means that the alignment variable ai indicates the English word
eai that should be aligned to French word fi . The variable e0 is a null word that connects to French
words not explained by any of e1 , . . . , e? . The graph expresses all three models ? the difference is
that, in Models 1 and 2, there are no edges between aj and aj+1 . In Model 1, p(aj = ?) is uniform
on the set {1, . . . , L}; in Model 2, the distribution over aj is a function only of its position j, and on
the English and French lengths ? and m respectively. In the M-HMM model, the ai variables form
a first order Markov chain.
l
e0
?
e1
e3
e2
e4
?0
?01
f1
a1
f2
a2
f3
u v
a3
m
(a) Models 1,2 and M-HMM
e1
e2
e3
?1
?2
?3
m?
?02
?11
?12
?13
?21
?02
?11
?12
?13
?21
f1
f2
f3
f4
f5
f6
a1
a2
a3
a4
a5
a6
?01
w
y
x
m
(b) Expanded M3 graph
Figure 1: Expanded 2-stream MDBN description of IBM Models 1 and 2, and the M-HMM model for
MT; and the expanded MDBN description of IBM Model 3 with fertility assignment ?0 = 2, ?1 =
3, ?2 = 1, ?3 = 0.
From the above, we see that it would be difficult to express this model graphically using a standard
DBN since L and M are unequal random variables. Indeed, there are two DBNs in operation, one
consisting of the English string, and the other consisting of the French string and its alignment.
Moreover, the fully connected structure of the graph in the figure can represent the appropriate
family of model, but it also represents models whose parameter space grows without bound ? the
switching function allows the model template to stay finite regardless of L and M .
With our MDBN descriptive abilities complete, it is now possible to describe the more complex IBM
models 3, and 4[3] (an MDBN for Model3 is depicted in fig. 1(b)). The top most random variable, ?,
is a hidden switching existence variable corresponding to the length of the English string. The box
abutting ? includes all the nodes whose existence depends on the value of ?. In the figure, ? = 3,
thus resulting in three English words e1 , e2 , and e3 connected using a second-order Markov chain.
To each English word ei corresponds a conditionally dependent fertility eb-node ?i , which indicates
how many times ei is used by words in the French string. Each ?i in turn controls the existence of a
set of variables under it. Given the fertilities (the figure depicts the case ?1 = 3, ?2 = 1, ?3 = 0), for
each word ei , ?i French word variables are granted existence and are denoted by ?i1 , ?i2 , . . . , ?i?i ,
what is called the tablet [3] of ei . The values taken by the ? variables need to match the actual
observed French sequence f1 , . . . , fm . This is represented as a shared constraint between all the
f , ?, and ? variables which have incoming edges into the observed variable v. v?s conditional
probability table is such that it is one only when the associated constraint is satisfied2 . The variable
2
This type of encoding of constraints corresponds to the standard mechanism used by Pearl [14]. A naive
implementation, however, would enumerate a number of configurations exponential in the number of constrained variables, while typically only a small fraction of the configurations would have positive probability.
?i,k ? {1, . . . , m} is a switching dependency parent with respect to the constraint variable v and
determines which fj participates in an equality constraint with ?i,k .
The bottom variable m is a switching existence node (observed to be 6 in the figure) with corresponding French word sequence and alignment variables. The French sequence participates in the v
constraint described above, while the alignment variables aj ? {1, . . . , ?}, j ? 1, . . . , m constrain
the fertilities to take their unique allowable values (for the given alignment). Alignments also restrict
the domain of permutation variables, ?, using the constraint variable x. Finally, the domain size of
each aj has to lie in the interval [0, ?] and that is enforced by the variable u. The dashed edges
connecting the alignment a variables represent an extension to implement an M3/M-HMM hybrid.
P?
The null submodel involving the deterministic node m? (= i=1 ?i ) and eb-node ?0 accounts for
French words that are not explained by any of the English words e1 , . . . , e? . In this submodel,
successive permutation variables are ordered and this constraint is implemented using the observed
child w of ?0i and ?0(i+1) .
Model 4 [3] is similar to Model 3 except that the former is based on a more elaborate distortion
model that uses relative instead of absolute positions both within and between tablets.
3
Inference, Parameter Estimation and MPE
Multi-dynamic Bayesian Networks are amenable to any type of inference that is applicable to regular
Bayesian networks as long as switching existence relationships are respected and all the constraints
(aggregation for example) are satisfied. Unfortunately DBN inference procedures that take advantage of the repeatable template and can preprocess it offline, are not easy to apply to MDBNs. A
case in point is the Junction Tree algorithm [11]. Triangulation algorithms exist that create an offline
triangulated version of the input graph and do not re-triangulate it for each different instance of the
input data [12, 2]. In MDBNs, due to the flexibility to unroll templates in several dimensions and to
specify dependencies and constraints spanning the entire unrolled graph, it is not obvious how we
can exploit any repetitive patterns in a Junction Tree-style offline triangulation of the graph template.
In section 4, we discuss sampling inference methods we have used. Here we discuss our extension to
a backtracking search algorithm with the same performance guarantees as the JT algorithm, but with
the advantage of easily handling determinism, existence uncertainty, and constraints, both learned
and explicitly stated.
Value Elimination (VE) ([1]), is a backtracking Bayesian network inference technique that caches
factors associated with portions of the search tree and uses them to avoid iterating again over the
same subtrees. We follow the notation introduced in [1] and refer the reader to that paper for details
about VE inference. We have extended the VE inference approach to handle explicitly encoded
constraints, existence uncertainty, and to perform approximate local domain pruning (see section 4).
We omit these details as well as others in the original paper and briefly describe the main data
structure required by VE and sketch the algorithm we refer to as FirstPass (fig. 1) since it constitutes
the first step of the learning procedure, our main contribution in this section.
A VE factor, F , is such that we can write the following marginal of the joint distribution
X
P (X = x, Y = y, Z) = F.val ? f (Z)
X=x
such that (X?Y)?Z = ?, F.val is a constant, and f (Z) a function of Z only. Y is a set of variables
previously instantiated in the current branch of search tree to the value vector y. The pair (Y, y) is
referred to as a dependency set (F.Dset). X is referred to as a subsumed set (F.Sset). By caching
the tuple (F.Dset, F.Sset, F.val), we avoid recomputing the marginal again whenever (1) F.Dset
is active, meaning all nodes stored in F.Dset are assigned their cached values in the current branch
of the search tree; and (2) none of the variables in F.Sset are assigned yet.
FirstPass (alg. 1) visits nodes in the graph in Depth First fashion. In line 7, we get the values of
all Newly Single-valued (NSV) CPTs i.e., CPTs that involve the current node, V , and in which all
We use a general directed domain pruning constraint. Deterministic relationships then become a special case
of our constraint whereby the domain of the child variable is constrained to a single value with probability one.
Variable traversal order: A, B, C, and D.
Factors are numbered by order of creation.
*Fi denotes the activation of factor i.
Tau values propagated recursively
F7: Dset={} Sset={A,B,C,D} val=P(E=e)
F7.tau = 1.0 = P(Evidence)/F7.val
A
F5: Dset={A=0} Sset={B,C,D}
F2
D
*F1 *F2
Factor values needed for c(A=0) and c(C=0,B=0) computation:
F5.val=P(B=0|A=0)*F3.val+P(B=1|A=0)*F4.val
F3.val=P(C=0|B=0)*F1.val+P(C=1|B=0)*F2.val
F4.val=P(C=0|B=1)*F1.val+P(C=1|B=1)*F2.val
F1.val=P(D=0|C=0)P(E=e|D=0)+P(D=1|C=0)P(E=e|D=1)
F2.val=P(D=0|C=1)P(E=e|D=0)+P(D=1|C=1)P(E=e|D=1)
First pass
C
*F3 *F4
Second pass
D
B
F4
C
F6.tau = F7.tau * P(A=1)
1
B
F3: Dset={B=0} Sset={C,D}
F1
F5.tau = F7.tau * P(A=0)
F6
0
F3.tau = F5.tau * P(B=0|A=0) + F6.tau * P(B=0|A=1) = P(B=0)
F4.tau = F5.tau * P(B=1|A=0) + F6.tau * P(B=1|A=1) = P(B=1)
F1.tau = F3.tau * P(C=0|B=0) + F4.tau * P(C=0|B=1) = P(C=0)
F2.tau = F3.tau * P(C=1|B=0) + F4.tau * P(C=1|B=1) = P(C=1)
c(A=0)=(1/P(e))*(F7.tau*P(A=0)*F5.val)=(1/P(e))(P(A=0)*P(E=e|A=0))=P(A=0|E=e)
c(C=0,B=0)=(1/P(e))*F3.tau*P(C=0|B=0)*F1.val
=(1/P(e) * (P(A=0,B=0)+P(A=1,B=0)) * P(C=0|B=0) * F1.val
=(1/P(e)) * P(B=0) * P(C=0|B=0) * F1.val
=(1/P(e)) * P(B=0) * P(C=0|B=0) * F1.val
=(1/P(e)) * P(C=0,B=0) * F1.val
=P(C=0,B=0,E=e)/P(e)=P(C=0,B=0|E=e)
Figure 2: Learning example using the Markov chain A ? B ? C ? D ? E, where E is observed.
In the first pass, factors (Dset, Sset and val) are learned in a bottom up fashion. Also, the normalization
constant P (E = e) (probability of evidence) is obtained. In the second pass, tau values are updated in a
top-down fashion and used to calculate expected counts c(F.head, pa(F.head)) corresponding to each F.head
(the figure shows the derivations for (A=0) and (C=0,B=0), but all counts are updated in the same pass).
other variables are already assigned (these variables and their values are accumulated into Dset). We
also check for factors that are active, multiply their values in, and accumulate subsumed vars in Sset
(to avoid branching on them). In line 10, we add V to the Sset. In line 11, we cache a new factor F
with value F.val = sum. We store V into F.head, a pointer to the last variable to be inserted into
F.Sset, and needed for parameter estimation described below. F.Dset consists of all the variables,
except V , that appeared in any NSV CPT or the Dset of an activated factor at line 6.
Regular Value Elimination is query-based, similar to variable elimination and recursive
conditioning?what this means is that to answer a query of the type P (Q|E = e), where Q is
query variable and E a set of evidence nodes, we force Q to be at the top of the search tree, run the
backtracking algorithm and then read the answers to the queries P (Q = q|E = e), q ? Dom[Q],
along each of the outgoing edges of Q. Parameter estimation would require running a number of
queries on the order of the number of parameters to estimate.
We extend VE into an algorithm that allows us to obtain Expectation Maximization sufficient statistics in a single run of Value Elimination plus a second pass, which can never take longer than the first
one (and in practice is much faster). This two-pass procedure is analogous to the collect-distribute
evidence procedure in the Junction Tree algorithm, but here we do this via a search tree.
Let ?X=x|pa(X)=y be a parameter associated with variable X with value x and parents Y = pa(X)
when they have value y. Assuming a maximum likelihood learning scenario3 , to estimate
?X=x|pa(X)=y , we need to compute
X
f (X = x, pa(X) = y, E = e) =
P (W, X = x, pa(X) = y, E = e)
W\{X,pa(X)}
which is a sum of joint probabilities of all configurations that are consistent with the assignment
{X = x, pa(X) = y}. If we were to turn off factor caching, we would enumerate all such variable
configurations and could compute the sum. When standard VE factors are used, however, this is no
longer possible whenever X or any of its parents becomes subsumed. Fig. 2 illustrates an example
of a VE tree and the factors that are learned in the case of a Markov chain with an evidence node
at the end. We can readily estimate the parameters associated with variables A and B as they are
not subsumed along any branch. C and D become subsumed, however, and we cannot obtain the
correct counts along all the branches that would lead to C and D in the full enumeration case.
To address this issue, we store a special value, F.tau, in each factor. F.tau holds the sum over
all path probabilities from the first level of the search tree to the level at which the factor F was
3
For Bayesian networks the likelihood function decomposes such that maximizing the expectation of the
complete likelihood is equivalent to maximizing the ?local likelihood? of each variable in the network.
either created or activated. For example, F 6.tau in fig. 2 is simply P (A = 1). Although we can
compute F 3.tau directly, we can also compute it recursively using F 5.tau and F 6.tau as shown in
the figure. This is because both F 5 and F 6 subsume F 3: in the context {F 5.Dset}, there exists a
(unique) value dsub of F 5.head4 s.t. F 3 becomes activable. Likewise for F 6. We cannot compute
F 1.tau directly, but we can, recursively, from F 3.tau and F 4.tau by taking advantage of a similar
subsumption relationship. In general, we can show that the following recursive relationship holds:
Q
X
Fact .val
pa
F.tau ?
F .tau ? N SVF pa .head=dsub ? Fact ?Fact
(1)
F.val
pa
pa
F
?F
where F pa is the set of factors that subsume F , Fact is the set of all factors (including F ) that become active in the context of {F pa .Dset, F pa .head = dsub } and N SVF pa .head=dsub is the product
of all newly single valued CPTs under the same context. For top-level factors (not subsumed by any
factor), F.tau = Pevidence /F.val, which is 1.0 when there is a unique top-level factor.
Alg. 2 is a simple recursive computation of eq. 1 for each factor. We visit learned factors in the
reverse order in which they were learned to ensure that, for any factor F ? , F ? .tau is incremented
(line 13) by any F that might have activated F ? (line 12). For example, in fig. 2, F 4 uses F 1 and
F 2, so F 4.tau needs to be updated before F 1.tau and F 2.tau. In line 11, we can increment the
counts for any NSV CPT entries since F.tau will account for the possible ways of reaching the
configuration {F.Dset, F.head = d} in an equivalent full enumeration tree.
Algorithm 1: FirstPass(level)
1
2
3
4
5
6
7
8
9
10
Input: Graph G
Output: A list of learned factors and Pevidence
Select var V to branch on
if V ==NONE then return
Sset={}, Dset={}
for d ? Dom[V ] do
V ?d
prod = productOfAllNSVsAndActiveFactors(Dset, Sset)
if prod != 0 then FirstPass(level+1)
sum += prod
Sset = Sset ? {V }
cacheNewFactor(F.head ? V ,F.val ? sum, F.Sset ? Sset, F.Dset ? Dset);
Algorithm 2: SecondPass()
1
2
3
4
5
6
7
8
9
10
11
12
13
Input: F : List of factors in the reverse order learned in the first pass and Pevidence .
Result: Updated counts
foreach F ? F do
if F.Dset = {} then
F.tau ? Pevidence /F.val
else
F.tau ? 0.0
Assign vars in F.Dset to their values
V ? F.head (last node to have been subsumed in this factor)
foreach d ? Dom[V ] do
prod = productOfAllNSVsAndActiveFactors()
prod? = F.tau
foreach newly single-valued CPT C do count(C.child,C.parents)+=prod/Pevidence
F ? =getListOfActiveFactors()
for F ? ? F ? do F ? .tau+ = prod/F ? .val
Most Probable Explanation We compute MPE using a very similar two-pass algorithm. In the
first pass, factors are used to store a maximum instead of a summation over variables in the Sset. We
also keep track of the value of F.head at which the maximum is achieved. In the second pass, we
recursively find the optimal variable configuration by following the trail of factors that are activated
when we assign each F.head variable to its maximum value starting from the last learned factor.
4
Recall, F.head is the last variable to be added to a newly created factor in line 10 of alg. 1
4
MACHINE TRANSLATION WORD ALIGNMENT EXPERIMENTS
A major motivation for pursuing the type of representation and inference described above is to
make it possible to solve computationally-intensive real-world problems using large amounts of data,
while retaining the full generality and expressiveness afforded by the MDBN modeling language. In
the experiments below we compare running times of MDBNs to GIZA++ on IBM Models 1 through
4 and the M-HMM model. GIZA++ is a special-purpose optimized MT word alignment C++ tool
that is widely used in current state-of-the-art phrase-based MT systems [10] and at the time of this
writing is the only publicly available software that implements all of the IBM Models. We test on
French-English 107 hand-aligned sentences5 from a corpus of the European parliament proceedings
(Europarl [9]) and train on 10000 sentence pairs from the same corpus and of maximum number of
words 40. The Alignment Error Rate (AER) [13] evaluation metric quantifies how well the MPE
assignment to the hidden alignment variables matches human-generated alignments.
Several pruning and smoothing techniques are used by GIZA and MDBNs. GIZA prunes low lexical
(P (f |e)) probability values and uses a default small value for unseen (or pruned) probability table
entries. For models 3 and 4, for which there is no known polynomial time algorithm to perform
the full E-step or compute MPE, GIZA generates a set of high probability alignments using an MHMM and hill-climbing and collects EM counts over these alignments using M3 or M4. For MDBN
models we use the following pruning strategy: at each level of the search tree we prune values which,
together, account for the lowest specified percentage of the total probability mass of the product of
all newly active CPTs in line 6 of alg. 1. This is a more effective pruning than simply removing
low-probability values of each CPD because it factors in the joint contribution of multiple active
variables.
Table 1 shows a comparison of timing numbers obtained GIZA++ and MDBNs. The runtime numbers shown are for the combined tasks of training and decoding; however, training time dominates
given the difference in size between train and test sets. For models 1 and 2 neither GIZA nor MDBNs
perform any pruning. For the M-HMM, we prune 60% of probability mass at each level and use a
Dirichlet prior over the alignment variables such that long-range transitions are exponentially less
likely than shorter ones.6 This model achieves similar times and AER to GIZA?s. Interestingly,
without any pruning, the MDBN M-HMM takes 160 minutes to complete while only marginally
improving upon the pruned model. Experimenting with several pruning thresholds, we found that
AER would worsen much more slowly than runtime decreases.
Models 3 and 4 have treewidth equal to the number of alignment variables (because of the global
constraints tying them) and therefore require approximate inference. Using Model 3, and a drastic
pruning threshold that only keeps the value with the top probability at each level, we were able to
achieve an AER not much higher than GIZA?s. For M4, it achieves a best AER of 31.7% while we
do not improve upon Model3, most likely because a too restrictive pruning. Nevertheless, a simple
variation on Model3 in the MDBN framework achieves a lower AER than our regular M3 (with
pruning still the same). The M3-HMM hybrid model combines the Markov alignment dependencies
from the M-HMM model with the fertility model of M3.
MCMC Inference Sampling is widely used for inference in high-treewidth models. Although
MDBNs support Likelihood Weighing, it is very inefficient when the probability of evidence is very
small, as is the case in our MT models. Besides being slow, Markov chain Monte Carlo can be
problematic when the joint distribution is not positive everywhere, in particular in the presence of
determinism and hard constraints. Techniques such as blocking Gibbs sampling [8] try to address the
problem. Often, however, one has to carefully choose a problem-dependent proposal distribution.
We used MCMC to improve training of the M3-HMM model. We were able to achieve an AER of
32.8% (down from 39.1%) but using 400 minutes of uniprocessor time.
5
CONCLUSION
The existing classes of graphical models are not ideally suited for representing SMT models because
?natural? semantics for specifying the latter combine flavors of different GM types on top of standard
directed Bayesian network semantics: switching parents found in Bayesian Multinets [6], aggregation relationships such as in Probabilistic Relational Models [5], and existence uncertainty [7]. We
5
Available at http://www.cs.washington.edu/homes/karim
French and English have similar word orders. On a different language pair, a different prior might be more
appropriate. With a uniform prior, the MDBN M-HMM has 36.0% AER.
6
Model
Init
M1
M2
M-HMM
M3
M4
M3-HMM
GIZA++
M1
M-HMM
1m45s (47.7%)
N/A
2m02s (41.3%)
N/A
4m05s (35.0%)
N/A
2m50 (45%)
5m20s (38.5%)
5m20s (34.8%) 7m45s (31.7%)
N/A
MDBN
M1
3m20s (48.0%)
5m30s (41.0%)
4m15s (33.0%)
12m (43.6%)
25m (43.6%)
9m30 (41.0%)
M-HMM
N/A
N/A
N/A
9m (42.5%)
23m (42.6%)
9m15s (39.1%)
MCMC 400m (32.8%)
Table 1: MDBN VE-based learning versus GIZA++ timings and %AER using 5 EM iterations. The columns
M1 and M-HMM correspond to the model that is used to initialize the model in the corresponding row. The
last row is a hybrid Model3-HMM model that we implemented using MDBNs and is not expressible using GIZA.
have introduced a generalization of dynamic Bayesian networks to easily and concisely build models
consisting of varying-length parallel asynchronous and interacting data streams. We have shown that
our framework is useful for expressing various statistical machine translation models. We have also
introduced new parameter estimation and decoding algorithms using exact and approximate searchbased probability computation. While our timing results are not yet as fast as a hand-optimized C++
program on the equivalent model, we have shown that even in this general-purpose framework of
MDBNs, our timing numbers are competitive and usable. Our framework can of course do much
more than the IBM and HMM models. One of our goals is to use this framework to rapidly prototype
novel MT systems and develop methods to statistically induce an interlingua. We also intend to use
MDBNs in other domains such as multi-party social interaction analysis.
References
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[4] T. Dean and K. Kanazawa. Probabilistic temporal reasoning. AAAI, pages 524?528, 1988.
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Uncertain Reasoning., 1995.
[9] P. Koehn.
Europarl:
A multilingual corpus for evaluation of machine
http://www.isi.edu/koehn/publications/europarl, 2002.
translation.
[10] P. Koehn, F. Och, and D. Marcu. Statistical phrase-based translation. In NAACL/HLT 2003, 2003.
[11] S. Lauritzen. Graphical Models. Oxford Science Publications, 1996.
[12] K. Murphy. Dynamic Bayesian Networks: Representation, Inference and Learning. PhD thesis, U.C.
Berkeley, Dept. of EECS, CS Division, 2002.
[13] F. J. Och and H. Ney. Improved statistical alignment models. In ACL, pages 440?447, Oct 2000.
[14] J. Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, 2nd printing edition, 1988.
[15] S. Vogel, H. Ney, and C. Tillmann. HMM-based word alignment in statistical translation. In Proceedings
of the 16th conference on Computational linguistics, pages 836?841, Morristown, NJ, USA, 1996.
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2,228 | 3,021 | Unsupervised Learning of a Probabilistic Grammar
for Object Detection and Parsing
Long (Leo) Zhu
Department of Statistics
University of California at Los Angeles
Los Angeles, CA 90095
[email protected]
Yuanhao Chen
Department of Automation
University of Science and Technology of China
Hefei, Anhui 230026 P.R.China
[email protected]
Alan Yuille
Department of Statistics
University of California at Los Angeles
Los Angeles, CA 90095
[email protected]
Abstract
We describe an unsupervised method for learning a probabilistic grammar of an
object from a set of training examples. Our approach is invariant to the scale and
rotation of the objects. We illustrate our approach using thirteen objects from the
Caltech 101 database. In addition, we learn the model of a hybrid object class
where we do not know the specific object or its position, scale or pose. This is illustrated by learning a hybrid class consisting of faces, motorbikes, and airplanes.
The individual objects can be recovered as different aspects of the grammar for the
object class. In all cases, we validate our results by learning the probability grammars from training datasets and evaluating them on the test datasets. We compare
our method to alternative approaches. The advantages of our approach is the speed
of inference (under one second), the parsing of the object, and increased accuracy
of performance. Moreover, our approach is very general and can be applied to a
large range of objects and structures.
1
Introduction
Remarkable progress in mathematics and computer science of probability is leading to a revolution
in the scope of probabilistic models. In particular, there are exciting new probability models [1,
3, 4, 5, 6, 11] defined on structured relational systems such as graphs or grammars. These new
formulations subsume more traditional models, such as Markov Random Fields (MRF?s) [2], and
have growing applications to natural languages, machine learning, and computer vision.
Although these models have enormous representational power, there are many practical drawbacks
which must be overcome before using them. In particular, we need efficient algorithms to learn the
models from training data and to perform inference on new examples. This problem is particularly
difficult when the structure of the representation is unknown and needs to be induced from the data.
In this paper we develop an algorithm called ?structure induction? (or ?structure pursuit?) which
we use to learn the probability model in an unsupervised manner from a set of training data. This
algorithm proceeds by building an AND-OR graph [5] in an iterative way. The form of the resulting
graph structure ensures that inference can be performed rapidly for new data.
Chair 90.9%
Cougar 90.9%
Piano 96.3%
Scissors 94.9%
Panda 90.0%
Rooster 92.1%
Stapler 90.5 %
Wheelchair 92.4%
Windsor Chair 92.4%
Wrench 84.6%
Figure 1: We have learnt probability grammars for these ten objects in the Caltech 101 database,
obtaining scores over 90 % for most objects. A score of 90.00 %, means that we have a detection
rate of 90 % and a false positive rate of 10 % (10 % = (100 - 90) %). The number of data examples
are 62, 69, 90, 39, 38, 49, 45, 59, 56, 39 ordered left-to-right and top-to-bottom.
Our application is to the detection, recognition, and parsing of objects in images. The training data
consists of a set of images where the target object is present but at an unknown location. This
topic has been much studied [16] (see technical report ? Zhu, Chen and Yuille 2006 ? for additional
references).
Our approach has the following four properties. Firstly, a wide range of applicability which we
demonstrate by learning models for 13 object categories from the Caltech-101 [16], Figure (1,5).
Secondly, the approach is invariant to rotation and a large range of scale of the objects. Thirdly, the
approach is able to deal with object classes, which we illustrate by learning a hybrid class consisting
of faces, motorbikes and airplane. Fourthly, the inference is performed rapidly in under a second.
2
2.1
Background
Representation, Inference and Learning
Structured models define a probability distribution on structured relational systems such as graphs
or grammars. This includes many standard models of probability distributions defined on graphs ?
for example, graphs with fixed structure, such as MRF?s [2] or Conditional Random Fields [3], or
Probabilistic Context Free Grammars (PCFG?s) [4] where the structure is variable. Attempts have
been made to unify these approaches under a common formulation. For example, Case-Factor Diagrams [1] have recently been proposed as a framework which subsumes both MRF?s and PCFG?s.
In this paper, we will be concerned with models that combine probabilistic grammars with MRF?s.
The grammars are based on AND-OR graphs [1, 5, 6], which relate to mixtures of trees [7]. This
merging of MRF?s with probabilistic grammars results in structured models which have great representational power.
There has been considerable interest in inference algorithms for these structured models, for example
McAllester et al [1] describe how dynamic programming algorithms (e.g. Viterbi and inside-outside)
can be used to rapidly compute properties of interest. Our paper is concerned with the task of unsupervised learning of structured models for applications to detecting, recognizing, and representing
visual objects. In this paper, we restrict ourselves to a special case of Probabilistic Grammars with
OR nodes, and MRF?s. This is simpler than the full cases studied by McAllester but is more complex
than the MRF models standardly used for this problem.
For MRF models, the number of graph nodes is fixed and structure induction consists of determining
the connections between the nodes and the forms of the corresponding potentials. For these graphs,
an effective strategy is feature induction [8] which is also known as feature pursuit [9]. A similar
strategy is also used to learn CRF?s [10]. In both cases, the learning is fully supervised. For Bayesian
network, there is work on learning the structure using the EM algorithm [12].
Learning the structure of grammars in an unsupervised way is more difficult. Klein and Manning
[4] have developed unsupervised learning of PCFG?s for parsing natural language, but here the
structure of grammar is specified. Zettlemoyer and Collins [11] perform similar work based on
lexical learning with lambda-calculus language.
In short, to our knowledge, there is no unsupervised learning algorithm for structure induction for
a Probabilistic Grammar-MRF model. Moreover, our vision application requires the ability to learn
the model of the target object in the presence of unknown background structure. Methods exist in
the computer vision literature for achieving this for an MRF model [16], but not for Probabilistic
Grammars.
2.2
Our Model: High-Level Description
Figure 2: Graphical Models.
In this paper, we consider a combination of PCFG and MRF. The leaf nodes of the graph will be
image features that are described by MRF?s. Instead of using the full PCFG, we restrict the grammar
to containing one OR-node.
Our model contains a restricted set of grammatical rules, see figure (2). The top, triangular node, is
an OR node. It can have an arbitrary number of child nodes. The simplest type of child node is a
histogram model (far left panel of figure (2)). We can obtain more complex models by adding MRF
models in the form of triples, see figure (2) left to right. Combination of triples can be expressed
in a junction tree representation, see the sixth and seventh panels of figure (2). This representation
enable rapid inference. The computation complexity of inference is bounded by the width and height
of the subtrees.
In more abstract terms, we define a set of rules R(x, y) for allowable parses of input x to a parse tree
y. These rules have potentials ?(x, r, t) for a production rule r ? R(x, y) and ?(x, wM , t) for the
MRF models (see details in the technical report), where t are nuisance parameters (e.g. geometric
transformations and missing data) and w = (wG , wM ) are model parameters. The wG are the
grammar parameters and the wM are the MRF parameters. We define a set W of model parameters
that are allowed to be non-zero (w = 0 if w ?
/ W ). The structure of the model is determined by the
set W .
The model is defined by:
P (x, y, w, t) = P (t)P (w)P (y)P (x|y, w, t),
where
P (x|y, w, t) =
1
e
Z
P
r?R(x,y)
wG ??(x,r,t)+
P
M RF
?M RF (x,t,wM )
(1)
,
(2)
where M RF denotes the cliques of the MRF. Z is the normalization constant.
We now face three tasks: (I) structure learning, (II) parameter learning to estimate w, and (III)
inference to estimate y.
Inference requires estimating the parse tree y from input x. The model parameters
P w are fixed.
The nuisance parameters are integrated out. This requires solving y ? = arg max t P (y, t|x, w)
by the EM algorithm using dynamic programming to estimate y ? efficiently. During the E step, we
approximate the sum over t by a saddle point approximation.
Parameter learning we specify a set W of parameters w which we estimate
P by MAP (the other
w?s are constrained to be zero). Hence we estimate w? = arg maxw?W y,t P (w, t, y|x). This
is performed by an EM algorithm, where the summation over y can be performed by dynamic
programming, the summation over t is again performed by a saddle point. The w can be calculated
by sufficient statistics.
Structure Learning corresponds to increasing the set of parameters w that can be non-zero. For
each structure we define a score given by its fit to the data. Formally we extend W to W 0 where
0
W ? W
P. (The allowed extensions are defined in thePnext section). We now compute P (x|w ?
W ) = w?W,t,y P (x, y, t|w) and P (x|w ? W 0 ) = w?W 0 ,t,y P (x, y, t|w). This requires EM,
dynamic programming, and a saddle point approximation. We refer to the model fits, P (x|w ? W )
and P (x|w ? W 0 ), as the scores for structure W and W 0 respectively.
3
Brief Details of Our Model
We now give a brief description of our model. A detailed description is given in our technical report
(Zhu, Chen, and Yuille 2006).
Figure 3: Triplets without Orientation (left two panels). Triplets with Orientation (right two panels).
3.1
The setup of the Model
We represent the images by features {xi : i = 1, .., N (? )}, where N (? ) is the number of features in
image ? . Each feature is represented by a pair xi = (zi , Ai ), where zi is the location of the feature
in the image and Ai is an appearance vector. The image features are detected by the Kadir-Brady
operator [13], and their appearance is calculated by the SIFT operator [14]. These operators ensure
that the features are invariant to scale, rotation, and some appearance variations.
The default background model for the image is to define a histogram model over the positions and
appearance of the image features, see first panel of figure (2).
Next we use triples of image features as the basic building blocks to construct a model. Our model
will be constructed by adding new triplets to the existing model, as shown in the first few panels of
figure (2). Each triplet will be represented by a triplet model which is given by Gaussian distributions
~ = ~1, T) = G(~z|T(~
~ ?A , ?A ), where
on spatial position and on appearance P (~x|M
?G , ?G )G(A|~
?G , ?A , ?G , ?A are the means and covariances of the positions and appearances. The {Mi } are
missing data index variables [15], and T denotes transformations due to rotation and scaling.
The major advantage of using triplets is that they have geometrical properties which are independent
of the scale and rotation of the triplet. These properties include the angles between the vertices, see
figure (3). Thus we can decompose the representation of the triplet into two types of properties:
(i) those which are independent of scale and rotation, (ii) those that depend explicitly on scale and
rotation. By using the invariant properties, we can perform rapid search over triplets when position,
scale, and rotation are unknown.
In addition, two triplets can be easily combined by a common edge to form a more complex model ?
see sixth panel of figure (2). This representation is suitable for the junction tree algorithm [2], which
enables rapid inference.
For structure learning, we face the task of how to expand the set W of non-zero parameters to a
new set W 0 . The problem is that there are many ways to expand the set, and it is computationally
impossible to evaluate all of them. Our strategy is to use a clustering method, see below, to make
proposals for expanding the structure. These proposals are then evaluated by model selection.
Our clustering method exploits the invariance properties of triplets. We perform clustering on both
the appearance and on the geometrical invariants of the triplets. This gives rise to a triplet vocabulary
consisting of triplets that frequently occur in the dataset. These are used to make proposals for which
triplets to include in the model, and hence for how to expand the set W of non-zero parameters.
Input: Training Image ? = 1, .., M and the triplet vocabulary {Ta : a ? ?}. Initialize G to be the root
node with the background model, and let G? = G.
Algorithm for Structure Induction:
? STEP 1:
? OR-NODE EXTENSION
For T ? {Ta : a ? ?}
S
? G0 = G T (ORing)
? Update parameters of G0 by EM algorithm
? If Score(G0 ) > Score(G? ) Then G? = G0
? AND-NODE EXTENSION
For Image ? = 1, .., M
? P = the highest probability parse for Image ? by G
? For each
T Triple T in Image ?
if T P 6= ?
S
? G0 = G T (ANDing)
? Update parameters of G0 by EM algorithm
? If Score(G0 ) > Score(G? ) Then G? = G0
? STEP 2: G = G? . Go to STEP 1 until Score(G) ? Score(G? ) < T hreshold
Output: G
Figure 4: Structure Induction Algorithm
3.2
Structure Induction: Learning the Probabilistic Grammar MRF
We now have the necessary background to describe our structure induction algorithm. The full
procedure is described in the pseudo code in figure (4). Figure (2) shows an example of the structure
being induced sequentially.
Initially we assume that all the data is generated by the background model. In the terminology of
section (2.2), this is equivalent to setting all of the model parameters w to be zero (except those
for the background model). We can estimate the parameters of this model and score the model as
described in section (2.2).
Next we seek to expand the structure of this model. To do this, we use the triplet vocabularies
to make proposals. Since the current model is the background model, the only structure change
allowed is to add a triplet model as one child of the root node (i.e. to create the background plus
triple model described in the previous section, see figure (2)). We consider all members of the triplet
vocabulary as candidates, using their cluster means and covariances as prior probabilities on their
geometry and attribute properties. Then, for all these triples we construct the background plus triplet
model, estimate their parameters and score them. We accept the one with highest score as the new
structure.
As the graph structure grows, we now have more ways to expand the graph. We can add a new
triplet as a child of the root node. This proceeds as in the previous paragraph. Or we can take two
members of an existing triplet, and use them to construct a new triplet. In this case, we first parse the
data using the current model. Then we use the triplet vocabulary to propose possible triplets, which
partially overlap with the current model (and give them prior probabilities on their parameters as
before). Then, for all possible extensions, we use the methods in section (2.2) to score the models.
We select the one with highest score as the new graph model. If the score increase is not sufficient,
we cease building the graph model. See the structured models in figure (5).
...
...
...
Figure 5: Individual Models learnt for Faces, Motorbikes and Airplanes.
Table 1: Performance Comparisons
4
4.1
Dataset
Size
Single Model
Hybrid Model
Constellation[16]
Faces
Motorbikes
Airplanes
Faces(Rotated)
Faces(Rotated+Scaled)
435
800
800
435
435
98.0
92.6
90.9
94.8
92.3
84.0
82.7
87.3
?
?
96.4
92.5
90.2
?
?
Experimental Results
Learning Individual Objects Models
In this section, we demonstrate the performance of our models for thirteen objects chosen from the
Caltech-101 dataset. Each dataset was randomly split into two sets with equal size(one for training
and the other for testing).
K-means clustering (typically, K is set to 150) was used to learn the triplet vocabularies (see Zhu,
Chen, Yuille 2006 for details). Each row in figure 3 corresponds to some triples in the same group.
In this experiment, we did not use orientation information from the feature detector.
We illustrate our results in figure (1) and Table (1). A score of 90 % means that we get a true positive
rate of 90 % and a false positive rate of 10 %. For comparison, we show the performance of the
Constellation Model [16]. (Further comparisons to alternative methods are reported in the technical
report).
The models for individual objects classes, learnt from the proposed algorithm, are illustrated in
figure (5). Observe that the generative models have different tree-width and depth. Each subtree
of the root node defines an Markov Random Field to describe one configuration of the object. The
computational cost of the inference, using dynamic programming, is proportional to the height of
the subtree and exponential to the maximum width(only three in our case). The detection time is
Figure 6: Parsed Results: Invariant to Rotation and Scale.
Figure 7: Hybrid Model learnt for Faces, Motorbikes and Airplanes.
less than one second (including the processing of features and inference) for the image with the size
of 320*240. The training time is around two hours for 250 training images.
4.2
Invariance to Rotation and Scale
This section shows that we can learn and detect objects even when the rotation (in the image) and the
scale are unknown (within a range). In this experiment, orientation information, output from feature
detector, is used to model the geometry distributions of the triplets. The relative angle between
the orientation of each feature and the orientation of the edge of tri-angle is calculated to make the
model invariant to rotation. See Figure (3).
We implemented the comparison experiment on face dataset. A face model is learnt from the training
images with normalized scale and orientation. We tested this model on the testing data with 360degree in-plane rotation and another testing data with rotation and scaling together. The scaling
range is from 60% of the original size to 150%(i.e. 180 ? 120 ? 450 ? 300). Table (1) shows the
comparison results. The parsing results (rotation+scale) are illustrated in Figure (6).
4.3
Learning Classes of Models
In this section, we show that we can learn a model for an object class. We use a hybrid class which
consists of faces, airplanes, and motorbikes. In other words, we know that one object is present
in each image but we do not know which. In the training stage, we randomly select images from
the datasets of faces, airplanes, and motorbikes. Similarly, we test the hybrid model on examples
selected randomly from these three datasets.
The learnt hybrid model is illustrated in Figure (7). It breaks down nicely into or?s of the models
for each object. Table (1) shows the performance for the hybrid model. This demonstrates that the
proposed method can learn a model for the class with extremely large variation. The parsed results
are shown in Figure (8).
5
Discussion
This paper showed that it is possible to perform unsupervised learning to determine a probabilistic
grammar combined with a Markov Random Fields. Our approach is based on structure pursuit where
the object model is built up in an iterative manner (similar to feature pursuit used for MRF?s and
CRF?s). The building blocks of our model are triplets of features, whose invariance properties can
be exploited for rapid computation.
Our application is to the detection and parsing of objects. We demonstrated: (a) that we can learn
probabilistic models for a variety of different objects, (b) that our approach is invariant to scale and
Figure 8: Parsed Results by Hybrid Model (left three panels). Parsed by Standard Model (right three
panels).
rotation, (c) that we can learn models for hybrid classes, and (d) that we can perform inference
rapidly in under one second.
Our approach can also be extended. By using a richer vocabulary of features we can learn a more
sophisticated generative grammar which will be able to represent objects in greater detail and deal
with significant variations in viewpoint and appearance.
Acknowledgements
We gratefully acknowledge support from the W.M. Keck Foundation, NSF grant number 0413214,
and NIH grant RO1 EY015261.
References
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2,229 | 3,022 | Learning Structural Equation Models for fMRI
Amos J. Storkey
School of Informatics
University of Edinburgh
Stephen Lawrie
Division of Psychiatry
University of Edinburgh
Enrico Simonotto
Division of Psychiatry
University of Edinburgh
Lawrence Murray
School of Informatics
University of Edinburgh
Heather Whalley
Division of Psychiatry
University of Edinburgh
David McGonigle
Centre for Functional Imaging Studies
University of Edinburgh
Abstract
Structural equation models can be seen as an extension of Gaussian belief networks to cyclic graphs, and we show they can be understood generatively as the
model for the joint distribution of long term average equilibrium activity of Gaussian dynamic belief networks. Most use of structural equation models in fMRI
involves postulating a particular structure and comparing learnt parameters across
different groups. In this paper it is argued that there are situations where priors
about structure are not firm or exhaustive, and given sufficient data, it is worth
investigating learning network structure as part of the approach to connectivity
analysis. First we demonstrate structure learning on a toy problem. We then show
that for particular fMRI data the simple models usually assumed are not supported.
We show that is is possible to learn sensible structural equation models that can
provide modelling benefits, but that are not necessarily going to be the same as a
true causal model, and suggest the combination of prior models and learning or
the use of temporal information from dynamic models may provide more benefits
than learning structural equations alone.
1
Introduction
Structural equation modelling (SEM) is a technique widely used in the behavioural sciences. It has
also appeared as a standard approach for analysis of what has become known as effective connectivity
in the functional magnetic resonance imaging (fMRI) literature and is still in common use despite the
increasing interest in dynamical methods such as dynamic causal models [6]. Simply put, effective
connectivity analysis involves looking at the possible causal influences between brain regions given
measurements of the activity of those regions. Structural equation models are a Gaussian modelling
tool, and are similar to Gaussian belief networks. In fact Gaussian belief networks can be seen as
a subset of valid structural equation models. However structural equation models do not have the
same acyclicity constraints as belief networks. It should be noted that the graphical form used in
this paper is at odds with traditional SEM representations, and consistent with that used for belief
networks, as those will be more familiar to the expected audience.
Within the fMRI context, the use of structural equation generally takes the following form. First
certain regions of interests (commonly called seeds) are chosen according to some understanding
of what brain regions might be of interest or of importance. Then neurobiological knowledge is
used to propose a connectivity model. This connectivity model states what regions are connected
to what other regions, and the direction of the connectivity. This connectivity model is used to
define a structural equation model. The parameters of this model are then typically estimated using
maximum likelihood methods, and then comparison of connection parameters is made across subject
classes.
In this paper we consider what can be done when it is hard to specify connectivity a priori, and ask
how much we can achieve by learning network structures from the fMRI data itself. The novel developments of this paper include the examination of various generative representations for structural
equation models which allow straightforward comparisons with belief networks and other models
such as dynamic causal models. We implement Bayesian Information Criterion approximations to
the evidence and use this in a Metropolis-Hastings sampling scheme for learning structural equation
models. These models are then applied to toy data, and to fMRI data, which allows the examination
of the types of assumptions typically made.
1.1 Related Work: Structural Equation Models
Structural equation models and path analysis have a long history. The methods were introduced in
the context of genetics in [20], and in econometrics in [7]. They have been used extensively in the
social sciences in a variety of ways. Linear Gaussian structural equation models can be split into the
case of path analysis [20], where the all the variables are directly measurable and structural equation
models with latent variables [1], where latent variable models are allowed. Factor analysis is another
special case of this latter. Furthermore structural equation models can also be characterised by the
inclusion of exogenous influences.
Structural equation models have been analysed and understood in Bayesian terms before. They form
a part of the causal modelling framework of Pearl [11], and have been discussed within that context,
as well as a number of others [11, 4, 13, 10]. Approaches to learning structural equation models
have not played a significant part in fMRI methods. One approach is described in [3] where they use
a genetic algorithm approach for the search. In [21], the authors look at learning Bayesian networks
but do not consider cyclic networks. For dynamic causal models (rather than structural equation
models) the issue of model comparison was dealt with in [12], but large scale structure learning was
not considered.
In fMRI literature, SEMs have generally been used to model ?effective connectivity?, or rather modelling the causal relationships between different brain regions. They were first applied to imaging
data by [9], and there have been many further applications [2, 5, 14]. The first analysis on data from
schizophrenia studies was detailed in [15]. In fact it seems SEMs have been the most widely used
model for connectivity analyses in neuroimaging. In all of the studies cited above the underlying
structure was presumed known or presumed to be one of a small number of possibilities. There has
been some discussion of how best to obtain reasonable structures from neuro-anatomical data, but
this approach is currently used only very rarely.
2
Why Learn SEMs?
The presumption in much fMRI connectivity analysis is that we can obtain models for activity
dependence from neuro-anatomical sources. The problem with this is that it fails to account for
the fact that connectivity analysis is usually done with a limited number of regions. It is highly
possible that a connection from one region to another is mediated via a third region, which is not
included in the SEM model. The strength of that mediation is unknown from neuro-anatomical data
and is generally ignored: most connectivity models focus only on direct anatomical connections,
with the accompanying implicit assumption that there are no other regions involved in the network
under study, or that these regions would contribute only minimally to the model. Furthermore, just
because regions are physically connected does not mean there is any actual functional influence in
a particular context. Hence it has to be accepted that neuro-anatomically derived connectivity is a
first guess at best.
It is not the purpose of this paper to propose that anatomical connectivity be ignored, but instead
it asks what happens if we go to the other extreme: can we say something about connectivity from
the data? In reality anatomical connectivity models are needed, and can be used to provide good
priors for the connections and even for the relative connection strengths. Statistically there are huge
equivalences in structural equation models that will not be determined by the data alone.
3
Understanding Structural Equation Models
In this section two generative views of structural equation modelling are presented. The idea behind
structural equation modelling is that it represents causal dependence between different variables.
The fact that cyclic structures are allowed in structural equation models could be seen as an implicit
assumption of some underlying dynamic which the structural equation model is an equilibrium rep-
resentation of. Indeed that is commonly how effective connectivity models are interpreted in an
fMRI context. Two linear models, both of which produce a structural equation model prior, are presented here. Though these models have the same statistical properties, they have different generative
motivations and different non-linear extensions, so they are both potentially instructive.
3.1 The Traditional Model
The standard SEM view is that the core SEM structure is a covariance produced by the solution to a
set of linear equations x = Ax + ? with Gaussian term ?. This does not have any direct generative
elucidation, but can instead be thought of as relating to a deterministic dynamical system subject to
uncertain fixed input. Suppose we have a dynamical system xt+1 = Axt + ?, subject to some input
?, where we presume the system input is unknown and Gaussian distributed. To generate from the
model, we sample ?, run the dynamical system to its fixed point, and use that fixed point as a sample
of x. This fixed point is given by x = (I ? A)?1 ? which produces the standard SEM covariance
structure for x. This requires A to be a contraction map to obtain stable fixed points. All the other
aspects of the general form of SEM are either inputs to or measurements from this system.
3.2 Average Activity Of A Gaussian Dynamic Bayesian Network
An alternative and potentially appealing view is that the the SEM represents the distribution of the
long term activity of the nodes in a Gaussian dynamic Bayesian network (Kalman filter). Suppose we have xt = Axt?1 + ? t , where ? t are IID Gaussian variables, and x0 , x1 , . . . is a series of real variables. This defines a Markov chain, and is the evolution equation of a Gaussian dynamic Bayesian network.
we are at the equilibrium distribution of this Markov
? Suppose
P
? = (1/ N ) N
x
for large N , we can use the Kalman filter to see that
chain. Then setting x
t
t=1
? PN
?
? PN
PN
(1/ N ) t=1 xt = ?
(1/ N )[A(x0 ? xN ) + A i=1 xt ] + (1/ N ) t=1 ? t . Presuming A is a
? ? A?
contraction map, (1/ N )[A(x0 ? xN )] becomes negligibly small and so x
x + ? where ? is
distributed identically to ? t due to the fact that the variance of a sum of Gaussians is the sum of the
variances. The approximation becomes an equality in the large N limit. Again this is the required
form for obtaining the covariance of the SEM.
This interpretation says that if we have some latent system running as a Gaussian dynamic Bayesian
network, but our measuring equipment is only capable of capturing longer term averages of the
network activity then our measurements are distributed according to an SEM. This generative interpretation is appealing in the context of fMRI acquisition. Note in both of these interpretations that it
is important that A is a contraction. By formulating the generative framework we see it is important
to restrict the form of connectivity model in this way.
4
Model Structure
The standard formalism for Structural Equation Models is now outlined. A structural equation
model for observational variables y, latent variables, x and sometimes for latent input variables ?
and observations of the input variables z is given by the following equations
x = (I ? A)?1 (R? + ?), y = Bx + ? and z = C? + ?
(1)
where ?, ?, ? and ? are Gaussian, and A is presumed to be zero diagonal.
For for S = I ? A, the resulting covariance for the observed variables (y, z) is given by
BS ?1 (RK? RT + K? )[S ?1 ]T + K?
CK? RT [S ?1 ]T B T
BS ?1 RK? C T
CK? C T + K?
.
(2)
where K? is the covariance of ?, K? the covariance of ? etc. There are a number of common
simplifications to this framework. The first case involves presuming no inputs and a fully visible
system. Hence we marginalise the observations of the input variables z, set K? = ?, C = 0
,R = 0, B = 1, ? = 0. Then the covariance K1 of y is K1 = (I ? A)?1 K? [(I ? A)?1 ]T .
The next simplest case would involve presuming once again that there are no inputs but that in fact
the observations are stochastic functions of the latent variables. This involves setting K? = ?,
C = 0, R = 0, B = 1. We then have K2 = (I ? A)?1 K? [(I ? A)?1 ]T + K? . If we view the
observations as noisy versions of the latent variables then K? is diagonal. This will be the most
general case considered in this paper. Adding any of the remaining components is not particularly
demanding as it simply uses a conditional rather than unconditional model.
Suppose we denote by K the covariance corresponding to the required model. For most of this
paper we presume K = K2 . We then have the following probability for the whole data Y =
{y1 , y2 , . . . , yN }.
?
?
Y
1 j
1
T
?1
j
?
?
?) =
(y
?
y
)
K
(y
?
y
)
(3)
exp
?
P (Y |K, y
2
(2?)m/2 |K|1/2
j
?=x
? +?
? and the latent mean is x
? = (I ? A)?1 ?,
? and where
where the observable model mean is y
? and ?, along with elements of the matrix A and covariances K? and K? , are parameters.
5
Priors, Maximum Posterior and Bayesian Information Criterion
The previous section outlines the basic model of the data given the various parameters. In this section
we provide prior distributions for the parameters of the structural equation model. Independent
Gaussian priors are put on the parameters:
?
?
1
T 1/2
2
?
(4)
exp ? T (Aij ? Aij )
P (Aij |T ) =
2
(2?)1/2
with regularisation parameter T . For the purposes of this paper we take A?ij = 0, presume we
have no particular a priori bias towards positive or negative connections and a uniform prior over
structures. An independent prior over connections seems reasonable as two separate connections
between different brain regions would have no a priori reason to be related. Any relationship is due
to functional purpose and is therefore a posteriori. The use of a uniform prior over all structures is
an extreme position, which we have taken in this paper to contrast with using only one structure. In
reality we would want to use neurobiologically guided priors over structures.
Inverse gamma priors were also specified for K? and K? originally, along with a prior for the
mean ?
? . It was found that these typically had no effects on the experiments and were dropped for
simplicity. Hence K? and K? will be optimised without regularisation, and ?
? is set to zero. T is
chosen by 10 fold cross-validation from a set of 10 possible values.
We can calculate all the relevant derivatives for the SEM straightforwardly, and adapt the parameters
to maximize the posterior of the structural equation model. In this paper we use a conjugate gradient
approach. By adding a Bayesian Information Criterion term [16], (?0.5m log N ) for m parameters
and N data points, to the log posterior at the maximum posterior solution, we can obtain an approximation of the evidence P (Y |M ) where M encodes the structural information we are interested in
and consists of indicator variables Mij indicating a connection for node j to node i. This will enable
us to sample from an approximate posterior distribution of structures to find a sample which best
represents the data.
6
Sampling From SEMs
In order to represent the posterior distribution over network structures, we resort to a sampling approach. Because there are no acyclicity constraints, MCMC proposals are simpler than the comparable situation for belief networks in that no acyclicity checking needs to be done for the proposals.
A simple proposal scheme is to randomly generate a swap matrix MS which is XORed with M . We
choose highly sparse swap matrices, but to reduce the possibility of transitioning randomly about the
larger graphs, without ever considering smaller networks we introduce a bias towards removing connections rather than adding connections in generating the swap matrix. This means the proposal is
no longer symmetric, and so a corresponding Hastings factor needs to be included in the acceptance
probability, so the result is still a sample from the original posterior.
7
Tests On A Toy Problem
We tested the approach on a toy problem with 8 variables. We sampled 800 data points from y =
(I?A)?1 ?+? for ? Gaussian with unit diagonal covariance, ? Gaussian with 0.2 diagonal covariance
and with A given by
? 0
?
0
0
?0.26
0
0
?0.03
0
?
?
?
?
0
0
0
0
?0.17
0.31
0.385
0
0
0
0
0.49
0.42
0
0.47
0
?0.36
0.27
0
?0.13
0
0
0
0
0
0
0
0
?0.36
0.34
0
0
?0.18
0
0.16
0.55
0
?0.03
0
0
0
0.5
0
0
?0.08
?0.25
0
0
0
0
0
0.25
0
0
?0.22
0
?
?
?
?
This connectivity matrix is represented graphically in Figure 11. In modelling this we used T =
10. This prior ensures that any A that is not of very low prior probability is a contraction. Also
contraction constraints were added to the optimisation. Priors on other parameters were set to be
broad. An annealed sampling procedure was used for the first 4000 samples from the MetropolisHastings Monte-Carlo procedure. After that a further 4000 samples burn-in was used. The next
4000 samples were used for analysis.
We assess what edges are common in the samples, and what the highest posterior sampled graphs
are. Figure 1b provides illustrations of edges which are present in more than 0.15 of the samples. It
can be seen that many of the critical edges are there in most samples (indeed some are always there).
Those which are missing in both cases tend to be either low magnitude connections or are due to
directional confusion.
(a)
(b)
(c)
(d)
Figure 1: (a) Graphical structure of the ground truth model for the toy data, and (b) Edges present in more
than 0.15 of the cases, (c) the highest posterior structure from the sample (d) a random sample.
The graphs for the maximum posterior sample and a random sample are shown in Figure 1. We can
see that again in the maximum posterior sample, there is a misplaced edge (the edge from 5 to 6 is
replaced by one from 5 to 1) and a number are missing or have exchanged direction. The samples
generally have likelihoods which are very similar to the likelihood for the true model.
We can conclude from this that we can gain some information from learning SEM structures, but as
with learning any graphical models there are many symmetries and equivalences, so it is vital not to
infer too much from the learnt structures.
8
Tests On fMRI Data
The approach of this paper was tested on two different fMRI datasets. The first dataset (D1) was
taken from a dataset that had previously used to examine inter-session variance in a single subject [8,
17]. We used the auditory paced finger-tapping task; briefly, a single subject tapped his right index
finger, paced by an auditory tone (1.5Hz). Each activation epoch was alternated with a rest epoch,
in which the pacing tone was delivered to control for auditory activation. Thirteen blocks were
collected per session (seven rest and six active). Each block was 24s/6 scans long, making 78 scans
in total for each of 33 sessions. The subject maintained fixation on a cross that was backprojected
onto a transparent screen by a LCD video projector as in previous experiments.
The subject was a healthy 23 year old right-handed male. The data were acquired on a Siemens
MAGNETOM Vision (Siemens, Erlangen, Germany) at 2T. Each BOLD-EPI volume scan consisted
of 48 transverse slices (inplane matrix 64x64; voxel size 3x3x3mm; TE=40ms; TR=4.1s). A T1weighted high-resolution MRI of the subject (1 x 1 x 1.5mm resolution) was acquired to facilitate
anatomical localisation of the functional data.
The data were processed with statistical parametric mapping (SPM) software SPM5 (Wellcome
Department of Cognitive Neurology; www.fil.ion.ucl.ac.uk/spm). After removal of the first two
volumes to account for T1 saturation effects, cerebral volumes were realigned to correct for both
within- and between-session subject motion). The data were filtered with a 128s high-pass filter,
and an AR(1)-model was used to account for serial correlation in the data Experimental effects were
estimated using session design matrices modeling the hemodynamically convolved time-course of
the active movement condition, and 6 subject movement parameters. Note that no spatial smoothing
was applied to this dataset, to attempt to preserve single-voxel timeseries.
Seeds were selected from significantly active voxels identified using a random effects analysis in
SPM5 (ones-sample t-test across 33 sessions; p < 0.05 FWE corrected for multiple comparsions).
For comparison with previous extant work, the most significant voxel in each cluster was chosen
as a seed, giving 13 seeds representing 13 separate anatomical regions. When it was obvious that
a given cluster encompassed more than one distinct anatomical region, seeds were also selected for
other regions covered by the cluster. 2000 data points were used for training, the remaining 574
were reserved as a test set.
The second dataset (D2) was from a long term study of subjects who are at genetically enhanced risk
of schizophrenia. Imaging was carried out on 90 subjects at the Brain Imaging Research Centre for
Scotland (Edinburgh, Scotland, UK) on a GE 1.5 T Signa scanner. A high resolution structural scan
was acquired using a 3D T1-weighted sequence (TI = 600 ms). Functional data was acquired using
an EPI sequence. A total of 204 volumes were acquired. The first four volumes of each acquisition
were discarded. Preliminary analysis was carried out using SPM99. Data were first realigned to
correct for head movement, normalized to the standard EPI template and smoothed.
The resulting data consists of a image-volume series of 200 time points for each of the remaining
90 patients. The voxel time courses were temporally filtered. In order to reduce task related effects,
we modelled the task conditions with standard block effects (Hayling), all convolved with canonical
hemodynamical response functions, and fitted a general linear model (which also included regressors
for the estimated head movement) to the time filtered data; the residuals of this procedure were used
as the data for all the work described in this paper. The full data set was split into two halves, a
training and a test set. Data from 45 of the subjects was used for training and 45 for testing.
For an effective connectivity analysis, a number of brain regions (seeds) were chosen on the basis
of the results of a functional connectivity study [19] and taking regard of areas which may be of
particular clinical interest. In total 14 regions were chosen, along with their 14 cross-hemisphere
counterparts. Hence we are interested in learning a 28 by 28 connectivity matrix.
8.1 Learning SEM Structure
For both datasets a similar procedure to the toy example was followed for learning structure for the
fMRI data. The stability of the log posterior along with estimations of cross-correlation against lag
were used as heuristics to determine convergence prior to obtaining 10000 sample points.
Assuming a fully visible path analysis (covariance K1 ) model, where no measurement noise is included is typical in fMRI analysis (e.g. [15] for a Schizophrenia studies), we found that samples
from the posterior of this model were in fact so highly connected that displaying them would infeasible. For D2 a connectivity of 350 of the 752 total possible connections was typical. However note
that only 376 connections are needed to fully specify a general covariance. Hence we can assume
that in this situation the data is not suggesting any particular structure in the data which is reasonably
amenable to path analysis.
We can generalise the path analysis model by making the region activities latent variables, and allow
the measurement variables to be noisy versions of those regions. In SEM terms this is equivalent to
assuming the covariance structure given by K2 . A repeat of the whole procedure with this covariance
results in much smaller structures. We focus on this approach.
For dataset D1, we sample posterior structures given the training data with T = 100. There is
notable variation in these structures although some key links (eg Left Motor Cortex (L M1) to Left
Posterior Parietal Cortex (L PPC) are included in most samples. In addition an a priori connectivity
(a)
(b)
Figure 2: Structure for (a) the hand specified model (b) the highest posterior sample.
(a)
(b)
(c)
Figure 3: Graphical structure for (a) the highest posterior structure from the sample (b) random sample. (c) a
sample from the two tier model. The regions are Inferior Frontal Gyrus, Medial Frontal Gyrus, Ant. Cingulate,
Frontal Operculum, Superior Frontal Gyrus, Middle Frontal Gyrus, Superior Temporal Gyrus, Middle Temporal
Gyrus, Insula, Thalamus, Amygdala Hippocampal Region, Cuneus/Precuneus, Inferior Parietal Lobule and
Posterior Cerebellum.
structure is proposed for the regions in the study, taking into account the task involved. This was obtained by using knowledge of neuroanatomical connectivity drawn from studies using tract-tracing in
non-human primates. It was produced independent of the connectivity analysis and without knowledge of its results, but taking into account the seed locations and their corresponding activities. Note
that this is a simple finger tapping motor task with seeds corresponding to the associated regions.
Though not trivial, we would expect the specification to be easier and more accurate here than for
more complicated cognitive tasks, due to the high number of papers using this task in functional
neuroimaging. Task D1 is also of note due to its focus on repeated scanning in a single individual,
thus negating any problems in seed selection that may arise from inter-subject spatial variance.
These two cases described above are specified as different hypothesised models. We denote the
hand-specified structure MH and we select the maximum a posteriori sample ML (for ?Learnt
Model?) as a potential alternative. The two structures are illustrated in Figure 2. The maximum
a posteriori parameters are then estimated for the two models using the same conjugate gradient
procedure on the same dataset. These two models are then used predictively on the remaining unseen test data. We compute the predictive log-likelihoods for each model. We find that the best
predictive log-likelihoods for each approach are the same (3SF) for both models. They are also the
same as the predictive likelihood using the full sample covariance, which given the large data sizes
used is well specified. Both these models perform better than other random models with equivalent
numbers of connections. In reality learnt models are going to be used in new situations and situations with less data. One test of the appropriateness of a model is to assess its predictive capability
when trained on little data. By estimating the model parameters on 100 data points, instead of 2000,
we find that the learnt model performs very slightly better than the hand specified model (log odds
ratio of 63 on a 574 point test set), and both perform better than the full covariance (log odds of 292).
This indicates that both MH and ML are providing salient reduced representations which capture
useful characteristics of the data.
We also ran tests on D2. Maximum posterior samples and a random sample are illustrated in Figure
3. Note that although these samples appear to still be quite highly connected, they in fact have about
130 connections. Even so this is significantly greater than the idealised connectivity structures
typically used in most studies.
One further approach is to assume a fully connected structure, but where the connectivity is in two
categories. We have priors on connectivity with the same values of Tij as before for the strong
connections and much larger values for the weaker connections. When this is added to the form
of the model (where we make the incorrect but practical assumption that the BIC assumption still
holds for the stronger connections) we obtain even simpler structures. Following this procedure
we find that models of the form of 3c are typical samples from the posterior where only the larger
connections are shown. Again connections such as those between the Cuneus/Precuneus and the
Superior Frontal Gyrus, the Thalamic connections, and some of the cross-hemispheric connections
are amongst those that would be expected. This approach is related to recent work on the use of
sparse priors for effective connectivity [18].
9
Future Directions
This work demonstrates that if we learn structural equation models from data, we find there is little
evidence for the simple forms of path analysis model which is in common use in the fMRI literature.
We suggest that learning connectivity can be a reasonable complement to current procedures where
prior specification is hard. Learning on its own does discover useful parameterised representations,
but these parameterisations are not the same as reasonable prior specifications. This is unsurprising
due to the statistical equivalence of many SEM structures. It should be expected that combining
learnt structures with prior anatomical models will help in the specification of more accurate connectivity assumptions, as it will reduce the number of equivalence and focus on more reasonable
structural forms. Furthermore future comparisons can be made using a sample of reasonable models
instead of a single a priori chosen model. We would also expect that the major gains in learning
models with come from the focus on dynamical networks which do not suffer from specificity problems. Even if the level of temporal information is small, any temporal information provides handles
for inferring causality that are unavailable with static equilibrium models.
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2,230 | 3,023 | Unified Inference for Variational Bayesian Linear
Gaussian State-Space Models
David Barber
IDIAP Research Institute
rue du Simplon 4, Martigny, Switzerland
[email protected]
Silvia Chiappa
IDIAP Research Institute
rue du Simplon 4, Martigny, Switzerland
[email protected]
Abstract
Linear Gaussian State-Space Models are widely used and a Bayesian treatment
of parameters is therefore of considerable interest. The approximate Variational
Bayesian method applied to these models is an attractive approach, used successfully in applications ranging from acoustics to bioinformatics. The most challenging aspect of implementing the method is in performing inference on the hidden
state sequence of the model. We show how to convert the inference problem so
that standard Kalman Filtering/Smoothing recursions from the literature may be
applied. This is in contrast to previously published approaches based on Belief
Propagation. Our framework both simplifies and unifies the inference problem, so
that future applications may be more easily developed. We demonstrate the elegance of the approach on Bayesian temporal ICA, with an application to finding
independent dynamical processes underlying noisy EEG signals.
1
Linear Gaussian State-Space Models
Linear Gaussian State-Space Models (LGSSMs)1 are fundamental in time-series analysis [1, 2, 3].
In these models the observations v1:T 2 are generated from an underlying dynamical system on h1:T
according to:
vt = Bht + ?tv , ?tv ? N (0V , ?V ),
ht = Aht?1 + ?th , ?th ? N (0H , ?H ) ,
where N (?, ?) denotes a Gaussian with mean ? and covariance ?, and 0X denotes an Xdimensional zero vector. The observation vt has dimension V and the hidden state ht has dimension
H. Probabilistically, the LGSSM is defined by:
p(v1:T , h1:T |?) = p(v1 |h1 )p(h1 )
T
Y
p(vt |ht )p(ht |ht?1 ),
t=2
with p(vt |ht ) = N (Bht , ?V ), p(ht |ht?1 ) = N (Aht?1 , ?H ), p(h1 ) = N (?, ?) and where
? = {A, B, ?H , ?V , ?, ?} denotes the model parameters. Because of the widespread use of these
models, a Bayesian treatment of parameters is of considerable interest [4, 5, 6, 7, 8].
An exact implementation of the Bayesian LGSSM is formally intractable [8], and recently a Variational Bayesian (VB) approximation has been studied [4, 5, 6, 7, 9]. The most challenging part of
implementing the VB method is performing inference over h1:T , and previous authors have developed their own specialized routines, based on Belief Propagation, since standard LGSSM inference
routines appear, at first sight, not to be applicable.
1
2
Also called Kalman Filters/Smoothers, Linear Dynamical Systems.
v1:T denotes v1 , . . . , vT .
A key contribution of this paper is to show how the Variational Bayesian treatment of the
LGSSM can be implemented using standard LGSSM inference routines. Based on the insight we
provide, any standard inference method may be applied, including those specifically addressed to
improve numerical stability [2, 10, 11]. In this article, we decided to describe the predictor-corrector
and Rauch-Tung-Striebel recursions [2], and also suggest a small modification that reduces computational cost.
The Bayesian LGSSM is particularly of interest when strong prior constraints are needed to find
adequate solutions. One such case is in EEG signal analysis, whereby we wish to extract sources
that evolve independently through time. Since EEG is particularly noisy [12], a prior that encourages
sources to have preferential dynamics is advantageous. This application is discussed in Section 4,
and demonstrates the ease of applying our VB framework.
2
Bayesian Linear Gaussian State-Space Models
In the Bayesian treatment of the LGSSM, instead of considering the model parameters ? as fixed,
? where ?
? is a set of hyperparameters. Then:
we define a prior distribution p(?|?),
Z
? =
? .
p(v1:T |?)
p(v1:T |?)p(?|?)
(1)
?
In a full Bayesian treatment we would define additional prior distributions over the hyperparameters
? Here we take instead the ML-II (?evidence?) framework, in which the optimal set of hyperpa?.
? with respect to ?
? [6, 7, 9].
rameters is found by maximizing p(v1:T |?)
For the parameter priors, here we define Gaussians on the columns of A and B 3 :
p(A|?, ?H ) ?
H
Y
e?
?j
2
(Aj ?A?j )
T
?
??1
H (Aj ?Aj )
,
p(B|?, ?V ) ?
j=1
H
Y
e?
?j
2
T
(Bj ?B?j )
?
??1
V (Bj ?Bj )
,
j=1
? The
which has the effect of biasing the transition and emission matrices to desired forms A? and B.
?1
?1
4
conjugate priors for general inverse covariances ?H and ?V are Wishart distributions [7] . In the
simpler case assumed here of diagonal covariances these become Gamma distributions [5, 7]. The
? = {?, ?}5 .
hyperparameters are then ?
Variational Bayes
? is difficult due to the intractability of the integrals. Instead, in
Optimizing Eq. (1) with respect to ?
VB, one considers the lower bound [6, 7, 9]6 :
D
E
? ? Hq (?, h1:T ) + log p(?|?)
?
L = log p(v1:T |?)
+ hE(h1:T , ?)iq(?,h1:T ) ? F,
q(?)
where
E(h1:T , ?) ? log p(v1:T , h1:T |?).
Hd (x) signifies the entropy of the distribution d(x), and h?id(x) denotes the expectation operator.
The key approximation in VB is q(?, h1:T ) ? q(?)q(h1:T ), from which one may show that, for
optimality of F,
? hE(h1:T ,?)iq(h1:T ) .
q(h1:T ) ? ehE(h1:T ,?)iq(?) ,
q(?) ? p(?|?)e
These coupled equations need to be iterated to convergence. The updates for the parameters q(?)
are straightforward and are given in Appendices A and B. Once converged, the hyperparameters are
? which lead to simple update formulae [7].
updated by maximizing F with respect to ?,
Our main concern is with the update for q(h1:T ), for which this paper makes a departure from
treatments previously presented.
3
More general Gaussian priors may be more suitable depending on the application.
For expositional simplicity, we do not put priors on ? and ?.
5
For simplicity, we keep the parameters of the Gamma priors fixed.
6
Strictly we should write throughout q(?|v1:T ). We omit the dependence on v1:T for notational convenience.
4
Unified Inference on q(h1:T )
3
Optimally q(h1:T ) is Gaussian since, up to a constant, hE(h1:T , ?)iq(?) is quadratic in h1:T 7 :
?
T ?
D
E
?
1X ?
T ?1
T ?1
?
(vt ?Bht ) ?V (vt ?Bht ) q(B,? ) + (ht ?Aht?1 ) ?H (ht ?Aht?1 )
.
V
2 t=1
q(A,?H )
(2)
In addition, optimally, q(A|?H ) and q(B|?V ) are Gaussians (see Appendix A), so we can easily
carry out the averages in Eq. (2). The further averages over q(?H ) and q(?V ) are also easy due
to conjugacy. Whilst this defines the distribution q(h1:T ), quantities such as q(ht ), required for
example for the parameter updates (see the Appendices), need to be inferred from this distribution.
Clearly, in the non-Bayesian case, the averages over the parameters are not present, and the above
simply represents the posterior distribution of an LGSSM whose visible variables have been clamped
into their evidential states. In that case, inference can be performed using any standard LGSSM
routine. Our aim, therefore, is to try to represent the averaged Eq. (2) directly as the posterior
distribution q?(h1:T |?
v1:T ) of an LGSSM , for some suitable parameter settings.
Mean + Fluctuation Decomposition
A useful decomposition is to write
?
?
? ?1 ?
T
(vt ? Bht )T ??1
?V (vt ? hBi ht ) + hTt SB ht ,
V (vt ? Bht ) q(B,?V )= (vt ? hBi ht )
| {z }
{z
}
|
mean
f luctuation
and similarly
?
?
? ?1 ?
T
(ht ?Aht?1 )T ??1
?H (ht ?hAi ht?1 ) +hTt?1 SA ht?1 ,
H (ht ?Aht?1 ) q(A,?H )= (ht ?hAi ht?1 )
{z
}
|
{z
} |
mean
?
f luctuation
?
T ? ?1 ?
?1
where the parameter covariances are SB ? B T ??1
?V hBi = V HB
and SA ?
V B ? hBi
? T ?1 ?
?
?
T
?1
?1
A ?H A ?hAi ?H hAi = HHA (for HA and HB defined in Appendix A). The mean terms
simply represent a clamped LGSSM with averaged parameters. However, the extra contributions
from the fluctuations mean that Eq. (2) cannot be written as a clamped LGSSM with averaged
parameters. In order to deal with these extra terms, our idea is to treat the fluctuations as arising
from an augmented visible variable, for which Eq. (2) can then be considered as a clamped LGSSM.
Inference Using an Augmented LGSSM
To represent Eq. (2) as an LGSSM q?(h1:T |?
v1:T ), we may augment vt and B as8 :
v?t = vert(vt , 0H , 0H ),
? = vert(hBi , UA , UB ),
B
where UA is the Cholesky decomposition of SA , so that UAT UA = SA . Similarly, UB is the Cholesky
decomposition of SB . The equivalent LGSSM q?(h1:T |?
v1:T ) is then completed by specifying9
?
?
?
?
? H ? ??1 ?1 , ?
? V ? diag( ??1 ?1 , IH , IH ), ?
? ? ?.
A? ? hAi , ?
? ? ?, ?
H
V
The validity of this parameter assignment can be checked by showing that, up to negligible constants,
the exponent of this augmented LGSSM has the same form as Eq. (2)10 . Now that this has been
written as an LGSSM q?(h1:T |?
v1:T ), standard inference routines in the literature may be applied to
compute q(ht |v1:T ) = q?(ht |?
v1:T ) [1, 2, 11]11 .
7
For simplicity of exposition, we ignore the first time-point here.
The notation vert(x1 , . . . , xn ) stands for vertically concatenating the arguments x1 , . . . , xn .
9
?t = B,
? for t = 1, . . . , T ? 1. For time T , B
?T has the
Strictly, we need a time-dependent emission B
Cholesky factor UA replaced by 0H,H .
10
There are several ways of achieving a similar augmentation. We chose this since, in the non-Bayesian limit
UA = UB = 0H,H , no numerical instabilities would be introduced.
11
Note that, since the augmented LGSSM q?(h1:T |?
v1:T ) is designed to match the fully clamped distribution
q(h1:T |v1:T ), the filtered posterior q?(ht |?
v1:t ) does not correspond to q(ht |v1:t ).
8
Algorithm 1 LGSSM: Forward and backward recursive updates. The smoothed posterior p(ht |v1:T )
? T and covariance P T .
is returned in the mean h
t
t
procedure F ORWARD
1a: P ? ?
?
??1 T
T
1b: P ? D?, where D ? I ? ?UAB I + UAB
?UAB
UAB
0
?
2a: h1 ? ?
? 0 ? D?
2b: h
1
?1 ? h
? 0 + K(vt ? B h
? 0)
3: K ? P B T (BP B T + ?V )?1 , P11 ? (I ? KB)P , h
1
1
1
for t ? 2, T do
t?1 T
4: Ptt?1 ? APt?1
A + ?H
t?1
5a: P ? Pt
?
??1 T
T
5b: P ? Dt Ptt?1 , where Dt ? I ? Ptt?1 UAB I + UAB
Ptt?1 UAB
UAB
? t?1 ? Ah
? t?1
6a: h
t
t?1
? t?1 ? Dt Ah
? t?1
6b: h
t
t?1
T
? t?1 + K(vt ? B h
? t?1 )
?t ? h
7: K ? P B (BP B T + ?V )?1 , Ptt ? (I ? KB)P , h
t
t
t
end for
end procedure
procedure BACKWARD
for t ? T ? 1, 1 do
?
?
t
At ? Ptt AT (Pt+1
)?1
?
? T
?
?
T
t
t
Pt ? Pt + At (Pt+1 ? Pt+1
)At T
?
?
?T ? h
? t + At (h
? T ? Ah
? t)
h
t
t
t
t+1
end for
end procedure
For completeness, we decided to describe the standard predictor-corrector form of the Kalman Filter, together with the Rauch-Tung-Striebel Smoother [2]. These are given in Algorithm 1, where
q?(ht |?
v1:T ) is computed by calling the FORWARD and BACKWARD procedures.
We present two variants of the FORWARD pass. Either we may call procedure FORWARD in
? B,
? ?
?H, ?
?V , ?
? and the augmented visible variables v?t in which
Algorithm 1 with parameters A,
?, ?
we use steps 1a, 2a, 5a and 6a. This is exactly the predictor-corrector form of a Kalman Filter [2].
Otherwise, in order to reduce the computational cost, we may call procedure FORWARD with the
?
?
? hBi , ?
? H , ??1 ?1 , ?
? and the original visible variable vt in which we use steps
parameters A,
?, ?
V
T
1b (where UAB UAB ? SA +SB ), 2b, 5b and 6b. The two algorithms are mathematically equivalent.
Computing q(ht |v1:T ) = q?(ht |?
v1:T ) is then completed by calling the common BACKWARD pass.
The important point here is that the reader may supply any standard Kalman Filtering/Smoothing
routine, and simply call it with the appropriate parameters. In some parameter regimes, or in very
long time-series, numerical stability may be a serious concern, for which several stabilized algorithms have been developed over the years, for example the square-root forms [2, 10, 11]. By
converting the problem to a standard form, we have therefore unified and simplified inference, so
that future applications may be more readily developed12 .
3.1
Relation to Previous Approaches
An alternative approach to the one above, and taken in [5, 7], is to write the posterior as
log q(h1:T ) =
T
X
?t (ht?1 , ht ) + const.
t=2
for suitably defined quadratic forms ?t (ht?1 , ht ). Here the potentials ?t (ht?1 , ht ) encode the averaging over the parameters A, B, ?H , ?V . The approach taken in [7] is to recognize this as a
12
The computation of the log-likelihood bound does not require any augmentation.
pairwise Markov chain, for which the Belief Propagation recursions may be applied. The approach
in [5] is based on a Kullback-Leibler minimization of the posterior with a chain structure, which is
algorithmically equivalent to Belief Propagation. Whilst mathematically valid procedures, the resulting algorithms do not correspond to any of the standard forms in the Kalman Filtering/Smoothing
literature, whose properties have been well studied [14].
4
An Application to Bayesian ICA
A particular case for which the Bayesian LGSSM is of interest is in
extracting independent source signals underlying a multivariate timeseries [5, 15]. This will demonstrate how the approach developed in
Section 3 makes VB easily to apply. The sources si are modeled as
independent in the following sense:
p(si1:T , sj1:T ) = p(si1:T )p(sj1:T ),
for i 6= j,
i, j = 1, . . . , C.
Independence implies block diagonal transition and state noise matrices A, ?H and ?, where each block c has dimension Hc . A one dimensional source sct for each independent dynamical subsystem is then
formed from sct = 1Tc hct , where 1c is a unit vector and hct is the state of
dynamical system c. Combining the sources, we can write st = P ht ,
where P = diag(1T1 , . . . , 1TC ), ht = vert(h1t , . . . , hC
t ). The resulting
emission matrix is constrained to be of the form B = W P , where
W is the V ? C mixing matrix. This means that the observations
are formed from linearly mixing the sources, vt = W st + ?tv . The
Figure 1: The structure of graphical structure of this model is presented in Fig 1. To encourage
redundant components to be removed, we place a zero mean Gaussian
the LGSSM for ICA.
prior on W . In this case, we do not define a prior for the parameters
?H and ?V which are instead considered as hyperparameters. More details of the model are given
in [15]. The constraint B = W P requires a minor modification from Section 3, as we discuss below.
Inference on q(h1:T )
A small modification of the mean + fluctuation decomposition for B occurs, namely:
?
?
T ?1
T T
(vt ? Bht )T ??1
V (vt ? Bht ) q(W ) = (vt ? hBi ht ) ?V (vt ? hBi ht ) + ht P SW P ht ,
?1
where hBi ? hW i P and SW = V HW
. The quantities hW i and HW are obtained as in Appendix
A.1 with the replacement ht ? P ht . To represent the above as a LGSSM, we augment vt and B as
v?t = vert(vt , 0H , 0C ),
? = vert(hBi , UA , UW P ),
B
where UW is the Cholesky decomposition of SW . The equivalent LGSSM is then completed by
? H ? ?H , ?
? V ? diag(?V , IH , IC ), ?
? ? ?, and inference for
specifying A? ? hAi, ?
? ? ?, ?
q(h1:T ) performed using Algorithm 1. This demonstrates the elegance and unity of the approach in
Section 3, since no new algorithm needs to be developed to perform inference, even in this special
constrained parameter case.
4.1
Demonstration
As a simple demonstration, we used an LGSSM to generate 3 sources sct with random 5?5 transition
matrices Ac , ? = 0H and ? ? ?H ? IH . The sources were mixed into three observations
vt = W st + ?tv , for W chosen with elements from a zero mean unit variance Gaussian distribution,
and ?V = IV . We then trained a Bayesian LGSSM with 5 sources and 7 ? 7 transition matrices Ac .
To bias the model to find the simplest sources, we used A?c ? 0Hc ,Hc for all sources. In Fig2a and Fig
2b we see the original sources and the noisy observations respectively. In Fig2c we see the estimated
sources from our method after convergence of the hyperparameter updates. Two of the 5 sources
have been removed, and the remaining three are a reasonable estimation of the original sources.
Another possible approach for introducing prior knowledge is to use a Maximum a Posteriori (MAP)
0
50
100
150
200
250
300 0
50
100
(a)
150
200
250
300 0
50
(b)
100
150
200
250
300 0
50
(c)
100
150
200
250
300
(d)
Figure 2: (a) Original sources st . (b) Observations resulting from mixing the original sources,
vt = W st + ?tv , ?tv ? N (0, I). (c) Recovered sources using the Bayesian LGSSM. (d) Sources
found with MAP LGSSM.
0
1
2
(a)
3s 0
1
2
(b)
3s 0
1
2
(c)
3s 0
1
2
(d)
3s 0
1
2
3s
(e)
Figure 3: (a) Original raw EEG recordings from 4 channels. (b-e) 16 sources st estimated by the
Bayesian LGSSM.
procedure by adding a prior term to the original log-likelihood log p(v1:T |A, W, ?H , ?V , ?, ?) +
log p(A|?) + log p(W |?). However, it is not clear how to reliably find the hyperparameters ? and ?
in this case. One solution is to estimate them by optimizing the new objective function jointly with
respect to the parameters and hyperparameters (this is the so-called joint map estimation ? see for
example [16]). A typical result of using this joint MAP approach on the artificial data is presented
in Fig 2d. The joint MAP does not estimate the hyperparameters well, and the incorrect number of
sources is identified.
4.2
Application to EEG Analysis
In Fig 3a we plot three seconds of EEG data recorded from 4 channels (located in the right hemisphere) while a person is performing imagined movement of the right hand. As is typical in EEG,
each channel shows drift terms below 1 Hz which correspond to artifacts of the instrumentation,
together with the presence of 50 Hz mains contamination and masks the rhythmical activity related
to the mental task, mainly centered at 10 and 20 Hz [17]. We would therefore like a method which
enables us to extract components in these information-rich 10 and 20 Hz frequency bands. Standard ICA methods such as FastICA do not find satisfactory sources based on raw ?noisy? data, and
preprocessing with band-pass filters is usually required. Additionally, in EEG research, flexibility
in the number of recovered sources is important since there may be many independent oscillators
of interest underlying the observations and we would like some way to automatically determine
their effective number. To preferentially find sources at particular frequencies, we specified a block
diagonal matrix A?c for each source c, where each block is a 2 ? 2 rotation matrix at the desired
frequency. We defined the following 16 groups of frequencies: [0.5], [0.5], [0.5], [0.5]; [10,11],
[10,11], [10,11], [10,11]; [20,21], [20,21], [20,21], [20,21]; [50], [50], [50], [50]. The temporal evolution of the sources obtained after training the Bayesian LGSSM is given in Fig3(b,c,d,e) (grouped
by frequency range). The Bayes LGSSM removed 4 unnecessary sources from the mixing matrix
W , that is one [10,11] Hz and three [20,21] Hz sources. The first 4 sources contain dominant low
frequency drift, sources 5, 6 and 8 contain [10,11] Hz, while source 10 contains [20,21] Hz centered
activity. Of the 4 sources initialized to 50 Hz, only 2 retained 50 Hz activity, while the Ac of the
other two have changed to model other frequencies present in the EEG. This method demonstrates
the usefulness and applicability of the VB method in a real-world situation.
5
Conclusion
We considered the application of Variational Bayesian learning to Linear Gaussian State-Space Models. This is an important class of models with widespread application, and finding a simple way to
implement this approximate Bayesian procedure is of considerable interest. The most demanding part of the procedure is inference of the hidden states of the model. Previously, this has been
achieved using Belief Propagation, which differs from inference in the Kalman Filtering/Smoothing
literature, for which highly efficient and stabilized procedures exist. A central contribution of this
paper is to show how inference can be written using the standard Kalman Filtering/Smoothing recursions by augmenting the original model. Additionally, a minor modification to the standard Kalman
Filtering routine may be applied for computational efficiency. We demonstrated the elegance and
unity of our approach by showing how to easily apply a Variational Bayes analysis of temporal ICA.
Specifically, our Bayes ICA approach successfully extracts independent processes underlying EEG
signals, biased towards preferred frequency ranges. We hope that this simple and unifying interpretation of Variational Bayesian LGSSMs may therefore facilitate the further application to related
models.
A
A.1
Parameter Updates for A and B
Determining q(B|?V )
By examining F, the contribution of q(B|?V ) can be interpreted as the negative KL divergence
between q(B|?
Hence,
?? V ) and a Gaussian.
??
?? optimally, q(B|?V ) is a Gaussian. The covariance
[?B ]ij,kl ? Bij ? hBij i Bkl ? hBkl i (averages wrt q(B|?V )) is given by:
?1
]jl [?V ]ik , where [HB ]jl ?
[?B ]ij,kl = [HB
T D
X
hjt hlt
t=1
?1
, where [NB ]ij ?
The mean is given by hBi = NB HB
E
q(ht )
D E
j
t=1 ht
PT
q(ht )
+ ?j ?jl .
?ij .
vti + ?j B
Determining q(A|?H )
Optimally, q(A|?H ) is a Gaussian with covariance
?1
]jl [?H ]ik , where [HA ]jl ?
[?A ]ij,kl = [HA
T
?1 D
X
hjt hlt
t=1
?1
, where [NA ]ij ?
The mean is given by hAi = NA HA
B
PT
t=2
D
E
q(ht )
hjt?1 hit
E
+ ?j ?jl .
q(ht?1:t )
+ ?j A?ij .
Covariance Updates
By specifying a Wishart prior for the inverse of the covariances, conjugate update formulae are
possible. In practice, it is more common to specify diagonal inverse covariances, for which the
corresponding priors are simply Gamma distributions [7, 5]. For this simple diagonal case, the
explicit updates are given below.
Determining q(?V )
For the constraint ??1
= diag(?), where each diagonal element follows a Gamma prior
V
Ga(b1 , b2 ) [7], q(?) factorizes and the optimal updates are
?
q(?i ) = Ga ?b1 +
where GB ?
?1 T
NB HB
NB .
?
T
X
T
1
, b2 + ? (vti )2 ? [GB ]ii +
2
2 t=1
X
j
??
2 ??
?ij
?j B
,
Determining q(?H )
Analogously, for ??1
H = diag(? ) with prior Ga(a1 , a2 ) [5], the updates are
?
??
?
T
X
X
? i 2?
T
?
1
1
, a2 + ?
(ht ) ? [GA ]ii +
?j A?2ij ?? ,
q(?i ) = Ga ?a1 +
2
2 t=2
j
?1 T
where GA ? NA HA
NA .
Acknowledgments
This work is supported by the European DIRAC Project FP6-0027787. This paper only reflects the
authors? views and funding agencies are not liable for any use that may be made of the information
contained herein.
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2,231 | 3,024 | Max-margin classification of incomplete data
2
Gal Chechik1 , Geremy Heitz2 ,
Gal Elidan1 , Pieter Abbeel 1 , Daphne Koller 1
1
Department of Computer Science, Stanford University, Stanford CA, 94305
Department of Electrical Engineering, Stanford University, Stanford CA, 94305
Email for correspondence: [email protected]
Abstract
We consider the problem of learning classifiers for structurally incomplete
data, where some objects have a subset of features inherently absent due to
complex relationships between the features. The common approach for handling missing features is to begin with a preprocessing phase that completes
the missing features, and then use a standard classification procedure. In
this paper we show how incomplete data can be classified directly without
any completion of the missing features using a max-margin learning framework. We formulate this task using a geometrically-inspired objective function, and discuss two optimization approaches: The linearly separable case
is written as a set of convex feasibility problems, and the non-separable case
has a non-convex objective that we optimize iteratively. By avoiding the
pre-processing phase in which the data is completed, these approaches offer
considerable computational savings. More importantly, we show that by elegantly handling complex patterns of missing values, our approach is both
competitive with other methods when the values are missing at random
and outperforms them when the missing values have non-trivial structure.
We demonstrate our results on two real-world problems: edge prediction in
metabolic pathways, and automobile detection in natural images.
1
Introduction
In the traditional formulation of supervised learning, data instances are viewed as vectors of
features in some high-dimensional space. However, in many real-world tasks, data instances
have a complex pattern of missing features. While features may sometimes be missing due
to measurement noise or corruption, different samples often have different sets of observable
features due to inherent properties of the instances. For example, in the case of recognizing
objects in natural images, an object is often classified using a set of image patches corresponding to parts of the object (like the license plate for cars); but some images may not
contain all parts, either because a part was not captured in the image or because the specific
instance does not have this part in the first place.
In other scenarios, some features cannot even be defined for all instances. Such situations
arise when the objects to be learned are organized based on a known graph structure, since
their features may rely on local properties of the graph. For example, we might wish to
classify the attributes of a web-page given the attributes of neighboring web-pages [8]. In
analyzing genomic data, we may wish to predict the edges in networks of interacting proteins
or chemical reactions [9, 15]. In these cases, the local neighborhood of an instance in the
graph often varies drastically, and it has already been observed that variation this could
introduce statistical biases [8]. In the web-page task, for instance, a useful feature is the
most common topic of other sites that point to a given page. When a page has no such
parents, however, this feature is meaningless and should be considered structurally absent.
The common approach for classification with missing features is imputation, a two phase
procedure where the values of the missing attributes are first filled in during a preprocessing
phase, after which a standard classifier is applied to the completed data [10]. Most Imputation techniques make most sense when the features are missing due to noise, especially
in the setting of missing at random (MAR, when the missingness pattern is conditionally
independent of the unobserved features given the observations), or missing completely at
random (MCAR, when it is independent of both observed and unobserved measurements).
In common practice of applying imputation, missing attributes in continuous data are often
filled with zeros, or with the average of all of the data instances, or using the k nearest
neighbors (kNN) of each instance to find a plausible value of its missing features. A second
family of imputation methods builds probabilistic generative models of the features using
raw maximum likelihood or algorithms such as expectation maximization (EM) [4]. Such
model-based methods allow the designer to introduce prior knowledge and are extremely
useful when priors can be explicitly modeled. These methods work very well for MAR data
settings, because they assume that the missing features are generated by the same model that
generates the observed features. However, model-based approaches can be computationally
expensive, and require significant prior knowledge about the data. More importantly, they
will produce meaningless completions for features that are structurally absent. As an extreme
example, consider two subpopulation of instances (e.g., animals and buildings) having no
overlapping features (e.g., body parts, and architectural aspects), in which filling missing
values (e.g., the body parts of buildings) is clearly meaningless and may harm classification
performance. As a result, for structurally absent features, it would be useful if we could avoid
unnecessary prediction of hypothetical undefined values, and classify instances directly.
We approach this problem directly from the geometric interpretation of the classification
task as finding a separating hyperplane in the feature space. We view instances with different feature sets as lying in subspaces of the full feature space, and suggest a modified
optimization objective within the framework of support vector machines (SVMs), that explicitly considers the subspace of each instance. We show how the linearly separable case
can be efficiently solved using convex optimization (second order cone programming, SOCP).
The objective of the non separable case is non-convex, and we propose an iterative procedure that is found to converge in practice. These approaches may be viewed as model-free
methods for handling missing data in the cases where the MAR assumption fails to hold.
We evaluate the performance of our approach in two real world applications: prediction of
missing enzymes in a metabolic network, and automobile detection in natural images. In
both tasks, features may be inherently absent due to the mechanisms described above, and
our methods are found superior to other simple imputation methods.
2
Max-Margin Formulation for Missing Features
Let x1 . . . xn be a set of samples with binary labels yi ? {?1, 1}. Each sample xi is characterized by a subset of features F(xi ), from a full set F of size d. A sample that has all
features F(xi ) = F, is viewed as a vector in Rd , where the ith coordinate corresponds to
the ith feature. A sample xi with partially valid features can be viewed as embedded in the
relevant subspace R|F (xi )| ? Rd . For simplicity of notation, we treat each xi as if it were a
vector in Rd where only its F(xi ) entries are valid and define the inner product with another
P
vector in Rd as wx = k:fk ?F (xi ) wk xk . Importantly, since instances share features, the
learned classifier must be consistent across instances, assigning the same weight to a given
feature in different samples, even if those instance do not lie in the same subspace.
In the classical SVM approach [14, 13], a linear classifier w is optimized to maximize the margin, defined as mini yi (wxi + b)/kwk, and the learning problem is reduced to the quadratic
constrained optimization problem
n
min
w,?,b
X
1
kwk2 + C
?i
2
i=1
s.t.
yi (wxi + b) ? 1 ? ?i ,
i = 1...n
(1)
where b is a threshold, the ??s are slack variables necessary for the case when the training
instances are not linearly separable, and C is the error penalty. Eq. (1) can be extended to
nonlinear classifiers using kernels [13].
Figure 1: The margin is incorrectly scaled when a sample
that has missing features is treated as if the missing features
have a value of zero. In this example, the margin of a sample
that only has one feature (the x dimension) is measured
both in the higher dimensional space (?2 ) and the lower one
(?1 ). If all features are assumed to exist, and we give the
missing feature (along the y axis) a value of zero, the margin
?2 measured in the higher dimensional space is shorter that
the margin measured in the relevant subspace ?1 .
?
2
?
1
Consider now learning such a classifier in the presence of missing data. At first glance, it may
appear that since the x?s only affect the optimization through inner products with w, missing
features can merely be skipped (or equivalently, replaced with zeros), thereby preserving the
values of the inner product. However, this does not properly normalize the different entries
in w, and damages classification accuracy. The reason is illustrated in Fig. 1 where a single
sample in R2 has one valid and one missing feature. Due to the missing feature, measuring
the margin in the full space ?2 , underestimates the correct geometric margin of the sample
in the valid space ?1 . This is different from the case where the feature exists but is unknown,
in which the sample?s margin could be either over- or under-estimated. In the next sections,
we explore how this Eq. (1) can be solved while properly taking this normalization into
account. We start by reminding the reader about the geometric interpretation of SVM.
3
Geometric interpretation
The derivation of the SVM classifier [14] is motivated by the goal of finding a hyperplane that
maximally separates the positive examples from the negative, as measured by the geometric
i wxi
margin ?(w) = mini ykwk
. The task of maximizing the margin ?(w),
yi wxi
max ?(w) = max min
w
w
i
kwk
(2)
is transformed into the quadratic programming problem of Eq. (1) in two steps. First, kwk,
1
is taken out of the minimization, yielding maxw kwk
(mini yi wxi ). Then, the following
invariance is used: for every solution, there exists a solution that achieves the same target
function value, but with a margin that equals 1. This allows us to write the SVM problem
?1
as a constrained optimization problem: maxw kwk s.t. yi (wxi ) ? 1. This is equivalent
2
to minimizing kwk with the same constraints, which equals the SVM problem of Eq. (1).
In the case of missing features, this derivation no longer optimizes the correct geometrical
margin (Fig. 1). To address this problem, we treat the margin of each instance in its own
(i)
xi
iw
subspace, by defining the instance margin for the ith instance as ?i (w) = ykw
where
(i) k
qP
(i)
2
kw k =
k:fk ?F (xi ) wk . The geometric margin is, as before, the minimum over all
instance margins, yielding a new optimization problem
yi w(i) xi
max min
.
(3)
w
i
kw(i) k
Unfortunately, since different margin terms are normalized by different norms kw (i) k, we
can no longer take the denominator out of the minimization as above. In addition, each
of the terms yi w(i) xi /kw(i) k is non-convex in w, which is difficult to solve directly in an
efficient way. We now discuss two approaches for solving this problem.
In the linearly separable case, the optimization problem of Eq. (3) is equivalent to
max
w,?
?
s.t.
yi w(i) xi ? ?kw(i) k
i = 1...n
,
(4)
This is a convex feasibility problem for any fixed value of ?, which is a real scalar that
corresponds to the margin. It can be solved efficiently using a bisection search over ? ? R + ,
where in each iteration we solve a convex second order cone program (SOCP) [11]. Unfortunately, extending this formulation to the non-separable while preserving the geometric
margin interpretation case makes the problem non-convex (this is discussed elsewhere).
A second approach for solving Eq. (3) is to treat each instance margin individually. We
represent each of the norms kw(i) k as a scaling of the full norm by defining scaling coefficients
si = kw(i) k/kwk, and rewriting Eq. (3) as
yi wxi
1
yi wxi
kw(i) k
max min
= max
min
, si =
.
(5)
w
w kwk
i si kwk
i
si
kwk
The si factors are scalars, and had we known them, we could have solved a standard SVM
problem. Unfortunately they depend on w (i) and are unknown. This formalism allows us to
use again the invariance to the rescaling of kwk and rewrite as a constrained optimization
problem over si and w. In the non-separable case, Eq. (5) becomes
min
w,b,?,s
X
1
kwk2 + C
?i
2
i
s.t.
1
(yi (wxi + b)) ? 1 ? ?i ,
si
si = kw(i) k/kwk ,
i = 1...n
(6)
i = 1...n
This constrained optimization problem is no longer a QP. In fact, due to the normalization
constraint it is not even convex in w. One solution is a projected gradient approach, in which
one iterates between steps in the direction of the gradient of the Lagrangian and projections
to the constrained space, by calculating si = kw(i) k/kwk. For the right choices of step sizes,
such approaches are guaranteed to converge to local minima [2].
We can use a faster iterative algorithm based on the fact that the problem is a QP for any
given set of si ?s, and iterate between (1) solve a QP for w given si , and (2) use the resulting
w to calculate new si ?s. This algorithm differs from a projected gradient approach in that
rather than taking a series of small gradient steps, it takes bigger leaps, and projects back to
the constrained space after each step. Since the convergence of this iterative algorithm is not
guaranteed, we used cross validation to choose an early stopping point and found that the
best solutions were obtained within 2-5 steps. Typically, the objective improved on the first
1-3 iterations, but then, in about 75% of the cases the objective oscillated. In the remaining
cases the algorithm converged to a fixed point. It is easy to see that a fixed point of this
iterative procedure achieves an optimal solution for Eq. (6), since it achieves a minimal kwk
while obeying the si constraints. As a result, when this algorithm converges, the solution is
also guaranteed to be a locally optimal solution of the original problem Eq. (3).
The power of the SVM approach can be largely attributed to the flexibility and efficiency
of nonlinear classification allowed through the use of kernels. The dual of the above QP can
be kernelized as in a standard SVM, yielding
n
n
X
1 X yi
yj
maxn
?i ?
?i K (xi , xj ) ?j
2
s
sj
i
??R i=1
i,j=1
s.t. 0 ? ?i ? C ;
n
X
?i yi = 0.
(7)
i=1
where K(xi , xj ) is the kernel function that simulates an inner product in the higher dimensional feature space. Classification
as in standard SVM by
Pof new samples are obtained
1
calculating the margin ?(xnew ) = j yj ?j s1j K(xj , xnew ) snew
.
Kernels in this formulation operate over vectors with missing features, hence we have to
develop kernels that handle them correctly. Fortunately, many kernels only depend on their
inputs through their inner product. In this case there is an easy procedure to construct a
modified kernel that takes such missing values into account. For example, for a polynomial
d
d
kernel K(xi , xj ) = (hxi , xj i + 1) , define K 0 (xi , xj ) = K(xi , xP
j ) = (hxi , xj iF + 1) , with
the inner product calculated over valid features hxi , xj iF =
k:fk ??(xj )?F (xi ) hxik , xjk i.
This can be easily proved to be a kernel.
(a)
(b)
(c)
0.5
classification error
0.4
0.3
0.2
0.1
0
geom
zero
mean flag?agg kNN 5
Figure 2: Car classification results. (a) An easy instance where all local features are approximately
in agreement. (b) A hard instance where local features are divided into two distinct groups. This
instance was correctly classified by the ?geometric margin? approach but misclassified by all other
methods. (c) Classification accuracy of the different methods for the task of object recognition in
real images. Error bars are standard errors of the mean (SEM) over the five cross validation sets.
4
Experiments
We evaluated our approaches in three different missingness scenarios. First, as a sanity
check, we explored performance when features are missing at random, in a series of five
standard UCI benchmarks, and also in a large digit recognition task using MNIST data. In
this setting our methods performed equally well as other approaches (or slightly better). The
full details of these experiments are provided in a longer version of this work. Second, we
study a visual object recognition application where some features are missing because they
cannot be located in the image. Finally, we apply our methods to a problem of biological
network completion, where missingness patterns of features is determined by the known
structure of the network.
For all applications, we compare our iterative algorithm with five common approaches for
completing missing features. 1. Zero: Missing values were set to zero. 2. Mean: Missing
values were set to the average feature values 3. Aggregated Flags: Features were annotated with an explicit additional feature noting whether a feature is valid or missing. To
reduce the number of added features, we added a single flag for each group of features that
were valid or invalid together across all instances. For example, In the vision application, all
features of a landmark candidate are grouped together since they are all invalid if the match
is wrong (see below). 4. kNN: Missing features were set with the mean value obtained
from the K nearest neighbors instances; neighborhood was measured using a Euclidean
distance in the subspace relevant to each two samples, number of neighbors was varied as
K = 3, 5, 10, 20, and the best result is the one reported. 5. EM: Generative model in
the spirit of [4]. A Gaussian mixture model is learned by iterating between (1) learning a
GMM model of the filled data (2) re-filling missing values using clusters means, weighted
by the posterior probability that a cluster generated the sample. Covariances were assumed
spherical. The number of clusters was varied as K = 3, 5, 10, 15, 20, and the best result is
the one reported. 6. Geometric margin: Our non-separable approach described in Sec. 3.
In all of the experiments, we used a 5-fold cross validation procedure and evaluated performance using a testing set that was not used during training. In addition, 20% of the training
set was used for choosing optimization parameters, such as the kernel type, its parameters,
and an early stopping point for the iterative algorithm.
4.1 Visual object recognition
We now consider a visual object recognition task where instances have structurally missing
features. In this task we attempt to determine if an object from a certain class (automobiles)
is present in a given input image. The task of classifying images based on the object class
that they contain has seen much work in recent years [1, 5],and discriminative approaches
have typically produced very good results [5, 12].
Features in these methods are commonly constructed from regions of interest (patches) in
the image. These patches typically cover ?landmarks? of the object, like the trunk or a
headlight for a car. A typical set of patches includes several candidates for any object part,
and some images may have more candidates for a given part than others. For example, a
trunk of a car may not be found in a picture of a hatch-back car, hence all its corresponding
features are considered to be structurally missing from that image. Our object model
contains a set of ?landmarks?, for which we find several matches in a given image (details
are omitted due to lack of space). Fig. 2 shows examples of matches for the front windshield
landmark. Because of the noisy matches, the highest scoring match often does not match
the true landmark, and the number of high-quality matches (features) varies in practice. It
is in precisely such a scenario that we expect our proposed algorithm to be effective.
In some cases, landmark models could provide confidence levels for each match. These
could in principle be used as additional features to help the classifiers give more weight to
better matches, and are expected to improve classification when the confidence measure is
reliable. While this is a potentially useful approach for the current application, this paper
takes a different approach: it does not use any soft confidence values but rather treats the
low-confidence matches as wrong, removing them from the data.
Concretely, we located up to 10 candidate patches (21 ? 21 pixels) that were promising
(likelihood above a given threshold) for each of the 19 landmarks in the car model. For
each candidate, we compute the first 10 principal component coefficients of the image patch
and concatenate these patches to form the image feature vector. If the number of patches
for a given landmark is less than ten, we consider the rest to be structurally absent. We
evaluated performance for this task using two levels of a 5-fold cross validation procedure
as explained above. We compared several kernels and report results using the kernel that
fared best on the validation set, which was usually a second order polynomial kernel.
Fig. 2c compares the accuracy of the different methods. We found the geometric approach
to be significantly superior to all other methods. To further evaluate our method, we qualitatively examined the classification results for several images across the various methods.
Fig. 2a shows the top 10 matches for the front windshield landmark for a representative
?easy? test instance where all local features are approximately in agreement. This instance
was correctly classified by all methods. In contrast, Fig. 2b shows a representative ?hard?
test instance where local features cluster into two different groups. In this case, the cluster
of bad matches was automatically excluded yielding missing features, and our geometric
approach was the only method able to classify the instance correctly.
4.2
Metabolic pathway reconstruction
As a final application, we consider the problem of predicting missing enzymes in metabolic
pathways, a long-standing and important challenge in computational biology [15, 9]. Instances in this task have missing features due to the structure of the biochemical network.
Cells use a complex network of chemical reactions to produce their chemical building blocks
(Fig. 3). Each reaction transforms a set of molecular compounds (called substrates) into
another set of molecules (products), and requires the presence of an enzyme to catalyze the
reaction. It is often unknown which enzyme catalyzes a given reaction, and it is desirable
to predict the identity of such missing enzymes computationally.
Our approach for predicting missing enzymes is based on the observation that enzymes in
local network neighborhoods usually participate in related functions. As a result, neighboring enzyme pairs have non trivial correlations over their features that reflect their functional
relations. Importantly, different types of neighborhood relations between enzyme pairs lead
to different relations of their properties. For example, an enzyme in a linear chain depends
on the preceding enzyme product as its substrate. Hence it is expected that the corresponding genes are co-expressed [9, 15]. On the other hand, enzymes in forking motifs (same
substrate, different products) often have anti-correlated expression profiles [7].
To preserve the distinction between different neighbor relations, we defined a set of network
motifs, including forks, funnels and linear chains. Each enzyme is represented as a vector
of features that measure its relatedness to each of its neighbors. A feature vector has
structurally missing entries if the enzyme does not have all types of neighbors. For example,
the enzyme PHA2 in Fig. 3 does not have a neighbor of type fork, and therefore all features
assigned to such a neighbor are absent in the representation of the reaction ?Prephanate ?
Phenylpyruvate?.
classification error
0.5
0.4
0.3
0.2
0.1
0
geom
zero
mean flag?agg kNN 10
1
1 ? False positives
0.8
0.2
0
0
geom
zero
mean
flag?agg
kNN 10
0.25
0.75
1
True positives
Figure 3: Left: A fragment of the full metabolic pathway network in S. cerevisiae. Chemical
reactions (arrows) transform a set of molecular compounds into other compounds. Small molecules
like CO2 were omitted from this drawing for clarity. Reactions are catalyzed by enzymes (boxed
names, e.g., ARO7), but in some cases these enzymes are unknown. The network imposes various
neighborhood relations between enzymes assigned to reactions, like linear chains (ARO7,PHA2),
forks (TRP2,ARO7) and funnels (ARO9,PHA2) Top Right: Classification accuracy for compared
methods. The classification task is to identify if a candidate enzyme is in the right ?neighborhood?.
Error bars are SEMs over 5 cross validation sets. Bottom right: ROC curves for the same task.
We used the metabolic network of S. cerevisiae, as reconstructed by Palsson and colleagues
[3], after removing 14 metabolic currencies and reactions with unknown enzymes, leaving
1265 directed reactions. We used three data types: (1) A compendium of 645 gene expression
experiments; each experimental condition k contributed one feature, the point-wise Pearson
xi (k)xj (k)
. xi is the vector of expression levels across conditions. (2) The proteincorrelation kx
i kkxj k
domain content of each enzyme as found by the Prosite database. Each domain k contributed
one feature, the point-wise symmetric DKL measure xi (k) (log(xi (k)/(xj (k) + xi (k))/2)) +
xj (k) (log(xj (k)/(xj (k) + xi (k))/2)). (3) The cellular localization of the protein [6]; each
cellular localization contributed one feature, the point-wise Hamming distance. In total,
the feature vector length was about 3900. Pathway reconstruction requires that we rank
candidate enzymes by their potential to match a reaction. As a first step towards this goal,
we train a binary classifier, to predict if an enzyme fits its neighborhood. We created a
set of positive examples from the reactions with known enzymes (? 520 reactions), and
also created negative examples by plugging impostor genes into ?wrong? neighborhoods. We
trained an SVM classifier using a 5-fold cross validation procedure as described above.
Figure 3 shows the classification error of the different methods in the gene filling task.
The geometric margin approach achieves significantly better performance in this task. kNN
achieved very poor performance compared to all other methods. One reason could be
that the Euclidean distance is inappropriate for the current task and that a more elaborate
distance measure needs to developed for this type of data. Learning metrics is a complicated
task in general, and more so in the current problem since the feature vectors contain entries
of several different types, making it unlikely that a naive distance measure would work well.
Finally, the resulting classifier is used for predicting missing enzymes, by ranking all candidate enzymes according to their match to a given neighborhood. Evaluating the quality
of ranking on known enzymes (cross validation), shows that it significantly outperforms
previous approaches [9] (not shown here due to space limitations). We attribute this to the
ability of the current approach to preserve different types of network-neighbors as separate
features in spite of creating missing values.
5
Discussion
We presented a novel method for max-margin training of classifiers in the presence of missing
features, where the pattern of missing features is an inherent part of the domain. Instead of
completing missing features as a preprocessing phase, we developed a max-margin learning
objective based on a geometric interpretation of the margin when different instances essentially lie in different spaces. Using two challenging real life problems we showed that our
method is significantly superior when the pattern of missing features has structure.
The standard treatment of missing features is based on the concept that missing features
exist but are unobserved. This assumption underlies the approach of completing features
before the data is used in classification. This paper focuses on a different scenario, in which
features are inherently absent. In such cases, it is not clear why we should guess hypothetical
values for undefined features, since the completed values are filled based on other observed
values, and do not add information to our classifiers. In fact, by completing features that
are not supposed to be part of an instance, we may be confusing the learning algorithm by
presenting it with problem which may be harder than the one we actually need to solve.
Interestingly, the problem of classifying with missing features is related to another problem,
where individual reliability measures are available for features at each instance separately.
This is a common case in analysis scientific measurements, where the reliability of each
experiment could be provided separately. For example, DNA micro-array experiments have
inherent measures of experimental noise levels, and biological variability is often estimated
using replicates. This problem can be viewed in the same framework described in this
paper: the geometric margin must be defined separately for each instance since the different
noise levels distort the relative scale of each coordinate of each instance separately. Relative
to this setting, the completely missing and fully valid features discussed in this paper are
extreme points on the spectrum of reliability. It will be interesting to see which aspects of
the geometric formulation discussed in this paper can be extended to this new problem.
Acknowledgement: This paper was supported by a NSF grant DBI-0345474.
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2,232 | 3,025 | Accelerated Variational Dirichlet Process Mixtures
Kenichi Kurihara
Dept. of Computer Science
Tokyo Institute of Technology
Tokyo, Japan
[email protected]
Max Welling
Bren School of Information and Computer Science
UC Irvine
Irvine, CA 92697-3425
[email protected]
Nikos Vlassis
Informatics Institute
University of Amsterdam
The Netherlands
[email protected]
Abstract
Dirichlet Process (DP) mixture models are promising candidates for clustering
applications where the number of clusters is unknown a priori. Due to computational considerations these models are unfortunately unsuitable for large scale
data-mining applications. We propose a class of deterministic accelerated DP
mixture models that can routinely handle millions of data-cases. The speedup is
achieved by incorporating kd-trees into a variational Bayesian algorithm for DP
mixtures in the stick-breaking representation, similar to that of Blei and Jordan
(2005). Our algorithm differs in the use of kd-trees and in the way we handle
truncation: we only assume that the variational distributions are fixed at their priors after a certain level. Experiments show that speedups relative to the standard
variational algorithm can be significant.
1
Introduction
Evidenced by three recent workshops1 , nonparametric Bayesian methods are gaining popularity
in the machine learning community. In each of these workshops computational efficiency was
mentioned as an important direction for future research. In this paper we propose computational
speedups for Dirichlet Process (DP) mixture models [1, 2, 3, 4, 5, 6, 7], with the purpose of improving their applicability in modern day data-mining problems where millions of data-cases are no
exception.
Our approach is related to, and complements, the variational mean-field algorithm for DP mixture
models of Blei and Jordan [7]. In this approach, the intractable posterior of the DP mixture is
approximated with a factorized variational finite (truncated) mixture model with T components, that
is optimized to minimize the KL distance to the posterior. However, a downside of their model is
that the variational families are not nested over T , and locating an optimal truncation level T may
be difficult (see Section 3).
In this paper we propose an alternative variational mean-field algorithm, called VDP (Variational
DP), in which the variational families are nested over T . In our model we allow for an unbounded
number of components for the variational mixture, but we tie the variational distributions after level
1
http://aluminum.cse.buffalo.edu:8079/npbayes/nipsws05/topics
http://www.cs.toronto.edu/? beal/npbayes/
http://www2.informatik.hu-berlin.de/? bickel/npb-workshop.html
T to their priors. Our algorithm proceeds in a greedy manner by starting with T = 1 and releasing
components when this improves (significantly) the KL bound. Releasing is most effectively done
by splitting a component in two children and updating them to convergence. Our approach essentially resolves the issue in [7] of searching for an optimal truncation level of the variational mixture
(see Section 4).
Additionally, a significant contribution is that we incorporate kd-trees into the VDP algorithm as a
way to speed up convergence [8, 9]. A kd-tree structure recursively partitions the data space into a
number of nodes, where each node contains a subset of the data-cases. Following [9], for a given tree
expansion we tie together the responsibility over mixture components of all data-cases contained in
each outer node of the tree. By caching certain sufficient statistics in each node of the kd-tree we
then achieve computational gains, while the variational approximation becomes a function of the
depth of the tree at which one operates (see Section 6).
The resulting Fast-VDP algorithm provides an elegant way to trade off computational resources
against accuracy. We can always release new components from the pool and split kd-tree nodes as
long as we have computational resources left. Our setup guarantees that this will always (at least
in theory) improve the KL bound (in practice local optima may force us to reject certain splits,
see Section 7). As we empirically demonstrate in Section 8, a kd-tree can offer significant speedups,
allowing our algorithm to handle millions of data-cases. As a result, Fast-VDP is the first algorithm entertaining an unbounded number of clusters that is practical for modern day data-mining
applications.
2
The Dirichlet Process Mixture in the Stick-Breaking Representation
A DP mixture model in the stick-breaking representation can be viewed as possessing an infinite
number of components with random mixing weights [4]. In particular, the generative model of a DP
mixture assumes:
? An infinite collection of components H = {?i }?
i=1 that are independently drawn from a
prior p? (?i |?) with hyperparameters ?.
? An infinite collection of ?stick lengths? V = {vi }?
i=1 , vi ? [0, 1], ?i, that are independently
drawn from a prior pv (vi |?) with hyperparameters ?. They define the mixing weights
Qi?1
{?i }?
i=1 of the mixture as ?i (V ) = vi
j=1 (1 ? vj ), for i = 1, . . . , ?.
? An observation model px (x|?) that generates a datum x from component ?.
Given a dataset X = {xn }N
n=1 , each data-case xn is assumed to be generated by first drawing a
component label zn = k ? {1, . . . , ?} from the infinite mixture with probability pz (zn = k|V ) ?
?k (V ), and then drawing xn from the corresponding observation model px (xn |?k ).
We will denote Z = {zn }N
n=1 the set of all labels, W = {H, V, Z} the set of all latent variables
of the DP mixture, and ? = {?, ?} the hyperparameters. In clustering problems we are mainly
interested in computing
the posterior
over data labels p(zn |X, ?), as well as the predictive density
R
R
p(x|X, ?) = H,V p(x|H, V ) Z p(W |X, ?), which are both intractable since p(W |X, ?) cannot be
computed analytically.
3
Variational Inference in Dirichlet Process Mixtures
For variational inference, the intractable posterior p(W |X, ?) of the DP mixture can be approximated with a parametric family of factorized variational distributions q(W ; ?) of the form
q(W ; ?) =
L h
Y
i=1
N
iY
qvi (vi ; ?vi ) q?i (?i ; ??i )
qzn (zn )
(1)
n=1
where qvi (vi ; ?vi ) and q?i (?i ; ??i ) are parametric models with parameters ?vi and ??i (one parameter
per i), and qzn (zn ) are discrete distributions over the component labels (one distribution per n).
Blei and Jordan [7] define an explicit truncation level L ? T for the variational mixture in (1) by
setting qvT (vT = 1) = 1 and assuming that data-cases assign zero responsibility to components
with index higher than the truncation level T , i.e., qzn (zn > T ) = 0. Consequently, in their model
only components of the mixture up to level T need to be considered. Variational inference then
consists in estimating a set of T parameters {?vi , ??i }Ti=1 and a set of N distributions {qzn (zn )}N
n=1 ,
collectively denoted by ?, that minimize the Kullback-Leibler divergence D[q(W ; ?)||p(W |X, ?)]
between the true posterior and the variational approximation, or equivalently that minimize the free
energy F (?) = Eq [log q(W ; ?)] ? Eq [log p(W, X|?)]. Since each distribution qzn (zn ) has nonzero
support only for zn ? T , minimizing F (?) results in a set of update equations for ? that involve
only finite sums [7].
However, explicitly truncating the variational mixture as above has the undesirable property that
the variational family with truncation level T is not contained within the variational family with
truncation level T + 1, i.e., the families are not nested.2 The result is that there may be an optimal
finite truncation level T for q, that contradicts the intuition that the more components we allow in
q the better the approximation should be (reaching its best when T ? ?). Moreover, locating a
near-optimal truncation level may be difficult since F as a function of T may exhibit local minima
(see Fig. 4 in [7]).
4
Variational Inference with an Infinite Variational Model
Here we propose a slightly different variational model for q that allows families over T to be nested.
In our setup, q is given by (1) where we let L go to infinity but we tie the parameters of all models after a specific level T (with T ? L). In particular, we impose the condition that for all
components with index i > T the variational distributions for the stick-lengths qvi (vi ) and the
variational distributions for the components q?i (?i ) are equal to their corresponding priors, i.e.,
qvi (vi ; ?vi ) = pv (vi |?) and q?i (?i ; ??i ) = p? (?i |?). In our model we define the free energy F as
the limit F = limL?? FL , where FL is the free energy defined by q in (1) and a truncated DP
mixture at level L (justified by the almost sure convergence of an L-truncated Dirichlet process to
an infinite Dirichlet process when L ? ? [6]). Using the parameter tying assumption for i > T , the
free energy reads
X
N
T
X
q? (?i ; ??i )
qzn (zn )
qv (vi ; ?vi )
+Eq?i log i
+
Eq log
Eqvi log i
. (2)
F =
pv (vi |?)
p? (?i |?)
pz (zn |V )px (xn |?zn )
n=1
i=1
In our scheme T defines an implicit truncation level of the variational mixture, since there are no free
parameters to optimize beyond level T . As in [7], the free energy F is a function of T parameters
{?vi , ??i }Ti=1 and N distributions {qzn (zn )}N
n=1 . However, contrary to [7], data-cases may now
assign nonzero responsibility to components beyond level T , and therefore each qzn (zn ) must now
have infinite support (which requires computing infinite sums in the various quantities of interest).
An important implication of our setup is that the variational families are now nested with respect
to T (since for i > T , qvi (vi ) and q?i (?i ) can always revert to their priors), and as a result it is
guaranteed that as we increase T there exist solutions that decrease F . This is an important result
because it allows for optimization with adaptive T starting from T = 1 (see Section 7).
?From the last term of (2) we directly see that the qzn (zn ) that minimizes F is given by
exp(Sn,i )
qzn (zn = i) = P?
j=1 exp(Sn,j )
where
Sn,i = EqV [log pz (zn = i|V )] + Eq?i [log px (xn |?i )].
?vi
??i
(3)
(4)
Minimization of F over
and
can be carried out by direct differentiation of (2) for particular
choices of models for qvi and q?i (see Section 5). Using qzn from (3), the free energy (2) reads
X
T
N
?
X
X
qvi (vi ; ?vi )
q?i (?i ; ??i )
F =
Eqvi log
+ Eq?i log
?
log
exp(Sn,i ). (5)
pv (vi |?)
p? (?i |?)
n=1
i=1
i=1
P?
Evaluation of F requires computing the infinite sum i=1 exp(Sn,i ) in (5). The difficult part is
P
?
i=T +1 exp(Sn,i ). Under the parameter tying assumption for i > T , most terms of Sn,i in (4)
2
We thank David Blei for pointing this out.
factor out of the infinite sum as constants (since they do not depend on i) except for the term
Pi?1
j=T +1 Epv [log(1 ? v)] = (i ? 1 ? T )Epv [log(1 ? v)]. From the above, the infinite sum can
be shown to be
?
X
Sn,T +1
.
exp(Sn,i ) =
(6)
1 ? exp Epv [log(1 ? v)]
i=T +1
Using the variational q(W ) as an approximation to the true posterior p(W |X, ?), the required posterior over data labels can be approximated by p(zn |X, ?) ? qzn (zn ). Although qzn (zn ) has infinite
support, in practice it suffices to use the individual qzn (zn = i) for the finite part i ? T , and the
cumulative qzn (zn > TP) for the infinite part. Finally, using the parameter tying assumption for
?
i > T , and the identity i=1 ?i (V ) = 1, the predictive density p(x|X, ?) can be approximated by
p(x|X, ?) ?
T
X
i=1
T
h
i
X
EqV [?i (V )]Eq?i [px (x|?i )] + 1 ?
Epv [?i (V )] Ep? [px (x|?)].
(7)
i=1
Note that all quantities of interest, such as the free energy (5) and the predictive distribution (7), can
be computed analytically even though they involve infinite sums.
5
Solutions for the exponential family
The results in the previous section apply independently of the choice of models for the DP mixture.
In this section we provide analytical solutions for models in the exponential family. In particular we
assume that pv (vi |?) = Beta(?1 , ?2 ) and qvi (vi ; ?vi ) = Beta(?vi,1 , ?vi,2 ), and that px (x|?), p? (?|?),
and q?i (?i ; ??i ) are given by
px (x|?) = h(x) exp{? T x ? a(?)}
p? (?|?) = h(?) exp{?1 ? + ?2 (?a(?)) ? a(?)}
(8)
(9)
q?i (?i ; ??i ) = h(?i ) exp{??i,1 ?i + ??i,2 (?a(?i )) ? a(??i )}.
(10)
In this case, the probabilities qzn (zn = i) are given by (3) with Sn,i computed from (4) using
Eqvi [log vi ] = ?(?vi,1 ) ? ?(?vi,1 + ?vi,2 )
Eqvj [log(1 ? vj )] =
?(?vi,2 )
?
?(?vi,1
+
(11)
?vi,2 )
(12)
Eq?i [log px (xn |?i )] = Eq?i [?i ]T xn ? Eq?i [a(?i )]
v
(13)
?
where ?(?) is the digamma function. The optimal parameters ? , ? can be found to be
?vi,1
= ?1 +
N
X
qzn (zn = i)
?vi,2
= ?2 +
n=1
??i,1
= ?1 +
N
X
N
?
X
X
qzn (zn = j)
(14)
n=1 j=i+1
qzn (zn = i)xn
??i,2
n=1
= ?2 +
N
X
qzn (zn = i).
(15)
n=1
The update equations are similar to those inP
[7] except that we have used Beta(?1 , ?2 ) instead of
?
Beta(1, ?), and ?vi,2 involves an infinite sum j=i+1 qzn (zn = j) which can be computed using (3)
and (6). In [7] the corresponding sum is finite since qzn (zn ) is truncated at T .
Note that the VDP algorithm operates in a space where component labels are distinguishable, i.e.,
if we permute the labels the total probability of the data changes. Since the average a priori mixture weights of the components are ordered by their size, the optimal labelling of the a posteriori
variational components is also ordered according to cluster size. Hence, we have incorporated a reordering step of components according to approximate size after each optimization step in our final
algorithm (a feature that was not present in [7]).
6
Accelerating inference using a kd-tree
In this section we show that we can achieve accelerated inference for large datasets when we store
the data in a kd-tree [10] and cache data sufficient statistics in each node of the kd-tree [8]. A kd-tree
is a binary tree in which the root node contains all points, and each child node contains a subset of
the data points contained in its father node, where points are separated by a (typically axis-aligned)
hyperplane. Each point in the set is contained in exactly one node, and the set of outer nodes of a
given expansion of the kd-tree form a partition of the data set.
Suppose the kd-tree containing our data X is expanded to some level. Following [9], to achieve
accelerated update equations we constrain all xn in outer node A to share the same qzn (zn ) ?
qzA (zA ). We can then show that, under this constraint, the qzA (zA ) that minimizes F is given by
exp(SA,i )
qzA (zA = i) = P?
j=1 exp(SA,j )
(16)
where SA,i is computed as in (4) using (11)?(13) with (13) replaced by Eq?i [?i ]T hxiA ?Eq?i [a(?i )],
and hxiA denotes average over all data xn contained in node A. Similarly, if |nA | is the number of
data in node A, the optimal parameters can be shown to be
?vi,1
= ?1 +
X
|nA |qzA (zA = i)
X
|nA |qzA (zA = i)hxiA
?vi,2
= ?2 +
A
??i,1 = ?1 +
|nA |
X
A
|nA |qzA (zA = i).
qzA (zA = j)
(17)
j=i+1
A
??i,2 = ?2 +
?
X
X
(18)
A
Finally, using qzA (zA ) from (16) the free energy (5) reads
X
T
?
X
X
q?i (?i ; ??i )
qvi (vi ; ?vi )
F =
Eqvi log
+ Eq?i log
?
|nA | log
exp(SA,i ). (19)
pv (vi |?)
p? (?i |?)
i=1
i=1
A
The infinite sums in (17) and (19) can be computed from (6) with Sn,T +1 replaced by SA,T +1 . Note
that the cost of each update cycle is O(T |A|), which can be a significant improvement over the
O(T N ) cost when not using a kd-tree. (The cost of building the kd-tree is O(N log N ) but this is
amortized by multiple optimization steps.) Note that by refining the tree (expanding outer nodes)
the free energy F cannot increase. This allows us to control the trade-off between computational
resources and approximation: we can always choose to descend the tree until our computational
resources run out, and the level of approximation will be directly tied to F (deeper levels will mean
lower F ).
7
The algorithm
The proposed framework is quite general and allows flexibility in the design of an algorithm. Below
we show in pseudocode the algorithm that we used in our experiments (for DP Gaussian mixtures).
Input is a dataset X = {xn }N
n=1 that is already stored in a kd-tree structure. Output is a set of
parameters {?vi , ??i }Ti=1 and a value for T . From that we can compute responsibilities qzn using (3).
1. Set T = 1. Expand the kd-tree to some initial level (e.g. four).
P
2. Sample a number of ?candidate? components c according to size A |nA |qzA (zA = c), and split the
component that leads to the maximal reduction of FT . For each candidate c do:
(a) Expand one-level deeper the outer nodes of the kd-tree that assign to c the highest responsibility
qzA (zA = c) among all components.
(b) Split c in two components, i and j, through the bisector of its principal component. Initialize
the responsibilities qzA (zA = i) and qzA (zA = j).
(c) Update only SA,i , ?vi , ??i and SA,j , ?vj , ??j for the new components i and j, keeping all other
parameters as well as the kd-tree expansion fixed.
3. Update SA,t , ?vt , ??t for all t ? T + 1, while expanding the kd-tree and reordering components.
4. If FT +1 > FT ? ? then halt, else set T := T + 1 and go to step 2.
In the above algorithm, the number of sampled candidate components in step 2 can be tuned according to the desired cost/accuracy tradeoff. In our experiments we used 10 candidate components.
In step 2b we initialized the responsibilities by qzA (zA = i) = 1 = 1 ? qzA (zA = j) if hxiA is
23
9
1
1
0.9
0.8
5000
1000 2000
free energy ratio
speedup factor
Fast-VDP
VDP
BJ
#data
Figure 1: Relative runtimes and free energies of Fast-VDP, VDP, and BJ.
Fast-VDP
VDP
15
10
5
1
160
40
1
5
3
1
1.02
1.02
1.01
1.01
1
10
100
1000
#data in thousands
1.02
1.01
1
1
16 32 64 128
dimensionality
1
2
3
c-separation
Figure 2: Speedup factors and free energy ratios between Fast-VDP and VDP. Top and bottom
figures show speedups and free energy ratios, respectively.
closer to i than to j (according to distance to the expected first moment). In order to speed up the
partial updates in step 2c, we additionally set qzA (zA = k) = 0 for all k 6= i, j (so all responsibility
is shared between the two new components). In step 3 we reordered components every one cycle
and expanded the kd-tree every three update cycles, controlling the expansion by the relative change
of qzA (zA ) between a node and its children (alternatively one can measure the change of FT +1 ).
Finally, in step 2c we monitored convergence of the partial updates through FT +1 which can be
efficiently computed by adding/subtracting terms involving the new/old components.
8
Experimental results
In this section we demonstrate VDP, and its kd-tree extension Fast-VDP, on synthetic and real
datasets. In all experiments we assumed a Gaussian observation model px (x|?) and a Gaussianinverse Wishart for p? (?|?) and q?i (?i ; ??i ).
Synthetic datasets. As argued in Section 4, an important advantage of VDP over the ?BJ? algorithm of [7] is that in VDP the variational families are nested over T , which ensures that the free
energy is a monotone decreasing function of T and therefore allows for an adaptive T (starting with
the trivial initialization T = 1). On the contrary, BJ optimizes the parameters for fixed T (and
potentially minimizes the resulting free energy over different values of T ), which requires a nontrivial initialization step for each T . Clearly, both the total runtime as well as the quality of the final
solution of BJ depend largely on its initialization step, which makes the direct comparison of VDP
with BJ difficult. Still, to get a feeling of the relative performance of VDP, Fast-VDP, and BJ, we
applied all three algorithms on a synthetic dataset containing 1000 to 5000 data-cases sampled from
10 Gaussians in 16 dimensions, in which the free parameters of BJ were set exactly as described
in [7] (20 initialization trials and T = 20). VDP and Fast-VDP were also executed until T = 20. In
Fig. 1 we show the speedup factors and free energy ratios3 among the three algorithms. Fast-VDP
3
Free energy ratio is defined as 1 + (FA ? FB )/|FB |, where A and B are either Fast-VDP, VDP or BJ.
Fast-VDP
VDP
Figure 3: Clustering results of Fast-VDP and VDP, with a speedup of 21. The clusters are ordered
according to size (from top left to bottom right).
was approximately 23 times faster than BJ, and three times faster than VDP on 5000 data-cases.
Moreover, Fast-VDP and VDP were always better than BJ in terms of free energy.
In a second synthetic set of experiments we compared the speedup of Fast-VDP over VDP. We
sampled data from 10 Gaussians in dimension D with component separation4 c. Using default
number of data-cases N = 10, 000, dimensionality D = 16, and separation c = 2, we varied
each of them, one at a time. In Fig. 2 we show the speedup factor (top) and the free energy ratio
(bottom) between the two algorithms. Note that the latter is always worse for Fast-VDP since it is an
approximation to VDP (ratio closer to one means better approximation). Fig. 2-left illustrates that
the speedup of Fast-VDP over VDP is at least linear in N , as expected from the update equations
in Section 6. The speedup factor was approximately 154 for one million data-cases, while the free
energy ratio was almost constant over N . Fig. 2-center shows an interesting dependence of speed on
dimensionality, with D = 64 giving the largest speedup. The three plots in Fig. 2 are in agreement
with similar plots in [8, 9].
Real datasets. In this experiment we applied VDP and Fast-VDP for clustering image data. We
used the MNIST dataset (http://yann.lecun.com/exdb/mnist/) which consists of 60, 000
images of the digits 0?9 in 784 dimensions (28 by 28 pixels). We first applied PCA to reduce the
dimensionality of the data to 50. Fast-VDP found 96 clusters in 3, 379 seconds with free energy
F = 1.759 ? 107 , while VDP found 88 clusters in 72, 037 seconds with free energy 1.684 ? 107 .
The speedup was 21 and the free energy ratio was 1.044. The mean images of the discovered
components are illustrated in Fig. 3. The results of the two algorithms seem qualitatively similar,
while Fast-VDP computed its results much faster than VDP.
In a second real data experiment we clustered documents from citeseer (http://citeseer.ist.
psu.edu). The dataset has 30, 696 documents, with a vocabulary size of 32, 473 words. Each
document is represented by the counts of words in its abstract. We preprocessed the dataset by
Latent Dirichlet Allocation [12] with 200 topics5 . We subsequently transformed these topic-counts
yj,k (count value of k?th topic in the j?th document) into xj,k = log(1 + yj,k ) to fit a normal
distribution better. In this problem the elapsed time of Fast-VDP and VDP were 335 seconds and
2,256 seconds, respectively, hence a speedup of 6.7. The free energy ratio was 1.040. Fast-VDP
found five clusters, while VDP found six clusters. Table 1 shows the three most frequent topics in
each cluster. Although the two algorithms found a different number of clusters, we can see that the
clusters B and F found by VDP are similar clusters, whereas Fast-VDP did not distinguish between
these two. Table 2 shows words included in these topics, showing that the documents are wellclustered.
9
Conclusions
We described VDP, a variational mean-field algorithm for Dirichlet Process mixtures, and its fast
extension Fast-VDP that utilizes kd-trees to achieve speedups. Our contribution is twofold: First,
4
A Gaussian mixture is c-separated if for each pair (i, j) of components we have ||mi ? mj ||2 ?
c D max(?max
, ?max
) , where ?max denotes the maximum eigenvalue of their covariance [11].
i
j
5
We thank David Newman for this preprocessing.
2
cluster
a
(in descending order)
the three most
1
frequent topics 2
3
81
102
59
Fast-VDP
b
c
d
73
174
40
35
50
110
49
92
94
e
A
B
76
4
129
81
102
59
73
40
174
VDP
C
D
35
50
110
76
4
129
E
F
49
92
94
73
174
40
Table 1: The three most frequent topics in each clusters. Fast-VDP found five clusters, a?e, while
VDP found six clusters, A?F.
cluster
the most
frequent topic
words
a, A
b, B, F
c, C
d, E
e, D
81
73
35
49
76
economic, policy, countries, bank, growth, firm, public, trade, market, ...
traffic, packet, tcp, network, delay, rate, bandwidth, buffer, end, loss, ...
algebra, algebras, ring, algebraic, ideal, field, lie, group, theory, ...
motion, tracking, camera, image, images, scene, stereo, object, ...
grammar, semantic, parsing, syntactic, discourse, parser, linguistic, ...
Table 2: Words in the most frequent topic of each cluster.
we extended the framework of [7] to allow for nested variational families and an adaptive truncation
level for the variational mixture. Second, we showed how kd-trees can be employed in the framework offering significant speedups, thus extending related results for finite mixture models [8, 9]. To
our knowledge, the VDP algorithm is the first nonparametric Bayesian approach to large-scale data
mining. Future work includes extending our approach to other models in the stick-breaking representation (e.g., priors of the form pvi (vi |ai , bi ) = Beta(ai , bi )), as well as alternative DP mixture
representations such as the Chinese restaurant process [3].
Acknowledgments
We thank Dave Newman for sharing code and David Blei for helpful comments. This material is
based upon work supported by ONR under Grant No. N00014-06-1-0734 and the National Science
Foundation under Grant No. 0535278
References
[1] T. Ferguson. A Bayesian analysis of some nonparametric problems. Ann. Statist., 1:209?230, 1973.
[2] C. Antoniak. Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann.
Statist., 2(6):1152?1174, 1974.
?
[3] D. Aldous. Exchangeability and related topics. In Ecole
d? e? t?e de Probabilit?e de Saint-Flour XIII, 1983.
[4] J. Sethuraman. A constructive definition of Dirichlet priors. Statist. Sinica, 4:639?650, 1994.
[5] C.E. Rasmussen. The infinite Gaussian mixture model. In NIPS 12. MIT Press, 2000.
[6] H. Ishwaran and M. Zarepour. Exact and approximate sum-representations for the Dirichlet process. Can.
J. Statist., 30:269?283, 2002.
[7] D.M. Blei and M.I. Jordan. Variational inference for Dirichlet process mixtures. Journal of Bayesian
Analysis, 1(1):121?144, 2005.
[8] A.W. Moore. Very fast EM-based mixture model clustering using multiresolution kd-trees. In NIPS 11.
MIT Press, 1999.
[9] J.J. Verbeek, J.R.J. Nunnink, and N. Vlassis. Accelerated EM-based clustering of large data sets. Data
Mining and Knowledge Discovery, 13(3):291?307, 2006.
[10] J.L. Bentley. Multidimensional binary search trees used for associative searching. Commun. ACM,
18(9):509?517, 1975.
[11] S. Dasgupta. Learning mixtures of Gaussians. In IEEE Symp. on Foundations of Computer Science, 1999.
[12] D.M. Blei, A.Y. Ng, and M.I. Jordan. Latent Dirichlet allocation. Journal of Machine Learning Research,
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2,233 | 3,026 | Blind source separation for over-determined delayed
mixtures
Lars Omlor, Martin Giese?
Laboratory for Action Representation and Learning
Department of Cognitive Neurology,
Hertie Institute for Clinical Brain Research
University of T?ubingen, Germany
Abstract
Blind source separation, i.e. the extraction of unknown sources from a set of given
signals, is relevant for many applications. A special case of this problem is dimension reduction, where the goal is to approximate a given set of signals by superpositions of a minimal number of sources. Since in this case the signals outnumber
the sources the problem is over-determined. Most popular approaches for addressing this problem are based on purely linear mixing models. However, many
applications like the modeling of acoustic signals, EMG signals, or movement
trajectories, require temporal shift-invariance of the extracted components. This
case has only rarely been treated in the computational literature, and specifically
for the case of dimension reduction almost no algorithms have been proposed. We
present a new algorithm for the solution of this problem, which is based on a timefrequency transformation (Wigner-Ville distribution) of the generative model. We
show that this algorithm outperforms classical source separation algorithms for
linear mixtures, and also a related method for mixtures with delays. In addition,
applying the new algorithm to trajectories of human gaits, we demonstrate that
it is suitable for the extraction of spatio-temporal components that are easier to
interpret than components extracted with other classical algorithms.
1
Introduction
Blind source separation techniques, such as Independent Components Analysis (ICA), have received
great interest in many domains including neuroscience [3; 19; 2], machine learning [12; 11], and
speech and signal processing [25].
A variety of algorithms have been proposed for different types of mixing models. Many studies have
focused on instantaneous mixing, where target signals are modeled by the linear superposition of a
number of source signals separately for each point in time. Another set of studies has treated convolutive mixing, where signals result from the superposition of filtered source signals (see [9] and [6]
for review). Much less explored are algorithms for anechoic mixing. In this case, signals are approximated by linear combinations of source signals with time delays. Classical cases of anechoic mixing
arise in electrical engineering, when signals from multiple antennas are received asynchronously, or
in acoustics when sound signals are recorded with multiple microphones resulting in different running times. A few algorithms have been proposed for the solution of under-determined anechoic
mixing problems, where the number of sources exceeds the number of signals [8; 4; 25; 22]. A
method that treats the case of equal numbers of signals and sources, which is based on joint diagonalization of spectral matrices, has been proposed by Yeredor [24]. Almost no work exists on
over-determined anechoic mixing problems, where the number of source signals is smaller than the
number of original signals ? the case that is most important for dimension reduction problems. Most
?
WWW home page: http://www.uni-tuebingen.de/uni/knv/arl/index.html
existing methods for the solution of under-determined problems cannot be transferred to the overdetermined case, because they involve additional assumptions about the data (e.g. specific spatial
structure [20]) or the solution (e.g. sparseness [4]). One approach employed for under-determined
anechoic mixtures is based on the assumption of small delays and a linearization of the mixture
model [5]. While this original method cannot be transferred to our problem, since it requires additional assumptions about the spatial structure of the data, preliminary work in [1] applies the same
basic approximation for the over-determined case.
In this paper we present a new algorithm for the solution of the over-determined anechoic mixing
problem, which makes no further assumptions about the size of the delays. The proposed method is
derived by applying methods from stochastic time-frequency analysis. We tested the novel algorithm
with two different test data sets, human movement trajectories and synthetic mixtures of acoustic
signals. We demonstrate that the method results in more accurate solutions with fewer sources than
classical methods (like PCA and normal ICA) for instantaneous mixing. Also, we demonstrate that
our algorithm outperforms the SOBIDS algorithm in [1] for anechoic mixtures. In addition, we
demonstrate that the method seems suitable for the extraction of biologically meaningful components from human movement data.
2
2.1
Source separation for over-determined delayed mixtures
Delayed mixture problem
In the following we assume that m signals xi (t), 1 ? i ? m have been observed. These signals
are approximated by a linear combination of n source signals sj (t) with 1 ? j ? n, with temporal
delays ?ij . In the case of anechoic mixing signals and sources obey the relationship:
xi (t) =
n
X
?ij ? sj (t ? ?ij ) i = 1, ? ? ? , m
(1)
j=1
In the over-determined case the signals outnumber the sources, i.e. m ? n.
Equation (1) is a special case of a convolutive mixture problem, where the filter kernels are given by
delta functions. However, the treatment as general deconvolution problem would neglect the special
structure of the convolutive kernel that is given by a weighted sum of delta pulses:
n
X
xi =
(?ij ?(t ? ?ij )) ? sj i = 1, ? ? ? , m
(2)
j=1
Nevertheless, this formulation suggests a treatment of this problem exploiting the framework of harmonic analysis. Since normal Fourier transformation of equation (1) results in frequency-dependent
mixtures of complex phase terms, time-frequency analysis turns out to be a more appropriate framework for the separation of sources in the above mixture models.
2.2
Wigner Ville Spectrum
In signal processing and acoustics a variety of time-frequency representations have been proposed,
ranging from linear and multilinear to nonlinear transformations. Due to their close connections to
energy and correlation measures, specifically bilinear or quadratic distributions seem very appealing.
A very popular quadratic representation is the Wigner distribution, and its modifications that are
included in Cohen?s class [7].
While the Wigner Ville Spectrum is usually defined as a deterministic integral transform, it can also
be extended for the analysis of (nonstationary) random processes resulting in the following definition
for the Wigner Ville spectrum (WVS): Assuming that x(t) is a random process and x? (t) = x
?(?t)
the reversed conjugated process, the WVS can be defined as [16]:
Z
n
? ?
? o ?2?i??
Wx (t, ?) = E x t +
x t?
e
d?
(3)
2
2
?
The WVS is basically a 2-D function defined over the time-frequency plain, which can loosely
be interpreted as a time-frequency distribution of the mean energy of x(t). The definition (3) implies many useful properties [16], three of which are particularly useful for the following derivation. If F denotes the Fourier Transform and T? , Mf the time and frequency shift operators, e.g.
(T? Mf x)(t) := e?2?if (t?? ) x(t ? ? ), then the WVS has the following properties:
1. Time-frequency shift covariance:
W(T? Mf x) (t, ?) = Wx (t ? ?, ? ? f )
(4)
Z
2. Marginal properties:
Wx (t, ?)dt = E{|Fx|2 }
(5)
Z
Wx (t, ?)df = E{|x|2 }
(6)
R
tWx (t, ?)dt
(7)
3. Mean group delay:
tx (?) := R
Wx (t, ?)dt
Since the group delay (7) is not uniquely defined in the stochastic case [15], the last property gives a
natural definition. Consistent with the standard definition for deterministic signals, the group delay
for the deterministic case can be rewritten:
R
tWx (t, ?)dt
?1 ?
arg((Fx)(?)) = tx (?) = R
2? ??
Wx (t, ?)dt
.
2.3
Application to the delayed mixture model
The original mixture model can be rewritten in the time-frequency domain by computing the WVS
of both sides of (1):
Z X
n
?
?
?
?ij ?ik sj (t + ? ?ij )sk (t ? ? ?ik ) e?2?i?? d?
Wxi (t, ?) = E
2
2
j,k=1
Z
n
X
?
?
=
?ij ?ik E sj (t + ? ?ij )s?k (t ? ? ?ik ) e?2?i?? d?
(8)
2
2
j,k=1
Assuming that the source signals sj are statistically independent equation (8) can be further simplified, since all cross terms of the form E{si sj } with i 6= j will vanish (assuming additionally
E{si } = 0). This together with the shift covariance of the distribution leads to the central equation:
n
X
|?|2ij Wsj (t ? ?ij , ?) i = 1, ? ? ? , m
(9)
Wxi (t, ?) =
j
Several existing algorithms for the solution of general convolutive or instantaneous mixture problems have exploited the 2-D structure of expressions equivalent or similar to equation (9), e.g. by
time-frequency masking [25] or joint diagonalization [24; 1]. However, the 2-D representation of a
1-D random process is redundant and sometimes may conceal the structure of the underlying data
due to interference [10]. Furthermore, the increased amount of data often results in prohibitive computational costs for large-scale problems.
This redundancy can be avoided by projecting the WVS back onto several 1-D random processes.
The most simple projections are obtained by computing the first- and zero-order moments of equation (9). These moments can be computed analytically making use of equations (5) and (7), resulting
in:
Z
Z
n
n
X
X
2
2
E{|Fxi | } = Wxi (t, ?)dt =
|?|ij Wsj (t ? ?ij , ?)dt =
|?|2ij E{|Fsj |2 }
(10)
j
j
and analogously
2
E{|Fxi (?)| } ? txi (?) =
n
X
|?|2ij
2
? E{|Fsi | } ? tsj (?) + ?ij
.
(11)
j
Equation (10) defines an instantaneous mixture problem with non-negativity constrains. Such problems can be treated with standard nonnegative matrix factorization [14] or ICA [11; 12] approaches.
After solving equation (10), the remaining unknowns in equation (11) are the delays and the group
delay (complex phase). By successive solution of these two equations the expected values of all
unknown parameters can be determined. So instead of resolving (9) it is sufficient to solve (10)
and (11) by successive iteration. This suggests the following two-step algorithm to estimate the
unknowns in the model (1).
2.4
Two-step algorithm
The last result implies the following algorithm for the estimation of the sources, delays, and the
mixing matrix in (1):
1. Compute |Fxi |2 and solve
|Fxi |2 (?) =
n
X
|?|2ij |Fsj |2 (?)
(12)
j
e.g. using non-negative matrix factorization. For our implementation we applied an algorithm for Bayesian non-negative ICA [11].
2. Initialize ?ij = 0 and iterate the following steps:
(a) Numerically solve :
n
|Fxi (?)|2 ?
X
?
arg{Fxi } =
|?|2ij ? |Fsi |2 ?
??
j
?
arg{Fsj } + ?ij
??
(13)
?
for the term ??
arg{Fsj }. Integrate the solution to obtain Fsj .
(b) Exploiting the knowledge of the sources sj it is possible to update the mixing and
the delay matrix through optimization of the following cost function, which is derived
from (1), where S(?~j ) = (sk (ti ? ?jk ))i,k , Aj = (?ij )i :
cj ] = argmin
[?~bj , A
~j )k2
[?~j ,Aj ] kxj ? Aj ? S(?
(14)
This minimization is accomplished following [21], assuming uncorrelatedness for the
sources and independence of the time delays.
(c) Update ?ij and go back to (a), until convergence is achieved.
3
Test data sets
For comparing the novel algorithm with other related methods we used two different types of data
sets. A first set of data was generated artificially by mixing different sound sources, varying the
mixing weights and the delays. This data was non-periodic and enabled us to validate the accuracy
of the reconstruction of the source signals. The second data set were human movement trajectories
of emotional gaits that were recorded using motion capture. The gait trajectories were periodic
and served for testing the suitability of the new method for extracting biologically interpretable
movement components.
Our first data set consisted of synthetically generated delayed mixtures generated from segments
of speech and sound signals taken from an ICA benchmark data set described in [6]. The data
basis contained in total 14 signals, with a length of 8000 time points each. In order to obtain
statistically representative results, data sets were recomputed 20 times with random selection of the
source signals, and/or of the mixing and delay matrices. Three types of mixtures were generated:
(I) Mixtures of 2 source segments with random mixing and delay matrices (2 ? 2), each resulting in two simulated signals x1,2 . This data set was used to compare the new method
with PCA and (fast) ICA [12]. Data set (I) was included to show that the new algorithm is
also able to address the even-determined case (n = m).
(II) Mixtures of 2 randomly selected segments from the speech data basis using the constant
mixing matrix A = [1, 2; 3, 1; 10, 5; 1, 2; 1, 1] and the constant delay matrix T = (?ij )ij =
[0, 4000; 2500, 5000; 100, 200; 1, 1; 500, 333]. This data set was used to compare the new
method with PCA and ICA, and the SOBIDS algorithm [1], which requires at least twice as
many signals as sources. Data set (II) with fixed mixing and delay matrices was included
since completely random generation sometimes produced degenerated anechoic mixtures
(instantaneous mixtures or ill-conditioned mixing matrices).
(III) A third data set was generated by mixing two randomly selected source segments with
random mixture matrices and random delay matrices.
To compare the performance of the different algorithms we used a performance measure M that was
defined by the maximum of the cross-correlations between extracted sources, sextract,j , and original
sources sorig,j (after appropriate matching of the individual sources, since the recovered sources are
not ordered in a specific manner):
M = (1/n)
n
X
j=1
max |E{sextract,j (t) ? sorig,j (t + ? )}|
?
The second test data set consisted of movement trajectories of human actors walking neutrally, or
with different emotional styles (happy, angry, sad and fearful). The movements were recorded using
a VICON 612 motion capture system with 7 cameras, obtaining the 3D positions of 41 passive
markers on the bodies of the actors. We recorded trajectories from 13 lay actors, repeating each
walking style three times per actor. A hierarchical kinematic body model (skeleton) with 17 joints
was fitted to the marker positions, and joint angles were computed. Rotations between adjacent
body segments were described as Euler angles, defining flexion, abduction and rotation about the
connecting joints. The data for the unsupervised learning procedure included only the flexion angles
of the hip, knee, elbow, shoulder and the clavicle, since the other angles had relatively high noise
levels. From each trajectory only one gait cycle was extracted, which was time normalized. This
resulted in a data set with 1950 samples with a length of 100 time points each.
3.1
Results
Delayed mixtures of sound sources: Figure 1 shows the results for the extraction of the sound
sources from the data sets (I)-(III). The bar plots show means and standard deviations of the performance measure M over twenty simulations. On all data sets our new method shows an overall
performance measure above 80%, while PCA and ICA show performances between 50 and 60%.
The SOBIDS algorithm reaches a performance level of 72%.
For a more accurate statistical comparison of the different methods we used a one-way repeated
measure ANOVA for the measure M . There was a significant effect for all three data sets. Post-hoc
comparison using the Least Significant Difference (LSD) [18] revealed that our new method was
significantly better than PCA and ICA for all three data sets. Our method significantly outperformed
the SOBIDS algorithm for data set (III). For data set (II) the difference between these two algorithms
was not significant, due to the increased overall variability in this data set.
The better performance of our algorithm compared to PCA and ICA results from the appropriate
modeling of time delays. The better performance compared to the SOBIDS algorithm might be
explained by the fact that this algorithm requires the assumption of small delays.
Figure 1: Comparison of different blind source separation algorithms for synthetic mixtures of sound
signals with delays (data sets I-III, see text). The stars indicate significant (p < 0.001) differences
compared to the new algorithm.
Human gait trajectories: With the second data set we tested whether the proposed novel algorithm is suitable for the extraction of interpretable source signals from human movement data.
By performing normal ICA separately on individual joint trajectories and comparing the extracted
sources, we had observed before that such sources are often very similar except for time shifts between different joints. This motivates the hypothesis that (1) might provide an appropriate generative
model for such gait trajectories.
This hypothesis is confirmed by the data presented in Figure 2 that shows the approximation quality (explained variance) for different numbers of extracted sources and comparing four different
algorithms: PCA, (fast) ICA [12], Bayesian positive ICA [11], SOBIDS, and the new algorithm.
The new method outperforms all other methods, and in particular the methods without time delays.
Specifically, the new algorithm is capable of approximating 97% of the trajectory data with only 3
sources, while PCA and ICA require more than 6 sources to achieve the same level of accuracy.
Figure 2: Comparison of different blind source separation algorithms. Explained variance is shown
for different numbers of extracted sources.
In order to test whether the novel algorithm results in source signals that provide useful interpretations of biological data we modeled all trajectories in our gait data sets by linear superpositions of
the extracted sources and analyzed the resulting mixture matrices A. To extract weight components
that are specific for individual emotional styles we modeled the mixture matrices applying sparse
linear regression. The (vectorized) weights of the individual gait trajectories for emotion j, defining
the vector aj , were approximated by the sum of a component a0 (containing the weights that characterize neutral walking) and an emotion-specific contribution. Formally, this multi-linear regression
model can be written as
aj ? a0 + C ? ej ,
(15)
where C is a weight matrix that determines the emotion-specific contributions to the mixing weights.
Its columns are given by the differences between the weights for the different emotional styles
(happy, sad, fearful and angry) and the weights for neutral walking. In order to obtain easily interpretable results, the matrix C was sparsified by L1 norm minimization. The solution of the linear
regression problem was obtained by minimizing the following cost function (with ? > 0) using
quadratic programming:
E(C) =
X
j
kaj ? a0 ? C ? ej k2 + ?
X
|Cij |
(16)
i,j
This regression basically computes the mean differences between the weights for neutral walking
and for emotional walking. The sparsification separates automatically important and less important
features. The concentration of the variance into a few important predictors simplifies the interpretation. Figure 3 shows a gray level plot of the matrix C, illustrating the weight differences compared
to neutral walking for the four different emotional styles and the different joint angles. Positive
elements of the matrix indicate cases where the joint amplitudes for the emotional gait are increased
compared to normal walking. Negative elements are indicated by white triangles in the lower left
corner of the individual cells of the plot. They correspond to cases where the joint angle amplitudes for the emotional walk are reduced compared to normal walking. The + and ? signs in the
figure summarize data from psychophysical experiments that have investigated kinematic features
that were important for the perception of emotional gaits [17; 23]. Plus signs indicate cases where
(perceived) increases of the joint amplitudes compared to normal walking were correlated with the
perception of the corresponding emotion, and minus signs to cases where a (perceived) reduction of
the joint angle amplitudes was correlated with the perception of the corresponding emotion.
Comparison between these psychophysical results and the elements of the matrix C (Figure 3a)
shows a very close match between the weight changes and the features that are important for the
perception of emotions from gaits. This implies that the novel algorithm extracts features that can
be meaningfully interpreted in a biological sense. Figure 3b shows the results of the same analysis
for sources that had been extracted with PCA, matching the numbers of non-zero elements of the
estimated matrix C. For gait trajectories the source signals by PCA and ICA are virtually identical.
Therefore, results in Figure 3 would be unchanged for ICA. In either case, the match is significantly
worse than for the sources extracted with the novel algorithm (panel a). In addition, the signs of
the matrix elements often do not match the signs of the amplitude changes in the psychophysical
experiments. This implies that the new algorithm extracts spatio-temporal components from human
gait data that are more easily interpretable than components extracted with PCA.
Figure 3: Elements of the weight matrix C, encoding emotion-specific deviations from neutral walking, for
different degrees of freedom. Negative elements are indicated by white triangles in the lower left corners of
the cells. Kinematic features that have been shown to be important for the perception of emotions from gait in
psychophysical experiments are indicated by the plus and minus signs. (Details see text.)
3.2
Conclusion
We present a new algorithm for the solution of over-determined blind source separation problems for
mixtures of sources with delays. The proposed method has been derived by application of a timefrequency transformation to the mixture model, resulting in a two-step algorithm that combines
positive ICA with another iterative optimization step. We demonstrate that the developed algorithm
outperforms other source separation algorithms with and without time delays on synthetic data sets
defined by delayed mixtures of speech signals, and also on real data sets obtained by motion capture
of human full-body movements. For human movements we also demonstrate that, at least for the
case of human gait, the new algorithm provides a more compact and interpretable representations
than the alternative methods we tested. To our knowledge the proposed algorithm is the first one
that solves over-determined delayed mixing problems without specific additional assumptions about
the structure of the delay matrix, e.g. limited sizes of the delays. In contrast to nonnegative matrix
factorization with delays [2], the proposed method is applicable to non-positive signals and sources.
Future work will focus on testing the algorithm with a broader range of data sets, also including
particularly non-periodic human movements. In addition, it seems possible to extend the proposed
method for multi-dimensional translation vectors (delays), making it applicable for the learning of
translation-invariant features in two-dimensional images.
Acknowledgments
This work was supported by HFSP, DFG, the Volkswagenstiftung and the EU FP6 Project ?COBOL?.
We thank C.L. Roether for help with the trajectory acquisition and the psychological interpretation
of the data, and W. Ilg for support with the motion capturing.
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2,234 | 3,027 | A Novel Gaussian Sum Smoother for Approximate
Inference in Switching Linear Dynamical Systems
David Barber and Bertrand Mesot
IDIAP Research Institute
Martigny 1920, Switzerland
david.barber/[email protected]
Abstract
We introduce a method for approximate smoothed inference in a class of switching
linear dynamical systems, based on a novel form of Gaussian Sum smoother. This
class includes the switching Kalman Filter and the more general case of switch
transitions dependent on the continuous latent state. The method improves on the
standard Kim smoothing approach by dispensing with one of the key approximations, thus making fuller use of the available future information. Whilst the only
central assumption required is projection to a mixture of Gaussians, we show that
an additional conditional independence assumption results in a simpler but stable
and accurate alternative. Unlike the alternative unstable Expectation Propagation
procedure, our method consists only of a single forward and backward pass and is
reminiscent of the standard smoothing ?correction? recursions in the simpler linear
dynamical system. The algorithm performs well on both toy experiments and in a
large scale application to noise robust speech recognition.
1 Switching Linear Dynamical System
The Linear Dynamical System (LDS) [1] is a key temporal model in which a latent linear process
generates the observed series. For complex time-series which are not well described globally by a
single LDS, we may break the time-series into segments, each modeled by a potentially different
LDS. This is the basis for the Switching LDS (SLDS) [2, 3, 4, 5] where, for each time t, a switch
variable st ? 1, . . . , S describes which of the LDSs is to be used. The observation (or ?visible?)
vt ? RV is linearly related to the hidden state ht ? RH with additive noise ? by
vt = B(st )ht + ? v (st )
?
p(vt |ht , st ) = N (B(st )ht , ?v (st ))
(1)
where N (?, ?) denotes a Gaussian distribution with mean ? and covariance ?. The transition
dynamics of the continuous hidden state ht is linear,
ht = A(st )ht?1 + ? h (st ),
?
p(ht |ht?1 , st ) = N A(st )ht?1 , ?h (st )
(2)
The switch st may depend on both the previous st?1 and ht?1 . This is an augmented SLDS
(aSLDS), and defines the model
p(v1:T , h1:T , s1:T ) =
T
Y
p(vt |ht , st )p(ht |ht?1 , st )p(st |ht?1 , st?1 )
t=1
The standard SLDS[4] considers only switch transitions p(st |st?1 ). At time t = 1, p(s1 |h0 , s0 )
simply denotes the prior p(s1 ), and p(h1 |h0 , s1 ) denotes p(h1 |s1 ).
The aim of this article is to address how to perform inference in the aSLDS. In particular we desire
the filtered estimate p(ht , st |v1:t ) and the smoothed estimate p(ht , st |v1:T ), for any 1 ? t ? T .
Both filtered and smoothed inference in the SLDS is intractable, scaling exponentially with time [4].
s1
s2
s3
s4
h1
h2
h3
h4
v1
v2
v3
v4
Figure 1: The independence structure of the aSLDS. Square nodes denote discrete variables, round
nodes continuous variables. In the SLDS links from h to s are not normally considered.
2 Expectation Correction
Our approach to approximate p(ht , st |v1:T ) mirrors the Rauch-Tung-Striebel ?correction? smoother
for the simpler LDS [1].The method consists of a single forward pass to recursively find the filtered
posterior p(ht , st |v1:t ), followed by a single backward pass to correct this into a smoothed posterior
p(ht , st |v1:T ). The forward pass we use is equivalent to standard Assumed Density Filtering (ADF)
[6]. The main contribution of this paper is a novel form of backward pass, based only on collapsing
the smoothed posterior to a mixture of Gaussians. Together with the ADF forward pass, we call the
method Expectation Correction, since it corrects the moments found from the forward pass. A more
detailed description of the method, including pseudocode, is given in [7].
2.1 Forward Pass (Filtering)
Readers familiar with ADF may wish to continue directly to Section (2.2). Our aim is to form a
recursion for p(st , ht |v1:t ), based on a Gaussian mixture approximation of p(ht |st , v1:t ). Without
loss of generality, we may decompose the filtered posterior as
p(ht , st |v1:t ) = p(ht |st , v1:t )p(st |v1:t )
(3)
The exact representation of p(ht |st , v1:t ) is a mixture with O(S t ) components. We therefore approximate this with a smaller I-component mixture
p(ht |st , v1:t ) ?
I
X
p(ht |it , st , v1:t )p(it |st , v1:t )
it =1
where p(ht |it , st , v1:t ) is a Gaussian parameterized with mean f (it , st ) and covariance F (it , st ). To
find a recursion for these parameters, consider
X
p(ht+1 |st+1 , v1:t+1 ) =
p(ht+1 |st , it , st+1 , v1:t+1 )p(st , it |st+1 , v1:t+1 )
(4)
st ,it
Evaluating p(ht+1 |st , it , st+1 , v1:t+1 )
We find p(ht+1 |st , it , st+1 , v1:t+1 ) by first computing the joint distribution
p(ht+1 , vt+1 |st , it , st+1 , v1:t ), which is a Gaussian with covariance and mean elements,
?hh = A(st+1 )F (it , st )AT (st+1 ) + ?h (st+1 ), ?vv = B(st+1 )?hh B T (st+1 ) + ?v (st+1 )
?vh = B(st+1 )F (it , st ),
?v = B(st+1 )A(st+1 )f (it , st ),
?h = A(st+1 )f (it , st )
(5)
and then conditioning on vt+1 1 . For the case S = 1, this forms the usual Kalman Filter recursions[1].
Evaluating p(st , it |st+1 , v1:t+1 )
The mixture weight in (4) can be found from the decomposition
p(st , it |st+1 , v1:t+1 ) ? p(vt+1 |it , st , st+1 , v1:t )p(st+1 |it , st , v1:t )p(it |st , v1:t )p(st |v1:t ) (6)
1
?1
p(x|y) is a Gaussian with mean ?x + ?xy ??1
yy (y ? ?y ) and covariance ?xx ? ?xy ?yy ?yx .
The first factor in (6), p(vt+1 |it , st , st+1 , v1:t ) is a Gaussian with mean ?v and covariance ?vv , as
given in (5). The last two factors p(it |st , v1:t ) and p(st |v1:t ) are given from the previous iteration.
Finally, p(st+1 |it , st , v1:t ) is found from
p(st+1 |it , st , v1:t ) = hp(st+1 |ht , st )ip(ht |it ,st ,v1:t )
(7)
where h?ip denotes expectation with respect to p. In the SLDS, (7) is replaced by the Markov
transition p(st+1 |st ). In the aSLDS, however, (7) will generally need to be computed numerically.
Closing the recursion
We are now in a position to calculate (4). For each setting of the variable st+1 , we have a mixture
of I ? S Gaussians which we numerically collapse back to I Gaussians to form
p(ht+1 |st+1 , v1:t+1 ) ?
I
X
p(ht+1 |it+1 , st+1 , v1:t+1 )p(it+1 |st+1 , v1:t+1 )
it+1 =1
Any method of choice may be supplied to collapse a mixture to a smaller mixture; our code
simply repeatedly merges low-weight components. In this way the new mixture coefficients
p(it+1 |st+1 , v1:t+1 ), it+1 ? 1, . . . , I are defined, completing the description of how to form a
recursion for p(ht+1 |st+1 , v1:t+1 ) in (3). A recursion for the switch variable is given by
X
p(st+1 |v1:t+1 ) ?
p(vt+1 |st+1 , it , st , v1:t )p(st+1 |it , st , v1:t )p(it |st , v1:t )p(st |v1:t )
st ,it
where all terms have been computed during the recursion for p(ht+1 |st+1 , v1:t+1 ).
The likelihood p(v1:T ) may be found by recursing p(v1:t+1 ) = p(vt+1 |v1:t )p(v1:t ), where
X
p(vt+1 |vt ) =
p(vt+1 |it , st , st+1 , v1:t )p(st+1 |it , st , v1:t )p(it |st , v1:t )p(st |v1:t )
it ,st ,st+1
2.2 Backward Pass (Smoothing)
The main contribution of this paper is to find a suitable way to ?correct? the filtered posterior
p(st , ht |v1:t ) obtained from the forward pass into a smoothed posterior p(st , ht |v1:T ). We derive
this for the case of a single Gaussian representation. The extension to the mixture case is straightforward and presented in [7]. We approximate the smoothed posterior p(ht |st , v1:T ) by a Gaussian
with mean g(st ) and covariance G(st ) and our aim is to find a recursion for these parameters. A
useful starting point for a recursion is:
X
p(st+1 |v1:T )p(ht |st , st+1 , v1:T )p(st |st+1 , v1:T )
p(ht , st |v1:T ) =
st+1
The term p(ht |st , st+1 , v1:T ) may be computed as
Z
p(ht |st , st+1 , v1:T ) =
p(ht |ht+1 , st , st+1 , v1:t )p(ht+1 |st , st+1 , v1:T )
(8)
ht+1
The recursion therefore requires p(ht+1 |st , st+1 , v1:T ), which we can write as
p(ht+1 |st , st+1 , v1:T ) ? p(ht+1 |st+1 , v1:T )p(st |st+1 , ht+1 , v1:t )
(9)
The difficulty here is that the functional form of p(st |st+1 , ht+1 , v1:t ) is not squared exponential
in ht+1 , so that p(ht+1 |st , st+1 , v1:T ) will not be Gaussian2 . One possibility would be to approximate the non-Gaussian p(ht+1 |st , st+1 , v1:T ) by a Gaussian (or mixture thereof) by minimizing
the Kullback-Leilbler divergence between the two, or performing moment matching in the case of
a single Gaussian. A simpler alternative (which forms ?standard? EC) is to make the assumption
p(ht+1 |st , st+1 , v1:T ) ? p(ht+1 |st+1 , v1:T ), where p(ht+1 |st+1 , v1:T ) is already known from the
previous backward recursion. Under this assumption, the recursion becomes
X
p(ht , st |v1:T ) ?
p(st+1 |v1:T )p(st |st+1 , v1:T ) hp(ht |ht+1 , st , st+1 , v1:t )ip(ht+1 |st+1 ,v1:T ) (10)
st+1
2
In the exact calculation, p(ht+1 |st , st+1 , v1:T ) is a mixture of Gaussians, see [7]. However, since in (9) the
two terms p(ht+1 |st+1 , v1:T ) will only be approximately computed during the recursion, our approximation to
p(ht+1 |st , st+1 , v1:T ) will not be a mixture of Gaussians.
Evaluating hp(ht |ht+1 , st , st+1 , v1:t )ip(ht+1 |st+1 ,v1:T )
hp(ht |ht+1 , st , st+1 , v1:t )ip(ht+1 |st+1 ,v1:T ) is a Gaussian in ht , whose statistics we will now compute. First we find p(ht |ht+1 , st , st+1 , v1:t ) which may be obtained from the joint distribution
p(ht , ht+1 |st , st+1 , v1:t ) = p(ht+1 |ht , st+1 )p(ht |st , v1:t )
(11)
which itself can be found from a forward dynamics from the filtered estimate p(ht |st , v1:t ). The
statistics for the marginal p(ht |st , st+1 , v1:t ) are simply those of p(ht |st , v1:t ), since st+1 carries no
extra information about ht . The remaining statistics are the mean of ht+1 , the covariance of ht+1
and cross-variance between ht and ht+1 , which are given by
hht+1 i = A(st+1 )ft (st ), ?t+1,t+1 = A(st+1 )Ft (st )AT (st+1 )+?h (st+1 ), ?t+1,t = A(st+1 )Ft (st )
Given the statistics of (11), we may now condition on ht+1 to find p(ht |ht+1 , st , st+1 , v1:t ). Doing
so effectively constitutes a reversal of the dynamics,
?
?
?
ht = A (st , st+1 )ht+1 + ?
? (st , st+1 )
?
?
?
?
? , s ), ?
where A (st , st+1 ) and ?
? (st , st+1 ) ? N (?
m(s
t t+1 ? (st , st+1 )) are easily found using
conditioning. Averaging the above reversed dynamics over p(ht+1 |st+1 , v1:T ), we find that
hp(ht |ht+1 , st , st+1 , v1:t )ip(ht+1 |st+1 ,v1:T ) is a Gaussian with statistics
?
?
?
?
?
?
?
? , s ), ? = ?
?t = A (st , st+1 )g(st+1 )+?
m(s
A (st , st+1 )G(st+1 ) A T (st , st+1 )+ ? (st , st+1 )
t t+1
t,t
These equations directly mirror the standard RTS backward pass[1].
Evaluating p(st |st+1 , v1:T )
The main departure of EC from previous methods is in treating the term
p(st |st+1 , v1:T ) = hp(st |ht+1 , st+1 , v1:t )ip(ht+1 |st+1 ,v1:T )
(12)
The term p(st |ht+1 , st+1 , v1:t ) is given by
p(ht+1 |st+1 , st , v1:t )p(st , st+1 |v1:t )
p(st |ht+1 , st+1 , v1:t ) = P
?
?
s? p(ht+1 |st+1 , st , v1:t )p(st , st+1 |v1:t )
(13)
t
Here p(st , st+1 |v1:t ) = p(st+1 |st , v1:t )p(st |v1:t ), where p(st+1 |st , v1:t ) occurs in the forward pass,
(7). In (13), p(ht+1 |st+1 , st , v1:t ) is found by marginalizing (11).
Computing the average of (13) with respect to p(ht+1 |st+1 , v1:T ) may be achieved by any numerical integration method desired. A simple approximation is to evaluate the integrand at the mean
value of the averaging distribution p(ht+1 |st+1 , v1:T ). More sophisticated methods (see [7]) such
as sampling from the Gaussian p(ht+1 |st+1 , v1:T ) have the advantage that covariance information
is used3 .
Closing the Recursion
We have now computed both the continuous and discrete factors in (8), which we wish to use to
write the smoothed estimate in the form p(ht , st |v1:T ) = p(st |v1:T )p(ht |st , v1:T ). The distribution
p(ht |st , v1:T ) is readily obtained from the joint (8) by conditioning on st to form the mixture
X
p(ht |st , v1:T ) =
p(st+1 |st , v1:T )p(ht |st , st+1 , v1:T )
st+1
which may then be collapsed to a single Gaussian (the mixture case is discussed in [7]). The
smoothed posterior p(st |v1:T ) is given by
X
p(st |v1:T ) =
p(st+1 |v1:T ) hp(st |ht+1 , st+1 , v1:t )ip(ht+1 |st+1 ,v1:T ) .
(14)
st+1
3
This is a form of exact sampling since drawing samples from a Gaussian is easy. This should not be
confused with meaning that this use of sampling renders EC a sequential Monte-Carlo scheme.
2.3 Relation to other methods
The EC Backward pass is closely related to Kim?s method [8]. In both EC and Kim?s method, the
approximation p(ht+1 |st , st+1 , v1:T ) ? p(ht+1 |st+1 , v1:T ), is used to form a numerically simple
backward pass. The other ?approximation? in EC is to numerically compute the average in (14).
In Kim?s method, however, an update for the discrete variables is formed by replacing the required
term in (14) by
hp(st |ht+1 , st+1 , v1:t )ip(ht+1 |st+1 ,v1:T ) ? p(st |st+1 , v1:t )
(15)
Since p(st |st+1 , v1:t ) ? p(st+1 |st )p(st |v1:t )/p(st+1 |v1:t ), this can be computed simply from the
filtered results alone. The fundamental difference therefore between EC and Kim?s method is that
the approximation, (15), is not required by EC. The EC backward pass therefore makes fuller use of
the future information, resulting in a recursion which intimately couples the continuous and discrete
variables. The resulting effect on the quality of the approximation can be profound, as we will see
in the experiments.
The Expectation Propagation (EP) algorithm makes the central assumption of collapsing the posteriors to a Gaussian family [5]; the collapse is defined by a consistency criterion on overlapping
marginals. In our experiments, we take the approach in [9] of collapsing to a single Gaussian. Ensuring consistency requires frequent translations between moment and canonical parameterizations,
which is the origin of potentially severe numerical instability [10]. In contrast, EC works largely
with moment parameterizations of Gaussians, for which relatively few numerical difficulties arise.
Unlike EP, EC is not based on a consistency criterion and a subtle issue arises about possible inconsistencies in the Forward and Backward approximations for EC. For example, under the conditional
independence assumption in the Backward Pass, p(hT |sT ?1 , sT , v1:T ) ? p(hT |sT , v1:T ), which
is in contradiction to (5) which states that the approximation to p(hT |sT ?1 , sT , v1:T ) will depend
on sT ?1 . Such potential inconsistencies arise because of the approximations made, and should not
be considered as separate approximations in themselves. Rather than using a global (consistency)
objective, EC attempts to faithfully approximate the exact Forward and Backward propagation routines. For this reason, as in the exact computation, only a single Forward and Backward pass are
required in EC.
In [11] a related dynamics reversed is proposed. However, the singularities resulting from incorrectly
treating p(vt+1:T |ht , st ) as a density are heuristically finessed.
In [12] a variational method approximates the joint distribution p(h1:T , s1:T |v1:T ) rather than the
marginal inference p(ht , st |v1:T ). This is a disadvantage when compared to other methods that
directly approximate the marginal.
Sequential Monte Carlo methods (Particle Filters)[13], are essentially mixture of delta-function approximations. Whilst potentially powerful, these typically suffer in high-dimensional hidden spaces,
unless techniques such as Rao-Blackwellization are performed. ADF is generally preferential to
Particle Filtering since in ADF the approximation is a mixture of non-trivial distributions, and is
therefore more able to represent the posterior.
3 Demonstration
Testing EC in a problem with a reasonably long temporal sequence, T , is important since numerical
instabilities may not be apparent in timeseries of just a few points. To do this, we sequentially
generate hidden and visible states from a given model, here with H = 3, S = 2, V = 1 ? see
Figure(2) for full details of the experimental setup. Then, given only the parameters of the model and
the visible observations (but not any of the hidden states h1:T , s1:T ), the task is to infer p(ht |st , v1:T )
and p(st |v1:T ). Since the exact computation is exponential in T , a simple alternative is to assume
that the original sample states s1:T are the ?correct? inferences, and compare how our most probable
posterior smoothed estimates arg maxst p(st |v1:T ) compare with the assumed correct sample st . We
chose conditions that, from the viewpoint of classical signal processing, are difficult, with changes
in the switches occurring at a much higher rate than the typical frequencies in the signal vt .
For EC we use the mean approximation for the numerical integration of (12). We included the Particle Filter merely for a point of comparison with ADF, since they are not designed to approximate
PF
RBPF
EP
ADFS
KimS
ECS
ADFM
KimM
ECM
1000
800
600
400
200
0
0
10
20
0
10
20
0
10
20
0
10
20
0
10
20
0
10
20
0
10
20
0
10
20
0
10
20
Figure 2: The number of errors in estimating p(st |v1:T ) for a binary switch (S = 2) over a time
series of length T = 100. Hence 50 errors corresponds to random guessing. Plotted are histograms
of the errors are over 1000 experiments. The x-axes are cut off at 20 errors to improve visualization
of the results. (PF) Particle Filter. (RBPF) Rao-Blackwellized PF. (EP) Expectation Propagation.
(ADFS) Assumed Density Filtering using a Single Gaussian. (KimS) Kim?s smoother using the
results from ADFS. (ECS) Expectation Correction using a Single Gaussian (I = J = 1). (ADFM)
ADF using a multiple of I = 4 Gaussians. (KimM) Kim?s smoother using the results from ADFM.
(ECM) Expectation Correction using a mixture with I = J = 4 components. S = 2, V = 1
(scalar observations), T = 100, with zero output bias. A(s) = 0.9999 ? orth(randn(H, H)),
B(s) = randn(V, H). H = 3, ?h (s) = IH , ?v (s) = 0.1IV , p(st+1 |st ) ? 1S?S + IS . At time
t = 1, the priors are p1 = uniform, with h1 drawn from N (10 ? randn(H, 1), IH ).
the smoothed estimate, for which 1000 particles were used, with Kitagawa resampling. For the RaoBlackwellized Particle Filter [13], 500 particles were used, with Kitagawa resampling. We found
that EP4 was numerically unstable and often struggled to converge. To encourage convergence, we
used the damping method in [9], performing 20 iterations with a damping factor of 0.5. Nevertheless, the disappointing performance of EP is most likely due to conflicts resulting from numerical
instabilities introduced by the frequent conversions between moment and canonical representations.
The best filtered results are given using ADF, since this is better able to represent the variance
in the filtered posterior than the sampling methods. Unlike Kim?s method, EC makes good use
of the future information to clean up the filtered results considerably. One should bear in mind
that both EC and Kim?s method use the same ADF filtered results. This demonstrates that EC
may dramatically improve on Kim?s method, so that the small amount of extra work in making a
numerical approximation of p(st |st+1 , v1:T ), (12), may bring significant benefits. We found similar
conclusions for experiments with an aSLDS[7].
4 Application to Noise Robust ASR
Here we briefly present an application of the SLDS to robust Automatic Speech Recognition (ASR),
for which the intractable inference is performed by EC, and serves to demonstrate how EC scales
well to a large-scale application. Fuller details are given in [14]. The standard approach to noise
robust ASR is to provide a set of noise-robust features to a standard Hidden Markov Model (HMM)
classifier, which is based on modeling the acoustic feature vector. For example, the method of Unsupervised Spectral Subtraction (USS) [15] provides state-of-the-art performance in this respect.
Incorporating noise models directly into such feature-based HMM systems is difficult, mainly because the explicit influence of the noise on the features is poorly understood. An alternative is to
model the raw speech signal directly, such as the SAR-HMM model [16] for which, under clean conditions, isolated spoken digit recognition performs well. However, the SAR-HMM performs poorly
under noisy conditions, since no explicit noise processes are taken into account by the model.
The approach we take here is to extend the SAR-HMM to include an explicit noise process, so that
the observed signal vt is modeled as a noise corrupted version of a clean hidden signal vth :
vt = vth + ??t
4
with
??t ? N (0, ?
?2)
Generalized EP [5], which groups variables together improves on the results, but is still far inferior to the
EC results presented here ? Onno Zoeter personal communication.
Noise Variance
0
10?7
10?6
10?5
10?4
10?3
SNR (dB)
26.5
26.3
25.1
19.7
10.6
0.7
HMM
100.0%
100.0%
90.9%
86.4%
59.1%
9.1%
SAR-HMM
97.0%
79.8%
56.7%
22.2%
9.7%
9.1%
AR-SLDS
96.8%
96.8%
96.4%
94.8%
84.0%
61.2%
Table 1: Comparison of the recognition accuracy of three models when the test utterances are corrupted by various levels of Gaussian noise.
The dynamics of the clean signal is modeled by a switching AR process
vth =
R
X
h
cr (st )vt?r
+ ?th (st ),
?th (st ) ? N (0, ? 2 (st ))
r=1
where st ? {1, . . . , S} denotes which of a set of AR coefficients cr (st ) are to be used at time t,
and ?th (st ) is the so-called innovation noise. When ? 2 (st ) ? 0, this model reproduces the SARHMM of [16], a specially constrained HMM. Hence inference and learning for the SAR-HMM are
tractable and straightforward. For the case ? 2 (st ) > 0 the model can be recast as an SLDS. To do
this we define ht as a vector which contains the R most recent clean hidden samples
T
h
ht = vth . . . vt?r+1
(16)
and we set A(st ) to be an R ? R matrix where the first row contains the AR coefficients ?cr (st )
and the rest is a shifted down identity matrix. For example, for a third order (R = 3) AR process,
"
#
?c1 (st ) ?c2 (st ) ?c3 (st )
1
0
0
A(st ) =
.
(17)
0
1
0
The hidden covariance matrix ?h (s) has all elements zero, except the top-left most which is set
to the innovation variance. To extract the first component of ht we use the (switch independent)
1 ? R projection matrix B = [ 1 0 . . . 0 ]. The (switch independent) visible scalar noise
variance is given by ?v ? ?v2 .
A well-known issue with raw speech signal models is that the energy of a signal may vary from one
speaker to another or because of a change in recording conditions. For this reason the innovation ?h
is adjusted by maximizing the likelihood of an observed sequence with respect to the innovation
covariance, a process called Gain Adaptation [16].
4.1 Training & Evaluation
Following [16], we trained a separate SAR-HMM for each of the eleven digits (0?9 and ?oh?) from
the TI-DIGITS database [17]. The training set for each digit was composed of 110 single digit
utterances down-sampled to 8 kHz, each one pronounced by a male speaker. Each SAR-HMM was
composed of ten states with a left-right transition matrix. Each state was associated with a 10thorder AR process and the model was constrained to stay an integer multiple of K = 140 time steps
(0.0175 seconds) in the same state. We refer the reader to [16] for a detailed explanation of the
training procedure used with the SAR-HMM.
An AR-SLDS was built for each of the eleven digits by copying the parameters of the corresponding trained SAR-HMM, i.e., the AR coefficients cr (s) are copied into the first row of the hidden
transition matrix A(s) and the same discrete transition distribution p(st | st?1 ) is used. The models
were then evaluated on a test set composed of 112 corrupted utterances of each of the eleven digits,
each pronounced by different male speakers than those used in the training set. The recognition
accuracy obtained by the models on the corrupted test sets is presented in Table 1. As expected, the
performance of the SAR-HMM rapidly decreases with noise. The feature-based HMM with USS
has high accuracy only for high SNR levels. In contrast, the AR-SLDS achieves a recognition accuracy of 61.2% at a SNR close to 0 dB, while the performance of the two other methods is equivalent
to random guessing (9.1%). Whilst other inference methods may also perform well in this case, we
found that EC performs admirably, without numerical instabilities, even for time-series with several
thousand time-steps.
5 Discussion
We presented a method for approximate smoothed inference in an augmented class of switching
linear dynamical systems. Our approximation is based on the idea that due to the forgetting which
commonly occurs in Markovian models, a finite number of mixture components may provide a
reasonable approximation. Clearly, in systems with very long correlation times our method may
require too many mixture components to produce a satisfactory result, although we are unaware of
other techniques that would be able to cope well in that case. The main benefit of EC over Kim
smoothing is that future information is more accurately dealt with. Whilst EC is not as general as
EP, EC carefully exploits the properties of singly-connected distributions, such as the aSLDS, to
provide a numerically stable procedure. We hope that the ideas presented here may therefore help
facilitate the practical application of dynamic hybrid networks.
Acknowledgements
This work is supported by the EU Project FP6-0027787. This paper only reflects the authors? views
and funding agencies are not liable for any use that may be made of the information contained herein.
References
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[15] G. Lathoud, M. Magimai-Doss, B. Mesot, and H. Bourlard. Unsupervised spectral subtraction for noiserobust ASR. In Proceedings of ASRU 2005, pages 189?194, November 2005.
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IEEE Signal Processing Letters, 12(2):166?169, February 2005.
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2,235 | 3,028 | A selective attention multi?chip system with dynamic
synapses and spiking neurons
Chiara Bartolozzi
Institute of neuroinformatics
UNI-ETH Zurich
Wintherthurerstr. 190, 8057, Switzerland
[email protected]
Giacomo Indiveri
Institute of neuroinformatics
UNI-ETH Zurich
Wintherthurerstr. 190, 8057, Switzerland
[email protected]
Abstract
Selective attention is the strategy used by biological sensory systems to solve
the problem of limited parallel processing capacity: salient subregions of the input stimuli are serially processed, while non?salient regions are suppressed. We
present an mixed mode analog/digital Very Large Scale Integration implementation of a building block for a multi?chip neuromorphic hardware model of selective attention. We describe the chip?s architecture and its behavior, when its is part
of a multi?chip system with a spiking retina as input, and show how it can be used
to implement in real-time flexible models of bottom-up attention.
1
Introduction
Biological systems interact with the outside world in real-time, reacting to complex stimuli in few
milliseconds. This is a highly demanding computational task, that requires either very high speed
sequential computation or fast massively parallel processing. Real systems however have to cope
with limited resources. Biological systems solve this issue by sequentially allocating computational
resources on small regions of the input stimuli, for analyzing them in parallel, with a strategy known
as Selective Attention, that takes advantage of both sequential and parallel processing.
A wise approach to the design of artificial systems that need to interact with the real world in real
time is to take inspiration from the strategies developed by biological systems.
The psychophysical study of selective attention distinguished two complementary strategies for the
selection of salient regions of the input stimuli, one depending on the physical (bottom-up) characteristic of the input, the other depending on its semantic (top-down) and task related properties.
Much of the applied research has focused on modeling the bottom-up aspect of selective attention.
As a consequence, several software [1, 2, 3] and hardware models [4, 5, 6, 7] based on the concept
of saliency map, winner-takes-all (WTA) competition, and inhibition of return (IOR) [8] have been
proposed.
We focus on HW implementation of such selective attention systems on compact, low-power, analogue VLSI chips. Previous implementations focused on either very abstract object/attention WTA
architectures with dedicated single-chip solutions [6], or on very detailed models of spike-based
competitive networks [9]; we propose a multi-chip solution that combines the advantages of spikebased solutions for communicating signals across chips, with a dedicated and compact WTA architecture for implementing competition among a large number of elements in parallel. Specifically
we present a new chip with 32 ? 32 cells, that can sequentially select the most active regions of the
input stimuli, the Selective Attention Chip (SAC). It is a transceiver chip employing a spike-based
representation (AER, Address-Event-Representation [10]). Its input signals topographically encode
the local conspicuousness of the input over the entire visual scene. Its output signals can be used in
real time to drive motors of active vision systems or to select subregions of images captured from
wide field-of-view cameras. The AER communication protocol and the 2D structure of the network
make it particularly suitable for processing signals from silicon spiking retinas.
The basic circuits of the chip we present have already been proposed in [11]. The chip we present
here comprises improvements in the basic circuits, and additional dynamic components that will
be described in Section 3. The chip?s improvements over previous implementations arise from the
design of new AER interfacing circuits, both for the input decoding stage and the output arbitration,
and new synaptic circuits: the Diff-Pair Integrator (DPI) described in [12]. The DPI is a log-domain
compact circuit that reproduces the time course of biological post-synaptic currents. Besides having
easily and independently tunable gain and time constant, it produces mean currents proportional
to the input frequencies, more suitable for the input of the current-mode WTA cell employed as
core computational unit in the SAC. This new circuit allows the analysis of the properties of the
chip, including the effect of the introduction of additional dynamic properties to the circuits, such as
Short-Term Depression (STD) [13, 14] in the input synapses and spike frequency adaptation in the
output Integrate and Fire (I&F ) neurons [15].
In the next sections we describe the chip?s architecture and present experimental results from a two
chip system comprising the SAC and a silicon ?transient? retina that produces spikes in response to
temporal changes in scene contrast.
2
The Selective Attention Chip
We fabricated a prototype of the SAC in standard AMS 0.35?m CMOS technology. The chip comprises an array of 32 ? 32 pixels, each one is 90 ? 45?m2 and the whole chip with AER digital
interface and pads occupies an area of 10mm2 . The basic functionality of the SAC is to scan the input in order of decreasing activity. The chip input and output signals are asynchronous digital pulses
(spikes) that use the Address Event Representation (AER) [16]. The input spikes to each pixel are
translated into a current (see Iex of Fig.1) by a circuit that models the dynamics of a biological excitatory synapse [12]. A current mode hysteretic Winner?Take?All (WTA) competitive cell compares
the input currents of each pixel; the winning cell sources a constant current to the correspondent output leaky Integrate and Fire (I&F) neuron [15]. The spiking neuron in the array then signals which
pixel is winning the competition for saliency, and therefore the pixel that receives the highest input
frequency. The output spikes of the I&F neuron are sent also to a feedback inhibitory synapse (see
Fig. 1), that subtracts current (Iior ) from the input node of the WTA cell; the net input current to
the winner pixel is then decreased, and a new pixel can eventually be selected. This self-inhibition
mechanism is known as Inhibition of Return (IOR) and allows the network to select sequentially the
most salient regions of input images, reproducing the attentional scan path.
To nearest
neighbors
AER
Input
A
E
R
Excitatory
Synapse
Iex
+
+
Inhibitory
Synapse
Hysteretic
WTA
A
E
R
Output
I&F
Neuron
AER
Output
Iior
IOR
To nearest
neighbors
Figure 1: Block diagram of a basic cell of the 32 ? 32 selective attention architecture.
This basic functionality of the SAC is augmented by the introduction of dynamic properties such
as Short-Term Depression (STD) in the input synapses and spike frequency adaptation in the output
neuron. STD is a property observed in physiological recordings[17] of synapses that decrease their
efficacy when they receive consecutive stimulations. In our synapse the effect of a single spike on
the integrated current depends on a voltage, the synaptic weight. The initial weight of the synapse is
set by an external voltage reference, then as the synapse receives spikes the effective synaptic weight
decreases. STD is a local gain control, that increases sensitivity to changes in the input and makes
the synapse insensitive to constant stimulation. Spiking frequency adaptation is another property of
neurons that when stimulated with constant input decrease their output firing rate with time. The
spiking frequency of the silicon I&F neuron is monotonic with its input current, the adaptation
neuron?s mechanism decreases the neuron?s firing rate with time [15]. We exploit this property to
decrease the output bandwidth of the SAC.
The SAC has been designed with tunable parameters that allow to modify the strength of synaptic
contributions, the dynamics of synaptic short term depression and of neuronal adaptation, as well as
the spatial extent of competition and the dynamics of IOR. All these parameters enrich the dynamics
of the network that can be exploited to model the complex selective attention scan path.
3
Multi?Chip Selective Attention System
The SAC uses the asynchronous AER SCX (Silicon Cortex) protocol, that allows multiple AER
chips to communicate using spikes, just like the cortex, and can be used in multi?chip systems, with
multiple senders and multiple receivers [18, 19]. Using this representation the SAC can exchange
data, while processing signals in parallel, in real time [20]. The communication protocol used
and the SAC?s bidimensional architecture make it particularly suitable for processing visual inputs
coming from artificial spiking retinas. We built a two chip system, connecting a silicon retina [21]
to the SAC input. The retina is an AER asynchronous imager that responds to contrast variations, it
has 64 ? 64 pixels that respond to on and off transients. A dedicated PCI-AER board [18] connects
the retina to the SAC, via a look-up table that maps the activity of the 64 ? 64 pixels of the retina to
the 32 ? 32 pixels of the SAC. In this setup the mapping is linear grouping 4 retina pixels to 1 SAC
pixel, more complex mappings, as for example the foveal mapping, will be tested in the future. The
board allows also to monitor the activity of both chips on a Linux desktop.
4
Experimental Data
We performed preliminary experiments with the two chips setup described in the previous section.
We stimulated the retina with two black squares flashing at 6Hz on a white background, on a LCD
screen, using the matlab PsychoToolbox [22] as shown in Fig. 2. In Fig. 3 we show the response of
the two chips to this stimulus: each dot represents the mean firing rate of the correspondent pixel in
the chips. The pixels of the retina that focus on the black squares are active and show a high mean
firing rate, some other pixels in the array have spontaneous activity. To show the mapping between
the retina and the SAC we performed a control experiment: we turned off the competition and the
IOR and also we disabled STD and the neuronal adaptation, in this way all the pixels that receive
an input activity will be active. All the pixels that receive the input from the pixels of the retina
that we stimulate with the black squares are active, more over the spontaneous activity (noise) of the
other pixels are ?cleaned?, thanks to the filtering property of the input synapses. In the next figures
we show the response of the system to the stimulus described above, while changing the settings
of the SAC. In all the figures the top and bottom boxes show raster plots, respectively of the retina
and the SAC: each dot corresponds to a spike emitted by a pixel (or neuron) (y axis) at a certain
time (x axis). The middle trace shows the voltage Vnet , that is proportional to the total input current
(Iex ? Iior of Fig. 1) to the WTA cell that receives input from one of the most active pixels of the
retina.
In Fig. 4(a) we show the same data of Fig. 3, the retina sends many spikes every time the black
squares appear and disappear from the screen, the WTA input node, with this settings, receives
only the excitatory current from the input synapse, as shown by the increase of the voltage Vnet in
correspondence of the retinal spikes. Since in our control experiment there is no competition, all
the stimulated pixels are active, as shown in the SAC raster plot. In Fig. 4(b) we show the effect of
10
5
20
10
Neuron Y
Neuron Y
Figure 2: Multi-chip system: The retina (top-right box) is stimulated with an LCD screen, its output
is sent to the SAC (bottom-right box) via the PCIAER board (bottom-left box). The activity of the
two chips is monitored via the PCIAER board on a Linux desktop.
30
40
15
20
25
50
30
60
10
20
30
40
Neuron X
(a)
50
60
5
10
15
20
Neuron X
25
30
(b)
Figure 3: Response of the two chips to an image. (a) The silicon retina is stimulated, via an LCD
screen, with two flashing (6Hz) black squares on a white background (see Fig. 2). We show the
mean firing output of each pixel of the retina. The pixels corresponding to the black squares in the
image have higher firing rate than the others, some of the pixels of the retina are spontaneously firing
at lower frequencies. (b) The activity of the retina is the input of the SAC: the 64 ? 64 pixels of the
retina are mapped with a ratio 4 : 1 to the 32 ? 32 pixels of the SAC. We show the mean firing rate
of the SAC pixels in response to the retinal stimulation, when the Winner-Takes-All competition is
disabled. In this case the SAC output reflects the input, with some suppression of the noisy pixels
due to the filtering properties of the input synaptic circuits.
introducing spike frequency adaptation: in this case the output frequency of each neuron decreases,
reducing the output bandwidth and the AER-bus traffic. In Fig. 5 we show the effect of competition
and Inhibition of Return. When we turn on the WTA competition only one pixel is selected at any
time, therefore only one neuron is firing, as shown in the raster plot of Fig. 5(a); on the node Vnet we
can observe that when the correspondent neuron is winning there is an extra input current, because
it doesn?t reset to its resting value when the synapse is not active. This positive current implements
a form of self-excitation that gives hysteretic properties to the network dynamics, and stabilizes the
WTA network. If we turn on the inhibitory synapse (Fig. 5(b)), as soon as the neuron starts to fire,
Neuron
4000
2000
Vnet (V)
0
0
0.2
0.4
0.6
0.8
1
1.2
0
0
1.4
2.2
0.2
0.4
0.6
0.8
1
1.2
0.4
0.6
0.8
1
1.2
1.4
0.2
0.4
0.6
0.8
1
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1.4
0.2
0.4
0.6
0.8
Time (sec)
1
1.2
1.4
2.2
1.4
1000
Neuron
1000
Neuron
0.2
2
2
500
0
0
2000
Vnet (V)
Neuron
4000
0.2
0.4
0.6
0.8
Time (sec)
(a)
1
1.2
1.4
500
0
0
(b)
Figure 4: Time response to black squares flashing on a white background: we use the same stimulation and setup described in Fig 3. The top figure shows the raster plot of the retina output, one dot
corresponds to a spike produced by one pixel at a specific time. The retina produces events every
time the squares appear on or disappear from the screen. The middle plot shows the voltage Vnet of
the input node of the WTA cell correspondent to the synapse that receives input from one of the most
active pixel of the retina. The bottom figure shows the raster plot of the SAC neurons. (a) We show
the ?control? experiment (same as in Fig 3): the competition, IOR, and all the other features of the
SAC are turned off, the output of the chip reproduces the input, with some suppression of the pixels
that receive very low activity from the retina, thanks to the input synapses filtering properties. In
the middle plot Vnet reflects the effect of the sole input current from the synapse, that integrates the
spikes received from the correspondent pixel of the retina. In this case, since the lateral inhibitory
connections are switched off, there is no competition and all the output I&F neurons correspondent
to the stimulated input synapses are active. (b) We add spike frequency adaptation to the previous
experiment settings, the output firing rate of the neurons is decreased, reducing the bandwidth of the
SAC output.
the inhibitory current decreases the total input current to the correspondent WTA cell: the voltage
Vnet reflects this mechanism as it is reset to its resting value even before the input from the retina
ceases. The WTA cell is then deselected and the output neuron stops firing, while another neuron is
selected and starts firing, as shown in the SAC raster plot. The inhibitory synapse time constant is
tunable and when it is slow the effect of inhibition lasts for hundreds of milliseconds after the I&F
stopped firing, in this way we prevent that pixel to be reselected immediately and we can have scan
path with many different pixels.
5
Conclusions
In this paper we presented a neuromorphic device implementing a Winner?Take?All network comprising dynamic synapses and adaptive neurons. This device is designed to be a part of a multi?chip
system for Selective Attention: via an AER communication system it can be interfaced to silicon
spiking retinas and to software implementations of associative memories.
We built a multi?chip system with the SAC and a silicon transient retina. The real time measurements allowed by the physical realization of the system are certainly a powerful method to explore
the network behavior by changing its parameters. Preliminary experiments confirmed the basic
functionality of the SAC and the robustness of the system; the analysis will be extended with the
systematic study of STD, IOR, adaptation and lateral excitatory coupling among the nearby cells.
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Acknowledgments
This work was inspired by interactions with the participants of the Neuromorphic Engineering Workshop
(http://www.ini.unizh.ch/telluride) and was supported by the EU grant ALAVLSI (IST?2001?38099) and
DAISY (FP6-2005-015803), and by the Italian CNR (Centro Nazionale delle Ricerche) fellowship 203.22.
The silicon retina was provided by Patrick Lichtsteiner. We further wish to thank Matthias Oster and Dylan
Muir for providing AER software tools.
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2,236 | 3,029 | Learning to Model Spatial Dependency:
Semi-Supervised Discriminative Random Fields
Chi-Hoon Lee
Department of Computing Science
University of Alberta
[email protected]
Feng Jiao
Department of Computing Science
University of Waterloo
[email protected]
Shaojun Wang ?
Department of Computer Science and Engineering
Wright State University
[email protected]
Dale Schuurmans, Russell Greiner
Department of Computing Science
University of Alberta
{dale, greiner}@cs.ualberta.ca
Abstract
We present a novel, semi-supervised approach to training discriminative random fields (DRFs) that efficiently exploits labeled and unlabeled training data to
achieve improved accuracy in a variety of image processing tasks. We formulate
DRF training as a form of MAP estimation that combines conditional loglikelihood on labeled data, given a data-dependent prior, with a conditional entropy
regularizer defined on unlabeled data. Although the training objective is no longer
concave, we develop an efficient local optimization procedure that produces classifiers that are more accurate than ones based on standard supervised DRF training. We then apply our semi-supervised approach to train DRFs to segment both
synthetic and real data sets, and demonstrate significant improvements over supervised DRFs in each case.
1 Introduction
Random field models are a popular probabilistic framework for representing complex dependencies
in natural image data. The two predominant types of random field models correspond to generative
versus discriminative graphical models respectively. Classical Markov random fields (MRFs) [2]
follow a traditional generative approach, where one models the joint probability of the observed
image along with the hidden label field over the pixels. Discriminative random fields (DRFs) [11,
10], on the other hand, directly model the conditional probability over the pixel label field given
an observed image. In this sense, a DRF is equivalent to a conditional random field [12] defined
over a 2-D lattice. Following the basic tenet of Vapnik [18], it is natural to anticipate that learning
an accurate joint model should be more challenging than learning an accurate conditional model.
Indeed, recent experimental evidence shows that DRFs tend to produce more accurate image labeling
models than MRFs, in many applications like gesture recognition [15] and object detection [11, 10,
19, 17].
Although DRFs tend to produce superior pixel labellings to MRFs, partly by relaxing the assumption
of conditional independence of observed images given the labels, the approach relies more heavily
on supervised training. DRF training typically uses labeled image data where each pixel label has
been assigned. However, it is considerably more difficult to obtain labeled data for image analysis
than for other classification tasks, such as document classification, since hand-labeling the individual
pixels of each image is much harder than assigning class labels to objects like text documents.
?
Work done while at University of Alberta
Recently, semi-supervised training has taken on an important new role in many application areas due
to the abundance of unlabeled data. Consequently, many researchers are now working on developing
semi-supervised learning techniques for a variety of approaches, including generative models [14],
self-learning [5], co-training [3], information-theoretic regularization [6, 8], and graph-based transduction [22, 23, 24]. However, most of these techniques have been developed for univariate classification problems, or class label classification with a structured input [22, 23, 24]. Unfortunately,
semi-supervised learning for structured classification problems, where the prediction variables are
interdependent in complex ways, have not been as widely studied, with few exceptions [1, 9].
Current work on semi-supervised learning for structured predictors [1, 9] has focused primarily on
simple sequence prediction tasks where learning and inference can be efficiently performed using
standard dynamic programming. Unfortunately, the problem we address is more challenging, since
the spatial correlations in a 2-D grid structure create numerous dependency cycles. That is, our
graphical model structure prevents exact inference from being feasible. Kumar et al [10] and Vishwanathan et al [19] argue that learning a model in the context of approximate inference creates a
greater risk of the over-fitting and over estimating.
In this paper, we extend the work on semi-supervised learning for sequence predictors [1, 9], particularly the CRF based approach [9], to semi-supervised learning of DRFs. There are several advantages of our approach to semi-supervised DRFs. (1) We inherit the standard advantage of discriminative conditional versus joint model training, while still being able to exploit unlabeled data. (2) The
use of unlabeled data enhances our ability to avoid parameter over-fitting and over-estimation in grid
based random fields even when using a learner that uses only approximate inference methods. (3)
We are still able to model spatial correlations in a 2-D lattice, despite the fact that this introduces
dependency cycles in the model. That is, our semi-supervised training procedure can be interpreted
as a MAP estimator, where the parameter prior for the model on labeled data is governed by the
conditional entropy of the model on unlabeled data. This allows us to learn local potentials that
capture spatial correlations while often avoiding local over-estimation. We demonstrate the robustness of our model by applying it to a pixel denoising problem on synthetic images, and also to a
challenging real world problem of segmenting tumor in magnetic resonance images. In each case,
we have obtained significant improvements over current baselines based on standard DRF training.
2 Semi-Supervised DRFs (SSDRFs)
We formulate a new semi-supervised DRF training principle based on the standard supervised formulation of [11, 10]. Let x be an observed input image, represented by x = {x i }i?S , where S
is a set of the observed image pixels (nodes). Let y = {yi }i?S be the joint set of labels over all
pixels of an image. For simplicity we assume each component y i ? y ranges over binary classes
Y = {?1, 1}. For example, x might be a magnetic resonance image of a brain and y is a realization
of a joint labeling over all pixels that indicates whether each pixel is normal or a tumor. In this case,
Y would be the set of pre-defined pixel categories (e.g. tumor versus non-tumor). A DRF is a conditional random field defined on the pixel labels, conditioned on the observation x. More explicitly,
the joint distribution over the labels y given the observations x is written
p? (y|x)
=
?X
?
XX
1
exp
?w (yi , x) +
?? (yi , yj , x)
Z? (x)
i?S
i?S j?N
(1)
i
Here Ni denotes the neighboring pixels of i. ?w (yi , x) = log ?(yi wT hi (x) denotes the node
potential at pixel i, which quantifies the belief that the class label is yi for the pre-defined feature
vextor hi (x), where ?(t) = 1+e1 ?t . ?? (yi , yj , x) = yi yj vT ?ij (x) is an edge potential that captures
spatial correlations among neighboring pixels (here, the ones at positions i and j), such that ? ij (x)
is the pre-defined feature vector associated with observation x. Z ? (x) is the normalizing factor, also
known as a (conditional) partition function, which is
Z? (x) =
X
y
exp
?X
i?S
?w (yi , x) +
X X
?? (yi , yj , x)
?
(2)
i?S j?Ni
Finally, ? = (w, ?) are the model parameters. When the edge potentials are set to zero, a DRF
yields a standard logistic regression classifier. The potentials in a DRF can use properties of the
observed image, and thereby relax the conditional independence assumption of MRFs. Moreover,
the edge potentials in a DRF can smooth discontinuities between heterogeneous class pixels, and
also correct errors made by the node potentials.
Assume we have a set of independent labeled images, D l = (x(1) , y(1) )), ? ? ? , (x(M ) , y(M ) ) , and a
?
?
set of independent unlabeled images, D u = x(M +1) , ? ? ? , x(T ) . Our goal is to build a DRF model
?
?
from the combined set of labeled and unlabeled examples, D l ? Du .
The standard supervised DRF training procedure is based on maximizing the log of the posterior
probability of the labeled examples in D l
CL(?)
=
M
X
log P (y(k) |x(k) ) ?
k=1
?T ?
2? 2
(3)
A Gaussian prior over the edge parameters ? is assumed and a uniform prior over parameters w.
Here p(?) = N (?; 0, ? 2 I), where I is the identity matrix. The hyperparameter ? 2 adds a regularization term. In effect, the Gaussian prior introduces a form of regularization to limit over-fitting on
rare features and avoid degeneracy in the case of correlated features.
There are a few issues regarding the supervised learning criteria (3). First, the value of ? 2 is critical
to the final result, and unfortunately selecting the appropriate ? 2 is a non-trivial task, which in turn
makes the learning procedures more challenging and costly [13]. Second, the Gaussian prior is
data-independent, and is not associated with either the unlabeled or labeled observations a priori.
Inspired by the work in [8] and [9], we propose a semi-supervised learning algorithm for DRFs that
makes full use of the available data by exploiting a form of entropy regularization as a prior over the
parameters on D u . Specifically, for a semi-supervised DRF, we attempt to find ? that maximizes the
following objective function
RL(?)
=
M
X
log p? (y(m) |x(m) ) + ?
m=1
T
X
m=M +1
X
p? (y|x(m) ) log p? (y|x(m) )
(4)
y
The first term of (4) is the conditional likelihood over the labeled data set D l , and the second term
is a conditional entropy prior over the unlabeled data set D u , weighted by a tradeoff parameter ?.
The resulting estimate is then formulated as a MAP estimate.
The goal of the objective (4) is to minimize the uncertainty on possible configurations over parameters. That is, minimizing the conditional entropy over unlabeled instances provides more confidence
to the algorithm that the hypothetical labellings for the unlabeled data are consistent with the supervised labels, as greater certainty on the estimated labellings coincides with greater conditional
likelihood on the supervised labels, and vice versa. This criterion has been shown to be effective for
univariate classification [8], and chain structured CRFs [9]; here we apply it to the 2-D lattice case.
3 Parameter Estimation
Several factors constrain the form of training algorithm: Because of overhead and the risk of divergence, it was not practical to employ a Newton method. Iterative scaling was not possible because
the updates no longer have a closed form. Although the criticism of the gradient descent?s principle
is well taken, it is the most practical approach we will adopt to optimize the semi-supervised MAP
formulation (4) and allows us to improve on standard supervised DRF training.
To formulate a local optimization procedure, we need to compute the gradient of the objective (4)
with respect to the parameters. Unfortunately, because of the nonlinear mapping function ?(.), we
are not able to represent the gradient of objective function as compactly as [9], which was able to
express the gradient as a product of the covariance matrix of features and the parameter vector ?.
Nevertheless, it is straightforward to show that the derivatives of objective function with respect to
the node parameters w is given by 1
1
Note that the derivatives of objective function with respect to the edge parameters ? are computed analogously.
?
RL(?) =
?w
0
1
M
?
? X
?
?
X
X
(m)
T
(m)
@y (m) 1 ? ?(y (m) wT hi (x(m) ) ?
p? (y|x
)yi 1 ? ?(yi w hi (x
) A hi (x(m) )
i
i
m=1 i?S m
+?
(5)
y
T
X
X
m=M +1 i?S m
0
? ?
?
?
X
X
T
(m)
(m)
@
?? (yi , yj , x) yi 1 ? ?(yi w hi (x
)
p? (y|x
) ?w (yi , x) +
j?Ni
y
?
hX
p? (y|x
(m)
?
) ?w (yi , x) +
?? (yi , yj , x)
?i
j?Ni
y
hX
X
?
p? (y|x(m) )yi 1 ? ?(yi wT hi (x(m) )
?i
y
1
A hi (x(m) ),
where the first term in (5) is the gradient of the supervised component of the DRF over labeled data,
and the second term is the gradient of conditional entropy prior of the DRF over unlabeled data.
Given the lattice structure of the joint labels, it is intractable to compute the exact expectation terms
in the above derivatives. It is also intractable to compute the conditional partition function Z ? (x).
Therefore, as in standard supervised DRFs, we need to incorporate some form of approximation.
Following [2, 11, 10], we incorporate the pseudo-likelihood approximation, which assumes that the
joint conditional distribution can be approximated as a product of the local posterior probabilities
given the neighboring nodes and the observation
p? (y|x)
Y
?
(6)
p? (yi |yNi , x)
i?S
p? (yi |yNi , x)
?
?
X
1
exp ?w (yi , x) +
?? (yi , yj , x)
zi (x)
j?N
=
(7)
i
Using the factored approximation in (7), we can reformulate the training objective as
m
RLP L (?)
S
M X
X
=
(m)
log p? (yi
(8)
(m)
|yNi , x(m) )
m=1 i=1
m
T
X
+?
S
X
X
p? (yi |yNi , x(m) ) log p? (yi |yNi x(m) )
m=M +1 i=1 yi
Here, the derivative of the second term in (8), with respect to the potential parameters w and ?, can
be reformulated as a factored conditional entropy, yielding
?
RLP L (?)
(9)
?w
0
1
M
?
? X
?
?
X X
@y (m) 1 ? ?(y (m) wT hi (x(m) ) ?
=
p? (yi |yNi , x(m) )yi 1 ? ?(yi wT hi (x(m) ) A hi (x(m) )
i
i
m=1 i?S m
+?
T
X
yi
X
m=M +1 i?S m
0
@
X
p? (yi |yNi , x
yi
?
hX
(m)
?
) ?w (yi , x) +
j?Ni
p? (yi |yNi x
(m)
?
yi
p? (yi |yNi , x
? ?
?
T
(m)
?? (yi , yj , x) yi 1 ? ?(yi w hi (x
)
) ?w (yi , x) +
yi
hX
X
X
?? (yi , yj , x)
?i
j?Ni
(m)
?
T
)yi 1 ? ?(yi w hi (x
(m)
)
?i
1
A hi (x(m) )
Note that ??? RLP L (?) is computed analogously. Assuming the factorization, the true conditional
entropy and feature expectations can be computed in terms of local conditional distributions. This allows us efficiently to approximate the global conditional entropy over unlabeled data. Note that there
may be an over-smoothing issue associated with the pseudo-likelihood approximation, as mentioned
in [10, 19]. However, due to the fast and stable performance of this approximation in the supervised
case [2, 10] we still employ it, but below show that the over-smoothing effect is mitigated by our
data-dependent prior in the MAP objective (4).
4 Inference
As a result of our formulation, the learning method is tightly coupled with the inference steps.
That is, for the unlabeled data, XU , each time we compute the local conditional covariance (9), we
perform inference steps for each node i and its neighboring nodes N i . Our inference is based on
iterative conditional modes (ICM) [2], and is given by
yi? = argmax P (yi |yNi , X)
(10)
yi ?Y
where, for each position i, we assume that the labels of all of its neighbors y 0P? Ni are fixed. We
could alternatively compute the marginal conditional probability P (y i |X) = yS\i P (yi , yS\i |X)
for each node using the sum-product algorithm (i.e. loopy belief propagation), which iteratively
propagates the belief of each node to its neighbors. Clearly, there are a range of approximation
methods available, each entailing different accuracy-complexity tradeoffs. However, we have found
that ICM yields good performance at our tasks below, and is probably one of the simplest possible
alternatives.
5 Experiments
Using standard supervised DRF models, Kumar and Hebert [11, 10] reported interesting experimental results for joint classification tasks on a 2-D lattice, which represents an image with a DRF
model. Since labeling image data is expensive and tedious, we believe that better results could be
further obtained by formulating a MAP estimation of DRFs by also using the abundant unlabeled
image data. In this section, we present a series of experiments on synthetic and real data sets using
our novel semi-supervised DRFs(SSDRFs). In order to evaluate our model, we compare the results
with those using maximum likelihood estimation of supervised DRFs [11]. There is a major reason that we consider the standard MLE DRF from [11] instead of the parameter regularized DRFs
from [10]: that is, we want to show the difference between the ML and MAP principles without
using any regularization term that can be problematic [10, 13].
To quantify the performance of each model, we used the Jaccard score J = (T P +FT PP +F N ) , where
TP denotes true positives, FP false positives, and FN false negatives. Although there are many
accuracy measures available, we used this score to penalize the false negatives since many imaging
tasks are very imbalanced: that is, only a small percentage of pixels are in the ?positive? class. The
tradeoff parameter, ?, was hand-tuned on one held out data set and then held fixed at 0.2 for all of
the experiments.
5.1 Synthetic image sets
Our primary goal in using synthetic data sets was to demonstrate how well different models classified pixels as a binary classification over a 2-D lattice in the presence of noise. We generated 18
synthetic data sets, each with its own shape. The intensities of pixels in each image were independently corrupted by noise generated from a Gaussian N (0, 1). Figure 1 shows the results of using
supervised DRFs, as well as semi-supervised DRFs. [10, 19] reported over-smoothing effects from
the local approximation approach of PL while our experiments indicate that the over-smoothing is
caused not only by PL approximation, but also by the sensitivity of the regularization to the parameters. However, using our semi-supervised DRF as a MAP formulation, we have dramatically
improved the performance over standard supervised DRF.
Note that the first row in Figure 1 shows good results from the standard DRF, while the oversmoothed
outputs are presented in the last row. Although the ML approach may learn proper parameters from
some of data sets, unfortunately its performance has not been consistent since the standard DRF?s
learning of the edge potential tends to be overestimated. For instance, the last row shows that
overestimating parameters of the DRF segment almost all pixels into a class due to the complicated
edges and structures containing non-target area within the target area, while semi-supervised DRF
performance is not degraded at all. Overall, by learning more statistics from unlabeled data, our
model dominates the standard DRF in most cases. This is because our MAP formulation avoids
the overestimate of potentials and uses the edge potential to correct the errors made by the node
potential. Figure 2(a) shows the results over 18 synthetic data sets. Each point above the diagonal
line in Figure 2(a) indicates SSDRF producing higher Jaccard scores for a data set. Note that our
model stably converged as we increased the ratio (nU/nL) of unlabeled data sets in our learning,
1
4500
0.9
J: 0.933890
4400
0.8
J: 0.933377
J: 0.729527
J: 0.957983
SSDRF
0.7
4300
0.6
0.5
4200
0.4
4100
0.3
0.2
4000
0.1
J: 0.008178
J: 0.923836
0
Figure 1: Outputs from synthetic data sets.
From left to right: Testing instance, Ground
Truth, Logistic Regression (LR), DRF, and
SSDRF.
0
(a)
0.2
0.4
DRF
0.6
0.8
Accuracy from DRF
and SSDRF for all 18 synthetic data sets
1
3900
1
2
3
4
5
6
7
8
9
10
(b)
Log likelihood values (Y axis) for a testing
image by increasing ratio
(X axis) of unlabeled instances for SSDRF
Figure 2: Accuracy and Convergency
as in Figure 2(b), where nU denotes the number of unlabeled images and nL the number of labeled
images. Similar results have also been reported in simple single variable classification task [8].
5.2 Brain Tumor Segmentation
We have applied our semi-supervised DRF model to the challenging real world problem of segmenting tumor in medical images. Our goal here is to classify each pixel of an magnetic resonance
(MR) image into a pre-defined category: tumor and non-tumor. This is a very important, yet notoriously difficult, task in surgical planning and radiation therapy which currently involves a significant
amount of manual work by human medical experts.
We applied three models to the classification of 9 studies from brain tumor MR images. For each
study2 , i, we divided the MR images into DiL , DiU , and DiS , where an MR image (a.k.a slice) has
three modalities available ? T1, T2, and T1 contrast. Note that each modality for each slice has
66, 564 pixels.
As with much of the related work on automatic brain tumor segmentation (such as [7, 21]), our
training is based on patient-specific data, where training MR images for a classifier are obtained
from the patient to be tested. Note that the training sets and testing sets for a classifier are disjoint.
Specifically, LR and DRF takes DiL as the training set and DiU and DiS for testing sets, while SSDRF
takes DiL and DiU for training and DiU and DiS for testing.
We segmented the ?enhancing? tumor area, the region that appears hyper-intense after injecting
the contrast agent (we also included non-enhancing areas contained within the enhancing contour).
Table 1 and 2 present Jaccard scores of testing DiU and DiS for each study, pi , respectively. While
the standard supervised DRF improves over its degenerate model LR by 1%, semi-supervised DRF
significantly improves over the supervised DRF by 11%, which is significant at p < 0.00566 using
a paired example t test. Considering the fact that MR images contain much noise and the three
modalities are not consistent among slices of the same patient, our improvement is considerable.
Figure 3 shows the segmentation results by overlaying the testing slices with segmented outputs
from the three models. Each row demonstrates the segmentation for a slice, where the white blob
areas for the slice correspond to the enhancing tumor area.
6 Conclusion
We have proposed a new semi-supervised learning algorithm for DRFs, which was formulated as
MAP estimation with conditional entropy over unlabeled data as a data-dependent prior regularization. Our approach is motivated by the information-theoretic argument [8, 16] that unlabeled
examples can provide the most benefit when classes have small overlap. We introduced a simple approximation approach for this new learning procedure that exploits the local conditional probability
to efficiently compute the derivative of objective function.
2
Each study involves a number (typically 21) of images of a single patient ? here parallel axial slices through
the head.
Table 1: Jaccard Scores for DiU .
Studies
p1
p2
p3
p4
p5
p6
p7
p8
p9
Average
Testing from DiU
LR
DRF SSDRF
53.84 59.81
59.81
83.24 83.65
84.67
30.72 30.17
75.76
72.04 76.16
79.02
73.26 73.59
75.25
88.39 89.61
87.01
69.33 69.91
75.60
58.49 58.89
73.03
60.85 56.49
83.91
65.57 66.48
77.12
Table 2: Jaccard Scores for DiS .
Studies
p1
p2
p3
p4
p5
p6
p7
p8
p9
Average
Testing from DiS
LR
DRF SSDRF
68.01 68.75
68.75
69.61 69.73
70.06
23.11 21.90
71.13
56.52 63.07
68.40
51.38 52.36
51.29
85.65 86.35
85.43
66.71 68.68
70.27
44.92 45.36
73.09
21.11 20.16
38.06
54.11 55.15
66.27
Figure 3: From Left to Right: Human Expert, LR, DRF, and SSDRF
We have applied this new approach to the problem of image pixel classification tasks. By exploiting
the availability of auxiliary unlabeled data, we are able to improve the performance of the state of
the art supervised DRF approach. Our semi-supervised DRF approach shares all of the benefits of
the standard DRF training, including the ability to exploit arbitrary potentials in the presence of
dependency cycles, while improving accuracy through the use of the unlabeled data.
The main drawback is the increased training time involved in computing the derivative of the conditional entropy over unlabeled data. Nevertheless, the algorithm is efficient to be trained on unlabeled
data sets, and to obtain a significant improvement in classification accuracy over standard supervised
training of DRFs as well as iid logistic regression classifiers. To further accelerate the performance
with respect to accuracy, we may apply loopy belief propagation [20] or graph-cuts [4] as an inference tool. Since our model is tightly coupled with inference steps during the learning, the proper
choice of an inference algorithm will most likely improve segmentation tasks.
Acknowledgments
This research is supported by the Alberta Ingenuity Centre for Machine Learning, Cross Cancer
Institute, and NSERC. We gratefully acknowledge many helpful suggestions from members of the
Brain Tumor Analysis Project, including Dr. A. Murtha and Dr. J Sander.
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2,237 | 303 | ART2/BP architecture for adaptive estimation of
dynamic processes
Einar S~rheim *
Department of Computer Science
UNIK, Kjeller
University of Oslo
N-2007 Norway
Abstract
The goal has been to construct a supervised artificial neural network that
learns incrementally an unknown mapping. As a result a network consisting of a combination of ART2 and backpropagation is proposed and
is called an "ART2/BP" network. The ART2 network is used to build
and focus a supervised backpropagation network. The ART2/BP network
has the advantage of being able to dynamically expand itself in response
to input patterns containing new information. Simulation results show
that the ART2/BP network outperforms a classical maximum likelihood
method for the estimation of a discrete dynamic and nonlinear transfer
function.
1
INTRODUCTION
Most current neural network architectures such as backpropagation require a cyclic
presentation of the entire training set to converge. They are thus not very well suited
for adaptive estimation tasks where the training vectors arrive one by one, and where
the network may never see the same training vector twice. The ART2/BP network
system is an attempt to construct a network that works well on these problems.
Main features of our ART2/BP are:
? implements incremental supervised learning
? dynamically self-expanding
*e-mail address:[email protected]@ifi.uio.no
169
170
Sorheim
? learning of a novel training pattern does not wash away memory of previous
training patterns
? short convergence time for learning a new pattern
2
BACKGROUND
Adaptive estimation of nonlinear functions requires some basic features of the estimation algorithm.
1. Incremental learning
The input/output pairs arrive to the estimation machine one by one. By accumulating the input/output pairs into a training set and rerun the training
procedure at every arrival of a new input/output pair, one could use a conventional method. Obvious disadvantages would however be
? huge learning time required as the size of the training set increases .
? an upper limit, N, on the number of elements in the training set will
have to be set. The training set will then be a gliding horizon of the N
last input/output pairs, and information prior to the N last input/output
pairs will be lost.
2. Plasticity
Learning of a new input/output pair should not wash away the memory of
previously learned nonconflicting input/output pairs. With most existing feedforward supervised nets this is hard to accomplish, though some efforts have
been made (Otwell 90). Some networks, like the ART-family and RCN (Ryan
1988) are plastic but they are self-organizing, not supervised.
To summarize:
Need a supervised network that learns incrementally the mapping of an unknown
system and that can be used to predict future outputs. The system in question
maps analog vectors to analog vectors.
3
COMBINED ARCHITECTURE
In the proposed network architecture an ART2 network controls a BP network, see
Figure 1.
The BP-network consists of many relatively small subnetworks where the subnets
are specialized on one particular domain of the input space. ART2 controls how the
input space is divided among the subnets and the total amount of sub nets needed.
The ART2 network analyzes the input part of the input/output pairs as they arrive
....
to the system. For a given input pattern i:r, ART2 finds the category G:r which has
the closest resemblance to ~. If this resemblance is good enough, ~ is of category
G:r and the LTM-weights of G:r are updated. The BP-subnetwork BP:r, connected
to G:r, is as a consequence activatedt and relearning of BP:r is done. The learning
set consists of a "representative" set of the neighbouring subnets patterns and a
small number of the previous patterns belonging to category G:r. To summarize the
ART2IBP Architecture for Adaptive Estimation of Dynamic Processes
algorithm goes as follows:
1. Send input vector to ART2 network
2. ART2 classification.
3. If in learning mode adjust ART2 LTM weights of the winning node.
4. Send input to the back propagation network connected to the winning ART2
node.
5. If in learning mode:
? find a representative training set.
? do epoch learning on training set.
Otherwise
? compute output of the selected back propagation network.
6. Go to 1. for new input vector.
The ART2/BP neural network can be used for adaptive estimation of nonlinear
dynamic processes. The mapping to be estimated then is
yet + ot)
u(t)
yet)
=
l( u(t), yet?~
(1)
f ~m
f ~n
The input/output pairs will be i7J [u(t) , yet), yet + ot)], denote the input part of
i7J: i [u(t) , yet)] and the output part of (0: 0 yet + ot).
=
4
=
ART2 MODIFIED
ART2 was developed by Carpenter& Grossberg see (Carpenter 1987) and (Carpenter 1988). ART2 categorizes arbitrary sequences of analog input patterns, and the
categories can be of arbitrary coarseness. For a detailed description of ART2, see
(Carpenter 1987).
4.1
MODIFICATION
In the standard ART2-algorithm input vectors (patterns) are normalized. For this
application it is not desired to classify parallel vectors of different magnitude as
belonging to the same category. By adding an extra element to the input vector
where this element is simply
(2)
the new input vector becomes
-
From a scaled vector of i: i
-
(3)
= a :{ the original vector i could easily be found as :
(4)
171
172
Sorheim
and by using the augmented i as the input to ART2 instead of i one can at any
point in Fl( representation layer) and F2( categorization layer) generate the corresponding non-normalized vector. The F2 node competition is modified so that the
node having bottom-up LTM weights with the smallest distance (distance being the
euclidean norm) to the Fl layer pattern code wins the competition. The distance
d J of F2 node J is given by:
IIv -
v
zj
(5)
zjll
being the 12 - norm
Fl pattern code.
bottom - up LTM weights of F2 node J
Reset is done by calculating the distance d between the Fl layer pattern code V and
~
J :
d = IIv- ~I
(6)
and comparing it to a largest acceptable bound p. If d > p the winning node is
inhibited and a new node will be created. If d ~ p LTM-patterns of the winning
node J are modified (learning).
5
BACK PROPAGATION NETWORK
The backpropagation network used in this work is of the standard feedforward type,
see (Rumelhart 1986) . The number of hidden layers and nodes should be kept low
in the subnetworks, for the problems in our simulations we used 1 hidden layer with
2 nodes. As for training algorithms several different kinds have been tried:
? Standard back propagation (SBP)
? A modified back propagation (MBP) method similar to the one used in the
BPS simulator from George Mason University.
? Quickprop (Q).
? A quasi-Newton method (BFGS).
All of these except SBP show similar performance in my test cases.
The BP-networks performs as an interpolator in this algorithm and any good interpolation algorithm can be used instead of BP. Approximation theory gives several
interesting techniques for approximation/interpolation of multidimensional functions such as Radial Basis Functions and Hyper Basis Functions, for further detail
see (Poggio 90). These methods requires a representative training set where the
input part determines the location of centers in the input space. The ART2 alg<r
rithm can be used for determining these centers in an adaptive way and thus making
possible an incremental version of the approximation theory techniques. This idea
has not been tested yet, but is an interesting concept for further research.
ART2IBP Architecture for Adaptive Estimation of Dynamic Processes
6
LEARNING
Learning in ART2/BP is a two stage process. First the input patterns is sent to
the ART2 network for categorizing and learning. ART2 will then activate the
BP subnetwork that is a local expert on patterns of the same category as the
input pattern, and learning of this subnetwork will occur. A training set that is
representative for the domain of the input space has to be found. Let a small
number of the last categorized input/output pairs be allocated to its corresponding
subnet to provide a part of the training set. Denote such a set as LJOc, (C being
the category). Define the location ofF2 node J to be its bottom-up weights ;J. Let
the current input i~ define an origin, then find the F2 nodes closest to origin in each
n-ant of the input space. Call this set of nodes N~ and the set oflast input/output
pairs stored in these nodes N JO~. The training set is then chosen to be:
T~
N _IO~ U LJO~
Before training, the elements in T~ are scaled to increase accuracy and to accelerate
learning. BP-Iearning is then performed, the stopping criteria being a fixed error
term or a maximum number of iterations.
=
7
ESTIMATION
In estimation mode learning in the network is turned off. Given an input thenetwork
will produce an output that hopefully will be close to the output of the real system.
The ART2-network selects a winning node in the same way as described before but
now the reset assembly is not activated. Then the input is fed to the corresponding
BP subnetwork and its output is used as an estimate of the original functions output.
Because each subnetwork is scaled to cover the domain of the input space made up
by the complex hull Co(T~) of its training set T~, the entire ART2/BP network will
cover the complex hull C o(T) C ~n+m where:
T=
{set of all previous fs used to train the network}
Good estimation/prediction can thus be expected if i ( Co(T). This means that if
the input vector i lies in a domain of the input space that has not been previously
explored by the elements in the training set, the network will generalize poorly.
8
EXAMPLE
The ART2/BP network has been used to estimate a dynamic model of a tank filled
with liquid. The liquid level is sampled every 6t time interval and the ART2/BP
network is used to estimate the discrete dynamic nonlinear transfer function of the
liquid level as a function of inlet liquid flow and previous liquid level. That is, we
want to find a good estimate j(.,.) of:
y(t
+ 6t)
u(t)
y(t)
f( u(t), y(t?
inlet liquid flow at time t
liquid level at time t
(7)
173
174
Sorheim
o. 2
0.1 5
o. 1
0.0 5
',J
,
'\
.1
An
.~
r VS] '
j
-0.0 5
\
w'
~J
WH
I,~
II!
.~,
.A.
e-
~Yce:::::::::J
150
2100
250
=
I
300
,
-0. 1
black line: ARMA model estimation error (y(t + 6t) - YARMA(t + 6t))
grey line: ART2/BP estimation error (y(t + 6t) - YART2/BP(t + 6t?
Figure 1: Comparison of the estimation error of the ARMA model and the
ART2/BP network
To increase the nonlinearities of the transfer function, the area of the tank varies
with a step function of the liquid level. The BP subnetworks have 2 input nodes,
1 hidden layer with 2 neurons and a single neuron output layer. In the simulations
p
0.04 and the last three categorized input/output pairs are stored at every
subnetwork. As the input space is 2-dimensional giving 4 neighbouring nodes the
maximum size of the training set 7 input/output pairs. After a learning period of
1000 samples with random inlet flow, three test cases are run with the network in
estimation mode. The network had then formed about 140 categories. The same
set of simulation data is also run through an offline maximum likelihood method to
estimate a linear ARMA model of the plant, see (Ljung 1983). /
=
Figure 1 shows the simulation results of the three test cases where :
samples 1-100 : random input flow.
samples 101-200 : constant input flow at a low level.
samples 201-300 : constant input flow at a high level.
In Figure 1, the estimation errors of the two methods are compared. For the
first 100 samples with stochastic input flow, the estimation error variance of the
ART2IBP Architecture for Adaptive Estimation of Dynamic ftocesses
ART2/BP network is roughly a factor 10 less than that of the ARMA-model. The
performance of ART2/BP is also significantly better for the constant input flow
cases, here the ARMA model has an error of -- 0.02 while the ART2/BP-error is
- 0.002. The overall improvement in estimation error is a reduction of roughly 0.1
. Also keep in mind that ART2/BP is compared to an offline maximum likelihood
method while ART2/BP clearly is an online method. The online version of the
maximum likelihood would most probably have given a worse performance than the
offline version.
9
CONCLUSION/COMMENTS
The proposed ART2/BP neural network architecture offers some unique features
compared to backpropagation. It provides incremental learning and can be applied
to truly adaptive estimation tasks. In our example it also outperforms a classical
maximum likelihood method for the estimation of a discrete dynamic nonlinear
transfer function. Future work will be the investigation of ART2/BP's properties for
multistep-ahead prediction of dynamic nonlinear transfer functions, and embedding
ART2/BP in a neural adaptive controller.
Acknow ledgments
Special thanks to Steve Lehar at Boston University for providing me with his ART2
simulation program. It proved to be crucial for getting a quick start on ART2 and
understanding the concept.
References
Carpenter, G.A. & Grossberg, S. (1987). ART2: Self-organization of stable category recognition codes for analog input patterns. Applied Optics pp 4919-4930.
Carpenter, G.A. & Grossberg, S. (1988). The ART of adaptive pattern recognition
by a self-organizing neural network. Computer 21 pp 77-88.
Fahlman, S.E. (1988). Faster-Learning Variations on Back-Propagation: An Empirical Study. Proceedings of the 1988 Connectionist Models Summer School. Morgan
Kaufmann.
Ljung, L. &
S~derstr~m
(1983). Theory and practice of recursive identification.
The MIT press, Cambridge, MA.
Otwell, K. (1990). Incremental backpropagation learning from novelty-based orthogonalization. Proceedings IJNN90 .
Poggio, T., Girosi, F. (1990). Networks for Approximation and Learning. Proceedings of the IEEE,Vol. 78, No.9.
Rumelhart, D.E., Hinton, G.E., & Williams, R.J. (1986). Parallel Distributed
Processing: Explorations in the microstructure of Cognition, Vol. 1. The MIT
Press,Cambridge, MA.
Ryan, T. W. (1988). The resonance correlation network. Proceedings IJNN88.
175
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2,238 | 3,030 | Approximate Correspondences in High Dimensions
Kristen Grauman
Department of Computer Sciences
University of Texas at Austin
[email protected]
Trevor Darrell
CS and AI Laboratory
Massachusetts Institute of Technology
[email protected]
Abstract
Pyramid intersection is an efficient method for computing an approximate partial
matching between two sets of feature vectors. We introduce a novel pyramid embedding based on a hierarchy of non-uniformly shaped bins that takes advantage
of the underlying structure of the feature space and remains accurate even for sets
with high-dimensional feature vectors. The matching similarity is computed in
linear time and forms a Mercer kernel. Whereas previous matching approximation algorithms suffer from distortion factors that increase linearly with the feature dimension, we demonstrate that our approach can maintain constant accuracy
even as the feature dimension increases. When used as a kernel in a discriminative classifier, our approach achieves improved object recognition results over a
state-of-the-art set kernel.
1
Introduction
When a single data object is described by a set of feature vectors, it is often useful to consider
the matching or ?correspondence? between two sets? elements in order to measure their overall
similarity or recover the alignment of their parts. For example, in computer vision, images are often
represented as collections of local part descriptions extracted from regions or patches (e.g., [11, 12]),
and many recognition algorithms rely on establishing the correspondence between the parts from
two images to quantify similarity between objects or localize an object within the image [2, 3, 7].
Likewise, in text processing, a document may be represented as a bag of word-feature vectors; for
example, Latent Semantic Analysis can be used to recover a ?word meaning? subspace on which
to project the co-occurrence count vectors for every word [9]. The relationship between documents
may then be judged in terms of the matching between the sets of local meaning features.
The critical challenge, however, is to compute the correspondences between the feature sets in an
efficient way. The optimal correspondences?those that minimize the matching cost?require cubic
time to compute, which quickly becomes prohibitive for sizeable sets and makes processing realistic
large data sets impractical. Due to the optimal matching?s complexity, researchers have developed
approximation algorithms to compute close solutions for a fraction of the computational cost [4, 8,
1, 7]. However, previous approximations suffer from distortion factors that increase linearly with
the dimension of the features, and they fail to take advantage of structure in the feature space.
In this paper we present a new algorithm for computing an approximate partial matching between
point sets that can remain accurate even for sets with high-dimensional feature vectors, and benefits
from taking advantage of the underlying structure in the feature space. The main idea is to derive a
hierarchical, data-dependent decomposition of the feature space that can be used to encode feature
sets as multi-resolution histograms with non-uniformly shaped bins. For two such histograms (pyramids), the matching cost is efficiently calculated by counting the number of features that intersect
in each bin, and weighting these match counts according to geometric estimates of inter-feature distances. Our method allows for partial matchings, which means that the input sets can have varying
numbers of features in them, and outlier features from the larger set can be ignored with no penalty
to the matching cost. The matching score is computed in time linear in the number of features per
set, and it forms a Mercer kernel suitable for use within existing kernel-based algorithms.
In this paper we demonstrate how, unlike previous set matching approximations (including our original pyramid match algorithm [7]), the proposed approach can maintain consistent accuracy as the
dimension of the features within the sets increases. We also show how the data-dependent hierarchical decomposition of the feature space produces more accurate correspondence fields than a previous
approximation that uses a uniform decomposition. Finally, using our matching measure as a kernel
in a discriminative classifier, we achieve improved object recognition results over a state-of-the-art
set kernel on a benchmark data set.
2
Related Work
Several previous matching approximation methods have also considered a hierarchical decomposition of the feature space to reduce matching complexity, but all suffer from distortion factors that
scale linearly with the feature dimension [4, 8, 1, 7]. In this work we show how to alleviate this
decline in accuracy for high-dimensional data by tuning the hierarchical decomposition according
to the particular structure of the data, when such structure exists.
We build on our pyramid match algorithm [7], a partial matching approximation that also uses
histogram intersection to efficiently count matches implicitly formed by the bin structures. However,
in contrast to [7], our use of data-dependent, non-uniform bins and a more precise weighting scheme
results in matchings that are consistently accurate for structured, high-dimensional data.
The idea of partitioning a feature space with vector quantization (VQ) is fairly widely used in practice; in the vision literature in particular, VQ has been used to establish a vocabulary of prototypical
image features, from ?textons? to the ?visual words? of [16]. A variant of the pyramid match applied to spatial features was shown to be effective for matching quantized features in [10]. More
recently, the authors of [13] have shown that a tree-structured vector quantization (TSVQ [5]) of image features provides a scalable means of indexing into a very large feature vocabulary. The actual
tree structure employed is similar to the one constructed in this work; however, whereas the authors
of [13] are interested in matching individual features to one another to access an inverted file, our
approach computes approximate correspondences between sets of features. Note the distinction between the problem we are addressing?approximate matchings between sets?and the problem of
efficiently identifying approximate or exact nearest neighbor feature vectors (e.g., via k-d trees): in
the former, the goal is a one-to-one correspondence between sets of vectors, whereas in the latter, a
single vector is independently matched to a nearby vector.
3
Approach
The main contribution of this work is a new very efficient approximate bipartite matching method
that measures the correspondence-based similarity between unordered, variable-sized sets of vectors, and can optionally extract an explicit correspondence field. We call our algorithm the
vocabulary-guided (VG) pyramid match, since the histogram pyramids are defined by the ?vocabulary? or structure of the feature space, and the pyramids are used to count implicit matches.
The basic idea is to first partition the given feature space into a pyramid of non-uniformly shaped regions based on the distribution of a provided corpus of feature vectors. Point sets are then encoded as
multi-resolution histograms determined by that pyramid, and an efficient intersection-based computation between any two histogram pyramids yields an approximate matching score for the original
sets. The implicit matching version of our method estimates the inter-feature distances based on
their respective distances to the bin centers. To produce an explicit correspondence field between
the sets, we use the pyramid construct to divide-and-conquer the optimal matching computation. As
our experiments will show, the proposed algorithm in practice provides a good approximation to the
optimal partial matching, but is orders of magnitude faster to compute.
Preliminaries: We consider a feature space F of d-dimensional vectors, F ? <d . The point sets
our algorithm matches will come from the input space S, which contains sets of feature vectors
drawn from F : S = {X|X = {x1 , . . . , xm }}, where each xi ? F , and the value m = |X| may
vary across instances of sets in S. Throughout the text we will use the terms feature, vector, and
point interchangeably to refer to the elements within a set.
(a) Uniform bins
(b) Vocabulary-guided bins
Figure 1: Rather than carve the feature space into uniformly-shaped partitions (left), we let the vocabulary
(structure) of the feature space determine the partitions (right). As a result, the bins are better concentrated on
decomposing the space where features cluster, particularly for high-dimensional feature spaces. These figures
depict the grid boundaries for two resolution levels for a 2-D feature space. In both (a) and (b), the left plot
contains the coarser resolution level, and the right plot contains the finer one. Features are red points, bin
centers are larger black points, and blue lines denote bin boundaries.
A partial matching between two point sets is an assignment that maps all points in the smaller set
to some subset of the points in the larger (or equally-sized) set. Given point sets X and Y, where
m = |X|, n = |Y|, and m ? n, a partial matching M (X, Y; ?) = {(x1 , y?1 ), . . . , (xm , y?m )}
pairs each point in X to some unique point in Y according to the permutation of indices specified
by ? = [?1 , . . . , ?m ], 1 ? ?i ? n, where ?i specifies which point y?i ? Y is matched to xi ? X,
for 1 ? i ? m. The cost
points:
P of a partial matching is the sum of the distances between matched
?
C (M(X, Y; ?)) =
||x
?
y
||
.
The
optimal
partial
matching
M(X,
Y;
?
)
uses
the
i
?
2
i
xi ?X
assignment ? ? that minimizes this cost: ? ? = argmin? C (M(X, Y; ?)). It is this matching that
we wish to efficiently approximate. In Section 3.2 we describe how our algorithm approximates
the cost C (M(X, Y; ? ? )); for a small increase in computational cost we can also extract explicit
correspondences to estimate ? ? itself.
3.1
Building Vocabulary-Guided Pyramids
The first step is to generate the structure of the vocabulary-guided (VG) pyramid to define the bin
placement for the multi-resolution histograms used in the matching. This is a one-time process
performed before any matching takes place. We would like the bins in the pyramid to follow the
feature distribution and concentrate partitions where the features actually fall. To accomplish this,
we perform hierarchical clustering on a sample of representative feature vectors from F .
We randomly select some example feature vectors from the feature type of interest to form the representative feature corpus, and perform hierarchical k-means clustering with the Euclidean distance to
build the pyramid tree. Other hierarchical clustering techniques, such as agglomerative clustering,
are also possible and do not change the operation of the method. For this unsupervised clustering
process there are two parameters: the number of levels in the tree L, and the branching factor k.
The initial corpus of features is clustered into k top-level groups, where group membership is determined by the Voronoi partitioning of the feature corpus according to the k cluster centers. Then the
clustering is repeated recursively L ? 1 times on each of these groups, filling out a tree with L total
levels containing k i bins (nodes) at level i, where levels are counted from the root (i = 0) to the
leaves (i = L ? 1). The bins are irregularly shaped and sized, and their boundaries are determined
by the Voronoi cells surrounding the cluster centers. (See Figure 1.) For each bin in the VG pyramid
we record its diameter, which we estimate empirically based on the maximal inter-feature distance
between any points from the initial feature corpus that were assigned to it.
Once we have constructed a VG pyramid, we can embed point sets from S as multi-resolution
histograms. A point?s placement in the pyramid is determined by comparing it to the appropriate k
bin centers at each of the L pyramid levels. The histogram count is incremented for the bin (among
the k choices) that the point is nearest to in terms of the same distance function used to cluster the
initial corpus. We then push the point down the tree and continue to increment finer level counts
only along the branch (bin center) that is chosen at each level. So a point is first assigned to one of
the top-level clusters, then it is assigned to one of its children, and so on recursively. This amounts
to a total of kL distances that must be computed between a point and the pyramid?s bin centers.
Given the bin structure of the VG pyramid, a point set X is mapped to its pyramid: ? (X) =
[H0 (X), . . . , HL?1 (X)], with Hi (X) = [hp, n, di1 , . . . , hp, n, diki ], and where Hi (X) is a k i dimensional histogram associated with level i in the pyramid, p ? Zi for entries in Hi (X), and
0 ? i < L. Each entry in this histogram is a triple hp, n, di giving the bin index, the bin count, and
the bin?s points? maximal distance to the bin center, respectively.
Storing the VG pyramid itself requires space for O(k L ) d-dimensional feature vectors, i.e., all of
the cluster centers. However, each point set?s histogram is stored sparsely, meaning only O(mL)
nonzero bin counts are maintained to encode the entire pyramid for a set with m features. This is
an important point: we do not store O(k L ) counts for every point set; Hi (X) is represented by at
most m triples having n > 0. We achieve a sparse implementation as follows: each vector in a set is
pushed through the tree as described above. At every level i, we record a hp, n, di triple describing
the nonzero entry for the current bin. The vector p = [p1 , . . . , pi ], pj ? [1, k] denotes the indices
of the clusters traversed from the root so far, n ? Z+ denotes the count for the bin (initially 1),
and d ? < denotes the distance computed between the inserted point and the current bin?s center.
Upon reaching the leaf level, p is an L-dimensional path-vector indicating which of the k bins were
chosen at each level, and every path-vector uniquely identifies some bin on the pyramid.
Initially, an input set with m features yields a total of mL such triples?there is one nonzero entry
per level per point, and each has n = 1. Then each of the L lists of entries is sorted by the index
vectors (p in the triple), and they are collapsed to a list of sorted nonzero entries with unique indices:
when two or more entries with the same index are found, they are replaced with a single entry with
the same index for p, the summed counts for n, and the maximum distance for d. The sorting is done
in linear time using integer sorting algorithms. Maintaining the maximum distance of any point in a
bin to the bin center will allow us to efficiently estimate inter-point distances at the time of matching,
as described in Section 3.2.
3.2 Vocabulary-Guided Pyramid Match
Given two point sets? pyramid encodings, we efficiently compute the approximate matching score
using a simple weighted intersection measure. The VG pyramid?s multi-resolution partitioning of
the feature space is used to direct the matching. The basic intuition is to start collecting groups of
matched points from the bottom of the pyramid up, i.e., from within increasingly larger partitions.
In this way, we will first consider matching the closest points (at the leaves), and as we climb to
the higher-level clusters in the pyramid we will allow increasingly further points to be matched. We
define the number of new matches within a bin to be a count of the minimum number of points either
of the two input sets contributes to that bin, minus the number of matches already counted by any of
its child bins. A weighted sum of these counts yields an approximate matching score.
Let nij (X) denote the element n from hp, n, dij , the j th bin entry of histogram Hi (X), and let
ch (nij (X)) denote the element n for the hth child bin of that entry, 1 ? h ? k. Similarly, let
dij (X) refer to the element d from the same triple. Given point sets X and Y, we compute the
matching score via their pyramids ?(X) and ?(Y) as follows:
C (?(X), ?(Y)) =
L?1
ki
XX
i=0 j=1
"
wij min (nij (X), nij (Y)) ?
k
X
h=1
#
min (ch (nij (X)) , ch (nij (Y))) .
(1)
The outer sum loops over the levels in the pyramids; the second sum loops over the bins at a given
level, and the innermost sum loops over the children of a given bin. The first min term reflects
the number of matchable points in the current bin, and the second min term tallies the number of
matches already counted at finer resolutions (in child bins). Note that as the leaf nodes have no
children, when i = L ? 1 the last sum is zero. All matches are new at the leaves. The matching
scores are normalized according to the size of the input sets in order to not favor larger sets.
The number of new matches calculated for a bin is weighted by wij , an estimate of the distance
between points contained in the bin.1 With a VG pyramid match there are two alternatives for the
distance estimate: (a) weights based on the diameters of the pyramid?s bins, or (b) input-dependent
weights based on the maximal distances of the points in the bin to its center. Option (a) is a conservative estimate of the actual inter-point distances in the bin if the corpus of features used to build
the pyramid is representative of the feature space; its advantages are that it provides a guaranteed
Mercer kernel (see below) and eliminates the need to store a distance d in the entry triples. Option
(b)?s input-specific weights estimate the distance between any two points in the bin as the sum of the
stored maximal to-center distances from either input set: wij = dij (X) + dij (Y). This weighting
1
To use our matching as a cost function, weights are set as the distance estimates; to use as a similarity
measure or kernel, weights are set as (some function of) the inverse of the distance estimates.
gives a true upper bound on the furthest any two points could be from one another, and it has the potential to provide tighter estimates of inter-feature distances (as we confirm experimentally below);
however, we cannot guarantee this weighting will yield a Mercer kernel.
Just as we encode the pyramids sparsely, we derive a means to compute intersections in Eqn. 1
without ever traversing the entire pyramid tree. Given two sparse lists Hi (X) and Hi (Y) which have
been sorted according to the bin indices, we obtain the minimum counts in linear time by moving
pointers down the lists and processing only those nonzero entries that share an index, making the
time required to compute a matching between two pyramids O(mL). A key aspect of our method is
that we obtain a measure of matching quality between two point sets without computing pair-wise
distances between their features?an O(m2 ) savings over sub-optimal greedy matchings. Instead,
we exploit the fact that the points? placement in the pyramid reflects their distance from one another.
The only inter-feature distances computed are the kL distances needed to insert a point into the
pyramid, and this small one-time cost is amortized every time we re-use a histogram to approximate
another matching against a different point set.
We first suggested the idea of using histogram intersection to count implicit matches in a multiresolution grid in [7]. However, in [7], bins are constructed to uniformly partition the space, bin
diameters exponentially increase over the levels, and intersections are weighted indistinguishably
across an entire level. In contrast, here we have developed a pyramid embedding that partitions
according to the distribution of features, and weighting schemes that allow more precise approximations of the inter-feature costs. As we will show in Section 4, our VG pyramid match remains accurate and efficient even for high-dimensional feature spaces, while the uniform-bin pyramid match is
limited in practice to relatively low-dimensional features.
For the increased accuracy our method provides, there are some complexity trade-offs versus [7],
which does not require computing any distances to place the points into bins; their uniform shape
and size allows points to be placed directly via division by bin size. On the other hand, sorting the
bin indices with the VG method has a lower complexity, since the values only range to k, the branch
factor, which is typically much smaller than the aspect ratio that bounds the range in [7]. In addition,
as we show in Section 4, in practice the cost of extracting an explicit correspondence field using the
uniform-bin pyramid in high dimensions approaches the cubic cost of the optimal measure, whereas
it remains linear with the proposed approach, assuming features are not uniformly distributed.
Our approximation can be used to compare sets of vectors in any case where the presence of lowcost correspondences indicates their similarity (e.g., nearest-neighbor retrieval). We can also employ
the measure as a kernel function for structured inputs. According to Mercer?s theorem, a kernel is
p.s.d if and only if it corresponds to an inner product in some feature space [15]. We can re-write
PL?1 Pki
Eqn. 1 as: C (?(X), ?(Y)) =
i=0
j=1 (wij ? pij ) min (nij (X), nij (Y)), where pij refers
to the weight associated with the parent bin of the j th node at level i. Since the min operation is
p.d. [14], and since kernels are closed under summation and scaling by a positive constant [15], we
have that the VG pyramid match is a Mercer kernel if wij ? pij . This inequality holds if every
child bin receives a similarity weight that is greater than its parent bin, or rather that every child
bin has a distance estimate that is less than that of its parent. Indeed this is the case for weighting
option (a), where wij is inversely proportional to the diameter of the bin. It holds by definition of the
hierarchical clustering: the diameter of a subset of points must be less than or equal to the diameter
of all those points. We cannot make this guarantee for weighting option (b).
In addition to scalar matching scores, we can optionally extract explicit correspondence fields
through the pyramid. In this case, the VG pyramid decomposes the required matching computation into a hierarchy of smaller matchings. Upon encountering a bin with a nonzero intersection,
the optimal matching is computed between only those features from the two sets that fall into that
particular bin. All points that are used in that per-bin matching are then flagged as matched and may
not take part in subsequent matchings at coarser resolutions of the pyramid.
4
Results
In this section, we provide results to empirically demonstrate our matching?s accuracy and efficiency
on real data, and we compare it to a pyramid match using a uniform partitioning of the feature
space. In addition to directly evaluating the matching scores and correspondence fields, we show
that our method leads to improved object recognition performance when used as a kernel within a
discriminative classifier.
Rank correlations, d=8(R=0.86)
0.95
9000
8000
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0.8
0.75
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Uniform bin pyramid match ranks
Rank correlations, d=8(R=0.92)
Rank correlations, d=128(R=0.95)
10000
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Uniform bin pyramid match ranks
Optimal ranks
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Optimal ranks
Spearman rank correlation
with optimal match (R)
1
Optimal ranks
Uniform bin pyramid
VG pyramid ? input?specific weights
VG pyramid ? global weights
10000
9000
Optimal ranks
Ranking quality over feature dimensions
1.05
Rank correlations, d=128(R=0.78)
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VG pyramid match (input?specific weights) ranks VG pyramid match (input?specific weights) ranks
Figure 2: Comparison of optimal and approximate matching rankings on image data. Left: The set rankings
produced with the VG pyramid match are consistently accurate for increasing feature dimensions, while the
accuracy with uniform bins degrades about linearly in the feature dimension. Right: Example rankings for both
approximations at d = [8, 128].
Approximate Matching Scores: In these experiments, we extracted local SIFT [11] features from
images in the ETH-80 database, producing an unordered set of about m = 256 vectors for every
example. In this case, F is the space of SIFT image features. We sampled some features from 300 of
the images to build the VG pyramid, and 100 images were used to test the matching. In order to test
across varying feature dimensions, we also used some training features to establish a PCA subspace
that was used to project features onto varying numbers of bases. For each feature dimension, we
built a VG pyramid with k = 10 and L = 5, encoded the 100 point sets as pyramids, and computed
the pair-wise matching scores with both our method and the optimal least-cost matching.
If our measure is approximating the optimal matching well, we should find the ranking we induce
to be highly correlated with the ranking produced by the optimal matching for the same data. In
other words, the images should be sorted similarly by either method. Spearman?s rank correlation
PN
coefficient R provides a good quantitative measure to evaluate this: R = 1 ? 6 1 D2 /N (N 2 ? 1),
where D is the difference in rank for the N corresponding ordinal values assigned by the two measures. The left plot in Figure 2 shows the Spearman correlation scores against the optimal measure
for both our method (with both weighting options) and the approximation in [7] for varying feature
dimensions for the 10,000 pair-wise matching scores for the 100 test sets. Due to the randomized
elements of the algorithms, for each method we have plotted the mean and standard deviation of the
correlation for 10 runs on the same data.
While the VG pyramid match remains consistently accurate for high feature dimensions (R = 0.95
with input-specific weights), the accuracy of the uniform bins degrades rapidly for dimensions
over 10. The ranking quality of the input-specific weighting scheme (blue diamonds) is somewhat
stronger than that of the ?global? bin diameter weighting scheme (green squares). The four plots
on the right of Figure 2 display the actual ranks computed for both approximations for two of the
26 dimensions summarized in the left plot. The black diagonals denote the optimal performance,
where the approximate rankings would be identical to the optimal ones; higher Spearman correlations have points clustered more tightly along this diagonal. For the low-dimensional features, the
methods perform fairly comparably; however, for the full 128-D features, the VG pyramid match
is far superior (rightmost column). The optimal measure requires about 1.25s per match, while our
approximation is about 2500x faster at 5 ? 10?4 s per match. Computing the pyramid structure from
the feature corpus took about three minutes in Matlab; this is a one-time offline cost.
For a pyramid matching to work well, the gradation in bin sizes up the pyramid must be such
that at most levels of the pyramid we can capture distinct groups of points to match within the
bins. That is, unless all the points in two sets are equidistant, the bin placement must allow us to
match very near points at the finest resolutions, and gradually add matches that are more distant
at coarser resolutions. In low dimensions, both uniform or data-dependent bins can achieve this.
In high dimensions, however, uniform bin placement and exponentially increasing bin diameters
fail to capture such a gradation: once any features from different point sets are close enough to
d=8
150
100
50
0
?50
0
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4
6
250
150
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Pyramid level
4
d = 128
d = 13
Vocabulary?guided bins
Uniform bins
200
6
8
Number of new matches formed
200
Number of new matches formed
Number of new matches formed
Number of new matches formed
d=3
Vocabulary?guided bins
Uniform bins
250
Vocabulary?guided bins
Uniform bins
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6
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Pyramid level
Figure 3: Number of new matches formed at each pyramid level for either uniform (dashed red) or VG (solid
blue) bins for increasing feature dimensions. Points represent mean counts per level for 10,000 matches. In low
dimensions, both partition styles gradually collect matches up the pyramid. In high dimensions with uniform
partitions, points begin sharing a bin ?all at once?; in contrast, the VG bins still accrue new matches consistently
across levels since the decomposition is tailored to where points cluster in the feature space.
x 10
Error in approximate correspondence fields
Mean error per match
1.4
1.2
1
0.8
Uniform bins, random per
Uniform bins, optimal per
Vocab.?guided bins, random per
Vocab.?guided bins, optimal per
0.6
0.4
0
20
40
60
80
Feature dimension (d )
100
120
Computation time
Mean time per match (s) (LOG SCALE)
5
1.6
Optimal
Uniform bins, random per
Uniform bins, optimal per
Vocab.?guided bins, random per
Vocab.?guided bins, optimal per
1
10
0
10
?1
10
?2
10
?3
10
0
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Feature dimension (d )
Figure 4: Comparison of correspondence field errors (left) and associated computation times (right). This
figure is best viewed in color. (Note that errors level out with d for all methods due to PCA.)
match (share bins), the bins are so large that almost all of them match. The matching score is then
approximately the number of points weighted by a single bin size. In contrast, when we tailor the
feature space partitions to the distribution of the data, even in high dimensions the match counts
increase gradually across levels, thereby yielding more discriminating implicit matches. Figure 3
confirms this intuition, again using the ETH-80 image data from above.
Approximate Correspondence Fields: For the same image data, we ran the explicit matching
variant of our method and compared the induced correspondences to those produced by the globally
optimal measure. For comparison, we also applied the same variant to pyramids with uniform bins.
We measure the error of an approximate
? by the sum of the errors at every link in the field:
Pmatching ?
E (M (X, Y; ?
? ) , M (X, Y; ? ? )) = xi ?X ||y??i ? y?i? ||2 . Figure 4 compares the correspondence
field error and computation times for the VG and uniform pyramids. For each approximation, there
are two variations tested: in one, an optimal assignment is computed for all points in the same bin;
for the other, a random assignment is made. The left plot shows the mean error per match for each
method, and the right plot shows the corresponding mean time required to compute those matches.
The computation times are as we would expect: the optimal matching is orders of magnitude more
expensive than the approximations. Using the random assignment variation, both approximations
have negligible costs, since they simply choose any combination of points within a bin. However, in
high dimensions, the time required by the uniform bin pyramid with the optimal per-bin matching
approaches the time required by the optimal matching itself. This occurs for similar reasons as the
poorer matching score accuracy exhibited by the uniform bins, both in the left plot and above in
Figure 2; since most or all of the points begin to match at a certain level, the pyramid does not help
to divide-and-conquer the computation, and for high dimensions, the optimal matching in its entirety
must be computed. In contrast, the expense of the VG pyramid matching remains steady and low,
even for high dimensions, since data-dependent pyramids better divide the matching labor into the
natural segments in the feature space.
For similar reasons, the errors are comparable for the optimal per-bin variation with either the VG
or uniform bins. The VG bins divide the computation so it can be done inexpensively, while the
uniform bins divide the computation poorly and must compute it expensively, but about as accurately. Likewise, the error for the uniform bins when using a per-bin random assignment is very
high for any but the lowest dimensions (red line on left plot), since such a large number of points
are being randomly assigned to one another. In contrast, the VG bins actually result in similar errors
whether the points in a bin are matched optimally or randomly (blue and pink lines on left plot).
Pyramid matching method
Vocabulary-guided bins
Uniform bins
Mean recognition rate/class (d=128 / d=10)
99.0 / 97.7
64.9 / 96.5
Time/match (s) (d=128 / d=10)
6.1e-4 / 6.2e-4
1.5e-3 / 5.7e-4
This again indicates that tuning the pyramid bins to the data?s distribution achieves a much more
suitable breakdown of the computation, even in high dimensions.
Realizing Improvements in Recognition: Finally, we have experimented with the VG pyramid
match within a discriminative classifier for an object recognition task. We trained an SVM with
our matching as the kernel to recognize the four categories in the Caltech-4 benchmark data set.
We trained with 200 images per class and tested with all the remaining images. We extracted features using both the Harris and MSER [12] detectors and the 128-D SIFT [11] descriptor. We also
generated lower-dimensional (d = 10) features using PCA. To form a Mercer kernel, the weights
were set according to each bin diameter Aij : wij = e?Aij /? , with ? set automatically as the mean
distance between a sample of features from the training set. The table shows our improvements
over the uniform-bin pyramid match kernel. The VG pyramid match is more accurate and requires
minor additional computation. Our near-perfect performance on this data set is comparable to that
reached by others in the literature; the real significance of the result is that it distinguishes what
can be achieved with a VG pyramid embedding as opposed to the uniform histograms used in [7],
particularly for high-dimensional features. In addition, here the optimal matching requires 0.31s per
match, over 500x the cost of our method.
Conclusion: We have introduced a linear-time method to compute a matching between point sets
that takes advantage of the underlying structure in the feature space and remains consistently accurate and efficient for high-dimensional inputs on real image data. Our results demonstrate the
strength of the approximation empirically, compare it directly against an alternative state-of-the-art
approximation, and successfully use it as a Mercer kernel for an object recognition task. We have
commented most on potential applications in vision and text, but in fact it is a generic matching
measure that can be applied whenever it is meaningful to compare sets by their correspondence.
Acknowledgments: We thank Ben Kuipers for suggesting the use of Spearman?s rank correlation.
References
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[5] A. Gersho and R. Gray. Vector Quantization and Signal Compression. Springer, 1992.
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[12] J. Matas, O. Chum, M. Urban, and T. Pajdla. Robust Wide Baseline Stereo from Maximally Stable
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[14] F. Odone, A. Barla, and A. Verri. Building Kernels from Binary Strings for Image Matching. IEEE Trans.
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[15] J. Shawe-Taylor and N. Cristianini. Kernel Methods for Pattern Analysis. Cambridge Univ. Press, 2004.
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Proc. IEEE Int. Conf. on Computer Vision, Nice, Oct 2003.
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2,239 | 3,031 | Predicting spike times from subthreshold dynamics of
a neuron
Ryota Kobayashi
Department of Physics
Kyoto University
Kyoto 606-8502, Japan
[email protected]
Shigeru Shinomoto
Department of Physics
Kyoto University
Kyoto 606-8502, Japan
[email protected]
Abstract
It has been established that a neuron reproduces highly precise spike response to
identical ?uctuating input currents. We wish to accurately predict the ?ring times
of a given neuron for any input current. For this purpose we adopt a model that
mimics the dynamics of the membrane potential, and then take a cue from its dynamics for predicting the spike occurrence for a novel input current. It is found
that the prediction is signi?cantly improved by observing the state space of the
membrane potential and its time derivative(s) in advance of a possible spike, in
comparison to simply thresholding an instantaneous value of the estimated potential.
1
Introduction
Since Hodgkin and Huxley [1] described the ionic ?ux across the neuronal membrane with four
nonlinear differential equations more than half a century ago, continuous efforts have been made
either to extract an essence of the nonlinear dynamical aspect by simplifying the model, or construct
ever more realistic models by including more ionic channels in the model.
In the simpli?cation proposed by FitzHugh [2] and Nagumo et al [3], the number of equations is
reduced to two: the fast and slow variables which minimally represent the excitable dynamics. The
leaky integrate-and-?re model [4], originally proposed far in advance of the Hodgkin-Huxley model,
consists of only one variable that corresponds to the membrane potential, with a voltage resetting
mechanism. Those simpli?ed models have been successful in not only extracting the essence of the
dynamics, but also in reducing the computational cost of studying the large-scale dynamics of an
assembly of neurons.
In contrast to such taste for simpli?cation, there are also a number of studies that pursue realism
by developing multi-compartment models and installing newly found ionic channels. User-friendly
simulation platforms, such as NEURON [5] and GENESIS [6], enable experimental neurophysiologists to reproduce casually their experimental results or to explore potentially interesting phenomena
for a new experiment to be performed.
Though those simulators have been successful in reproducing qualitative aspects of neuronal responses to various conditions, quantitative reproduction as well as prediction for novel experiments
appears to be dif?cult to realize [7]. The dif?culty is due to the complexity of the model accompanied with a large number of undetermined free parameters. Even if a true model of a particular
neuron is included in the family of models, it is practically dif?cult to explore the true parameters in
the high-dimensional space of parameters that dominate the nonlinear dynamics.
Recently it was suggested by Kistler et al [8, 9] to extend the leaky integrate-and-?re model so that
real membrane dynamics of any neuron can be adopted. The so called ?spike response model? has
been successful in not only reproducing the data but also in predicting the spike timing for a novel
input current [8, 9, 10, 11]. The details of an integration kernel are learned easily from the sample
data. The fairly precise prediction achieved by such a simple model indicates that the spike occurrence is determined principally by the subthreshold dynamics. In other words, the highly nonlinear
dynamics of a neuron can be decomposed into two simple, predictable processes: a relatively simple
subthreshold dynamics, and the dynamics of an action potential of a nearly ?xed shape (Fig.1).
action potential
(nearly fixed)
V
subthreshold
(predictable)
dV/dt
Figure 1: The highly nonlinear dynamics of a neuron is decomposed into two simple, predictable
processes.
In this paper, we propose a framework of improving the prediction of spike times by paying close
attention to the transfer between the two predictable processes mentioned above. It is assumed in
the original spike response model that a spike occurs if the membrane potential exceeds a certain
threshold [9]. We revised this rule to maximally utilize the information of a higher-dimensional state
space, consisting of not only the instantaneous membrane potential, but also its time derivative(s).
Such a subthreshold state can provide cues for the occurrence of a spike, but with a certain time
difference. For the purpose of exploring the optimal time shift, we propose a method of maximizing
the mutual information between the subthreshold state and the occurrence of a spike.
By employing the linear ?lter model [12] and the spike response model [9] for mimicking the subthreshold voltage response of a neuron, we examine how much the present framework may improve
the prediction for simulation data of the fast-spiking model [13].
2
Methods
The response of a neuron is precisely reproduced when presented with identical ?uctuating input
currents [14]. This implies that the neuronal membrane potential V (t) is determined by the past
input current {I(t)}, or
V (t) = F ({I(t)}),
(1)
where F ({I(t)}) represents a functional of a time-dependent current I(t). A rapid swing in the
polarity of the membrane potential is called a ?spike.? The occurrence of a spike could be de?ned
practically by measuring the membrane potential V (t) exceeding a certain threshold,
V (t) > Vth .
(2)
The time of each spike could be de?ned either as the ?rst time the threshold is exceeded, or as the
peak of the action potential that follows the crossing.
Kistler et al [8] and Jolivet et al [10, 11] proposed a method of mimicking the membrane dynamics of
a target neuron with the simple spike response model in which an input current is linearly integrated.
The leaky integrate-and-?re model can be regarded as an example of the spike response model [9];
the differential equation can be rewritten as an integral equation in which the membrane potential
is given as the integral of the past input current with an exponentially decaying kernel. The spike
response model is an extension of the leaky integrate-and-?re model, where the integrating kernel is
adaptively determined by the data, and the after hyperpolarizing potential is added subsequently to
every spike. It is also possible to further include terms that reduce the responsiveness and increase
the threshold after an action potential takes place.
Even in the learning stage, no model is able to perfectly reproduce the output V (t) of a target neuron
for a given input I(t). We will denote the output of the model (in the lower case) as
v(t) = fk ({I(t)}),
(3)
where k represents a set of model parameters. The model parameters are learned by mimicking
sample input-output data. This is achieved by minimizing the integrated square error,
?
2
Ek = (V (t) ? v(t)) dt.
(4)
2.1
State space method
As the output of the model v(t) is not identical to the true membrane potential of the target neuron
V (t), a spike occurrence cannot be determined accurately by simply applying the same threshold
rule Eq.(2) to v(t). In this paper, we suggest revising the spike generation rule so that a spike
occurrence is best predicted from the model potential v(t).
Suppose that we have adjusted the parameters of fk ({I(t)}) so that the output of the model {v(t)}
best approximates the membrane potential {V (t)} of a target neuron for a given set of currents
{I(t)}. If the sample data set {I(t), V (t)} employed in learning is large enough, the spike occurrence can be predicted by estimating an empirical probability of a spike being generated at the time
t, given a time-dependent orbit of an estimated output, {v(t)}, as
pspike (t|{v(t)}).
(5)
In a practical experiment, however, the amount of collectable data is insuf?cient for estimating the
spiking probability with respect to any orbit of v(t). In place of such exhaustive examination, we
suggest utilizing the state space information such as the time derivatives of the model potential at a
certain time. The spike occurrence at time t could be predicted from the m-dimensional state space
information ?v ? (v, v ? , ? ? ? , v (m?1) ), as observed at a time s before t, as
pspike (t|?vt?s ),
(6)
where ?vt?s ? (v(t ? s), v ? (t ? s), ? ? ? , v (m?1) (t ? s)).
2.2
Determination of the optimal time shift
The time shift s introduced in the spike time prediction, Eq.(6), is chosen to make the prediction
more reliable. We propose optimizing the time shift s by maximizing the mutual information between the state space information ?vt?s and the presence or absence of a spike at a time interval
(t ? ?t/2, t + ?t/2], which is denoted as zt = 1 or 0. The mutual information [15] is given as
M I(zt ; ?vt?s ) = M I(?vt?s ; zt ) = H(?vt?s ) ? H(?vt?s |zt ),
where
(7)
?
H(?vt?s )
H(?vt?s |zt )
= ?
d?vt?s p(?vt?s ) log p(?vt?s ),
? ?
= ?
d?vt?s p(?vt?s |zt )p(zt ) log p(?vt?s |zt ).
(8)
(9)
zt ?{0,1}
Here, p(?vt?s |zt ) is the probability, given the presence or absence of a spike at time t ? (t??t/2, t+
?t/2], of the state being ?vt?s , a time s before the spike.
With the time difference s optimized, we then obtain the empirical probability of the spike occurrence at the time t, given the state space information at the time t ? s, using the Bayes theorem,
pspike (t|?vt?s ) ? p(zt = 1|?vt?s ) =
3
p(?vt?s |zt )p(zt )
.
p(?vt?s )
(10)
Results
We evaluated our state space method of predicting spike times by applying it to target data obtained
for a fast-spiking neuron model proposed by Erisir et al [13] (see Appendix). In this virtual experiment, two ?uctuating currents characterized by the same mean and standard deviation are injected
to the (model) fast-spiking neuron to obtain two sets of input-output data {I(t), V (t)}. A predictive model was trained using one sample data set, and then its predictive performance for the other
sample data was evaluated.
Each input current is generated by the Ornstein-Uhlenbeck process. We have tested two kinds of
?uctuating currents characterized with different means and standard deviations: (Currents I) the
mean ? = 1.5 [?A], the standard deviation ? = 1.0 [?A] and the time scale of the ?uctuation ? = 2
[msec]; (Currents II) the mean ? = 0.0 [?A], the standard deviation ? = 4.0 [?A] and the time
scale of the ?uctuation ? = 2 [msec]. For each set with these statistics, we derived two independent
sequences of I(t). In this study we adopted the linear ?lter model and the spike response model as
prediction models.
A
B
0.03
2
K(t)
1
MI
0.02
0.01
0
0
20
t (ms)
40
0
0
4
s (ms)
8
Figure 2: A: The estimated kernel. B: The mutual information between the estimated potential and
the occurrence of a spike.
We brie?y describe here the results for the linear ?lter model [12],
? ?
v(t) =
K(t? )I(t ? t? ) dt? + v0 .
(11)
0
The model parameters k consist of the shape of the kernel K(t) and the constant v0 . In learning
the target sample data {I(t), V (t)}, these parameters are adjusted to minimize the integrated square
error, Eq.(4). Figure 2A depicts the shape of the kernel K(t) estimated from the target sample data
{I(t), V (t)} obtained from the virtual experiment of the fast-spiking neuron model.
Based on the voltage estimation v(t) with respect to sample data, we compute the empirical
probabilities, p(?vt?s ), p(?vt?s |zt ) and p(zt ) for two-dimensional state space information ?vt?s ?
(v(t ? s), v ? (t ? s)). In computing empirical probabilities, we quantized the two-dimensional phase
space ?v ? (v, v ? ), and the time. In the discretized time, we counted the occurrence of a spike,
zt = 1, for the bins in which the true membrane potential V (t) exceeds a reasonable threshold Vth .
With a suf?ciently small time step (we adopted ?t = 0.1 [msec]), a single spike is transformed into
a succession of spike occurrences zt = 1. The mutual information computed according to Eq.(7) is
depicted in Fig. 2B whose maximum position of s ? 2 [msec] determines the optimal time shift.
The spike is predicted if the estimated probability pspike (t|?vt?s ) of Eq.(10) exceeds a certain threshold value. Though it would be more ef?cient to use the systematic method suggested by Paninski et
al [16], we determined the threshold value empirically so that the coincidence factor ? described in
the following is maximized.
Figure 3 compares a naive thresholding method and our state space method, in reference to the
original spike times. It is observed from this ?gure that the prediction of the state space method is
more accurate and robust than that of thresholding method.
50
V(t)
0
50
v(t)
0
pspike(t)
3000
3500
t (ms)
Figure 3: Comparison of the spike time predictions. (Top): The target membrane potential V (t).
(Middle): Prediction by thresholding the model potential. (Bottom): Prediction by the present state
space method. Vertical arrows represent the predicted spikes.
Figure 4 depicts an orbit in the state space of (V, V ? ) of a target neuron for an instance of the spike
generation, and the orbit of the predictive model in the state space of (v, v ? ) that mimics it. The
predictive model can mimic the target orbit in the subthreshold region, but fails to catch the spiking
orbit in the suprathreshold region. The spike occurrence is predicted by estimating the conditional
probability, Eq.(10), given the state (v, v ? ) of the predictive model.
Contour lines of the probability in Figure 4C are not parallel to the dv/dt-axis. The contour lines for
higher probabilities of spiking resemble an ad hoc ?dynamic spike threshold? introduced by Azouz
and Gray [17]. Namely, v drops with dv/dt along the contour lines. Contrastingly, the contour lines
for lower probabilities of spiking are inversely curved: v increases with dv/dt along the contour
lines. In the present framework, the state space information corresponding to the relatively low
probability of spiking is effectively used for predicting spike times.
Prediction performance is compared with the benchmark of Kistler et al [8], the ?coincidence factor,?
?(?) =
Ncoinc ? ?Ncoinc ?
1
2 (Ndata + Nmodel )
1
,
1 ? 2??
(12)
where Ndata and Nmodel respectively represent the numbers of spikes in the original data and prediction model, Ncoinc is the number of coincident spikes with the precision of ?, ?Ncoinc ? = 2??Ndata
is the expected number of coincidences of the data and the Poisson spikes with rate ?. ? is chosen
as 2 [msec] in accordance with Jolivet et al [10].
Table 1: The coincidence factors evaluated for two methods of prediction based on the linear ?lter
model.
method
Currents I Currents II
thresholding
0.272
0.567
state space
0.430
0.666
B
C
v
V
c
14
14
b
A
c
12
12
10
10
8
8
b
V
120
80
40
a
a
6
0
-500 0 500
dV/dt
6
-2
0
dV/dt
2
-2
0
dv/dt
2
Figure 4: A: An orbit in the state space of (V, V ? ) of a target neuron for an instance of the spike
generation (from 3240 to 3270 [msec] of Fig. 3). B: magni?ed view. C: The orbit in the state space
of (v, v ? ) of the predictive model that mimics the target neuron. Contours represent the probability of
spike occurrence computed with the Bayes formula, Eq.(10).The dashed lines represent the threshold
adopted in the naive thresholding method (Fig 3 Middle). Three points a, b, and c in the spaces of
(V, V ? ) and (v, v ? ) represent the states of identical times, respectively, t = 3242, 3252 and 3253
[msec].
Table 2: The coincidence factors evaluated for two methods of prediction based on the spike response model.
method
thresholding
state space
Currents I
0.501
0.641
Currents II
0.805
0.842
The coincidence factors evaluated for a simple thresholding method and the state space method
based on the linear ?lter model are summarized in Table 1, and those based on the spike response
model are summarized in Table 2. It is observed that the prediction is signi?cantly improved by our
state space method.
It should be noted, however, that a model with the same set of parameters does not perform well
over a range of inputs generated with different mean and variance: The model parameterized with
the Currents I does not effectively predict the spikes of the neuron for the Current II, and vice
versa. Nevertheless, our state-space method exhibits the better prediction than the naive thresholding
strategy, if the statistics of the different inputs are relatively similar.
4
Summary
We proposed a method of evaluating the probability of the spike occurrence by observing the state
space of the membrane potential and its time derivative(s) in advance of the possible spike time.
It is found that the prediction is signi?cantly improved by the state space method compared to the
prediction obtained by simply thresholding an instantaneous value of the estimated potential. It is
interesting to apply our method to biological data and categorize neurons based on their spiking
mechanisms.
The state space method developed here is a rather general framework that may be applicable to
any nonlinear phenomena composed of locally predictable dynamics. The generalization of linear ?lter analysis developed here has a certain similarity to the Linear-Nonlinear-Poisson (LNP)
model [18, 19]. It would be interesting to generalize the present method of analysis to a wider range
of phenomena such as the analysis of the coding of visual system [19, 20].
Acknowledgments
This study is supported in part by Grants-in-Aid for Scienti?c Research to SS from the Ministry of
Education, Culture, Sports, Science and Technology of Japan (16300068, 18020015) and the 21st
Century COE ?Center for Diversity and Universality in Physics? and to RK from Foundation For C
& C Promotion.
Appendix: Fast-spiking neuron model
The fast-spiking neuron model proposed by Erisir et al [13] was used in this contribution as a (virtual) target experiment. The details of the model were adjusted to Jolivet et al [10] to allow the direct
comparison of the performances. Speci?cally, the model is described as
du(t)
C
= ?[INa + IK1 + IK2 + IL ] + I ext (t) ,
(13)
dt
INa = gNa m3 h(u ? ENa ) ,
(14)
IK1 = gK1 n41 (u ? EK ) ,
IL = gL (u ? EL ) ,
IK2 = gK2 n22 (u ? EK ) ,
where the gate variables x ? n1 , n2 , m and h obey the differential equations of the form,
dx
= ?x (u)(1 ? x) ? ?x (u)x,
dt
whose parameters ?x (u) and ?x (u) are functions of u, as listed in Table 3.
(15)
(16)
(17)
Table 3: The parameters for the fast-spiking model. The membrane capacity is C = 1.0[?F/cm2 ].
Channel
Variable
?
?
gx (mS/cm2 )
Ex (mV)
Na
m
?3020+40u
1?exp(? u?75.5
13.5 )
1.2262
u
exp( 42.248
)
112.5
74
??
h
0.0035
u
exp( 24.186
)
0.8712+0.017u
1?exp(? 51.25+u
)
5.2
??
??
K1
n1
0.014(44+u)
1?exp(? 44+u
2.3 )
0.0043
exp( 44+u
34 )
0.225
?90.0
K2
n2
u?95
1?exp(? u?95
11.8 )
0.025
u
exp( 22.22
)
225.0
?90.0
L
??
??
??
0.25
?70
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corresponds:1 determines:1 lewis:1 hines:1 ma:1 conditional:1 leonard:1 installing:1 absence:2 included:1 determined:5 neurophysiologists:1 reducing:1 called:2 experimental:2 m3:1 uzzell:1 categorize:1 tested:1 phenomenon:3 ex:1 |
2,240 | 3,032 | Learning Time-Intensity Profiles of Human Activity
using Non-Parametric Bayesian Models
Alexander T. Ihler
Padhraic Smyth
Donald Bren School of Information and Computer Science
U.C. Irvine
[email protected]
[email protected]
Abstract
Data sets that characterize human activity over time through collections of timestamped events or counts are of increasing interest in application areas as humancomputer interaction, video surveillance, and Web data analysis. We propose a
non-parametric Bayesian framework for modeling collections of such data. In
particular, we use a Dirichlet process framework for learning a set of intensity
functions corresponding to different categories, which form a basis set for representing individual time-periods (e.g., several days) depending on which categories
the time-periods are assigned to. This allows the model to learn in a data-driven
fashion what ?factors? are generating the observations on a particular day, including (for example) weekday versus weekend effects or day-specific effects corresponding to unique (single-day) occurrences of unusual behavior, sharing information where appropriate to obtain improved estimates of the behavior associated
with each category. Applications to real?world data sets of count data involving
both vehicles and people are used to illustrate the technique.
1
Introduction
As sensor and storage technologies continue to improve in terms of both cost and performance,
increasingly rich data sets are becoming available that characterize the rhythms of human activity
over time. Examples include logs of radio frequency identification (RFID) tags, freeway traffic
over time (loop-sensor data), crime statistics, email and Web access logs, and many more. Such
data can be used to support a variety of different applications, such as classification of human or
animal activities, detection of unusual events, or to support the broad understanding of behavior in
a particular context such as the temporal patterns of Web usage.
To ground the discussion, consider data consisting of a collection of individual or aggregated events
from a single sensor, e.g., a time-stamped log recording every entry and exit from a building, or the
timing and number of highway traffic accidents. For example, Figure 1 shows several days worth of
data from a building log, smoothed so that the similarities in patterns are more readily visible.
Of interest is the modeling of the underlying intensity of the process generating the data, where
intensity here refers to the rate at which events occur. These processes are typically inhomogeneous
in time (as in Figure 1), as they arise from the aggregated behavior of individuals, and thus exhibit
a temporal dependence linked to the rhythms of the underlying human activity. The complexity
of this temporal dependence is application-dependent and generally unknown before observing the
data, suggesting that non- or semi-parametric methods (methods whose complexity is capable of
growing as the number of observations increase) may be particularly appropriate.
Formulating the underlying event generation as an inhomogeneous Poisson process is a common first
step (see, e.g., [1, 4]), as it allows the application of various classic density estimation techniques to
estimate the time-dependent intensity function (a normalized version of the rate function; see Sec-
tion 2). Techniques used in this context include kernel density estimation [2], wavelet analysis [3],
discretization [1], and nonparametric Bayesian models [4, 5].
Among these, nonparametric Bayesian approaches have a number of appealing advan60
tages. First, they allow us to represent and reason about uncertainty in the intensity function,
50
providing not just a single estimate but a distribution over functions. Second, the Bayesian
40
framework provides natural methods for model
selection, allowing the data to be naturally ex30
plained by a parsimonious set of intensity functions, rather than using the most complex expla20
nation (though similar effects may be achieved
using penalized likelihood functions [3]). Fi10
nally, Bayesian methods generalize to multiple
or hierarchical models, which allow informa0
0
6
12
18
24
tion to be shared among several related but differing sets of observations (e.g., multiple days Figure 1: Count data from a building entry log
of data). This second point is crucial for many observed on ten Mondays, each smoothed using a
problems, as we rarely obtain many observa- kernel function [2, 6] to enable visual comparison.
tions of exactly the same process under exactly
the same conditions; instead, we observe multiple instances which are thought to be similar, but may in fact represent any number of slightly
differing circumstances. For example, behavior may be dependent on not only time of day but also
day of week, type of day (weekend or weekday), unobserved factors such as the weather, or other
unusual circumstances. Sharing information allows us to improve our model, but we should only
do so where appropriate (itself best indicated by similarity in the data). By being Bayesian, we can
remain agnostic of what data should be shared and reason over our uncertainty in this structure.
In what follows we propose a non-parametric Bayesian framework for modeling intensity functions
for event data over time. In particular, we describe a Dirichlet process framework for learning the
unknown rate functions, and learn a set of such functions corresponding to different categories. Individual time-periods (e.g., individual days) are then represented as additive combinations of intensity
functions, depending on which categories are assigned to each time-period. This allows the model
to learn in a data-driven fashion what ?factors? are generating the observations on a particular day,
including (for example) weekday versus weekend effects as well as day-specific effects corresponding to unusual behavior present only on a single day. Applications to two real?world data sets, a
building access log and accident statistics, are used to illustrate the technique.
We will discuss in more detail in the sections that follow how our proposed approach is related to
prior work on similar topics. Broadly speaking, from the viewpoint of modeling of inhomogeneous
time-series of counts, our work extends the work of [4] to allow sharing of information among
multiple, related processes (e.g., different days). Our approach can also be viewed as an alternative
to the hierarchical Dirichlet process (HDP, [7]) for problems where the patterns across different
groups are much more constrained than would be expected under an HDP model.
2
Poisson processes
A common model for continuous-time event (counting) data is the Poisson process [8]. As the
discrete Poisson distribution is characterized by a rate parameter ?, the Poisson process1 is characterized by a rate function ?(t); it has the property that over any given
R time interval T , the number of
events occurring within that time is Poisson with rate given by ? = T ?(t). We shall use a Bayesian
semi-parametric model for ?(t), described next.
Let us suppose that we have a single collection of event times {?i } arising from a Poisson process
with rate function ?(t), i.e.,
{?i } ? P[? ; ?(t)]
(1)
1
Here, we shall use the term Poisson process interchangeably with inhomogeneous Poisson process, meaning that the rate is a non-constant function of time t.
R?
where ?(t) is defined on t ? [??, ?]. We may write ?(t) = ?f (t), where ? = ?? ?(t) and f (t)
is the intensity function, a normalized version of the rate function. A Bayesian model places prior
distributions on these quantities; by selecting a parametric prior for ? and a nonparametric prior for
f (t), we obtain a semi-parametric prior for ?(t). Specifically, we choose
Z
? ? ?(a, b)
f (t) = K(t; ?)dG(?)
G ? DP[?G0 ]
where ? is the gamma distribution, K is a kernel function (for example a Gaussian distribution)
and DP is a Dirichlet process [9] with parameter ? and base distribution G0 . The Dirichlet process
provides a nonparametric prior for f (t), such thatP
(with probability one) f has the form of a mixture
model with infinitely many components: f (t) = j wj K(t; ?j ). If desired we may also place prior
distributions on some or all of these quantities (e.g., ?, {a, b}, or the parameters of G0 ) as well.
Dirichlet processes and their variations [7, 9?11] have gained recent attention for their ability to
provide representations consisting of arbitrarily large mixture models. In particular, they have been
the subject of recent work in modeling intensity functions for Poisson processes defined over time [4]
and space?time [5].
2.1
Monte Carlo Inference
For the Poisson process model just described, the likelihood of the data {?i }, i = 1 . . . N at some
time T is given by
? Z
!
T
Y
p({?i }; ?, f (?)) = exp ?
?f (t) ? N
f (?i )
??
i
which, as T ? ? (i.e., as we observe a complete data set) becomes
"
#
?
? Y
N
p({?i }; ?, f (?)) = exp(??)?
f (?i )
(2)
i
The rightmost term (term involving f ) has the same form as the likelihood of the ?i as i.i.d. samples
from the mixture model distribution defined by f . As in many mixture model applications, it will be
helpful to create auxiliary assignment variables zi for each ?i , indicating with which of the mixture
components the sample ?i is associated. The complete data likelihood is then
"
#
Y
?
?
p({?i , zi }; ?, f (?)) = exp(??)? N
wzi K(?i ; ?zi ) .
i
Inference is typically accomplished using Markov chain Monte Carlo (MCMC) sampling [9].
Specifically, although the posterior for ? has a simple closed form, p(?|{?i }) ? ?(N + a, 1 + b),
sampling from f is more complicated. Samples from f can be drawn in a variety of ways. One
of the most common methods is the so-called ?Chinese Restaurant Process? (CRP, [7, 9]), in which
the relative weights wj are marginalized out while drawing the assignment variables zi . Such exact
sampling approaches work by exploiting the fact that only a finite number of the mixture components are occupied by the data; by treating the unoccupied clusters as a single group, the infinite
number of potential associations can be treated as a finite number. The operations involved (such as
sampling values for ?j given a collection of associated event times ?i ) are easier for certain choices
of K and G than others; for example using a Gaussian kernel and normal-Wishart distribution, the
necessary quantities have convenient closed forms [9].
Another, more brute-force way around the issue of having infinitely many mixture components is to
perform approximate sampling using a ?truncated? Dirichlet process representation [12, 13]. As described in [12], for a given ?, data set size N , and tolerance ?, one can compute a maximum number
of components M necessary to approximate the Dirichlet process with a Dirichlet distribution using
the relation
? ? 4 N exp[?(M ? 1)/?]
and in this manner, can work with finite numbers of mixture components. This representation will
prove useful in Section 3.
The truncated DP approximation is helpful primarily because it allows us to sample the (complete)
function f (t) (as compared to only the ?occupied? part in the CRP formulation). Given a set of
assignments {zi } occupying (arbitrarily numbered) clusters 1 . . . J, we can sample the weights wj
in two steps.P
First, we sample the occupied mixture weights, wj (j ? J), and the total unoccupied
?
weight w
? = J+1 wj , by drawing independent, Gamma-distributed random variables according to
?(Nj , 1) and ?(?, 1), respectively, and normalizing them to sum to one. The values of weights wj in
the unoccupied clusters (j > J) can then be sampled given w
? using the stick?breaking representation
of Sethuraman [14].
Note that the truncated DP approximation highlights the importance of also sampling ? if we hope
for our representation to act non-parametric in the sense that it may grow more complex as the data
increase, since for a fixed ? and ? the number of components M is quite insensitive to N . For more
details on sampling such hyper-parameters see e.g. [10].
2.2
Finite Time Domains
Our description of non-parametric Bayesian techniques for Poisson processes has so far made implicit use of the fact that the domain of f (t) is infinite. When the domain of f is finite, for example
[0, 1], a few minor complications arise. For example, the kernel functions K(?) should properly
be defined as positive only on this interval. One possible solution to this issue is to use an alternate
kernel function, such as the Beta distribution [4]. However, this means that posterior sampling of the
parameters ? is no longer possible in closed form. Although methods such as Metropolis-Hastings
may be used [4], they can be highly dependent on the choice of proposal density.
Here, we take a slightly different approach, drawing truncated Gaussian kernels with parameters
sampled from a truncated Normal-Wishart distribution. Specifically, we define
N (t; ?, ? 2 )?1 (?)
K(t; ? = [?, ? 2 ]) = R 1
N (x; ?, ? 2 )dx
0
[?, ? 2 ] ? ?1 (?) ?1 (?) N W(?, ? 2 )
where ?1 (t) is one on [0, 1] and zero elsewhere and N W is the normal-Wishart distribution. Sampling in this model turns out to be relatively simple and efficient using rejection methods. Given the
R1
restrictions imposed on ? and ?, one can show that the normalizing quantity Z = 0 N (x; ?, ? 2 ) is
always greater than one-third. Thus, to sample from the posterior we simply draw from the original,
closed form posterior distribution, discarding (and re-sampling) if ? 6? [0, 1], ? 6? [0, 1], or with
probability 1 ? (3Z)?1 .
3
Categorical Models
As mentioned in the introduction, we often have several collections d = 1 . . . D of observations,
{?di } with i = 1 . . . Nd , arising from D instances of the same or similar processes. If these processes
are known to be identical and independent, sharing information among them is relatively easy?we
obtain D observations Nd with which to estimate ?, and the ?di are collectively used to estimate
f (t). However, if these processes are not necessarily identical, sharing information becomes more
difficult.
Yet it is just this situation which is most typical. Again consider Figure 1, which shows event data
from ten different Mondays. Clearly, there is a great deal of consistency in both size and shape,
although not every day is exactly the same, and one or two stand out as different. Were we to
also look at, for example, Sundays or Tuesdays (as we do in Section 4), we would see that although
Sunday and Monday appear quite different and, one suspects, have little shared information, Monday
and Tuesday appear relatively similar and this similarity can probably be used to improve our rate
estimates for both days.
In this example, we might reasonably assume that the category memberships are known (for example, whether a given day is a weekday or weekend, or a Monday or Tuesday), though we shall
relax this assumption in later sections. Then, given a structure of potential relationships, what is a
reasonable model for sharing information among categories? There are, of course, many possible
choices; we use a simple additive model, described in the next section.
3.1
Additive Models
The intuition behind an additive model is that the data arises from the superposition of several
underlying causes present during the period of interest. Again, we initially assume that the category
memberships are known; thus, if a category is associated with a particular day, the activity profile
associated with that category will be observed, along with additional activity arising from each of
the other categories present.
Let us associate a rate function ?c (t) = ?c fc (t) with each category in our model. We define the rate
function of a given day d to be the sum of the rate functions of each category to which d belongs.
Denoting by sdc the (binary-valued)
membership indicator, i.e., that category c is present during day
P
d, we have that ?d (t) = c:sdc =1 ?c (t).
At first, this model might seem quite restrictive. However, it matches our intuition of how the data
is generated, stemming from the presence or absence of a particular behavioral pattern associated
with some underlying cause (such as it being a work day). In fact, we do not want a model which
is too flexible, such as a linear combination of patterns, since it is not physically meaningful to say,
for example, that a day is only ?part? Monday. To learn the profiles associated with a given cause
(e.g., things that happen every day versus only on weekdays or only on Mondays), it makes sense
to take an ?all or nothing? model where the pattern is either present, or not. This also suggests
that other methods of coupling Dirichlet processes, such as the hierarchical Dirichlet process [7],
may be too flexible. The HDP couples the parameters of components across levels, but only loosely
relates the actual shape of the profile, since it allows components to be larger or smaller (or even
disappear completely). In [7], this is a desirable quality, but in our application it is not. Using an
additive model allows both a consistent size and shape to emerge for each category, while associating
deviations from that profile to categories further down in the hierarchy.
Inference in this system is not significantly more difficult than in the single rate function case (Section 2). We define the association as [ydi , zdi ], where ydi indicates which of the categories
generated
P
event ?di . It is easy to sample ydi according to p(ydi = c|{?c (t)}) ? [?c (?di )] / [ c0 ?c0 (?di )].
3.2
Sampling Membership
Of course, it is frequently the case that the membership(s) of each collection of data are not known
precisely. In an extreme case, we may have no idea which collections are similar and should be
grouped together and wish to find profiles in an unsupervised manner. More commonly, however,
we have some prior knowledge and interpretation of the profiles but do not wish to strictly enforce
a known membership. For example, if we create categories with assigned meanings (weekdays,
weekends, Sundays, Mondays, and so on), a day which is nominally a Monday but also happens
to be a holiday, closure, or other unusual circumstances may be completely different from other
Monday profiles. Similarly, a day with unusual extra activity (receptions, talks, etc.) may see
behavior unique to its particular circumstances and warrant an additional category to represent it.
We can accommodate both these possibilities by also sampling the values of the membership indicator variables sdc , i.e., the binary indicator that day d sees behavior from category c. To this end,
let us assume we have some prior knowledge of these membership probabilities, pdc (sdc ); we may
then re-sample from their posterior distributions at each iteration of MCMC.
This sampling step is difficult to do outside the truncated representation. Although up until this point
we could easily have elected to use, for example, the CRP formulation for sampling, the association
variables {ydi , zdi } are tightly coupled with the memberships sdc since if any ydi = c we must have
that sdc = 1. Instead, to sample the sdc we condition on the truncated rate functions ?c (t), with
truncation depth M chosen to provide arbitrarily high precision. The likelihood of the data under
these rate functions for any values of {sdc } can then be computed directly via (2) where
X
X
sdc ?c (t).
sdc ?c
and
f (t) = ? ?1
?=
c
c
In practice, we propose changing the value of each membership variable sdc individually given
the others, though more complex moves could also be applied. This gives the following sequence of
MCMC sampling: (1) given a truncated representation of the {?c (t)}, sample membership variables
{sdc }; (2) given {?c (t)} and {sdc }, sample associations {zdi }; (3) given associations {zdi }, sample
3
50
50
40
40
30
30
20
20
2
1
10
0
0
6
12
18
24
(a) Sundays
0
0
10
6
12
18
(b) Mondays
24
0
0
6
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18
24
(c) Tuesdays
Figure 2: Posterior mean estimates of rate functions for building entry log data, estimated individually for each day (dotted) and learned by sharing information among multiple days (solid) for (a)
Sundays, (b) Mondays, and (c) Tuesdays. Sharing information among similar days gives greatly
improved estimates of the rate functions, resolving otherwise obscured features such as the decrease
during and increase subsequent to lunchtime.
category magnitudes {?c } and a truncated representation of each fc (t) consisting of weights {wj }
and parameters {?j }.
4
Experiments
In this section we consider the application of our model to two data sets, one (mentioned previously)
from the entry log of people entering a large campus building (produced by optical sensors at the
front door), and the other from a log of vehicular traffic accidents. By design, both data sets contain
about ten weeks worth of observations. In both cases, we have a plausible prior structure for and
interpretation of the categories, i.e., that similar days will have similar profiles. To this end, we create
categories for ?all days?, ?weekends?, ?weekdays?, and ?Sundays? through ?Saturdays?. Each of
these categories has a high probability (pdc = .99) of membership for each eligible day. To account
for the possibility of unusual increases in activity, we also add categories unique to each day, with
lower prior probability (pdc = .20) of membership. This allows but discourages each day to add a
new category if there is evidence of unusual activity.
4.1
Building Entry Data
To see the improvement in the estimated rate functions when information is shared among similar
days, Figure 2 shows results from three different days of the week (Sunday, Monday, Tuesday).
Each panel shows the estimated profiles of each of the ten days estimated individually (using only
that day?s observations) under a Dirichlet process mixture model (dotted lines). Superimposed in
each panel is a single, black curve corresponding to the total profile for that day of week estimated
using our categorical model; so, (a) shows the sum of the rate functions for ?all days?, ?weekends?,
and ?Sundays?, while (b) shows the sum of ?all days?, ?weekdays?, and ?Mondays?. We use the
same prior distributions for both the individual estimates and the shared estimate.
Several features are worth noting. First, by sharing several days worth of observations, the model can
produce a much more accurate estimate of the profiles. In this case, no single day contains enough
observations to be confident about the details of the rate function, so each individually?estimated
rate function appears relatively smooth. However, when information from other days is included,
the rate function begins to resolve into a clearly bi-modal shape for weekdays. This ?bi-modal? rate
behavior is quite real, and corresponds to the arrival of occupants in the morning (first mode), a lull
during lunchtime, and a larger, narrower second peak as most occupants return from lunch.
Second, although Monday and Tuesday profiles appear similar, they also have distinct behavior, such
as increased activity late Tuesday morning. This behavior too has some basis in reality, corresponding to a regular weekly meeting held around lunchtime over most (though not quite all) of the weeks
in question. The breakdown of a particular day (the first Tuesday) into its component categories is
shown in Figure 3. As we might expect, there is little consistency between weekdays and weekends,
quite a bit of similarity among weekdays and among just Tuesdays, and (for this particular day) very
little to set it apart from other Tuesdays.
We can also check to see that the category memberships sdc are being used effectively. One of
the Mondays in our data set fell on a holiday (the individual profile very near zero). If we average
the probabilities computed during MCMC to estimate the posterior probability of the sdc for that
40
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40
30
30
30
30
20
20
20
20
10
10
10
0
0
6
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24
6
(a) All Days
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18
10
0
0
24
6
(b) Weekdays
12
18
0
0
24
6
(c) Tuesdays
12
18
24
(d) Unique
Figure 3: Posterior mean estimates of the rate functions for each category to which the first Tuesday
data might belong. For comparison, the total rate (sum of all categories) is shown as the dotted
line. (a) The ?all days? category is small, indicating little consistency in the data between weekdays
and weekends; (b) the ?weekdays? category is larger, and contains a component which appears
to correspond to the occupants? return from lunch; (c) the ?Tuesday? category has modes in the
morning and afternoon, perhaps capturing regular meetings or classes; (d) the ?unique? category (a
category unique to this particular day) shows little or no activity.
15
20
0.6
15
10
0.4
10
5
0.2
0
0
5
6
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24
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6
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18
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24
6
12
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24
Figure 4: Profiles associated with individual-day categories in the entry log data for several days
with known events (periods between dashed vertical lines). The model successfully learns which
days have significant unusual activity and associates reasonable profiles with that activity (note that
increases in entrance count rate typically occurs shortly before or at the beginning of the event time).
particular day, we find that it has near-zero probability of belonging to either the weekday or Monday
categories, and uses only the all-day and unique categories.
We can also examine days which have high probability of requiring their own category (indicating unusual activity). For this data set, we also have partial ground truth, consisting of a number
of dates and times when activities were scheduled to take place in the building. Figure 4 shows
three such days, and the corresponding rate profiles associated with their single-day categories.
Again, all three days are estimated to have additional activity, and the period of time for that activity
corresponds well with the actual start and end time shown in the schedule (dashed vertical lines).
60
4.2
Vehicular Accident Data
Our second data set consists of a database of vehicular accident times recorded by North Carolina police departments. As we might expect
of driving patterns, there is still less activity on
weekends, but far more than was observed in
the campus building log.
40
20
0
0
6
12
18
24
As before, sharing information allows us to Figure 5: Posterior mean and uncertainty for a
decrease our posterior uncertainty on the rate single day of accident data, estimated individually
for any particular day. Figure 5 quantifies this (red) and with data sharing (black). Sharing data
idea by showing the posterior means and (point- considerably reduces the posterior uncertainty in
wise) two-sigma confidence intervals for the the profile shape.
rate function estimated for the same day (the
first Monday in the data set) using that day?s
data only (red curves) and using the category-based additive model (black). The additive model
leverages the additional data to produce much tighter estimates of the rate profile.
As with the previous example, the additional data also helps resolve detailed features of each day?s
profile, as seen in Figure 6. For example, the weekday profiles show a tri-modal shape, with one
mode corresponding to the morning commute, a small mode around noon, and another large mode
100
100
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80
80
80
60
60
60
40
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(a) Sundays
18
24
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(b) Mondays
24
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(c) Fridays
Figure 6: Posterior mean estimates of rate functions for vehicular accidents, estimated individually
for each day (dotted) and with sharing among multiple days (solid) for (a) Sundays, (b) Mondays,
and (c) Fridays. As in Figure 2, sharing information helps resolve features which the individual days
do not have enough data to reliably estimate.
around the evening commute. This also helps make the pattern of deviation on Friday clear, showing
(as we would expect) increased activity at night.
5
Conclusions
The increasing availability of logs of ?human activity? data provides interesting opportunities for the
application of statistical learning techniques. In this paper we proposed a non-parametric Bayesian
approach to learning time-intensity profiles for such activity data, based on an inhomogeneous Poisson process framework. The proposed approach allows collections of observations (e.g., days) to
be grouped together by category (day of week, weekday/weekend, etc.) which in turn leverages
data across different collections to yield higher quality profile estimates. When the categorization
of days is not a priori certain (e.g., days that fall on a holiday or days with unusual non-recurring
additional activity) the model can infer the appropriate categorization, allowing (for example) automated detection of unusual events. On two large real-world data sets the model was able to infer
interpretable activity profiles that correspond to real-world phenomena. Directions for further work
in this area include richer models that allow for incorporation of observed covariates such as weather
and other exogenous phenomena, as well as modeling of multiple spatially-correlated sensors (e.g.,
loop sensor data for freeway traffic).
References
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[3] R. Willett and R. Nowak. Multiscale Poisson intensity and density estimation. submitted to IEEE Trans.
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[4] A. Kottas. Bayesian nonparametric mixture modeling for the intensity function of non-homogeneous
Poisson processes. Technical Report ams2005-02, Department of Applied Math and Statistics, U.C.
Santa Cruz, Santa Cruz, CA, 2005.
[5] A. Kottas and B. Sanso. Bayesian mixture modeling for spatial Poisson process intensities, with applications to extreme value analysis. Technical Report ams2005-19, Dept. of Applied Math and Statistics,
U.C. Santa Cruz, Santa Cruz, CA, 2005.
[6] B.W. Silverman. Density Estimation for Statistics and Data Analysis. Chapman & Hall, NY, 1986.
[7] Y.W. Teh, M.I. Jordan, M.J. Beal, and D.M. Blei. Hierarchical Dirichlet processes. In NIPS 17, 2004.
[8] D.R. Cox. Some statistical methods connected with series of events. J. R. Stat. Soc. B, 17:129?164, 1955.
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[10] M.D. Escobar and M. West. Bayesian density estimation and inference using mixtures. J. Amer. Stat.
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[11] L.F. James. Functionals of Dirichlet processes, the Cifarelli-Reganzzini identity and Beta-Gamma
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[12] H. Ishwaran and L.F. James. Gibbs sampling methods for stick-breaking priors. J. Amer. Stat. Assoc.,
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2,241 | 3,033 | Modeling Dyadic Data with Binary Latent Factors
Edward Meeds
Department of Computer Science
University of Toronto
[email protected]
Zoubin Ghahramani
Department of Engineering
Cambridge University
[email protected]
Radford Neal
Department of Computer Science
University of Toronto
[email protected]
Sam Roweis
Department of Computer Science
University of Toronto
[email protected]
Abstract
We introduce binary matrix factorization, a novel model for unsupervised matrix decomposition. The decomposition is learned by fitting a non-parametric
Bayesian probabilistic model with binary latent variables to a matrix of dyadic
data. Unlike bi-clustering models, which assign each row or column to a single
cluster based on a categorical hidden feature, our binary feature model reflects the
prior belief that items and attributes can be associated with more than one latent
cluster at a time. We provide simple learning and inference rules for this new
model and show how to extend it to an infinite model in which the number of
features is not a priori fixed but is allowed to grow with the size of the data.
1 Distributed representations for dyadic data
One of the major goals of probabilistic unsupervised learning is to discover underlying or hidden
structure in a dataset by using latent variables to describe a complex data generation process. In this
paper we focus on dyadic data: our domains have two finite sets of objects/entities and observations are made on dyads (pairs with one element from each set). Examples include sparse matrices
of movie-viewer ratings, word-document counts or product-customer purchases. A simple way to
capture structure in this kind of data is to do ?bi-clustering? (possibly using mixture models) by
grouping the rows and (independently or simultaneously) the columns[6, 13, 9]. The modelling assumption in such a case is that movies come in K types and viewers in L types and that knowing
the type of movie and type of viewer is sufficient to predict the response. Clustering or mixture
models are quite restrictive ? their major disadvantage is that they do not admit a componential or
distributed representation because items cannot simultaneously belong to several classes. (A movie,
for example, might be explained as coming from a cluster of ?dramas? or ?comedies?; a viewer as
a ?single male? or as a ?young mother?.) We might instead prefer a model (e.g. [10, 5]) in which
objects can be assigned to multiple latent clusters: a movie might be a drama and have won an Oscar and have subtitles; a viewer might be single and female and a university graduate. Inference in
such models falls under the broad area of factorial learning (e.g. [7, 1, 3, 12]), in which multiple
interacting latent causes explain each observed datum.
In this paper, we assume that both data items (rows) and attributes (columns) have this kind of
componential structure: each item (row) has associated with it an unobserved vector of K binary
features; similarly each attribute (column) has a hidden vector of L binary features. Knowing the
features of the item and the features of the attribute are sufficient to generate (before noise) the
response at that location in the matrix. In effect, we are factorizing a real-valued data (response)
> , where and are binary feature
matrix into (a distribution defined by) the product
is a real-valued weight matrix. Below, we develop this binary matrix factorization
matrices, and
X
W
UWV
U
V
K
, w o
k
wkl
l
uik
xij
vjl
I
L
X
=f
W
U
V>
J
(A)
(B)
Figure 1: (A) The graphical model representation of the linear-Gaussian BMF model. The concentration parameter and Beta weights for the columns of are represented by the symbols and l .
(B) BMF shown pictorally.
X
(BMF) model using Bayesian non-parametric priors over the number and values of the unobserved
binary features and the unknown weights.
2 BMF model description
X
Binary matrix factorization is a model of an I J dyadic data matrix with exchangeable rows
and columns. The entries of
can be real-valued, binary, or categorial; BMF models suitable
for each type are described below. Associated with each row is a latent binary feature vector i ;
similarly each column has an unobserved binary vector j . The primary parameters are represented
of interaction weights.
is generated by a fixed observation process f () applied
by a matrix
(elementwise) to the linear inner product of the features and weights, which is the ?factorization? or
approximation of the data:
X
W
u
v
X
X j U; V; W f (UWV> ; )
(1)
where are extra parameters specific to the model variant. Three possible parametric forms for
> and covariance (1=) ;
the noise (observation) distribution f are: Gaussian, with mean
>
>.
logistic, with mean 1= (1 + exp(
)); and Poisson, with mean (and variance)
Other parametric forms are also possible. For illustrative purposes, we will use the linear-Gaussian
model throughout this paper; this can be thought of as a two-sided version of the linear-Gaussian
model found in [5].
UWV
UWV
UWV
I
To complete the description of the model, we need to specify prior distributions over the feature
matrices ; and the weights
. We adopt the same priors over binary matrices as previously
described in [5]. For finite sized matrices
with I rows and K columns, we generate a bias k
independently for each column k using a Beta prior (denoted B ) and then conditioned on this bias
generate the entries in column k independently from a Bernoulli with mean k .
UV
W
k j ; K
B ( =K; )
Uj
P
U
I Y
K
Y
i=1 k=1
j a ; b
kuik (1 k )1 uik
=
G (a ; b )
K
Y
k=1
knk (1 k )I nk
where nk = i uik . The hyperprior on the concentration is a Gamma distribution (denoted G ),
whose shape and scale hyperparameters control the expected fraction of zeros/ones in the matrix.
The biases are easily integrated out, which creates dependencies between the rows, although
they remain exchangeable. The resulting prior depends only on the number nk of active features
in each column. An identical prior is used on , with J rows and L columns, but with different
concentration prior . The variable was set to 1 for all experiments.
V
The appropriate prior distribution over weights depends on the observation distribution f (). For
is a matrix normal with prior mean o and
the linear-Gaussian variant, a convenient prior on
W
W
I
covariance (1= ) . The scale
hyperpriors:
of the weights and output precision
(if needed) have Gamma
W j Wo; N (Wo; (1= ) I)
j a ; b
j a ; b
G (a ; b )
G ( a ; b )
In certain cases, when the prior on the weights is conjugate to the output distribution model f , the
weights may be analytically integrated out, expressing the marginal distribution of the data j ;
only in terms of the binary features. This is true, for example, when we place a Gaussian prior on
the weights and use a linear-Gaussian output process.
XU V
U
V
Remarkably, the Beta-Bernoulli prior distribution over (and similarly ) can easily be extended
to the case where K ! 1, creating a distribution over binary matrices with a fixed number I of
exchangeable rows and a potentially infinite number of columns (although the expected number of
columns which are not entirely zero remains finite). Such a distribution, the Indian Buffet Process
(IBP) was described by [5] and is analogous to the Dirichlet process and the associated Chinese
restaurant process (CRP) [11]. Fortunately, as we will see, inference with this infinite prior is not
only tractable, but is also nearly as efficient as the finite version.
3 Inference of features and parameters
U
As with many other complex hierarchical Bayesian models, exact inference of the latent variables
and in the BMF model is intractable (ie there is no efficient way to sample exactly from the posterior nor to compute its exact marginals). However, as with many other non-parametric Bayesian
models, we can employ Markov Chain Monte Carlo (MCMC) methods to create an iterative procedure which, if run for sufficiently long, will produce correct posterior samples.
V
3.1 Finite binary latent feature matrices
U V
The posterior distribution of a single entry in (or ) given all other model parameters is proportional to the product of the conditional prior and the data likelihood. The conditional prior comes
from integrating out the biases in the Beta-Bernoulli model and is proportional the number of
active entries in other rows of the same column plus a term for new activations. Gibbs sampling for
single entries of (or ) can be done using the following updates:
U V
P (uik = 1jU ik ; V; W; X) = C ( =K + n i;k ) P (XjU ik ; uik = 1; V; W)
(2)
P (uik = 0jU ik ; V; W; X) = C ( + (I 1) n i;k ) P (XjU ik ; uik = 0; V; W) (3)
P
where n i;k = h6=i uhk , U ik excludes entry ik , and C is a normalizing constant. (Conditioning
on ; K and is implicit.) When conditioning on W, we only need to calculate the ratio of likelihoods corresponding to row i. (Note that this is not the case when
the weights are integrated out.)
+ = P uih vjl whl (when uik = 1) and
^
This ratio is a simple function of the model?s predictions x
ij
hl
P
x^ij = hl uih vjl whl (when uik = 0). In the linear-Gaussian case:
log
P (u
P (u
ik
ik
jU ; V; W; X) = log ( =K + n )
= 0jU
; V ; W ; X)
( + (I
1)
n
=1
ik
1
i;k
i;k )
ik
X
2
ij
x
(
ij
x^+ )2
ij
x
(
ij
x^ )2
ij
j
In the linear-Gaussian case, we can easily derive analogous Gibbs sampling updates for the weights
and hyperparameters. To simplify the presentation, we consider a ?vectorized? representation of
our variables. Let be an IJ column vector taken column-wise from , be a KL column vector
taken column-wise from
and
be a IJ KL binary matrix which is the kronecker product
. (In ?Matlab notation?, = (:); = (:) and = kron( ; ).) In this notation, the
data distribution is written as: j ; ; N (
; (1=) ). Given values for and , samples
can be drawn for , , and using the following posterior distributions (where conditioning on
o ; ; ; a ; b ; a ; b is implicit):
j ; N ( > + ) 1 ( > + o ) ; ( > + ) 1
W
V U
w
x
w
w xA
W
A
x X w W
xA w
Aw
AA
I
Xw
VU
A
I
Ax
w
AA
U
I
V
G a + KL=2; b + 12 (w wo )> (w wo )
1
j x; A; w G a + IJ=2; b + (x Aw)> (x Aw)
2
jw
A
AA
Note that we do not have to explicitly compute the matrix . For computing the posterior of linearGaussian weights, the matrix > can be computed as > = kron( > ; > ). Similarly,
the expression > is constructed by computing >
and taking the elements column-wise.
Ax
AA
U XV
V VU U
3.2 Infinite binary latent feature matrices
One of the most elegant aspects of non-parametric Bayesian modeling is the ability to use a prior
which allows a countably infinite number of latent features. The number of instantiated features is
automatically adjusted during inference and depends on the amount of data and how many features
it supports. Remarkably, we can do MCMC sampling using such infinite priors with essentially no
computational penalty over the finite case. To derive these updates (e.g. for row i of the matrix ),
it is useful to consider partitioning the columns of into two sets as shown below.
Let set A have at least one non-zero entry
in rows other than i. Let set B be all other
set A
set B
columns, including the set of columns where
0 1 0 0 1
0 0 0 0 0
the only non-zero entries are found in row i
0 0 1 0 0
0 0 0 0 0
and the countably infinite number of all-zero
1 1 0 0 1
0 0 0 0 0
1 0 0 1 1
0 0 0 0 0
columns. Sampling values for elements in row
1 1 0 0 1
0 1 0 1 0 row i
i of set A given everything else is straightfor0 1 0 0 0
0 0 0 0 0
ward, and involves Gibbs updates almost iden0 0 0 1 0
0 0 0 0 0
tical to those in the finite case handled by equa1
0
0
0
1
0
0
0
0
0
tions (2) and (3); as K !1 and k in set A we
get:
U
U
P (uik = 1jU ik ; V; W)
P (uik = 0jU ik ; V; W)
= C n i;k P (XjU ik ; uik = 1; V; W)
= C ( + I 1 n i;k ) P (XjU ik ; uik = 0; V; W)
(4)
(5)
When sampling new values for set B, the columns are exchangeable, and so we are really only
interested in the number of entries n?B in set B which will be turned on in row i. Sampling
the number of entries set to 1 can be done with Metropolis-Hastings updates. Let J (n?B jnB ) =
Poisson (n?B j = ( + I 1)) be the proposal distribution for a move which replaces the current nB
active entries with n?B active entries
in set B. The reverse proposal is J (nB jn?B ). The acceptance
probability is min 1; rnB !n?B , where rnB !n?B is
P (n?B j ) J (nB jn?B ) P ( jn?B ) Poisson(n?B j = ( + I 1))J (nB jn?B ) P ( jn?B )
=
=
(6)
P (nB j ) J (n?B jnB ) P ( jnB ) Poisson(nB j = ( + I 1))J (n?B jnB ) P ( jnB )
X
X
X
X
X
X
W
This assumes a conjugate situation in which the weights
are explicitly integrated out of the
model to compute the marginal likelihood P ( jn?B ). In the non-conjugate case, a more complicated proposal is required. Instead of proposing n?B , we jointly propose n?B and associated feature
? from their prior distributions. In the linear-Gaussian model, where ? is a set of
parameters B
B
weights for features in set B, the proposal distribution is:
J (n?B ; B? jnB ; B ) = Poisson (n?B j = ( + I 1)) Normal ( B? jn?B ; )
(7)
? where uik = 1. As in the conjugate case, the
We need actually sample only the finite portion of B
acceptance ratio reduces to the ratio of data likelihoods:
P ( jn?B ; B? )
rnB ;wB !n?B ;wB? =
(8)
P ( jn B ; B )
X
w
w
w
w
w
w
X
X
w
w
3.3 Faster mixing transition proposals
UV
W
are the simplest moves we could
The Gibbs updates described above for the entries of , and
make in a Markov Chain Monte Carlo inference procedure for the BMF model. However, these
limited local updates may result in extremely slow mixing. In practice, we often implement larger
moves in indicator space using, for example, Metropolis-Hastings proposals on multiple features
for row i simultaneously. For example, we can propose new values for several columns in row i
of matrix
by sampling feature values independently from their conditional priors. To compute
the reverse proposal, we imagine forgetting the current configuration of those features for row i
and compute the probability under the conditional prior of proposing the current configuration. The
acceptance probability of such a proposal is (the maximum of unity and) the ratio of likelihoods
between the new proposed configuration and the current configuration.
U
Split-merge moves may also be useful for efficiently sampling from the posterior distribution of
the binary feature matrices. Jain and Neal [8] describe split-merge algorithms for Dirichlet process
mixture models with non-conjugate component distributions. We have developed and implemented
similar split-merge proposals for binary matrices with IBP priors. Due to space limitations, we
present here only a sketch of the procedure. Two nonzero entries in
are selected uniformly at
random. If they are in the same column, we propose splitting that column; if they are in different
columns, we propose merging their columns. The key difference between this algorithm and the Jain
and Neal algorithm is that the binary features are not constrained to sum to unity in each row. Our
split-merge algorithm also performs restricted Gibbs scans on columns of to increase acceptance
probability.
U
U
3.4 Predictions
A major reason for building generative models of data is to be able to impute missing data values
given some observations. In the linear-Gaussian model, the predictive distribution at each iteration
of the Markov chain is a Gaussian distribution. The interaction weights can be analytically integrated
out at each iteration, also resulting in a Gaussian posterior, removing sampling noise contributed by
having the weights explicitly represented. Computing the exact predictive distribution, however,
conditional only on the model hyperparameters, is analytically intractable: it requires integrating
over all binary matrices and , and all other nuisance parameters (e.g., the weights and precisions). Instead we integrate over these parameters implicitly by averaging predictive distributions
from many MCMC iterations. This posterior, which is conditional only on the observed data and hyperparameters, is highly complex, potentially multimodal, and non-linear function of the observed
variables.
U
V
U
V
and . In our
By averaging predictive distributions, our algorithm implicitly integrates over
experiments, we show samples from the posteriors of and to help explain what the model is
doing, but we stress that the posterior may have significant mass on many possible binary matrices.
The number of features and their degrees of overlap will vary over MCMC iterations. Such variation
will depend, for example, on the current value of and (higher values will result in more features)
and precision values (higher weight precision results in less variation in weights).
U
V
4 Experiments
4.1 Modified ?bars? problem
A toy problem commonly used to illustrate additive feature or multiple cause models is the bars
problem ([2, 12, 1]). Vertical and horizontal bars are combined in some way to generate data samples. The goal of the illustration is to show recovery of the latent structure in the form of bars. We
have modified the typical usage of bars to accommodate the linear-Gaussian BMF with infinite features. Data consists of I vectors of size 82 where each vector can be reshaped into a square image.
The generation process is as follows: since has the same number of rows as the dimension of the
images, is fixed to be a set of vertical and horizontal bars (when reshaped into an image).
is
sampled from the IBP, and global precisions and are set to 1=2. The weights
are sampled
from zero mean Gaussians. Model estimates of and were initialized from an IBP prior.
V
V
U
W
V
U
In Figure 2 we demonstrate the performance of the linear-Gaussian BMF on the bars data. We train
the BMF with 200 training examples of the type shown in the top row in Figure 2. Some examples
have their bottom halves labeled missing and are shown in the Figure with constant grey values. To
handle this, we resample their values at each iteration of the Markov chain. The bottom row shows
. Despite the relatively high
the expected reconstruction using MCMC samples of , , and
UV
W
noise levels in the data, the model is able to capture the complex relationships between bars and
weights. The reconstruction of vertical bars is very good. The reconstruction of horizontal bars is
good as well, considering that the model has no information regarding the existence of horizontal
bars on the bottom half.
(A) Data samples
(B) Noise-free data
(C) Initial reconstruction
(D) Mean reconstruction
(E) Nearest neighbour
Figure 2: Bars reconstruction. (A) Bars randomly sampled from the complete dataset. The bottom
half of these bars were removed and labeled missing during learning. (B) Noise-free versions of the
same data. (C) The initial reconstruction. The missing values have been set to their expected value,
0, to highlight the missing region. (D) The average MCMC reconstruction of the entire image. (E)
Based solely on the information in the top-half of the original data, these are the noise-free nearest
neighbours in pixel space.
V
VW>
V
WV
> used to generate the data. The
Figure 3: Bars features. The top row shows values of and
>
> can be thought of as a
second row shows a sample of and
from the Markov chain.
set of basis images which can be added together with binary coefficients ( ) to create images.
V
WV
WV
U
By examining the features captured by the model, we can understand the performance just described.
> along with one sample of those
In Figure 3 we show the generating, or true, values of and
> basis
features from the Markov chain. Because the model is generated by adding multiple
images shown on the right of Figure 3, multiple bars are used in each image. This is reflected in the
> are fairly similar to the generating
> , but the former are
captured features. The learned
composed of overlapping bar structure (learned ).
V
WV
WV
WV
WV
V
4.2 Digits
In Section 2 we briefly stated that BMF can be applied to data models other than the linear-Gaussian
model. We demonstrate this with a logistic BMF applied to binarized images of handwritten digits.
We train logistic BMF with 100 examples each of digits 1, 2, and 3 from the USPS dataset. In
the first five rows of Figure 4 we again illustrate the ability of BMF to impute missing data values.
The top row shows all 16 samples from the dataset which had their bottom halves labeled missing.
Missing values are filled-in at each iteration of the Markov chain. In the third and fourth rows we
show the mean and mode (P (xij = 1) > 0:5) of the BMF reconstruction. In the bottom row we
have shown the nearest neighbors, in pixel space, to the training examples based only on the top
halves of the original digits.
In the last three rows of Figure 4 we show the features captured by the model. In row F, we show
the average image of the data which have each feature in on. It is clear that some row features
> are shown.
have distinct digit forms and others are overlapping. In row G, the basis images
By adjusting the features that are non-zero in each row of , images are composed by adding basis
images together. Finally, in row H we show . These pixel features mask out different regions in
V
U
U
WV
pixel space, which are weighted together to create the basis images. Note that there are K features
in rows F and G, and L features in row H.
(A)
(B)
(C)
(D)
(E)
(F)
(G)
(H)
Figure 4: Digits reconstruction. (A) Digits randomly sampled from the complete dataset. The
bottom half of these digits were removed and labeled missing during learning. (B) The data shown
to the algorithm. The top half is the original data value. (C) The mean of the reconstruction for
the bottom halves. (D) The mode reconstruction of the bottom halves. (E) The nearest neighbours
of the original data are shown in the bottom half, and were found based solely on the information
from the top halves of the images. (F) The average of all digits for each feature. (G) The feature
> reshaped in the form of digits. By adding these features together, which the features do,
reconstructions of the digits is possible. (H)
reshaped into the form of digits. The first image
represents a bias feature.
WV
U
V
U
4.3 Gene expression data
Gene expression data is able to exhibit multiple and overlapping clusters simultaneously; finding
models for such complex data is an interesting and active research area ([10], [13]). The plaid
model[10], originally introduced for analysis of gene expression data, can be thought of as a nonBayesian special case of our model in which the matrix
is diagonal and the number of binary features is fixed. Our goal in this experiment is merely to illustrate qualitatively the ability
of BMF to find multiple clusters in gene expression data, some of which are overlapping, others
non-overlapping. The data in this experiment consists of rows corresponding to genes and columns
corresponding to patients; the patients suffer from one of two types of acute Leukemia [4]. In Figure
5 we show the factorization produced by the final state in the Markov chain. The rows and columns
of the data and its expected reconstruction are ordered such that contiguous regions in were observable. Some of the many feature pairings are highlighted. The BMF clusters consist of broad,
overlapping clusters, and small, non-overlapping clusters. One of the interesting possibilities of using BMF to model gene expression data would be to fix certain columns of or with knowledge
gained from experiments or literature, and to allow the model to add new features that help explain
the data in more detail.
W
X
U V
5 Conclusion
We have introduced a new model, binary matrix factorization, for unsupervised decomposition of
dyadic data matrices. BMF makes use of non-parametric Bayesian methods to simultaneously discover binary distributed representations of both rows and columns of dyadic data. The model explains each row and column entity using a componential code composed of multiple binary latent
features along with a set of parameters describing how the features interact to create the observed
responses at each position in the matrix. BMF is based on a hierarchical Bayesian model and can be
naturally extended to make use of a prior distribution which permits an infinite number of features,
at very little extra computational cost. We have given MCMC algorithms for posterior inference
of both the binary factors and the interaction parameters conditioned on some observed data, and
(A)
(B)
X
Figure 5: Gene expression results. (A) The top-left is sorted according to contiguous features in
the final and in the Markov chain. The bottom-left is > and the top-right is . The bottom> . We have highlighted
right is . (B) The same as (A), but the expected value of , ^ =
regions that have both uik and vjl on. For clarity, we have only shown the (at most) two largest
contiguous regions for each feature pair.
U
W
V
V
X X UWV
U
demonstrated the model?s ability to capture overlapping structure and model complex joint distributions on a variety of data. BMF is fundamentally different from bi-clustering algorithms because of
its distributed latent representation and from factorial models with continuous latent variables which
interact linearly to produce the observations. This allows a much richer latent structure, which we
believe makes BMF useful for many applications beyond the ones we outlined in this paper.
References
[1] P. Dayan and R. S. Zemel. Competition and multiple cause models. Neural Computation, 7(3), 1995.
[2] P. Foldiak. Forming sparse representations by local anti-Hebbian learning. Biological Cybernetics, 64,
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[3] Z. Ghahramani. Factorial learning and the EM algorithm. In NIPS, volume 7. MIT Press, 1995.
[4] T. R. Golub, D. K. Slonim, P. Tamayo, C. Huard, M. Gaasenbeek, J. P. Mesirov, H. Coller, M. L. Loh,
J. R. Downing, M. A. Caligiuri, C. D. Bloomfield, and E. S. Lander. Molecular classification of cancer:
Class discovery and class prediction by gene expression monitoring. Science, 286(5439), 1999.
[5] T. Griffiths and Z. Ghahramani. Infinite latent feature models and the Indian buffet process. In NIPS,
volume 18. MIT Press, 2005.
[6] J. A. Hartigan. Direct clustering of a data matrix. Journal of the American Statistical Association, 67,
1972.
[7] G. Hinton and R. S. Zemel. Autoencoders, minimum description length, and Helmholtz free energy. In
NIPS, volume 6. Morgan Kaufmann, 1994.
[8] S. Jain and R. M. Neal. Splitting and merging for a nonconjugate Dirichlet process mixture model. To
appear in Bayesian Analysis.
[9] C. Kemp, J. B. Tenebaum, T. L. Griffiths, T. Yamada, and N. Ueda. Learning systems of concepts with an
infinite relational model. Proceedings of the Twenty-First National Conference on Artificial Intelligence,
2006.
[10] L. Lazzeroni and A. Owen. Plaid models for gene expression data. Statistica Sinica, 12, 2002.
[11] J. Pitman. Combinatorial stochastic processes. Lecture Notes for St. Flour Course, 2002.
[12] E. Saund. A multiple cause mixture model for unsupervised learning. Neural Computation, 7(1), 1994.
[13] R. Tibshirani, T. Hastie, M. Eisen, D. Ross, D. Botstein, and P. Brown. Clustering methods for the analysis
of DNA microarray data. Technical report, Stanford University, 1999. Department of Statistics.
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2,242 | 3,034 | Implicit Surfaces with Globally Regularised and
Compactly Supported Basis Functions
?
Christian Walder?? , Bernhard Sch?olkopf? & Olivier Chapelle?
Max Planck Institute for Biological Cybernetics, 72076 T?ubingen, Germany
?
The University of Queensland, Brisbane, Queensland 4072, Australia
[email protected]
Abstract
We consider the problem of constructing a function whose zero set is to represent
a surface, given sample points with surface normal vectors. The contributions
include a novel means of regularising multi-scale compactly supported basis
functions that leads to the desirable properties previously only associated with
fully supported bases, and show equivalence to a Gaussian process with modified
covariance function. We also provide a regularisation framework for simpler
and more direct treatment of surface normals, along with a corresponding
generalisation of the representer theorem. We demonstrate the techniques on 3D
problems of up to 14 million data points, as well as 4D time series data.
1
Introduction
The problem of reconstructing a surface from a set of points frequently arises in computer graphics.
Numerous methods of sampling physical surfaces are now available, including laser scanners,
optical triangulation systems and mechanical probing methods. Inferring a surface from millions
of points sampled with noise is a non-trivial task however, for which a variety of methods have been
proposed. The class of implicit or level set surface representations is a rather large one, however
other methods have also been suggested ? for a review see [1]. The implicit surface methods closest
to the present work are those that construct the implicit using regularised function approximation [2],
such as the ?Variational Implicits? of Turk and O?Brien [3], which produce excellent results, but at a
cubic computational fitting cost in the number of points. The effectiveness of this type of approach
is undisputed however, and has led researchers to look for ways to overcome the computational
problems. Two main options have emerged.
The first approach uses compactly supported kernel functions (we define and discuss kernel
functions in Section 2), leading to fast algorithms that are easy to implement [4]. Unfortunately
however these methods are suitable for benign data sets only. As noted in [5], compactly supported
basis functions ?yield surfaces with many undesirable artifacts in addition to the lack of extrapolation
across holes?. A similar conclusion was reached in [6] which states that local processing methods
are ?more sensitive to the quality of input data [than] approximation and interpolation techniques
based on globally-supported radial basis functions? ? a conclusion corroborated by the results within
a different paper from the same group [7]. The second means of overcoming the aforementioned
computational problem does not suffer from these problems however, as demonstrated by the
FastRBFTM algorithm [5], which uses the the Fast Multipole Method (FMM) [8] to overcome the
computational problems of non-compactly supported kernels. The resulting method is non-trivial to
implement however and to date exists only in the proprietary FastRBFTM package.
We believe that by applying them in a different manner, compactly supported basis functions can
lead to high quality results, and the present work is an attempt to bring the reader to the same
conclusion. In Section 3 we introduce a new technique for regularising such basis functions which
Figure 1: (a) Rendered implicit
surface model of ?Lucy?,
constructed from 14 million
points with normals. (b) A
planar slice that cuts the nose ?
the colour represents the value
of the embedding function and
the black line its zero level.
(c) A black dot at each of the
364,982 compactly supported
basis function centres which,
along with the corresponding
dilations and magnitudes,
define the implicit.
(a)
(b)
(c)
allows high quality, highly scalable algorithms that are relatively easy to implement. We also
show that the approximation can be interpreted as a Gaussian process with modified covariance
function. Before doing so however, we present in Section 2 the other main contribution of the
present work, which is to show how surface normal vectors can be incorporated directly into
the regularised regression framework that is typically used for fitting implicit surfaces, thereby
avoiding the problematic approach of constructing ?off-surface? points for the regression problem.
To demonstrate the effectiveness of the method we apply it to various problems in Section 4 before
summarising in the final Section 5.
2
Implicit Surface Fitting by Regularised Regression
Here we discuss the use of regularised regression [2] for the problem of implicit surface fitting. In
Section 2.1 we motivate and introduce a clean and direct means of making use of normal vectors.
The following Section 2.2 extends on the ideas in Section 2.1 by formally generalising the important
representer theorem. The final Section 2.3 discusses the choice of regulariser (and associated kernel
function), as well as the associated computational problems that we overcome in Section 3.
2.1
Regression Based Approaches and the Use of Normal Vectors
Typically implicit surface has been done by solving a regularised regression problem [5, 4]
2
arg min kf kH + C
f
m
X
2
(f (xi ) ? yi ) ,
(1)
i=1
where the yi are some estimate of the signed distance function at the xi , and f is the embedding
function which takes on the value zero on the implicit surface. The norm kf kH is a regulariser
which takes on larger values for less ?smooth? functions. We take H to be a reproducing kernel
Hilbert space (RKHS) with representer of evaluation (kernel function) k(?, ?), so that we have the
reproducing property, f (x) = hf, k(x, ?)iH . The solution to this problem has the form
f (x) =
m
X
?i k (xi , x) .
(2)
i
Note as a technical aside that the thin-plate kernel case ? which we will adopt ? requires a somewhat
more technical interpretatiosn, as it is only conditionally positive definite. We discuss the positive
definite case for clarity only, as it is simpler and yet sufficient to demonstrate the ideas involved.
Choosing the (xi , yi ) pairs for (2) is itself a non-trivial problem, and heuristics are typically used
to prevent contradictory target values (see e.g. [5]). We now propose more direct method, novel in
the context of implicit fitting, which avoids these problems. The approach is suggested by the fact
that the normal direction of the implicit surface is given by the gradient of the embedding function
? thus normal vectors can be incorporated by regression with gradient targets. The function that we
seek is the minimiser of:
m
m
X
X
2
2
2
k(?f ) (xi ) ? ni kRd ,
(f (xi )) + C2
(3)
kf kH + C1
i=1
i=1
which uses the given surface point/normal pairs (xi , ni ) directly. By imposing stationarity and using
the reproducing property we can solve for the optimal f . A detailed derivation of this procedure is
given in [1]. Here we provide only the result, which is that we have to solve for m coefficients ?i as
well as a further md coefficients ?lj to obtain the optimal solution
f (x) =
m
X
?i k (xi , x) +
m X
d
X
i
i
?li kl (xi , x) ,
(4)
l
.
where we define kl (xi , x) = [(?k) (xi , x)]l , the partial derivative of k in the l-th component of its
first argument.1 The coefficients ? and ?l of the solution are found by solving the system given by
X
0 = (K + I/C1 )? +
Kl ?l
(5)
l
Nm
= Km ? + (Kmm + I/C2 )?k +
X
Klm ?l , m = 1 . . . d
(6)
l6=m
where, writing klm for the second derivatives of k(?, ?) (defined similarly to the first), we?ve defined
[Nl ]i = [ni ]l ;
[?l ]i = ?li ;
[Kl ]i,j = kl (xi , xj ) ;
[?]i = ?i
[K]i,j = k(xi , xj )
[Klm ]i,j = klm (xi , xj ).
In summary, minimum norm approximation in an RKHS with gradient target values is optimally
solved by a function in the span of the kernels and derivatives thereof as per Equation 4 (cf. Equation
2), and the coefficients of the solution are given by Equations (5) and (6). It turns out, however, that
we can make a more general statement, which we do briefly in the next sub-Section.
2.2
The Representer Theorem with Linear Operators
The representer theorem, much celebrated in the Machine Learning community, says that the
function minimising an RKHS norm along with some penalties associated with the function value
at various points (as in Equation 1 for example) is a sum of kernel functions at those points (as in
Equation 2). As we saw in the previous section however, if gradients also appear in the risk function
to be minimised, then gradients of the kernel function appear in the optimal solution. We now
make a more general statement ? the case in the previous section corresponds to the following if we
choose the linear operators Li (which we define shortly) as either identities or partial derivatives.
The theorem is a generalisation of [9] (using the same proof idea) with equivalence if we choose all
Li to be identity operators. The case of general linear operators was in fact dealt with already in [2]
(which merely states the earlier result in [10]) ? but only for the case of a specific loss function c.
The following theorem therefore combines the two frameworks:
Theorem 1
Denote by X a non-empty set, by k a reproducing kernel with reproducing kernel Hilbert space H,
by ? a strictly monotonic increasing real-valued function on [0, ?), by c : Rm ? R ? {?} an
arbitrary cost function, and by L1 , . . . Lm a set of linear operators H ? H. Each minimiser f ? H
of the regularised risk functional
c((L1 f )(x1 ), . . . (Lm f )(xm )) + ?(||f ||2H )
admits the form
f=
m
X
?i L?i kxi ,
i=1
where kx , k(?, x) and L?i denotes the adjoint of Li .
1
Square brackets with subscripts indicate matrix elements: [a]i is the i-th element of the vector a.
(7)
(8)
Pm
Proof. Decompose f into f = i=1 ?i L?i kxi + f? , with ?i ? R and hf? , L?i kxi iH = 0, for each
i = 1 . . . m. Due to the reproducing property we can write, for j = 1 . . . m,
= h(Lj f ), k(?, xj )iH
m
X
=
?i hLj L?i kxi , k(?, xj )iH + h(Lj f? ), k(?, xj )iH
(Lj f )(xj )
i=1
=
m
X
?i hLj L?i kxi , k(?, xj )iH .
i=1
Thus, the first term in Equation 7 is independent of f? . Moreover, it is clear due to orthogonality
that if f? 6= 0 then
?
?
2 ?
2 ?
m
m
X
X
? ?
?i L?i kxi + f?
? > ? ?
?i L?i kxi
? ,
i=1
H
i=1
H
so that for any fixed ?i ? R, Equation 7 is minimised when f? = 0.
2.3
Thin Plate Regulariser and Associated Kernel
As is well known (see e.g. [2]), the choice of regulariser (the function norm in Equation 3) leads to
a particular kernel function k(?, ?) to be used in Equation 4. For geometrical problems, an excellent
regulariser is the thin-plate energy, which for arbitrary order m and dimension d is given by [2]:
kf k2H
=
=
h?f, ?f iL2
d
X
i1 =1
???
(9)
d Z
X
im =1
?
x1 =??
Z
?
???
xd =??
?
?
?
f
???
?xi1
?xim
??
?
?
f
???
?xi1
?xim
?
dx1 . . . dxd , (10)
where ? is a regularisation operator taking all partial derivatives of order m, which corresponds to
a ?radial? kernel function of the form k(x, y) = t(||x ? y||), where [11]
2m?d
r
ln(r) if 2m > d and d is even,
t(r) =
r2m?d
otherwise.
There are a number of good reasons to use this regulariser rather than those leading to compactly
supported kernels, as we touched on in the introduction. The main problem with compactly
supported kernels is that the corresponding regularisers are somewhat poor for geometrical problems
? they always draw the function towards some nominal constant as one moves away from the data,
thereby implementing the non-intuitive behaviour of regularising the constant function and making
interpolation impossible ? for further discussion see [1] as well as [5, 6, 7]. The scheme we propose
in Section 3 solves these problems, previously associated with compactly supported basis functions,
by defining and computing the regulariser separately from the function basis.
3
A Fast Scheme using Compactly Supported Basis Functions
Here we present a fast approximate scheme for solving the problem of the previous Section, in
which we restrict the class of functions to the span of a compactly supported, multi-scale basis, as
described in Section 3.1, and minimise the thin-plate regulariser within this span as per Section 3.2.
3.1
Restricting the Set of Available Functions
Computationally, using the thin-plate spline leads to the problem that the linear system we need to
solve (Equations 5 and 6), which is of size m(d + 1), is dense in the sense of having almost all
non-zero entries. Since solving such a system na??vely has a cubic time complexity in m, we propose
forcing f (?) to take the form:
p
X
f (?) =
?k fk (?),
(11)
k=1
where the individual basis functions are fk (?) = ?(||vk ??||/sk ) for some function ? : R+ ? R with
support [0, 1). The vk and sk are the basis function centres and dilations (or scales), respectively.
For ? we choose the B3 -spline function:
4
X
(?1)n
n
d+1
?(r) =
? n))d+ ,
(12)
(r + (
d!
2
d
+
1
n=0
although this choice is rather inconsequential since, as we shall ensure, the regulariser is unrelated
to the function basis ? any smooth compactly supported basis function could be used. In order
to achieve the same interpolating properties as the thin-plate spline, we wish to minimise our
regularised risk function given by Equation 3 within the span of Equation 11. The key to doing
this is to note that as given before in Equation 9, the regulariser (function norm) can be written as
2
kf kH = h?f, ?f iL2 . Given this fact, a straightforward calculation leads to the following system
for the optimal ?k (in the sense of minimising Equation 3):
!
d
d
X
X
T
T
Kxvl Nl ,
(13)
Kreg + C1 Kxv Kxv + C2
Kxvl Kxvl ? = C2
l=1
l=1
where we have defined the following matrices:
[Kreg ]k,k0 = h?fk , ?fk0 iL2 ; [Kxv ]i,k = fk (xi ); [Kxvl ]i,k = [(?fk )(xi )]l ;
[?]k = ?k ; [Nl ]i = [ni ]l .
The computational advantage is that the coefficients that we need are now given by a sparse pdimensional positive semi-definite linear system, which can be constructed efficiently by simple
code that takes advantage of software libraries for fast nearest neighbour type searches (see e.g.
[12]). The system can then be solved efficiently using conjugate gradient type methods. In [1] we
describe how we construct a basis with p m that results in a highly sparse linear system, but that
still contains good solutions. The critical matter of computing Kreg is dealt with next.
3.2
Computing the Regularisation Matrix
We now come to the crucial point of calculating Kreg , which can be thought of as the regularisation
matrix. The present Section is highly related to [13], however there numerical methods were
resorted to for the calculation of Kreg ? presently we shall derive closed form solutions. Also worth
comparing to the present Section is [14], where a prior over the expansion coefficients (here the
?) is used to mimic a given regulariser within an arbitrary basis, achieving a similar result but
without the computational advantages we are aiming for. As we have already noted we can write
2
kf kH = h?f, ?f iL2 [2], so that for the function given by Equation 11 we have:
? p
?2
?X
?
?
?
?j fj (?)?
?
?
?
j=1
*
=
?
p
X
?j fj (?), ?
j=1
H
=
p
X
p
X
+
?k fk (?)
k=1
L2
.
?j ?k h?fj (?), ?fk (?)iL2 = ? T Kreg ?.
j,k=1
To build the sparse matrix Kreg , a fast range search library (e.g. [12]) can be used to identify the
non-zero entries ? that is, all those [Kreg ]i,j for which i and j satisfy kvi ? vj k ? si + sj . In order
to evaluate h?fj (?), ?fk (?)iL2 , it is necessary to solve the integral of Equation 10, the full derivation
of which we relegate to [1] ? here we just provide the main results. It turns out that since the fi are
all dilations and translations of the same function ?(k?k), then it is sufficient solve for the following
.
function of si , sj and d = vi ? vj :
h??((?)si ? d), ??((?)sj )iL2 ,
which it turns out is given by
"
F??1
2m
(2?j k?k)
|s1 s2 |
#
?
?
?( )?( ) (d),
s1
s2
(14)
Figure 2: Various values of the regularisation
parameters lead to various amounts of ?smoothing? ?
here we set C1 = C2 in Equation 3 to an increasing
value from top-left to bottom-right of the figure.
Figure 3: Ray traced three dimensional
implicits, ?Happy Buddha? (543K points
with normals) and the ?Thai Statue? (5
million points with normals).
where j 2 = ?1, ? = Fx [?(x)], and by F (and F ?1 ) we mean the Fourier (inverse Fourier)
transform operators in the subscripted variable. Computing Fourier transforms in d dimensions
difficult in general, but for radial functions g(x) = gr (||x||) it may be made easier by the fact that
the Fourier transform in d dimensions (as well as its inverse) can be computed by the single integral:
d
Z ?
d
(2?) 2
r 2 gr (r)J d?2 (||?||r)dr,
Fx [gr (kxk)] (k?k) =
d?2
2
||?|| 2 0
where J? (r)is the ?-th order Bessel function of the first kind.
Unfortunately the integrals required to attain Equation 14 in closed form cannot be solved for general
dimensionality d, regularisation operator ? and basis function form ?, however we did manage to
solve them for arguably the most useful case: d = 3 with the m = 2 thin plate energy and the B3 spline basis function of Equation 12. The resulting expressions are rather unwieldy however, so we
give only an implementation in the C language in the Appendix of [1], where we also show that for
the cases that cannot be solved analytically the required integral can at worst always be transformed
to a two dimensional integral for which one can use numerical methods.
3.3
Interpretation as a Gaussian Process
Presently we use ideas from [15] to demonstrate that the approximation described in this Section 3
is equivalent to inference in an exact Gaussian Process with covariance function depending on the
choice of function basis. Placing a multivariate Gaussian prior over the coefficients in (11), namely
?1
), we see that f obeys a zero mean Gaussian process prior ? writing [fx ]i = f (xi )
? ? N (0, Kreg
and denoting expectations by E [?] we have for the covariance
T
E [fx fxT ] = Kxz E [?? T ] Kxz
?1 T
= Kxz Kreg
Kxz
Now, assuming an iid Gaussian noise model with variance ? 2 and defining Kxt etc. similarly to
Kxz we can immediately
write the joint distribution between the observation at a test point
t, that is
yt ? N f (t), ? 2 and the vector of observations at the xi , namely yx ? N fx , ? 2 I , which is
?1
?1
Kxz Kreg
Kzx + ? 2 I
Kxz Kreg
Kzt
p(yx , yt ) = N 0,
.
?1
?1
Ktz Kreg
Kzx
Ktz Kreg
Kzt + ? 2 I
The posterior distribution is therefore itself Gaussian, p(yt |yx ) ? N ?yt |yx , ?yt |yx , and we
can employ a well known expression2 for the marginals of a multivariate Gaussian followed by the
Matrix inversion lemma to derive an expression for the mean of the posterior,
T
?1
?1
?1
?t|y = Kxz Kreg
Kzt
Kxz Kreg
Kzx + ? 2 I
y
?1
T
T
= Ktz ? 2 Kreg + Kxz
Kxz
Kxz
y.
2
??
x
y
?
??
?N
a
b
? ?
A
,
CT
C
B
???
`
`
??
? x|y ? N a + CB ?1 (y ? b), A ? CB ?1 C T
Name
Bunny
Face
Armadillo
Dragon
Buddha
Asian Dragon
Thai Statue
Lucy
# Points
34834
75970
172974
437645
543197
3609455
4999996
14027872
# Bases
9283
7593
45704
65288
105993
232197
530966
364982
Basis
0.4
0.7
6.6
14.4
117.4
441.6
3742.0
1425.8
Kreg
2.4
1.9
8.5
16.3
27.4
60.9
197.5
170.5
Kxv , Kzv?
3.7
7.0
37.0
70.9
99.4
608.3
1575.6
3484.1
Multiply
11.7
20.3
123.4
322.8
423.7
1885.0
3121.2
9367.7
Solve
20.4
16.0
72.3
1381.4
2909.3
1009.5
2569.5
1340.5
Total
38.7
46.0
247.9
1805.7
3577.2
4005.2
11205.7
15788.5
Table 1: Timing results with a 2.4GHz AMD Opteron 850 processor, for various 3D data sets.
Column one is the number of points, each of which has an associated normal vector, and column
two is the number of basis vectors (the p of Section 3.1). The remaining columns are all in units of
seconds: column three is the time taken to construct the function basis, columns four and five are
the times required to construct the indicated matrices, column six is the time required to multiply
the matrices as per Equation 13, column seven is the time required to solve that same equation for
? and the final column is the total fitting time.
By comparison with (11) and (13) (but with C1 = 1/? 2 , C2 = 0 and y = 0) we can see that
the mean of the posterior distribution is identical to our approximate regularised solution based on
compactly supported basis functions. For the corresponding posterior variance we have
?1
?1
?1
?1
?1
Kzt + ? 2 ? Ktz Kreg
?yt |yx = Ktz Kreg
Kzx Kxz Kreg
Kzx + ? 2 I
Kxz Kreg
Kzt
?1
T
= ? 2 Ktz ? 2 Kreg + Kxz
Kxz
Kzt + ? 2 .
4
Experiments
We fit models to 3D data sets of up to 14 million data points ? timings are given in Table 1, where
we also see that good compression ratios are attained, in that relatively few basis functions represent
the shapes. Also note that the fitting time scales rather well, from 38 seconds for the Stanford Bunny
(35 thousand points with normals) to 4 hours 23 minutes for the Lucy statue (14 million points with
normals ? 14?106 ?(1 value term + 3 gradient terms ) ? 56 million ?regression targets?). Taking
account of the different hardware the times seem to be similar to those of the FMM approach [5].
Some rendered examples are given in Figures 1 and 3, and the well-behaved nature of the implicit
over the entire 3D volume of interest is shown for the Lucy data-set in the accompanying video.
In practice the system is extremely robust and produces excellent results without any parameter
adjustment ? smaller values of C1 and C2 in Equation 3 simply lead to the smoothing effect shown
in Figure 2. The system also handles missing and noisy data gracefully, as demonstrated in [1].
Higher dimensional implicit surfaces are also possible, interesting being a 4D representation (3D
+ ?time?) of a moving 3D shape ? one use for this being the construction of animation sequences
from a time series of 3D point cloud data ? in this case both spatial and temporal information
can help to resolve noise or missing data problems within individual scans. We demonstrate this
in the accompanying video, which shows that 4D surfaces yield superior 3D animation results
in comparison to a sequence of 3D models. Also interesting are interpolations in 4D ? in the
accompanying video we effectively interpolate between two three dimensional shapes.
5
Summary
We have presented ideas both theoretically and practically useful for the computer graphics and
machine learning communities, demonstrating them within the framework of implicit surface fitting.
Many authors have demonstrated fast but limited quality results that occur with compactly supported
function bases. The present work differs by precisely minimising a well justified regulariser within
the span of such a basis, achieving fast and high quality results. We also showed how normal vectors
can be incorporated directly into the usual regression based implicit surface fitting framework, giving
a generalisation of the representer theorem. We demonstrated the algorithm on 3D problems of up
to 14 million data points and in the accompanying video we showed the advantage of constructing a
4D surface (3D + time) for 3D animation, rather than a sequence of 3D surfaces.
Figure 4: Reconstruction of the Stanford bunny after adding Gaussian noise with standard deviation,
from left to right, 0, 0.6, 1.5 and 3.6 percent of the radius of the smallest enclosing sphere ?
the normal vectors were similarly corrupted assuming they had length equal to this radius. The
parameters C1 and C2 were chosen automatically using five-fold cross validation.
References
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[12] C. Merkwirth, U. Parlitz, and W. Lauterborn. Fast nearest neighbor searching for nonlinear signal
processing. Phys. Rev. E, 62(2):2089?2097, 2000.
[13] Christian Walder, Olivier Chapelle, and Bernhard Sch?olkopf. Implicit surface modelling as an eigenvalue
problem. Proceedings of the 22nd International Conference on Machine Learning, 2005.
[14] M. O. Franz and P. V. Gehler. How to choose the covariance for gaussian process regression independently
of the basis. In Proc. Gaussian Processes in Practice Workshop, 2006.
[15] J. Quionero Candela and C. E. Rasmussen. A unifying view of sparse approximate gaussian process
regression. Journal of Machine Learning Research, 6:1935?1959, 12 2005.
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yoo:1 avoiding:1 |
2,243 | 3,035 | Learning to Traverse Image Manifolds
Piotr Doll?ar, Vincent Rabaud and Serge Belongie
University of California, San Diego
{pdollar,vrabaud,sjb}@cs.ucsd.edu
Abstract
We present a new algorithm, Locally Smooth Manifold Learning (LSML), that
learns a warping function from a point on an manifold to its neighbors. Important
characteristics of LSML include the ability to recover the structure of the manifold
in sparsely populated regions and beyond the support of the provided data. Applications of our proposed technique include embedding with a natural out-of-sample
extension and tasks such as tangent distance estimation, frame rate up-conversion,
video compression and motion transfer.
1
Introduction
A number of techniques have been developed for dealing with high dimensional data sets that fall
on or near a smooth low dimensional nonlinear manifold. Such data sets arise whenever the number
of modes of variability of the data are much fewer than the dimension of the input space, as is the
case for image sequences. Unsupervised manifold learning refers to the problem of recovering the
structure of a manifold from a set of unordered sample points. Manifold learning is often equated
with dimensionality reduction, where the goal is to find an embedding or ?unrolling? of the manifold
into a lower dimensional space such that certain relationships between points are preserved. Such
embeddings are typically used for visualization, with the projected dimension being 2 or 3.
Image manifolds have also been studied in the context of measuring distance between images undergoing known transformations. For example, the tangent distance [20, 21] between two images is
computed by generating local approximations of a manifold from known transformations and then
computing the distance between these approximated manifolds. In this work, we seek to frame the
problem of recovering the structure of a manifold as that of directly learning the transformations
a point on a manifold may undergo. Our approach, Locally Smooth Manifold Learning (LSML),
attempts to learn a warping function W with d degrees of freedom that can take any point on the
manifold and generate its neighbors. LSML recovers a first order approximation of W, and by making smoothness assumptions on W can generalize to unseen points.
We show that LSML can recover the structure of the manifold where data is given, and also in
regions where it is not, including regions beyond the support of the original data. We propose a
number of uses for the recovered warping function W, including embedding with a natural out-ofsample extension, and in the image domain discuss how it can be used for tasks such as computation
of tangent distance, image sequence interpolation, compression, and motion transfer. We also show
examples where LSML is used to simultaneously learn the structure of multiple ?parallel? manifolds,
and even generalize to data on new manifolds. Finally, we show that by exploiting the manifold
smoothness, LSML is robust under conditions where many embedding methods have difficulty.
Related work is presented in Section 2 and the algorithm in Section 3. Experiments on point sets
and results on images are shown in Sections 4 and 5, respectively. We conclude in Section 6.
2
Related Work
Related work can be divided into two categories. The first is the literature on manifold learning,
which serves as the foundation for this work. The second is work in computer vision and computer
graphics addressing image warping and generative models for image formation.
A number of classic methods exist for recovering the structure of a manifold. Principal component
analysis (PCA) tries to find a linear subspace that best captures the variance of the original data.
Traditional methods for nonlinear manifolds include self organizing maps, principal curves, and
variants of multi-dimensional scaling (MDS) among others, see [11] for a brief introduction to these
techniques. Recently the field has seen a number of interesting developments in nonlinear manifold
learning. [19] introduced a kernelized version of (PCA). A number of related embedding methods
have also been introduced, representatives include LLE [17], I SOMAP [22], and more recently SDE
[24]. Broadly, such methods can be classified as spectral embedding techniques [24]; the embeddings they compute are based on an eigenvector decomposition of an n ? n matrix that represents
geometrical relationships of some form between the original n points. Out-of-sample extensions
have been proposed [3]. The goal of embedding methods (to find structure preserving embeddings)
differs from the goals of LSML (learn to traverse the manifold).
Four methods that we share inspiration with are [6, 13, 2, 16]. [6] employs a novel charting based
technique to achieve increased robustness to noise and decreased probability of pathological behavior vs. LLE and I SOMAP; we exploit similar ideas in the construction of LSML but differ in motivation and potential applicability. [2] proposed a method to learn the tangent space of a manifold and
demonstrated a preliminary illustration of rotating a small bitmap image by about 1? . Work by [13]
is based on the notion of learning a model for class specific variation, the method reduces to computing a linear tangent subspace that models variability of each class. [16] shares one of our goals
as it addresses the problem of learning Lie groups, the infinitesimal generators of certain geometric
transformations.
In image analysis, the number of dimensions is usually reduced via approaches like PCA [15], epitomic representation [12], or generative models like in the realMOVES system developed by Di
Bernardo et al. [1]. Sometimes, a precise model of the data, like for faces [4] or eyes [14], is even
used to reduce the complexity of the data. Another common approach is simply to have instances of
an object in different conditions: [5] start by estimating feature correspondences between a novel input with unknown pose and lighting and a stored labeled example in order to apply an arbitrary warp
between pictures. The applications range from video texture synthesis [18] and facial expression
extrapolation [8, 23] to face recognition [10] and video rewrite [7].
3
Algorithm
Let D be the dimension of the input space, and assume the data lies on a smooth d-dimensional
manifold (d D). For simplicity assume that the manifold is diffeomorphic with a subset of Rd ,
meaning that it can be endowed with a global coordinate system (this requirement can easily be
relaxed) and that there exists a continuous bijective mapping M that converts coordinates y ? Rd
to points x ? RD on the manifold. The goal of most dimensionality reduction techniques given a
set of data points xi is to find an embedding yi = M?1 (xi ) that preserves certain properties of the
original data like the distances between all points (classical MDS) or the distances or angles between
nearby points (e.g. spectral embedding methods).
Instead, we seek to learn a warping function W that can take a point on the manifold and return
any neighboring point on the manifold, capturing all the modes of variation of the data. Let us use
W(x, ) to denote the warping of x, with ? Rd acting on the degrees of freedom of the warp
according to the formula M: W(x, ) = M(y + ), where y = M?1 (x). Taking the first order
approximation of the above gives: W(x, ) ? x + H(x), where each column H?k (x) of the matrix
H(x) is the partial derivative of M w.r.t. yk : H?k (x) = ?/?yk M(y). This approximation is valid
given small enough, hence we speak of W being an infinitesimal warping function.
We can restate our goal of learning to warp in terms of learning a function H? : RD ? RD?d
parameterized by a variable ?. Only data points xi sampled from one or several manifolds are given.
For each xi , the set N i of neighbors is then computed (e.g. using variants of nearest neighbor such
(a)
(b)
(c)
(d)
(e)
(f)
Figure 1: Overview. Twenty points (n=20) that lie on 1D curve (d=1) in a 2D space (D=2) are shown in (a).
Black lines denote neighbors, in this case the neighborhood graph is not connected. We apply LSML to train H
(with f = 4 RBFs). H maps points in R2 to tangent vectors; in (b) tangent vectors computed over a regularly
spaced grid are displayed, with original points (blue) and curve (gray) overlayed. Tangent vectors near original
points align with the curve, but note the seam through the middle. Regularization fixes this problem (c), the
resulting tangents roughly align to the curve along its entirety. We can traverse the manifold by taking small
steps in the direction of the tangent; (d) shows two such paths, generated starting at the red plus and traversing
outward in large steps (outer curve) and finer steps (inner curve). This generates a coordinate system for the
curve resulting in a 1D embedding shown in (e). In (f) two parallel curves are shown, with n=8 samples each.
Training a common H results in a vector field that more accurately fits each curve than training a separate H
for each (if the structure of the two manifolds was very different this need not be the case).
as kNN or NN), with the constraint that two points can be neighbors only if they come from the
same manifold. To proceed, we assume that if xj is a neighbor of xi , there then exists an unknown
ij such that W(xi , ij ) = xj to within a good approximation. Equivalently: H? (xi )ij ? xj ? xi .
We wish to find the best ? in the squared error sense (the ij being additional free parameters that
must be optimized over). The expression of the error we need to minimize is therefore:
error1 (?) = min
ij
{ }
n X
X
H? (xi )ij ? (xj ? xi )
2
2
(1)
i=1 j?N i
Minimizing the above error function can be interpreted as trying to find a warping function that can
transform a point into its neighbors. Note, however, that the warping function has only d degrees of
freedom while a point may have many more neighbors. This intuition allows us to rewrite the error
in an alternate form. Let ?i be the matrix where each column is of the form (xj ? xi ) for each
>
neighbor of xi . Let ?i = U i ?i V i be the thin singular value decomposition of ?i . Then, one can
show [9] that error1 is equivalent to the following:
error2 (?) = min
i
{E }
n
X
H? (xi )E i ? U i ?i
2
(2)
F
i=1
Here, the matrices E i are the additional free parameters. Minimizing the above can be interpreted
as searching for a warping function that directly explains the modes of variation at each point. This
form is convenient since we no longer have to keep track of neighbors. Furthermore, if there is no
noise and the linearity assumption holds there are at most d non- zero singular values. In practice we
use the truncated SVD, keeping at most 2d singular values, allowing for significant computational
savings.
We now give the remaining details of LSML for the general case [9]. For the case of images, we
present an efficient version in Section 5 which uses some basic domain knowledge to avoid solving
a large regression. Although potentially any regression technique is applicable, a linear model is
particularly easy to work with. Let f i be f features computed over xi . We can then define H? (xi ) =
[?1 f i ? ? ? ?D f i ]> , where each ?k is a d ? f matrix. Re- arranging error2 gives:
errorlin (?) = min
i
{E }
D
n X
2
X
i> k > i
i
?i
f ? E ? Uk?
i=1 k=1
2
(3)
Solving simultaneously for E and ? is complex, but if either E or ? is fixed, solving for the
remaining variable becomes a least squares problem (an equation of the form AXB = C can be
rewritten as B > ? A ? vec(X) = vec(C), where ? denotes the Kronecker product and vec the
(a)
(b)
(c)
(d)
Figure 2: Robustness.
LSML used to recover the embedding of the S- curve under a number of sampling
conditions. In each plot we show the original points along with the computed embedding (rotated to align
vertically), correspondence is indicated by coloring/shading (color was determined by the y- coordinate of the
embedding). In each case LSML was run with f = 8, d = 2, and neighbors computed by NN with = 1
(the height of the curve is 4). The embeddings shown were recovered from data that was: (a) densely sampled
(n=500) (b) sparsely sampled (n=100), (c) highly structured (n=190), and (d) noisy (n=500, random Gaussian
noise with ? = .1). In each case LSML recovered the correct embedding. For comparison, LLE recovered good
embeddings for (a) and (c) and I SOMAP for (a),(b), and (c). The experiments were repeated a number of times
yielding similar results. For a discussion see the text.
matrix vectorization function). To solve for ?, we use an alternating minimization procedure. In all
experiments in this paper we perform 30 iterations of the above procedure, and while local minima
do not seem to be to prevalent, we randomly restart the procedure 5 times. Finally, nowhere in
the construction have we enforced that the learned tangent vectors be orthogonal (such a constraint
would only be appropriate if the manifold was isometric to a plane). To avoid numerically unstable
solutions we regularize the error:
error0lin (?) = errorlin (?) + ?E
n
D
X
X
i
2
k
2
E
+ ??
?
F
F
i=1
(4)
k=1
For the features we use radial basis functions (RBFs) [11], the number of basis functions, f , being an
additional parameter. Each basis function is of the form f j (x) = exp(?kx ? ?j k22 /2? 2 ) where the
centers ?j are obtained using K- means clustering on the original data with f clusters and the width
parameter ? is set to be twice the average of the minimum distance between each cluster and its
nearest neighbor center. The feature vectors are then simply defined as f i = [f 1 (xi ) ? ? ? f p (xi )]> .
The parameter f controls the smoothness of the final mapping H? ; larger values result in mappings
that better fit local variations of the data, but whose generalization abilities to other points on the
manifold may be weaker. This is exactly analogous to the standard supervised setting and techniques
like cross validation could be used to optimize over f .
4
Experiments on Point Sets
We begin with a discussion on the intuition behind various aspects of LSML. We then show experiments demonstrating the robustness of the method, followed by a number of applications. In the
figures that follow we make use of color/shading to indicate point correspondences, for example
when we show the original point set and its embedding.
LSML learns a function H from points in RD to tangent directions that agree, up to a linear combination, with estimated tangent directions at the original training points of the manifold. By constraining
H to be smooth (through use of a limited number of RBFs), we can compute tangents at points not
seen during training, including points that may not lie on the underlying manifold. This generalization ability of H will be central to the types of applications considered. Finally, given multiple
non- overlapping manifolds with similar structure, we can train a single H to correctly predict the
tangents of each, allowing information to be shared. Fig. 1 gives a visual tutorial of these different
concepts.
LSML appears quite robust. Fig. 2 shows LSML successfully applied for recovering the embedding of
the ?S- curve? under a number of sampling conditions (similar results were obtained on the ?Swissroll?). After H is learned, the embedding is computed by choosing a random point on the manifold
(a)
(b)
(c)
(d)
Figure 3: Reconstruction. Reconstruction examples are used to demonstrate quality and generalization of
H. (a) Points sampled from the Swiss- roll manifold (middle), some recovered tangent vectors in a zoomedin region (left) and embedding found by LSML (right). Here n = 500 f = 20, d = 2, and neighbors were
computed by NN with = 4 (height of roll is 20). Reconstruction of Swiss- roll (b), created by a backprojection
from regularly spaced grid points in the embedding (traversal was done from a single original point located at
the base of the roll, see text for details). Another reconstruction (c), this time using all points and extending
the grid well beyond the support of the original data. The Swiss- roll is extended in a reasonable manner both
inward (occluded) and outward. (d) Reconstruction of unit hemisphere (LSML trained with n = 100 f = 6,
d = 2, NN with = .3) by traversing outward from topmost point, note reconstruction in regions with no
points.
and establishing a coordinate system by traversing outward (the same procedure can be used to
embed novel points, providing a natural out- of- sample extension). Here we compare only to LLE
and I SOMAP using published code. The densely sampled case, Fig. 2(a), is comparatively easy and
a number of methods have been shown to successfully recover an embedding. On sparsely sampled
data, Fig. 2(b), the problem is more challenging; LLE had problems for n < 250 (lowering LLE?s
regularization parameter helped somewhat). Real data need not be uniformly sampled, see Fig. 2(c).
In the presence of noise Fig. 2(d), I SOMAP and LLE performed poorly. A single outlier can distort
the shortest path computed by I SOMAP, and LLE does not directly use global information necessary
to disambiguate noise. Other methods are known to be robust [6], and in [25] the authors propose a
method to ?smooth? a manifold as a preprocessing step for manifold learning algorithms; however
a full comparison is outside the scope of this work.
Having learned H and computed an embedding, we can also backproject from a point y ? Rd to a
point x on the manifold by first finding the coordinate of the closest point yi in the original data,
then traversing from xi by j = yj ? yji along each tangent direction j (see Fig. 1(d)). Fig. 3(a)
shows tangents and an embedding recovered by LSML on the Swiss- roll. In Fig. 3(b) we backproject from a grid of points in R2 ; by linking adjacent sets of points to form quadrilaterals we can
display the resulting backprojected points as a surface. In Fig. 3(c), we likewise do a backprojection
(this time keeping all the original points), however we backproject grid points well below and above
the support of the original data. Although there is no ground truth here, the resulting extension of
the surface seems ?natural?. Fig. 3(d) shows the reconstruction of a unit hemisphere by traversing
outward from the topmost point. There is no isometric mapping (preserving distance) between a
hemisphere and a plane, and given a sphere there is actually not even a conformal mapping (preserving angles). In the latter case an embedding is not possible, however, we can still easily recover
H for both (only hemisphere results are shown).
5
Results on Images
Before continuing, we consider potential applications of H in the image domain, including tangent
distance estimation, nonlinear interpolation, extrapolation, compression, and motion transfer. We refer to results on point- sets to aid visualization. Tangent distance estimation: H computes the tangent
and can be used directly in invariant recognition schemes such as [21]. Compression: Fig. 3(b,d)
suggest how given a reference point and H nearby points can be reconstructed using d numbers
(with distortion increasing with distance). Nonlinear interpolation and extrapolation: points can be
generated within and beyond the support of given data (cf . Fig. 3); of potential use in tasks such as
frame rate up- conversion, reconstructing dropped frames and view synthesis. Motion transfer: for
certain classes of manifolds with ?parallel? structure (cf . Fig. 1(f)), a recovered warp may be used
on an entirely novel image. These applications will depend not only on the accuracy of the learned
H but also on how close a set of images is to a smooth manifold.
(a)
(b)
(c)
(d)
Figure 4: The translation manifold. Here F i = X i ; s = 17, d = 2 and 9 sets of 6 translated images each
were used (not including the cameraman). (a) Zero padded, smoothed test image x. (b) Visualization of learned
?, see text for details. (c) H? (x) computed via convolution. (d) Several transformations obtained after multiple
steps along manifold for different linear combinations of H? (x). Some artifacts due to error propagation start
to appear in the top figures.
The key insight to working with images is that although images can live in very high dimensional
spaces (with D ? 106 quite common), we do not have to learn a transformation with that many
parameters. Let x be an image and H?k (x), k ? [1, d], be the d tangent images. Here we assume
that each pixel in H?k (x) can be computed based only on the information in s ? s patch centered
on the corresponding pixel in x. Thus, instead of learning a function RD ? RD?d we learn a
2
function Rs ? Rd , and to compute H we apply the per patch function at each of the D locations
in the image. The resulting technique scales independently of D, in fact different sized images can
be used. The per patch assumption is not always suitable, most notably for transformations that are
based only on image coordinate and are independent of appearance.
The approach of Section 3 needs to be slightly modified to accommodate patches. We rewrite each
image xi ? RD as a s2 ? D matrix X i where each row contains pixels from one patch in xi (in
training we sub-sample patches). Patches from all the images are clustered to obtain the f RBFs;
each X i is then transformed to a f ? D matrix F i that contains the features computed for each
patch. The per patch linear model can now be written as H? (xi ) = (?F i )> , where ? is a d ? f
matrix (compare with the D ?s needed without the patch assumption). The error function, which is
minimized in a similar way [9], becomes:
errorimg (?) = min
i
{E }
n
2
X
i> > i
F ? E ? U i ?i
i=1
F
(5)
We begin with the illustrative example of translation (Fig. 4). Here, RBFs were not used, instead
F i = X i . The learned ? is a 2 ? s2 matrix, which can be visualized as two s ? s images as in Fig.
4(b). These resemble derivative of Gaussian filters, which are in fact the infinitesimal generates for
translation [16]. Computing the dot product of each column of ? with each patch can be done using
a convolution. Fig. 4 shows applications of the learned transformations, which resemble translations
with some artifacts.
Fig. 5 shows the application of LSML for learning out-of-plane rotation of a teapot. On this size
problem training LSML (in M ATLAB) takes a few minutes; convergence occurs within about 10
iterations of the minimization procedure. H? (x) for novel x can be computed with f convolutions
(to compute cross correlation) and is also fast. The outer frames in Fig. 5 highlight a limitation
of the approach: with every successive step error is introduced; eventually significant error can
accumulate. Here, we used a step size which gives roughly 10 interpolated frames between each
pair of original frames. With out-of-plane rotation, information must be created and the problem
becomes ambiguous (multiple manifolds can intersect at a single point), hence generalization across
images is not expected to be good.
In Fig. 6, results are shown on an eye manifold with 2 degrees of freedom. LSML was trained on
sparse data from video of a single eye; H? was used to synthesize views within and also well outside
the support of the original data (cf . Fig. 6(c)). In Fig. 6(d), we applied the transformation learned
from one person?s eye to a single image of another person?s eye (taken under the same imaging
conditions). LSML was able to start from the novel test image and generate a convincing series of
?~
?~
?
(a)
?~
?~
?
(b)
(c)
(d)
(e)
Figure 5: Manifold generated by out- of- plane rotation of a teapot (data from [23], sub- sampled and smoothed).
Here, d = 1, f = 400 and roughly 3000 patches of width s = 13 were sampled from 30 frames. Bottom row shows the
?
ground truth
dashed
box contains
of 30
images, representing
? 8(data
of physical
rotation.
top rowand
Figure
5: images;
Manifold
generated
by 3outof-training
plane rotation
of a teapot
from [24],
sub-The
sampled
shows
the
learned
transformation
applied
to
the
central
image.
By
observing
the
tip,
handle
the two from
white 30
blobs
on
smoothed). Here, d = 1, f = 400 and roughly 3000 patches of width s = 13
wereand
sampled
frames.
the teapot, and comparing to ground truth data, we can observe the quality of the learned transformation on seen data (b) ?
Bottom
row
shows
the
ground
truth
images;
dashed
box
contains
3
of
30
training
images,
representing
?
8
and unseen data (d), both starting from a single frame (c). The outmost figures (a)(e) shows failure for large rotations.
of physical rotation. The top row shows the learned transformation applied to the central image. By observing
the tip, handle and the two white blobs on the teapot, and comparing to ground truth data, we can observe the
quality of the learned transformation on seen data (b) and unseen data (d), both starting from a single frame (c).
Fig. outmost
5 showsfigures
the application
of LSML
out- of- plane rotation of a teapot. On this size problem
The
(a)(e) shows
failurefor
forlearning
large rotations.
training LSML (in M ATLAB) takes a few minutes; convergence occurs within about 10 iterations of the minimization procedure. H? (x) for novel x can be computed with f convolutions (to compute cross correlation)
and is also fast. The outer frames in Fig. 5 highlight a limitation of the approach: with every successive
step
error is introduced; eventually significant error can
accumulate. Here, we used a step size which gives
Q
Q 10 interpolated frames between each pair of original
roughly
frames. With out- of- plane rotation, information
Q
must beQcreated
and the problem becomes ambiguous
(multiple manifolds can intersect at a single point),
Q
3
k
Q
6
Q
hence generalization
across images
is not expected
to be good.
Q
Q
Q
Q
Q
Q
QQ
Q
Q
Q
Q
Q
Q
Q
3
k
Q
6
Q
Q
Q
Q
Q
QQ
+
sQ
?
Q
Q
Q
Q
Q
QQ
Q
Q
Q
Q
(a) QQ
(b)
(c)
(d)
Q
Figure
Q
1: Traversing the eye manifold. LSML
+
sQtrained on one eye moving along five different lines (3 vertical
2 horizontal). Here d?= 2, f = 600, Q
and
s =Q19 and around 5000 patches were sampled; 2 frames were
neighbors the
if they
were
adjacent in
time.trained
Figure (a)
images
generated
from
central image.
considered
Q
Figure
6:
Traversing
eye
manifold.
LSML
onshows
one eye
moving
along
fivethedifferent
lines (3 vertical
The inner 8 frames lie just outside the support of the Q
training
data (not shown), the outer 8 are extrapolated
Q
and beyond
2 horizontal).
Here
d
=
2,
f
=
600,
s
=
19
and
around
5000
patches
were
sampled;
frames were
its support. Figure (b) details H? (x) for two images in a warping sequence: a linear combination2can
considered
they in
were
adjacent
in time.
shows
images
from(c)the
central image.
lead the neighbors
iris/eyelid toifmove
different
directions
(e.g.Figure
the sum(a)
would
make
the irisgenerated
go up). Figure
shows
The extrapolation
inner 8 frames
lie just
ofwide
the open
training
dataclosed.
(not(c)
shown),
the outer
8 (d)
are
extrapolated
far beyond
the
training the
data,support
i.e. an eye
and
Finally, Figure(d)
shows
how
(a)outside
(b)fully
the eye
learned
one eyeH
can
be applied
a novel in
eyeanot
seen during
training.a linear combination can
beyond
its manifold
support. we
Figure
(b)ondetails
for twoonimages
warping
sequence:
? (x)
Traversing
eyeinmanifold.
trained(e.g.
on one
moving
vertical
2
Figure
lead
the6:iris/eyelid
to the
move
differentLSML
directions
theeye
sum
wouldalong
makefive
thedifferent
iris go lines
up). (3Figure
(c)and
shows
horizontal). Here d = 2, f = 600, s = 19 and around 5000 patches were sampled; 2 frames were considered neighbors
extrapolation
far beyond the training data, i.e. an eye wide open and fully closed. Finally, Figure(d) shows how
if they were adjacent in time. Figure (a) shows images generated from the central image. The inner 8 frames lie just
the
eye the
manifold
learned
on data
one eye
be applied
novel
eye not beyond
seen during
training.
outside
supportwe
of the
training
(not can
shown),
the outeron8 aare
extrapolated
its support.
Figure (b) details
H? (x) for two images in a warping sequence: a linear combination can lead the iris/eyelid to move in different directions
(e.g. the sum would make the iris go up). Figure (c) shows extrapolation far beyond the training data, i.e. an eye wide
open and fully closed. Finally, Figure(d) shows how the eye manifold we learned on one eye can be applied on a novel
transformations.
Thus, motion transfer was possible - H? trained on one series of images generalized
eye not seen during training.
to a different set of images.
6
Conclusion
In this work we presented an algorithm, Locally Smooth Manifold Learning, for learning the structure of a manifold. Rather than pose manifold learning as the problem of recovering an embedding,
we posed the problem in terms of learning a warping function for traversing the manifold. Smoothness assumptions on W allowed us to generalize to unseen data. Proposed uses of LSML include
tangent distance estimation, frame rate up- conversion, video compression and motion transfer.
We are currently engaged in scaling the implementation to handle large datasets; the goal is to
integrate LSML into recognition systems to provide increased invariance to transformations.
Acknowledgements
This work was funded by the following grants and organizations: NSF Career Grant #0448615,
Alfred P. Sloan Research Fellowship, NSF IGERT Grant DGE-0333451, and UCSD Division of
Calit2. We would like to thank Sameer Agarwal, Kristin Branson, Matt Tong, and Neel Joshi for
valuable input and Anna Shemorry for helping us make it through the deadline.
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2,244 | 3,036 | A Bayesian Approach to Diffusion Models of
Decision-Making and Response Time
Michael D. Lee?
Department of Cognitive Sciences
University of California, Irvine
Irvine, CA, 92697-5100.
[email protected]
Ian G. Fuss
Defence Science and Technology Organisation
PO Box 1500, Edinburgh, SA 5111, Australia
[email protected]
Daniel J. Navarro
School of Psychology
University of Adelaide, SA 5005, Australia
[email protected]
Abstract
We present a computational Bayesian approach for Wiener diffusion models,
which are prominent accounts of response time distributions in decision-making.
We first develop a general closed-form analytic approximation to the response
time distributions for one-dimensional diffusion processes, and derive the required
Wiener diffusion as a special case. We use this result to undertake Bayesian modeling of benchmark data, using posterior sampling to draw inferences about the
interesting psychological parameters. With the aid of the benchmark data, we
show the Bayesian account has several advantages, including dealing naturally
with the parameter variation needed to account for some key features of the data,
and providing quantitative measures to guide decisions about model construction.
1
Introduction
In the past decade, modern computational Bayesian methods have been productively applied to
the modeling of many core psychological phenomena. These areas include similarity modeling
and structure learning [1], concept and category learning [2, 3], inductive inference and decisionmaking [4], language processes [5], and individual differences [6]. One central area that has been
less affected is the modeling of response times in decision-making.
Nevertheless, the time people take to produce behavior is a basic and ubiquitious measure that can
constrain models and theories of human cognitive processes [7]. There is a large and well-developed
set of competing models that aim to account for accuracy, response time distributions and (sometimes) confidence in decision-making. However, besides the effective application of hierarchical
Bayesian methods to models that assume response times follow a Weibull distribution [8], most of
the inference remains frequentist. In particular, sequential sampling models of response time, which
are the dominant class in the field, have not adopted modern Bayesian methods for inference. The
prominent recent review paper by Ratcliff and Smith, for example, relies entirely on frequentist
methods for parameter estimation, and does not go beyond the application of the Bayesian Information Criterion for model selection [9].
?
Address correspondence to: Michael D. Lee, Department of Cognitive Sciences, 3151 Social Sciences
Plaza, University of California, Irvine, CA 92697-5100. Telephone: (949) 824 5074. Facsimile: (949) 824
2307. URL: www.socsci.uci.edu/?mdlee.
f (t??,?,?,?)
Evidence (x)
?
?
0
?
?
?
f?(t??,?,?,?)
Response Time (t)
Figure 1: A diffusion model for response time distributions for both decisions in a two-choice
decision-making task. See text for details.
Much of the utility, however, in using sequential sampling models to understand decision-making,
and their application to practical problems [10], requires making inferences about variations in parameters across subjects, stimuli, or experimental conditions. These inferences would benefit from
the principled representation of uncertainty inherent in the Bayesian approach. In addition, many
of the competing models have many parameters, are non-linear, and are non-nested. This means
their comparison would benefit from Bayesian methods for model selection that do not approximate
model complexity by counting the number of free parameters, as the Bayesian Information Criterion
does.
In this paper, we present a computational Bayesian approach for Wiener diffusion models [11],
which are the most widely used special case of the sequential sampling approach. We apply our
Bayesian method to the benchmark data of Ratcliff and Rouder [12], using posterior sampling to
draw inferences about the interesting psychological parameters. With the aid of this application,
we show that adopting the Bayesian perspective has several advantages, including dealing naturally
with the parameter variation needed to account for some key features of the data, and providing
quantitative measures to guide decisions about model construction.
2
The Diffusion Model and its Application to Benchmark Data
2.1 The Basic Model
The basic one-dimensional diffusion model for accuracy and response time distributions in a twochoice decision-making task is shown in Figure 2.1. Time, t, progresses from left to right, and
includes an fixed offset ? that parameterizes the time taken for the non-decision component of response time, such as the time taken to encode the stimulus and complete a motor response.
The decision-making component itself is driven by independent samples from an stationary distribution that represents the evidence the stimulus provides in favor of the two alternative decisions. In
the Wiener diffusion, this distribution is assumed to be Gaussian, with mean ?. Evidence sampled
from this distribution is accrued over time, leading to a diffusion process that is finally absorbed by
boundaries above and below at distances ? and ? from the origin. The response time distribution is
then given by the first-passage distribution p (t | ?, ?, ?, ?) = f? (t | ?, ?, ?, ?) + f? (t | ?, ?, ?, ?),
with the areas under f? and f? giving the proportion of decisions at each boundary.
A natural reparameterization is to consider the starting point of evidence accrual z =
(? ? ?) / (? + ?), which is considered a measure of bias, and the boundary separation a = ? + ?,
which is considered a measure of caution. In either case, this basic form of the model has four free
parameters: ?, ? and either ? and ? or z and a.
2.2 Previous Application to Benchmark Data
The evolution of Wiener diffusion models of decision-making has involved a series of additional
assumptions to address shortcomings in its ability to capture basic empirical regularities. This evolution is well described by Ratcliff and Rouder [12], who, in their Experiment 1 present a diffusion
model analysis of a benchmark data set [8, 13].
In this experiment, three observers completed ten 35 minute sessions, each consisting of ten blocks
with 102 trials per block. The task of observers was to decide between ?bright? and ?dark? responses
for simple visual stimuli with different proportions of white dots, given noisy feedback about the
accuracy of the responses. There were 33 different types of stimuli, ranging from 0% to 100% white
in equal increments. In addition, the subjects were required to switch between adherence to ?speed?
instructions and ?accuracy? instructions every two blocks. In accord with the experimental design,
Ratcliff and Rouder fitted separate drift rates ?i for each of the i = 1, . . ., 33 stimuli; fitted separate
boundaries for speed and accuracy instructions, but assumed the boundaries were symmetric (i.e.,
there was no bias), so that ?j = ?j for j = 1, 2; and fitted one offset ? for all stimuli and instructions.
The data from this experiment are considered benchmark because they show a cross-over effect,
whereby errors are faster than correct decisions for easy stimulus conditions under speed instructions, but errors are as slow or slower than correct decisions for hard stimulus conditions under
accuracy instructions. As Ratcliff and Rouder point out, these trends are not accommodated by the
basic model without allowing for variation in the parameters. Accordingly, to predict fast errors,
the basic model is extended by assuming that the starting point is subject to between-trial variation,
and so is convolved with a Gaussian or uniform distribution. Similarly, to predict slow errors, it is
assumed that the mean drift rate is also subject to between-trial variation, and so is convolved with a
Gaussian distribution. Both of these noise processes are parameterized with the standard sufficient
statistics, which become additional parameters of the model.
3
Closed-form Response Time Distributions for Diffusion Models
One practical reason diffusion models have resisted a Bayesian treatment is that the evaluation of
their likelihood function through standard methods is computationally intensive, typically requiring
the estimation of an oscillating but convergent infinite sum for each datum [14]. Instead, we use
a new closed-form approximation to the required response time distribution. We give a very brief
presentation of the approximation here. A more detailed technical note is available from the first
author?s web page.
The key assumption in our approximation is that the evolving diffusion distributions always assume
a limiting form f. Given this form, we define the required limit for a sampling distribution with
respect to an arbitrary time dependent mean ? (t, ?) and variance ?2 (t, ?), both which depend on
parameters ? of the sampling distribution, in terms of the evidence accumulated x, as
f x; ? (t, ?) , ?2 (t, ?) ,
(1)
from which the cumulative function at an upper boundary of one unit is obtained as
Z ?
F1 ? (t, ?) , ?2 (t, ?) =
f x; ? (t, ?) , ?2 (t, ?) dx.
(2)
1
Differentiation, followed by algebraic manipulation gives the general result
f1 ? (t, ?) , ?2 (t, ?)
=
=
d
F1 ? (t, ?) , ?2 (t, ?)
dt
?
? (t, ?) [1 ? ? (t, ?)] + ? (t, ?)
?t
?2 (t, ?)
?
? (t, ?)
?t
f
1 ? ? (t, ?)
? (t, ?)
.(3)
Weiner diffusion from the origin to boundaries ? and ? with mean drift rate ? and variance ?2 (t) = t
can be represented in this model by defining f (y) = exp ? 12 y2 , rescaling to a variance ?2 (t) =
1
1
a2 t and setting ? (t, ?) = a ?t + z. Thus the response time distributions for Weiner diffusion in this
?=?=10
?=?=15
?=15,?=10
Response Time Distribution
?=0.1
?=0.05
?=0.01
Response Time
Figure 2: Comparison between the closed form approximation (dark broken lines) and the infinite
sum distributions (light solid lines) for nine realistic combinations of drift rate and boundaries.
approximation are
f? (t | ?, ?, ?, ?) =
2? + ? (t ? ?)
3
2 (t ? ?) 2
f? (t | ?, ?, ?, ?) =
2? ? ? (t ? ?)
3
2 (t ? ?) 2
(2? ? ? (t ? ?))
exp ?
2 (t ? ?)
(2? + ? (t ? ?))
exp ?
2 (t ? ?)
2
2
!
!
,
.
(4)
3.1 Adequacy of the Wiener Approximation
Figure 2 shows the relationship between the response time distributions found by the previous infinite sum method, and those generated by our closed form approximation. For every combination
of drift rates ? = 0.01, 0.05 and 0.10, and boundary combinations ? = ? = 10, ? = ? = 15 and
? = 15, ? = 10 we found the best (least-squares) match between the infinite-sum distribution and
those distributions indexed by our approximation. These generating parameter combinations were
chosen because they cover the range of the posterior distributions we infer from data later. Figure 2
shows that the approximation provides close matches across these parameterizations, although we
note the approximation distributions do seem to use slightly (and apparently systematically) different parameter combinations to generate the best-matching distribution. While additional work is
required to understand the exact relationship between the infinite-sum method and our approximation, the approximation is sufficiently accurate over the range of parameterizations of interest to be
used as the basis for beginning to apply Bayesian methods to diffusion models.
4
Bayesian Modeling of Benchmark Data
4.1 General Model
Our log likelihood function evaluates the density of each response time at the boundary corresponding to its associated decision, and assumes independence, so that
X
X
ln f? (t | ?, ?, ?, ?) +
ln f? (t | ?, ?, ?, ?) ,
(5)
ln L (T | ?, ?, ?, ?) =
t?D?
t?D?
?j
?i
tijk
k = 1, . . ., n
i = 1, . . . , 33
?
?j
j = 1, 2
Figure 3: Graphical model for the benchmark data analysis.
where D? and D? are the set of all response times at the upper and lower boundaries respectively.
We threshold at 10?30 to guard against degeneracy to negative values inherent in the approximation.
The graphical model representation of the benchmark data is shown in Figure 3, where the observed
response time data are now denoted T = {tijk }, with i = 1, . . ., 33 indexing the presented stimulus, j = 1, 2 indexing speed or accuracy instructions, and k = 1, . . ., n indexing all of the trials
with this stimulus and instruction combination. We place proper approximations to non-informative
distributions on all the parameters, so that they are all essentially flat over the values of interest.
Specifically we assume the 33 drift rates are independent and each have a zero mean Gaussian prior
with very small precision: ?i ? Gaussian (0, ? ), with ? = 10?6. The boundary parameters ? and ?
are given the same priors, but because they are constrained to be positive, their sampling is censored
accordingly: ?j , ?j ? Gaussian (0, ? ) ; ?j , ?j > 0. Since ? is bounded by the minimum time
observed, we use the Uniform prior distribution: ? ? Uniform (0, min T ). This is a data-dependent
prior, but the same results could be achieved with a fixed prior, and scaling of the time data, which
are arbitrary up to scalar multiplication.
4.2 Formalizing Model Construction
The Bayesian approach allows us to test the intuitively plausible model construction decisions made
previously by Ratcliff and Rouder. Using the data from one of the observers (N.H.), we considered the marginal likelihoods, denoted simply L, based on the harmonic mean approximation [15],
calculated from three chains of 105 samples from the posterior obtained using Winbugs.
? The full model described by Figure 3, with asymmetric boundaries, varying across speed
and accuracy instructions. This model had ln L = ?48, 416.
? The restricted model with symmetric boundaries, still varying across instructions, as assumed by Ratcliff and Rouder. This model had ln L = ?48, 264.
? The restricted model with asymmetric boundaries not varying across instructions. This
model had ln L = ?48, 964.
? The restricted model with symmetric boundaries not varying across instructions. This
model had ln L = ?48, 907.
These marginal log likelihoods make it clear that different boundaries are needed for the speed and
accuracy instructions, but it is overly complicated to allow them to be asymmetric (i.e., there is no
need to parameterize bias). We tested the robustness of these values by halving and doubling the
prior variances, and using an adapted form of the ?informative? priors collated in [16], all of which
lead to similar quantitative and identical qualitative conclusions. These results formally justify the
model construction decisions made by Ratcliff and Rouder, and the remainder of our analysis applies
to this restricted model.
4.3 Posterior Distributions
Figure 4 shows the posterior distributions for the symmetric boundaries under both speed and accuracy instructions. These distributions are consistent with traditional analyses and the speed boundary
is clearly significantly smaller than the accuracy boundary. We note that, for historical reasons only,
2
Wiener diffusion models have assumed ?2 (t) = (0.1t) , and so our scale is 100 times larger.
p(??T)
Speed
Accuracy
0
5
10
Boundary Values
15
20
Figure 4: Posterior distributions for the boundaries under speed and accuracy instructions.
The main panel of Figure 5 shows the posterior distributions for all 33 drift rate parameters. The posteriors are shown against the vertical axes, with wider bars corresponding to greater density, and are
located according to their proportion of white dots on the horizontal axis. The approximately monotonic relationship between drift rate and proportion shows that the model allows stimulus properties
to be inferred from the behavioral decision time data, as found by previous analyses.
The right hand panel of Figure 5 shows the projection of three of the posterior distributions, labelled
4, 17 and 22. It is interesting to note that the uncertainty about the drift rate of stimuli 4 and 17 both
take a Gaussian form, but with very different variances. More dramatically, the uncertainty about
the drift rate for stimulus 22 is clearly bi-modal.
0.15
Drift Rate (?i)
0.1
0.05
22
0
17
?0.05
?0.1
?0.15
4
0
0.1
17
0.2
4
22
0.3 0.4 0.5 0.6 0.7
Stimulus Proportion White
0.8
0.9
1
p(?i?T)
Figure 5: Posterior distributions for the 33 drift rates, in terms of the proportion of white dots in
their associated stimuli.
4.4 Accuracy and Fast and Slow Errors
Figure 6 follows previous analyses of these data, and shows the relationship between the empirical
proportions, and the model prediction of decision proportions for each stimulus type. For both the
speed and accuracy instructions, there is close agreement between the model and data.
Figure 7 shows the posterior predictive distribution of the model for two cases, analogous to those
highlighted previously [13, Figure 6]. The left panel involves a relatively easy decision, corresponding to stimulus number 22 under speed instructions, and shows the models predictions for the
response time for both correct (upper) and error (lower) decisions, together with the data, indicated
by short vertical lines. For this easy decision, it can be seen the model predicts relatively fast errors.
The right panel of Figure 7 involves a harder decision, corresponding to stimulus number 18 under
Accuracy
1
0.8
0.8
0.6
0.6
Data
Data
Speed
1
0.4
0.2
0.4
0.2
0
0
0
0.2
0.4 0.6
Model
0.8
1
0
0.2
0.4 0.6
Model
0.8
1
Figure 6: Relationship between modeled and empirical accuracy, for the speed instructions (left
panel) and accuracy instructions (right panel). Each marker corresponds to one of the 33 stimuli.
accuracy instructions. Here the model predicts much slower errors, with a heavier tail than for the
easy decision.
Response Time Distribution
These are the basic qualitative properties of prediction that motivated the introduction of betweentrial variability through noise processes in the traditional account. In the present Bayesian treatment,
the required predictions are achieved because the posterior predictive automatically samples from a
range of values for the drift and boundary parameters. By representing this variation in parameters
as uncertainty about fixed values, we are making different basic assumptions from the traditional
Wiener diffusion model. It is interesting to speculate that, if Bayesian results like those in Figure 7 had always been available, the introduction of additional variability processes described in
[12] might never have eventuated. These processes seem solely designed to account for empirical
effects like the cross-over effect; in particular, we are not aware of the parameters of the additional
variability processes being used to draw substantive psychological inferences from data.
0
1500
Response Time (ms)
0
6000
Response Time (ms)
Figure 7: Posterior predictive distributions for both correct (solid line) and error (broken line) responses, for two stimuli corresponding to easy (left panel) and hard (right panel) decisions. The
density for error decisions in the easy responses has been scaled to allow its shape to be visible.
5
Conclusions
Our analyses of the benchmark data confirm many of the central conclusions of previous analyses,
but also make several new contributions. The posterior distributions shown in Figure 5 suggest
that current parametric assumptions about drift rate variability may not be entirely appropriate. In
particular, there is the intriguing possibility of multi-modalities evident in the drift rate of stimulus
22, and the associated raw data in Figure 7.
Figure 5 also suggests a hierarchical account of the benchmark data, modeling the 33 drift rates ?i
in terms of, for example, a low-dimensional psychometric function. This would be easily achieved
in the current Bayesian framework. It should also be possible to introduce contaminant distributions
in a mixture model, following previous suggestions [8, 14], using latent variable assignments for
each response time. If it was desirable to replicate the current assumptions of starting-point and
drift-rate variability, that would also easily be done in an extended hierarchical account. Finally, the
availability of marginal likelihood measures, accounting for both model fit and complexity, offer the
possibility of rigorous quantitative comparisons of alternative sequential sampling accounts, such as
the Ornstein-Uhlenbeck and accumulator models [9], of response times in decision-making.
Acknowledgments
We thank Jeff Rouder for supplying the benchmark data, and Scott Brown, E.-J. Wagenmakers, and
the reviewers for helpful comments.
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[15] A. E. Raftery. Hypothesis testing and model selection. In W. R. Gilks, S. Richardson, and
D. J. Spiegelhalter, editors, Markov chain Monte Carlo in Practice, pages 163?187. Chapman
& Hall/CRC, Boca Raton (FL), 1996.
[16] E.-J. Wagenmakers, H. J. L. van der Maas, and R. P. P. P. Grasman. An EZ?diffusion model
for response time and accuracy. Psychonomic Bulletin & Review, in press.
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2,245 | 3,037 | Bayesian Image Super-resolution, Continued
Lyndsey C. Pickup, David P. Capel? , Stephen J. Roberts Andrew Zisserman
Information Engineering Building, Dept. of Eng. Science, Parks Road, Oxford, OX1 3PJ, UK
{elle,sjrob,az}@robots.ox.ac.uk
?
2D3, [email protected]
Abstract
This paper develops a multi-frame image super-resolution approach from a
Bayesian view-point by marginalizing over the unknown registration parameters
relating the set of input low-resolution views. In Tipping and Bishop?s Bayesian
image super-resolution approach [16], the marginalization was over the superresolution image, necessitating the use of an unfavorable image prior. By integrating over the registration parameters rather than the high-resolution image, our
method allows for more realistic prior distributions, and also reduces the dimension of the integral considerably, removing the main computational bottleneck of
the other algorithm. In addition to the motion model used by Tipping and Bishop,
illumination components are introduced into the generative model, allowing us
to handle changes in lighting as well as motion. We show results on real and
synthetic datasets to illustrate the efficacy of this approach.
1 Introduction
Multi-frame image super-resolution refers to the process by which a group of images of the same
scene are fused to produce an image or images with a higher spatial resolution, or with more visible
detail in the high spatial frequency features [7]. Such problems are common, with everything from
holiday snaps and DVD frames to satellite terrain imagery providing collections of low-resolution
images to be enhanced, for instance to produce a more aesthetic image for media publication [15],
or for higher-level vision tasks such as object recognition or localization [5].
Limits on the resolution of the original imaging device can be improved by exploiting the relative
sub-pixel motion between the scene and the imaging plane. No matter how accurate the registration
estimate, there will be some residual uncertainty associated with the parameters [13]. We propose a
scheme to deal with this uncertainty by integrating over the registration parameters, and demonstrate
improved results on synthetic and real digital image data.
Image registration and super-resolution are often treated as distinct processes, to be considered sequentially [1, 3, 7]. Hardie et al. demonstrated that the low-resolution image registration can be
updated using the super-resolution image estimate, and that this improves a Maximum a Posteriori
(MAP) super-resolution image estimate [5]. More recently, Pickup et al. used a similar joint MAP
approach to learn more general geometric and photometric registrations, the super-resolution image,
and values for the prior?s parameters simultaneously [12]. Tipping and Bishop?s Bayesian image
super-resolution work [16] uses a Maximum Likelihood (ML) point estimate of the registration parameters and the camera imaging blur, found by integrating the high-resolution image out of the
registration problem and optimizing the marginal probability of the observed low-resolution images
directly. This gives an improvement in the accuracy of the recovered registration (measured against
known truth on synthetic data) compared to the MAP approach.
The image-integrating Bayesian super-resolution method [16] is extremely costly in terms of computation time, requiring operations that scale with the cube of the total number of high-resolution
pixels, severely limiting the size of the image patches over which they perform the registration (they
use 9 ? 9 pixel patches). The marginalization also requires a form of prior on the super-resolution
image that renders the integral tractable, though priors such as Tipping and Bishop?s chosen Gaussian form are known to be poor for tasks such as edge preservation, and much super-resolution work
has employed other more favorable priors [2, 3, 4, 11, 14].
It is generally more desirable to integrate over the registration parameters rather than the superresolution image, because it is the registration that constitutes the ?nuisance parameters?, and the
super-resolution image that we wish to estimate. We derive a new view of Bayesian image superresolution in which a MAP high-resolution image estimate is found by marginalizing over the
uncertain registration parameters. Memory requirements are considerably lower than the imageintegrating case; while the algorithm is more costly than a simple MAP super-resolution estimate, it
is not infeasible to run on images of several hundred pixels in size.
Sections 2 and 3 develop the model and the proposed objective function. Section 4 evaluates results on synthetically-generated sequences (with ground truth for comparison), and on a real data
example. A discussion of this approach and concluding remarks can be found in section 5.
2 Generative model
The generative model for multi-frame super-resolution assumes a known scene x (vectorized, size
N ? 1), and a given registration vector ?(k) . These are used to generate a vectorized low-resolution
image y(k) with M pixels through a system matrix W(k) . Gaussian i.i.d. noise with precision ? is
then added to y(k) ,
(k)
(k)
y(k) = ?(k)
x + ?? + (k)
(1)
? W ?
(k) ? N 0, ? ?1 I .
(2)
Photometric parameters ?? and ?? provide a global affine correction for the scene illumination, and
?? is simply an M ? 1 vector filled out with the value of ?? . Each row of W(k) constructs a single
pixel in y(k) , and the row?s entries are the vectorized and point-spread function (PSF) response
for each low-resolution pixel, in the frame of the super-resolution image [2, 3, 16]. The PSF is
usually assumed to be an isotropic Gaussian on the imaging plane, though for some motion models
(e.g. planar projective) this does not necessarily lead to a Gaussian distribution on the frame of x.
For an individual low-resolution image, given registrations and x, the data likelihood is
M2
2
?
?
(k)
(k)
exp ? y(k) ? ?(k)
W
?
x
?
?
. (3)
p y(k) x, ?(k) , ?(k)
=
?
?
2?
2
2
When the registration is known approximately, for instance by pre-registering inputs, the uncertainty
? (k) for each image?s parameter
can be modeled as a Gaussian perturbation about the mean estimate ?
set, with covariance C, which we restrict to be a diagonal matrix,
?
?
? (k) ?
?
?(k)
?
? (k) ?
? ? (k) ?
(k)
(4)
? ?? ? = ? ?? ? + ?
(k)
(k)
?
??
??
? (k)
p ?(k) , ?(k)
?
=
N (0, C)
?1 21
|C |
1 (k)T ?1 (k)
exp ? ?
C ?
.
(2?)n
2
(5)
(6)
A Huber prior is assumed for the directional image gradients Dx in the super-resolution image x
(in the horizontal, vertical, and two diagonal directions),
n ?
o
1
exp ? ? (Dx, ?)
(7)
p (x) =
Z
2
x 2
z
if |z| < ?
?(z, ?) =
(8)
2?|z| ? ?2 otherwise
where ? is a parameter of the Huber potential function, and ? is the prior strength parameter. This
belongs to a family of functions often favored over Gaussians for super-resolution image priors [2,
3, 14] because the Huber distribution?s heavy tails mean image edges are penalized less severely.
The difficulty in computing the partition function Zx is a consideration when marginalizing over x
as in [16], though for the MAP image estimate, a value for this scale factor is not required.
Regardless of the exact forms of these probability distributions, probabilistic super-resolution algorithms can usually be interpreted in one of the following ways.
The most popular approach to super-resolution
is to
obtain a MAP estimate, typically using an
iterative scheme to maximize p x y( k), ?(k) , ?(k) with respect to x, where
QK
n
o
p (x) k=1 p y(k) x, ?(k) , ?(k)
(
(k)
(k)
p x y k), ? , ?
,
(9)
=
p y(k) ?(k) , ?(k)
and the denominator is an unknown scaling factor.
Tipping and Bishop?s approach takes an ML estimate of the registration by marginalizing over x,
then calculates the super-resolution estimate as in (9). While Tipping and Bishop did not include a
photometric model, the equivalent expression to be maximized with respect to ? and ? is
Z
K
n
o n
o
Y
p y(y) ? (k) , ?(k)
=
p (x)
p y(y) x, ?(k) , ?(k) dx.
(10)
k=1
Note that Tipping and Bishop?s work does employ the same data likelihood expression as in (3),
which forced them to select a Gaussian form for p (x), rather than a more suitable image prior, in
order to keep the integral tractable.
Finally, in this paper we find x through
marginalizing
over ? and ?, so that a MAP estimate of x can
be obtained by maximizing p x y(k) directly with respect to x. This is achieved by finding
Z Y
K
n
o
p(x)
(k)
(k)
(k)
(k)
(k)
=
p
?
,
?
p
y
?
,
?
d {?, ?} , (11)
p x y(k)
x,
p y(k)
k=1
which is developed further in the next section. Note that the integral does not involve the prior, p (x).
3 Marginalizing over registration parameters
In order to obtain an expression for p x| y(k)
from expressions (3), (6) and (7) above, the
(k)
? (k) , ?
??
parameter variations ?
must be integrated out of the problem. Registration estimates ?
?
and ?? can be obtained using classical registration methods, either intensity-based [8] or estimation
from image points [6], and the diagonal matrix C is constructed to reflect the confidence in each
parameter estimate. This might mean a standard deviation of a tenth of a low-resolution pixel on
image translation parameters, or a few gray levels? shift on the illumination model, for instance.
The integral performed is
KM
Kn
n ?
o
2
2
1
?
b
1
exp
?
?
(Dx,
?)
2?
2?
Zx
2
p y(k)
(
)
Z
K
2 1
X ?
(k)
(k)
(k)
(k)
(k) (k)?1 (k)
? exp ?
x ? ?? + ? C
?
d?, (12)
y ? ?? W ?
2
2
2
k=1
where ? T = ? (1)T , ? (2)T , . . . , ? (K)T and all the ? and ? parameters are functions of ? as in
(4). Expanding the data error term in the exponent for each low-resolution image as a second-order
Taylor series about the estimated geometric registration parameter yields
2
(k)
e(k) (?) = y(k) ? ?? ?(k) W(k) ?(k) x ? ?? ? (k)
(13)
n
o
p x| y(k)
=
2
1
= F (k) + G(k)T ? + ? (k)T H(k) ? (k) ,
2
(14)
Values for F , G and H can be found numerically
(for geometric
n
o registrations) or analytically (for
(k)
(k)
the photometric parameters) from x and y(k) , ?(k) , ?? , ?? . Thus the whole exponent of (12),
f , becomes
f
K
X
? (k) ? (k)T (k) 1 (k)T ? (k)
(k)
?1
?
? ? ?
H +C
=
? F ? G
2
2
2
2
k=1
?
?
1
?
= ? F ? GT ? ? ? T
H + V?1 ?,
2
2
2
2
(15)
(16)
where the omission of image superscripts indicates stacked matrices, and H is therefore a blockdiagonal nK ? nK sparse matrix, and V is comprised of the repeated diagonal of C.
Finally, letting S =
Z
?
2H
+ V?1 ,
exp {f } d?
=
=
Z
?
1 T
? T
exp ? F
exp ? G ? ? ? S? d?
2
2
2
2
nK
1
?
?
?2
T ?1
2
G S G .
exp ? F (2?) |S| exp
2
8
(17)
(18)
The objective function, L, to be minimized with respect to x is obtained by taking the negative log
of (12), using the result from (18), and neglecting the constant terms:
L =
?
?
1
? 2 T ?1
? (Dx, ?) + F + log |S| ?
G S G.
2
2
2
8
(19)
This can be optimized using Scaled Conjugate Gradients (SCG) [9], noting that the gradient can be
expressed
dL
dx
=
? T d
? dF
? 2 T ?1 dG
D
? (Dx) +
?
G S
2 dx
2 dx
4
dx
3
dvecH
?
?
T
+
vec S?1 ?
GT S?1 ? GT S?1
,
4
16
dx
(20)
where derivatives of F , G and H with respect to x can be found analytically for photometric parameters, and numerically (using the analytic gradient of e(k) ? (k) with respect to x) with respect
to the geometric parameters.
3.1 Implementation notes
Notice that the value F from (16) is simply the reprojection error of the current estimate of x at
the mean registration parameter values, and that gradients of this expression with respect to the ?
parameters, and with respect to x can both be found analytically. To find the gradient with respect to
(k)
a geometric registration parameter ?i , and elements of the Hessian involving it, a central difference
scheme involving only the k th image is used.
Mean values for the registration are computed by standard registration techniques, and x is initialized
using around 10 iterations of SCG to find the maximum likelihood solution evaluated at these mean
parameters. Additionally, pixel values are scaled to lie between ? 21 and 12 , and the ML solution is
bounded to lie within these values in order to curb the severe overfitting usually observed in ML
super-resolution results.
In our implementation, the parameters representing the ? values are scaled so that they share the
same standard deviations as the ? parameters, which represent the sub-pixel geometric registration
shifts, which makes the matrix V a multiple of the identity. The scale factors are chosen so that one
standard deviation in ?? gives a 10-gray-level shift, and one standard deviation in ?? varies pixel
values by around 10 gray levels at mean image intensity.
4 Results
The first experiment takes a sixteen-image synthetic dataset created from an eyechart image. Data is
generated at a zoom factor of 4, using a 2D translation-only motion model, and the two-parameter
global affine illumination model described above, giving a total of four registration parameters per
low-resolution image. Gaussian noise with standard deviation equivalent to 5 gray levels is added
to each low-resolution pixel independently. The sub-pixel perturbations are evenly spaced over a
grid up to plus or minus one half of a low-resolution pixel, giving a similar setup to that described
in [10], but with additional lighting variation. The ground truth image and two of the low-resolution
images appear in the first row of Figure 1.
Geometric and photometric registration parameters were initialized to the identity, and the images
were registered using an iterative intensity-based scheme. The resulting parameter values were used
to recover two sets of super-resolution images: one using the standard Huber MAP algorithm, and
the second using our extension integrating over the registration uncertainty. The Huber parameter ?
was fixed at 0.01 for all runs, and ? was varied over a range of possible values representing ratios
between ? and the image noise precision ?.
The images giving lowest RMS error from each set are displayed in the second row of Figure 1.
Visually, the differences between the images are subtle, though the bottom row of letters is better
defined in the output from the new algorithm. Plotting the RMSE as a function of ? in Figure 2,
we see that the proposed registration-integrating approach achieves a lower error, compared to the
ground truth high-resolution image, than the standard Huber MAP algorithm for any choice of prior
strength, ? in the optimal region.
(a) ground truth high?res
(d) best Huber (err = 15.6)
(b) input 1/16
(c) input 16/16
(e) best int???? (err = 14.8)
Figure 1: (a) Ground truth image. Only the central recoverable part is shown; (b,c) low-resolution
images. The variation in intensity is clearly visible, and the sub-pixel displacements necessary for
multi-frame image super-resolution are most apparent on the ?D? characters to the right of each image; (d) The best (?.e. minimum MSE ? see Figure 2) image from the regular Huber MAP algorithm,
having super-resolved the dataset multiple times with different prior strength settings; (e) The best
result using out approach of integrating over ? and ?. As well as having a lower RMSE, note the
improvement in black-white edge detail on some of the letters on the bottom line.
The second experiment uses real data with a 2D translation motion model and an affine lighting
model exactly as above. The first and last images appear on the top row of Figure 3. Image registration was carried out in the same manner as before, and the geometric parameters agree with the
provided homographies to within a few hundredths of a pixel. Super-resolution images were created
RMSE comparison
23
Standard Huber MAP
Integrating over registrations and illumination
22
RMSE in gray levels
21
20
19
18
17
16
15
14
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
ratio of prior strength parameter, ?, and noise precision, ?
0.09
0.1
Figure 2: Plot showing the variation of RMSE with prior strength for the standard Huber-prior MAP
super-resolution method and our approach integrating over ? and ?. The images corresponding to
the minima of the two curves are shown in Figure 1
for a number of ? values, the equivalent values to those quoted in [3] were found subjectively to be
the most suitable.
The covariance of the registration values was chosen to be similar to that used in the synthetic
experiments. Finally, Tipping and Bishop?s method was extended to cover the illumination model
and used to register and super-resolve the dataset, using the same PSF standard deviation (0.4 lowresolution pixels) as the other methods.
The three sets of results on the real data sequence are shown in the middle and bottom rows of
Figure 3. To facilitate a better comparison, a sub-region of each is expanded to make the letter
details clearer. The Huber prior tends to make the edges unnaturally sharp, though it is very successful at regularizing the solution elsewhere. Between the Tipping and Bishop image and the
registration-integrating approach, the text appears more clear in our method, and the regularization
in the constant background regions is slightly more successful.
5 Discussion
It is possible to interpret the extra terms introduced into the objective function in the derivation
of this method as an extra regularizer term or image prior. Considering (19), the first two terms
are identical to the standard MAP super-resolution problem using a Huber image prior. The two
additional terms constitute an additional distribution over x in the cases where S is not dominated
by V; as the distribution over ? and ? tightens to a single point, the terms tend to constant values.
The intuition behind the method?s success is that this extra prior resulting from the final two terms
of (19) will favor image solutions which are not acutely sensitive to minor adjustments in the image
registration. The images of figure 4 illustrate the type of solution which would score poorly. To
create the figure, one dataset was used to produce two super-resolved images, using two independent
sets of registration parameters which were randomly perturbed by an i.i.d. Gaussian vector with a
standard deviation of only 0.04 low-resolution pixels. The checker-board pattern typical of ML
super-resolution images can be observed, and the difference image on the right shows the drastic
contrast between the two image estimates.
(a) input 1/10
(c) integrating ?, ?
(e) regular Huber (detailed region)
(b) input 10/10
(d) integrating ?, ? (detailed region)
(f) Tipping & Bishop (detailed region)
Figure 3: (a,b) First and last images from a real data sequence containing 10 images acquired on a
rig which constrained the motion to be pure translation in 2D. (c) The full super-resolution output
from our algorithm. (d) Detailed region of the central letters, again with our algorithm. (e) Detailed
region of the regular Huber MAP super-resolution image, using parameter values suggested in [3],
which are also found to be subjectively good choices. The edges are slightly artificially crisp, but the
large smooth regions are well regularized. (f) Close-up of letter detail for comparison with Tipping
and Bishop?s method of marginalization. The Gaussian form of their prior leads to a more blurred
output, or one that over-fits to the image noise on the input data if the prior?s influence is decreased.
5.1 Conclusion
This work has developed an alternative approach for Bayesian image super-resolution with several
advantages over Tipping and Bishop?s original algorithm. These are namely a formal treatment of
registration uncertainty, the use of a much more realistic image prior, and the computational speed
and memory efficiency relating to the smaller dimension of the space over which we integrate.
The results on real and synthetic images with this method show an advantage over the popular
MAP approach, and over the result from Tipping and Bishop?s method, largely owing to our more
favorable prior over the super-resolution image.
It will be a straightforward extension of the current approach to incorporate learning for the pointspread function covariance, though it will result in a less sparse Hessian matrix H, because each
row and column associated with the PSF parameter(s) has the potential to be full-rank, assuming a
common camera configuration is shared across all the frames.
Finally, the best way of learning the appropriate covariance values for the distribution over ? given
the observed data, and how to assess the trade-off between its ?prior-like? effects and the need for a
standard Huber-style image prior, are still open questions.
Acknowledgements
The real dataset used in the results section is due to Tomas Pajdla and Daniel Martinec, CMP, Prague,
and is available at http://www.robots.ox.ac.uk/?vgg/data4.html.
(a) truth
(b) ML image 1
(c) ML image 2
(d) difference
Figure 4: An example of the effect of tiny changes in the registration parameters. (a) Ground truth
image from which a 16-image low-resolution dataset was generated. (b,c) Two ML super-resolution
estimates. In both cases, the same dataset was used, but the registration parameters were perturbed
by an i.i.d. vector with standard deviation of just 0.04 low-resolution pixels. (d) The difference
between the two solutions. In all these images, values outside the valid image intensity range have
been rounded to white or black values.
This work was funded in part by EC Network of Excellence PASCAL.
References
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2,246 | 3,038 | Implicit Online Learning with Kernels
Li Cheng
S.V. N. Vishwanathan
National ICT Australia
[email protected]
[email protected]
Shaojun Wang
Department of Computer Science and Engineering
Wright State University
[email protected]
Dale Schuurmans
Department of Computing Science
University of Alberta, Canada
[email protected]
Terry Caelli
National ICT Australia
[email protected]
Abstract
We present two new algorithms for online learning in reproducing kernel Hilbert
spaces. Our first algorithm, ILK (implicit online learning with kernels), employs
a new, implicit update technique that can be applied to a wide variety of convex
loss functions. We then introduce a bounded memory version, SILK (sparse ILK),
that maintains a compact representation of the predictor without compromising
solution quality, even in non-stationary environments. We prove loss bounds and
analyze the convergence rate of both. Experimental evidence shows that our proposed algorithms outperform current methods on synthetic and real data.
1
Introduction
Online learning refers to a paradigm where, at each time t, an instance xt ? X is presented to a
learner, which uses its parameter vector ft to predict a label. This predicted label is then compared
to the true label yt , via a non-negative, piecewise differentiable, convex loss function L(xt , yt , ft ).
The learner then updates its parameter vector to minimize a risk functional, and the process repeats.
Kivinen and Warmuth [1] proposed a generic framework for online learning where the risk functional, Jt (f ), to be minimized consists of two terms: a Bregman divergence between parameters
?G (f, ft ) := G(f ) ? G(ft ) ? hf ? ft , ?f G(ft )i, defined via a convex function G, and the instantaneous risk R(xt , yt , f ), which is usually given by a function of the instantaneous loss L(xt , yt , f ).
The parameter updates are then derived via the principle
ft+1 = argmin Jt (f ) := argmin{?G (f, ft ) + ?t R(xt , yt , f )},
(1)
f
f
where ?t is the learning rate. Since Jt (f ) is convex, (1) is solved by setting the gradient (or, if
necessary, a subgradient) to 0. Using the fact that ?f ?G (f, ft ) = ?f G(f ) ? ?f G(ft ), one obtains
?f G(ft+1 ) = ?f G(ft ) ? ?t ?f R(xt , yt , ft+1 ).
(2)
Since it is difficult to determine ?f R(xt , yt , ft+1 ) in closed form, an explicit update, as opposed to
the above implicit update, uses the approximation ?f R(xt , yt , ft+1 ) ? ?f R(xt , yt , ft ) to arrive at
the more easily computable expression [1]
?f G(ft+1 ) = ?f G(ft ) ? ?t ?f R(xt , yt , ft ).
(3)
In particular, if we set G(f ) = 21 ||f ||2 , then ?G (f, ft ) = 12 ||f ? ft ||2 and ?f G(f ) = f , and we
obtain the familiar stochastic gradient descent update
ft+1 = ft ? ?t ?f R(xt , yt , ft ).
(4)
We are interested in applying online learning updates in a reproducing kernel Hilbert space (RKHS).
To lift the above update into an RKHS, H, one typically restricts attention to f ? H and defines [2]
?
R(xt , yt , f ) := ||f ||2H + C ? L(xt , yt , f ),
(5)
2
where || ? ||H denotes the RKHS norm, ? > 0 is a regularization constant, and C > 0 determines the
penalty imposed on point prediction violations.
Recall that if H is a RKHS of functions on X ? Y, then its defining kernel k : (X ? Y)2 ? R satisfies the reproducing property; namely that hf, k((x, y), ?)iH = f (x, y) for all f ? H. Therefore, by
making the standard assumption that L only depends on f via its evaluations at f (x, y), one reaches
the conclusion that ?f L(x, y, f ) ? H, and in particular
X
?f L(x, y, f ) =
?y?k((x, y?), ?),
(6)
y??Y
for some ?y? ? R. Since ?f R(xt , yt , ft ) = ?ft + C ? ?f L(xt , yt , ft ), one can use (4) to obtain an
explicit update ft+1 = (1 ? ?t ?)ft ? ?t C ? ?f L(xt , yt , ft ), which combined with (6) shows that
there must exist coefficients ?i,?y fully specifying ft+1 via
t X
X
ft+1 =
?i,?y k((xi , y?), ?).
(7)
i=1 y??Y
In this paper we propose an algorithm ILK (implicit online learning with kernels) that solves (2)
directly, while still expressing updates in the form (7). That is, we derive a technique for computing
the implicit update that can be applied to many popular loss functions, including quadratic, hinge,
and logistic losses, as well as their extensions to structured domains (see e.g. [3])?in an RKHS. We
also provide a general recipe to check if a new convex loss function is amenable to these implicit
updates. Furthermore, to reduce the memory requirement of ILK, which grows linearly with the
number of observations (instance-label pairs), we propose a sparse variant SILK (sparse ILK) that
approximates the decision function f by truncating past observations with insignificant weights.
2
Implicit Updates in an RKHS
As shown in (1), to perform an implicit update one needs to minimize ?G (f, ft ) + R(xt , yt , f ). By
replacing R(xt , yt , f ) with (5), and setting G(f ) = 12 ||f ||2H , one obtains
?
1
ft+1 = arg min J(f ) = argmin ||f ? ft ||2H + ?t
||f ||2H + C ? L(xt , yt , f ) . (8)
f
2
2
f
Since L is assumed convex with respect to f , setting ?f J = 0 and using an auxiliary variable
?t ?
yields
?t = 1+?
t?
ft+1 = (1 ? ?t )ft ? (1 ? ?t )?t C?f L(xt , yt , ft+1 ).
(9)
On the other hand, from the form (7) it follows that ft+1 can also be written as
ft+1 =
t?1 X
X
?i,?y k((xi , y?), ?) +
i=1 y??Y
X
?t,?y k((xt , y?), ?),
(10)
y??Y
for some ?j,?y ? R and j = 1, . . . , t. Since
?f L(xt , yt , ft+1 ) =
X
?t,?y k((xt , y?), ?),
y??Y
and for ease of exposition, we assume a fixed step size (learning rate) ?t = 1, consequently ?t = ? ,
it follows from (9) and (10) that
?i,?y = (1 ? ? )?i,?y
?t,?y = ?(1 ? ? )C?t,?y
for i = 1, . . . , t ? 1, and y? ? Y,
for all y? ? Y .
(11)
(12)
Note that sophisticated step size adaptation algorithms (e.g. [3]) can be modified in a straightforward
manner to work in our setting.
The main difficulty in performing the above update arises from the fact that ?t,?y depends on ft+1
(see e.g. (13)) which in turn depends on ?t,?y via ?t,?y . The general recipe to overcome this problem
is to first use (9) to write ?t,?y as a function of ?t,?y . Plugging this back into (12) yields an equation
in ?t,?y alone, which sometimes can be solved efficiently. We now elucidate the details for some
well-known loss functions.
Square Loss In this case, k((xt , yt ), ?) = k(xt , ?). That is, the kernel does not depend on the
value of y. Furthermore, we assume that Y = R, and write
1
1
L(xt , yt , f ) = (f (xt ) ? yt )2 = (hf (?), k(xt , ?)iH ? yt )2 ,
2
2
which yields
?f L(xt , yt , f ) = (f (xt ) ? yt ) k(xt , ?).
(13)
Substituting into (12) and using (9) we have
?t = ?(1 ? ? )C((1 ? ? )ft (xt ) + ?t k(xt , xt ) ? yt ).
After some straightforward algebraic manipulation we obtain the solution
?t =
C(1 ? ? )(yt ? (1 ? ? )ft (xt ))
.
1 + C(1 ? ? )k(xt , xt )
Binary Hinge Loss As before, we assume k((xt , yt ), ?) = k(xt , ?), and set Y = {?1}. The hinge
loss for binary classification can be written as
L(xt , yt , f ) = (? ? yt f (xt ))+ = (? ? yt hf, k(xt , ?)iH )+ ,
(14)
where ? > 0 is the margin parameter, and (?)+ := max(0, ?). Recall that the subgradient is a
set, and the function is said to be differentiable at a point if this set is a singleton [4]. The binary
hinge loss is not differentiable at the hinge point, but its subgradient exists everywhere. Writing
?f L(xt , yt , f ) = ?t k(xt , ?) we have:
yt f (xt ) > ? =? ?t = 0;
yt f (xt ) = ? =? ?t ? [0, ?yt ];
yt f (xt ) < ? =? ?t = ?yt .
(15a)
(15b)
(15c)
We need to balance between two conflicting requirements while computing ?t . On one hand we
want the loss to be zero, which can be achieved by setting ? ? yt ft+1 (xt ) = 0. On the other
hand, the gradient of the loss at the new point ?f L(xt , yt , ft+1 ) must satisfy (15). We satisfy both
constraints by appropriately clipping the optimal estimate of ?t .
Let ?
? t denote the optimal estimate of ?t which leads to ? ? yt ft+1 (xt ) = 0. Using (9) we have
? ? yt ((1 ? ? )ft (xt ) + ?
? t k(xt , xt )) = 0, which yields
?
?t =
yt (? ? (1 ? ? )yt ft (xt ))
? ? (1 ? ? )yt ft (xt )
=
.
yt k(xt , xt )
k(xt , xt )
On the other hand, by using (15) and (12) we
scenarios, we arrive at the final update
?
?t
??
?t = 0
?
yt (1 ? ? )C
have ?t yt ? [0, (1 ? ? )C]. By combining the two
if yt ?
? t ? [0, (1 ? ? )C];
if yt ?
? t < 0;
if yt ?
? t > (1 ? ? )C.
(16)
The updates for the hinge loss used in novelty detection are very similar.
Graph Structured Loss The graph-structured loss on label domain can be written as
.
L(xt , yt , f ) = ?f (xt , yt ) + max(?(yt , y?) + f (xt , y?))
y?6=yt
(17)
+
Here, the margin of separation between labels is given by ?(yt , y?) which in turn depends on the
graph structure of the output space. This a very general loss, which includes binary and multiclass
hinge loss as special cases (see e.g. [3]). We briefly summarize the update equations for this case.
Let y ? = argmaxy?6=yt {?(yt , y?) + ft (xt , y?)} denote the best runner-up label for current instance
xt . Then set ?t,yt = ??t,y? = ?t , use kt (y, y 0 ) to denote k((xt , y), (xt , y 0 )) and write
?
?t =
?(1 ? ? )ft (xt , yt ) + ?(yt , y ? ) + (1 ? ? )ft (xt , y ? )
.
(kt (yt , yt ) + kt (y ? , y ? ) ? 2kt (yt , y ? ))
The updates are now given by
?
?0
?t = ?
?
? t
(1 ? ? )C
if ?
? t < 0;
if ?
? t ? [0, (1 ? ? )C];
if ?
? t > (1 ? ? )C.
(18)
Logisitic Regression Loss The logistic regression loss and its gradient can be written as
?yt k(xt , ?)
.
L(xt , yt , f ) = log (1 + exp(?yt f (xt ))) , ?f L(xt , yt , f ) =
1 + exp(yt f (xt ))
respectively. Using (9) and (12), we obtain
(1 ? ? )Cyt
?t =
.
1 + exp(yt (1 ? ? )ft (xt ) + ?t yt k(xt , xt ))
Although this equation does not give a closed-form solution, the value of ?t can still be obtained by
using a numerical root-finding routine, such as those described in [5].
2.1
ILK and SILK Algorithms
We refer to the algorithm that performs implicit updates as ILK, for ?implicit online learning with
kernels?. The update equations of ILK enjoy certain advantages. For example, using (11) it is easy to
see that an exponential decay term can be naturally incorporated to down-weight past observations:
t X
X
ft+1 =
(1 ? ? )t?i ?i,?y k((xi , y?), ?).
(19)
i=1 y??Y
Intuitively, the parameter ? ? (0, 1) (determined by ? and ?) trades off between the regularizer and
the loss on the current sample. In the case of hinge losses?both binary and graph structured?the
weight |?t | is always upper bounded by (1 ? ? )C, which ensures limited influence from outliers (cf.
(16) and (18)).
A major drawback of the ILK algorithm described above is that the size of the kernel expansion
grows linearly with the number of data points up to time t (see (10)). In many practical domains,
where real time prediction is important (for example, video surveillance), storing all the past observations and their coefficients is prohibitively expensive. Therefore, following Kivinen et al. [2] and
Vishwanathan et al. [3] one can truncate the function expansion by storing only a few relevant past
observations. We call this version of our algorithm SILK, for ?sparse ILK?.
Specifically, the SILK algorithm maintains a buffer of size ?. Each new point is inserted into the
buffer with coefficient ?t . Once the buffer limit ? is exceeded, the point with the lowest coefficient
value is discarded to maintain a bound on memory usage. This scheme is more effective than the
straightforward least recently used (LRU) strategy proposed in Kivinen et al. [2] and Vishwanathan
et al. [3]. It is relatively straightforward to show that the difference between the true predictor and
its truncated version obtained by storing only ? expansion coefficients decreases exponentially as
the buffer size ? increases [2].
3
Theoretical Analysis
In this section we will primarily focus on analyzing the graph-structured loss (17), establishing
relative loss bounds and analyzing the rate of convergence of ILK and SILK. Our proof techniques
adopt those of Kivinen et al. [2]. Due to the space constraints, we leave some details and analysis to
the full version of the paper. Although the bounds we obtain are similar to those obtained in [2], our
experimental results clearly show that ILK and SILK are stronger than the NORMA strategy of [2]
and its truncated variant.
3.1 Mistake Bound
We begin with a technical definition.
Definition 1 A sequence of hypotheses {(f1 , . . . , fT ) :Pft ? H} is said to be (T, B,PD1 , D2 )
bounded if it satisfies ||ft ||2H ? B 2 ?t ? {1, . . . , T }, t ||ft ? ft+1 ||H ? D1 , and t ||ft ?
ft+1 ||2H ? D2 for some B, D1 , D2 ? 0. The set of all (T, B, D1 , D2 ) bounded hypothesis sequences is denoted as F(T, B, D1 , D2 ).
Given a fixed sequence of observations {(x1 , y1 ), . . . , (xT , yT )}, and a sequence of hypotheses
{(f1 , . . . , fT ) ? F}, the number of errors M is defined as
M := |{t : ?f (xt , yt , yt? ) ? 0}| ,
where ?f (xt , yt , yt? ) = f (xt , yt ) ? f (xt , yt? ) and yt? is the best runner-up label. To keep the equations succinct, we denote ?kt ((yt , y), ?) := k((xt , yt ), ?)?k((xt , y), ?), and ?kt ((yt , y), (yt , y)) :=
k?kt ((yt , y), ?)k2H = kt (yt , yt ) ? 2kt (yt , y) + kt (y, y). In the following we bound the number
of mistakes M made by ILK by the cumulative loss of an arbitrary sequence of hypotheses from
F(T, B, D1 , D2 ).
Theorem 2 Let {(x1 , y1 ), . . . , (xT , yT )} be an arbitrary sequence of observations such that
?kt ((yt , y), (yt , y)) ? X 2 holds for any t, any y, and for some X > 0. For an arbitrary
P average margin ? =
P sequence ofg? hypotheses ?(g1 , ? ? ? , gT ) ? F(T, B, D1 , D2 ) with
1
)
?
?(y
,
y
)
,
and
bounded
cumulative
loss
K
:=
?(y
,
y
t t
t t
t L(xt , yt , gt ), the numt?E
|E |
ber of mistakes q
of the sequence of hypotheses (f1 , ? ? ? , fT ) generated by ILK with learning rate
?t = ?, ? =
1
B?
D2
T
is upper-bounded by
1 1
K
2S
S 2 S 2
K
M ?
,
(20)
+ 2 +2
+ 2
?
?
?
?
?2
?
2
where S = X4 (B 2 + BD1 + B T D2 ), ? > 0, and ytg? denotes the best runner-up label with
hypothesis gt .
When considering the stationary distribution in a separable (noiseless) scenario, this theorem allows
us to obtain a mistake bound that is reminiscent of the Perceptron convergence theorem. In particular, if we assume the sequence of hypotheses (g1 , ? ? ? , gT ) ? F(T, B, D1 = 0, D2 = 0) and the
cumulative loss K = 0, we obtain a bound on the number of mistakes
B2X 2
.
?2
M ?
3.2
(21)
Convergence Analysis
PT
The following theorem asserts that under mild assumptions, the cumulative risk t=1 R(xt , yt , ft )
of the hypothesis sequence produced by ILK converges to the minimum risk of the batch learning
PT
counterpart g ? := argming?H t=1 R(xt , yt , g) at a rate of O(T ?1/2 ).
Theorem 3 Let {(x1 , y1 ), . . . , (xT , yT )} be an arbitrary sequence of observations such that
?kt ((yt , yt ), (yt , yt )) ? X 2 holds for any t, any y. Denote (f1 , . . . , fT ) the sequence of hypotheses
PT
produced by ILK with learning rate ?t = ?t?1/2 , t=1 R(xt , yt , ft ) the cumulative risk of this
PT
sequence, and t=1 R(xt , yt , g) the batch cumulative risk of (g, . . . , g), for any g ? H. Then
T
X
R(xt , yt , ft ) ?
t=1
where U =
CX
? ,
2
T
X
t=1
2
a = 4?C X +
2U 2
? ,
and b =
U2
2?
g ? = arg min
g?H
we obtain
R(xt , yt , g) + aT 1/2 + b,
are constants. In particular, if
T
X
R(xt , yt , g),
t=1
T
T
1X
1X
R(xt , yt , ft ) ?
R(xt , yt , g ? ) + O(T ?1/2 ).
T t=1
T t=1
(22)
Essentially the same theorem holds for SILK, but now with a slightly larger constant a =
2
4 ?(1 + ?2 )C 2 X 2 + 2U? . In addition, denote g ? the minimizer of the batch learning cumulaP
tive risk t R(xt , yt , g), and f ? the minimizer of the minimum expected risk with R(f ? ) :=
minf E(x,y)?P (x,y) R(x, y, f ). As stated in [6] for the structured risk minimization framework, as
3500
3000
NORMA vs. ILK
180
2000
180
1500
170
1000
160
Mistakes of NORMA
2500
NORMA
160
150
500
0
-400
ILK
140
-200
0
200
400
600
800
SILK
140
140
ILK(0)
Trunc.
NORMA(0)
160
Mistakes of ILK
180
Figure 1: The left panel depicts a synthetic data sequence containing two classes (blue crosses and red
diamonds, see the zoomed-in portion in bottom-left corner), with each class being sampled from a mixture of two drifting Gaussian distributions. Performance comparison of ILK vs NORMA and truncated
NORMA on this data: Average cumulative error over 100 trials (middle), and average cumulative error
each trial (right).
the sample size T grows, T ? ?, we obtain g ? ? f ? in probability. This subsequently guarantees
the convergence of the average regularized risk of ILK and SILK to R(f ? ).
The upper bound in the above theorem can be directly plugged into Corollary 2 of Cesa-Bianchi
et al. [7] to obtain bounds on the generalization error of ILK. Let f? denote the average hypothesis
produced by averaging over all hypotheses f1 , . . . , fT . Then for any ? ? (0, 1), with probability
at least 1 ? ?, the expected risk of f? is upper
bounded by the risk of the best hypothesis chosen in
q
hindsight plus a term which grows as O
4
1
T
.
Experiments
We evaluate the performance of ILK and SILK by comparing them to NORMA [2] and its truncated
variant. On OCR data, we also compare our algorithms to SVMD, a sophisticated step-size adaptation algorithm in RKHS presented in [3]. For a fair comparison we tuned the parameters of each
algorithm separately and report the best results. In addition, we fixed the margin to ? = 1 for all our
loss functions.
Binary Classification on Synthetic Sequences The aim here is to demonstrate that ILK is better
than NORMA in coping with non-stationary distributions. Each trial of our experiment works with
2000 two-dimensional instances sampled from a non-stationary distribution (see Figure 1) and the
task is to classify the sampled points into one of two classes. The central panel of Figure 1 compares
the number of errors made by various algorithms, averaged over 100 trials. Here, ILK and SILK
make fewer mistakes than NORMA and truncated NORMA. We also tested two other algorithms,
ILK(0) obtained by setting the decay factor ? to zero, and similarly for NORMA(0). As expected,
both these variants make more mistakes because they are unable to forget the past, which is crucial
for obtaining good performance in a non-stationary environment. To further compare the performance of ILK and NORMA we plot the relative errors of these two algorithms in the right panel of
Figure 1. As can be seen, ILK out-performs NORMA on this simple non-stationary problem.
Novelty Detection on Video Sequences As a significant application, we applied SILK to a background subtraction problem in video data analysis. The goal is to detect the moving foreground
objects (such as cars, persons, etc) from relatively static background scenes in real time. The challenge in this application is to be able to cope with variations in lighting as well as jitter due to
shaking of the camera. We formulate the problem as a novelty detection task using a network of
classifiers, one for each pixel. For this task we compare the performance of SILK vs. truncated
NORMA. (The ILK and NORMA algorithms are not suitable since their storage requirements grow
linearly). A constant buffer size ? = 20 is used for both algorithms in this application. We report
further implementation details in the full version of this paper.
The first task is to identify people, under varying lighting conditions, in an indoor video sequence
taken with a static camera. The left hand panel of Figure 2 plots the ROC curves of NORMA and
SILK, which demonstrates the overall better performance of SILK. We sampled one of the initial
frames after the light was switched off and back on. The results are shown in the right panel of
Figure 2. As can be seen, SILK is able to recover from the change in lighting condition better than
NORMA, and is able to identify foreground objects reasonably close to the ground truth.
1
0.9
True Positive
0.8
Frame 1353
0.7
Ground Truth
SILK
NORMA
0.6
0.5
0.4
0
1
2
3
4
False Positive
5
6
?3
x 10
NORMA
SILK
Figure 2: Performance comparison of SILK vs truncated NORMA on a background subtraction (moving
object detection) task, with varying lighting conditions. ROC curve (left) and a comparison of algorithms
immediately after the lights have been switched off and on (right).
Figure 3: Performance of SILK on a road traffic sequence (moving car detection) task, with a jittery
camera. Two random frames and the performance of SILK on those frames are depicted.
Our second experiment is a traffic sequence taken by a camera that shakes irregularly, which creates
a challenging problem for any novelty detection algorithm. As seen from the randomly chosen
frames plotted in Figure 3 SILK manages to obtain a visually plausible detection result. We cannot
report a quantitative comparison with other methods in this case, due to the lack of manually labeled
ground-truth data.
Binary and Multiclass Classification on OCR data We present two sets of experiments on the
MNIST dataset. The aim of the first set experiment is to show that SILK is competitive with
NORMA and SVMD on a simple binary task. The data is split into two classes comprising the
digits 0 ? 4 and 5 ? 9, respectively. A polynomial kernel of degree 9 and a buffer size of ? = 128 is
employed for all three algorithms. Figure 4 (a) plots current average error rate, i.e., the total number
of errors on the examples seen so far divided by the iteration number. As can be seen, after the
initial oscillations have died out, SILK consistently outperforms SVMD and NORMA, achieving a
lower average error after one pass through the dataset. Figure 4 (b) examines the effect of buffer
size on SILK. As expected, smaller buffer sizes result in larger truncation error and hence worse
performance. With increasing buffer size the asymptotic average error decreases. For the 10-way
multiclass classification task we set ? = 128, and used a Gaussian kernel following [3]. Figure 4 (c)
shows that SILK consistently outperforms NORMA and SVMD, while the trend with the increasing
buffer size is repeated, as shown in Figure 4 (d). In both experiments, we used the parameters for
NORMA and SVMD reported in [3], and set ? = 0.00005 and C = 100 for SILK.
5
Outlook and Discussion
In this paper we presented a general recipe for performing implicit online updates in an RKHS.
Specifically, we showed that for many popular loss functions these updates can be computed efficiently. We then presented a sparse version of our algorithm which uses limited basis expansions
to approximate the function. For graph-structured loss we also showed loss bounds and rates of
convergence. Experiments on real life datasets demonstrate that our algorithm is able to track nonstationary targets, and outperforms existing algorithms.
For the binary hinge loss, when ? = 0 the proposed update formula for ?t (16) reduces to the
PA-I algorithm of Crammer et al. [8]. Curiously enough, the motivation for the updates in both
cases seems completely different. While we use an implicit update formula Crammer et al. [8] use
(a)
(b)
(c)
(d)
Figure 4: Performance comparison of different algorithms over one run of the MNIST dataset. (a) Online
binary classification. (b) Performance of SILK using different buffer sizes. (c) Online 10-way multiclass
classification. (d) Performance of SILK on three different buffer sizes.
a Lagrangian formulation, and a passive-aggressive strategy. Furthermore, the loss functions they
handle are generally linear (hinge loss and its various generalizations) while our updates can handle
other non-linear losses such as quadratic or logistic loss.
Our analysis of loss bounds is admittedly straightforward given current results. The use of more
sophisticated analysis and extending our bounds to deal with other non-linear loss functions is ongoing. We are also applying our techniques to video analysis applications by exploiting the structure
of the output space.
Acknowledgements
We thank Xinhua Zhang, Simon Guenter, Nic Schraudolph and Bob Williamson for carefully proof
reading the paper, pointing us to many references, and helping us improving presentation style.
National ICT Australia is funded by the Australian Government?s Department of Communications,
Information Technology and the Arts and the Australian Research Council through Backing Australia?s Ability and the ICT Center of Excellence program. This work is supported by the IST
Program of the European Community, under the Pascal Network of Excellence, IST-2002-506778.
References
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2,247 | 3,039 | Unsupervised Regression with Applications to
Nonlinear System Identification
Ali Rahimi
Intel Research Seattle
Seattle, WA 98105
[email protected]
Ben Recht
California Institute of Technology
Pasadena, CA 91125
[email protected]
Abstract
We derive a cost functional for estimating the relationship between highdimensional observations and the low-dimensional process that generated them
with no input-output examples. Limiting our search to invertible observation
functions confers numerous benefits, including a compact representation and no
suboptimal local minima. Our approximation algorithms for optimizing this cost
functional are fast and give diagnostic bounds on the quality of their solution. Our
method can be viewed as a manifold learning algorithm that utilizes a prior on the
low-dimensional manifold coordinates. The benefits of taking advantage of such
priors in manifold learning and searching for the inverse observation functions
in system identification are demonstrated empirically by learning to track moving targets from raw measurements in a sensor network setting and in an RFID
tracking experiment.
1
Introduction
Measurements from sensor systems typically serve as a proxy for latent variables of interest. To
recover these latent variables, the parameters of the sensor system must first be determined. When
pairs of measurements and their corresponding latent variables are available, fully supervised regression techniques may be applied to learn a mapping between latent states and measurements.
In many applications, however, latent states cannot be observed and only a diffuse prior on them is
available. In such cases, marginalizing over the latent variables and searching for the model parameters using Expectation Maximization (EM) has become a popular approach [3,9,19]. Unfortunately,
such algorithms are prone to local minima and require very careful initialization in practice.
Using a simple change-of-variable model, we derive an approximation algorithm for the Unsupervised Regression problem ? estimating the nonlinear relationship between latent-states and their
observations when no example pairs are available, when the observation function is invertible, and
when the measurement noise is small. Our method is not susceptible to local minima and provides a
guarantee on the quality of the recovered observation function. We identify conditions under which
our estimate of the mapping is asymptotically consistent and empirically evaluate the quality of our
solutions and their stability under variations of the prior. Because our algorithm takes advantage of
an explicit prior on the latent variables, it recovers latent variables more accurately than manifold
learning algorithms when applied to similar tasks.
Our method may be applied to estimate the observation function in nonlinear dynamical systems
by enforcing a Markovian dynamics prior over the latent states. We demonstrate this approach
to nonlinear system identification by learning to track a moving object in a field of completely
uncalibrated sensor nodes whose measurement functions are unknown. Given that the object moves
smoothly over time, our algorithm learns a function that maps the raw measurements from the sensor
network to the target?s location. In another experiment, we learn to track Radio Frequency ID (RFID)
tags given a sequence of voltage measurements induced by the tag in a set of antennae . Given only
these measurements and that the tag moves smoothly over time, we can recover a mapping from
the voltages to the position of the tag. These results are surprising because no parametric sensor
model is available in either scenario. We are able to recover the measurement model up to an affine
transform given only raw measurement sequences and a diffuse prior on the state sequence.
2
A diffeomorphic warping model for unsupervised regression
We assume that the set X = {xi }1???N of latent variables is drawn (not necessarily iid) from a known
distribution, pX (X) = pX (x1 , ? ? ? , xN ). The set of measurements Y = {yi }1???N is the output of
an unknown invertible nonlinearity applied to each latent variable, yi = f0 (xi ). We assume that
observations, yi ? RD , are higher dimensional than latent variables xi ? Rd . Computing a MAP
estimate of f0 requires marginalizing over X and maximizing over f . EM, or some other form of
coordinate ascent on a Jensen bound of the likelihood, is a common way of estimating the parameters
of this model, but such methods suffer from local minima.
Because we have assumed that f0 is invertible and that there is no observation noise, this process
describes a change of variables. The true distribution pY (Y) over Y can be computed in closed form
using a generalization of the standard change of variables formula (see [14, thm 9.3.1] and [7, chap
11]):
1
N
Y
0
?2
pY (Y) = pY (Y; f0 ) = pX (f0?1 (y1 ), ? ? ? , f0?1 (yN ))
det ?f f0?1 (yi ) ?f f0?1 (yi )
.
i=1
(1)
The determinant corrects the warping of each infinitesimal volume element around f0?1 (yi ) by accounting for the stretching induced by the nonlinearity. The change of variables formula immediately yields a likelihood over f , circumventing the need for integrating over the latent variables.
We assume f0 diffeomorphically maps a ball in Rd containing the data onto its image. In this case,
there exists a function g defined on an open set containing the image of f such that g(f (x)) = x
and ?g?f = I for all x in the open set [5]. Consequently, we can substitute g for f ?1 in (1) and,
taking advantage of the identity det(?f 0 ?f )?1 = det ?g?g 0 , write its log likelihood as
N
lY (Y; g) = log pY (Y; g) = log pX (g(y1 ), . . . , g(yN )) +
1X
log det (?g(yi )?g(yi )0 ) .
2 i=1
(2)
For many common priors pX , the maximum likelihood g yields an asymptotically consistent estimate of the true distribution pY . When certain conditions on pX are met (including stationarity,
ergodicity, and kth-order Markov approximability), a generalized version of the Shannon-McMillanBreiman theorem [1] guarantees that log pY (Y; g) asymptotically converges to the relative entropy
rate between the true pY (Y) and pY (Y; g). This quantity is maximized when these two distributions are equal. Therefore, if the true pY follows the change of variable model (1), the recovered g
converges to the true f0?1 in the sense that they both describe a change of variable from the prior
distribution pX to the distribution pY .
Note that although our generative model assumes no observation noise, some noise in Y can be
tolerated if we constrain our search over smooth functions g. This way, small perturbations in y due
to observation noise produce small perturbations in g(y).
3
Approximation algorithms for finding the inverse mapping
We constrain our search for g to a subset of smooth functions by requiring that g have a finite
representation as a weighted sum of positive definite kernels k centered on observed data, g(y) =
PN
d
i=1 ci k(y, yi ), with the weight vectors ci ? R . Accordingly, applying g to the set of observations
gives g(Y) = CK, where C = [ c1 ???cN ] and K is the kernel matrix with Kij = k(yi , yj ). In
i ,y)
addition, ?g(y) = C?(y), where ?(y) is an N ? D matrix whose ith row is ?k(y
. We tune
?y
the smoothness of g by regularizing (2) with the RKHS norm [17] of g. This norm has the form
kgk2k = tr CKC0 , and the regularization parameter is set to ?2 .
For simplicity, we require pX to be a Gaussian with mean zero and inverse covariance ?X , but we
note our methods can be extended to any log-concave distribution. Substituting into (2) and adding
the smoothness penalty on g, we obtain:
max
C
0
?vec (KC0 ) ?X vec (KC0 ) ? ?trCKC0 +
N
X
log det C?(yi )?(yi )0 C0 ,
(3)
i=1
where the vec (?) operator stacks up the columns of its matrix argument into a column vector.
Equation (3) is not concave in C and is likely to be hard to maximize exactly. This is because
log det(A0 A) is not concave for A ? Rd?D . Since the cost is non-concave, gradient descent
methods may converge to local minima. Such local minima, in addition to the burdensome time and
storage requirements, rule out descent strategies for optimizing (3).
Our first algorithm for approximately solving this optimization problem constructs a semidefinite
relaxation using a standard approach that replaces outer products of vectors with positive definite
matrices. Rewrite (3) as
N
X
0
0
0 0
max ? tr Mvec (C0 ) vec (C0 ) +
log det
tr Jkl
vec
(C
)
vec
(C
)
(4)
i
C
i=1
M = (Id ? K)?X (Id ? K) + ?(Id ? K)
,
lk
0
Jkl
i = E ? ?(yi )?(yi )
(5)
where the klth entry of the matrix argument of the logdet is as specified, and the matrix Eij is zero
everywhere except for 1 in its ijth entry. This optimization is equivalent to
max
Z0
? tr (MZ) +
N
X
log det
tr Jkl
i Z ,
(6)
i=1
subject to the additional constraint that rank(Z) = 1. Dropping the rank constraint yields a concave
relaxation for (3). Standard interior point methods [20] or subgradient methods [2] can efficiently
compute the optimal Z for this relaxed problem. A set of coefficients C can then be extracted from
the top eigenvectors of the optimal Z, yielding an approximate solution to (3). Since (6) without
the rank constraint is a relaxation of (3), the optimum of (6) is an upper bound on that of (3). Thus
we can bound the difference in the value of the extracted solution and that of the global maximum
of (3). As we will see in the following section, this method produces high quality solutions for a
diverse set of learning problems.
In practice, standard algorithms for (6) run slowly for large data sets, so we have developed an
intuitive algorithm that also provides good approximations and runs much more quickly. The nonconcave logdet term serves to prevent the optimal solution of (2) from collapsing to g(y) = 0,
since X = 0 is the most likely setting for the zero-mean Gaussian prior pX . To circumvent the
non-concavity of the logdet term, we replace it with constraints requiring that the sample mean and
covariance of g(Y) match the expected mean and covariance of the random variables X. These
moment constraints prevent the optimal solution from collapsing to zero while remaining in the
? X , can be computed by averaging
typical set of pX . The expected covariance of X, denoted by ?
? X only influences the final solution
the block diagonals of ??1
.
However,
the
particular
choice
of
?
X
up to a scaling and rotation on g, so in practice, we set it to the identity matrix. We thus obtain the
following optimization problem:
0
min vec (KC0 ) ?X vec (KC0 ) + ?trCKC0
C
s.t.
1
?X
CK(CK)0 = ?
N
1
CK1 = 0,
N
(7)
(8)
(9)
where 1 is a column vector of 1s. This optimization problem searches for a g that transforms
observations into variables that are given high probability by pX and match its stationary statistics.
This is a quadratic minimization with a single quadratic constraint and, after eliminating the linear
constraints with a change of variables, can be solved as a generalized eigenvalue problem [4].
4
Related Work
Manifold learning algorithms and unsupervised nonlinear system identification algorithms solve
variants of the unsupervised regression problem considered here.
Our method provides a statistical model that augments manifold learning algorithms with a prior on
latent variables. Our spectral algorithm from Section 3 reduces to a variant of KPCA [15] when X
are drawn iid from a spherical Gaussian. By adopting a nearest-neighbors form for g instead of the
RBF form, we obtain an algorithm that is similar to embedding step of LLE [12, chap 5]. In addition
to our use of dynamics, a notable difference between our method and principal manifold methods
[16] is that instead of learning a mapping from states to observations, we learn mappings from
observations to states. This reduces the storage and computational requirements when processing
high-dimensional data. As far as we are aware, in the manifold learning literature, only Jenkins and
Mataric [6] explicitly take temporal coherency into account, by increasing the affinity of temporally
adjacent points and applying Isomap [18].
State-of-the-art nonlinear system identification techniques seek to recover all the parameters of a
continuous hidden Markov chain with nonlinear state transitions and observation functions given
noisy observations [3, 8, 9, 19]. Because these models are so rich and have so many unknowns, these
algorithms resort to coordinate ascent (for example, via EM), making them susceptible to local minima. In addition, each iteration of coordinate ascent requires some form of nonlinear smoothing
over the latent variables, which is itself both computationally costly and becomes prone to local
minima when the estimated observation function becomes non-invertible during the iterations. Further, because mappings from low-dimensional to high-dimensional vectors require many parameters
to represent, existing approaches tend to be unsuitable for large-scale sensor network or image analysis problems. Our algorithms do not have local minima and represent the more compact inverse
observation function where high-dimensional observations appear only in pairwise kernel evaluations.
Comparisons with a semi-supervised variant of these algorithms [13] show that weak priors on the
latent variables are extremely informative and that additional labeled data is often only necessary to
fix the coordinate system.
5
Experiments
The following experiments show that latent states and observation functions can be accurately and
efficiently recovered up to a linear coordinate transformation given only raw measurements and a
generic prior over the latent variables. We compare against various manifold learning and nonlinear
system identification algorithms. We also show that our algorithm is robust to variations in the
choice of the prior.
As a measure of quality, we report the affine registration error, the average residual per data point
after registering the recovered
latent variables with their ground truth values using an affine transqP
N
0
0
2
formation: err = minA,b N1
t=1 kAxt ? xt + bk2 , where xt is the ground truth setting for xt .
All of our experiments use a spherical Gaussian kernel. To define the Gaussian prior pX , we start
with a linear Gaussian Markov chain st = Ast?1 + ?t , where A and the covariance of ? are block
diagonal and define d Markov chains that evolve independently from each other according to Newtonian motion. The latent variables xt extract the position components of st . The inverse covariance
matrix corresponding to this process can be obtained in closed form. More details and additional
experiments can be found in [12].
We begin with a low-dimensional data set to simplify visualization and comparison to systems
that do not scale well with the dimensionality of the observations. Figure 1(b) shows the embedding of a 1500 step 2D random walk shown in Figure 1(a) into R3 by the function f (x, y) =
(x, y cos(2y), y sin(2y)). Note that the 2D walk was not generated by a linear Gaussian model, as
it bounces off the edges of its bounding box. Lifted points were passed to our algorithm, which
returned the 2D variables shown in Figure 1(c). The true 2D coordinates are recovered up to a scale,
a flip, and some shrinking in the lower left corner. Therefore the recovered g is close the inverse
of the original mapping, up to a linear transform. Figure 1(d) shows states recovered by the algorithm of Roweis and Ghahramani [3]. Smoothing with the recovered function simply projects the
4
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Figure 1: (a) 2D ground truth trajectory. Brighter colors indicate greater distance to the origin. (b) Embedding
of the trajectory into R3 . (c) Latent variables are recovered up to a linear transformation and minor distortion.
Roweis-Ghahramani (d), Isomap (e), and Isomap+temporal coherence (f) recovered low-dimensional coordinates that exhibit folding and other artifacts that cannot be corrected by a linear transformation.
observations without unrolling the roll. The joint-max version of this algorithm took about an hour
to converge on a 1Ghz Pentium III and converges only when started at solutions that are sufficiently
close to the true solution. Our spectral algorithm took about 10 seconds. Isomap (Figure 1(e)) performs poorly on this data set due to the low sampling rate on the manifold and the fact that the true
mapping f is not isometric. Including temporal neighbors into Isomap?s neighborhood structure (as
per ST-Isomap) creates some folding, and the true underlying walk is not recovered (Figure 1(f)).
KPCA (not shown) chooses a linear projection that simply eliminates the first coordinate. We found
the optimal parameter settings for Isomap, KPCA, and ST-Isomap by a fine grid search over the
parameter space of each algorithm.
The upper bound on the log-likelihood returned by the relaxation (6) serves as a diagnostic on the
quality of our approximations. This bound was ?3.9 ? 10?3 for this experiment. Rounding the
result of the relaxation returned a g with log likelihood ?5.5 ? 10?3 . The spectral approximation
(7) also returned a solution with log likelihood ?5.5 ? 10?3 , confirming our experience that these
algorithms usually return similar solutions. For comparison, log-likelihood of KPCA?s solution was
?1.69 ? 10?2 , significantly less likely than our solutions, or the upper bound.
5.1
Learning to track in an uncalibrated sensor network
We consider an artificial distributed sensor network scenario where many sensor nodes are deployed
randomly in a field in order to track a moving target (Figure 2(a)). The location of the sensor
nodes is unknown, and the sensors are uncalibrated, so that it is not known how the position of the
target maps to the reported measurements. This situation arises when it is not feasible to calibrate
each sensor prior to deployment or when variations in environmental conditions affect each sensor
differently. Given only the raw measurements produced by the network from watching a smoothly
moving target, we wish to learn a mapping from these measurements to the location of the target,
even though no functional form for the measurement model is available. A similar problem was
considered by [11], who sought to recover the location of sensor nodes using off-the-shelf manifold
learning algorithms.
Each latent state xt is the unknown position of the target at time t. The unknown function f (xt )
gives the set of measurements yt reported by the sensor network at time t. Figure 2(b) shows the
time series of measurements from observing the target. In this case, measurements were generated
by having each sensor s report its true distance dst to the target at time t and passing it through a
random nonlinearity of the form ?s exp(?? s dst ). Note that only f , not the measurement function
of each sensor, needs be invertible. This is equivalent to requiring that a memoryless mapping from
measurements to positions must exist.
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Figure 2: (a) A target followed a smooth trajectory (dotted line) in a field of 100 randomly placed uncalibrated
sensors with random and unknown observation functions (circle). (b) Time series of measurements produced
by the sensor network in response to the target?s motion. (c) The recovered trajectory given only raw sensor
measurements, and no information about the observation function (other than smoothness and invertibility). It
is recovered up to scaling and a rotation. (d) To test the recovered mapping further, the target was made to
follow a zigzag pattern. (e) Output of g on the resulting measurements. The resulting trajectory is again similar
to the ground truth zigzag, up to minor distortion. (f) The mapping obtained by KPCA cannot recover the
zigzag, because KPCA does not utilize the prior on latent states.
Assuming only that the target vaguely follows linear-Gaussian dynamics, and given only the time series of the raw measurements from the sensor network, our learning algorithm finds a transformation
that maps observations from the sensor network to the position of the target up to a linear coordinate
transform (Figure 2(c)). The recovered function g implicitly performs all the triangulation necessary
for recovering the position of the target, even though the position or characteristics of the sensors
were not known a priori. The bottom row of Figure 2 tests the recovered g by applying it to a new
measurement set. To show that this sensor network problem is not trivial, the figure also shows the
output of the mapping obtained by KPCA.
5.2
Learning to Track with the Sensetable
The Sensetable is a hardware platform for tracking the position of radio frequency identification
(RFID) tags. It consists of 10 antennae woven into a flat surface 30 ? 30 cm. As an RFID tag moves
along the flat surface, the strength of the RF signal induced by RFID tag in each antenna is reported,
producing a time series of 10 numbers. We wish to learn a mapping from these 10 voltage measurements to the 2D position of the RFID tag. Previously, such a mapping was recovered by hand, by
meticulous physical modeling of this system, followed by trial-and-error to refine these mappings;
a process that took about 3 months in total [10]. We show that it is possible to recover this mapping
automatically, up to an affine transformation, given only the raw time series of measurements generated by moving the RFID tag by hand on the Sensetable for about 5 minutes. This is a challenging
task because the relationship between the tag?s position and the observed measurements is highly
oscillatory. (Figure 3(a)). Once it is learned, we can use the mapping to track RFID tags. This
experiment serves as a real-world instantiation of the sensor network setup of the previous section
in that each antenna effectively acts as an uncalibrated sensor node with an unknown and highly
oscillatory measurement function.
Figure 3(b) shows the ground truth trajectory of the RFID tag in this data set. Given only the 5
minute-long time series of raw voltage measurements, our algorithm recovered the trajectory shown
in Figure 3(c). These recovered coordinates are scaled down and flipped about both axes as compared to the ground truth coordinates. There is also some additional shrinkage in the upper right
corner, but the coordinates are otherwise recovered accurately, with an affine registration error of
1.8 cm per pixel.
Figure 4 shows the result of LLE, KPCA, Isomap and ST-Isomap on this data set under their best
parameter settings (again found by a grid search on each algorithm?s search space). None of these
algorithms recover low-dimensional coordinates that resemble the ground truth. LLE, in addition to
collapsing the coordinates to one dimension, exhibits severe folding, obtaining an affine registration
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Figure 3: (a) The output of the Sensetable over a six second period, while moving the tag from the left edge
of the table to the right edge. The observation function is highly complex and oscillatory. (b) The ground truth
trajectory of the tag. Brighter points have greater ground truth y-value. (c) The trajectory recovered by our
spectral algorithm is correct up to flips about both axes, a scale change, and some shrinkage along the edge.
Isomap K=7
LLE k=15
KPCA
Figure 4: From left to right, the trajectories recovered by LLE, KPCA, Isomap, ST-Isomap. All of these
trajectories exhibit folding and severe distortions.
error of 8.5 cm. KPCA also exhibited folding and large holes, with an affine registration error of
7.2 cm. Of these, Isomap performed the best with an affine registration error of 3.4 cm, though it
exhibited some folding and a large hole in the center. Isomap with temporal coherency performed
similarly, with a best affine registration error of 3.1 cm. Smoothing the output of these algorithms
using the prior sometimes improves their accuracy by a few millimeters, but more often diminishes
their accuracy by causing overshoots.
To further test the mapping recovered by our algorithm, we traced various trajectories with an RFID
tag and passed the resulting voltages through the recovered g. Figure 5 plots the results (after a
flip about the y-axis). These shapes resemble the trajectories we traced. Noise in the recovered
coordinates is due to measurement noise.
The algorithm is robust to perturbations in pX . To demonstrate this, we generated 2000 random perturbations of the parameters of the inverse covariance of X used to generate the Sensetable results,
and evaluated the resulting affine registration error. The random perturbations were generated by
scaling the components of A and the diagonal elements of the covariance of ? over four orders of
magnitude using a log uniform scaling. The affine registration error was below 3.6 cm for 38% of
these 2000 perturbations. Typically, only the parameters of the kernel need to be tuned. In practice,
we simply choose the kernel bandwidth parameter so that the minimum entry in K is approximately
0.1.
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Figure 5: Tracking RFID tags using the recovered mapping.
6
Conclusions and Future Work
We have shown how to recover the latent variables in a dynamical system given an approximate prior
on the dynamics of these variables and observations of these states through an unknown invertible
nonlinearity. The requirement that the observation function be invertible is similar to the requirement in manifold learning algorithms that the manifold not intersect itself. Our algorithm enhances
manifold learning algorithms by leveraging a prior on the latent variables. Because we search for a
mapping from observations to unknown states (as opposed to from states to observations), we can
devise algorithms that are stable and avoid local minima. We applied this methodology to learning
to track objects given only raw measurements from sensors with no constraints on the observation
model other than invertibility and smoothness.
We are currently evaluating various ways to relax the invertibility requirement on the observation
function by allowing invertibility up to a linear subspace. We are also exploring different prior
models, and experimenting with ways to jointly optimize over g and the parameters of pX .
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2,248 | 304 | Reconfigurable Neural Net Chip with 32K
Connections
H.P. Graf, R. Janow, D. Henderson, and R. Lee
AT&T Bell Laboratories, Room 4G320, Holmdel, NJ 07733
Abstract
We describe a CMOS neural net chip with a reconfigurable network architecture. It contains 32,768 binary, programmable connections arranged in
256 'building block' neurons. Several 'building blocks' can be connected to
form long neurons with up to 1024 binary connections or to form neurons
with analog connections. Single- or multi-layer networks can be implemented with this chip. We have integrated this chip into a board system
together with a digital signal processor and fast memory. This system is
currently in use for image processing applications in which the chip extracts
features such as edges and corners from binary and gray-level images.
1
INTRODUCTION
A key problem for a hardware implementation of neural nets is to find the proper
network architecture. With a fixed network structure only few problems can be
solved efficiently. Therefore, we opted for a programmable architecture that can be
changed under software control. A large, fully interconnected network can, in principle, implement any architecture, but this usually wastes a lot of the connections
since many have to be set to zero. To make better use of the silicon, other designs
implemented a programmable architecture - either by connecting several chips with
switching blocks (Mueller89), or by placing switches between blocks of synapses on
one chip (Satyanarayana90). The present design (GramO) consists of building blocks
that can be connected to form many different network configurations. Single-layer
1032
Reconfigurable Neural Net Chip with 32K Connections
nets or multi-layer nets can be implemented. The connections can be binary or can
have an analog depth of up to four bits.
We designed this neural net chip mainly for pattern recognition applications, which
tYp"ically require nets far too large for a single chip. However, the nets can often be
structured so that the neurons have local receptive fields, and many neurons share
the same receptive field. Such nets can be split into smaller parts that fit onto a
chip, and the small nets are then scanned sequentially over an image. The circuit
has been optimized for this type of network by adding shift registers for the data
transport.
The neural net chip implementation uses a mixture of analog and digital electronics.
The weights, the neuron states and all the control signals are digital, while summing
the contributions of all the weights is performed in analog form. All the data going
on and off the chip are digital, which makes the integration of the network into a
digital system straight-forward.
2
2.1
THE CIRCUIT ARCHITECTURE
The Building Block
INPUT DATA REGISTER
128 BITS
CONNECT OTHER
128 CONNECTIONS
SUMMING WIRE
Figure I: One of the building blocks, a "neuron"
Figure 1 shows schematically one of the building blocks. It consists of an array of
128 connections which receive input signals from other neurons or from external
sources. The weights as well as the inputs have binary values, +1 or -1. The output
of a connection is a current representing the result of the multiplication of a weight
with a state, and on a wire the currents from all the connections are summed. This
sum is multiplied with a programmable factor and can be added to the currents
of other neurons. The result is compared with a reference and is thresholded in a
comparator. A total of 256 such building blocks are contained on the chip.
Up to 8 of the building blocks can be connected to form a single neuron with up to
1024 connections. The network is not restricted to binary connections. Connections
with four bits of analog depth are obtained by joining four building blocks and by
setting each of the multipliers to a different value: 1, 1/2, 1/4, 1/8 (see Figure
1033
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Graf, Janow, Henderson, and Lee
INPUT DATA REGISTER
128 BITS
128 CONNECOONS
128 CONNECTIONS
Figure 2: Connecting four building blocks to form connections with four bits of
resolution
Figure 3: Photo micrograph of the neural net chip
Reconfigurable Neural Net Chip with 32K Connections
2). In this case four binary connections, one in each building block, form one
connection with an analog depth of four bits. Alternatively, the network can be
configured for two-bit input signals and two-bit weights or four-bit inputs and onebit weights. The multiplications of the input signals with the weights are fourquadrant multiplications, whether binary signals are used or multi-bit signals. With
this approach only one scaling multiplier is needed per neuron, instead of one per
connection as would be the case if connections were implemented with multiplying
Dj A converters.
The transfer function of a neuron is provided by the comparator. With a single comparator the transfer function has a hard threshold. Other types of transfer functions
can be obtained when several building blocks are connected. Then, several comparators receive the same analog input and for each comparator a different reference
can be selected (compare figure 2). In this way, for example, eight comparators may
work as a three-bit AjD converter. Other transfer functions, such as sigmoids, can
be approximated by selecting appropriate thresholds for the comparators.
The neurons are arranged in groups of 16. For each group there is one register of
128 bits providing the input data. The whole network contains 16 such groups, split
in two halfs, each with eight groups. These groups of neurons can be recognized in
Figure 3 that shows a photomicrograph of the circuit. The chip contains 412,000
transistors and measures 4.5mm x 7mm. It is fabricated in 0.9J.lm CMOS technology
with one level of poly and two levels of metal.
2.2
Moving Data Through The Circuit
From a user's point of view the chip consists of the four different types of registers
listed in table 1. Programming of the chip consists in moving the data over a
high-speed bus of 128 bits width between these registers. Results produced by the
network can be loaded directly into data-input registers and can be used for a next
computation. In this way some multi-layer networks can be implemented without
loading data off chip between layers.
Table 1: Registers in the neural net chip
REGISTER
FUNCTION
Shift register
Data-input registers
Configuration registers
Result registers
Input and output of the data
Provide input to the connections
Determine the connectivity of the network
Contain the output of the network
In a typical operation 16 bits are loaded from the outside into a shift register. From
that register the main bus distributes the data through the whole circuit. They are
loaded into one or several of the data-input registers. The analog computation is
then started and the results are latched into the result registers. These results are
loaded either into data-input registers if a network with feedback or a multi-layer
network is implemented, or they are loaded into the output shift register and off
chip.
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Graf, Janow, Henderson, and Lee
In addition to the main bus, the chip contains two 128 bit wide shift registers, one
through each half of the connection matrix. All the shift registers were added to
speed up the operation when networks with local fields of view are scanned over a
signal. In such an application shift registers drastically reduce the amount of new
data that have to be loaded into the chip from one run to the next. For example,
when an input field of 16 x 16 pixels is scanned over an image, only 16 new data
values have to be loaded for each run instead of 256. Loading the data on and off
the chip is often the speed-limiting operation. Therefore, it is important to provide
some support in hardware.
3
TEST RESULTS
The speed of the circuit is limited by the time it takes the analog computation
to settle to its final value. This operation requires less than 50 ns. The chip can
be operated with instruction cycles of lOOns, where in the first 50 ns the analog
computation settles down and during the following 50 ns the results are read out.
Simultaneously with reading out the results, new data are loaded into the datainput registers. In this way 32k one-bit multiply-accumulates are executed every
lOOns, which amounts to 320 billion connections/second.
The accuracy of the analog computation is about ?5%. This means, for example,
that a comparator whose threshold is set to a value of 100 connections may already
turn on when it receives the current from 95 connections. This limited accuracy is
due to mismatches of the devices used for the analog computation. However, the
threshold for each comparator may be individually adjusted at the cost of dedicating
neurons to the task. Then a threshold can be adjusted to ?1 %. The operation of
the digital part of the network and the analog part has been synchronized in such a
way that the noise generated by the digital part has a minimal effect on the analog
computation.
4
THE BOARD SYSTEM
A system was developed to use the neural net chip as a coprocessor of a workstation
with a VME bus. A schematic of this system is shown in figure 4. Beside the neural
net chip, the board contains a digital signal processor to control the whole system
and 256k of fast memory. Pictures are loaded from the host into the board's memory
and are then scanned with the neural net chip. The results are loaded back into
the board memory and from there to the host.
Loading pictures over the VME bus limits the overall speed of this system to about
one frame of 512 x 512 pixels per second. Although this corresponds to less than
10% of the chips maximum data throughput, operations such as scanning an image
with 32 16 x 16 kernels can be done in one second. The same operation would take
around 30 minutes on the workstation alone. Therefore, this system represents a
very useful tool for image processing, in particular for developing algorithms. Its
architecture makes it very flexible since part of a problem can be solved by the
digital signal processor and the computationally intensive parts on the neural net
chip. An extra data path for the signals will be added later to take full advantage
of the neural net's speed.
Reconfigurable Neural Net Chip with 32K Connections
ADDRESS BUS
,I
,
EPROM
G
I
256k
SRAM
I
,
DSP32C
~
I VME INTERFACE I
NET32K ~
DATA BUS
VME BUS
Figure 4: Schematic of the board system for the neural net chip
Figure 5: Result of a feature extraction application. Left image: Edges extracted
from the milling cutter. Right image: The crosses mark where corners were detected.
A total of 16 features were extracted simultaneously with detectors of 16 x 16 pixels
In sIze.
1037
1038
Graf,Janow, Henderson, and Lee
5
APPLICATIONS
Figure 5 shows the result of an application, where the net is used for extracting
simultaneously edges and corners from a gray-level image. The network actually
handles only a small resolution in the pixel values. Therefore, the picture is first
half-toned with a standard error-diffusion algorithm and then, the halftoned image
is scanned with the network. To extract these features, kernels with three levels in
the weights are loaded into the network. One neuron with 256 two-bit connections
represents one kernel. There are a total of 16 such kernels in the network, each one
tuned to a corner or an edge of a different orientation. For each comparator an
extra neuron is used to set the threshold. This whole task fills 50% of the chip.
Edges and corners are important features that are often used to identify objects
or to determine their positions and orientations. We are applying them now to
segment complex images. Convolutional algorithms have long been recognized as
reliable methods for extracting features. However, they are computationally very
expensive so that often special-purpose hardware is required. To our knowledge, no
other circuit can extract such a large number of features at a comparable rate.
This application demonstrates, how a large number of connections can compensate
for a limited resolution in the weights and the states. We took a gray level image
and clipped its pixels to binary values. Despite this coarse quantization of the
signal the relevant information can be extracted reliably. Since many connections
are contributing to each result, uncorrelated errors due to quantization are averaged
out. The key to a good result is to make sure that the quantization errors are indeed
uncorrelated, at least approximately.
This circuit has been designed with pattern matching applications in mind. However, its flexibility makes it suitable for a much wider range of applications. In
particular, since its connections as well as its architecture can be changed fast, in
the order of lOOns, it can be integrated in an adaptive or a learning system.
Acknowledgements
We acknowledge many stimulating discussions with the other members of the neural
network group at AT&T in Holmdel. Part of this work was supported by the
USASDC under contract #DASG60-88-0044.
References
H. P. Graf & D. Henderson. (1990) A Reconfigurable CMOS Neural Network. in
Digest IEEE Int. Solid State Circuits Con/. , 144-145.
P. Mueller, J. van der Spiegel, D. Blackman, T. Chiu, T. Clare, J. Dao, Ch. Donham,
T.P. Hsieh & M. Loinaz. (1989) A Programmable Analog Neural Computer and
Simulator. In D.S. Touretzky (ed.), Advances in Neural Information Processing
Systems 1, 712 - 719. San Mateo, CA: Morgan Kaufmann.
S. Satyanarayana, Y. Tsividis & H. P. Graf. (1990) A Reconfigurable Analog VLSI
Neural Network Chip. In D.S. Touretzky (ed.), Advances in Neural Information
Processing Systems 2, 758 - 768. San Mateo, CA: Morgan Kaufmann.
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2,249 | 3,040 | Analysis of Contour Motions
Ce Liu William T. Freeman Edward H. Adelson
Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology
Cambridge, MA 02139, USA
{celiu,billf,adelson}@csail.mit.edu
Abstract
A reliable motion estimation algorithm must function under a wide range of conditions. One regime, which we consider here, is the case of moving objects with
contours but no visible texture. Tracking distinctive features such as corners can
disambiguate the motion of contours, but spurious features such as T-junctions
can be badly misleading. It is difficult to determine the reliability of motion from
local measurements, since a full rank covariance matrix can result from both real
and spurious features. We propose a novel approach that avoids these points altogether, and derives global motion estimates by utilizing information from three
levels of contour analysis: edgelets, boundary fragments and contours. Boundary
fragment are chains of orientated edgelets, for which we derive motion estimates
from local evidence. The uncertainties of the local estimates are disambiguated
after the boundary fragments are properly grouped into contours. The grouping
is done by constructing a graphical model and marginalizing it using importance
sampling. We propose two equivalent representations in this graphical model, reversible switch variables attached to the ends of fragments and fragment chains,
to capture both local and global statistics of boundaries. Our system is successfully applied to both synthetic and real video sequences containing high-contrast
boundaries and textureless regions. The system produces good motion estimates
along with properly grouped and completed contours.
1 Introduction
Humans can reliably analyze visual motion under a diverse set of conditions, including textured
as well as featureless objects. Computer vision algorithms have focussed on conditions of texture,
where junction or corner-like image structures are assumed to be reliable features for tracking [5, 4,
17]. But under other conditions, these features can generate spurious motions. T-junctions caused by
occlusion can move in an image very differently than either of the objects involved in the occlusion
event [11]. To properly analyze motions of featureless objects requires a different approach.
The spurious matching of T-junctions has been explained in [18] and [9]. We briefly restate it using
the simple two bar stimulus in Figure 1 (from [18]). The gray bar is moving rightward in front of
the leftward moving black bar, (a). If we analyze the motion locally, i.e. match to the next frame
in a local circular window, the flow vectors of the corner and line points are as displayed in Figure
1 (b). The T-junctions located at the intersections of the two bars move downwards, but there is no
such a motion by the depicted objects.
One approach to handling the spurious motions of corners or T-junctions has been to detect such
junctions and remove them from the motion analysis [18, 12]. However, T-junctions are often very
difficult to detect in a static image from local, bottom-up information [9]. Motion at occluding
boundaries has been studied, for example in [1]. The boundary motion is typically analyzed locally,
(a)
(b)
(c)
Figure 1: Illustration of the spurious T-junction motion. (a) The front gray bar is moving to the right and the
black bar behind is moving to the left [18]. (b) Based on a local window matching, the eight corners of the bars
show the correct motion, whereas the T-junctions show spurious downwards motion. (c) Using the boundarybased representation our system is able to correctly estimate the motion and generate the illusory boundary as
well.
which can again lead to spurious junction trackings. We are not aware of an existing algorithm that
can properly analyze the motions of featureless objects.
In this paper, we use a boundary-based approach which does not rely on motion estimates at corners
or junctions. We develop a graphical model which integrates local information and assigns probabilities to candidate contour groupings in order to favor motion interpretations corresponding to the
motions of the underlying objects. Boundary completion and discounting the motions of spurious
features result from optimizing the graphical model states to explain the contours and their motions.
Our system is able to automatically detect and group the boundary fragments, analyze the motion
correctly, as well as exploit both static and dynamic cues to synthesize the illusory boundaries (c).
We represent the boundaries at three levels of grouping: edgelets, boundary fragments and contours,
where a fragment is a chain of edgelets and a contour is a chain of fragments. Each edgelet within
a boundary fragment has a position and an orientation and carries local evidence for motion. The
main task of our model is then to group the boundary fragments into contours so that the local motion
uncertainties associated with the edgelets are disambiguated and occlusion or other spurious feature
events are properly explained. The result is a specialized motion tracking algorithm that properly
analyzes the motions of textureless objects.
Our system consists of four conceptual steps, discussed over the next three sections (the last two
steps happen together while finding the optimal states in the graphical model):
(a) Boundary fragment extraction: Boundary fragments are detected in the first frame.
(b) Edgelet tracking with uncertainties: Boundary fragments are broken into edgelets, and,
based on local evidence, the probability distribution is found for the motion of each edgelet
of each boundary fragment.
(c) Grouping boundary fragments into contours: Boundary fragments are grouped, using
both temporal and spatial cues.
(d) Motion estimation: The final fragment groupings disambiguate motion uncertainties and
specify the final inferred motions.
We restrict the problem to two-frame motion analysis though the algorithm can easily be extended
to multiple frames.
2 Boundary Fragment Extraction
Extracting boundaries from images is a nontrivial task by itself. We use a simple algorithm for
boundary extraction, analyzing oriented energy using steerable filters [3] and tracking the boundary
in a manner similar to that of the Canny edge detector [2]. A more sophisticated boundary detector
can be found in [8]; occluding boundaries can also be detected using special cameras [13]. However,
for our motion algorithm designed to handle the special case of textureless objects, we find that our
simple boundary detection algorithm works well.
Mathematically, given an image I, we seek to obtain a set of fragments B = {b i }, where each
i
fragment bi is a chain of edgelets b i = {eik }nk=1
. Each edgelet e ik = {pik , ?ik } is a particle which
2
embeds both location p ik ? R and orientation ? ik ? [0, 2?) information.
(a)
(b)
(c)
(d)
Figure 2: The local motion vector is estimated for each contour in isolation by selectively comparing orientation energies across frames. (a) A T-junction of the two bar example showing the contour orientation for this
motion analysis. (b) The other frame. (c) The relevant orientation energy along the boundary fragment, both for
the 2nd frame. A Gaussian pdf is fit to estimate flow, weighted by the oriented energy. (d) Visualization of the
Gaussian pdf. The possible contour motions are unaffected by the occluding contour at a different orientation
and no spurious motion is detected at this junction.
We use H4 and G4 steerable filters [3] to filter the image and obtain orientation energy per pixel.
These filters are selected because they describe the orientation energies well even at corners. For
each pixel we find the maximum energy orientation and check if it is local maximum within a slice
perpendicular to this orientation. If that is true and the maximum energy is above a threshold T 1 we
call this point a primary boundary point. We collect a pool of primary boundary points after running
this test for all the pixels.
We find the primary boundary point with the maximum orientation energy from the pool and do
bidirectional contour tracking, consisting of prediction and projection steps. In the prediction step,
the current edgelet generates a new one by following its orientation with a certain step size. In
the projection step, the orientation is locally maximized both in the orientation bands and within a
small spatial window. The tracking is stopped if the energy is below a threshold T 2 or if the turning
angle is above a threshold. The primary boundary points that are close to the tracked trajectory are
removed from the pool. This process is repeated until the pool is empty. The two thresholds T 1 and
T2 play the same roles as those in Canny edge detection [2]. While the boundary tracker should
stop at sharp corners, it can turn around and continue tracking. We run a postprocess to break the
boundaries by detecting points of curvature local maxima which exceed a curvature threshold.
3 Edgelet Tracking with Uncertainties
We next break the boundary contours into very short edgelets and obtain the probabilities, based
on local motion of the boundary fragment, for the motion vector at each edgelet. We cannot use
conventional algorithms, such as Lucas-Kanade [5], for local motion estimation since they rely on
corners. The orientation ? ik for each edgelet was obtained during boundary fragment extraction.
We obtain the motion vector by finding the spatial offsets of the edgelet which match the orientation
energy along the boundary fragment in this orientation. We fit a Gaussian distribution N (? ik , ?ik )
of the flow weighted by the orientation energy in the window. The mean and covariance matrix is
added to the edgelet: e ik = {pik , ?ik , ?ik , ?ik }. This procedure is illustrated in Figure 2.
Grouping the boundary fragments allows the motion uncertainties to be resolved. We next discuss
the mathematical model of grouping as well as the computational approach.
4 Boundary Fragment Grouping and Motion Estimation
4.1 Two Equivalent Representations for Fragment Grouping
The essential part of our model is to find the connection between the boundary fragments. There are
two possible representations for grouping. One representation is the connection of each end of the
boundary fragment. We formulate the probability of this connection to model the local saliency of
contours. The other equivalent representation is a chain of fragments that forms a contour, on which
global statistics are formulated, e.g. structural saliency [16]. Similar local and global modeling of
contour saliency was proposed in [14]; in [7], both edge saliency and curvilinear continuity were
used to extract closed contours from static images. In [15], contour ends are grouped using loopy
belief propagation to interpret contours.
The connections between fragment ends are modeled by switch variables. For each boundary frag(0)
ment bi , we use a binary variable {0, 1} to denote the two ends of the fragment, i.e. b i = ei1 and
(t )
(1)
(ti )
bi = ei,ni . Let switch variable S(i, ti ) = (j, tj ) denote the connection from b i to bj j . This
b3
b3
b2
b1
b1
(a)
b1( 0)
(b)
b2
b2
b1
b3
b3
b3
(c)
b2
b1
(d)
b2
b1
(e)
Figure 3: A simple example illustrating switch variables, reversibility and fragment chains. The color arrows
show the switch variables. The empty circle indicates end 0 and the filled indicates end 1. (a) Shows three
(0)
boundary fragments. Theoretically b1 can connect to any of the other ends including itself, (b). However, the
(0)
(0)
switch variable is exclusive, i.e. there is only one connection to b1 , and reversible, i.e. if b1 connects to
(0)
(0)
(0)
b3 , then b3 should also connect to b1 , as shown in (c). Figures (d) and (e) show two of the legal contour
groupings for the boundary fragments: two open contours and a closed loop contour.
connection is exclusive, i.e. each end of the fragment should either connect to one end of the other
fragment, or simply have no connection. An exclusive switch is further called reversible, i.e.
if S(i, ti ) = (j, tj ), then S(j, tj ) = (i, ti ),
or in a more compact form
S(S(i, ti )) = (i, ti ).
(1)
(t )
bi i ,
When there is no connection to
we simply set S(i, ti ) = (i, ti ). We use the binary function
(t )
(t )
?[S(i, ti )?(j, tj )] to indicate whether there is a connection between b i i and bj j . The set of all
the switches are denoted as S = {S(i, t i )|i = 1 : N, ti = 0, 1}. We say S is reversible if every switch
variable satisfies Eqn. (1). The reversibility of switch variables is shown in Figure 3 (b) and (c).
From the values of the switch variables we can obtain contours, which are chains of
boundary fragments. A fragment chain is defined as a series of the end points c =
(x )
(x )
(x )
(x )
{(bi1 1 , bi1 1 ), ? ? ? , (bimm , bimm )}. The chain is specified by fragment label {i 1 , ? ? ? , im } and
end label {x1 , ? ? ? , xm }. It can be either open or closed. The order of the chain is determined by the
switch variable. Each end appears in the chain at most once. The notation of a chain is not unique.
Two open chains are identical if the fragment and end labels are reversed. Two closed chains are
identical if they match each other by rotating one of them. These identities are guaranteed from the
reversibility of the switch variables. A set of chains C = {c i } can be uniquely extracted based on
the values of the reversible switch variables, as illustrated in Figure 3 (d) and (e).
4.2 The Graphical Model
Given the observation O, the two images, and the boundary fragments B, we want to estimate the
flow vectors V = {vi } and vi = {vik }, where each vik associates with edgelet eik , and the grouping
variables S (switches) or equivalently C (fragment chains). Since the grouping variable S plays an
essential role in the problem, we shall first infer S and then infer V based on S.
4.2.1 The Graph for Boundary Fragment Grouping
We use two equivalent representations for boundary grouping, switch variables and chains. We use
?[S(S(i, ti )) ? (i, ti )] for each end to enforce the reversibility. Suppose otherwise S(i 1 , ti1 ) =
S(i2 , ti2 ) = (j, tj ) for i1 = i2 . Let S(j, tj ) = (i1 , ti1 ) without loss of generality, then
?[S(S(i2 , ti2 )) ? (i2 , ti2 )] = 0, which means that the switch variables are not reversible.
(ti )
We use a function ?(S(i, t i ); B, O) to measure the distribution of S(i, t i ), i.e. how likely b i
connects to the end of other fragments. Intuitively, two ends should be connected if
Motion similarity the distributions of the motion of the two end edgelets are similar;
Curve smoothness the illusory boundary to connect the two ends is smooth;
Contrast consistency the brightness contrast at the two ends consistent with each other.
We write ?(?) as a product of three terms, one enforcing each criterion. We shall follow the example
in Figure 4 to simplify the notation, where the task is to compute ?(S(1, 0) = (2, 0)). The first term
b (20 )
( ? 21 , ? 21 )
b2
h21
h22
b2
b2
r
b1
b1( 0 )
( ?11 , ?11 )
b1
(a)
h11
b1
b1
(b)
b2
(c)
h12
(d)
Figure 4: An illustration of local saliency computation. (a) Without loss of generalization we assume the two
(0)
(0)
ends to be b1 and b2 . (b) The KL divergence between the distributions of flow vectors are used to measure
the motion similarity. (c) An illusory boundary ? is generated by minimizing the energy of the curve. The sum
of square curvatures are used to measure the curve smoothness. (d) The means of the local patches located at
(0)
(0)
the two ends are extracted, i.e. h11 and h12 from b1 , h21 and h22 from b2 , to compute contrast consistency.
is the KL divergence between the two Gaussian distributions of the flow vectors
exp{??KL KL(N (?11 , ?11 ), N (?21 , ?21 ))},
(2)
where ?KL is a scaling factor. The second term is the local saliency measure on the illusory boundary ? that connects the two ends. The illusory boundary is simply generated by minimizing the
energy of the curve. The saliency is defined as
2
d?
exp ???
ds ,
(3)
ds
?
where ?(s) is the slope along the curve, and d?
ds is local curvature [16]. ? ? is a scaling factor. The
third term is computed by extracting the mean of local patches located at the two ends
dmax
dmin
,
(4)
exp ? 2
? 2
2?max
2?min
where d1 = (h11 ? h21 )2 , d2 = (h12 ? h22 )2 , and dmax = max(d1 , d2 ), dmin = min(d1 , d2 ).
?max > ?min are the scale parameters. h 11 , h12 , h21 , h22 are the means of the pixel values of
the four patches located at the two end points. For self connection we simply set a constant value:
?(S(i, ti ) = (i, ti )) = ? .
We use a function ?(ci ; B, O) to model the structural saliency of contours. It was discovered in [10]
that convex occluding contours are more salient, and additional T-junctions along the contour may
increase or decrease the occlusion perception. Here we simply enforce that a contour should have
no self-intersection. ?(c i ; B, O) = 1 if there is no self intersection and ?(c i ; B, O) = 0 otherwise.
Thus, the (discrete) graphical model favoring the desired fragment grouping is
Pr(S; B, O) =
N
1
M
1
?(S(i, ti ); B, O)?[S(S(i, ti )) ? (i, ti )] ?
?(cj ; B, O),
ZS i=1 t =0
j=1
(5)
i
where ZS is a normalization constant. Note that this model measures both the switch variables
S(i, ti ) for local saliency and the fragment chains c i to enforce global structural saliency.
4.2.2 Gaussian MRF on Flow Vectors
Given the fragment grouping, we model the flow vectors V as a Gaussian Markov random field
(GMRF). The edgelet displacement within each boundary fragment should be smooth and match
the observation along the fragment. The probability density is formulated as
?(vi ; bi ) =
ni
k=1
exp{?(vik ? ?ik )
T
??1
ik (vik
? ?ik )}
n
i ?1
k=1
exp{?
1
vik ? vi,k+1 2 },
2? 2
(6)
where ?ik and ?ik are the motion parameters of each edgelet estimated in Sect 3.
We use V(i, ti ) to denote the flow vector of end t i of fragment b i . We define V(S(i, ti )) = V(j, tj )
if S(i, ti ) = (j, tj ). Intuitively the flow vectors of the two ends should be similar if they are connected, or mathematically
1
ifS(i, ti ) = (i, ti ),
1
?(V(i, ti ), V(S(i, ti ))) =
(7)
2
exp{? 2 V(i, ti )?V(S(i, ti )) } otherwise.
2?
The (continuous) graphical model of the flow vectors is therefore defined as
Pr(V|S; B) =
N
1
1
?(vi ; bi )
?(V(i, ti ), V(S(i, ti )))
ZV i=1
t =0
(8)
i
where ZV is a normalization constant. When S is given it is a GMRF which can be solved by least
squares.
4.3 Inference
Having defined the graphical model to favor the desired motion and grouping interpretations, we
need to find the state parameters that best explain the image observations. The natural decomposition
of S and V in our graphical model
Pr(V, S; B, O) = Pr(S; B, O) ? Pr(V|S; B, O),
(9)
(where Pr(S; B, O) and Pr(V|S; B, O) are defined in Eqn. (5) and (8) respectively) lends itself
to performing two-step inference. We first infer the boundary grouping B, and then infer V based
on B. The second step is simply to solve least square problem since Pr(V|S; B, O) is a GMRF.
This approach does not globally optimize Eqn. (9) but results in reasonable solution because V
strongly depends on S. The density function Pr(S; B, O) is not a random field, so we use importance
sampling [6] to obtain the marginal distribution Pr(S(i, t i ); B, O). The proposal density of each
switch variable is set to be
1
q (S(i, ti ) = (j, tj )) ?
? (S(i, ti ) = (j, tj )) ? (S(j, tj ) = (i, ti ))
(10)
Zq
where ?(?) has been normalized to sum to 1 for each end. We found that this bidirectional measure
is crucial to take valid samples. To sample the proposal density, we first randomly select a boundary
fragment, and connect to other fragments based on q(S(i, t i )) to form a contour (a chain of boundary
fragments). Each end is sampled only once, to ensure reversibility. This procedure is repeated
until no fragment is left. In the importance step we run the binary function ?(c i ) to check that
each contour has no self-intersection. If ?(c i ) = 0 then this sample is rejected. The marginal
distributions are estimated from the samples. Lastly the optimal grouping is obtained by replacing
random sampling with selecting the maximum-probability connection over the estimated marginal
distributions. The number of samples needed depends on the number of the fragments. In practice
we find that n2 samples are sufficient for n fragments.
5 Experimental Results
Figure 6 shows the boundary extraction, grouping, and motion estimation results of our system for
both real and synthetic examples 1. All the results are generated using the same parameter settings.
The algorithm is implemented in MATLAB, and the running time varies from ten seconds to a few
minutes, depending on the number of the boundary fragments found in the image.
The two-bar examples in Figure 1(a) yields fourteen detected boundary fragments in Figure 6(a)
and two contours in (b). The estimated motion matches the ground truth at the T-junctions. The
fragments belonging to the same contour are plotted in the same color and the illusory boundaries
are synthesized as shown in (c). The boundaries are warped according to the estimated flow and
displayed in (d). The hallucinated illusory boundaries in frame 1 (c) and 2 (d) are plausible amodal
completions.
The second example is the Kanizsa square where the frontal white square moves to the right bottom.
Twelve fragments are detected in (a) and five contours are grouped in (b). The estimated motion and
generated illusory boundary also match the ground truth and human perception. Notice that the arcs
tend to connect to other ones if we do not impose the structural saliency ?(?).
We apply our system to a video of a dancer (Figure 5 (a) and (b)). In this stimulus the right leg
moves downwards, but there is weak occluding boundary at the intersection of the legs. Eleven
1
The results can be viewed online http://people.csail.mit.edu/celiu/contourmotions/
(a) Dancer frame 1
(b) Dancer frame 2
(c) Chair frame 1
(d) Chair frame 2
Figure 5: Input images for the non-synthetic examples of Figure 6. The dancer?s right leg is moving downwards and the chair is rotating (note the changing space between the chair?s arms).
boundary fragments are extracted in (a) and five contours are extracted in (b). The estimated motion
(b) matches the ground truth. The hallucinated illusory boundary in (c) and (d) correctly connect the
occluded boundary of the right leg and the invisible boundary of the left leg.
The final row shows challenging images of a rotating chair (Figure 5 (c) and (d)), also showing
proper contour completion and motion analysis. Thirty-seven boundary fragments are extracted
and seven contours are grouped. To complete the occluded contours of this image would be nearly
impossible working only from a static image. Exploiting motion as well as static information, our
system is able to complete the contours properly.
Note that the traditional motion analysis algorithms fail at estimating motion for these examples
(see supplementary videos) and would thus also fail at correctly grouping the objects based on the
motion cues.
6 Conclusion
We propose a novel boundary-based representation to estimate motion under the challenging visual conditions of moving textureless objects. Ambiguous local motion measurements are resolved
through a graphical model relating edgelets, boundary fragments, completed contours, and their
motions. Contours are grouped and their motions analyzed simultaneously, leading to the correct
handling of otherwise spurious occlusion and T-junction features. The motion cues help the contour
completion task, allowing completion of contours that would be difficult or impossible using only
low-level information in a static image. A motion analysis algorithm such as this one that correctly
handles featureless contour motions is an essential element in a visual system?s toolbox of motion
analysis methods.
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(a) Extracted boundaries
(b) Estimated flow
(c) Frame 1
(d) Frame 2
Figure 6: Experimental results for some synthetic and real examples. The same parameter settings were used
for all examples. Column (a): Boundary fragments are extracted using our boundary tracker. The red dots are
the edgelets and the green ones are the boundary fragment ends. Column (b): Boundary fragments are grouped
into contours and the flow vectors are estimated. Each contour is shown in its own color. Columns (c): the
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2,250 | 3,041 | Parameter Expanded Variational Bayesian Methods
Tommi S. Jaakkola
MIT CSAIL
32 Vassar street
Cambridge, MA 02139
[email protected]
Yuan (Alan) Qi
MIT CSAIL
32 Vassar street
Cambridge, MA 02139
[email protected]
Abstract
Bayesian inference has become increasingly important in statistical machine
learning. Exact Bayesian calculations are often not feasible in practice, however.
A number of approximate Bayesian methods have been proposed to make such
calculations practical, among them the variational Bayesian (VB) approach. The
VB approach, while useful, can nevertheless suffer from slow convergence to the
approximate solution. To address this problem, we propose Parameter-eXpanded
Variational Bayesian (PX-VB) methods to speed up VB. The new algorithm is inspired by parameter-expanded expectation maximization (PX-EM) and parameterexpanded data augmentation (PX-DA). Similar to PX-EM and -DA, PX-VB expands a model with auxiliary variables to reduce the coupling between variables
in the original model. We analyze the convergence rates of VB and PX-VB and
demonstrate the superior convergence rates of PX-VB in variational probit regression and automatic relevance determination.
1
Introduction
A number of approximate Bayesian methods have been proposed to offset the high computational
cost of exact Bayesian calculations. Variational Bayes (VB) is one popular method of approximation. Given a target probability distribution, variational Bayesian methods approximate the target
distribution with a factored distribution. While factoring omits dependencies present in the target
distribution, the parameters of the factored approximation can be adjusted to improve the match.
Specifically, the approximation is optimized by minimizing the KL-divergence between the factored
distribution and the target. This minimization can be often carried out iteratively, one component
update at a time, despite the fact that the target distribution may not lend itself to exact Bayesian
calculations. Variational Bayesian approximations have been widely used in Bayesian learning (e.g.,
(Jordan et al., 1998; Beal, 2003; Bishop & Tipping, 2000)).
Variational Bayesian methods nevertheless suffer from slow convergence when the variables in the
factored approximation are actually strongly coupled in the original model. The same problem arises
in popular Gibbs sampling algorithm. The sampling process converges slowly in cases where the
variables are strongly correlated. The slow convergence can be alleviated by data augmentation (van
Dyk & Meng, 2001; Liu & Wu, 1999), where the idea is to identify an optimal reparameterization
(within a family of possible reparameterizations) so as to remove coupling. Similarly, in a deterministic context, Liu et al. (1998) proposed over-parameterization of the model to speed up EM
convergence. Our work here is inspired by DA sampling and PX-EM. Our approach uses auxiliary
parameters to speed up the deterministic approximation of the target distribution.
Specifically, we propose Parameter-eXpanded Variational Bayesian (PX-VB) method. The original
model is modified by auxiliary parameters that are optimized in conjunction with the variational
approximation. The optimization of the auxiliary parameters corresponds to a parameterized joint
optimization of the variational components; the role of the new updates is to precisely remove otherwise strong functional couplings between the components thereby facilitating fast convergence.
2
An illustrative example
Consider a toy Bayesian model, which has been considered by Liu and Wu (1999) for sampling.
p(y|w, z) = N (y | w + z, 1),
p(z) = N (z | 0, D)
(1)
where D is a know hyperparameter and p(w) ? 1. The task is to compute the posterior distribution of w. Suppose we use a VB method to approximate p(w|y), p(z|y) and p(w, z|y) by
q(w), q(z) and q(w, z) = q(w)q(z), respectively. The approximation is optimized by minimizing
KL(q(w)q(z)kp(y|w, z)p(z)) (the second argument need not be normalized). The general forms of
the component updates are given by
q(w) ? exp(hln p(y|w, z)p(z)iq(z) )
(2)
q(z) ? exp(hln p(y|w, z)p(z)iq(w) )
It is easy to derive the updates in this case:
y ? hwi
1
q(w) = N (w|y ? hzi, 1)
q(z) = N (z|
,
)
1 + D?1 1 + D?1
(3)
(4)
Now let us analyze the convergence of the mean parameter of q(w), hwi = y ? hzi. Iteratively,
hwi
D?1
y+
= D?1 (1 + D?1 )?1 y + (1 + D?1 )?2 y + ? ? ? = y.
hwi =
?1
?1
1+D
1+D
The variational estimate hwi converges to y, which actually is the true posterior mean (For this toy
problem, p(w|y) = N (w|y, 1+D)). Furthermore, if D is large, hwi converges slowly. Note that the
variance parameter of q(w) converges to 1 in one iteration, though underestimates the true posterior
variance 1 + D.
Intuitively, the convergence speed of hwi and q(w) suffers from strong coupling between the updates
of w and z. In other words, the update information has to go through a feedback loop w ? z ?
w ? ? ? . To alleviate the coupling, we expand the original model with an additional parameter ?:
p(y|w, z) = N (y | w + z, 1)
p(z|?) = N (z | ?, D)
(5)
The expanded model reduces to the original one when ? equals the null value ?0 = 0.
Now having computed q(z) given ? = 0, we minimize KL(q(w)q(z)kp(y|w, z)p(z|?)) over ? and
obtain the minimizer ? = hzi. Then, we reduce the expanded model to the original one by applying
the reduction rule
z new = z ? ? = z ? hzi,
wnew = w + ? = w + hzi.
Correspondingly, we change the measures of q(w) and q(z):
1
q(w + hzi) ? q(wnew ) = N (wnew |y, 1)
q(z ? hzi) ? q(z new ) = N (z new |0,
) (6)
1 + D?1
Thus, the PX-VB method converges. Here ? breaks the update loop between q(w) and q(z) and
plays the role of a correction force; it corrects the update trajectories of q(w) and q(z) and makes
them point directly to the convergence point.
3
The PX-VB Algorithm
In the general PX-VB formulation, we over-parameterize the model p(?
x, D) to get p? (x, D), where
the original model is recovered for some default values of the auxiliary parameters ? = ?0 . The
algorithm consists of the typical VB updates relative to p? (x, D), the optimization of auxiliary
parameters ?, as well as a reduction step to turn the model back to the original form where ? = ?0 .
This last reduction step has the effect of jointly modifying the components of the factored variational
approximation. Put another way, we push the change in p? (x, D), due to the optimization of ?,
into the variational approximation instead. Changing the variational approximation in this manner
permits us to return the model into its original form and set ? = ?0 .
Specifically, we first expand p(?
x, D) to obtain p? (x, D). Then at the tth iteration,
Q
1. q(xs ) are updated sequentially. Note that the approximate distribution q(x) = s q(xs ).
2. We minimize KL(q(x)kp? (x, D)) over the auxiliary parameters ?. This optimization can
be done jointly with some components of the variational distribution, if feasible.
3. The expanded model is reduced to the original model through reparameterization. Accordingly, we change q (t+1) (x) to q (t+1) (?
x) such that
KL(q (t+1) (?
x)kp?0 (?
x, D)) = KL(q(x)kp?(t+1) (x, D))
(7)
where q (t+1) (?
x) are the modified components of the variational approximation.
4. Set ? = ?0 .
Since each update of PX-VB decreases or maintains the KL divergence KL(q(x)kp(x, D)), which
is lower bounded, PX-VB reaches a stationary point for KL(q(x)kp(x, D)). Empirically, PX-VB
often achieves solution similar to what VB achieves, with faster convergence.
A simple strategy to implement PX-VB is to use a mapping S? , parameterized by ?, over the vari? . After sequentially optimizing over the components {q(xs )}, we maximize hln p? (x)iq(x)
ables x
over ?. Then, we reduce p? (x, D) to p(?
x, D) and q(x) to q(?
x) through the inverse mapping of S? ,
?1
M? ? S ?
. Since we optimize ? after optimizing {q(x?s }, the mapping S should change at least
two components of x. Otherwise, the optimization over ? will do nothing since we have already
optimized over each q(x?s ). If we jointly optimize ? and one component q(xs ), it suffices (albeit
need not be optimal) for the mapping S? to change only q(xs ).
Algorithmically, PX-VB bears a strong similarity to PX-EM (Liu et al., 1998). They both expand the
original model and both are based on lower bounding KL-divergence. However, the key difference
is that the reduction step in PX-VB changes the lower-bounding distributions {q(xs )}, while in PXEM the reduction step is performed only for the parameters in p(x, D). We also note that the PX-VB
reduction step via M? leaves the KL-divergence (lower bound on the likelihood) invariant, while in
PX-EM the likelihood of the observed data remains the same after the reduction. Because of these
differences, general EM acceleration methods (e.g., (Salakhutdinov et al., 2003)) can not be directly
applied to speed up VB convergence.
In the following sections, we present PX-VB methods for two popular Bayesian models: Probit regression for data classification and Automatic Relevance Determination (ARD) for feature selection
and sparse learner.
3.1
Bayesian Probit regression
Probit regression is a standard classification technique (see, e.g., (Liu et al., 1998) for the maximum
likelihood estimation). Here we demonstrate the use of variational Bayesian methods to train Probit
models.
The data likelihood for Probit regression is
p(t|X, w) =
Y
?(tn wT xn ),
n
where X = [x1 , . . . , xN ] and ? is the standard normal cumulative distribution function. We can
rewrite the likelihood in an equivalent form
p(zn |w, xn ) = N (zn |wT xn , 1)
p(tn |zn ) = sign(tn zn )
(8)
Given a Gaussian prior over the parameter, p(w) = N (w|0, v0 I),
Q we wish to approximate the Q
posterior distribution p(w, z|X, t) by q(w, z) = q(w) n q(zn ). Minimizing
KL(q(w) n q(zn )kp(w, z, t|X)), we obtain the following VB updates:
q(zn ) = T N (zn |hwiT xn , 1, tn zn )
T
q(w) = N (w|(XX +
v0?1 I)?1 Xhzi, (XXT
(9)
+
v0?1 I)?1 )
where T N (zn |hwiT xn , 1, tn zn ) stands for a truncated Gaussian such
T N (zn |hwiT xn , 1, tn zn ) = N (zn |hwiT xn , 1) when tn zn > 0, and it equals 0 otherwise.
(10)
that
To speed up the convergence of the above iterative updates, we apply the PX-VB method. First, we
? z
?, t|X) to pc (w, z, t|X) with the mapping
expand the orginal model p(w,
?
?c
w = wc
z=z
(11)
such that
pc (zn |w, xn ) = N (zn |wT xn , c2 )
p(w) = N (w|0, c2 v0 I)
(12)
Setting c = c0 = 1 in the expanded model, we update
q(z
)
and
q(w)
as
before,
via
(9)
and
(10).
n
Then, we minimize KL q(z)q(w)kpc (w, z, t|X) over c, yielding
X
1
?1
T
T
c2 =
(hzn2 i ? 2hzn ihwiT xn + xT
(13)
n hww ixn ) + v0 hww i
N +M n
where M is the dimension of w. In the degenerate case where v0 = ?, the denominator of the
above equation becomes N instead of N + M . Since this equation can be efficiently calculated, the
extra computational cost induced by the auxiliary variable is therefore small. We omit the details.
The transformation back to pc0 can be made via the inverse map
b = w/c
b
w
z = z/c.
(14)
b
Accordingly, we change q(w) to obtain a new posterior approximation qc (w):
?1 ?1
?1 ?1 2
T
T
b = N (w|(XX
b
qc (w)
+ v0 I) Xhzi/c, (XX + v0 I) /c )
We do not actually need to compute qc (zn ) if this component will be optimized next.
(15)
b through (14), the KL divergence between the approximate and exact
By changing variables w to w
b and q(?
posteriors remains the same. After obtaining new approximations qc (w)
zn ), we reset c =
c0 = 1 for the next iteration.
Though similar to the PX-EM updates for the Probit regression problem (Liu et al., 1998), the PXVB updates are geared towards providing an approximate posterior distribution.
We use both synthetic data and a kidney biopsy data (van Dyk & Meng, 2001) as numerical examples
for probit regression. We set v0 = ? in the experiment. The comparison of convergence speeds for
VB and PXVB is illustrated in figure 1.
?2
log(||wt+1?wt||)
log(||wt+1?wt||)
0
VB
PX?VB
0
?4
?6
?8
VB
PX?VB
?2
?4
?6
?8
0
1000 2000 3000 4000 5000
Number of iterations
(a)
0
2000
4000
6000
Number of iterations
(b)
Figure 1: Comparison between VB and PX-VB for probit regression on synthetic (a) and kidneybiospy data sets (b). PX-VB converges significantly faster than VB. Note that the Y axis shows the
difference between two consecutive estimates of the posterior mean of the parameter w.
For the synthetic data, we randomly sample a classifier and use it to define the data labels for sampled
inputs. We have 100 training and 500 test data points, each of which is 20 features. The kidney
data set has 55 data points, each of which is a 3 dimensional vector. On the synthetic data, PXVB converges immediately while VB updates are slow to converge. Both PX-VB and VB trained
classifiers achieve zero test error. On the kidney biopsy data set, PX-VB converges in 507 iterations,
while VB converges in 7518 iterations. In other words, PX-VB requires 15 times fewer iterations
than VB. In terms of CPU time, which reflects the extra computational cost induced by the auxiliary
variables, PX-VB is 14 times more efficient. Among all these runs, PX-VB and VB achieve very
similar estimates of the model parameters and the same prediction results. In sum, with a simple
modification of VB updates, we significantly improve the convergence speed of variational Bayesian
estimation for probit model.
3.2
Automatic Relevance Determination
Automatic relevance determination (ARD) is a powerful Bayesian sparse learning technique
(MacKay, 1992; Tipping, 2000; Bishop & Tipping, 2000). Here, we focus on variational ARD
proposed by Bishop and Tipping (2000) for sparse Bayesian regression and classification.
The likelihood for ARD regression is
p(t|X, w, ? ) =
Y
N (tn |wT ?n , ? ?1 )
n
where ?n is a feature vector based on xn , such as [k(x1 , xn ), . . . , [k(xN , xn )]T where k(xi , xj )
is a nonlinear basis function. For example, we can choose a radial basis function k(xi , xj ) =
exp(?kxi ? xj k/(2?2 ), where ? is the kernel width.
QM
?1
In ARD, we assign a Gaussian prior on the model parameters w: p(w|?) = m=0 N (wm |0, ?m
),
where the inverse variance diag(?) follows a factorized Gamma distribution:
Y
Y
a?1 ?b?m
/?(a)
(16)
p(?) =
Gamma(?m |a, b) =
ba ?m
e
m
m
where a and b are hyperparameters of the model. The posterior does not have a closed form. Let
us approximate p(w, ?, ? |X, t) by a factorized distribution q(w, ?, ? ) = q(w)q(?)q(? ). The
sequential VB updates on q(? ), q(w) and q(?) are described by Bishop and Tipping (2000).
The variational RVM achieves good generalization performance as demonstrated by Bishop and
Tipping (2000). However, its training based on the VB updates can be quite slow. We apply PX-VB
to address this issue.
b ?,
? ??|X, t) via
First, we expand the original model p(w,
b
w = w/r
(17)
? and ?? unchanged. Consequently, the data likelihood and the prior on w become
while maintaining ?
pr (t|w, X, ? ) =
Y
N (tn |rwT ?n , ? ?1 )
pr (w|?) =
n
M
Y
?1
N (wm |0, r?2 ?m
)
(18)
m=0
Setting r = r0 = 1, we update q(? ) and q(?) as in the regular VB. Then, we want to joint optimize
over q(w) and r. Instead of performing a fully joint optimization, we optimize q(w) and r separately
at the same time. This gives
p
g + g 2 + 16M f
r=
(19)
4f
P
P
P
T
2
T
T
where f = h? i n xT
n hww ixn +
m hwm ih?m i and g = 2h? i
m hw ixn tn . where hw i and
T
T
hww i are the first and second order moments of the previous q(w). Since both f and X hwi has
been computed previously in VB updates, the added computational cost for r is negligible overall.
The separate optimization over q(w) and r often decreases the KL divergence. But it cannot guarantee to achieve a smaller KL divergence than what optimization only over q(w) would achieves. If
the regular update over q(w) achieves a smaller KL divergence, we reset r = 1.
b = rw to reduce the expanded model to the original one. CorGiven r and q(w), we use w
? =
respondingly, we change q(w) = N (w|?w , ?w ) via this reduction rule to obtain qr (w)
2
b
N (w|r?
w , r ?w ).
?
We can also introduce another auxiliary variable s such that ? = ?/s.
Similar to the above procedure, we optimize over s the expected log joint probability of the expanded model, and at the same
? using the inverse mapping ?
? = s?. Due to
time update q(?). Then we change q(?) back to qs (?)
the space limitation, we skip the details here.
The auxiliary variables r and s change the individual approximate posteriors q(w) and q(?) separately. We can combine these two variables into one and use it to adjust q(w) and q(?) jointly.
Specifically, we introduce the variable c:
b
w = w/c
b
? = c2 ?.
VB
PX?VB
?6
?8
t
?w ||)
?4
?6
500
1000
1500
2000
Number of iterations
2500
?6
?7
?8
?8
0
?5
t+1
?4
?2
VB
PX?VB
?4
log(||w
?2
VB
PX?VB
0
log(||wt+1?wt||)
log(||wt+1?wt||)
0
0
500
(a)
1000
1500
2000
Number of iterations
2500
(b)
0
500
1000
1500
2000
Number of iterations
2500
(c)
Figure 2: Convergence comparison between VB and PX-VB for ARD regression on synthetic data
(a,b) and gene expression data (c). The PX-VB results in (a) and (c) are based on independent auxiliar variables on w and ?. The PX-VB result in (b) is based on the auxiliar variable that correlates
both w and ?. The added computational cost for PX-VB in each iteraction is negligible overall.
Setting c = c0 = 1, we perform the regular updates over q(? ), q(w) and q(?). Then we optimize
over c the expected log joint probablity of the expanded model. We cannot find a closed-form solution for the maximization. But we can efficiently compute its gradient and Hessian. Therefore, we
perform a few steps of Newton updates to partially optimize c. Again, the additional computational
cost for calculating c is small. Then using the inverse mapping, we reduce the expanded model to
the original one and adjust both q(w) and q(?) accordingly. Empirically, this approach can achieve
faster convergence than using auxiliary variables on q(w) and q(?) separately. This is demonstrated
in figure 2(a) and (b).
We compare the convergence speed of VB and PX-VB for the ARD model on both synthetic data
and gene expression data. The synthetic data are sampled from the function sinc(x) = (sinx)/x for
x ? (?10, 10) with added Gaussian noise. We use RBF kernels for the feature expansion ?n with
kernel width 3. VB and PX-VB provide basically identical predictions. For gene expression data,
we apply ARD to analyze the relationship between binding motifs and the expression of their target
genes. For this task, we use 3 order polynomial kernels.
The results of convergence comparison are shown in figure 2. With a little modification of VB
updates, we increase the convergence speed significantly. Though we demonstrate PX-VB improvement only for ARD regression, the same technique can be used to speed up ARD classification.
4
Convergence properties of VB and PX-VB
In this section, we analyze convergence of VB and PX-VB, and their convergence rates.
Define the mapping q(t+1) = M (q(t) ) as one VB update of all the approximate distributions.
Define an objective function as the unnormalized KL divergence:
Q
Z Y
Z
Z Y
qi (x)
Q(q) =
qi (x) log
) + ( p(x)dx ?
qi (x)dx).
(20)
p(x)
It is easy to check that minimizing Q(q) gives the same updates as VB which minimizes KL divergence.
Based on Theorem 2.1 by Luo and Tseng (1992), an iterative application of this mapping to minimize
Q(q) results in at least linear convergence to an element q? in the solution set.
Define the mapping q(t+1) = Mx (q(t) ) as one PX-VB update of all the approximate distributions. The convergence of PX-VB follows from similar arguments. i.e.,? = [qT ?T ]T converges to
T
[q? T ?T
0 ] , where ? ? ? are the expanded model parameters, ?0 are the null value in the original
model.
4.1
Convergence rate of VB and PX-VB
The matrix rate of convergence DM (q):
q(t+1) ? q? = DM (q)T (q(t) ? q? )
(21)
where DM (q) =
?Mj (q)
?qi
.
(t+1)
?
?q k
Define the global rate of convergence for q:
r
=
limt?? kqkq(t) ?q
?k .
Under certain regularity conditions, r = the largest eigenvalue of DM (q). The smaller r is,
the faster the algorithm converges.
Define the constraint set gs as the constraints for the sth update. Then the following theorem holds:
Theorem 4.1 The matrix convergence rate for VB is:
S
Y
DM (q? ) =
Ps
(22)
s=1
2
? ?1
2
? ?1
and Bs = ?gs (q? ).
Bs ]?1 BT
where Ps = Bs [BT
s D Q(q )
s D Q(q )
Proof: Define ? as the current approximation q. Let Gs (?) be qs that maximizes the objective
function Q(q) under the constraint gs (q) = gs (?) = [? \s ].
Let M0 (q) = q and
for all 1 ? s ? S.
Ms (q) = Gs (Ms?1 (q))
Then by construction of VB, we have q(t+s/S) = Ms (q(t) ),
DMS (q? ). At the stationary points, q? = DMs (q? ) for all s.
(23)
s = 1, . . . , S and DM (q? ) =
We differentiate both sides of equation (23) and evaluate them at q = q? :
DMs (q) = DMs?1 (q)DGS (Ms?1 (q? )) = DMs?1 (q? )DGS (q? )
QS
It follows that DM (q? ) = s=1 DGS (q? ).
(24)
To calculate DGS (q? ), we differentiate the constraint gs (Gs (?)) = gs (?) and evaluate both sides
at ? = q? , such that
DGs (q? )Bs = Bs .
(25)
Similarly, we differentiate the Lagrange equation DQs (G(?)) ? ?gs (G(?))?s (?) = 0 and evaluate
both sides at ? = q? . This yields
DGs (q? )D2 Qs (q? ) ? D?s (q? )BT
s =0
Equation (26) holds because
? 2 gs
?qi ?qj
(26)
= 0.
Combining (25) and (26) yields
2
? ?1
2
? ?1
DGs (q? ) = Bs [BT
Bs ]?1 BT
.2
s D Qs (q )
s D Qs (q )
(27)
In the s update we fix q\s , i.e., gs (q) = q\s . Therefore, Bs is an identity matrix with its sth column
removed Bs = I:,s , where I is the identity matrix and s, : means without the sth column.
?1
Denote Cs = D2 Qs (q? )
. Without the loss of generality, we set s = S. It is easy to obtain
BT
S CBS = C\S,\S
(28)
where \S, \S means without row S and column S.
Inserting (28) into (27) yields
Id?1
?
PS = DGS (q ) =
0
?1
C\S,\S
C\S,S
0
=
Id?1
0
?D2 Q\S,S (D2 QS,S )?1
0
where Id?1 is a (d ? 1) by (d ? 1) identity matrix, and D2 Q\S,S =
? 2 Q(qq (x)kp(x))
?q\S T ?qS
(29)
and D2 QS,S =
2
? Q(qq (x)kp(x))
.
?qS T ?qS
Notice that we use Schur complements to obtain (29). Similar to the calculation of
PS via (29), we can derive Ps for s = 1, . . . , S ? 1 with structures similar to PS .
The above results help us understand the convergence speed of VB. For example, we have
q(t+1) ? q? = PST ? ? ? P1T (q(t) ? q? ).
(t+1)
For qS , qS
? q?S = ? (D2 QS,S )?1 D2 QS,\S 0 (q(t+(S?1)/S) ? q? ).
(30)
Clearly, if we view D2 QS,\S as the correlation between qS and q\S , then the smaller ?correlation?,
the faster the convergence. In the extreme case, if there is no correlation between qS and q\S , then
(t+1)
qS
? q?S = 0 after the first iteration. Since the global convergence rate is bounded by the
maximal component convergence rate and generally there are many components with convergence
rate same as the global rate. Therefore, the instant convergence of qS could help increase the global
convergence rate.
For PX-VB, we can compute the matrix rate of convergence similarly. In the toy example in
Section 2, PX-VB introduces an auxiliary variable ? which has zero correlation with w, leading an instant convergence of the algorithm. This suggests that PX-VB improves the convergence by reducing the correlation among {qs }. Rigorously speaking, the reduction step in PXVB implictly defines a mapping between q to q?0 through the auxiliary variables ?: (q, p?0 ) ?
(q, p? ) ? (q? , p?0 ). Denote this mapping as M? such as q? = M? (q). Then we have
DMx (q? ) = DG1 (q? ) ? ? ? DG? (q? ) ? ? ? DGS (q? )
It is known that the spectral norm has the following submultiplicative property kEF k <= kEkkF k,
where E and F are two matrices. Thus, as long as the largest eigenvalue of M? is smaller than 1,
PX-VB converges faster than VB. The choice of ? affects the convergence rate by controlling the
eigenvalue of this mapping. The smaller the largest eigenvalue of M? , the faster PX-VB converges.
In practice, we can check this eigenvalue to make sure the constructed PX-VB algorithm enjoys a
fast convergence rate.
5
Discussion
We have provided a general approach to speeding up convergence of variational Bayesian learning.
Faster convergence is guaranteed theoretically provided that the Jacobian of the transformation from
auxiliary parameters to variational components has spectral norm bounded by one. This property
can be verified in each case separately. Our empirical results show that the performance gain due to
the auxiliary method is substantial.
Acknowledgments
T. S. Jaakkola was supported by DARPA Transfer Learning program.
References
Beal, M. (2003). Variational algorithms for approximate Bayesian inference. Doctoral dissertation, Gatsby
Computational Neuroscience Unit, University College London.
Bishop, C., & Tipping, M. E. (2000). Variational relevance vector machines. 16th UAI.
Jordan, M. I., Ghahramani, Z., Jaakkola, T. S., & Saul, L. K. (1998). An introduction to variational methods in
graphical models. Learning in Graphical Models. http://www.ai.mit.edu/?tommi/papers.html.
Liu, C., Rubin, D. B., & Wu, Y. N. (1998). Parameter expansion to accelerate EM: the PX-EM algorithm.
Biometrika, 85, 755?770.
Liu, J. S., & Wu, Y. N. (1999). Parameter expansion for data augmentation. Journal of the American Statistical
Association, 94, 1264?1274.
Luo, Z. Q., & Tseng, P. (1992). On the convergence of the coordinate descent method for convex differentiable
minimization. Journal of Optimization Theory and Applications, 72, 7?35.
MacKay, D. J. (1992). Bayesian interpolation. Neural Computation, 4, 415?447.
Salakhutdinov, R., Roweis, S. T., & Ghahramani, Z. (2003). Optimization with EM and Expectation-ConjugateGradient. Proceedings of International Conference on Machine Learning.
Tipping, M. E. (2000). The relevance vector machine. NIPS (pp. 652?658). The MIT Press.
van Dyk, D. A., & Meng, X. L. (2001). The art of data augmentation (with discussion). Journal of Computational and Graphical Statistics, 10, 1?111.
| 3041 |@word polynomial:1 norm:2 c0:3 d2:9 thereby:1 moment:1 reduction:9 liu:8 recovered:1 current:1 luo:2 dx:2 alanqi:1 numerical:1 remove:2 update:31 hwit:4 stationary:2 leaf:1 fewer:1 parameterization:1 accordingly:3 dissertation:1 pc0:1 probablity:1 c2:4 constructed:1 become:2 yuan:1 consists:1 combine:1 introduce:2 manner:1 theoretically:1 expected:2 inspired:2 salakhutdinov:2 cpu:1 little:1 becomes:1 provided:2 xx:3 bounded:3 maximizes:1 factorized:2 null:2 what:2 minimizes:1 transformation:2 guarantee:1 expands:1 biometrika:1 classifier:2 qm:1 unit:1 omit:1 orginal:1 before:1 negligible:2 despite:1 vassar:2 id:3 meng:3 interpolation:1 doctoral:1 suggests:1 practical:1 acknowledgment:1 practice:2 implement:1 procedure:1 empirical:1 significantly:3 alleviated:1 word:2 radial:1 regular:3 get:1 cannot:2 selection:1 put:1 context:1 applying:1 optimize:7 equivalent:1 deterministic:2 map:1 demonstrated:2 www:1 go:1 kidney:3 convex:1 qc:4 immediately:1 factored:5 rule:2 q:22 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2,251 | 3,042 | Statistical Modeling of Images with
Fields of Gaussian Scale Mixtures
Siwei Lyu
Eero. P. Simoncelli
Howard Hughes Medical Institute
Center for Neural Science, and
Courant Institute of Mathematical Sciences
New York University, New York, NY 10003
Abstract
The local statistical properties of photographic images, when represented in a
multi-scale basis, have been described using Gaussian scale mixtures (GSMs).
Here, we use this local description to construct a global field of Gaussian scale
mixtures (FoGSM). Specifically, we model subbands of wavelet coe?cients as
a product of an exponentiated homogeneous Gaussian Markov random field
(hGMRF) and a second independent hGMRF. We show that parameter estimation for FoGSM is feasible, and that samples drawn from an estimated FoGSM
model have marginal and joint statistics similar to wavelet coe?cients of photographic images. We develop an algorithm for image denoising based on the
FoGSM model, and demonstrate substantial improvements over current state-ofthe-art denoising method based on the local GSM model.
Many successful methods in image processing and computer vision rely on statistical models for
images, and it is thus of continuing interest to develop improved models, both in terms of their ability
to precisely capture image structures, and in terms of their tractability when used in applications.
Constructing such a model is di?cult, primarily because of the intrinsic high dimensionality of the
space of images. Two simplifying assumptions are usually made to reduce model complexity. The
first is Markovianity: the density of a pixel conditioned on a small neighborhood, is assumed to be
independent from the rest of the image. The second assumption is homogeneity: the local density is
assumed to be independent of its absolute position within the image. The set of models satisfying
both of these assumptions constitute the class of homogeneous Markov random fields (hMRFs).
Over the past two decades, studies of photographic images represented with multi-scale multiorientation image decompositions (loosely referred to as ?wavelets?) have revealed striking nonGaussian regularities and inter and intra-subband dependencies. For instance, wavelet coe?cients
generally have highly kurtotic marginal distributions [1, 2], and their amplitudes exhibit strong correlations with the amplitudes of nearby coe?cients [3, 4]. One model that can capture the nonGaussian marginal behaviors is a product of non-Gaussian scalar variables [5]. A number of authors
have developed non-Gaussian MRF models based on this sort of local description [6, 7, 8], among
which the recently developed fields of experts model [7] has demonstrated impressive performance
in denoising (albeit at an extremely high computational cost in learning model parameters).
An alternative model that can capture non-Gaussian local structure is a scale mixture model [9, 10,
11]. An important special case is Gaussian scale mixtures (GSM), which consists of a Gaussian random vector whose amplitude is modulated by a hidden scaling variable. The GSM model provides
a particularly good description of local image statistics, and the Gaussian substructure of the model
leads to e?cient algorithms for parameter estimation and inference. Local GSM-based methods
represent the current state-of-the-art in image denoising [12]. The power of GSM models should be
substantially improved when extended to describe more than a small neighborhood of wavelet coe?cients. To this end, several authors have embedded local Gaussian mixtures into tree-structured
MRF models [e.g., 13, 14]. In order to maintain tractability, these models are arranged such that
coe?cients are grouped in non-overlapping clusters, allowing a graphical probability model with no
loops. Despite their global consistency, the artificially imposed cluster boundaries lead to substantial
artifacts in applications such as denoising.
In this paper, we use a local GSM as a basis for a globally consistent and spatially homogeneous field
of Gaussian scale mixtures (FoGSM). Specifically, the FoGSM is formulated as the product of two
mutually independent MRFs: a positive multiplier field obtained by exponentiating a homogeneous
Gaussian MRF (hGMRF), and a second hGMRF. We develop a parameter estimation procedure,
and show that the model is able to capture important statistical regularities in the marginal and joint
wavelet statistics of a photographic image. We apply the FoGSM to image denoising, demonstrating
substantial improvement over the previous state-of-the-art results obtained with a local GSM model.
1 Gaussian scale mixtures
A GSM random vector x is formed as the product of a zero-mean Gaussian random vector u and an
d ?
d
independent random variable z, as x = zu, where = denotes equality in distribution. The density of
x is determined by the covariance of the Gaussian vector, ?, and the density of the multiplier, p z (z),
through the integral
T ?1
x ? x
1
pz (z)dz.
(1)
exp ?
p(x) = Nx (0, z?)pz (z)dz ?
?
2z
z|?|
z
z
A key property of GSMs is that when z determines the scale of the conditional variance of x given z,
which
? is a Gaussian variable with zero mean and covariance z?. In addition, the normalized variable
x z is a zero mean Gaussian with covariance matrix ?.
The GSM model has been used to describe the marginal and joint densities of local clusters of
wavelet coe?cients, both within and across subbands [9], where the embedded Gaussian structure
a?ords simple and e?cient computation. This local GSM model has been be used for denoising, by
independently estimating each coe?cient conditioned on its surrounding cluster [12]. This method
achieves state-of-the-art performances, despite the fact that treating overlapping clusters as independent does not give rise to a globally consistent statistical model that satisfies all the local constraints.
2 Fields of Gaussian scale mixtures
In this section, we develop fields of Gaussian scale mixtures (FoGSM) as a framework for modeling wavelet coe?cients of photographic images. Analogous to the local GSM model, we use a
latent multiplier field to modulate a homogeneous Gaussian MRF (hGMRF). Formally, we define a
FoGSM x as the product of two mutually independent MRFs,
?
d
x = u ? z,
(2)
where u is a zero-mean hGMRF, and z is a field of positive multipliers that control the local coe?cient variances. The operator ? denotes element-wise multiplication, and the square root operation
is applied to each component. Note that x has a one-dimensional GSM marginal distributions, while
its components have dependencies captured by the MRF structures of u and z.
Analogous to the local GSM, when conditioned on z, x is an inhomogeneous GMRF
? ?1
? T
?
1 T ? ?1
1
|Qu |
|Qu |
p(x|z) ? exp ? x D z Qu D z x = exp ? (x z) Qu (x z) ,
2
2
i zi
i zi
(3)
where Qu is the inverse covariance matrix of u (also known as the precision matrix), and D(?)
denotes the operator that form a diagonal
matrix from an input vector. Note also that the element?
wise division of the two fields, x z, yields a hGMRF with precision matrix Q u .
To complete the FoGSM model, we need to specify the structure of the multiplier field z. For
tractability, we use another hGMRF as a substrate, and map it into positive values by exponentiation,
x
u
log z
Fig. 1. Decomposition of a subband from image ?boat? (left) into the normalized subband u (middle)
and the multiplier field z (right, in the logarithm domain). Each image is rescaled individually to fill
the full range of grayscale intensities.
as was done in [10]. To be more specific, we model log(z) as a hGMRF with mean ? and precision
matrix Qz , where the log operator is applied element-wise, from which the density of z follows as:
|Qz |
1
exp ? (log z ? ?)T Qz (log z ? ?) .
(4)
pz (z) ?
2
i zi
This is a natural extension of the univariate lognormal prior used previously for the scalar multiplier
in the local GSM model [12].
The restriction to hGMRFs greatly simplifies computation with FoGSM. Particularly, we take advantage of the fact that a 2D hGMRF with circular boundary handling has a sparse block-circulant
precision matrix with a generating kernel ? specifying its nonzero elements. A block-circulant matrix is diagonalized by the Fourier transform, and its multiplication with a vector corresponds to
convolution with the kernel ?. The diagonalizability with a fixed and e?ciently computed transform
makes the parameter estimation, sampling, and inference with a hGMRF substantially more tractable
than with a general MRF. Readers are referred to [15] for a detailed description of hGMRFs.
Parameter estimation: The estimation of the latent multiplier field z and the model parameters
(?, Qz , Qu ) may be achieved by maximizing log p(x, z; Q u, Qz , ?) with an iterative coordinate-ascent
method, which is guaranteed to converge. Specifically, based on the statistical dependency structures
in the FoGSM model, the following three steps are repeated until convergence:
(i)
z(t+1) = argmaxz log p(x|z; Q(t)u ) + log p(z; Q(t)z , ?(t) )
(ii)
Q(t+1)
= argmaxQu log p(x|z(t+1); Qu )
u
(t+1)
(t+1)
(iii) (Qz , ? ) = argmaxQz ,? log p(z(t+1) ; Qz , ?)
(5)
According to the FoGSM model structure,
correspond to maximum likelihood
steps
? (ii) and (iii)
estimates of the parameters of hGMRFs, x z(t+1) and log z(t+1) , respectively. Because of this,
both steps may be e?ciently implemented by exploiting the diagonalization of the precision matrices
with 2D Fourier transforms [15].
Step (i) in (5) may be implemented with conjugate gradient ascent [16]. To simplify description and
computation,
? we introduce a new variable for the element-wise inverse square root of the multiplier:
s = 1 z. The likelihood in (3) is then changed to:
1
1 T
T
si exp ? (x ? s) Qu (x ? s) =
si exp ? s D(x)Qu D(x)s .
p(x|s) ?
(6)
2
2
i
i
The joint density of s is obtained from (4), using the relations between densities of transformed
variables, as
1
1
(7)
p(s) ? exp ? (2 log s + ?)T Qz (2 log s + ?) .
2
i si
Combining . (6) and (7), step (i) in (5) is equivalent to computing s? = argmaxs log p(x|s; Qu ) +
log p(s; Qz , ?), which is further simplified into:
1
1
(8)
argmin sT D (x) Qu D (x) s + (2 log s + ?)T Qz (2 log s + ?) .
2
2
s
boat
house
peppers
x
x
x
x
log p(x)
Barbara
Fig. 2. Empirical marginal log distributions of coe?cients from a multi-scale decomposition of photographic images (blue dot-dashed line), synthesized FoGSM samples from the same subband (red solid
line), and a Gaussian with the same standard deviation (red dashed line).
and the optimal z? is then recovered as z? = 1 (?s ? s? ). We the optimize (8) with conjugate gradient
ascent [16]. Specifically, the negative gradient of the objective function in (8) with respect to s is
?
? log p(x|s)p(s)
?s
= D (x) Qu D (x) s + 2 D(s)?1 Qz (2 log s + ?)
= x ? (?u (x ? s)) + 2(?z (2 log s + ?)) s,
and the multiplication of any vector h with the Hessian matrix can be computed as:
?2 log p(x|s)p(s)
h = x ? (?u (x ? h)) + 4 (?z (h s)) s ? 2 ?z (log s + ?) ? h (s ? s).
?s2
Both operations can be expressed entirely in terms of element-wise operations ( and ?) and 2D
convolutions () with the generating kernels of the two precision matrices ? u and ?z , which allows
for e?cient implementation.
3 Modeling photographic images
We have applied the FoGSM model to subbands of a multi-scale image representation known as
a steerable pyramid [17]. This decomposition is a tight frame, constructed from oriented multiscale derivative operators, and is overcomplete by a factor of 4K/3, where K is the number of
orientation bands. Note that the marginal and joint statistics we describe are not specific to this
decomposition, and are similar for other multi-scale oriented representations. We fit a FoGSM
model to each subband of a decomposed photographic image, using the algorithms described in the
previous section. For precision matrices Q u and Qz , we assumed a 5 ? 5 Markov neighborhood
(corresponding to a 5 ? 5 convolution kernel), which was loosely chosen to optimize the tradeo?
between accuracy and overfitting.
Figure 1 shows the result of fitting a FoGSM model to an example subband from the ?boat? image
(left panel). The subband is decomposed into the product of the u field (middle panel) and the
z field (right panel, in the logarithm domain), along with model parameters Q u , ? and Qz (not
shown). Visually, the changing spatial variances are represented in the estimated log z field, and the
estimated u is much more homogeneous than the original subband and has a marginal distribution
close to Gaussian.1 However, the log z field still has a non-Gaussian marginal distribution and
is spatially inhomogeneous, suggesting limitations of FoGSM for modeling photographic image
wavelet coe?cients (see Discussion).
The statistical dependencies captured by the FoGSM model can be further revealed by examining
marginal and joint statistics of samples synthesized with the estimated
? model parameters. A sample
from FoGSM can be formed by multiplying samples of u and z. The former is obtained by
sampling from hGMRF u, and the latter is obtained from the element-wise exponentiation followed
by a element-wise square root operation of a sample of hGMRF log z. This procedure is again
e?cient for FoGSM due to the use of hGMRFs as building blocks [15].
Marginal distributions: We start by comparing the marginal distributions of the samples and the
original subband. Figure 2 shows empirical histograms in the log domain of a particular subband
1
This ?Gaussianizing? behavior was first noted in photographic images by Ruderman [18], who observed
that image derivative measurements that were normalized by a local estimate of their standard deviation had
approximately Gaussian marginal distributions.
close ? = 1
near ? = 4
far ? = 32
orientation
scale
real
sim
real
sim
Fig. 3. Examples of empirically observed distributions of wavelet coe?cient pairs, compared with
distributions from synthesized samples with the FoGSM model. See text for details.
from four di?erent photographic images (blue dot-dashed line), and those of the synthesized samples
of FoGSM models learned from each corresponding subband (red solid line). For comparison, a
Gaussian with the same standard deviation as the image subband is also displayed (red dashed line).
Note that the synthesized samples have conspicuous non-Gaussian characteristics similar to the real
subbands, exemplified by the high peak and heavy tails in the marginal distributions. On the other
hand, they are typically less kurtotic than the real subbands. We believe this arises from the imprecise
Gaussian approximation of log z (see Discussion).
Joint distributions: In addition to one-dimensional marginal statistics, the FoGSM model is capable of capturing the joint behavior of wavelet coe?cients. As described in [4, 9], wavelet coe?cients
of photographic images present non-Gaussian dependencies. Shown in the first and the third rows
in Fig. 3 are empirical joint and conditional histograms for one subband of the ?boat? image, for
five pairs of coe?cients, corresponding to basis functions with spatial separations of ? = {1, 4, 32}
samples, two orthogonal orientations and two adjacent scales. Contour lines in the joint histogram
are drawn at equal intervals of log probability. Intensities in the conditional histograms correspond
to probability, except that each column is independently rescaled to fill the full range of intensity.
For a pair of adjacent coe?cients, we observe an elliptical joint distribution and a ?bow-tie? shaped
conditional distribution. The latter is indicative of strong non-Gaussian dependencies. For coe?cients that are distant, the dependency becomes weaker and the corresponding joint and conditional
histograms become more separable, as would be expected for two independent random variables.
Random samples drawn from a FoGSM model, with parameters fitted to the corresponding subband,
have statistical characteristics consistent with the general description of wavelet coe?cients of photographic images. Shown in the second and the fourth rows of Fig. 3 are the joint and conditional
histograms of synthesized samples from the FoGSM model estimated from the same subband as
in the first and the third rows. Note that the joint and conditional histograms of the synthesized
samples have similar transition of spatial dependencies as the separation increases (column 1,2 and
3), suggesting that the FoGSM accounts well for pairwise joint dependencies of coe?cients over a
full range of spatial separations. On the other hand, the dependencies between subbands of di?erent
orientations and scales are not properly modeled by FoGSM (column 4 and 5). This is especially
true for subbands at di?erent scales, which exhibit strong dependencies. The current FoGSM model
original image
noisy image (? = 50)
(PSNR = 14.15dB)
GSM-BLS
(PSNR = 26.34dB)
FoGSM
(PSNR = 27.01dB)
Fig. 4. Denoising results using local GSM [12] and FoGSM. Performances are evaluated in peaksignal-to-noise-ratio (PSNR), 20 log10 (255/?e ), where ?e is the standard deviation of the error.
does not exhibit those dependencies as only spatial neighbors are used to make use the 2D hGMRFs
(see Discussion).
4 Application to image denoising
Let y = x+w be a wavelet subband of an image that has been corrupted with white Gaussian noise of
known variance. In an overcomplete wavelet domain such as steerable pyramid, the white Gaussian
noise is transformed into correlated Gaussian noise w ? N w (0, ?w ), whose covariance ? w can be
derived from the basis functions of the pyramid transform. With FoGSM as prior over x, commonly
used denoising methods involve expensive high-dimensional integration: for instance, maximum
a posterior estimate, x? MAP = argmaxx log p(x|y), requires a high-dimensional integral over z, and
the Bayesian least square estimation, x? BLS = E(x|y) requires a double high-dimensional integral
over x and z. Although it is possible to optimize with these criteria using Monte-Carlo Markov
sampling or other approximations, we instead develop a more e?cient deterministic algorithm that
takes advantage of the hGMRF structure in the FoGSM model. Specifically, we compute
(?x, z? , Q?u , Q? z , ?)
? = argmaxx,z,Qu ,Qz ,? log p(x, z|y; Q u, Qz , ?)
(9)
and take x? as the optimal denoised subband. Note that the model parameters are learned within the
inference process rather than in a separate parameter learning step. This strategy, known as partial
optimal solution [19], greatly reduces the computational complexity.
We optimize (9) with coordinate ascent, iterating between maximizing each of (x, z, Q u , Qz , ?) while
fixing the others. With fixed estimates of (z, Q u , Qz , ?), the optimization of x is
argmaxx log p(x, z|y; Q u, Qz , ?) = argmaxx log p(x|z, y; Q u, Qz , ?) + log p(z|y; Q u, Qz , ?) ,
which reduces to argmax x log p(x|z, y; Q u), with the second term independent of x and can be
dropped from optimization. Given the Gaussian structure of x given z, this step is then equivalent to
a Wiener filter (linear in y). Fixing (x, Q u , Qz , ?), the optimization of z is
argmaxz log p(x, z|y; Q u, Qz , ?)= argmaxz log p(y|x, z; Q u)+log p(x, z; Q u, Qz , ?)?log p(y; Qu , Qz , ?) ,
which is further reduced to argmax z log p(x, z; Qu , Qz , ?). Here, the first term was dropped since y is
independent of z when conditioned on x. The last term was also dropped since it is also independent
of z. Therefore, optimizing z given (x, Q u , Qz , ?) is equivalent to the first step of the algorithm in section 2, which can be implemented with e?cient gradient descent.
given (x, z), the FoGSM
? Finally,
(t+1)
and log z(t+1) , similar to the
model parameters (Q u , Qz , ?) are estimated from hGMRFs x z
second and third step in the algorithm of section 2. However, to reduce the overall computation
time, instead of a complete maximum likelihood estimation, these parameters are estimated with a
maximum pseudo-likelihood procedure [20], which finds the parameters maximizing the product of
all conditional distributions (which are 1D Gaussians in the GMRF case), followed by a projection
to the subspace of FoGSM parameters that results in positive definite precision matrices.
We tested this denoising method on a standard set of test images [12]. The noise corrupted images
were first decomposed these into a steerable pyramid with multiple levels (5 levels for a 512 ? 512
image and 4 levels for a 256 ? 256 image ) and 8 orientations. We assumed a FoGSM model for
each subband, with a 5 ? 5 neighborhood for field u and a 1 ? 1 neighborhood for field log z. These
sizes were chosen to provide a reasonable combination of performance and computational e?ciency.
We then estimate the optimal x with the algorithm described previously, with the initial values of
x and z computed from subband denoised with the local GSM model [12]. Shown in Fig. 4 is an
example of denoising the ?boat? image corrupted with simulated additive white Gaussian noise of
strength ? = 50, corresponding to a peak-signal-to-noise-ratio (PSNR), of 14.15 dB. We compare
this with the local GSM method in [12], which, assuming a local GSM model for the neighborhood
consisting of 3 ? 3 spatial neighbors plus parent in the next coarsest scale, computes a Bayes least
squares estimate of each coe?cient conditioned on its surrounding neighborhood. The FoGSM
denoising achieves substantial improvement (+0.68 in PSNR) and is seen to exhibit better contrast
and continuation of oriented features (see Fig. 4). On the other hand, FoGSM introduces some
noticeable artifacts in low contrast areas, which is caused by numerical instability at locations with
small z. We find that the improvement in terms of PSNR is consistent across photographic images
and noise levels, as reported in Table 1. But even with a restricted neighborhood for the multiplier
field, this PSNR improvement does come at a substantial computational cost. As a rough indication,
running on a PowerPC G5 workstation with 2.3 Ghz processor and 16 Gb RAM memory, using
unoptimized MATLAB (version R14) code, denoising a 512 ? 512 image takes on average 4.5 hours
(results averaging over 5 images), and denoising a 256?256 image takes on average 2.4 hours (result
averaging over 2 images), to a convergence precision producing the reported results. Our preliminary
investigation indicates that the slow running time is mainly due to the nature of coordinate ascent
and the landscape of (9), which requires many iterations to converge.
5 Discussion
We have introduced fields of Gaussian scale mixtures as a flexible and e?cient tool for modeling the
statistics of wavelet coe?cients of photographic images. We developed a feasible (although admittedly computationally costly) parameter estimation method, and showed that samples synthesized
from the fitted FoGSM model are able to capture structures in the marginal and joint wavelet statistics of photographic images. Preliminary results of applying FoGSM to image denoising indicate
substantial improvements over the state-of-the-art methods based on the local GSM model.
Although FoGSM has a structure that is similar to the local scale mixture model [9, 10], there
is a fundamental di?erence between them. In FoGSM, hGMRF structures are enforced in u and
log z, while the local scale mixture models impose minimal statistical structure on these variables.
Because of this, our model easily extends to images of arbitrary size, while the local scale mixture
models are essentially confined to describing small image patches (the curse of dimensionality, and
the increase in computational cost prevent one from scaling the patch size up). On the other hand,
the close relation to Gaussian MRF makes the analysis and computation of FoGSM significantly
easier than other non-Gaussian MRF based image models [6, 7, 5].
We envision, and are currently working on, a number of model improvements. First, the model
should benefit from the introduction of more general Markov neighborhoods, including wavelet
coe?cients from subbands at other scales and orientations [4, 12], since the current model is clearly
not accounting for these dependencies (see Fig. 3). Secondly, the log transformation used to derive
the multiplier field from a hGMRF is somewhat ad hoc, and we believe that substitution of another
nonlinear transformation (e.g., a power law [14]) might lead to a more accurate description of the
image statistics. Thirdly, the current denoising method estimates model parameter during the process
of denoising, which produces image adaptive model parameters. We are exploring the possibility of
using a set of generic model parameter learned a priori on a large set of photographic images, so that
a generic statistical model for all photographic images based on FoGSM can be built. Finally, there
exist residual inhomogeneous structures in the log z field (see Fig. 1) that can likely be captured by
explicitly incorporating local orientation [21] or phase into the model. Finding tractable models and
algorithms for handling such circular variables is challenging, but we believe their inclusion will
result in substantial improvements in modeling and in denoising performance.
?/PSNR
10/28.13
25/20.17
50/14.15
100/8.13
?/PSNR
10/28.13
25/20.17
50/14.15
100/8.13
Barbara
35.01 (34.01)
30.10 (29.07)
26.40 (25.45)
23.01 (22.61)
Flintstones
32.47 (31.78)
28.29 (27.48)
24.82 (24.02)
21.24 (20.49)
barco
35.05 (34.42)
30.44 (29.73)
27.36 (26.63)
24.44 (23.84)
house
35.63 (35.27)
31.64 (31.32)
28.51 (28.23)
25.33 (25.31)
boat
34.12 (33.58)
30.03 (29.34)
27.01 (26.35)
24.20 (23.79)
Lena
35.94 (35.60)
32.11 (31.70)
29.12 (28.62)
26.12 (25.77)
fingerprint
33.28 (32.45)
28.45 (27.44)
25.11 (24.13)
21.78 (21.21)
peppers
34.38 (33.73)
29.78 (29.18)
26.43 (25.93)
23.17 (22.80)
Table 1. Denoising results with FoGSM on di?erent images and di?erent noise levels. Shown in the
table are PSNRs (20 log10 (255/?e ), where ?e is the standard deviation of the error) of the denoised images, and in the parenthesis are the PSNRs of the same images denoised with a local GSM model [12].
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2,252 | 3,043 | Hyperparameter Learning for Graph Based
Semi-supervised Learning Algorithms
Xinhua Zhang?
Statistical Machine Learning Program
National ICT Australia, Canberra, Australia
and CSL, RSISE, ANU, Canberra, Australia
[email protected]
Wee Sun Lee
Department of Computer Science
National University of Singapore
3 Science Drive 2, Singapore 117543
[email protected]
Abstract
Semi-supervised learning algorithms have been successfully applied in many applications with scarce labeled data, by utilizing the unlabeled data. One important
category is graph based semi-supervised learning algorithms, for which the performance depends considerably on the quality of the graph, or its hyperparameters. In
this paper, we deal with the less explored problem of learning the graphs. We propose a graph learning method for the harmonic energy minimization method; this
is done by minimizing the leave-one-out prediction error on labeled data points.
We use a gradient based method and designed an efficient algorithm which significantly accelerates the calculation of the gradient by applying the matrix inversion
lemma and using careful pre-computation. Experimental results show that the
graph learning method is effective in improving the performance of the classification algorithm.
1
Introduction
Recently, graph based semi-supervised learning algorithms have been used successfully in various
machine learning problems including classification, regression, ranking, and dimensionality reduction. These methods create graphs whose vertices correspond to the labeled and unlabeled data
while the edge weights encode the similarity between each pair of data points. Classification is
performed using these graphs by labeling unlabeled data in such a way that instances connected by
large weights are given similar labels. Example graph based semi-supervised algorithms include
min-cut [3], harmonic energy minimization [11], and spectral graphical transducer [8].
The performance of the classifier depends considerably on the similarity measure of the graph, which
is normally defined in two steps. Firstly, the weights are defined locally in a pair-wise parametric
form using functions that are essentially based on a distance metric such as radial basis functions
(RBF). It is argued in [7] that modeling error can degrade performance of semi-supervised learning.
As the distance metric is an important part of graph based semi-supervised learning, it is crucial to
use a good distance metric. In the second step, smoothing is applied globally, typically, based on
the spectral transformation of the graph Laplacian [6, 10].
There have been only a few existing approaches which address the problem of graph learning. [13]
learns a nonparametric spectral transformation of the graph Laplacian, assuming that the weight and
distance metric are given. [9] learns the spectral parameters by performing evidence maximization
using approximate inference and gradient descent. [12] uses evidence maximization and Laplace
approximation to learn simple parameters of the similarity function. Instead of learning one single
good graph, [4] proposed building robust graphs by applying random perturbation and edge removal
?
This work was done when the author was at the National University of Singapore.
from an ensemble of minimum spanning trees. [1] combined graph Laplacians to learn a graph.
Closest to our work is [11], which learns different bandwidths for different dimensions by minimizing the entropy on unlabeled data; like the maximum margin motivation in transductive SVM, the
aim here is to get confident labeling of the data by the algorithm.
In this paper, we propose a new algorithm to learn the hyperparameters of distance metric, or more
specifically, the bandwidth for different dimensions in the RBF form. In essence, these bandwidths
are just model parameters and normal model selection methods include k-fold cross validation or
leave-one-out (LOO) cross validation in the extreme case can be used for selecting the bandwidths.
Motivated by the same spirit, we base our learning algorithm on the aim of achieving low LOO
prediction loss on labeled data, i.e., each labeled data can be correctly classified by the other labeled
data in a semi-supervised style with as high probability as possible. This idea is similar to [5] which
learns multiple parameters for SVM. Since most LOO style algorithms are plagued with prohibitive
computational cost, an efficient algorithm is designed. With a simple regularizer, the experimental
results show that learning the hyperparameters by minimizing the LOO loss is effective.
2
Graph Based Semi-supervised Learning
Suppose we have a set of labeled data points {(xi , yi )} for i ? L , {1, ..., l}. In this paper, we
only consider binary classification, i.e., yi ? {1 (positive), 0 (negative)}. In addition, we also have
a set of unlabeled data points {xi } for i ? U , {l + 1, ..., l + u}. Denote n , l + u. Suppose the
dimensionality of input feature vectors is m.
2.1
Graph Based Classification Algorithms
One of the earliest graph based semi-supervised learning algorithms is min-cut by [3], which miniX
mizes:
E(f ) ,
wij (fi ? fj )2
(1)
i,j
where the nonnegative wij encodes the similarity between instance i and j. The label fi is fixed to
yi ? {1, 0} if i ? L. The optimization variables fi (i ? U ) are constrained to {1, 0}. This combinatorial optimization problem can be efficiently solved by the max-flow algorithm. [11] relaxed the
constraint fi ? {1, 0} (i ? U ) to real numbers. The optimal solution of the unlabeled data?s soft
labels can be written neatly as:
fU = (DU ? WU U )?1 WU L fL = (I ? PU U )?1 PU L fL
(2)
where fL is the vector of soft labels (fixed to yi ) for L. D , diag(di ), where di , j wij and DU
is the submatrix of D associated with unlabeled data. P , D?1 W . WU U , WU L , PU U , and PU L
are defined by:
WLL WLU
PLL PLU
W =
,P =
.
WU L WU U
PU L PU U
P
The solution (2) has a number of interesting properties pointed out by [11]. All fi (i ? U ) are
automatically bounded by [0, 1], so it is also known as square interpolation. They can be interpreted
by using Markov random walk on the graph. Imagine a graph with n nodes corresponding to the n
data points. Define the probability of transferring from xi to xj as pij , which is actually row-wise
normalization of wij . The random walk starts from any unlabeled points, and stops once it hits any
labeled point (absorbing boundary). Then fi is the probability of hitting a positive labeled point. In
this sense, the labeling of each unlabeled point is largely based on its neighboring labeled points,
which helps to alleviate the problem of noisy data. (1) can also be interpreted as a quadratic energy
function and its minimizer is known to be harmonic: fi (i ? U ) equals the average of fj (j 6= i)
weighted by pij . So we call this algorithm Harmonic Energy Minimization (HEM). By (1), fU is
independent of wii (i = 1, ..., n), so henceforth we fix wii = pii = 0.
Finally, to translate the soft labels fi to hard labels pos/neg, the simplest way is by thresholding at
0.5, which works well when the two classes are well separated. [11] proposed another approach,
called Class Mass Normalization (CMN), to make use of prior information such as class ratio in
unlabeled
labels to fi+ ,
.P data, estimated by that in labeled data. Specifically, they normalize the.soft
P
n
n
?
fi
j=1 (1 ? fj ) as
j=1 fj as the probabilistic score of being positive, and to fi , (1 ? fi )
the score of being negative. Suppose there are r+ positive points and r? negative points in the
labeled data, then we classify xi to positive iff fi+ r+ > fi? r? .
2.2
Basic Hyperparameter Learning Algorithms
One of the simplest parametric form of wij is RBF:
X
wij = exp ?
(xi,d ? xj,d )2 ?d2
d
(3)
th
where xi,d is the d component of xi , and likewise the meaning of fU,i in (4). The bandwidth ?d
has considerable influence on the classification accuracy. HEM uses one common bandwidth for all
dimensions, which can be easily selected by cross validation. However, it will be desirable to learn
different ?d for different dimensions; this allows a form of feature selection. [11] proposed learning
the hyperparameters ?d by minimizing the entropy on unlabeled data points (we call it MinEnt):
Xu
H (fU ) = ?
(fU,i log fU,i + (1 ? fU,i ) log(1 ? fU,i ))
(4)
i=1
The optimization is conducted by gradient descent. To prevent numerical problems, they replaced
P with P? = U + (1 ? )P , where ? [0, 1), and U is the uniform matrix with Uij = n?1 .
3
Leave-one-out Hyperparameter Learning
In this section, we present the formulation and efficient calculation of our graph learning algorithm.
3.1
Formulation and Efficient Calculation
We propose a graph learning algorithm which is similar to minimizing the leave-one-out cross validation error. Suppose we hold out a labeled example xt and predict its label by using the rest of the
labeled and unlabeled examples. Making use of the result in (2), the soft label for xt is s> fUt (the
first component of fUt ), where
t
s , (1, 0, ..., 0)> ? Ru+1 , fUt , (f0t , fl+1
, ..., fnt )> .
Here, the value of fUt can be determined by fUt = (I ? P?Ut U )?1 P?Ut L fLt , where
ptt ptU
>
fLt , (f1 , .., ft?1 , ft+1 , ..., fl ) , p?ij , (1 ? ?)pij + ?/n , PUt U ,
,
pU t P U U
pU t , (pl+1,t , ..., pn,t )> , ptU , (pt,l+1 , ..., pt,n ) ,
?
?
pt,1
???
pt,t?1
pt,t+1
???
pt,l
? ? ? pl+1,t?1 pl+1,t+1 ? ? ? pl+1,l ?
? ? p
PUt L = ? l+1,1
.
???
???
???
???
???
??? ?
pn,1 ? ? ?
pn,t?1
pn,t+1 ? ? ?
pn,l
t
If xt is positive, then we hope that fU,1
is as close to 1 as possible. Otherwise, if xt is negative, we
t
hope that fU,1
is as close to 0 as possible. So the cost function to be minimized can be written as:
Xl
Xl
t
ht s> (I ? P?Ut U )?1 P?Ut L fLt
(5)
Q=
ht fU,1
=
t=1
t=1
where ht (x) is the cost function for instance t. We denote ht (x) = h+ (x) for yt = 1 and ht (x) =
h? (x) for yt = 0. Possible choices of h+ (x) include 1 ? x, (1 ? x)a , a?x , and ? log x with a > 1.
t
Possible choices for h? (x) include x, xa , ax?1 , and ? log(1?x). Let Loo loss(xt , yt ) , ht fU,1
.
To minimize Q, we use gradient-based optimization methods.
.The gradient is: .
>
Pl
0
t
t
?1
t
?
?
?Q/??d = t=1 ht fU,1 s (I ? PU U )
? PU U ??d ? fUt + ? P?Ut L ??d ? fLt ,
t
using matrix property dX ?1 = ?X ?1 (dX)X ?1 . Denoting (? t )> , h0t (fU,1
)s> (I ? P?Ut U )?1
and noting P? = ?U + (1 ? ?)P, we have
Xl
?Q/??d = (1 ? ?)
(? t )> ?PUt U ??d ? fUt + ?PUt L ??d ? fLt .
(6)
t=1
Since in both PUt U and PUt L , the first row corresponds to xt , and the ith row (i ? 2) corresponds
to xi+l?1 , denoting PUt N , (PUt L PUt U ) makes sense as each row of PUt N corresponds to a well
defined P
single data point. Let all
about P carry over to the corresponding W . We now use
Pnnotations
n
t
t
swit , k=1 wU
(i,
k)
and
?w
N
U N (i, k)/??d (i = 1, ..., u + 1) to denote the sum of these
k=1
corresponding rows. Now (6) can be rewritten in ground terms by the following ?two? equations:
Xn
t
t
t
?PUt ? (i, j) ??d = (swit )?1 ?wU
?wU
? (i, j) ??d ? pU ? (i, j)
N (i, k) ??d ,
k=1
3
2
where ? can be U or L. ?wij /??d = 2wij (xi,d ? xj,d ) ?d by (3).
The na??ve way to calculate the function value Q and its gradient is presented in Algorithm 1. We
call it leave-one-out hyperparameter learning (LOOHL).
Algorithm 1 na??ve form of LOOHL
1: function value Q ? 0, gradient g ? (0, ..., 0)> ? Rm
2: for each t = 1, ..., l (leave-one-out loop for each labeled point) do
3:
fLt ? (f1 , .., ft?1 , ft+1 , ..., fl )> ,
fUt ? (I ? P?Ut U )?1 P?Ut L fLt ,
t
t
Q ? Q + ht fU,1 ,
(? t )> ? h0t (fU,1
)s> (I ? P?Ut U )?1
4:
for each d = 1, ..., m(for all feature dimensions) do
n
t
t
t
P
?wU
?PU
?wU
t
1
U (i,j)
N (i,k)
U (i,j)
?
?
p
(i,
j)
5:
UU
??d
??d
??d
swit
k=1
P
t
where swit = n
i, j = 1, ..., u + 1
k=1 wU N (i, k),
n
t
t
t
P
?wU N (i,k)
?PU L (i,j)
?wU L (i,j)
t
1
? swt
? pU L (i, j)
i = 1, ..., u + 1, j = 1, ..., l ? 1
6:
??d
??d
??d
i
k=1
t
t
?P
?P
7:
gd ? gd + (1 ? )(? t )> ??UdU fUt + ??UdL fLt
8:
end for
9: end for
The computational complexity of the na??ve algorithm is expensive: O(lu(mn+u2 )), just to calculate
the gradient once. Here we assume the cost of inverting a u ? u matrix is O(u3 ). We reduce the two
terms in the cost by means of using matrix inversion lemma and careful pre-computation.
One part of the cost, O(lu3 ), stems from inverting I ? P?Ut U , a (u + 1) ? (u + 1) matrix, for l times in
(5). We note that for different t, I ? P?Ut U differs only by the first row and first column. So there exist
two vectors ?, ? ? Ru+1 such that I ? P?Ut1U = (I ? P?Ut2U ) + e?> + ?e> , where e = (1, 0, ..., 0)>
? Ru+1 . With I ? P? t expressed in this form, we are ready to apply matrix inversion lemma:
UU
A + ?? >
?1
= A?1 ? A?1 ? ? ? > A?1
1 + ?> A? .
(7)
We only need to invert I ? P?Ut U for t = 1 from scratch, and thenapply (7) twice for each t > 2. The
new total complexity related to matrix inversion is O u3 + lu2 .
The other part of the cost, O(lumn) , can be reduced by using careful pre-computation. Written in
detail, we have:
l
u+1
X X ?it
?Q
=
??d
swit
t=1 i=1
?
n
X
k=1
u+1
X
j=1
t
?wU
N (i, k)
??d
l?1
t
X ?wt (i, j) t
?wU
U (i, j) t
UL
fU,j +
fL,j
??d
??d
j=1
u+1
X
j=1
ptU U
t
(i, j) fU,j
+
l?1
X
ptU L
t
(i, j) fL,j
j=1
!!
,
n
n X
X
i=1 j=1
?ij
?wij
??d
The crucial observation is the existence of ?ij , which are independent of dimension index d. Therefore, they can be pre-computed efficiently. The Algorithm 2 below presents the efficient approach
to gradient calculation.
Algorithm 2 Efficient algorithm to gradient calculation
1: for i, j = 1, ..., n do
2:
for all feature dimension d on which either xi or xj is nonzero do
3:
gd = gd + ?ij ? ?wij /??d
4:
end for
5: end for
Figure 1: Examples of degenerative graphs learned by pure LOOHL.
Letting swi ,
?ij =
swi?1
?ij =
Pn
l
X
k=1
wik and ?(?) be Kroneker delta, we derive the form of ?ij as:
t
?i?l+1
t=1
l
X
swi?1
t=1
?ij =
t
fU,j?l+1
?
n
X
t
pik fU,k?l+1
t
?pit fU,1
k=l+1
?
t
t
t
?fU,1
?i?l+1
?(t = j) + fj ?(t 6= j) ? pit fU,1
?
swi?1 ?1i
i
fU,j?l+1
?
n
X
i
pik fU,k?l+1
k=l+1
?ij =
swi?1 ?1i
fj ?
n
X
i
pik fU,k?l+1
k=l+1
?
l
X
?
l
X
pik fk
k=1
?
l
X
k=1
pik fk ? for i > l and j > l;
?
l
X
t
pik fU,k?l+1
?
pik fk ?
k=1:k6=t
n
X
k=l+1
pik fk
!
?
!
k=1:k6=t
for i > l and j 6 l;
for i 6 l and j > l;
for i 6 l and j 6 l,
and ?ii are fixed to 0 for all i since we fix wii = pii = 0.
All ?ij can be computed in O(u2 l) time and Algorithm 2 can be completed in O(n2 m)
? time, where
X
m
? , 2n?1 (n ? 1)?1 ?
|{ d ? 1...m| xi or xj is not zero on feature d}|.
16i<j6n
In many applications such as text classification and image pattern recognition, the data is very sparse
and m
? m. In sum, the computational cost has been reduced from O(lu(mn + u2 )) to O(lnu +
2
n m
? + u3 ) . The space cost is mild at O(n2 + nm).
?
4
Regularizing the Graph Learning
Similar to the MinEnt method, purely applying LOOHL can lead to degenerative graphs. In this
section, we show two such examples and then propose a simple approach which regularizes the
graph learning process.
Two degenerative graphs are shown in Figure 1. In example (a), the points with the same xv coordinate are from the same classes. For each labeled point, there is another labeled point from the
opposite class which has the same xh coordinate. So the leave-one-out hyperparameter learning will
push 1/?h to zero and 1/?v to infinity, i.e., all points can transfer only horizontally. Therefore the
graph will effectively split into six disconnected sub-graphs, each sharing the same xv coordinate as
showed in (a). So the desired gradual change of label from positive to negative along dimension xv
cannot appear. As a result, the point at question mark cannot hit any labeled points and cannot be
classified. One way to preventP
such degenerate graphs is to prevent 1/?v from growing too large,
e.g., with a regularizer such as d (1/?d )2 .
In example (b), although the negative points will encourage both horizontal and vertical walk, horizontal walk will make the leave-one-out error large on positive points. So the learned 1/?v will
be far smaller than 1/?h , i.e., the result strongly encourages walking in vertical direction and ignoring the information from the horizontal direction. As a result, the point at the question mark
will be labeled as positive, although by nearest neighbor intuition, it should be labeled as negative.
We notice that the four negativePpoints will be partitioned into two groups as shown in the figure.
In such a case, the regularizer d (1/?d )2 will not be helpful with utilizing dimensions that are
informative. A different regularizer that encourages the use of more dimensions may be better in
this case. One
is to minimize the variance of the inverse
P
P simple regularizer that has this property
bandwidth d (1/?d ? ?)2 , where ? = m?1 d 1/?d , assuming that the mean is non-zero. It
Table 1: Dataset properties. Sparsity is the average frequency of features to be zero in the whole
dataset. The rightmost column gives the size of the whole dataset from which the labeled data in
experiment is sampled. Some data in text dataset has unknown label, thus always used as unlabeled.
is a priori unclear which regularizer will be better empirically, but for the datasets in our experiments, the minimum variance regularizer is overwhelmingly better, even when useless features are
intentionally added to the datasets.
Since the gradient based optimization can get stuck in local minima, it is advantageous to test several
different parameter initialization. With this in mind, we implement a simple approximation to the
minimum variance regularizer that tests different parameter initialization as well. We discretize ?
P
2
and minimize the leave-one-out loss plus d (1/?d ? 1/?
? ) , where ?
? is fixed a priori to several
different possible values. We run with different ?
? and set all initial ?d to ?
? . Then we choose the
function produced by the value of ?
? that has the smallest regularized cost function value. This
process is similar to restarting from various values to avoid local minima, but now we are also trying
with different mean of estimated optimal bandwidth at the same time. A similar way to regularize is
P
2
by using a Gaussian prior with mean ??1 and minimizing Q + C d (1/?d ? 1/?) with respect
to ?d and ? simultaneously.
5
Experimental Results
Using HEM as a basis for classification, we compare the test accuracy of three model selection
methods: LOOHL, 5-CV (tying all bandwidths and choose by 5-fold cross validation), and MinEnt,
each with both thresholding and CMN. Since the topic of this paper is how to learn the hyperparameters of a graph, we pay more attention to how the performance of a given recognized classifier
can be improved by means of learning the graph, than to the comparison between different classifers? performance, i.e., comparing with other semi-supervised or supervised learning algorithms.
Ionosphere is from UCI repository. The other four datasets used in the experiment are from NIPS
2003 Workshop on feature selection challenge. Each of them has two versions: original version
and probe version which adds useless probing features in order to investigate the algorithm?s performance in the presence of useless features, though at current stage we do not use the algorithm
as feature selector. Since the workshop did not provide the original datasets, we downloaded the
original datasets from other sites. Our original intention was to use original versions that we downloaded and to reproduce the probe version ourself using the pre-processing described in NIPS 2003
workshop, so that we can check the performance of the algorithms on datasets with and without
redundant features. Unfortunately, we find that with our own effort at pre-processing, the datasets
with probes yield far different accuracies compared with the datasets with probes downloaded from
the workshop web site. Thus we are using the original version and the probe version downloaded
from difference sources, and the comparison between them should be done with care, though the
demonstration of LOOHL?s efficacy is not affected.
The properties of the five datasets are summarized in Table 1. We randomly pick the labeled subset
L from all labeled data available under the constraint that both classes must be present in L. The
remaining labeled and unlabeled data are used as unlabeled data. For example, by saying |L| = 20
for text dataset, we mean randomly picking 20 points from the 600 labeled data as labeled, and label
the other 1980 points by using our algorithm. Finally we calculate the prediction accuracy on the
580 (originally) labeled points. For other datasets, say cancer, testing is on 180 points since we
know the label of all points. For each fixed |L|, this random test is conducted for 10 times and the
average accuracy is reported. Then |L| is varied. We normalized all input feature vectors to have
length 1.
(a) 4 vs 9 (original)
(b) cancer (original)
(c) text (original)
(d) thrombin (original)
(e) 4 vs 9 (probe)
(f) cancer (probe)
(g) text (probe)
(h) thrombin (probe)
Figure 2: Accuracy of original and probe versions in percentage vs. number of labeled data.
The initial common bandwidth and smoothing factor in MinEnt are selected by five fold cross
validation. For LOOHL, We fix h+ (x) = (1 ? x)2 and h? (x) = x2 . The final objective
function is:
.
P
2
C1 ? Loo loss N ormal + C2 ? d (1/?d ? 1/?
? ) m,
X
X
Loo loss N ormal , (2r+ )?1
Loo loss(xi , yi )+(2r? )?1
Loo loss(xi , yi ), (8)
yi =1
yi =0
and there are r+ positive labeled examples and r? negative labeled examples. For each C1 :C2 ratio,
we run on ?
? = 0.05, 0.1, 0.15, 0.2, 0.25, 0.3 for all datasets and select the function that corresponds
to the smallest objective function value for use in cross validation testing. The final C1 :C2 value
was picked by five fold cross validation, with discrete levels at 10?i , where i = 1, 2, 3, 4, 5, since
strong regularizer is needed given the large number of features (variables) and much fewer labeled
points. The optimization solver we use is the Toolkit for Advanced Optimization [2].
From the results in Figure 2 and Figure 3, we can make the following observations and conclusions:
1. LOOHL generally outperforms 5-CV and MinEnt. Both LOOHL+Thrd and LOOHL+CMN
outperform 5-CV and MinEnt (regardless of Thrd or CMN) on all datasets except thrombin and
ionosphere, where either LOOHL+CMN or LOOHL+Thrd finally performs best.
2. For 5-CV, CMN is almost always better than thresholding, except on the original form of cancer
and thrombin dataset, where CMN hurts 5-CV. In [11], it is claimed that although the theory of
HEM is sound, CMN is still necessary to achieve reasonable performance because the underlying
graph is often poorly estimated and may not reflect the classification goal, i.e., one should not rely
exclusively on the graph. Now that our LOOHL is aimed at learning a good graph, the ideal case is
that the graph learned is suitable for our classification such that the improvement by CMN will not
be large. In other words, the difference between LOOHL+CMN and LOOHL+Thrd, compared with
the difference between 5-CV+CMN and 5-CV+Thrd, can be viewed as an approximate indicator of
how well the graph is learned by LOOHL.
The efficacy of LOOHL can be clearly observed in datasets 4vs9, cancer, text, ionosphere and original version of thrombin. In these cases, we see that LOOHL+Thrd is already achieving high accuracy and LOOHL+CMN does not offer much improvement then or even hurts performance due to
inaccurate class ratio estimation. In fact, LOOHL+Thrd performs reliably well on all datasets. It is
thus desirable to learn the bandwidth for each dimension of the feature vector, and there is no longer
any need to post-process by using class ratio information.
3. The performance of MinEnt is generally inferior to 5-CV and LOOHL. MinEnt+Thrd has equal
chance of out-performing or losing to 5-CV+Thrd, while 5-CV+CMN is almost always better than
MinEnt+CMN. Most of the time, MinEnt+CMN performs significantly better than MinEnt+Thrd,
so we can conclude that MinEnt fails to learn a good graph. This may be due to converging to a poor
local minimum, or that the idea of minimizing the entropy on unlabeled data is by itself insufficient.
Figure 3: Accuracy of Ionosphere
in percentage vs. number of labeled
data.
Figure 4: Accuracy comparison of priors in percentage
between
P ?2 minimizing sum of square inverse bandwidth
d ?d and minimizing variance of inverse bandwidth.
4. For these datasets, assuming low variance of inverse bandwidth with discretization as regularizer
is more reasonable than assuming that many features are irrelevant to the classification. This is even
true for probe versions of the datasets. Figure 4 shows the comparison.
6
Conclusions
In this paper, we proposed learning the graph for graph based semi-supervised learning by minimizing the leave-one-out prediction error, with a simple regularizer. Efficient gradient calculation
algorithms are designed and the empirical result is encouraging.
Acknowledgements
This work is partially funded by the Singapore-MIT Alliance. National ICT Australia is funded
through the Australian Government?s Backing Australia?s Ability initiative, in part through the Australian Research Council.
References
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2,253 | 3,044 | Cross-Validation Optimization for Large Scale
Hierarchical Classification Kernel Methods
Matthias W. Seeger
Max Planck Institute for Biological Cybernetics
P.O. Box 2169, 72012 T?ubingen, Germany
[email protected]
Abstract
We propose a highly efficient framework for kernel multi-class models with a
large and structured set of classes. Kernel parameters are learned automatically
by maximizing the cross-validation log likelihood, and predictive probabilities are
estimated. We demonstrate our approach on large scale text classification tasks
with hierarchical class structure, achieving state-of-the-art results in an order of
magnitude less time than previous work.
1
Introduction
In many real-world statistical problems, we would like to fit a model with a large number of dependent variables to a training sample with very many cases. For example, in multi-way classification
problems with a structured label space, modern applications demand predictions on thousands of
classes, and very large datasets become available. If n and C denote dataset size and number of
classes respectively, nonparametric kernel methods like SVMs or Gaussian processes typically scale
superlinearly in n C, if dependencies between the latent class functions are properly represented.
Furthermore, most large scale kernel methods proposed so far refrain from solving the problem of
learning hyperparameters (kernel or loss function parameters). The user has to run cross-validation
schemes, which require frequent human interaction and are not suitable for learning more than a few
hyperparameters.
In this paper, we propose a general framework for learning in probabilistic kernel classification
models. While the basic model is standard, a major feature of our approach is the high computational
efficiency with which the primary fitting (for fixed hyperparameters) is done, allowing us to deal
with hundreds of classes and thousands of datapoints within a few minutes. The primary fitting
scales linearly in C, and depends on n mainly via a fixed number of matrix-vector multiplications
(MVM) with n ? n kernel matrices. In many situations, these MVM primitives can be computed
very efficiently, as will be demonstrated. Furthermore, we optimize hyperparameters automatically
by minimizing the cross-validation log likelihood, making use of our primary fitting technology as
inner loop in order to compute the CV criterion and its gradient. Our approach can be used to learn
a large number of hyperparameters and does not need user interaction.
Our framework is generally applicable to structured label spaces, which we demonstrate here for
hierarchical classification of text documents. The hierarchy is represented through an ANOVA
setup. While the C latent class functions are fully dependent a priori, the scaling of our method
stays within a factor of two compared to unstructured classification. We test our framework on the
same tasks treated in [1], achieving comparable results in at least an order of magnitude less time.
Our method estimates predictive probabilities for each test point, which can allow better predictions
w.r.t. loss functions different from zero-one.
The primary fitting method is given in Section 2, the extension to hierarchical classification in Section 3. Hyperparameter learning is discussed in Section 4. Computational details are provided in
Section 5. We present experimental results in Section 6. Our highly efficient implementation is
publicly available, as project klr in the LHOTSE1 toolbox for adaptive statistical models.
2
Penalized Multiple Logistic Regression
Our problem is to predict y ? {1, . . . , C} from x ? X , given some i.i.d. data D = {(xi , y i ) | i =
1, . . . , n}. We use zero-one coding, i.e. y i ? {0, 1}C , 1T y i = 1. We elpoy the multiple logistic regression model, consisting of C latent (unobserved) class functions
P uc feeding into the
multiple logistic (or softmax) likelihood P (yi,c = 1|xi , ui ) = euc (xi ) /( c0 euc0 (xi ) ). We write
uc = fc + bc for intercept parameters bc ? R and functions fc living in a reproducing kernel Hilbert space (RKHS) with kernel K (c) , and consider the penalized negative log likelihood
Pn
PC
? = ? i=1 log P (y i |ui ) + (1/2) c=1 kfc k2c + (1/2)? ?2 kbk2 , which we minimize for primary
fitting. k ? kc is the RKHS norm for kernel K (c) . Details on such setups can be found in [4].
Our notation for n C vectors2 (and matrices) uses the ordering y = (y1,1 , y2,1 , . . . , yn,1 , y1,2 , . . . ).
We set u = (uc (xi )) ? RnC . ? denotes the Kronecker product, 1 is the vector of all ones. Selection
indexes I are applied to i only: y I = (yi,c )i?I,c ? R|I|C .
Since the likelihood
on the fc only through fc (xi ), every minimizer of ? must be a kernel
P depends
(c)
expansion: fc = i ?i,c K (?, xi ) (representer theorem, see [4]). Plugging this in, the regularizer
becomes (1/2)?T K ? + (1/2)? ?2 kbk2 . K (c) = (K (c) (xi , xj ))i,j ? Rn,n , K = diag(K (c) )c
is block-diagonal. We refer to this setup as flat classification model. The bc may be eliminated as
? = K + ? 2 (I ? 1)(I ? 1T ), then ? becomes
b = ? 2 (I ? 1T )?. Thus, if K
1
? ?, ?lh = ?y T u + 1T l, li = log 1T exp(ui ), u = K
? ?.
? = ?lh + ?T K
(1)
2
? is strictly convex in ? (because the likelihood is log-concave),
so it has a unique minimum point
P
(c)
? The corresponding kernel expansions are u
?.
?c =
?
?
(K
(?, xi ) + ? 2 ). Estimates of the
i i,c
conditional probability on test points x? are obtained by plugging u
?c (x? ) into the likelihood.
We note that this setup can also be seen as MAP approximation to a Bayesian model, where the fc
are given independent Gaussian process priors, e.g.[7]. It is also related to the multi-class SVM [2],
where ? log P (yi |ui ) is replaced by the margin loss ?uyi (xi ) + maxc {uc (xi ) + 1 ? ?c,yi }. The
negative log multiple logistic likelihood has similar properties, but is smooth as a function of u, and
the primary fitting of ? does not require constrained convex optimization.
We minimize ? using the Newton-Raphson (NR) algorithm, the details are provided in Section 5.
The complexity of our fitting algorithm is dominated by k1 (k2 + 2) matrix-vector multiplications
with K , where k1 is the number of NR iterations, k2 the number of linear conjugate gradient (LCG)
steps for computing each Newton direction. Since NR is a second-order convergent method, k1 can
be chosen small. k2 determines the quality of each Newton direction, for both fairly small values
are sufficient (see Section 6.2).
3
Hierarchical Classification
So far we dealt with flat classification, the classes being independent a priori, with block-diagonal
kernel matrix K . However, if the label set has a known structure3 , we can benefit from representing
it in the model. Here we focus on hierarchical classification, the label set {1, . . . , C} being the leaf
nodes of a tree. Classes with lower common ancestor should be more closely related. In this Section,
we propose a model for this setup and show how it can be dealt with in our framework with minor
modifications and minor extra cost.
In flat classification, the latent class functions uc are modelled as a priori independent, in that the
regularizer (which plays the role of a log prior) is a sum of individual terms for each uc , without any
1
See www.kyb.tuebingen.mpg.de/bs/people/seeger/lhotse/.
In Matlab, reshape(y,n,C) would give the matrix (yi,c ) ? Rn,C .
3
Learning an unknown label set structure may be achieved by expectation maximization techniques, but this
is subject to future work.
2
interaction terms. Analysis of variance (ANOVA) models go beyond this independent design, they
have previously been applied to text classification by [1]. Let {0, . . . , P } be the nodes of the tree, 0
being the root, and the numbers are assigned breadth first (1, 2, . . . are the root?s children). The tree
is determined by P and np , p = 0, . . . , P , the number of children of node p. Let L be the set of leaf
nodes, |L| = C. Assign a pair of latent functions up , u
?p to each node, except the root. The u
?p are
assumed a priori independent, as in flat classification. up is the sum of u
?p0 , p0 running over the nodes
(including p) on the path from the root to p. The class functions to be fed into the likelihood are the
uL(c) of the leafs. This setup represents similarities conditioned on the hierarchy. For example, if
leafs L(c), L(c0 ) have the common parent p, then uL(c) = up + u
?L(c) , uL(c0 ) = up + u
?L(c0 ) , so
the class functions share the effect up . Since regularization forces all independent effects u
?p0 to be
smooth, the classes c, c0 are urged to behave similarly a priori.
? = (?
? ? ?
Let u = (up (xi ))i,p , u
up (xi ))i,p ? RnP . The vectors are related as u = (? ? I)u,
P,P
{0, 1} . Importantly, ? has a simple structure which allows MVM with ? or ?T to be computed
easily in O(P ), without having to compute or store ? explicitly. MVM with ? is described in
Algorithm 1, and MVM with ?T works in a similar manner [8].
Under the hierarchical model, the class functions uL(c) are strongly dependent a priori. We may
represent this prior coupling in our framework by simply plugging in the implied kernel matrix K :
? (?T ? I),
(2)
K = (?L,? ? I)K
L,?
? is block-diagonal. K is not sparse and certainly not block-diagonal, but the
where the inner K
important point is that we are still able to do kernel MVMs efficiently: pre- and postmultiplying by
? is block-diagonal just as in the flat case.
? is cheap, and K
We note that the step from flat to hierarchical
classification requires minor modifications of Algorithm 1: Matrix-vector multiplication
existing code only. If code for representing a y = ?x
block-diagonal K is available, we can use it
y ? (). y0 := 0. s := 0.
? , just replacing C by
to represent the inner K
for p = 0, . . . , P do
P . This simplicity carries through to the hyperif np > 0 (p not a leaf node) then
parameter learning case (see Section 4). The
Let J(p) = {s + 1, . . . , s + np }.
cost of a kernel MVM is increased by a factor
y ? (y T , yp 1T + xTJ(p) )T . s ? s + np .
P/C < 2, which in most hierarchies in pracend if
tice is close to 1. However, it would be wrong
end for
to claim that hierarchical classification in general comes as cheap as flat classification.
The subtle issue is that the primary fitting becomes more costly, precisely because there is more
coupling between the variables. In the flat case, the Hessian of ? is close to block-diagonal. The
LCG algorithm to compute Newton directions converges quickly, because it nearly decomposes into
C independent ones, and fewer NR steps are required (see Section 5). In the hierarchical case,
both LCG and NR need more iterations to attain the same accuracy. In numerical mathematics,
much work has been done to approximately decouple linear systems by preconditioning. In some
of these strategies, knowledge about the structure of the system matrix (in our case: the hierarchy)
can be used to drive preconditioning. An important point for future research is to find a good
preconditioning strategy for the system of Eq. 5. However, in all our experiments so far the fitting
of the hierarchical model took less than twice the time required for the flat model on the same task.
Some further extensions, such as learning with incomplete label information, are discussed in [8].
4
Hyperparameter Learning
In any model of interest, there will be free hyperparameters h, for example parameters of the kernels K (c) . These were assumed to be fixed in the primary fitting method introduced in Section 2.
In this Section, we describe a scheme for learning h which makes use of the primary fitting algorithm as inner loop. Note that such nested strategies are commonplace in Bayesian Statistics, where
(marginal) inference is typically used as subroutine for parameter learning.
Recall that primary fitting consists of minimizing ? of Eq. 1 w.r.t. ?. If we minimize ? w.r.t. h as
well, we run into the problem of overfitting. A common remedy is to minimize the negative cross-
validation log likelihood ? instead. Let {Ik } be a partition of {1, . . . , n}, with Jk = {1, . . . , n}\Ik ,
and let ?Jk = uT[Jk ] ((1/2)?[Jk ] ? y Jk ) + 1T l[Jk ] be the primary criterion on the subset Jk of the
? J ?[J ] . The ?[J ] are independent variables, not part of a common ?. The
data. Here, u[Jk ] = K
k
k
k
CV criterion is
X
? I ,J ?[J ] ,
?=
?Ik , ?Ik = ?y TIk u[Ik ] + 1T l[Ik ] , u[Ik ] = K
(3)
k
k k
k
where ?[Jk ] minimizes ?Jk . Since for each k, we fit and evaluate on disjoint parts of y, ? is
an unbiased estimator of the test negative log likelihood, and minimizing ? should be robust to
overfitting.
In order to select h, we pick a fixed partition at random, then do gradient-based minimization of ?
w.r.t. h. To this end, we keep the set {?[Jk ] } of primary variables, and iterate between re-fitting those
for each fold Ik , and computing ? and ?h ?. The latter can be determined analytically, requiring
? J V [J ] already encountered during
us to solve a linear system with the Hessian matrix I + V T[Jk ] K
k
k
primary fitting (see Section 5). This means that the same LCG code used to compute Newton
directions there can be applied here in order to compute the gradient of ?. The details are given
in Section 5. As for the complexity, suppose there are q folds. The update of the ?[Jk ] requires
q primary fitting applications, but since they are initialized with the previous values ?[Jk ] , they do
converge very rapidly, especially during later outer iterations. Computing ? based on the ?[Jk ]
comes basically for free. The gradient computation decomposes into two parts: accumulation, and
kernel derivative MVMs. The accumulation part requires solving q systems of size ((q ? 1)/q)n C,
? J if linear conjugate gradients (LCG) is used, k3 being the number
thus q k3 kernel MVMs on the K
k
of LCG steps. We also need two buffer matrices E , F of q n C elements each. Note that the
accumulation step is independent of the number of hyperparameters. The kernel derivative MVM
part consists of q derivative MVM calls for each independent component of h, see Section 5.1. As
opposed to the accumulation part, this part consists of a simple large matrix operation and can be
run very efficiently using specialized numerical linear algebra code.
As shown in Section 5, the extension of hyperparameter learning to the hierarchical case of Section 3
is simply done by wrapping the accumulation part, the coding and additional memory effort being
minimal. Given a method for computing ? and ?h ?, we plug these into a custom optimizer such
as Quasi-Newton in order to learn h.
5
Computational Details
In this Section, we provide details for the general plan laid out above. It is precisely these which
characterize our framework and allow us to apply a standard model to domains beyond its usual
applications, but of interest to Machine Learning.
Recall Section 2. We minimize ? by choosing search directions s, and doing line minimizations
? ?. We have:
along ? + ?s, ? > 0. For the latter, we maintain the pair (?, u), u = K
?u ? = ? ? y + ?, ? = exp(u ? 1 ? l), i .e. ?i,c = P (yi,c = 1|ui ).
(4)
Given (?, u), ? and ?u ? can be computed in O(n C), without requiring MVMs. This suggests to
? s, the corresponding ? can be constructed
perform the line search in u along the direction s? = K
from the final ?. Since kernel MVMs are significantly more expensive than these O(n C) operations,
the line searches basically come for free!
We choose search directions by Newton-Raphson (NR)4 , since the Hessian of ? is required anyway
for hyperparameter learning. Let D = diag ?, P = (1 ? I)(1T ? I), and W = D ? DP D. We
? )?0 =
have ??u ?lh = W , and g = ?u ?lh = ? ? y from Eq. 4. The NR system is (I + W K
1/2
0
W u ? g, with the NR direction being s = ? ? ?. If V = (I ? DP )D , then W = V V T ,
because (1T ? I)D = I. We see that ?0 = V ? (using (1T ? I)g = 0), and we can obtain it from
the equivalent symmetric system
? V ? = V T u ? D ?1/2 (? ? y), ?0 = V ?
I + V TK
(5)
4
Initial experiments with conjugate gradients in ? gave very slow convergence, due to poor conditioning,
but experiments with a different dual criterion are in preparation.
P
0
(details are in [8]). Note that P x = ( c0 x(c ) )c , so that MVM with V can be done in O(n C).
The NR direction is obtained by solving this system approximately by the linear conjugate gradients
(LCG) method, requiring a MVM with the system matrix in each iteration, thus a single MVM with
K . Our implementation includes diagonal preconditioning and numerical stability safeguards [8].
The NR system need not be solved to high accuracy (see Section 6.2). Initially, ? = D ?1/2 ?,
because then V ? = ? if only (1T ? I)? = 0, which is true if the initial ? fulfils it.
We now show how to compute the gradient ?h ? for the CV criterion ? (Eq. 3). Note that
?[J] is determined by the stationary equation ?[J] + g [J] = 0. Taking the derivative gives
? J (d?[J] )). We obtain a system for d?[J] which is symd?[J] = ?W [J] ((dK J )?[J] + K
T ?
metrized as above: (I + V [J] K J V [J] )? = ?V T[J] (dK J )?[J] , d?[J] = V [J] ?. Also,
? I,J (d?[J] )). With s = I ?,I (? [I] ? y ) ? I ?,J V [J] (I +
d?I = (? [I] ? y I )T ((dK I,J )?[J] + K
I
T ?
?1 T ?
V K J V [J] ) V K J,I (? [I] ?y I ), we have that d?I = (I ?,J ?[J] )T (dK )s. If we collect these
[J]
[J]
vectors as columns of E , F ? RnC,q , we have that d? = tr E T (dK )F . In the hierarchical setup,
? = (?T ? I)E ? RnP,q , F? accordingly, then d? = tr E
? T (dK
? )F? . Here, we
we use Eq. 2: E
L,?
? , F? , then transform them later in place.
build E , F in the buffers allocated for E
We finally mention some of the computational ?tricks?, without which we could not have dealt with
the largest tasks in Section 6.2 (for section B, a single n C vector requires 88M of memory). For
the linear kernel (see Section 5.1), the main primitive A 7? X X T A can be coded very efficiently
using a standard sparse matrix format for X . If A is stored row-major (a1,1 , a1,2 , . . . ), the computation becomes faster by a factor of 4 to 6 compared to the standard column-major format5 . For
? J . ?Covariance reprehyperparameter learning, we work on subsets Jk and need MVMs with K
k
?
sentation shuffling? permutes the representation s.t. K Jk sits in the upper left part, and MVM can
use flat rather than indexed code, which is many times faster. We also share memory blocks of size
n C between LCG, gradient accumulation, line searches, keeping the overall memory requirements
at r n C for a small constant r, and avoiding frequent reallocations.
5.1
Matrix-Vector Multiplication
MVM with K is the bottleneck of our framework, and all efforts should be concentrated on this
primitive. We can tap into much prior work in numerical mathematics. With many classes C, we
may share kernels: K (c) = vc M (lc ) , vc > 0 variance parameters, M (l) independent correlation
functions. Our generic implementation stores two symmetric matrices M (l) in a n ? n buffer.
The linear kernel K (c) (x, x0 ) = vc xT x0 is frequently used for text classification (see Section 6.2).
If the data matrix X is sparse, kernel MVM can be done in much less than the generic O(C n2 ),
typically in O(C n), requiring O(n) storage for X only, even if the dimension of x is way beyond
n.
If the K (c) are isotropic kernels (depending on kx ?x0 k only) and the x are low-dimensional, MVM
with K (c) can be approximated using specialized nearest neighbour data structures such as KD trees
[12, 9]. Again, the MVM cost is typically O(C n) in this case. For general kernels whose kernel
matrices have a rapidly decaying eigenspectrum, one can approximate MVM by using low-rank
matrices instead of the K (c) [10], whence MVM is O(C n d), d the rank.
In Section 4 we also need MVM with the derivatives (?/?hj )K (c) . Note that (?/? log vc )K (c) =
K (c) , reducing to kernel MVM. For isotropic kernels, K (c) = f (A), ai,j = kxi ? xj k, so
(?/?hj )K (c) = gj (A). If KD trees are used to approximate A, they can be used equivalently (and
with little additional cost) for computing derivative MVMs.
5
The innermost vector operations work on contiguous chunks of memory, rather than strided ones, thus
supporting cacheing or vector functions of the processor.
6
Experiments
In this Section, we provide experimental results for our framework on data from remote sensing, and
on a set of large text classification tasks with very many classes, the latter are hierarchical.
6.1
Flat Classification: Remote Sensing
We use the satimage remote sensing task from the statlog repository.6 This task has been used in
the extensive SVM multi-class study of [5], where it is among the datasets on which the different
methods show the most variance. It has n = 4435 training, m = 2000 test cases, and C = 6 classes.
We use the isotropic Gaussian (RBF) kernel
w
c
K (c) (x, x0 ) = vc exp ? kx ? x0 k2 , vc , wc > 0, x, x0 ? Rd .
(6)
2d
We compare the methods mc-sep (ours with separate kernels for each class; 12 hyperparameters),
mc-tied (ours with a single shared kernel; 2 hyperparameters), 1rest (one-against-rest: C binary
classifiers are trained separately to discriminate c from the rest, they are voted by log probability
upon prediction; 12 hyperparameters). Note that 1rest is arguably the most efficient method which
can be used for multi-class, because its binary classifiers can be fitted separately and in parallel. Even
if run sequentially, 1rest requires less memory by a factor of C than a joint multi-class method.
We use our 5-fold CV criterion ? for each method. Results here are averaged over ten randomly
drawn 5-partitions of the training set (the same partitions are used for the different methods). The
test error (in percent) of mc-sep is 7.81 vs. 8.01 for 1rest. The result for mc-sep is state-of-the-art,
for example the best SVM technique tested in [5] attained 7.65, and SVM one-against-rest attained
8.30 in this study. Note that while 1rest also may choose 12 independent kernel parameters, it does
not make good use of this possibility, as opposed to mc-sep. mc-tied has test error 8.37, suggesting
that tying kernels leads to significant degradation. ROC curves for the different methods are given
in [8], showing that mc-sep also profits from estimating the predictive probabilities in a better way.
6.2
Hierarchical Classification: Patent Text Classification
We use the WIPO-alpha collection7 previously studied in [1], where patents (title and claim text)
are to be classified w.r.t. the standard taxonomy IPC, a tree with 4 levels and 5229 nodes. Sections
A, B,. . . , H. form the first level. As in [1], we concentrate on the 8 subtasks rooted at the sections,
ranging from D (n = 1140, C = 160, P = 187) to B (n = 9794, C = 1172, P = 1319).
We use linear kernels (see Section 5.1) with variance parameters vc . All experiments are averaged
over three training/test splits, different methods using the same ones. ? is used with a different
5-partition per section and split, the same across all methods. Our method outputs a predictive
pj ? RC for each test case xj . The standard prediction y(xj ) = argmaxc pj,c maximizes expected
accuracy, classes are ranked
as rj (c) ? rj (c0 ) iff pj,c ? pj,c0 .P
The test scores are the same as in [1]:
P
?1
?1
?1
accuracy (acc) m
I
,
precision
(prec)
m
, parent accuracy (pacc)
j {y(xj )=yj }
j rj (yj )
P
?1
m
c0 ) be half the length of
j I{par(y(xj ))=par(yj )} , par(c) being the parent of L(c). Let ?(c,P
0
?1
the shortest path between leafs L(c), L(c ). The taxo-loss (taxo) is m
j ?(y(xj ), yj ). These
scores are motivated in [1]. For taxo-loss and parent accuracy, we better choose y(xj ) to minimize
expected loss8 , different from the standard prediction.
We compare methods F1, F2, H1, H2 (F: flat; H: hierarchical). F1: all vc shared (1); H1: vc shared
across each level of the tree (3). F2, H2: vc shared across each subtree rooted at root?s children (A:
15, B: 34, C: 17, D: 7, E: 7, F: 17, G: 12, H: 5). Recall that there are 3 accuracy parameters. For
hyperparameter learning: k1 = 8, k2 = 4, k3 = 15 (F1, F2); k1 = 10, k2 = 4, k3 = 25 (H1, H2)9 .
6
Available at http://www.niaad.liacc.up.pt/old/statlog/.
Raw data from www.wipo.int/ibis/datasets.
Label hierarchy described at
www.wipo.int/classifications/en. Thanks to L. Cai, T. Hofmann for providing us with the
count data and dictionary. We did Porter stemming, stop word removal, and removal of empty categories. The
attributes are bag-of-words over the dictionary of occuring words. All cases xi were scaled to unit norm.
8
For parent accuracy, let p(j) be the node with maximal mass (under pj ) of its children which are leafs,
then y(xj ) must be a child of p(j).
9
Except for section C, where k1 = 14, k2 = 6, k3 = 35.
7
A
B
C
D
E
F
G
H
F1
40.6
32.0
33.7
40.0
33.0
31.4
40.1
39.3
A
B
C
D
E
F
G
H
F1
1.28
1.54
1.33
1.20
1.43
1.43
1.32
1.19
acc (%)
H1
F2
41.9 40.5
32.9 31.7
34.7 34.1
40.6 39.7
34.2 32.8
32.4 31.4
40.7 40.2
39.6 39.4
taxo[0-1]
H1
F2
1.19 1.29
1.44 1.56
1.26 1.32
1.12 1.22
1.33 1.44
1.34 1.44
1.26 1.32
1.16 1.19
H2
41.9
32.7
34.5
40.8
34.1
32.5
40.7
39.7
F1
51.6
41.8
45.2
52.4
45.1
42.8
51.2
52.4
H2
1.18
1.44
1.26
1.12
1.34
1.34
1.26
1.15
F1
58.9
53.6
58.9
64.6
56.0
56.8
58.0
61.6
prec (%)
H1
F2
53.4 51.4
43.8 41.6
46.6 45.4
54.1 52.2
47.1 45.0
44.9 42.8
52.5 51.3
53.3 52.5
pacc (%)
H1
F2
61.6 58.2
56.4 52.7
62.6 58.5
67.0 64.4
59.1 56.2
59.7 56.8
59.7 57.6
62.5 61.8
H2
53.4
43.7
46.4
54.3
47.1
45.0
52.5
53.4
F1
1.27
1.52
1.34
1.19
1.39
1.43
1.32
1.17
H2
61.5
56.6
62.0
67.1
59.2
59.8
59.6
62.5
F1
57.2
51.9
58.6
63.5
54.0
54.9
56.8
59.9
taxo
H1
F2
1.19 1.29
1.44 1.55
1.26 1.35
1.11 1.18
1.31 1.38
1.34 1.43
1.26 1.32
1.15 1.17
pacc[0-1] (%)
H1
F2
61.3 56.9
55.9 51.4
61.8 58.9
67.1 62.6
58.2 53.5
58.7 54.6
59.2 56.6
61.6 60.0
H2
1.19
1.44
1.27
1.11
1.31
1.34
1.26
1.14
H2
61.4
55.9
61.6
67.0
57.9
58.9
58.9
61.8
Table 1: Results on tasks A-H. Methods F1, F2 flat, H1, H2 hierarchical. taxo[0-1], pacc[0-1] for
argmaxc pj,c rule, rather than minimize expected loss.
A
B
C
D
Final NR (s)
F1
H1
2030 3873
3751 8657
4237 7422
56.3 118.5
CV Fold (s)
F1
H1
573
598
873 1720
719 1326
9.32 20.2
E
F
G
H
Final NR (s)
F1
H1
131.5 203.4
1202 2871
1342 2947
971.7 1052
CV Fold (s)
F1
H1
32.2 49.6
426
568
232
579
146
230
Table 2: Running times for tasks A-H. Method F1 flat, H1 hierarchical. CV Fold: Re-optimization
of ?[J] , gradient accumulation for single fold.
For final fitting: k1 = 25, k2 = 12 (F1, F2); k1 = 30, k2 = 17 (H1, H2). The optimization is started
from vc = 5 for all methods. Results are given in Table 1.
The hierarchical model outperforms the flat one consistently. While the differences in accuracy
and precision are hardly significant (as also found in [1]), they (partly) are in taxo-loss and parent
accuracy. Also, minimizing expected loss is consistently better than using the standard rule for the
latter, although the differences are very small. H1 and H2 do not perform differently: choosing
many different vc in the linear kernel seems no advantage here (but see Section 6.1). The results are
very similar to the ones of [1]. However, for our method, the recommendation in [1] to use vc = 1
leads to significantly worse results in all scores, the vc chosen by our methods are generally larger.
In Table 2, we present running times10 for the final fitting and for a single fold during hyperparameter
optimization (5 of them are required for ?, ?h ?). Cai and Hofmann [1] quote a final fitting time of
2200s on the D section, while we require 119s (more than 18 times faster). It is precisely this high
efficiency of primary fitting which allows us to use it as inner loop for hyperparameter learning.
7
Discussion
We presented a general framework for very efficient large scale kernel multi-way classification with
structured label spaces and demonstrated its features on hierarchical text classification tasks with
many classes. As shown for the hierarchical case, the framework is easily extended to novel struc10
Processor time on 64bit 2.33GHz AMD machines.
tural priors or covariance functions, and while not shown here, it is also easy to extend it to different
likelihoods (as long as they are log-concave). We solve the kernel parameter learning problem by
optimizing the CV log likelihood, whose gradient can be computed within the framework. Our
method provides estimates of the predictive distribution at test points, which may result in better
predictions for non-standard losses or ROC curves. Efficient and easily extendable code is publicly
available (see Section 1).
An extension to multi-label classification is planned. More advanced label set structures can be
adressed, noting that Hessian vector products can often be computed in about the same way as gradients. An application to label sequence learning is work in progress, which may even be combined
with a hierarchical prior. Infering a hierarchy from data is possible in principle, using expectation
maximization techniques (note that the primary fitting can deal with target distributions y i ), as well
as incorporating uncertain data.
Empirical Bayesian methods or approximate CV scores for hyperparameter learning have been proposed in [11, 3, 6], but they are orders of magnitude more expensive than our proposal here, and
do not apply to a massive number of classes. Many multi-class SVM techniques are available (see
[2, 5] for references). Here, fitting is a constrained convex problem, and often fairly sparse solutions
(many zeros in ?) are found. However, if the degree of sparsity is not large, first-order conditional
gradient methods typically applied can be slow11 . SVM methods typically do not come with efficient automatic kernel parameter learning schemes, and they do not provide estimates of predictive
probabilities which are asymptotically correct.
Acknowledgments
Thanks to Olivier Chapelle for many useful discussions. Supported in part by the IST Programme
of the European Community, under the PASCAL Network of Excellence, IST-2002-506778.
References
[1] L. Cai and T. Hofmann. Hierarchical document categorization with support vector machines. In CIKM
13, pages 78?87, 2004.
[2] K. Crammer and Y. Singer. On the algorithmic implementation of multiclass kernel-based vector machines. J. M. Learn. Res., 2:265?292, 2001.
[3] P. Craven and G. Wahba. Smoothing noisy data with spline functions: Estimating the correct degree of
smoothing by the method of generalized cross-validation. Numerische Mathematik, 31:377?403, 1979.
[4] P.J. Green and B. Silverman. Nonparametric Regression and Generalized Linear Models. Monographs
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[5] C.-W. Hsu and C.-J. Lin. A comparison of methods for multi-class support vector machines. IEEE
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See
www.kyb.tuebingen.mpg.de/bs/people/seeger.
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[11] C. K. I. Williams and D. Barber.
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11
These methods solve a very large number of small problems iteratively, as opposed to ours which does few
expensive Newton steps. The latter kind, if feasible at all, often makes better use of hardware features such as
cacheing and vector operations, and therefore is the preferred approach in numerical optimization.
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2,254 | 3,045 | Multiple timescales and uncertainty in motor
adaptation
Konrad P. Ko? rding
Rehabilitation Institute of Chicago
Northwestern University, Dept. PM&R
Chicago, IL 60611
[email protected]
Joshua B. Tenenbaum
Massachusetts Institute of Technology
Cambridge, MA 02139
[email protected]
Reza Shadmehr
Johns Hopkins University
Baltimore, MD 21205
[email protected]
Abstract
Our motor system changes due to causes that span multiple timescales. For example, muscle response can change because of fatigue, a condition where the
disturbance has a fast timescale or because of disease where the disturbance is
much slower. Here we hypothesize that the nervous system adapts in a way
that reflects the temporal properties of such potential disturbances. According
to a Bayesian formulation of this idea, movement error results in a credit assignment problem: what timescale is responsible for this disturbance? The adaptation
schedule influences the behavior of the optimal learner, changing estimates at different timescales as well as the uncertainty. A system that adapts in this way
predicts many properties observed in saccadic gain adaptation. It well predicts
the timecourses of motor adaptation in cases of partial sensory deprivation and
reversals of the adaptation direction.
1 Introduction
Saccades are rapid eye movements that shift the direction of gaze from one target to another. The
eyes move so fast [1] that visual feedback can not usually be used during the movement. For that
reason, without adaptation any changes in the properties of the oculomotor plant would lead to inaccurate saccades [2]. Motor gain is the ratio of actual and desired movement distances. If the motor
gain decreases to below one then the nervous system must send a stronger command to produce
a movement of the same size. Indeed, it has been observed that if saccades overshoot the target,
the gain tends to decrease and if they undershoot, the gain tends to increase. The saccadic jump
paradigm [3] is often used to probe such adaptation [4]: while the subject moves its eyes towards a
target, the target is moved. This is not distinguishable to the subject from a change in the properties of the oculomotor plant [5]. Using this paradigm it is possible to probe the mechanism that is
normally used to adapt to ongoing changes of the oculomotor plant.
1.1 Disturbances to the motor plant
Properties of the oculomotor plant may change due to a variety of disturbances, such as various
kinds of fatigue and disease. The fundamental characteristic of these disturbances is that their effects
unfold over a wide range of timescales. Here we model each disturbance as a random walk with a
characteristic timescale (Figures 1A and B) over which the disturbance is expected to go away.
disturbance? (t + ?) = (1 ? 1/? )disturbance? (t) + ?
(1)
where ? is drawn from a mean zero normal distribution of width ? ? , and ? is the timescale. The
larger ? the closer (1 ? 1/? is to 1 and the longer does a disturbance typically last.
1.2 Parameter choice
For the experiments that we want to explain only those timescales will matter that are not much
longer than the overall time of the experiments (because they would already have been integrated
out) and that are not much shorter than the time of an individual saccade (because they would average
out). For that reason we chose the distribution of ? to be 30 values exponentially scaled between
1 and 33333 saccades. The distribution of expected gains thus only depends on the distribution of
?? , a characterization of how important disturbances are at various timescales. It seems plausible
that disturbances that have a short timescale tend to be more variable than those that have a long
timescale, and we choose: ? ? = c/? where c is one of the two free parameters of our model.
Moreover, as we expect each disturbance to be relatively small, we assume linearity and that the
motor gain is simply one plus the sum of all the disturbances:
disturbance? (t)
(2)
gain(t) = 1 +
?
If the motor plant underwent such changes in its properties, and if the nervous system produced the
same motor commands without adaptation, then saccade gain would differ from one, resulting in
motor error. However, with each saccade, the brain observes consequences of the motor commands.
We assume that this observation is corrupted by noise:
observation(t) = gain(t) + w
(3)
where w is the second free parameter of our model, the observation noise with a width ? w . Throughout this paper we choose ? w = 0.05 which we estimated from the spread of saccade gains over typical periods of 200 saccades and c = 0.002 because that yielded good fits to the data by Hopp and
Fuchs [2]. We chose to model all data using the same set of parameters to avoid issues of overfitting.
1.3 Inference
Given this explicit model, Bayesian statistics allows deriving an optimal adaptation strategy. We
observe that the system is equivalent to the generative model of the Kalman filter [6] with a diagonal
transition matrix M = diag(1 ? 1/? ) and an observation matrix H that is a vector consisting of one
1 for each of the 30 potential disturbances, and a diagonal process noise matrix of Q = diag(? ?1 ).
Process noise is what is driving the changes of each of the disturbances. We obtain the solution that
is well known from the Kalman Filter literature. We use the Kalman filter toolbox written by Kevin
Murphy to numerically solve these equations.
An optimally adapting system needs to explicitly represent contribution of each timescale. Because
the contribution of each timescale can never be known precisely, the Bayesian learner represents
what it knows as a probability distribution. As the model is linear and the noises are Gaussian, it
is sufficient to keep first and second order statistics. And so the learner represents what it knows
about the contribution of each timescale as a best estimate, but also keeps a measure of uncertainty
around this estimate (Fig 1C). Any point along the +0% gain line is a point where the fast and slow
timescale cancel each other. There is a line associated with any possible gain (e.g. +30% and -30%).
Every timestep the system starts with its belief that it has from the previous timestep (sketched in
yellow) and combines this with information from the current saccade (sketched in blue) to come up
with a new estimate (sketched in red). Two important changes happen to the belief of the learner
over time. (1) When time passes, disturbances can be expected to get smaller but at the same time
our uncertainty about them increases. (2) when a movement error is observed then this biases the
sum of the disturbances to the observed error value and it also decreases the uncertainty. These
effects are sketched in Figure 1D. Normally the adaptation mechanism is responding to the small
drifts that happen to the oculomotor plant and the estimate from the saccade is largely overlapping
with the prior belief and with the new belief. When the light is turned off the estimate of each of the
B
0
-45-45
error
gain
error
gain
-0.1
gain
fast disturbance [%]
d ?3
0
motor gain
=0
in
ga
d ?3
fast (e.g. fatigue)
0.1
0
=3
d ?2
0
- 0.1
30
=in
ga
d ?2
0.1
in
t+1
d ?1
45
C
slow (e.g. disease)
ga
t
d ?1
disturbance
A
0
45
slow disturbance [%]
1.1
state estimate
1
0.9
prior belief
4000
0
evidence from saccade
time [saccades]
D
Normal saccades
In the dark
Saccadic jump
+30 saccades
Washout
Reversal
Figure 1: A generative model for changes in the motor plant and the corresponding optimal inference. A) Various disturbances d evolve over time as independent random walks. The gain is a
linear function of all these random walks. The observed error is a noisy version of the gain. B)
An example of a system with two timescales (fast and slow), and the resulting gain. C) Optimal
inference during a saccade adaptation experiment. For illustrative purposes, here we assume only
two timescales. The yellow cloud represents the learners belief about the current combination of
disturbances (prior). The system observes a saccade with an error of +30%. The region about the
blue line is the uncertainty about the observation (i.e., the likelihood). Combining this information
with the prior belief (yellow) leads to the posterior estimate (red). After a single observation of the
+30% condition the most probable estimate thus is that it is a fast disturbance. D) The changes
of estimates under various perturbations. Here we simulated a saccade on every 10th time step of
the model. Each column shows three consecutive trials (top to bottom). Only in the darkness case
saccades 1 3 and 50 are shown. In the dark, parameter uncertainties increase because the learner is
not allowed to make observations (sensory noise is effectively infinite). In a gain increase paradigm,
initially most of the error is associated with the fast perturbations. After 30 saccades in the gain increase paradigm, most of the error is associated with slow perturbations. Washout trials that follow
gain increase do not return the system to a naive state. Rather, estimates of fast and slow perturbations cancel each other. Gain decrease following gain increase training will mostly affect the fast
system.
D
1400
Saccade Number
2800
Day 3
0.8
Timecourse [saccades]
p<0.06
800
400
0
Day 1 Day 2
F
Day 5
Bayesian
adaptation
1200
0.6
0.4
0
Robinson et al
Darkness
Day 4
Darkness
Day 2
Darkness
Darkness
Saccade Gain
Gain
1.1
1.0
0.9
0.8
0.7
-200 0
1
E
1200
Bayesian adaptation
Day 1
Optimal adaptation
Darkness
B
Darkness
Monkey experiment: Hopp and Fuchs
Darkness
Darkness
Monkey Experiment: Robinson et al
Timecourse [saccades]
C
A
2000
4000
Saccade Numb
er
6000
800
400
0
Day 1 Day 2
Figure 2: Saccadic gain adaptation in a target jump paradigm. A) Data replotted from Hopp and
Fuchs [2] with permission. Each dot is one saccade, the thick lines are exponential fits to the intervals
[0 1400] and [1400:2800]. Starting at saccade number 0 the target is jumping 30% short of the target
giving the impression of muscles that are too strong. The gain then decreases until the manipulation
is ended at saccade number 1400. B) The same plot is shown for the optimal Bayesian learner.
Changes without feedback. C) Data reprinted from [7]. Normal saccadic gain change paradigm as
in Figure 2, however now the monkey spends its nights without vision and the paradigm is continued
for many days. D) The same plot as in C) but for the Bayesian learner. E) Comparison of the
saccadic gain change timecourses obtained by fitting an exponential. F) the same figure as in E) for
the Bayesian learner
disturbances slowly creeps towards zero. At the same time, however the uncertainty increases a lot
and larger uncertainty allows faster learning because the new information is more precise than the
prior information. In the saccadic jump paradigm the error is much larger than it would be during
normal life and this is first interpreted by the learner as a fast change and as it persists progressively
interpreted as a slow change. When the saccadic jumps ends then the fast timescale goes negative
fast and the slow timescale slowly approaches zero. In a reversal setting the fast timescale becomes
very negative and the slow timescale goes towards zero. Already with two timescales the optimal
learner can thus exhibit a large number of interesting properties.
2 Results: Comparison with experimental data
2.1 Saccadic gain adaptation
In an impressive range of experiments started by Mclaughlin [3], investigators have examined how
monkeys adapt their saccadic gain. Figure 2A shows how the gain changes over time so that saccades
progressively become more precise. The rate of adaptation typically starts fast and then progressively gets slower. This is a classic pattern that is reflected in numerous motor adaptation paradigms
[8, 9]. The same patterns are seen for the Bayesian multiscale learner (Figure 2B). Fast timescale
disturbances are assumed to increase and decrease faster than slow timescale disturbances. Therefore, when the gain rapidly changes, it is a priori most likely that it will go away fast. (Fig. 1D,
saccadic jump). Between trials, the estimates of the fast disturbances decay fast, but this decay is
smaller in the slower timescales. If the gain change is maintained, the relative contribution of the
fast timescales diminishes in comparison to the slow timescales (Fig. 1D, +30 saccades). As fast
timescales adapt fast but decay fast as well and slow timescales adapt and decay slowly, this implies
that the gain change is driven by progressively slower timescales resulting in the transition from
initial fast adapting to a progressively slower adapting.
2.2 Saccadic gain adaptation after sensory deprivation
The effects of a wide range of timescales and uncertainty about the causes of changes of the oculomotor plant will largely be hidden if experiments are of a relatively short duration and no uncertainty
is produced. However, in a recent experiment Robinson et al analyzed saccadic gain adaptation [7]
in a way that allowed insight into many timescales as well as insight into the way the nervous system
deals with uncertainty. The adaptation target was set to -50%. The monkey adapted for about 1500
saccades every day for 21 consecutive days. Because of the long duration many different timescales
are involved in this process. Interestingly, during the rest of the day the monkey wore goggles that
blocked vision. During these breaks monkeys will accumulate uncertainty about the state of their
oculomotor plant. Figure 2C shows results from such an experiment and figure 2D shows the results
we are getting from the Bayesian learner. The results are surprisingly similar given that we used
the same parameters that we had used the model parameters inferred from the Hopp and Fuchs data.
Two effects are visible in the data. (1) There are several timescales during adaptation: there is a
fast (100 saccades) and a slow (10 days) timescale. Closer examination of the data reveals a wide
spectrum of timescales. (2) The state estimate is affected by the periods of darkness. During the
breaks that are paired with darkness the system is decaying back to a gain of zero, as predicted by
the model. Moreover, darkness leads to increased uncertainty. Increased uncertainty means that new
information is relatively more precise than old information which in turn leads to faster learning.
Consequently monkeys learn faster during the second day (after spending a night without feedback)
than during the first (quantified in figure 2E and F). The finding that the Bayesian learner seems to
change faster than the monkey may be related to the context being somewhat different than in the
Hopp and Fuchs experiment. The system seems to represent uncertainty and clearly represents the
way the motor plant is expected to change in the absence of feedback. It has been proposed that
the nervous system may use a set of integrators where one is learning fast and the other is learning
slowly [10, 11]. The Bayesian learner, however, keeps a measure of uncertainty about its estimates.
For that reason only the Bayesian learner can explain the fact that sensory deprivation appears to
enhance learning rates.
2.3 Gain adaptation with reversals
Kojima et al [12] reported a host of surprising behavioral results during saccade adaptation. In these
experiments the adaptation direction was changed 3 times. The saccadic gain was initially increased,
then decreased until it reached unity, and finally increased again (Figure 3A). The saccadic gain
increased faster during the second gain-up session than during the first(Figure 3B). Therefore, the
reversal learning did not washout the system. The Bayesian learner shows a similar phenomenon and
provides a rationale: At the end of the first gain-up session for the Bayesian learner, most of the gain
change is associated with a slow timescale (Figure 3C). In the subsequent gain-down session, errors
produce rapid changes in the fast timescales so that by the time the gain estimate reaches unity, the
fast and slow timescales have opposite estimates. Therefore, the gain-down session did not reset the
system, but the latent variables store the history of adaptation. In the subsequent gain-up session,
the rate of re-adaptation is faster than initial adaptation because the fast timescales decay upwards
in between trials (Figure 3D). After about 100 saccades the speed gain from the low frequencies is
over and is turned into a slowed increase due to the decreased error term.
In a second experiment, Kojima et al [12] found that saccade gains could change despite the fact
that the animal was provided with no feedback to guide its performance. In this experiment the
monkeys were again trained in a gain-up following by a gain-down session. Afterwards they spent
some time in the dark. When they come out of the dark their gain had spontaneously increased
(Figure 3E). The same effect is seen for the Bayesian learner (Figure 3F). In the dark period, the
system makes no observations and therefore cannot learn from error. However, the estimates are
still affected by their timescales of change: the estimate moves up fast along the fast timescales but
slowly along the slow timescales. At the start of the darkness period there is a positive upward and
a negative downward disturbance inferred by the system (Figure 1C, reversal). Consequently, by
the end of the dark period, the estimate has become gain-up, the gain learned in the initial session.
This produces the apparent spontaneous recovery observed in Figure 3F. Updating without feedback
leads the system to infer unobserved dynamics of the oculomotor plant and these dynamics lead to
the observed changes.
Kojima et al
B
C
Optimal
D
Kojima et al
E
Optimal
F
Kojima et al
Gain
1.2
1.0
0.8
0
500
1000
Saccade number
1500
Optimal
10
Dark
1.4
Gain
6.4
4.9
1.4
Gain change rate [x 10 -4 /saccade]
Gain change rate [x 10 -4 /saccade]
A
1.2
5
0.05
1.0
1
control
test
0
1056
Saccade number
2000
Figure 3: The double reversal paradigm. A) The gain is first adapted up until it reaches about 1.2
with a negative target jump of 35%. Then it is adapted down with a positive target jump of 35%.
Once the gain reaches 1 again it is adapted up with a positive target jump again. Data reprinted from
[12] with permission. B) The speed of adaptation is compared between the first adaptation and the
second positive adaptation. C) the same as in A) for the Bayesian learner. D) the same as in B for
the Bayesian learner. E) Double reversal paradigm with darkness, reprinted from [12]. The gain
used by the monkey is changing during this interval. F) The same graph is shown for the Bayesian
learner.
3 Discussion
Traditional models of adaptation simply change motor commands to reduce prediction errors [13].
Our approach differs from traditional approaches in three major ways. (1) The system represents
its knowledge of the properties of the motor system at different timescales and explicitly models
how these disturbances evolve over time. (2) It represents the uncertainty it has about the magnitude
of the disturbances. (3) It formulates the computational aim of adaptation in terms of optimally
predicting ongoing changes in the properties of the motor plant.
Multiple studies address each single of these points on its own. Multi-timescale learning is a classical
phenomenon described frequently [14, 8]. Two timescales had been proposed in the context of
connectionist learning theory [11]. In the context of motor adaptation Smith et al. [10] proposed
a model where the motor system responds to error with two systems: one that is highly sensitive
to error but rapidly forgets and another that has poor sensitivity to error but has strong retention.
In the context of classical conditioning, it has been proposed that the nervous system should keep
a measure of uncertainty about its current parameter estimates to allow an optimal combination of
new information with current knowledge [15]. Even the earliest studies of oculomotor adaptation
realized that the objective of adaptation is to allow precise movement with a relentlessly changing
motor plant [3]. Our approach unifies these ideas in a consistent computational framework and
explains a wide range of experiments.
Multi timescale adaptation and learning is a near universal phenomenon [14, 8, 16, 17]. Within
the area of psychology it was found that learning follows multiscale behavior [17]. It has been
proposed that multiscale learning may arise from chunking effects [14, 18]. The work presented
here suggests a different interpretation. Multiscale learning in cognitive systems may be a result of a
system that has originally evolved to deal with ever changing motor problems. Multiscale adaptation
can also be seen in the way visual neurons adapt to changing visual stimuli [16]. The phenomenon
of spontaneous recovery in classical conditioning [19, 20] is largely equivalent to the findings of
Kojima et al [12] and can also be explained within the Bayesian multiscale learner framework.
The presented model obviously does not explain all known effects in motor or even saccadic gain
adaptation. For example it has been found that adapting up usually has a somewhat different timecourse to adapting down [21, 16, 12]. Moreover it seems that adaptation speed of monkeys can be
very different on one day than the other and from one experimental setting to the other (e.g. Figure
2E and F). In learning reach control, there is more direct evidence that people can actually modify
their rates of adaptation as a function of the auto-correlations of the perturbation [22]. This can be
seen as the system learning about the size of the change parameter ? ? in this theory. Moreover, we
certainly estimate the uncertainty we have about a visual stimulus in a continuous fashion: uncertainty is smallest for a high contrast stimulus in our fovea and progressively larger with decreasing
contrast and increasing eccentricity.
An important question for further enquiry is how the nervous system solves problems that require
multiple timescale adaptation. The necessary effects could potentially be implemented directly by
synapses that could exhibit LTP with powerlaw characteristics [23, 24]. Alternatively, small groups
of neurons may jointly represent the estimates along with their uncertainties.
In summary, if we begin with the assumption that the nervous system optimally solves the problem of
producing reliable movements with a motor plant that is affected by perturbations that have multiple
timescales, then the learner will exhibit numerous properties that appear to match those reported in
saccade and reach adaptation experiments.
References
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2,255 | 3,046 | Approximate inference using planar graph
decomposition
Amir Globerson Tommi Jaakkola
Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology
Cambridge, MA 02139
gamir,[email protected]
Abstract
A number of exact and approximate methods are available for inference calculations in graphical models. Many recent approximate methods for graphs with
cycles are based on tractable algorithms for tree structured graphs. Here we base
the approximation on a different tractable model, planar graphs with binary variables and pure interaction potentials (no external field). The partition function for
such models can be calculated exactly using an algorithm introduced by Fisher and
Kasteleyn in the 1960s. We show how such tractable planar models can be used
in a decomposition to derive upper bounds on the partition function of non-planar
models. The resulting algorithm also allows for the estimation of marginals. We
compare our planar decomposition to the tree decomposition method of Wainwright et. al., showing that it results in a much tighter bound on the partition
function, improved pairwise marginals, and comparable singleton marginals.
Graphical models are a powerful tool for modeling multivariate distributions, and have been successfully applied in various fields such as coding theory and image processing. Applications of
graphical models typically involve calculating two types of quantities, namely marginal distributions, and MAP assignments. The evaluation of the model partition function is closely related to
calculating marginals [12]. These three problems can rarely be solved exactly in polynomial time,
and are provably computationally hard in the general case [1]. When the model conforms to a tree
structure, however, all these problems can be solved in polynomial time. This has prompted extensive research into tree based methods. For example, the junction tree method [6] converts a graphical
model into a tree by clustering nodes into cliques, such that the graph over cliques is a tree. The
resulting maximal clique size (cf. tree width) may nevertheless be prohibitively large.
Wainwright et. al. [9, 11] proposed an approximate method based on trees known as tree reweighting (TRW). The TRW approach decomposes the potential vector of a graphical model into
a mixture over spanning trees of the model, and then uses convexity arguments to bound various
quantities, such as the partition function. One key advantage of this approach is that it provides
bounds on partition function value, a property which is not shared by approximations based on
Bethe free energies [13].
In this paper we focus on a different class of tractable models: planar graphs. A graph is called
planar if it can be drawn in the plane without crossing edges. Works in the 1960s by physicists
Fisher [5] and Kasteleyn [7], among others, have shown that the partition function for planar graphs
may be calculated in polynomial time. This, however, is true under two key restrictions. One is that
the variables xi are binary. The other is that the interaction potential depends only on xi xj (where
xi ? {?1}), and not on their individual values (i.e., the zero external field case).
Here we show how the above method can be used to obtain upper bounds on the partition function
for non-planar graphs. As in TRW, we decompose the potential of a non-planar graph into a sum
over spanning planar models, and then use a convexity argument to obtain an upper bound on the log
partition function. The bound optimization is a convex problem, and can be solved in polynomial
time. We compare our method with TRW on a planar graph with an external field, and show that it
performs favorably with respect to both pairwise marginals and the bound on the partition function,
and the two methods give similar results for singleton marginals.
1
Definitions and Notations
Given a graph G with n vertices and a set of edges E, we are interested in pairwise Markov Random
Fields (MRF) over the graph G. A pairwise MRF [13] is a multivariate distribution over variables
x = {x1 , . . . , xn } defined as
1 P
p(x) = e ij?E fij (xi ,xj )
(1)
Z
where fij are a set of |E| functions, or interaction
potentials, defined over pairs of variables. The
P P
fij (xi ,xj )
ij?E
partition function is defined as Z = x e
. Here we will focus on the case where
xi ? {?1}. Furthermore, we will be interested in interaction potentials which only depend on
agreement or disagreement between the signs of their variables. We define those by
1
?ij (1 + xi xj ) = ?ij I(xi = xj )
(2)
2
so that fij (xi , xj ) is zero if xi 6= xj and ?ij if xi = xj . The model is then defined via the set of
parameters ?ij . We use ? to denote the vector of parameters ?ij , and denote the partition function
by Z(?) to highlight its dependence on these parameters.
f (xi , xj ) =
A graph G is defined as planar if it can be drawn in the plane without any intersection of edges [4].
With some abuse of notation, we define E as the set of line segments in <2 corresponding to the
edges in the graph. The regions of <2 \ E are defined as the faces of the graph. The face which
corresponds to an unbounded region is called the external face. Given a planar graph G, its dual
graph G? is defined in the following way: the vertices of G? correspond to faces of G, and there is
an edge between two vertices in G? iff the two corresponding faces in G share an edge. If the graph
G is weighted, the weight on an edge in G? is the weight on the edge shared by the corresponding
faces in G.
A plane triangulation of a planar graph G is obtained from G by adding edges such that all the faces
of the resulting graph have exactly three vertices. Thus a plane triangulated graph has a dual where
all vertices have degree three. It can be shown that every plane graph can be plane triangulated [4].
We shall also need the notion of a perfect matching on a graph. A perfect matching on a graph G is
defined as a set of edges H ? E such that every vertex in G has exactly one edge in H incident on
it. If the graph is weighted, the weight of the matching is defined as the product of the weights of
the edges in the matching.
Finally, we recall the definition of a marginal polytope of a graph [12]. Consider an MRF over a
graph G where fij are given by Equation 2. Denote the probability of the event I(xi = xj ) under
p(x) by ?ij . The marginal polytope of G, denoted by M(G), is defined as the set of values ?ij that
can be obtained under some assignment to the parameters ?ij . For a general graph G the polytope
M(G) cannot be described using a polynomial number of inequalities. However, for planar graphs,
it turns out that a set of O(n3 ) constraints, commonly referred to as triangle inequalities, suffice to
describe M(G) (see [3] page 434). The triangle inequalities are defined by 1
TRI(n) = {?ij : ?ij + ?jk ? ?ik ? 1, ?ij + ?jk + ?ik ? 1, ?i, j, k ? {1, . . . , n}}
(3)
Note that the above inequalities actually contain variables ?ij which do not correspond to edges in
the original graph G. Thus the equality M(G) = TRI(n) should be understood as referring only
to the values of ?ij that correspond to edges in the graph. Importantly, the values of ?ij for edges
not in the graph need not be valid marginals for any MRF. In other words M(G) is a projection of
TRI(n) on the set of edges of G. It is well known that the marginal polytope for trees is described
via pairwise constraints. It is thus interesting that for planar graphs, it is triplets, rather than pairwise
1
The definition here is slightly different from that in [3], since here we refer to agreement probabilities,
whereas [3] refers to disagreement probabilities. This polytope is also referred to as the cut polytope.
constraints, that characterize the polytope. In this sense, planar graphs and trees may be viewed as
a hierarchy of polytope complexity classes. It remains an interesting problem to characterize other
structures in this hierarchy and their related inference algorithms.
2
Exact calculation of partition function using perfect matching
The seminal works of Kasteleyn [7] and Fisher [5] have shown how one can calculate the partition
function for a binary MRF over a planar graph with pure interaction potentials. We briefly review
Fisher?s construction, which we will use in what follows. Our interpretation of the method differs
somewhat from that of Fisher, but we believe it is more straightforward. The key idea in calculating
the partition function is to convert the summation over values of x to the problem of calculating the
sum of weights of all perfect matchings in a graph constructed from G, as shown below.
In this section, we consider weighted graphs (graphs with numbers assigned to their edges). For the
graph G associated with the pairwise MRF, we assign weights wij = e2?ij to the edges. The first
step in the construction is to plane triangulate the graph G. Let us call the resulting graph GT . We
define an MRF on GT by assigning a parameter ?ij = 0 to the edges that have been added to G,
and the corresponding weight wij = 1. Thus GT essentially describes the same distribution as G,
and therefore has the same partition function. We can thus restrict our attention to calculating the
partition function for the MRF on GT .
As a first step in calculating a partition function over GT , we introduce the following definition: a
? in GT is an agreement edge set (or AES) if for every triangle face F in GT one of the
set of edges E
? or exactly one of the edges in F is in E.
? The weight
following holds: The edges in F are all in E,
?
?
of a set E is defined as the product of the weights of the edges in E.
It can be shown that there exists a bijection between pairs of assignments {x, ?x} and agreement
edge sets. The mapping from x to an edge set is simply the set of edges such that xi = xj . It is easy
to see that this is an agreement edge set. The reverse mapping is obtained by finding an assignment
x such that xi = xj iff the corresponding edge is in the agreement edge set. The existence of this
mapping can be shown by induction on the number of (triangle) faces.
P
The contribution of a given assignment x to the partition function is e
? it is easy to see that
sponds to an AES denoted by E
P
e
ij?E
?ij I(xi =xj )
= e?
P
ij?E
?ij
P
e
?
ij?E
2?ij
= ce
P
?
ij?E
ij?E
2?ij
?ij I(xi =xj )
=c
Y
wij
. If x corre(4)
?
ij?E
P
where c = e? ij?E ?ij . P
Define Q
the superset ? as the set of agreement edge sets. The above then
implies that Z(?) = 2c E??
?
? wij , and is thus proportional to the sum of AES weights.
ij?E
To sum over agreement edge sets, we use the following elegant trick introduced by Fisher [5].
Construct a new graph GPM from the dual of GT by introducing new vertices and edges according to
the following rule: Replace each original vertex with three vertices that are connected to each other,
and assign a weight of one to the new edges. Next, consider the three neighbors of the original
vertex 2 . Connect each of the three new vertices to one of these three neighbors, keeping the original
weights on these edges. The transformation is illustrated in Figure 1.
The new graph GPM has O(3n) vertices, and is also planar. It can be seen that there is a one to one
correspondence between perfect matchings in GPM andP
agreement
Q edge sets in GT . Define ? to be
the set of perfect matchings in GPM . Then Z(?) = 2c M ?? ij?M wij where we have used the
fact that all the new weights have a value of one. Thus, the partition function is a sum over the
weights of perfect matchings in GPM . Finally, we need a way of summing over the weights of the set
of perfect matchings in a graph. Kasteleyn [7] proved that for a planar graph GPM , this sum may be
obtained using the following sequence of steps:
? Direct the edges of the graph GPM such that for every face (except possibly the external
face), the number of edges on its perimeter oriented in a clockwise manner is odd. Kasteleyn showed that such a so called Pfaffian orientation may be constructed in polynomial
time for a planar graph (see also [8] page 322).
2
Note that in the dual of GT all vertices have degree three, since GT is plane triangulated.
1.2
0.7
0.6
1
1 1
0.8
0.6
0.8
1.5
1.4
1.5
1
1
1.2
1
1 1
1
0.7
1.4
1 1
1
Figure 1: Illustration of the graph transformations in Section 2 for a complete graph with four vertices. Left
panel shows the original weighted graph (dotted edges and grey vertices) and its dual (solid edges and black
vertices). Right panel shows the dual graph with each vertex replaced by a triangle (the graph GPM in the text).
Weights for dual graph edges correspond to the weights on the original graph.
? Define the matrix P (GPM ) to be a skew symmetric matrix such that Pij = 0 if ij is not an
edge, Pij = wij if the arrow on edge ij runs from i to j and Pij = ?wij otherwise.
p
? The sum over weighted matchings can then be shown to equal |P (GPM )|.
p
The partition function is thus given by Z(?) = 2c |P (GPM )|. To conclude this section we reiterate
the following two key points: the partition function of a binary MRF over a planar graph with
interaction potentials as in Equation 2 may be calculated in polynomial time by calculating the
determinant of a matrix of size O(3n). An important outcome of this result is that the functional
relation between Z(?) and the parameters ?ij is known, a fact we shall use in what follows.
3
Partition function bounds via planar decomposition
Given a non-planar graph G over binary variables with a vector of interaction potentials ?, we wish
to use the exact planar computation to obtain a bound on the partition function of the MRF on G.
We assume for simplicity that the potentials on the MRF for G are given in the form of Equation 2.
Thus, G violates the assumptions of the previous section only in its non-planarity. Define G(r) as a
set of spanning planar subgraphs of G, i.e., each graph G(r) is planar and contains all the vertices of
G and some its edges. Denote by m the number of such graphs. Introduce the following definitions:
(r)
? ? (r) is a set of parameters on the edges of G(r) , and ?ij is an element in this set. Z(? (r) )
is the partition function of the MRF on G(r) with parameters ? (r) .
? (r) is a set of parameters on the edges of G such that if edge (ij) is in G(r) then ??(r) =
? ?
ij
(r)
(r)
?ij , and otherwise ??ij = 0.
P
Given a distribution ?(r) on the graphs G(r) (i.e., ?(r) ? 0 for r = 1, . . . , m and r ?(r) = 1),
assume that the parameters for G(r) are such that
X
? (r)
?=
?(r)?
(5)
r
Then, by the convexity of the log partition function, as a function of the model parameters, we have
X
log Z(?) ?
?(r) log Z(? (r) ) ? f (?, ?, ? (r) )
(6)
r
Since by assumption the graphs G(r) are planar, this bound can be calculated in polynomial time.
Since this bound is true for any set of parameters ? (r) which satisfies the condition in Equation 5
and for any distribution ?(r), we may optimize over these two variables to obtain the tightest bound
possible. Define the optimal bound for a fixed value of ?(r) by g(?, ?) (optimization is w.r.t. ? (r) )
g(?, ?) =
f (?, ?, ? (r) )
min
? (r) :
P
?
?(r)?
(r)
=?
(7)
Also, define the optimum of the above w.r.t. ? by h(?).
h(?) =
min
g(?, ?)
P
?(r) ? 0, ?(r) = 1
(8)
Thus, h(?) is the optimal upper bound for the given parameter vector ?. In the following section we
argue that we can in fact find the global optimum of the above problem.
4
Globally Optimal Bound Optimization
First consider calculating g(?, ?) from Equation 7. Note that since log Z(? (r) ) is a convex function
of ? (r) , and the constraints are linear, the overall optimization is convex and can be solved efficiently. In the current implementation, we use a projected gradient algorithm [2]. The gradient of
f (?, ?, ? (r) ) w.r.t. ? (r) is given by
h
i
(r)
?f (?, ?, ? (r) )
(r)
(r)
?ij
?1
P
(G
=
?(r)
1
+
e
)
Sign(P
(G
(9)
k(i,j)
PM
PM ))
(r)
k(i,j)
??ij
where k(i, j) returns the row and column indices of the element in the upper triangular matrix of
(r)
(r)
P (GPM ), which contains the element e2?ij .
Since the optimization in Equation 7 is convex, it has an equivalent convex dual. Although we do not
use this dual for optimization (because of the difficulty of expressing the entropy of planar models
solely in terms of triplet marginals), it nevertheless allows some insight into the structure of the
problem. The dual in this case is closely linked to the notion of the marginal polytope defined in
Section 1. Using a derivation similar to [11], we arrive at the following characterization of the dual
X
g(?, ?) = max ? ? ? +
?(r)H(? (r) (? ))
(10)
? ?TRI(n)
r
where ? (r) (? ) denotes the parameters of an MRF on G(r) such that its marginals are given by the
restriction of ? to the edges of G(r) , and H(? (r) (? )) denotes the entropy of the MRF over G(r)
with parameters ? (r) (? ). The maximized function in Equation 10 is linear in ? and thus g(?, ?) is a
pointwise maximum over (linear) convex functions in ? and is thus convex in ?. It therefore has no
(r)
local minima. Denote by ?min (?) the set of parameters that minimizes Equation 7 for a given value
of ?. Using a derivation similar to that in [11], the gradient of g(?, ?) can be shown to be
?g(?, ?)
(r)
= H(?min (?))
??(r)
(11)
Since the partition function for G(r) can be calculated efficiently, so can the entropy. We can now
summarize the algorithm for calculating h(?)
? Initialize ?0 . Iterate:
? For ?t , find ? (r) which solves the minimization in Equation 7.
? Calculate the gradient of g(?, ?) at ?t using the expression in Equation 11
? Update ?t+1 = ?t + ?v where v is a feasible search direction calculated from the
gradient of g(?, ?) and the simplex constraints on ?. The step size ? is calculated via
an Armijo line search.
? Halt when the change in g(?, ?) is smaller than some threshold.
Note that the minimization w.r.t. ? (r) is not very time consuming since we can initialize it with the
minimum from the previous step, and thus only a few iterations are needed to find the new optimum,
provided the change in ? is not too big. The above algorithm is guaranteed to converge to a global
optimum of ? [2], and thus we obtain the tightest possible upper bound on Z(?) given our planar
graph decomposition.
The procedure described here is asymmetric w.r.t. ? and ? (r) . In a symmetric formulation the
minimizing gradient steps could be carried out jointly or in an alternating sequence. The symmetric
P
? (r) = ?.
formulation can be obtained by decoupling ? and ? (r) in the bi-linear constraint
?(r)?
Field
Figure 2: Illustration of planar subgraph construction for a rectangular lattice with external field. Original
graph is shown on the left. The field vertex is connected to all vertices (edges not shown). The graph on the
right results from isolating the 4th ,5th columns of the original graph (shown in grey), and connecting the field
vertex to the external vertices of the three disconnected components. Note that the resulting graph is planar.
Specifically, we introduce ??(r) = ? (r) ?(r) and perform the optimization w.r.t. ? and ??(r) . It can be
shown that a stationary point of f (?, ?, ??(r) ) with the relevant (de-coupled) constraint is equivalent
to the procedure described above. The advantage of this approach is that the exact minimization w.r.t
? (r) is not required before modifying ?. Our experiments have shown, however, that the methods
take comparable times to converge, although this may be a property of the implementation.
5
Estimating Marginals
The optimization problem as defined above minimizes an upper bound on the partition function.
However, it may also be of interest to obtain estimates of the marginals of the MRF over G. To
obtain marginal estimates, we follow the approach in [11]. We first characterize the optimum of
Equation 7 for a fixed value of ?. Deriving the Lagrangian of Equation 7 w.r.t. ? (r) we obtain the
(r)
following characterization of ?min (?):
Marginal Optimality Criterion: For any two graphs G(r) , G(s) such that the edge (ij) is in both
(r)
(s)
graphs, the optimal parameter vector satisfies ?ij (?min (?)) = ?ij (?min (?)).
Thus, the optimal set of parameters for the graphs G(r) is such that every two graphs agree on the
marginals of all the edges they share. This implies that at the optimum, there is a well defined set of
marginals over all the edges. We use this set as an approximation to the true marginals.
A different method for estimating marginals uses the partition function
bound directly. We first
P
P
calculate partition function bounds on the sums: ?i (1) = x:xi =1 e ij?E fij (xi ,xj ) and ?i (?1) =
P
P
?i (1)
ij?E fij (xi ,xj ) and then normalize
x:xi =?1 e
?i (1)+?i (?1) to obtain an estimate for p(xi = 1).
This method has the advantage of being more numerically stable (since it does not depend on derivatives of log Z). However, it needs to be calculated separately for each variable, so that it may be
time consuming if one is interested in marginals for a large set of variables.
6
Experimental Evaluation
We study the application of our Planar Decomposition (PDC) P
method to a binary
MRF on a square
P
lattice with an external field. The MRF is given by p(x) ? e ij?E ?ij xi xj + i?V ?i xi where V are
the lattice vertices, and ?i and ?ij are parameters. Note that this interaction does not satisfy the conditions for exact calculation of the partition function, even though the graph is planar. This problem
is in fact NP hard [1]. However, it is possible to obtain the desired interaction form by introducing
an additional variable xn+1 that is connected to all the original
variables.PDenote the correspondP
?ij xi xj + i?V ?i,n+1 xi xn+1
ij?E
ing graph by Gf . Consider the distribution p(x, xn+1 ) ? e
, where
?i,n+1 = ?i . It is easy to see that any property of p(x) (e.g., partition function, marginals) may be
calculated from the corresponding property of p(x, xn+1 ). The advantage of the latter distribution
is that it has the desired interaction form. We can thus apply PDC by choosing planar subgraphs of
the non-planar graph Gf .
0.25
0.15
0.1
0.05
0.5
1
1.5
Interaction Strength
0.03
Singleton Marginal Error
Z Bound Error
Pairwise Marginals Error
0.08
PDC
TRW
0.2
0.07
0.06
0.05
0.04
0.03
0.02
2
0.5
1
1.5
Interaction Strength
0.025
0.02
0.015
0.01
0.005
2
0.5
1
1.5
Interaction Strength
2
!3
x 10
0.025
0.02
0.015
0.5
1
Field Strength
1.5
2
Singleton Marginal Error
Pairwise Marginals Error
Z Bound Error
0.03
0.03
0.025
0.02
0.015
0.5
1
Field Strength
1.5
2
9
8
7
6
5
4
3
0.5
1
Field Strength
1.5
2
Figure 3: Comparison of the TRW and Planar Decomposition (PDC) algorithms on a 7?7 square lattice. TRW
results shown in red squares, and PDC in blue circles. Left column shows the error in the log partition bound.
Middle column is the mean error for pairwise marginals, and right column is the error for the singleton marginal
of the variable at the lattice center. Results in upper row are for field parameters drawn from U[?0.05, 0.05] and
various interaction parameters. Results in the lower row are for interaction parameters drawn from U [?0.5, 0.5]
and various field parameters. Error bars are standard errors calculated from 40 random trials.
There are clearly many ways to choose spanning planar subgraphs of Gf . Spanning subtrees are one
option, and were used in [11]. Since our optimization is polynomial
in the number of subgraphs,
?
we preferred to use a number of subgraphs that is linear in n. The key idea in generating these
planar subgraphs is to generate disconnected components of the lattice and connect xn+1 only to the
external vertices of these components. Here we generate three disconnected components by isolating
two neighboring columns (or rows) from
? the rest of the graph, resulting in three components. This is
illustrated in Figure 2. To this set of 2 n graphs, we add the independent variables graph consisting
only of edges from the field node to all the other nodes.
We compared the performance of the PDC and TRW methods 3 4 on a 7 ? 7 lattice . Since the exact
partition function and marginals can be calculated for this case, we could compare both algorithms to
the true values. The MRF parameters were set according to the two following scenarios: 1) Varying
Interaction - The field parameters ?i were drawn uniformly from U[?0.05, 0.05], and the interaction
?ij from U[??, ?] where ? ? {0.2, 0.4, . . . , 2}. This is the setting tested in [11]. 2) Varying Field ?i was drawn uniformly from U[??, ?], where ? ? {0.2, 0.4, . . . , 2} and ?ij from U[?0.5, 0.5].
For each scenario, we calculated the following measures: 1)
log partition error
PNormalized
1
1
alg
true
alg
(log
Z
?
log
Z
).
2)
Error
in
pairwise
marginals
|p
(x
= 1, xj = 1) ?
i
ij?E
49
|E|
true
p
(xi = 1, xj = 1)|. Pairwise marginals were calculated jointly using the marginal optimality
criterion of Section 5. 3) Error in singleton marginals. We calculated the singleton marginals for
the innermost node in the lattice (i.e., coordinate [3, 3]), which intuitively should be the most difficult for the planar based algorithm. This marginal was calculated using two partition functions,
as explained in Section 5 5 . The same method was used for TRW. The reported error measure is
|palg (xi = 1) ? ptrue (xi = 1)|. Results were averaged over 40 random trials.
Results for the two scenarios and different evaluation measures are given in Figure 3. It can be seen
that the partition function bound for PDC is significantly better than TRW for almost all parameter
settings, although the difference becomes smaller for large field values. Error for the PDC pairwise
3
TRW and PDC bounds were optimized over both the subgraph parameters and the mixture parameters ?.
In terms of running time, PDC optimization for a fixed value of ? took about 30 seconds, which is still
slower than the TRW message passing implementation.
5
Results using the marginal optimality criterion were worse for PDC, possibly due to its reduced numerical
precision.
4
marginals are smaller than those of TRW for all parameter settings. For the singleton parameters,
TRW slightly outperforms PDC. This is not surprising since the field is modeled by every spanning
tree in the TRW decomposition, whereas in PDC not all the structures model a given field.
7
Discussion
We have presented a method for using planar graphs as the basis for approximating non-planar
graphs such as planar graphs with external fields. While the restriction to binary variables limits
the applicability of our approach, it remains relevant in many important applications, such as coding theory and combinatorial optimization. Moreover, it is always possible to convert a non-binary
graphical model to a binary one by introducing additional variables. The resulting graph will typically not be planar, even when the original graph over k?ary variables is. However, the planar
decomposition method can then be applied to this non-planar graph.
The optimization of the decomposition is carried out explicitly over the planar subgraphs, thus limiting the number of subgraphs that can be used in the approximation. In the TRW method this problem
is circumvented since it is possible to implicitly optimize over all spanning trees. The reason this
can be done for trees is that the entropy of an MRF over a tree may be written as a function of
its marginal variables. We do not know of an equivalent result for planar graphs, and it remains a
challenge to find one. It is however possible to combine the planar and tree decompositions into one
single bound, which is guaranteed to outperform the tree or planar approximations alone.
The planar decomposition idea may in principle be applied to bounding the value of the MAP assignment. However, as in TRW, it can be shown that the solution is not dependent on the decomposition
(as long as each edge appears in some structure), and the problem is equivalent to maximizing a linear function over the marginal polytope (which can be done in polynomial time for planar graphs).
However, such a decomposition may suggest new message passing algorithms, as in [10].
Acknowledgments
The authors acknowledge support from the Defense Advanced Research Projects Agency (Transfer Learning
program). Amir Globerson is also supported by the Rothschild Yad-Hanadiv fellowship. The authors also wish
to thank Martin Wainwright for providing his TRW code.
References
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[2] D. P. Bertsekas, editor. Nonlinear Programming. Athena Scientific, Belmont, MA, 1995.
[3] M.M. Deza and M. Laurent. Geometry of Cuts and Metrics. Springe-Verlag, 1997.
[4] R. Diestel. Graph Theory. Springer-Verlag, 1997.
[5] M.E. Fisher. On the dimer solution of planar ising models. J. Math. Phys., 7:1776?1781, 1966.
[6] M.I. Jordan, editor. Learning in graphical models. MIT press, Cambridge, MA, 1998.
[7] P.W. Kasteleyn. Dimer statistics and phase transitions. Journal of Math. Physics, 4:287?293, 1963.
[8] L. Lovasz and M.D. Plummer. Matching Theory, volume 29 of Annals of discrete mathematics. NorthHolland, New-York, 1986.
[9] M. J. Wainwright, T. Jaakkola, and A. S. Willsky. Tree-based reparameterization framework for analysis
of sum-product and related algorithms. IEEE Trans. on Information Theory, 49(5):1120?1146, 2003.
[10] M. J. Wainwright, T. Jaakkola, and A. S. Willsky. Map estimation via agreement on trees: messagepassing and linear programming. IEEE Trans. on Information Theory, 51(11):1120?1146, 2005.
[11] M. J. Wainwright, T. Jaakkola, and A. S. Willsky. A new class of upper bounds on the log partition
function. IEEE Trans. on Information Theory, 51(7):2313?2335, 2005.
[12] M.J. Wainwright and M.I. Jordan. Graphical models, exponential families, and variational inference.
Technical report, UC Berkeley Dept. of Statistics, 2003.
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2,256 | 3,047 | Multi-Instance Multi-Label Learning with
Application to Scene Classification
Zhi-Hua Zhou
Min-Ling Zhang
National Laboratory for Novel Software Technology
Nanjing University, Nanjing 210093, China
{zhouzh,zhangml}@lamda.nju.edu.cn
Abstract
In this paper, we formalize multi-instance multi-label learning, where each training example is associated with not only multiple instances but also multiple class
labels. Such a problem can occur in many real-world tasks, e.g. an image usually
contains multiple patches each of which can be described by a feature vector, and
the image can belong to multiple categories since its semantics can be recognized
in different ways. We analyze the relationship between multi-instance multi-label
learning and the learning frameworks of traditional supervised learning, multiinstance learning and multi-label learning. Then, we propose the M IML B OOST
and M IML S VM algorithms which achieve good performance in an application to
scene classification.
1
Introduction
In traditional supervised learning, an object is represented by an instance (or feature vector) and
associated with a class label. Formally, let X denote the instance space (or feature space) and Y
the set of class labels. Then the task is to learn a function f : X ? Y from a given data set
{(x1 , y1 ), (x2 , y2 ), ? ? ? , (xm , ym )}, where xi ? X is an instance and yi ? Y the known label of xi .
Although the above formalization is prevailing and successful, there are many real-world problems
which do not fit this framework well, where a real-world object may be associated with a number of
instances and a number of labels simultaneously. For example, an image usually contains multiple
patches each can be represented by an instance, while in image classification such an image can
belong to several classes simultaneously, e.g. an image can belong to mountains as well as Africa.
Another example is text categorization, where a document usually contains multiple sections each of
which can be represented as an instance, and the document can be regarded as belonging to different
categories if it was viewed from different aspects, e.g. a document can be categorized as scientific
novel, Jules Verne?s writing or even books on travelling. Web mining is a further example, where
each of the links can be regarded as an instance while the web page itself can be recognized as news
page, sports page, soccer page, etc.
In order to deal with such problems, in this paper we formalize multi-instance multi-label learning
(abbreviated as M IML). In this learning framework, a training example is described by multiple
instances and associated with multiple class labels. Formally, let X denote the instance space and
Y the set of class labels. Then the task is to learn a function fM IM L : 2X ? 2Y from a given data
(i)
(i)
(i)
set {(X1 , Y1 ), (X2 , Y2 ), ? ? ? , (Xm , Ym )}, where Xi ? X is a set of instances {x1 , x2 , ? ? ? , xni },
(i)
(i) (i)
(i)
(i)
xj ? X (j = 1, 2, ? ? ? , ni ), and Yi ? Y is a set of labels {y1 , y2 , ? ? ? , yli }, yk ? Y (k =
1, 2, ? ? ? , li ). Here ni denotes the number of instances in Xi and li the number of labels in Yi .
After analyzing the relationship between M IML and the frameworks of traditional supervised learning, multi-instance learning and multi-label learning, we propose two M IML algorithms, M IML -
B OOST and M IML S VM. Application to scene classification shows that, solving some real-world
problems in the M IML framework can achieve better performance than solving them in existing
frameworks such as multi-instance learning and multi-label learning.
2
Multi-Instance Multi-Label Learning
We start by investigating the relationship between M IML and the frameworks of traditional supervised learning, multi-instance learning and multi-label learning, and then we develop some solutions.
Multi-instance learning [4] studies the problem where a real-world object described by a number of
instances is associated with one class label. Formally, the task is to learn a function fM IL : 2X ?
{?1, +1} from a given data set {(X1 , y1 ), (X2 , y2 ), ? ? ? , (Xm , ym )}, where Xi ? X is a set of
(i)
(i)
(i)
(i)
instances {x1 , x2 , ? ? ? , xni }, xj ? X (j = 1, 2, ? ? ? , ni ), yi ? {?1, +1} is the label of Xi .1
Multi-instance learning techniques have been successfully applied to diverse applications including
scene classification [3, 7].
Multi-label learning [8] studies the problem where a real-world object described by one instance is
associated with a number of class labels. Formally, the task is to learn a function fM LL : X ? 2Y
from a given data set {(x1 , Y1 ), (x2 , Y2 ), ? ? ? , (xm , Ym )}, where xi ? X is an instance and Yi ? Y
(i) (i)
(i)
(i)
a set of labels {y1 , y2 , ? ? ? , yli }, yk ? Y (k = 1, 2, ? ? ? , li ).2 Multi-label learning techniques
have also been successfully applied to scene classification [1].
In fact, the multi- learning frameworks result from the ambiguity in representing real-world objects.
Multi-instance learning studies the ambiguity in the input space (or instance space), where an object
has many alternative input descriptions, i.e. instances; multi-label learning studies the ambiguity
in the output space (or label space), where an object has many alternative output descriptions, i.e.
labels; while M IML considers the ambiguity in the input and output spaces simultaneously. We
illustrate the differences among these learning frameworks in Figure 1.
(a) Traditional supervised learning
(b) Multi-instance learning
(c) Multi-label learning
(d) Multi-instance multi-label learning
Figure 1: Four different learning frameworks
Traditional supervised learning is evidently a degenerated version of multi-instance learning as well
as a degenerated version of multi-label learning, while traditional supervised learning, multi-instance
learning and multi-label learning are all degenerated versions of M IML. Thus, we can tackle M IML
by identifying its equivalence in the traditional supervised learning framework, using multi-instance
learning or multi-label learning as the bridge.
1
According to notions used in multi-instance learning, (Xi , yi ) is a labeled bag while Xi an unlabeled bag.
Although most works on multi-label learning assume that an instance can be associated with multiple valid
labels, there are also works assuming that only one of the labels associated with an instance is correct [6]. We
adopt the former assumption in this paper.
2
Solution 1: Using multi-instance learning as the bridge: We can transform a M IML learning task,
i.e. to learn a function fM IM L : 2X ? 2Y , into a multi-instance learning task, i.e. to learn a
function fM IL : 2X ? Y ? {?1, +1}. For any y ? Y, fM IL (Xi , y) = +1 if y ? Yi and
?1 otherwise. The proper labels for a new example X ? can be determined according to Y ? =
{y| argy?Y [fM IL (X ? , y) = +1]}. We can transform this multi-instance learning task further into
a traditional supervised learning task, i.e. to learn a function fSISL : X ? Y ? {?1, +1}, under
(i)
a constraint specifying how to derive fM IL (Xi , y) from fSISL (xj , y) (j = 1, ? ? ? , ni ). For any
(i)
y ? Y, fSISL (xj , y) = +1 if y ? Yi and ?1 otherwise. Here the constraint can be fM IL (Xi , y) =
Pni
(i)
sign[ j=1
fSISL (xj , y)] which has been used in transforming multi-instance learning tasks into
traditional supervised learning tasks [9].3 Note that other kinds of constraint can also be used here.
Solution 2: Using multi-label learning as the bridge: We can also transform a M IML learning task,
i.e. to learn a function fM IM L : 2X ? 2Y , into a multi-label learning task, i.e. to learn a function
fM LL : Z ? 2Y . For any zi ? Z, fM LL (zi ) = fM IM L (Xi ) if zi = ?(Xi ), ? : 2X ? Z.
The proper labels for a new example X ? can be determined according to Y ? = fM LL (?(X ? )). We
can transform this multi-label learning task further into a traditional supervised learning task, i.e. to
learn a function fSISL : Z ? Y ? {?1, +1}. For any y ? Y, fSISL (zi , y) = +1 if y ? Yi and
?1 otherwise. That is, fM LL (zi ) = {y| argy?Y [fSISL (zi , y) = +1]}. Here the mapping ? can be
implemented with constructive clustering which has been used in transforming multi-instance bags
into traditional single-instances [11]. Note that other kinds of mapping can also be used here.
3
Algorithms
In this section, we propose two algorithms for solving M IML problems: M IML B OOST works along
the first solution described in Section 2, while M IML S VM works along the second solution.
3.1
M IML B OOST
Given any set ?, let |?| denote its size, i.e. the number of elements in ?; given any predicate ?, let
[[?]] be 1 if ? holds and 0 otherwise; given (Xi , Yi ), for any y ? Y, let ?(Xi , y) = +1 if y ? Yi
and ?1 otherwise, where ? is a function ? : 2X ? Y ? {?1, +1}. The M IML B OOST algorithm is
presented in Table 1.
In the first step, each M IML example (Xu , Yu ) (u = 1, 2, ? ? ? , m) is transformed into a set of |Y|
number of multi-instance bags, i.e. {[(Xu , y1 ), ?(Xu , y1 )], [(Xu , y2 ), ?(Xu , y2 )], ? ? ? , [(Xu , y|Y| ),
?(Xu , y|Y| )]}. Note that [(Xu , yv ), ?(Xu , yv )] (v = 1, 2, ? ? ? , |Y|) is a labeled multi-instance
(u)
(u)
bag where (Xu , yv ) is a bag containing nu number of instances, i.e. {(x1 , yv ), (x2 , yv ), ? ? ? ,
(u)
(xnu , yv )}, and ?(Xu , yv ) ? {+1, ?1} is the label of this bag.
Thus, the original M IML data set is transformed into a multi-instance data set containing m ? |Y|
number of bags, i.e. {[(X1 , y1 ), ?(X1 , y1 )], ? ? ? , [(X1 , y|Y| ), ?(X1 , y|Y| )], [(X2 , y1 ), ?(X2 , y1 )],
? ? ? , [(Xm , y|Y| ), ?(Xm , y|Y| )]}. Let [(X (i) , y (i) ), ?(X (i) , y (i) )] denote the ith of these m ? |Y|
number of bags, that is, (X (1) , y (1) ) denotes (X1 , y1 ), ? ? ? , (X (|Y|) , y (|Y|) ) denotes (X1 , y|Y| ), ? ? ? ,
(X (m?|Y|) , y (m?|Y|) ) denotes (Xm , y|Y| ), where (X (i) , y (i) ) contains ni number of instances, i.e.
(i)
(i)
(i)
{(x1 , y (i) ), (x2 , y (i) ), ? ? ? , (xni , y (i) )}.
Then, from the data set a multi-instance learning function fM IL can be learned, which can accomplish the desired M IML function because fM IM L (X ? ) = {y| argy?Y (sign[fM IL (X ? , y)] = +1)}.
Here we use M I B OOSTING [9] to implement fM IL .
For convenience, let (B, g) denote the bag [(X, y), ?(X, y)]. Then, here the goal is to learn a function F(B) minimizing the bag-level exponential loss EB EG|B [exp(?gF(B))], which ultimately
3
This constraint assumes that all instances contribute equally and independently to a bag?s label, which is
different from the standard multi-instance assumption that there is one ?key? instance in a bag that triggers
whether the bag?s class label will be positive or negative. Nevertheless, it has been shown that this assumption
is reasonable and effective [9]. Note that the standard multi-instance assumption does not always hold, e.g. the
label Africa of an image is usually triggered by several patches jointly instead of by only one patch.
Table 1: The M IML B OOST algorithm
1
Transform each M IML example (Xu , Yu ) (u = 1, 2, ? ? ? , m) into |Y| number of multiinstance bags {[(Xu , y1 ), ?(Xu , y1 )], ? ? ? , [(Xu , y|Y| ), ?(Xu , y|Y| )]}. Thus, the original
data set is transformed into a multi-instance data set containing m ? |Y| number of
multi-instance bags, denoted by {[(X (i) , y (i) ), ?(X (i) , y (i) )]} (i = 1, 2, ? ? ? , m ? |Y|).
2
Initialize weight of each bag to W (i) =
1
m?|Y|
(i = 1, 2, ? ? ? , m ? |Y|).
Repeat for t = 1, 2, ? ? ? , T iterations:
3
(i)
3a
Set Wj = W (i) /ni (i = 1, 2, ? ? ? , m ? |Y|), assign the bag?s label ?(X (i) , y (i) )
(i)
to each of its instances (xj , y (i) ) (j = 1, 2, ? ? ? , ni ), and build an instance-level
(i)
predictor ht [(xj , y (i) )] ? {?1, +1}.
3b
3c
For the ith bag, compute the error rate e(i) ? [0, 1]P
by counting the number of
3d
3e
Compute ct = arg minct
If ct ? 0, go to Step 4.
3f
Set W (i) = W (i) exp[(2e(i) ? 1)ct ] (i = 1, 2, ? ? ? , m ? |Y|) and re-normalize such
Pm?|Y|
that 0 ? W (i) ? 1 and i=1 W (i) = 1.
ni
misclassified instances within the bag, i.e. e(i) =
(i)
If e
(i)
[[ht [(x j ,y (i) )]6=?(X (i) ,y (i) )]]
ni
.
< 0.5 for all i ? {1, 2, ? ? ? , m ? |Y|}, go to Step 4.
Return Y ? = {y| argy?Y sign
4
j=1
Pm?|Y|
W (i) exp[(2e(i) ? 1)ct ].
?P P
ct ht [(x?j , y)] = +1} (x?j is X ? ?s jth instance).
i=1
j
t
?
r(g=1|B)
estimates the bag-level log-odds function 12 log PPr(g=?1|B)
. In each boosting round, the aim is to
expand F(B) into F(B) + cf (B), i.e. adding a new weak classifier, so that the exponential loss
is minimized. Assuming
P all instances in a bag contribute equally and independently to the bag?s
label, f (B) = n1B j h(bj ) can be derived, where h(bj ) ? {?1, +1} is the prediction of the
instance-level classifier h(?) for the jth instance in bag B, and nB is the number of instances in B.
It has been shown by [9] that the best f (B) to be added can be achieved by seeking h(?) which
P Pni 1 (i) (i)
(i)
maximizes i j=1
[ ni W g h(bj )], given the bag-level weights W = exp(?gF(B)). By
assigning each instance the label of its bag and the corresponding weight W (i) /ni , h(?) can be
learned by minimizing the weighted instance-level classification error. This actually corresponds to
the Step 3a of M IML B OOST. When f (B) is found, the best multiplier c > 0 can be got by directly
optimizing the exponential loss:
?
!
P
(i)
(i)
X
g
h(b
)
j
j
EB EG|B [exp(?gF(B) + c(?gf (B)))] =
W (i) exp[c ?
]
i
ni
X
=
W (i) exp[(2e(i) ? 1)c]
i
(i)
1
ni
P
(i)
j [[(h(bj )
(i)
where e =
6= g )]] (computed in Step 3b). Minimization of this expectation actually corresponds to Step 3d, where numeric optimization techniques such as quasi-Newton method
can be used. Finally, the bag-level weights are updated in Step 3f according to the additive structure
of F(B).
3.2
M IML S VM
Given (Xi , Yi ) and zi = ?(Xi ) where ? : 2X ? Z, for any y ? Y, let ?(zi , y) = +1 if y ? Yi
and ?1 otherwise, where ? is a function ? : Z ? Y ? {?1, +1}. The M IML S VM algorithm is
presented in Table 2.
In the first step, the Xu of each M IML example (Xu , Yu ) (u = 1, 2, ? ? ? , m) is collected and put
into a data set ?. Then, in the second step, k-medoids clustering is performed on ?. Since each
Table 2: The M IML S VM algorithm
1
For M IML examples (Xu , Yu ) (u = 1, 2, ? ? ? , m), ? = {Xu |u = 1, 2, ? ? ? , m}.
2
Randomly select k elements from ? to initialize the medoids Mt (t = 1, 2, ? ? ? , k),
repeat until all Mt do not change:
2a ?t = {Mt } (t = 1, 2, ? ? ? , k).
2b Repeat for each Xu ? (? ? {Mt |t = 1, 2, ? ? ? , k}):
index = arg mint?{1,???,k} dH (Xu , Mt ), ?index = ?index ? {Xu }.
P
2c Mt = arg min
dH (A, B) (t = 1, 2, ? ? ? , k).
A??t B??
t
3
Transform (Xu , Yu ) into a multi-label example (zu , Yu ) (u = 1, 2, ? ? ? , m), where
zu = (zu1 , zu2 , ? ? ? , zuk ) = (dH (Xu , M1 ), dH (Xu , M2 ), ? ? ? , dH (Xu , Mk )).
4
For each y ? Y, derive a data set Dy = {(zu , ? (zu , y)) |u = 1, 2, ? ? ? , m}, and then
train an S VM hy = SV M T rain(Dy ).
5
Return Y ? = {arg max hy (z ? )} ? {y|hy (z ? ) ? 0, y ? Y}, where z ? = (dH (X ? , M1 ),
y?Y
dH (X ? , M2 ), ? ? ? , dH (X ? , Mk )).
data item in ?, i.e. Xu , is an unlabeled multi-instance bag instead of a single instance, we employ
Hausdorff distance [5] to measure the distance. In detail, given two bags A = {a1 , a2 , ? ? ? , anA }
and B = {b1 , b2 , ? ? ? , bnB }, the Hausdorff distance between A and B is defined as
dH (A, B) = max{max min ka ? bk, max min kb ? ak}
a?A b?B
b?B a?A
where ka ? bk measures the distance between the instances a and b, which takes the form of
Euclidean distance here.
After the clustering process, we divide the data set ? into k partitions whose medoids are Mt (t =
1, 2, ? ? ? , k), respectively. With the help of these medoids, we transform the original multi-instance
example Xu into a k-dimensional numerical vector zu , where the ith (i = 1, 2, ? ? ? , k) component
of zu is the distance between Xu and Mi , that is, dH (Xu , Mi ). In other words, zui encodes some
structure information of the data, that is, the relationship between Xu and the ith partition of ?.
This process reassembles the constructive clustering process used by [11] in transforming multiinstance examples into single-instance examples except that in [11] the clustering is executed at the
instance level while here we execute it at the bag level. Thus, the original M IML examples (Xu , Yu )
(u = 1, 2, ? ? ? , m) have been transformed into multi-label examples (zu , Yu ) (u = 1, 2, ? ? ? , m),
which corresponds to the Step 3 of M IML S VM. Note that this transformation may lose information,
nevertheless the performance of M IML S VM is still good. This suggests that M IML is a powerful
framework which has captured more original information than other learning frameworks.
Then, from the data set a multi-label learning function fM LL can be learned, which can accomplish the desired M IML function because fM IM L (X ? ) = fM LL (z ? ). Here we use M L S VM [1] to
implement fM LL .
Concretely, M L S VM decomposes the multi-label learning problem into multiple independent binary
classification problems (one per class), where each example associated with the label set Y is regarded as a positive example when building S VM for any class y ? Y , while regarded as a negative
example when building S VM for any class y ?
/ Y , as shown in the Step 4 of M IML S VM. In making
predictions, the T-Criterion [1] is used, which actually corresponds to the Step 5 of the M IML S VM
algorithm. That is, the test example is labeled by all the class labels with positive S VM scores, except that when all the S VM scores are negative, the test example is labeled by the class label which
is with the top (least negative) score.
4
Application to Scene Classification
The data set consists of 2,000 natural scene images belonging to the classes desert, mountains, sea,
sunset, and trees, as shown in Table 3. Some images were from the C OREL image collection while
some were collected from the Internet. Over 22% images belong to multiple classes simultaneously.
Table 3: The image data set (d: desert, m: mountains, s: sea, su: sunset, t: trees)
label
d
m
s
su
t
4.1
# images
340
268
341
216
378
label
d+m
d+s
d + su
d+t
m+s
# images
19
5
21
20
38
label
m + su
m+t
s + su
s+t
su + t
# images
19
106
172
14
28
label
d + m + su
d + su + t
m+s+t
m + su + t
s + su + t
# images
1
3
6
1
4
Comparison with Multi-Label Learning Algorithms
Since the scene classification task has been successfully tackled by multi-label learning algorithms [1], we compare the M IML algorithms with established multi-label learning algorithms A D A B OOST.MH [8] and M L S VM [1]. The former is the core of a successful multi-label learning system
B OOS T EXTER [8], while the latter has achieved excellent performance in scene classification [1].
For M IML B OOST and M IML S VM, each image is represented as a bag of nine instances generated
by the S BN method [7]. Here each instance actually corresponds to an image patch, and better
performance can be expected with better image patch generation method. For A DA B OOST.MH and
M L S VM, each image is represented as a feature vector obtained by concatenating the instances of
M IML B OOST or M IML S VM. Gaussian kernel L IBSVM [2] is used to implement M L S VM, where
the cross-training strategy is used to build the classifiers while the T-Criterion is used to label the
images [1]. The M IML S VM algorithm is also realized with a Gaussian kernel, while the parameter
k is set to be 20% of the number of training images.4 Note that the instance-level predictor used in
Step 3a of M IML B OOST is also a Gaussian kernel L IBSVM (with default parameters).
Since A DA B OOST.MH and M L S VM make multi-label predictions, here the performance of the
compared algorithms are evaluated according to five multi-label evaluation metrics, as shown in
Tables 4 to 7, where ??? indicates ?the smaller the better? while ??? indicates ?the bigger the better?.
Details of these evaluation metrics can be found in [8]. Tenfold cross-validation is performed and
?mean ? std? is presented in the tables, where the best performance achieved by each algorithm
is bolded. Note that since in each boosting round M IML B OOST performs more operations than
A DA B OOST.MH does, for fair comparison, the boosting rounds used by A DA B OOST.MH are set to
ten times of that used by M IML B OOST such that the time cost of them are comparable.
Table 4: The performance of M IML B OOST with different boosting rounds
boosting
rounds
5
10
15
20
25
evaluation metric
hamm. loss
.202?.011
.197?.010
.195?.009
.193?.008
.189?.009
?
?
one-error
.373?.045
.362?.040
.361?.034
.355?.037
.351?.039
coverage ?
1.026?.093
1.013?.109
1.004?.101
.996?.102
.989?.103
rank. loss ?
.208?.028
.191?.027
.186?.025
.183?.025
.181?.026
ave. prec. ?
.764?.027
.770?.026
.772?.023
.775?.024
.777?.025
Table 5: The performance of A DA B OOST.MH with different boosting rounds
boosting
rounds
50
100
150
200
250
4
evaluation metric
hamm. loss ?
.228?.013
.234?.019
.233?.020
.232?.012
.231?.018
one-error ?
.473?.031
.465?.042
.465?.053
.453?.031
.451?.046
coverage ?
1.299?.099
1.292?.138
1.279?.140
1.269?.107
1.258?.137
rank. loss ?
.263?.022
.259?.030
.255?.032
.253?.022
.250?.031
ave. prec. ?
.695?.022
.698?.033
.700?.033
.706?.020
.708?.030
In preliminary experiments, several percentage values have been tested ranging from 20% to 100% with an
interval of 20%. The results show that these values do not significantly affect the performance of M IML S VM.
Table 6: The performance of M IML S VM with different ? used in Gaussian kernel
Gaussian
kernel
? = .1
? = .2
? = .3
? = .4
? = .5
evaluation metric
hamm. loss
.181?.017
.180?.017
.188?.016
.193?.014
.196?.014
?
?
one-error
.332?.036
.327?.033
.344?.032
.358?.030
.370?.033
coverage ?
1.024?.089
1.022?.085
1.065?.094
1.080?.099
1.109?.101
rank. loss ?
.187?.018
.187?.018
.196?.020
.202?.022
.209?.023
ave. prec. ?
.780?.021
.783?.020
.772?.020
.764?.021
.757?.023
Table 7: The performance of M L S VM with different ? used in Gaussian kernel
Gaussian
kernel
?=1
?=2
?=3
?=4
?=5
evaluation metric
hamm. loss ?
.200?.014
.196?.013
.195?.015
.196?.016
.202?.015
one-error ?
.379?.032
.368?.032
.370?.034
.372?.034
.388?.032
coverage ?
1.125?.115
1.115?.122
1.129?.113
1.151?.122
1.181?.128
rank. loss ?
.214?.020
.211?.023
.214?.022
.220?.024
.229?.026
ave. prec. ?
.751?.022
.756?.022
.754?.023
.751?.023
.741?.023
Comparing Tables 4 to 7 we can find that both M IML B OOST and M IML S VM are apparently better
than A DA B OOST.MH and M L S VM. Impressively, pair-wise t-tests with .05 significance level reveal
that the worst performance of M IML B OOST (with 5 boosting rounds) is even significantly better than
the best performance of A DA B OOST.MH (with 250 boosting rounds) on all the evaluation metrics,
and is significantly better than the best performance of M L S VM (with ? = 2) in terms of coverage
while comparable on the remaining metrics; the worse performance of M IML S VM (with ? = .5)
is even comparable to the best performance of M L S VM and is significantly better than the best
performance of A DA B OOST.MH on all the evaluation metrics. These observations confirm that formalizing the scene classification task as a M IML problem to solve by M IML B OOST or M IML S VM is
better than formalizing it as a multi-label learning problem to solve by A DA B OOST.MH or M L S VM.
4.2
Comparison with Multi-Instance Learning Algorithms
Since the scene classification task has been successfully tackled by multi-instance learning algorithms [7], we compare the M IML algorithms with established multi-instance learning algorithms
D IVERSE D ENSITY [7] and E M - DD [10]. The former is one of the most influential multi-instance
learning algorithm and has achieved excellent performance in scene classification [7], while the
latter has achieved excellent performance on multi-instance benchmark tests [10].
Here all the compared algorithms use the same input representation. That is, each image is represented as a bag of nine instances generated by the S BN method [7]. The parameters of D IVERSE
D ENSITY and E M - DD are set according to the settings that resulted in the best performance [7, 10].
The M IML B OOST and M IML S VM algorithms are implemented as described in Section 4.1, with 25
boosting rounds for M IML B OOST while ? = .2 for M IML S VM.
Since D IVERSE D ENSITY and E M - DD make single-label predictions, here the performance of the
compared algorithms are evaluated according to predictive accuracy, i.e. classification accuracy
on test set. Note that for M IML B OOST and M IML S VM, the top ranked class is regarded as the
single-label prediction. Tenfold cross-validation is performed and ?mean ? std? is presented in
Table 8, where the best performance on each image class is bolded. Note that besides the predictive
accuracies on each class, the overall accuracy is also presented, which is denoted by ?overall?.
We can find from Table 8 that M IML B OOST achieves the best performance on image classes desert
and trees while M IML S VM achieves the best performance on the remaining image classes. Overall,
M IML S VM achieves the best performance. Pair-wise t-tests with .05 significance level reveal that
the overall performance of M IML S VM is comparable to that of M IML B OOST, both are significantly
better than that of D IVERSE D ENSITY and E M - DD. These observations confirm that formalizing the
scene classification task as a M IML problem to solve by M IML B OOST or M IML S VM is better than
formalizing it as a multi-instance learning problem to solve by D IVERSE D ENSITY or E M - DD.
Table 8: Compare predictive accuracy of M IML B OOST, M IML S VM, D IVERSE D ENSITY and E M - DD
Image
class
desert
mountains
sea
sunset
trees
overall
5
M IML B OOST
.869?.014
.791?.024
.729?.026
.864?.033
.801?.015
.811?.022
Compared algorithms
M IML S VM D IVERSE D ENSITY
.868?.026
.768?.037
.820?.022
.721?.030
.730?.030
.587?.038
.883?.023
.841?.036
.798?.017
.781?.028
.820?.024
.739?.034
E M - DD
.751?.047
.717?.036
.639?.063
.815?.063
.632?.060
.711?.054
Conclusion
In this paper, we formalize multi-instance multi-label learning where an example is associated with
multiple instances and multiple labels simultaneously. Although there were some works investigating the ambiguity of alternative input descriptions or alternative output descriptions associated
with an object, this is the first work studying both these ambiguities simultaneously. We show that
an M IML problem can be solved by identifying its equivalence in the traditional supervised learning framework, using multi-instance learning or multi-label learning as the bridge. The proposed
algorithms, M IML B OOST and M IML S VM, have achieved good performance in the application to
scene classification. An interesting future issue is to develop M IML versions of other popular machine learning algorithms. Moreover, it remains an open problem that whether M IML can be tackled
directly, possibly by exploiting the connections between the instances and the labels. It is also interesting to discover the relationship between the instances and labels. By unravelling the mixed
connections, maybe we can get deeper understanding of ambiguity.
Acknowledgments
This work was supported by the National Science Foundation of China (60325207, 60473046).
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2,257 | 3,048 | Greedy Layer-Wise Training of Deep Networks
Yoshua Bengio, Pascal Lamblin, Dan Popovici, Hugo Larochelle
Universit?e de Montr?eal
Montr?eal, Qu?ebec
{bengioy,lamblinp,popovicd,larocheh}@iro.umontreal.ca
Abstract
Complexity theory of circuits strongly suggests that deep architectures can be much
more efficient (sometimes exponentially) than shallow architectures, in terms of
computational elements required to represent some functions. Deep multi-layer
neural networks have many levels of non-linearities allowing them to compactly
represent highly non-linear and highly-varying functions. However, until recently
it was not clear how to train such deep networks, since gradient-based optimization
starting from random initialization appears to often get stuck in poor solutions. Hinton et al. recently introduced a greedy layer-wise unsupervised learning algorithm
for Deep Belief Networks (DBN), a generative model with many layers of hidden
causal variables. In the context of the above optimization problem, we study this algorithm empirically and explore variants to better understand its success and extend
it to cases where the inputs are continuous or where the structure of the input distribution is not revealing enough about the variable to be predicted in a supervised
task. Our experiments also confirm the hypothesis that the greedy layer-wise unsupervised training strategy mostly helps the optimization, by initializing weights in a
region near a good local minimum, giving rise to internal distributed representations
that are high-level abstractions of the input, bringing better generalization.
1 Introduction
Recent analyses (Bengio, Delalleau, & Le Roux, 2006; Bengio & Le Cun, 2007) of modern nonparametric machine learning algorithms that are kernel machines, such as Support Vector Machines
(SVMs), graph-based manifold and semi-supervised learning algorithms suggest fundamental limitations of some learning algorithms. The problem is clear in kernel-based approaches when the kernel
is ?local? (e.g., the Gaussian kernel), i.e., K(x, y) converges to a constant when ||x ? y|| increases.
These analyses point to the difficulty of learning ?highly-varying functions?, i.e., functions that have
a large number of ?variations? in the domain of interest, e.g., they would require a large number of
pieces to be well represented by a piecewise-linear approximation. Since the number of pieces can be
made to grow exponentially with the number of factors of variations in the input, this is connected with
the well-known curse of dimensionality for classical non-parametric learning algorithms (for regression, classification and density estimation). If the shapes of all these pieces are unrelated, one needs
enough examples for each piece in order to generalize properly. However, if these shapes are related
and can be predicted from each other, ?non-local? learning algorithms have the potential to generalize
to pieces not covered by the training set. Such ability would seem necessary for learning in complex
domains such as Artificial Intelligence tasks (e.g., related to vision, language, speech, robotics).
Kernel machines (not only those with a local kernel) have a shallow architecture, i.e., only two
levels of data-dependent computational elements. This is also true of feedforward neural networks
with a single hidden layer (which can become SVMs when the number of hidden units becomes
large (Bengio, Le Roux, Vincent, Delalleau, & Marcotte, 2006)). A serious problem with shallow
architectures is that they can be very inefficient in terms of the number of computational units (e.g.,
bases, hidden units), and thus in terms of required examples (Bengio & Le Cun, 2007). One way to
represent a highly-varying function compactly (with few parameters) is through the composition of
many non-linearities, i.e., with a deep architecture. For example, the parity function with d inputs
requires O(2d ) examples and parameters to be represented by a Gaussian SVM (Bengio et al., 2006),
O(d2 ) parameters for a one-hidden-layer neural network, O(d) parameters and units for a multi-layer
network with O(log2 d) layers, and O(1) parameters with a recurrent neural network. More generally,
boolean functions (such as the function that computes the multiplication of two numbers from their
d-bit representation) expressible by O(log d) layers of combinatorial logic with O(d) elements in
each layer may require O(2d ) elements when expressed with only 2 layers (Utgoff & Stracuzzi, 2002;
Bengio & Le Cun, 2007). When the representation of a concept requires an exponential number of
elements, e.g., with a shallow circuit, the number of training examples required to learn the concept
may also be impractical. Formal analyses of the computational complexity of shallow circuits can be
found in (Hastad, 1987) or (Allender, 1996). They point in the same direction: shallow circuits are
much less expressive than deep ones.
However, until recently, it was believed too difficult to train deep multi-layer neural networks. Empirically, deep networks were generally found to be not better, and often worse, than neural networks with
one or two hidden layers (Tesauro, 1992). As this is a negative result, it has not been much reported in
the machine learning literature. A reasonable explanation is that gradient-based optimization starting
from random initialization may get stuck near poor solutions. An approach that has been explored with
some success in the past is based on constructively adding layers. This was previously done using a
supervised criterion at each stage (Fahlman & Lebiere, 1990; Lengell?e & Denoeux, 1996). Hinton,
Osindero, and Teh (2006) recently introduced a greedy layer-wise unsupervised learning algorithm for
Deep Belief Networks (DBN), a generative model with many layers of hidden causal variables. The
training strategy for such networks may hold great promise as a principle to help address the problem
of training deep networks. Upper layers of a DBN are supposed to represent more ?abstract? concepts
that explain the input observation x, whereas lower layers extract ?low-level features? from x. They
learn simpler concepts first, and build on them to learn more abstract concepts. This strategy, studied
in detail here, has not yet been much exploited in machine learning. We hypothesize that three aspects
of this strategy are particularly important: first, pre-training one layer at a time in a greedy way; second, using unsupervised learning at each layer in order to preserve information from the input; and
finally, fine-tuning the whole network with respect to the ultimate criterion of interest.
We first extend DBNs and their component layers, Restricted Boltzmann Machines (RBM), so that
they can more naturally handle continuous values in input. Second, we perform experiments to better
understand the advantage brought by the greedy layer-wise unsupervised learning. The basic question
to answer is whether or not this approach helps to solve a difficult optimization problem. In DBNs,
RBMs are used as building blocks, but applying this same strategy using auto-encoders yielded similar
results. Finally, we discuss a problem that occurs with the layer-wise greedy unsupervised procedure
when the input distribution is not revealing enough of the conditional distribution of the target variable
given the input variable. We evaluate a simple and successful solution to this problem.
2 Deep Belief Nets
Let x be the input, and gi the hidden variables at layer i, with joint distribution
P (x, g1 , g2 , . . . , g` ) = P (x|g1 )P (g1 |g2 ) ? ? ? P (g`?2 |g`?1 )P (g`?1 , g` ),
where all the conditional layers P (g i |gi+1 ) are factorized conditional distributions for which computation of probability and sampling are easy. In Hinton et al. (2006) one considers the hidden layer g i
a binary random vector with ni elements gji :
i
i
P (g |g
i+1
)=
n
Y
P (gji |gi+1 )
with
P (gji
= 1|g
i+1
)=
sigm(bij
j=1
+
i+1
n
X
i i+1
Wkj
gk )
(1)
k=1
where sigm(t) = 1/(1 + e?t ), the bij are biases for unit j of layer i, and W i is the weight matrix for
layer i. If we denote g0 = x, the generative model for the first layer P (x|g 1 ) also follows (1).
2.1 Restricted Boltzmann machines
The top-level prior P (g`?1 , g` ) is a Restricted Boltzmann Machine (RBM) between layer ` ? 1
and layer `. To lighten notation, consider a generic RBM with input layer activations v (for visible units) and hidden layer activations h (for hidden units). It has the following joint distribution:
0
0
0
P (v, h) = Z1 eh W v+b v+c h , where Z is the normalization constant for this distribution, b is the vector of biases for visible units, c is the vector of biases for the hidden units, and W is the weight matrix
for the layer. Minus the argument of the exponential is called the energy function,
energy(v, h) = ?h0 W v ? b0 v ? c0 h.
(2)
We denote the RBM parameters together with ? = (W, b, c). We denote Q(h|v) and P (v|h)
the layer-to-layer conditional distributions associated with the above RBM joint distribution.
The layer-to-layer conditionals
P associated with the RBM factorize likePin (1) and give rise to
P (vk = 1|h) = sigm(bk + j Wjk hj ) and Q(hj = 1|v) = sigm(cj + k Wjk vk ).
2.2 Gibbs Markov chain and log-likelihood gradient in an RBM
To obtain an estimator of the gradient on the log-likelihood of an RBM, we consider a Gibbs Markov
chain on the (visible units, hidden units) pair of variables. Gibbs sampling from an RBM proceeds by
sampling h given v, then v given h, etc. Denote vt for the t-th v sample from that chain, starting at
t = 0 with v0 , the ?input observation? for the RBM. Therefore, (v k , hk ) for k ? ? is a sample from
the joint P (v, h). The log-likelihood of a value v0 under the model of the RBM is
X
X
X
log P (v0 ) = log
P (v0 , h) = log
e?energy(v0 ,h) ? log
e?energy(v,h)
h
h
v,h
and its gradient with respect to ? = (W, b, c) is
X
X
? log P (v0 )
?energy(v0 , h0 )
?energy(vk , hk )
=?
Q(h0 |v0 )
+
P (vk , hk )
??
??
??
h0
vk ,hk
?energy(v0 , h0 )
?energy(vk , hk )
+ E hk
|vk ,
for k ? ?. An unbiased sample is ?
??
??
where h0 is a sample from Q(h0 |v0 ) and (vk , hk ) is a sample of the Markov chain, and the expectation can be easily computed thanks to P (hk |vk ) factorizing. The idea of the Contrastive Divergence
algorithm (Hinton, 2002) is to take k small (typically k = 1). A pseudo-code for Contrastive Divergence training (with k = 1) of an RBM with binomial input and hidden units is presented in the
Appendix (Algorithm RBMupdate(x, , W, b, c)). This procedure is called repeatedly with v 0 = x
sampled from the training distribution for the RBM. To decide when to stop one may use a proxy for
the training criterion, such as the reconstruction error ? log P (v 1 = x|v0 = x).
2.3 Greedy layer-wise training of a DBN
A greedy layer-wise training algorithm was proposed (Hinton et al., 2006) to train a DBN one layer at
a time. One first trains an RBM that takes the empirical data as input and models it. Denote Q(g 1 |g0 )
the posterior over g1 associated with that trained RBM (we recall that g 0 = x with x the observed
1
0
input). This gives rise to an ?empirical? distribution pb1 over
X the first layer g , when g is sampled
1 1
0
1 0
from the data empirical distribution pb: we have pb (g ) =
pb(g )Q(g |g ).
g0
Note that a 1-level DBN is an RBM. The basic idea of the greedy layer-wise strategy is that after
training the top-level RBM of a `-level DBN, one changes the interpretation of the RBM parameters
to insert them in a (` + 1)-level DBN: the distribution P (g `?1 |g` ) from the RBM associated with
layers ` ? 1 and ` is kept as part of the DBN generative model. In the RBM between layers ` ? 1
and `, P (g ` ) is defined in terms on the parameters of that RBM, whereas in the DBN P (g ` ) is defined
in terms of the parameters of the upper layers. Consequently, Q(g ` |g`?1 ) of the RBM does not
correspond to P (g` |g`?1 ) in the DBN, except when that RBM is the top layer of the DBN. However,
we use Q(g` |g`?1 ) of the RBM as an approximation of the posterior P (g ` |g`?1 ) for the DBN.
The samples of g`?1 , with empirical distribution
pb`?1 , are converted stochastically into samples of g `
P
`
` `
`?1 `?1
with distribution pb through pb (g ) =
b (g )Q(g` |g`?1 ). Although pb` cannot be repg`?1 p
resented explicitly it is easy to sample unbiasedly from it: pick a training example and propagate it
stochastically through the Q(g i |gi?1 ) at each level. As a nice side benefit, one obtains an approximation of the posterior for all the hidden variables in the DBN, at all levels, given an input g 0 = x.
Mean-field propagation (see below) gives a fast deterministic approximation of posteriors P (g ` |x).
Note that if we consider all the layers of a DBN from level i to the top, we have a smaller DBN,
which generates the marginal distribution P (g i ) for the complete DBN. The motivation for the greedy
procedure is that a partial DBN with ` ? i levels starting above level i may provide a better model for
P (gi ) than does the RBM initially associated with level i itself.
The above greedy procedure is justified using a variational bound (Hinton et al., 2006). As a consequence of that bound, when inserting an additional layer, if it is initialized appropriately and has
enough units, one can guarantee that initial improvements on the training criterion for the next layer
(fitting pb` ) will yield improvement on the training criterion for the previous layer (likelihood with
respect to pb`?1 ). The greedy layer-wise training algorithm for DBNs is quite simple, as illustrated by
the pseudo-code in Algorithm TrainUnsupervisedDBN of the Appendix.
2.4 Supervised fine-tuning
As a last training stage, it is possible to fine-tune the parameters of all the layers together. For example Hinton et al. (2006) propose to use the wake-sleep algorithm (Hinton, Dayan, Frey, & Neal, 1995)
to continue unsupervised training. Hinton et al. (2006) also propose to optionally use a mean-field approximation of the posteriors P (g i |g0 ), by replacing the samples gji?1 at level i ? 1 by their bit-wise
i
i
i i?1
mean-field expected value ?i?1
). According to these propagation
j , with ? = sigm(b + W ?
rules, the whole network now deterministically computes internal representations as functions of the
network input g0 = x. After unsupervised pre-training of the layers of a DBN following Algorithm
TrainUnsupervisedDBN (see Appendix) the whole network can be further optimized by gradient
descent with respect to any deterministically computable training criterion that depends on these representations. For example, this can be used (Hinton & Salakhutdinov, 2006) to fine-tune a very deep
auto-encoder, minimizing a reconstruction error. It is also possible to use this as initialization of all
except the last layer of a traditional multi-layer neural network, using gradient descent to fine-tune the
whole network with respect to a supervised training criterion.
Algorithm DBNSupervisedFineTuning in the appendix contains pseudo-code for supervised
fine-tuning, as part of the global supervised learning algorithm TrainSupervisedDBN. Note that
better results were obtained when using a 20-fold larger learning rate with the supervised criterion
(here, squared error or cross-entropy) updates than in the contrastive divergence updates.
3 Extension to continuous-valued inputs
With the binary units introduced for RBMs and DBNs in Hinton et al. (2006) one can ?cheat? and
handle continuous-valued inputs by scaling them to the (0,1) interval and considering each input continuous value as the probability for a binary random variable to take the value 1. This has worked
well for pixel gray levels, but it may be inappropriate for other kinds of input variables. Previous
work on continuous-valued input in RBMs include (Chen & Murray, 2003), in which noise is added
to sigmoidal units, and the RBM forms a special form of Diffusion Network (Movellan, Mineiro, &
Williams, 2002). We concentrate here on simple extensions of the RBM framework in which only the
energy function and the allowed range of values are changed.
Linear energy: exponential or truncated exponential
Consider a unit with value y of an RBM, connected to units z of the other layer. p(y|z) can be
obtained from the terms in the exponential that contain y , which can be grouped in ya(z) for linear
energy functions as in (2), where a(z) = b + w 0 z with b the bias of unit y , and w the vector of weights
connecting unit y to units z. If we allow y to take any value in interval I , the conditional density
exp(ya(z))1y?I
of y becomes p(y|z) = R
. When I = [0, ?), this is an exponential density
v exp(va(z))1v?I dv
with parameter a(z), and the normalizing integral equals ?1/a(z), but only exists if ?z, a(z) < 0
Computing the density, computing the expected value (= ?1/a(z)) and sampling would all be easy.
Alternatively, if I is a closed interval (as in many applications of interest), or if we would like to
use such a unit as a hidden unit with non-linear expected value, the above density is a truncated
exponential. For simplicity we consider the case I = [0, 1] here, for which the normalizing integral,
exp(?a(z))?1
which always exists, is
. The conditional expectation of u given z is interesting because
a(z)
1
1
it has a sigmoidal-like saturating and monotone non-linearity: E[y|z] = 1?exp(?a(z))
? a(z)
.A
sampling from the truncated exponential is easily obtained from a uniform sample U , using the inverse
cumulative F ?1 of the conditional density y|z: F ?1 (U ) = log(1?U ?(1?exp(a(z))))
. In both truncated
a(z)
and not truncated cases, the Contrastive Divergence updates have the same form as for binomial units
(input value times output value), since the updates only depend on the derivative of the energy with
respect to the parameters. Only sampling is changed, according to the unit?s conditional density.
Quadratic energy: Gaussian units
P
To obtain Gaussian-distributed units, one adds quadratic terms to the energy. Adding i d2i yi2 gives
rise to a diagonal covariance matrix between units of the same layer, where y i is the continuous value
of a Gaussian unit and d2i is a positive parameter that is equal to the inverse of the variance of y i . In
classification error on training set
0.6
Deep Network with no pre?training
DBN with partially supervised pre?training
DBN with unsupervised pre?training
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0
1.
2.
3.
4.
5.
6.
50
100
150
200
250
300
350
Figure 1: Training classification error vs training
iteration, on the Cotton price task, for deep network without pre-training, for DBN with unsupervised pre-training, and DBN with partially supervised pre-training. Illustrates optimization difficulty of deep networks and advantage of partially
supervised training.
400
Deep Network with no pre-training
Logistic regression
DBN, binomial inputs, unsupervised
DBN, binomial inputs, partially supervised
DBN, Gaussian inputs, unsupervised
DBN, Gaussian inputs, partially supervised
train.
4.23
?
4.59
4.39
4.25
4.23
Abalone
valid.
4.43
?
4.60
4.45
4.42
4.43
test.
4.2
?
4.47
4.28
4.19
4.18
train.
45.2%
44.0%
44.0%
43.3%
35.7%
27.5%
Cotton
valid.
42.9%
42.6%
42.6%
41.1%
34.9%
28.4%
test.
43.0%
45.0%
45.0%
43.7%
35.8%
31.4%
Table 1: Mean squared prediction error on Abalone task and classification error on Cotton task,
showing improvement with Gaussian units.
this case the variance is unconditional, whereas the mean depends on the inputs of the unit: for a unit
y with inputs z and inverse variance d2 , E[y|z] = a(z)
2d2 .
The Contrastive Divergence updates are easily obtained by computing the derivative of the energy
with respect to the parameters. For the parameters in the linear terms of the energy function (e.g., b
and w above), the derivatives have the same form (input unit value times output unit value) as for the
case of binomial units. For quadratic parameter d > 0, the derivative is simply 2dy 2 . Gaussian units
were previously used as hidden units of an RBM (with binomial or multinomial inputs) applied to an
information retrieval task (Welling, Rosen-Zvi, & Hinton, 2005). Our interest here is to use them for
continuous-valued inputs.
Using continuous-valued hidden units
Although we have introduced RBM units with continuous values to better deal with the representation of input variables, they could also be considered for use in the hidden layers, in replacement or
complementing the binomial units which have been used in the past. However, Gaussian and exponential hidden units have a weakness: the mean-field propagation through a Gaussian unit gives rise
to a purely linear transformation. Hence if we have only such linear hidden units in a multi-layered
network, the mean-field propagation function that maps inputs to internal representations would be
completely linear. In addition, in a DBN containing only Gaussian units, one would only be able
to model Gaussian data. On the other hand, combining Gaussian with other types of units could be
interesting. In contrast with Gaussian or exponential units, remark that the conditional expectation
of truncated exponential units is non-linear, and in fact involves a sigmoidal form of non-linearity
applied to the weighted sum of its inputs.
Experiment 1
This experiment was performed on two data sets: the UCI repository Abalone data set (split in 2177
training examples, 1000 validation examples, 1000 test examples) and a financial data set. The latter
has real-valued input variables representing averages of returns and squared returns for which the binomial approximation would seem inappropriate. The target variable is next month?s return of a Cotton
futures contract. There are 13 continuous input variables, that are averages of returns over different
time-windows up to 504 days. There are 3135 training examples, 1000 validation examples, and 1000
test examples. The dataset is publicly available at http://www.iro.umontreal.ca/?lisa/
fin_data/. In Table 1 (rows 3 and 5), we show improvements brought by DBNs with Gaussian
inputs over DBNs with binomial inputs (with binomial hidden units in both cases). The networks have
two hidden layers. All hyper-parameters are selected based on validation set performance.
4 Understanding why the layer-wise strategy works
A reasonable explanation for the apparent success of the layer-wise training strategy for DBNs is that
unsupervised pre-training helps to mitigate the difficult optimization problem of deep networks by
better initializing the weights of all layers. Here we present experiments that support and clarify this.
Training each layer as an auto-encoder
We want to verify that the layer-wise greedy unsupervised pre-training principle can be applied when
using an auto-encoder instead of the RBM as a layer building block. Let x be the input vector
with xi ? (0, 1). For a layer with weights matrix W , hidden biases column vector b and input
biases column vector c, the reconstruction probability for bit i is p i (x), with the vector of probabilities p(x) = sigm(c + W sigm(b + W 0 x)). The training criterion for the layer is the average
of negative log-likelihoods for predicting x from p(x). For example, if x is interpreted either as a
sequence
Pof bits or a sequence of bit probabilities, we minimize the reconstruction cross-entropy:
R = ? i xi log pi (x) + (1 ? xi ) log(1 ? pi (x)). We report several experimental results using this
training criterion for each layer, in comparison to the contrastive divergence algorithm for an RBM.
Pseudo-code for a deep network obtained by training each layer as an auto-encoder is given in Appendix (Algorithm TrainGreedyAutoEncodingDeepNet).
One question that arises with auto-encoders in comparison with RBMs is whether the auto-encoders
will fail to learn a useful representation when the number of units is not strictly decreasing from one
layer to the next (since the networks could theoretically just learn to be the identity and perfectly minimize the reconstruction error). However, our experiments suggest that networks with non-decreasing
layer sizes generalize well. This might be due to weight decay and stochastic gradient descent, preventing large weights: optimization falls in a local minimum which corresponds to a good transformation
of the input (that provides a good initialization for supervised training of the whole net).
Greedy layer-wise supervised training
A reasonable question to ask is whether the fact that each layer is trained in an unsupervised way is
critical or not. An alternative algorithm is supervised, greedy and layer-wise: train each new hidden
layer as the hidden layer of a one-hidden layer supervised neural network NN (taking as input the
output of the last of previously trained layers), and then throw away the output layer of NN and use the
parameters of the hidden layer of NN as pre-training initialization of the new top layer of the deep net,
to map the output of the previous layers to a hopefully better representation. Pseudo-code for a deep
network obtained by training each layer as the hidden layer of a supervised one-hidden-layer neural
network is given in Appendix (Algorithm TrainGreedySupervisedDeepNet).
Experiment 2.
We compared the performance on the MNIST digit classification task obtained with five algorithms:
(a) DBN, (b) deep network whose layers are initialized as auto-encoders, (c) above described supervised greedy layer-wise algorithm to pre-train each layer, (d) deep network with no pre-training
(random initialization), (e) shallow network (1 hidden layer) with no pre-training.
The final fine-tuning is done by adding a logistic regression layer on top of the network and training the whole network by stochastic gradient descent on the cross-entropy with respect to the target
classification. The networks have the following architecture: 784 inputs, 10 outputs, 3 hidden layers
with variable number of hidden units, selected by validation set performance (typically selected layer
sizes are between 500 and 1000). The shallow network has a single hidden layer. An L2 weight
decay hyper-parameter is also optimized. The DBN was slower to train and less experiments were
performed, so that longer training and more appropriately chosen sizes of layers and learning rates
could yield better results (Hinton 2006, unpublished, reports 1.15% error on the MNIST test set).
DBN, unsupervised pre-training
Deep net, auto-associator pre-training
Deep net, supervised pre-training
Deep net, no pre-training
Shallow net, no pre-training
Experiment 2
train.
valid. test
0%
1.2% 1.2%
0%
1.4% 1.4%
0%
1.7% 2.0%
.004% 2.1% 2.4%
.004% 1.8% 1.9%
Experiment 3
train. valid. test
0%
1.5% 1.5%
0%
1.4% 1.6%
0%
1.8% 1.9%
.59% 2.1% 2.2%
3.6% 4.7% 5.0%
Table 2: Classification error on MNIST training, validation, and test sets, with the best hyperparameters according to validation error, with and without pre-training, using purely supervised or
purely unsupervised pre-training. In experiment 3, the size of the top hidden layer was set to 20.
On MNIST, differences of more than .1% are statistically significant. The results in Table 2 suggest
that the auto-encoding criterion can yield performance comparable to the DBN when the layers are
finally tuned in a supervised fashion. They also clearly show that the greedy unsupervised layer-wise
pre-training gives much better results than the standard way to train a deep network (with no greedy
pre-training) or a shallow network, and that, without pre-training, deep networks tend to perform worse
than shallow networks. The results also suggest that unsupervised greedy layer-wise pre-training can
perform significantly better than purely supervised greedy layer-wise pre-training. A possible explanation is that the greedy supervised procedure is too greedy: in the learned hidden units representation
it may discard some of the information about the target, information that cannot be captured easily by
a one-hidden-layer neural network but could be captured by composing more hidden layers.
Experiment 3
However, there is something troubling in the Experiment 2 results (Table 2): all the networks, even
those without greedy layer-wise pre-training, perform almost perfectly on the training set, which
would appear to contradict the hypothesis that the main effect of the layer-wise greedy strategy is
to help the optimization (with poor optimization one would expect poor training error). A possible
explanation coherent with our initial hypothesis and with the above results is captured by the following
hypothesis. Without pre-training, the lower layers are initialized poorly, but still allowing the top two
layers to learn the training set almost perfectly, because the output layer and the last hidden layer
form a standard shallow but fat neural network. Consider the top two layers of the deep network
with pre-training: it presumably takes as input a better representation, one that allows for better
generalization. Instead, the network without pre-training sees a ?random? transformation of the input,
one that preserves enough information about the input to fit the training set, but that does not help to
generalize. To test that hypothesis, we performed a second series of experiments in which we constrain
the top hidden layer to be small (20 hidden units). The Experiment 3 results (Table 2) clearly confirm
our hypothesis. With no pre-training, training error degrades significantly when there are only 20
hidden units in the top hidden layer. In addition, the results obtained without pre-training were found
to have extremely large variance indicating high sensitivity to initial conditions. Overall, the results
in the tables and in Figure 1 are consistent with the hypothesis that the greedy layer-wise procedure
essentially helps to better optimize the deep networks, probably by initializing the hidden layers so that
they represent more meaningful representations of the input, which also yields to better generalization.
Continuous training of all layers of a DBN
With the layer-wise training algorithm for DBNs (TrainUnsupervisedDBN in Appendix), one
element that we would like to dispense with is having to decide the number of training iterations for
each layer. It would be good if we did not have to explicitly add layers one at a time, i.e., if we
could train all layers simultaneously, but keeping the ?greedy? idea that each layer is pre-trained to
model its input, ignoring the effect of higher layers. To achieve this it is sufficient to insert a line
in TrainUnsupervisedDBN, so that RBMupdate is called on all the layers and the stochastic
hidden values are propagated all the way up. Experiments with this variant demonstrated that it works
at least as well as the original algorithm. The advantage is that we can now have a single stopping
criterion (for the whole network). Computation time is slightly greater, since we do more computations
initially (on the upper layers), which might be wasted (before the lower layers converge to a decent
representation), but time is saved on optimizing hyper-parameters. This variant may be more appealing
for on-line training on very large data-sets, where one would never cycle back on the training data.
5 Dealing with uncooperative input distributions
In classification problems such as MNIST where classes are well separated, the structure of the input
distribution p(x) naturally contains much information about the target variable y . Imagine a supervised learning task in which the input distribution is mostly unrelated with y . In regression problems,
which we are interested in studying here, this problem could be much more prevalent. For example
imagine a task in which x ? p(x) and the target y = f (x) + noise (e.g., p is Gaussian and f = sinus)
with no particular relation between p and f . In such settings we cannot expect the unsupervised
greedy layer-wise pre-training procedure to help in training deep supervised networks. To deal with
such uncooperative input distributions, we propose to train each layer with a mixed training criterion
that combines the unsupervised objective (modeling or reconstructing the input) and a supervised objective (helping to predict the target). A simple algorithm thus adds the updates on the hidden layer
weights from the unsupervised algorithm (Contrastive Divergence or reconstruction error gradient)
with the updates from the gradient on a supervised prediction error, using a temporary output layer, as
with the greedy layer-wise supervised training algorithm. In our experiments it appeared sufficient to
perform that partial supervision with the first layer only, since once the predictive information about
the target is ?forced? into the representation of the first layer, it tends to stay in the upper layers. The
results in Figure 1 and Table 1 clearly show the advantage of this partially supervised greedy training
algorithm, in the case of the financial dataset. Pseudo-code for partially supervising the first (or later
layer) is given in Algorithm TrainPartiallySupervisedLayer (in the Appendix).
6 Conclusion
This paper is motivated by the need to develop good training algorithms for deep architectures, since
these can be much more representationally efficient than shallow ones such as SVMs and one-hiddenlayer neural nets. We study Deep Belief Networks applied to supervised learning tasks, and the principles that could explain the good performance they have yielded. The three principal contributions
of this paper are the following. First we extended RBMs and DBNs in new ways to naturally handle
continuous-valued inputs, showing examples where much better predictive models can thus be obtained. Second, we performed experiments which support the hypothesis that the greedy unsupervised
layer-wise training strategy helps to optimize deep networks, but suggest that better generalization is
also obtained because this strategy initializes upper layers with better representations of relevant highlevel abstractions. These experiments suggest a general principle that can be applied beyond DBNs,
and we obtained similar results when each layer is initialized as an auto-associator instead of as an
RBM. Finally, although we found that it is important to have an unsupervised component to train each
layer (a fully supervised greedy layer-wise strategy performed worse), we studied supervised tasks in
which the structure of the input distribution is not revealing enough of the conditional density of y
given x. In that case the DBN unsupervised greedy layer-wise strategy appears inadequate and we
proposed a simple fix based on partial supervision, that can yield significant improvements.
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2,258 | 3,049 | Doubly Stochastic Normalization for Spectral
Clustering
Ron Zass
and Amnon Shashua ?
Abstract
In this paper we focus on the issue of normalization of the affinity matrix in spectral clustering. We show that the difference between N-cuts and Ratio-cuts is
in the error measure being used (relative-entropy versus L1 norm) in finding the
closest doubly-stochastic matrix to the input affinity matrix. We then develop a
scheme for finding the optimal, under Frobenius norm, doubly-stochastic approximation using Von-Neumann?s successive projections lemma. The new normalization scheme is simple and efficient and provides superior clustering performance
over many of the standardized tests.
1
Introduction
The problem of partitioning data points into a number of distinct sets, known as the clustering
problem, is central in data analysis and machine learning. Typically, a graph-theoretic approach to
clustering starts with a measure of pairwise affinity Kij measuring the degree of similarity between
points xi , xj , followed by a normalization step, followed by the extraction of the leading eigenvectors
which form an embedded coordinate system from which the partitioning is readily available. In this
domain there are three principle dimensions which make a successful clustering: (i) the affinity
measure, (ii) the normalization of the affinity matrix, and (iii) the particular clustering algorithm.
Common practice indicates that the former two are largely responsible for the performance whereas
the particulars of the clustering process itself have a relatively smaller impact on the performance.
In this paper we focus on the normalization of the affinity matrix. We first show that the existing
popular methods Ratio-cut (cf. [1]) and Normalized-cut [7] employ an implicit normalization which
corresponds to L1 and Relative Entropy based approximations of the affinity matrix K to a doubly
stochastic matrix. We then introduce a Frobenius norm (L2 ) normalization algorithm based on a
simple successive projections scheme (based on Von-Neumann?s [5] successive projection lemma
for finding the closest intersection of sub-spaces) which finds the closest doubly stochastic matrix
under the least-squares error norm. We demonstrate the impact of the various normalization schemes
on a large variety of data sets and show that the new normalization algorithm often induces a significant performance boost in standardized tests. Taken together, we introduce a new tuning dimension
to clustering algorithms allowing better control of the clustering performance.
2
The Role of Doubly Stochastic Normalization
It has been shown in the past [11, 4] that K-means and spectral clustering are intimately related
where in particular [11] shows that the popular affinity matrix normalization such as employed
by Normalized-cuts is related to a doubly-stochastic constraint induced by K-means. Since this
background is a key to our work we will briefly introduce the relevant arguments and derivations.
Let xi ? RN , i = 1, ..., n,Pbe points arranged in k (mutually exclusive) clusters ?1 , .., ?k with
nj points in cluster ?j and j nj = n. Let Kij = ?(xi , xj ) be a symmetric positive-semi-definite
?
School of Engineering and Computer Science, Hebrew University of Jerusalem, Jerusalem 91904, Israel.
affinity function, e.g. Kij = exp?kxi ?xj k
by maximizing:
max
2
?1 ,...,?k
/? 2
. Then, the problem of finding the cluster assignments
k
X
1
n
j=1 j
X
Kr,s ,
(1)
(r,s)??j
is equivalent to minimizing the ?kernel K-means? problem:
min
c1 ,...,ck ?1 ,...,?k
k X
X
k?(xi ) ? cj k2 ,
j=1 i??j
where P
?(xi ) is a mapping associated with the kernel ?(xi , xj ) = ?(xi )> ?(xj ) and cj =
(1/nj ) i??j ?(xi ) are the class centers. After some algebraic manipulations it can be shown
that the optimization setup of eqn. 1 is equivalent to the matrix form:
max tr(G> KG) s.t G ? 0, GG> 1 = 1, G> G = I
G
(2)
?
where G is the desired assignment matrix with Gij = 1/ nj if i ? ?j and zero otherwise, and
1 is a column vector of ones. Note that the feasible set of matrices satisfying the constraints G ?
0, GG> 1 = 1, G> G = I are of this form for some partitioning ?1 , ..., ?k . Note also that the matrix
F = GG> must be doubly stochastic (F is non-negative, symmetric and F 1 = 1).
Taken together, we see that the desire is to find
P a doubly-stochastic matrix F as close as possible to the input matrix K (in the sense that ij Fij Kij is maximized over all feasible F ), such
that the symmetric decomposition F = GG> satisfies non-negativity (G ? 0) and orthonormality
constraints (G> G = I).
To see the connection with spectral clustering, and N-cuts in particular, relax the non-negativity
condition of eqn. 2 and define a two-stage approach: find the closest doubly stochastic matrix K 0 to
K and we are left with a spectral decomposition problem:
max tr(G> K 0 G) s.t G> G = I
G
(3)
where G contains the leading k eigenvectors of K 0 . We will refer to the process of transforming K to
K 0 as a normalization step. In N-cuts, the normalization takes the form K 0 = D?1/2 KD?1/2 where
D = diag(K1) (a diagonal matrix containing the row sums of K) [9]. In [11] it was shown that
repeating the N-cuts normalization, i.e., setting up the iterative step K (t+1) = D?1/2 K (t) D?1/2
where D = diag(K (t) 1) and K (0) = K converges to a doubly-stochastic matrix (a symmetric
version of the well known ?iterative proportional fitting procedure? [8]).
The conclusion of this brief background is to highlight the motivation for seeking a doubly-stochastic
approximation to the input affinity matrix as part of the clustering process. The open issue is under
what error measure is the approximation to take place? It is not difficult to show that repeating
the N-cuts normalization converges to the global optimum under the relative entropy measure (see
Appendix). Noting that spectral clustering optimizes the Frobenius norm it seems less natural to
have the normalization step optimize a relative entropy error measure.
We will derive in this paper the normalization under the L1 norm and under the Frobenius norm. The
purpose of the L1 norm is to show that the resulting scheme is equivalent to a ratio-cut clustering
? thereby not introducing a new clustering scheme but only contributing to the unification and
better understanding the differences between the N-cuts and Ratio-cuts schemes. The Frobenius
norm normalization is a new formulation and is based on a simple iterative scheme. The resulting
normalization provides a new clustering performance which proves quite practical and boosts the
clustering performance in many of the standardized tests we conducted.
3
Ratio-cut and the L1 Normalization
Given that our desire is to find a doubly stochastic approximation K 0 to the input affinity matrix K,
we begin with the L1 norm approximation:
Proposition 1 (ratio-cut) The closest doubly stochastic matrix K 0 under the L1 error norm is
K 0 = K ? D + I,
which leads to the ratio-cut clustering algorithm, i.e., the partitioning of the data set into two clusters
is determined by the second smallest eigenvector of the Laplacian D ? K, where D = diag(K1).
P
Proof: Let r = minF kK ? F k1 s.t. F 1 = 1, F = F > , where kAk1 = ij abs(Aij ) is the L1
norm. Since kK ? F k1 ? k(K ? F )1k1 for any matrix F , we must have:
r ? k(K ? F )1k1 = kD1 ? 1k1 = kD ? Ik1 .
Let F = K ? D + I, then
kK ? (K ? D + I)k1 = kD ? Ik1 .
If v is an eigenvector of the Laplacian D ? K with eigenvalue ?, then v is also an eigenvector of
K 0 = K ? D + I with eigenvalue 1 ? ? and since (D ? K)1 = 0 then the smallest eigenvector
v = 1 of the Laplacian is the largest of K 0 , and the second smallest eigenvector of the Laplacian
(the ratio-cut result) corresponds to the second largest eigenvector of K 0 .
What we have so far is that the difference between N-cuts and Ratio-cuts as two popular spectral
clustering schemes is that the former uses the relative entropy error measure in finding a doubly
stochastic approximation to K and the latter uses the L1 norm error measure (which turns out to be
the negative Laplacian with an added identity matrix).
4
Normalizing under Frobenius Norm
Given that spectral clustering optimizes the Frobenius norm, there is a strong argument in favor of
finding a Frobenius-norm optimum doubly stochastic approximation to K. The optimization setup
is that of a quadratic linear programming (QLP). However, the special circumstances of our problem
render the solution to the QLP to consist of a very simple iterative computation, as described next.
The closest doubly-stochastic matrix K 0 under Frobenius norm is the solution to the following QLP:
K 0 = argminF kK ? F k2F s.t. F ? 0, F 1 = 1, F = F > ,
(4)
P
2
where kAk2F =
ij Aij is the Frobenius norm. We define next two sub-problems, each with a
closed-form solution, and have our QLP solution derived by alternating successively between the
two until convergence. Consider the affine sub-problem:
P1 (X) = argminF kX ? F k2F s.t. F 1 = 1, F = F >
(5)
and the convex sub-problem:
P2 (X) = argminF kX ? F k2F s.t. F ? 0
(6)
We will use the Von-Neumann [5] successive projection lemma stating that P1 P2 P1 P2 ...P1 (K) will
converge onto the projection of K onto the intersection of the affine and conic subspaces described
above1 . Therefore, what remains to show is that the projections P1 and P2 can be solved efficiently
(and in closed form).
We begin with the solution for P1 . The Lagrangian corresponding to eqn. 5 takes the form:
>
>
L(F, ?1 , ?2 ) = trace(F > F ? 2X > F ) ? ?>
1 (F 1 ? 1) ? ?2 (F 1 ? 1),
where from the condition F = F > we have that ?1 = ?2 = ?. Setting the derivative with respect
to F to zero yields:
F = X + ?1> + 1?> .
1
actually, the Von-Neumann lemma applies only to linear subspaces. The extension to convex subspaces
involves a ?deflection? component described by Dykstra [3]. However, it is possible to show that for this
specific problem the deflection component is redundant and the Von-Neumann lemma still applies.
4000
4
Projection
Matlab QP
2000
1000
0
L1
Frobenius
Relative Entropy
3
seconds
seconds
3000
2
1
10
20
30
# of data?points
40
50
0
500
1000
1500
# of data?points
(a)
2000
(b)
Figure 1: Running times of the normalization algorithms. (a) the Frobenius scheme compared to a general
Matlab QLP solver, (b) running time of the three normalization schemes.
Isolate ? by multiplying by 1 on both sides: ? = (nI + 11> )?1 (I ? X)1. Noting that (nI +
11> )?1 = (1/n)(I ? (1/2n)11> ) we obtain a closed form solution:
!
1> X1
1
1
1
I+
I ? X 11> ? 11> X.
(7)
P1 (X) = X +
n
n2
n
n
The projection P2 (X) can also be described in a simple closed form manner. Let I+ be the set of
indices corresponding to non-negative entries of X and I? the set of negative entries of X. The
criterion function kX ? F k2F becomes:
X
X
kX ? F k2F =
(Xij ? Fij )2 +
(Xij ? Fij )2 .
(i,j)?I+
(i,j)?I?
Clearly, the minimum energy over F ? 0 is obtained when Fij = Xij for all (i, j) ? I+ and zero
otherwise. Let th?0 (X) stand for the operator that zeroes out all negative entries of X. Then,
P2 (X) = th?0 (X).
To conclude, the global optimum of eqn. 4 which returns the closest doubly stochastic matrix K 0 in
Frobenius error norm to the input affinity matrix K is obtained by repeating the following steps:
Algorithm 1 (Frobenius-optimal Doubly Stochastic Normalization) finds the closest doubly
stochastic approximation in Frobenius error norm to a given matrix K (global optimum of eqn. 4).
1. Let X (0) = K.
2. Repeat t = 0, 1, 2, ...
(a) X (t+1) = P1 (X (t) )
(b) If X (t+1) ? 0 then stop and set K 0 = X (t+1) , otherwise set X (t+1) = th?0 (X (t+1) ).
This algorithm is simple and very efficient. Fig. 1a shows the running time of the algorithm compared to an off-the-shelf QLP Matlab solver over random matrices of increasing size ? one can see
that the run-time of our algorithm is a fraction of the standard QLP solver and scales very well with
dimension. In fact the standard QLP solver can handle only small problem sizes. In Fig. 1b we plot
the running times of all three normalization schemes: the L1 norm (computing the Laplacian), the
relative-entropy (the iterative D?1/2 KD?1/2 ), and the Frobenius scheme presented in this section.
The Frobenius is more efficient than the relative-entropy normalization (which is the least efficient
among the three).
5
Experiments
For the clustering algorithm into k ? 2 clusters we experimented with the spectral algorithms
described in [10] and [6]. The latter uses the N-cuts normalization D?1/2 KD?1/2 followed by
K-means on the embedded coordinates (the leading k eigenvectors of the normalized affinity) and
the former uses a certain discretization scheme to turn the k leading eigenvectors into an indicator matrix. Both algorithms produced similar results thus we focused on [10] while replacing the
normalization with the three schemes presented above. We refer to ?Ncuts? as the original normalization D?1/2 KD?1/2 , by ?RE? to the iterative application of the original normalization (which is
proven to converge to a doubly stochastic matrix [11]), by ?L1? to the L1 doubly-stochastic normalization (which we have shown is equivalent to Ratio-cuts) and by ?Frobenius? to the iterative
Frobenius scheme based on Von-Neumann?s lemma described in Section 4. We also included a
?None? field which corresponds to no normalization being applied.
Dataset
SPECTF heart
Pima
Wine
SpamBase
BUPA
WDBC
Kernel
RBF
RBF
RBF
RBF
Poly
Poly
k
Size
2
2
3
2
2
2
267
768
178
4601
345
569
Dim.
44
8
13
57
6
30
L1
27.5
36.2
38.8
36.1
37.4
18.8
Lowest Error Rate
Frobenius RE NCuts
19.2
27.5
27.5
35.2
34.9
35.2
27.0
34.3
29.2
30.3
37.7
31.8
37.4
41.7
41.7
11.1
37.4
37.4
None
29.5
35.4
27.5
30.4
37.4
18.8
Table 1: UCI datasets used, together with some characteristics and the best result achieved using the different
methods.
Dataset
Leukemia
Lung
Prostate
Prostate Outcome
Kernel
Poly
Poly
RBF
RBF
k
2
2
2
2
Size
72
181
136
21
#PC
5
5
5
5
L1
27.8
15.5
40.4
28.6
Lowest Error Rate
Frobenius RE NCuts
16.7
36.1
38.9
9.9
16.6
15.5
19.9
43.4
40.4
4.8
23.8
28.6
None
30.6
15.5
40.4
28.6
Table 2: Cancer datasets used, together with some characteristics and the best result achieved using the different methods.
We begin with evaluating the clustering quality obtained under the different normalization methods
taken over a number of well studied datasets from the UCI repository2 . The data-sets are listed
in Table 1 together with some of their characteristics. The best performance (lowest error rate)
kxi ?xj k2
is presented in Boldface. With the first four datasets we used an RBF kernel e ?2
for the
affinity matrix, while for the latter two a polynomial kernel (xTi xj + 1)d was used. The kernel
parameters were calibrated independently for each method and for each dataset. In most cases the
best performance was obtained with the Frobenius norm approximation, but as a general rule the type
of normalization depends on the data. Also worth noting are instances, such as Wine and SpamBase,
when the RE or Ncuts actually worsen the performance. In that case the RE performance is worse
the Ncuts as the entire normalization direction is counter-productive. When RE outperforms None
it also outperforms Ncuts (as can be expected since Ncuts is the first step in the iterative scheme of
RE).
With regard to tuning the affinity measure, we show in Fig. 2 the clustering performance of each
dataset under each normalization scheme under varying kernel setting (? and d values). Generally,
the performance of the Frobenius normalization behaves in a smoother manner and is more stable
under varying kernel settings than the other normalization schemes.
Our next set of experiments was over some well studied cancer data-sets3 . The data-sets are listed
in Table 2 together with some of their characteristics. The column ?#PC? refers to the number of
principal components used in a PCA pre-processing for the purpose of dimensionality reduction
prior to clustering. Note that better results can be achieved when using a more sophisticated preprocessing, but since the focus is on the performances of the clustering algorithms and not on the
datasets, we prefer not to use the optimal pre-processing and leave the data noisy. The AML/ALL
2
3
http://www.ics.uci.edu/? mlearn/MLRepository.html
All cancer datasets can be found at http://sdmc.i2r.a-star.edu.sg/rp/
50
40
45
% errors
% errors
50
30
20
10
40
35
20
40
60
sigma
80
30
100
1
2
3
sigma
(SPECTF)
4
5
6
250
300
(Pima)
60
45
50
% errors
% errors
40
40
35
30
20
200
400
600
sigma
30
800
50
100
(Wine)
150
200
sigma
(SpamBase)
50
50
40
% errors
% errors
45
30
40
20
35
10
20
30
40
degree
(BUPA)
50
60
10
1
2
3
degree
4
5
6
(WDBC)
Figure 2: Error rate vs. similarity measure, for the UCI datasets listed in Table 1
L1 in magenta +; Forbenius in blue o; Relative Entropy in black ?; and Normalized-Cuts in red
50
50
40
% errors
% errors
40
30
20
10
30
20
10
2
4
6
degree
8
0
10
2
4
(AML/ALL Leukemia)
10
50
40
% errors
40
% errors
8
(Lung Cancer)
50
30
20
10
6
degree
30
20
10
50
sigma
100
(Prostate)
150
0
200
400
sigma
600
800
(Prostate Outcome)
Figure 3: Error rate vs. similarity measure, for the cancer datasets listed in Table 2.
L1 in magenta +; Forbenius in blue o; Relative Entropy in black ?; and Normalized-Cuts in red
Leukemia dataset is a challenging benchmark common in the cancer community, where the task is
to distinguish between two types of Leukemia. The original dataset consists of 7129 coordinates
probed from 6817 human genes, and we perform PCA to obtain 5 leading principal components
prior to clustering using a polynomial kernel. Lung Cancer (Brigham and Women?s Hospital, Harvard Medical School) dataset is another common benchmark that describes 12533 genes sampled
from 181 tissues. The task is to distinguish between malignant pleural mesothelioma (MPM) and
adenocarcinoma (ADCA) of the lung. The Prostate dataset consists of 12,600 coordinates representing different genes, where the task is to identify prostate samples as tumor or non-tumor. We use the
first five principal components as input for clustering using an RBF kernel. The Prostate Outcome
dataset uses the same genes from another set of prostate samples, where the task is to predict the
clinical outcome (relapse or non-relapse for at least four years). Finally, Fig. 3 shows the clustering
performance of each dataset under each normalization scheme under varying kernel settings (? and
d values).
6
Summary
Normalization of the affinity matrix is a crucial element in the success of spectral clustering. The
type of normalization performed by N-cuts is a step towards a doubly-stochastic approximation of
the affinity matrix under relative entropy [11]. In this paper we have extended the normalization
via doubly-stochasticity in three ways: (i) we have shown that the difference between N-Cuts and
Ratio-cuts is in the error measure used to find the closest doubly stochastic approximation to the
input affinity matrix, (ii) we have introduced a new normalization scheme based on Frobenius norm
approximation. The scheme involves a succession of simple computations, is very simple to implement and is efficient computation-wise, and (iii) throughout extensive experimentation on standard
data-sets we have shown the importance of normalization to the performance of spectral clustering.
In the experiments we have conducted the Frobenius normalization had the upper-hand in most
cases. We have also shown that the relative-entropy normalization is not always the right approach
as in some data-sets the performance worsened after the relative-entropy but never worsened when
the Frobenius normalization was applied.
References
[1] P. K. Chan, M. D. F. Schlag, and J. Y. Zien. Spectral k-way ratio-cut partitioning and clustering.
IEEE Transactions on Computer-aided Design of Integrated Circuits and Systems, 13(9):1088?
1096, 1994.
[2] I. Csiszar. I-divergence geometry of probability distributions and minimization problems. The
Annals of Probability, 3(1):146?158, 1975.
[3] R.L. Dykstra. An algorithm for restricted least squares regression. J. of the Amer. Stat. Assoc.,
78:837?842, 1983.
[4] I.S.Dhillon, Y.Guan, and B.Kulis. Kernel k-means, spectral clustering and normalized cuts. In
International Conference on Knowledge Discovery and Data Mining(KDD), pages 551?556,
Aug. 2004.
[5] J. Von Neumann. Functional Operators Vol. II. Princeton University Press, 1950.
[6] A.Y. Ng, M.I. Jordan, and Y. Weiss. On spectral clustering: Analysis and an algorithm. In
Proceedings of the conference on Neural Information Processing Systems (NIPS), 2001.
[7] J. Shi and J. Malik. Normalized cuts and image segmentation. IEEE Transactions on Pattern
Analysis and Machine Intelligence, 22(8), 2000.
[8] R. Sinkhorn and P. Knopp. Conerning non-negative matrices and doubly stochastic matrices.
Pacific J. Math., 21:343?348, 1967.
[9] Y. Weiss. Segmentation using eigenvectors: a unifying view. In Proceedings of the IEEE
Conference on Computer Vision and Pattern Recognition, 1999.
[10] S.X. Yu and J. Shi. Multiclass spectral clustering. In Proceedings of the International Conference on Computer Vision, 2003.
[11] R. Zass and A. Shashua. A unifying approach to hard and probabilistic clustering. In Proceedings of the International Conference on Computer Vision, Beijing, China, Oct. 2005.
A
Normalized Cuts and Relative Entropy Normalization
The following proposition is an extension (symmetric version) of the claim about the iterative proportional fitting procedure converging in relative entropy error measure [2]:
Proposition 2 The closest doubly-stochastic matrix F under the relative-entropy error measure to
a given symmetric matrix K, i.e., which minimizes:
min RE(F ||K) s.t. F ? 0, F = F > , F 1 = 1, F > 1 = 1
F
has the form F = DKD for some (unique) diagonal matrix D.
Proof: The Lagrangian of the problem is:
X
X
X X
X
X
fij X
L() =
fij ln
+
kij ?
fij ?
?i (
fij ? 1) ?
?j (
fij ? 1)
kij
ij
ij
ij
i
j
j
i
The derivative with respect to fij is:
?L
= ln fij + 1 ? ln kij ? 1 ? ?i ? ?j = 0
?fij
from which we obtain:
fij = e?i e?j kij
Let D1 = diag(e?1 , ..., e?n ) and D2 = diag(e?1 , ..., e?n ), then we have:
F = D1 KD2
>
Since F = F and K is symmetric we must have D1 = D2 .
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2,259 | 3,050 | Nonlinear physically-based models for decoding
motor-cortical population activity
Gregory Shakhnarovich Sung-Phil Kim Michael J. Black
Department of Computer Science
Brown University
Providence, RI 02912
{gregory,spkim,black}@cs.brown.edu
Abstract
Neural motor prostheses (NMPs) require the accurate decoding of motor cortical
population activity for the control of an artificial motor system. Previous work
on cortical decoding for NMPs has focused on the recovery of hand kinematics.
Human NMPs however may require the control of computer cursors or robotic
devices with very different physical and dynamical properties. Here we show
that the firing rates of cells in the primary motor cortex of non-human primates
can be used to control the parameters of an artificial physical system exhibiting
realistic dynamics. The model represents 2D hand motion in terms of a point mass
connected to a system of idealized springs. The nonlinear spring coefficients are
estimated from the firing rates of neurons in the motor cortex. We evaluate linear
and a nonlinear decoding algorithms using neural recordings from two monkeys
performing two different tasks. We found that the decoded spring coefficients
produced accurate hand trajectories compared with state-of-the-art methods for
direct decoding of hand kinematics. Furthermore, using a physically-based system
produced decoded movements that were more ?natural? in that their frequency
spectrum more closely matched that of natural hand movements.
1
Introduction
Neural motor prostheses (NMPs) aim to restore lost motor function to people with intact cerebral
motor areas who, through disease or injury, have lost the ability to control their limbs. Central
to these devices is a method for decoding the firing activity of motor cortical neurons to produce
a voluntary control signal. A number of groups have recently demonstrated the real-time neural
control of 2D or 3D computer cursors or simple robotic limbs in monkeys [1, 13, 18, 20, 22] and
humans [6]. Previous work on decoding motor cortical signals however has focused on modeling
the relationship between neural firing rates and simple hand kinematics including hand direction,
speed, position, velocity, or acceleration [2, 4, 8, 10].
While the relationship between neural firing rates and hand kinematics is well established in ablebodied monkeys, the situation of a human NMP is quite different. For a paralyzed human, the NMP
represents an artificial motor system with different physical properties than the intact human motor
system. In particular, a human NMP may involve the control of devices as different as computer
cursors or robotic wheelchairs. It remains an open question whether motor cortical neurons can
successfully control such varied systems with dynamics that are quite different from human limbs.
Here we propose a model that makes a first step toward neural control of novel artificial motor
systems. We show that motor cortical firing rates can be nonlinearly related to the parameters of
an idealized physical system. This provides an important proof-of-concept for human NMPs. Our
model decodes the dynamics of hand movement directly from the neural activity. Ultimately, such a
model should reflect the actuator being controlled. For a biological actuator this means the activation
of individual muscles; for a robotic one, the forces and torques produced by the motors in the system.
A model incorporating direct cortical control of dynamics has been proposed in [19]. There are two
major distinctions between that work and ours. First, we consider the task of controlling an artificial
system, rather than the subject?s real limb. Second, applying the model in [19] in practice would require constructing a very complex biomechanical model and controlling its many degrees of freedom
with a limited neural bandwidth. Here we propose a much simpler approach, that does not attempt
to accurately model the musculoskeletal structure of the arm. Instead, it provides a computationally
effective framework to model the dynamics of the limb moving in two dimensional plane. Our approach is inspired by the recent work of Hinton and Nair [5], that suggested a generative model for
images of handwritten digits. In that work, observed images were assumed to have been generated
by a pen connected to a set of springs, the trajectory of the pen controlled by varying the stiffness of
the springs according to a digit-specific ?motor program?. The goal was to infer the motor program
from an observed image, in order to classify the digit. In the context of neural decoding, the image
observation is replaced with the recorded neural signal, from which we need to recover the ?motor
program?, and thus the intended movement. This is where the parallels between our work and [5]
end. One particularly important difference is that the neural decoder may be learned in a supervised
procedure, where the groud truth for the movement associated with a given neural signal is known.
An advantage of this spring-based model (SBM) over previous kinematics-based decoding methods
is that the realistic dynamics of the model produce smoother recovered movement. We show that
the motions are more natural in that they better match the power spectrum of true hand movements.
This suggests that the control of a physical system (even an artificial one) may prove more natural
for a human NMP.
The experimental setup we consider in this paper involves an electrode array, implanted in the
arm/hand area of the MI cortex of a behaving monkey [17]. The animals are trained to control
the cursor by moving the endpoint of a two-link manipulandum constrained to a plane, much like
a human would use a computer mouse [11, 13]. Neural data and hand kinematics were recorded
from two monkeys performing two different tasks. The data was separated into training and testing
segments and we quantitatively compared a variety of popular linear and nonlinear algorithms for
decoding hand kinematics and the spring coefficients of our SBM. As expected, nonlinear methods tend to outperform linear ones. Moreover, movement reconstructed with the SBM has a power
spectrum significantly closer to that of natural movement. These results suggest that the control of
idealized physical systems with real-time nonlinear decoding algorithms may form the basis for a
practical human NMP.
D
y2
x2
B
A
ax = kA x2 ? kB x1 ? ?vx
]
ay = kC y1 ? kD y2 ? ?vy
y
y1
,a
[ax
C
=
?L
a
x1
L
Figure 1: Sketch of the spring-based
model. The outer endpoints of the
springs are assumed to slide without
friction, so that A and B are always orthogonal to C and D. The rest length
is assumed to be zero for all springs.
Movement is controlled by varying the
stiffness coefficients kA , kB , kC and
kD .
2
The spring-based model
Decoding neural activity in N cells involves estimating the values of a hidden state X(t) at time
t given an observed sequence of firing rates Z(0) . . . Z(t) up to time t, with each Z(i) being a
1 ? N vector. The state here is typically taken to be either hand position, velocity, etc. Methods
described in the literature can be roughly divided into two classes. Generative methods formulate the
likelihood of the observed firing rates conditioned on the state and use Bayesian inference methods
such as the Kalman filter [21] or particle filter [3] to estimate the system state from observations.
In contrast, direct (or discriminative) methods learn a function that maps firing rates over some
preceding temporal window into hand kinematics. Various methods have been explored including
linear regression [1, 13], support-vector regression [15] and neural network algorithms [12, 20].
All these previous methods have focused on direct decoding of kinematic properties of the hand
movement and have ignored the arm dynamics.
2.1
Parametrization of dynamics
Our approach to incorporating dynamics into the decoding process has been inspired by the following model of [5], sketched out in Figure 1. Without loss of generality, let the work area (that fully
contains the movement range) be an axis-aligned square [?L, L] ? [?L, L]. The endpoint of the
limb (wrist) is assumed to be connected to one end of four imaginary springs, the other end of which
is sliding with no friction along rails forming the boundaries of the ?work area?. Thus, at every time
instance each spring is parallel to one of the axes. The analysis of dynamics therefore can be easily
decomposed to x and y components. Below we focus on the x component.
All four springs are assumed to have rest length of zero. Suppose that the position of the wrist at
time t is [x(t), y(t)]. Then the springs A and B apply forces determined by Hooke?s law, namely,
kA (t) (L ? x(t))) and ?kB (t) (x(t) + L), where kA (t) and kB (t) are the stiffness coefficients of A
and B at time t. To reflect physical constraints on movement in the real world, the model presumes
a point mass m in the center of the wrist (i.e. at the cursor location). Furthermore, it is assumed that
the movement is damped by a viscous force proportional to the instantaneous velocity, ??vx (t).
The viscosity is meant to represent both the medium resistance and the elasticity of the muscles. In
summary, according to Newton?s second law the acceleration of the hand at time t is given by
m ? ax (t) = kA (t) ? (L ? x(t)) ? kB (t) ? (L + x(t)) ? ? ? vx (t),
(1)
where vx (t) is the instantaneous velocity of the wrist at time t along the x axis.
Control of movement in this model is realized through varying the stiffness coefficients of the
springs: given the current position of the wrist x, the desired acceleration a is achieved by setting kA (t) and kB (t) so as to solve (1). This solution is not unique, in general. We note, however,
that the physiological meaning of the k?s requires them to be non-negative, since the muscles can
not ?push?. This motivates us to introduce the total stiffness constraint
kA + kB = ?,
(2)
where ? is a constant chosen so that no feasible acceleration would yield negative kA or kB .
We can now recover the underlying parameters K = [kA , kB , kC , kD ] for the observed movement
by applying (1) at each time step as follows. First we estimate the velocities v?x (t) = x(t + 1) ? x(t)
and accelerations a
?x (t) = v?x (t + 1) ? v?x (t). Then, we substitute (2) into (1), yielding
m?a
?x (t) + v?x (t) + ? ? (L + x(t))
.
k?A (t) =
2L
(3)
The value of kB (t) is then uniquely determined from (2). Repeating these calculations for the y-axis
produces the coefficients for springs C and D.
2.2
Decoding neural activity
We now turn to our main goal: inferring the desired movement from a recorded neural signal. We
treat this as a supervised learning task. In the training stage, we take a data set in which we have
both the recorded neural signal Z(t) and the observed trajectory of hand positions X(t) associated
with that signal. From this, we can learn a mapping g from the neural signal to the desired representation of movement. For direct kinematic decoding this means inference g : Z(t) ? X(t). For
decoding with the SBM, this means inference of spring coefficients in the SBM, g : Z(t) ? K(t),
followed by the calculation K(t) ? X(t) as described above. The SBM formulation also requires
a preprocessing step for the training data: we need to convert the observed position trajectory X to
the trajectory through K, acording to (3).
We have focused on two ways of constructing g, described below.
Linear filter. The linear filter (LF) approach [13] consists of modeling the mapping from firing
rate to movement by a linear transformation W that is applied on a concatenated firing rate vector
for a fixed history depth l:
?
X(t) = x0 + WZ(t),
(4)
where x0 is a constant (bias) term and
T
?
Z(t)
= ZT (t ? l), . . . , ZT (t) .
(5)
? for a recording from N channels, is 1 ? lN . The transformation W is fit to
The dimension of Z(t)
the training data by solving the least squares problem, and then used at the decoding stage to predict
values of X. Application of the LF to the SBM is straightforward: the target of the mapping is in
the space of coefficients K, rather than position X.
Support vector regression. Support Vector Machines (SVM) are a widely popular learning architecture that relies on two key ideas: mapping the data into a (possibly infinite-dimensional) feature
space using a kernel function, and optimizing the bound on generalization error. In the context of
regression [16] this means using an -insensitive loss function, that does not penalize training errors
up to , to fit a linear function in the feature space. SVMs also aim at reducing model complexity
by penalizing the objective for the norm of the resulting function. The solution is finally expressed
in terms of kernel functions involving a subset of the training examples (the support vectors). The
key parameters that affect the performance of SVMs are the value of , the tradeoff c that governs
the penalty of training error, and parameters of the kernel function.
SVMs have been widely successful in many applications of machine learning. However, their application to the task of neural decoding has been limited to the directional center-out task [15]. Here
we evaluate SV regression as a method for decoding more general 2D movement. Again, the SVM
formulation is readily extended to the SBM (with the target functions being components of K).
Alternative decoders. A variety of other decoding approaches has been proposed in the literature.
We conducted experiments with three additional algorithms: Kalman filter [21], Multilayer Perceptrons [20] and Echo-state Networks, a recurrent neural network architecture [7]. The Kalman filter
uses a linear model of the mapping of neural signals to movement, while the models underlying the
other two methods are nonlinear. Our findings can be summarized as follows, for both kinematic
decoding and decoding with the spring-based model. First, nonlinear methods perform significantly
better than linear ones. Second, there was a trend for the Kalman filter to perform better than the
linear filter. Third, among nonlinear methods SVM tended to perform better than the two neural network architectures. However, these latter differences could not be established with significance. In
the following section, we focus on experiments with the linear filter (the de-facto standard decoding
method today) and SVM, which achieved the overall best results in our experiments.
3
Experiments
We evaluated the performance of the proposed approach on data sets obtained from two behaving
monkeys (Macaca Mulatta). The neural signal was obtained with a Cyberkinetics microelectrode
array [9] (96 electrodes) implanted in the arm/hand area of MI cortex. The experimental animals
performed the tasks described below.
Sequential reaching movement , described in [13] Reach targets and a hand position feedback
cursor were presented on a video screen in front of the monkey. When a reach target was presented
Table 1: Details of experiments. Units: number of
distinct units identified after spike sorting. Train, test:
length of train and test sequences in seconds.
Session
CL-sequential
LA-continuous
CL-continuous
Units
49
96
55
Train
623
244
448
Test
140
165
140
the animal?s task was to move a manipulandum so that the feedback cursor moved into the target
and remained in the target for 500ms, at which time that target was extinguished and a new reach
target was presented in a different location. Target locations were drawn i.i.d. from the uniform
distribution over the screen surface. This was repeated for up to 10 targets per trial. Upon successful
completion of a trial the animal received a juice reward. Hand kinematics and neural activity were
simultaneously recorded while the animal performed the task.
Continuous tracking , described in [14] Monkey was viewing a computer screen on which a
visual target appeared in a random, but smooth, sequence of locations. The monkey was trained to
follow the target?s position with a cursor, using a manipulandum, and received a reward for each
successful trial (i.e. when the cursor remained within the target for a duration drawn for each target
randomly between 3 and 10 seconds).
The recorded neural activity was converted to spike trains by computer-assisted spike-sorting software, and the spike counts were calculated in non-overlapping 70ms windows. The hand kinematics
(obtained by recording the 2D position of the manipulandum) were averaged within each window,
to produce an aligned representation.
3.1
Evaluation protocol
In each of the data sets, we selected a segment of the recording to train all the decoders, and a
subsequent segment to test the decoding accuracy. Tuning of parameters (the kernel parameters
of the SVM or the mass and viscosity of the spring model) was done on a held-out portion of the
training segment. We built the firing rate history matrix by concatenating for each time step the firing
rates for 15 bins. For instance, for monkey CL, continuous tracking, the dimension of the neural
signal representation was 825 (55 channels ? 15 history bins). This firing rates were then normalized
so that all values would be within [-1,1]. Basic statistics of the data used in the experiments are given
in Table 1.
We considered three evaluation criteria:
Correlation coefficients (CC): between the estimated and true value for each of the two spatial
coordinates over the entire trajectory:
P
??t )
(xt ? x
?t )(?
xt ? x
.
CC = qP t
P
2
??t )2
(x
?
x
?
)
(?
x
?
x
t
t
t
t
t
Mean absolute error (MAE): in the estimated position versus the ground truth: MAE =
PN
1
?
t=1 kX(t) ? X(t)k.
N
Power spectrum reconstruction : One of the objectives of a practical decoding algorithm, especially in the context of assistive technology, is to produce movement that appears ?natural?. As a criterion for evaluating the degree of ?naturalness? we use the similarity between
power spectrum densities of the true movement and the reconstructed one. Specifically, we
calculated the L1 norm between the energy distributions over normalized angular frequencies, taken in the log domain (see Figure 2 for illustration).
3.2
Results
The reported results for SVM were obtained with quadratic kernel, k(x, y) = (x ? y + 1)2 ; the tradeoff term c was fixed to 100, and the insensitivity parameter was set to 5 for the spring coefficient
and 2 for direct position decoding. The number of support vectors was between 20% and 65% of
the training set size.
Decoder
Linear-kinematics
Linear-SBM
SVM-kinematics
SVM-SBM
CL/sequential
MAE CCx CCy
5.3
0.69 0.79
5.7
0.64 0.74
4.45 0.80 0.85
4.91 0.76 0.81
LA/continuous
MAE CCx CCy
5.03
0.5
0.75
5.26 0.46 0.72
4.44 0.60 0.82
4.69 0.55 0.80
CL/continuous
MAE CCx CCy
6.66 0.80 0.83
6.82 0.77 0.81
3.82 0.86 0.86
4.05 0.83 0.84
Table 2: Summary of results on the three datasets. MAE is given in cm, over workspace of roughly
30?30 cm.
Table 2 summarizes the MAE and CC measured on the test segment for each method. One observation is that SVM tends to outperform the linear filter, in line with previous observations [12, 15].
We believe that this is due to inherent nonlinearity in the underlying relationship, which is better
captured by the SVM. Moreover, it is apparent that the decoding accuracy of the SBM is on par
with that of the conventional kinematic decoding (the observed differences were not significant at
the 0.05 level, measured over the per-bin position errors).
Power/freq. unit, db/sample
40
20
Figure 2: Example of power spectrum densities for true hand trajectory (dotted black), reconstruction with
SVM-kinematics (dashed blue) and reconstruction with SVM-SBM (solid
red). Estimated using Burg?s algorithm
(pburg in Matlab, order 4). Data from
x coordinate, LA-continuous.
0
?20
?40
?60
0
0.2
0.4
0.6
0.8
Normalized frequency, ? rad/sample
1
Figure 3: A 1.5 second path segment, true (circles) and reconstructed (squares). Left: SVM on
kinematics, right: SVM with SBM. Markers show position averaged in each 70ms bin. Note the
ragged form of the SVM-kinematics trajectory.
Results in Table 2, however, tell only a part of the story. Figure 3 shows, for a segment of 4.2 sec,
a typical example of the movement reconstructed with SVM on kinematics versus SVM on spring
coefficients. The accuracy in terms of deviation from ground truth is similar, however the estimate
produced by the direct kinematic decoding is significantly more ?ragged?. Such discrepancy is not
necessarily reflected in the standard measures of accuracy such as CC or MAE. Quantitavely, this
can be assessed by calculating the L1 norm between the power spectrum densities of the true and
reconstructed hand trajectories. The estmated values of this quantity in our experiments are shown
in Table 3. These results reflect the relationship shown in Figure 2 (a typical case).
Table 3: Estimated L1 norm between power spectrum density of true and reconstructed trajectories.
Decoder
CL-sequential
LA-continuous
CL-continuous
x
y
x
y
x
y
Linear-kinematics 147.41 154.80 199.24 206.61 49.68 44.37
Linear-SBM
71.58
68.24
72.99
80.37 35.68 43.72
SVM-kinematics
143.78 151.35 188.65 196.14 33.96 28.44
SVM-SBM
51.45
52.31
53.05
66.20 20.83 21.15
4
Discussion
The spring-based model proposed in this paper represents a first attempt to directly incorporate
realistic physical constraints into a neural decoding model. Our experiments illustrate that the coefficients of an idealized physical system can be decoded from motor cortical firing rates, without
sttistically significant loss of decoding accuracy compared to more standard direct decoding of kinematics. An advantage of such an approach is that the physical properties of the system damp high
frequency motions resulting in decoded movements that inherently have the properties of natural
movement, with no ad-hoc smoothing.
Future work should consider more sophisticated physical models such as a simulated robotic arm
and a biophysically motivated musculoskeletal system. With the current state of the art in neural
recording and decoding, recovering the parameters of such models may be challenging. In contrast,
the approach presented here ?summarizes? the effect of a more complicated system with just a few
idealized muscle-like elements.
Additional experiments are also warranted. In particular using a robotic feedback device we can
simulate the physical system of springs presented here such that the monkeys control a device with
the properties of our model. We hypothesize that the accuracy of decoding spring coefficients from
motor cortical activity in this condition will improve. This would suggest that matching the decoding
model to the physical system being controlled will improve decoding accuracy.
Finally, the real test of physically-based models will come in human NMP experiments. We plan
to test human cursor control with kinematic and physically-based decoders. We hypothesize that
the dynamics of the physically-based model will make it easier to control accurately (and perhaps
provide a more satisfying experience for the user). This could be a first step toward the neural control
of mechanical actuators in the physical world.
Acknowledgments
This work is partially supported by NIH-NINDS R01 NS 50867-01 as part of the NSF/NIH Collaborative Research in Computational Neuroscience Program and by the Office of Naval Research
(award N0014-04-1-082). We also thank the European Neurobotics Program FP6-IST-001917. We
thank Matthew Fellows and John Donoghue for providing data, and Reza Shadmehr for helpful
conversations.
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2,260 | 3,051 | Large Margin Hidden Markov Models for
Automatic Speech Recognition
Fei Sha
Computer Science Division
University of California
Berkeley, CA 94720-1776
[email protected]
Lawrence K. Saul
Department of Computer Science and Engineering
University of California (San Diego)
La Jolla, CA 92093-0404
[email protected]
Abstract
We study the problem of parameter estimation in continuous density hidden
Markov models (CD-HMMs) for automatic speech recognition (ASR). As in support vector machines, we propose a learning algorithm based on the goal of margin maximization. Unlike earlier work on max-margin Markov networks, our
approach is specifically geared to the modeling of real-valued observations (such
as acoustic feature vectors) using Gaussian mixture models. Unlike previous discriminative frameworks for ASR, such as maximum mutual information and minimum classification error, our framework leads to a convex optimization, without
any spurious local minima. The objective function for large margin training of
CD-HMMs is defined over a parameter space of positive semidefinite matrices.
Its optimization can be performed efficiently with simple gradient-based methods that scale well to large problems. We obtain competitive results for phonetic
recognition on the TIMIT speech corpus.
1 Introduction
As a result of many years of widespread use, continuous density hidden Markov models (CDHMMs) are very well matched to current front and back ends for automatic speech recognition
(ASR) [21]. Typical front ends compute real-valued feature vectors from the short-time power spectra of speech signals. The distributions of these acoustic feature vectors are modeled by Gaussian
mixture models (GMMs), which in turn appear as observation models in CD-HMMs. Viterbi decoding is used to solve the problem of sequential classification in ASR?namely, the mapping of
sequences of acoustic feature vectors to sequences of phonemes and/or words, which are modeled
by state transitions in CD-HMMs.
The simplest method for parameter estimation in CD-HMMs is the Expectation-Maximization (EM)
algorithm. The EM algorithm is based on maximizing the joint likelihood of observed feature vectors
and label sequences. It is widely used due to its simplicity and scalability to large data sets, which
are common in ASR. A weakness of this approach, however, is that the model parameters of CDHMMs are not optimized for sequential classification: in general, maximizing the joint likelihood
does not minimize the phoneme or word error rates, which are more relevant metrics for ASR.
Noting this weakness, many researchers in ASR have studied alternative frameworks for parameter estimation based on conditional maximum likelihood [11], minimum classification error [4] and
maximum mutual information [20]. The learning algorithms in these frameworks optimize discriminative criteria that more closely track actual error rates, as opposed to the EM algorithm for maximum likelihood estimation. These algorithms do not enjoy the simple update rules and relatively fast
convergence of EM, but carefully and skillfully implemented, they lead to lower error rates [13, 20].
Recently, in a new approach to discriminative acoustic modeling, we proposed the use of ?large
margin GMMs? for multiway classification [15]. Inspired by support vector machines (SVMs), the
learning algorithm in large margin GMMs is designed to maximize the distance between labeled examples and the decision boundaries that separate different classes [19]. Under mild assumptions, the
required optimization is convex, without any spurious local minima. In contrast to SVMs, however,
large margin GMMs are very naturally suited to problems in multiway (as opposed to binary) classification; also, they do not require the kernel trick for nonlinear decision boundaries. We showed
how to train large margin GMMs as segment-based phonetic classifiers, yielding significantly lower
error rates than maximum likelihood GMMs [15]. The integrated large margin training of GMMs
and transition probabilities in CD-HMMs, however, was left as an open problem.
We address that problem in this paper, showing how to train large margin CD-HMMs in the more
general setting of sequential (as opposed to multiway) classification. In this setting, the GMMs
appear as acoustic models whose likelihoods are integrated over time by Viterbi decoding. Experimentally, we find that large margin training of HMMs for sequential classification leads to significant
improvement beyond the frame-based and segment-based discriminative training in [15].
Our framework for large margin training of CD-HMMs builds on ideas from many previous studies
in machine learning and ASR. It has similar motivation as recent frameworks for sequential classification in the machine learning community [1, 6, 17], but differs in its focus on the real-valued acoustic feature representations used in ASR. It has similar motivation as other discriminative paradigms
in ASR [3, 4, 5, 11, 13, 20], but differs in its goal of margin maximization and its formulation
of the learning problem as a convex optimization over positive semidefinite matrices. The recent
margin-based approach of [10] is closest in terms of its goals, but entirely different in its mechanics;
moreover, its learning is limited to the mean parameters in GMMs.
2 Large margin GMMs for multiway classification
Before developing large margin HMMs for ASR, we briefly review large margin GMMs for multiway classification [15]. The problem of multiway classification is to map inputs x ? ?d to labels
y ? {1, 2, . . . , C}, where C is the number of classes. Large margin GMMs are trained from a set of
labeled examples {(xn , yn )}N
n=1 . They have many parallels to SVMs, including the goal of margin
maximization and the use of a convex surrogate to the zero-one loss [19]. Unlike SVMs, where
classes are modeled by half-spaces, in large margin GMMs the classes are modeled by collections
of ellipsoids. For this reason, they are more naturally suited to problems in multiway as opposed to
binary classification. Sections 2.1?2.3 review the basic framework for large margin GMMs: first,
the simplest setting in which each class is modeled by a single ellipsoid; second, the formulation
of the learning problem as a convex optimization; third, the general setting in which each class is
modeled by two or more ellipsoids. Section 2.4 presents results on handwritten digit recognition.
2.1 Parameterization of the decision rule
The simplest large margin GMMs model each class by a single ellipsoid in the input space. The
ellipsoid for class c is parameterized by a centroid vector ?c ? ?d and a positive semidefinite
matrix ?c ? ?d?d that determines its orientation. Also associated with each class is a nonnegative
scalar offset ?c ? 0. The decision rule labels an example x ? ?d by the class whose centroid yields
the smallest Mahalanobis distance:
y = argmin (x??c )T ?c (x??c ) + ?c .
(1)
c
The decision rule in eq. (1) is merely an alternative way of parameterizing the maximum a posterior
(MAP) label in traditional GMMs with mean vectors ?c , covariance matrices ??1
c , and prior class
probabilities pc , given by y = argminc { pc N (?c , ??1
)
}.
c
The argument on the right hand side of the decision rule in eq. (1) is nonlinear in the ellipsoid
parameters ?c and ?c . As shown in [15], however, a useful reparameterization yields a simpler
expression. For each class c, the reparameterization collects the parameters {?c , ?c , ?c } in a single
enlarged matrix ?c ? ?(d+1)?(d+1) :
?c
??c ?c
?c =
.
(2)
??T
?T
c ?c
c ?c ?c + ?c
Note that ?c is positive semidefinite. Furthermore, if ?c is strictly positive definite, the parameters
{?c , ?c , ?c } can be uniquely recovered from ?c . With this reparameterization, the decision rule in
eq. (1) simplifies to:
T
x
y = argmin z ?c z
where z =
.
(3)
1
c
The argument on the right hand side of the decision rule in eq. (3) is linear in the parameters ?c . In
what follows, we will adopt the representation in eq. (3), implicitly constructing the ?augmented?
vector z for each input vector x. Note that eq. (3) still yields nonlinear (piecewise quadratic) decision boundaries in the vector z.
2.2 Margin maximization
Analogous to learning in SVMs, we find the parameters {?c } that minimize the empirical risk on
the training data?i.e., parameters that not only classify the training data correctly, but also place
the decision boundaries as far away as possible. The margin of a labeled example is defined as its
distance to the nearest decision boundary. If possible, each labeled example is constrained to lie at
least one unit distance away from the decision boundary to each competing class:
?c 6= yn , z T
n (?c ? ?yn ) z n ? 1.
(4)
Fig. 1 illustrates this idea. Note that in the ?realizable? setting where these constraints can be
simultaneously satisfied, they do not uniquely determine the parameters {?c }, which can be scaled
to yield arbitrarily large margins. Therefore, as in SVMs, we propose a convex optimization that
selects the ?smallest? parameters that satisfy the large margin constraints in eq. (4). In this case, the
optimization is an instance of semidefinite programming [18]:
P
min
c trace(?c )
(5)
s.t. 1 + z T
?c 6= yn , n = 1, 2, . . . , N
n (?yn ? ?c )z n ? 0,
?c ? 0, c = 1, 2, . . . , C
Note that the trace of the matrix ?c appears in the above objective function, as opposed to the trace
of the matrix ?c , as defined in eq. (2); minimizing the former imposes the scale regularization only
on the inverse covariance matrices of the GMM, while the latter would improperly regularize the
mean vectors as well. The constraints ?c ? 0 restrict the matrices to be positive semidefinite.
The objective function must be modified for training data that lead to infeasible constraints in eq. (5).
As in SVMs, we introduce nonnegative slack variables ?nc to monitor the amount by which the
margin constraints in eq. (4) are violated [15]. The objective function in this setting balances the
margin violations versus the scale regularization:
P
P
min
nc ?nc + ?
c trace(?c )
s.t. 1 + z T
(?
?
?
y
c )z n ? ?nc ,
n
n
(6)
?nc ? 0, ?c 6= yn , n = 1, 2, . . . , N
?c ? 0, c = 1, 2, . . . , C
where the balancing hyperparameter ? > 0 is set by some form of cross-validation. This optimization
is also an instance of semidefinite programming.
2.3 Softmax margin maximization for multiple mixture components
Lastly we review the extension to mixture modeling where each class is represented by multiple
ellipsoids [15]. Let ?cm denote the matrix for the mth ellipsoid (or mixture component) in class
c. We imagine that each example xn has not only a class label yn , but also a mixture component
label mn . Such labels are not provided a priori in the training data, but we can generate ?proxy?
labels by fitting GMMs to the examples in each class by maximum likelihood estimation, then for
each example, computing the mixture component with the highest posterior probability.
In the setting where each class is represented by multiple ellipsoids, the goal of learning is to ensure
that each example is closer to its ?target? ellipsoid than the ellipsoids from all other classes. Specifically, for a labeled example (xn , yn , mn ), the constraint in eq. (4) is replaced by the M constraints:
?c 6= yn , ?m, z T
n (?cm ? ?yn mn )z n ? 1,
(7)
a
d
e
c
i
s
i
o
n
b
o
u
n
d
a
r
y
r
g
i
n
m
mixture
1
2
4
8
Figure 1: Decision boundary in a large margin GMM:
labeled examples lie at least one unit of distance away.
EM
4.2%
3.4%
3.0%
3.3%
margin
1.4%
1.4%
1.2%
1.5%
Table 1: Test error rates on MNIST
digit recognition: maximum likelihood versus large margin GMMs.
where M is the number of mixture components (assumed, for simplicity, to be the same for each
class). We fold these multiple
constraints into a single one by appealing to the ?softmax? inequalP
ity: minm am ? ? log m e?am . Specifically, using the inequality to derive a lower bound on
minm z T
n ?cm z n , we replace the M constraints in eq. (7) by the stricter constraint:
X
T
(8)
?c 6= yn , ? log
e?zn ?cm zn ? z T
n ?yn mn z n ? 1.
m
We will use a similar technique in section 3 to handle the exponentially many constraints that arise
in sequential classification. Note that the inequality in eq. (8) implies the inequality of eq. (7) but not
vice versa. Also, though nonlinear in the matrices {?cm }, the constraint in eq. (8) is still convex.
The objective function in eq. (6) extends straightforwardly to this setting. It balances a regularizing
term that sums over ellipsoids versus a penalty term that sums over slack variables, one for each
constraint in eq. (8). The optimization is given by:
P
P
min
)
nc ?nc + ?
cm trace(?
Pcm ?zT ?cm zn
T
s.t. 1 + z n ?yn mn z n + log m e n
? ?nc ,
(9)
?nc ? 0, ?c 6= yn , n = 1, 2, . . . , N
?cm ? 0, c = 1, 2, . . . , C, m = 1, 2, . . . , M
This optimization is not an instance of semidefinite programming, but it is convex. We discuss how
to perform the optimization efficiently for large data sets in appendix A.
2.4 Handwritten digit recognition
We trained large margin GMMs for multiway classification of MNIST handwritten digits [8]. The
MNIST data set has 60000 training examples and 10000 test examples. Table 1 shows that the large
margin GMMs yielded significantly lower test error rates than GMMs trained by maximum likelihood estimation. Our best results are comparable to the best SVM results (1.0-1.4%) on deskewed
images [8] that do not make use of prior knowledge. For our best model, with four mixture components per digit class, the core training optimization over all training examples took five minutes on a
PC. (Multiple runs of this optimization on smaller validation sets, however, were also required to set
two hyperparameters: the regularizer for model complexity, and the termination criterion for early
stopping.)
3 Large margin HMMs for sequential classification
In this section, we extend the framework in the previous section from multiway classification to
sequential classification. Particularly, we have in mind the application to ASR, where GMMs are
used to parameterize the emission densities of CD-HMMs. Strictly speaking, the GMMs in our
framework cannot be interpreted as emission densities because their parameters are not constrained
to represent normalized distributions. Such an interpretation, however, is not necessary for their use
as discriminative models. In sequential classification by CD-HMMs, the goal is to infer the correct
hidden state sequence y = [y1 , y2 , . . . , yT ] given the observation sequence X = [x1 , x2 , . . . , xT ].
In the application to ASR, the hidden states correspond to phoneme labels, and the observations are
acoustic feature vectors. Note that if an observation sequence has length T and each label can belong
to C classes, then the number of incorrect state sequences grows as O(C T ). This combinatorial
explosion presents the main challenge for large margin methods in sequential classification: how to
separate the correct hidden state sequence from the exponentially large number of incorrect ones.
The section is organized as follows. Section 3.1 explains the way that margins are computed for sequential classification. Section 3.2 describes our algorithm for large margin training of CD-HMMs.
Details are given only for the simple case where the observations in each hidden state are modeled
by a single ellipsoid. The extension to multiple mixture components closely follows the approach
in section 2.3 and can be found in [14, 16]. Margin-based learning of transition probabilities is
likewise straightforward but omitted for brevity. Both these extensions were implemented, however,
for the experiments on phonetic recognition in section 3.3.
3.1 Margin constraints for sequential classification
We start by defining a discriminant function over state (label) sequences of the CD-HMM. Let
a(i, j) denote the transition probabilities of the CD-HMM, and let ?s denote the ellipsoid parameters of state s. The discriminant function D(X, s) computes the score of the state sequence
s = [s1 , s2 , . . . , sT ] on an observation sequence X = [x1 , x2 , . . . , xT ] as:
D(X, s) =
X
log a(st?1 , st ) ?
t
T
X
zT
t ?st z t .
(10)
t=1
This score has the same form as the log-probability log P (X, s) in a CD-HMM with Gaussian emission densities. The first term accumulates the log-transition probabilities along the state sequence,
while the second term accumulates ?acoustic scores? computed as the Mahalanobis distances to
each state?s centroid. In the setting where each state is modeled by multiple mixture components,
the acoustic scores from individual Mahalanobis distances are replaced with ?softmax? distances of
P
?z T
t ?st m z t , as described in section 2.3 and [14, 16].
the form log M
m=1 e
We introduce margin constraints in terms of the above discriminant function. Let H(s, y) denote
the Hamming distance (i.e., the number of mismatched labels) between an arbitrary state sequence s
and the target state sequence y. Earlier, in section 2 on multiway classification, we constrained each
labeled example to lie at least one unit distance from the decision boundary to each competing class;
see eq. (4). Here, by extension, we constrain the score of each target sequence to exceed that of each
competing sequence by an amount equal to or greater than the Hamming distance:
?s 6= y,
D(X, y) ? D(X, s) ? H(s, y)
(11)
Intuitively, eq. (11) requires that the (log-likelihood) gap between the score of an incorrect sequence s and the target sequence y should grow in proportion to the number of individual label
errors. The appropriateness of such proportional constraints for sequential classification was first
noted by [17].
3.2 Softmax margin maximization for sequential classification
The challenge of large margin sequence classification lies in the exponentially large number of
constraints, one for each incorrect sequence s, embodied by eq. (11). We will use the same softmax
inequality, previously introduced in section 2.3, to fold these multiple constraints into one, thus
considerably simplifying the optimization required for parameter estimation. We first rewrite the
constraint in eq. (11) as:
?D(X, y) + max{H(s, y) + D(X, s)} ? 0
s6=y
(12)
We obtain a more manageable constraint by substituting a softmax upper bound for the max term
and requiring that the inequality still hold:
X
?D(X, y) + log
eH(s,y)+D(X,s) ? 0
(13)
s6=y
Note that eq. (13) implies eq. (12) but not vice versa. As in the setting for multiway classification,
the objective function for sequential classification balances two terms: one regularizing the scale of
the GMM parameters, the other penalizing margin violations. Denoting the training sequences by
{X n , y n }N
n=1 and the slack variables (one for each training sequence) by ?n ? 0, we obtain the
following convex optimization:
P
P
min
n ?n + ?
cm trace(?
P cm )
s.t. ?D(X n , y n ) + log s6=yn eH(s,yn )+D(X n ,s) ? ?n ,
(14)
?n ? 0, n = 1, 2, . . . , N
?cm ? 0, c = 1, 2, . . . , C, m = 1, 2, . . . , M
It is worth emphasizing several crucial differences between this optimization and previous ones [4,
11, 20] for discriminative training of CD-HMMs for ASR. First, the softmax large margin constraint
in eq. (13) is a differentiable function of the model parameters, as opposed to the ?hard? maximum
in eq. (12) and the number of classification errors in the MCE training criteria [4]. The constraint and
its gradients with respect to GMM parameters ?cm and transition parameters a(?, ?) can be computed
efficiently using dynamic programming, by a variant of the standard forward-backward procedure in
HMMs [14]. Second, due to the reparameterization in eq. (2), the discriminant function D(X n , y n )
and the softmax function are convex in the model parameters. Therefore, the optimization eq. (14)
can be cast as convex optimization, avoiding spurious local minima [14]. Third, the optimization not
only increases the log-likelihood gap between correct and incorrect state sequences, but also drives
the gap to grow in proportion to the number of individually incorrect labels (which we believe leads
to more robust generalization). Finally, compared to the large margin framework in [17], the softmax
handling of exponentially large number of margin constraints makes it possible to train on larger data
sets. We discuss how to perform the optimization efficiently in appendix A.
3.3 Phoneme recognition
We used the TIMIT speech corpus [7, 9, 12] to perform experiments in phonetic recognition. We
followed standard practices in preparing the training, development, and test data. Our signal processing front-end computed 39-dimensional acoustic feature vectors from 13 mel-frequency cepstral
coefficients and their first and second temporal derivatives. In total, the training utterances gave rise
to roughly 1.2 million frames, all of which were used in training.
We trained baseline maximum likelihood recognizers and two different types of large margin recognizers. The large margin recognizers in the first group were ?low-cost? discriminative CD-HMMs
whose GMMs were merely trained for frame-based classification. In particular, these GMMs were
estimated by solving the optimization in eq. (8), then substituted into first-order CD-HMMs for sequence decoding. The large margin recognizers in the second group were fully trained for sequential
classification. In particular, their CD-HMMs were estimated by solving the optimization in eq. (14),
generalized to multiple mixture components and adaptive transition parameters [14, 16]. In all the
recognizers, the acoustic feature vectors were labeled by 48 phonetic classes, each represented by
one state in a first-order CD-HMM.
For each recognizer, we compared the phonetic state sequences obtained by Viterbi decoding to the
?ground-truth? phonetic transcriptions provided by the TIMIT corpus. For the purpose of computing error rates, we followed standard conventions in mapping the 48 phonetic state labels down to
39 broader phone categories. We computed two different types of phone error rates, one based on
Hamming distance, the other based on edit distance. The former was computed simply from the
percentage of mismatches at the level of individual frames. The latter was computed by aligning the
Viterbi and ground truth transcriptions using dynamic programming [9] and summing the substitution, deletion, and insertion error rates from the alignment process. The ?frame-based? phone error
rate computed from Hamming distances is more closely tracked by our objective function for large
margin training, while the ?string-based? phone error rate computed from edit distances provides a
more relevant metric for ASR.
Tables 2 and 3 show the results of our experiments. For both types of error rates, and across all
model sizes, the best performance was consistently obtained by large margin CD-HMMs trained
for sequential classification. Moreover, among the two different types of large margin recognizers,
utterance-based training generally yielded significant improvement over frame-based training.
Discriminative learning of CD-HMMs is an active research area in ASR. Two types of algorithms
have been widely used: maximum mutual information (MMI) [20] and minimum classification er-
mixture
(per state)
1
2
4
8
baseline
(EM)
45%
45%
42%
41%
margin
(frame)
37%
36%
35%
34%
margin
(utterance)
30%
29%
28%
27%
Table 2: Frame-based phone error rates, from
Hamming distance, of different recognizers.
See text for details.
mixture
(per state)
1
2
4
8
baseline
(EM)
40.1%
36.5%
34.7%
32.7%
margin
(frame)
36.3%
33.5%
32.6%
31.0%
margin
(utterance)
31.2%
30.8%
29.8%
28.2%
Table 3: String-based phone error rates, from
edit distance, of different recognizers. See text
for details.
ror [4]. In [16], we compare the large margin training proposed in this paper to both MMI and MCE
systems for phoneme recognition trained on the exact same acoustic features. There we find that the
large margin approach leads to lower error rates, owing perhaps to the absence of local minima in
the objective function and/or the use of margin constraints based on Hamming distances.
4 Discussion
Discriminative learning of sequential models is an active area of research in both ASR [10, 13, 20]
and machine learning [1, 6, 17]. This paper makes contributions to lines of work in both communities. First, in distinction to previous work in ASR, we have proposed a convex, margin-based
cost function that penalizes incorrect decodings in proportion to their Hamming distance from the
desired transcription. The use of the Hamming distance in this context is a crucial insight from the
work of [17] in the machine learning community, and it differs profoundly from merely penalizing
the log-likelihood gap between incorrect and correct transcriptions, as commonly done in ASR.
Second, in distinction to previous work in machine learning, we have proposed a framework for sequential classification that naturally integrates with the infrastructure of modern speech recognizers.
Using the softmax function, we have also proposed a novel way to monitor the exponentially many
margin constraints that arise in sequential classification. For real-valued observation sequences, we
have shown how to train large margin HMMs via convex optimizations over their parameter space of
positive semidefinite matrices. Finally, we have demonstrated that these learning algorithms lead to
improved sequential classification on data sets with over one million training examples (i.e., phonetically labeled frames of speech). In ongoing work, we are applying our approach to large vocabulary
ASR and other tasks such as speaker identification and visual object recognition.
A
Solver
The optimizations in eqs. (5), (6), (9) and (14) are convex: specifically, in terms of the matrices
that parameterize large margin GMMs and HMMs, the objective functions are linear, while the constraints define convex sets. Despite being convex, however, these optimizations cannot be managed
by off-the-shelf numerical optimization solvers or generic interior point methods for problems as
large as the ones in this paper. We devised our own special-purpose solver for these purposes.
For simplicity, we describe our solver for the optimization of eq. (6), noting that it is easily extended
to eqs. (9) and (14). To begin, we eliminate the slack variables and rewrite the objective function in
terms of the hinge loss function: hinge(z) = max(0, z). This yields the objective function:
X
X
L=
hinge 1 + z T
trace(?c ),
(15)
n (?yn ??c )z n + ?
n,c6=yn
c
which is convex in terms of the positive semidefinite matrices ?c . We minimize L using a projected
subgradient method [2], taking steps along the subgradient of L, then projecting the matrices {?c }
back onto the set of positive semidefinite matrices after each update. This method is guaranteed to
converge to the global minimum, though it typically converges very slowly. For faster convergence,
we precede this method with an unconstrained conjugate gradient optimization in the square-root
matrices {?c }, where ?c = ?c ?T
c . The latter optimization is not convex, but in practice it rapidly
converges to an excellent starting point for the projected subgradient method.
Acknowledgment
This work was supported by the National Science Foundation under grant Number 0238323. We
thank F. Pereira, K. Crammer, and S. Roweis for useful discussions and correspondence. Part of this
work was conducted while both authors were affiliated with the University of Pennsylvania.
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[17] B. Taskar, C. Guestrin, and D. Koller. Max-margin markov networks. In S. Thrun, L. Saul, and
B. Sch?olkopf, editors, Advances in Neural Information Processing Systems (NIPS 16). MIT Press, Cambridge, MA, 2004.
[18] L. Vandenberghe and S. P. Boyd. Semidefinite programming. SIAM Review, 38(1):49?95, March 1996.
[19] V. Vapnik. Statistical Learning Theory. Wiley, N.Y., 1998.
[20] P. C. Woodland and D. Povey. Large scale discriminative training of hidden Markov models for speech
recognition. Computer Speech and Language, 16:25?47, 2002.
[21] S. J. Young. Acoustic modelling for large vocabulary continuous speech recognition. In K. Ponting,
editor, Computational Models of Speech Pattern Processing, pages 18?39. Springer, 1999.
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2,261 | 3,052 | Logarithmic Online Regret Bounds for Undiscounted
Reinforcement Learning
Peter Auer
Ronald Ortner
University of Leoben, Franz-Josef-Strasse 18,
8700 Leoben, Austria
{auer,rortner}@unileoben.ac.at
Abstract
We present a learning algorithm for undiscounted reinforcement learning. Our
interest lies in bounds for the algorithm?s online performance after some finite
number of steps. In the spirit of similar methods already successfully applied
for the exploration-exploitation tradeoff in multi-armed bandit problems, we use
upper confidence bounds to show that our UCRL algorithm achieves logarithmic
online regret in the number of steps taken with respect to an optimal policy.
1
1.1
Introduction
Preliminaries
Definition 1. A Markov decision process (MDP) M on a finite set of states S with a finite set
of actions A available in each state ? S consists of (i) an initial distribution ?0 over S, (ii) the
transition probabilities p(s, a, s0 ) that specify the probability of reaching state s0 when choosing
action a in state s, and (iii) the payoff distributions with mean r(s, a) and support in [0, 1] that
specify the random reward for choosing action a in state s.
A policy on an MDP M is a mapping ? : S ? A. We will mainly consider unichain MDPs, in
which under any policy any state can be reached (after a finite number of transitions) from any state.
For a policy ? let ?? be the stationary distribution induced by ? on M .1 The average reward of ?
then is defined as
X
?(M, ?) :=
?? (s)r(s, ?(s)).
(1)
s?S
A policy ? ? is called optimal on M , if ?(M, ?) ? ?(M, ? ? ) =: ?? (M ) =: ?? for all policies ?.
Our measure for the quality of a learning algorithm is the total regret after some finite number of
steps. When a learning algorithm A executes action at in state st at step t obtaining reward rt , then
PT ?1
RT := t=0 rt ? T ?? denotes the total regret of A after T steps. The total regret RT? with respect
to an ?-optimal policy (i.e. a policy whose return differs from ?? by at most ?) is defined accordingly.
1.2
Discussion
We would like to compare this approach with the various PAC-like bounds in the literature as given
for the E3 -algorithm of Kearns, Singh [1] and the R-Max algorithm of Brafman, Tennenholtz [2] (cf.
also [3]). Both take as inputs (among others) a confidence parameter ? and an accuracy parameter
1
Every policy ? induces a Markov chain C? on M . If C? is ergodic with transition matrix P , then there
exists a unique invariant andP
strictly positive distribution ?? , such that independent of ?0 one has ?n =
j
?0 P?n ? ?? , where P?n = n1 n
j=1 P . If C? is not ergodic, ?? will depend on ?0 .
?. The algorithms then are shown to yield ?-optimal return after time polynomial in 1? , 1? (among
others) with probability 1??. In contrast, our algorithm has no such input parameters and converges
to an optimal policy with expected logarithmic online regret in the number of steps taken.
Obviously, by using a decreasing sequence ?t , online regret bounds for E3 and R-Max can be
achieved. However, it is not clear whether such a procedure can give logarithmic online regret
bounds. We rather conjecture that these bounds either will be not logarithmic in the total number of
steps (if ?t decreases quickly) or that the dependency on the parameters of the MDP ? in particular
on the distance between the reward of the best and a second best policy ? won?t be polynomial (if ?t
decreases slowly).
Moreover, although our UCRL algorithm shares the ?optimism under uncertainty? maxim with RMax, our mechanism for the exploitation-exploration tradeoff is implicit, while E3 and R-Max have
to distinguish between ?known? and ?unknown? states explicitly. Finally, in their original form both
E3 and R-Max need a policy?s ?-return mixing time T? as input parameter. The knowledge of this
parameter then is eliminated by calculating the ?-optimal policy for T? = 1, 2, . . ., so that sooner
or later the correct ?-return mixing time is reached. This is sufficient to obtain polynomial PACbounds, but seems to be intricate for practical purposes. Moreover, as noted in [2], at some time step
the assumed T? may be exponential in the true T? , which makes policy computation exponential in
T? . Unlike that, we need our mixing time parameter only in the analysis. This makes our algorithm
rather simple and intuitive.
Recently, more refined performance measures such as the sample complexity of exploration [3] were
introduced. Strehl and Littman [4] showed that in the discounted setting, efficiency in the sample
complexity implies efficiency in the average loss. However, average loss is defined in respect to
the actually visited states, so that small average loss does not guarantee small total regret, which is
defined in respect to the states visited by an optimal policy. For this average loss polylogarithmic online bounds were shown for for the MBIE algorithm [4], while more recently logarithmic bounds for
delayed Q-learning were given in [5]. However, discounted reinforcement learning is a bit simpler
than undiscounted reinforcement learning, as depending on the discount factor only a finite number
of steps is relevant. This makes discounted reinforcement learning similar to the setting with trials
of constant length from a fixed initial state [6]. For this case logarithmic online regret bounds in the
number of trials have already been given in [7].
Since we measure performance during exploration, the exploration vs. exploitation dilemma becomes an important issue. In the multi-armed bandit problem, similar exploration-exploitation
tradeoffs were handled with upper confidence bounds for the expected immediate returns [8, 9].
This approach has been shown to allow good performance during the learning phase, while still converging fast to a nearly optimal policy. Our UCRL algorithm takes into account the state structure
of the MDP, but is still based on upper confidence bounds for the expected return of a policy. Upper confidence bounds have been applied to reinforcement learning in various places and different
contexts, e.g. interval estimation [10, 11], action elimination [12], or PAC-learning [6]. Our UCRL
algorithm is similar to Strehl, Littman?s MBIE algorithm [10, 4], but our confidence bounds are
different, and we are interested in the undiscounted case.
Another paper with a similar approach is Burnetas, Katehakis [13]. The basic idea of their rather
complex index policies is to choose the action with maximal return in some specified confidence
region of the MDP?s probability distributions. The online-regret of their algorithm is asymptotically
logarithmic in the number of steps, which is best possible. Our UCRL algorithm is simpler and
achieves logarithmic regret not only asymptotically but uniformly over time. Moreover, unlike in
the approach of [13], knowledge about the MDP?s underlying state structure is not needed.
More recently, online reinforcement learning with changing rewards chosen by an adversary was
considered under the presumption that the learner has full knowledge
of the transition probabilities
?
[14]. The given algorithm achieves best possible regret of O( T ) after T steps.
In the subsequent Sections 2 and 3 we introduce our UCRL algorithm and show that its expected
online regret in unichain MDPs is O(log T ) after T steps. In Section 4 we consider problems that
arise when the underlying MDP is not unichain.
2
The UCRL Algorithm
To select good policies, we keep track of estimates for the average rewards and the transition probabilities. For each step t let
Nt (s, a)
Rt (s, a)
= |{0 ? ? < t : s? = s, a? = a}|,
X
=
r? ,
0?? <t:
s? =s, a? =a
0
Pt (s, a, s )
= |{0 ? ? < t : s? = s, a? = a, s? +1 = s0 }|,
be the number of steps when action a was chosen in state s, the sum of rewards obtained when
choosing this action, and the number of times the transition was to state s0 , respectively. From these
numbers we immediately get estimates for the average rewards and transition probabilities,
Rt (s, a)
,
Nt (s, a)
Pt (s, a, s0 )
p?t (s, a, s0 ) :=
,
Nt (s, a)
r?t (s, a) :=
provided that the number of visits in (s, a), Nt (s, a) > 0. In general, these estimates will deviate
from the respective true values. However, together with appropriate confidence intervals they may
be used to define a set Mt of plausible MDPs. Our algorithm then chooses an optimal policy ?
?t for
?
? ?
?
an MDP Mt with maximal average reward ??t := ? (Mt ) among the MDPs in Mt . That is,
?
?t
?t
M
:= arg max{?(M, ?) : M ? Mt },
?
and
:= arg max {?(M, ?
?t )}.
M ?Mt
More precisely, we want Mt to be a set of plausible MDPs in the sense that
P {?? > ???t } < t??
(2)
for some ? > 2. Essentially, condition (2) means that it is unlikely that the true MDP M is not
in Mt . Actually, Mt is defined to contain exactly those unichain MDPs M 0 whose transition
probabilities p0 (?, ?, ?) and rewards r0 (?, ?) satisfy for all states s, s0 and actions a
q
? |S||A|)
, and
(3)
r0 (s, a) ? r?t (s, a) + log(2t
2Nt (s,a)
q
log(4t? |S|2 |A|)
|p0 (s, a, s0 ) ? p?t (s, a, s0 )| ?
.
(4)
2Nt (s,a)
Conditions (3) and (4) describe confidence bounds on the rewards and transition probabilities of the
true MDP M such that (2) is implied (cf. Section 3.1 below). The intuition behind the algorithm is
that if a non-optimal policy is followed, then this is eventually observed and something about the
MDP is learned. In the proofs we show that this learning happens sufficiently fast to approach an
optimal policy with only logarithmic regret.
As switching policies too often may be harmful, and estimates don?t change very much after few
steps, our algorithm discards the policy ?
?t only if there was considerable progress concerning the estimates p?(s, ?
?t (s), s0 ) or r?(s, ?
?t (s)). That is, UCRL sticks to a policy until the length of some of the
confidence intervals given by conditions (3) and (4) is halved. Only then a new policy is calculated.
We will see below (cf. Section 3.3) that this condition limits the number of policy changes without
paying too much for not changing to an optimal policy earlier. Summing up, Figure 1 displays our
algorithm.
Remark 1. The optimal policy ?
? in the algorithm can be efficiently calculated by a modified version
of value iteration (cf. [15]).
3
3.1
Analysis for Unichain MDPs
An Upper Bound on the Optimal Reward
We show that with high probability the true MDP M is contained in the set Mt of plausible MDPs.
Notation:
n q
o
n q
o
? |S|2 |A|)
? |S||A|)
and confr (t, s, a) := min 1, log(2t
.
Set confp (t, s, a) := min 1, log(4t
2Nt (s,a)
2Nt (s,a)
Initialization:
? Set t = 0.
? Set N0 (s, a) := R0 (s, a) := P0 (s, a, s0 ) = 0 for all s, a, s0 .
? Observe first state s0 .
For rounds k = 1, 2, . . . do
Initialize round k:
1. Set tk := t.
2. Recalculate estimates r?t (s, a) and p?t (s, a, s0 ) according to
r?t (s, a) :=
Rt (s,a)
Nt (s,a) ,
and
p?t (s, a, s0 ) :=
Pt (s,a,s0 )
Nt (s,a) ,
provided that Nt (s, a) > 0. Otherwise set r?t (s, a) := 1 and p?t (s, a, s0 ) :=
1
|S| .
3. Calculate new policy
?
?tk := arg max{?(M, ?) : M ? Mt },
?
where Mt consists of plausible unichain MDPs M 0 with rewards
r0 (s, a) ? r?t (s, a) ? confr (t, s, a)
and transition probabilities
|p0 (s, a, s0 ) ? p?t (s, a, s0 )| ? confp (t, s, a).
Execute chosen policy ?
?tk :
4. While
confr (t, S, A) > confr (tk , S, A)/2 and
confp (t, S, A) > confp (tk , S, A)/2
do
(a) Choose action at := ?
?tk (st ).
(b) Observe obtained reward rt and next state st+1 .
(c) Update:
? Set Nt+1 (st , at ) := Nt (st , at ) + 1.
? Set Rt+1 (st , at ) := Rt (st , at ) + rt .
? Set Pt+1 (st , at , st+1 ) := Pt (st , at , st+1 ) + 1.
? All other values Nt+1 (s, a), Rt+1 (s, a), and Pt+1 (s, a, s0 ) are set to
Nt (s, a), Rt (s, a), and Pt (s, a, s0 ), respectively.
(d) Set t := t + 1.
Figure 1: The UCRL algorithm.
Lemma 1. For any t, any reward r(s, a) and any transition probability p(s, a, s0 ) of the true MDP
M we have
q
n
o
? |S||A|)
P r?t (s, a) < r(s, a) ? log(2t
2Nt (s,a)
<
q
n
o
? |S|2 |A|)
P |?
pt (s, a, s0 ) ? p(s, a, s0 )| > log(4t
<
2Nt (s,a)
Proof. By Chernoff-Hoeffding?s inequality.
t??
,
2|S||A|
t??
.
2|S|2 |A|
(5)
(6)
Using the definition of Mt as given by (3) and (4) and summing over all s, a, and s0 , Lemma
1 shows that M ? Mt with high probability. This implies that the maximal average reward ???t
assumed by our algorithm when calculating a new policy at step t is an upper bound on ?? (M ) with
high probability.
Corollary 1. For any t:
3.2
P {?? > ???t } < t?? .
Sufficient Precision and Mixing Times
In order to upper bound the loss, we consider the precision needed to guarantee that the policy
calculated by UCRL is (?-)optimal. This sufficient precision will of course depend on ? or ? in case
one wants to compete with an optimal policy ? the minimal difference between ?? and the average
reward of some suboptimal policy,
? :=
min
?:?(M,?)<??
?? ? ?(M, ?).
? t, ?
It is sufficient that the difference between ?(M
?t ) and ?(M, ?
?t ) is small in order to guarantee
?
that ?
?t is an (?-)optimal policy. For if |?(Mt , ?
?t ) ? ?(M, ?
?t )| < ?, then by Corollary 1 with high
probability
? t, ?
? > |?(M
?t ) ? ?(M, ?
?t )| ? |?? (M ) ? ?(M, ?
?t )|,
(7)
so that ?
?t is already an ?-optimal policy on M . For ? = ?, (7) implies the optimality of ?
?t .
Thus, we consider bounds on the deviation of the transition probabilities and rewards for the assumed
? t from the true values, such that (7) is implied. This is handled in the subsequent proposition,
MDP M
where we use the notion of the MDP?s mixing time, which will play an essential role throughout the
analysis.
Definition 2. Given an ergodic Markov chain C, let Ts,s0 be the first passage time for two states s,
s0 , that is, the time needed to reach s0 when starting in s. Furthermore let Ts,s the return time to
max
0
E(T
0
)
s 6=s
s ,s
state s. Let TC := maxs,s0 ?S E(Ts,s0 ), and ?C := maxs?S
. Then the mixing time
2E(Ts,s )
of a unichain MDP M is TM := max? TC? , where C? is the Markov chain induced by ? on M .
Furthermore, we set ?M := max? ?C? .
Our notion of mixing time is different from the notion of ?-return mixing time given in [1, 2], which
depends on an additional parameter ?. However, it serves a similar purpose.
Proposition 1. Let p(?, ?), p?(?, ?) and r(?), r?(?) be the transition probabilities and rewards of the
? under the policy ?
MDPs M and M
? , respectively. If for all states s, s0
|?
r(s) ? r(s)| < ?r :=
?
2
and
|?
p(s, s0 ) ? p(s, s0 )| < ?p :=
?
,
2?M |S|2
?,?
then |?(M
? ) ? ?(M, ?
? )| < ?.
The proposition is an easy consequence of the following result about the difference in the stationary
distributions of ergodic Markov chains.
Theorem 1 (Cho, Meyer[16]). Let C, C? be two ergodic Markov chains on the same state space
S with transition probabilities p(?, ?), p?(?, ?) and stationary distributions ?, ?
?. Then the difference
in the distributions ?, ?
? can be upper bounded by the difference in the transition probabilities as
follows:
X
max |?(s) ? ?
?(s)| ? ?C max
|p(s, s0 ) ? p?(s, s0 )|,
(8)
s?S
s?S
s0 ?S
where ?C is as given in Definition 2.
Proof of Proposition 1. By (8),
X
X
|?(s) ? ?
?(s)| ? |S|?M max
|?
p(s, s0 ) ? p(s, s0 )| ? ?M |S|2 ?p .
s?S
s?S
s0 ?S
As the rewards are ? [0, 1] and
P
?(s) = 1, we have by (1)
X
X
?,?
|?(M
? ) ? ?(M, ?
? )| ?
|?
?(s) ? ?(s)|?
r(s) +
|?
r(s) ? r(s)|?(s)
s
s?S
<
s?S
2
?M |S| ?p + ?r = ?.
Since ?r > ?p and the confidence intervals for rewards are smaller than for transition probabilities
(cf. Lemma 1), in the following we only consider the precision needed for transition probabilities.
3.3
Bounding the Regret
As can be seen from the description of the algorithm, we split the sequence of steps into rounds,
where a new round starts whenever the algorithm recalculates its policy. The following facts follow
immediately from the form of our confidence intervals and Lemma 1, respectively.
Proposition 2. For halving a confidence interval of a reward or transition probability for some
(s, a) ? S ? A, the number Nt (s, a) of visits in (s, a) has to be at least doubled.
Corollary 2. The number of rounds after T steps cannot exceed |S||A| log2
Proposition 3. If Nt (s, a) ?
?.
log(4t? |S|2 |A|)
,
2? 2
T
|S||A| .
then the confidence intervals for (s, a) are smaller than
We need to consider three sources of regret: first, by executing a suboptimal policy in a round of
length ? , we may lose reward up to ? within this round; second, there may be some loss when changing policies; third, we have to consider the error probabilities with which some of our confidence
intervals fail.
3.3.1 Regret due to Suboptimal Rounds Proposition 3 provides an upper bound on the number
of visits needed in each (s, a) in order to guarantee that a newly calculated policy is optimal. This
can be used to upper bound the total number of steps in suboptimal rounds.
Consider all suboptimal rounds with |?
ptk (s, a, s0 ) ? p(s, a, s0 )| ? ?p for some s0 , where a policy
?
?tk with ?
?tk (s) = a is played. Let m(s, a) be the number of these rounds and ?i (s, a) (i =
1, . . . , m(s, a)) their respective lengths. The mean passage time between any state s00 and s is upper
bounded by TM . Then by Markov?s inequality, the probability that it takes more than 2TM steps
(s,a)
intervals
to reach s from s00 is smaller than 21 . Thus we may separate each round i into ?i2T
M
of length ? 2TM , in each of which the probability of visiting state s is at least 12 . Thus we may
lower bound the number of visits Ns,a (n) in (s, a) within n such intervals by an application of
Chernoff-Hoeffding?s inequality:
o
n
1
n p
? n log T ? 1 ? .
(9)
P Ns,a (n) ?
2
T
?
Since by Proposition 3, Nt (s, a) < 2 log(4T?p|S|
2
m(s,a)
X
i=1
?i (s, a)
2TM
2
|A|)
, we get
< c
log(4T ? |S|2 |A|)
?p 2
with probability 1 ? T1 for a suitable constant c < 11. This gives for the expected regret in these
rounds
m(s,a)
X
log(4T ? |S|2 |A|)
1
E
?i (s, a) < 2 c ? TM
+ 2 m(s, a) TM + T.
?
2
T
p
i=1
Applying Corollary 2 and summing up over all (s, a), one sees that the expected regret due to
suboptimal rounds cannot exceed
2 c |S||A|TM
log(4T ? |S|2 |A|)
T
+ 2TM |S|2 |A|2 log2
+ |S||A|.
?p 2
|S||A|
3.3.2 Loss by Policy Changes For any policy ?
?t there may be some states from which the expected average reward for the next ? steps is larger than when starting in some other state. This does
not play a role if ? ? ?. However, as we are playing our policies only for a finite number of steps
before considering a change, we have to take into account that every time we switch policies, we
may need a start-up phase to get into such a favorable state. In average, this cannot take more than
TM steps, as this time is sufficient to reach any ?good? state from some ?bad? state. This is made
more precise in the following lemma. We omit a detailed proof.
Lemma 2. For all policies ?, all starting states s0 and all T ? 0
?1
TX
E
r(st , ?(st )) ? T ?(?, M ) ? TM .
t=0
By Corollary 2, the corresponding expected regret after T steps is ? |S||A|TM log2
T
|S||A| .
3.3.3 Regret if Confidence Intervals Fail Finally, we have to take into account the error probabilities, with which in each round a transition probability or a reward, respectively, is not contained
in its confidence interval. According to Lemma 1, the probability that this happens at some step t
t??
t??
t??
for a given state-action pair is < 2|S||A|
+ |S| 2|S|
2 |A| = |S||A| . Now let t1 = 1, t2 , . . . , tN ? T
be the steps in which a new round starts. As the regret in each round can be upper bounded by its
length, one obtains for the regret caused by failure of confidence intervals
N
?1
X
i=1
N
?1
?
X
X
t1??
t??
t??
i
i
c
(ti+1 ? ti ) ?
cti <
< c0 ,
|S||A|
|S||A|
|S||A|
t=1
i=1
using that ti+1 ? ti < cti for a suitable constant c = c(|S|, |A|, TM ) and provided that ? > 2.
3.3.4 Putting Everything Together Summing up over all the sources of regret and replacing for
?p yields the following theorem, which is a generalization of similar results that were achieved for
the multi-armed bandit problem in [8].
Theorem 2. On unichain MDPs, the expected total regret of the UCRL algorithm with respect to an
(?-)optimal policy after T > 1 steps can be upper bounded by
T
|A|TM ?2M |S|5
log T + 3TM |S|2 |A|2 log2
, and
?2
|S||A|
|A|TM ?2M |S|5
T
E(RT ) < const ?
log T + 3TM |S|2 |A|2 log2
.
?2
|S||A|
E(RT? ) < const ?
4
Remarks and Open Questions on Multichain MDPs
In a multichain MDP a policy ? may split up the MDP into ergodic subchains Si? . Thus it may
happen during the learning phase that one goes wrong and ends up in a part of the MDP that gives
suboptimal return but cannot be left under no policy whatsoever. As already observed by Kearns,
Singh [1], in this case it seems fair to compete with ?? (M ) := max? minSi? ?(Si? , ?).
Unfortunately, the original UCRL algorithm may not work very well in this setting, as it is impossible for the algorithm to distinguish between a very low probability for a transition and its
impossibility. Here the ?optimism in the face of uncertainty? idea fails, as there is no way to falsify
the wrong belief in a possible transition.
Obviously, if we knew for each policy which subchains it induces on M (the MDP?s ergodic struc? t and a policy ?
ture), UCRL could choose an MDP M
?t that maximizes the reward among all plausible MDPs with the given ergodic structure. However, only the empiric ergodic structure (based
on the observations so far) is known. As the empiric ergodic structure may not be reliable, one may
additionally explore the ergodic structures of all policies. Alas, the number of additional exploration
steps will depend on the smallest positive transition probability. If the latter is not known, it seems
that logarithmic online regret bounds can be no longer guaranteed.
However, we conjecture that for a slightly modified algorithm the logarithmic online regret bounds
still hold for communicating MDPs, in which for any two states s, s0 there is a suitable policy ? such
that s is reachable from s0 under ? (i.e., s, s0 are contained in the same subchain Si? ). As Theorem
1 does not hold for communicating MDPs in general, a proof would need a different analysis.
5
Conclusion and Outlook
Beside the open problems on multichain MDPs, it is an interesting question whether our results
also hold when assuming for the mixing time not the slowest policy for reaching any state but
the fastest. Another research direction is to consider value function approximation and continuous
reinforcement learning problems.
For practical purposes, using the variance of the estimates will reduce the width of the upper confidence bounds and will make the exploration even more focused, improving learning speed and regret
bounds. In this setting, we have experimental results comparable to those of the MBIE algorithm
[10], which clearly outperforms other learning algorithms like R-Max or ?-greedy.
Acknowledgements.
This work was supported in part by the the Austrian Science Fund FWF (S9104-N04 SP4) and the
IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002506778. This publication only reflects the authors? views.
References
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2,262 | 3,053 | The Robustness-Performance Tradeoff in Markov
Decision Processes
Huan Xu, Shie Mannor
Department of Electrical and Computer Engineering
McGill University
Montreal, Quebec, Canada, H3A2A7
[email protected]
[email protected]
Abstract
Computation of a satisfactory control policy for a Markov decision process when
the parameters of the model are not exactly known is a problem encountered in
many practical applications. The traditional robust approach is based on a worstcase analysis and may lead to an overly conservative policy. In this paper we consider the tradeoff between nominal performance and the worst case performance
over all possible models. Based on parametric linear programming, we propose a
method that computes the whole set of Pareto efficient policies in the performancerobustness plane when only the reward parameters are subject to uncertainty. In
the more general case when the transition probabilities are also subject to error,
we show that the strategy with the ?optimal? tradeoff might be non-Markovian
and hence is in general not tractable.
1
Introduction
In many decision problems the parameters of the problem are inherently uncertain. This uncertainty,
termed parameter uncertainty, can be the result of estimating the parameters from a finite sample or
a specification of the parameters that itself includes uncertainty. The standard approach in decision
making to circumvent the adverse effect of the parameter uncertainty is to find a solution that performs best under the worst possible parameters. This approach, termed the ?robust? approach, has
been used in both single stage ([1]) and multi-stage decision problems (e.g., [2]).
In robust optimization problems, it is usually assumed that the constraint parameters are uncertain.
By requiring the solution to be feasible to all possible parameters within the uncertainty set, Soyester
([1]) solved the column-wise independent uncertainty case, and Ben-Tal and Nemirovski ([3]) solved
the row-wise independent case. In robust MDP problems, there may be two different types of
parameter uncertainty, namely, the reward uncertainty and the transition probability uncertainty.
Under the assumption that the uncertainty is state-wise independent (an assumption made by all
papers to date, to the best of our knowledge), the optimality principle holds and this problem can be
decomposed as a series of step by step mini-max problems solved by backward induction ([2, 4, 5]).
The above cited results focus on worst-case analysis. This implies that the vector of nominal parameters (the parameters used as an approximation of the true one regardless of the uncertainty) is not
treated in a special way and is just an element of the set of feasible parameters. The objective of the
worst-case analysis is to eliminate the possibility of disastrous performance. There are several disadvantages to this approach. First, worst-case analysis may lead to an overly conservative solution,
i.e., a solution which provides mediocre performance under all possible parameters. Second, the
desirability of the solution highly depends on the precise modeling of the uncertainty set which is
often based on some ad-hoc criterion. Third, it may happen that the nominal parameters are close to
the real parameters, so that the performance of the solution under nominal parameters may provide
important information for predicting the performance under the true parameters. Finally, there is
a certain tradeoff relationship between the worst-case performance and the nominal performance,
that is, if the decision maker insists on maximizing one criterion, the other criterion may decrease
dramatically. On the other hand, relaxing both criteria may lead to a well balanced solution with
both satisfactory nominal performance and also reasonable robustness to parameter uncertainty.
In this paper we capture the Robustness-Performance (RP) tradeoff explicitly. We use the worstcase behavior of a solution as the function representing its robustness, and formulate the decision
problem as an optimization of both the robustness criterion and the performance under nominal
parameters simultaneously. Here, ?simultaneously? is achieved by optimizing the weighted sum of
the performance criterion and the robustness criterion. To the best of our knowledge, this is the first
attempt to address the overly conservativeness of worst-case analysis in robust MDP.
Instead of optimizing the weighted sum of the robustness and performance for some specific weights,
we show how to efficiently find the solutions for all possible weights. We prove that the set of these
solutions is in fact equivalent to the set of all Pareto efficient solutions in the robustness-performance
space. Therefore, we solve the tradeoff problem without choosing a specific tradeoff parameter, and
leave the subjective decision of determining the exact tradeoff to the decision maker. Instead of
arbitrarily claiming that a certain solution is a good tradeoff, our algorithm computes the whole
tradeoff relationship so that the decision maker can choose the most desirable solution according to
her preference, which is usually complicated and an explicit form is not available. Our approach
thus avoids the tuning of tradeoff parameters, where generally no good a-priori method exists. This
is opposed to certain relaxations of the worst-case robust optimization approach like [6] (for single
stage only) where some explicit tradeoff parameters have to be chosen. Unlike risk sensitive learning
approaches [7, 8, 9] which aim to tune a strategy online, our approach compute a robust strategy offline without trial and error.
The paper is organized as follows. Section 2 is devoted to the RP tradeoff for Linear Programming.
In Section 3 and Section 4 we discuss the RP tradeoff for MDP with uncertain rewards, and uncertain transition probabilities, respectively. In Section 5 we present a computational example. Some
concluding remarks are offered in Section 6.
2
Parametric linear programming and RP tradeoffs in optimization
In this section, we briefly recall Parametric Linear Programming (PLP) [10, 11, 12], and show
how it can be used to find the whole set of Pareto efficient solutions for RP tradeoffs in Linear
Programming. This serves as the base for the discussion of RP tradeoffs in MDPs.
2.1
Parametric Linear Programming
A Parametric Linear Programming is the following set of infinitely many optimization problems:
For all ? ? [0, 1]: Minimize:
Subject to:
>
>
>
?c(1) x + (1 ? ?)c(2) x
Ax = b
x ? 0.
(1)
>
We call c(1) x the first objective, and c(2) x the second objective. We assume that the Linear
Program (LP) is feasible and bounded for both objectives. Although there are uncountably many
possible ?, Problem (1) can be solved by a simplex-like algorithm. Here, ?solve? means that for each
?, we find at least one optimal solution. An outline of the PLP algorithm is described in Algorithm
1, which is essentially a tableau simplex algorithm while the entering variable is determined in a
specific way. See [10] for a precise description.
Algorithm 1.
1. Find a basic feasible optimal solution for ? = 0. If multiple solutions exist,
>
choose one among those with minimal c(1) x.
2. Record current basic feasible solution. Check the reduced cost (i.e., the zero row in the
(1)
simplex table) of the first objective, denoted as c?j . If none of them is negative, end.
(1)
(1)
(2)
3. Among all columns with negative c?j , choose the one with largest ratio |?
cj /?
cj | as the
entering variable.
4. Pivot the base, go to 2.
This algorithm is based on the observation that for any ?, there exists an optimal basic feasible
solution. Hence, by finding a suitable subset of all vertices of the feasible region, we can solve
the PLP. Furthermore, we can find this subset by sequentially pivoting among neighboring extreme
points like the simplex algorithm does. This algorithm terminates after finitely many iterations. It
is also known that the optimal value for PLP is a continuous piecewise linear function of ?. The
theoretical computational cost is exponential, although practically it works well. Such property is
shared by all simplex based algorithm. A detailed discussion on PLP can be found in [10, 11, 12].
2.2
RP tradeoffs in Linear Programming
Consider the following LP:
NOMINAL PROBLEM : Minimize: c> x
Subject to: Ax ? b
Here A ? Rn?m , x ? Rm , b ? Rn , c ? Rm .
(2)
Suppose that the constraint matrix A is only a guess of the unknown true parameter Ar which is
known to belonging to set A (we call A the uncertainty set). We assume
Qn that A is constraint-wise
independent and polyhedral for each of the constraints.
That
is,
A
=
i=1 Ai , and for each i, there
exists a matrix T (i) and a vector v(i) such that Ai = a(i)> |T (i)a(i) ? v(i) .
To quantify how a solution x behaves with respect to the parameter uncertainty, we define the following criterion to be minimized as its robustness measure (more accurately, non-robustness measure).
h
i+
?
p(x) , sup
Ax ? b
?
1
A?A
("
#
)
(3)
n
n
X
X
>
>
max
sup
?
a(i) x ? bi , 0 .
max ?
a(i) x ? bi , 0 =
= sup
?
A?A
i=1
?
a(i):T (i)?
a(i)?v(i)
i=1
? and bi is the
Here [?] stands for the positive part of a vector, ?
a(i) is the ith row of the matrix A,
ith element of b. In words, the function p(x) is the largest possible sum of constraint violations.
+
>
Using the weighted sum of the performance and robustness objective as the minimizing objective,
we formulate the explicit tradeoff between robustness and performance as:
GENERAL PROBLEM : ? ? [0, 1]
Minimize: ?c> x + (1 ? ?)p(x)
Subject to: Ax ? b.
(4)
Here A ? Rn?m , x ? Rm , b ? Rn , c ? Rm .
By duality theorem, for a given x, sup?a(i):T (i)?a(i)?v(i) ?
a(i)> x equals to the optimal value of the
following LP on y(i):
Minimize: v(i)> y(i)
Subject to:
T (i)> y(i) = x
y(i) ? 0.
Thus, by adding slack variables, we rewrite GENERAL PROBLEM as the following PLP and solve
it using Algorithm 1:
GENERAL PROBLEM (PLP) : ? ? [0, 1]
Minimize: ?c> x + (1 ? ?)1> z
Subject to: Ax ? b,
T (i)> y(i) = x,
v(i)> y(i) ? bi ? zi ,
z ? 0,
y(i) ? 0; i = 1, 2, ? ? ? , n.
(5)
Here, 1 stands for a vector of ones of length n, zi is the ith element of z, and x, y(i), z are the
optimization variables.
3
The robustness-performance tradeoff for MDPs with uncertain rewards
A (finite) MDP is defined as a 5-tuple < T, S, As , p(?|s, a), r(s, a) > where: T is the (possibly
infinite) set of decision stages; S is the state set; As is the action set of state s; p(?|s, a) is the
transition probability; and r(s, a) is the expected reward of state s with action a ? As . We use
r to denote the vector combining the reward for all state-action pairs and rs to denote the vector
combining all reward of state s. Thus, r(s, a) = rs (a). Both S and As are assumed finite. Both p
and r are time invariant.
In this section, we consider the case where r is not known exactly. More specifically, we have a
nominal parameter r(s, a) which is believed to be a reasonably good guess of the true reward. The
reward r is known to belong to a bounded set R. We further assume
Q that the uncertainty set R is
state-wise independent and a polytope for each state. That is, R = s?S Rs , and for each s ? S,
there exists a matrix Cs and a vector ds such that Rs = {rs |Cs rs ? ds }. We assume that for
different visits of one state, the realization of the reward need not be identical and may take different
values within the uncertainty set. The set of admissible control policies for the decision maker is the
set of randomized history dependent policies, which we denote by ?HR .
In the following three subsections we discuss different standard reward criteria: cumulative reward
with a finite horizon, discounted reward with infinite horizon, and limiting average reward with
infinite horizon under a unichain assumption.
3.1
Finite horizon case
In the finite horizon case (T = {1, ? ? ? , N }), we assume without loss of generality that each state
belongs to only one stage, which is equivalent to the assumption of non-stationary reward realization,
and use Si to denote the set of states at the ith stage. We also assume that the first stage consists of
only one state s1 , and that there are no terminal rewards. We define the following two functions as
the performance measure and the robustness measure of a policy ? ? ?HR :
N
?1
X
P (?) , E? {
r(si , ai )},
i=1
N
?1
X
R(?) , min E? {
r?R
(6)
r(si , ai )}.
i=1
The minimum is attainable, since R is compact and the total expected reward is a continuous function of r. We say that a strategy ? is Pareto efficient if it obtains the maximum of P (?) among all
strategies that have a certain value of R(?). The following result is straightforward; the proof can
be found in the full version of the paper.
Proposition 1.
1. If ? ? is a Pareto efficient strategy, then there exists a ? ? [0, 1] such that
? ? ? arg max???HR {?P (?) + (1 ? ?)R(?)}.
2. If ? ? ? arg max???HR {?P (?) + (1 ? ?)R(?)} for some ? ? (0, 1). Then ? ? is a Pareto
efficient strategy.
For 0 ? t ? N , s ? St , and ? ? [0, 1] define:
(N ?1
)
X
Pt (?, s) , E?
r(si , ai )|st = s
i=t
Rt (?, s) , min E?
r?R
c?t (s)
(N ?1
X
i=t
)
r(si , ai )|st = s
, max {?Pt (?, s) + (1 ? ?)Rt (?, s)} .
???HR
(7)
We set PN ? RN ? cN ? 0, and note that c?1 (s1 ) is the optimal RP tradeoff with weight ?. The
following theorem shows that the principle of optimality holds for c. The proof is omitted since it
follows similarly to standard backward induction in finite horizon robust decision problems.
Theorem 1. For s ? St , t < N , let ?s be the probability simplex on As , then
n
P
P
c?t (s) = max min
? a?As r(s, a)q(a) + (1 ? ?) a?As r(s, a)q(a) +
q??s rs ?Rs
o
P
P
0
?
0
s0 ?St+1
a?As p(s |s, a)q(a)ct+1 (s ) .
We now consider the maximin problem in each state and show how to find the solutions for all ?
in one pass. We also prove that c?t (s) is piecewise linear in ?. Let St+1 = {s1 , ? ? ? , sk }. Assume
for all j ? {1, ? ? ? , k}, c?t+1 (sj ) are continuous piece-wise linear functions. Thus, we can divide
[0, 1] into finite (say n) intervals [0, ?1 ], ? ? ? [?n?1 , 1] such that in each interval, all ct+1 functions
are linear. That is, there exist constants lij and mji such that c?t+1 (sj ) = lij ? + mji , for ? ? [?i?1 , ?i ].
By the duality theorem, we have that c?t (s) equals to the optimal value of the following LP on y and
q.
>
Maximize: (1 ? ?)d>
s y + ?rs q +
Subject to:
Cs> y
>
= q,
k X
X
j=1 a?As
p(sj |s, a)q(a)c?t+1 (sj )
(8)
1 q = 1,
q, y ? 0.
Observe that the feasible set is the same for all ?. Substituting c?t+1 (sj ) and rearranging, it follows
that for ? ? [?i?1 , ?i ] the objective function equals to
nP
hP
i
o
k
j
j
>
(1 ? ?)
a?As
j=1 p(s |s, a)mi q(a) + ds y
i
o
nP
h
Pk
j
j
j
r(s,
a)
+
p(s
|s,
a)(l
+
m
)
q(a)
.
+?
i
i
j=1
a?As
Thus, for ? ? [?i?1 , ?i ], from the optimal solution for ?i?1 , we can solve for all ? using Algorithm 1. Furthermore, we need not to re-initiate for each interval, since the optimal solution for
the end of ith interval is also the optimal solution for the begin of the next interval. It is obvious
that the resulting c?t (s) is also continuous, piecewise linear. Thus, since cN = 0, the assumption of
continuous and piecewise linear value functions holds by backward induction.
3.2
Discounted reward infinite horizon case
In this section we address the RP tradeoff for infinite horizon MDPs with a discounted reward
criterion. For a fixed ?, the problem is equivalent to a zero-sum game, with the decision maker
trying to maximize the weighted sum and Nature trying to minimize it by selecting an adversarial
reward realization. A well known result in discounted zero-sum stochastic games states that, even if
non-stationary policies are admissible, a Nash equilibrium in which both players choose a stationary
policy exists; see Proposition 7.3 in [13].
Given an initial state distribution ?(s), it is also a known result [14]
that there exists a one-toP?
one correspondence relationship between the state-action frequencies i=1 ? i?1 E(1si =s,ai =a ) for
stationary strategies and vectors belonging to the following polytope X :
X
X X
x(s0 , a) ?
?p(s0 |s, a)x(s, a) = ?(s0 ), x(s, a) ? 0, ?s, ?a ? As .
(9)
a?As0
s?S a?As
Since it suffices to consider a stationary policy for Nature, the tradeoff problem becomes:
X X
Maximize: inf
[?r(s, a)x(s, a) + (1 ? ?)r(s, a)x(s, a)]
r?R
s?S a?As
Subject to: x ? X .
(10)
By duality of LP, Equation (10) could be rewritten as the following PLP and solved by Algorithm 1.
X X
X
Maximize:?
r(s, a)x(s, a) + (1 ? ?)
d>
s ys
s?S a?As
Subject to:
X
a?As0
s?S
0
x(s , a) ?
x(s, a) ? 0,
X X
s?S a?As
0
?p(s |s, a)x(s, a) = ?(s0 ),
?s0 ,
(11)
?s, ?a,
Cs> ys = xs ?s,
ys ? 0, ?s.
3.3
Limiting average reward case (unichain)
In the unichain
the set of limiting average
state-action frequency vectors (that is, all limit points
case,
PT
of sequences T1 n=1 E? [1sn =s,an =a ] , for ? ? ?HR ) is the following polytope X :
X
X X
x(s0 , a) ?
p(s0 |s, a)x(s, a) = 0, ?s0 ? S,
a?As0
s?S a?As
X X
(12)
x(s, a) = 1,
s?S a?As
x(s, a) ? 0, ?s, ?a ? As .
As before, there exists an optimal maximin stationary policy. By a similar argument as for the
discounted case, the tradeoff problem can be converted to the following PLP:
X X
X
Maximize:?
r(s, a)x(s, a) + (1 ? ?)
d>
s ys
s?S a?As
Subject to:
X
a?As0
0
x(s , a) ?
X X
s?S
X X
s?S a?As
0
p(s |s, a)x(s, a) = 0,
?s0 ,
x(s, a) = 1,
(13)
s?S a?As
Cs> ys = xs , ?s,
ys ? 0, ?s,
x(s, a) ? 0, ?s, ?a.
4
The RP tradeoff in MDPs with uncertain transition probabilities
In this section we provide a counterexample which demonstrates that the weighted sum criterion
in the most general case, i.e., the uncertain transition probability case, may lead to non-Markovian
optimal policies.
In the finite horizon MDP shown in the Figure 1, S = {s1, s2, s3, s4, s5, t1, t2, t3, t4}; As1 =
{a(1, 1)}; As2 = {a(2, 1)}; As3 = {a(3, 1)}; As4 = {a(4, 1)} and As5 = {a(5, 1), a(5, 2)}.
Rewards are only available at the final stage, and are perfectly known. The nominal transition probabilities are p (s2|s1, a(1, 1)) = 0.5, p (s4|s2, a(2, 1)) = 1, and p (t3|s5, a(5, 2)) = 1. The set of
possible realization is p (s2|s1, a(1, 1)) ? {0.5}, p (s4|s2, a(2, 1)) ? [0, 1], and p (t3|s5, a(5, 2)) ?
[0, 1]. Observe that the worst parameter realization is p(s4|s2, a(2, 1)) = p(t3|s5, a(5, 2)) = 0.
We look for the strategy that maximizes the sum of the nominal reward and the worst-reward (i.e.,
? = 0.5). Since multiple actions only exist in state s5, a strategy is determined by the action chosen
on s5. Let the probability of choosing action a(5, 1) and a(5, 2) be p and 1 ? p, respectively.
Consider the history ?s1 ? s2?. In this case, with the nominal transition probability, this trajectory
will reach t1 with a reward of 10, regardless of the choice of p. The worst transition is that action
a(2, 1) leads to s5 and action a(5, 2) leads to t4, hence the expected reward is 5p + 4(1 ? p).
Therefore the optimal p equals to 1, i.e., the optimal action is to choose a(5, 1) deterministically.
0.5 (0.5)
s1
s2 a(2,1) 1 (0)
s4 a(4,1)
t1
r=10
a(1,1)
0 (1)
a(5,1)
0.5 (0.5)
s3 a(3,1)
s5
a(5,2)
1 (0)
0 (1)
t2
r=5
t3
r=8
t4
r=4
Figure 1: Example of non-Markovian best strategy
Consider the history ?s1 ? s3?. In this case, the nominal reward is 5p + 8(1 ? p), and the worst
case reward is 5p + 4(1 ? p). Thus p = 0 optimize the weighted sum, i.e., the optimal strategy is to
choose a(5, 2).
The unique optimal strategy for this example is thus non-Markovian. This non-Markovian property
implies a possibility that past actions affect the choice of future actions, and hence could render the
problem intractable. The optimal strategy is non-Markovian because we are taking expectation over
two different probability measures, hence the smoothing property of conditional expectation cannot
be used in finding the optimal strategy.
5
A computational example
We apply our algorithm to a T -stage machine maintenance problem. Let S , {1, ? ? ? , n} denote the
state space for each stage. In state h, the decision maker can choose either to replace the machine
which will lead to state 1 deterministically, or to continue running, which with probability p will
lead to state h + 1. If the machine is in state n, then the decision maker has to replace it. The
replacing cost is perfectly known to be cr , and the nominal running cost in state h is ch . We?
assume
that the realization of the running cost lies in the interval [ch ? ?h , ch + ?h ]. We set ch = h ? 1
and ?h = 2h/n. The objective is to minimize the total cost, in a risk-averse attitude. Figure 2(a)
shows the tradeoff of this MDP.
For each solution found, we sample the reward 300 times according to a uniform distribution. We
normalize the cost for each simulation, i.e., we divide the cost by the smallest expected nominal
cost. Denoting the normalized cost of the ith simulation for strategy j as si (j), we use the following
function to compare the solutions:
s
P300
?
?
i=1 |si (j)|
vj (?) =
.
300
Note that ? = 1 is the mean of the simulation cost, whereas larger ? puts higher penalty on deviation
representing a risk-averse decision maker. Figure 2(b) shows that, the solutions that focus on nominal parameters (i.e., ? close to 1) achieve good performance for small ?, but worse performance for
large ?. That is, if the decision maker is risk neutral, then the solutions based on nominal parameters
are good. However, these solutions are not robust and are not good choices for risk-averse decision
makers. Note that, in this example, the nominal cost is the expected cost for each stage, i.e., the
parameters are exactly formulated. Even in such case, we see that risk-averse decision makers can
benefit from considering the RP tradeoff.
6
Concluding remarks
In this paper we proposed a method that directly addresses the robustness versus performance tradeoff by treating the robustness as an optimization objective. Based on PLP, for MDPs where only
(a)
(b)
29.28
29
?=1
1.4
Normalized Modified Mean
1.35
Worst Case Performance
28.5
28
27.5
27
26.5
26
25.68
16.79
17.2
17.4
17.6
17.8
18
18.2
18.4
18.6
1.2
1.15
1.1
1.05
?=1
?=10
?=100
?=1000
1
0.95
?=0
17
1.3
1.25
18.76
0.9
0
Norminal Performance
0.2
0.4
?
0.6
0.8
1
Figure 2: The machine maintenance problem: (a) the PR tradeoff; (b) normalized mean of the
simulation for different values of ?.
rewards are uncertain, we presented an efficient algorithm that computes the whole set of optimal
RP tradeoffs for MDPs with finite horizon, infinite horizon discounted reward, and limiting average
reward (unichain). For MDPs with uncertain transition probabilities, we showed an example where
the solution may be non-Markovian and hence may in general be intractable.
The main advantage of the presented approach is that it addresses robustness directly. This frees
the decision maker from the need to make probabilistic assumptions on the problems parameters. It
also allows the decision maker to determine the desired robustness-performance tradeoff based on
observing the whole curve of possible tradeoffs rather than guessing a single value.
References
[1] A. L. Soyster. Convex programming with set-inclusive constraints and applications to inexact linear
programming. Oper. Res., 1973.
[2] A. Bagnell, A. Ng, and J. Schneider. Solving uncertain markov decision processes. Technical Report
CMU-RI-TR-01-25, Carnegie Mellon University, August 2001.
[3] A. Ben-Tal and A. Nemirovski. Robust solutions of uncertain linear programs. Oper. Res. Lett., 25(1):1?
13, August 1999.
[4] C. C. White III and H. K. El-Deib. Markov decision process with imprecise transition probabilities. Oper.
Res., 42(4):739?748, July 1992.
[5] A. Nilim and L. El Ghaoui. Robust control of markov decision processes with uncertain transition matrices. Oper. Res., 53(5):780?798, September 2005.
[6] D. Bertsimas and M. Sim. The price of robustness. Oper. Res., 52(1):35?53, January 2004.
[7] M. Heger. Consideration of risk in reinforcement learning. In Proc. 11th International Conference on
Machine Learning, pages 105?111. Morgan Kaufmann, 1994.
[8] R. Neuneier and O. Mihatsch. Risk sensitive reinforcement learning. In Advances in Neural Information
Processing Systems 11, pages 1031?1037, Cambridge, MA, USA, 1999. MIT Press.
[9] P. Geibel. Reinforcement learning with bounded risk. In Proc. 18th International Conf. on Machine
Learning, pages 162?169. Morgan Kaufmann, San Francisco, CA, 2001.
[10] D. Bertsimas and J. N. Tsitsiklis. Introduction to Linear Optimization. Athena Scientific, 1997.
[11] M. Ehrgott. Multicriteria Optimization. Springer-Verlag Berlin Heidelberg, 2000.
[12] K. G. Murty. Linear Programming. John Wiley & Sons, 1983.
[13] D. P. Bertsekas and J. N. Tsitsiklis. Neuro-Dynamic Programming. Athena Scientific, 1996.
[14] M. L. Puterman. Markov Decision Processes. John Wiley & Sons, INC, 1994.
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2,263 | 3,054 | Clustering appearance and shape by learning jigsaws
Anitha Kannan, John Winn, Carsten Rother
Microsoft Research Cambridge
[ankannan, jwinn, carrot]@microsoft.com
Abstract
Patch-based appearance models are used in a wide range of computer vision applications. To learn such models it has previously been necessary to specify a
suitable set of patch sizes and shapes by hand. In the jigsaw model presented
here, the shape, size and appearance of patches are learned automatically from
the repeated structures in a set of training images. By learning such irregularly
shaped ?jigsaw pieces?, we are able to discover both the shape and the appearance
of object parts without supervision. When applied to face images, for example,
the learned jigsaw pieces are surprisingly strongly associated with face parts of
different shapes and scales such as eyes, noses, eyebrows and cheeks, to name
a few. We conclude that learning the shape of the patch not only improves the
accuracy of appearance-based part detection but also allows for shape-based part
detection. This enables parts of similar appearance but different shapes to be distinguished; for example, while foreheads and cheeks are both skin colored, they
have markedly different shapes.
1
Introduction
Many computer vision tasks require the use of appearance and shape models to represent objects
in the scene. The choices for appearance models range from histogram-based representations that
throws away spatial information, to template-based representations that try to capture the entire spatial layout of the objects but cope poorly with articulation, deformation or variation in appearance.
In the middle of this spectrum lie patch-based models that aim to find the right balance between the
two extremes.
However, a central problem with existing patch-based models is that there is no way to choose the
shape and size of a patch; typically a predefined set of patch sizes and shapes (often rectangles or
circles) are used. We believe that natural images can provide enough cues to allow patches to be
discovered of varying shape and size corresponding to the shape and size of object parts present
in the images. Indeed, we will show that the patches discovered by the jigsaw model can become
strongly associated with semantic object parts.
With this motivation, we introduce a generative model for a set of images that learns to extract irregularly shaped and sized patches from a latent image which are combined to generate each training
image. We call this latent image a jigsaw as it contains all the necessary ?jigsaw pieces? that can be
used to generate the target image set. We present an inference algorithm for learning the jigsaw and
for finding the jigsaw pieces that make up each image.
As our proposed jigsaw model is a generative model for an image, it can be readily used as a component in many computer vision applications for both image understanding and image synthesis. These
include object recognition, detection, image segmentation and image classification, object synthesis, image de-noising, super resolution, texture transfer between images and image in-painting. In
fact, the jigsaw model is likely to be useable as a direct replacement for a fixed patch model in any
existing patch-based system.
2
Related work
The closest work to ours is the epitome model of Jojic et al. [1]. This is a generative model for
image patches, or alternatively a model for images if patches that share coordinates in the image
are averaged together (although this averaging often leads to a blurry result). Epitomes are learned
using a set of fixed shaped patches over a small range of sizes. In contrast, in the jigsaw model,
the inference process chooses appropriately shaped and sized pieces from the training images when
learning the jigsaw. The difference between these two models is illustrated in section 4.
Our work also closely relates to the seminal work of Freeman et al. [2] that proposed a general
machinery for inferring underlying scenes from images, with goals such as in optical flow estimation
and super-resolution. They define a Markov random field over image patches and infer the hidden
scene representation using belief propagation. Again, they use a set of fixed size image patches,
hoping to reach a reasonable trade-off between capturing sufficient statistics in each patch, and
disambiguating different kinds of features. Along these lines, Markov random field with larger
cliques have also been used to capture the statistic of natural images, such as the field of experts
model proposed in [3] which represents the field potentials as non-linear functions of linear filters.
Again, the underlying linear filters are applied to patches of a fixed size.
In the domain of image synthesis the work of Freeman et al. [2] has inspired many patch-based
synthesis algorithms including super resolution, texture transfer, image in-painting or photo synthesis. They can be viewed as a data-driven way of sampling from the Markov random field with
high-order cliques given by the overlapping patches. The texture synthesis and transfer algorithm of
Efros et al. [4] constructs a new image by greedily selecting overlapping patches so that the seam
transition is not visible. Whilst this work does allow different patch shapes, it does not learn patch
appearance since it works from a supplied texture image. Recently a similar approach has been
proposed in [5] for synthesising a collage image from a given set of input images, although in this
case a probabilistic model is defined and optimised.
Patch-based models are also widely applied in object recognition research [6, 7, 8, 9, 10]. These
models use hand-selected patch shapes (typically rectangles) which can lead to poor results given
that different object parts have different sizes and shapes. In fact, the use of fixed patches reduces
accuracy when the object part is of different size and shape than the chosen patch; in this case, the
patch model has to cope with the variability outside the object part. This effect is particularly evident
when the part is at the edge of the object as the model then has to try and capture the variability of
the background. In addition, such models ignore the shape of the object part which is frequently
much more discriminative than appearance alone.
The paper is structured as follows: In section 3 we introduce the probabilistic model and describe
a method for performing learning and inference in the model. In section 4 we show results for
synthetic and real data and present a comparison to the epitome model. Finally, in section 5, we
discuss possible extensions to the model.
3
Probabilistic model
This section describes the probabilistic model that we use to learn a jigsaw from a set of training
images. We aim to learn a jigsaw such that, given an image set, pieces of the jigsaw image satisfy
the following criteria:
? each piece is similar in appearance and shape to several regions of the training images;
? any of the training images can be approximately reconstructed using only pieces from the
jigsaw (a piece may be used more than once in a single image);
? pieces are as large as possible for a particular accuracy of reconstruction.
Thus, while allowing the jigsaw pieces to have arbitrary shape, we ensure that such pieces are shared
across the entire image set, exhaustively explain the input image set, and are also large enough to
be discriminative. By meeting these criteria, we can capture both the appearance and the shape of
repeated image structures, for example, eyes, noses and mouths in a set of face images.
We define a jigsaw J to be an image such that each pixel z in J has an intensity value ?(z) and an
associated variance ??1 (z) (so ? is the inverse variance, also called the precision). A set of spatially
Figure 1: Graphical model showing how the jigsaw J is used to generate a set of images I1 . . . IN
by combining the jigsaw pieces in different ways. Each image has a corresponding offset map L
which defines the jigsaw pieces used to generate that image (see text for details). Notice that several
jigsaw pieces can overlap and hence share parts of their appearance.
grouped pixels in J is a jigsaw piece. We can combine many of these pieces to generate images,
noting that pixels in the jigsaw be re-used in multiple pieces.
Our probabilistic model is a generative image model which generates an image by joining together
pieces of the jigsaw and then adding Gaussian noise of variance given by the jigsaw. For each image
I, we have an associated offset map L of the same size which determines the jigsaw pieces used
to make that image. This offset map defines a position in the jigsaw for each pixel in the image
(more than one image pixel can map to the same jigsaw pixel). Each entry in the offset map is a
two-dimensional offset li = (lx , ly ), which maps a 2D point i in the image to a 2D point z in the
jigsaw using z = (i ? li ) mod |J|, where |J| = (width, height) are the dimensions of the jigsaw.
Notice that if two adjacent pixels in the image have the same offset label, then they map to adjacent
pixels in the jigsaw. Figure 1 provides a schematic view of the overall probabilistic model, as it is
used to generate a set of N face images.
Given this mapping and the jigsaw, the probability distribution of an image is assumed to be independent for each pixel and is given by
Y
P (I | J, L) =
N (I(i); ?(i ? li ), ?(i ? li )?1 )
(1)
i
where the product is over image pixel positions and both subtractions are modulo |J|.
We want the images to consist of coherent pieces of the jigsaw, and so we define a Markov random
field on the offset map to encourage neighboring pixels to have the same offsets.
?
?
X
1
?(li , lj )?
(2)
P (L) = exp ??
Z
(i,j)?E
where E is the set of edges in a 4-connected grid. The interaction potential ? defines a Pott?s model
on the offsets:
?(li , lj ) = ? ?(li 6= lj )
(3)
where ? is a parameter which influences the typical size of the learned jigsaw pieces. Currently, ? is
set to give the largest pieces whilst maintaining reasonable quality when the image is reconstructed
from the jigsaw.
When learning the jigsaw, it is possible for regions of the jigsaw to be unused, that is, to have no
image pixels mapped to them. To allow for this case, we define a Normal-Gamma prior on ? and ?
for each jigsaw pixel z,
P (J) =
Y
N (?(z); ?0 , (??(z))?1 ) Gamma(?(z); a, b).
(4)
z
This prior means that the behaviour of the model is well defined for unused regions. For our experiments, we fix the hyperparameters ? to .5, ? to 1, b to three times the inverse data variance and a to
the square of b. The local interaction strength ? is set to 5 per channel.
Inference and learning: The model defines the joint probability distribution on a jigsaw J, a set of
images I1 . . . IN , and their offset maps L1 . . . LN to be
N
Y
?
?
P J, {I, L}N
=
P
(J)
P (In |J, Ln )P (L).
1
(5)
n=1
When learning a jigsaw, the image set I1 . . . IN is known and we aim to achieve MAP learning of
the remaining variables. In other words, our goal is to find the jigsaw J and offset maps L1 . . . LN
that maximise the joint probability (5).
We achieve this in an iterative manner. First, the jigsaw is initialised by setting the precisions ? to
the expected value under the prior b/a and the means ? to Gaussian noise with the same mean and
variance as the data. Given this initialisation, the offset maps are updated for each image by applying
the alpha-expansion graph-cut algorithm of [11] (note that our energy is submodular, also known as
regular). Whilst this process will not necessarily find the most probable offset map, it is guaranteed
to find at least a strong local minimum such that no single expansion move can increase (5).
?
?
can be found analytiGiven the inferred offset maps, the jigsaw J that maximises P J, {I, L}N
1
cally. This is achieved for a jigsaw pixel z, the optimal mean ?? and precision ?? by using
P
??0 + x?X(z) I(x)
?
(6)
? =
? + |X(z)|
P
b + ??20 ? (? + |X(z)|)(?? )2 + x?X(z) I(x)2
?1?
(7)
?
=
a + |X(z)|
where X(z) is the set of image pixels that are mapped to the jigsaw pixel z across all images. We
iterate between finding the offset maps holding the jigsaw fixed, and updating the jigsaw using the
recently updated offset maps.
When inference has converged, we apply a clustering step to determine the jigsaw pieces (in future
we plan to extend the model so that this clustering arises directly during learning). Regions of the
image are placed in clusters according to the degree of overlap they have in the jigsaw. The degree
of overlap is measured as the ratio of the intersection to the union of the two regions of the jigsaw
the image regions map to. This has the effect of clustering image regions by both appearance and
shape. Each cluster then corresponds to a region of the jigsaw with an (approximately) consistent
shape that explains a large number of image regions.
4
Results
A toy example: In this experiment, we applied our model to the hand-crafted 150x150 RGB image shown in Fig. 2a. This image was constructed by placing four distinct objects (star, triangle,
square and circle), at random positions on a black background image, with the pixels from the more
recently placed object replacing the previously drawn pixels. Hence, we can see substantial amount
of occlusion of parts of these objects. Using this image as the only input, we would like our model
to automatically infer the appearances and shapes of the objects present in the image.
Existing patch-based models are not well-suited to analyzing this image for two reasons: first, there
is no clear way to choose the appropriate patch shapes and sizes, and secondly, even if such a choice
is known, it is difficult for these existing methods (such as epitome [1]) to learn the shape as they
cannot allow for occlusion boundaries without having an explicit occlusion model. For instance, in
[1], a separate shape epitome is learned in conjunction with the appearance epitome so that image
patches can be explained as a two-layered composition of appearance patches using the shape patch.
However, this type of image is difficult to model with a small number of layers due to the large
(a) Input image
(b) Input image showing segmentation into patches
(c) Jigsaw mean
(d) Jigsaw variance
Figure 2: Toy example: (a) The input image (b) Input image with segmentation boundaries superimposed. Red boundary lines have been drawn on the edge of neighboring pixels that have differing
offsets. This segmentation illustrates the different shaped jigsaw pieces found when learning the jigsaw shown in (c)-(d). (c) Jigsaw mean with the four most-used jigsaw pieces are outlined in white.
(d) The jigsaw variance summed across the RGB channels; white is high, black is low.
number of objects present. In contrast, our model can infer any number of overlapping objects,
without any explicit modelling of layers or depth. This is because our learning algorithm has the
freedom to appropriately adjust a patch?s shape and size to explain only a portion of an object without
explicitly having to represent a global layer ordering. Moreover, we have the potential to infer the
relative depth ordering of neighboring patches by treating rare transitions as occlusions.
Fig. 2b-d shows the results of learning a jigsaw of this toy image. In fig. 2b, we show how the
image decomposes into jigsaw pieces. When two neighboring pixels have different labels, they map
to non-neighboring locations in the jigsaw. With this understanding, we can look at the change in
the labels of the adjacent pixels and plot such a change as a red line. Hence, each region bounded
by the red lines indicates a region from the input image being mapped to the jigsaw. From Fig. 2b,
we can see that the model has discovered well-defined parts (in this example, objects) present in the
image. This is further illustrated in the 36 ? 36 learned jigsaw whose mean and variance are shown
in Fig. 2c,d. The learned jigsaw has captured the shapes and appearances of the four objects and
a black region for modelling the background. Under our Bayesian model, pixels in the jigsaw that
have never been used in explaining the observation are set to ?0 , which we have fixed to .5 (gray).
We can obtain jigsaw pieces by doing the clustering step outlined in Section. 3. In Fig. 2c, we also
show the four most-used jigsaw pieces thus obtained by outlining them in white.
Comparison to epitome model: In this section, we compare the jigsaw model with the epitome
model [1], as applied to the dog image in Fig. 3a. We learned a 32 ? 32 epitome (Fig. 3d) using
all the possible 7 ? 7 patches from the input image. We then learned a 32 ? 32 jigsaw (Fig. 3c)
such that the average patch area was 49 pixels, the same as in the epitome model. This was achieved
by modifying the compatibility parameter ?. Fig. 3b shows the segmentation of the image after
(a) Input image
(b) Image showing segmentation
(c) Jigsaw mean
Reconstructions from:
(e) Jigsaw
Mean squared error: .0537
(f) Epitome (no averaging)
Mean squared error: .0711
(d) Epitome mean
(g) Epitome (averaging 49 patches)
Mean squared error: : .0541
Figure 3: Comparison between jigsaw and epitome. (a) The input image (b) The segmentation of
the image given by the jigsaw model (c,d) The means of the learned jigsaw and epitome models
(e) Reconstruction of the image using the jigsaw (f) Reconstruction from the epitome where each
image pixel is reconstructed using only one fixed-size patch (g) Reconstruction from the epitome
where each image pixel is the average of 49 patches. While this reconstruction has similar mean
squared error to the jigsaw reconstruction, it is more blurry and less visually pleasing.
learning, with patch boundaries overlaid in red. We can see that the pieces correspond to meaningful
regions such as flowers, and also that patch boundaries tend to follow object boundaries.
Comparing Figs. 3c & d, we find that the jigsaw is much less blurred than the epitome and also
doesn?t have the epitome?s artificial ?block? structure. Instead, the boundaries between different
textures are placed to allocate the appropriate amount of jigsaw space to each texture, for example,
entire flowers are represented as one coherent region. Epitome models can use multi-resolution
learning to reduce, but not eliminate, block artifacts. However, whilst this technique can also be
applied to jigsaw learning, it has not been found to be necessary in order to obtain a good solution.
In Fig. 3e-g, we compare reconstructions of the input image from the learned jigsaw and epitome
models. Since the jigsaw is a generative model for an image, we can reconstruct the image by
mapping pixel colors from the jigsaw according to the offset map. When reconstructing from the
epitome, we can choose to either use one patch per pixel, or to average a number of patches per pixel.
The first approach is most comparable to the jigsaw reconstruction, as it requires only one offset per
pixel. However, we find that, as shown in Fig. 3f, the reconstruction is very blocky. When we
reconstruct the each pixel from the 49 overlapping patches (Fig. 3g), we find that the reconstruction
is overly blurry compared to the jigsaw, despite having a similar mean squared reconstruction error.
In addition, this method requires 49 parameters per pixel rather than one and hence is a significantly
less compact representation of the image. The reconstruction from the jigsaw is noticeably less
blurry and is more visibly pleasing as there is no averaging in the generative process and patch
boundaries tend to occur at actual object boundaries.
Modelling face images: We next applied the jigsaw model to a set of 100 face images from the
Olivetti database at AT&T consisting of 10 different images of 10 people. Each of these grayscale
images are of size 64?64 pixels. We set the jigsaw size to 128?128 pixels so that the jigsaw has only
1/25 of the area of the input images combined. Figure 4a shows the inferred segmentation of the
images into different shaped and sized pieces (each row contains the images of one person). When
the faces depict the same person with similar pose, the resulting segmentations for these images are
typically similar, showing that similar jigsaw pieces are being used to explain each image. This can
be seen, for instance, from the first row of images in that figure.
Figure 4b shows the mean of the learned jigsaw which can be seen to contain a number of face
?elements? such as eyes, noses etc. To obtain the jigsaw pieces, we applied the clustering step
outlined in Section. 3. The obtained clusters are shown in Figure 5(left), which also shows the
(a) 100 face images showing learned segmentation
(b) Jigsaw mean
Figure 4: Face images: (a) A set of 100 images, each row containing ten different images of the
same person, with the segmentation given by the jigsaw model shown in red. (b) Jigsaw learned
from these face images, see Figure 5 for clustering results.
sharing of these jigsaw pieces. With the jigsaw pieces known, we can now retrieve the regions from
the image set that correspond to each jigsaw piece. In Figure 5 (right), we show a random selection
of image regions corresponding to several of the most common jigsaw pieces (shown color-coded).
What is surprising is that a particular jigsaw piece becomes very strongly associated with a particular
face part (far more so than when clustering by appearance alone). Thus, by learning the shape of
each jigsaw piece, our model has effectively identified small and large face parts of widely different
Figure 5: Left: The learned face jigsaw of Fig. 4 with overlaid white outlines showing different
overlapping jigsaw pieces. For clarity, pieces used five or fewer times are not shown. Areas of the
jigsaw not used by the remaining pieces have been blacked out. Seven of the most frequently used
jigsaw pieces are shown colored. Right: Unsupervised part learning. For each color-coded jigsaw
piece in the left image, a column shows randomly chosen images from the image set, for which that
piece was selected. Notice how these pieces are very strongly associated with different face parts ?
the model has achieved unsupervised discovery of two different nose shapes, eyes, eyebrows, cheeks
etc, despite their widely different shapes and sizes.
shapes and aspect ratios. We can also see from that figure that certain jigsaw pieces are conserved
across different people ? for example, the nose piece shown in the first column of that figure.
5
Discussion
We have presented a generative jigsaw model which is capable of learning the shape, size and appearance of repeated regions in a set of images. We have also shown that, for a set of face images,
the learned jigsaw pieces are strongly associated with particular face parts.
Currently, we apply a post-hoc clustering step to learn the jigsaw pieces. This process can be incorporated into the model by extending the pixel offset to include a cluster label and learning the region
of jigsaw used by each cluster. We are investigating how best to achieve this.
While we chose a Gaussian as the model for pixel appearance, alternative models can be used, such
as histograms, whilst retaining the ability to achieve translation-invariant clustering. Indeed, by
using appearance models of other forms, we believe that our model could be used to find repeated
structures in other domains such as audio and biology, as well as in images.
Other transformations, such as rotation, scalings and flip, can be incorporated in the model with
cost increasing linearly with the number of transformations. We can also extend the model to allow
the jigsaw pieces to undergo deformation by favoring neighboring offsets that are similar as well as
being identical, using a scheme similar to that of [12].
A practical issue with learning jigsaws is the computational requirement. Every iteration of learning
involves solving as many binary graph cuts as there are pixels in the jigsaw. For instance, for the
toy example, it took about 30 minutes to learn a 36 ? 36 jigsaw from a 150 ? 150 image. We have
since developed a significantly faster inference algorithm based on sparse belief propagation. This
speed-up allows the model to be applied to larger image sets and to learn larger jigsaws.
Currently, our model does not explicitly account for multiple sources of appearance variability, such
as illumination. This means that the same object under different illuminations, for example, will
be modelled by different parts of the jigsaw. To account for this, we are investigating factored
variants of the jigsaw which separate out different latent causes of appearance variability. Despite
this limitation, however, we are already achieving very promising results when using the jigsaw for
image synthesis, motion segmentation and object recognition.
Acknowledgments: We acknowledge helpful discussions with Nebojsa Jojic, and thank the reviewers for their valuable feedback.
References
[1] N. Jojic, B. Frey, and A. Kannan. Epitomic analysis of appearance and shape. In ICCV, 2003.
[2] W. Freeman, E. Pasztor, and O. Carmichael. Learning low-level vision. IJCV, 40(1), 2000.
[3] S. Roth and M. J. Black. Fields of experts: A framework for learning image priors. In Proceedings of
IEEE CVPR, 2005.
[4] A. Efros and W. Freeman. Image quilting for texture synthesis and transfer. In ACM Transactions on
Graphics (Siggraph), 2001.
[5] C. Rother, S. Kumar, V. Kolmogorov, and A. Blake. Digital tapestry. In Proc. Conf. Computer Vision and
Pattern Recognition, 2005.
[6] R. Fergus, P. Perona, and A. Zisserman. Object class recognition by unsupervised scale-invariant learning.
In CVPR, volume 2, pages 264?271, June 2003.
[7] B. Leibe and B. Schiele. Interleaved object categorization and segmentation. In BMVC, 2003.
[8] E. Borenstein, E. Sharon, and S. Ullman. Combining top-down and bottom-up segmentation. In Proceedings IEEE workshop on Perceptual Organization in Computer Vision, CVPR 2004, 2004.
[9] E. Borenstein and S. Ullman. Class-specific, top-down segmentation. In Proceedings of ECCV, 2003.
[10] D. Huttenlocher, D. Crandall, and P. Felzenszwalb. Spatial priors for part-based recognition using statistical models. In Proceedings of IEEE CVPR, 2005.
[11] Y Boykov, O. Veksler, and R. Zabih. Fast approximate energy minimization via graph cuts. PAMI, 23(11),
2001.
[12] J. Winn and J. Shotton. The layout consistent random field for recognizing and segmenting partially
occluded objects. In Proceedings of IEEE CVPR, 2006.
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2,264 | 3,055 | Theory and Dynamics of Perceptual Bistability
Paul R. Schrater?
Departments of Psychology and Computer Sci. & Eng.
University of Minnesota
Minneapolis, MN 55455
[email protected]
Rashmi Sundareswara
Department of Computer Sci. & Eng.
University of Minnesota
[email protected]
Abstract
Perceptual Bistability refers to the phenomenon of spontaneously switching between two or more interpretations of an image under continuous viewing. Although switching behavior is increasingly well characterized, the origins remain
elusive. We propose that perceptual switching naturally arises from the brain?s
search for best interpretations while performing Bayesian inference. In particular,
we propose that the brain explores a posterior distribution over image interpretations at a rapid time scale via a sampling-like process and updates its interpretation
when a sampled interpretation is better than the discounted value of its current interpretation. We formalize the theory, explicitly derive switching rate distributions
and discuss qualitative properties of the theory including the effect of changes in
the posterior distribution on switching rates. Finally, predictions of the theory are
shown to be consistent with measured changes in human switching dynamics to
Necker cube stimuli induced by context.
1 Introduction
Our visual system is remarkably good at producing consistent, crisp percepts of the world around
us, in the process hiding interpretation uncertainty. Perceptual bistability is one of the few circumstances where ambiguity in the visual processing is exposed to conscious awareness. Spontaneous
switching of perceptual states frequently occurs during continuously viewing an ambiguous image,
and when a new interpretation of a previously stable stimuli is revealed (as in the sax/girl in figure ??a), spontaneous switching begins to occur[?]. Moreover, although perceptual switching can
be modulated by conscious effort[?, ?], it cannot be completely controlled.
(a)
(b)
Figure 1: Examples of ambiguous figures: (a) can be interpreted as a woman?s face or a saxophone player. (b)
can be interpreted as a cube viewed from two different viewpoints.
Stimuli that produce bistability are characterized by having several distinct interpretations that are
in some sense equally plausible. Given the successes of Bayesian inference as a model of perception
?
http://www.schrater.org
(for instance [?, ?, ?]), these observations suggest that bistability is intimately connected with making perceptual decisions in the presence of a multi-modal posterior distribution, as previously noted
by several authors[?, ?]. However, typical Bayesian models of perceptual inference have no dynamics, and probabilistic inference per se provides no reason for spontaneous switching, raising the
possibility that switching stems from idiosyncracies in the brain?s implementation of probabilistic
inference, rather than from general principles.
In fact, most explanations of bistability have been historically rooted in proposals about the nature
of neural processing of visual stimuli, involving low-level visual processes like retinal adaptation
and neural fatigue[?, ?, ?]. However, the abundance of behavioral and brain imaging data that show
high level influences on switching (like intentional control which can produce 3-fold changes in
alternation rate[?]) have revised current views toward neural hypotheses involving combinations of
both sensory and higher order cortical processing[?]. The goal of this paper is to provide a simple
explanation for the origins of bistability based on general principles that can potentially handle both
top-down and bottom-up effects.
2 Basic theory
The basic ideas that constitute our theory are simple and partly form standard assumptions about
perceptual processing. The core assumptions are:
1. Perception performs Bayesian inference by exploring and updating the posterior distribution across time by a kind of sampling process (e.g. [?]).
2. Conscious percepts result from a decision process that picks the interpretations by finding
sample interpretations with the highest posterior probability (possibly weighted by the cost
of making errors).
3. The results of these decisions and their associated posterior probabilities are stored in memory until a better interpretation is sampled.
4. The posterior probability associated with the interpretation in memory decays with time.
The intuition behind the model is that most percepts of objects in a scene are built up across a series
of fixations. When an object previously fixated is eccentrically viewed or occluded, the brain should
store the previous interpretation in memory until better data comes along or the memory becomes too
old to be trusted. Finally, the interpretation space required for direct Bayesian inference is too large
for even simple images, but sampling schemes may provide a simple way to perform approximate
inference.
The theory provides a natural interface to interpret both high-level and low-level effects on bistability, because any event that has an impact on the relative heights or positions of the modes in the
posterior can potentially influence durations. For example, patterns of eye fixations have long been
known to influence the dominant percept[?]. Because eye movement events create sudden changes
in image information, it is natural that they should be associated with changes in the dominant
mode. Similarly, control of information via selective attention and changes in decision thresholds
offer concrete loci for intentional effects on bistability.
3 Analysis
To analyze the proposed theory, we need to develop temporal distributions for the maxima of a multimodal posterior based on a sampling process and describe circumstances under which a current
sample will produce an interpretation better than the one in memory. We proceed as follows. First
we develop a general approximation to multi-modal posterior distributions that can vary over time,
and analyze the probability that a sample from the posterior are close to maximal. We then describe
how the samples close to the max interact with a sample in memory with decay.
A tractable approximation to a multi-modal distribution can be formed using a mixture of uni-modal
distributions centered at each maxima.
#maxima
X
P (Dt |?t )P (?t |D0:t??t )
?
?
?
pi (Dt |?t ; ?t,i
)Pi (?|D0:t??t ; ?t,i
) (1)
P (?t |D0:t ) =
P (Dt |D0:t??t )
i=1
?
where ?t is the vector of unknown parameters (e.g. shape for Necker Cube) at time t, ?t,i
is the
location of the maxima of the ith mode, Dt is the most recent data, D0:t??t is the data history,
?
and Pi (?|D0:t??t ; ?t,i
) is the predictive distribution (prior) for the current data based on recent
1
experience .
Near the maxima, the negative log of the uni-modal distributions can be expanded into a secondorder Taylor series:
d2i + ki
?Li (?t |Dt ) ?
=
(?t ?
(2)
? T
?
?t,i
) Ii (?t,i
|D0:t )(?
?
?
?t,i
)
+
1/2 log(|Ii?1 |)
+ ci
(3)
2
(?t |D0:t ))
?
?
where Ii (?t,i
|D0:t ) = ? log(P
is the observed information matrix and ci =
|?t,i
???? T
?
log P (?t,i |D0:t ) represents the effect of the predictive prior on the posterior height at the ith
mode. Thus, samples from a posterior mode will be approximately ?2 distributed near the maximum with effective degrees of freedom n given by the number of significant eigenvalues of Ii?1 .
Essentially n encodes the effective degrees of freedom in interpretation space. 2
3.1 Distribution of transition times
We assume that the perceptual interpretation is selected by a decision process that updates the interpretation in memory m? (t) whenever the posterior probability of the most recent sample both
exceeds a decision threshold and the discounted probability of the sample in memory. Given these
assumptions, we can approximate the probability distribution for update events. Assuming the sampling forms a locally stationary process d2i (t), update events involving entry into mode i are first
passage times Ti of d2i (t) below both the minimum of the current memory sample ?t and the decision threshold ?:
Ti (?, ?t ) = min{t : ?ti ? min{?, ?t }}
where ?ti = d2i (t) ? ki , time t is the duration since the last update event and ?t =
log(P (m? (t)|D0:t )) is the log posterior of the sample in memory at time t. Let Mti = inf 0?s?t ?si .
The probability of waiting at least t for an update event is related to the minima of the process by:
P (Ti (?, ?t ) < t) = P (Mti < min{?, ?t })
This probability can be expressed as:
P (Mti < min{?, ?t }) =
Z t
p(??i < ?t )
0
(4)
p(?t < ?)P (i|? )d? +
Rt
0
p(??i < ?) (1 ? P (?t < ?)) P (i|? )d?
where P (i|t) = P (?t ? Si ) denotes the probability that a sample drawn between times 0 and t is
in the support Si of the ith mode. To generate tractable expressions from equation ??, we make the
following assumptions.
Memory distribution Assume that the memory decay process is slow relative to the sampling
events, and that thePdecay process can be modeled as a random walk in the interpretation space
m? (t) = m? (0) + 0??i ?t ?? (?i ), where ?i are sample times, and ?? are small disturbances with
zero mean and variance ? we assume to be small. Because variances add, the average effect on the
distance ?t is a linear increase: ?t = ?0 + ??t, where ? is the sampling rate. These disturbances
could represent changes in the local of the maxima of the posterior due to the incorporation of new
data, neural noise, or even discounting (note that linearly increasing ?t corresponds to exponential
or multiplicative discounting in probability).
1
Because time is critical to our arguments, we assume that the posterior is updated across time (and hence
new data) using a process that resembles Bayesian updating.
2
For the Necker cube, the interpretation space can be thought of as the depths of the vertices. A strong
prior assumption that world angles between vertices are close to 90deg produces two dominate modes in the
posterior that correspond to the typical interpretations. Within a mode, the brain must still decide whether the
vertices conform exactly to a cube. Thus for the Necker cube, n might be as high as 8 (one depth value per
vertex) or as low as 1 (all vertices fixed once the front corner depth is determined).
To understand the behavior of this memory process, notice that every m? (0) must be within distance ? of the maximum of the posterior for an update to occur. Due to the properties of extrema of
distributions of ?2 random variables, an m? (0) will be (in expectation) a characteristic distance
?m (?) below ? and for t > 0 drifts with linear dynamics3. This suggests the approximation,
p(?t < ?) ? ?(?m + ??t ? ?t ), which can be formally justified because p(?t < ?) will be
highly peaked with respect to the distribution of the sampling process p(??i ). Finally assuming slow
drift means (1 ? P (?t < ?)) ? 0 on the time scale that transitions occur4. Under these assumptions,
equation ?? reduces to:
Z t
P (Mti < min{?, ?t }) =
p(??i < ?t )?(?m + ??t ? ?t )P (i|? )d?
(5)
0
? P (Mti < ?m + ??t)P (i)
where P (i) is the average frequency of sampling from the i
th
(6)
mode.
Extrema of the posterior sampling process If the sampling process has no long-range temporal
dependence, then under mild assumptions the distribution of extrema converge in distribution5 to
one of three characteristic forms that depend only on the domain of the random variable[?]. For ?2
samples, the distribution of minima converges to
P (Mti ? b) = 1 ? exp(?cN ba?1 )
?a+1
where N is the number of samples, c(n) = 2?(a) , a(n) = n2 . Set N = ?t and let ? = 1 for
convenience, where ? is the effective sampling rate, and equation ?? can be written as:
P (Ti < t) = P (Mti ? min{?, ?t }) ?
=
P (Mti < ?m + ?t)P (i)
(7)
a?1
1 ? exp ?c t(?m + ?t)
P (i)
(8)
The probability distribution shows a range of behavior depending on the values of a = n/2 and
?m (?). Note that the time scale for switching. In particular, for n > 4 and ?m (?) relatively small,
the distribution has a gamma-like behavior, where new memory update transitions are suppressed
near recent transitions. For n = 2, or for ?m (?) large, the above equation reduces to exponential.
This behavior shows the effect of the decision threshold, as without a decision threshold the asymptotic behavior of simple sampling schemes will generate approximately exponentially distributed
update event times, as a consequence of extreme value theory. Finally, for n = 1and small ?m (?),
the distribution becomes Cauchy-like with extremely long tails. See figure ?? for example distributions. Note that the time scale of events can be arbitrarily controlled by appropriately selecting ?
(controls the time scale of the sampling process) and ? (controls the time scale of the memory decay
process).
Effects of posterior parameters on update events The memory update distributions are effected
primarily by two factors, the log posterior heights and their difference ?kij = ki ? kj , and the
effective number of degrees of freedom per mode n.
Effect of ki , ?kij The variable ?kij has possible effects both on the probability that a mode is
sampled, and the temporal distributions. When the modes are strongly peaked (and the
sampling procedure is unbiased) log P (i) ? ?kij . Secondly, ?kij effectively sets different
thresholds for each mode, because memory update events occur when:
?ti = d2i (t) ? ki > min{?t , ?}
Increasing the effective threshold for mode i makes updates of type i more frequent, and
should drive the temporal dynamics of the dominant mode toward exponential. Finally,
if the posterior becomes more peaked while the threshold remains fixed, the update rates
should increase and the temporal distributions will move toward exponential. If we assume increased viewing time makes the posterior more peaked, then our model predicts the
common finding of increased transition rates with viewing duration.
3
In the simulations, ?m is chosen as the expected value of the set of events below the threshold xi
Conversely fast drift in the limit means P (?t < ?) ? 0, which results in transitions entirely determined
by the minima of the sampling process and ?.
5
Corresponds to the limit assertion supb |P (Mt ? b) ? exp(??(b, t)t)| ? 0 as t ? ?
4
Prob. Time to next update > t (P(T>t))
1
Effective degrees of freedom
n=8
n=4
n=2
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
500
1000
1500
2000
2500
3000
3500
4000
Time(t) (in number of samples)
Figure 2: Examples of cumulative distribution functions of memory update times. Solid curves
are generated by simulating the sampling with decision process described in the text. Dashed lines
represent theoretical curves based on the approximation in equation ??, showing the quality of the
approximation.
Effect of n One of the surprising aspects of the theory above is the strong dependence on the effective number of degrees of freedom. The theory makes a strong prediction that stimuli that
have more interpretation degrees of freedom will have longer durations between transitions,
which appears to be qualitatively true across both rivalry and bistability experiments[?, ?]
(of course, depending how you interpret the number of degrees of freedom).
Relating theory to behavioral data via induced Semi-Markov Renewal Process Assuming that
memory update events involving transitions to the same mode are not perceptually accessible, only
update events that switch modes are potentially measurable. However, the process described above
fits the description of a generator for a semi-Markov renewal process. A semi-Markov renewal
process involves Markov transitions between discrete states i and j determined by a matrix with
entries Pij , coupled with random durations spent in that state sampled from time distributions Fij (t).
The product of these distributions Qij (t) = Pij Fij (t) is the generator of the process, that describes
the conditional probability of first transition between states i ? j in time less than t, given first
entry into state i occurs at time t = 0. In the theory above, Fij (t) = Fii (t) = P (Ti < t), while
Pij = Pjj = P (j)6
The main reason for introducing the notion of a renewal process is that they can be used to express
the relationship between the theoretical distributions and observable quantities. The most commonly
collected data are times between transitions and (possibly contingent) percept frequencies. Here we
present results found in Ross[?]. Let the state s(t) = i refer to when the memory process is in the
support of mode i: m? (t) ? Si at time t. The distribution of first transition times from state s = i
can be expressed formally as a cumulative probability of first transition:
Gij (t) = P (Nj (t) > 0|s(0) = i) = P (Tj < t|s(0) = i)
where Nj (t) is the number of transitions into state j in time <= t, Tj is the time until first memory
update of type j. For two state processes, only G01 (t) and G10 (t) are measurable. Let P (0), denote
the probability of sampling from mode 0. The relationship between the generating process and the
distribution of first transitions is given by:
Z t
G01 (t) =
G01 (t ? ? )dQ00 (? ) + Q01 (t)
(9)
0
G01 (t)
= P (0)
Z
0
t
G01 (t ? ? )
dP (T0 < ? )
d? + P (1)P (T0 < t)
dt
(10)
which appears only to be solvable numerically for the general form of our memory update transition
functions, however, for the case in which P (T0 < t) is exponential, G01 (t) is as well. Moreover,
6
The independence relations are a consequence of an assumption of independence in the sampling procedure, and relaxing that assumption can produce state contingencies in Qij (t). Therefore, we do not consider
this to be a prediction of the theory. For example, mild temporal dependence (e.g. MCMC-like sampling
with large steps) can create contingencies in the frequency of sampling from the ith mode that will produce a
non-independent transition matrix Pij = P (?t ? Si |?t???t ? Sj ).
for gamma-like distributions, the convolution integral tends to increase the shape parameter, which
means that gamma parameter estimates produced by fitting transition durations will overestimate
the amount of ?memory? in the process7 . Finally note the limiting behavior as P (0) ? 0, G01 (t) =
P (T0 < t), so that direct measurement of the temporal distributions is possible but only for the
(almost) supressed perceptual state. Similar relationships exist for survival probabilities, defined as
Sij (t) = P (s(t) = j|s(0) = i)
4 Experiments
In this section we investigate simple qualitative predictions of the theory, that biasing perception
toward one of the interpretations will produce a coupled set of changes in both percept frequencies and durations, under the assumption that perceptual biases result from differences in posterior
heights . To bias perception of a bistable stimuli, we had observers view a Necker cube flanked
with ?fields of cubes? that are perceptually unambiguous and match one of the two percepts (see
figure ??). Subjects are typically biased toward seeing the Necker cube in the ?looking down? state
(65-70% response rates), and the context stimuli shown in figure ??a) have little effect on Necker
cube reversals. We found that the looking up context, boosts ?looking up? response rates from 30%
to 55%.
4.1 Methods
Subject?s perceptual state while viewing the stimuli in fig. ?? were collected using the methods
described in[?]. Eye movement effects[?] were controlled by having observers focus on a tiny
sphere in the center of the Necker cube, and attention was controlled using catch trials. Base rates
for reversals were established for each observer (18 total) in a training phase. Each observer viewed
100 randomly generated context stimuli and each stimulus was viewed long enough to acquire 10
responses (taking 10-12 sec on average). For ease of notation, we represent the ?Looking down?
condition as state 0 and the ?Looking Up? as state 1.
(a) An instance of the ?Looking down? con- (b) An instance of the ?Looking up? context
text with the Necker cube in the middle
with the Necker cube in the middle
Figure 3: The two figures are examples of the ?Looking down? and ?Looking up? context conditions.
4.2 Results
We measured the effect of context on estimates of perceptual switching rates, Ri = P (s(t) = i),
first transition durations Gij , and survival probabilities Pii = P (s(t) = i|s(0) = i) by counting
the number of events of each type. Additionally, we fit a semi-Markov renewal process Qij (t) =
Pij Fij (t) to the data using a sampling based procedure. The procedure is too complex to fully
describe in this paper, so a brief description follows. For ease of sampling, Fij (t) were gamma with
separate parameters for each of the four conditionals {00, 01, 10, 11}, resulting in 10 parameters
7
gamma shape parameters are frequently interpreted as the number of events in some abstract Poisson
process that must occur before transition
overall. The process was fit by iteratively choosing parameter values for Qij (t), simulating response
data and measuring the mismatch between the simulated and human Gij and Pii distributions.
The effect of context on Gij and Pii is shown in Fig.?? and Fig.?? for the contexts ?Looking
Down? and ?Looking Up? respectively. The figures also show the maximum likelihood fitted gamma
functions. Testable predictions generated by simulating the memory process described above were
verified, including changes in mean durations of about 2sec, coupling of the duration distributions,
and an increase in the underlying renewal process shape parameters when the percepts are closer to
equally probable.
Shown "Down" context
P(1st transition|S(O)=?Up?)
Probability
P( 1st transition|S(O)="Down")
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
P(Surv. |S(0)="Down")
0.3
0.2
Human data
Max. Likelihood fit
0.1
0
P(Survival |S(0)="Up")
0
5
10
0.3
0.2
Human data
Max. Likelihood fit
0.1
0
0
5
Time
10
Time
Figure 4: Data pooled across subjects for the ?Looking Down? context condition. (a) Prob. of first transition
and the survival probability of the ?Looking down? percept. (b)Prob. of first transition and conditional survival probability of the ?Looking Up? percept. A semi-Markov renewal process with transition paramters Pij ,
gamma means mij and gamma variances vij was fit to all the data via max. likelihood. The best fit curves are
superimposed on the data.
Shown "Up" context
Probability
1
P( 1st transition|S(O)="Down")
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
P(Survival |S(0)="Down")
0.2
P(Survival |S(0)="Up")
0.3
0.2
Human data
Max. Likelihood fit
0.1
0
P( 1st transition|S(O)="Up" )
0
5
10
Human data
Max. Likelihood fit
0.1
0
0
5
10
Figure 5: Same as figure ??, but for the ?Looking Up? context condition.
5 Discussion/Conclusions
Although [?] also presents a theory for average transitions times in bistability based on random processes and a multi-modal posterior distribution, their theory is fundamentally different as it derives
switching events from tunneling probabilities that arise from input noise. Moreover, their theory
predicts increasing transition times as the posterior becomes increasingly peaked, exactly opposite
our predictions.
In conclusion, we have presented a novel theory of perceptual bistability based on simple assumptions about how the brain makes perceptual decisions. In addition, results from a simple experiment
show that manipulations which change the dominance of a percept produce coupled changes in the
probability of transition events as predicted by theory. However, we do not regard the experiment as
a strong test of the theory. We believe the strength of the theory is that it can make a large set of qualitative predictions about the distribution of transition events by coupling transition times to simple
properties of the posterior distribution. Our theory suggests that the basic descriptive model sufficient to capture perceptual bistability is a semi-Markov renewal process, which we showed could
successfully simulate the temporal dynamics of human data for the Necker cube.
References
[1] Aldous, D .(1989) Probability approximations via the Poisson clumping heuristic. Applied Math. Sci, 77.
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[2] Bialek, W., DeWeese, M. (1995) Random Switching and Optimal Processing in the Perception of Ambiguous Signals. Physics Review Letters 74(15) 3077-80.
[3] Brascamp, J. W., van Ee, R., Pestman, W. R., & van den Berg, A. V. (2005). Distributions of alternation
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[12] Rock, I. and Mitchener, K.(1992) Further evidence of the failure of reversal of ambiguous figures by
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2,265 | 3,056 | Kernels on Structured Objects Through Nested
Histograms
Marco Cuturi
Institute of Statistical Mathematics
Minami-azabu 4-6-7, Minato ku,
Tokyo, Japan.
Kenji Fukumizu
Institute of Statistical Mathematics
Minami-azabu 4-6-7, Minato ku,
Tokyo, Japan.
Abstract
We propose a family of kernels for structured objects which is based on the bag-ofcomponents paradigm. However, rather than decomposing each complex object
into the single histogram of its components, we use for each object a family of
nested histograms, where each histogram in this hierarchy describes the object
seen from an increasingly granular perspective. We use this hierarchy of histograms to define elementary kernels which can detect coarse and fine similarities
between the objects. We compute through an efficient averaging trick a mixture
of such specific kernels, to propose a final kernel value which weights efficiently
local and global matches. We propose experimental results on an image retrieval
experiment which show that this mixture is an effective template procedure to be
used with kernels on histograms.
1 Introduction
Kernel methods have shown to be competitive with other techniques in classification or regression
tasks where the input data lie in a vector space. Arguably, this success rests on two factors: first,
the good ability of kernel algorithms, such as the support vector machine, to generalize and provide a sparse formulation for the underlying learning problem; second, the capacity of nonlinear
kernels, such as the polynomial and gaussian kernels, to quantify meaningful similarities between
vectors, notably non-linear correlations between their components. Using kernel machines with
non-vectorial data (e.g., in bioinformatics, image and text analysis or signal processing) requires
more arbitrary choices, both to represent the objects in a malleable form, and to choose suitable
kernels on these representations. The challenge of using kernel methods on real-world data has thus
recently fostered many proposals for kernels on complex objects, notably strings, trees, images or
graphs to cite a few.
In common practice, most of these objects can be regarded as structured aggregates of smaller components, and the coarsest approach to study such aggregates is to consider them directly as bags
of components. In the field of kernel methods, such a representation has not only been widely
adopted (Haussler, 1999; Joachims, 2002; Sch?olkopf et al., 2004), but it has also spurred the proposal of kernels better suited to the geometry of the underlying histograms (Kondor & Jebara, 2003;
Lafferty & Lebanon, 2005; Hein & Bousquet, 2005; Cuturi et al., 2005). However, one of the drawbacks of the bag-of-components representation is that it implicitly assumes that each component
sampled in the object has been generated independently from an identical distribution. While this
viewpoint may translate into adequate properties for some learning tasks, such as translation or rotation invariance when using histograms of colors to manipulate images (Chapelle et al., 1999), it
may however appear too restrictive when such a strong invariance may just be too coarse to be of
practical use.
A possible way to cope with this limitation is to expand artificially the size of the components?
space, either by considering families of larger components to take into account more contextual information, or by considering histograms which index both components and their possible location
in the object (R?atsch & Sonnenburg, 2004). As one would expect, these histograms are usually
sparse and need to be regularized using ad-hoc rules and prior knowledge (Leslie et al., 2003) before being directly compared using kernels on histograms. For sequential data, other state-of-the-art
methods compute an optimal alignment between the sequences based on elementary operations such
as substitutions, deletions and insertions of components. Such alignment scores may yield positive
definite (p.d.) kernels if particular care is taken to adapt them (Vert et al., 2004) and have shown very
competitive performances. However, their computational cost can be prohibitive when dealing with
large datasets, and can only be applied to sequential data. Following these contributions, we propose
t1
t2
t1
t2.1
t2.1
t2.2
t2.2
t2
Figure 1: From the bag of components representation to a set of nested bags, using a set of labels.
in this paper new families of kernels which can be easily tuned to detect both coarse and fine similarities between the objects, in a range spanned from kernels which only consider coarse histograms
to kernels which only detect strict local matches. To size such types of similarities between two
objects, we elaborate on the elementary bag-of-components perspective to consider instead families
of nested histograms (indexed by a set of hierarchical labels to be defined) to describe each object.
In this framework, the root label corresponds to the global representation introduced before, while
longer labels represent a specific condition under which the components have been sampled. We
then define kernels that take into account mixtures of similarities, spanning from detailed resolutions which only compare the smallest bags to the coarsest one. This trade-off between fine and
coarse perspectives sets an averaging framework to define kernels, which we introduce formally in
Section 2. This theoretical framework would not be tractable without an efficient factorization detailed in Section 3 which yields computations which grow linearly in time and space with respect
to the number of labels to evaluate the value of the kernel. We then provide experimental results in
Section 4 on an image retrieval task which shows that the methodology improves the performance
of kernel based state-of-the art techniques in this field with a low extra computational cost.
2 Kernels Defined through Hierarchies of Histograms
In the kernel literature, structured objects are usually represented as histograms of components, e.g.,
images as histograms of colors and/or features, texts as bags of words and sequences as histograms
of letters or n-grams. The obvious drawback of this representation is that it usually loses all the
contextual information which may be useful to characterize each sampled component in the original
object. One may instead create families of histograms, indexed by specific sampling conditions:
? In image analysis, create color or feature histograms following a prior partition of the image
into predefined patches, as in (Grauman & Darrell, 2005). Another possibility would be to
define families of histograms, all for the same image, which would consider increasingly
granular discretizations of the color space.
? In sequence analysis, extract local histograms which may correspond to predefined regions
of the original sequence, as in (Matsuda et al., 2005). A different option would be to associate to each histogram a context of arbitrary length, e.g. by considering the 26 histogram
of letters sampled just after the letters {A, B, ? ? ? , Z}, or the 26 ? 26 histograms of letters
after contexts {AA, AB, ? ? ? , ZZ}.
? In text analysis, use histograms of words found after grammatical categories of increasing
complexity, such as verbs, nouns, articles or adverbs.
? For synchronous time series (e.g. financial time series or gene expression profiles), define
a reference series (e.g. an index or a specific gene) and decompose each of the subsequent
series into histograms of values conditioned to the value of the reference series.
We write L for an arbitrary index set to label such specific histograms. Structured objects are thus
def
b
represented as a family ? of ML (X ) = (M+
(X ))L , that is ? = {?t }t?L where for each t ? L, ?t
P
b
is a bounded measure of M+ (X ). We write |?| for t?L |?t |.
2.1 Local Similarities Between Measures
To compare two objects under the light of any sampling condition t, that is comparing their respecb
tive decompositions as measures ?t and ?0t , we make use of an arbitrary p.d. kernel k on M+
(X )
to which we will refer as the base kernel throughout the paper. For interpretation purposes only, we
will assume in the following sections that k is an infinitely divisible kernel which can be written
1
b
as k = e? ? ? , ? > 0, where ? is a negative definite (Berg et al., 1984) kernel on M+
(X ), or
equivalently ?? is a conditionally p.d. kernel. Note also that k has to be p.d. not only on probability measures, but on any bounded measure. For two elements ?, ?0 of ML (X ) and a given element
t ? L, the kernel
def
kt (?, ?0 ) = k(?t , ?0t )
0
quantifies the similarity of ? and ? by measuring how similarly their components were observed
with respect to label t. For two different labels s and t of L, ks and kt can be associated through
polynomial combinations with positive coefficients to result in new kernels, notably their sum ks +kt
or their product ks kt . This is particularly adequate if some complementarity is assumed between s
and t, so that their combination can provide new insights for a given learning task. If on the contrary
these labels are assumed to be similar, then they can be regarded as a grouped label {s} ? {t} and
result in the kernel
def
k{s}?{t} (?, ?0 ) = k(?s + ?t , ?0s + ?0t ),
which will measure the similarity of m and m0 under both s or t labels. Let us give an intuition
for this definition by considering two texts A, B built up with words from a dictionary D. As an
b
alternative to the general histograms of words ?A and ?B of M+
(D), one may consider for instance
A
A
B
B
?can , ?may and ?can , ?may , the respective histograms of words that follow the words can and may in
texts A and B respectively. If one considers that can and may are different words, then the following
kernel quantifies the similarity of A and B taking advantage of this difference:
B
A
B
A
).
, ?may
k{can},{may} (A, B) = k(?can
, ?can
) ? k(?may
If on the contrary one decides that can and may are equivalent, an adequate kernel would first
merge the histograms, and then compare them:
A
A
B
B
k{can,may} (A, B) = k(?can
+ ?may
, ?can
+ ?may
).
The previous formula can be naturally extended to define kernels indexed on a set T ? L of grouped
labels, through
X
X
def
def
def
kT (?, ?0 ) = k (?T , ?0T ) , where ?T =
?t and ?0T =
?0t .
t? T
t? T
2.2 Resolution Specific Kernels
Having defined a family of kernels {kT , T ? L} which can detect conditional similarities between
two elements of ML (X ) given a subset T of L, we define in this section different ways to combine
them to obtain a kernel which can take into account all of their histograms. Let P be a finite partition
of L,Sthat is a finite family P = (T1 , ..., Tn ) of sets of L, such that Ti ? Tj = ? if 1 ? i < j ? n
n
and i=1 Ti = L. We write P(L) for the set of all partitions of L. Consider now the kernel defined
by a partition P as
n
Y
def
kP (?, ?0 ) =
kTi (?, ?0 ).
(1)
i=1
The kernel kP quantifies the similarity between two objects by detecting their joint similarity under
all possible labels of L, assuming a priori that certain labels can be grouped together, following the
subsets Ti enumerated in the partition P . Note that there is some arbitrary in this definition since
a simple multiplication of base kernels kTi is used to define kP , rather than any other polynomial
combination. We follow in that sense the convolution kernels (Haussler, 1999) approach, and indeed,
for each partition P , kP can be regarded as a convolution kernel. More precisely, the multiplicative
structure of Equation (1) quantifies how similar two objects are given a partition P , in a way that
imposes for the objects to be similar according to all subsets Ti . If the base kernel k can be written
1
as k = e? ? ? , where ? is a negative definite kernel, then kP can be expressed as the exponential of
minus
n
n
X
X
def
?P (?, ?0 ) =
?Ti (?, ?0 ) =
?(?Ti , ?0Ti ),
i=1
i=1
0
a quantity which penalizes local differences between the decompositions
P
P 0of ? and ? over L, as
opposed to the coarsest approach where P = {L} and only ?( t ?t , t ?t ) is considered.
Figure 2: A useful set of labels L for images which would focus on pixel localization can be represented by a grid, such as the 8 ? 8 one represented above. In this case P3 corresponds to the 43
windows presented in the left image, P2 to the 16 larger squares obtained when grouping 4 small
windows, P1 to the image divided into 4 equal parts and P0 is simply the whole image. Any partition
P of the image which complies with the hierarchy P03 in the example above, can in turn be used to
represent an image as a family of sub-probability measures, which reduces in the case of two-color
images to binary histograms as illustrated in the right-most image. For two images, these respective
histograms can be directly compared through the kernel kP .
As illustrated in Figure 2, where images are summarized through histograms indexed by patches, a
partition of L reflects a given belief on how patches may or may not be associated or split to focus
on local dissimilarities. Hence, all partitions contained in the set P(L) of all possible partitions1
are not likely to be equally meaningful given that some labels may a natural form of grouping. If
the index is built to highlight differences in locations, one would naturally favor mergers between
neighboring indexes. If one uses a Markovian analysis, that is consider histograms of components
conditioned by contexts, a natural way to group contexts would be to group them according to their
semantic or grammatical content for text analysis or according to their suffix for sequence analysis.
Such meaningful partitions can be intuitively obtained when a hierarchical structure which groups
elements of L together is known a priori. A hierarchy on L, such as the triadic hierarchy shown in
Figure 3, is a family
(Pd )D
d=0 = {P0 = {L}, .., PD = {{t}, t ? L}}
of partitions of L. To provide a hierarchical information, the family (Pd )D
d=0 is such that any subset
present in a partition Pd is strictly included in a (unique by definition of a partition) subset from
the coarser partition Pd?1 . This is equivalent to stating that each subset T in a partition Pd is
divided in Pd+1 as a partition of T which is not T itself. We write s(T ) for this partition (e.g.,
in Figure 3, s(1) = {11 , ? ? ? , 19 }) and name its elements the siblings of T . Consider now the
subset PD ? P(L) of all partitions of L obtained by using only sets contained in the collection
def SD
def
P0D = d=0 Pd , namely PD = {P ? P(L) s.t. ? T ? P, T ? P0D }. The set PD contains both the
coarsest and the finest resolutions, respectively P0 and PD , but also all variable resolutions for sets
enumerated in P0D , as can be seen for instance in the third image of Figure 2.
1
P
P(L) is quite a big space, since if L is a finite set of cardinal r, the cardinal of the set of partitions is known
ur
as the Bell Number of order r with Br = 1e ?
? er ln r .
u=1 u!
r??
11
1
2
3
4
5
6
7
8
9
19
61
0
73
P0
P1
99
P2
Figure 3: A hierarchy generated by two successive triadic partitions.
2.3 Averaging Resolution Specific Kernels
Each partition P contained in PD provides a resolution to compare two objects, which generates
a large family of kernels kP when P spans PD . Some partitions are likely to be better suited for
certain tasks, which may call for an efficient estimation scheme to select an optimal partition for
a given task. This would be similar in spirit to estimating a maximum a posteriori model for the
data and use it consequently to compare the objects. We take in this section a different direction
which has a more Bayesian flavor by considering an averaging of such kernels based on a prior
on the set of partitions. In practice, this averaging favours objects which share similarities under a
large collection of resolutions, and may also be interpreted as a Bayesian averaging of convolution
kernels (Haussler, 1999).
Definition 1 Let L be an index set endowed with a hierarchy (Pd )D
d=0 , ? be a prior measure on the
b
b
corresponding set of partitions PD and k a base kernel on M+
(X ) ? M+
(X ). The averaged kernel
k? on ML (X ) ? ML (X ) is defined as
X
k? (?, ?0 ) =
?(P ) kP (?, ?0 ).
(2)
P ? PD
As can be observed in Equation (2), the kernel automatically detects in the range of all partitions
the ones which provide a good match between the compared objects, to increase subsequently the
resulting similarity score. Also note that in an image-analysis context, the pyramid-matching kernel proposed in (Grauman & Darrell, 2005) only considers the original partitions of the hierarchy
(Pd )D
d=0 , while Equation (2) considers all possible partitions of PD . This can be carried out with
little cost if an adequate set of priors ? is selected as seen below.
3 Kernel Computation
We provide in this section hierarchies (Pd )D
d=0 and priors ? for which the computation of k? is both
meaningful and tractable, yielding namely a computational time to calculate k? which is loosely
upperbounded by D ? card L ? c(k) where c(k) is the time required to compute the base kernel.
3.1 Partitions Generated by Branching Processes
All partitions P of PD can be generated through the following rule, starting from the initial root
partition P := P0 = {L}. For each set T of P :
1. either leave the set as it is in P with probability 1 ? ?T ,
2. either replace it by its siblings in s(T ) with probability ?T , and reapply this rule to each
sibling unless they belong to the finest partition PD .
The resulting prior for PD depends on the overall coarseness of the considered partitions, and can be
tuned through parameters ?T to favor adaptively coarse or fine partitions. For a partition P ? PD ,
?
Q
Q
D
?(P ) = T ? P (1 ? ?T )
? (?T ), where the set P = {T ? P0 s.t. ?V ? P, V ( T } gathers
T?P
all coarser sets belonging to coarser resolutions than P , and can be regarded as the set of all ancestors
in P0D of sets enumerated in P .
3.2 Factorization of k?
We use the branching-process prior can be used to factorize the formula in Equation (2):
Proposition 2 For two elements ?, ?0 of ML (X ), define for T spanning recursively all sets contained in PD , PD?1 , ..., P0 the quantity KT below; then k? (?, ?0 ) = KL .
Y
KT = (1 ? ?T )kT (?, ?0 ) + ?T
KU .
U ? s(T )
Proof
The proof follows from a factorization which
uses the branching process prior used for the
tree generation, and can be derived from the
proof of (Catoni, 2004, Proposition 5.2). The
opposite figure underlines the importance of
incorporating to each node KT a weighted
product of the sibling kernel evaluations KU .
The update rule for the computation of k?
takes into account the branching process prior
by weighting the kernel kT with all values kti
obtained for finer resolutions ti in s(T ).
K t1
?t1
?0t1
K t2
?t2
?0t2
K t3
KT = (1 ? ?T )k(?T , ?0T ) + ?T
Q
K ti
P
?T = P ?ti
0
?T = ?0ti
?t3
?0t3
If the hierarchy of L is such that the cardinality of s(T ) is fixed to a constant ? for any set T , typically
? = 4 for images in the case described in Figure 2, then the computation of k? is upperbounded
by (?D+1 ? 1)c(k). This complexity is also upperbounded by the total amount of components
considered in the compared objects, as in (Cuturi & Vert, 2005) for instance.
3.3 Choosing the Base Kernel
b
Any kernel on M+
(X ) can be used to comply with the terms of Definition 1 and apply an average
scheme on families of measures. We also note that an even more general formulation can be obtained
by using a different kernel kt for each label t of L, without altering the overall applicability of the
factorization above. However, we only consider in this discussion a unique choice k for all t ? L.
First, one can note that kernels such as the information diffusion kernel (Lafferty & Lebanon, 2005)
and variance based kernels (Kondor & Jebara, 2003; Cuturi et al., 2005) may not work in this
b
setting since they are not p.d., nor sometimes defined, on the whole of M+
(X ). The most adequate
b
geometry of M+ (X ), following the denormalization scheme proposed in (Amari & Nagaoka, 2001,
?
p.47), may arguably be derived from the Riemannian embedding ? 7? ?, where the Euclidian
distance between two measures in this representation is equal to the geodesic distance between ?
b
and ? 0 in M+
(X ) endowed with the Fisher metric, as expressed in ?H2 below. More generally,
one can consider the whole family of kernels for bounded measures described in (Hein & Bousquet,
1
2005) to choose the base kernel k, namely the family of Hilbertian metrics ? such that k = e? ? ? .
We thus use in our experiments the Jensen divergence, the ?2 distance, the total variation, and two
variations of the Hellinger distance:
X (?i ? ?0 )2
? + ?0
h(?) + h(?0 )
i
0
?JD (?, ? ) = h
?
, ??2 (?, ?0 ) =
0 ,
2
2
?
+
?
i
i
i
X
X p
X p
p
p
| ?i ? ?i0 |.
?T V (?, ?0 ) =
|?i ? ?i0 |, ?H2 (?, ?0 ) =
| ?i ? ?i0 |2 , ?H1 (?, ?0 ) =
i
i
i
4 Experiments in Image Retrieval
We present in this section experiments inspired by the image retrieval task first considered
in (Chapelle et al., 1999) and reused in (Hein & Bousquet, 2005). Our dataset was also extracted
from the Corel Stock database and includes 12 families of labeled images, each class containing
2
log2 ( ?1 )
0
-6
-12
0
1/2
?
1
0.23
0.22
0.21
0.2
0.19
0.18
0.17
0.16
0.15
0.14
0.13
Figure 4: Misclassification rate on the corel experiment, using the Hellinger H1 distance between
histograms coupled with one-vs-all SVM classification (C = 100) as a function of ? and ?. ?1
is taken in {2?12 , ? ? ? , 22 } while ? spans {0, 0.1, ? ? ? , 0.9, 1}. ? controls the granularity of the
averaging kernel, ranging from the coarsest perspective (? = 0) when only the global histogram is
used, to the finest one (? = 1) when only the finest histograms are considered. Dark values represent
error rates which are greater or equal to 24%. The central values are roughly 14.5% while the best
value obtained in the columns ? = 0 and ? = 1 are 18.4% and 17.3% respectively
100 color images of 256 ? 384 pixels. The families depict images of bears, African specialty animals, monkeys, cougars, fireworks, mountains, office interiors, bonsais, sunsets, clouds, apes and
rocks and gems. The database is randomly split into balanced sets of 800 training images and 400
test images. The task consists in classifying the test images with the rule learned by training 12
one-versus-all SVM?s on the learning fold. Note that previous work conducted in (Chapelle et al.,
1999) illustrates the competitiveness of SVM?s in this context over other algorithms such as nearest
neighbors. Our results are averaged over 3 random splits, using the Spider toolbox.
We used 9 bits for the color of each pixel to reduce the size of the RGB color space to 83 = 512
from the original set of 2563 = 16, 777, 216 colors, and we defined centered grids of 4, 42 = 16
and 43 = 64 local patches. We provide results for each of the 5 considered kernels and for each
considered depth D ranging from 1 to 3. Figure 5 presents 15 = 5?3 plots, where each plot displays
the misclassification rate as a function of the width parameter ?1 and the branching process prior ?
set over all nodes of the tree. The constant C is set to 100, but other choices for C (1000 and 10)
gave comparable plots, although a bit different in shape. By considering values of ? ranging from 0
to 1, we aim at giving a sketch of the robustness of the averaging approach, since the SVM?s seem
to perform better when 0 < ? < 1 for a large span of ? values. For a better understanding of these
plots, the reader may refer to Figure 4 which focuses on ?H1 and D = 2, noting that the color scales
used for Figures 4 and 5 are the same. Finally, the Gaussian kernel was also tested but its very poor
performance (with error rate above 22% for all parameters) illustrates once more that the Gaussian
kernel is usually a poor choice to compare histograms directly.
5 Discussion
The computation of averaged kernels can be performed almost as fast as kernels which only rely
on fine resolutions, which along with their robustness and improved performance might advocate
their use, notably as an extension of kernels based on arbitrary partitions (Grauman & Darrell, 2005;
Matsuda et al., 2005). Principled ways of estimating in a semi-supervised setting both ? and ?, or
preferably localized priors ?T and ?T , T ? P0D , might give them an additional edge. This is a topic
of current research, and we suggest to set these parameters through cross-validation at the moment,
while H1 seems to be a reasonable choice to define the base kernel. Our approach is related to
the Multiple Kernel Learning framework (Lanckriet et al., 2004), although we do not aim here at
learning linear combinations of the kernels kT , but rather start from an hierarchical belief on them
to propose an algebraic combination.
Acknowledgments: This research was supported by the Function and Induction Research Project,
Transdisciplinary Research Integration Center - Research Organization of Information and Systems.
H1
H2
TV
Xi2
JD
D=1
D=2
D=3
Figure 5: Error-rate results for different kernels and depths are displayed in the same way that in
Figure 4, using the same colorscale across experiments.
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2,266 | 3,057 | Inferring Network Structure from Co-Occurrences
Michael G. Rabbat
Electrical and Computer Eng.
University of Wisconsin
Madison, WI 53706
[email protected]
M?ario A.T. Figueiredo
Instituto de Telecomunicac?o? es
Instituto Superior T?ecnico
Lisboa, Portugal
[email protected]
Robert D. Nowak
Electrical and Computer Eng.
University of Wisconsin
Madison, WI 53706
[email protected]
Abstract
We consider the problem of inferring the structure of a network from cooccurrence data: observations that indicate which nodes occur in a signaling pathway but do not directly reveal node order within the pathway. This problem is
motivated by network inference problems arising in computational biology and
communication systems, in which it is difficult or impossible to obtain precise
time ordering information. Without order information, every permutation of the
activated nodes leads to a different feasible solution, resulting in combinatorial
explosion of the feasible set. However, physical principles underlying most networked systems suggest that not all feasible solutions are equally likely. Intuitively, nodes that co-occur more frequently are probably more closely connected.
Building on this intuition, we model path co-occurrences as randomly shuffled
samples of a random walk on the network. We derive a computationally efficient
network inference algorithm and, via novel concentration inequalities for importance sampling estimators, prove that a polynomial complexity Monte Carlo version of the algorithm converges with high probability.
1
Introduction
The study of complex networked systems is an emerging field impacting nearly every area of engineering and science, including the important domains of biology, cognitive science, sociology, and
telecommunications. Inferring the structure of signalling networks from experimental data precedes
any such analysis and is thus a basic and fundamental task. Measurements which directly reveal network structure are often beyond experimental capabilities or are excessively expensive. This paper
addresses the problem of inferring the structure of a network from co-occurrence data: observations
which indicate nodes that are activated in each of a set of signaling pathways but do not directly reveal the order of nodes within each pathway. Co-occurrence observations arise naturally in a number
of interesting contexts, including biological and communication networks, and networks of neuronal
colonies.
Biological signal transduction networks describe fundamental cell functions and responses to environmental stress [1]. Although it is possible to test for individual, localized interactions between
gene pairs, this approach (called genetic epistatic analysis) is expensive and time-consuming. Highthroughput measurement techniques such as microarrays have successfully been used to identify
the components of different signal transduction pathways [2]. However, microarray data only reflects order information at a very coarse, unreliable level. Developing computational techniques for
inferring pathway orders is a largely unexplored research area [3].
A similar problem has been studied in telecommunication networks [4]. In this context, each path
corresponds to a transmission between an origin and destination. The origin and destination are observed, in addition to the activated switches/routers carrying the transmission through the network.
However, due to the geographically distributed nature of the measurement infrastructure and the rapidity at which transmissions are completed, it is not possible to obtain precise ordering information.
Another exciting potential application arises in neuroimaging [5, 6]. Functional magnetic resonance
imaging provides images of brain activity with high spatial resolution but has relatively poor temporal resolution. Treating distinct brain regions as nodes in a functional brain network that co-activate
when a subject performs different tasks may lead to a similar network inference problem.
Given a collection of co-occurrences, a feasible network (consistent with the observations) is easily
obtained by assigning an order to the elements of each co-occurrence, thereby specifying a path
through the hypothesized network. Since any arbitrary order of each co-occurrence leads to a feasible network, the number of feasible solutions is proportional to the number of permutations of all the
co-occurrence observations. Consequently we are faced with combinatorial explosion of the feasible
set, and without additional assumptions or side information there is no reason to prefer one particular
feasible network over the others. See the supplementary document [7] for further discussion.
Despite the apparent intractability of the problem, physical principles governing most networks
suggest that not all feasible solutions are equally plausible. Intuitively, nodes that co-occur more
frequently are more likely to be connected in the underlying network. This intuition has been used
as a stepping stone by recent approaches proposed in the context of telecommunications [4], and in
learning networks of collaborators [8]. However, because of their heuristic nature, these approaches
do not produce easily interpreted results and do not readily lend themselves to analysis or to the
incorporation of side information.
In this paper, we model co-occurrences as randomly permuted samples of a random walk on the
underlying network. The random permutation accounts for lack of observed order. We refer to this
process as the shuffled Markov model. In this framework, network inference amounts to maximum
likelihood estimation of the parameters governing the random walk (initial state distribution and
transition matrix). Direct maximization is intractable due to the highly non-convex log-likelihood
function and exponential feasible set arising from simultaneously considering all permutations of all
co-occurrences. Instead, we derive a computationally efficient EM algorithm, treating the random
permutations as hidden variables. In this framework the likelihood factorizes with respect to each
pathway/observation, so that the computational complexity of the EM algorithm is determined by
the E-step which is only exponential in the longest path. In order to handle networks with long
paths, we propose a Monte Carlo E-step based on a simple, linear complexity importance sampling
scheme. Whereas the exact E-step has computational complexity which is exponential in path length,
we prove that a polynomial number of importance samples suffices to retain desirable convergence
properties of the EM algorithm with high probability. In this sense, our Monte Carlo EM algorithm
breaks the curse of dimensionality using randomness.
It is worth noting that the approach described here differs considerably from that of learning the
structure of a directed graphical model or Bayesian network [9, 10]. The aim of graphical modelling is to find a graph corresponding to a factorization of a high-dimensional distribution which
predicts the observations well. These probabilistic models do not directly reflect physical structures,
and applying such an approach to co-occurrences would ignore physical constraints inherent to the
observations: co-occurring vertices must lie along a path in the network.
2
2.1
Model Formulation and EM Algorithm
The Shuffled Markov Model
We model a network as a directed graph G = (V, E), where V = {1, . . . , |V |} is the vertex (node)
set and E ? V 2 is the set of edges (direct connections between vertices). An observation, y ? V ,
is a subset of vertices co-activated when a particular stimulus is applied to the network (e.g., collection of signaling proteins activated in response to an environmental stress). Given a set of T
(m)
(m)
observations, Y = {y(1), . . . , y(T ) }, each corresponding to a path, where y(m) = {y1 , . . . , yNm },
we say that a graph (V, E) is feasible w.r.t. Y if for each y(m) ? Y there is an ordered path
(m)
(m)
(m)
(m)
(m)
(m)
z(m) = (z1 , . . . , zNm ) and a permutation ? (m) = (?1 , . . . , ?Nm ) such that zt = y (m) , and
?t
(zt?1 , zt ) ? E, for t = 2, ..., Nm .
The (unobserved) ordered paths, Z = {z(1) , ..., z(T ) }, are modelled as T independent samples of
a first-order Markov chain with state set V . The Markov chain is parameterized by the initial state
distribution ? and the (stochastic) transition matrix A. We assume that the support of the transition
matrix is determined by the adjacency structure of the graph; i.e., Ai,j > 0 ? (i, j) ? E. Each
observation y(m) results from shuffling the elements of z(m) via an unobserved permutation ? (m) ,
(m)
(m)
drawn uniformly from SNm (the set of all permutations of Nm objects); i.e., zt
= y (m) , for
?t
t = 1, . . . , Nm . All the ? (m) are assumed mutually independent and independent of all the z(m) .
Under this model, the log-likelihood of the set of observations Y is
?
?
? ?
T
X
X
?log ?
log P [Y|A, ?] =
P [y(m) |? , A, ?]? ? log(Nm !)? .
(1)
m=1
? ?SNm
QN
where P [y|? , A, ?] = ?y?1 t=2 Ay?t?1 ,y?t , and network inference consists in computing the
maximum likelihood (ML) estimates (AML , ? ML ) = arg maxA,? log P [Y|A, ?]. With the ML
estimates in hand, we may determine the most likely permutation for each y(m) and obtain a feasible reconstruction from the ordered paths. In general, log P [Y|A, ?] is a non-concave function of
(A, ?), so finding (AML , ? ML ) is not easy. Next, we derive an EM algorithm for this purpose, by
treating the permutations as missing data.
2.2
EM Algorithm
(m)
(m)
(m)
Let w(m) = (w1 , ..., wNm ) be a binary representation of z(m) , defined by wt
(m)
(m)
..., wt,|V | ) ? {0, 1}|V | , with (wt,i
(m)
= 1) ? (zt
(m)
= (wt,1 ,
= i); let W = {w(1) , ..., w(T ) }. Let
X = {x(1) , . . . , x(T ) } be the binary representation for Y, defined in a similar way: x(m) =
(m)
(m)
(m)
(m)
(m)
(m)
(m)
(x1 , ..., xNm ), where xt
= (xt,1 , ..., xt,|V | ) ? {0, 1}|V | , with (xt,i = 1) ? (yt
= i).
Finally, let R = {r(1) , . . . , r(T ) } be the collection of permutation matrices corresponding to
(m)
(m)
= t0 ). With this notation in place, the comT = {? (1) , . . . , ? (T ) }; i.e., (rt,t0 = 1) ? (?t
plete log-likelihood can be written as log P [X , R|A, ?] = log P [X |R, A, ?] + log P [R], where
log P [X |R, A, ?]
=
T
X
log P [x(m) |r(m) , A, ?]
m=1
=
|V |
T
X
X
Nm
X
Nm
X
m=1 i,j=1 t0 ,t00 =1
(m) (m)
(m) (m)
rt,t0 rt?1,t00 xt00 ,i xt0 ,j
log Ai,j +
|V | Nm
T X
X
X
(m) (m)
r1,t0 xt0 ,i log ?i ,
(2)
m=1 i=1 t0 =1
t=2
and P [R] is the probability of the set of permutations, which is constant and thus dropped, since the
permutations are independent and equiprobable.
The
by (the E-step) computing Q A, ?; Ak , ? k
=
EM algorithm proceeds
E log P [X , R|A, ?] X , Ak , ? k , the expected value of log P [X , R|A, ?] w.r.t. the missing
R, conditioned on the observations and on the current model estimate (Ak , ? k ). Examining
log P [X , R|A, ?] reveals that it is linear w.r.t. simple functions of R: (a) the first row of each
PNm (m) (m)
(m)
(m)
r(m) , i.e., r1,t0 ; (b) sums of transition indicators, i.e., ?t0 ,t00 ? t=2
rt,t0 rt?1,t00 . Consequently,
(m)
(m)
(m)
the E-step reduces to computing the conditional expectations of r1,t0 and ?t0 ,t00 , denoted r?1,t0
(m)
and ?
? t0 ,t00 , respectively, and plugging them into the complete log-likelihood (2), which yields
Q A, ?; Ak , ? k .
(m)
Since the permutations are (a priori) equiprobable, we have P [r(m) ] = (Nm !)?1 , P r1,t0 = 1] =
(m)
(Nm ? 1)!/Nm ! = 1/Nm , and P [r(m) |r1,t0 = 1] = 1/(Nm ? 1)!. Using these facts, the mutual
independence among different observations, and Bayes law, it is not hard to show that
(m)
?0
(m)
r?1,t0 = PN t
m
(m)
t0 =1 ?t0
with
(m)
?t0
=
X
r: r1,t0 =1
P x(m) r, Ak , ? k ,
(3)
where each term P x(m) r, Ak , ? k is easily computed after using r to ?unshuffle? x(m) :
N
m
Y
(m)
(m)
k
k
k
k
k
P x
r, A , ? = P y
? , A , ? = ?y(m)
Aky(m)
?1
(m)
t=2
(m)
?t?1 ,y?t
(m)
.
(m) (m)
The computation of ?
? t0 ,t00 is similar to that of r?1,t0 ; the key observations are that P [rt,t0 rt?1,t00 =
(m) (m)
1] = (Nm ? 2)!/Nm ! and P [r(m) |rt,t0 rt?1,t00 = 1] = 1/(Nm ? 2)!, leading to
(m)
(m)
?
? t0 ,t00
?t0 ,t00
= PN
m
t0 =1
(m)
,
with
?t0
(m)
?t0 ,t00
=
X
(m)
P [x
k
k
|r, A , ? ]
r
Nm
X
rt,t0 rt?1,t00 .
(4)
t=2
(m)
(m)
Computing {?
r1,t0 } and {?
?t0 ,t00 } requires O Nm ! operations. For large Nm , this is a heavy load; in
Section 3, we describe a sampling approach for computing approximations to r?1,t0 and ?
? t0 ,t00 .
Maximization of Q A, ?; Ak , ? k w.r.t. A and ?, under the normalization constraints, leads to the
M-step:
PT PNm
PT
PNm (m) (m)
(m) (m) (m)
? t0 ,t00 xt00 ,i xt0 ,j
?1,t0 xt0 ,i
m=1
t0 ,t00 =1 ?
m=1
t0 =1 r
k+1
k+1
and ?i = P|S| PT
Ai,j = P|S| PT PN
PNm (m) (m) .
(m) (m) (m)
m
? t0 ,t00 xt00 ,i xt0 ,j
?1,t0 xt0 ,i
j=1
i=1
m=1
m=1
t0 ,t00 =1 ?
t0 =1 r
(5)
Standard convergence results for the EM algorithm due to Boyles and Wu [11,12] guarantee that the
sequence {(Ak , ? k )} converges monotonically to a local maximum of the likelihood.
2.3
Handling Known Endpoints
In some applications, (one or both of) the endpoints of each path are known and only the internal
nodes are shuffled. For example, in telecommunications problems, the origin and destination of
each transmission are known, but not the network connectivity. In estimating biological signal
transduction pathways, a physical stimulus (e.g., hypotonic shock) causes a sequence of protein
interactions, resulting in another observable physical response (e.g., a change in cell wall structure);
in this case, the stimulus and response act as fixed endpoints, the goal is to infer the order of the
sequence of protein interactions. Knowledge of the endpoints of each path imposes the constraints
(m)
(m)
r1,1 = 1 and rNm ,Nm = 1. Under the first constraint, estimates of the initial state probabilities
PT
(m)
are simply given by ?i = T1 m=1 x1,i . Thus, EM only needs to be used to estimate A. In this
setup, the E-step has a similar form as (4) but with sums over r replaced by sums over permutation
matrices satisfying r1,1 = 1 and rN,N = 1. The M-step update for Ak+1 remains unchanged.
3
Large Scale Inference via Importance Sampling
For long paths, the combinatorial nature of the exact E-step ? summing over all permutations of
each sequence in (3) and (4) ? may render exact computation intractable. This section presents a
Monte Carlo importance sampling (see, e.g., [13]) version of the E-step, along with finite sample
bounds guaranteeing that a polynomial complexity Monte Carlo EM algorithm retains desirable
convergence properties of the EM algorithm; i.e., monotonic convergence to a local maximum.
3.1
Monte Carlo E-Step by Importance Sampling
To lighten notation in this section we drop the superscripts from (Ak , ? k ), using simply (A, ?)
(m)
(m)
for the current parameter estimates. Moreover, since the statistics ?
? t0 ,t00 and r?1,t0 depend only
on the mth co-activation observation, y(m) , we focus on a particular length-N path observation
y = (y1 , y2 , . . . , yN ) and drop the superscript (m).
A na??ve Monte Carlo approximation would be based on random permutations sampled from the
uniform distribution on SN . However, the reason we resort to approximation techniques in the first
place is that SN is large, but typically only a small fraction of its elements have non-negligible
posterior probability, P [? |y, A, ?]. Although we would ideally sample directly from the posterior, this would require determining its value for all N ! permutations. Instead, we propose the
following sequential scheme for sampling a permutation using the current parameter estimates,
(A, ?). To ensure the same element is not sampled twice we introduce a vector of binary flags,
f = (f1 , f2 , . . . , f|V | ) ? {0, 1}|V | . Given a probability distribution p = (p1 , p2 , . . . , p|V | ) on the
vertex set, V , denote by p|f the restriction of p to those elements i ? V for which fi = 1; i.e.,
pi fi
(p|f )i = P|V |
,
j=1 pj fj
for i = 1, 2, . . . , |V |.
(6)
Our sampling scheme proceeds as follows:
Step 1: Initialize f so that fi = 1 if yt = i for some t = 1, . . . , N , and fi = 0 otherwise.
Sample an element v from V according to the distribution ?|f on V .
Find t such that yt = v. Set ?1 = t.
Set fv = 0 to prevent yt from being sampled again (ensure ? is a permutation). Set i = 2.
Step 2: Let Av denote the vth row of the transition matrix.
Sample an element v 0 from V according to the distribution Av |f on V .
Find t such that yt = v 0 . Set ?i = t. Set fv0 = 0.
Step 3: While i < N , update v ? v 0 and i ? i + 1 and repeat Step 2; otherwise, stop.
Repeating this sampling procedure L times yields a collection of iid permutations ? 1 , ? 2 , . . . , ? L ,
where the superscript now identifies the sample number; the corresponding permutation matrices
are r1 , r2 , . . . , rL . Samples generated according to the scheme described above are drawn from a
distribution R[? |x, A, ?] on SN which is different from the posterior P [? |x, A, ?]. Importance
sample estimates correct for this disparity and are given by the expressions
PL
PL
PN ` `
`
`=1 u` r1,t0
`=1 u`
t=2 rt,t0 rt?1,t00
0
00
0
rb1,t = PL
and ?
bt ,t =
,
(7)
PL
`=1 u`
`=1 u`
where the correction factor (or weight) for sample r` is given by
N
u` =
N
P [? ` |y, A, ?] Y X
P [r` |x, A, ?]
=
=
Ay? ` ,y? ` .
R[r` |x, A, ?]
R[? ` |y, A, ?] t=2 0
t?1
t0
(8)
t =t
A detailed derivation of the exact form of the induced distribution, R, and the correction factor, u` ,
based on the sequential nature of the sampling scheme, along with further discussion and comparison
with alternative sampling schemes can be found in the supplementary document [7]. In fact, terms
in the product (8) are readily available as a byproduct of Step 2 (denominator of Av |f ).
3.2
Monotonicity and Convergence
Standard EM convergence results directly apply when the exact E-step is used [11, 12]. Let
? k = (Ak , ? k ). By choosing ? k+1 according to (5) we have ? k+1 = arg max? Q(?; ? k ), and
the monotonicity property, Q(? k+1 ; ? k ) ? Q(? k ; ? k ), is satisfied. Together with the fact that the
marginal log-likelihood (1) is continuous in ? and bounded above, the monotonicity property guarantees that the exact EM iterates converge monotonically to a local maximum of log P [Y|?].
When the Monte Carlo E-step is used, we no longer have monotonicity since now the M-step solves
(m)
(m)
bk+1 = arg max Q(?;
bk ), where Q
b ?
b is defined analogously to Q but with ?
?
? t0 ,t00 and r?1,t0 replaced
?
(m)
(m)
bk+1 ; ?
bk ) ? Q(?
bk ; ?
bk ). To assure the Monte Carlo
by ?
bt0 ,t00 and rb1,t0 ; for monotonicity we need Q(?
EM algorithm (MCEM) converges, the number of importance samples, L, must be chosen carefully
b approximates Q well enough; otherwise the MCEM may be swamped with error.
so that Q
Recently, Caffo et al. [14] have proposed a method, based on central limit theorem-like arguments,
for automatically adapting the number of Monte Carlo samples used at each EM iteration. They
guarantee what we refer to as an (, ?)-probably approximately monotonic (PAM) update, stating
bk+1 ; ?
bk ) ? Q(?
bk ; ?
bk ) ? , with probability at least 1 ? ?.
that Q(?
Rather than resorting to asymptotic approximations, we take advantage of the specific form of Q
bk+1 ; ?
bk ) involves terms
b ?
in our problem to obtain the finite-sample PAM result below. Because Q(
bk and log ?
bk and ?
b does not
log A
bik , in practice we bound A
bik away from zero to ensure that Q
i,j
i,j
k
b
blow up. Specifically, we assume a small positive constant ?min so that A ? ?min and ?
bk ? ?min .
i,j
i
Theorem 1 Let , ? > 0 be given. There exist finite constants bm > 0, independent of Nm , so that
if
4
2
2b2m T 2 Nm
| log ?min |2
2Nm
Lm =
log
(9)
2
1 ? (1 ? ?)1/T
b
importance samples are used for the mth observation, then Q(?
probability greater than 1 ? ?.
k+1
k
k
k
b ) ? Q(?
b ;?
b ) ? , with
;?
The proof involves two key steps. First, we derive finite sample concentration-style bounds for
(m)
(m)
the importance sample estimates showing, e.g., that ?
bt0 ,t00 converges to ?
? t0 ,t00 at a rate which is
exponential in the number of importance samples used. These bounds are based on rather novel
concentration inequalities for importance sampling estimators, which may be of interest in their
own right (see the supplementary document [7] for details). Then, accounting for the explicit form
of Q in our problem, the result follows from application of the union bound and the assumptions that
k
bk , ?
A
i,j bi ? ?min . In fact, by making a slightly stronger assumption it can be shown that the MCEM
update is probably monotonic (i.e., (0, ?)-PAM, not approximately monotonic) if L0m importance
samples are used for the mth observation, where L0m also depends polynomially on Nm and T . See
the supplementary document [7] for further discussion and for the full proof of Theorem 1.
Recall that exact E-step computation requires Nm ! operations for the mth observation (enumerating
all permutations). The bound above stipulates that the number of importance samples required for a
2
4
. Generating one importance sample using the sequential
log Nm
PAM update is on the order of Nm
procedure described above requires Nm operations. In contrast to the (exponential complexity) exact
EM algorithm, this clearly demonstrates that the MCEM converges with high probability while only
having polynomial computational complexity, and, in this sense, the MCEM meaningfully breaks
the curse of dimensionality by using randomness to preserve the monotonic convergence property.
4
Experimental Results
The performance of our algorithm for network inference from co-occurrences (NICO, pronounced
?nee-koh?) has been evaluated on both simulated data and on a biological data set. In these experiments, network structure is inferred by first executing the EM algorithm to infer the parameters
(A, ?) of a Markov chain. Then, inserting edges in the inferred graph based on the most likely order
of each path according to (A, ?) ensures the resulting graph is feasible with respect to the observations. Because the EM algorithm is only guaranteed to converge to a local maximum, we rerun
the algorithm from multiple random initializations and chose the mostly likely of these solutions.
To gauge the performance of our algorithm we use the edge symmetric difference error: the total
number of false positives (edges in the inferred network which do not exist in the true network) plus
the number of false negatives (edges in the true network not appearing in the inferred network).
We simulate co-occurrence observations in the following fashion. A random graph on 50 vertices
is sampled. Disjoint sets of vertices are randomly chosen as path origins and destinations, paths
are generated between each origin-destination pair using the shortest path algorithm with either unit
weight per edge (?shortest path?) or a random weight on each edge (?random routing?), and then
co-occurrence observations are formed from each path. We keep the number of origins fixed at 5
and vary the number of destinations between 5 and 40 to see how the number of observations effects
performance. NICO performance is compared against the frequency method (FM) described in [4].
Figure 1 plots the edge error for synthetic data generated using (a) shortest path routing, and (b)
random routing. Each curve is the average performance over 100 different network and path real-
7
5
4
3
2
1
0
5
Freq. Method (Sparsest)
Freq. Method (Best)
NICO (ML)
6
Edge Symmetric Difference
6
Edge Symmetric Difference
7
Freq. Method (Sparsest)
Freq. Method (Best)
NICO (ML)
5
4
3
2
1
10
15
20
25
Num. Destinations
30
(a) Shortest path routes
35
40
0
5
10
15
20
25
Num. Destinations
30
35
40
(b) Random routes
Figure 1: Edge symmetric differences between inferred networks and the network one would obtain
using co-occurrence measurements arranged in the correct order. Performance is averaged over
100 different network realizations. For each configuration 10 NICO and FM solutions are obtained
via different initializations. We then choose the NICO solution yielding the largest likelihood, and
compare with both the sparsest (fewest edges) and clairvoyant best (lowest error) FM solution.
izations. For each network/path realization, the EM algorithm is executed with 10 random initializations. Exact E-step calculation is used for observations with Nm ? 12, and importance sampling
is used for longer paths. The longest observation in our data has Nm = 19. The FM uses simple
pairwise frequencies of co-occurrence to assign an order independently to each path observation. Of
the 10 NICO solutions (different random initializations), we use the one based on parameter estimates yielding the highest likelihood score which also always gives the best performance. Because
it is a heuristic, the FM does not provide a similar mechanism for ranking solutions from different
initializations. We plot FM performance for two schemes; one based on choosing the sparsest FM
solution (the one with the fewest edges), and one based on clairvoyantly choosing the FM solution
with lowest error. NICO consistently outperforms even the clairvoyant best FM solution.
Our method has also been applied to infer the stress-activated protein kinease (SAPK)/Jun N terminal kinase (JNK) and NF?B signal transduction pathways1 (biological networks). The clustering procedure described in [2] is applied to microarray data in order to identify 18 co-occurrences
arising from different environmental stresses or growth factors (path source) and terminating in the
production of SAPK/JNK or NF?B proteins. The reconstructed network (combined SAPK/JNK and
NF?B signal transduction pathways) is depicted in Figure 2. This structure agrees with the signalling
pathways identified using traditional experimental techniques which test individually for each possible edge (e.g., ?MAPK? and ?NF-?B Signaling? on http://www.cellsignal.com).
5
Conclusion
This paper describes a probabilistic model and statistical inference procedure for inferring network
structure from incomplete ?co-occurrence? measurements. Co-occurrences are modelled as samples
of a first-order Markov chain subjected to a random permutation. We describe exact and Monte Carlo
EM algorithms for calculating maximum likelihood estimates of the Markov chain parameters (initial state distribution and transition matrix), treating the random permutations as hidden variables.
Standard results for the EM algorithm guarantee convergence to a local maximum. Although our
exact EM algorithm has exponential computational complexity, we provide finite-sample bounds
guaranteeing convergence of the Monte Carlo EM variation to a local maximum with high probability and with only polynomial complexity. Our algorithm is easily extended to compute maximum a
posteriori estimates, applying a Dirichlet prior to the initial state distribution and to each row of the
Markov transition matrix.
1
NF?B proteins control genes regulating a broad range of biological processes including innate and adaptive
immunity, inflammation and B cell development. The NF?B pathway is a collection of paths activated by
various environmental stresses and growth factors, and terminating in the production of NF?B.
LT
Ag
NIK
PI3K
ArtCot
AgMHC
PLCgamma2
PKC
MALT1
TRAF6
RHO
CS2
TAK1
IKK
IL1
RAC
NFkappaBC1
dsRNA
GF
RAS
NFkappaBC2
NFKappaB
PKR
CS1
bTrCP
CDC42
MEKK
MKK
JNK
HPK
GCKs
UV
FAS
TNF
ASK1
OS
Figure 2: Inferred topology of the combined SAPK/JNK and NF?B signal transduction pathways.
Co-occurrences are obtained from gene expression data via the clustering algorithm described in [2],
and then network is inferred using NICO.
Acknowledgments
The authors of this paper would like to thank D. Zhu and A.O. Hero for providing the data and collaborating on the biological network experiment reported in Section 4. This work was supported in
part by the Portuguese Foundation for Science and Technology grant POSC/EEA-SRI/61924/2004,
the Directorate of National Intelligence, and National Science Foundation grants CCF-0353079 and
CCR-0350213.
References
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2,267 | 3,058 | Tighter PAC-Bayes Bounds
Amiran Ambroladze
Dep. of Mathematics
Lund University/LTH
Box 118, S-221 00 Lund, SWEDEN
[email protected]
Emilio Parrado-Hern?andez
Dep. of Signal Processing and Communications
University Carlos III of Madrid
Legan?es, 28911, SPAIN
[email protected]
John Shawe-Taylor
Dep. of Computer Science
University College London
Gower Street,
London WC1E 6BT, UK
[email protected]
Abstract
This paper proposes a PAC-Bayes bound to measure the performance of Support
Vector Machine (SVM) classifiers. The bound is based on learning a prior over
the distribution of classifiers with a part of the training samples. Experimental
work shows that this bound is tighter than the original PAC-Bayes, resulting in an
enhancement of the predictive capabilities of the PAC-Bayes bound. In addition,
it is shown that the use of this bound as a means to estimate the hyperparameters
of the classifier compares favourably with cross validation in terms of accuracy of
the model, while saving a lot of computational burden.
1
Introduction
Support vector machines (SVM) implement linear classifiers in a high-dimensional feature space
using the kernel trick to enable a dual representation and efficient computation.
The danger of overfitting in such high-dimensional spaces is countered by maximising the margin
of the classifier on the training examples. For this reason there has been considerable interest in
bounds on the generalisation in terms of the margin.
Early bounds have relied on covering number computations [7], while later bounds have considered
Rademacher complexity. The tightest bounds for practical applications appear to be the PAC-Bayes
bound [4, 5]. In particular the form given in [3] is specially attractive for margin classifiers, like
SVM. The PAC-Bayesian bounds are also present in other Machine Learning models such as Gaussian Processes [6].
The aim of this paper is to consider a refinement of the PAC-Bayes approach and investigate whether
it can improve on the original PAC-Bayes bound and uphold its capabilities of delivering reliable
model selection.
The standard PAC-Bayes bound uses a Gaussian prior centred at the origin in weight space. The
key to the new bound is to use part of the training set to compute a more informative prior and then
compute the bound on the remainder of the examples relative to this prior. The bounds are tested
experimentally in several classification tasks, including the model selection, on common benchmark
datasets.
The rest of the document is organised as follows. Section 2 briefly reviews the PAC-Bayes bound for
SVMs obtained in [3]. The new bound obtained by means of the refinement of the prior is presented
in Section 3. The experimental work, included in Section 4, compares the tightness of the new bound
with the original one and indicates about its usability in a model selection task. Finally, the main
conclusions of this work are outlined in Section 5.
2
PAC-Bayes Bound
This section is devoted to a brief review of the PAC-Bayes Bound Theorem of [3]. Let us consider a
distribution D of patterns x lying in a certain input space X , with their corresponding output labels
y, y ? {?1, 1}. In addition, let us also consider a distribution Q over the classifiers c. For every
classifier c, the following two error measures are defined:
Definition (True error) The true error cD of a classifier c is defined as the probability of misclassifying a pair pattern-label (x, y) selected at random from D
cD ? Pr(x,y)?D (c(x) 6= y)
Definition (Empirical error) The empirical error c?S of a classifier c on a sample S of size m is
defined as the rate of errors on a set S
m
c?S ? Pr(x,y)?S (c(x) 6= y) =
1 X
I(c(xi ) 6= yi )
m i=1
where I(?) is a function equal to 1 if the argument is true and equal to 0 if the argument is false.
Now we can define two error measures on the distribution of classifiers: the true error, QD ?
Ec?Q cD , as the probability of misclassifying an instance x chosen from D with a classifier c chosen
? S ? Ec?Q c?S , as the probability of classifier c chosen
according to Q; and the empirical error Q
according to Q misclassifying an instance x chosen from a sample S.
For these two quantities we can derive the PAC-Bayes Bound on the true error of the distribution of
classifiers:
Theorem 2.1 (PAC-Bayes Bound) For all prior distributions P (c) over the classifiers c, and for
any ? ? (0, 1]
KL(Q(c)||P (c)) + ln( m+1
? )
?
m
PrS?D
?Q(c) : KL(QS ||QD ) ?
? 1 ? ?,
m
1?q
and
where KL is the Kullback-Leibler divergence, KL(p||q) = q ln pq + (1 ? q) ln 1?p
KL(Q(c)||P (c)) = Ec?Q ln Q(c)
P (c) .
The proof of the theorem can be found in [3].
This bound can be particularised for the case of linear classifiers in the following way. The m
training patterns define a linear classifier that can be represented by the following equation1 :
c(x) = sign(wT ?(x))
(1)
where ?(x) is a nonlinear projection to a certain feature space where a linear classification actually
takes place, and w is a vector from that feature space that determines the separating plane.
For any vector w we can define a stochastic classifier in the following way: we choose the distribution Q = Q(w, ?) to be a spherical Gaussian with identity covariance matrix centred on the
direction given by w at a distance ? from the origin. Moreover, we can choose the prior P (c) to be
a spherical Gaussian with identity covariance matrix centred on the origin. Then, for classifiers of
the form in equation (1) performance can be bounded by
1
We are considering here unbiased classifiers, i.e., with b = 0.
Corollary 2.2 (PAC-Bayes Bound for margin classifiers [3]) For all distributions D, for all classifiers given by w and ? > 0, for all ? ? (0, 1], we have
!
?2
m+1
+
ln(
)
2
?
? S (w, ?)||QD (w, ?)) ?
? 1 ? ?.
Pr KL(Q
m
It can be shown (see [3]) that
? S (w, ?) = Em [F? (??(x, y))]
Q
(2)
where Em is the average over the m training examples, ?(x, y) is the normalised margin of the
training patterns
ywT ?(x)
?(x, y) =
(3)
k?(x)kkwk
and F? = 1 ? F , where F is the cumulative normal distribution
Z x
2
1
? e?x /2 dx.
F (x) =
(4)
2?
??
Note that the SVM is a thresholded linear classifier expressed as (1) computed by means of the
kernel trick [2]. The generalisation error of such a classifier can be bounded by at most twice the
true (stochastic) error QD (w, ?) in Corollary 2.2, (see [4]);
Pr(x,y)?D sign(wT ?(x)) 6= y ? 2QD (w, ?)
for all ?.
3
Choosing a prior for the PAC-Bayes Bound
Our first contribution is motivated by the fact that the PAC-Bayes bound allows us to choose the
prior distribution, P (c). In the standard application of the bound this is chosen to be a Gaussian
centred at the origin. We now consider learning a different prior based on training an SVM on a
subset R of the training set comprising r training patterns and labels. In the experiments this is
taken as a random subset but for simplicity of the presentation we will assume these to be the last r
examples {xk , yk }m
k=m?r+1 in the description below.
With these r examples we can determine an SVM classifier, wr and form a prior P (w|wr ) consisting of a Gaussian distribution with identity covariance matrix centred on wr .
The introduction of this prior P (w|wr ) in Theorem 2.1 results in the following new bound.
Corollary 3.1 (Single Prior based PAC-Bayes Bound for margin classifiers) Let us consider a prior
on the distribution of classifiers consisting in a spherical Gaussian with identity covariance centred
along the direction given by wr at a distance ? from the origin. Then, for all distributions D, for all
classifiers wm and ? > 0, for all ? ? (0, 1], we have
!
||?wr ??wm ||2
m?r+1
+
ln(
)
2
?
? S\R (wm , ?)||QD (wm , ?)) ?
?1??
PrS?D KL(Q
m?r
? S\R is a stochastic measure of the error of the classifier on the m ? r samples not used to
where Q
learn the prior. This stochastic error is computed as indicated in equation (2) averaged over S\R.
Proof Since we separate r instances to learn the prior, the actual size of the training set to which we
apply the bound is m ? r. In addition, the stochastic error must be computed only on the instances
not used to learn the prior, i.e. the subset S\R.
The KL divergence between prior and posterior is computed as follows:
KL(Q(w)||P (w)) = Ew?Q ln Q(w)
P (w)
exp(? 12 (w??wm )T (w??wm ))
= Ew?Q ln exp ? 1 (w??w )T (w??w )
( 2
r
r )
1
T
T
= Ew?Q ? 12 (w ?
?w
)
(w
?
?w
m
m ) + 2 (w ? ?w
r ) (w ? ?wr )
T
T
= Ew?Q ?wm
w ? 21 ?2 wm
wm ? Ew?Q ?wT wr + 21 ? 2 wrT wr
Taking expectations using Ew?Q w = ?wm we arrive at
1
||?wm ? ?wr ||2
2
Intuitively, if the selection of the prior is appropriate, the bound can be tighter than the one given in
Corollary 2.2 when applied to the SVM weight vector on the whole training set. It is perhaps worth
stressing that the bound holds for all wm and so can be applied to the SVM trained on the whole set.
This might at first appear as ?cheating?, but the critical point is that the bound is evaluated on the set
S\R not involved in generating the prior. The experimental work illustrates how in fact this bound
can be tighter than the standard PAC-Bayes bound.
Moreover, the selection of the prior may be further refined in exchange for a very small increase in
the penalty term. This can be achieved with the application of the following result.
Theorem 3.2 (Bound for several priors) Let {Pj (c)}Jj=1 be a set of possible priors that can be sePJ
lected with positive weights {?j }Jj=1 so that j=1 ?j = 1. Then, for all priors P (c) ? {Pj (c)}Jj=1 ,
for all posterior distributions Q(c), for all ? ? (0, 1],
!
1
KL(Q(c)||Pj (c)) + ln m+1
? + ln ?j
?
? 1 ? ?,
PrS?Dm ?Q(c), ?j : KL(QS ||QD ) ?
m
Proof The bound in Theorem 2.1 can be particularised for a certain Pj (c) with associated weight
?j and with confidence ??j
PrS?Dm
? S ||QD ) >
?Q(c) : KL(Q
KL(Q(c)||Pj (c)) + ln( m+1
??j )
m
!
< ??j ,
Now let us combine the bounds for all the priors {Pj (c)}Jj=1 with the union operation (we use the
fact that P (a ? b) ? P (a) + P (b)).
!
?Q(c), ?P (c) ? {Pj (c)}Jj=1 :
1
< ?,
(5)
PrS?Dm
KL(Q(c)||Pj (c))+ln m+1
? +ln ?j
? S ||QD ) >
KL(Q
m
Finally, let us take the negation of (5) to arrive at the final result.
This result can be also particularised for the case of SVM classifiers. The set of priors is constructed
by allocating Gaussian distributions with identity covariance matrix along the direction given by wr
at distances {?j }Jj=1 from the origin where {?j }Jj=1 are real numbers. In such a case, we obtain
Corollary 3.3 (Multiple Prior PAC-Bayes Bound for linear classifiers) Let us consider a set
{Pj (w|wr , ?j )}Jj=1 of prior distributions of classifiers consisting in spherical Gaussian distributions with identity covariance matrix centred on ?j wr , where {?j }Jj=1 are real numbers. Then, for
all distributions D, for all classifiers w, for all ? > 0, for all ? ? (0, 1], we have
!
||?j wr ??w||2
m?r+1
+
ln(
)
+
ln
J
2
?
? S\R (w, ?)||QD (w, ?)) ?
?1??
PrS?D KL(Q
m?r
Proof The proof is straightforward, substituting ?j = J1 for all j in Theorem 3.2 and computing
the KL divergence between prior and posterior as in the proof of Corollary 3.1.
Note that the {?j }Jj=1 must be chosen before we actually compute the posterior. However, the bound
holds for all ?. Therefore, a linear search can be implemented for the value of ? that leads to the
tightest bound. In the case of several priors, the search is repeated for every prior and the reported
value of the bound is the tightest. In Section 4 we present experimental results comparing this new
bound to the standard PAC-Bayes bound and using it to guide model selection.
4
Experiments
The tightness of the new bound is evaluated in a model selection and classification task using some
UCI [1] datasets (see their description in terms of number of instances, input dimension and number
of positive/negative examples in Table 1).
Problem
Wdbc
Image
Waveform
Ringnorm
# samples
569
2310
5000
7400
input dim.
30
18
21
20
Pos/Neg
357 / 212
1320 / 990
1647 / 3353
3664 / 3736
Table 1: Description of the datasets: for every set we give the number of patterns, number of input
variables and number of positive/negative examples.
For every dataset, we obtain 50 different training/test set partitions with 80% of the samples forming
the training set and the remaining 20% forming the test set.
With each of the partitions we learn a SVM classifier with Gaussian RBF kernel preceded by a model
selection. The model selection consists in the determination of an optimal pair of hyperparameters
(C, ?). C is the SVM trade-off between the maximisation of the margin and the minimisation of
the hinge loss of the training samples, while ? is the width of the Gaussian kernel. The best pair is
sought in a 15 ? 15 grid of parameters
0.2, ?
0.5, 1,?2, 5, ?
10, 20,
?
? where
? C?
? {0.02,
? 0.05,
? 0.1,?
?50, 100,
?
200, 500, 1000} and ? ? { 81 d, 17 d, 16 d, 15 d, 14 d, 13 d, 12 d, d, 2 d, 3 d, 4 d, 5 d,
?
?
?
6 d, 7 d, 8 d}, where d is the input space dimension.
For completeness, this model selection is guided by the PAC-Bayes bound: we select the model
corresponding to the pair that yields a lower value of QD in the bound. Table 2 shows the value of
the PAC-Bayes Bound averaged over the 50 training/test partitions. For every partition we use the
minimum value of the bound resulting from all the pairs (C, ?) of the grid. Note that this procedure
is computationally less costly than the commonly used N -fold cross validation model selection,
since it saves the training of N classifiers (one for each fold) for each parameter combination.
Problem
Wdbc
Image
Waveform
Ringnorm
PAC-Bayes Bound
0.334 ? 0.005
0.254 ? 0.003
0.198 ? 0.002
0.212 ? 0.002
Test error rate
0.073 ? 0.021
0.074 ? 0.014
0.089 ? 0.008
0.026 ? 0.005
Table 2: Averaged PAC-Bayes Bound and Test Error Rate obtained by the model that yielded the
lowest bound in each of the 50 training/test partitions.
We repeated this experiment using the Prior PAC-Bayes Bound with different configurations for
learning the prior distribution of classifiers. These configurations are defined by variations on the
percentage of training patterns separated to compute the prior and on the number of scalings of
the magnitude of that prior. The scalings represent different lengths ? of ||wr || equally spaced
between ? = 1 and ? = 100. To summarize, for every training/test partition and for every pair (%
patterns, # of scalings) we look at the pair (C, ?) that outputs the smaller value of QD .
In this case, the use of the Prior PAC-Bayes Bound to perform the model selection increases the
computational burden of using the PAC-Bayes one in the training of one classifier (the one used to
learn the prior), in comparison to the extra N classifiers needed by N -fold cross validation.
Table 3 displays both the average value and the sample standard deviation over the 50 realisations.
It seems that ten scalings of the prior are enough to obtain tighter bounds, since the use of 100 or
500 scalings does not improve the best results. With respect to the percentage of training instances
left out to learn the prior, something close to 50% of the training set works well in the considered
problems. It is worth mentioning that we treat each position in the Table as a separate experiment.
Winsconsin Database of Breast Cancer (PAC-Bayes Bound = 0.334?0.005)
Percentage of training set used to compute the prior
10 %
20%
30%
40%
50%
1
0.341 ? 0.006 0.351 ? 0.007 0.364 ? 0.009 0.379 ? 0.011 0.398 ? 0.013
10
0.337 ? 0.010 0.323 ? 0.012 0.314 ? 0.012 0.310 ? 0.013 0.306 ? 0.018
100
0.319 ? 0.007 0.315 ? 0.010 0.313 ? 0.011 0.315 ? 0.013 0.315 ? 0.017
500
0.324 ? 0.007 0.320 ? 0.009 0.319 ? 0.011 0.321 ? 0.013 0.322 ? 0.017
Image Segmentation (PAC-Bayes Bound = 0.254?0.003)
Scalings
Percentage of training set used to compute the prior
10 %
20%
30%
40%
50%
1
0.255 ? 0.003 0.262 ? 0.005 0.274 ? 0.003 0.284 ? 0.005 0.300 ? 0.008
10
0.215 ? 0.004 0.203 ? 0.006 0.200 ? 0.005 0.188 ? 0.007 0.184 ? 0.010
100
0.217 ? 0.004 0.203 ? 0.007 0.196 ? 0.005 0.187 ? 0.007 0.186 ? 0.009
500
0.218 ? 0.004 0.204 ? 0.007 0.198 ? 0.005 0.189 ? 0.007 0.188 ? 0.009
Waveform (PAC-Bayes Bound = 0.198?0.002)
Scalings
Percentage of training set used to compute the prior
10 %
20%
30%
40%
50%
1
0.197 ? 0.003 0.201 ? 0.003 0.207 ? 0.003 0.214 ? 0.004 0.222 ? 0.005
10
0.161 ? 0.004 0.156 ? 0.004 0.153 ? 0.004 0.150 ? 0.005 0.151 ? 0.005
100
0.161 ? 0.004 0.155 ? 0.004 0.153 ? 0.004 0.152 ? 0.005 0.153 ? 0.005
500
0.162 ? 0.004 0.157 ? 0.004 0.155 ? 0.004 0.154 ? 0.005 0.155 ? 0.005
Ringnorm (PAC-Bayes Bound = 0.212?0.002)
Scalings
Percentage of training set used to compute the prior
10 %
20%
30%
40%
50%
1
0.216 ? 0.001 0.225 ? 0.002 0.236 ? 0.002 0.249 ? 0.004 0.265 ? 0.002
10
0.172 ? 0.068 0.140 ? 0.047 0.126 ? 0.037 0.116 ? 0.030 0.109 ? 0.024
100
0.173 ? 0.068 0.139 ? 0.047 0.126 ? 0.037 0.117 ? 0.030 0.110 ? 0.024
500
0.173 ? 0.068 0.140 ? 0.047 0.127 ? 0.037 0.117 ? 0.030 0.110 ? 0.024
Scalings
Table 3: Averaged Prior PAC-Bayes bound for different settings of percentage of training instances
reserved to compute the prior and of number of scalings of the normalised prior.
However, one could have included the tuning of the pair (% patterns, # of scalings) in
the model selection. This would have involved a further application of the union bound with the 20
entries of the Table for each problem, at the cost of adding an extra ln(20)/m (0.0053 for Wdbc and
less for the other datasets) in the right part of Theorem 3.2. We decided to fix the number of scalings
and the amount of training patterns to compute the prior since to perform all of the different options
would augment the computational burden of the model selection.
In order to evaluate the predictive capabilities of the Prior PAC-Bayes bound as a means to select
models with low test error rate, Table 4 displays the averaged test error corresponding to the models selected in the previous experiment (note that in this case the computational burden involved in
determining the model is increased by the training of the SVM that learns the prior wr ). Table 5
displays the test error rate obtained by SVMs with their hyperparameters tuned on the above mentioned grid by means of ten-fold cross-validation, that serves as a baseline method for comparison
purposes.
According to the values shown in the tables, the Prior PAC-Bayes bound achieves tighter predictions
of the generalization error of the randomized classifier in almost all cases.
Notice how the length of the prior is not so critical in comparison with its direction. The goodness
of the latter relying on the subset of samples left out for the purpose of learning the prior classifier.
Moreover it has to be remarked that this tightening of the bound does not appear to deliver any
reduction in the capabilities to select a good model (such a case would imply that we can predict
more accurately a bigger error rate, but our bound is able to predict accurately the same error rate as
the PAC-Bayes Bound).
Winsconsin Database of Breast Cancer (PAC-Bayes Test Error = 0.073?0.021)
Scalings
Percentage of training set used to compute the prior
10 %
20%
30%
40%
50%
1
0.076 ? 0.020 0.076 ? 0.021 0.076 ? 0.021 0.076 ? 0.021 0.076 ? 0.021
10
0.075 ? 0.021 0.076 ? 0.021 0.075 ? 0.021 0.074 ? 0.021 0.072 ? 0.021
100
0.076 ? 0.021 0.076 ? 0.021 0.074 ? 0.021 0.074 ? 0.020 0.072 ? 0.021
500
0.076 ? 0.020 0.076 ? 0.021 0.074 ? 0.020 0.073 ? 0.020 0.072 ? 0.021
Image Segmentation (PAC-Bayes Test Error = 0.074?0.014)
Scalings
Percentage of training set used to compute the prior
10 %
20%
30%
40%
50%
1
0.078 ? 0.011 0.078 ? 0.011 0.078 ? 0.011 0.083 ? 0.019 0.100 ? 0.019
10
0.064 ? 0.011 0.066 ? 0.011 0.063 ? 0.014 0.054 ? 0.010 0.056 ? 0.011
100
0.064 ? 0.011 0.063 ? 0.011 0.061 ? 0.011 0.059 ? 0.011 0.057 ? 0.012
500
0.064 ? 0.011 0.063 ? 0.011 0.061 ? 0.011 0.059 ? 0.011 0.057 ? 0.012
Waveform (PAC-Bayes Test Error = 0.089?0.008)
Scalings
Percentage of training set used to compute the prior
10 %
20%
30%
40%
50%
1
0.089 ? 0.008 0.089 ? 0.008 0.090 ? 0.009 0.091 ? 0.009 0.091 ? 0.009
10
0.089 ? 0.008 0.089 ? 0.008 0.089 ? 0.008 0.089 ? 0.008 0.089 ? 0.009
100
0.089 ? 0.008 0.089 ? 0.008 0.089 ? 0.008 0.089 ? 0.008 0.089 ? 0.009
500
0.089 ? 0.008 0.089 ? 0.008 0.089 ? 0.008 0.089 ? 0.008 0.089 ? 0.009
Ringnorm (PAC-Bayes Test Error = 0.026?0.005)
Scalings
Percentage of training set used to compute the prior
10 %
20%
30%
40%
50%
1
0.025 ? 0.004 0.030 ? 0.007 0.038 ? 0.005 0.036 ? 0.007 0.038 ? 0.005
10
0.020 ? 0.007 0.021 ? 0.007 0.021 ? 0.007 0.025 ? 0.008 0.026 ? 0.008
100
0.020 ? 0.007 0.021 ? 0.007 0.021 ? 0.007 0.025 ? 0.008 0.026 ? 0.008
500
0.020 ? 0.007 0.021 ? 0.007 0.021 ? 0.007 0.025 ? 0.008 0.025 ? 0.005
Table 4: Averaged Test Error Rate corresponding to the model determined by the bound for the
different settings of Table 3.
Problem
Wdbc
Image
Waveform
Ringnorm
Cross-validation error rate
0.060 ? 0.006
0.022 ? 0.002
0.079 ? 0.011
0.015 ? 0.001
Test error rate
0.072 ? 0.024
0.024 ? 0.008
0.085 ? 0.009
0.017 ? 0.004
Table 5: Averaged test error rate. For every partition we select the test error rate corresponding to
the model reporting the smaller cross-validation error.
However, the comparison with Table 5 points out that the PAC-Bayes bound is not as accurate
as Ten Fold cross-validation when it comes to selecting a model that yields a low test error rate.
Nevertheless, in two out of the four problems (waveform, and wdbc) the bound provided a model
as good as the one found by cross-validation, added to the fact that in ringnorm the error bars
overlap. We conclude the discussion by pointing that the Cross-validation error rate cannot be used
directly as a prediction on the expected test error rate in the sense of worse case performances. Of
course the values of the cross-validation error rate and the test error rate are close, but it is difficult
to predict how close they are going to be.
5
Conclusions and ongoing research
In this paper we have presented a version of the PAC-Bayes bound for linear classifiers that introduces the learning of the prior distribution over the classifiers. This prior distribution is a Gaussian
with identity covariance matrix. The mean weight vector is learnt in the following way: its direction
is determined from a separate subset of the training examples, while its length has to be chosen from
an a priori fixed set of lengths.
The experimental work shows that this new version of the bound achieves tighter predictions of the
generalization error of the stochastic classifier, compared to the original PAC-Bayes bound predictions. Moreover, if the model selection is driven by the bound, the Prior PAC-Bayes does not degrade
the quality of the model selected by the original bound. Nevertheless, it has to be said that in some
of our experiments the model selected by the bounds resulted as accurate as the ones selected by
ten-fold cross-validation in terms of test error rate on a separate test. This fact is remarkable since
to include the model selection in the training of the classifier roughly multiplies by ten the computational burden of the training when using ten-fold cross-validation but roughly by two when using the
prior PAC-Bayes bound. Of course the original PAC-Bayes provides with a cheaper model selection,
but its predictions about the generalization capabilities are more pessimistic.
The amount of training patterns used to learn the prior seems to be a key aspect in the goodness of
this prior and thus in the tightness of the bound. Therefore, ongoing research includes methods to
systematically determine an amount of patterns that provides with suitable priors. Another line of
research explores the use of these bounds to reinforce different properties of the design of classifiers,
such as sparsity. Finally, a deeper study about which dataset structure causes differences among the
performances of cross-validation and bound-driven model selections is also being carried out.
Acknowledgments
This work has been supported by the IST Programme of the European Community under the PASCAL Network of Excellence IST2002-506788. E. P-H. acknowledges support from Spain CICYT
grant TEC2005-04264/TCM.
References
[1] C L Blake and C J Merz.
UCI Repository of machine learning databases.
University of California, Irvine, Dept. of Information and Computer Sciences,
[http://www.ics.uci.edu/?mlearn/MLRepository.html], 1998.
[2] Bernhard E. Boser, Isabelle Guyon, and Vladimir Vapnik. A training algorithm for optimal
margin classifiers. In Computational Learing Theory, pages 144?152, 1992.
[3] J Langford. Tutorial on practical prediction theory for classification. Journal of Machine Learning Research, 6(Mar):273?306, 2005.
[4] J Langford and J Shawe-Taylor. PAC-Bayes & Margins. In Advances in Neural Information
Processing Systems, volume 14, Cambridge MA, 2002. MIT Press.
[5] D McAllester. Pac-bayesian stochastic model selection. Machine Learning, 51(1):5?21, 2003.
[6] M Seeger. PAC-Bayesian Generalization Error Bounds for Gaussian Process Classification.
Journal of Machine Learning Research, 3:233?269, 2002.
[7] J Shawe-Taylor, P L Bartlett, R C Williamson, and M Anthony. Structural risk minimization
over data-dependent hierarchies. IEEE Trans. Information Theory, 44(5):1926 ? 1940, 1998.
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2,268 | 3,059 | An Efficient Method for Gradient-Based Adaptation
of Hyperparameters in SVM Models
S. Sathiya Keerthi
Vikas Sindhwani
Olivier Chapelle
Yahoo! Research
3333 Empire Avenue
Burbank, CA 91504
Department of Computer Science
University of Chicago
Chicago, IL 60637
MPI for Biological Cybernetics
Spemannstra?e 38
72076 T?
ubingen
[email protected]
[email protected]
[email protected]
Abstract
We consider the task of tuning hyperparameters in SVM models based on minimizing a smooth performance validation function, e.g., smoothed k-fold crossvalidation error, using non-linear optimization techniques. The key computation
in this approach is that of the gradient of the validation function with respect to
hyperparameters. We show that for large-scale problems involving a wide choice
of kernel-based models and validation functions, this computation can be very efficiently done; often within just a fraction of the training time. Empirical results
show that a near-optimal set of hyperparameters can be identified by our approach
with very few training rounds and gradient computations.
.
1
Introduction
Consider the general SVM classifier model in which, given n training examples {(x i , yi )}ni=1 , the
primal problem consists of solving the following problem:
n
X
1
min kwk2 + C
l(oi , yi )
(1)
(w,b) 2
i=1
where l denotes a loss function over labels yi ? {+1, ?1} and the outputs
Pn oi on the training set. The
machine?s output o for any example x is given as o = w ? ?(x) ? b = j=1 ?j yj k(x, xi ) ? b where
the ?i are the dual variables, b is the threshold parameter and, as usual, computations involving ?
are handled using the kernel function: k(x, z) = ?(x) ? ?(z). For example, the Gaussian kernel is
given by
k(x, z) = exp(??kx ? zk2 )
(2)
The regularization parameter C and kernel parameters such as ? comprise the vector h of hyperparameters in the model. h is usually chosen by optimizing a validation measure (such as the k-fold
cross validation error) on a grid of values (e.g. a uniform grid in the (log C, log ?) space). Such
a grid search is usually expensive. Particularly, when n is large, this search is so time-consuming
that one usually resorts to either default hyperparameter values or crude search strategies. The problem becomes more acute when there are more than two hyperparameters. For example, for feature
weighting/selection purposes one may wish to use the following ARD-Gaussian kernel:
X
k(x, z) = exp(?
? t kxt ? z t k2 )
(3)
t
where ? t = weight on the tth feature. In such cases, a grid based search is ruled out. In Figure 1 (see
section 5) we show contour plots of performance of an SVM on the log C ? log ? plane for a realworld binary classification problem. These plots show that learning performance behaves ?nicely? as
a function of hyperparameters. Intuitively, as C and ? are varied one expects the SVM to smoothly
transition from providing underfitting solutions to overfitting solutions. Given that this phenomenon
seems to occur routinely on real-world learning tasks, a very appealing and principled alternative to
grid search is to consider a differentiable version of the performance validation function and invoke
non-linear gradient-based optimization techniques for adapting hyperparameters. Such an approach
requires the computation of the gradient of the validation function with respect to h.
Chapelle et al. (2002) give a number of possibilities for such an approach. One of their most promising methods is to use a differentiable version of the leave-one-out (LOO) error. A major disadvantage of this method is that it requires the expensive computation and storage of the inverse of a
kernel sub-matrix corresponding to the support vectors. It is worth noting that, even if, on some
large scale problems, the support vector set is of a manageable size at the optimal hyperparameters,
the corresponding set can be large when the hyperparameter vector is away from the optimal; on
many problems, such a far-off region in the hyperparameter space is usually traversed during the
adaptation process!
We highlight the contributions of this paper.
(1) We consider differentiable versions of validation-set-based objective functions for model selection (such as k-fold error) and give an efficient method for computing the gradient of this function
with respect to h. Our method does not require the computation of the inverse of a large kernel
sub-matrix. Instead, it only needs a single linear system of equations to be solved, which can be
done either by decomposition or conjugate-gradient techniques. In essence, the cost of computing
the gradient with respect to h is about the same, and usually much lesser than the cost of solving (1)
for a given h.
(2) Our method is applicable to a wide range of validation objective functions and SVM models that
may involve many hyperparameters. For example, a variety of loss functions can be used together
with multiclass classification, regression, structured output or semi-supervised SVM algorithms.
(3) Large-scale empirical results show that with BFGS optimization, just trying about 10-20 hyperparameter points leads to the determination of optimal hyperparameters. Moreover, even as
compared to a fine grid search, the gradient procedure provides a more precise placement of hyperparameters leading to better generalization performance. The benefit in terms of efficiency over
the grid approach is evident even with just two hyperparameters. We also show the usefulness of our
method for tuning more than two hyperparameters when optimizing validation functions such as the
F measure and weighted error rate. This is particularly useful for imbalanced problems.
This paper is organized as follows: In section 2, we discuss the general class of SVM models to
which our method can be applied. In section 3, we describe our framework and provide the details
of the gradient computation for general validation functions. In section 4, we discuss how to develop
differentiable versions of several common performance validation functions. Empirical results are
presented in section 5. We conclude this paper in section 6. Due to space limitations, several details
have been omitted but can be found in the technical report (Keerthi et al. (2006)).
2
SVM Classification Models
In this section, we discuss the assumptions required for our method to be applicable. Consider SVM
classification models of the form in (1). We assume that the kernel function k is a continuously
differentiable function of h. Three commonly used SVM loss functions are: (1) hinge loss; (2)
squared hinge loss; and (3) squared loss. In each of these cases, the solution of (1) is obtained by
computing the vector ? that solves a dual problem. The solution usually leads to a linear system
relating ? and b:
?
P
=q
(4)
b
where P and q are, in general, functions of h. We make the following assumption: Locally around
h (at which we are interested in calculating the gradient of the validation function to be defined
soon) P and q are continuously differentiable functions of h. We write down P and q for the hinge
loss function and discuss the validity of the above assumption. Details for other loss functions are
similar.
Hinge loss. l(oi , yi ) = max{0, 1 ? yi oi }. After the solution of (1), the training set indices get
partitioned into three sets: I0 = {i : ?i = 0}, Ic = {i : ?i = C} and Iu = {i : 0 < ?i < C}. Let
?0 , ?c , ?u , yc , yu , ec , eu , ?uc , ?uu etc be appropriately defined vectors and matrices. Then (4) is
given by
eu ? ?uc ?c
?uu ?yu
?u
=
(5)
?0 = 0, ?c = Cec ,
b
ycT ?c
?yuT
0
If the partitions I0 , Ic and Iu do not change locally around a given h then assumption 2 holds.
Generically, this happens for almost all h.
The modified Huber loss function can also be used, though the derivation of (4) for it is more
complex than for the three loss functions mentioned above. Recently, weighted hinge loss with
asymmetric margins (Grandvalet et al., 2005) has been explored for treating imbalanced problems.
Weighted Hinge loss. l(oi , yi ) = Ci max{0, mi ? yi oi }. where Ci = C+ , mi = m+ if yi = 1
and Ci = C? , mi = m? if yi = ?1. Because C+ and C? are present, the hyperparameter C in (1)
can be omitted. The SVM model with weighted hinge loss has four extra hyperparameters, C + , C? ,
m+ and m? , apart from the kernel hyperparameters. Our methods in this paper allow the possibility
of efficiently tuning all these parameters together with kernel parameters.
The method described in this paper is not special to classification models only. It extends to a
wide class of kernel methods for which the optimality conditions for minimizing a training objective
function can be expressed as a linear system (4) in a continuously differentiable manner 1 . These
include many models for multiclass classification, regression, structured output and semi-supervised
learning (see Keerthi et al. (2006)).
3
The gradient of a validation function
Suppose that for the purpose of hyperparameter tuning, we are given a validation scheme involving
a small number of (training set, validation set) partitions, such as: (1) using a single validation set,
(2) k-fold cross validation, or (3) averaging over k randomly chosen (training set, validation set)
partitions. Our method applies to any of these three schemes. To keep notations simple, we explain
the ideas only for scheme (1) and expand on the other schemes towards the end of this section. Note
that throughout the hyperparameter optimization process, the training-validation splits are fixed.
?
? li = k(?
Let {?
xl , y?l }nl=1
denote the validation set. Let K
xl , xi ) involving a kernel calculation between
an element of a validation
set
with
an
element
of
the
training set. The output on the l th validation
P
?
example is o?l = i ?i yi Kli ? b which, for convenience, we will rewrite as
o?l = ?lT ?
(6)
? li , i = 1, . . . , n and ?1 as
where ? is a vector containing ? and b, and ?l is a vector containing yi K
the last element (corresponding to b). Let us suppose that the model selection problem is formulated
as a non-linear optimization problem:
h? = argmin f (?
o1 , . . . , o?n? )
(7)
h
where f is a differentiable validation function of the outputs o?l which implicitly depend on h. In the
next section, we will outline the construction of such functions for criteria like error rate, F measure
etc. We now discuss the computation of ?h f . Let ? denote a generic parameter in h and let us
represent partial derivative of some quantity, say v, with respect to ? as v.
? Before writing down
?
?
expressions for f , let us discuss how to get ?. Differentiating (4) with respect to ? gives
P ?? + P? ? = q?
Now let us write down f?.
f? =
?
n
?
X
?? = P ?1 (q? ? P? ?)
(?f /? o?l )o?? l
(8)
(9)
l=1
1
Infact, the main ideas easily extend when the optimality conditions form a non-linear system in (?, b) (e.g.,
in Kernel Logistic Regression).
where o?? l is obtained by differentiating (6):
o?? l = ?lT ?? + ?? lT ?
(10)
The computation of ?? in (8) is the most expensive step, mainly because it requires P ?1 . Note that,
for hinge loss, P ?1 can be computed in a somewhat cheaper way: only a matrix of the dimension
of Iu needs to be inverted. Even then, in large scale problems the dimension of the matrix to be
inverted can become so large that even storing it may be a problem; even when large storage is
possible, the inverse can be very expensive. Most times, the effective rank of P is much smaller
than its dimension. Thus, instead of computing P ?1 in (8), we can instead solve
P ?? = (q? ? P? ?)
(11)
for ?? approximately using decomposition methods or iterative methods such as conjugate-gradients.
This can improve efficiency as well as take care of memory issues by storing P only partially and
computing the remaining parts of P as and when needed. Since the right-hand-side vector (q? ? P? ?)
in (11) changes for each different ? with respect to which we are differentiating, we need to solve
(11) for each element of h. If the number of elements of h is not small (say, we want to use (3) with
the MNIST dataset which has more than 700 features) then, even with (11), the computations can
still remain very expensive.
We now give a simple trick that shows that if the gradient calculations are re-organized, then obtaining the solution of just a single linear system suffices for computing the full gradient of f with
respect to all elements of h. Let us denote the coefficient of o?? l in the expression for f? in (9) by ?l ,
i.e.,
?l = ?f /? o?l
(12)
Using (10) and plugging the expression for ?? from (8) into (9) gives
X
X
X
f? =
?l o?? l =
?l (?lT P ?1 (q? ? P? ?) + ?? lT ?) = dT (q? ? P? ?) + (
?l ?? l )T ?
(13)
l
where d is the solution of
l
l
PTd = (
X
? l ?l )
(14)
l
The beauty of the reorganization in (13) is that d is the same for all variables ? in h about which the
differentiation is being done. Thus (14) needs to be solved only once. In concurrent work (Seeger,
2006) has used a similar idea for kernel logistic regression.
As a word of caution, note that P may not be symmetric. See, e.g., the P arising from (5) for
the hinge loss case. Also, the parts corresponding to zero components should be omitted from
calculations and the special structure of P should be utilized,e.g., for hinge loss when computing
P? ? the parts of P? corresponding to ?0 (see (5)) can be ignored. The linear system in the above
equation can be efficiently solved using conjugate gradient techniques.
The sequence of steps for the computation of the full gradient of f with respect to h is as follows.
First compute ?l from (12). For various choices of validation function, we outline this computation
in the next section. Then solve (14) for d. Then, for each ? use (13) to get all the derivatives of
f . The computation of P? ? has to be performed for each hyperparameter separately. In problems
with many hyperparameters, this is the most expensive part of the gradient computation. Note that
in some cases, e.g., ? = C, P? ? is immediately obtained. For ? = ? or ?t , when using (2,3), one
can cache pairwise distance computations while computing the kernel matrix. We have found (see
section 5) that the cost of computing the gradient of f with respect to h to be usually much less than
the cost of solving (1) and then obtaining f .
We can also employ the above ideas in a validation scheme where one uses k training-validation
splits (e.g in k-fold cross-validation). In this case, for each partition one obtains the linear system
(4), corresponding validation outputs (6) and the linear system in (14). The gradient is simply
Pk
computed by summing over the k partitions, i.e., f? =
f?(k) where f?(k) is given by (13) using
the quantities P, q, d etc associated with the k th partition.
j=1
The model selection problem (7) may now be solved using, e.g., Quasi-Newton methods such as
BFGS which only require function value and gradient at a hyperparameter setting. In particular,
reaching the minimizer of f too closely is not important. In our implementations we terminate
optimization iterations when the following loose termination criterion is met: |f (h k+1 ) ? f (hk )| ?
10?3 |f (hk )|, where hk+1 and hk are consecutive iterates in the optimization process.
A general concern with descent methods is the presence of local minima. In section 5, we make
some encouraging empirical observations in this regard, e.g., local minima problems did not occur
for the C, ? tuning task; for several other tasks, starting points that work surprisingly well could be
easily obtained.
4
Smooth validation functions
We consider validation functions that are general functions of the confusion matrix, of the form
f (tp, f p) where tp is the number of true positives and f p is the number of false positives. Let u(z)
denote the unit step function which is 0 when z < 0 and 1 otherwise. Denote u
? l = u(?
yl o?l ), which
evaluates to 1 ifP
the lth example is P
correctly classified and 0 otherwise. Then, tp and f p can be
?l ). Let n
? + and n
? ? be the number of validation
?l , f p = l:?yl =?1 (1 ? u
written as tp = l:?yl =+1 u
examples in the positive and negative classes. The most commonly used validation function is error
rate.
Error rate (er) is simply the percentage of incorrect predictions, i.e., er = (?
n + ? tp + f p)/?
n.
For classification problems with imbalanced classes it is usual to consider either weighted error rate
or a function of precision and recall such as the F measure.
Weighted Error rate (wer) is given by wer = (?
n+ ? tp + ?f p)/(?
n+ + ??
n? ), where ? is the ratio
of the cost of misclassifications of the negative class to that of the positive class.
F measure (F ) is the harmonic mean of precision and recall: F = 2tp/(?
n+ + tp + f p)
Alternatively, one may want to maximize precision under a recall constraint, or maximize the area
under the ROC Curve or maximize the precision-recall breakeven point. See Keerthi et al. (2006)
for a discussion on how to treat these cases.
It is common practice to evaluate measures like precision, recall and F measure while varying the
threshold on the real-valued classifier output, i.e., at any given threshold ?0 , tp and f p can be
redefined in terms of the following,
u
?l = u (?
yl (?
ol ? ?0 ))
(15)
For imbalanced problems one may wish to maximize a score such as the F measure over all values
of ?0 . In such cases, it is appropriate to incorporate ?0 as an additional hyperparameter that needs
to be tuned. Such bias-shifting is particularly also useful as a compensation mechanism for the
mismatch between training objective function and validation function; often one uses an SVM as
the underlying classifier even though it is not explicitly trained to minimize the validation function
that the practitioner truly cares about. In section 5, we make some empirical observations related to
this point.
The validation functions discussed above are based on discrete counts. In order to use gradient-based
methods smooth functions of h are needed. To develop smooth versions of validation functions, we
define s?l , which is a sigmoidal approximation to u
?l (15) of the following form:
s?l = 1/[1 + exp (??1 y?l (?
ol ? ?0 ))]
(16)
where ?1 > 0 is a sigmoidal scale factor. In general, ?0 , ?1 may be functions of the validation outputs. (As discussed above, one may alternatively wish to treat ?0 as an additional hyperparameter.)
The scale factor ?1 influences how closely s?l approximates the step function u
?l and hence controls
the degree of smoothness in building the sigmoidal approximation. As the hyperparameter space
is probed, the magnitude of the outputs can vary quite a bit. ?1 takes the scale of the outputs into
account. Below we discuss various methods to set ?0 , ?1 .
We build a differentiable version of such a function by simply replacing u
? l by s?l . Thus, we have
f = f (?
s1 . . . s?n? ). The value of ?l (12) is given by:
!
!
X ?f ??
X ?f ??
?f ??
sl
sr ??0
sr ??1
?l =
+
+
(17)
??
sl ? o?l
??
sr ??0 ? o?l
??
sr ??1 ? o?l
r
r
Smooth Val Error Rate (er)
4
2
0
?2
0
Log gamma
2
Test Error Rate
6
Log C
Log C
6
4
2
0
?2
0
Log gamma
2
Figure 1: Performance contours for IJCNN with 2000 training points. The sequence of points generated by Grad are shown by (best is in red). The point chosen by Grid is shown by ? in red.
where the partial derivatives of s?l with respect to o?l , ?0 , ?1 can be easily derived from (16) and
(?f /??
sl ) = (?f /?tp)(?tp/??
sl ) + (?f /?f p)(?f p/??
sl ).
We now discuss three methods to compute the sigmoidal parameters ?0 , ?1 . For each of these
methods the partial derivatives of ?0 , ?1 with respect to o?l can be obtained (Keerthi et al. (2006))
and used for computing (17).
Direct Method. Here, we simply set, ?0 = 0, ?1 = t/?, where ? denotes standard deviation
of the outputs {?
ol } and t is a constant which is heuristically set to some fixed value in order to
well-approximate the step function. In our implementation we use t = 10.
Hyperparameter Bias Method. Here, we treat ?0 as a hyperparameter and set ?1 as above.
Minimization Method. In this method, we obtain ?0 , ?1 by performing sigmoidal fitting based
on unconstrained minimization of some smooth criterion N , i.e., (?0 , ?1 ) = argminR2 N . A
natural choice of N is based on Platt?s method (Platt (1999)) where s?l is interpreted as the posterior
probability that the class of P
l th validation example is y?l , and we take N to be the Negative-LogLikelihood: N = Nnll = ? l log(?
sl ). Sigmoidal fitting based on NP
nll has also been previously
proposed in Chapelle et al. (2002). The probabilistic error rate: per = l (1 ? s?l )/?
n and f = Nnll
are suitable validation functions which go well with the choice N = Nnll .
5
Empirical Results
We demonstrate the effectiveness of our method on several binary classification problems. The SVM
model with hinge loss was used. SVM training was done using the SMO algorithm. Five fold cross
validation was used to form the validation functions. Four datasets were used: Adult, IJCNN, Vehicle
and Splice. The first three were taken from http://www.csie.ntu.edu.tw/?cjlin/libsvmtools/datasets/
and Splice was taken from http://ida.first.fraunhofer.de/?raetsch/. The number of examples/features
in these datasets are: Adult: 32561/123; IJCNN: 141691/22; Vehicle: 98528/100; and Splice:
3175/60. For each dataset, training sets of different sizes were chosen in a class-wise stratified
fashion; the remaining examples formed the test set.
The Gaussian kernel (2) and the ARD-Gaussian kernel (3) were used. For (C, ?) tuning with the
Gaussian Kernel, we also tried the popular Grid over a 15 ? 15 grid of values. For C, ? tuning with
the gradient method, the starting point C = ? = 1 was used.
Comparison of validation functions. Figure 1 shows the contours of the smoothed validation
error rate and the actual test error rate for the IJCNN dataset with 2000 training examples on the
(log C, log ?) plane. Grid and Grad respectively denote the grid and the gradient methods applied
to the (C, ?) tuning task. We used f = er smoothed with the direct method for Grad. It can be seen
that the contours are quite similar. We also generated corresponding contours (omitted) for f = per
and f = Nnll (see end of section 4) and found that the validation er with the direct method better
represents the test error rate. Figure 1 also shows that the gradient method very quickly plunges into
the high-performance region in the (C, ?) space.
Comparison of Grid and Grad methods. For various training set sizes of IJCNN, in Table 1, we
compare the speed and generalization performance of Grid and Grad, Clearly Grad is much more
efficient than Grid. The good speed improvement is seen even at small training set sizes. Although
the efficiency of Grid can be improved in certain ways (say, by performing a crude search followed
by a refined search, by avoiding unnecessary exploration of difficult regions in the hyperparameter space etc) Grad determines the optimal hyperparameters more precisely. Table 2 compares
Grid and Grad on Adult and Vehicle datasets for various training sizes. Though the generalization
performance of the two methods are close, Grid is much slower.
Table 1: Comparison of Grid, Grad & Grad-ARD on IJCNN & Splice. nf= number of hyperparameter vectors tried. (For Grid, nf= 225.) cpu= cpu time in minutes. erate=% test error rate.
ntrg
cpu
Grid
erate
nf
2000
4000
8000
16000
32000
10.03
38.77
218.92
1130.37
5331.15
2.95
2.42
1.76
1.24
0.91
11
12
14
12
9
2000
11.42
9.19
13
Grad
cpu
IJCNN
4.58
11.40
68.58
127.03
382.20
Splice
7.57
erate
nf
Grad-ARD
cpu
erate
2.87
2.42
1.77
1.26
0.91
28
13
17
20
7
5.63
8.40
38.58
154.03
269.16
2.65
2.14
1.50
1.08
0.82
8.17
37
35.04
3.49
Table 2: Comparison of Grad & Grid methods on Adult & Vehicle. Definitions of nf, cpu & erate
are as in Table 1. For Vehicle and ntrg =16000, Grid was discontinued after 5 days of computation.
ntrg
2000
4000
8000
16000
nf
9
16
10
6
Grad
cpu
3.62
15.98
52.17
256.40
Adult
erate
16.21
15.64
15.69
15.40
Grid
cpu
8.66
37.53
306.25
3667.90
erate
16.14
15.95
15.59
15.37
nf
7
5
9
6
Grad
cpu
2.50
8.60
83.10
360.88
Vehicle
erate
13.58
13.29
12.84
12.58
Grid
cpu
erate
15.25
13.84
135.28
13.30
1458.12 12.82
?
?
Feature Weighting Experiments. To study the effectiveness of our gradient-based approach when
many hyperparameters are present, we use the ARD-Gaussian kernel in (3) and tune C together
with all the ? t ?s. As before, we used f = er smoothed with the direct method. The solution for
Gaussian kernel was seeded as the starting point for the optimization. Results are reported in Table
1 as Grad-ARD where cpu denotes the extra time for this optimization. We see that Grad-ARD
achieves significant improvements in generalization performance over Grad without increasing the
computational cost by much even though a large number of hyperparameters are being tuned.
Maximizing F-measure by threshold adjustment. In section 4 we mentioned about the possible value of threshold adjustment when the validation/test function of interest is a quantity that is
different from error rate. We now illustrate this by taking the Adult dataset, with F measure. The
size of the training set is 2000. Gaussian kernel (2) was used. We implemented two methods: (1)
we set ?0 = 0 and tuned only C and ?; (2) we tuned the three hyperparameters C, ? and ? 0 . We
ran the methods on ten different random training set/test set splits. Without ?0 , the mean (standard
deviation) of F measure values on 5-fold cross validation and on the test set were: 0.6385 (0.0062)
and 0.6363 (0.0081). With ?0 , the corresponding values improved: 0.6635 (0.0095) and 0.6641
(0.0044). Clearly, the use of ?0 yields a very significant improvement on the F-measure. The ability
to easily include the threshold as an extra hyperparameter is a very useful advantage for our method.
Optimizing weighted error rate in imbalanced problems. In imbalanced problems where the
proportion of examples in the positive class is small, one usually minimizes weighted error rate
wer (see section 4) with a small value of ?. One can think of four possible methods in which,
apart from the kernel parameter ? and threshold ?0 (we used the Hyperparameter bias method for
smoothening), we include other parameters by considering sub-cases of the weighted hinge loss
model (see section 2) ? (1) Usual SVM: Set m+ = m? = 1, C+ = C, C? = C and tune C. (2)
Set m+ = m? = 1, C+ = C, C? = ?C and tune C. (3) Set m+ = m? = 1 and tune C+ and
C? treating them as independent parameters. (4) Use the full Weighted Hinge loss model and tune
C+ , C? , m+ and m? . To compare the performance of these methods we took the IJCNN dataset,
randomly choosing 2000 training examples and keeping the remaining examples as the test set. Ten
such random splits were tried. We take ? = 0.01. The top half of Table 3 reports weighted error
rates associated with validation and test. The weighted hinge loss model performs best.
Table 3: Mean (standard deviation) of weighted (? = 0.01) error rate values on the IJCNN dataset.
C+ = C, C? = C
Validation
Test
0.0571 (0.0183)
0.0638 (0.0160)
Validation
Test
0.1953 (0.0557)
0.1861 (0.0540)
C+ = C, C? = ?C
C+ , C? tuned
With ?0
0.0419 (0.0060)
0.0490 (0.0104)
0.0549 (0.0098)
0.0571 (0.0136)
Without ?0
0.1051 (0.0164)
0.1008 (0.0607)
0.0897 (0.0154)
0.0969 (0.0502)
Full Weighted Hinge
0.0357 (0.0063)
0.0461 (0.0078)
0.0364 (0.0061)
0.0469 (0.0076)
The presence of the threshold parameter ?0 is important for the first three methods. The bottom half
of Table 3 gives the performance statistics of the methods when threshold is not tuned. Interestingly,
for the weighted hinge loss method, tuning of threshold has little effect. Grandvalet et al. (2005)
also make the observation that this method appropriately sets the threshold on its own.
Cost Break-up. In the gradient-based solution process, each step of the optimization requires the
evaluation of f and ?h f . In doing this, there are three steps that take up the bulk of the computational cost: (1) training using the SMO algorithm; (2) the solution of the linear system in (14); and
(3) the remaining computations associated with the gradient, of which the computation of P? ? in (13)
is the major part. We studied the relative break-up of the costs for the IJCNN dataset (training set
sizes ranging from 2000 to 32000), for solution by Grad and Grad-ARD methods. On an average,
the cost of solution by SMO forms 85 to 95% of the total computational time. Thus, the gradient
computation is very cheap. We also found that the P? ? cost of Grad-ARD doesn?t become large in
spite of the fact that 23 hyperparameters are tuned there. This is mainly due to the efficient reusage
of terms in the ARD-Gaussian calculations that we mentioned in section 4.
6
Conclusion
The main contribution of this paper is a fast method of computing the gradient of a validation function with respect to hyperparameters for a range of SVM models; together with a nonlinear optimization technique it can be used to efficiently determine the optimal values of many hyperparameters.
Even in models with just two hyperparameters our approach is faster and offers a more precise hyperparameter placement than the Grid approach. Our approach is particularly of great value for
large scale problems. The ability to tune many hyperparameters easily should be used with care.
On a text classification problem involving many thousands of features we placed an independent
feature weight for each feature and optimized all these weights (together with C) only to find severe
overfitting taking place. So, for a given problem it is important to choose the set of hyperparameters
carefully, in accordance with the richness of the training set.
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2,269 | 306 | Cholinergic Modulation May Enhance Cortical
Associative Memory Function
Michael E. Hasselmo?
Computation and
Neural Systems
Caltech 216-76
Pasadena, CA 91125
Brooke P. Andersont
Computation and
Neural Systems
Caltech 139-74
Pasadena, CA 91125
James M. Bower
Computation and
Neural Systems
Caltech 216-76
Pasadena, CA 91125
Abstract
Combining neuropharmacological experiments with computational modeling, we have shown that cholinergic modulation may enhance associative
memory function in piriform (olfactory) cortex. We have shown that the
acetylcholine analogue carbachol selectively suppresses synaptic transmission between cells within piriform cortex, while leaving input connections
unaffected. When tested in a computational model of piriform cortex,
this selective suppression, applied during learning, enhances associative
memory performance.
1
INTRODUCTION
A wide range of behavioral studies support a role for the neurotransmitter acetylcholine in memory function (Kopelman, 1986; Hagan and Morris, 1989). However,
the role of acetylcholine in memory function has not been linked to the specific
neuropharmacological effects of this transmitter within cerebral cortical networks.
For several years, we have explored cerebral cortical associative memory function
using the piriform cortex as a model system (Wilson and Bower, 1988, Bower, 1990;
Hasselmo et a/., 1991). The anatomical structure of piriform cortex (represented
schematically in figure 1) shows the essential features of more abstract associative
? e-mail: [email protected]
t e-mail: [email protected]
46
Cholinergic Modulation May Enhance Cortical Associative Memory Function
matrix memory models (Haberly and Bower, 1989) 1. Afferent fibers in layer la
provide widely distributed input, while intrinsic fibers in layer Ib provide extensive
excitatory connections between cells within the cortex. Computational models of
piriform cortex demonstrate a theoretical capacity for associative memory function
(Wilson and Bower, 1988; Bower, 1990; Hasselmo et at., 1991). Recently, we have
investigated differences in the physiological properties of the afferent and intrinsic
fiber systems, using modeling to test how these differences affect memory function.
In the experiments described below, we found a selective cholinergic suppression of
intrinsic fiber synaptic transmission. When tested in a simplified model of piriform
cortex, this modulation enhances associative memory performance.
Afferent fiber
synapses (layer la)
- -- -- ------
}
-
- - _.
Afferent input =
}
Intrinsic fiber
synapses (layer lb) =
A .I
BIJ..
+- Neuron activation = ai(t)
}
Lateral inhibition
(via interneurons) =
HIJ..
Neuron output = g( ai (t?
Figure 1: Schematic representation of piriform cortex, showing afferent input Ai
(layer la) and intrinsic connections Bij (layer Ib)
2
EXPERIMENTS
To study differences in the effect of acetylcholine on afferent and intrinsic fiber
systems, we applied the pharmacological agent carbachol (a chemical analogue
of acetylcholine) to a brain slice preparation of piriform cortex while monitoring
changes in the strength of synaptic transmission associated with each fiber system.
In these experiments, both extracellular and intracellular recordings demonstrated
clear differences in the effects of carbachol on synaptic transmission (Hasselmo and
Bower, 1991). The results in figure 2 show that synaptic potentials evoked by
activating intrinsic fibers in layer 1b were strongly suppressed in the presence of
1 For descriptions of standard associative memory models, see for example (Anderson
et al., 1977; Kohonen et al., 1977).
47
48
Hasselmo, Anderson, and Bower
100llM carbachol, while at the same concentration, synaptic potentials evoked by
stimulation of afferent fibers in layer la showed almost no change.
1x 10? o M Carbachol
Control
Washout
3
2
V V V
LAYER lA
1
V
LAYER 1B
O.SmV
3
2
V
V.5mv
20m.
I
Figure 2: Synaptic potentials recorded in layer la and layer Ib before, during, and
after perfusion with the acetylcholine analogue carbachol. Carbachol selectively
suppresses layer Ib (intrinsic fiber) synaptic transmission.
These experiments demonstrate that there is a substantial difference in the neurochemical modulation of synapses associated with the afferent and intrinsic fiber systems within piriform cortex. Cholinergic agents selectively suppress intrinsic fiber
synaptic transmission without affecting afferent fiber synaptic transmission. While
interesting in purely pharmacological terms, these differential effects are even more
intriguing when considered in the context of our computational models of memory
function in this region.
3
MODELING
To investigate the effects of cholinergic suppression of intrinsic fiber synaptic transmission on associative memory function, we developed a simplified model of the
piriform cortex. This simplified model is shown schematically in figure 1. At each
time step, a neuron was picked at random, and its activation was updated as
N
ai(t
+ 1) = Ai(t) + 2)(1- C)Bij -
Hij]g(aj(t?.
j=1
where N = the number of neurons; t = time E {O, 1,2, ... }; c = a parameter
representing the amount of acetylcholine present. c E [0,1]; ai = the activation or
membrane potential of neuron i; g(ai) = the output or firing frequency of neuron
i given ai; Ai
the input to neuron i, representing the afferent input from the
olfactory bulb; Bij = the weight matrix or the synaptic strength from neuron j
to neuron i; and Hij = the inhibition matrix or the amount that neuron j inhibits
neuron i. To account for the local nature of inhibition in the piriform cortex, Hij = 0
=
Cholinergic Modulation May Enhance Cortical Associative Memory Function
for Ii - il > r and Hi; = h for Ii - il < r, where r is the inhibition radius. The
function g(ai) was set to 0 if at < (Ja, where (Ja = a firing threshold; otherwise, it
was set to "Yatanh(ai - (Ja), where "Ya = a firing gain.
The weights were updated every N time steps according to the following hebbian
learning rule.
Bij = !(Wij)
AWij = Wij(t + N) - Wij(t) = (1- c)"Yl(ai - (Jl)g(aj)
The function !(.) is a saturating function, similar to g( .), used so that the weights
could not become negative or grow arbitrarily large (representing a restriction on
how effective synapses could become). "Yl is a parameter that adjusts learning
speed, and (Jl is a learning threshold. The weights were updated every N time steps
to account for the different time scales between synapse modification and neuron
settling.
3.1
TRAINING OF THE MODEL
During learning, the model was presented with various vectors (taken to represent
odors) at the input (Ai(t?. The network was then allowed to run and the weights
to adapt.
The procedure for creating the set of vectors {Am 1m E {I, ... , M} } was: set Ai" =
max{O, G(J.l,
where G = gaussian with average J.l and standard deviation 0', and
normalize the whole vector so that IIA m 112 = N ( 0'2 + J.l2). M = number of memories
or odors presented to network during training, and Ai" the input to neuron i while
odor m is present.
un,
=
=
During learning, in the asynchronous update equation, Ai(t) At for T time steps,
then Ai(t)
for the next T time steps, and so on; i.e., the various odors were
presented cyclically.
= Al
3.2
PERFORMANCE MEASURE FOR THE MODEL
The piriform cortex gets inputs from the olfactory bulb and sends outputs to other
areas of the brain. Assuming that during recall the network receives noisy versions
of the learned input patterns (or odors), we presume the piriform cortex performs
a useful service if it reduces the chance of error in deciding which odor is present
at the input. One way to quantify this is by using the minimum probability of
classification error, Pe (from the field of pattern recognition 2).
=
For the case of 2 odors corrupted by gaussian noise, Pe
the area underneath
the intersection of the gaussians. For spherically symmetric gaussians with mean
vectors J.ll and J.l2 and identical standard deviations 0',
2
Pe = .;:;
=
1
*,
00
.L
e -u2du
where d 1IJ.l1 - J.l211. Thus, the important parameter is the amount of overlap as
quantified by dIu - the larger the dIu, the lower the overlap and Pe .
2See, for example, (Duda and Hart, 1973).
49
50
Hasselmo, Anderson, and Bower
For more than 2 odors and for non-gaussian noise or non-spherically-symmetric
gaussian noise, the equation for Pe becomes less tractable. But keeping with the
above calculations, an analogue of dIu was developed as follows. ul was set =
(IIx -lJiIl 2 ), and then f3 was defined as
f3
=L
i<i
!llJi -lJj II
~(Ui + Uj)
where i,j E {I, ... , M}. Here, f3 is the analogue of dIu in the previous paragraph
and is similar to an average over all odor pairs of dIu.
For the model, if f3 is larger for the output vectors than for the input vectors, there
is less overlap in the outputs, classification of the outputs is easier than classification
of the inputs, and the model is serving a useful purpose. Thus, we use p f3out! f3in
as the performance measure.
=
3.3
TESTING THE MODEL
The model was designed to show whether the presence of acetylcholine has any
influence on learning performance. To that end, the model was allowed to learn
for a time with various levels of acetylcholine present, and then acetylcholine was
turned off and the model was tested.
For testing, weight adaptation was turned off, acetylcholine influence was turned
off (c
0), noisy versions of the various odors presented during learning were
presented at the input, and the network was allowed to settle. From these noisy
input/output pairs, u's could be estimated, f3in and f30ut could be calculated, and
finally p could be calculated. Then, the state of the network could either be reset
(for a new learning run) or be set to what it was before the test (so that learning
could continue as if uninterrupted).
=
3.4
RESULTS OF TESTING
A typical example of a test run is shown in figure 3. There, c was varied from 0 (no
acetylcholine) to .9 (a large concentration), and the various other parameters were:
N
lO, M
10, r
2, h .3, 'Ya
1, 00
1, 'Yl
lO-3, Ol
1, and T
10. In
the figure, large dark rectangles represent larger values of p. Small or non-existant
rectangles represent values of p :5 1.
=
=
=
=
=
=
=
=
=
Notice that, for a fixed amount of acetylcholine, the model's performance rises and
then falls over time. Ideally, the performance should rise and then flatten out, as
further learning should not degrade performance. The weight adaptation equation
used in the model was not optimized to preclude overlearning (where all of the
weights being reinforced have saturated to the largest allowed value). In principle,
the function 1(?) could be used for this, perhaps in conjunction with a weight decay
term. This was not of great concern since the peak performance is what indicates
whether or not acetylcholine has a useful effect. Also, the more acetylcholine present,
the longer the learning took. This is reasonable as, before saturation, AW ex: (I-C).
Figure 4 shows maximum average performance for various values of acetylcholine.
Averages were calculated by doing many tests like the one above. This is useful as
Cholinergic Modulation May Enhance Cortical Associative Memory Function
the odor inputs and the individual tests are stochastic in nature. Obviously, the
larger values of acetylcholine enhance performance.
0.9
I.
u
.. ............ . .. .... . ........ .. .... .... ... ........... .. ..... ?????1 .. "1" .. ?? 11 ... ? ... 111.11111?111111 ... 111111
? ....... , . .. .. . .. .. ...... ' ??1 ? ? ??' ?IIIII.IoI'lIl.I'IIIIIIUII".IIIIII'lIl1l1qlll"II'II~III'III? ?1I1111
? ...... .... ... ........ '" "'1,111 " ,111111111111111111111111""11111".. l1li1? ?1)???1.1.
.. .. .. . ?????? ? ? 11 ?IIIIIIIIII... U..........
1d
'11,11.11'.111 11'11'
,
I
" ......... I. I.lIlilllllllJlil .. . . . . . . ? ? ? ?1'1111111'. " ? 1????????????1.......... 111111.11.11 111.11
? . '., ..... ....... I....II ...
I~? ?ql.IIIIII .. I...I..?. ??1 111."... ,11 .....1 ?.?!I?I .... 1111.... 11II111n1I11Jt1J1.U.
... . ?? ... IIIIIIIIIIII.I...U _ "?I????? .??1II11411111??I?II.1411110 .. ??1 ." ..... ....... .... .......... .. .. . ... ??1 .'
?........ ?10 11.. 111 1? ?1,.111.. ..1.. ...... 1.' ???1111 .? 11?............ .. ...... ...... ... ..... .. ..................... ...... ..... ... ..
???????? ?IIIIII14Ulqlll,II ?? .?.?IIIII?????.? ???? ????????? .. ........................ ??? ???? ?? ??1?????? ???? ?? ??? ???? ???? ??? ???? ?? ... ... .... ..... .
. ...... ,11111111.11.1 .. ?lil??1?? .... . ... . ..... ..... .... ' " ...... . . ... . .... ... . ..... . . . . . ....... . . .. . . . .. ................ . ........... . . " ..... .
.... ?'?ldl.IIII ?..I?? ???? ?? ??? ???.. ?? ??????????? ????????? .. .................. ................ ???? ?? ???? ??????1????? ?????????? ......... '" ..
<l)
? ... ?111111111??????????? ....... .... ........ ........ .... .... . ......... .. ... ... .. ............. .... . .... ...... ..............?....... ... .. ..... ?1??????
.---...
u
'--"
.-....
t::
u....
0
ro
I I I III. III ??
a......
....t::
;.....
<l)
u
t::
0
-
t::
.........
.. ' 11'11'" .... ... ...
. ............ .. . .. . .
........................... ... ... ..... .. ... ....... .
..... 111 11 .. ...... ...... .................... ..... ....... .......................................... ............................................... .... .
0
...c:
u
????11?1?1?????????????? ??????? ???? ??????? ??? ???? ????? ............................... ...... ......... ................ ?? ???????????? ??????1??????????
????111?????? ??? ???????? ????? ?? ???? ????????????????? ????????????????????????1???????????????????????? ????????????????????? ???1????? ................ .
........
>-.
....
?????1??????????? ???? ???????????? ????????????? ????? ???.... ....... ........ .. ...... .......... ....... ... .... .. .... .. ....... .. ??? ??????? ?.. ????????? ???1
<l)
<.)
? ?? 111 ?? .. ? .. ??? ? .. ? ??? ??????? ?? ? ? .. ?? ???? ?? ?????? .... ? ?? ??? ? ?????? ?? ...... ??????? .. ?? .. ? ?? ?? .. ?? ..
~
?????????1.. ?.. ???????.... ?.....................
??.. 1????? ?? ????????? ? .. ?????? ??????? .. ?? .. ???? .. ??? ? .. ?.. ?? .... ?? ..?.. .. ? .. ??. ...... ??? ........ ? ? .. ????.. ? .. ? 1 .. ?? .. ?????????? .. ?? . ......... ... ....
0
.
???1??? ??? .. ? .. ??? .. ??? .. ? .. ??? .. ??????? .. ?? .... ?? .. ? .. ? .. ? .. ? .. ?? ?? ......... .. ..................... .. .............. ........ ... .. ....... ... .... .. .. .
o
4.5x105
Time (t)
Figure 3: Sample test run, with time on the horizontal axis and acetylcholine level
on the vertical axis. Larger black rectangles indicate better performance.
1.7
~
~
t..
::s
1.6
eJ.l ~
e'3
~
e'3
~
(j
e
e;
::s 5
e
.>< c8
1.5
~~
1.2
~
>
t..
e'3 t..
1.4
1.3 .
1.1
0.0
0.2
0.4
0.6
0.8
1.0
Acetylcholine concentration (c)
Figure 4: Maximum average performance vs. acetylcholine level. Acetylcholine
increases the performance level attained.
51
52
Hasselmo, Anderson, and Bower
4
CONCLUSION
The results from the model show that suppression of connections between cells
within the piriform cortex during learning enhances the performance during recall.
Thus, acetylcholine released in the cortex during learning may enhance associative
memory function. These results may explain some of the behavioral evidence for
the role of acetylcholine in memory function and predict that acetylcholine may
be released in cortical structures preferentially during learning. Further biological
experiments are necessary to confirm this prediction.
Acknowledgements
This work was supported by ONR contracts NOOOl4-88-K-0513 and NOOOl4-87-K0377 and NIH postdoctoral training grant NS0725l.
References
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Part II
N euro- Dynal11.ics
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2,270 | 3,060 | Graph Laplacian Regularization for Large-Scale
Semidefinite Programming
Fei Sha
Computer Science Division
UC Berkeley, CA 94720
[email protected]
Kilian Q. Weinberger
Dept of Computer and Information Science
U of Pennsylvania, Philadelphia, PA 19104
[email protected]
Qihui Zhu
Dept of Computer and Information Science
U of Pennsylvania, Philadelphia, PA 19104
[email protected]
Lawrence K. Saul
Dept of Computer Science and Engineering
UC San Diego, La Jolla, CA 92093
[email protected]
Abstract
In many areas of science and engineering, the problem arises how to discover low
dimensional representations of high dimensional data. Recently, a number of researchers have converged on common solutions to this problem using methods
from convex optimization. In particular, many results have been obtained by constructing semidefinite programs (SDPs) with low rank solutions. While the rank
of matrix variables in SDPs cannot be directly constrained, it has been observed
that low rank solutions emerge naturally by computing high variance or maximal
trace solutions that respect local distance constraints. In this paper, we show how
to solve very large problems of this type by a matrix factorization that leads to
much smaller SDPs than those previously studied. The matrix factorization is derived by expanding the solution of the original problem in terms of the bottom
eigenvectors of a graph Laplacian. The smaller SDPs obtained from this matrix
factorization yield very good approximations to solutions of the original problem.
Moreover, these approximations can be further refined by conjugate gradient descent. We illustrate the approach on localization in large scale sensor networks,
where optimizations involving tens of thousands of nodes can be solved in just a
few minutes.
1
Introduction
In many areas of science and engineering, the problem arises how to discover low dimensional representations of high dimensional data. Typically, this high dimensional data is represented in the form
of large graphs or matrices. Such data arises in many applications, including manifold learning [12],
robot navigation [3], protein clustering [6], and sensor localization [1]. In all these applications, the
challenge is to compute low dimensional representations that are consistent with observed measurements of local proximity. For example, in robot path mapping, the robot?s locations must be inferred
from the high dimensional description of its state in terms of sensorimotor input. In this setting,
we expect similar state descriptions to map to similar locations. Likewise, in sensor networks, the
locations of individual nodes must be inferred from the estimated distances between nearby sensors.
Again, the challenge is to find a planar representation of the sensors that preserves local distances.
In general, it is possible to formulate these problems as simple optimizations over the low dimensional representations !xi of individual instances (e.g., robot states, sensor nodes). The most straight-
forward formulations, however, lead to non-convex optimizations that are plagued by local minima.
For this reason, large-scale problems cannot be reliably solved in this manner.
A more promising approach reformulates these problems as convex optimizations, whose global
minima can be efficiently computed. Convexity is obtained by recasting the problems as optimizations over the inner product matrices Xij = !xi ? !xj . The required optimizations can then be relaxed
as instances of semidefinite programming [10], or SDPs. Two difficulties arise, however, from this
approach. First, only low rank solutions for the inner product matrices X yield low dimensional
representations for the vectors !xi . Rank constraints, however, are non-convex; thus SDPs and other
convex relaxations are not guaranteed to yield the desired low dimensional solutions. Second, the
resulting SDPs do not scale very well to large problems. Despite the theoretical guarantees that follow from convexity, it remains prohibitively expensive to solve SDPs over matrices with (say) tens
of thousands of rows and similarly large numbers of constraints.
For the first problem of ?rank regularization?, an apparent solution has emerged from recent work in
manifold learning [12] and nonlinear dimensionality reduction [14]. This work has shown that while
the rank of solutions from SDPs cannot be directly constrained, low rank solutions often emerge naturally by computing maximal trace solutions that respect local distance constraints. Maximizing the
trace of the inner product matrix X has the effect of maximizing the variance of the low dimensional
representation {!xi }. This idea was originally introduced as ?semidefinite embedding? [12, 14], then
later described as ?maximum variance unfolding? [9] (and yet later as ?kernel regularization? [6, 7]).
Here, we adopt the name maximum variance unfolding (MVU) which seems to be currently accepted [13, 15] as best capturing the underlying intuition.
This paper addresses the second problem mentioned above: how to solve very large problems in
MVU. We show how to solve such problems by approximately factorizing the large n?n matrix X
as X ? QYQ! where Q is a pre-computed n?m rectangular matrix with m # n. The factorization
leaves only the much smaller m ? m matrix Y to be optimized with respect to local distance constraints. With this factorization, and by collecting constraints using the Schur complement lemma,
we show how to rewrite the original optimization over the large matrix X as a simple SDP involving
the smaller matrix Y. This SDP can be solved very quickly, yielding an accurate approximation to
the solution of the original problem. Moreover, if desirable, this solution can be further refined [1]
by (non-convex) conjugate gradient descent in the vectors {!xi }.
The main contribution of this paper is the matrix factorization that makes it possible to solve large
problems in MVU. Where does the factorization come from? Either implicitly or explicitly, all
problems of this sort specify a graph whose nodes represent the vectors {!xi } and whose edges
represent local distance constraints. The matrix factorization is obtained by expanding the low
dimensional representation of these nodes (e.g., sensor locations) in terms of the m # n bottom
(smoothest) eigenvectors of the graph Laplacian. Due to the local distance constraints, one expects
the low dimensional representation of these nodes to vary smoothly as one traverses edges in the
graph. The presumption of smoothness justifies the partial orthogonal expansion in terms of the
bottom eigenvectors of the graph Laplacian [5]. Similar ideas have been widely applied in graphbased approaches to semi-supervised learning [4]. Matrix factorizations of this type have also been
previously studied for manifold learning; in [11, 15], though, the local distance constraints were
not properly formulated to permit the large-scale applications considered here, while in [8], the
approximation was not considered in conjunction with a variance-maximizing term to favor low
dimensional representations.
The approach in this paper applies generally to any setting in which low dimensional representations are derived from an SDP that maximizes variance subject to local distance constraints. For
concreteness, we illustrate the approach on the problem of localization in large scale sensor networks, as recently described by [1]. Here, we are able to solve optimizations involving tens of
thousands of nodes in just a few minutes. Similar applications to the SDPs that arise in manifold
learning [12], robot path mapping [3], and protein clustering [6, 7] present no conceptual difficulty.
This paper is organized as follows. Section 2 reviews the problem of localization in large scale
sensor networks and its formulation by [1] as an SDP that maximizes variance subject to local
distance constraints. Section 3 shows how we solve large problems of this form?by approximating
the inner product matrix of sensor locations as the product of smaller matrices, by solving the smaller
SDP that results from this approximation, and by refining the solution from this smaller SDP using
local search. Section 4 presents our experimental results on several simulated networks. Finally,
section 5 concludes by discussing further opportunities for research.
2
Sensor localization via maximum variance unfolding
The problem of sensor localization is best illustrated by example; see Fig. 1. Imagine that
sensors are located in major cities throughout
the continental US, and that nearby sensors can
estimate their distances to one another (e.g., via
radio transmitters). From only this local information, the problem of sensor localization is to
compute the individual sensor locations and to
identify the whole network topology. In purely
mathematical terms, the problem can be viewed
as computing a low rank embedding in two or
three dimensional Euclidean space subject to
local distance constraints.
We assume there are n sensors distributed
in the plane and formulate the problem as
an optimization over their planar coordinates Figure 1: Sensors distributed over US cities. Dis!x1 , . . . , !xn ? %2 . (Sensor localization in three tances are estimated between nearby cities within
dimensional space can be solved in a similar a fixed radius.
way.) We define a neighbor relation i ? j if
the ith and jth sensors are sufficiently close to
estimate their pairwise distance via limited-range radio transmission. From such (noisy) estimates
of local pairwise distances {dij }, the problem of sensor localization is to infer the planar coordinates {!xi }. Work on this problem has typically focused on minimizing the sum-of-squares loss
function [1] that penalizes large deviations from the estimated distances:
!"
#2
(1)
'!xi ? !xj '2 ? d2ij
min
!
x1 ,...,!
xn
i?j
In some applications, the locations of a few sensors are also known in advance. For simplicity, in
this work we consider the scenario where no such ?anchor points? are available as prior knowledge,
and the goal is simply to position the sensors up to a global rotation, reflection, and translation.
Thus, to the above optimization, without loss of generality we can add the centering constraint:
$! $2
$
$
(2)
!xi $ = 0.
$
i
It is straightforward to extend our approach to incorporate anchor points, which generally leads to
even better solutions. In this case, the centering constraint is not needed.
The optimization in eq. (1) is not convex; hence, it is likely to be trapped by local minima. By relaxing the constraint that the sensor locations !xi lie in the %2 plane, we obtain a convex optimization
that is much more tractable [1]. This is done by rewriting the optimization in eqs. (1?2) in terms of
the elements of the inner product matrix Xij =!xi ? !xj . In this way, we obtain:
! "
#2
Minimize:
Xii ? 2Xij + Xjj ? d2ij
i?j
!
(3)
subject to:
(i)
Xij = 0
and
(ii) X ) 0.
ij
The first constraint centers the sensors on the origin, as in eq. (2), while the second constraint
specifies that X is positive semidefinite, which is necessary to interpret it as an inner product matrix
in Euclidean space. In this case, the vectors {!xi } are determined (up to rotation) by singular value
decomposition.
The convex relaxation of the optimization in eqs. (1?2) drops the constraint that that the vectors
!xi lie in the %2 plane. Instead, the vectors will more generally lie in a subspace of dimensionality
equal to the rank of the solution X. To obtain planar coordinates, one can project these vectors into
their two dimensional subspace of maximum variance, obtained from the top two eigenvectors of
X. Unfortunately, if the rank of X is high, this projection loses information. As the error of the
projection grows with the rank of X, we would like to enforce that X has low rank. However, the
rank of a matrix is not a convex function of its elements; thus it cannot be directly constrained as
part of a convex optimization.
Mindful of this problem, the approach to sensor localization in [1] borrows an idea from recent work
in unsupervised learning [12, 14]. Very simply, an extra term is added to the loss function that favors
solutions with high variance, or equivalently, solutions with high trace. (The trace is proportional
%
to the variance assuming that the sensors are centered on the origin, since tr(X) =
xi '2 .)
i '!
The extra variance term in the loss function favors low rank solutions; intuitively, it is based on
the observation that a flat piece of paper has greater diameter than a crumpled one. Following this
intuition, we consider the following optimization:
! "
#2
Maximize:
tr(X) ? ?
Xii ? 2Xij + Xjj ? d2ij
i?j
!
(4)
subject to:
(i)
Xij = 0
and
(ii) X ) 0.
ij
The parameter ? > 0 balances the trade-off between maximizing variance and preserving local
distances. This general framework for trading off global variance versus local rigidity has come to
be known as maximum variance unfolding (MVU) [9, 15, 13].
As demonstrated in [1, 9, 6, 14], these types of optimizations can be written as semidefinite programs
(SDPs) [10]. Many general-purpose solvers for SDPs exist in the public domain (e.g., [2]), but even
for systems with sparse constraints, they do not scale very well to large problems. Thus, for small
networks, this approach to sensor localization is viable, but for large networks (n ? 104 ), exact
solutions are prohibitively expensive. This leads us to consider the methods in the next section.
3 Large-scale maximum variance unfolding
Most SDP solvers are based on interior-point methods whose time-complexity scales cubically in
the matrix size and number of constraints [2]. To solve large problems in MVU, even approximately,
we must therefore reduce them to SDPs over small matrices with small numbers of constraints.
3.1 Matrix factorization
To obtain an optimization involving smaller matrices, we appeal to ideas in spectral graph theory [5].
The sensor network defines a connected graph whose edges represent local pairwise connectivity.
Whenever two nodes share an edge in this graph, we expect the locations of these nodes to be
relatively similar. We can view the location of the sensors as a function that is defined over the
nodes of this graph. Because the edges represent local distance constraints, we expect this function
to vary smoothly as we traverse edges in the graph. The idea of graph regularization in this context is
best understood by analogy. If a smooth function is defined on a bounded interval of %1 , then from
real analysis, we know that it can be well approximated by a low order Fourier series. A similar type
of low order approximation exists if a smooth function is defined over the nodes of a graph. This
low-order approximation on graphs will enable us to simplify the SDPs for MVU, just as low-order
Fourier expansions have been used to regularize many problems in statistical estimation.
Function approximations on graphs are most naturally derived from the eigenvectors of the graph
Laplacian [5]. For unweighted graphs, the graph Laplacian L computes the quadratic form
!
(fi ? fj )2
(5)
f ! Lf =
i?j
on functions f ? % defined over the nodes of the graph. The eigenvectors of L provide a set of
basis functions over the nodes of the graph, ordered by smoothness. Thus, smooth functions f can
be well approximated by linear combinations of the bottom eigenvectors of L.
n
Expanding the sensor locations !xi in terms of these
eigenvectors yields a compact factorization
%m
for the inner product matrix X. Suppose that !xi ? ?=1 Qi? !y? , where the columns of the n?m
rectangular matrix Q store the m bottom eigenvectors of the graph Laplacian (excluding the uniform
eigenvector with zero eigenvalue). Note that in this approximation, the matrix Q can be cheaply precomputed from the unweighted connectivity graph of the sensor network, while the vectors !y? play
the role of unknowns that depend in a complicated way on the local distance estimates dij . Let Y
denote the m ? m inner product matrix of these vectors, with elements Y?? = !y? ? !y? . From the
low-order approximation to the sensor locations, we obtain the matrix factorization:
X ? QYQ! .
(6)
Eq. (6) approximates the inner product matrix X as the product of much smaller matrices. Using this
approximation for localization in large scale networks, we can solve an optimization for the much
smaller m?m matrix Y, as opposed to the original n?n matrix X.
The optimization for the matrix Y is obtained by substituting eq. (6) wherever the matrix X appears
in eq. (4). Some simplifications occur due to the structure of the matrix Q. Because the columns
of Q store mutually orthogonal eigenvectors, it follows that tr(QYQ! ) = tr(Y). Because we do not
include the uniform eigenvector in Q, it follows that QYQ! automatically satisfies the centering
constraint, which can therefore be dropped. Finally, it is sufficient to constrain Y ) 0, which implies
that QYQ! ) 0. With these simplifications, we obtain the following optimization:
! &
'2
Maximize:
tr(Y) ? ?
(QYQ! )ii ?2(QYQ! )ij +(QYQ! )jj ? d2ij
i?j
(7)
subject to:
Y)0
Eq. (6) can alternately be viewed as a form of regularization, as it constrains neighboring sensors
to have nearby locations even when the estimated local distances dij suggest otherwise (e.g., due to
noise). Similar forms of graph regularization have been widely used in semi-supervised learning [4].
3.2 Formulation as SDP
As noted earlier, our strategy for solving large problems in MVU depends on casting the required
optimizations as SDPs over small matrices with few constraints. The matrix factorization in eq. (6)
leads to an optimization over the m ? m matrix Y, as opposed to the n ? n matrix X. In this
section, we show how to cast this optimization as a correspondingly small SDP. This requires us to
reformulate the quadratic optimization over Y ) 0 in eq. (4) in terms of a linear objective function
with linear or positive semidefinite constraints.
We start by noting that the objective function in eq. (7) is a quadratic function of the elements of the
2
matrix Y. Let Y ? %m denote the vector obtained by concatenating all the columns of Y. With
this notation, the objective function (up to an additive constant) takes the form
2
b! Y ? Y ! AY,
2
(8)
where A ? %m ?m is the positive semidefinite matrix that collects all the quadratic coefficients in
2
the objective function and b ? %m is the vector that collects all the linear coefficients. Note that
the trace term in the objective function, tr(Y), is absorbed by the vector b.
With the above notation, we can write the optimization in eq. (7) as an SDP in standard form. As in
[8], this is done in two steps. First, we introduce a dummy variable # that serves as a lower bound
on the quadratic piece of the objective function in eq. (8). Next, we express this bound as a linear
matrix inequality via the Schur complement lemma. Combining these steps, we obtain the SDP:
Maximize:
subject to:
b! Y ? #
(i) Y ) 0
and
(ii)
(
I
1
(A 2 Y)!
1
A2 Y
#
)
) 0.
(9)
1
In the second constraint of this SDP, we have used I to denote the m2 ?m2 identity matrix and A 2
to denote the matrix square root. Thus, via the Schur lemma, this constraint expresses the lower
bound # ? Y ! AY, and the SDP is seen to be equivalent to the optimization in eqs. (7?8).
The SDP in eq. (9) represents a drastic reduction in complexity from the optimization in eq. (7).
The only variables of the SDP are the m(m + 1)/2 elements of Y and the unknown scalar #. The
only constraints are the positive semidefinite constraint on Y and the linear matrix inequality of size
m2 ? m2 . Note that the complexity of this SDP does not depend on the number of nodes or edges in
the network. As a result, this approach scales very well to large problems in sensor localization.
In the above formulation, it is worth noting the important role played by quadratic penalties. The
use of the Schur lemma in eq. (9) was conditioned on the quadratic form of the objective function
in eq. (7). Previous work on MVU has enforced the distance constraints as strict equalities [12],
as one-sided inequalities [9, 11], and as soft constraints with linear penalties [14]. Expressed as
SDPs, these earlier formulations of MVU involved as many constraints as edges in the underlying
graph, even with the matrix factorization in eq. (6). Thus, the speed-ups obtained here over previous
approaches are not merely due to graph regularization, but more precisely to its use in conjunction
with quadratic penalties, all of which can be collected in a single linear matrix inequality via the
Schur lemma.
3.3 Gradient-based improvement
While the matrix factorization in eq. (6) leads to much more tractable optimizations, it only provides
an approximation to the global minimum of the original loss function in eq. (1). As suggested in [1],
we can refine the approximation from eq. (9) by using it as initial starting point for gradient descent
in eq. (1). In general, gradient descent on non-convex functions can converge to undesirable local
minima. In this setting, however, the solution of the SDP in eq. (9) provides a highly accurate
initialization. Though no theoretical guarantees can be made, in practice we have observed that this
initialization often lies in the basin of attraction of the true global minimum.
Our most robust results were obtained by a two-step process. First, starting from the m-dimensional
solution of eq. (9), we used conjugate gradient methods to maximize the objective function in eq. (4).
Though this objective function is written in terms of the inner product matrix X, the hill-climbing in
this step was performed in terms of the vectors !xi ? %m . While not always necessary, this first step
was mainly helpful for localization in sensor networks with irregular (and particularly non-convex)
boundaries. It seems generally difficult to representation such boundaries in terms of the bottom
eigenvectors of the graph Laplacian. Next, we projected the results of this first step into the %2
plane and use conjugate gradient methods to minimize the loss function in eq. (1). This second step
helps to correct patches of the network where either the graph regularization leads to oversmoothing
and/or the rank constraint is not well modeled by MVU.
4 Results
We evaluated our algorithm on two simulated sensor networks of different size and topology. We
did not assume any prior knowledge of sensor locations (e.g., from anchor points). We added white
noise to each local distance measurement with a standard deviation of 10% of the true local distance.
0.6
0.6
0.4
0.4
0.2
0.2
0
0
!0.2
!0.2
!0.4
!0.4
!0.6
!0.6
!0.8
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
0.8
!0.8
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
0.8
Figure 2: Sensor locations inferred for n = 1055 largest cities in the continental US. On average,
each sensor estimated local distances to 18 neighbors, with measurements corrupted by 10% Gaussian noise; see text. Left: sensor locations obtained by solving the SDP in eq. (9) using the m = 10
bottom eigenvectors of the graph Laplacian (computation time 4s). Despite the obvious distortion,
the solution provides a good initial starting point for gradient-based improvement. Right: sensor
locations after post-processing by conjugate gradient descent (additional computation time 3s).
Figure 3: Results on a simulated network with n = 20000 uniformly distributed nodes inside a
centered unit square. See text for details.
The first simulated network, shown in Fig. 1, placed nodes at scaled locations of the n = 1055 largest
cities in the continental US. Each node estimated the local distance to up to 18 other nodes within
a radius of size r = 0.09. The SDP in eq. (9) was solved using the m = 10 bottom eigenvectors
of the graph Laplacian. Fig. 2 shows the solution from this SDP (on the left), as well as the final
result after gradient-based improvement (on the right), as described in section 3.3. From the figure,
it can be seen that the solution of the SDP recovers the general topology of the network but tends to
clump nodes together, especially near the boundaries. After gradient-based improvement, however,
the inferred locations differ very little from the true locations. The construction and solution of the
SDP required 4s of total computation time on a 2.4 GHz Pentium 4 desktop computer, while the
post-processing by conjugate gradient descent took an additional 3s.
computation time (in sec)
objective value
The second simulated network, shown in Fig. 3,
placed nodes at n = 20000 uniformly sampled
objective
points inside the unit square. The nodes were then
time
2.0
480
centered on the origin. Each node estimated the local distance to up to 20 other nodes within a radius of
size r = 0.06. The SDP in eq. (9) was solved using
1.0
240
the m = 10 bottom eigenvectors of the graph Laplacian. The computation time to construct and solve
the SDP was 19s. The follow-up conjugate gradi0
5
10
15
20
ent optimization required 52s for 100 line searches.
number of eigenvectors
Fig. 3 illustrates the absolute positional errors of the
sensor locations computed in three different ways:
the solution from the SDP in eq. (8), the refined so- Figure 4: Left: the value of the loss funclution obtain by conjugate gradient descent, and the tion in eq. (1) from the solution of the SDP in
?baseline? solution obtained by conjugate gradient eq. (8). Right: the computation time to solve
descent from a random initialization. For these plots, the SDP. Both are plotted versus the number
the sensors were colored so that the ground truth of eigenvectors, m, in the matrix factorizapositioning reveals the word CONVEX in the fore- tion.
ground with a radial color gradient in the background. The refined solution in the third panel is seen
to yield highly accurate results. (Note: the representations in the second and fourth panels were
scaled by factors of 0.50 and 1028, respectively, to have the same size as the others.)
We also evaluated the effect of the number of eigenvectors, m, used in the SDP. (We focused on the
role of m, noting that previous studies [1, 7] have thoroughly investigated the role of parameters such
as the weight constant ?, the sensor radius r, and the noise level.) For the simulated network with
nodes at US cities, Fig. 4 plots the value of the loss function in eq. (1) obtained from the solution of
eq. (8) as a function of m. It also plots the computation time required to create and solve the SDP.
The figure shows that more eigenvectors lead to better solutions, but at the expense of increased
computation time. In our experience, there is a ?sweet spot? around m ? 10 that best manages
this tradeoff. Here, the SDP can typically be solved in seconds while still providing a sufficiently
accurate initialization for rapid convergence of subsequent gradient-based methods.
Finally, though not reported here due to space constraints, we also tested our approach on various
data sets in manifold learning from [12]. Our approach generally reduced previous computation
times of minutes or hours to seconds with no noticeable loss of accuracy.
5 Discussion
In this paper, we have proposed an approach for solving large-scale problems in MVU. The approach
makes use of a matrix factorization computed from the bottom eigenvectors of the graph Laplacian.
The factorization yields accurate approximate solutions which can be further refined by local search.
The power of the approach was illustrated by simulated results on sensor localization. The networks
in section 4 have far more nodes and edges than could be analyzed by previously formulated SDPs
for these types of problems [1, 3, 6, 14]. Beyond the problem of sensor localization, our approach
applies quite generally to other settings where low dimensional representations are inferred from
local distance constraints. Thus we are hopeful that the ideas in this paper will find further use in
areas such as robotic path mapping [3], protein clustering [6, 7], and manifold learning [12].
Acknowledgments
This work was supported by NSF Award 0238323.
References
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2,271 | 3,061 | A Probabilistic Algorithm Integrating Source
Localization and Noise Suppression of MEG and
EEG Data
Johanna M. Zumer
Biomagnetic Imaging Lab
Department of Radiology
Joint Graduate Group in Bioengineering
University of California, San Francisco
San Francisco, CA 94143-0628
[email protected]
Hagai T. Attias
Golden Metallic, Inc.
San Francisco, CA
[email protected]
Kensuke Sekihara
Dept. of Systems Design and Engineering
Tokyo Metropolitan, University
Tokyo, 191-0065 Japan
[email protected]
Srikantan S. Nagarajan
Biomagnetic Imaging Lab
Department of Radiology
Joint Graduate Group in Bioengineering
University of California, San Francisco
San Francisco, CA 94143-0628
[email protected]
Abstract
We have developed a novel algorithm for integrating source localization and
noise suppression based on a probabilistic graphical model of stimulus-evoked
MEG/EEG data. Our algorithm localizes multiple dipoles while suppressing noise
sources with the computational complexity equivalent to a single dipole scan,
and is therefore more efficient than traditional multidipole fitting procedures. In
simulation, the algorithm can accurately localize and estimate the time course of
several simultaneously-active dipoles, with rotating or fixed orientation, at noise
levels typical for averaged MEG data. Furthermore, the algorithm is superior to
beamforming techniques, which we show to be an approximation to our graphical
model, in estimation of temporally correlated sources. Success of this algorithm
for localizing auditory cortex in a tumor patient and for localizing an epileptic
spike source are also demonstrated.
1
Introduction
Mapping functional brain activity is an important problem in basic neuroscience research as well as
clinical use. Clinically, such brain mapping procedures are useful to guide neurosurgical planning,
navigation, and tumor and epileptic spike removal, as well as guiding the surgeon as to which areas
of the brain are still relevant for cognitive and motor function in each patient.
Many non-invasive techniques have emerged for functional brain mapping, such as functional magnetic resonance imaging (fMRI) and electromagnetic source imaging (ESI). Although fMRI is the
most popular method for functional brain imaging with high spatial resolution, it suffers from poor
temporal resolution since it measures blood oxygenation level signals with fluctuations in the order
of seconds. However, dynamic neuronal activity has fluctuations in the sub-millisecond time-scale
that can only be directly measured with electromagnetic source imaging (ESI). ESI refers to imaging of neuronal activity using magnetoencephalography (MEG) and electroencephalography (EEG)
data. MEG refers to measurement of tiny magnetic fields surrounding the head and EEG refers to
measurement of voltage potentials using an electrode array placed on the scalp.
The past decade has shown rapid development of whole-head MEG/EEG sensor arrays and of algorithms for reconstruction of brain source activity from MEG and EEG data. Source localization
algorithms, which can be broadly classified as parametric or tomographic, make assumptions to
overcome the ill-posed inverse problem. Parametric methods, including equivalent current dipole
(ECD) fitting techniques, assume knowledge about the number of sources and their approximate locations. A single dipolar source can be localized well, but ECD techniques poorly describe multiple
sources or sources with large spatial extent. Alternatively, tomographic methods reconstruct an estimate of source activity at every grid point across the whole brain. Of many tomographic algorithms,
the adaptive beamformer has been shown to have the best spatial resolution and zero localization
bias [1, 2].
All existing methods for brain source localization are hampered by the many types of noise present
in MEG/EEG data. The magnitude of the stimulus-evoked neural sources are on the order of noise on
a single trial, and so typically 50-200 trials are needed to average in order to distinguish the sources
above noise. This can be time-consuming and difficult for a subject or patient to hold still or pay
attention through the duration of the experiment. Gaussian thermal noise is present at the sensors
themselves. Background room interference such as from powerlines and electronic equipment can
be problematic. Biological noise such as heartbeat, eyeblink or other muscle artifact can also be
present. Ongoing brain activity itself, including the drowsy-state alpha (?10Hz) rhythm can drown
out evoked brain sources. Finally, most localization algorithms have difficulty in separating neural
sources of interest that have temporally overlapping activity.
Noise in MEG and EEG data is typically reduced by a variety of preprocessing algorithms before
being used by source localization algorithms. Simple forms of preprocessing include filtering out
frequency bands not containing a brain signal of interest. Additionally and more recently, ICA
algorithms have been used to remove artefactual components, such as eyeblinks. More sophisticated
techniques have also recently been developed using graphical models for preprocessing prior to
source localization [3, 4].
This paper presents a probabilistic modeling framework for MEG/EEG source localization that is
robust to interference and noise. The framework uses a probabilistic hidden variable model that describes the observed sensor data in terms of activity from unobserved brain and interference sources.
The unobserved source activities and model parameters are inferred from the data by a VariationalBayes Expectation-Maximization algorithm. The algorithm then creates a spatiotemporal image of
brain activity by scanning the brain, inferring the model parameters and variables from sensor data,
and using them to compute the likelihood of a dipole at each grid location in the brain. We also
show that an established source localization method, the minimum variance adaptive beamformer
(MVAB), is an approximation of our framework.
2
Probabilistic model integrating source localization and noise suppression
This section describes the generative model for the data. We assume that the MEG/EEG data has
been collected such that stimulus onset or some other experimental marker indicated the ?zero? time
point. Ongoing brain activity, biological noise, background environmental noise, and sensor noise
are present in both pre-stimulus and post-stimulus periods; however, the evoked neural sources of
interest are only present in the post-stimulus time period. We therefore assume that the sensor data
can be described as coming from four types of sources: (1) evoked source at a particular voxel (grid
point), (2) all other evoked sources not at that voxel, (3) all background noise sources with spatial
covariance at the sensors (including brain, biological, or environmental sources), and (4) sensor
noise. We first infer the model describing source types (3) and (4) from the pre-stimulus data, then
fix certain quantities (described in section 2.2) and infer the full model describing the remaining
source types (1) and (2) from the post-stimulus data (described in section 2.1). After inference of
the model, a map of the source activity is created as well as a map of the likelihood of activity across
voxels.
Let yn denote the K ? 1 vector of sensor data for time point n, where K is the number of sensors
(typically 200). Time ranges from ?Npre : 0 : Npost ? 1 where Npre (Npost ) indicates the number
?
?
?
x
F
s
B
u
?
y
Figure 1: (Left) Graphical model for proposed algorithm. Variables are inside dotted box, parameters outside dotted box. Values in circles unknown and learned from the model, and values in
squares known. (Right) Representation of factors influencing the data recorded at the sensors. In
orange, a post-stimulus source at the voxel of interest, focused on by the lead field F. In red, other
post-stimulus sources not at that particular voxel. In green, all background sources, including ongoing brain activity, eyeblinks, heartbeat, and electrical noise. In blue, thermal noise present in each
sensor.
of time samples in the pre-(post-)stimulus period. The generative model for data y n is
Bun + vn
n = ?Npre , . . . , ?1
yn =
F r srn + Ar xrn + Bun + vn
n = 0, . . . , Npost ? 1
(1)
The K ? 3 forward lead field matrix F r represents the physical (and linear) relationship between
a dipole source at voxel r for each orientation, and its influence on sensor k = 1 : K [5]. The
lead field F r is calculated from knowing the geometry of the source location to the sensor location,
as well as the conducting medium in which the source lies: the human head is most commonly
approximated as a single-shell sphere volume conductor. The source activity s rn is a 3 ? 1 vector
of dipole strength in each of the three orientations at time n for the voxel r. The K ? L matrix A
and the L ? 1 vector xn represent the post-stimulus mixing matrix and evoked non-localized factors,
respectively, corresponding to source type (2) discussed above. The K ? M matrix B and the M ? 1
vector un represent the background mixing matrix and background factors, respectively. The K ? 1
vector vn represents the sensor-level noise. All quantities depend on r in the post-stimulus period
except for B, un and ? (the sensor precision), which will be learned from the pre-stimulus data and
fixed as the other quantities are learned for each voxel. Note however the posterior update for u
?n
does depend on the voxel r. The graphical model is shown in Fig. 1. This generative model becomes
a probabilistic model when we specify prior distributions, as described in the next two sections.
2.1
Localization of evoked sources learned from post-stimulus data
In the stimulus-evoked paradigm, the source strength at each voxel is learned from the post-stimulus
data. The background mixing matrix B and sensor noise precision ? are fixed, after having been
learned from the pre-stimulus data, described in section 2.2. We assume those quantities remain constant through the post-stimulus period and are independent of source location. We assume Gaussian
prior distributions on the source factors and interference factors. We further make the assumption
that the signals are independent and identically distributed (i.i.d.) across time. The source factors
have prior precision given by the 3 ? 3 matrix ?, which relates to the strength of the dipole in each
of 3 orientations. All Normal distributions specified in this paper are defined by their mean and
precision (inverse covariance).
p(s) =
Y
n
p(sn );
p(sn ) =
3
Y
p(sjn ) = N (0, ?)
(2)
j=1
The interference and background factors are assumed to have identity precision. To complete specification of this model, we need to specify prior distributions on the model parameters. We use a
conjugate prior for the interference mixing matrix A, where the ?j is a hyperparameter over the jth
column of A and ?i is the precision of the ith sensor. The hyperparameter ? (a diagonal matrix)
provides a robust mechanism for automatic model order selection, so that the optimal size of A is
inferred from the data through ?.
L
Y
Y
p(x) =
p(xn ); p(xn ) =
p(xjn ) = N (0, I);
n
p(u) =
Y
j=1
p(un ); p(un ) =
n
M
Y
p(ujn ) = N (0, I)
j=1
p(A) =
Y
(3)
N (Aij |0, ?i ?j )
ij
We now specify the full model:
p(y|s, x, u, A, B) =
Y
p(yn |sn , xn , un , A, B);
n
(4)
p(yn |sn , xn , un , A, B, ?) = N (yn |F sn + Axn + Bun , ?)
Exact inference on this model is intractable using the joint posterior over the interference factors and
interference mixing matrix; thus the following variational-Bayesian approximation for the posteriors
is used:
(5)
p(s, x, A|y) ? q(s, x, A|y) = q(s, x|y)q(A|y)
We learn the hidden variables and parameters from the post-stimulus data, iterating through each
voxel in the brain, using a variational-Bayesian Expectation-Maximization (EM) algorithm. All
variables, parameters and hyperparameters are hidden and are learned from the data. In place of
maximizing the logp(y), which would be mathematically intractable, we maximize a lower bound
to logp(y) defined by F in the following equation
Z
F = dx ds dA q(s, x, A|y) [logp(y, s, x, A) ? logq(s, x, A|y)]
(6)
= log p(y) ? KL[q(s, x, A|y)||p(s, x, A|y)]
where KL(q||p) is the Kullback-Leibler divergence between distributions q and p. F is equal to
logp(y) when the approximation in Eq. 5 is true, thus making the KL-distance zero. We use a
variational-Bayesian EM algorithm which alternately maximizes the function F with respect to the
posteriors q(s, x|y) and q(A|y). In the E-step, F is maximized w.r.t. q(s, x|y), keeping q(A|y)
constant, and the sufficient statistics of the hidden variables are computed. In the M-step, F is
maximized w.r.t. q(A|y), keeping q(s, x|y) constant, and the MAP estimate of the parameters and
hyperparameters are computed. In the E-step, the posterior distribution of the background factors
given the data is computed:
q(x0n |yn ) = N (?
x0n , ?);
x
?0n = ??1 A?0T ?yn ;
? = A?0T ?A?0 + K? + I 0
where we define:
!
s?n
0
x
?n
; A?0 =
x
?n =
u
?n
?
F A? B
0
; I =
?
0
0
0
I
0
0
0
I
!
; ?=
(7)
0
0
0
0
?AA
0
0
0
0
!
(8)
In the M-step, we maximize the function F w.r.t. q(A|y) holding q(s, x|y) fixed. We update the
posterior distribution of the interference mixing matrix A including its precision ? AA . Note that the
? was learned
lead field F is fixed based on the geometry of the sensors relative to the head, and B
and fixed from the pre-stimulus data. The sensor noise precision ? is also kept fixed from the prestimulus period. The MAP values of the hyperparameter ? and source factor precision ? are learned
here from the post-stimulus data.
? ux )?AA ; ?AA = (Rxx + ?)?1 ;
A? = (Ryx ? F Rsx ? BR
1
1 ?T ?
??1 = Rss ; ??1 = diag( A
?A + ?AA )
N
K
(9)
The matrices, such as Ryx , represent the posterior covariance between the two subscripts and explicit
definitions are omitted for space. In each iteration of EM, the marginal likelihood is increased.
The variational likelihood function (the lower bound on the exact marginal likelihood) is given as
follows:
N
Lr =
K
0
N
|?||?r | 1 X T
T r r 0r
?
x
?
log
?
?y
?
x
?
log|?r ||?r |
y
n
n +
n
2
|?r |
2 n=1 n
2
(10)
This likelihood function is dependent on the source voxel r and thus a map of the likelihood across
the brain can be displayed. Furthermore, we can also plot an image of the source power estimates
and the time course of activity at each voxel.
2.2 Separation
of background sources learned from pre-stimulus data
Localization error of proposed model (blue)
relative to beamforming (green)
20
Algortihm, sim. interference
Algorithm, real brain noise
MVAB, sim. inteference
MVAB, real brain noise
15
Error (mm)
We note that the computational complexity of the proposed
algorithm is on the order O(KLN S), roughly equivalent to
a single dipole scan, which is of order O(N (K 2 +S)). These
are much smaller than the complexity of a multi-dipole scan
which is order O(N S P ) where P is the number of dipoles,
and if S represents roughly several thousand voxels. We further note that the number of hidden variables to be estimated
is less than the number of data points observed, thus not posing significant problems for estimation accuracy.
10
5
0
-10
-5
0
5
SNIR (dB)
We learn the background mixing matrix and sensor noise pre- Figure 2: Performance of algorithm
cision from the pre-stimulus data using a variational-Bayes relative to beamforming for simufactor analysis model. We assume Gaussian prior distribu- lated datasets. See text for details.
tions on the background factors and sensor noise, with zero
mean and identity precision; we assume a flat prior on the
sensor precision. We again use a conjugate prior for the background mixing matrix B, where ?j is a hyperparameter, similar to the expression for the interference
mixing matrix. All variables, parameters and hyperparameters are hidden and are learned from the
pre-stimulus data. We make the variational-Bayesian approximation for the background mixing matrix and background factors p(u, B|y) ? q(u, B|y) = q(u|y)q(B|y). In the E-step, we maximize
the function F w.r.t. q(u|y) holding q(B|y) fixed. We update the posterior distribution of the factors:
q(u|y) =
Y
q(un |yn ); q(un |yn ) = N (?
un , ?)
n
? T ?B
? + K? ?1 + I
u
?n = ? ?1 B T ?yn ; ? = B
In the M-step, we compute the full posterior distribution of the background mixing matrix B, including its precision matrix ?, and the MAP estimates of the noise precision ? and the hyperparameter
?. We assume the noise precision is diagonal.
? = Ryu ?; ? = (Ruu + ?)?1
B
? ?1 = diag(
2.3
1 ?T ?
B ?B + ?);
K
??1 =
1
? T
diag(Ryy ? BR
yu )
N
(11)
Relationship to minimum-variance adaptive beamforming
Minimum variance adaptive beamforming (MVAB) is one of the best performing source localization techniques. MVAB estimates the dipole source time series by s?n = WM V AB yn , where
?1
?1
WM V AB = (F T Ryy
F )?1 F T Ryy
and Ryy is the measured data covariance matrix. Thus, MVAB
also has computational complexity equivalent to a single-dipole scan, on the order O(K 2 + S).
MVAB attempts to suppress interference, but recent studies have shown the MVAB is ineffective in
cancellation of interference from other brain sources, especially if there are many such sources. In
this section, we derive that MVAB is an approximation to inference on our model.
1
0
z (mm)
z (mm)
-1
-600
1
0.5
40
40
20
0
20
-50
0
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Figure 3: Example of algorithm and MVAB for correlated source simulation. See text for details
We start by rewriting Eq. (1) as yn = F sn + zn , where zn is termed the total noise and is given
by zn = Axn + Bun + vn . It has mean zero and precision matrix ? = (AAT + BB T + ??1 )?1 .
Assuming we have estimated the model parameters A, B, ?, ?, the MAP estimate of the dipole
source time series is s?n = W yn , where W = ??1 F T ? and ? = F T ?F + ?. It can be shown that
this expression is equivalent to Eq. 8.
In the infinite data limit, the data covariance satisfies Ryy = F ??1 F T + ??1 . Its inverse is found,
?1
= ? ? ?F ??1 F T ?. Hence, we obtain
using the matrix inversion lemma, to be Ryy
(12)
?1
F T Ryy
= (I ? F T ?F ??1 )F T ? = ???1 F T ?
where the last step used the expression for ?. Next, we approximate ? ? F T ?F . We then use Eq.
(12) to obtain:
?1
?1
W ? (F T ?F )?1 F T ? = (F T ?F )?1 ???1 ???1 F T ? = (F T Ryy
F )?1 F T Ryy
= WM V AB
3.1
Results
Simulations
The proposed method was tested in a variety of realistic source configurations reconstructed on a 5mm
voxel grid. A single-shell spherical volume conductor model was used to calculate the forward lead
field [5]. Simulated datasets were constructed by
placing Gaussian-damped sinusoidal time courses at
specific locations inside a voxel grid based on realistic head geometry. Sources were assumed to be
present in the post-stimulus period with 437 samples
along with a pre-stimulus period of 263 samples.
Performance of SAKETINI (blue) relative to
beamforming (green) in ability to estimate source time course
NA
Correlation of estimated with true source
3
IN
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In the ?noise-alone (NA)? cases, Gaussian noise
SNIR (dB)
SNIR (dB)
SNIR (dB)
only was added to all time points at the sensors. In
Algorithm, uncorrelated sources
the ?interference (IN)? cases, Gaussian noise time
MVAB, uncorrelated sources
courses occurring in both pre- and post-stimulus
Algorithm, correlated sources
periods, representing simulated ?ongoing? activity,
MVAB, correlated sources
were placed at 50 random locations throughout the
brain voxel grid, and their activity was projected Figure 4: Performance of algorithm relative
onto the sensors and added to both the sensor noise to beamforming for simulated datasets. See
and source activity. Finally, in the ?real (RE)? cases, text for details.
700 samples of real MEG sensor data averaged over
100 trials collected while a human subject was alert but not performing tasks or receiving stimuli.
This real background data thus includes real sensor noise plus real ?ongoing? brain activity that
could interfere with evoked sources and adds spatial correlation to the sensor data. We varied the
Signal-to-Noise Ratio (SNR) and the corresponding Signal to Noise-plus-Interference Ratio (SNIR).
SNIR is calculated from the ratio of the sensor data resulting from sources only to sensor data from
noise plus interference. The first performance figure (Fig. 2) shows the localization error of the
proposed method relative to the MVAB. For this data, a single dipole was placed randomly within
Normalized Intensity (-1000 to 1000)
1000
500
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-1000
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400
Figure 5: Algorithm applied to auditory MEG dataset in patient with temporal lobe tumor.
the voxel grid space. The largest peak in the likelihood map was found and the distance from this
point to the true source was recorded. Each datapoint is an average of 20 realizations of the source
configuration, with error bars showing standard error. This simulation was performed for a variety
of SNIR?s and for all three cases of noise described above. The results from NA were omitted since
both the proposed method and MVAB performed perfectly (zero error). This figure clearly shows
that the error in localization is smaller for the proposed method (black) than for MVAB (green).
The next set of simulations examines the proposed method?s ability to estimate the source time
course sn . Three sources were placed in the brain voxel grid. The locations of these sources were
fixed, but the orientation and time courses were allowed to vary across realizations of the simulations. In half the cases, two of the three sources were forced to be perfectly correlated in time (a
scenario where the MVAB is known to fail), while the time course of the third source was random
relative to the other two. An example of the likelihood map and estimated time courses are shown in
Fig. 3. The likelihood map from the proposed method (on the left) has peaks near all three sources,
including the two that were perfectly correlated (depicted by squares). However, the MVAB (middle
plot) largely misses the source on the left. On the right plot, the estimated time courses from the
proposed method (dashes) and MVAB (dots) are plotted relative to the true time course (solid). The
top and middle plots correspond to the (square) correlated sources. While both methods estimate
the time courses well, MVAB underestimates the overall strength of the source on the top plot, and
exhibits extra noise in the pre-stimulus period for the middle plot.
The performance of the proposed model on the same set of simulations of correlated sources, compared to beamforming, are shown in Fig. 4. This figure shows the correlation of the estimated
with the true time course, for three cases of NA, IN, and RE, and for both correlated and uncorrelated sources, as a function of SNIR. The proposed method consistently out-performs the MVAB
whether the simulated sources are highly correlated with each other (dashed lines), or uncorrelated
(solid), and especially in the RE case. Each datapoint represents an average of 10 realizations of the
simulation, with standard errors on the order of 0.05 (not shown).
3.2
Real data
Stimulus-evoked data was collected in a 275-channel CTF System MEG device from a patient with
temporal lobe tumor near auditory cortex. The stimulus was a noise burst presented binaurally in
120 trials. A large peak is typically seen around 100ms after presentation of an auditory stimulus,
termed the M100 peak. Figure 5 shows the results of the proposed method applied to this dataset.
On the right, the likelihood map show a spatial peak in auditory cortex near the tumor. At that peak
voxel, the time course was extracted and plotted on the left, showing the clear M100 peak. This
information can be useful to the neurosurgical team for guiding the location of surgical lesion and
for providing knowledge of the patient?s auditory processing abilities.
We next tested the proposed method on its ability to localize interictal spikes obtained from a patient
with epilepsy. No sensory stimuli were presented to this patient in this dataset, which was collected
in the same MEG device described above. A Registered EEG/Evoked Potential Technologist marked
segments of the continuously-collected dataset which contained spontaneous spikes, as well as segments that clearly contained no spikes. One segment of data with a spike marked at 400ms was used
here as the ?post-stimulus? period and a separate, spike-free, segment of equal length was used as
the ?pre-stimulus? period. Figure 6 shows the proposed method?s performance on this dataset. The
top left subplot shows the raw sensor data for the segment containing the marked spike. The bottom
left shows the location of the equivalent-current dipole (ECD) fit to several spikes from this patient;
this location from the ECD fit would normally be used clinically. The middle bottom figure shows
the likelihood map from the proposed model; the peak is in clear agreement with the standard ECD
localization. The middle top figure shows the time course estimated for the likelihood spatial peak.
12
RMS = 311.2 fT
Normalized Intensity (-1000 to 1000)
0.6
Magnetic Field (T
)
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x 10
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Figure 6: Performance of algorithm applied to data from an epileptic patient. See text for details.
The spike at 400ms is clearly seen; this cleaned waveform could be of use to the clinician in analyzing peak shape. Finally, the top right plot shows a source time course from a randomly selected
location far from the epileptic spike source (shown with cross-hairs on bottom right plot), in order
to show the low noise level and to show lack of cross-talk onto source estimates elsewhere.
4
Extensions
We have described a novel probabilistic algorithm which performs source localization while robust
to interference and demonstrated its superior performance over a standard method in a variety of
simulations and real datasets. The model takes advantage of knowledge of when sources of interest
are not occurring (such as in the pre-stimulus period of a evoked response paradigm). This model
currently assumes averaged data from an evoked response paradigm, but could be extended to examine variations from the average in individual trials, only involving a few extra parameters to estimate.
Furthermore, the model could be extended to take advantage of temporal smoothness in the data as
well as frequency content. Additionally, spatial smoothness or spatial priors from other modalities,
such as structural or functional MRI, could be incorporated. Furthermore, one is not limited to s n
in a single voxel; the above formulation holds for any P arbitrarily chosen dipole components, no
matter which voxels they belong to, and for any value of P . Of course, as P increases the inferred
value of ? becomes less accurate, and one might choose to restrict it to a diagonal or block-diagonal
form.
References
[1] K. Sekihara, M. Sahani, and S.S. Nagarajan, ?Localization bias and spatial resolution of adaptive
and non-adaptive spatial filters for MEG source reconstruction,? NeuroImage, vol. 25, pp. 1056?
1067, 2005.
[2] K. Sekihara, S.S. Nagarajan, D. Poeppel, and A. Marantz, ?Performance of an MEG adaptivebeamformer technique in the presence of correlated neural activities: Effects on signal intensity
and time-course estimates,? IEEE Trans Biomed Eng, vol. 49, pp. 1534?1546, 2002.
[3] Srikantan S. Nagarajan, Hagai T. Attias, Kenneth E. Hild, and Kensuke Sekihara, ?A graphical
model for estimating stimulus-evoked brain responses from magnetoencephalography data with
large background brain activity,? Neuroimage, vol. 30, pp. 400?416, 2006.
[4] S.S. Nagarajan, H.T. Attias, K.E. Hild, and K. Sekihara, ?Stimulus evoked independent factor
analysis of MEG data with large background activity,? in Adv. Neur. Info. Proc. Sys., 2005.
[5] J. Sarvas, ?Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem,? Phys Med Biol, vol. 32, pp. 11?22, 1987.
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2,272 | 3,062 | Combining causal and similarity-based reasoning
Charles Kemp, Patrick Shafto, Allison Berke & Joshua B. Tenenbaum
Department of Brain and Cognitive Sciences, MIT, Cambridge, MA 02139
{ckemp,shafto,berke,jbt}@mit.edu
Abstract
Everyday inductive reasoning draws on many kinds of knowledge, including
knowledge about relationships between properties and knowledge about relationships between objects. Previous accounts of inductive reasoning generally focus
on just one kind of knowledge: models of causal reasoning often focus on relationships between properties, and models of similarity-based reasoning often
focus on similarity relationships between objects. We present a Bayesian model
of inductive reasoning that incorporates both kinds of knowledge, and show that
it accounts well for human inferences about the properties of biological species.
1
Introduction
Will that berry taste good? Is that table strong enough to sit on? Predicting whether an object has an
unobserved property is among the most basic of all inductive problems. Many kinds of knowledge
appear to be relevant: different researchers emphasize the role of causal knowledge, similarity, category judgments, associations, analogical mappings, scripts, and intuitive theories, and each of these
approaches accounts for an important subset of everyday inferences. Taken in isolation, however,
each of these approaches is fundamentally limited. Humans draw on multiple kinds of knowledge
and integrate them ?exibly when required, and eventually our models should attempt to match this
ability [1]. As an initial step towards this goal, we present a model of inductive reasoning that
is sensitive both to causal relationships between properties and to similarity relationships between
objects.
The inductive problem we consider can be formalized as the problem of ?lling in missing entries
in an object-property matrix (Figure 1). Previous accounts of inductive reasoning generally address
some version of this problem. Models of causal reasoning [2] usually focus on relationships between properties (Figure 1a): if animal A has wings, for instance, it is likely that animal A can ?y.
Similarity-based models [3, 4, 5] usually focus on relationships between objects (Figure 1b): if a
duck carries gene X, a goose is probably more likely than a pig to carry the same gene. Previous
models, however, cannot account for inferences that rely on similarity and causality: if a duck carries gene X and gene X causes enzyme Y to be expressed, it is likely that a goose expresses enzyme
Y (Figure 1c). We develop a unifying model that handles inferences like this, and that subsumes
previous probabilistic approaches to causal reasoning [2] and similarity-based reasoning [5, 6].
Our formal framework overcomes some serious limitations of the two approaches it subsumes. Approaches that rely on causal graphical models typically assume that the feature vectors of any two
objects (any two rows of the matrix in Figure 1a) are conditionally independent given a causal network over the features. Suppose, for example, that the rows of the matrix correspond to people and
the causal network states that smoking leads to lung cancer with probability 0.3. Suppose that Tim,
Tom and Zach are smokers, that Tim and Tom are identical twins, and that Tim has lung cancer. The
assumption of conditional independence implies that Tom and Zach are equally likely to suffer from
long cancer, a conclusion that seems unsatisfactory. The assumption is false because of variables
that are unknown but causally relevant?variables capturing unknown biological and environmental factors that mediate the relationship between smoking and disease. Dealing with these unknown
a)
f1
f2
f3
o1
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c)
f1
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..
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.
.
.
.
Figure 1: (a) Models of causal reasoning generally assume that the rows of an object-feature matrix
are conditionally independent given a causal structure over the features. These models are often
used to make predictions about unobserved features of novel objects. (b) Models of similaritybased reasoning generally assume that columns of the matrix are conditionally independent given a
similarity structure over the objects. These models are often used to make predictions about novel
features. (c) We develop a generative model for object-feature matrices that incorporates causal
relationships between features and similarity relationships between objects. The model uses both
kinds of information to make predictions about matrices with missing entries.
variables is dif?cult, but we suggest that knowledge about similarity between objects can help. Since
Tim is more similar to Tom than Zach, our model correctly predicts that Tom is more likely to have
lung cancer than Zach.
Previous models of similarity-based reasoning [5, 6] also suffer from a restrictive assumption of
conditional independence. This time the assumption states that features (columns of the matrix in
Figure 1b) are conditionally independent given information about the similarity between objects.
Empirical tests of similarity-based models often attempt to satisfy this assumption by using blank
properties?subjects, for example, might be told that coyotes have property P, and asked to judge
the probability that foxes have property P [3]. To a ?rst approximation, inferences in tasks like this
conform to judgments of similarity: subjects conclude, for example, that foxes are more likely to
have property P than mice, since coyotes are more similar to foxes than mice. People, however,
?nd it natural to reason about properties that are linked to familiar properties, and that therefore
violate the assumption of conditional independence. Suppose, for instance, you learn that desert
foxes have skin that is resistant to sunburn. It now seems that desert rats are more likely to share this
property than arctic foxes, even though desert foxes are more similar in general to arctic foxes than
to desert rats. Our model captures inferences like this by incorporating causal relationships between
properties: in this case, having sunburn-resistant skin is linked to the property of living in the desert.
Limiting assumptions of conditional independence can be avoided by specifying a joint distribution
on an entire object-property matrix. Our model uses a distribution that is sensitive both to causal
relationships between properties and to similarity relationships between objects. We know of no
previous models that attempt to combine causality and similarity, and one set of experiments that
has been taken to suggest that people ?nd it dif?cult to combine these sources of information [7].
After introducing our model, we present two experiments designed to test it. The results suggest
that people are able to combine causality with similarity, and that our model accounts well for this
capacity.
2
A generative model for object-feature matrices
Consider ?rst a probabilistic approach to similarity-based reasoning. Assume that So is an object
structure: a graphical model that captures relationships between a known set of objects (Figure 1b).
Suppose, for instance, that the objects include a mouse, a rat, a squirrel and a sheep (o1 through o4 ).
So can be viewed as a graphical model that captures phylogenetic relationships, or as a formalization
of the intuitive similarity between these animals. Given some feature of interest, the feature values
for all objects can be collected into an object vector vo and So speci?es a distribution P (vo ) on these
vectors.
We work with the case where (So , ?) is a tree-structured graphical model of the sort previously used
by methods for Bayesian phylogenetics [8] and cognitive models of property induction [5, 6]. The
objects lie at the leaves of the tree, and we assume that object vectors are binary vectors generated
by a mutation process over the tree. This process has a parameter, ?, that represents the base rate of
a novel feature?the expected proportion of objects with that feature. For instance, if ? is low, the
model (So , ?) will predict that a novel feature will probably not be found in any of the animals, but
if the feature does occur in exactly two of the animals, the mouse and the rat are a more likely pair
than the mouse and the sheep.
The mutation process can be formalized as a continuous-time Markov process with two states (off
and on) and with in?nitesimal matrix:
??
?
Q=
1 ? ? ?(1 ? ?)
We can generate object vectors from this model by imagining a binary feature spreading out over the
tree from root to leaves. The feature is on at the root with probability ?, and the feature may switch
states at any point along any branch. The parameter ? determines how easy it is to move between
the on state and the off state. If ? is high, it will be easy for the Markov process to enter the on state,
and dif?cult for it to leave once it is there.
Consider now a probabilistic approach to causal reasoning. Assume that Sf is a feature structure:
a graphical model that captures relationships between a known set of features (Figure 1a). The
features, for instance, may correspond to enzymes, and Sf may capture the causal relationships
between these enzymes. One possible structure states that enzyme f1 is involved in the production
of enzyme f2 , which is in turn involved in the production of enzyme f3 . The feature values for any
given object can be collected into a feature vector vf and Sf speci?es a distribution P (vf ) on these
vectors.
Suppose now that we are interested in a model that combines the knowledge represented by Sf and
So (Figure 1c). Given that the mouse expresses enzyme f1 , for instance, a combined model should
predict that rats are more likely than squirrels to express enzyme f2 . Formally, we seek a distribution
P (M ), where M is an object-feature matrix, and P (M ) is sensitive to both the relationships between
features and the relationships between animals. Given this distribution, Bayesian inference can be
used to make predictions about the missing entries in a partially observed matrix.
If the features in Sf happen to be independent (Figure 1b), we can assume that column i of the
matrix is generated by (So , ?i ), where ?i is the base rate of fi . Consider then the case where
Sf captures causal relationships between the features (Figure 1c). These causal relationships will
typically depend on several hidden variables. Causal relationships between enzymes, for instance,
are likely to depend on other biological variables, and the causal link between smoking and lung
cancer is mediated by many genetic and environmental variables. Often little is known about these
hidden variables, but to a ?rst approximation we can assume that they respect the similarity structure
So . In Figure 1c, for example, the unknown variables that mediate the relationship between f1 and
f2 are more likely to take the same value in o1 and o2 than in o1 and o4 .
We formalize these intuitions by converting a probabilistic model Sf (Figure 2a) into an equivalent
model SfD (Figure 2b) that uses a deterministic combination of independent random events. These
random events will include hidden but causally relevant variables. In Figure 2b, for example, the
model SfD indicates that the effect e is deterministically present if the cause c is present and the
transmission mechanism t is active, or if there is a background cause b that activates e. The model
SfD is equivalent to Sf in the sense that both models induce the same distribution over the variables
that appear in Sf . In general there will be many models SfD that meet this condition, and there
are algorithms which convert Sf into one of these models [9]. For some applications it might be
desirable to integrate over all of these models, but here we attempt to choose the simplest?the
model SfD with the fewest variables.
Given a commitment to a speci?c deterministic model, we assume that the root variables in SfD are
independently generated over So . More precisely, suppose that the base rate of the ith variable in SfD
is ?i . The distribution P (M ) we seek must meet two conditions (note that each candidate matrix M
now has a column for each variable in SfD ). First, the marginal distribution on each row must match
a) c = 0
0.9
c=1
0.1 c
e
b) c = 0
0.9
c=1 c
0.1
e
t=0
0.2
t=1
0.8 t
b=0
0.9
b=1 b
0.1
c
0
1
c
0
0
0
0
1
1
1
1
e=0 e=1
0.9
0.1
0.18 0.72
t
0
0
1
1
0
0
1
1
b
0
1
0
1
0
1
0
1
e=0 e=1
1
0
0
1
1
0
0
1
1
0
0
1
0
1
0
1
c)
c
t
b
e
o1
c1
t1
b1
e1
o2
c2
t2
b2
e2
o3
c3
t3
b3
e3
Figure 2: (a) A graphical model Sf that captures a probabilistic relationship between a cause c and
an effect e. (b) A deterministic model SfD that induces the same joint distribution over c and e. t
indicates whether the mechanism of causal transmission between c and e is active, and b indicates
whether e is true owing to a background cause independent of c. All of the root variables (c, t
and b) are independent, and the remaining variables (e) are deterministically speci?ed once the
root variables are ?xed. (c) A graphical model created by combining SfD with a tree-structured
representation of the similarity between three objects. The root variables in SfD (c, t, and b) are
independently generated over the tree. Note that the arrows on the edges of the combined model
have been suppressed.
the distribution speci?ed by SfD . Second, if fi is a root variable in SfD , the marginal distribution on
column i must match the distribution speci?ed by (So , ?i ).
There is precisely one distribution P (M ) that satis?es these conditions, and we can represent it
using a graphical model that we call the combined model. Suppose that there are n objects in So . To
create the combined model, we ?rst introduce n copies of SfD . For each root variable i in SfD , we
now connect all copies of variable i according to the structure of So (Figure 2c). The resulting graph
provides the topology of the combined model, and the conditional probability distributions (CPDs)
are inherited from So and SfD . Each node that belongs to the ith copy of So inherits a CPD from
(So , ?i ), and all remaining nodes inherit a (deterministic) CPD from Sf . Now that the distribution
P (M ) is represented as a graphical model, standard inference techniques can be used to compute
the missing entries in a partially-observed matrix M . All results in this paper were computed using
the implementation of the junction tree algorithm included in the Bayes Net toolbox [10].
3
Experiments
When making inductive inferences, a rational agent should exploit all of the information available, including causal relationships between features and similarity relationships between objects.
Whether humans are able to meet this normative standard is not clear, and almost certainly varies
from task to task. On one hand, there are motivating examples like the case of the three smokers
where it seems natural to think about causal relationships and similarity relationships at the same
time. On the other hand, Rehder [7] argues that causal information tends to overwhelm similarity information, and supports this conclusion with data from several tasks involving arti?cial categories.
To help resolve these competing views, we designed several tasks where subjects were required
to simultaneously reason about causal relationships between enzymes and similarity relationships
between animals.
3.1
Experiment 1
Materials and Methods. 16 adults participated in this experiment. Subjects were asked to reason
about the presence of enzymes in a set of four animals: a mouse, a rat, a sheep, and a squirrel.
Each subject was trained on two causal structures, each of which involved three enzymes. Pseudobiological names like ?dexotase? were used in the experiment, but here we will call the enzymes
f1 , f2 and f3 . In the chain condition, subjects were told that f3 is known to be produced by several
a)
Task 1
2
f1
f2
f3
r=0.89
r=0.94
r=0.83
r=0.74
r=0.91
r=0.93
r=0.86
r=0.93
r=0.86
r=0.82
0
Task 2
f1
Task 1
r=0.68
1
?1
2
1
0
?1
?2
b)
r=0.97
f2
2
1
0
f3
?1
Task 2
2
1
0
?1
mouse rat sqrl sheep mouse rat sqrl sheep mouse rat sqrl sheep mouse rat sqrl sheep
Human
Combined
Causal
Similarity
Figure 3: Experiment 1: Behavioral data (column 1) and predictions for three models. (a) Results
for the chain condition. Known test results are marked with arrows: in task 1, subjects were told
only that the mouse had tested positive for f1 , and in task 2 they were told in addition that the rat
had tested negative for f2 . Error bars represent the standard error of the mean. (b) Results for the
common-effect condition.
pathways, and that the most common pathway begins with f1 , which stimulates production of f2 ,
which in turn leads to the production of f3 . In the common-effect condition, subjects were told
that f3 is known to be produced by several pathways, and that one of the most common pathways
involves f1 and the other involves f2 .
To reinforce each causal structure, subjects were shown 20 cards representing animals from twenty
different mammal species (names of the species were not supplied). The card for each animal
represented whether that animal had tested positive for each of the three enzymes. The cards were
chosen to be representative of the distribution captured by a causal network with known structure
(chain or common-effect) and known parameterization. In the chain condition, for example, the
network was a noisy-or network with the form of a chain, where leak probabilities were set to 0.4
(f1 ) or 0.3 (f2 and f3 ), and the probability that each causal link was active was set to 0.7. After
subjects had studied the cards for as long as they liked, the cards were removed and subjects were
asked several questions about the enzymes (e.g. ?you learn about a new mammal?how likely is it
that the mammal produces f3 ??) The questions in this training phase were intended to encourage
subjects to re?ect on the causal relationships between the enzymes.
In both conditions, subjects were told that they would be testing the four animals (mouse, rat, sheep
and squirrel) for each of the three enzymes. Each condition included two tasks. In the chain condition, subjects were told that the mouse had tested positive for f1 , and asked to predict the outcome
of each remaining test (Figure 1c). Subjects were then told in addition that the rat had tested negative for f2 , and again asked to predict the outcome of each remaining test. Note that this second
task requires subjects to integrate causal reasoning with similarity-based reasoning: causal reasoning predicts that the mouse has f2 , and similarity-based reasoning predicts that it does not. In the
common-effect condition, subjects were told that the mouse had tested positive for f3 , then told in
addition that the rat had tested negative for f2 . Ratings were provided on a scale from 0 (very likely
to test negative) to 100 (very likely to test positive).
Results. Subjects used the 100 point scale very differently: in task 1 of each condition, some
subjects chose numbers between 80 and 100, and others chose numbers between 0 and 100. We
therefore converted each set of ratings to z-scores. Average z-scores are shown in the ?rst column
of Figure 3, and the remaining columns show predictions for several models. In each case, model
predictions have been converted from probabilities to z-scores to allow a direct comparison with the
human data.
Our combined model uses a tree over the four animals and a causal network over the features. We
used the tree shown in Figure 1b, where objects o1 through o4 correspond to the mouse, the rat, the
squirrel and the sheep. The tree component of our model has one free-parameter?the total path
length of the tree. The smaller the path length, the more likely that all four animals have the same
feature values, and the greater the path length, the more likely that distant animals in the tree (e.g.
the mouse and the sheep) will have different feature values. All results reported here use the same
value of this parameter?the value that maximizes the average correlation achieved by our model
across Experiments 1 and 2. The causal component of our model includes no free parameters, since
we used the parameters of the network that generated the cards shown to subjects during the training
phase. Comparing the ?rst two columns of Figure 3, we see that our combined model accounts well
for the human data.
Columns 3 and 4 of Figure 3 show model predictions when we remove the similarity component
(column 3) or the causal component (column 4) from our combined model. The model that uses the
causal network alone is described by [2], among others, and the model that uses the tree alone is
described by [6]. Both of these models miss qualitative trends evident in the human data. In task
1 of each condition, the causal model makes identical predictions about the rat, the squirrel and the
sheep: in task 1 of the chain condition, for example, it cannot use the similarity between the mouse
and the rat to predict that the rat is also likely to test positive for f1 . In task 1 of each condition the
similarity model predicts that the unobserved features (f2 and f3 for the chain condition, and f1 and
f2 for the common-effect condition) are distributed identically across the four animals. In task 1 of
the chain condition, for example, the similarity model does not predict that the mouse is more likely
than the sheep to test positive for f2 and f3 .
The limitations of the causal and similarity models suggest that some combination of causality and
similarity is necessary to account for our data. There are likely to be approaches other than our
combined model that account well for our data, but we suggest that accurate predictions will only
be achieved when the causal network and the similarity information are tightly integrated. Simply
averaging the predictions for the causal model and the similarity model will not suf?ce: in task 1 of
the chain condition, for example, both of these models predict that the rat and the sheep are equally
likely to test positive for f2 , and computing an average across these models will result in the same
prediction.
3.2
Experiment 2
Our working hypothesis is that similarity and causality should be combined in most contexts. An
alternative hypothesis?the root-variables hypothesis?was suggested to us by Bob Rehder, and
states that similarity relationships are used only if some of the root variables in a causal structure Sf
are unobserved. For instance, similarity might have in?uenced inferences in the chain condition of
Experiment 1 only because the root variable f1 was never observed for all four animals.
The root-variables hypothesis should be correct in cases where all root variables in the true causal
structure are known. In Figure 2c, for instance, similarity no longer plays a role once the root variables are observed, since the remaining variables are deterministically speci?ed. We are interested,
however, in cases where Sf may not contain all of the causally relevant variables, and where similarity can help to make predictions about the effects of unobserved variables. Consider, for example,
the case of the three smokers, where Sf states that smoking causes lung cancer. Even though the
root variable is observed for Tim, Tom and Zach (all three are smokers), we still believe that Tom is
more likely to suffer from lung cancer than Zach having discovered that Tim has lung cancer. The
case of the three smokers therefore provides intuitive evidence against the root-variables hypothesis,
and we designed a related experiment to explore this hypothesis empirically.
Materials and Methods. Experiment 2 was similar to Experiment 1, except that the common-effect
condition was replaced by a common-cause condition. In the ?rst task for each condition, subjects
were told only that the mouse had tested positive for f1 . In the second task, subjects were told in
addition that the rat, the squirrel and the sheep had tested positive for f1 , and that the mouse had
a)
Task 1
2
f1 f2
f3
r=0.55
r=0.92
r=0.98
r=0.82
r=0.97
r=0.99
r=0.79
r=0.84
r=0.98
r=0.89
r=0.98
1
0
?1
1
Task 2
r=0.94
0
?1
?2
b)
f1
Task 1
2
f2
1
f3
0
?1
Task 2
1
0
?1
?2
mouse rat sqrl sheep mouse rat sqrl sheep mouse rat sqrl sheep mouse rat sqrl sheep
Human
Combined
Causal
Similarity
Figure 4: Experiment 2: Behavioral data and predictions for three models. In task 2 of each condition, the root variable in the causal network (f1 ) is observed for all four animals.
tested negative for f2 . Note that in the second task, values for the root variable (f1 ) were provided
for all animals. 18 adults participated in this experiment.
Results. Figure 4 shows mean z-scores for the subjects and for the four models described previously.
The judgments for the ?rst task in each condition replicate the ?nding from Experiment 1 that
subjects combine causality and similarity when just one of the 12 animal-feature pairs is observed.
The results for the second task rule out the root-variables hypothesis. In the chain condition, for
example, the causal model predicts that the rat and the sheep are equally likely to test positive for
f2 . Subjects predict that the rat is less likely than the sheep to test positive for f2 , and our combined
model accounts for this prediction.
4
Discussion
We developed a model of inductive reasoning that is sensitive to causal relationships between features and similarity relationships between objects, and demonstrated in two experiments that it provides a good account of human reasoning. Our model makes three contributions. First, it provides an
integrated view of two inductive problems?causal reasoning and similarity-based reasoning?that
are usually considered separately. Second, unlike previous accounts of causal reasoning, it acknowledges the importance of unknown but causally relevant variables, and uses similarity to constrain
inferences about the effects of these variables. Third, unlike previous models of similarity-based
reasoning, our model can handle novel properties that are causally linked to known properties.
For expository convenience we have emphasized the distinction between causality and similarity,
but the notion of similarity needed by our approach will often have a causal interpretation. A treestructured taxonomy, for example, is a simple representation of the causal process that generated
biological species?the process of evolution. Our combined model can therefore be seen as a causal
model that takes both relationships between features and evolutionary relationships between species
into account. More generally, our framework can be seen as a method for building sophisticated
causal models, and our experiments suggest that these kinds of models will be needed to account for
the complexity and subtlety of human causal reasoning.
Other researchers have proposed strategies for combining probabilistic models [11], and some of
these methods may account well for our data. In particular, the product of experts approach [12]
should lead to predictions that are qualitatively similar to the predictions of our combined model.
Unlike our approach, a product of experts model is not a directed graphical model, and does not
support predictions about interventions. Neither of our experiments explored inferences about interventions, but an adequate causal model should be able to handle inferences of this sort.
Causal knowledge and similarity are just two of the many varieties of knowledge that support inductive reasoning. Any single form of knowledge is a worthy topic of study, but everyday inferences
often draw upon multiple kinds of knowledge. We have not provided a recipe for combining arbitrary forms of knowledge, but our work illustrates two general themes that may apply quite broadly.
First, different generative models may capture different aspects of human knowledge, but all of these
models use a common language: the language of probability. Probabilistic models are modular, and
can be composed in many different ways to build integrated models of inductive reasoning. Second,
the stochastic component of most generative models is in part an expression of ignorance. Using one
model (e.g. a similarity model) to constrain the stochastic component of another model (e.g. a causal
network) may be a relatively general method for combining probabilistic knowledge representations.
Although we have focused on human reasoning, integrated models of induction are needed in many
scienti?c ?elds. Our combined model may ?nd applications in computational biology: predicting
whether an organism expresses a certain gene, for example, should rely on phylogenetic relationships between organisms and causal relationships between genes. Related models have already
been explored: Engelhardt et al. [13] develop an approach to protein function prediction that combines phylogenetic relationships between proteins with relationships between protein functions, and
several authors have explored models that combine phylogenies with hidden Markov models. Combining two models is only a small step towards a fully integrated approach, but probability theory
provides a lingua franca for combining many different representations of the world.
Acknowledgments We thank Bob Rehder and Brian Milch for valuable discussions. This work was supported
in part by AFOSR MURI contract FA9550-05-1-0321, the William Asbjornsen Albert memorial fellowship
(CK) and the Paul E. Newton Chair (JBT).
References
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2,273 | 3,063 | Detecting Humans via Their Pose
Alessandro Bissacco
Computer Science Department
University of California, Los Angeles
Los Angeles, CA 90095
[email protected]
Ming-Hsuan Yang
Honda Research Institute
800 California Street
Mountain View, CA 94041
[email protected]
Stefano Soatto
Computer Science Department
University of California, Los Angeles
Los Angeles, CA 90095
[email protected]
Abstract
We consider the problem of detecting humans and classifying their pose from a
single image. Specifically, our goal is to devise a statistical model that simultaneously answers two questions: 1) is there a human in the image? and, if so, 2) what
is a low-dimensional representation of her pose? We investigate models that can
be learned in an unsupervised manner on unlabeled images of human poses, and
provide information that can be used to match the pose of a new image to the ones
present in the training set. Starting from a set of descriptors recently proposed for
human detection, we apply the Latent Dirichlet Allocation framework to model
the statistics of these features, and use the resulting model to answer the above
questions. We show how our model can efficiently describe the space of images
of humans with their pose, by providing an effective representation of poses for
tasks such as classification and matching, while performing remarkably well in
human/non human decision problems, thus enabling its use for human detection.
We validate the model with extensive quantitative experiments and comparisons
with other approaches on human detection and pose matching.
1
Introduction
Human detection and localization from a single image is an active area of research that has witnessed
a surge of interest in recent years [9, 18, 6]. Simply put, given an image, we want to devise an
automatic procedure that locates the regions that contain human bodies in arbitrary pose. This is
hard because of the wide variability that images of humans exhibit. Given that it is impractical
to explicitly model nuisance factors such as clothing, lighting conditions, viewpoint, body pose,
partial and/or self-occlusions, one can learn a descriptive model of human/non human statistics.
The problem then reduces to a binary classification task for which we can directly apply general
statistical learning techniques. Consequently, the main focus of research on human detection so far
has been on deriving a suitable representation [9, 18, 6], i.e. one that is most insensitive to typical
appearance variations, so that it provides good features to a standard classifier.
Recently local descriptors based on histograms of gradient orientations such as [6] have proven
to be particularly successful for human detection tasks. The main idea is to use distributions of
gradient orientations in order to be insensitve to color, brightness and contrast changes and, to some
extent, local deformations. However, to account for more macroscopic variations, due for example
to changes in pose, a more complex statistical model is warranted. We show how a special class of
hierarchical Bayesian processes can be used as generative models for these features and applied to
the problem of detection and pose classification.
This work can be interpreted as an attempt to bridge the gap between the two related problems of
human detection and pose estimation in the literature. In human detection, since a simple yes/no
answer is required, there is no need to introduce a complex model with latent variables associated
to physical quantities. In pose estimation, on the other hand, the goal is to infer these quantities and
therefore a full generative model is a natural approach. Between these extremes lies our approach.
We estimate a probabilistic model with a set of latent variables, which do not necessarily admit a
direct interpretation in terms of configurations of objects in the image. However, these quantities are
instrumental to both human detection and the pose classification problem.
The main difficulty is in the representation of the pose information. Humans are highly articulated
objects with many degrees of freedom, which makes defining pose classes a remarkably difficult
problem. Even with manual labeling, how does one judge the distance between two poses or cluster
them? In such situations, we believe that the only avenue is an unsupervised method. We propose an approach which allows for unsupervised clustering of images of humans and provides a
low dimensional representation encoding essential information on their pose. The chief difference
with standard clustering or dimensionality reduction techniques is that we derive a full probabilistic
framework, which provides principled ways to combine and compare different models, as required
for tasks such as human detection, pose classification and matching.
2
Context and Motivation
The literature on human detection and pose estimation is too broad for us to review here. So we focus
on the case of a single image, neglecting scenarios where temporal information or a background
model are available and effective algorithms based on silhouettes [20, 12, 1] or motion patterns [18]
can be applied.
Detecting humans and estimating poses from single images is a fundamental problem with a range
of sensible applications, such as image retrieval and understanding. It makes sense to tackle this
problem as we know humans are capable of telling the locations and poses of people from the visual
information contained in photographs. The question is how to represent such information, and the
answer we give constitutes the main novelty of this work.
Numerous representation schemes have been exploited for human detection, e.g., Haar wavelets
[18], edges [9], gradient orientations [6], gradients and second derivatives [19] and regions from
image segmentation [15]. With these representations, algorithms have been applied for the detection
process such as template matching [9], support vector machine [19, 6], Adaboost [18], and grouping
[15], to name a few. Most approaches to pose estimation are based on body part detectors, using
either edge, shape, color and texture cues [7, 21, 15], or learned from training data [19]. The optimal
configuration of the part assembly is then computed using dynamic programming as first introduced
in [7], or by performing inference on a generative probabilistic model, using either Data Driven
Markov Chain Monte Carlo, Belief Propagation or its non-Gaussian extensions [21].
These works focus on only one of the two problems, either detection or pose estimation. Our approach is different, in that our goal is to extract more information than a simple yes/no answer, while
at the same time not reaching the full level of detail of determining the precise location of all body
parts. Thus we want to simultaneously perform detection and pose classification, and we want to
do it in an unsupervised manner. In this aspect, our work is related to the constellation models of
Weber et al. [23], although we do not have an explicit decomposition of the object in parts.
We start from the representation [6] based on gradient histograms recently applied to human detection with excellent results, and derive a probabilistic model for it. We show that with this model
one can successfully detect humans and classify their poses. The statistical tools used in this work,
Latent Dirichlet Allocation (LDA) [3] and related algorithms [5, 4], have been introduced in the
text analysis context and recently applied to the problem of recognition of object and action classes
[8, 22, 2, 16]. Contrary to most approaches (all but [8]) where the image is treated as a ?bag of
features? and all spatial information is lost, we encode the location and orientation of edges in the
basic elements (words) so that this essential information is explicitly represented by the model.
3
A Probabilistic Model for Gradient Orientations
We first describe the features that we use as the basic representations of images, and then propose a
probabilistic model with its application to the feature generation process.
3.1 Histogram of Oriented Gradients
Local descriptors based on gradient orientations are one of the most successful representations for
image-based detection and matching, as was firstly demonstrated by Lowe in [14]. Among the various approaches within this class, the best performer for humans appears to be [6]. This descriptor is
obtained by computing weighted histograms of gradient orientations over a grid of spatial neighborhoods (cells), which are then grouped in overlapping regions (blocks) and normalized for brightness
and contrast changes.
Assume that we are given a patch of 64 ? 128 pixels, we divide the patch into cells of 8 ? 8
pixels, and for each cell a gradient orientation histogram is computed. The histogram represents
a quantization in 9 bins of gradient orientations in the range 0? ? 180? . Each pixel contributes
to the neighboring bins, both in orientation and space, by an amount proportional to the gradient
magnitude and linearly decreasing with the distance from the bin center. These cells are grouped in
2 ? 2 blocks, and the contribution of each pixel is also weighted by a Gaussian kernel with ? = 8,
centered in the block. Finally the vectors v of cell histograms within one block are normalized in
L2 norm: v
? = v/(||v||2 + ). The final descriptor is a collection of histograms from overlapping
blocks (each cell shared by 4 blocks).
The main characteristic of such a representation is robustness to local deformations, illumination
changes and, to a limited extent, viewpoint and pose changes due to coarsening of the histograms.
In order to handle the larger variations typical of human body images, we need to complement this
representation with a model. We propose a probabilistic model that can accurately describe the
generation process of these features.
3.2 Latent Dirichlet Allocation
Latent Dirichlet Allocation (LDA) [3] is a hierachical model for sparse discrete mixture distributions, where the basic elements (words) are sampled from a mixture of component distributions, and
each component defines a discrete distribuition over the set of words.
We are given a collection of documents where words w, the basic units of our data, take values in
a dictionary of W unique elements w ? { 1, ? ? ? , W }. A document w = ( w1 , w2 , ? ? ? , wW )
PW
is a collection of word counts wj : j=1 wj = N . The standard LDA model does not include the
distribution of N , so it can be omitted in what follows. The corpus D = { w1 , w2 , ? ? ? , wM } is a
collection of M documents.
The LDA model introduces a set of K latent variables, called topics. Each word in the document
is assumed to be generated by one of the topics. Under this model, the generative process for each
document w in the corpus is as follows:
1. Choose ? ? Dirichlet(?).
2. For each word j = 1, ? ? ? , W in the dictionary, choose a word count wj ? p(wj |?, ?).
where the word counts wj are drawn from a discrete distribution conditioned on the topic proportions ?: p(wj |?, ?) = ?j. ?. Recently several variants to this model have been developed, notably
the Multinomial PCA [4], where the discrete distributions are replaced by multinomials, and the
Gamma-Poisson process [5], where the number of words ?i from each component are independent
Gamma samples and p(wj |?, ?) is Poisson. The hyperparameter ? ? RK
+ represents the prior on
W ?K
the topic distribution, ? ? RK
are the parameters of the
+ are the topic proportions, and ? ? R+
word distributions conditioned on topics. In the context of this work, words correspond to oriented
gradients, and documents as well as corpus correspond to images and a set of images respectively.
The topic derived by the LDA model is the pose of interest in this work. Here we can safely assume
that the topic distributions ? are deterministic parameters, later for the purpose of inference we will
treat them as random variables and assign them a Dirichlet prior: ?.k ? Dirichlet(?) , where ?.k
denotes the k-th column of ?.
Then the likelihood of a document w is:
Z
p(w|?, ?) =
p(?|?)
W
Y
p(wn |?, ?)d?
(1)
n=1
where documents are represented as a continuous mixture distribution. The advantage over standard
mixture of discrete distributions [17], is that we allow each document to be generated by more than
one topic.
3.3 A Bayesian Model for Gradient Orientation Histograms
Now we can show how the described two-level Bayesian process finds a natural application in modeling the spatial distribution of gradient orientations. Here we consider the histogram of oriented
gradients [6] as the basic feature from which we build our generative model, but let us point out
that the framework we introduce is more general and can be applied to any descriptor based on histograms1 . In this histogram descriptor, we have that each bin represents the intensity of the gradient
at a particular location, defined by a range of orientations and a local neighborhood (cell). Thus the
bin height denotes the strength and number of the edges in the cell.
The first thing to notice in deriving a generative models for this class of features is that, since they
represent a weighted histogram, they have non-negative elements. Thus a proper generative model
for these descriptors imposes non-negativity constraints. As we will see in the experiments, a linear approach such as Non-negative Matrix Factorization [13] leads to extremely poor performance,
probably due to the high curvature of the space. On the opposite end, representing the nonlinearity
of the space with a set of samples by Vector Quantization is feasible only using a large number of
samples, which is against our goal of deriving an economical representation of the pose.
We propose using the Latent Dirichlet Allocation model to represent the statistics of the gradient
orientation features. In order to do so we need to quantize feature values. While not investigated
in the original paper [6], quantization is common practice for similar histogram-based descriptors,
such as [14]. We tested the effect of quantization on the performance of the human detector based
on Histogram of Oriented Gradient descriptors and linear Support Vector Machines described in [6].
As evident in Figure 1, with 16 or more discrete levels we practically obtain the same performance
as with the original continuous descriptors. Thus in what follows we can safely assume that the
basic features are collections of small integers, the histogram bin counts wj .
Thus, if we quantize histogram bins and assign a unique word to each bin, we obtain a representation for which we can directly apply the LDA framework. Analogous to document analysis, an
orientation histogram computed on an image patch is a document w represented as a bag of words
( w1 , ? ? ? , wW ), where the word counts wj are the bin heights. We assume that such a histogram
is generated by a mixture of basic components (topics), where each topic z induces a discrete distribution p(r|?.z ) on bins representing a typical configuration of edges common to a class of elements
in the dataset. By summing the contributions from each topic we obtain the total count wj for each
bin, distributed according to p(wj |?, ?).
The main property of such feature formation process, desirable for our applications, is the fact that
topics combine additively. That is, the same bin may have contributions from multiple topics, and
this models the fact that the bin height is the count of edges in a neighborhood which may include
parts generated by different components. Finally, let us point out that by assigning a unique word
to each bin we model spatial information, encoded in the word identity, whereas most previous
approaches (e.g. [22]) using similar probabilistic models for object class recognition did not exploit
this kind of information.
4
Probabilistic Detection and Pose Estimation
The first application of our approach is human detection. Notice that our main goal is to develop a
model to represent the statistics of images for human pose classification. We use the human detection
problem as a convenient testbed for validating the goodness of our representation, since for this
application large labelled datasets and efficient algorithms are available. By no means we intend
to compete with state-of-the-art discriminative approaches for human detection alone, which are
optimized to represent the decision boundary and thus are supposed to perform better than generative
approaches in binary classification tasks. However, if the generative model is good at capturing the
statistics of human images we expect it to perform well also in discriminating humans from the
background.
In human detection, given a set of positive and negative examples and a previously unseen image
Inew , we are asked to choose between two hypotheses: either it contains a human or it is a background image. The first step is to compute the gradient histogram representation w(I) for the test
and training images. Then we learn a model for humans and background images and use a threshold
1
Notice that, due to the particular normalization procedure applied, the histogram features we consider here
do not have unit norm (in fact, they are zero on uniform regions).
on the likelihood ratio2 for detection:
L=
P (w(Inew )|Human)
(2)
P (w(Inew )|Background)
For the the LDA (and related models [5, 4], the likelihoods p(w(I)|?, ?) are computed as in (1),
where ?, ? are model parameters and can be learned from data. In practice, we can assume ?
is known and compute an estimate of ? from the training corpus. In doing so, we can choose
from two main inference algorithms: mean field or variational inference [3] and Gibbs sampling
[10]. Mean field algorithms provide a lower bound on the likelihood, while Gibbs sampling gives
statistics based on a sequential sampling scheme. As shown in Figure 1, in our experiments Gibbs
sampling exhibited superior performance over mean field in terms of classification accuracy. We
have experimented with two variations, a direct method and Rao-Blackwellised sampling (see [4]
for details). Both methods gave similar performance, here we report the results obtained using the
direct method, whose main iteration is as follows:
1. For each document wi = (wi,1 , ? ? ? , wi,W ):
(i)
First sample ?(i) ? p(?|wi , ?, ?), and then sample vj. ? Multinomial(?j. ?(i) , wi,j )
2. For each topic k:
P (i)
Sample ?.k ? Dirichlet( i v.k + ?)
In pose classification, we start from a set of unlabeled training examples of human poses and learn
the topic distribution ?. This defines a probabilistic mapping to the topic variables, which can be
seen as an economical representation encoding essential information of the pose. That is, from a
?
image Inew , we estimate the topic proportions
Z ?(Inew ) as:
? new ) = ?p(?|w(Inew ), ?, ?)d?
?(I
(3)
Pose information can be recovered by matching the new image Inew to an image I in the training set. For matching, ideally we would like to compute the matching score as Sopt (I, Inew ) =
P (w(Inew )|w(I), ?, ?), i.e. the posterior probability of the test image Inew given the training image I and the model ?, ?. However this would be computationally expensive as for each pair I, Inew
it requires computing an expectation of the form (3), thus we opted for a suboptimal solution. For
?
each training document I, in the learning step we compute the posterior topic proportions ?(I)
as
in (3). Then the matching score S between Inew and I is given by the dot product between the two
? and ?(I
? new ):
vectors ?(I)
?
? new ) >
S(I, Inew ) =< ?(I),
?(I
(4)
?
The computation of this score requires only a dot product between low dimensional unit vectors ?,
so our approach represent an efficient method for matching and clustering poses in large datasets.
5
Experiments
We first tested the efficacy of our model for the human detection task. We used the dataset provided
by [6], consisting of 2340 64 ? 128 images of pedestrians in various configurations and 1671 images
of outdoor scenes not containing humans. We collected negative examples by random sampling 10
patches from each of the first 1218 non-human images. These, together with 1208 positive examples
and their left-right reflections, constituted our first training set. We used the learned model to classify
remaining 1132 positive and on 5889 patches randomly extracted from the residual background
images.
We first computed the histograms of oriented gradients from the image patches following the procedure outlined in Section 3.1. These feature are quantized so that they can be represented by our
discrete stochastic model.
We tested the effect of different quantization levels on the performances of the boosted SVM classifier [6]: a initial training on the provided dataset is followed by a boosting round where the trained
classifier is applied to the background images to find false positive; these hard examples are then
added to for a second training of the classifier. As Figure 1 shows, the effect of quantization is significant only if we use less than 4 bits. Therefore, we chose to discretize the features to 16 quantization
levels.
2
Ideally we would like to use the posterior ratio R = P (Human|Inew )/P (Background|Inew ). However
notice that R is equal to (2) if we assume equal priors P (Human) = P (Background).
Given the number of topics K and the prior hyperparameters ?, ?, we learned topic distributions ?
? using either Gibbs sampling or Mean Field. We tested both Gamma [5]
and topic proportions ?(I)
and Dirichlet [3, 4] distributions for topic priors, obtaining best results with the multinomial model
[4] with scalar priors ?i = a, ?i = b, in these experiments a = 2/K and b = 0.5.
The number of topics K is an important parameter that should be carefully chosen based on considerations on modeling power and complexity. With a higher number of topics we can more accurately
fit the data, which can be measured by the increase in the likelihood of the training set. This does
not come for free: we have a larger number of parameters and an increased computational cost for
learning. Eventually, an excessive topic number causes overfitting, which can be measured as the
likelihood in the test dataset decreases. For the INRIA data, experimental evaluations suggested that
a good tradeoff is obtained with K = 24.
We learned two models, one for positive and one for negative examples. For learning we run the
Gibbs sampling algorithm described in Section 4 for a total number of 300 samples per document,
including 50 samples to compute the likelihoods (1). We also trained the model using the Mean
Field approximation, but as we can see in Figures 1 and 4 the results using Gibbs sampling are
better. For details on the implementation we refer to [4]. We then obtain a detector by computing
the likelihood ratio (2) and comparing it with a threshold.
In Figure 1 we show the performances of our detector on the INRIA dataset, where for the sake of
comparison with other approaches boosting is not performed. We show the results for:
? Linear SVM classifier: Trained as described, using the SVMLight software package.
? Vector Quantization: Positive and negative models learned as collections of K clusters
using the K-Means algorithm. Then the decision rule is Nearest Neighbor, that is whether
the closest cluster belongs to positive or negative model.
? Non-negative Matrix Factorization: Feature vectors are collected in a matrix V , and the factorization that minimizes ||Y ? W H||22 with W, H nonnegative is computed using the multiplicative update algorithm of [13]. Using an analogy with the LDA model, the columns
of W contain the topic distributions, while the columns of H represent the component
weights. A classifier is obtained as the difference of the residuals of the feature projections
on the positive and negative models.
From the plot we see how the results of our approach are comparable with the performance of the
Linear SVM, while being far superior to the other generative approaches. We would like to stress
that a sole comparison on detection performance with state-of-the discriminative classifiers would
be inappropriate, since our model targets pose classification which is harder than binary detection.
A fair comparison should divide the dataset in classes and compare our model with a multiclass
classifier. But then we would face the difficult problem of how to label human poses.
For the experiments on pose classification and matching, we used the CMU Mobo dataset [11]. It
consists of sequences of subjects performing different motion patterns, each sequence taken from 6
different views. In the experiments we used 22 sequences of fast walking motion, picking the first
100 frames from each sequence.
In the first experiment we trained the model with all the views and set the number of topics equal
to the number of views, K = 6. As expected, each topic distribution represents a view and by
assigning every image I to the topic k with highest proportion k = arg maxk ??k (I) we correctly
associated all the images from the same view to the same topic.
To obtain a more challenging setup, we restricted to a single view and tested the classification performance of our approach in matching poses. We learned a model with K = 8 topics from 16 training
sequences, and used the remaining 6 for testing. In Figure 2 we show sample topics distributions
from this model. In Figure 3, for each test sequence we display a sample frame and the associated top ten matches from the training data according to the score (4). We can see how the pose is
matched against change of appearance and motion style, specifically a test subject pose is matched
to similar poses of different subjects in the training set. This shows how the topic representation
factors out most of the appearance variations and retains only essential information on the pose.
In order to give a quantitative evaluation of the pose matching performance and compare with other
approaches, we labeled the dataset by mapping the set of walking poses to the interval [0, 1]. We
manually assigned 0 to the frames at the beginning of the double support phase, when the swinging
Effect of Histogram Quantization on Human Detection
Detector Performance Comparison
0.2
0.5
0.1
miss rate
miss rate
0.2
0.05
0.1
0.05
0.02
0.01 ?6
10
Continous
32 Levels
16 Levels
8 Levels
4 Levels
?5
10
0.02
?4
?3
?2
10
10
10
false positives per window (FPPW)
?1
10
0.01 ?3
10
NMF
VQ
LDA Gibbs
LDA MF
Linear SVM
?2
?1
10
10
false positives per window (FPPW)
Figure 1: Human detection results. (Left) Effect on human detection performances of quantizing the histogram of oriented gradient descriptor [6] for a boosted linear SVM classifier based on these features. Here
we show false positive vs. false negative curves on log scale. We can see that for 16 quantization levels or
more the differences are negligible, thus validating our discrete approach. (Right) Performances of five detectors using HOG features trained without boosting and tested on the INRIA dataset: LDA detectors learned by
Gibbs Sampling and Mean Field, Vector Quantization, Non-negative Matrix Factorization - all with K = 24
components/codewords - and Linear SVM. We can see how the Gibbs LDA outperform by far the other unsupervised clustering techniques and scores comparably with the Linear SVM, which is specifically optimized
for the simpler binary classification problem.
Figure 2: Topics distributions and clusters. We show sample topics (2 out of 8) from the LDA model trained
on the single view Mobo sequences. For each topic k, we show 12 images in 2 rows. The first column shows the
distribution of local orientations associated with topic k: (top) visualization of the orientations and (bottom)
average gradient intensities for each cell. The right 5 columns show the top ten images in the dataset with
highest topic proportion ??k , shown below each image. We can see that topics are tightly related to pose classes.
foot touches the ground, and 1 to the frames where the legs are approximately parallel. We labeled
the remaining frames automatically using linear interpolation between keyframes. The average interval between keyframes is 8.1 frames, this motivates our choice of the number of topics K = 8.
For each test frame, we computed the pose error as the difference between the associated pose value
and the average pose of the best top 10 matches in the training dataset. We obtained an average error
of 0.16, corresponding to 1.3 frames. In Figure 4 we show the average pose error per test sequence
obtained with our approach compared with Vector Quantization, where the pose is obtained as average of labels associated with the closest clusters, and Non-negative Matrix Factorization, where as in
LDA similarity of poses is computed as dot product of the component weights. In all the models we
set equal number of components/clusters to K = 8. We can see that our approach performs best in
all testing sequences. In Figure 4 we also show the average pose error when matching test frames to
a single train sequence. Although the different appearance affects the matching performance, overall the results shows how our approach can be successfully applied to automatically match poses of
different subjects.
6
Conclusions
We introduce a novel approach to human detection, pose classification and matching from a single
image. Starting from a representation robust to a limited range of variations in the appearance of
humans in images, we derive a generative probabilistic model which allows for automatic discovery
of pose information. The model can successfully perform detection and provides a low dimensional
representation of the pose. It automatically clusters the images using representative distributions and
allows for an efficient approach to pose matching. Our experiments show that our approach matches
or exceeds the state of the art in human detection, pose classification and matching.
Figure 3: Pose matching examples. On the left one sample frame from test sequences, on the right the top
10 matches in the training set based on the similarity score (4), reported below the image. We can see how
our approach allows to match poses even despite large changes in appearance, and the same pose is correctly
matched across different subjects.
Average Pose Error
0.5
LDA Gibbs
LDA MF
NMF
VQ
0.4
0.3
0.2
0.1
0
Figure 4: Pose matching error. (Left) Average pose error in matching test sequences to the training set, for
our model (both Gibbs and Mean Field learning), Non-Negative Matrix Factorization and Vector Quantization.
We see how our model trained with Gibbs sampling model clearly outperforms the other approaches. (Right)
Average pose error in matching test and training sequence pairs with our approach, where each row is a test
sequence and each column a training sequence. The highest error corresponds to about 2 frames, while the
mean error is 0.16 and amounts to approximately 1.3 frames.
Acknowledgments
This work was conducted while the first author was an intern at Honda Research Institute in 2005.
Work at UCLA was supported by AFOSR F49620-03-1-0095 and ONR N00014-03-1-0850:P0001.
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relevance:1 tested:6 |
2,274 | 3,064 | Information Bottleneck for Non Co-Occurrence Data
Yevgeny Seldin?
Noam Slonim?
Naftali Tishby??
?
School of Computer Science and Engineering
Interdisciplinary Center for Neural Computation
The Hebrew University of Jerusalem
?
The Lewis-Sigler Institute for Integrative Genomics
Princeton University
{seldin,tishby}@cs.huji.ac.il, [email protected]
?
Abstract
We present a general model-independent approach to the analysis of data in cases
when these data do not appear in the form of co-occurrence of two variables X, Y ,
but rather as a sample of values of an unknown (stochastic) function Z(X, Y ). For
example, in gene expression data, the expression level Z is a function of gene X
and condition Y ; or in movie ratings data the rating Z is a function of viewer X
and movie Y . The approach represents a consistent extension of the Information
Bottleneck method that has previously relied on the availability of co-occurrence
statistics. By altering the relevance variable we eliminate the need in the sample of
joint distribution of all input variables. This new formulation also enables simple
MDL-like model complexity control and prediction of missing values of Z. The
approach is analyzed and shown to be on a par with the best known clustering
algorithms for a wide range of domains. For the prediction of missing values
(collaborative filtering) it improves the currently best known results.
1
Introduction
In the situation of information explosion that characterizes todays world, the need for automatic tools
for data analysis is more than obvious. Here, we focus on an unsupervised analysis of data that can be
organized in matrix form. Clearly, this broad definition covers various types of data. For instance, in
text analysis data, the rows of a matrix correspond to words, the columns to different documents, and
entries indicate the number of occurrences of a particular word in a specific document. In a matrix
of gene expression data, rows correspond to genes, columns to various experimental conditions,
and entries indicate expression levels of given genes in given conditions. In movie rating data,
rows correspond to viewers, columns to movies, and entries indicate ratings made by the viewers.
Finally, for financial data, rows correspond to stocks, columns to different time points, and each
entry indicates a price change of a particular stock at a given time point.
While the text analysis case is a classical example of co-occurrence data, the remaining examples
are not naturally interpreted that way. Typically, a normalized words-documents table is used as
an estimator of a words-documents joint probability distribution, where each entry estimates the
probability of finding a given word in a given document, whereas the words are assumed to be independent of each other [1, 2]. By contrast, values in a financial data matrix are a general function of
stocks and days and in particular might include negative numerical values. Here, the data cannot be
regarded as a sample from a joint probability distribution of stocks and days, even if a normalization
is applied. Though it can be argued that each entry of the matrix is a sample from a joint probability
distribution of three variables: stock, day, and price change - the degenerate nature of this sample
must be taken into account: the rate change of a given stock on a given day occurs only once and no
statistics exist. Therefore the joint probability distribution of the three variables cannot be estimated
by direct sampling. A similar argument applies to survey data like movie ratings. In this case the
sample might be even more degenerate, with many ?missing values?, since not all the viewers rate
all the movies. Although gene expression data can be considered as a repeatable experiment, very
often different experimental conditions correspond to only one single column in the matrix. Thus
once again a single data point represents the joint statistics of three variables: gene, condition, and
expression level.
Nevertheless, in most such cases there is a statistical relationship within rows and/or within columns
of a matrix. For instance, people with similar interests typically give similar ratings to movies
and movies with similar characteristics give rise to similar rating profiles. Such relationships can be
exploited by clustering algorithms that group together similar rows and/or columns, and furthermore
make it possible to complete the missing entries in the matrix [3].
The existing clustering techniques can be classified into three major categories. (i) Similarity, or
distance-based methods, require a pre-defined similarity measure that can be applied to all data
points and possibly to new points as well. The nature of the distance measure is crucial to these
techniques and inherently requires expert knowledge of the application domain, which is often unavailable. (ii) Generative modeling techniques in which a specific class of statistical models is
chosen to describe the data. As before, an a priori choice of an appropriate model is far from obvious for most real world applications. (iii) An alternative line of study, relevant to our work, is the
Information Bottleneck (IB) approach [4] and its extensions. Instead of defining the clustering objective through a distortion measure or data generation process, the approach suggests using relevant
variables. A tradeoff between compression of the irrelevant and prediction of the relevant variables
is then optimized using information theoretic principles. Importantly, the definition of the relevant
variable is often natural and obvious for the task at hand, and in turn the method yields the optimal
relevant distortion for the problem.
Since the original work in [4], multiple studies have highlighted the theoretical and practical importance of the IB method, in particular in the context of cluster analysis [2, 5, 6, 7, 8]. However,
the original formulation is based on the availability of co-occurrence data. In practice a given cooccurrence table is treated as a finite sample out of a joint distribution of the variables; namely
row and column indices. Unfortunately, as mentioned above, this assumption does do not fit many
realistic datasets, thus preventing a direct application of the IB approach in various domains.
To address this issue, in [9] a random walk over the data points is defined, serving to transform
non co-occurrence data into a transition probability matrix that can be further analyzed via an IB
algorithm. In a more recent work [10] the suggestion is made to use the mutual information between
different data points as part of a general information-theoretic treatment to the clustering problem.
The resulting algorithm, termed the Iclust algorithm, was demonstrated to be superior or comparable
to 18 other commonly used clustering techniques over a wide range of applications where the input
data cannot be interpreted as co-occurrences [10]. However, both of the approaches have limitations
? Iclust requires a sufficient amount of columns in the matrix for reliable estimation of the information relations between rows, and the Markovian Relaxation algorithm involves various non-trivial
steps in the data pre-process [9].
Here, we suggest an alternative approach, inspired by the multivariate IB framework [8]. The multivariate IB principle expands the original IB work to handle situations where multiple systems of
clusters are constructed simultaneously with respect to different variables and the input data may
correspond to more than two variables. While the multivariate IB was originally proposed for cooccurrence data, we argue here that this framework is rich enough to be rigorously applicable in
the new situation. The idea is simple and intuitive: we look for a compact grouping of rows and/or
columns such that the product space defined by the resulting clusters is maximally informative about
the matrix content, i.e. the matrix entries. We show that this problem can be posed and solved within
the original multivariate IB framework. The new choice of relevance variable eliminates the need
to know the joint distribution of all the input variables (which is inaccessible in all the applications
presented here). Moreover, when missing values are present, the analysis suggests an information
theoretic technique for their completion.
We explore the application of this approach to various domains. For gene expression data and financial data we obtain clusters of comparable quality (measured as coherence with manual labeling) to
those obtained by state-of-the-art methods [10]. For movie rating matrix completion, performance
is superior to the best known alternatives in the collaborative filtering literature [11].
2
2.1
Theory
Problem Setting
We henceforth denote the rows of a matrix by X, the columns by Y , and the matrix entries by
Z (and small x, y and z for specific instances). The number of rows is denoted by n, and the
number of columns by m. We regard X and Y as discrete coordinate space and Z(X, Y ) as a
function. Generalization to higher dimensions and continuous coordinates is readily possible, but
not discussed here. For a given matrix, a row x, and a column y, the value of z(x, y) is assumed to
be deterministic. This can be relaxed as well.
The objective is to find ?good? partitions of the X-Y space that will be informative with respect
to the function values Z(X, Y ). The partitions are defined by grouping of rows into clusters of
rows C and grouping of columns into clusters of columns D. The complexity of such partitions
is measured by the sum of the weighted mutual information values nI(X; C) + mI(Y ; D). For
the hard partitions considered in the paper this sum is the number of bits required to describe the
partition (see [12]). The informativeness of the partition is measured by another mutual information,
I(C, D; Z). In these terms, the goal is to find minimally complex partitions that preserve a given
information level about the matrix values Z. This can be expressed via the minimization of the
following functional:
min
q(c|x),q(d|y)
nI(X; C) + mI(Y ; D) ? ?I(C, D; Z),
(1)
where q(c|x) is the mapping of rows x to row clusters c, q(d|y) is the mapping of columns y to
column clusters d, and ? is a Lagrange multiplier controlling the tradeoff between compression and
accuracy.
We first derive the relations between the quantities in the above optimization problem and then
describe a sequential algorithm for its minimization. We will stick to the following notation conventions: p is used for distributions that involve only input parameters and hence do not change
during the analysis, p? is used for empirical distributions, q for the sought mapping distributions and
q? for empirical distributions dependent on the sought mappings. By the definition of the mutual
information [12]:
X
q(c|x) X
q(c|x)
I(X; C) =
p(x)q(c|x) log
?
p?(x)q(c|x) log
.
q(c)
q?(c)
x,c
x,c
We define the indicator function:
1, if the entry (x, y) is present in the matrix
1x,y =
0, if the entry (x, y) is absent in the matrix
and denote the total number of populated entries (which is our sample size) by: N =
Then:
P
N umber of populated entries in row x
y 1x,y
p?(x) =
=
,
N
T otal number of populated entries
X
q?(c) =
p?(x)q(c|x),
P
x,y
x
I(Y ; D), p?(y), and q?(d) are defined similarly.
X
X
q(z|c, d)
q?(z|c, d)
I(C, D; Z) =
q(c, d)q(z|c, d) log
?
q?(c, d)?
q (z|c, d) log
.
p(z)
p?(z)
c,d,z
c,d,z
We assume Z is a categorical variable, thus:
P
N umber of entries equal to z
x,y:z(x,y)=z 1
p?(z) =
=
,
N
T otal number of populated entries
1x,y .
P
q(c|x)q(d|y)1x,y
N umber of populated entries in section c, d
,
T otal number of populated entries
N umber of entries equal to z in section c, d
x,y:z(x,y)=z q(c|x)q(d|y)
q?(z|c, d) = P
=
.
q(c|x)q(d|y)1
N
umber of populated entries in section c, d
x,y
x,y
In the special case of complete data matrices 1x,y is identically 1 and q?(c, d) may be decomposed
1
as: q?(c, d) = q?(c)?
q (d). In addition p?(x) and p?(y) accept the form p?(x) = n1 and p?(y) = m
. But in
the general case considered in this paper X and Y (and thus C and D) are not independent.
q?(c, d) =
P
2.2
x,y
N
=
Sequential Optimization
Given q(c|x) and q(d|y), one can calculate all the quantities defined above, and in particular the
minimization functional Lmin = nI(X; C) + mI(Y ; D) ? ?I(C, D; Z) defined in equation (1). To
minimize Lmin (using hard partitions) we can use the sequential (greedy) optimization algorithm
suggested in [13]. This algorithm is quite simple:
1. Start with a random (hard) partition q(c|x), q(d|y).
2. Iteratively until convergence (no changes at step (b) are done) traverse all rows x and
columns y of a matrix in a random order. For each row/column:
(a) Draw x (or y) from its cluster.
(b) Reassign it to a new cluster c? (or d? ), so that Lmin is minimized. The new cluster
may appear to be the old cluster, and then no change is counted.
Due to monotonic decrease in Lmin , which is lower bounded by ??H(Z) the algorithm is guaranteed to converge to some local minima of (1). Multiple random initializations may be used to
improve the result. This simple algorithm is by far not the only way to optimize (1), but in practice
it was shown to achieve very good results on similar optimization problems [2].
The complexity of the algorithm is analyzed in the complementary material, where it is shown to be
O(M (n + m)|C||D|), when M is the number of iterations required for convergence (usually 10-40)
and |C|, |D| are cardinalities of the corresponding variables.
2.3
Minimal Description Length (MDL) Formulation
The minimization functional Lmin has three free parameters that have to be externally determined:
the tradeoff (or resolution) parameter ?, and the cardinalities |C|, and |D|. Whereas in some applications they may be given (e.g. the desired number of clusters), there are cases when they also
require optimization (as in the example of matrix completion in the next section). To perform such
optimization, the Minimum Description Length (MDL) principle [14] is used. The idea behind MDL
is that models achieving better compression of the training data - when the compression includes a
model description - also achieve better generalization on the test data.
The following compression scheme is defined: |C| row and |D| column clusters define |C||D| secN
tions each getting roughly |C||D|
samples. The corresponding distributions q?(z|c, d) over categorical
N
variable Z may be described by |Z||C||D|
log |C||D|
bits (see [14]). As already mentioned, the ma2
trix partition itself may be described by nI(X; C) + mI(Y ; D) bits. And given the partition and the
distributions q?(z|c, d) the number of bits required to code the matrix entries is N H(Z|C, D) [12].
N
Thus the total description length is nI(X; C) + mI(Y ; D) + N H(Z|C, D) + |Z||C||D|
log |C||D|
.
2
Since H(Z|C, D) = H(Z) ? I(C, D; Z) and H(Z) is constant the latter can be omitted from
optimization, which results in total minimization functional
|Z||C||D|
N
Fmdl = nI(X; C) + mI(Y ; D) ? N I(C, D; Z) +
log
.
(2)
2
|C||D|
Observe that constrained on |C|, and |D|, Lmin corresponding to Fmdl accepts the form of
Lmin = nI(X; C) + mI(Y ; D) ? N I(C, D; Z),
(3)
i.e. the optimal tradeoff ? = N is uniquely determined. Since in practice Fmdl is roughly convex in
both C and D, the optimal values for these two parameters may be easily determined by scanning.
C
Z
X
D
C
Y
Z
X
(a) Gin
D
Y
(b) Gout
Figure 1: Gin and Gout in Multivariate IB formulation.
2.4
Relation with the Multivariate Information Bottleneck
Multivariate Information Bottleneck (IB) [8] is an unsupervised approach for structured data exploration. Its core lies in combining the Bayesian networks formalism [15] with the Information
Bottleneck method [4]. Multivariate IB searches for a meaningful structured partition of the data,
defined by compression variables (in our case these are C and D). Two graphs, Gin and Gout , are
defined. The former specifies the relations between the input (data) variables ? in our case, X, Y ,
and Z ? and the compression variables. The latter specifies the information terms that are to be
preserved by the partition. A tradeoff between the multi-information preserved in the input structure I Gin (which we want to minimize) and the multi-information expressed by the target structure
I Gout (which we want to maximize) is then optimized. For a set of variables V = V1 , .., Vn , a
directed acyclic graph (Bayesian network) G, and a joint probability distribution over V, p(V), the
multi-information I G (V) is defined as:
n
X
I G [p(V)] =
I(Vi ; P aG (Vi )),
i=1
where P aG (Vi ) are the parents of node Vi in G.
The graphs Gin and Gout corresponding to our case are given in Figure 1. (The dashed link between
X and Y in Gin appears when missing values are present in the matrix and may be chosen in any
direction.) The corresponding optimization functional is:
min
q(c|x),q(d|y)
=
min
q(c|x),q(d|y)
I Gin (X; Y ; Z; C; D) ? ?I Gout (X; Y ; Z; C; D)
I(X; Y ) + I(X, Y ; Z) + I(X; C) + I(Y ; D) ? ?I(C, D; Z).
By observing that I(X, Y ; Z) and I(X; Y ) are independent of q(c|x) and q(d|y) we can eliminate
them from the above optimization and obtain exactly the optimization functional defined in (1), only
with equal weighting of I(X; C) and I(Y ; D).
Thus the approach may be seen as a special case of the multivariate IB for the graphs Gin and Gout
defined in Figure 1. An important distinction should be made though: unlike the multivariate IB
we do not require the existence of the joint probability distribution p(x, y, z). This is achieved by
excluding the term I(X, Y ; Z) from the optimization functional.
2.5
Relation with Information Based Clustering
A recent information-theoretic approach for cluster analysis is given in [10], and is known as information based clustering, abbreviated as Iclust. In contrast to the original IB method, Iclust is equally
applicable to co-occurrence as well as non co-occurrence data. In the following we highlight the relation of our work and this earlier contribution.
By changing the notations used in [10] to those used here we can write the similarity measure
s(x1 , x2 ) used in [10] as:
XX
p(y)p(z1 , z2 |x1 , x2 , y)
P
s(x1 , x2 ) =
p(y)p(z1 , z2 |x1 , x2 , y) log P
p(y
)p(z
1
1 |x1 , y1 )
y1
y2 p(y2 )p(z2 |x2 , y2 )
z ,z
y
1
2
Table 1: Clusters coherence for the ESR and S&P stock datasets. The table provides the coherence of the achieved solutions for Nc = 5, 10, 15 and 20 row clusters. The results achieved by the
Iclust algorithm at the same settings are shown in brackets alongside the results of our algorithm.
For the ESR data an average coherence according to the three GOs is shown. Separate results for
each GO are provided in the supplementary material1 .
Dataset
Nc = 5
Nc = 10
Nc = 15
Nc = 20
ESR
S&P
69 (79)
94 (88)
53 (49)
83 (91)
50 (52)
92 (93)
42 (42)
86 (86)
= I(Z1 ; Z2 |x1 , x2 ).
Substituting this in the optimization functional of [10], changing maximization to minimization by
flipping the sign, and substituting T = ?1 we obtain:
X
X
min I(X; C) ? ?
q(c)
q(x1 |c)q(x2 |c)s(x1 , x2 )
q(c|x)
c
= min I(X; C) ? ?
q(c|x)
x1 ,x2
X
q(c, x1 , x2 )I(Z1 ; Z2 |x1 , x2 ),
c,x1 ,x2
which is reminiscent of equation (1) if no column (Y ) grouping is done and cluster variance is
measured through pairwise distances and not based on a centroid model.
Importantly, in order to be able to evaluate I(Z1 ; Z2 |x1 , x2 ), I-Clust requires a sufficient amount
of columns to be available, whereas our approach can operate with any amount of columns given.
Alternately, even when the data contain many columns, but are relatively sparse, i.e., have many
missing values, evaluating I(Z1 ; Z2 |x1 , x2 ) might be prohibitive as it requires a large enough intersection of non-missing observations for z1 and z2 . In our approach it is not a limitation. On
the contrary, the approach is designed to cope with this kind of data and resolves the problem by
simultaneous grouping of rows and columns of the matrix to amplify statistics.
3
Applications
We first compare our algorithm to I-Clust, as it was shown to be superior/comparable to 18 other
commonly used clustering techniques over a wide range of application domains [10]. We then
describe an experiment on matrix completion. Another application to a small dataset is provided in
the supplementary material1 . In the last two cases Iclust is not directly applicable. The multivariate
IB is not directly applicable to all the provided examples.
3.1
One Dimensional Clustering - Comparison to I-Clust
We focus on two applications reported in [10]. For purposes of comparison we restrict our algorithm
to cluster only the rows dimension of the matrix by setting the number of column clusters, |D|,
equal to the number of columns, m. This simplifies the objective functional defined in equation
(1) to Lmin = I(X; C) ? ?I(C, Y ; Z). (To have a similar form to [10] we incorporate factor n
multiplying I(X; C) in ?.) For both applications we use exactly the same setting as [10], including
row-wise quantization of the input data into five equally populated bins and choosing the same
values for the ? parameter.
The first dataset consists of gene expression levels of yeast genes in 173 various forms of environmental stress [16]. Previous analysis identified a group of ? 300 stress-induced and ? 600
stress-repressed genes with ?nearly identical but opposite patterns of expression in response to the
environmental shifts? [17]. These 900 genes were termed the yeast environmental stress response
(ESR) module. Following [10] we cluster the genes into |C| = 5, 10, 15, and 20 clusters. To assess
1
Supplementary material is available at http://www.cs.huji.ac.il/?seldin
the biological significance of the results we consider the coherence [18] of the obtained clusters
with respect to three Gene Ontologies (GOs) [19]. Specifically, the coherence of a cluster is defined
as the percentage of elements within this cluster that are given an annotation that was found to be
significantly enriched in the cluster [18]. The results achieved by our algorithm on this dataset are
comparable to the results achieved by I-Clust in all the verified settings - see Table 1.
The second dataset is the day-to-day fractional changes in the price of the stocks in the Standard &
Poor (S&P) 500 list2 , during 273 trading days of 2003. As with the gene expression data we take
exactly the same setting used by [10] and cluster the stocks into |C| = 5, 10, 15 and 20 clusters. To
evaluate the coherence of the ensuing clusters we use the Global Industry Classification Standard3 ,
which classifies companies into four different levels, organized in a hierarchical tree: sector, industry
group, industry, and subindustry. As with the ESR dataset our results are comparable with the results
of I-Clust for all the configurations - see Table 1.
3.2
Matrix Completion and Collaborative Filtering
Here, we explore the full power of our algorithm in simultaneous grouping of rows and columns of
a matrix. A highly relevant application is matrix completion - given a matrix with missing values
we would like to be able to complete it by utilizing similarities between rows and columns. This
problem is at the core of collaborative filtering applications, but may also appear in other fields.
We test our algorithm on the publicly available MovieLens 100K dataset4 . The dataset consists of
100,000 ratings on a five-star scale for 1,682 movies by 943 users. We take the five non-overlapping
splits of the dataset into 80% train on 20% test size provided at the MovieLens web site. We stress
that with this division the training data are extremely sparse - only 5% of the training matrix entries
are populated, whereas 95% of the values are missing.
To find a ?good? bi-clustering of the ratings matrix, minimization of Fmdl defined in (2) is done
by scanning cluster cardinalities |C| and |D| and optimizing Lmin as defined in (3) for each fixed
pair of |C|, |D|. The minimum of Fmdl is obtained at |C| ? 13 and |D| ? 6 with beyond 1%
sensitivity to small changes in |C| and in |D| both in Fmdl values and in prediction accuracy. See
supplementary material1 for visualization of the solution at |C| = 4 and |D| = 3.
To measure the accuracy of our algorithm we use mean absolute error (MAE) metrics, which
is commonly used for evaluation on this dataset [11]. The mean absolute error is defined as:
PN
M AE = N1 i=1 |zi ? ri |, where zi -s are the predicted and ri -s are the actual ratings. To convert
the distributions q?(z|c, d) we obtained in our clustering procedure to concrete predictions we take
the median of z values within each section c, d.
Note that our algorithm is general and does not directly optimize the MAE error functional. Nevertheless we obtain 0.72 MAE (with a deviation of less than 0.01 over multiple experiments). This
confidently beats the ?magic barrier? of 0.73 reported in the collaborative filtering literature [11].
The root mean squared error (RMSE) measured for the same clustering with a mean of z values
within each section c, d taken for prediction yields 0.96 (with a deviation below 0.01). This is much
better than 1.165 RMSE reported for a dataset 20 times larger [20] and quite close to 0.9525 RMSE
reported by Netflix for a dataset 1000 times larger of a similar nature5 .
4
Discussion
A new model independent approach to the analysis of data given in the form of samples of a function
Z(X, Y ) rather than samples of co-occurrence statistics of X and Y is introduced. From a theoretical viewpoint the approach is a much required extension of the Information Bottleneck method
that allows for its application to entirely new domains. The approach also provides a natural way for
bi-clustering and matrix completion. From a practical viewpoint the major contribution of the paper
is the achievement of the best known results for a wide range of applications with a single algorithm.
As well, we improve on the results of prediction of missing values (collaborative filtering).
2
Available at http://www.standardpoors.com
Available at http://wrds.wharton.upenn.edu
4
Available at http://www.grouplens.org
5
See http://www.netflixprize.com/rules
3
Possible directions for further research include generalization to continuous data values, such as
those obtained in gene expression and stock price data, and relaxation of the algorithm to ?soft?
clustering solutions. Another interesting extension would be to dimensionality reduction, rather
than clustering, as occurs in IB when applied to continuous variables [21].
The proposed framework also provides a natural platform for derivation of generalization bounds
for missing values prediction that will be discussed elsewhere.
References
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[2] Noam Slonim. The Information Bottleneck: Theory and Applications. PhD thesis, The Hebrew University
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method. In Advances in Neural Information Processing Systems 16.
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method. In NIPS, 2000.
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[14] A. Barron, J. Rissanen, and B. Yu. The minimum description length principle in coding and modeling.
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[15] J. Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. San Mateo,
CA: Morgan Kaufman Publishers, 1988.
[16] Gasch A.P., Spellman P.T., Kao C.M., Carmel-Harel O., Eisen M.B., Storz G., Botstein D., and Brown
P.O. Genomic expression programs in the response of yeast cells to environmental changes. Molecular
Biology. Cell, 11(12):4241?57, December 2000.
[17] A.P. Gasch. The environmental stress response: a common yeast response to environmental stresses.
Topics in Current Genetics (series editor S. Hohmann), 1:11?70, 2002.
[18] Segal E., Shapira M., Regev A., Pe?er D., Botstein D., Koller D., and Friedman N. Module networks:
identifying regulatory modules and their condition-specific regulators from gene expression data. Natural
Genetics, 34(2):166?76, 2003.
[19] M. Ashburner, C. A. Ball, J. A. Blake, D. Botstein, H. Butler, J. M. Cherry, A. P. Davis, K. Dolinski,
S. S. Dwight, J. T. Eppig, M. A. Harris, D. P. Hill, L. Issel-Tarver, A. Kasarskis, S. Lewis, J. C. Matese,
J. E. Richardson, M. Ringwald, G. M. Rubin, and G. Sherlock. Gene ontology: tool for the unification of
biology. Nature Genetics, 25:25?29, May 2000.
[20] Thomas Hoffman. Latent semantic models for collaborative filtering. In ACM Transactions on Information Systems, volume 22, January 2004.
[21] Gal Chechik, Amir Globerson, Naftali Tishby, and Yair Weiss. Gaussian information bottleneck. Journal
of Machine Learning Research, 6:165?188, January 2005.
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2,275 | 3,065 | Data Integration for Classification Problems
Employing Gaussian Process Priors
Mark Girolami
Department of Computing Science
University of Glasgow
Scotland, UK
[email protected]
Mingjun Zhong
IRISA, Campus de Beaulieu
F-35042 Rennes Cedex
France
[email protected]
Abstract
By adopting Gaussian process priors a fully Bayesian solution to the problem
of integrating possibly heterogeneous data sets within a classification setting is
presented. Approximate inference schemes employing Variational & Expectation
Propagation based methods are developed and rigorously assessed. We demonstrate our approach to integrating multiple data sets on a large scale protein fold
prediction problem where we infer the optimal combinations of covariance functions and achieve state-of-the-art performance without resorting to any ad hoc
parameter tuning and classifier combination.
1
Introduction
Various emerging quantitative measurement technologies in the life sciences are producing genome,
transcriptome and proteome-wide data collections which has motivated the development of data integration methods within an inferential framework. It has been demonstrated that for certain prediction tasks within computational biology synergistic improvements in performance can be obtained
via the integration of a number of (possibly heterogeneous) data sources. In [2] six different data
representations of proteins were employed for fold recognition of proteins using Support Vector
Machines (SVM). It was observed that certain data combinations provided increased accuracy over
the use of any single dataset. Likewise in [9] a comprehensive experimental study observed improvements in SVM based gene function prediction when data from both microarray expression and
phylogentic profiles were manually combined. More recently protein network inference was shown
to be improved when various genomic data sources were integrated [16] and in [1] it was shown that
superior prediction accuracy of protein-protein interactions was obtainable when a number of diverse data types were combined in an SVM. Whilst all of these papers exploited the kernel method
in providing a means of data fusion within SVM based classifiers it was initially only in [5] that
a means of estimating an optimal linear combination of the kernel functions was presented using
semi-definite programming. However, the methods developed in [5] are based on binary SVM?s,
whilst arguably the majority of important classification problems within computational biology are
inherently multiclass. It is unclear how this approach could be extended in a straightforward or
practical manner to discrimination over multiple-classes. In addition the SVM is non-probabilistic
and whilst post hoc methods for obtaining predictive probabilities are available [10] these are not
without problems such as overfitting. On the other hand Gaussian Process (GP) methods [11], [8]
for classification provide a very natural way to both integrate and infer optimal combinations of
multiple heterogeneous datasets via composite covariance functions within the Bayesian framework
an idea first proposed in [8].
In this paper it is shown that GP?s can indeed be successfully employed on general classification
problems, without recourse to ad hoc binary classification combination schemes, where there are
multiple data sources which are also optimally combined employing full Bayesian inference. A
large scale example of protein fold prediction [2] is provided where state-of-the-art predictive performance is achieved in a straightforward manner without resorting to any extensive ad hoc engineering
of the solution (see [2], [13]). As an additional important by-product of this work inference employing Variational Bayesian (VB) and Expectation Propagation (EP) based approximations for GP
classification over multiple classes are studied and assessed in detail. It has been unclear whether EP
based approximations would provide similar improvements in performance in the multi-class setting
over the Laplace approximation and this work provides experimental evidence that both Variational
and EP based approximations perform as well as a Gibbs sampler consistently outperforming the
Laplace approximation. In addition we see that there is no statistically significant practical advantage of EP based approximations over VB approximations in this particular setting.
2
Integrating Data with Gaussian Process Priors
Let us denote each of J independent (possibly heterogeneous) feature representations, Fj (X), of
an object X by xj ? j = 1 ? ? ? J . For each object there is a corresponding polychotomous response target variable, t, so to model this response we assume an additive generalized regression
model. Each distinct data representation of X, Fj (X) = xj , is nonlinearly transformed such that
fj (xj ) : Fj 7? R and a linear
P model is employed in this new space such that the overall nonlinear
transformation is f (X) = j ?j fj (xj ).
2.1
Composite Covariance Functions
Rather than specifying an explicit functional form for each of the functions fj (xj ) we assume that
each nonlinear function corresponds to a Gaussian process (GP) [11] such that fj (xj ) ? GP (?j )
where GP (?j ) corresponds to a GP with trend and covariance functions mj (xj ) and Cj (xj , x0j ; ?j )
where ?j denotes a set of hyper-parameters associated with the covariance function. Due to the
assumed independence of the feature representations the overall nonlinear function will also be a
realization of a Gaussian process defined as f (X)
J , ?1 ? ? ? ?J ) where now the
P ? GP (?1 ? ? ? ?P
overall trend and covariance functions follow as j ?j mj (xj ) and j ?j2 Cj (xj , x0j ; ?j ). For N
object samples, X1 ? ? ? XN , each defined by the J feature representations, x1j ? ? ? xN
j , denoted by
Xj , with associated class specific response fk = [fk (X1 ) ? ? ? fk (XN )]T the overall GP prior is a
multivariate Normal such that
X
fk | Xj=1???J , ?1k , ? ? ? ?J k , ?1k ? ? ? ?J k ? Nfk 0,
?jk Cjk (?jk )
(1)
j
2
?jk
The positive random variables
are denoted by ?jk , zero-trend GP functions have been assumed
n
and each Cjk (?jk ) is an N ? N matrix with elements Cj (xm
j , xj ; ?jk ). A GP functional prior, over
all possible responses (classes), is now available where possibly heterogeneous data sources are
integrated via the composite covariance function. It is then, in principle, a straightforward matter
to perform Bayesian inference with this model and no further recourse to ad hoc binary classifier
combination methods or ancillary optimizations to obtain the data combination weights is required.
2.2
Bayesian Inference
As we are concerned with classification problems over possibly multiple classes we employ a multinomial probit likelihood rather than a multinomial logit as it provides a means of developing a
Gibbs sampler, and subsequent computationally efficient approximations, for the GP random variables. The Gibbs sampler is to be preferred over the Metropolis scheme as no tuning of a proposal
distribution is required. As in [3] the auxiliary variables ynk = fk (Xn ) + nk , nk ? N (0, 1) are
introduced and the N ? 1 dimensional vector of target class values associated with each Xn is given
as t where each element tn ? {1, ? ? ? , K}. The N ? K matrix of GP random variables fk (Xn ) is
denoted by F. We represent the N ? 1 dimensional columns of F by F?,k and the corresponding
K ? 1 dimensional vectors, Fn,? , which are formed by the indexed rows of F . The N ? K matrix
of auxiliary variables ynk is represented as Y, where the N ? 1 dimensional columns are denoted
by Y?,k and the corresponding K ? 1 dimensional vectors are obtained from the rows of Y as Yn,? .
The multinomial probit likelihood [3] is adopted which follows as
tn = j if ynj = argmax {ynk }
1?k?K
(2)
and this has the effect of dividing RK into K non-overlapping K-dimensional cones Ck = {y :
yk > yi , k 6= i} where RK = ?k Ck and so each P (tn = i|Yn,? ) can be represented as ?(yni >
ynk ? k 6= i). Class specific independent Gamma priors, with parameters ?k , are placed on each
?jk and the individual components of ?jk (denote ?k = {?jk , ?jk }j=1???J ), a further Gamma prior
is placed on each element of ?k with overall parameters a and b so this defines the full model
likelihood and associated priors.
2.3
MCMC Procedure
Samples from the full posterior P (Y, F, ?1???K , ?1???K |X1???N , t, a, b) can be obtained from the
following Metropolis-within-Blocked-Gibbs Sampling scheme indexing over all n = 1 ? ? ? N and
k = 1 ? ? ? K.
(i+1)
Yn,?
(i)
|Fn,? , tn
(i+1)
(i+1)
(i)
F?,k |Y?,k , ?k , X1,??? ,N
(i+1)
?1
(i+1)
|F?,1
(i+1)
, Y?,k
(i)
, ?1 , X1,??? ,N
(i+1)
(i+1)
?k
|?k
, ak , bk
(i)
?
T N (Fn,? , I, tn )
(3)
?
(i) (i+1)
(i)
N (?k Y?,k , ?k )
(4)
(i+1)
? P (?k
?
)
(5)
(i+1)
P (?k
)
(6)
where T N (Fn,? , I, tn ) denotes a conic truncation of a multivariate Gaussian with location parameters Fn,? and dispersion parameters I and the dimension indicated by the class value of tn
will be the largest. An accept-reject strategy can be employed in sampling from the conic truncated Gaussian however this will very quickly become inefficient for problems with moderately
large numbers of classes and as such a further Gibbs sampling scheme may be required. Each
P
(i)
(i)
(i)
(i)
(i)
(i)
(i)
?k = Ck (I + Ck )?1 and Ck = j=1 ?jk Cjk (?jk ) with the elements of Cjk (?jk ) defined
(i)
n
as Cj (xm
j , xj ; ?jk ). A Metropolis sub-sampler is required to obtain samples for the conditional
(i+1)
(i+1)
distribution over the composite covariance function parameters P (?k
) and finally P (?k
)
is a simple product of Gamma distributions. The predictive likelihood of a test sample X? is
P (t? = k|X? , X1???N , t, a, b) which can be obtained by integrating over the posterior and predictive
prior such that
Z
P (t? = k|f? )p(f? |?, X? , X1???N )p(?|X1???N , t, a, b)df? d?
(7)
where ? = Y, ?1???K
. A Monte-Carlo estimate is obtained by using samples drawn from the full
PS R
posterior S1 s=1 P (t? = k|f? )p(f? |?(s) , X? , X1???N )df? and the integral over the predictive prior
(l|s)
requires further conditional samples, f? , to be drawn from each p(f? |?(s) , X? , X1???N ) finally
yielding a Monte Carlo approximation of P (t? = k|X? , X1???N , t, a, b)
?
?
L
S
L X
S
?Y
?
X
1 XX
1
(l|s)
(l|s)
(l|s)
P t? = k|f?
=
Ep(u)
? u + f?,k ? f?,j
(8)
?
?
LS
LS
l=1 s=1
l=1 s=1
j6=k
MCMC procedures for GP classification have been previously presented in [8] and whilst this provides a practical means to perform Bayesian inference employing GP?s the computational cost incurred and difficulties associated with monitoring convergence and running multiple-chains on reasonably sized problems are well documented and have motivated the development of computationally less costly approximations [15]. A recent study has shown that EP is superior to the Laplace
approximation for binary classification [4] and that for multi-class classification VB methods are
superior to the Laplace approximation [3]. However the comparison between Variational and EP
based approximations for the multi-class setting have not been considered in the literature and so we
seek to address this issue in the following sections.
2.4
Variational Approximation
From the conditional probabilities which appear in the Gibbs sampler it can be seen that a mean field
approximation gives a simple iterative scheme which provides a computationally efficient alternative
to the full sampler (including the Metropolis sub-sampler for the covariance function parameters),
details of which are given in [3]. However given the excellent performance of EP on a number of
approximate Bayesian inference problems it is incumbent on us to consider an EP solution here.
We should point out that only the top level inference on the GP variables is considered here and
the composite covariance function parameters will be obtained using another appropriate type-II
maximum likelihood optimization scheme if possible.
2.5
Expectation Propagation with Full Posterior Covariance
The required posterior can also be approximated by EP [7]. In this case the multinomial probit
by a multivariate Gaussian such that p(F|t, X1???N ) ? Q(F) =
Q likelihood is approximated
Q
1
p(F
|X
)
g
(F
)
where gn (Fn,? ) = NFn,? (?n , ?n ), ?n is a K ? 1 vector and
?,k
1???N
n,?
k
n n
?
is
a
full
K
?
K
dimensional
covariance matrix. Denoting the cavity density as Q\n (F) =
Qn
Q
p(F
|X
)
g
(F
),
?,k
1???N
i,? EP proceeds by iteratively re-estimating the moments ?n , ?n
k
i,i6=n i
by moment matching [7] giving the following
?new
= Ep?n {Fn,? } and ?new
= Ep?n {Fn,? FTn,? } ? Ep?n {Fn,? }Ep?n {Fn,? }T ,
n
n
(9)
where p?n = Zn?1 Q\n (Fn,? )p(tn |Fn,? ), and Zn is the required normalizing (partition) function
which is required to obtain the above mean and covariance estimates. To proceed an analytic form
for the partition function Zn is required. Indeed for binary classification employing a binomial probit likelihood an elegant EP solution follows due to the analytic form of the partition function [4].
However for the case of multiple classes with a multinomial probit likelihood the partition function
no longer has a closed analytic form and further approximations are required to make any progress.
There are two strategies which we consider, the first retains the full posterior coupling in the covariance matrices ?n by employing Laplace Propagation (LP) [14] and the second assumes no posterior
coupling in ?n by setting this as a diagonal covariance matrix. The second form of approximation
has been adopted in [12] when developing a multi-class version of the Informative Vector Machine
(IVM) [6]. In the first case where we employ LP an additional significant O(K 3 N 3 ) computational
scaling will be incurred however it can be argued that the retention of the posterior coupling is important. For the second case clearly we lose this explicit posterior coupling but, of course, do not
incur the expensive computational overhead required of LP. We observed in unreported experiments
that there is little of statistical significance lost, in terms of predictive performance, when assuming
a factorable form for each p?n . LP proceeds by propagating the approximate moments such that
?1
? 2 log p?n
new
new
argmax
?n ?
log p?n and ?n ? ?
(10)
Fn,?
?Fn,? ?FTn,?
The required derivatives follow straightforwardly and details are included in the accompanying material. The approximate predictive distribution for a new data point x? requires a Monte Carlo
estimate employing samples drawn from a K-dimensional multivariate Gaussian for which details
are given in the supplementary material2 .
2.6
Expectation Propagation with Diagonal Posterior Covariance
By assuming a factorable approximate posterior, as in the variational approximation [3], a distinct
Q simplification of the problem setting follows, where now we assume that gn (Fn,? ) =
k NFn,k (?n,k , ?n,k ) i.e. is a factorable distribution. This assumption has already been made
in [12] in developing an EP based multi-class IVM. Now significant computational simplificanew
tion follows where the required moment matching amounts to ?new
nk = Ep?nk {Fn,k } and ?nk =
2
2
Ep?nk {Fn,k } ? Ep?nk {Fn,k } where the density p?nk has a partition function which now has the
analytic form
q
?
?
??
\n
\n
\n
K
? Y
u + v ?ni + ?ni ? ?nj ?
?
q
Zn = Ep(u)p(v)
??
(11)
?
?
\n
1
+
?
j=1,j6=i
nj
1
Conditioning on the covariance function parameters and associated hyper-parameters is implicit
Supplementary
material
http://www.dcs.gla.ac.uk/people/personal/girolami/
pubs_2006/NIPS2006/index.htm
2
where u and v are both standard Normal random variables (v
\n
q
\n
\n
\n
?ni = Fn,i ? ?ni ) with ?ni and
?ni having the usual meanings (details in accompanying material). Derivatives of this partition
function follow in a straightforward way now allowing the required EP updates to proceed (details
in supplementary material). The approximate predictive distribution for a new data point X? in this
case takes a similar form to that for the Variational approximation [3]. So we have
?
!?
p
K
? Y
u + v ??k + ??k ? ??j ?
p
P (t? = k|X? , X1???N , t) = Ep(u)p(v)
?
(12)
?
?
1 + ??j
j=1,j6=k
where the predictive mean and variance follow in standard form.
?1
??j = (C?j )T (Cj + ?j )
?j and ??j = c?j ? (C?j )T (Cj + ?j )
?1
C?j
(13)
It should be noted here that the expectation over p(u) and p(v) could be computed by using either
Gaussian quadrature or a simple Monte Carlo approximation which is straightforward as sampling
from a univariate standardized Normal only is required. The VB approximation [3] however only
requires a 1-D Monte Carlo integral rather than the 2-D one required here.
3
Experiments
Before considering the main example of data integration within a large scale protein fold prediction problem we attempt to assess a number of approximate inference schemes for GP multi-class
classification. We provide a short comparative study of the Laplace, VB, and both possible EP
approximations by employing the Gibbs sampler as the comparative gold standard. For these experiments six multi-class data sets are employed 3 , i.e., Iris (N = 150, K = 3), Wine (N = 178, K =
3), Soybean (N = 47, K = 4), Teaching (N = 151, K = 3), Waveform (N = 300, K = 3) and ABE
(N = 300, K = 3, which is a subset of the Isolet dataset using the letters ?A?, ?B? and ?E?,). A single
radial basis covariance function with one length scale parameter is used in this comparative study.
Ten-fold cross validation (CV) was used to estimate the predictive log-likelihood and the percentage
predictive error. Within each of the ten folds a further 10 CV routine was employed to select the
length-scale of the covariance function. For the Gibbs sampler, after a burn-in of 2000 samples,
the following 3000 samples were used for inference, and the predictive error and likelihood were
computed from the 3000 post-burn-in samples. For each data set and each method the percentage
predictive error and the predictive log-likelihood were estimated in this manner. The summary results given as the mean and standard deviation over the ten folds are shown in Table 1. The results
which cannot be distinguished from each other, under a Wilcoxon rank sum test with a 5% significance level, are highlighted in bold. From those results, we can see that across most data sets used,
the predictive log-likelihood obtained from the Laplace approximation is lower than those of the
three other methods. In our observations, the predictive performance of VB and the IEP approximation are consistently indistinguishable from the performance achieved from the Gibbs sampler.
From the experiments conducted there is no evidence to suggest any difference in predictive performance between IEP & VB methods in the case of multi-way classification. As there is no benefit
in choosing an EP based approximation over the Variational one we now select the Variational approximation in that inference over the covariance parameters follows simply by obtaining posterior
mean estimates using an importance sampler.
As a brief illustration of how the Variational approximation compares to the full Metropolis-withinBlocked-Gibbs Sampler consider a toy dataset consisting of three classes formed by a Gaussian
surrounded by two annular rings having ten features only two of which are predictive of the class
labels [3]. We can compare the compute time taken to obtain reasonable predictions from the full
MCMC and the approximate Variational scheme [3]. Figure 1 (a) shows the samples of the covariance function parameters ? drawn from the Metropolis subsampler4 and overlaid in black the
corresponding approximate posterior mean estimates obtained from the variational scheme [3]. It
3
http://www.ics.uci.edu/?mlearn/MPRepository.html
It should be noted that multiple Metropolis sub-chains had to be run in order to obtain reasonable sampling
of the ? ? R10
+
4
Table 1: Percentage predictive error (PE) and predictive log-likelihood (PL) for six data sets from
UCI computed using Laplace, Variational Bayes (VB), independent EP (IEP), as well as MCMC
using Gibbs sampler. Best results which are statistically indistinguishable from each other are highlighted in bold.
Laplace
VB
Gibbs
IEP
Laplace
VB
Gibbs
IEP
Laplace
VB
Gibbs
IEP
ABE
PE
4.000?3.063
2.000?2.330
3.333?3.143
5.333?5.019
Wine
PE
3.889?5.885
2.222?3.884
4.514?5.757
3.889?5.885
Teach
PE
39.24?15.74
41.12?9.92
42.41?6.22
42.54?11.32
PL
-0.290?0.123
-0.164?0.026
-0.158?0.037
-0.139?0.050
PL
-0.258?0.045
-0.182?0.057
-0.177?0.054
-0.133?0.047
PL
-0.836?0.072
-0.711?0.125
-0.730?0.113
-0.800?0.072
Iris
PE
3.333?3.513
3.333?3.513
3.333?3.513
3.333?3.513
Soybean
PE
0.000?0.000
0.000?0.000
0.000?0.000
0.000?0.000
Wave
PE
17.50?9.17
18.33?9.46
15.83?8.29
17.50?10.72
70
PL
-0.132?0.052
-0.087?0.056
-0.079?0.056
-0.063?0.059
PL
-0.359?0.040
-0.158?0.034
-0.158?0.039
-0.172?0.037
PL
-0.430?0.085
-0.410?0.100
-0.380?0.116
-0.383?0.107
?0.2
60
?2
10
?0.4
Predictive Likelihood
Percentage Error
0
10
50
40
30
20
?0.6
?0.8
?1
?4
10
10
0
10
1
2
10
10
(a)
0 0
10
5
Time (Seconds ? Log)
10
?1.2 0
10
(b)
Time (Seconds ? Log)
5
10
(c)
Figure 1: (a) Progression of MCMC and Variational methods in estimating covariance function
parameters, vertical axis denotes each ?d , horizontal axis is time (all log scale) (b) percentage error under the MCMC (gray) and Variational (black) schemes, (c) predictive likelihood under both
schemes.
is clear that after 100 calls to the sub-sampler the samples obtained reflect the relevance of the features, however the deterministic steps taken in the variational routine achieve this in just over ten
computational steps of equal cost to the Metropolis sub-sampler. Figure 1 (b) shows the predictive
error incurred by the classifier and under the MCMC scheme 30,000 CPU seconds are required to
achieve the same level of predictive accuracy under the variational approximation obtained in 200
seconds (a factor of 150 times faster). This is due, in part, to the additional level of sampling from
the predictive prior which is required when using MCMC to obtain predictive posteriors. Because of
these results we now adopt the variational approximation for the following large scale experiment.
4
Protein Fold Prediction with GP Based Data Fusion
To illustrate the proposed GP based method of data integration a substantial protein fold classification problem originally studied in [2] and more recently in [13] is considered. The task is to devise a
predictor of 27 distinct SCOP classes from a set (N = 314) of low homology protein sequences. Six
0.2
50
40
30
20
10
0
AA HP PT PY SS VP MA MF
(a)
2.5
0.15
Alpha Weight
Predictive Likelihood
Percent Accuracy
60
0.1
0.05
2
1.5
1
0.5
0
AA HP PT PY SS VP MA MF
(b)
0
AA
HP
PT
SS
VP
PZ
(c)
Figure 2: (a) The prediction accuracy for each individual data set and the corresponding combinations, (MA) employing inferred weights and (MF) employing a fixed weighting scheme (b) The
predictive likelihood achieved for each individual data set and with the integrated data (c) The posterior mean values of the covariance function weights ?1 ? ? ? ?6 .
different data representations (each comprised of around 20 features) are available characterizing
(1) Amino Acid composition (AA); (2) Hydrophobicity profile (HP); (3) Polarity (PT); (4) Polarizability (PY); (5) Secondary Structure (SS); (6) Van der Waals volume profile of the protein (VP).
In [2] a number of classifier and data combination strategies were employed in devising a multiway
classifier from a series of binary SVM?s. In the original work of [2] the best predictive accuracy
obtained on an independent set (N = 385) of low sequence similarity proteins was 53%. It was
noted after extensive careful manual experimentation by the authors that a combination of Gaussian
kernels each composed of the (AA), (SS) and (HP) datasets significantly improved predictive accuracy. More recently in [13] a heavily tuned ad hoc ensemble combination of classifiers raised this
performance to 62% the best reported on this problem. We employ the proposed GP based method
(Variational approximation) in devising a classifier for this task where now we employ a composite
covariance function (shared across all 27 classes), a linear combination of RBF functions for each
data set. Figure (2) shows the predictive performance of the GP classifier in terms of percentage
prediction accuracy (a) and predictive likelihood on the independent test set (b). We note a significant synergistic increase in performance when all data sets are combined and weighted (MA)
where the overall performance accuracy achieved is 62%. Although the 0-1 loss test error is the
same for an equal weighting of the data sets (MF) and that obtained using the proposed inference
procedure (MA) for (MA) there is an increase in predictive likelihood i.e. more confident correct
predictions being made. It is interesting to note that the weighting obtained (posterior mean for ?)
Figure (2.c) weights the (AA) & (SS) with equal importance whilst other data sets play less of a role
in performance improvement.
5
Conclusions
In this paper we have considered the problem of integrating data sets within a classification setting, a
common scenario within many bioinformatics problems. We have argued that the GP prior provides
an elegant solution to this problem within the Bayesian inference framework. To obtain a computationally practical solution three approximate approaches to multi-class classification with GP priors,
i.e. Laplace, Variational and EP based approximations have been considered. It is found that EP and
Variational approximations approach the performance of a Gibbs sampler and indeed their predictive
performances are indistinguishable at the 5% level of significance. The full EP (FEP) approximation
employing LP has an excessive computational cost and there is little to recommend it in terms of
predictive performance over the independent assumption (IEP). Likewise there is little to distinguish
between IEP and VB approximations in terms of predictive performance in the multi-class classification setting though further experiments on a larger number of data sets is desirable. We employ
VB to infer the optimal parameterized combinations of covariance functions for the protein fold
prediction problem over 27 possible folds and achieve state-of-the-art performance without recourse
to any ad hoc tinkering and tuning and the inferred combination weights are intuitive in terms of
the information content of the highest weighted data sets. This is a highly practical solution to the
problem of heterogenous data fusion in the classification setting which employs Bayesian inferen-
tial semantics throughout in a consistent manner. We note that on the fold prediction problem the
best performance achieved is equaled without resorting to complex and ad hoc data and classifier
weighting and combination schemes.
5.1
Acknowledgements
MG is supported by the Engineering and Physical Sciences Research Council (UK) grant number
EP/C010620/1, MZ is supported by the National Natural Science Foundation of China grant number
60501021.
References
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2,276 | 3,066 | An Oracle Inequality for Clipped Regularized Risk
Minimizers
Ingo Steinwart, Don Hush, and Clint Scovel
Modelling, Algorithms and Informatics Group, CCS-3
Los Alamos National Laboratory
Los Alamos, NM 87545
{ingo,dhush,jcs}@lanl.gov
Abstract
We establish a general oracle inequality for clipped approximate minimizers of
regularized empirical risks and apply this inequality to support vector machine
(SVM) type algorithms. We then show that for SVMs using Gaussian RBF kernels
for classification this oracle inequality leads to learning rates that are faster than
the ones established in [9]. Finally, we use our oracle inequality to show that a
simple parameter selection approach based on a validation set can yield the same
fast learning rates without knowing the noise exponents which were required to
be known a-priori in [9].
1
Introduction
The theoretical understanding of support vector machines (SVMs) and related kernel-based methods has been substantially improved in recent years. For example using Talagrand?s concentration
inequality and local Rademacher averages it has recently been shown that SVMs for classification
can learn with rates up to n?1 under somewhat realistic assumptions on the data-generating distribution (see [9, 11] and the related work [2]). However, the so-called ?shrinking technique? of [9, 11]
for establishing such rates, requires the free parameters to be chosen a-priori, and in addition, the
optimal values of these parameters depend on features of the data-generating distribution which are
typically unknown. Consequently, [9, 11] do not provide a practical method for learning with fast
rates. On the other hand, the oracle inequality in [2] only holds for distributions having Tsybakov
noise exponent ?, and hence it describes a situation which is rarely met in practice.
The goal of this work is to overcome these shortcomings by establishing a general oracle inequality
(see Theorem 3.1) for regularized empirical risk minimizers. The key ingredient of this oracle
inequality is the observation that for most commonly used loss functions it is possible to ?clip?
the decision function of the algorithm before beginning with the theoretical analysis. In addition,
a careful choice of the weighted empirical process Talagrand?s inequality is applied to, makes the
?shrinking technique? superfluous. Finally, by explicitly dealing with -approximate minimizers of
the regularized risk our results also apply to actual SVM algorithms.
With the help of the general oracle inequality we then establish an oracle inequality for SVM type
algorithms (see Theorem 2.1) as well as a simple oracle inequality for model selection (see Theorem
4.2). For the former, we show that it leads to improved rates for e.g. binary classification under
the assumptions considered in [9] and a-priori known noise exponents. Using the model selection
theorem we then show how our new oracle inequality for SVMs can be used to analyze a simple
parameter selection procedure based on a validation set that achieves the same learning rates without
prior knowledge on the noise exponents.
The rest of this work is organized as follows: In Section 2 we present our oracle inequality for
SVM type algorithms. We then discuss its implications and analyze the simple parameter selection
procedure when using Gaussian RBF kernels. In Section 3 we then present and prove the general
oracle inequality. The proof of Theorem 2.1 as well as the oracle inequality for model selection can
be found in Section 4.
2
Main Results
Throughout this work we assume that X is compact metric space, Y ? [?1, 1] is compact, P
is a Borel probability measure on X ? Y , and F is a set of functions over X such that 0 ? F.
Often F is a reproducing kernel Hilbert space (RKHS) H of continuous functions over X with
closed unit ball BH . It is well-known that H can then be continuously embedded into the space
of continuous functions C(X) equipped with the usual maximum-norm k.k? . In order to avoid
constants we always assume that this embedding has norm 1, i.e. k.k? ? k.kH . Furthermore,
L : Y ? R ? [0, ?) always denotes a continuous function which is convex in its second variable
such that L(y, 0) ? 1. The functions L will serve as loss functions and consequently let us recall
that the associated L-risk of a measurable function f : X ? R is defined by
RL,P (f ) = E(x,y)?P L y, f (x) .
Note that the assumption L(y, 0) ? 1 immediately gives RL,P (0) ? 1. Furthermore, the minimal
L-risk is denoted by R?L,P , i.e. R?L,P = inf{RL,P (f ) | f : X ? R measurable}, and a function
?
?
exists.
. We always assume that such an fL,P
attaining this infimum is denoted by fL,P
The learning schemes we are mainly interested in are based on an optimization problem of the form
fP,? := arg min ?kf k2H + RL,P (f ) ,
(1)
f ?H
where ? > 0. Note that if we identify a training set T = ((x1 , y1 ), . . . , (xn , yn )) ? (X ? Y )n with
its empirical measure, then fT,? denotes the empirical estimators of the above learning scheme.
Obviously, support vector machines (see e.g. [5]) and regularization networks (see e.g. [7]) are both
learning algorithms which fall into the above category. One way to describe the approximation error
of these learning schemes is the approximation error function
a(?) := ?kfP,? k2H + RL,P (fP,? ) ? R?L,P ,
? > 0,
which has been discussed in some detail in [10]. Furthermore in order to deal with the complexity
of the used RKHSs let us recall that for a subset A ? E of a Banach space E the covering numbers
are defined by
n
n
o
[
N (A, ?, E) := min n ? 1 : ?x1 , . . . , xn ? E with A ?
(xi + ?BE ) ,
? > 0,
i=1
where BE denotes the closed unit ball of E. Given a finite sequence T = ((x1 , y1 ), . . . , (xn , yn )) ?
(X ? Y )n we write TX := (x1 , . . . , xn ). For our main results we are particularly interested in
covering numbers in the Hilbert space L2 (TX ) which consists of all equivalence classes of functions
f : X ? Y ? R and which is equipped with the norm
kf kL2 (TX )
n
1 X
1
f (xi ) 2 2 .
:=
n i=1
(2)
In other words, L2 (TX ) is a L2 -space with respect to the empirical measure of (x1 , . . . , xn ).
Learning schemes of the form (1) typically produce functions fP,? with lim??0 kfP,? k? = ?
(see e.g. [10] for a precise statement). Unfortunately, this behaviour has a serious negative impact
on the learning rates when directly employing standard tool?s such as Hoeffding?s, Bernstein?s or
Talagrand?s inequality. On the other hand, when dealing with e.g. the hinge loss it is obvious that
clipping the function fP,? at ?1 and 1 does not worsen the corresponding risks. Following this
simple observation we will consider loss functions L that satisfy the clipping condition
L(y, 1)
if t ? 1
L(y, t) ?
(3)
L(y, ?1) if t ? ?1 ,
for all y ? Y . Recall that this type of loss function was already considered in [4, 11], but the clipping
idea actually goes back to [1]. Moreover, it is elementary to check that most commonly used loss
functions including the hinge loss and the least squares loss satisfy (3). Given a function f : X ? R
we now define its clipped version f? : X ? [?1, 1] by
?
if f (x) > 1
?1
f?(x) := f (x) if f (x) ? [?1, 1]
?
?1
if f (x) < ?1 .
It is clear from (3) that we always have L(y, f?(x)) ? L(y, f (x)) and consequently we obtain
RL,P (f?) ? RL,P (f ) for all distributions P . Finally, we also need the following Lipschitz condition
|L|1 :=
|L(y, t1 ) ? L(y, t2 )|
? 2.
|t1 ? t2 |
y?Y,?1?t1 ,t2 ?1
sup
(4)
With the help of these definitions we can now state our main result which establishes an oracle
inequality for clipped versions of fT,? :
Theorem 2.1 Let P be a distribution on X ? Y and let L be a loss function which satisfies (3) and
(4). Let H be a RKHS of continuous functions on X. For a fixed element f0 ? H we define
:= ?kf0 k2H + RL,P (f0 ) ? R?L,P
B(f0 ) :=
sup L(y, f0 (x)) .
a(f0 )
(5)
x?X,y?Y
In addition, we assume that we have a variance bound of the form
2
?
?
?
EP L ? f? ? L ? fL,P
? v EP (L ? f? ? L ? fL,P
)
(6)
for constants v ? 1, ? ? [0, 1] and all measurable f : X ? R. Moreover, suppose that H satisfies
sup
log N BH , ?, L2 (TX ) ? a??2p ,
? > 0,
(7)
T ?(X?Y )n
for some constants p ? (0, 1) and a ? 1. For fixed ? > 0 let fT,? ? H be a function that minimizes
f 7? ?kf k2H + RL,T (f ) up to some > 0. Then there exists a constant Kp,v depending only on p
and v such that for all ? ? 1 we have with probability not less than 1 ? 3e?? that
1
1
32v? 2??
Kp,v a 2??+p(??1)
Kp,v a
140?
14B(f0 )?
+
+
5
+
+
RL,P (f?T,? ) ? R?L,P ?
?p n
?p n
n
n
3n
+ 8a(f0 ) + 4.
(8)
The above oracle inequality has some interesting consequences as the following examples illustrate.
We begin with an example that deals with a fixed kernel:
Example 2.2 (Learning rates for single kernel) Assume that in Theorem 2.1 we have a Lipschitz
continuous loss function such as the hinge loss. In addition assume that the approximation error
function satisfies a(?) ? c?? , ? > 0, for some constants c > 0 and ? ? (0, 1]. Setting f0 := fP,?
and optimizing (8) with respect to ? then shows that the corresponding SVM learns with rate n?? ,
where
n
?
2? o
? := min
,
.
? 2 ? ? + p(? ? 1) + p ? + 1
Recall that this learning rate has already been obtained in [11].
The next example investigates SVMs that use a Gaussian RBF kernel whose width may vary with
the sample size:
Example 2.3 (Classification with several Gaussian kernels) Let X be the unit ball in Rd and
Y := {?1, 1}. Furthermore assume that we are interested in binary classification using the hinge
2
2
loss and the Gaussian RKHSs H? that belong to the RBF kernels k? (x1 , x2 ) := e?? kx1 ?x2 k with
width ? > 0. If P has geometric noise exponent ? ? (0, ?) in the sense of [9] then it was shown in
[9] that there exists a function f0 ? H? with kf0 k? ? 1 and
a? (f0 ) ? c ? d ? + ? ??d ,
? > 0, ? > 0,
where c > 0 is a constant independent of ? and ?. Moreover, [9, Thm. 2.1] shows that H? satisfies
(7) for all p ? (0, 1) with
a := cp,d,? ? (1?p)(1+?)d
where ? > 0 can be arbitrarily chosen and cp,d,? is a suitable constant. Now assume that P has
Tsybakov noise exponent q ? [0, ?] in the sense of [9]. It was then shown in [9] that (6) is satisfied
q
for ? := q+1
. Minimizing (8) with respect to ? and ? and choosing p and ? sufficiently small then
yields that the corresponding SVM can learn with rate n??+? , where
? :=
?(q + 1)
,
?(q + 2) + q + 1
and ? > 0 can be chosen arbitrarily small. Note that these rates are superior to those obtained in
[9, Theorem 2.8].
In the above examples the optimal parameters ? and ? depend on the sample size n but not on the
training samples T . However, these optimal parameters require us to know certain characteristics
of the distribution such as the approximation exponent ? or the noise exponents ? and q. The
following example shows that the oracle inequality of Theorem 2.1 can be used to find these optimal
parameters in a data-dependent fashion which does not require any a-priori knowledge:
Example 2.4 In this example we assume that our training set T consists of 2n samples. We write
T0 for the first n samples and T1 for the last n samples. Let fT0 ,?,? be the SVM solution using
a Gaussian kernel with width ?. Moreover, let ? ? [1, n1/d ) and ? ? (0, 1] be finite sets with
cardinality m? and m? , respectively. Under the assumptions of Example 2.3 the oracle inequality
(8) then shows that with probability not less than 1 ? 3m? m? e?? we have
d q+1
? q+1
?
q+2??
q+2
?
d
??d
?
RL,P (fT0 ,?,? ) ? RL,P ? Kd,q,?,?
+
+? ?+?
?? n
n
simultaneously for all ? ? ? and ? ? ?, where ? ? (0, 1] is arbitrarily but fixed and Kd,q,?,? is
a suitable constant. Now using a simple model selection approach (see e.g. Theorem 4.2) for the
second half T1 of our training set we find that with probability not less than 1 ? e?? we have
q+1
? + log(m? m? ) q+2
?
?
RL,P (fT0 ,?T? ,??T ) ? RL,P ? C
1
1
n
d q+1
?
q+2?
d
??d
+ C min
+
?
?
+
?
,
???,???
?? n
where C is a constant only depending on d, q, ?, and ?, and (?T?1 , ??T1 ) ? ? ? ? is a pair that
minimizes the empirical risk RL,T1 (.) over ? ? ?.
Now assume that ?n and ?n are 1/n- and 1/n2 -nets of [1, n1/d ) and (0, 1], respectively. Obviously,
we can choose ?n and ?n such that m?n ? n2 and m?n ? n2 , respectively. With such parameter
sets it is then easy to check that we obtain exactly the rates we have found in Example 2.3, but
without knowing the noise exponents ? and q a-priori.
3
An oracle inequality for clipped penalized ERM
Theorem 2.1 is a consequence of a far more general oracle inequality on clipped penalized empirical
risk minimizers. Since this result is of its own interest we now present it together with its proof in
detail. To ?
this end recall that a subroot is a nondecreasing function ? : [0, ?) ? [0, ?) such
that ?(r)/ r is nonincreasing in r. Moreover, for a Rademacher sequence ? := (?1 , . . . , ?n )
n
with respect to the measure ? and
a function h : Z ? R we define R? h : Z ? R by R? h :=
?1
n
?1 h(z1 ) + ? ? ? + ?n h(zn ) . Now the general oracle inequality is:
Theorem 3.1 Let P 6= ? be a set of (hyper)-parameters, F be a set of measurable functions f :
X ? R with 0 ? F, and ? : P ? F ? [0, ?] be a function. Let P be a distribution on X ? Y and
L be a loss function which satisfies (3) and (4). For a fixed pair (p0 , f0 ) ? P ? F we define
a? (p0 , f0 ) := ?(p0 , f0 ) + RL,P (f0 ) ? R?L,P .
Moreover, let us assume that the quantity B(f0 ) defined in (5) is finite. In addition, we assume
that we have a variance bound of the form (6) for constants v ? 1, ? ? [0, 1] and all measurable
f : X ? R. Furthermore, suppose that there exists a subroot ?n with
?
R? (L ? f? ? L ? fL,P
ET ?P n E???
r > 0. (9)
sup
) ? ?n (r) ,
(p,f )?P?F
?
?(p,f )+EP (L?f??L?fL,P
)?r
Finally, let (pT,? , fT,? ) be an -approximate minimizer of (p, f ) 7? ?(p, f ) + RL,T (f ). Then for
all ? ? 1 and all r satisfying
1
n
32v? 2??
28? o
r ? max 120?n (r),
,
(10)
n
n
we have with probability not less than 1 ? 3e?? that
?(pT,? , fT,? ) + RL,P (f?T,? ) ? R?L,P ? 5r +
14B(f0 )?
+ 8a? (p0 , f0 ) + 4.
3n
Proof: We write B for B(f0 ). For T ? (X ? Y )n we now observe ?(pT,? , fT,? ) + RL,T (f?T,? ) ?
?(p0 , f0 ) ? RL,T (f0 ) ? by the definition of (pT,? , fT,? ), and hence we find
?(pT,? , fT,?)+RL,P (f?T,?)?R?L,P
? RL,P (f?T,? ) ? RL,T (f?T,? ) + RL,T (f0 ) ? RL,P (f0 ) + a? (p0 , f0 ) +
= RL,P (f?T,?)?RL,P (f ? )?RL,T (f?T,?)+RL,T (f ? )
L,P
L,P
+RL,T (f0 )?RL,T (f?0 )?RL,P (f0 )+RL,P (f?0 )
?
?
+RL,T (f?0 )?RL,T (fL,P
)?RL,P (f?0 )+RL,P (fL,P
)
(11)
(12)
(13)
+a? (p0 , f0 ) + .
Let us first estimate the term in line (12). To this end we write h1 := L ? f0 ? L ? f?0 . Then our
assumption on L guarantees h1 ? 0, and since we also have kh1 k? ? B, we find kh1 ?EP h1 k? ?
B. In addition, we obviously have EP (h1 ?EP h1 )2 ? EP h21 ? BEP h1 . Consequently, Bernstein?s
inequality [6, Thm. 8.2] shows that with probability not less than 1 ? e?? we have
r
2? B EP h1
2B?
+
.
ET h1 ? EP h1 <
n
3n
?
?
1
Now using ab ? a2 + 2b we find 2? BEP h1 ? n? 2 ? EP h1 + B?
2n , and consequently we have
7B?
? 1?e?? . (14)
P n T ? Z n : RL,T (f0 )?RL,T (f?0 )?RL,P (f0 )+RL,P (f?0 ) < EP h1 +
6n
Let us now estimate the term in line (13). To this end we write h2 := L? f?0 ?L?f ? . Then we have
L,P
kh2 k? ? 3 and kh2 ? EP h2 k? ? 6. In addition, our variance bound gives EP (h2 ? EP h2 )2 ?
EP h22 ? v(EP h2 )? , and consequently, Bernstein?s inequality shows that with probability not less
than 1 ? e?? we have
r
2? v(EP h2 )?
4?
ET h2 ? EP h2 <
+
.
n
n
0
?1
?1
Now, for q ?1 + (q 0 ) = 1 the elementary inequality ab ? aq q ?1 + bq (q 0 ) holds, and hence for
?
1
?/2
2
q := 2??
, q 0 := ?2 , a := 21?? ?? ? v ? n? 2 , and b := 2EP? h2
we obtain
r
1
2? v(EP h2 )?
? 21?? ?? v? 2??
? 1?
+ EP h2 .
n
2
n
1
Since elementary calculations show that 2?? ?? 2?? ? 1 we obtain
r
1
2? v(EP h2 )?
? 2v? 2??
? 1?
+ EP h2 .
n
2
n
Therefore we have with probability not less than 1 ? e?? that
1
? 2v? 2??
4?
?
?
RL,T (f?0 ) ? RL,T (fL,P
) ? RL,P (f?0 ) + RL,P (fL,P
) < EP h2 + 1 ?
+
. (15)
2
n
n
?
Let us finally estimate the term in line (11). To this end we write hf := L ? f? ? L ? fL,P
, f ? F.
Moreover, for r > 0 we define
n
o
EP hf ? hf
Gr :=
: (p, f ) ? P ? F .
?(p, f ) + EP (hf ) + r
Then for gp,f :=
EP hf ?hf
?(p,f )+EP (hf )+r
kgp,f k? = sup
z?Z
? Gr we have EP gp,f = 0 and
EP hf ? hf (z)
kEP hf ? hf k?
6
? .
=
?(p, f ) + EP (hf ) + r
?(p, f ) + EP (hf ) + r
r
In addition, the inequality a? b2?? ? (a + b)2 and the variance bound assumption (6) implies that
2
EP gp,f
?
EP h2f
EP h2f
v
?
? 2?? .
(EP (hf ) + r)2
r2?? (EP hf )?
r
Now define
?(r) := ET ?P n
sup
(p,f )?P?F
EP hf ? ET hf
.
?(p, f ) + EP (hf ) + r
Standard symmetrization then yields
ET ?P n
sup
|EP hf ? ET hf | ? 2ET ?P n E???
(p,f )?P?F
?(p,f )+EP (hf )?r
sup
|R? hf | ,
(p,f )?P?F
?(p,f )+EP (hf )?r
and hence Lemma 3.2 proved below together with (9) shows ?(r) ? 10?n (r)r?1 , r > 0. Therefore
applying Talagrand?s inequality in the version of [3] to the class Gr we obtain
r
30?n (r)
2? v
7?
n
n
+
+
? 1 ? e?? .
P T ? Z : sup ET g ?
r
nr2??
nr
g?Gr
1/2 7?
v
Let us define ?r := 30?rn (r) + nr2?2??
+ nr . Then the above inequality gives with probability
not less than 1 ? e?? that for all (p, f ) ? P ? F we have
r
7?
2? vr?
+
,
EP hf ? ET hf ? ?r ? ?(p, f ) + EP hf + 30?n (r) +
n
n
and consequently we have with probability not less than 1 ? e?? that
?
?
RL,P (f?T,? ) ? RL,P (fL,P
) ? RL,T (f?T,? ) + RL,T (fL,P
)
r
2? vr?
7?
+
.
(16)
n
n
Now observe that for the functions h1 and h2 which we defined when estimating (12) and (13) we
have
EP g + EP h = RL,P (f0 ) ? R?L,P ,
(17)
and hence we can combine our estimates (16), (14), and (15) of the terms (11), (12), and (13) to
obtain that with probability not less than 1 ? 3e?? we have
(1??r ) ?(pT,? , fT,? ) + RL,P (f?T,? ) ? R?L,P
r
1
2? vr?
? 2v? 2??
(66+7B)?
? 30?n (r)+
+ (1? )
+
+ a? (p0 , f0 )+RL,P (f0 )?R?L,P +.
n
2
n
6n
?
? ?r ? ?(pT,? , fT,? ) + RL,P (f?T,? ) ? RL,P (fL,P
) + 30?n (r) +
1/2
v
In particular, for r satisfying the assumption (10) we have 30?rn (r) ? 14 , nr2?2??
? 14 , and
1
1
7?
??
that
nr ? 4 . This shows 1 ? ?r ? 4 , and hence we obtain with probability not less than 1 ? 3e
r
1
2v? 2??
44?
32? vr?
?(pT,? , fT,? ) + RL,P (f?T,? ) ? R?L,P ? 120?n (r) +
+ 2(2 ? ?)
+
n
n
n
14B?
+
+ 4a? (p0 , f0 ) + 4 RL,P (f0 )?R?L,P + 4.
3n
1
? 1/2
5r
2v? 2??
? r, 44?
?
However we also have 120?n (r) ? r, 32?nvr
n ? 3 , and 2(2 ? ?) n
2(2 ? ?) 4r ? r, and hence we find the assertion.
For the proof of Theorem 3.1 it remains to show the following lemma:
Lemma 3.2 Let P and F be as in Theorem 3.1. Furthermore, let W : F ? R and a : P ? F ?
[0, ?). Define
ET W (f ) ? EP W (f )
?(r) := ET ?P n sup
a(p, f ) + r
f ?P?F
and suppose that there exists a subroot ? such that
ET W (f ) ? EP W (f ) ? ?(r) ,
ET ?P n
sup
r > 0.
(p,f )?P?F
a(p,f )?r
Then we have ?(r) ? 5r ?(r) for all r > 0.
Proof: For x > 1, r > 0, and T ? (X ? Y )n we obtain by a standard peeling approach that
sup
(p,f )?P?F
|EP W (f ) ? ET W (f )|
a(p, f ) + r
?
?
sup
(p,f )?P?F
a(p,f )?r
|EP W (f ) ? ET W (f )| X
+
a(p, f ) + r
i=0
sup
(p,f )?P?F
a(p,f )?rxi
a(p,f )?rxi+1
?
?
|EP W (f ) ? ET W (f )| X
+
r
(p,f )?P?F
i=0
sup
sup
a(p,f )?r
(p,f )?P?F
a(p,f )?rxi
a(p,f )?rxi+1
|EP W (f ) ? ET W (f )|
a(p, f ) + r
|EP W (f ) ? ET W (f )|
rxi + r
?
?
1
1X 1
sup |EP W (f ) ? ET W (f )| +
r (p,f )?P?F
r i=0 xi + 1
a(p,f )?r
=
1
r
?(r) +
?
X
?(rxi+1 )
i=0
xi + 1
|EP W (f ) ? ET W (f )|
.
However since ? is a subroot we obtain that ?(rxi+1 ) ? x
by setting x := 4.
4
sup
(p,f )?P?F
a(p,f )?rxi+1
i+1
2
?(r) so that we obtain the assertion
Proof of Theorem 2.1
Before we begin the proof of Theorem 2.1 let us state the following proposition which follows
directly from [8] (see also [9, Prop. 5.7]) together with simple considerations on covering numbers:
Proposition 4.1 Let F := H be a RKHS, P := {p0 } be a singleton, and ?(p0 , f ) := ?kf k2 . If (7)
is satisfied then there exists a constant cp depending only on p such that (9) is satisfied for
p
1
r p2 a 12 r 1+p
1
?
a 1+p
,
.
?n (r) := cp max v 2 (1?p) r 2 (1?p)
?
n
?
n
Proof of Theorem 2.1: From the covering bound assumption we observe that Proposition 4.1 implies we have the bound (9) with ?n (r) defined by the righthand side of Proposition 4.1 and therefore
Theorem 3.1 implies that Condition (10) becomes
p
1
1
n
p 1
r 1+p
32v? 2??
1
a 1+p
28? o
(1?p) ?
(1?p) r 2 a 2
2
2
r ? max 120cp v
(18)
, 120cp
,
,
r
?
n
?
n
n
n
and solving with respect to r yields the conclusion.
Finally, for the parameter selection approach in Example 2.4 we need the following oracle inequality
for model selection:
Theorem 4.2 Let P be a distribution on X ? Y and let L be a loss function which satisfies (3),
(4), and the variance bound (6). Furthermore, let F := {f1 , . . . , fm } be a finite set of functions
mapping X into [?1, 1]. For T ? (X ? Y )n we define
fT := arg min RL,T (f ) .
f ?F
Then there exists a universal constant K such that for all ? ? 1 we have with probability not less
than 1 ? 3e?? that
1
1
K log m 2??
32v? 2??
5K log m + 154?
RL,P (fT ) ? R?L,P ? 5
+5
+
n
n
n
+8 min(RL,P (f ) ? R?L,P ) .
f ?F
Proof: Since all functions fi already map into [?1, 1] we do not have to consider the clipping operator. For r > 0 we now define Fr := {f ? F : RL,P (f ) ? R?L,P ? r}. Then the cardinality of
?
, ?, L2 (T )) ? m for all
Fr is smaller than or equal to m and hence we have N (L ? Fr ? L ? fL,P
? > 0. Using the technique of [8] (cf. also [9, Prop. 5.7]) we hence obtain that (9) is satisfied for
p
c
log m
?n (r) := ? max
v log m r?/2 , ?
,
n
n
where c is a universal constant. Applying Theorem 3.1 then yields the assertion.
References
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is more important than the size of the network. IEEE Trans. Inform. Theory, 44:525?536, 1998.
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[5] N. Cristianini and J. Shawe-Taylor. An Introduction to Support Vector Machines. Cambridge University
Press, 2000.
[6] L. Devroye, L. Gy?orfi, and G. Lugosi. A Probabilistic Theory of Pattern Recognition. Springer, New
York, 1996.
[7] F. Girosi, M. Jones, and T. Poggio. Regularization theory and neural networks architectures. Neural
Computation, 7:219?269, 1995.
[8] S. Mendelson. Improving the sample complexity using global data. IEEE Trans. Inform. Theory,
48:1977?1991, 2002.
[9] I. Steinwart and C. Scovel. Fast rates for support vector machines using Gaussian kernels. Annals of
Statistics, to appear.
[10] I. Steinwart and C. Scovel. Fast rates for support vector machines. In Proceedings of the 18th Annual
Conference on Learning Theory, COLT 2005, pages 279?294. Springer, 2005.
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| 3066 |@word version:3 norm:3 p0:11 rkhs:3 scovel:3 realistic:1 girosi:1 half:1 beginning:1 math:1 prove:1 consists:2 combine:1 multi:1 gov:1 actual:1 equipped:2 cardinality:2 becomes:1 begin:2 estimating:1 moreover:7 substantially:1 minimizes:2 guarantee:1 exactly:1 k2:1 classifier:2 unit:3 yn:2 appear:2 before:2 t1:7 local:1 consequence:2 acad:1 establishing:2 clint:1 lugosi:1 equivalence:1 bennet:1 practical:1 practice:1 procedure:2 kh2:2 universal:2 suprema:1 empirical:9 orfi:1 word:1 selection:9 operator:1 bh:2 risk:9 applying:2 measurable:5 map:1 go:1 convex:1 immediately:1 bep:2 estimator:1 embedding:1 annals:1 pt:8 suppose:3 element:1 satisfying:2 particularly:1 recognition:1 ep:55 ft:13 complexity:4 cristianini:1 depend:2 solving:1 serve:1 tx:5 fast:4 shortcoming:1 describe:1 kp:3 hyper:1 choosing:1 whose:1 statistic:1 gp:3 nondecreasing:1 obviously:3 sequence:2 net:1 h22:1 fr:3 kx1:1 kh:1 los:2 rademacher:2 produce:1 generating:2 help:2 depending:3 illustrate:1 p2:1 implies:3 met:1 require:2 behaviour:1 f1:1 proposition:4 elementary:3 hold:2 sufficiently:1 considered:2 k2h:4 mapping:1 achieves:1 vary:1 a2:1 symmetrization:1 establishes:1 tool:1 weighted:1 gaussian:7 always:4 avoid:1 zhou:2 modelling:1 check:2 mainly:1 sense:2 dependent:1 minimizers:5 typically:2 kep:1 interested:3 arg:2 classification:6 colt:1 denoted:2 exponent:9 priori:5 equal:1 having:1 jones:1 t2:3 report:1 serious:1 simultaneously:1 national:1 n1:2 ab:2 interest:1 righthand:1 superfluous:1 nonincreasing:1 implication:1 poggio:1 bq:1 taylor:1 theoretical:2 minimal:1 soft:1 assertion:3 rxi:8 zn:1 clipping:4 nr2:3 subset:1 alamo:2 gr:4 probabilistic:1 informatics:1 together:3 continuously:1 nm:1 satisfied:4 choose:1 hoeffding:1 singleton:1 attaining:1 gy:1 b2:1 blanchard:1 satisfy:2 explicitly:1 h1:13 closed:2 analyze:2 sup:18 hf:26 worsen:1 square:1 variance:5 characteristic:1 yield:5 identify:1 kh1:2 kfp:2 cc:1 inform:2 definition:2 kl2:1 obvious:1 proof:9 associated:1 proved:1 recall:5 knowledge:2 lim:1 organized:1 hilbert:2 actually:1 back:1 nvr:1 kgp:1 improved:2 furthermore:7 talagrand:4 hand:2 steinwart:3 infimum:1 former:1 hence:9 regularization:2 laboratory:1 deal:2 width:3 covering:4 cp:6 consideration:1 recently:1 fi:1 superior:1 rl:60 banach:1 discussed:1 belong:1 h2f:2 cambridge:1 rd:1 jcs:1 shawe:1 aq:1 f0:34 own:1 recent:1 optimizing:1 inf:1 certain:1 inequality:33 binary:2 arbitrarily:3 somewhat:1 technical:1 faster:1 calculation:1 impact:1 metric:1 kernel:12 addition:8 rest:1 massart:1 bernstein:3 easy:1 architecture:1 fm:1 idea:1 knowing:2 t0:1 bartlett:1 york:1 clear:1 tsybakov:2 clip:1 svms:5 category:1 write:6 group:1 key:1 year:1 clipped:6 throughout:1 wu:2 decision:1 investigates:1 fl:15 bound:7 oracle:22 annual:1 x2:2 bousquet:2 min:6 ball:3 kd:2 describes:1 smaller:1 erm:1 remains:1 discus:1 know:1 end:4 apply:2 observe:3 rkhss:2 denotes:3 cf:1 hinge:4 establish:2 already:3 quantity:1 concentration:2 usual:1 nr:3 sci:1 devroye:1 minimizing:1 ying:2 unfortunately:1 statement:1 negative:1 unknown:1 observation:2 ingo:2 finite:4 situation:1 precise:1 y1:2 rn:2 reproducing:1 thm:2 pair:2 required:1 lanl:1 paris:1 z1:1 established:1 hush:1 trans:2 below:1 pattern:2 fp:5 including:1 max:4 suitable:2 regularized:5 scheme:4 prior:1 understanding:1 l2:5 geometric:1 kf:4 embedded:1 loss:14 interesting:1 ingredient:1 validation:2 h2:15 penalized:2 last:1 free:1 side:1 fall:1 overcome:1 xn:5 commonly:2 employing:1 far:1 approximate:3 compact:2 dealing:2 global:1 xi:4 don:1 continuous:5 learn:2 improving:1 ft0:3 main:3 noise:8 n2:3 x1:6 borel:1 fashion:1 vr:4 shrinking:2 learns:1 peeling:1 theorem:20 r2:1 svm:7 exists:7 mendelson:1 margin:1 chen:1 springer:2 minimizer:1 satisfies:6 prop:2 goal:1 consequently:7 rbf:4 careful:1 lipschitz:2 kf0:2 lemma:3 called:1 rarely:1 h21:1 support:8 |
2,277 | 3,067 | Denoising and Dimension Reduction in Feature Space
Mikio L. Braun
Fraunhofer Institute1
FIRST.IDA
Kekul?estr. 7, 12489 Berlin
[email protected]
Joachim Buhmann
Inst. of Computational Science
ETH Zurich
CH-8092 Z?urich
[email protected]
2,1
?
Klaus-Robert Muller
Technical University of Berlin2
Computer Science
Franklinstr. 28/29, 10587 Berlin
[email protected]
Abstract
We show that the relevant information about a classification problem in feature
space is contained up to negligible error in a finite number of leading kernel PCA
components if the kernel matches the underlying learning problem. Thus, kernels not only transform data sets such that good generalization can be achieved
even by linear discriminant functions, but this transformation is also performed
in a manner which makes economic use of feature space dimensions. In the best
case, kernels provide efficient implicit representations of the data to perform classification. Practically, we propose an algorithm which enables us to recover the
subspace and dimensionality relevant for good classification. Our algorithm can
therefore be applied (1) to analyze the interplay of data set and kernel in a geometric fashion, (2) to help in model selection, and to (3) de-noise in feature space
in order to yield better classification results.
1
Introduction
Kernel machines use a kernel function as a non-linear mapping of the original data into a highdimensional feature space; this mapping is often referred to as empirical kernel map [6, 11, 8, 9].
By virtue of the empirical kernel map, the data is ideally transformed such that a linear discriminative
function can separate the classes with low generalization error, say via a canonical hyperplane with
large margin. The latter is used to provide an appropriate mechanism of capacity control and thus to
?protect? against the high dimensionality of the feature space.
The idea of this paper is to add another aspect, not covered by this picture. We will show theoretically
that if the learning problem matches the kernel well, the relevant information of a supervised learning
data set is always contained in a finite number of leading kernel PCA components (that is, the label
information projected to the kernel PCA directions), up to negligible error. This result is based on
recent approximation bounds dealing with the eigenvectors of the kernel matrix which show that if
a function can be reconstructed using only a few kernel PCA components asymptotically, then the
same already holds in a finite sample setting, even for small sample sizes.
Consequently, the use of a kernel function not only greatly increases the expressive power of linear
methods by non-linearly transforming the data, but it does so ensuring that the high dimensionality
of the feature space will not become overwhelming: the relevant information for classification will
stay confined within a comparably low dimensional subspace. This finding underlines the efficient
use of data that is made by kernel machines using a kernel suited to the problem. While the number
of data points stays constant for a given problem, a smart choice of kernel permits to make better
use of the available data at a favorable ?data point per effective dimension?-ratio, even for infinitedimensional feature spaces. Furthermore we can use de-noising techniques in feature space, much
in the spirit of Mika et al. [8, 5] and thus regularize the learning problem in an elegant manner.
Let us consider an example. Figure 1 shows the first six kernel PCA components for an example
data set. Above each plot, the variance of the data along this direction and the contribution of this
Figure 1: Although the data set is embedded into a high-dimensional manifold, not all directions
contain interesting information. Above the first six kernel PCA components are plotted. Of these,
only the fourths is highly relevant for the learning problem. Note, however, that this example is
atypical in having a single relevant component. In general, several components will have to be
combined to construct the decision boundary.
component to the class labels are plotted (normalized such that the maximal possible contribution is
one1 ). Of these six components, only the fourth contributes significantly to the class memberships.
As we will see below, the contributions in the other directions is mostly noise. This is true especially for components with small variance. Therefore, after removing this noise, a finite number of
components suffice to represent the optimal decision boundary.
The dimensionality of the data set in feature space is characteristic for the relation between a data
set and a kernel. Roughly speaking, the relevant dimensionality of the data set corresponds to the
complexity of the learning problem when viewed through the lens of the kernel function. This notion
of complexity relates the number of data points required by the learning problem and the noise, as a
small relevant dimensionality enables the de-noising of the data set to obtain an estimate of the true
class labels, making the learning process much more stable. This combination of dimension and
noise estimate allows us to distinguish among data sets showing weak performance which might
either be complex or noisy.
To summarize the main contributions of this paper: (1) We provide theoretical bounds showing that
the relevant information (defined in section 2) is actually contained in the leading projected kernel
principal components under appropriate conditions. (2) We propose an algorithm which estimates
the relevant dimensionality of the data set and permits to analyze the appropriateness of a kernel for
the data set, and thus to perform model selection among different kernels. (3) We show how the
dimension estimate can be used in conjunction with kernel PCA to perform effective de-noising. We
analyze some well-known benchmark data sets and evaluate the performance as a de-noising tool
in Section 5. Note that we do not claim to obtain better performance within our framework when
compared to, for example, cross-validation techniques. Rather, we are on par. Our contribution
is to foster an understanding about a data set and to gain better insights of whether a mediocre
classification result is due to intrinsic high dimensionality of the data or overwhelming noise level.
2
The Relevant Information and Kernel PCA Components
In this section, we will define the notion of the relevant information contained in the class labels,
and show that the location of this vector with respect to the kernel PCA components is linked to the
scalar products with the eigenvectors of the kernel matrix.
1
Note, however, that these numbers do not simply add up, instead the contribution of a and b is
?
a2 + b2 .
Let us start to formalize the ideas introduced so far. As usual, we will consider a data set (X1 , Y1 ),
. . . , (Xn , Yn ) where the X lie in some space X and the Y are in Y = {?1}. We assume that the
(Xi , Yi ) are drawn i.i.d. from PX ?Y . In kernel methods, the data is non-linearly mapped into some
feature space F via the feature map ?. Scalar products in F can be computed by the kernel k in
closed form: h?(x), ?(x0 )i = k(x, x0 ). Summarizing all the pairwise scalar products results in the
(normalized) kernel matrix K with entries k(Xi , Xj )/n.
We wish to summarize the information contained in the class label vector Y = (Y1 , . . . , Yn )
about the optimal decision boundary. We define the relevant information vector as the vector
G = (E(Y1 |X1 ), . . . , E(Yn |Xn )) containing the expected class labels for the objects in the training
set. The idea is that since E(Y |X) = P (Y = 1|X) ? P (Y = ?1|X), the sign of G contains the
relevant information on the true class membership by telling us which class is more probable. The
observed class label vector can be written as Y = G ? N with N = G ? Y denoting the noise in
the class labels. We want to study the relation of G with respect to the kernel PCA components. The
following lemma relates projections of G to the eigenvectors of the kernel matrix K:
Lemma 1 The kth kernel PCA component fk evaluated on the Xi s is equal to the kth eigenvector2
of the kernel matrix K: (fk (X1 ), . . . , fk (Xn )) = uk . Consequently, the projection of a vector
Pd
Y ? Rn to the leading d kernel PCA components is given by ?d (Y ) = k=1 uk u>k Y.
Pn
Proof The kernel PCA directions are given as (see [10]) vk =
i=1 ?i ?(Xi ), where ?i =
[uk ]i /lk , [uk ]i denoting the ith component of uk , and lk , uk being the eigenvalues and eigenvectors of the kernel matrix K. Thus, the kth PCA component for a point Xj in the training set is
fk (Xj ) = h?(Xj ), vk i =
n
n
1 X
1 X
h?(Xj ), ?(Xi )i[uk ]i =
k(Xj , Xi )[uk ]i .
lk i=1
lk i=1
The sum computes the jth component of Kuk = lk uk , because uk is an eigenvector of K. Therefore
fk (Xj ) =
1
[lk uk ]j = [uk ]j .
lk
Since the uk are orthogonal (K is a symmetric matrix), the projection of Y to the space spanned by
Pd
the first d kernel PCA components is given by i=1 ui u>i Y .
3
A Bound on the Contribution of Single Kernel PCA Components
As we have just shown, the location of G is characterized by its scalar products with the eigenvectors
of the kernel matrix. In this section, we will apply results from [1, 2] which deal with the asymptotic
convergence of spectral properties of the kernel matrix to show that the decay rate of the scalar
products are linked to the decay rate of the kernel PCA principal values.
It is clear that we cannot expect G to generally locate favorably with respect to the kernel PCA
components, but only when there is some kind of match between G and the chosen kernel. This
link will be established by asymptotic considerations. Kernel PCA is closely linked to the spectral
properties of the kernel matrix, and it is known [3, 4] that the eigenvalues and the projections to
eigenspaces converge. Their asymptotic
R limits are given as the eigenvalues ?i and eigenfunctions
?i of the integral operator Tk f = k( ? , x)f (x)PX (dx) defined on L2 (PX ), where PX is the
marginal measure of PX ?Y which generates our samples. The eigenvalues and
also
Peigenfunctions
?
occur in the well-known Mercer?s formula: By Mercer?s theorem, k(x, x0 ) = i=1 ?i ?i (x)?i (x0 ).
The asymptotic counterpart of G is given by the function g(x) = E(Y |X = x).
We will encode fitness between k and g by requiring that g lies P
in the image of Tk . This is equivalent
?
to saying that there exists a sequence (?i ) ? `2 such that g = i=1 ?i ?i ?i .3 Under this condition,
the scalar products decay as quickly as the eigenvalues, because hg, ?i i = ?i ?i = O(?i ). Because
of the known convergence of spectral projections, we can expect the same behavior asymptotically
2
As usual, the eigenvectors are arranged in descending order by corresponding eigenvalue.
A different condition is that g lies in the RKHS generated by k. This amounts to saying that g lies in the
1/2
image of Tk . Therefore, the condition used here is slightly more restrictive.
3
from the finite sample case. However, the convergence speed is the crucial question. This question is
not trivial, because eigenvector stability is known to be linked to the gap between the corresponding
eigenvalues, which will be fairly small for small eigenvalues. In fact, for example, the results from
[14] do not scale properly with the corresponding eigenvalue, such that the bounds are too loose. A
number of recent results on the spectral properties of the kernel matrix [1, 2] specifically deal with
error bounds for small eigenvalues and their associated spectral projections. Using these results, we
obtain the following bound on u>i G.4
Theorem 1 Let g =
with high probability.
P?
i=1
?i ?i ?i as explained above, and let G = (g(X1 ), . . . , g(Xn )). Then,
1
? |u>i G| < 2li ar ci (1 + O(rn?1/4 ))
n
p
p
+ rar ?r O(1) + Tr + ATr O(n?1/4 ) + rar ?r O(n?1/2 ),
where r balancesP
the different terms (1 ? r ? n), ci measures the size of the eigenvalue cluster
r
around li , ar = i=1 |?i | is a measure of the size of the first r components, ?r is the sum of all
eigenvalues smaller than ?r , A is the supremum norm of g, and Tr is the error of projecting g to the
space spanned by the first r eigenfunctions.
The bound consists of a part which scales with li (first term) and a part which does not (remaining
terms). Typically, the bound initially scales with li until the non-scaling part dominates the bound
for larger i. These two parts are balanced by r. However, note that all terms which do not scale with
li will typically be small: for smooth kernels, the eigenvalues quickly decay to zero as r ? ?. The
related quantities ?r , and Tr , will also decay to zero at slightly slower rates. Therefore, by adjusting
r (as n ? ?), the non-scaling part can be made arbitrarily small, leading to a small bound on |u>i G|
for larger i.
Put differently, the bound shows that the relevant information vector G (as introduced in Section 2)
is contained in a number of leading PCA components up to a negligible error. The number of dimensions depends on the asymptotic coefficients ?i and the decay rate of the asymptotic eigenvalues of
k. Since this rate is related to the smoothness of the kernel function, the dimension will be small for
smooth kernels whose leading eigenfunctions ?i permit good approximation of g.
4
The Relevant Dimension Estimation Algorithm
In this section, we will propose the relevant dimension estimation (RDE) algorithm which estimates
the dimensionality of the relevant information from a finite sample, allowing us to analyze the fit
between a kernel function and a data set in a practical way.
Dimension Estimation We propose an approach which is motivated by the geometric findings
explained above. Since G is not known, we can only observe the contributions of the kernel PCA
components to Y , which can be written as Y = G + N (see Section 2). The contributions u>i Y
will thus be formed as a superposition of u>i Y = u>i G + u>i N . Now, by Theorem 1, we know that
G will be very close to zero for the latter coefficients, while on the other hand, the noise N will be
equally distributed over all coefficients. Therefore, the kernel PCA coefficients s = u>i Y will have
the shape of an evenly distributed noise floor u>i N from which the coefficients u>i G of the relevant
information protrude (see Figure 2(b) for an example).
We thus propose the following algorithm: Given a fixed kernel k, we estimate the true dimension
by fitting a two component model to the coordinates of the label vector. Let s = (u>1 Y, . . . , u>n Y ).
Then, assume that
N (0, ?12 ) 1 ? i ? d
si ?
N (0, ?22 ) d < i ? n.
4
We have tried to reduce the bound to its most prominent features. For a more detailed explanation of the
quantities and the proof, see the appendix. Also, the confidence ? of the ?with high probability? part is hidden
in the O( ? ) notation. We have used the O( ? ) notation rather deliberately to exhibit the dominant constants.
(a)
(b)
(c)
Figure 2: Further plots on the toy example from the introduction. (a) contains the kernel PCA
component contributions (dots), and the training and test error by projecting the data set to the
given number of leading kernel PCA components. (b) shows the negative log-likelihood of the two
component model used to estimate the dimensionality of the data. (c) The resulting fit when using
only the first four components.
We select the d minimizing the negative log-likelihood, which is proportional to
`(d) =
n?d
d
log ?12 +
log ?22 ,
n
n
with
?12 =
d
n
X
1X 2 2
1
si , ?2 =
s2i .
d i=1
n?d
(1)
i=d+1
Model Selection for Kernel Choice For different kernels, we again use the likelihood and select
the kernel which leads to the best fit in terms of the likelihood. If the kernel width does not match
the scale of the structure of the data set, the fit of the two component model will be inferior: for very
small or very large kernels, the kernel PCA coefficients of Y have no clear structure, such that the
likelihood will be small. For example, for Gaussian kernels, for very small kernel widths, noise is interpreted as relevant information, such that there appears to be no noise, only very high-dimensional
data. On the other hand, for very large kernel widths, any structure will be indistinguishable from
noise such that the problem appears to be very noisy with almost no structure. In both cases, fitting
the two component model will not work very well, leading to large values of `.
Experimental Error Estimation The estimated dimension can be used to estimate the noise
level present in the data set. The idea is to measure the error between the projected label
? = ?d (Y ), which approximates the true label information G. The resulting number
vector G
P
1
? i 6= Yi } is an estimate of the fraction of misclassified examples in the training
? = n ni=1 1{[G]
err
set, and therefore an estimate for the noise level in the class labels.
? and the noise level are consistent if the estimated
A Note on Consistency Both the estimate of G
dimension d scales sub-linearly with n. The argument can be sketched as follows: since the kernel
PCA components do not depend on Y , the noise N contained in Y is projected to a random subspace
of dimension d. Therefore, n1 k?d (N )k2 ? nd ( n1 kN k2 ) ? 0 as n ? ?, since d/n ? 0 and
1
2
2
n kN k ? E(N ). Empirically, d was found to be rather stable, but in principle, the condition
on d could even be enforced by adding a small sub-linear term (for example,
estimated dimension d).
5
?
n, or log n, to the
Experiments
Toy Data Set Returning to the toy example from the introduction, let us now take a closer look at
this data set. In Figure 2(a), the spectrum for the toy data set is plotted. We can see that every kernel
PCA component contributes to the observed class label vector. However, most of these contributions
are noise, since the classes are overlapping. The RDE method estimates that only the first four
components are relevant. This behavior of the algorithm can also be seen from the training and
independent test error measured on a second data set of size 1000 which can also be found in this
plot. In Figure 2(b), the log-likelihoods from (1) are shown, and one observes a well pronounced
minimum. Finally, in Figure 2(c), the resulting fit is shown.
Benchmark data sets We performed experiments on the classification learning sets from [7]. For
each of the data sets, we de-noise the data set using a family of rbf kernels by projecting the class
labels to the estimated number of leading kernel PCA components. The kernel width is also selected
automatically using the achieved log-likelihood as described above. The width of the rbf kernel is
selected from 20 logarithmically spaced points between 10?2 and 104 for each data set.
For the dimension estimation task, we compare our RDE method to a dimensionality estimate based
Pd
on cross-validation. More concretely, the matrix S = i=1 ui u>i computes the projection to the
leading d kernel PCA components. Interpreting the matrix S as a linear fit matrix, the leave-oneout cross-validation error can be computed in closed form (see [12])5 , since S is diagonal with
respect to the eigenvector basis ui . Evaluating the cross-validation error for all dimensions and for
a number of kernel parameters, one can select the best dimension and kernel parameter. Since the
cross-validation can be computed efficiently, the computational demands of both methods are equal.
Table 3 shows the resulting dimension estimates. We see that both methods perform on par, which
shows that the strong structural prior assumption underlying RDE is justified.
For the de-noising task, we have compared a (unregularized) least-squares fit in the reduced feature
space (kPCR) against kernel ridge regression (KRR) and support vector machines (SVM) on the
same data set. The resulting test errors are plotted also in Table 3. We see that a relatively simple
method on the reduced features leads to classification which is on par with the state-of-the-art competitors. Also note that the estimated error rates match the actually observed error rates quite well,
although there is a tendency to under-estimate the true error.
Finally, inspecting the estimated dimension and noise level reveals that the data sets breast-cancer,
diabetis, flare-solar, german, and titanic all have only moderately large dimensionalities. This suggest that these data sets are inherently noisy and better results cannot be expected, at least within
the family of rbf kernels. On the other hand, the data set image seems to be particularly noise free,
given that one can achieve a small error in spite of the large dimensionality. Finally, the splice data
set seems to be a good candidate to benefit from more data.
6
Conclusion
Both in theory and on practical data sets, we have shown that the relevant information in a supervised
learning scenario is contained in the leading projected kernel PCA components if the kernel matches
the learning problem. The theory provides a consistent estimation for the expected class labels and
the noise level. This behavior complements the common statistical learning theoretical view on kernel based learning with insight on the interaction of data and kernel: A well chosen kernel (a) makes
the model estimate efficiently and generalize well, since only a comparatively low dimensional representation needs to be learned for a fixed given data size and (b) permits a de-noising step that
discards some void projected kernel PCA directions and thus provides a regularized model.
Practically, our RDE algorithm automatically selects the appropriate kernel model for the data and
extracts as additional side information an estimate of the effective dimension and estimated expected
5
This applies only to the 2-norm. However, as the performance of 2-norm based methods like kernel ridge
regression on classification problems show, the 2-norm is also informative on the classification performance.
data set
banana
breast-cancer
diabetis
flare-solar
german
heart
image
ringnorm
splice
thyroid
titanic
twonorm
waveform
dim
24
2
9
10
12
4
272
36
92
17
4
2
14
dim (cv)
26
2
9
10
12
5
368
37
89
18
6
2
23
est. error rate
8.8 ? 1.5
25.6 ? 2.1
21.5 ? 1.3
32.9 ? 1.2
22.9 ? 1.1
15.8 ? 2.5
1.7 ? 1.0
1.9 ? 0.7
9.2 ? 1.3
2.0 ? 1.0
20.8 ? 3.8
2.3 ? 0.7
8.4 ? 1.5
kPCR
11.3 ? 0.7
27.0 ? 4.6
23.6 ? 1.8
33.3 ? 1.8
24.1 ? 2.1
16.7 ? 3.8
4.2 ? 0.9
4.4 ? 1.2
13.8 ? 0.9
5.1 ? 2.1
22.9 ? 1.6
2.4 ? 0.1
10.8 ? 0.9
KRR
10.6 ? 0.5
26.5 ? 4.7
23.2 ? 1.7
34.1 ? 1.8
23.5 ? 2.2
16.6 ? 3.5
2.8 ? 0.5
4.7 ? 0.8
11.0 ? 0.6
4.3 ? 2.3
22.5 ? 1.0
2.8 ? 0.2
9.7 ? 0.4
SVM
11.5 ? 0.7
26.0 ? 4.7
23.5 ? 1.7
32.4 ? 1.8
23.6 ? 2.1
16.0 ? 3.3
3.0 ? 0.6
1.7 ? 0.1
10.9 ? 0.6
4.8 ? 2.2
22.4 ? 1.0
3.0 ? 0.2
9.9 ? 0.4
Figure 3: Estimated dimensions and error rates for the benchmark data sets from [7]. ?dim? shows
the medians of the estimated dimensionalities over the resamples. ?dim (cv)? shows the same quantity, but this time, the dimensions have been estimated by leave-one-out cross-validation. ?est. error
rate? is the estimated error rate on the training set by comparing the de-noise class labels to the true
class labels. The last three columns show the test error rates of three algorithms: ?kPCR? predicts
using a simple least-squares hyperplane on the estimated subspace in feature space, ?KRR? is kernel
ridge regression with parameters estimated using leave-one-out cross-validation, and ?SVM? are the
original error rates from [7].
error for the learning problem. Compared to common cross-validation techniques one could argue
that all we have achieved is to find a similar model as usual at a comparable computing time. However, we would like to emphasize that the side information extracted by our procedure contributes
to a better understanding of the learning problem at hand: Is the classification result limited due to
intrinsic high dimensional structure or are we facing noise and nuisance dimensions? Simulations
show the usefulness of our RDE algorithm.
An interesting future direction lies in combining these results with generalization bounds which
are also based on the notion of an effective dimension, this time, however, with respect to some
regularized hypothesis class (see, for example, [13]). Linking the effective dimension of the data set
with the dimension of a learning algorithm, one could obtain data dependent bounds in a natural way
with the potential to be tighter than bounds which are based on the abstract capacity of a hypothesis
class.
References
[1] ML Braun. Spectral Properties of the Kernel Matrix and Their Application to Kernel Methods in Machine Learning. PhD thesis, University of Bonn, 2005. Available electronically at http://hss.ulb.unibonn.de/diss online/math nat fak/2005/braun mikio.
[2] ML Braun. Accurate error bounds for the eigenvalues of the kernel matrix. Journal of Machine Learning
Research, 2006. To appear.
[3] V Koltchinskii and E Gin?e. Random matrix approximation of spectra of integral operators. Bernoulli,
6(1):113?167, 2000.
[4] VI Koltchinskii. Asymptotics of spectral projections of some random matrices approximating integral
operators. Progress in Probability, 43:191?227, 1998.
[5] S Mika, B Sch?olkopf, A Smola, K-R M?uller, M Scholz, and Gunnar R?athsch. Kernel PCA and de-noising
in feature space. In Advances in Neural Information Processing Systems 11. MIT Press, 1999.
[6] K-R M?uller, S Mika, G R?atsch, K Tsuda, and B Sch?olkopf. An introduction to kernel-based learning
algorithms. IEEE Transaction on Neural Networks, 12(2):181?201, May 2001.
[7] G R?atsch, T Onoda, and K-R M?uller. Soft margins for AdaBoost. Machine Learning, 42(3):287?320,
March 2001.
[8] B Sch?olkopf, S Mika, CJC Burges, P Knirsch, K-R M?uller, G R?atsch, and AJ Smola. Input space vs. feature space in kernel-based methods. IEEE Transactions on Neural Networks, 10(5):1000?1017, September 1999.
[9] B Sch?olkopf and AJ Smola. Learning with Kernels. MIT Press, 2001.
[10] B Sch?olkopf, AJ Smola, and K-R M?uller. Nonlinear component analysis as a kernel eigenvalue problem.
Neural Computation, 10:1299?1319, 1998.
[11] V Vapnik. Statistical Learning Theory. Wiley, 1998.
[12] G Wahba. Spline Models For Observational Data. Society for Industrial and Applied Mathematics, 1990.
[13] T Zhang. Learning bounds for kernel regression using effective data dimensionality. Neural Computation,
17:2077?2098, 2005.
[14] L Zwald and G Blanchard. On the convergence of eigenspaces in kernel principal components analysis.
In Advances in Neural Information Processing Systems (NIPS 2005), volume 18, 2006.
A
Proof of Theorem 1
First, let usP
collect the definitions concerning kernel functions. Let k be a Mercer kernel with
?
k(x, x0 ) = i=1 ?i ?i (x)?i (x0 ), and k(x, x) ? K < ?. The kernel matrix of k for an n-sample
X1 , . . . , Xn is [K]ij = k(Xi , Xj )/n. Eigenvalues of K are li , and P
its eigenvectors are ui . The
r
kernel k is approximated by the truncated kernel matrix is kr (x, x0 ) = i=1 ?i ?i (x)?i (x0 ), and its
kernel matrix is denoted by Kr , whose eigenvalues are mi . The approximation error is measured
by Er = Kr ? K. We will measure the amount of clustering ci of the eigenvalues by the number
of eigenvalues of Kr between li /2 and 2li . The matrix ?
containing the sample vectors of the first
r eigenfunctions ?i of k is given by [?r ]i` = ?` (Xi )/ n, 1 ? i ? n, 1 ? ` ? r. Since the
eigenfunctions are orthogonal asymptotically, we can expect that the sample vectors of the eigenfunctionsP
converge to either 0 or 1. The error is measured by the matrix Cr = ?>r ?r ? I. Finally,
?
let ?r = i=r+1 ?i .
P?
Next, we collect definitions concerning some function f . Let f =
i=1 ?i ?i ?i with (?i ) ?
2
`P
, and |f | ? A < ?. The size of the contribution of the first r terms is measured by ar =
r
|?i |. Define the error of truncating f to the first r elements of its series expansion by Tr =
Pi=1
?
( i=r+1 ?2i ?i2 )1/2 .
The proof of Theorem 1 is based on performing rough estimates of the bound from Theorem 4.92
in [1]. The bound is
1
? |u>i f (X)| < min li D(r, n) + E(r, n) + T (r, n)
1?r?n
n
where the three terms are given by
r
p
1
4
+
+
r
,
D(r, n) = 2ar k?r kci ,
E(r, n) = 2rar k?r kkE k,
T (r, n) = Tr + F Tr
n?
>
1/2
It holds that k?+
= (1 ? kCr k)?1/2 ([1], Lemma 4.44). Furthermore,
r k ? (1 ? k?r ?r ? Ik)
kCr k ? 1 as n ? ? for fixed r. For kernel with p
bounded diagonal, it holds that with probability
larger than 1?? ([1], Lemma 3.135) that kCr k ? r r(r + 1)K/?r n? = r2 O(n?1/2 ) with a rather
?1/2
large constant, especially, if ?r is small. Consequently, k?+
= 1+O(rn?1/4 ).
r k ? (1?kCr k)
Now, Lemma 3.135 in [1] bounds kEr k from which we can derive the asymptotic
r
p
1
2K?r
kEr k ? ?r + ?r +
= ?r + ?r O(n? 2 ),
n?
assuming that K will be reasonably small (for example, for rbf-kernels, K = 1). Combining this
with our rate for k?+
k, we obtain
r p
p
1
1
1
1
E(r, n) = 2rar ?r + ?r O(n? 2 ) (1+O(rn? 4 )) = 2rar ?r (1+O(rn? 4 ))+rar ?r O(n? 2 ).
Finally, we obtain
1
1
? |u>i f (X)| = 2li ar ci (1 + O(rn? 4 ))
n
1
+ 2rar ?r (1 + O(rn? 4 )) + rar
p
p
1
1
?r O(n? 2 ). + Tr + ATr O(n? 2 ).
1
If we assume that ?r will be rather small, we replace 1 + O(rn? 4 ) by O(1) for the second term and
obtain the claimed rate.
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2,278 | 3,068 | Learnability and the Doubling Dimension
Yi Li
Genome Institute of Singapore
[email protected]
Philip M. Long
Google
[email protected]
Abstract
Given a set F of classifiers and a probability distribution over their domain, one
can define a metric by taking the distance between a pair of classifiers to be the
probability that they classify a random item differently. We prove bounds on the
sample complexity of PAC learning in terms of the doubling dimension of this
metric. These bounds imply known bounds on the sample complexity of learning
halfspaces with respect to the uniform distribution that are optimal up to a constant
factor. We prove a bound that holds for any algorithm that outputs a classifier with
zero error whenever this is possible; this bound is in terms of the maximum of the
doubling dimension and the VC-dimension of F , and strengthens the best known
bound in terms of the VC-dimension alone. We show that there is no bound on
the doubling dimension in terms of the VC-dimension of F (in contrast with the
metric dimension).
1 Introduction
A set F of classifiers and a probability distribution D over their domain induce a metric in which the
distance between classifiers is the probability that they disagree on how to classify a random object.
(Let us call this metric D .) Properties of metrics like this have long been used for analyzing the
generalization ability of learning algorithms [11, 32]. This paper is about bounds on the number of
examples required for PAC learning in terms of the doubling dimension [4] of this metric space.
The doubling dimension of a metric space is the least d such that any ball can be covered by 2d
balls of half its radius. The doubling dimension has been frequently used lately in the analysis of
algorithms [13, 20, 21, 17, 29, 14, 7, 22, 28, 6].
In the PAC-learning model, an algorithm is given examples (x1 ; f (x1 )); :::; (xm ; f (xm )) of the behavior of an arbitrary member f of a known class F . The items x1 ; :::; xm are chosen independently
at random according to D. The algorithm must, with probability at least 1 ? (w.r.t. to the random
choice of x1 ; :::; xm ), output a classifier whose distance from f is at most .
We show that if
using
(F; D ) has doubling dimension d, then F
can be PAC-learned with respect to
D
d + log ?1
(1)
examples. If in addition the VC-dimension of F is d, we show that any algorithm that outputs a
classifier with zero training error whenever this is possible PAC-learns F w.r.t. D using
O
0 q
1
d log 1 + log 1?
A
O@
examples.
(2)
We show that if F consists of halfspaces through the origin, and D is the uniform distribution over
the unit ball in Rn, then the doubling dimension of (F; D ) is O(n). Thus (1) generalizes the
n+log ?1
known bound of O
for learning halfspaces with respect to the uniform distribution [25],
matching a knownlower bound for this problem [23] up to a constant factor. Both upper bounds
n log 1 +log 1?
improve on the O
bound that follows from the traditional analysis; (2) is the first
such improvement for a polynomial-time algorithm.
Some previous analyses of the sample complexity of learning have made use of the fact that the
?metric dimension? [18] is at most the VC-dimension [11, 15]. Since using the doubling dimension
can sometimes lead to a better bound, a natural question is whether there is also a bound on the
doubling dimension in terms of the VC-dimension. We show that this is not the case: it is possible
to pack (1= )(1=2 o(1))d classifiers in a set F of VC-dimension d so that the distance between every
pair is in the interval [ ; 2 ]. Our analysis was inspired by some previous work in computational
geometry [19], but is simpler.
Combining our upper bound analysis with established techniques (see [33, 3, 8, 31, 30]), one can
perform similar analyses for the more general case in which no classifier in F has zero error. We
have begun with the PAC model because it is a clean setting in which to illustrate the power of
the doubling dimension for analyzing learning algorithms. The doubling dimension appears most
useful when the best achievable error rate (the Bayes error) is of the same order as the inverse of the
number of training examples (or smaller).
Bounding the doubling dimension is useful for analyzing the sample complexity of learning because
it limits the richness of a subclass of F near the classifier to be learned. For other analyses that
exploit bounds on such local richness, please see [31, 30, 5, 25, 26, 34]. It could be that stronger
results could be obtained by marrying the techniques of this paper with those. In any case, it appears
that the doubling dimension is an intuitive yet powerful way to bound the local complexity of a
collection of classifiers.
2 Preliminaries
2.1 Learning
For some domain X , an example consists of a member of X , and its classification in f0; 1g. A
classifier is a mapping from X to f0; 1g. A training set is a finite collection of examples. A learning
algorithm takes as input a training set, and outputs a classifier.
Suppose D is a probability distribution over X . Then define
D (f; g) = PrxD (f (x) 6= g(x)):
A learning algorithm A PAC learns F w.r.t. D with accuracy and confidence ? from m examples
if, for any f 2 F , if
domain elements x ; :::; xm are drawn independently at random according to D, and
(x ; f (x )); :::; (xm ; f (xm )) is passed to A, which outputs h,
1
1
then
1
Pr(D (f; h) > ) ?:
If F is a set of classifiers, a learning algorithm is a consistent hypothesis finder for F if it outputs an
element of F that correctly classifies all of the training data whenever it is possible to do so.
2.2 Metrics
Suppose = (Z; ) is a metric space.
An -cover for is a set T Z such that every element of
distance at most (with respect to ).
Z has a counterpart in T
that is at a
An -packing for is a set T
than (again, with respect to ).
Z such that every pair of elements of T
are at a distance greater
2 Z consists of all t 2 Z for which (z; t) .
Denote the size of the smallest -cover by N ( ; ). Denote the size of the largest -packing by
M( ; ).
The -ball centered at z
Lemma 1 ([18]) For any metric space = (Z; ), and any > 0,
M(2 ; ) N ( ; ) M( ; ):
The doubling dimension of is the least d such that, for all radii > 0, any -ball in can be
covered by at most 2d =2-balls. That is, for any > 0 and any z 2 Z , there is a C Z such that
jC j 2d, and
ft 2 Z : (z; t) g [c2C ft 2 Z : (c; t) =2g.
2.3 Probability
For a function and a probability distribution D, let ExD ( (x)) be the expectation of
Pm w.r.t. D.
1
We will shorten this to ED ( ), and if u = (u1 ; :::; um ) 2 X m , then Eu ( ) will be m
i=1 (ui ).
We will use PrxD , PrD , and Pru similarly.
3 The strongest upper bound
Theorem 2 Suppose
d is thedoubling dimension of
=?) examples.
learns F from O d+log(1
(F; D ).
There is an algorithm
A that PAC-
The key lemma limits the extent to which points that are separated from one another can crowd
around some point in a metric space with limited doubling dimension.
Lemma 3 (see [13]) Suppose = (Z; ) is a metric space with doubling dimension d and z
For > 0, let B (z; ) consist of the elements of u
centered at z ). Then
2 Z.
2 Z such that (u; z ) (that is, the -ball
d
M( ; B (z; )) 8 :
(In other words, any -packing must have at most (8 = )d elements within distance of z .)
Proof: Since has doubling dimension d, the set B (z; ) can be covered by 2d balls of radius =2.
Each of these can be covered by 2d balls of radius =4, and so on. Thus, B (z; ) can be covered by
2ddlog2 = e (4 = )d balls of radius =2. Applying Lemma 1 completes the proof.
Now we are ready to prove Theorem 2. The proof is an application of the peeling technique [1] (see
[30]).
Proof of Theorem 2: Construct an =4 packing G greedily, by repeatedly adding an element of F
to G for as long as this is possible. This packing is also an =4-cover, since otherwise we could add
another member to G.
Consider the algorithm that outputs the element of G with minimum error on the training set. What
ever the target, some element of G has error at most =4. Applying Chernoff bounds, O log(1 =?)
examples are sufficient that, with probability at least 1 ?=2, this classifier is incorrect on at most
a fraction =2 of the training data. Thus, the training error of the hypothesis output by A is at most
=2 with probability at least 1 ?=2.
Choose an arbitrary function f , and let S be the random training set resulting from drawing m examples according to D, and classifying them using f . Define S (g; h) to be the fraction of examples
in S on which g and h disagree. We have
Pr(9g 2 G; D (g; f ) > and S (g; f ) =2)
X=)
log(1
k=0
X=)
log(1
k=0
Pr(9g 2 G; 2k < D (g; f ) 2k+1 and S (g; f ) =2)
2(k+5)d e
2k
m=8
by Lemma 3 and the standard Chernoff bound.
Each of the following steps is a straightforward manipulation:
X=)
log(1
k=0
32d
2(k+5)d e
1
X
k=0
22k de
2k
m=8
2k
= 32d
m=8
X=)
log(1
2kd e
2k
m=8
k=0
d
64 e m=8 :
1 2de m=8
Since m = O((d + log(1=? ))=) is sufficient for
completes the proof.
64de
m=8
32d
X=)
log(1
k=0
22k de
?=2 and 2de
2k
m=8
m=8
1=2, this
4 A bound for consistent hypothesis finders
In this section we analyze algorithms that work by finding hypotheses with zero training error. This
is one way to achieve computational efficiency, as is the case when F consists of halfspaces. This
analysis will use the notion of VC-dimension.
Definition 4 The VC-dimension of a set F of f0; 1g-valued functions with a common domain is the
size of the largest set x1 ; :::; xd of domain elements such that
f(f (x1 ); :::; f (xd )) : f 2 F g = f0; 1gd:
The following lemma generalizes the Chernoff bound to hold uniformly over a class of random
variables; it concentrates on a simplified consequence of the Chernoff bound that is useful when
bounding the probability that an empirical estimate is much larger than the true expectation.
Lemma 5 (see [12, 24]) Suppose F is a set of f0; 1g-valued functions with a common domain X .
Let d be the VC-dimension of F . Let D be a probability distribution over X . Choose > 0 and
K 1. Then if
c(d log 1 + log 1? )
m
;
where c is an absolute constant, then
K log(1 + K )
PruDm (9f; g 2 F; PrD (f 6= g) but Pru (f 6= g) > (1 + K ) ) ?:
Now we are ready for the main analysis of this section.
Theorem 6 Suppose the doubling dimension of (F; D ) and the VC-dimension
of F are
both at most
q
d. Any consistent hypothesis finder for F
PAC learns F from O 1
Proof: Assume without loss of generality that
1=100, we have =8.
1=100.
Let
d log 1 + log 1?
= exp(
q
examples.
ln 1 ); since
Choose a target function f . For each h 2 F , define `h : X ! f0; 1g by `h (x) = 1 , h(x) 6= f (x).
Let `F = f`h : h 2 F g. Since `g (x) 6= `h (x) exactly when g (x) 6= h(x), the doubling dimension
of `F is the same as the doubling dimension of F ; the VC-dimension of `F is also known to be the
same as the VC-dimension of F (see [32]).
Construct an packing G greedily, by repeatedly adding an element of `F to G for as long as this
is possible. This packing is also an -cover.
2 `F , let (g) be its nearest neighbor in G. Since =8, by the triangle inequality,
ED (g) > and Eu(g) = 0 ) ED ( (g)) > 7=8 and Eu (g) = 0:
(3)
For each g
The triangle inequality also yields
Eu (g) = 0 ) (Eu ( (g)) =4 or Pru ( (g) 6= g) > =4):
Combining this with (3), we have we have
Pr(9g 2 `F ; ED (g) > but Eu (g) = 0)
Pr(9g 2 `F ; ED ( (g)) > 7=8 but Eu ( (g)) =4)
+Pr(9g 2 `F ; Pru ( (g) 6= g) > =4):
(4)
We have
Pr(9g 2 `F ; ED ( (g)) > 7=8 but Eu ( (g)) =4)
Pr(9g 2 G; ED (g) > 7=8 but Eu (g) =4)
= Pr(9g 2 G; D (f; g) > 7=8 but Pru (f 6= g) =4)
=
X
log(8 (7 ))
k=0
=
X
Pr(9g 2 G; 2k (7=8) < D (g; f ) 2k+1 (7=8) and Pru (f 6= g) =4)
log(8 (7 ))
82k+1
d
k
e c2 m;
k=0
where c > 0 is an absolute constant, by Lemma 3 and the standard Chernoff bound.
Computing a geometric sum exactly as in the proof of Theorem 2, we have that m = O(d=) suffices
for
d
Pr(9g 2 `F ; ED ( (g)) > 7=8 but Eu( (g)) =4) c1 e c2m ;
> 0.
By plugging in the value of and solving, we can see that
for absolute constants c1 ; c2
r
1 d log 1 + log 1
m=O
?
!!
suffices for
Pr(9g 2 `F ; PrD ( (g)) > 7=8 but Pru ( (g)) =4) ?=2:
(5)
Since PrD ( (g ) 6= g ) =8 for all g 2 `F , applying Lemma 5 with K = =(4 ) 1, we get
that there is an absolute constant c > 0 such that
c d log 1 + log 1?
m
(6)
(=4 ) log( 4 )
also suffices for
Pr(9g 2 `F ; Pru ( (g) 6= g) > =4) ?=2:
Substituting the value into (6), it is sufficient that
m
q
c d(log 1 + log 1 ) + log 1?
q
:
(=8) log 1 log 4
Putting this together with (5) and (4) completes the proof.
5 Halfspaces and the uniform distribution
Proposition 7 If Un is the uniform distribution over the unit ball in Rn , and Hn is the set of
halfspaces that go through the origin, then the doubling dimension of (Hn ; Un ) is O(n).
Proof: Choose h 2 Hn and > 0. We will show that the ball of radius
covered by O(n) balls of radius =2.
centered at h can be
Suppose UHn is the probability distribution over Hn obtained by choosing a normal vector w uniformly from the unit ball, and outputting fx : w x 0g. The argument will be a ?volume
argument? using UHn .
It is known (see Lemma 4 of [25]) that
PrgUHn (Un (g; h) =4) (c1 )n
1
> 0 is an absolute constant independent of and n. Furthermore,
PrgUHn (Un (g; h) 5 =4) (c2 )n 1
where c2 > 0 is another absolute constant.
Suppose we choose arbitrarily choose g1 ; g2 ; ::: 2 Hn that are at a distance at most from h, but
=2 far from one another. By the triangle inequality, =4-balls centered at g1 ; g2 ; ::: are disjoint.
Thus, the probability that an random element of Hn is in a ball of radius =4 centered at one of
g1 ; :::; gN is at least N (c1 )n 1 . On the other hand, since each g1 ; :::; gN has distance at most
from h, any element of an =4 ball centered at one of them is at most + =4 far from h. Thus,
the union of the =4 balls centered at g1 ; :::; gN is contained in the 5 =4 ball centered at h. Thus
N (c1 )n 1 (c2 )n 1 , which implies N (c2 =c1 )n 1 = 2O(n) , completing the proof.
where c1
6 Separation
Theorem 8 For all 2 [0; 1=2] and positive integers d there is a set F of classifiers and a probability distribution D over their common domain with the following properties:
the VC-dimension of F is d
jF j b e d= c
for each f; g 2 F , D (f; g) 2 .
1
2
2
1
2
This proof uses the probabilistic method. We begin with the following lemma.
Lemma 9 Choose positive integers s and d. Suppose A is chosen uniformly at random from among
the subsets of f1; :::; sg of size d. Then, for any B > 1,
Pr(jA \ f1; :::; dgj (1 + B )E (jA \ f1; :::; dgj)) 1 +e B
(1+B )E (jA\f1;:::;dgj)
:
Proof: in Appendix A.
Now we?re ready for the proof of Theorem 8, which uses the deletion technique (see [2]).
s d=2 c.
Proof (of Theorem 8): Set the domain X to be f1; :::; sg, where s = dd= e. Let N = b 2ed
Suppose f1 ; :::; fN are chosen independently, uniformly at random from among the classifiers that
evaluate to 1 on exactly d elements of X . For any distinct i; j , suppose fi 1 (1) is fixed, and we think
of the members of fj 1 (1) as being chosen one at a time. The probability that any of the elements
of fj 1 (1) is also in fi 1 (1) is d=s. Applying the linearity of expectation, and averaging over the
different possibilities for fi 1 (1), we get
E(jfi 1 (1) \ fj 1 (1)j) = ds :
2
Applying Lemma 9,
Pr(jfi (1) \ fj (1)j d=2)
1
1
=
2ed
( 2sd )( ds2 )
s
2ed d=2 :
s
Thus, the expected number of pairs i; j such that Pr(jfi 1 (1) \ fj
d=2
(N 2 =2) 2ed
. This implies that there exist f1 ; :::; fN such that
s
1
(1)j d=2)
jffi; j g : jfi (1) \ fj (1)j d=2gj (N =2) 2sed
1
1
2
d=2
is at most
:
If we delete one element from each such pair, and form G from what remains, then each pair g; h of
elements in G satisfies
jg 1 (1) \ h 1 (1)j < d=2:
(7)
f1; :::;
sg, then (7) implies D (g; h) > . The number of
d=
ed
elements of G is at least N (N =2) s
N=2.
Since each g 2 G has g (1) = d, no function in G evaluates to 1 on each element of any set of
If
D
is the uniform distribution over
2
2
2
1
d + 1 elements of X . Thus, the VC-dimension of G is at most d.
Theorem 8 implies that there is no bound on the doubling dimension of (G; D ) in terms of the
VC-dimension of G. For any constraint on the VC-dimension, a set G satisfying the constraint can
have arbitrarily large doubling dimension by setting the value of in Theorem 8 arbitrarily small.
Acknowledgement
We thank G?abor Lugosi and Tong Zhang for their help.
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A
Proof of Lemma 9
Definition 10 ([16]) A collection X1 ; :::; Xn of random variables are negatively associated if for every disjoint
jI j
pair I; J
1; :::; n of index sets, and for every pair f : R
R and g : RjJ j R of non-decreasing
functions, we have
f
g
!
!
E(f (Xi ; i 2 I )g(Xj ; j 2 J )) E(f (Xi ; i 2 I ))E(g(Xj ; j 2 J )):
Lemma 11 ([10]) If A is chosen uniformly at random from among the subsets of f1; :::; sg with exactly d
elements, and Xi = 1 if i 2 A and 0 otherwise, then X1 ; :::; Xs are negatively associated.
Lemma 12 ([9]) Collections
Pn X1 ; :::; XnQofn negatively associated random variables satisfy Chernoff bounds:
for any > 0, E(exp( i=1 Xi ))
i=1 E(exp(Xi )):
2f g
gj
2
Proof of Lemma 9: Let Xi
0; 1 indicate whether i
A. By Lemma 11, X1 ; :::; Xd are negatively
Pd
associated. We have A 1; :::; d = i=1 Xi . Combining Lemma 12 with a standard Chernoff-Hoeffding
bound (see Theorem 4.1 of [27]) completes the proof.
jf \f
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2,279 | 3,069 | Fundamental Limitations of Spectral Clustering
Boaz Nadler?, Meirav Galun
Department of Applied Mathematics and Computer Science
Weizmann Institute of Science, Rehovot, Israel 76100
boaz.nadler,[email protected]
Abstract
Spectral clustering methods are common graph-based approaches to clustering of
data. Spectral clustering algorithms typically start from local information encoded
in a weighted graph on the data and cluster according to the global eigenvectors of
the corresponding (normalized) similarity matrix. One contribution of this paper
is to present fundamental limitations of this general local to global approach. We
show that based only on local information, the normalized cut functional is not a
suitable measure for the quality of clustering. Further, even with a suitable similarity measure, we show that the first few eigenvectors of such adjacency matrices
cannot successfully cluster datasets that contain structures at different scales of
size and density. Based on these findings, a second contribution of this paper is
a novel diffusion based measure to evaluate the coherence of individual clusters.
Our measure can be used in conjunction with any bottom-up graph-based clustering method, it is scale-free and can determine coherent clusters at all scales. We
present both synthetic examples and real image segmentation problems where various spectral clustering algorithms fail. In contrast, using this coherence measure
finds the expected clusters at all scales.
Keywords: Clustering, kernels, learning theory.
1
Introduction
Spectral clustering methods are common graph-based approaches to (unsupervised) clustering of
data. Given a dataset of n points {xi }ni=1 ? Rp , these methods first construct a weighted graph
G = (V, W ), where the n points are the set of nodes V and the weighted edges Wi,j are computed
by some local symmetric and non-negative similarity measure. A common choice is a Gaussian
kernel with width ?, where k ? k denotes the standard Euclidean metric in Rp
kxi ? xj k2
Wi,j = exp ?
(1)
2? 2
In this framework, clustering is translated into a graph partitioning problem. Two main spectral
approaches for graph partitioning have been suggested. The first is to construct a normalized cut
(conductance) functional to measure the quality of a partition of the graph nodes V into k clusters[1,
2]. Specifically, for a 2-cluster partition V = S ? (V \ S) minimizing the following functional is
suggested in [1]
X
1
1
?(S) =
Wi,j
+
(2)
a(S) a(V \ S)
i?S,j?V \S
P
where a(S) = i?S,j?V Wi,j . While extensions of this functional to more than two clusters are
possible, both works suggest a recursive top-down approach where additional clusters are found by
?
Corresponding author. www.wisdom.weizmann.ac.il/?nadler
minimizing the same clustering functional on each of the two subgraphs. In [3] the authors also
propose to augment this top-down approach by a bottom-up aggregation of the sub-clusters.
As shown in [1] minimization ofP(2) is equivalent to maxy (y T W y)/(y T Dy), where D is a diagonal n ? n matrix with Di,i = j Wi,j , and y is a vector of length n that satisfies the constraints
y T D1 = 0 and y i ? {1, ?b} with b some constant in (0, 1). Since this maximization problem is
NP-hard, both works relax it by allowing the vector y to take on real values. This approximation
leads to clustering according to the eigenvector with second largest eigenvalue of the normalized
graph Laplacian, W y = ?Dy. We note that there are also graph partitioning algorithms based on a
non-normalized functional leading to clustering according to the second eigenvector of the standard
graph Laplacian matrix D ? W, also known as the Fiedler vector [4].
A second class of spectral clustering algorithms does not recursively employ a single eigenvector,
but rather proposes to map the original data into the first k eigenvectors of the normalized adjacency
matrix (or a matrix similar to it) and then apply a standard clustering algorithm such as k-means
on these new coordinates, see for example [5]-[11] and references therein. In recent years, much
theoretical work was done to justify this approach. Belkin and Niyogi [8] showed that for data uniformly sampled from a manifold, these eigenvectors approximate the eigenfunctions of the Laplace
Beltrami operator, which give an optimal low dimensional embedding under a certain criterion.
Optimality of these eigenvectors, including rotations, was derived in [9] for multiclass spectral clustering. Probabilistic interpretations, based on the fact that these eigenvectors correspond to a random
walk on the graph were also given by several authors [11]-[15]. Limitations of spectral clustering
in the presence of background noise and multiscale data were noted in [10, 16], with suggestions to
replace the uniform ? 2 in eq. (1) with a location dependent scale ?(xi )?(xj ).
The aim of this paper is to present fundamental limitations of spectral clustering methods, and
propose a novel diffusion based coherence measure to evaluate the internal consistency of individual clusters. First, in Section 2 we show that based on the isotropic local similarity measure (1),
the NP-hard normalized cut criterion may not be a suitable global functional for data clustering.
We construct a simple example with only two clusters, where we prove that the minimum of this
functional does not correspond to the natural expected partitioning of the data into its two clusters.
Further, in Section 3 we show that spectral clustering suffers from additional limitations, even with
a suitable similarity measure. Our theoretical analysis is based on the probabilistic interpretation
of spectral clustering as a random walk on the graph and on the intimate connection between the
corresponding eigenvalues and eigenvectors and the characteristic relaxation times and processes of
this random walk. We show that similar to Fourier analysis, spectral clustering methods are global
in nature. Therefore, even with a location dependent ?(x) as in [10], these methods typically fail to
simultaneously identify clusters at different scales. Based on this analysis, we present in Section 4
simple examples where spectral clustering fails. We conclude with Section 5, where we propose a
novel diffusion based coherence measure. This quantity measures the coherence of a set of points as
all belonging to a single cluster, by comparing the relaxation times on the set and on its suggested
partition. Its main use is as a decision tool whether to divide a set of points into two subsets or leave
it intact as a single coherent cluster. As such, it can be used in conjunction with either top-down or
bottom-up clustering approaches and may overcome some of their limitations. We show how use of
this measure correctly clusters the examples of Section 4, where spectral clustering fails.
2
Unsuitability of normalized cut functional with local information
As reported in the literature, clustering by approximate minimization of the functional (2) performs
well in many cases. However, a theoretical question still remains: Under what circumstances is
this functional indeed a good measure for the quality of clustering ? Recall that the basic goal of
clustering is to group together highly similar points while setting apart dissimilar ones. Yet this
similarity measure is typically based only on local information as in (1). Therefore, the question can
be rephrased - is local information sufficient for global clustering ?
While this local to global concept is indeed appealing, we show that it does not work in general. We
construct a simple example where local information is insufficient for correct clustering according
to the functional (2). Consider data sampled from a mixture of two densities in two dimensions
1
p(x) = p(x1 , x2 ) = [pL,? (x1 , x2 ) + pG (x1 , x2 )]
(3)
2
Normalized cut ? = 0.05
Original Data
3
3
2
2
1
1
0
0
?1
?1
?2
?2
2
4
(a)
6
2
4
(b)
6
Figure 1: A dataset with two clusters and result of normalized cut algorithm [2]. Other spectral
clustering algorithms give similar results.
where pL,? denotes uniform density in a rectangular region ? = {(x1 , x2 ) | 0 < x1 < L, ?? <
x2 < 0} of length L and width ?, and pG denotes a Gaussian density centered at (?1 , ?2 ) with
diagonal covariance matrix ?2 I. In fig. 1(a) a plot of n = 1400 points from this density is shown
with L = 8, ? = 0.05 L, (?1 , ?2 ) = (2, 0.2) and ? = 0.1. Clearly, the two clusters are the
Gaussian ball and the rectangular strip ?.
However, as shown in fig. 1(b), clustering based on the second eigenvector of the normalized graph
Laplacian with weights Wi,j given by (1) partitions the points somewhere along the long strip instead of between the strip and the Gaussian ball. We now show that this result is not due to the
approximation of the NP-hard problem but rather a feature of the original functional (2). Intuitively,
the failure of the normalized cut criterion is clear. Since the overlap between the Gaussian ball and
the rectangular strip is larger than the width of the strip, a cut that separates them has a higher penalty
than a cut somewhere along the thin strip.
To show this mathematically, we consider the penalty of the cut due to the numerator in (2) in the
limit of a large number of points n ? ?. In this population setting, as n ? ? each point has an
infinite number of neighbors, so we can consider the limit ? ? 0. Upon normalizing the similarity
measure (1) by 1/2?? 2 , the numerator is given by
Z Z
2
2
1 X X
1
Cut(?1 ) = lim
p(x)p(y)e?kx?y k /2? dxdy (4)
Wi,j =
2
n?? |V |
2?? ?1 ?2
x??1 y ??2
where ?1 , ?2 ? R2 are the regions of the two clusters. For ? L, a vertical cut of the strip at
location x = x1 far away from the ball (|x1 ? x0 | ?) gives
Z ?Z 0
1 1 ?(x?x0 )2 /2?2
1
e
dxdx0 =
(5)
Cut(x > x1 ) ' lim
2 2?? 2
2
??0 0
L
2?L
??
A similar calculation shows that for a horizontal cut at y = 0,
2
2
1 e??2 /2?
?
Cut(y > 0) '
L
8??
(6)
Finally, note that for a vertical cut far from the rectangle boundary ??, the denominators of the
two cuts in eq. (2) have the same order of magnitude. Therefore, if L ? and ?2 /? = O(1) the
horizontal cut between the ball and the strip has larger normalized penalty than a vertical cut of the
strip. This analysis explains the numerical results in fig. 1(b). Other spectral clustering algorithms
that use two eigenvectors, including those that take a local scale into account, also fail to separate
the ball from the strip and yield similar results to fig.1(b). A possible solution to this problem is to
introduce multiscale anisotropic features that capture the geometry and dimensionality of the data
in the similarity metric. In the context of image and texture segmentation, the need for multiscale
features is well known [17, 18, 19]. Our example highlights its importance in general data clustering.
3
Additional Limitations of Spectral Clustering Methods
An additional problem with recursive bi-partitioning is the need of a saliency criterion when required
to return k > 2 clusters. Consider, for example a dataset which contains k = 3 clusters. After
the first cut, the recursive algorithm should decide which subgraph to further partition and which
to leave intact. A common approach that avoids this decision problem is to directly find three
clusters by using the first three eigenvectors of W v = ?Dv. Specifically, denote by {?j , v j } the
set of eigenvectors of W v = ?Dv with eigenvalues sorted in decreasing order, and denote by
v j (xi ) the i-th entry (corresponding to the point xi ) in the j-th eigenvector v j . Many algorithms
propose to map each point xi ? Rp into ?(xi ) = (v 1 (xi ), . . . , v k (xi )) ? Rk , and apply simple
? j of
clustering algorithms to the points ?(xi ) [8, 9, 12]. Some works [6, 10] use the eigenvectors v
? j = D1/2 v j .
D?1/2 W D?1/2 instead, related to the ones above via v
We now show that spectral clustering that uses the first k eigenvectors for finding k clusters also
suffers from fundamental limitations. Our starting point is the observation that vj are also eigenvectors of the Markov matrix M = D?1 W [13, 12]. Assuming the graph is connected, the largest
eigenvalue is ?1 = 1 with |?j | < 1 for j > 1. Therefore, regardless of the initial condition
P the
random walk converges to the unique equilibrium distribution ?s , given by ?s (i) = Di,i / j Dj,j .
Moreover, as shown in [13], the Euclidean distance between points mapped to these eigenvectors is
equal to a so called ?diffusion distance? between points on the graph,
X
2
2
?2t
(7)
j (v j (x) ? v j (y)) = kp(z, t | x) ? p(z, t | y)kL2 (1/?s )
j
where p(z, t | x) is the probability distribution of a random walk at time t given that it started at
x, ?s is the equilibrium distribution, and k ? kL2 (w) is the weighted L2 norm with weight w(z).
Therefore, the eigenvalues and eigenvectors {?j , v j } for j > 1, capture the characteristic relaxation
times and processes of the random walk on the graph towards equilibrium. Since most methods use
the first few eigenvector coordinates for clustering, it is instructive to study the properties of these
relaxation times and of the corresponding eigenvectors.
We perform this analysis under the following statistical model: we assume that the points {xi } are
random samples from a smooth density p(x) in a smooth domain ? ? Rp . We write the density in
Boltzmann form p(x) = e?U (x)/2 and denote U (x) as the potential. As described in [13], in the
limit n ? ?, ? ? 0, the random walk with transition matrix M on the graph of points sampled
from this density converges to a stochastic differential equation (SDE)
?
?
?
x(t)
= ??U (x) + 2w(t)
(8)
where w(t) is standard white noise (Brownian motion), and the right eigenvectors of the matrix M
converge to the eigenfunctions of the following Fokker-Planck operator
L?(x) ? ?? ? ?? ? ?U = ???(x)
(9)
defined for x ? ? with reflecting boundary conditions on ??. This operator is non-positive and its
eigenvalues are ?1 = 0 < ?2 ? ?3 ? . . .. The eigenvalues ??j of L and the eigenvalues ?j of M
are related by ?j = limn??,??0 (1 ? ?j )/?. Therefore the top eigenvalues of M correspond to the
smallest of L. Eq. (7) shows that these eigenfunctions and eigenvalues capture the leading characteristic relaxation processes and time scales of the SDE (8). These have been studied extensively in
the literature [20], and can give insight into the success and limitations of spectral clustering [13].
For example, if ? = Rp and the density p(x) consists of k highly separated Gaussian clusters of
roughly equal size (k clusters), then there are exactly k eigenvalues very close or equal to zero, and
their corresponding eigenfunctions are approximately piecewise constant in each of these clusters.
Therefore, in this setting spectral clustering with k eigenvectors works very well.
To understand the limitations of spectral clustering, we now explicitly analyze situations with clusters at different scales of size and density. For example, consider a density with three isotropic
Gaussian clusters: one large cloud (cluster #1) and two smaller clouds (clusters 2 and 3). These correspond to one wide well and two narrow wells in the potential U (x). A representative 2-D dataset
drawn from such a density is shown in fig. 2 (top left).
The SDE (8) with this potential has a few characteristic time scales which determine the structure of
its leading eigenfunctions. The slowest one is the mean passage time between cluster 1 and clusters
2 or 3, approximately given by [20]
2?
?1,2 = p 00
e(U (xmax )?U (xmin ))
00
|Umin Umax
|
(10)
00
00
where xmin is the bottom of the deepest well, xmax is the saddle point of U (x), and Umin
, Umax
are the second derivatives at these points. Eq. (10), also known as Arrhenius or Kramers formula
of chemical reaction theory, shows that the mean first passage time is exponential in the barrier
height [20]. The corresponding eigenfunction ?2 is approximately piecewise constant inside the
large well and inside the two smaller wells with a sharp transition near the saddle point xmax . This
eigenfunction easily separates cluster 1 from clusters 2 and 3 (see top center panel in fig. 2).
A second characteristic time is ?2,3 , the mean first passage time between clusters 2 and 3, also given
by a formula similar to (10). If the potential barrier between these two wells is much smaller than
between wells 1 and 2, then ?2,3 ?1,2 . A third characteristic time is the equilibration time inside
cluster 1. To compute it we consider a diffusion process only inside cluster 1, e.g. with an isotropic
parabolic potential of the form U (x) = U (x1 )+U100 kx?x1 k2 /2, where x1 is the bottom of the well.
In 1-D the eigenvalues and eigenfunctions are given by ?k = (k ? 1)U100 , with ?k (x) a polynomial
of degree k ? 1. The corresponding intra-well relaxation times are given by ?kR = 1/?k+1 (k ? 1).
The key point in our analysis is that if the equilibration time inside the wide well is slower than
the mean first passage time between the two smaller wells, ?1R > ?2,3 , then the third eigenfunction
of L captures the relaxation process inside the large well and is approximately constant inside the
two smaller wells. This eigenfunction cannot separate between clusters 2 and 3. Moreover, if
?2R = ?1R /2 is still larger than ?2,3 then even the next leading eigenfunction captures the equilibration
process inside the wide well, see a plot of ?3 , ?4 in fig. 2 (rows 1,2). Therefore, even this next
eigenfunction is not useful for separating the two small clusters. In the example of fig. 2, only ?5
separates these two clusters.
This analysis shows that when confronted with clusters of different scales, corresponding to a multiscale landscape potential, standard spectral clustering which uses the first k eigenvectors to find k
clusters will fail. We present explicit examples in Section 4 below. The fact that spectral clustering
with a single scale ? may fail to correctly cluster multiscale data was already noted in [10, 16]. To
overcome this failure, [10] proposed replacing the uniform ? 2 in eq. (1) with ?(xi )?(xj ) where
?(x) is proportional to the local density at x. Our analysis can also provide a probabilistic interpretation to their method. In a nutshell, the effect of this scaling is to speed up the diffusion process at
regions of low density, thus changing some of its characteristic times. If the larger cluster has low
density, as in the examples in their paper, this approach is successful as it decreases ?1R . However, if
the large cluster has a high density (comparable to the density of the small clusters), this approach is
not able to overcome the limitations of spectral clustering, see fig. 3. Moreover, this approach may
also fail in the case of uniform density clusters defined solely by geometry (see fig. 4).
4
Examples
We illustrate the theoretical analysis of Section 3 with three examples, all in 2-D. In the first two examples, the n points {xi } ? R2 are random samples from the following mixture of three Gaussians
?1 N (x1 , ?12 I) + ?2 N (x2 , ?22 I) + ?3 N (x3 , ?32 I)
(11)
P
with centers xi isotropic standard deviations ?i and weights ?i ( i ?i = 1). Specifically, we
consider one large cluster with ?1 = 2 centered at x1 = (?6, 0), and two smaller clusters with
?2 = ?3 = 0.5 centered at x2 = (0, 0) and x3 = (2, 0). We present the results of both the NJW
algorithm [6] and the ZP algorithm [10] for two different weight vectors.
Example I: Weights (?1 , ?2 , ?3 ) = (1/3, 1/3, 1/3). In the top left panel of fig. 2, n = 1000
random points from this density clearly show the difference in scales between the large cluster and
the smaller ones. The first few eigenvectors of M with a uniform ? = 1 are shown in the first
two rows of the figure. The second eigenvector ?2 is indeed approximately piecewise constant and
easily separates the larger cluster from the smaller ones. However, ?3 and ?4 are constant on the
smaller clusters, capturing the relaxation process in the larger cluster (?3 captures relaxation along
the y-direction, hence it is not a function of the x-coordinate). In this example, only ?5 can separate
the two small clusters. Therefore, as predicted theoretically, the NJW algorithm [6] fails to produce
reasonable clusterings for all values of ?. In this example, the density of the large cluster is low, and
therefore as expected and shown in the last row of fig. 2, the ZP algorithm clusters correctly.
Example II: Weights (?1 , ?2 , ?3 ) = (0.8, 0.1, 0.1). In this case the density of the large cluster is
high, and comparable to that of the small clusters. Indeed, as seen in fig. 3 and predicted theoretically
?
Original Data
3
10
8
2
6
0
y
?
2
4
0
4
?2
2
?4
?10
0
?10
?5
0
?10
x
?5
0
?10
x
?4
NJW clustering, ?= 1
5
?5
0
x
?5
20
8
6
4
2
0
10
0
0
?5
?10
?5
?10
?10
0
?5
0
?10
x
ZP ? ?2
ZP clustering, kNN = 7
5
?5
?10
?5
0
0
x
ZP ? ?3
8
6
4
2
0
?2
8
6
4
2
0
?2
0
?5
?10
?5
x
0
?10
?5
x
0
Figure 2: A three cluster dataset corresponding to example I (top left), clustering results of NJW
and ZP algorithms [6, 10] (center and bottom left, respectively), and various eigenvectors of M vs.
the x coordinate (blue dots in 2nd and 3rd columns). The red dotted line is the potential U (x, 0).
y
ZP ? ?2
ZP results kNN = 7
Original Data
5
5
10
0
0
5
?5
?5
0
ZP ? ?3
10
5
0
?10
?5
0
x
5
?10
?5
0
5
?10
?5
x
0
?5
?10
?5
x
0
Figure 3: Dataset corresponding to example II and result of ZP algorithm.
the ZP algorithm fails to correctly cluster this data for all values of the parameter kN N in their
algorithm. Needless to say, the NJW algorithm also fails to correctly cluster this example.
Example III: Consider data {xi } uniformly sampled from a domain ? ? R2 , which consists of
three clusters, one a large rectangular container and two smaller disks, all connected by long and
narrow tubes (see fig. 4 (left)). In this example the container is so large that the relaxation time inside
it is slower than the characteristic time to diffuse between the small disks, hence NJW algorithm fails
to cluster correctly. Since density is uniform, the ZP algorithm fails as well, fig. 4 (right).
Note that spectral clustering with the eigenvectors of the standard graph Laplacian has similar limitations, since the Euclidean distance between these eigenvectors is equal to the mean commute time
on the graph [11]. Therefore, these methods may also fail when confronted with multiscale data.
5
Clustering with a Relaxation Time Coherence Measure
The analysis and examples of Sections 3 and 4 may suggest the use of more than k eigenvectors
in spectral clustering. However, clustering with k-means using 5 eigenvectors on the examples
of Section 4 produced unsatisfactory results (not shown). Moreover, since the eigenvectors of the
matrix M are orthonormal under a specific weight function, they become increasingly oscillatory.
Therefore, it is quite difficult to use them to detect a small cluster, much in analogy to Fourier
analysis, where it is difficult to detect a localized bump in a function from its Fourier coefficients.
ZP k
Original Data
NN
10
10
5
5
0
0
?5
?5
?10
?10
?10
?5
0
5
?10
10
?5
0
=7
5
10
Figure 4: Three clusters defined solely by geometry, and result of ZP clustering (Example III).
Original Image
Coherence Measure
Ncut with 4 clusters
(a)
(b)
(c)
Figure 5: Normalized cut and coherence measure segmentation on a synthetic image.
Based on our analysis, we propose a different approach to graph-based clustering. Given the importance of relaxation times on the graph as indication of clusters, we propose a novel and principled
measure for the coherence of a set of points as belonging to a single cluster. Our coherence measure can be used in conjunction with any clustering algorithm. Specifically, let G = (V, W ) be
a weighted graph of points and let V = S ? (V \ S) be a possible partition (computed by some
clustering algorithm). Our aim is to construct a meaningful measure to decide whether to accept
or reject this partition. To this end, let ?2 denote the second largest eigenvalue of the Markov
matrix M corresponding to the full graph G. We define ?V = 1/(1 ? ?2 ) as the characteristic
relaxation time of this graph. Similarly, ?1 and ?2 denote the characteristic relaxation times of the
two subgraphs corresponding to the partitions S and V \ S. If V is a single coherent cluster, then
we expect ?V = O(?1 + ?2 ). If, however, V consists of two weakly connected clusters defined
by S and V \ S, then ?1 and ?2 measure the characteristic relaxation times inside these two clusters while ?V measures the overall relaxation time. If the two sub-clusters are of comparable size,
then ?V (?1 + ?2 ). If however, one of them is much smaller than the other, then we expect
max(?1 , ?2 )/ min(?1 , ?2 ) 1. Thus, we define a set V as coherent if either ?V < c1 (?1 + ?2 ) or if
max(?1 , ?2 )/ min(?1 , ?2 ) < c2 . In this case, V is not partitioned further. Otherwise, the subgraphs
S and V \ S need to be further partitioned and similarly checked for their coherence. While a theoretical analysis is beyond the scope of this paper, reasonable numbers that worked in practice are
c1 = 1.8 and c2 = 10. We note that other works have also considered relaxation times for clustering
with different approaches [21, 22].
We now present use of this coherence measure with normalized cut clustering on the third example
of Section 4. The first partition of normalized cut on this data with ? = 1 separates between the large
container and the two smaller disks. The relaxation times of the full graph and the two subgraphs
are (?V , ?1 , ?2 ) = (1350, 294, 360). These numbers indicate that the full dataset is not coherent, and
indeed should be partitioned. Next, we try to partition the large container. Normalized cuts partitions
the container roughly into two parts with (?V , ?1 , ?2 ) = (294, 130, 135), which according to our
coherence measure means that the big container is a single structure that should not be split. Finally,
normalized cut on the two small disks correctly separates them giving (?V , ?1 , ?2 ) = (360, 18, 28),
which indicates that indeed the two disks should be split. Further analysis of each of the single disks
by our measure shows that each is a coherent cluster. Thus, combination of our coherence measure
with normalized cut not only clusters correctly, but also automatically finds the correct number of
clusters, regardless of cluster scale. Similar results are obtained for the other examples in this paper.
Finally, our analysis also applies to image segmentation. In fig. 5(a) a synthetic image is shown.
The segmentation results of normalized cuts [24] and of the coherence measure combined with
[23] appear in panels (b) and (c). Results on a real image are shown in fig. 6. Each segments is
Original Image
Coherence Measure
Ncut 6 clusters
Ncut 20 clusters
Figure 6: Normalized cut and coherence measure segmentation on a real image.
represented by a different color. With a small number of clusters normalized cut cannot find the
small coherent segments in the image, whereas with a large number of clusters, large objects are
segmented. Implementing our coherence measure with [23] finds salient clusters at different scales.
Acknowlegments: The research of BN was supported by the Israel Science Foundation (grant 432/06), by the
Hana and Julius Rosen fund and by the William Z. and Eda Bess Novick Young Scientist fund.
References
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[2] R. Kannan, S. Vempala, A. Vetta, On clusterings: good, bad and spectral, J. ACM, 51(3):497-515, 2004.
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SIGMOD/PODS, 2005.
[4] F.R.K. Chung, Spectral Graph Theory, Regional Conference Series in Mathematics Vol. 92, 1997.
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[6] A.Y. Ng, M.I. Jordan, Y. Weiss, On Spectral Clustering: Analysis and an algorithm, NIPS Vol. 14, 2002.
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14th Annual Machine Learning Conference of Belgium and the Netherlands, pp. 21-28, 2005.
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Comp. Vis. 43(1):7-27, 2001.
[18] E. Sharon, A. Brandt, R. Basri, Segmentation and Boundary Detection Using Multiscale Intensity Measurements, CVPR, 2001.
[19] M. Galun, E. Sharon, R. Basri and A. Brandt, Texture Segmentation by Multiscale Aggregation of Filter
Responses and Shape Elements, ICCV, 2003.
[20] C.W. Gardiner, Handbook of stochastic methods, third edition, Springer NY, 2004.
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NIPS, 2000.
[22] C. Chennubhotla, A.J. Jepson, Half-lives of eigenflows for spectral clustering, NIPS, 2002.
[23] E. Sharon, A. Brandt, R. Basri, Fast multiscale image segmentation, ICCV, 2000.
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2,280 | 307 | An Attractor Neural Network Model of Recall
and Recognition
Eytan Ruppin
Yechezkel Yeshurun
Department of Computer Science Department of Computer Science
School of Mathematical Sciences School of Mathematical Sciences
Sackler Faculty of Exact Sciences Sackler Faculty of Exact Sciences
Tel Aviv University
Tel Aviv University
69978, Tel Aviv, Israel
69978, Tel Aviv, Israel
Abstract
This work presents an Attractor Neural Network (ANN) model of Recall and Recognition. It is shown that an ANN model can qualitatively
account for a wide range of experimental psychological data pertaining
to the these two main aspects of memory access. Certain psychological
phenomena are accounted for, including the effects of list-length, wordfrequency, presentation time, context shift, and aging. Thereafter, the
probabilities of successful Recall and Recognition are estimated, in order
to possibly enable further quantitative examination of the model.
1
Motivation
The goal of this paper is to demonstrate that a Hopfield-based [Hop82] ANN model
can qualitatively account for a wide range of experimental psychological data pertaining to the two main aspects of memory access, Recall and Recognition. Recall
is defined as the ability to retrieve an item from a list of items (words) originally
presented during a previous learning phase, given an appropriate cue (cued RecalQ,
or spontaneously (free RecalQ. Recognition is defined as the ability to successfully
acknowledge that a certain item has or has not appeared in the tutorial list learned
before.
The main prospects of ANN modeling is that some parameter values, that in former,
'classical' models of memory retrieval (see e.g. [GS84]) had to be explicitly assigned,
can now be shown to be emergent properties of the model.
642
An Attractor Neural Network Model of Recall and Recognition
2
The Model
The model consists of a Hopfield ANN, in which distributed patterns representing
the learned items are stored during the learning phase, and are later presented as
inputs during the test phase. In this framework, successful Recall and Recognition
is defined. Some additional components are added to the basic Hopfield model to
enable the modeling of the relevant psychological phenomena.
2.1
The Hopfield Model
The Hopfield model's dynamics are composed of a non-linear, iterative, asynchronous transformation of the network state [Hop82]. The process may include
a stochastic noise which is analogous to the 'temperature' T in statistical mechanics. Formally, the Hopfield model is described as follows: Let neuron's i state be a
binary variable Si, taking the values ? 1 denoting a firing or a resting state, correspondingly. Let the network's state be a vector S specifying the binary values of
all its neurons. Let Jij be the synaptic strength between neurons i and j. Then,
hi, the input 'field' of neuron i is given by hi
L:f# JijSj. The neuron's dynamic
behavior is described by
=
Si (t
+ 1) = {
1,
-1,
with probability ~(1 + tgh( ~?
with probability ~(1 - tgh( ~?
Storing a new memory pattern eJ.' in the network is performed by modifying every
ij element of the syna.ptic connection matrix according to JlY w
Ji d + -keJ.' ieJ.' j.
= 1
A Hopfield network will always converge to a stable state, and every stored memory
is an attractor having an area surrounding it termed its basin of attraction [Hop82].
In addition to the stored memories, also other, non-memory states exist as stable
states (local minima) of the network [AGS85]. The maximal number m of (randomly
generated) memory patterns which can be stored in the basic Hopfield network of
n neurons is m = eke ? n, eke :::::: 0.14 [AGS85].
2.2
2.2.1
Recall and Recognition in the model's framework
Recall
Recall is considered successful when upon starting from an initial cue the network
converges to a stable state which corresponds to the learned memory nearest to
the input pattern. Inter-pattern distance is measured by the Hamming distance
between the input and the learned item encodings. If the network converges to a
non-memory stable state, its output will stand for a 'failure of recall' response. 1.
IThe question of "How do such non-memory states bear the meaning of 'recall failure'?"
is out of the scope of this work. However, a possible explanation is that during the learning
phase 'meaning' is assigned to the stored patterns via connections formed with external
patterns, and since non-memory states lack such associations with external patterns, they
are 'meaningless', yielding the 'recall failure' response. Another possible mechanism is that
every output pattern generated in the recall process passes also a recognition phase so that
non-memory states are rejected, (see the following paragraph describing recognition in our
model).
643
644
Ruppin and Yeshurun
2.2.2
Recognition
Recognition is considered successful when the network arrives at a stable state during a time interval A, beginning from input presentation. In general, the shorter
the distance between an input and its nearest memory, the faster is its convergence
[AM88, KP88, RY90]. Since non-memory (non-learned) stable states have higher
energy levels and much shallower basins of attraction than memorized stable states
[AGS85, LN89], convergence to such states takes significantly longer timer. Therefore, there exists a range of possible values of A that enable successful recognition
only of inputs similar to one of the stored memories.
2.3
Other features of the model
? The context of the psychological experiments is represented as a substring of
the input's encoding. In order to minimize inter-pattern correlation, the size of
the context encoding relative to the total size of the memory encoding is kept
small .
? The total associational linkage of a learned item, is modeled as an external
field vector E. When a learned memory pattern eJ.' is presented to the network,
the value of the external field vector generated is Ei
h . ~J.', where h is an
'orientation' coefficient, expressing the association strength.
=
Additional features, including a modified storage equation accounting for learning
taking place at the test phase, and a storage decay parameter, are described in
[RY90].
3
The Modeling of experimental data.
Regarding every phenomenon discussed, a brief description of the psychological
findings is followed by an account of its modeling. We rely on the known results
pertaining to Hopfield models to show that qualitatively, the psychological phenomena reviewed are emergent properties of the model. When such analytical evidence
is lacking, simulations were performed in order to account for the experimental
data. For a review of the psychological literature supporting the findings modeled
see [GS84].
The List-Length Effect: It is known that the probability of successful Recall or
Recognition of a particular item decreases as the length of list of learned items
lllcreases.
List length is expressed in memory load. Since It has been shown that the
width of the memories basins of attraction monotonically decreases following
an approximately inverse parabolic curve [Wei85], Recall performance should
decrease as memory load is increased. We have examined the convergence time
of the same set of input patterns at different values of memory load. As demonstrated in Fig. 1, it was found tha.t, as the memory load is increased, successful
convergence has occurred (on the average) only after an increasingly growing
number of asynchronous iterations. Hence, convergence takes more time and
can result in Recognition failure, although memories' stability is maintained
till the critical capacity a c is reached.
An Attractor Neural Network Model of Recall and Recognition
4000.0
3000.0
en
c
?2
0
2000.0
'-
~
1000.0
0.0 L - - - - ' - _ - - ' - - _ ' - - - - L _ '--......1.._....1....-----1_-'---1
10.0
20.0
30.0
40.0
50.0
60.0
Memory load
Figurc I: Ilccogllitioll speed (No. of a.<;Ylldll?OIlOIlS iterations) a.c; a (1Il1ction of IIlcllIory
load (No. of storcd memories). The nclwork's parameters arc n
500, T = 0.28
=
The word-frequency effect: The more frequent a word is in language, the probabilit.y of recalling it increases, while the probability of recognizing it decreases.
A word's frequency in the language is assumed to effect its retrieval through
the stored word's semantic relations and associa.tions [Kat85, NCBK87]. It
is assumed, that relative to low frequency words, high frequency words have
more semantic relations and therefore more connections between the patterns
representing them and other patterns stored in the memory (i.e., in other networks). This one-ta-many relationship is assumed to be reciprocal, i.e., each..~f
the externally stored patterns has also connections projected to several of th?:
stored patterns in the allocated network.
The process leading to the formation of the external field E (acting upon the
allocated network), generated by an input pattern n~arest to some stored memory pattern {IJ is assumed to be characterized as follows:
1. There is a threshold degree of overlap &ntin, such that E > 0 only when
the allocated network's state overiap H IJ is higher than Omin.
2. At overlap values HIJ which are only moderately larger than Omin, hI-' is
monotonically increasing, but as HI-' continues to rise, a certain 'optimal'
point is reached, beyond which h IJ is monotonically decreasing.
3. High-frequency words have lower Omin values than low-frequency words.
Recognition tests are characterized by a high initial value of overlap HI-', to
some memory {IJ. The value of h IJ and EJ.' generated is post-optimal and
therefore smaller than in the case of low-frequency words which have higher
Omin values.
In Recall tests the initial situation is characterized by low values of overlap HI-'
to some nearest memory {I-'. only the overlap value of high-frequency words is
sufficient for activating associated it.ems, i.e. HJ.' > Omin.
645
646
Ruppin and Yeshurun
Presentation Time: Increasing the presentation time of learned words is known
to improve both their Recall and Recognition.
This is explained by the phenomenon of maintenance rehearsal; The memories'
basins of attraction get deeper, since the 'energy' E of a given state equals
to 2.:;:1 H/J 2 ? Deeper basins of attraction are also wider [HFP83, KPKP90].
Therefore, the probability of successful Recall and Recognition of rehearsed
items is increased. The effect of a uniform rehearsal is equivalent to a temperature decrease. Hence, increasing presentation time will attenuate and delay the
List length phenomenon, till a certain limit. In a similar way, the Test Delay
phenomenon is accounted for [RY90].
Context Shift: The term Context Shift refers to the change in context from the
tutorial period to the test period. Studies examining the effect of context shift
have shown a decrement in Recall performance with context shift, but little
change in Recognition performance.
As demonstrated in [RY90], when a context shift is simulated by flipping some
of the context string's bits, Recall performance severely deteriorates while memories stability remains intact. No significant increase in the time (i.e. number of
asynchronous iterations) required for convergence was found , thus maintaining
the pre-shift probability of successful Recognition.
Age differences in Recall and Recognition: It was found that older people
perform more poorly on Recall tasks than they do on Recognition tasks [eM87].
These findings can be accounted for by the assumption that synapses are being weakened and deleted with aging, which although being controversial has
gained some experimental support (see [RY90]) . 'Ve have investigated the retrieval performance as a function of the input's initial overlap, various levels
of synaptic dilution, and memory load: As demonstrated in Fig. 2, when
the synaptic dilution is increased, a 'critical' phase is reached where memory
retrieval of far-away input patterns is decreased but the retrieval of input patterns with a high level of ihitial overlap remains intact. As the memory load
is increased, this 'critical' phase begins at lower levels of synaptic dilution. On
the other hand, only a mild increase (of 15%) in recognition speed was found.
?;-
'i.
u
1
?J
2
n.
.
'0
u
a:
Figure? Til ... I'I"I'AJ,ilily (If I'1I(rr;s"ful n:Lriev:ol ,~rfllrrllAtlre IlS II runcliotl or memory
lu,ullltlclll'e illput ,,,,II'CIII'I; iuitialll\'Crlal', lIt lwo clifTcJcuL cl ...grccs of "YIIAplic llilu.
liun; ~O% ill lhe right?si,lc-J figllle, AI"I 5J';f, in lIlc Idl?sitlc<1 figure. Tlrc nclwork's
parlllllc\crj; IIrC " = 500, l' = O.OS.
An Attractor Neural Network Model of Recall and Recognition
The interested reader can find a description of the modeling of additional phenomena, including test position, word fragment completion, and distractor similarity, in [RY90].
4
4.1
On a quantitative test of the model.
Estimating Recall performance
In a given network, with n neurons and m memories, the radius r of the basins
of attraction of the memories decreases as the memory load parameter (a = min)
is increased. According to [MPRV87], n, m, and r are related according to the
expression m = (1-2.r)l. n
4
logn'
The concept of the basins of attraction implies a non-linear probability function
with low probability when input vectors are further than the radius of attraction
and high probability otherwise. The slope of this non-linearity increases as the noise
level T is decreased.
The probability Pc that a random input vector will converge to one of the stored
memories can be estimated by Pc ~ l::~~ (~) . m. It is interesting to note that
the rates of change of r and of Pc have distinct forms; Recall tests beginning from
randomly generated cues would yield a very low rate of successful Recall (Pc). Yet,
if one examines Recall by picking a stored memory, flipping some of its encoding
bits, and presenting it as an input to the network (determining r), 'reasonable'
levels of successful Recall can still be obtained even when a 'considerable' number
of encoding bits are flipped. Pc can be also estimated by considering the context
representation [RY90].
4.2
Estimating Recognition performance
The probability of correct Recognition depends mainly on the the length of the
interval ~; assume that after an input pattern is presented to a network of n neurons,
during the time interval ~, k iterations steps of a Monte Carlo simulation are
performed: In each such step, a neuron is randomly selected, and then it examines
whether or not it should flip its state, according to its input.
We show that the probability Pg { d} that an input pattern will be successfully
recognized is is bounded by Pg {d} ~ 1 - d . e -nlc ? It can be seen that Recognition's
success depends strongly on the initial input proximity to a stored memory, and
even more strongly dependent on the number of allowed asynchronous iteration k,
determined by the length of~. For a selection of k = n(ln(d) + e), one obtains
Pg ~ 1-e- c . The expected number of iterations, (denoted as Exp( X? till successful
convergence is achieved is E(X) = l:t=l E(Xd = n . l:t=l ~ n .In(d).
+
In the more general case, Let 0 denote the Hamming distance (between the network's
state S and a stored memory) below which retrieval is considered successful. Then,
the corrected estimations of retrieval performance are Pg ~ 1 - (!) . e -~.o, and
E(X) ~ n .In( ~). In simulations we have performed, (n 500, d 20, 0
10), the
=
=
=
647
648
Ruppin and Yeshurun
average number of iterations until successful convergence was in the range of 300 400, in excellent correspondence with the predicted expectation, E(X) = 500?ln(2).
References
[AGS85]
[AM88]
[CM87]
[GS84]
[HFP83]
[Hop82]
[Kat85]
[KP88]
D. J. Amit, H. Gutfreund, and H. Sompolinsky. Storing infinite numbers
of patterns in a spin-glass model of neural networks. Phys. Rev. Lett.,
55:1530, 1985.
S. I. Amari and K. Maginu. Statistical neurodynamics of associative
memory. Neural Networks, 1:63, 1988.
F.I.M. Craik and J .M. McDowd. Age differences in recall and recognition. Journal of Experimental Psychology; Learning, Memory, and
Cognition, 13(3):474, 1987.
G. Gillund and M. Shiffrin. A retrieval model for both recognition and
recall. Psychological Review, 91:1, 1984.
J.J. Hopfield, D. I. Fienstien, and R. G. Palmer. Unlearning' has a
stabilizing effect in collective memories. Nature, 304:158, 1983.
J.J. Hopfield. Neural networks and physical systems with emergent
collective abilities. Proc. Nat. Acad. Sci. USA, 79:2554, 1982.
T. Kato. Semantic-memory sources of episodic retrieval failure. Memory
& Cognition, 13(5):442, 1985.
J. Komios and R. Paturi. Convergence results in an associative memory
model. Neural Networks, 1:239, 1988.
[KPKP90] B. Kagmar-Parsi and B. Kagmar-Parsi. On problem solving with hopfield neural networks. Bioi. Cybern., 62:415, 1990.
M. Lewenstein and A. Nowak. Fully connected neural networks with
[LN89]
self-control of noise levels. Phys. Rev. Lett., 62(2):225, 1989.
[MPRV87] R.J. McEliece, E.C. Posner, E.R. Rodemich, and S.S. Venkatesh. The
capacity of the hopfield associative memory. IEEE Transactions on
Information theory, IT-33( 4):461, 1987.
[NCBK87] D.L Nelson, J.J. Canas, M.T. Bajo, and P.D. Keelan. Comparing word
fragment completion and cued recall with letter cues. Journal of Experimental Psychology: Learning, Memory and Cognition, 13(4) :542, 1987.
E. Ruppin and Y. Yeshurun. Recall and recognition in an attractor
[RY90]
neural network model of memory retrieval. Technical report, Dept. of
Computer Science, Tel-Aviv University, 1990.
G. Weisbuch . Scaling laws for the attractors of hopfield networks. J.
[Wei85]
Physique Lett., 46:L-623, 1985.
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2,281 | 3,070 | Information Bottleneck Optimization and
Independent Component Extraction with Spiking
Neurons
Stefan Klampfl, Robert Legenstein, Wolfgang Maass
Institute for Theoretical Computer Science
Graz University of Technology
A-8010 Graz, Austria
{klampfl,legi,maass}@igi.tugraz.at
Abstract
The extraction of statistically independent components from high-dimensional
multi-sensory input streams is assumed to be an essential component of sensory
processing in the brain. Such independent component analysis (or blind source
separation) could provide a less redundant representation of information about the
external world. Another powerful processing strategy is to extract preferentially
those components from high-dimensional input streams that are related to other
information sources, such as internal predictions or proprioceptive feedback. This
strategy allows the optimization of internal representation according to the information bottleneck method. However, concrete learning rules that implement these
general unsupervised learning principles for spiking neurons are still missing. We
show how both information bottleneck optimization and the extraction of independent components can in principle be implemented with stochastically spiking
neurons with refractoriness. The new learning rule that achieves this is derived
from abstract information optimization principles.
1
Introduction
The Information Bottleneck (IB) approach and independent component analysis (ICA) have both
attracted substantial interest as general principles for unsupervised learning [1, 2]. A hope has been,
that they might also help us to understand strategies for unsupervised learning in biological systems.
However it has turned out to be quite difficult to establish links between known learning algorithms
that have been derived from these general principles, and learning rules that could possibly be implemented by synaptic plasticity of a spiking neuron. Fortunately, in a simpler context a direct link
between an abstract information theoretic optimization goal and a rule for synaptic plasticity has
recently been established [3]. The resulting rule for the change of synaptic weights in [3] maximizes the mutual information between pre- and postsynaptic spike trains, under the constraint that
the postsynaptic firing rate stays close to some target firing rate. We show in this article, that this
approach can be extended to situations where simultaneously the mutual information between the
postsynaptic spike train of the neuron and other signals (such as for example the spike trains of other
neurons) has to be minimized (Figure 1). This opens the door to the exploration of learning rules
for information bottleneck analysis and independent component extraction with spiking neurons that
would be optimal from a theoretical perspective.
We review in section 2 the neuron model and learning rule from [3]. We show in section 3 how this
learning rule can be extended so that it not only maximizes mutual information with some given
spike trains and keeps the output firing rate within a desired range, but simultaneously minimizes
mutual information with other spike trains, or other time-varying signals. Applications to infor-
A
B
Figure 1: Different learning situations analyzed in this article. A In an information bottleneck task
the learning neuron (neuron 1) wants to maximize the mutual information between its output Y1K
and the activity of one or several target neurons Y2K , Y3K , . . . (which can be functions of the inputs
X K and/or other external signals), while at the same time keeping the mutual information between
the inputs X K and the output Y1K as low as possible (and its firing rate within a desired range). Thus
the neuron should learn to extract from its high-dimensional input those aspects that are related to
these target signals. This setup is discussed in sections 3 and 4. B Two neurons receiving the
same inputs X K from a common set of presynaptic neurons both learn to maximize information
transmission, and simultaneously to keep their outputs Y1K and Y2K statistically independent. Such
extraction of independent components from the input is described in section 5.
mation bottleneck tasks are discussed in section 4. In section 5 we show that a modification of
this learning rule allows a spiking neuron to extract information from its input spike trains that is
independent from the component extracted by another neuron.
2
Neuron model and a basic learning rule
We use the model from [3], which is a stochastically spiking neuron model with refractoriness,
where the probability of firing in each time step depends on the current membrane potential and the
time since the last output spike. It is convenient to formulate the model in discrete time with step
size ?t. The total membrane potential of a neuron i in time step tk = k?t is given by
ui (tk ) = ur +
N X
k
X
wij ?(tk ? tn )xnj ,
(1)
j=1 n=1
where ur = ?70mV is the resting potential and wij is the weight of synapse j (j = 1, . . . , N ).
An input spike train at synapse j up to the k-th time step is described by a sequence Xjk =
(x1j , x2j , . . . , xkj ) of zeros (no spike) and ones (spike); each presynaptic spike at time tn (xnj = 1)
evokes a postsynaptic potential (PSP) with exponentially decaying time course ?(t ? tn ) with time
constant ?m = 10ms. The probability ?ki of firing of neuron i in each time step tk is given by
?ki = 1 ? exp[?g(ui (tk )Ri (tk )?t] ? g(ui (tk ))Ri (tk )?t,
(2)
where g(u) = r0 log{1 + exp[(u ? u0 )/?u]} is a smooth increasing function of the membrane
potential u (u0 = ?65mV, ?u = 2mV, r0 = 11Hz). The approximation is valid for sufficiently
t?i ??abs )2
small ?t (?ki ? 1). The refractory variable Ri (t) = ? 2 (t?
?(t ? t?i ? ?abs ) assumes
+(t?t? ??
)2
ref r
i
abs
values in [0, 1] and depends on the last firing time t?i of neuron i (absolute refractory period ?abs =
3ms, relative refractory time ?ref r = 10ms). The Heaviside step function ? takes a value of 1 for
non-negative arguments and 0 otherwise.
This model from [3] is a special case of the spike-response model, and with a refractory variable
R(t) that depends only on the time since the last postsynaptic event it has renewal properties [4].
The output of neuron i at the k-th time step is denoted by a variable yik that assumes the value 1 if a
postsynaptic spike occurred and 0 otherwise. A specific spike train up to the k-th time step is written
as Yik = (yi1 , yi2 , . . . , yik ).
The information transmission between an ensemble of input spike trains XK and the output spike
train YiK can be quantified by the mutual information1 [5]
X
P (YiK |X K )
.
(3)
I(XK ; YiK ) =
P (X K , YiK ) log
P (YiK )
K
K
X
,Yi
The idea in [3] was to maximize the quantity I(XK ; YiK ) ? ?DKL (P (YiK )||P? (YiK )), where
P
K
K
? K
DKL (P (YiK )||P? (YiK )) =
YiK P (Yi ) log(P (Yi )/P (Yi )) denotes the Kullback-Leibler divergence [5], imposing the additional constraint that the firing statistics P (Yi ) of the neuron should
stay as close as possible to a target distribution P? (Yi ). This distribution was chosen to be that of a
constant target firing rate g? accounting for homeostatic processes. An online learning-rule performk
ing gradient ascent on this quantity was derived for the weights wij of neuron i, with ?wij
denoting
the weight change during the k-th time step:
k
?wij
k k
= ?Cij
Bi (?),
(4)
?t
k
k
which consists of the ?correlation term? Cij
and the ?postsynaptic term? Bik [3]. The term Cij
measures coincidences between postsynaptic spikes at neuron i and PSPs generated by presynaptic
action potentials arriving at synapse j,
? X
?
k
?
g 0 (u1 (tk )) ? k
?t
k?1
k
+
?(tk ? tn )xnj
y1 ? ?k1 ,
(5)
C1j = C1j
1?
k
?C
g(u1 (t ))
n=1
in an exponential time window with time constant ?C = 1s and g 0 (ui (tk )) denoting the derivative
of g with respect to u. The term
?
?
?? ?
g(u1 (tk ))
g?
yk
B1k (?) = 1 log
?t
g?1 (tk )
g?1 (tk )
?
?
? (1 ? y1k )R1 (tk ) g(u1 (tk )) ? (1 + ?)?
g1 (tk ) + ??
g ,
(6)
compares the current firing rate g(ui (tk )) with its average firing rate2 g?i (tk ), and simultaneously
the running average g?i (tk ) with the constant target rate g?. The argument indicates that this term also
depends on the optimization parameter ?.
3
Learning rule for multi-neuron interactions
We extend the learning rule presented in the previous section to a more complex scenario, where the
mutual information between the output spike train Y1K of the learning neuron (neuron 1) and some
target spike trains YlK (l > 1) has to be maximized, while simultaneously minimizing the mutual
information between the inputs X K and the output Y1K . Obviously, this is the generic IB scenario
applied to spiking neurons (see Figure 1A). A learning rule for extracting independent components
with spiking neurons (see section 5) can be derived in a similar manner.
For simplicity, we consider the case of an IB optimization for only one target spike train Y2K , and
derive an update rule for the synaptic weights w1j of neuron 1. The quantity to maximize is therefore
L = ?I(XK ; Y1K ) + ?I(Y1K ; Y2K ) ? ?DKL (P (Y1K )||P? (Y1K )),
(7)
where ? and ? are optimization constants. To maximize this objective function, we derive the weight
k
change ?w1j
during the k-th time step by gradient ascent on (7), assuming that the weights w1j can
change between some bounds 0 ? w1j ? wmax (we assume wmax = 1 throughout this paper).
1
We use boldface letters (Xk ) to distinguish random variables from specific realizations (X k ).
The rate g?i (tk ) = hg(ui (tk ))iXk |Y k?1 denotes an expectation of the firing rate over the input distribution
i
given the postsynaptic history and is implemented as a running average with an exponential time window (with
a time constant of 10ms).
2
Note that all three terms of (7) implicitly depend on w1j because the output distribution P (Y1K )
changes if we modify the weights w1j . Since the first and the last term of (7) have already been
considered (up to the sign) in [3], we will concentrate here on the middle term L12 := ?I(Y1K ; Y2K )
k
and denote the contribution of the gradient of L12 to the total weight change ?w1j
in the k-th time
k
step by ?w
?1j .
In order to get an expression for the weight change in a specific time step tk we write the probabilities
P (YiK ) and P (Y1K , Y2K ) occurring in (7) as products over individual time bins, i.e., P (YiK ) =
QK
QK
k?1
k
) and P (Y1K , Y2K ) = k=1 P (y1k , y2k |Y1k?1 , Y2k?1 ), according to the chain rule
k=1 P (yi |Yi
of information theory [5]. Consequently, we rewrite L12 as a sum over the contributions of the
PK
individual time bins, L12 = k=1 ?Lk12 , with
+
*
k?1
k?1
k k
P
(y
,
y
|Y
,
Y
)
1
2 1
2
.
(8)
?Lk12 = ? log
P (y1k |Y1k?1 )P (y2k |Y2k?1 ) Xk ,Yk ,Yk
1
2
k
The weight change ?w
?1j
is then proportional to the gradient of this expression with respect to the
k
weights w1j , i.e., ?w
?1j = ?(??Lk12 /?w1j ), with some learning rate ? > 0. The evaluation of the
? k
?
k
k
k
gradient yields ?w
?1j = ? C1j
?F12
with a correlation term C1j
as in (5) and a term
Xk ,Y1k ,Y2k
?
?
g?12 (tk )
g?12 (tk )
k
k
k
k
?
g
?
(t
)
?
?
y
(1
?
y
)R
(t
)?t
2
2
1
2
g?1 (tk )?
g?1 (tk )
g2 (tk )
?
?
g?12 (tk )
k
?
g
?
(t
)
+
? (1 ? y1k )y2k R1 (tk )?t
1
g?2 (tk )
?
?
+ (1 ? y1k )(1 ? y2k )R1 (tk )R2 (tk )(?t)2 g?12 (tk ) ? g?1 (tk )?
g2 (tk ) .
k
F12
= y1k y2k log
(9)
Here, g?i (tk ) = hg(ui (tk ))iXk |Y k?1 denotes the average firing rate of neuron i and g?12 (tk ) =
i
hg(u1 (tk ))g(u2 (tk ))iXk |Y k?1 ,Y k?1 denotes the average product of firing rates of both neurons. Both
2
1
quantities are implemented online as running exponential averages with a time constant of 10s.
Under the assumption of a small learning rate ? we can approximate the expectation h?iXk ,Y1k ,Y2k
by averaging over a single long trial. Considering now all three terms in (7) we finally arrive at an
online rule for maximizing (7)
k
? k
?
?w1j
k
k
= ??C1j
B1 (??) ? ??tB12
.
?t
(10)
k
which consists of a term C1j
sensitive to correlations between the output of the neuron and its
k
that characterize the
presynaptic input at synapse j (?correlation term?) and terms B1k and B12
postsynaptic state of the neuron (?postsynaptic terms?). Note that the argument of B1k is different
from (4) because some of the terms of the objective function (7) have a different sign. In order to
k
to have
compensate the effect of a small ?t, the constant ? has to be large enough for the term B12
an influence on the weight change.
k
The factors C1j
and B1k were described in the previous section. In addition, our learning rule
k
k
/(?t)2 that is sensitive to the statistical dependence between
contains an extra term B12
= F12
the output spike train of the neuron and the target. It is given by
?
?
g?12 (tk )
g?12 (tk )
yk yk
y1k
k
k
k
k
(1
?
y
)R
(t
)
?
g
?
(t
)
B12
= 1 22 log
?
2
2
2
(?t)
g?1 (tk )?
g?1 (tk )
g2 (tk ) ?t
?
?
g?12 (tk )
yk
? g?1 (tk )
? 2 (1 ? y1k )R1 (tk )
?t
g?2 (tk )
?
?
+ (1 ? y1k )(1 ? y2k )R1 (tk )R2 (tk ) g?12 (tk ) ? g?1 (tk )?
g2 (tk ) .
(11)
This term basically compares the average product of firing rates g?12 (which corresponds to the joint
probability of spiking) with the product of average firing rates g?1 g?2 (representing the probability
of independent spiking). In this way, it measures the momentary mutual information between the
output of the neuron and the target spike train.
For a simplified neuron model without refractoriness (R(t) = 1), the update rule (4) resembles the
BCM-rule [6] as shown in [3]. With the objective function (7) to maximize, we expect an ?antiHebbian BCM? rule with another term accounting for statistical dependencies between Y1K and
Y2K . Since there is no refractoriness, the postsynaptic rate ?1 (tk ) is given directly by the current
value of g(u(tk )), and the update rule (10) reduces to the rate model3
? ? k ? k ?? ?
k
?w1j
??1
?
= ???jpre,k f (?1k ) log 1k
?t
g?
??1
? k
???
?
? k ?
??12
??12
k
k
?1
, (12)
???t ?2 log k k ? ??2
??1 ??2
??1k ??2k
Pk
where the presynaptic rate at synapse j at time tk is denoted by ?jpre,k = a n=1 ?(tk ? tn )xnj
k
with a in units (Vs)?1 . The values ??1k , ??2k , and ??12
are running averages of the output rate ?1k , the
k
rate of the target signal ?2 and of the product of these values, ?1k ?2k , respectively. The function
f (?1k ) = g 0 (g ?1 (?1k ))/a is proportional to the derivative of g with respect to u, evaluated at the
current membrane potential. The first term in the curly brackets accounts for the homeostatic process
(similar to the BCM rule, see [3]), whereas the second term reinforces dependencies between Y1K
and Y2K . Note that this term is zero if the rates of the two neurons are independent.
It is interesting to note that if we rewrite the simplified rate-based learning rule (12) in the following
way,
k
?w1j
= ???jpre,k ?(?1k , ?2k ),
(13)
?t
we can view it as an extension of the classical Bienenstock-Cooper-Munro (BCM) rule [6] with a
two-dimensional synaptic modification function ?(?1k , ?2k ). Here, values of ? > 0 produce LTD
whereas values of ? < 0 produce LTP. These regimes are separated by a sliding threshold, however,
in contrast to the original BCM rule this threshold does not only depend on the running average of
the postsynaptic rate ??1k , but also on the current values of ?2k and ??2k .
4
Application to Information Bottleneck Optimization
We use a setup as in Figure 1A where we want to maximize the information which the output Y1K of
a learning neuron conveys about two target signals Y2K and Y3K . If the target signals are statistically
independent from each other we can optimize the mutual information to each target signal separately.
This leads to an update rule
k
? k
? k
??
?w1j
k
k
= ??C1j
B1 (??) ? ??t B12
+ B13
,
?t
(14)
k
k
are the postsynaptic terms (11) sensitive to the statistical dependence between
and B13
where B12
the output and target signals 1 and 2, respectively. We choose g? = 30Hz for the target firing rate,
and we use discrete time with ?t = 1ms.
In this experiment we demonstrate that it is possible to consider two very different kinds of target
signals: one target spike train has has a similar rate modulation as one part of the input, while the
other target spike train has a high spike-spike correlation with another part of the input. The learning
neuron receives input at 100 synapses, which are divided into 4 groups of 25 inputs each. The first
two input groups consist of rate modulated Poisson spike trains4 (Figure 2A). Spike trains from the
remaining groups 3 and 4 are correlated with a coefficient of 0.5 within each group, however, spike
trains from different groups are uncorrelated. Correlated spike trains are generated by the procedure
described in [7].
The first target signal is chosen to have the same rate modulation as the inputs from group 1, except
that Gaussian random noise is superimposed with a standard deviation of 2Hz. The second target
spike train is correlated with inputs from group 3 (with a coefficient of 0.5), but uncorrelated to
inputs from group 4. Furthermore, both target signals are silent during random intervals: at each
3
In the absence of refractoriness we use an alternative gain function galt (u) = [1/gmax + 1/g(u)]?1 in
order to pose an upper limit of gmax = 100Hz on the postsynaptic firing rate.
C
evolution of weights
0
0
2500
5000
50
0
0
2500
t [ms]
D
20
40
0.5
60
80
100
5000
1
20
40
t [min]
60
0
E
?3
0.04
50
0
0
2500
5000
2500
t [ms]
5000
50
0
0
F
MI/KLD of neuron 1
0.01
target 1 [Hz] output [Hz]
B
50
synapse idx
input 2 [Hz]
input 1 [Hz]
A
1
x 10
I(output;targets)
correlation with targets
0.5
0.4
0.02
0.005
target 1
target 2
0.3
0.5
0.2
0.1
0
0
20
40
t [min]
0
60
0
0
20
40
t [min]
60
0
0
20
40
60
t [min]
Figure 2: Performance of the spike-based learning rule (10) for the IB task. A Modulation of input
rates to input groups 1 and 2. B Evolution of weights during 60 minutes of learning (bright: strong
synapses, wij ? 1, dark: depressed synapses, wij ? 0.) Weights are initialized randomly between
0.10 and 0.12, ? = 10?4 , ? = 2 ? 103 , ? = 50. C Output rate and rate of target signal 1 during 5
seconds after learning. D Evolution of the average mutual information per time bin (solid line, left
scale) between input and output and the Kullback-Leibler divergence per time bin (dashed line, right
scale) as a function of time. Averages are calculated over segments of 1 minute. E Evolution of the
average mutual information per time bin between output and both target spike trains as a function
of time. F Trace of the correlation between output rate and rate of target signal 1 (solid line) and
the spike-spike correlation (dashed line) between the output and target spike train 2 during learning.
Correlation coefficients are calculated every 10 seconds.
time step, each target signal is independently set to 0 with a certain probability (10?5 ) and remains
silent for a duration chosen from a Gaussian distribution with mean 5s and SD 1s (minimum duration
is 1s). Hence this experiment tests whether learning works even if the target signals are not available
all of the time.
Figure 2 shows that strong weights evolve for the first and third group of synapses, whereas the
efficacies for the remaining inputs are depressed. Both groups with growing weights are correlated
with one of the target signals, therefore the mutual information between output and target spike trains
increases. Since spike-spike correlations convey more information than rate modulations synaptic
efficacies develop more strongly to group 3 (the group with spike-spike correlations). This results in
an initial decrease in correlation with the rate-modulated target to the benefit of higher correlation
with the second target. However, after about 30 minutes when the weights become stable, the
correlations as well as the mutual information quantities stay roughly constant.
An application of the simplified rule (12) to the same task is shown in Figure 3 where it can be
seen that strong weights close to wmax are developed for the rate-modulated input. To some extent
weights grow also for the inputs with spike-spike correlations in order to reach the constant target
firing rate g?. In contrast to the spike-based rule the simplified rule is not able to detect spike-spike
correlations between output and target spike trains.
4
The rate of the first 25 inputs is modulated by a Gaussian white-noise signal with mean 20Hz that has been
low pass filtered with a cut-off frequency of 5Hz. Synapses 26 to 50 receive a rate that has a constant value
of 2Hz, except that a burst is initiated at each time step with a probability of 0.0005. Thus there is a burst on
average every 2s. The duration of a burst is chosen from a Gaussian distribution with mean 0.5s and SD 0.2s,
the minimum duration is chosen to be 0.1s. During a burst the rate is set to 50Hz. In the simulations we use
discrete time with ?t = 1ms.
A
B
synapse idx
evolution of weights
0.5
3
0.03
0.4
2
0.02
0.3
1
0.01
0.2
4
20
0.8
40
0.6
60
0.4
80
MI/KLD of neuron 1
0.04
1
100
C
?3
x 10
correlation with target 1
0.2
10
20
t [min]
30
0
0
10
20
0
30
0.1
0
10
t [min]
20
30
t [min]
Figure 3: Performance of the simplified update rule (12) for the IB task. A Evolution of weights
during 30 minutes of learning (bright: strong synapses, wij ? 1, dark: depressed synapses, wij ?
0.) Weights are initialized randomly between 0.10 and 0.12, ? = 10?3 , ? = 104 , ? = 10. B
Evolution of the average mutual information per time bin (solid line, left scale) between input and
output and the Kullback-Leibler divergence per time bin (dashed line, right scale) as a function of
time. Averages are calculated over segments of 1 minute. C Trace of the correlation between output
rate and target rate during learning. Correlation coefficients are calculated every 10 seconds.
5
Extracting Independent Components
With a slight modification in the objective function (7) the learning rule allows us to extract statistically independent components from an ensemble of input spike trains. We consider two neurons
receiving the same input at their synapses (see Figure 1B). For both neurons i = 1, 2 we maximize
information transmission under the constraint that their outputs stay as statistically independent from
each other as possible. That is, we maximize
? i = I(XK ; YiK ) ? ?I(Y1K ; Y2K ) ? ?DKL (P (YiK )||P? (YiK )).
L
(15)
Since the same terms (up to the sign) are optimized in (7) and (15) we can derive a gradient ascent
rule for the weights of neuron i, wij , analogously to section 3:
k
?
? k
?wij
k
k
= ?Cij
.
Bi (?) ? ??tB12
?t
(16)
Figure 4 shows the results of an experiment where two neurons receive the same Poisson input with a
rate of 20Hz at their 100 synapses. The input is divided into two groups of 40 spike trains each, such
that synapses 1 to 40 and 41 to 80 receive correlated input with a correlation coefficient of 0.5 within
each group, however, any spike trains belonging to different input groups are uncorrelated. The
remaining 20 synapses receive uncorrelated Poisson input. Weights close to the maximal efficacy
wmax = 1 are developed for one of the groups of synapses that receives correlated input (group 2 in
this case) whereas those for the other correlated group (group 1) as well as those for the uncorrelated
group (group 3) stay low. Neuron 2 develops strong weights to the other correlated group of synapses
(group 1) whereas the efficacies of the second correlated group (group 2) remain depressed, thereby
trying to produce a statistically independent output. For both neurons the mutual information is
maximized and the target output distribution of a constant firing rate of 30Hz is approached well.
After an initial increase in the mutual information and in the correlation between the outputs, when
the weights of both neurons start to grow simultaneously, the amounts of information and correlation
drop as both neurons develop strong efficacies to different parts of the input.
6
Discussion
Information Bottleneck (IB) and Independent Component Analysis (ICA) have been proposed as
general principles for unsupervised learning in lower cortical areas, however, learning rules that
can implement these principles with spiking neurons have been missing. In this article we have
derived from information theoretic principles learning rules which enable a stochastically spiking
neuron to solve these tasks. These learning rules are optimal from the perspective of information
theory, but they are not local in the sense that they use only information that is available at a single
A
B
C
weights of neuron 1
?4
weights of neuron 2
20
40
0.5
60
80
100
10
20
t [min]
30
0.5
60
2
80
10
20
t [min]
30
30
correlation between outputs
0.016
0.04
0.012
0.03
0.012
0.03
0.008
0.02
0.008
0.02
0.004
0.01
0.004
0.01
0
30
20
F
MI/KLD of neuron 2
0.04
20
10
t [min]
0.016
t [min]
0
0
0
E
10
I(output 1;output2)
4
40
MI/KLD of neuron 1
0
0
x 10
20
100
0
D
6
1
synapse idx
synapse idx
1
0
0
10
20
t [min]
0
30
0.6
0.4
0.2
0
0
10
20
30
t [min]
Figure 4: Extracting independent components. A,B Evolution of weights during 30 minutes of
learning for both postsynaptic neurons (red: strong synapses, wij ? 1, blue: depressed synapses,
wij ? 0.) Weights are initialized randomly between 0.10 and 0.12, ? = 10?3 , ? = 100, ? = 10.
C Evolution of the average mutual information per time bin between both output spike trains as
a function of time. D,E Evolution of the average mutual information per time bin (solid line, left
scale) between input and output and the Kullback-Leibler divergence per time bin for both neurons
(dashed line, right scale) as a function of time. Averages are calculated over segments of 1 minute.
F Trace of the correlation between both output spike trains during learning. Correlation coefficients
are calculated every 10 seconds.
synapse without an auxiliary network of interneurons or other biological processes. Rather, they tell
us what type of information would have to be ideally provided by such auxiliary network, and how
the synapse should change its efficacy in order to approximate a theoretically optimal learning rule.
Acknowledgments
We would like to thank Wulfram Gerstner and Jean-Pascal Pfister for helpful discussions. This paper
was written under partial support by the Austrian Science Fund FWF, # S9102-N13 and # P17229N04, and was also supported by PASCAL, project # IST2002-506778, and FACETS, project #
15879, of the European Union.
References
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[2] A. Hyv?arinen, J. Karhunen, and E. Oja. Independent Component Analysis. Wiley, New York, 2001.
[3] T. Toyoizumi, J.-P. Pfister, K. Aihara, and W. Gerstner. Generalized Bienenstock-Cooper-Munro rule for
spiking neurons that maximizes information transmission. Proc. Natl. Acad. Sci. USA, 102:5239?5244,
2005.
[4] W. Gerstner and W. M. Kistler. Spiking Neuron Models. Cambridge University Press, Cambridge, 2002.
[5] T. M. Cover and J. A. Thomas. Elements of Information Theory. Wiley, New York, 1991.
[6] E. L. Bienenstock, L. N. Cooper, and P. W. Munro. Theory for the development of neuron selectivity:
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[7] R. G?utig, R. Aharonov, S. Rotter, and H. Sompolinsky. Learning input correlations through non-linear
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2,282 | 3,071 | A Small World Threshold
for Economic Network Formation
Eyal Even-Dar
Computer and Information Science
University of Pennsylvania
Philadelphia, PA 19104
[email protected]
Michael Kearns
Computer and Information Science
University of Pennsylvania
Philadelphia, PA 19104
[email protected]
Abstract
We introduce a game-theoretic model for network formation inspired by earlier
stochastic models that mix localized and long-distance connectivity. In this model,
players may purchase edges at distance d at a cost of d? , and wish to minimize
the sum of their edge purchases and their average distance to other players. In
this model, we show there is a striking ?small world? threshold phenomenon: in
two dimensions, if ? < 2 then every Nash equilibrium results in a network of
constant diameter (independent of network size), and if ? > 2 then every Nash
equilibrium results in a network whose diameter grows as a root of the network
size, and thus is unbounded. We contrast our results with those of Kleinberg [8] in
a stochastic model, and empirically investigate the ?navigability? of equilibrium
networks. Our theoretical results all generalize to higher dimensions.
1
Introduction
Research over the last decade from fields as diverse as biology, sociology, economics and computer
science has established the frequent empirical appearance of certain structural properties in naturally occurring networks. These properties include small diameter, local clustering of edges, and
heavy-tailed degree distributions [11]. Not content to simply catalog such apparently ?universal?
properties, many researchers have proposed stochastic models of decentralized network formation
that can explain their emergence. A typical such model is known as preferential attachment [3], in
which arriving vertices are probabilistically more likely to form links to existing vertices with high
degree; this generative process is known to form networks with power law degree distributions.
In parallel with these advances, economists and computer scientists have examined models in which
networks are formed due to ?rational? or game-theoretic forces rather than probabilistic ones. In
such models networks are formed via the self-interested behavior of individuals who benefit from
participation in the network [7]. Common examples include models in which a vertex or player can
purchase edges, and would like to minimize their average shortest-path distance to all other vertices
in the jointly formed network. A player?s overall utility thus balances the desire to purchase few
edges yet still be ?well-connected? in the network. While stochastic models for network formation
define a (possibly complex) distribution over possible networks, the game-theoretic models are typically equated with their (possibly complex) set of (Nash) equilibrium networks. It is also common
to analyze the so-called Price of Anarchy [9] in such models, which measures how much worse an
equilibrium network can be than some measure of social or centralized optimality [6, 2, 5, 1].
In this paper we introduce and give a rather sharp analysis of a network formation model of the
game-theoretic variety, but which was inspired by a striking result of Kleinberg [8] in a stochastic
model, and thus forms a bridge between these two lines of thought. In Kleinberg?s stochastic model,
the network formation process begins on an underlying substrate network that is highly regular ?
for instance, a grid in two dimensions. This regular substrate is viewed as a coarse model of ?local?
connectivity, such as one?s geographically close neighbors. The stochastic process then adds ?longdistance? edges to the grid, in an attempt to model connections formed by travel, chance meetings,
and so on. Kleinberg?s model assumes that the probability that an edge connecting two vertices
whose grid distance is d is proportional to 1/d? for some ? > 0 ? thus, longer-distance edges are
less likely, but will still appear in significant numbers due to the long tail of the generating distribution. An interesting recent empirical study [4] of the migration patterns of dollar bills provides
evidence for the validity of such a model. In a theoretical examination of the ?six degrees of separation? or ?small world? folklore first popularized by the pioneering empirical work of Travers and
Milgram [10], Kleinberg proved that only for ? = 2 will the resulting network be likely to support
the routing of messages on short paths using a natural distributed algorithm. For larger values of ?
the network simply does not have short paths (small diameter), and for smaller values the diameter is
quite small, but the long-distance edges cannot be exploited effectively from only local topological
information.
Our model and result can be viewed as an ?economic? contrast to Kleinberg?s. We again begin with
a regular substrate like the grid in two dimensions; these edges are viewed as being provided free of
charge to the players or vertices. A vertex u is then free to purchase an edge to a vertex v at grid
distance d = ?(u, v) at a cost of d? for ? > 0. Thus, longer-distance edges now have higher cost
rather than lower probability, but again in a power law form. We analyze the networks that are Nash
equilibria of a game in which each player?s payoff is the negative of the sum of their edge purchases
and average distances to the other vertices.
Our main result is a precise analysis of the diameter (longest shortest path between any pair of
vertices) of equilibrium networks in this model. In particular, we show a sharp threshold result:
for any ? < 2, every pure Nash equilibrium network has only constant diameter (that is, diameter
independent of the network size n); and for any ? > 2, every pure Nash equilibrium has diameter
that grows as a root of the network size (that is, unbounded and growing rapidly with n). In the full
version, we show in addition that the threshold phenomenon occurs in mixed Nash equilibrium as
well.
Despite the outward similarity, there are some important differences between our results and Kleinberg?s. In addition to the proofs being essentially unrelated (since one requires a stochastic and the
other an equilibrium analysis), Kleinberg?s result establishes a ?knife?s edge? (fast routing only at ?
exactly 2), while ours is a threshold or phase transition ? there is a broad range of ? values yielding
constant diameter, which sharply crosses over to polynomial growth at ? = 2. On the other hand,
for ? = 2 Kleinberg establishes that in his model not only that there is small (though order log(n)2
rather than constant) diameter, but that short paths can be navigated by a naive greedy routing algorithm. However, simulation results discussed in Section 5 suggest that the equilibrium networks of
our model do support fast routing as well. Like Kleinberg?s results, all of ours generalize to higher
dimensions as well, with the threshold occurring at ? = r in r-dimensional space.
The outline of the paper is as follows. In Section 2 we define our game-theoretic model and introduce
the required equilibrium concepts. In Section 3 we provide the constant diameter upper bound for
r = 2 when ? < 2, and also even better constants for ? ? 1. Section 4 provides the diameter
lower bound for ? > 2, while in Section 5 we explore greedy routing in equilibrium networks via
simulation.
2
Preliminaries
We devote this section to a formal definition of the model. We assume that the players are
? located on
a grid, so each player v is uniquely identified with a grid point (a, b), where 1 ? a, b ? n; thus the
total number of players is n. The action of player vi is a vector si ? {0, 1}n indicating which edges
to other players vi has purchased. We let s = s1 ? ? ? ? ? sn be the joint action of all the players,
v1 , ..., vn . We also use s?i to denote the joint action of all players except player vi .
The Graph. The joint action s defines an undirected graph G(s) as follows. The nodes of G(s) are
the players V = {v1 , . . . , vn }. An edge (vi , vj ) is bought by player vi if and only if si (j) = 1. Let
Ei (si ) = {(vi , vj ) | si (j) = 1} be the set of edges bought by player vi and let E(s) = ?i?V Ei (si ).
The graph induced by s is G(s) = (V, E(s)).
Distances and Costs. The grid defines a natural distance ?. Let vi be the player identified with the
grid point (a, b) and vi0 with (a0 , b0 ); then their grid distance is ?(vi , vi0 ) = |a ? a0 | + |b ? b0 |. Next
we define a natural family of edge cost functions in which the cost of an edge is a function of the
grid distance:
?
0
?(vi , vj ) = 1
c(vi , vj ) =
a?(vi , vj )? otherwise
where a, ? > 0 are parameters of the model. Thus, grid edges are free to the players, and longer
edges have a cost polynomial in their grid distance.
The Game. We are now ready to define the formal network formation game we shall analyze. The
overall cost function ci of player vi is defined as
ci (s) = ci (si , s?i ) =
X
e?Ei (si )
c(e) +
n
X
?G(s) (vi , vj )
j=1
where ?G(s) (u, v) is the shortest distance between u and v in G(s). Thus, in this game player
i wishes to minimize ci (s), which requires balancing edge costs and shortest paths. We emphasize that players benefit from edge purchases by other players, since shortest paths are measured
with respect to the overall graph formed by all edges purchased. The graph diameter is defined as
maxi,j ?G(s) (vi , vj ).
Equilibrium Concepts. A joint action s = s1 ? ? ? ? ? sn is said to be a Nash equilibrium if for
every player i and any alternative action s?i ? {0, 1}n , we have ci (si , s?i ) ? ci (?
si , s?i ). If s is a
Nash equilibrium we say that its corresponding graph G(s) is an equilibrium graph. A joint action
s = s1 ? ? ? ? ? sn is said to be link stable if for every player i and any alternative action s?i ? {0, 1}n
that differs from si in exactly one coordinate (i.e. one edge), we have ci (si , s?i ) ? ci (?
si , s?i ). If s is
link stable we say that its corresponding graph G(s) is a stable graph. Note that an equilibrium graph
implies a link stable graph. Link stability means that the graph is stable under single-edge unilateral
deviations (as opposed to Nash, which permits arbitrary unilateral deviations), and is a private case
of the pairwise stability given notion given in [7]. The popularity of the link stable notion is due to its
simplicity and due to the fact that it is easily computable, as opposed to computing best responses
which in similar problems is known to be NP-Hard [6]. Note?that as the grid edges are free, the
diameter of an equilibrium or link stable graph is bounded by 2 n.
3
Constant Diameter at Equilibrium for ? ? [0, 2]
In this section we analyze the diameter of equilibrium networks when ? ? [0, 2]. Our results actually
hold under the more general notion of link stability as well. The following is the first of our two
main theorems.
Theorem 3.1 For any constant ? > 0, if ? = 2 ? ?, then there exists a constant c(?) such that for
any n, all Nash equilibria or link stable graphs over n players have diameter at most c(?).
The proof of this theorem has a number of technical subtleties, so we first provide its intuition, which
is illustrated in Figure 1(B). We analyze an equilibrium (or link stable) graph in stages, and focus on
the distance of vertices to some focal player u. In each stage we argue that more grid-distant players
have an incentive to purchase an edge to u due to the centrality of u in the graph.
We start with the following simple fact: for every nodes v and w we have that if ?(v, w) ? d
then ?G(s) (v, w) ? d since all grid edges are free. We would like to show that even a stronger
property holds ? namely, that if ?(v, w) ? d? then ?G(s) (w, v) ? d for some ? > 1. Since
this property is no longer simply implied by the grid edges, it requires arguing that grid-distant
vertices have an incentive to purchase edges to each other. Suppose there are nodes u and v such
that ?G(s) (u, v) ? d. We first define a ?close? graph neighborhood of u, Su = {w|?G(s) (u, w) ?
d/3}. Note that for every w ? Su we have that ?G(s) (v, w) ? 2d/3. Next we would like to claim
that the cardinality of Su is large ? thus u?s neighborhood is densely populated. For this we define
Su? = {w|?(u, w) ? d/3} ? Su . Using the grid topology (see Figure 1(A)) we see that |Su? | is of
order d2 .
All distances are on the grid
?=d
?
Su
u
v
The grid center
Nodes in distance 1
Nodes in distance 2
Nodes in distance 3
v?
(A)
? = d3/?
(B)
Figure 1: (A) The number of nodes at exact distance k is exactly 4k, while the number within k is
order k 2 . (B) Illustration of the main argument of Theorem 3.1. Here u and v are vertices at grid
distance d, while u and v 0 are at grid distance d0 , where d ? d3/? ? d0 . In the proof we use the
size of Su? to show that v benefits by purchasing an edge to u and thus must be distance 1 to u in the
equilibrium graph; this in turn allows us to argue that v 0 wishes to purchase an edge to u as well.
Now consider the benefit to v of buying the edge (v, u) (which is not in the graph since
?G(s) (u, v) ? d). Since the distance from v to every node in Su is reduced by at least d/3 and
the set size is at least order d2 , we have that the benefit is of order d3 . The fact that this edge was
not bought implies that ?(v, u)? = ?(d3 ). Therefore, we have that ?G(s) (u, v) ? d implies that
?(u, v) = ?(d3/? ), which is the contrapositive of ?(u, v) = O(d3/? ) implies ?G(s) (u, v) ? d.
In other words, for ?small enough? values of ? (quantified in the full proof), vertices quite distant
from u in the grid have an incentive to buy an edge to u, by virtue of the dense population in Su? .
But this in turn argues that the size of Su is even larger than Su? ; we then ?bootstrap? this argument
to show that yet further vertices have an incentive to connect to u, and so on. We now proceed with
the formal proof based on this argument.
Lemma 3.2 Let G(s) be an equilibrium or link stable graph and u be the ?
grid center. Suppose
that for every node v such that ?(u, v) ? d? (where ? ? 1 and d? < n/2), we have that
0
?G(s) (u, v) ? d. Then for every d, and for every node v such that ?(u, v) ? 21/? (d/3)? , where
? 0 = 2?+1
? , we have that ?G(s) (u, v) ? d.
Proof: Let v be a node such that ?G(s) (u, v) = d and let Su = {w|?G(s) (u, w) ? d/3}; observe
that d0 = minw?Su ?G(s) (w, v) is at least 2d
3 and thus v?s benefit of buying the edge (v, u) is at
least d3 |Su |. Next we would like to bound the size of Su from below. Using the topology of the
grid, the grid the center node has 4k nodes (See Figure 1(A)) in exact grid distance k (if k ? n/2),
which implies that the center node has 2k 2 nodes in grid distance at most k. The set Su contains all
nodes such that ?G(s) (u, w) ? d/3 by definition which implies by our assumption that it includes
all nodes w such that ?(u, w) ? (d/3)? . Therefore, the size of Su is at least 2(d/3)2? . Now since
G(s) is an equilibrium or link stable graph, it means that v would not like to buy the edge (u, v) and
thus
d2?+1
?(u, v)? > 2(d/3)2? ? d/3 = 2 2?+1
3
(2?+1)/?
Taking the ? root, we have that ?G(s) (u, v) > d implies ?(u, v) ? 21/? d3(2?+1)/? , which is the
(2?+1)/?
contrapositive of ?(u, v) ? 21/? d3(2?+1)/? implies ?G(s) (u, v) ? d, as required.
Equipped with this lemma we can prove rather strong results regarding the case where ? = 2??, for
? > 0. In the previous lemma there are two parts in the change of the radius ? one is the exponent,
which grows, and the second is that instead of having d in the base we have only d/3. The next
lemma shows that as long as d is large enough we can ignore the fact that the base decreases from d
to d/3 ? and thus ?amplify? the exponent ? in the preceding analysis to a larger exponent (1+?1 )?.
Su2
v
u
Figure 2: A graph with diameter of 6.
Lemma 3.3 (Amplification Lemma) Let G(s) be an equilibrium or link stable graph. Let ? = 2 ? ?
for some ? > 0. Let c(?) be a constant determined by subsequent analysis. Suppose
that for every
?
d > c(?), for every node v such that ?(u, v) ? d? (where ? ? 1, d? < n/2, and u is the
grid center), we have that ?G(s) (u, v) ? d. Then for every d > c(?), for every node v such that
0
?
.
?(u, v) ? d? , where ? 0 = ?(1 + ?1 ), we have that ?G(s) (u, v) ? d, where ?1 = 2(2??)
Proof: Set c(?) = 3
1+2?1
?1
1+2?1
. By Lemma 3.2 we have that for every d > 3 ?1 for every nodes u and
?
(d/3)?
v such that ?(u, v) ? 2 , where ?? = 2?+1
? , we have that ?G(s) (u, v) ? d.
?
2?+1
?
?
d(1+ 2?? )?
(d/3) ?
d(1+ 2?? )?+1/?
>
=
=
?
?
2
2 ? 3(1+ 2?? )?+1/?
3(1+ 2?? )?
d(1+?1 )? d?1 ?
? d(1+?1 )?
=
3(1+2?1 )?
where both inequalities hold for d ? c(?)
(d/3)?
2
Now we are ready to prove the main theorem of this section.
1+2?1
?
and let u be the grid center. For every
Proof: (Theorem 3.1) Let c0 (?) = 3 ?1 , where ?1 = 2(2??)
node v such that ?(u, v) ? c0 (?), we must have ?G(s) (u, v) ? c0 (?), since all grid edges are part
?
of G(s). Next we prove that all nodes within grid distance n/2 are within graph distance c0 (?).
Since ?G(s) (u, v) ? ?(u, v), we can apply Lemma 3.3 to obtain that in radius c0 (?) of u in G, are
all nodes v such that ?(u, v) ? c0 (?)1+?1 . We repeat this argument recursively and obtain after the
k
k-th time, that all nodes v such that ?(u, v) ? c0 (?)(1+?1 ) satisfy ?G(s) (u, v) ? c0 (?). Taking
?
k = log1+?1 ( n/2), this implies that there are n/2 nodes within c0 (?) from u. Now suppose there
exists a node v such that ?G(s) (u, v) ? 3c0 (?). Then by buying the edge (v, u), u?s benefit is at
least 2c0 (?)n/2 (we know that
are at least n/2 nodes within graph distance of c0?
(?) from u),
? there
?
while its cost is bounded by n < n (since any node grid distance from u is at most n). Setting
c(?) = 6c0 (?), we obtain the theorem.
3.1
Even Smaller Constant Diameter at Equilibrium for ? ? 1
The constant diameter bound c(?) in Theorem 3.1 blows up as ? approaches 0. In this section we
show that for ? < 1, rather?
small constant bounds hold. Note that when ? ? 1, the most expensive
edge cost is bounded by 2a n, where a is the edge cost constant. We will use this fact to show that
every equilibrium graph G(s) has a small constant diameter.
Let u, v ? V , we let TG(s) (u, v) be the set of all nodes that u can reach on a shortest path that
includes v. Formally, TG(s) (u, v) = {w | ?G(s) (u, w) = ?G(s) (u, v) + ?G(s) (v, w)}. We start by
providing a technical lemma.
Lemma 3.4 Let G(s) be an equilibrium or link stable graph. Let u, v ? V be an arbitrary pair of
?(u,v)?
players. If (u, v) ?
/ E(s) then | TG(s) (u, v) | ? ?G(s)
(u,v)?1 .
Proof: Buying the edge (u, v) (at a cost of ?(u, v)? ) makes the distance from u to every w ?
TG(s) (u, v) shorter by ?G(s) (u, v) ? 1. However, s is a Nash equilibrium, thus we know that the
edge (u, v) was not bought. This implies that the benefit (?G(s) (u, v) ? 1) ? |TG(s) (u, v)| from
buying the edge is bounded by ?(u, v)? .
Lemma 3.5 Let G(s) = (V, E(s)) be an equilibrium graph and let u, v ? V .
? If ? < 1 then ?G(s) (u, v) ? 5.
? If ? = 1 then ?G(s) (u, v) ? 2da2 + 4e
Proof: We prove for the case that the cost functions is a?(u, v) and omit the proof for the case where
? < 1 which is similar. Assume for contradiction that there exist a node v such that ?G(s) (u, v) ?
2
da
where u is the grid center node (note that the grid distance from u is bounded by
? + 4e + 1,
n). Let Su2 = {w|?G(s) (u, w) ? 2} be the set of nodes at a distance of at most 2 from u
(See Figure 3.1) including u. We first bound the size of Su2 . For every node w ? Su2 we have
?G(s) (u, w) ? da2 +4e?1. Buying the edge (v, u) makes the distance between v and every w ? Su2
at most 3. Thus, the benefit from buying the edge (v, u) is at least (da2 +4e?1?3)|Su2 | = da2 e|Su2 |.
However, the edge (v, u) ?
/ E(s) and is not part of the equilibrium graph.?Therefore, the benefit
2
2
from buying it is at most ?(v, u). This implies that
u | ? ?(v, u) ? a n. Now we look on a
? da e|S
2
n/da
e
shortest paths tree rooted at
u.
There
are
at
most
?
2 nodes at a distance of 2 from u. Each
?
n
descendants
one
of
them
has
at
most
a
by
Lemma
3.4.
Since
the graph is connected, we get that
?
?
?
a n/da2 e(a n ? 2) + a n/da2 e ? n, which is a contradiction.
3.2
The Case ? = 2
In this case we obtain neither a constant upper ?
bound nor a polynomial lower bound. We show that
? 2/ logn
? c
for ? = 2 the diameter is bounded by O( n
), which is bounded by n for every constant
c (i.e. this bound is very small as well); however it bounds from above any polylogarithmic function.
Theorem 3.6 Let the edge cost be c((u, v) = ?(u, v)2 , and let G(s) =?(V, E(s)) be an equilibrium
? 2/ logn
or link stable graph . Then the graph diameter is bounded by O( n
).
Proof: We again apply Lemma 3.2 repeatedly, but now with ? = 2. After applying it for the first
time we have that all nodes which are in grid distance ( d3 )3/2 from u the grid center are within graph
distance d. Recall that Su = {w|?G(s) (u, w) ? d/3}. Using the same arguments we construct a
series of distances xk , such that if ?(u, v) ? xk then ?G(s) (u, v) ? d. We begin with x1 as ( d3 )3/2
and now compute x2 :
d d3/2
d
x22 > |Su | = ( 3/2 /33/2 )2
3
3 3
d4/2
Solving it we obtain that x2 = 37/2 . Suppose that after repeating the argument for the kth time we
have that xk is at least dak /3bk . Using this bound we derive a lower bound on the size of Su and
obtain the following bound for the k + 1 iteration:
d dak /3ak 2
d
)
x2k+1 > |Su | = (
3
3
3bk
ak +1/2
Thus we obtain that xk+1 = 3bdk +ak +1/2 and
ak+1 = ak + 1/2
bk+1 = bk + ak + 1/2.
Our next goal is to estimate ak and bk . The estimation of ak is straight forward and ak = k/2 + 1.
For bk it is enough for our needs to consider an upper bound; since we have bk+1 = bk + k/2 + 3/2,
one can easily verify that k 2 /2 is an upper bound for k ? 3. Therefore, in order to provide an upper
bound on the distance form the center grid u we would like to find an initial d such that
?
dk/2
?k such that k2 /2 ? n
3?
?
and d is minimal. This clearly holds for d = n2/ logn and k = logn and can be shown to be
the minimal value for which it holds. Now using similar arguments to previous proofs we show that
every other node cannot be further away from u.
4
Polynomial Diameter at Equilibrium for ? > 2
We now give our second main result, which states that for ? > 2 the diameter grows as a root of n
and is thus unbounded.
? ??2
Theorem 4.1 For any ?, the diameter of any Nash equilibrium or link stable graph is ?( n ?+1 ).
Before giving the proof we note that this bound implies a trivial lower bound of a constant for ? ? 2,
? 1/4
and a polynomial for ? > 2. For instance, setting ? = 3 we obtain a lower bound of ?( n ).
We first provide a simple lemma (stated without proof) regarding the influence of one edge on a
connected graph?s diameter.
S
Lemma 4.2 Let G = (V, E) be a connected graph with diameter C, and Let G0 = (V, E {e}) for
0
any edge e then the diameter of G is at least C/2.
Proof: (Theorem 4.1) Let D be the diameter of an equilibrium graph, and d be the grid distance
of (w, v) the most expensive edge bought in G, note that the most expensive
edge corresponds to
?
n/d,
the
longest
edge
in
grid
distance
terms.
First
we
observe
that
D
?
2
as
the grid diameter is
?
2 n and the fastest way to traverse it is through edges of maximal length which is d. By Lemma
4.2 the benefit of buying an edge (u, v) is at most 2D(n ? 3), since the diameter before was at
most 2D and the distance to your two neighbor and yourself has not been changed. Therefore, have
?(u, v)? = d? ? 2D(n ? 3). Next we use the two simple bounds
d? ? 2Dn
?
2 n/d ? D
(1)
(2)
Substituting the bound of d in Equation 2 into equation 1 we obtain that
?
(2 n/D)? ? 2Dn
?
(2 n)?
? D1+?
2n
? ??2
c( n 1+? ) ? D
as required.
5
Simulations
The analyses we have considered so far examine static properties of equilibrium and link stable
graphs, and as such do not shed light on natural dynamics that might lead to them. In this section we
briefly describe dynamical simulations on a 100 ? 100 grid (which has 108 possible edges). At each
iteration a random vertex u is selected. With probability 1/2, an existing edge of u (grid or longdistance) is selected at random, and we compute whether (given the current global configuration of
the graph), u would prefer not to purchase this edge, in which case it is deleted. With probability 1/2,
we instead select a second random vertex v, and compute whether (again given the global graph) u
would like to purchase the edge (u, v), in which case it is added. Note that if this dynamic converges,
it is to a link stable graph and not necessarily a Nash equilibrium, since only single-edge deviations
are considered.
The left panel of Figure 3 shows the worst-case diameter as a function of the number of iterations,
and demonstrates the qualitative validity of our theory for this dynamic. For ? = 1, 2 the diameter
quickly falls to a rather small value (less than 10). The asymptotes for ? = 3, 4 are considerably
higher.
The right panel revisits the question that was the primary interest of Kleinberg?s work [8], namely
the efficiency of ?naive? or greedy navigation or routing. If we wish to route a message from the
grid center to a randomly chosen destination, and the message is always forwarded from its current
vertex to the graph neighbor whose grid address is closest to the destination, how long will it take?
Kleinberg was the first to observe and explain the fact that the mere existence of short paths (small
diameter) may not be sufficient for such greedy local routing algorithms to find the short paths. In the
right panel of Figure 3 we show that the routing efficiency does in fact seem to echo our theoretical
results ? for the aforementioned dynamic, very short paths (only slightly higher than the diameter)
are found for small ?, much longer paths for larger ?.
Dynamics
Greedy routing and dynamics
60
60
?=1
?=2
?=3
?=4
50
Average greedy routing distance
50
Average distance
40
30
20
10
0
?=1
?=2
?=3
?=4
40
30
20
10
0
5
10
Iterations
15
0
0
5
x 10
5
10
Iterations
15
5
x 10
Figure 3: Left panel: graph diameter vs. iterations for a simple dynamic. Right panel: greedy
routing efficiency vs. iterations for the same dynamic.
6
Extensions
We conclude by briefly mentioning generalizations of our theoretical results that we omit detailing.
All of the results carry over higher dimensions, where the threshold phenomenon takes place at ?
equaling the grid dimension. We can also easily handle the case where the grid wraps around rather
than having boundaries. We can also generalize to the pairwise link stability notion of [7], in which
that the cost of each link is shared between the end points of the edge. Finally, we can construct
network that are stable.
References
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stability for network design with fair cost allocation. In Proc. of FOCS, pages 295?304, 2004.
[3] Albert-L?aszl?o Barab?asi and R. Albert. Emergence of scaling in random networks. Science, 286:509?512,
1999.
[4] D. Brockmann, L. Hufnagel, and T. Geisel. The scaling laws of human travel. Nature, 439:462?465,
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2,283 | 3,072 | Generalized Maximum Margin Clustering and
Unsupervised Kernel Learning
Hamed Valizadegan
Computer Science and Engineering
Michigan State University
East Lansing, MI 48824
[email protected]
Rong Jin
Computer Science and Engineering
Michigan State University
East Lansing, MI 48824
[email protected]
Abstract
Maximum margin clustering was proposed lately and has shown promising
performance in recent studies [1, 2]. It extends the theory of support vector machine to unsupervised learning. Despite its good performance, there
are three major problems with maximum margin clustering that question
its efficiency for real-world applications. First, it is computationally expensive and difficult to scale to large-scale datasets because the number of
parameters in maximum margin clustering is quadratic in the number of
examples. Second, it requires data preprocessing to ensure that any clustering boundary will pass through the origins, which makes it unsuitable for
clustering unbalanced dataset. Third, it is sensitive to the choice of kernel
functions, and requires external procedure to determine the appropriate
values for the parameters of kernel functions. In this paper, we propose
?generalized maximum margin clustering? framework that addresses
the above three problems simultaneously. The new framework generalizes the maximum margin clustering algorithm by allowing any clustering
boundaries including those not passing through the origins. It significantly
improves the computational efficiency by reducing the number of parameters. Furthermore, the new framework is able to automatically determine
the appropriate kernel matrix without any labeled data. Finally, we show a
formal connection between maximum margin clustering and spectral clustering. We demonstrate the efficiency of the generalized maximum margin
clustering algorithm using both synthetic datasets and real datasets from
the UCI repository.
1
Introduction
Data clustering, the unsupervised classification of samples into groups, is an important research area in machine learning for several decades. A large number of algorithms have
been developed for data clustering, including the k-means algorithm [3], mixture models [4],
and spectral clustering [5, 6, 7, 8, 9]. More recently, maximum margin clustering [1, 2] was
proposed for data clustering and has shown promising performance. The key idea of maximum margin clustering is to extend the theory of support vector machine to unsupervised
learning. However, despite its success, the following three major problems with maximum
margin clustering has prevented it from being applied to real-world applications:
? High computational cost. The number of parameters in maximum margin clustering
is quadratic in the number of examples. Thus, it is difficult to scale to large-scale
datasets. Figure 1 shows the computational time (in seconds) of the maximum
margin clustering algorithm with respect to different numbers of examples. We
Time comparision
1600
1400
Generalized Maxmium Marging Clustering
Maximum Margin Clustering
Time (seconds)
1200
1000
800
600
400
200
0
40
60
80
100
120
140
160
Number of Samples
180
200
220
Figure 1: The scalability of the original maximum margin clustering algorithm versus the
generalized maximum margin clustering algorithm
50
2
45
1.8
40
35
Clustering error
1.6
1.4
1.2
1
30
25
20
15
10
0.8
5
0.6
0
10
0.4
0.4
0.6
0.8
1
1.2
1.4
1.6
(a) Data distribution
1.8
2
2.2
20
30
40
50
60
70
80
90
100
Kernel Width (% of data range) in RBF function
(b) Clustering error versus kernel width
Figure 2: Clustering error of spectral clustering using the RBF kernel with different kernel
width. The horizonal axis of Figure 2(b) represents the percentage of the distance range
(i.e., the difference between the maximum and the minimum distance) that is used for kernel
width.
clearly see that the computational time increases dramatically when we apply the
maximum margin clustering algorithm to even modest numbers of examples.
? Requiring clustering boundaries to pass through the origins. One important assumption made by the maximum margin clustering in [1] is that the clustering boundaries
will pass through the origins. To this end, maximum margin clustering requires
centralizing data points around the origins before clustering data. It is important
to note that centralizing data points at the origins does not guarantee clustering
boundaries to go through origins, particularly when cluster sizes are unbalanced
with one cluster significantly more popular than the other.
? Sensitive to the choice of kernel functions. Figure 2(b) shows the clustering error of
maximum margin clustering for the synthesized data of two overlapped Gaussians
clusters (Figure 2(a)) using the RBF kernel with different kernel width. We see that
the performance of maximum margin clustering depends critically on the choice
of kernel width. The same problem is also observed in spectral clustering [10].
Although a number of studies [8, 9, 10, 6] are devote to automatically identifying
appropriate kernel matrices in clustering, they are either heuristic approaches or
require additional labeled data.
In this paper, we propose ?generalized maximum margin clustering? framework that
resolves the above three problems simultaneously. In particular, the proposed framework
reformulates the problem of maximum margin clustering to include the bias term in the
classification boundary, and therefore remove the assumption that clustering boundaries
have to pass through the origins. Furthermore, the new formulism reduces the number
of parameters to be linear in the number of examples, and therefore significantly reduces
the computational cost. Finally, it is equipped with the capability of unsupervised kernel
learning, and therefore, is able to determine the appropriate kernel matrix and clustering
memberships simultaneously. More interestingly, we will show that spectral clustering, such
as the normalized cut algorithm, can be viewed as a special case of the generalized maximum
margin clustering.
The remainder of the paper is organized as follows: Section 2 reviews the work of maximum
margin clustering and kernel learning. Section 3 presents the framework of generalized
maximum margin clustering. Our empirical studies are presented in Section 4. Section 5
concludes this work.
2
Related Work
The key idea of maximum margin clustering is to extend the theory of support vector
machine to unsupervised learning. Given the training examples D = (x1 , x2 , . . . , xn ) and
their class labels y = (y1 , y2 , . . . , yn ) ? {?1, +1}n , the dual problem of support vector
machine can be written as:
1
maxn ?> e ? ?> diag(y)Kdiag(y)?
??R
2
s. t. 0 ? ? ? C, ?> y = 0
(1)
where K ? Rn?n is the kernel matrix and diag(y) stands for the diagonal matrix that
uses the vector y as its diagonal elements. To apply the above formulism to unsupervised
learning, the maximum margin clustering approach relaxes class labels y to continuous
variables, and searches for both y and ? that maximizes the classification margin. This
leads to the following optimization problem:
min
y,?,?,?
s. t.
t
?
(yy> ) ? K
(e + ? ? ? + ?y)>
? ? 0, ? ? 0
e + ? ? ? + ?y
t ? 2C? > e
?
?0
where ? stands for the element wise product between two matrices. To convert the above
problem into a convex programming problem, the authors of [1] makes two important relaxations. The first one relaxes yy> into a positive semi-definitive (PSD) matrix M ? 0 whose
diagonal elements are set to be 1. The second relaxation sets ? = 0, which is equivalent to
assuming that there is no bias term b in the expression of classification boundaries, or in
other words, classification boundaries have to pass through the origins of data. These two
assumption simplify the above optimization problem as follows:
min
M,?,?
s. t.
t
?
M ?K
(e + ? ? ?)>
e+???
t ? 2C? > e
? ? 0, ? ? 0, M ? 0
?
?0
(2)
Finally, a few additional constraints of M are added to the above optimization problem to
prevent skewed clustering sizes [1]. As a consequence of these two relaxations, the number
of parameters is increased from n to n2 , which will significantly increase the computational
cost. Furthermore, by setting ? = 0, the maximum margin clustering algorithm requires
clustering boundaries to pass through the origins of data, which is unsuitable for clustering
data with unbalanced clusters.
Another important problem with the above maximum margin clustering is the difficulty
in determining the appropriate kernel similarity matrix K. Although many kernel based
clustering algorithms set the kernel parameters manually, there are several studies devoted
to automatic selection of kernel functions, in particular the kernel width for the RBF kernel,
?
?
kx ?x k2
i.e., ? in exp ? i2?2j 2 . Shi et al. [8] recommended choosing the kernel width as 10% to
20% of the range of the distance between samples. However, in our experiment, we found
that this is not always a good choice, and in many situations it produces poor results. Ng
et al. [9] chose kernel width which provides the least distorted clusters by running the same
clustering algorithm several times for each kernel width. Although this approach seems to
generate good results, it requires running seperate experiments for each kernel width, and
therefore could be computationally intensive. Manor et al. in [10] proposed a self-tuning
spectral clustering algorithm that computes a different local kernel width for each data point
xi . In particular, the local kernel width for each xi is computed as the distance of xi to
its kth nearest neighbor. Although empirical study seems to show the effectiveness of this
approach, it is unclear how to find the optimal k in computing the local kernel width. As
we will see in the experiment section, the clustering accuracy depends heavily on the choice
of k.
Finally, we will briefly overview the existing work on kernel learning. Most previous work
focus on supervised kernel learning. The representative approaches in this category include
the kernel alignment [11, 12], semi-definitive programming [13], and spectral graph partitioning [6]. Unlike these approaches, the proposed framework is designed for unsupervised
kernel learning.
3
Generalized Maximum Margin Clustering and Unsupervised
Kernel Learning
We will first present the proposed clustering algorithm for hard margin, followed by the
extension to soft margin and unsupervised kernel learning.
3.1
Hard Margin
In the case of hard margin, the dual problem of SVM is almost identical to the problem in
Eqn. (1) except that the parameter ? does not have the upper bound C. Following [13], we
further convert the problem in (1) into its dual form:
1
(e + ? + ?y)T diag(y)K ?1 diag(y)(e + ? + ?y)
min
? ,y,? 2
s. t. ? ? 0, y ? {+1, ?1}n
(3)
where e is a vector with all its elements being one. Unlike the treatment in [13], which
rewrites the above problem as a semi-definitive programming problem, we introduce variables z that is defined as follows:
z = diag(y)(e + ?)
Given that ? ? 0, the above expression for z is essentially equivalent to the constraint
|zi | ? 1 or zi2 ? 1 for i = 1, 2, . . . , n. Then, the optimization problem in (3) is rewritten as
follows:
1
(z + ?e)T K ?1 (z + ?e)
min
z,?
2
s. t. zi2 ? 1, i = 1, 2, . . . , n
(4)
Note that the above problem may not have unique solutions for z and ? due to the translation
invariance of the objective function. More specifically, given an optimal solution z and ?,
we may be able to construct another solution z0 and ?0 such that:
z0 = z + ?e, ?0 = ? ? ?.
Evidently, both solutions result in the same value for the objective function in (4). Furthermore, with appropriately chosen ?, the new solution z0 and ?0 will be able to satisfy the
constraint zi2 ? 1. Thus, z0 and ?0 is another optimal solution for (3). This is in fact related
to the problem in SVM where the bias term b may not be unique [14]. To remove the translation invariance from the objective function, we introduce an additional term Ce (z> e)2
into the objective function, i.e.
1
(z + ?e)T K ?1 (z + ?e) + Ce (z> e)2
min
z,?
2
s. t. zi2 ? 1, i = 1, 2, . . . , n
(5)
where constant Ce weights the important of the punishment factor against the original
objective. It is set to be 10, 000 in our experiment. For the simplicity of our expression, we
further define
w = (z; ?) and P = (In , e).
Then, the problem in (4) becomes
2
min
wT P T K ?1 P w + Ce (e>
0 w)
n+1
w?R
s. t.
wi2 ? 1, i = 1, 2, . . . , n
(6)
where e0 is a vector with all its elements being 1 except its last element which is zero. We
then construct the Lagrangian as follows
n
X
2
i
L(w, ?) = wT P T K ?1 P w + Ce (e>
?i (w> In+1
w ? 1)
0 w) ?
i=1
?
= w
>
T
P K
?1
P+
C e e0 e>
0
?
n
X
i
?i In+1
!
w+
i=1
n
X
?i
i=1
i
where In+1
is an (n + 1) ? (n + 1) matrix with all the elements being zero except the ith
diagonal element which is 1. Hence, the dual problem of (6) is
n
X
?i
maxn
??R
s. t.
i=1
T
P K
?1
P+
Ce e0 e>
0
?
n
X
i
?i In+1
?0
i=1
?i ? 0, i = 1, 2, . . . , n
Finally, the solution?w can be computed using the KKT
! condition, i.e.,
n
X
i
P T K ?1 P + Ce e0 e>
?i In+1
w = 0n+1
0 ?
(7)
i=1
In
words, the Psolution w
? is proportional to the eigenvector of matrix
? T other
n
i
?
?
I
P K ?1 P + Ce e0 e>
i
n+1 for the zero eigenvalue. Since wi = (1 + ?i )yi , i =
0
i=1
1, 2, . . . , n and ?i ? 0, the class labels {yi }ni=1 can be inferred directly from the sign of
{wi }ni=1 .
Remark I It is important to realize that the problem in (5) is non-convex due to the nonconvex constraint wi2 ? 1. Thus, the optimal solution found by the dual problem in (7) is
not necessarily the optimal solution for the prime problem in (5). Our hope is that although
the solution found by the dual problem is not optimal for the prime problem, it is still a
good solution for the prime problem in (5). This is similar to the SDP relaxation made by
the maximum margin clustering algorithm in (2) that relaxes a non-convex programming
problem into a convex one. However, unlike the relaxation made in (2) that increases the
number of variables from n to n2 , the new formulism of maximum margin does not increase
the number of parameters (i.e., ?), and therefore will be computational more efficient. This
is shown in Figure 1, in which the computational time of generalized maximum margin
clustering is increased much slower than that of the maximum margin algorithm.
Remark II To avoid the high computational cost in estimating K ?1 , we replace K ?1 with
1/2
its normalized graph Laplacian L(K) [15], which is defined as L(K) = I ?D1/2
where
PKD
n
D is a diagonal matrix whose diagonal elements are computed as Di,i = j=1 Ki,j , i =
? = L(K)? where ? stands for the
1, 2, . . . , n. This is equivalent to defining a kernel matrix K
operator of pseudo inverse. More interesting, we have the following theorem showing the
relationship between generalized maximum margin clustering and the normalized cut.
Theorem 1. The normalized cut algorithm is a special case of the generalized maximum
margin clustering in (7) if the following conditions hold, i.e., (1) K ?1 is set to be the
?
normalized Laplacian L(K),
(2) all the ?s are enforced to be the same, i.e., ?i = ?0 , i =
1, 2, . . . , n, and (3) Ce ? 1.
Proof sketch: Given the conditions 1 to 3 in the theorem, the new objective function in (7)
?
becomes: max ? s.t. L(K)
? ?In and the solution for this problem is the largest eigenvector
?
of L(K).
??0
3.2
Soft Margin
We extend the formulism in (7) to the case of soft margin by considering the following
problem:
n
X
1
(e + ? ? ? + ?y)T diag(y)K ?1 diag(y)(e + ? ? ? + ?y) + C?
min
?i2
? ,y,?,? 2
i=1
s. t. ? ? 0, ? ? 0, y ? {+1, ?1}n
(8)
where C? weights the importance of the clustering errors against the clustering margin.
Similar to the previous derivation, we introduce the slack variable z and simplify the above
problem as follows:
n
X
1
(z + ?e)T K ?1 (z + ?e) + Ce (z> e)2 + C?
min
?i2
z,?,?
2
i=1
s. t. (zi + ?i )2 ? 1, ?i ? 0, i = 1, 2, . . . , n
(9)
By approximating (zi + ?i )2 as zi2 + ?i2 , we have the dual form of the above problem written
as:
n
X
maxn
?i
??R
s. t.
i=1
P > K ?1 P + Ce e0 e>
0 ?
n
X
i
?0
?i In+1
i=1
0 ? ?i ? C? , i = 1, 2, . . . , n
(10)
The main difference between the above formulism and the formulism in (7) is the introduction of the upper bound C? for ? in the case of soft margin. In the experiment, we set the
parameter C? to be 100, 000, a very large value.
3.3
Unsupervised Kernel Learning
As already pointed out, the performance of many clustering algorithms depend on the right
choice of the kernel similarity matrix. To address this problem, we extend the formulism
in (10) by including the kernel learning mechanism. In particular, we assume that a set of
m kernel similarity matrices K1 , K2 , . . . , Km
available. Our goal is to identify the linear
Pare
m
combination of kernel matrices, i.e., K = i=1 ?i Ki , that leads to the optimal clustering
accuracy. More specifically, we need to solve the following optimization problem:
n
X
max
?i
?,?
s. t.
i=1
P
>
?m
X
!?1
?i Ki
P + C e e0 e>
0 ?
i=1
n
X
i
?i In+1
?0
i=1
0 ? ?i ? C? , i = 1, 2, . . . , n,
m
X
?i = 1, ?i ? 0, i = 1, 2, . . . , m
(11)
i=1
Unfortunately,
it is difficult to solve the above problem due to the complexity introduced
Pm
by ( i=1 ?i Ki )?1 . Hence, we consider an alternative problem to the above one. We first
?1, L
?2, . . . , L
? m . Each Laplacian Li is conintroduce a set of normalized graph Laplacian L
structed from the P
kernel similarity matrix Ki . We then defined the inverse of the combined
m
? i . Then, we have the following optimization problem
matrix as K ?1 = i=1 ?i L
n
X
max
?i
?,?
s. t.
i=1
m
X
i=1
? i P + C e e0 e>
?i P > L
0 ?
n
X
i
?i In+1
?0
i=1
0 ? ?i ? C? , i = 1, 2, . . . , n,
m
X
i=1
?i = 1, ?i ? 0, i = 1, 2, . . . , m
(12)
2
0.6
1
1.8
0.4
1.6
0.5
0.2
1.4
0
0
1.2
?0.2
1
?0.5
0.8
?0.4
?1
0.6
?0.6
0.4
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
(a) Overlapped Gaussian
?1
?0.5
0
0.5
(b) Two Circles
1
?0.6
?0.4
?0.2
0
0.2
0.4
0.6
(c) Two Connected Circles
Figure 3: Data distribution of the three synthesized datasets
By solving the above problem, we are able to resolve both ? (corresponding to clustering
memberships) and ? (corresponding to kernel learning) simultaneously.
4
Experiment
We tested the generalized maximum margin clustering algorithm on both synthetic datasets
and real datasets from the UCI repository. Figure 3 gives the distribution of the synthetic
datasets. The four UCI datasets used in our study are ?Vote?, ?Digits?, ?Ionosphere?, and
?Breast?. These four datasets comprise of 218, 180, 351, and 285 examples, respectively,
and each example in these four datasets is represented by 17, 64, 35, and 32 features. Since
the ?Digits? dataset consists of multiple classes, we further decompose it into four datasets
of binary classes that include pairs of digits difficult to distinguish. Both the normalized cut
algorithm [8] and the maximum margin clustering algorithm [1] are used as the baseline.
The RBF kernel is used throughout this study to construct the kernel similarity matrices.
In our first experiment, we examine the optimal performance of each clustering algorithm
by using the optimal kernel width that is acquired through an exhaustive search. The optimal clustering errors of these three algorithms are summarized in the first three columns of
Table 1. It is clear that generalized maximum margin clustering algorithm achieve similar
or better performance than both maximum margin clustering and normlized cut for most
datasets when they are given the optimal kernel matrices. Note that the results of maximum margin clustering are reported for a subset of samples(including 80 instances) in UCI
datasets due to the out of memory problem.
Table 1: Clustering error (%) of normalized cut (NC), maximum margin clustering (MMC),
generalized maximum margin clustering (GMMC) and self-tuning spectral clustering (ST).
Dataset
Optimal Kernel Width
Unsupervised Kernel Learning
NC MMC GMMC GMMC ST (Best k) ST(Worst k)
Two Circles
2
0
0
0
0
50
7
6.25
0
0
1
45
Two Jointed Circles
1.25
2.5
1.25
3.75
5
7.5
Two Gaussian
25
15
9.6
11.90
11
40
Vote
35
10
5.6
5.6
5
50
Digits 3-8
45
31.25
2.2
3
0
47
Digits 1-7
34
1.25
.5
5.6
1.5
50
Digits 2-7
48
3.75
16
12
9
48
Digits 8-9
25
21.25
23.5
27.3
26.5
48
Ionosphere
36.5 38.75
36.1
37
37.5
41.5
Breast
In the second experiment, we evaluate the effectiveness of unsupervised kernel learning. Ten
kernel matrices are created by using the RBF kernel with the kernel width varied from 10%
to 100% of the range of distance between any two examples. We compare the proposed
unsupervised kernel learning to the self-tuning spectral clustering algorithm in [10]. One
of the problem with the self-tuning spectral clustering algorithm is that its clustering error
usually depends on the parameter k, i.e., the number of nearest neighbor used for computing
the kernel width. To provide a full picture of the self-tuning spectral clustering, we vary k
from 1 and 15 , and calculate both best and worst performance using different k. The last
three columns of Table 1 summarizes the clustering errors of generalized maximum margin
clustering and self-tuning spectral clustering with both best and worst k. First, observe
the big gap between best and worst performance of self-tuning spectral clustering with
different choice of k, which implies that this algorithm is sensitive to parameter k. Second,
for most datasets, generalized maximum margin clustering achieves similar performance as
self-tuning spectral clustering with the best k. Furthermore, for a number of datasets, the
unsupervised kernel learning method achieves the performance close to the one using the
optimal kernel width. Both results indicate that the proposed algorithm for unsupervised
kernel learning is effective in identifying appropriate kernels.
5
Conclusion
In this paper, we proposed a framework for the generalized maximum margin clustering.
Compared to the existing algorithm for maximum margin clustering, the new framework
has three advantages: 1) it reduces the number of parameters from n2 to n, and therefore
has a significantly lower computational cost, 2) it allows for clustering boundaries that do
not pass through the origin, and 3) it can automatically identify the appropriate kernel
similarity matrix through unsupervised kernel learning. Our empirical study with three
synthetic datasets and four UCI datasets shows the promising performance of our proposed
algorithm.
References
[1] L. Xu, J. Neufeld, B. Larson, and D. Schuurmans. Maximum margin clustering. In
Advances in Neural Information Processing Systems (NIPS) 17, 2004.
[2] L. Xu and D. Schuurmans. Unsupervised and semi-supervised multi-class support vector machines. In Proceedings of the 20th National Conference on Artificial Intelligence
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[4] R. A. Redner and H. F. Walker. Mixture densities, maximum likelihood and the em
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[5] C. Ding, X. He, H. Zha, M. Gu, and H. Simon. A min-max cut algorithm for graph
partitioning and data clustering. In Proc. IEEE Int?l Conf. Data Mining, 2001.
[6] F. R. Bach and M. I. Jordan. Learning spectral clustering. In Advances in Neural
Information Processing Systems 16, 2004.
[7] R. Jin, C. Ding, and F. Kang. A probabilistic approach for optimizing spectral clustering. In Advances in Neural Information Processing Systems 18, 2006.
[8] J. Shi and J. Malik. Normalized cuts and image segmentation. IEEE Transactions on
Pattern Analysis and Machine Intelligence, 22(8):888?905, 2000.
[9] A. Ng, M. Jordan, and Y. Weiss. On spectral clustering: Analysis and an algorithm.
In Advances in Neural Information Processing Systems 14, 2001.
[10] L. Zelnik-Manor and P. Perona. Self-tuning spectral clustering. In Advances in Neural
Information Processing Systems 17, pages 1601?1608, 2005.
[11] N. Cristianini, J. Shawe-Taylor, A. Elisseeff, and J. S. Kandola. On kernel-target
alignment. In NIPS, pages 367?373, 2001.
[12] X. Zhu, J. Kandola, Z. Ghahramani, and J. Lafferty. Nonparametric transforms of
graph kernels for semi-supervised learning. In Advances in Neural Information Processing Systems 17, pages 1641?1648, 2005.
[13] G. R. G. Lanckriet, N. Cristianini, P. L. Bartlett, Laurent El Ghaoui, and Michael I.
Jordan. Learning the kernel matrix with semidefinite programming. Journal of Machine
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[15] F.R.K. Chung. Spectral Graph Theory. Amer. Math. Society, 1997.
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2,284 | 3,073 | Simplifying Mixture Models
through Function Approximation
Kai Zhang
James T. Kwok
Department of Computer Science and Engineering
The Hong Kong University of Science and Technology
Clear Water Bay, Kowloon, Hong Kong
{twinsen, jamesk}@cse.ust.hk
Abstract
Finite mixture model is a powerful tool in many statistical learning problems.
In this paper, we propose a general, structure-preserving approach to reduce its
model complexity, which can bring significant computational benefits in many
applications. The basic idea is to group the original mixture components into
compact clusters, and then minimize an upper bound on the approximation error
between the original and simplified models. By adopting the L2 norm as the distance measure between mixture models, we can derive closed-form solutions that
are more robust and reliable than using the KL-based distance measure. Moreover,
the complexity of our algorithm is only linear in the sample size and dimensionality. Experiments on density estimation and clustering-based image segmentation
demonstrate its outstanding performance in terms of both speed and accuracy.
1 Introduction
In many statistical learning problems, it is useful to obtain an estimate of the underlying probability
density given a set of observations. Such a density model can facilitate discovery of the underlying
data structure in unsupervised learning, and can also yield, asymptotically, optimal discriminant
procedures [7]. In this paper, we focus on the finite mixture model,
P which describes the distribution
by a mixture of simple parametric functions ?(?)?s as f (x) = nj=1 ?j ?(x, ?j ). Here, ?j is the
Pn
parameter for the jth component, and the mixing parameters ?j ?s satisfy j=1 ?j = 1. The most
common parametric form of ? is the Gaussian, leading to the well-known Gaussian mixtures.
The mixture model has been widely used in clustering and density estimation, where the model
parameters can be estimated by the standard Expectation-Maximization (EM) algorithm. However,
the EM can be prohibitively expensive on large problems [12]. On the other hand, note that in many
learning processes using mixture models (such as particle filtering [6] and non-parametric belief
propagation [13]), the computational requirement is also very demanding due to the large number
of components involved in the model. In this situation, our interest is more on reducing the number
of components for prospective computational saving. Previous works typically employ spatial data
structures, such as the kd-tree [8, 9], for acceleration. Recently, [5] proposes to reduce a large
Gaussian mixture into a smaller one by minimizing a KL-based distance between the two mixtures.
This has been applied with success on hierarchical clustering of scenery images and handwritten
digits.
In this paper, we propose a new algorithm for simplifying a given finite mixture model while preserving its component structures, with application to nonparametric density estimation and clustering.
The idea is to minimize an upper bound on the approximation error between the original and simplified mixture models. By adopting the L2 norm as the error criterion, we can derive closed-form
solutions that are more robust and reliable than using the KL-based distance measures. At the same
time, our algorithm can be applied to general Gaussian kernels, and the complexity is only linear in
the sample size and dimensionality.
The rest of the paper is organized as follows. In Section 2 we describe the proposed approach
in detail, and illustrate its advantages compared with the existing ones. In Section 3, we report
experimental results on simplifying the Parzen window estimator, and color image segmentation
through the mean shift clustering procedure. Section 4 gives some concluding remarks.
2 Approximation Algorithm
Given a mixture model
f (x) =
n
X
?j ?j (x),
(1)
j=1
we assume that the jth component ?j (x) is of the form
?j (x) = |Hj |?1/2 KHj (x ? xj ) ,
(2)
with weight ?j , center xj and covariance matrix Hj . Here, KH (x) = K(H?1/2 x) where K(x)
is the kernel that is bounded and has compact support. Note that for radially symmetric kernels, it
suffices to define K by the profile k such that K(x) = k(kxk2 ). With this notion, the gradient of
the kernel function, KH (x), can be conveniently written as ?x KH (x) = k ? (r)?x r = 2k ? (r)H?1 x,
where r = xH?1 x. Our task is to approximate f with a simpler mixture model
g(x) =
m
X
wi gi (x),
(3)
i=1
with m ? n, where each component gi also takes the form
? i |?1/2 K ? (x ? ti ) ,
gi (x) = |H
Hi
(4)
? i.
with weight wi , center ti , and covariance matrix H
Note that direct approximation of f by g is not feasible, because they involve a large number of
components. Given a distance measure D(?, ?) between functions, the approximation error
?
?
n
m
X
X
E = D(f, g) = D ?
?j ?j ,
wi gi ?
(5)
j=1
i=1
is usually difficult to optimize. However, the problem can be
by minimizing
R very much simplified
2
an upper bound of E. Consider the L2 distance D(?, ?? ) = (?(x) ? ?? (x)) dx, and suppose that
the mixture components {?j }nj=1 are divided into disjoint clusters S1 , . . . , Sm . Then, it is easy to
see that the approximation error E is bounded by
?
?2
?
?2
Z X
n
m
m Z
X
X
X
?wi gi (x) ?
E= ?
?j ?j (x) ?
wi gi (x)? dx ? m
?j ?j (x)? dx.
j=1
i=1
i=1
Denote this upper bound by E = m
Ei =
Z
Pm
i=1
?
j?Si
E i , where
?wi gi (x) ?
X
j?Si
?2
?j ?j (x)? dx.
(6)
Note that E is the sum of the ?local? approximation errors E i ?s. Hence, if we can find a good
representative wi gi for each cluster by minimizing the local approximation error E i , the overall
approximation performance can also be guaranteed. This suggests partitioning the original mixture
components into compact clusters, wherein approximation can then be done much more easily. Our
basic algorithm proceeds as follows:
1. (Section 2.1.1) Partition the set of mixture components (?j ?s) into m clusters where m ? n.
Let Si be the set that indexes all components belonging to the ith cluster.P
2. (Section 2.1.2) For each cluster, approximate the local mixture model j?Si ?j ?j by a single
component wi gi , where gi is defined in (4).
Pm
3. The simplified model g is obtained by g(x) = i=1 wi gi (x).
These steps will be discussed in more detail in the following sections.
2.1 Procedure
2.1.1 Partitioning of Components
In this section, we consider how to group similar components into the same cluster, so that the
subsequent local approximation can be more accurate. A useful algorithm for this task is the classic
vector quantization (VQ) [4], where one iterates between partitioning a set of vectors and finding
the best prototype for each partition until the distortion error converges. By defining a distance
D(?, ?) between mixture components ?j ?s, we can partition the mixture components in a similar
way. However, vector quantization is sensitive to the initial partitioning. So we first introduce a
simple but highly efficient partitioning method called sequential sampling (SS):
1. Randomly select a ?j and add it to the set of representatives R.
2. For all the components (j = 1, 2, . . . , n), do the following
? Compute the distance D (?j , Ri ), where Ri ? R.
? Once if D (?j , Ri ) ? r, where r is a predefined threshold, assign ?j to the representative
Ri , and then process the next component.
? If D (?j , Ri ) > r for all Ri ? R, add ?j as a new representative of R.
3. Terminate when all the components have been processed.
This procedure partitions the components by choosing those ?j ?s that are enough far away as representatives, with a user-defined resolution r. So it is less sensitive to initialization. In practice, we
will first initialize by sequential sampling, and then perform the iterative VQ procedure to further
refine the partition, i.e., find the best representative Ri for each
Pcluster, reassign each component ?j
to the closest representative R?(j) , and iterate until the error j ?j D(?j , R?(j) ) converges.
2.1.2 Local Approximation
In this part, we consider how to obtain a good representative, wi gi in (4), for each local cluster Si .
? i associated with gi . Using the L2 norm,
The task is to determine the unknown variables wi , ti and H
the upper bound (6) of the local approximation error can be written as
?
?2
Z
X
?wi gi (x) ?
Ei =
?j ?j (x)? dx
j?Si
X 2CK ?j k(rij )
CK
? wi
+ ci .
? i |1/2
? i |1/2
|2H
|Hj + H
j?Si
R
R P
Here, CK = k(x? x)dx is a kernel-dependent constant, ci = ( j?Si ?j ?2j (x))2 dx is a data? i )?1 (ti ?xj ).
dependent constant (irrelevant to the unknown variables), and rij = (ti ?xj )? (Hj + H
Here we have assumed that k(a) ? k(b) = k(a + b), which is valid for the Gaussian and negative
exponential kernels. Without this assumption, solutions can still be obtained but are less compact.
=
wi2
? i , one can set the corresponding partial derivatives of E i to
To minimize E i w.r.t. wi , ti and H
zero. However, this leads to a nonlinear system that is quite difficult to solve. Here, we decouple
the relations among these three parameters. First, observe that E i is a quadratic function of wi .
? i and ti , the minimum value of E i can be easily obtained as
Therefore, given H
?
?2
?1/2
X
1
min
? i| 2 ?
? i
? .
Ei
= |H
?j k(rij ) Hj + H
(7)
j?Si
min
The remaining task is to minimize E i
ti = M?1
i
where
Mi =
min
? i . By setting ?t E
w.r.t. ti and H
i i
= 0, we have
X ?j k ? (rij ) (Hj + H
? i )?1 xj
,
? i |1/2
|Hj + H
j?Si
(8)
X ?j k ? (rij ) (Hj + H
? i )?1
.
? i |1/2
|Hj + H
j?Si
? i is fixed, we can obtain ti by starting with an initial
This is an iterative contraction mapping. If H
(0)
? i , we set ? ? E min = 0 and obtain
ti , and then iterate (8) until convergence. Now, to solve H
Hi i
? i = P?1
H
i
X ?j (H
? i + Hj )?1
? i + Hj )?1 H
?i ,
k(rij )Hj + 4(?k ? (rij ))(xj ? ti )(xj ? ti )? (H
? i |1/2
|Hj + H
j?Si
(9)
where
Pi =
X (H
? i + Hj )?1
? k(r ).
? i |1/2 j ij
|Hj + H
j?Si
In summary, we first initialize
P
P
(0)
ti
=
j /( j?Si ?j ),
j?Si ?j x
P
P
(0)
(0)
? (0) =
H
?j Hj + (t ? xj )(t ? xj )? /(
i
i
j?Si
i
j?Si
?j ),
? i are substituted
and then iterate (8) and (9) until convergence. The converged values of ti and H
into ?wi E i = 0 to obtain wi as
1
? i| 2
wi = |2H
X
j?Si
?j k(rij )
.
? i |1/2
|Hj + H
(10)
2.2 Complexity
In the partitioning step, sequential sampling has a complexity of O(dmn), where n is original model
size, m is the number of clusters, and d the dimension. By using a hierarchical scheme [2], this
can be reduced to O(dn
Pm log(m)). The VQ takes O(dnm) time. In the local approximation step,
the complexity is l i=1 ni d3 = lnd3 , where l is the maximum number of iterations needed. In
? i ?s while still obtaining a
practice, we can enforce a diagonal structure on the covariance matrix H
closed-form solution. Hence, the complexity becomes linear in the dimension d instead of cubic.
Summing up these three terms, the overall complexity is O(dn log(m) + dnm + lnd) = O(dn(m +
l)), which is linear in both the data size and dimension (in practice m and l are quite small).
2.3 Remarks
In this section, we discuss some interesting properties of the approximation scheme proposed in
Section 2.1.2. To have better intuitions, we examine the special case of a Parzen window density
estimator [11], where all ?j ?s have the same weights and bandwidths (Hj = H for j = 1, 2, . . . , n).
Equation (9) then reduces to
? i = H + 4H
? i (H
? i + H)?1 Vi ,
H
where
Vi =
P
j?Si
?j (?k ? (rij ))(xj ? ti )(xj ? ti )?
P
.
j?Si ?j k(rij )
(11)
? i of gi can be decomposed into two parts: the bandwidth H of the
It shows that the bandwidth H
original kernel density estimator, and the covariance Vi of the local cluster Si with an adjusting
? i (H
? i + H)?1 . As an illustration, consider the 1-D case where H = h2 , H
? i = h2 .
matrix ?i = 4H
i
4h2
2
2
2
2
i
Then ?i = h2 +h
2 , and hi = h + ?i Vi . Since Vi ? 0 and ?i ? 2, we can see that hi ? h + Vi .
i
Moreover, hi is closely related to the spread of the local cluster. If all the points in Si are located at
the same position (i.e., Vi = 0), then h2i = h2 . Otherwise, the larger the spread of the local cluster,
? i ?s are adaptive to the local data distribution.
the larger is hi . In other words, the bandwidths H
? i = H + Cov[Si ].
Related works in simplifying the mixture models (such as [5]) simply choose H
In comparison, our covariance term Vi is more reliable in that it incorporates distance-based weighting. Interestingly, this is somewhat similar to the bandwidth matrix used in the manifold Parzen
windows [14], which is designed for handling sparse, high-dimensional data more robustly. Note
? i is derived rigorously by minimizing the L2 approximation error. Therefore,
that our choice of H
this coincidence naturally indicates the robustness of the L2 -norm based distance measures. Moreover, note that the adjusting matrix ?i changes not only the scale of the bandwidth, but also its
eigen-structures in an iterative manner. This will be very beneficial in multivariate cases.
Second, in determining the center of gi , (8) can be reduced to
P
?
? i (xj ? ti ) xj
j?Si ?j kH+H
.
ti = P
?
? (xj ? ti )
j?Si ?j kH+H
(12)
i
This can be regarded a mean-shift procedure [1] in the d-dimensional space with kernel K. It
is easy to verify that this iterative procedure is indeed locating the peak of the density function
? i |? 21 P
pi (x) = |H + H
? i (x ? xj ). Note, on the other hand, that what we want to
j?Si KH+H
1 P
approximate originally is the local density fi (x) = |H|? 2 j?Si KH (x ? xj ). In the 1-D case
? i = h2 ), the bandwidth of pi (i.e., h2 + h2 ) is larger than that of fi (i.e., h2 ).
(with H = h2 , and H
i
i
It appears intriguing that on fitting a kernel density fi (x) estimated on the sample set {xj }j?Si ,
one needs to locate the maximum of another density function pi (x), instead of the maximum of
fi (x) itself or simply, the mean of the sample set {xj }j?Si as chosen in [5]. Indeed, these three
choices coincide when the distribution of Si is symmetric and uni-modal, but will differ otherwise.
Intuitively, when the data is asymmetric, the center ti should be biased towards the heavier side of
the data distribution. The maximum of fi (x) thus fails to meet this requirement. On the other hand,
the mean of Si , though biased towards the heavier side, still lacks an accurate control on the degree
of bias. In comparison, our method provides a principled way of selecting the center. Note that
pi (x) has a larger bandwidth than the original fi (x). Therefore, its maximum will move towards the
heavier side of the distribution compared with that of fi (x), with the degree of bias automatically
controlled by the mean shift iterations in (12).
Here, we give an illustration on the performance of the three center selection schemes. Figure 1(a)
shows the histogram of a local cluster Si , whose Parzen window estimator (fi ) is asymmetric. Figure 1(b) plots the corresponding approximation error E i (6) at different bandwidths hi (the remaining
parameter, wi , is set to the optimal value by (10) ). As can be seen, the approximation error of our
method is consistently lower than those of the other two. Moreover, the resultant optimum is also
much lower.
10
35
8
approximation error
histogram
25
20
15
10
7
6
5
4
3
2
1
5
0
0
local maximu
local mean
our method
9
30
1.5
2
2.5
3
x
(a) The histogram of a local cluster
Si and its density fi .
0.05
0.1
0.15
0.2
0.25
0.3
h2i
(b) Approximation error.
Figure 1: Approximation of an asymmetric density using different center selection schemes.
3 Experiments
In this section, we perform experiments to evaluate the performance of our mixture simplification
scheme. We focus on the Parzen window estimator which, on given a set of samples S = {xi }ni=1 in
1 Pn
Rd , can be written as f?(x) = n1 |H|? 2 j=1 KH (x ? xj ) . Note that the Parzen window estimator
is a limiting form of the mixture model, where the number of components equals the data size
and can be quite huge. In Section 3.1, we use the proposed approach to reduce the number of
components in the kernel density estimator, and compare its performance with the algorithm in [5].
Then, in Section 3.2, we perform color image segmentation by running the mean shift clustering
algorithm on the simplified density model.
3.1 Simplifying Nonparametric Density Models
In this section, we reduce the number of kernels in the Parzen window estimator by using the proposed approach and the method in [5]. Experiments are performed on a 1-D set with 1800 samples
6
4
8
N (?2.6, 0.09) + 18
N (?0.8, 0.36) + 18
N (1.7, 0.64), where
drawn from the Gaussian mixture 18
2
2
N (?, ? ) denotes the normal distribution with mean ? and variance ? . The Gaussian kernel with
fixed bandwidth h = 0.3 is used for density estimation. To make the problem more challenging,
we choose m = 5, i.e., only 5 kernels are used to approximate the density. The k-means algorithm
is used for initialization. As can be seen from Figure 2(b), the third Gaussian component has been
broken into two by the method in [5]. In comparison, our result in Figure 2(c) is more reliable.
180
0.45
160
0.4
0.45
0.4
140
0.35
0.35
120
0.3
0.3
100
0.25
0.25
80
0.2
0.2
60
0.15
0.15
40
0.1
0.1
20
0.05
0.05
0
?4
?3
?2
?1
0
1
2
3
4
0
?4
?3
?2
?1
x
(a) Histogram.
0
1
2
3
4
5
x
(b) Result by [5].
0
?4
?3
?2
?1
0
1
2
3
4
5
x
(c) Our result.
Figure 2: Approximate the Parzen window estimator by simplifying mixture models. Green: Parzen
window estimator; black: simplified mixture model; blue-dashed: components of the mixture model.
To have a quantitative evaluation, we randomly generate the 3-Gaussian data 100 times, and compare
the two algorithms (ours and [5]) using the following error criteria: 1) the L2 error (5); 2) the
standard KL distance;
3) the local KL-distance
used in [5]. The local KL-distance between two
Pn
Pm
mixtures, f = j=1 ?j ?j and g = i=1 wi gi , is defined as
d(f, g) =
n
X
?j KL(?j ||g?(j) ),
j=1
where ?(j) is the function that maps each component ?j to the closest representative component
g?(j) such that ?(j) = arg mini=1,2,...,m KL(?j ||gi ).
Results are plotted in Figure 3, where for clarity we order the results in increasing error obtained
by [5]. We can see that under the L2 norm, the error of our algorithm is significantly lower than
that of [5]. Quantitatively, our error is only about 36.61% of that by [5]. On using the standard
KL-distance, our error is about 87.34% of that by [5], where the improvement is less significant.
This is because the KL-distance is sensitive to the tail of the distribution, i.e., a small difference in
the low-density regions may induce a huge KL-distance. As for the local KL-distance, our error is
about 99.35% of that by [5].
3.2 Image Segmentation
The Parzen window estimator can be used to reveal important clustering information, namely that its
modes (or local maxima) correspond to dominant clusters in the data. This property is utilized in the
?5
5
?3
x 10
7
x 10
method in [5]
our method
4.5
3600
method in [5]
our method
method in [5]
our method
3500
6
4
3
2.5
2
1.5
local KL error
KL distance
2
L ?norm error
3400
5
3.5
4
3
2
3300
3200
3100
3000
1
1
2900
0.5
0
0
20
40
60
80
100
number of tests
(a) The L2 distance error.
0
0
20
40
60
x
80
100
2800
0
20
40
60
80
100
number of tests
(b) Standard KL-distance. (c) Local KL-distance defined by [5]
Figure 3: Quantitative comparison of the approximation errors.
mean shift clustering algorithm [1, 3], where every data point is moved along the density gradient
until it reaches the nearest local density maximum. The mean shift algorithm is robust, and can
identify arbitrarily-shaped clusters in the feature space.
Recently, mean shift is applied in color image segmentation and has proven to be quite successful [1].
The idea is to identify homogeneous image regions through clustering in a properly selected feature
space (such as color, texture, or shape). However, mean shift can be quite expensive due to the large
number of kernels involved in the density estimator. To reduce the computational requirement, we
first reduce the density estimator f?(x) to a simpler model g(x) using our simplification scheme, and
then apply the iterative mean shift procedure on the simplified model g(x).
Experiments are performed on a number of benchmark images1 used in [1]. We use the Gaussian
kernel with bandwidth h = 20. The partition parameter is r = 25. For comparison, we also implement the standard mean shift and its fast version using kd-trees (using the ANN library [10]). The
codes are written in C++ and run on a 2.26GHz Pentium-III machine. As the ?true? segmentation
of an image is subjective, so only a visual comparison is intended here.
Table 1: Total wall time (in seconds) on various segmentation tasks, and the number of components in g(x).
image
squirrel
hand
house
lake
data size
60,192 (209?288)
73,386 (243?302)
48,960 (192?255)
262,144 (512?512)
mean shift
standard kd-tree
1215.8
11.94
1679.7
12.92
1284.5
5.16
3343.0
85.65
our method
# components time consumption
81
0.18
120
0.35
159
0.22
440
3.67
Segmentation results are shown in Figures 4. The rows, from top to bottom, are: the original image,
segmentation results by standard mean shift and our approach. We can see that our results are closer
to those by the standard mean shift (applied on the original density estimator), with the number of
components (Table 1) dramatically smaller than the data size n. This demonstrates the success of
our approximation scheme in maintaining the structure of the data distribution using highly compact
models. Our algorithm is also much faster than the standard mean shift and its fast version using kdtrees. The reason is that kd-trees only facilitates range searching but does not reduce the expensive
computations associated with the large number of kernels.
4 Conclusion
Finite mixture is a powerful model in many statistical learning problems. However, the large model
size can be a major hindrance in many applications. In this paper, we reduce the model complexity
by first grouping the components into compact clusters, and then perform local function approximation that minimizes an upper bound of the approximation error. Our algorithm has low complexity,
and demonstrates more reliable performance compared with methods using KL-based distances.
1
http://www.caip.rutgers.edu/?comanici/MSPAMI/msPamiResults.html
Figure 4: Image segmentation by standard mean shift (2nd row), and ours (bottom).
References
[1] D. Comaniciu and P. Meer. Mean shift: A robust approach toward feature space analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24(5):603?619, 2002.
[2] T. Feder and D. Greene. Optimal algorithms for approximate clustering. In Proceedings of ACM Symposium on Theory of Computing, pages 434?444, 1988.
[3] K. Fukunaga and L. Hostetler. The estimation of the gradient of a density function, with applications in
pattern recognition. IEEE Transactions on Information Theory, 21:32?40, 1975.
[4] A. Gersho and R.M. Gray. Vector Quantization and Signal Compression. Kluwer Academic Press, Boston,
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[5] J. Goldberger and S. Roweis. Hierarchical clustering of a mixture model. In Advances in Neural Information Processing Systems 17, pages 505?512. 2005.
[6] B. Han, D. Comaniciu, Y. Zhu, and L. Davis. Incremental density approximation and kernel-based
Bayesian filtering for object tracking. In Proceedings of the International Conference on Computer Vision
and Pattern Recognition, pages 638?644, 2004.
[7] A.J. Izenman. Recent developments in nonparametric density estimation. Journal of the American Statistical Association, 86(413):205?224, 1991.
[8] T. Kanungo, D.M. Mount, N.S. Netanyahu, C.D. Piatko, R. Silverman, and A.Y. Wu. An efficient kmeans clustering algorithm: Analysis and implementation. IEEE Transactions on Pattern Analysis and
Machine Intelligence, 24(7):881?892, 2002.
[9] A.W. Moore. Very fast EM-based mixture model clustering using multiresolution kd-trees. In Advances
in Neural Information Processing Systems 11, pages 543?549, 1998.
[10] D.M. Mount and S. Arya. ANN: A library for approximate nearest neighbor searching. In Proceedings
of Center for Geometric Computing Second Annual Fall Workshop Computational Geometry (available
from http://www.cs.umd.edu/?mount/ANN), 1997.
[11] E. Parzen. On estimation of a probability density function and mode. Annals of Mathematical Statistics,
33:1065?1075, 1962.
[12] K. Popat and R.W. Picard. Cluster-based probability model and its application to image and texture
processing. IEEE Transactions on Image Processing, 6(2):268?284, 1997.
[13] E.B. Sudderth, A. Torralba, W.T. Freeman, and A.S. Willsky. Describing visual scenes using transformed
Dirichlet processes. In Advances in Neural Information Processing Systems 19, 2006.
[14] P. Vincent and Y. Bengio. Manifold Parzen windows. In Advances in Neural Information Processing
Systems 15, 2003.
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2,285 | 3,074 | On Transductive Regression
Corinna Cortes
Google Research
76 Ninth Avenue
New York, NY 10011
[email protected]
Mehryar Mohri
Courant Institute of Mathematical Sciences
and Google Research
251 Mercer Street
New York, NY 10012
[email protected]
Abstract
In many modern large-scale learning applications, the amount of unlabeled data
far exceeds that of labeled data. A common instance of this problem is the transductive setting where the unlabeled test points are known to the learning algorithm. This paper presents a study of regression problems in that setting. It
presents explicit VC-dimension error bounds for transductive regression that hold
for all bounded loss functions and coincide with the tight classification bounds of
Vapnik when applied to classification. It also presents a new transductive regression algorithm inspired by our bound that admits a primal and kernelized closedform solution and deals efficiently with large amounts of unlabeled data. The
algorithm exploits the position of unlabeled points to locally estimate their labels
and then uses a global optimization to ensure robust predictions. Our study also
includes the results of experiments with several publicly available regression data
sets with up to 20,000 unlabeled examples. The comparison with other transductive regression algorithms shows that it performs well and that it can scale to large
data sets.
1 Introduction
In many modern large-scale learning applications, the amount of unlabeled data far exceeds that of
labeled data. Large amounts of digitized data are widely available but the cost of labeling is often
prohibitive since it typically requires human assistance. Semi-supervised learning or transductive
inference leverage unlabeled data to achieve better predictions and are thus particularly relevant to
modern applications. Semi-supervised learning consists of using both labeled and unlabeled data
to find a hypothesis that accurately labels unseen examples. Transductive inference uses the same
information but only aims at predicting the labels of the known unlabeled examples.
This paper deals with regression problems in the transductive setting, which arise in a variety of
contexts. This may be to predict the real-valued labels of the nodes of a known graph in computational biology, or the scores associated to known documents in information extraction problems.
The problem of transduction inference was originally formulated and analyzed by Vapnik [1982]
who described it as a simpler task than the traditional induction treated in machine learning. A
number of recent publications have dealt with the topic of transductive inference [Vapnik, 1998,
Joachims, 1999, Bennett and Demiriz, 1998, Chapelle et al., 1999, Graepel et al., 1999, Schuurmans
and Southey, 2002, Corduneanu and Jaakkola, 2003, Zhu et al., 2004, Lanckriet et al., 2004, Derbeko et al., 2004, Belkin et al., 2004, Zhou et al., 2005]. But, with the exception of [Chapelle et al.,
1999], [Schuurmans and Southey, 2002], and [Belkin et al., 2004], this work has primarily dealt
with classification problems.
We present a specific study of transductive regression. We give new error bounds for transductive
regression that hold for all bounded loss functions and coincide with the tight classification bounds of
Vapnik [1998] when applied to classification. Our results also include explicit VC-dimension bounds
for transductive regression. This contrasts with the original regression bound given by Vapnik [1998]
which assumes a specific condition of global regularity on the class of functions and is based on a
complicated and implicit function of the samples sizes and the confidence parameter. As stated by
Vapnik [1998], this function must be ?tabulated by a computer?.
We also present a new algorithm for transductive regression inspired by our bound which first exploits the position of unlabeled points to locally estimate their labels, and then uses a global optimization to ensure robust predictions. We show that our algorithm admits both a primal and a kernelized closed-form solution. Existing algorithms for the transductive setting require the inversion
of a matrix whose dimension is either the total number of unlabeled and labeled examples [Belkin
et al., 2004], or the total number of unlabeled examples [Chapelle et al., 1999]. This may be prohibitive for many real-world applications with very large amounts of unlabeled examples. One of the
original motivations for our work was to design algorithms dealing precisely with such situations.
When the dimension of the feature space N is not too large, our algorithm provides a very efficient
solution whose cost is dominated by the construction and inversion of an N ? N -matrix. Similarly,
when the number of training points m is small compared to the number of unlabeled points, using
an empirical kernel map, our algorithm requires only constructing and inverting an m ? m-matrix.
Our study also includes the results of our experiments with several publicly available regression data
sets with up to 20,000 unlabeled examples, limited only by the size of the data sets. We compared
our algorithm with those of Belkin et al. [2004] and Chapelle et al. [1999], which are among the
very few algorithms described in the literature dealing specifically with the problem of transductive
regression. The results show that our algorithm performs well in several data sets compared to these
algorithms and that it can scale to large data sets.
The paper is organized as follows. Section 2 describes in more detail the transductive regression
setting we are studying. New generalization error bounds for transductive regression are presented
in Section 3. Section 4 describes and analyzes both the primal and dual versions of our algorithm
and the experimental results of our study are reported in Section 5.
2 Definition of the Problem
Assume that a full sample X of m + u examples is given. The learning algorithm further receives
the labels of a random subset of X of size m which serves as a training sample:
(x1 , y1 ), . . . , (xm , ym ) ? X ? R.
(1)
The remaining u unlabeled examples, xm+1 , . . . , xm+u ? X , serve as test data. The learning
problem that we consider consists of predicting accurately the labels ym+1 , . . . , ym+u of the test
examples. No other test examples will ever be considered. This is a transduction regression problem
[Vapnik, 1998].1 It differs from the standard (induction) regression estimation problem by the fact
that the learning algorithm is given the unlabeled test examples beforehand. Thus, it may exploit
that information and achieve a better result than via the standard induction.
In what follows, we consider a hypothesis space H of real-valued functions for regression estimation. For a hypothesis h ? H, we denote by R0 (h) its mean squared error on the full sample, by
b
R(h)
its error on the training data, and by R(h) the error of h on the test examples:
m+u
1 X
(h(xi )?yi )2 .
u i=m+1
(2)
For convenience, we will sometimes denote by yx = yi the label of a point x = xi ? X .
R0 (h) =
m+u
X
1
(h(xi )?yi )2
m + u i=1
m
1 X
b
R(h)
=
(h(xi )?yi )2
m i=1
R(h) =
3 Transductive Regression Generalization Error
This section presents explicit generalization error bounds for transductive regression.
Vapnik [1998] introduced and analyzed the problem of transduction and presented transductive inference bounds for both classification and regression. His regression bound assumes however a
specific regularity condition on the hypothesis functions leading in particular to a surprising bound
where no error on the training data implies zero generalization error. The bound has the multiplicab
tive form: R(h) ? ?(m, u, d, ?)R(h),
where d is the VC-dimension of the class of hypotheses used
and ? is the confidence parameter. Furthermore, for certain values of the parameters, for example
larger ds or smaller ?s, ? becomes infinite and the bound is ineffective [Vapnik, 1998, page 349]. ?
is also based on a complicated and implicit function of m, u, and ?, which makes its interpretation
difficult. For example, it is hard to analyze the asymptotic behavior of the bound for large u.
1
This is in fact one of the two transduction settings discussed by [Vapnik, 1998], but, under some general
conditions, the results proved with this setting carry over to the other.
Instead, our bounds simply hold for general bounded loss functions and, when applied to classification, coincide with the tight classification bounds of Vapnik [1998]. Our results also include explicit
VC-dimension bounds for transductive regression. To the best of our knowledge, these are the first
general explicit bounds for transductive regression.
? defined as follows. Let ?(?, k) be defined by:
Our first bound uses the function ?
k m+u?k
X
r
m?r
,
?? ? 0, ?k ? N, u? ? k ? m(1 ? ?) + u, ?(?, k) =
(3)
m+u
m
r?I(m,u,?)
r
where I(m, u, k, ?) is the set of integers r such that: k?r
u ? m > ? and max(0, k ? u) ? r ?
min(m, k). ?(?, k) represents the probability of observing a difference in error rate of more than
? between the training and test set when the total
q number of errors is k (see [Cortes and Mohri,
k
? is used in the transductive classi?
?
?, k). ?
2006]). Then ? is defined as ?(?) = maxk ?(
m+u
fication bound of Vapnik [1998] (see [Cortes and Mohri, 2006][Theorem 2]). [Cortes and Mohri,
?
2006][Corollary 2] gives an upper bound on ?.
For any subset X ? ? X , any non-negative real number t ? 0, and hypothesis h ? H, let ?(h, t, X ? )
denote the fraction of the points xi ? X ? , i = 1, . . . , k, such that (h(xi ) ? yi )2 ? t > 0. Thus,
?(h, t, X ? ) represents the error rate over the sample X ? of the classifier that associates to a point x
the value zero if (h(x) ? yx )2 ? t, one otherwise.
Two classifiers associated in this way to ?(h, t, X ) and ?(h? , t? , X ) can be viewed as equivalent if
they label X in an identical way. Since X is finite, there is a finite number of equivalence classes of
such classifiers, we will denote that number by N (m + u).
?
Theorem 1 Let ? > 0, and let ?0 > 0 be the minimum value of ? such that N (m + u)?(?)
? ?, and
assume that the loss function is bounded: for all h ? H and x ? X , (h(x) ? yx )2 ? B 2 , where
B ? R+ . Then, with probability at least 1 ? ?, for all h ? H,
s
2
2 2
u?
B
u?0 B
0
b
b
R(h) ? R(h)
+
+
+ ?0 B R(h)
.
(4)
2(m + u)
2(m + u)
Proof.
For any h ? H, let R1 (h) be defined by:
Z B2 p
R1 (h) =
?(h, t, X ) dt.
(5)
0
By the Cauchy-Schwarz inequality,
R1 (h) ?
Z
B2
0
?(h, t, X ) dt
!1/2
Z
B2
0
1dt
!1/2
=B
Z
B2
0
?(h, t, X ) dt
!1/2
.
(6)
1
Let D denote the uniform probability distribution associated to the sample X . Thus, D(x) = m+u
for all x ? X . Let Prx?D [Ex ] denote the probability of event Ex when x is randomly drawn
according to D. By definition of R0 and the Lebesgue integral, for all h ? H,
Z
Z ?
Z B2
2
2
R0 (h) =
(h(x) ? yx ) D(x) dx =
Pr [(h(x) ? yx ) > t] dt =
?(h, t, X ) dt. (7)
0
X
x?D
0
Similarly, setting Xm = {xi ? X : i ? [1, m]} and Xu = {xi ? X : i ? [m + 1, m + u]}, we have
Z B2
Z B2
b
R(h) =
?(h, t, Xm ) dt and R(h) =
?(h, t, Xu ) dt.
(8)
0
0
In view of Equation 7, Inequality 6 can be rewritten as: R1 (h) ? B
2006][Theorem 2], for all ? > 0 and for any t ? 0,
Pr[ sup
h?H
p
R0 (h). By [Cortes and Mohri,
?(h, t, Xu ) ? ?(h, t, Xm )
?
p
> ?] ? N (m + u)?(?).
?(h, t, X )
(9)
?
Fix ? > 0. Then, with probability at least 1 ? N (m + u)?(?),
for all integers n > 1 and i ? 0,
2
2
?(h, iBn , Xu ) ? ?(h, iBn , Xm )
q
? ?.
2
?(h, iBn , X )
Then, the convergence of the Riemann sums to the integral ensures that
n
n
1X
1X
iB 2
iB 2
b
R(h) ? R(h)
= lim
?(h,
, Xu ) ?
?(h,
, Xm )
n?? n
n
n i=0
n
i=0
r
n
p
1X
iB 2
? ? lim
, X ) = ?R1 (h) ? ?B R0 (h).
?(h,
n?? n
n
i=0
(10)
(11)
(12)
?
Let ? > 0 and select ? = ?0 as the minimum value of ? such that N (m + u)?(?)
? ?, then with
probability at least 1 ? ?,
p
b
R(h) ? R(h)
? ?0 B R0 (h).
(13)
b
Plugging in the following expression of R0 (h) with respect to R(h) and R(h)
m b
u
R(h) +
R(h),
(14)
R0 (h) =
m+u
m+u
and solving the second-degree equation in R(h) yields directly the statement of the theorem.
Theorem 1 provides a general bound on the regression error within the transduction setting. The
theorem can also be used to derive a bound in the classification case by simply setting B = 1. The
resulting bound coincides with the tight classification bound given by Vapnik [1998]. The bound
? and is implicit. The following provides a general and
given by Theorem 1 depends on the function ?
explicit error bound for transduction regression directly expressed in terms of the empirical error,
the number of equivalence N (m + u) or the VC-dimension d, and the sample sizes m and u.
Corollary 1 Let H be a set of hypotheses with VC-dimension d. Assume that the loss function is
bounded: for all h ? H and x ? X , (h(x) ? yx )2 ? B 2 , where B ? R+ . Then, with probability at
least 1 ? ?, for all h ? H,
s
2
2 2
u? B
u?B
b
b
R(h) ? R(h)
+
+
+ ?B R(h)
,
(15)
2(m + u)
2(m + u)
r
q
2(m+u)
2(m+u)
(m+u)e
1
1
with ? =
N
(m
+
u)
+
log
+
log
log
?
d
log
mu
?
mu
d
? .
?
?(?)
? ?. By [Cortes
Proof. By Theorem 1, Inequality 15 holds for all ? >
0 such that N (m+ u)
1 mu
?
and Mohri, 2006][Corollary 2], log N (m + u) ?(?) ? log N (m + u) ? 2 m+u ?2 . Setting log ?
to match this upper bound yields the expression of ? given above. Since N (m + u) is bounded by
the shattering coefficient of H of order m + u, by Sauer?s lemma, log N (m + u) ? d log (m+u)e
.
d
This gives the upper bound on ? in terms of the VC-dimension.
The bound is explicit and can be readily used within the Structural Risk Minimization (SRM) framework, either by using the expression of ? in terms of the VC-dimension, or the tighter expression
with respect to the number of equivalence classes N . In the latter case, a structure of increasing
number of equivalence classes can be constructed as in [Vapnik, 1998, page 360]. A more practical
algorithm inspired by these concepts is described in the next section.
4 Transductive Regression Algorithm
This section presents an algorithm for the transductive regression problem.
Before presenting this algorithm, let us first emphasize that the algorithms introduced for transductive classification problems, e.g., transductive SVMs [Vapnik, 1998, Joachims, 1999], cannot be
readily used for regression. These algorithms typically select the hypothesis h, out of a hypothesis
space H, that minimizes the following optimization function
m
u
X
1 X
?1
?
L
(h(x
),
y
)
+
C
L h(xm+i ), ym+i
,
(16)
min
?(h)
+
C
i
i
?
ym+i
,i=1,...,u
m i=1
u i=1
where ?(h) is a capacity measure term, L is the loss function used, C ? 0 and C ? ? 0 regularization
?
?
parameters, and where the minimum is taken over all possible labels ym+1
, . . . , ym+u
for the test
points. In regression, this scheme would lead to a trivial solution not exploiting the transduction
setting. Indeed, let h0 be the hypothesis minimizing the first two terms, that is the solution of
?
the induction problem. For the particular choice ym+i
= h0 (xm+i ), i = 1, . . . , u, the third term
vanishes. Thus, h0 is also minimizing the sum of all three terms. In two-group classification,
the trivial solution is typically not the solution of the minimization problem because in general
h0 (xm+i ) is not in {0, 1}.
The main idea behind the design of our algorithm is to exploit the additional information provided in
transduction, that is the position of the unlabeled examples. Our algorithm has two stages. The first
stage is based on the position of unlabeled points. For each unlabeled point xi , i = m+1, . . . , m+u,
a local estimate label y?i is determined using the labeled points in the neighborhood of xi . In the
second stage, a global hypothesis h is found that best matches all labels, those of the training data
and the estimate labels y?i .
This second stage is critical and distinguishes our method from other suggested ones. While using
local information to determine labels is important (see for example the discussion of Vapnik [1998]),
it is not sufficient for a robust prediction. A global estimate of all labels is needed to make predictions
less vulnerable to noise.
4.1 Local Estimates
Let ? be a feature mapping from X to a vector space F provided with a norm. We fix a radius
r ? 0 and consider for all x? ? Xu , the ball of radius r centered in ?(x? ), denoted by B(?(x? ), r).
This defines the neighborhood of the image of each unlabeled point. A single radius r is used for all
neighborhoods to limit the number of parameters for the algorithm. Labeled points x ? Xm whose
images ?(x) fall within the neighborhood of ?(x? ), x? ? Xu , help determine an estimate label of x? .
With a very large radius r, the labels of all training examples contribute to the definition of the local
estimates. But, with smaller radii, only a limited number of computations are needed. When no such
labeled point exists in the neighborhood of x? ? Xu , which depends on the radius r selected, x? is
disregarded in both training stages of the algorithm.
There are many possible ways to define the estimate label of x? ? Xu based on the neighborhood points. One simple way consists of defining it as the weighted average of the neighborhood labels yx , where the weights may be defined as the inverse of distances of ?(x) to
?(x? ), or as similarity measures K(x, x? ) when a positive definite kernel K is associated to ?.
Thus, when the set of labeled points with images in the neighborhood of ?(x? ) is not empty,
I = {i ? [1, m] : ?(xi ) ? B(?(x? ), r)} 6= ?, the estimate label y?x? of x? ? Xu can be given by:
X wi y?i
P
with wi?1 = k?(x? ) ? ?(xi )k ? r or wi = K(x? , xi ).
(17)
y?x? =
i wi
i?I
The estimate labels can also be obtained as the solution of a local linear or kernel ridge regression,
which is what we used in most of our experiments.
In practice, with a relatively small radius r, the computation of an estimated label y?i depends only
on a limited number of labeled points and their labels, and is quite efficient.
4.2 Global Optimization
The second stage of our algorithm consists of selecting a hypothesis h that fits best the labels of
the training points and the estimate labels provided in the first stage. As suggested by Corollary 1,
hypothesis spaces with a smaller number of equivalence classes guarantee a better generalization
error. The bound also suggests reducing the empirical error. This leads us to consider the following
objective function
2
G = ||w|| + C
m
X
i=1
2
(h(xi ) ? yi ) + C
?
m+u
X
(h(xi ) ? y?i )2 ,
(18)
i=m+1
where h is as a linear function with weight vector w ? F : ?x ? X , h(x) = w ? ?(x), and where
C ? 0 and C ? ? 0 are regularization parameters. The first two terms of the objective function
coincide with those used in standard (kernel) ridge regression. The third term, which restricts the
estimate error, can be viewed as imposing a smaller number of equivalence classes on the hypothesis
space as suggested by the error bound of Corollary 1. The constraint explicitly exploits knowledge
about the location of all the test points, and limits the range of the hypothesis at these locations,
thereby reducing the number of equivalence classes. Our algorithm can be viewed as a generalization
of (kernel) ridge regression to the transductive setting. In the following, we will show that this
generalized optimization problem admits a closed-form solution and a natural kernel-based solution.
4.2.1 Primal solution
Let N be the dimension of the feature space and let W ? RN ?1 denote the column matrix whose
components are the coordinates of w, Y ? Rm?1 the column matrix whose components are the
labels yi of the training examples, and Y? ? Ru?1 the column-matrix whose components are the
estimated labels y?i of the test examples. Let X = [?(x1 ), . . . , ?(xm )] ? RN ?m denote the matrix
whose columns are the components of the images by ? of the training examples, and similarly
X? = [?(xm+1 ), . . . , ?(xm+u )] ? RN ?u the matrix corresponding to the test examples. G can
then be rewritten as:
G = kWk2 + CkX? W ? Yk2 + C ? kX?? W ? Y? k2 .
(19)
G is convex and differentiable and its gradient is given by
?G = 2W + 2C X(X? W ? Y) + 2C ? X? (X?? W ? Y? ).
(20)
?
The matrix W minimizing G is the unique solution of ?G = 0. Since (IN + C XX + C X X?? )
is invertible, it is given by the following expression
W = (IN + C XX? + C ? X? X?? )?1 (C XY + C ? X? Y? ).
?
?
(21)
N ?N
This gives a closed-form solution in the primal space based on the inversion of a matrix in R
.
Let T (N ) be the time complexity of computing the inverse of a matrix in RN ?N . T (N ) = O(N 3 )
using standard methods or T (N ) = O(N 2.376 ) with the method of Coppersmith and Winograd. The
time complexity of the computation of W from X, X? , Y, and Y? is thus in O(T (N )+(m+u)N 2).
When the dimension N of the feature space is small compared to the number of examples m + u,
which is typical in modern learning applications where u is large, this method remains practical and
leads to a very efficient computation. The use of the so-called empirical kernel map [Sch?olkopf and
Smola, 2002] also makes this method very attractive. Given a kernel K, the empirical kernel feature
vector associated to x is the m-dimensional vector ?(x) = [K(x, x1 ), . . . , K(x, xm )]? . Thus, the
dimension of the feature space is then N = m. For relatively small m, even for very large values of
u with respect to m, the solution is efficiently computable and yet benefits from the use of kernels.
This computational advantage is not shared by other methods such as the manifold regularization
techniques [Belkin et al., 2004], or even by the regression technique described by [Chapelle et al.,
1999], despite it is based on a primal method (we have derived a dual version of that method as well,
see Section 5) since it requires among other things the inversion of a matrix in Ru?u .
Once W is computed, prediction can be done by computing X?? W in time O(uN ).
4.2.2 Dual solution
The computation can also be done in the dual space, which is useful in the case of very highdimensional feature spaces. Let MX ? RN ?(m+u) and MY ? R(m+u)?1 be the matrices defined
by:
?
?
?
CY
?
?
?
M
=
.
(22)
MX =
CX
C X
Y
C ? Y?
?1
Then, Equation 21 can be rewritten as: W = (IN + MX M?
MX MY . To determine the dual
X)
solution, observe that
?
?1
?1
M?
= (M?
M?
X (MX MX + ?IN )
X MX + ?Im+u )
X,
(23)
(m+u)?(m+u)
where Im+u denotes the identity matrix of R
. This can be derived without difficulty
?1
from a series expansion of (MX M?
+
?I
)
.
Thus,
W
can also be computed via:
N
X
W = MX (Im+u + K)?1 MY ,
(24)
where K is the Gram matrix K = M?
Let K21 ? Ru?m and K22 ? Ru?u
X MX .
the sub-matrices of the Gram K defined by: K21 = (K(xm+i , xj )1?i?u,1?j?m ) and K22
(K(xm+i , xm+j )1?i,j?u ) and let K2 ? Ru?(m+u) be the matrix defined by:
K2 =
?
C K21
?
C ? K22 = X?? MX .
be
=
(25)
Dataset
Boston Housing [13]
California Housing [8]
kin-32fh [32]
Elevators [18]
No. of unlab.
points
25
500
2,500
5,000
20,000
2,500
8,000
500
2500
15,000
Relative improvement in MSE (%)
Our algorithm Chapelle et al. [1999] Belkin et al. [2004]
20.2?14.7
4.3?11.3
2.4?5.4
8.4?6.9
2.7?3.0
3.9?12.3
25.9?8.3
0.2?0.3
0.0?0.0
17.2?8.7
0.0?0.0
0.0?0.0
22.0?11.0
?
?
9.4?3.7
2.2?2.6
2.7?3.1
18.4?5.9
0.5?0.5
0.9?0.7
14.4?10.4
1.5?2.7
2.6 ?7.7
9.0?6.9
2.2?2.9
0.0?0.0
9.7?5.8
?
?
Table 1: Transductive regression experiments. The number in brackets after the name indicates the input
dimensionality of the data set. The number of training examples was m = 481 for the Boston Housing data
set, m = 25 for the other tasks. The number of unlabeled examples was u = 25 for the Boston Housing data
set and varied from u = 500 to the maximum of 20,000 examples for the California Housing data set. For
u ? 10,000, the algorithms of Chapelle et al. [1999] and Belkin et al. [2004] did not terminate within the time
period of our experiments.
Then, predictions can be made using kernel functions alone since X ?? W can be computed by:
X?? W = X?? MX (Im+u + K)?1 MY = K2 (Im+u + K)?1 MY .
(26)
When the dimension of the feature space N is very large with respect to the total number of examples, this can lead to a faster computation of the solution. (Im+u + K)?1 MY can be computed in
O(T (m + u) + (m + u)2 tK ) and predictions are computed in time O(u (m + u)), where tK is the
time complexity of the computation of K(x, x), x, x? ? X . As already pointed in the description of
the local estimates, in practice, some unlabeled points are disregarded in the training phases because
no labeled point falls in their neighborhood. Thus, instead of u, a smaller number of unlabeled
examples u? ? u determines the computational cost.
5 Experimental Results
This section reports the results of our experiments with the transductive regression algorithm just
presented with several data sets. For comparison, we also implemented the algorithm of Chapelle
et al. [1999] and that of Belkin et al. [2004], which are among the very few algorithms described
in the literature dealing specifically with the problem of transductive regression. For the algorithm
of Chapelle et al. [1999], we in fact derived and implemented a dual solution not described in the
original paper. With the notation used in that paper, it can be shown that
?K
? ? (K
?K
? ? + ?I)?1 .
C=I?K
(27)
Our comparisons were made using several publicly available regression data sets: Boston Housing,
kin-32fh a data set in the Kinematics family with high unpredictability or noise, California Housing,
and Elevators [Torgo, 2006]. For the Boston Housing data set, we used the same partitioning of the
training and test sets as in [Chapelle et al., 1999]: 481 training examples and 25 test examples. The
input variables were normalized to have mean zero and a variance one. For the kin-32fh, California
Housing, and Elevators data sets, 25 training examples were used with varying (large) amounts of
test examples: 2,500 and 8,000 for kin-32fh; from 500 up to 20,000 for California Housing; and
from 500 to 15,000 for Elevators. The experiments were repeated for 100 random partitions of
training and test sets.
The kernels used with all algorithms were Gaussian kernels. To measure the improvement produced
by the transductive inference algorithms, we used kernel ridge regression as a baseline. The optimal
values for the width of the Gaussian ? and the ridge C1 were determined using cross-validation.
These parameters were then fixed at these values. The remaining parameters for our algorithm, r
and C ? , were determined using a grid search and cross-validation. The parameters of the algorithms
of Chapelle et al. [1999] and Belkin et al. [2004] were determined in the same way. Alternatively, the
parameters could be selected using the explicit VC-dimension generalization bound of Corollary 1.
For our algorithm, we found the best values of r to be typically among the 2.5% smallest distances
between training and test points. Thus, each estimate label was determined by only a small number
of labeled points.
For our algorithm, we experimented both with the dual solution using Gaussian kernels, and the
primal solution with an empirical Gaussian kernel map as described in Section 4.2.1. The results
obtained were very similar, however the primal method was dramatically faster since it required the
inversion of relatively small-dimensional matrices even for a large number of unlabeled examples.
For consistency, all the results reported for our method relate to the dual solution, except from those
with very large u, e.g. u ? 10,000, where the dual method was too time-consuming.
Table 1 shows the results of our experiments. For each data set and each algorithm, the relative
improvement in mean squared error (MSE) with respect to the baseline averaged over the random
partitions is indicated, followed by its standard deviation. Some improvements were small or not
statistically significant. In general, we observed no significant performance improvement over the
baseline on any of these data sets using the Laplacian regularized least squares method of Belkin
et al. [2004]. We note that, while positive classification results have been previously reported for this
algorithm, no transductive regression experimental result seems to have been published for it. Our
results for the method of Chapelle et al. [1999] match those reported by the authors for the Boston
Housing data set (both absolute and relative MSE).
Our algorithm achieved a significant improvement of the MSE in all data sets and for different
amounts of unlabeled data and was shown to be practical for large data sets of 20,000 test examples.
This matches many real-world situations where amount of unlabeled data is orders of magnitude
larger than that of labeled data.
6 Conclusion
We presented a general study of transductive regression. We gave new and general explicit error
bounds for transductive regression and described a simple and general algorithm inspired by our
bound that can scale to relatively large data sets. The results of experiments show that our algorithm
achieves a smaller error in several tasks compared to other previously published algorithms for
transductive regression.
The problem of transductive regression arises in a variety of learning contexts, in particular for
learning node labels of a very large graphs such as the web graph. This leads to computational
problems that may require approximations or new algorithms. We hope that our study will be useful
for dealing with these and other similar transduction regression problems.
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2,286 | 3,075 | Correcting Sample Selection Bias by Unlabeled Data
Jiayuan Huang
School of Computer Science
Univ. of Waterloo, Canada
[email protected]
Alexander J. Smola
NICTA, ANU
Canberra, Australia
[email protected]
Karsten M. Borgwardt
Ludwig-Maximilians-University
Munich, Germany
[email protected]
Arthur Gretton
MPI for Biological Cybernetics
T?ubingen, Germany
[email protected]
Bernhard Sch?olkopf
MPI for Biological Cybernetics
T?ubingen, Germany
[email protected]
Abstract
We consider the scenario where training and test data are drawn from different
distributions, commonly referred to as sample selection bias. Most algorithms
for this setting try to first recover sampling distributions and then make appropriate corrections based on the distribution estimate. We present a nonparametric
method which directly produces resampling weights without distribution estimation. Our method works by matching distributions between training and testing
sets in feature space. Experimental results demonstrate that our method works
well in practice.
1 Introduction
The default assumption in many learning scenarios is that training and test data are independently
and identically (iid) drawn from the same distribution. When the distributions on training and test
set do not match, we are facing sample selection bias or covariate shift. Specifically, given a domain
of patterns X and labels Y, we obtain training samples Z = {(x1 , y1 ), . . . , (xm , ym )} ? X ? Y from
?
a Borel probability distribution Pr(x, y), and test samples Z ? = {(x?1 , y1? ), . . . , (x?m? , ym
? )} ? X?Y
?
drawn from another such distribution Pr (x, y).
Although there exists previous work addressing this problem [2, 5, 8, 9, 12, 16, 20], sample selection
bias is typically ignored in standard estimation algorithms. Nonetheless, in reality the problem
occurs rather frequently : While the available data have been collected in a biased manner, the test is
usually performed over a more general target population. Below, we give two examples; but similar
situations occur in many other domains.
1. Suppose we wish to generate a model to diagnose breast cancer. Suppose, moreover, that most
women who participate in the breast screening test are middle-aged and likely to have attended the
screening in the preceding 3 years. Consequently our sample includes mostly older women and
those who have low risk of breast cancer because they have been tested before. The examples do not
reflect the general population with respect to age (which amounts to a bias in Pr(x)) and they only
contain very few diseased cases (i.e. a bias in Pr(y|x)).
2. Gene expression profile studies using DNA microarrays are used in tumor diagnosis. A common
problem is that the samples are obtained using certain protocols, microarray platforms and analysis
techniques. In addition, they typically have small sample sizes. The test cases are recorded under
different conditions, resulting in a different distribution of gene expression values.
In this paper, we utilize the availability of unlabeled data to direct a sample selection de-biasing
procedure for various learning methods. Unlike previous work we infer the resampling weight directly by distribution matching between training and testing sets in feature space in a non-parametric
manner. We do not require the estimation of biased densities or selection probabilities [20, 2, 12], or
the assumption that probabilities of the different classes are known [8]. Rather, we account for the
difference between Pr(x, y) and Pr? (x, y) by reweighting the training points such that the means
of the training and test points in a reproducing kernel Hilbert space (RKHS) are close. We call this
reweighting process kernel mean matching (KMM). When the RKHS is universal [14], the population solution to this miminisation is exactly the ratio Pr? (x, y)/ Pr(x, y); however, we also derive a
cautionary result, which states that even granted this ideal population reweighting, the convergence
of the empirical means in the RKHS depends on an upper bound on the ratio of distributions (but
not on the dimension of the space), and will be extremely slow if this ratio is large.
The required optimisation is a simple QP problem, and the reweighted sample can be incorporated straightforwardly into several different regression and classification algorithms. We apply our
method to a variety of regression and classification benchmarks from UCI and elsewhere, as well as
to classification of microarrays from prostate and breast cancer patients. These experiments demonstrate that KMM greatly improves learning performance compared with training on unweighted data,
and that our reweighting scheme can in some cases outperform reweighting using the true sample
bias distribution.
Key Assumption 1: In general, the estimation problem with two different distributions Pr(x, y)
and Pr? (x, y) is unsolvable, as the two terms could be arbitrarily far apart. In particular, for arbitrary Pr(y|x) and Pr? (y|x), there is no way we could infer a good estimator based on the training
sample. Hence we make the simplifying assumption that Pr(x, y) and Pr? (x, y) only differ via
Pr(x, y) = Pr(y|x) Pr(x) and Pr(y|x) Pr? (x). In other words, the conditional probabilities of y|x
remain unchanged (this particular case of sample selection bias has been termed covariate shift
[12]). However, we will see experimentally that even in situations where our key assumption is not
valid, our method can nonetheless perform well (see Section 4).
2 Sample Reweighting
We begin by stating the problem of regularized risk minimization. In general a learning method
minimizes the expected risk
R[Pr, ?, l(x, y, ?)] = E(x,y)?Pr [l(x, y, ?)]
(1)
of a loss function l(x, y, ?) that depends on a parameter ?. For instance, the loss function could
be the negative log-likelihood ? log Pr(y|x, ?), a misclassification loss, or some form of regression
loss. However, since typically we only observe examples (x, y) drawn from Pr(x, y) rather than
Pr? (x, y), we resort to computing the empirical average
m
1 X
Remp [Z, ?, l(x, y, ?)] =
l(xi , yi , ?).
(2)
m i=1
To avoid overfitting, instead of minimizing Remp directly we often minimize a regularized variant
Rreg [Z, ?, l(x, y, ?)] := Remp [Z, ?, l(x, y, ?)] + ??[?], where ?[?] is a regularizer.
2.1 Sample Correction
The problem is more involved if Pr(x, y) and Pr? (x, y) are different. The training set is drawn from
Pr, however what we would really like is to minimize R[Pr? , ?, l] as we wish to generalize to test
examples drawn from Pr? . An observation from the field of importance
is thati
h sampling
?
(x,y)
R[Pr ? , ?, l(x, y, ?)] = E(x,y)?Pr? [l(x, y, ?)] = E(x,y)?Pr Pr
l(x,
y, ?)
(3)
Pr(x,y)
| {z }
:=?(x,y)
= R[Pr, ?, ?(x, y)l(x, y, ?)],
(4)
provided that the support of Pr? is contained in the support of Pr. Given ?(x, y), we can thus
compute the risk with respect to Pr? using Pr. Similarly, we can estimate the risk with respect to
Pr? by computing Remp [Z, ?, ?(x, y)l(x, y, ?)].
The key problem is that the coefficients ?(x, y) are usually unknown, and we need to estimate them
from the data. When Pr and Pr? differ only in Pr(x) and Pr? (x), we have ?(x, y) = Pr? (x)/Pr(x),
where ? is a reweighting factor for the training examples. We thus reweight every observation
(x, y) such that observations that are under-represented in Pr obtain a higher weight, whereas overrepresented cases are downweighted.
Now we could estimate Pr and Pr? and subsequently compute ? based on those estimates. This is
closely related to the methods in [20, 8], as they have to either estimate the selection probabilities
or have prior knowledge of the class distributions. Although intuitive, this approach has two major
problems: first, it only works whenever the density estimates for Pr and Pr? (or potentially, the selection probabilities or class distributions) are good. In particular, small errors in estimating Pr can
lead to large coefficients ? and consequently to a serious overweighting of the corresponding observations. Second, estimating both densities just for the purpose of computing reweighting coefficients
may be overkill: we may be able to directly estimate the coefficients ?i := ?(xi , yi ) without having
to estimate the two distributions. Furthermore, we can regularize ?i directly with more flexibility,
taking prior knowledge into account similar to learning methods for other problems.
2.2 Using the sample reweighting in learning algorithms
Before we describe how we will estimate the reweighting coefficients ?i , let us briefly discuss how
to minimize the reweighted regularized risk
m
1 X
Rreg [Z, ?, l(x, y, ?)] :=
?i l(xi , yi , ?) + ??[?],
(5)
m i=1
in the classification and regression settings (an additional classification method is discussed in the
accompanying technical report [7]).
Support Vector Classification: Utilizing the setting of [17]we can have the following minimization
problem (the original SVMs can be formulated in the same way):
m
X
1
2
?i ?i
(6a)
minimize k?k + C
?,?
2
i=1
subject to h?(xi , yi ) ? ?(xi , y), ?i ? 1 ? ?i /?(yi , y) for all y ? Y, and ?i ? 0.
(6b)
?
Here, ?(x, y) is a feature map from X ? Y into a feature space F, where ? ? F and ?(y, y ) denotes
a discrepancy function between y and y ? . The dual of (6) is given by
m
m
X
X
1
minimize
?iy ?jy? k(xi , y, xj , y ? ) ?
?iy
(7a)
?
2
i=1;y?Y
i,j=1;y,y ? ?Y
X
subject to ?iy ? 0 for all i, y and
?iy /?(yi , y) ? ?i C.
(7b)
y?Y
?
?
?
?
Here k(x, y, x , y ) := h?(x, y), ?(x , y )i denotes the inner product between the feature maps. This
generalizes the observation-dependent binary SV classification described in [10]. Modifications of
existing solvers, such as SVMStruct [17], are straightforward.
2
2
Penalized LMS Regression: Assume l(x, y, ?) = (y ? h?(x), ?i) and ?[?] = k?k . Here we
m
minimize
X
2
?i (yi ? h?(xi ), ?i)2 + ? k?k .
(8)
i=1
Denote by ?? the diagonal matrix with diagonal (?1 , . . . , ?m ) and let K ? Rm?m be the kernel
? ?
matrix Kij = k(xi , xj ). In this case minimizing (8) is equivalent to minimizing (y ? K?)? ?(y
?
?
K?) + ?? K? with respect to ?. Assuming that K and ? have full rank, the minimization yields
? = (????1 + K)?1 y. The advantage of this formulation is that it can be solved as easily as solving
the standard penalized regression problem. Essentially, we rescale the regularizer depending on the
pattern weights: the higher the weight of an observation, the less we regularize.
3 Distribution Matching
3.1 Kernel Mean Matching and its relation to importance sampling
Let ? : X ? F be a map into a feature space F and denote by ? : P ? F the expectation operator
?(Pr) := Ex?Pr(x) [?(x)] .
(9)
Clearly ? is a linear operator mapping the space of all probability distributions P into feature space.
Denote by M(?) := {?(Pr) where Pr ? P} the image of P under ?. This set is also often referred
to as the marginal polytope. We have the following theorem (proved in [7]):
Theorem 1 The operator ? is bijective if F is an RKHS with a universal kernel k(x, x? ) =
h?(x), ?(x? )i in the sense of Steinwart [15].
The use of feature space means to compare distributions is further explored in [3]. The practical
consequence of this (rather abstract) result is that if we know ?(Pr? ), we can infer a suitable ? by
solving the following minimization problem:
minimize
?(Pr ? ) ? Ex?Pr(x) [?(x)?(x)]
subject to ?(x) ? 0 and Ex?Pr(x) [?(x)] = 1. (10)
?
This is the kernel mean matching (KMM) procedure. For a proof of the following (and further
results in the paper) see [7].
Lemma 2 The problem (10) is convex. Moreover, assume that Pr? is absolutely continuous with
respect to Pr (so Pr(A) = 0 implies Pr? (A) = 0). Finally assume that k is universal. Then the
solution ?(x) of (10) is P r? (x) = ?(x)P r(x).
3.2 Convergence of reweighted means in feature space
Lemma 2 shows that in principle, if we knew Pr and ?[Pr? ], we could fully recover Pr? by solving
a simple quadratic program. In practice, however, neither ?(Pr? ) nor Pr is known. Instead, we only
have samples X and X ? of size m and m? , drawn iid from Pr and Pr? respectively.
Naively we could just replace the expectations in (10) by empirical averages and hope that the
resulting optimization problem provides us with a good estimate of ?. However, it is to be expected
that empirical averages will differ from each other due to finite sample size effects. In this section,
we explore two such effects. First, we demonstrate that in the finite sample case, for a fixed ?, the
empirical estimate of the expectation of ? is normally distributed:
this provides a natural limit on
R
the precision with which we should enforce the constraint ?(x)d Pr(x) = 1 when using empirical
expectations (we will return to this point in the next section).
Lemma 3 If ?(x) ? [0, B] is some fixed function of x ? X,P
then given xi ? Pr iid such that ?(xi )
1
has finite mean and non-zero variance, the sample mean m
i ?(xi ) converges in distribution to a
R
Gaussian with mean ?(x)d Pr(x) and standard deviation bounded by 2?Bm .
This lemma is a direct consequence of the central limit theorem [1, Theorem 5.5.15].
Alternatively,
?
it is straightforward to get a large deviation bound that likewise converges as 1/ m [6].
Our second result demonstrates the deviation between the empirical means of Pr? and ?(x) Pr in
feature space, given ?(x) is chosen perfectly in the population sense. In particular, this result shows
that convergence of these two means will be slow if there is a large difference in the probability mass
of Pr? and Pr (and thus the bound B on the ratio of probability masses is large).
Lemma 4 In addition to the Lemma 3 conditions, assume that we draw X ? := {x?1 , . . . , x?m? } iid
from X using Pr? = ?(x) Pr, and k?(x)k ? R for all x ? X. Then with probability at least 1 ? ?
m
m?
1 X
p
p
1 X
?(xi )?(xi ) ? ?
?(x?i )
? 1 + ?2 log ?/2 R B 2 /m + 1/m?
m i=1
m i=1
(11)
Note that this lemma shows that for a given ?(x), which is correct in the population sense, we can
bound the deviation between the feature space mean of Pr? and the reweighted feature space mean
of Pr. It is not a guarantee that we will find coefficients ?i that are close to ?(xi ), but it gives us a
useful upper bound on the outcome of the optimization.
p
Lemma 4 implies that we have O(B 1/m + 1/m? B 2 ) convergence in m, m? and B. This means
that, for very different distributions we need a large equivalent sample size to get reasonable convergence. Our result also implies that it is unrealistic to assume that the empirical means (reweighted
or not) should match exactly.
3.3 Empirical KMM optimization
To find suitable values of ? ? RmPwe want to minimize the discrepancy between means subject
m
1
to constraints ?i ? [0, B] and | m
i=1 ?i ? 1| ? ?. The former limits the scope of discrepancy
between Pr and Pr? whereas the latter ensures that the measure ?(x) Pr(x) is close to a probability
distribution. The objective function is given by the discrepancy term between the two empirical
P m?
m
?
means. Using Kij := k(xi , xj ) and ?i := m
?
j=1 k(xi , xj ) one may check that
m
m?
1 X
2
1
1 X
2
?i ?(xi ) ? ?
?(x?i )
= 2 ? ? K? ? 2 ?? ? + const.
m i=1
m i=1
m
m
We now have all necessary ingredients to formulate a quadratic problem to find suitable ? via
m
X
1
?i ? m ? m?.
(12)
minimize ? ? K? ? ?? ? subject to ?i ? [0, B] and
?
2
i=1
?
In accordance with Lemma 3, we conclude that a good choice of ? should be O(B/ m). Note
that (12) is a quadratic program which can be solved efficiently using interior point methods or any
other successive optimization procedure. We also point out that (12) resembles Single Class SVM
[11] using the ?-trick. Besides the approximate equality constraint, the main difference is the linear
correction term by means of ?. Large values of ?i correspond to particularly important observations
xi and are likely to lead to large ?i .
4 Experiments
4.1 Toy regression example
Our first experiment is on toy data, and is intended mainly to provide a comparison with the approach
of [12]. This method uses an information criterion to optimise the weights, under certain restrictions
on Pr and Pr? (namely, Pr? must be known, while Pr can be either known exactly, Gaussian with
unknown parameters, or approximated via kernel density estimation).
Our data is generated according to the polynomial regression example from [12, Section 2], for
which Pr ? N(0.5, 0.52 ) and Pr? ? N(0, 0.32 ) are two normal distributions. The observations are
generated according to y = ?x + x3 , and are observed in Gaussian noise with standard deviation
0.3 (see Figure 1(a); the blue curve is the noise-free signal).
We sampled 100 training (blue circles) and testing (red circles) points from Pr and Pr? respectively.
We attempted to model the observations with a degree 1 polynomial. The black dashed line is a
best-case scenario, which is shown for reference purposes: it represents the model fit using ordinary
least squared (OLS) on the labeled test points. The red line is a second reference result, derived
only from the training data via OLS, and predicts the test data very poorly. The other three dashed
lines are fit with weighted ordinary least square (WOLS), using one of three weighting schemes: the
ratio of the underlying training and test densities, KMM, and the information criterion of [12]. A
summary of the performance over 100 trials is shown in Figure 1(b). Our method outperforms the
two other reweighting methods.
0.6
0.4
1
0.2
Sum of square loss
0
?0.2
?0.4
?0.6
?0.8
?1
?1.2
?1.4
?0.4
(a)
x from q0
true fitting model
OLS fitting x
q0
x from q1
OLS fitting xq1
0
0.2
0.6
0.4
0.2
WOLS by ratio
WOLS by KMM
WOLS by min IC
?0.2
0.8
0
0.4
0.6
0.8
1
1.2
ratio
KMM
IC
OLS
(b)
Figure 1: (a) Polynomial models of degree 1 fit with OLS and WOLS;(b) Average performances of three
WOLS methods and OLS on the test data in (a). Labels are Ratio for ratio of test to training density; KMM for
our approach; min IC for the approach of [12]; and OLS for the model trained on the labeled test points.
4.2 Real world datasets
We next test our approach on real world data sets, from which we select training examples using a
deliberately biased procedure (as in [20, 9]). To describe our biased selection scheme, we need to
define an additional random variable si for each point in the pool of possible training samples, where
si = 1 means the ith sample is included, and si = 0 indicates an excluded sample. Two situations
are considered: the selection bias corresponds to our assumption regarding the relation between
the training and test distributions, and P (si = 1|xi , yi ) = P (si |xi ); or si is dependent only on
yi , i.e. P (si |xi , yi ) = P (si |yi ), which potentially creates a greater challenge since it violates our
key assumption 1. In the following, we compare our method (labeled KMM) against two others: a
baseline unweighted method (unweighted), in which no modification is made, and a weighting by
the inverse of the true sampling distribution (importance sampling), as in [20, 9]. We emphasise,
however, that our method does not require any prior knowledge of the true sampling probabilities.
2
In our experiments, we used a Gaussian kernel
? exp(??kx
? i ? xj k ) in our kernel classification and
regression algorithms, and parameters ? = ( m ? 1)/ m and B = 1000 in the optimization (12).
0.2
0.35
unweighted
importance sampling
KMM
0.18
unweighted
importance sampling
KMM
0.3
0.16
0.25
0.12
test error
test error
0.14
0.1
0.08
0.06
0.2
0.15
0.1
0.04
0.05
0.02
0
1
2
3
4
5
6
biased feature
7
8
0
9
(a) Simple bias on features
0.2
0.3
0.4
training set proportion
0.5
(b) Joint bias on features
12
0.07
unweighted
importance sampling
KMM
0.06
0.1
optimal weights
inverse of true sampling
probabilites
10
0.05
test error
8
0.04
6
0.03
4
0.02
2
0.01
0
1
2
3
4
training set proportion
(c) Bias on labels
0
0
5
(d)
?
10
20
30
40
50
vs inverse sampling prob.
Figure 2: Classification performance analysis on breast cancer dataset from UCI.
4.2.1 Breast Cancer Dataset
This dataset is from the UCI Archive, and is a binary classification task. It includes 699 examples
from 2 classes: benign (positive label) and malignant (negative label). The data are randomly split
into training and test sets, where the proportion of examples used for training varies from 10% to
50%. Test results are averaged over 30 trials, and were obtained using a support vector classifier with
kernel size ? = 0.1. First, we consider a biased sampling scheme based on the input features, of
which there are nine, with integer values from 0 to 9. Since smaller feature values predominate in the
unbiased data, we sample according to P (s = 1|x ? 5) = 0.2 and P (s = 1|x > 5) = 0.8, repeating
the experiment for each of the features in turn. Results are an average over 30 random training/test
splits, with 1/4 of the data used for training and 3/4 for testing. Performance is shown in Figure 2(a):
we consistently outperform the unweighted method, and match or exceed the performance obtained
using the known distribution ratio. Next, we consider a sampling bias that operates jointly across
multiple features. We select samples less often when they are further from the sample mean x over
the training data, i.e. P (si |xi ) ? exp(??kxi ? xk2 ) where ? = 1/20. Performance of our method
in 2(b) is again better than the unweighted case, and as good as or better than reweighting using the
sampling model. Finally, we consider a simple biased sampling scheme which depends only on the
label y: P (s = 1|y = 1) = 0.1 and P (s = 1|y = ?1) = 0.9 (the data has on average twice as
many positive as negative examples when uniformly sampled). Average performance for different
training/testing split proportions is in Figure 2(c); remarkably, despite our assumption regarding the
difference between the training and test distributions being violated, our method still improves the
test performance, and outperforms the reweighting by density ratio for large training set sizes. Fig-
ure 2(d) shows the weights ? are proportional to the inverse of true sampling probabilities: positive
examples have higher weights and negative ones have lower weights.
4.2.2 Further Benchmark Datasets
We next compare the performance on further benchmark datasets1 by selecting training data via
various biased sampling schemes. Specifically, for the sampling distribution bias on labels, we
use P (s = 1|y) = exp(a + by)/(1 + exp(a + by)) (datasets 1 to 5), or the simple step distribution P (s = 1|y = 1) = a, P (s = 1|y = ?1) = b (datasets 6 and 7). For the remaining
datasets, we generate biased sampling schemes over their features. We first do PCA, selecting the
first principal component of the training data and the corresponding projection values. Denoting
the minimum value of the projection as m and the mean as m, we apply a normal distribution with
mean m + (m ? m)/a and variance (m ? m)/b as the biased sampling scheme. Please refer to
[7] for detailed parameter settings. We use penalized LMS for regression problems and SVM for
classification problems. To evaluate generalization performance, we utilize the normalized mean
P
i ??i )
square error (NMSE) given by n1 ni=1 (yvar
y for regression problems, and the average test error
for classification problems. In 13 out of 23 experiments, our reweighting approach is the most accurate (see Table 1), despite having no prior information about the bias of the test sample (and, in some
cases, despite the additional fact that the data reweighting does not conform to our key assumption
1). In addition, the KMM always improves test performance compared with the unweighted case.
Two additional points should be borne in mind: first, we use the same ? for the kernel mean matching and the SVM, as listed in Table 1. Performance might be improved by decoupling these kernel
sizes: indeed, we employ kernels that are somewhat large, suggesting that the KMM procedure is
helpful in the case of relatively smooth classification/regresssion functions. Second, we did not find
a performance improvement in the case of data sets with smaller sample sizes. This is not surprising,
since a reweighting would further reduce the effective number of points used for training, resulting
in insufficient data for learning.
Table 1: Test results for three methods on 18 datasets with different sampling schemes. The results are
averages over 10 trials for regression problems (marked *) and 30 trials for classification problems. We used a
Gaussian kernel of size ? for both the kernel mean matching and the SVM/LMS regression, and set B = 1000.
DataSet
1. Abalone*
2. CA Housing*
3. Delta Ailerons(1)*
4. Ailerons*
5. haberman(1)
6. USPS(6vs8)(1)
7. USPS(3vs9)(1)
8. Bank8FM*
9. Bank32nh*
10. cpu-act*
11. cpu-small*
12. Delta Ailerons(2)*
13. Boston house*
14. kin8nm*
15. puma8nh*
16. haberman(2)
17. USPS(6vs8) (2)
18. USPS(6vs8) (3)
19. USPS(3vs9)(2)
20. Breast Cancer
21. India diabetes
22. ionosphere
23. German credit
?
1e ? 1
1e ? 1
1e3
1e ? 5
1e ? 2
1/128
1/128
1e ? 1
1e ? 2
1e ? 12
1e ? 12
1e3
1e ? 4
1e ? 1
1e ? 1
1e ? 2
1/128
1/128
1/128
1e ? 1
1e ? 4
1e ? 1
1e ? 4
ntr
2000
16512
4000
7154
150
500
500
4500
4500
4000
4000
4000
300
5000
4499
150
500
500
500
280
200
150
400
selected
853
3470
1678
925
52
260
252
654
740
1462
1488
634
108
428
823
90
156
104
252
96
97
64
214
ntst
2177
4128
3129
6596
156
1042
1145
3692
3692
4192
4192
3129
206
3192
3693
156
1042
1042
1145
419
568
201
600
unweighted
1.00 ? 0.08
2.29 ? 0.01
0.51 ? 0.01
1.50 ? 0.06
0.50 ? 0.09
0.13 ? 0.18
0.016 ? 0.006
0.5 ? 0.1
23 ? 4.0
10 ? 1
9?2
2?2
0.8 ? 0.2
0.85 ? 0.2
1.1 ? 0.1
0.27 ? 0.01
0.23 ? 0.2
0.54 ? 0.0002
0.46 ? 0.09
0.05 ? 0.01
0.32 ? 0.02
0.32 ? 0.06
0.283 ? 0.004
NMSE / Test err.
importance samp.
1.1 ? 0.2
1.72 ? 0.04
0.51 ? 0.01
0.7 ? 0.1
0.37 ? 0.03
0.1 ? 0.2
0.012 ? 0.005
0.45 ? 0.06
19 ? 2
4.0 ? 0.2
4.0 ? 0.2
1.5 ? 1.5
0.74 ? 0.09
0.81 ? 0.1
0.77 ? 0.05
0.39 ? 0.04
0.23 ? 0.2
0.5 ? 0.2
0.5 ? 0.2
0.036 ? 0.005
0.30 ? 0.02
0.31 ? 0.07
0.282 ? 0.004
KMM
0.6 ? 0.1
1.24 ? 0.09
0.401 ? 0.007
1.2 ? 0.2
0.30 ? 0.05
0.1 ? 0.1
0.013 ? 0.005
0.47 ? 0.05
19 ? 2
1.9 ? 0.2
2.0 ? 0.5
1.7 ? 0.9
0.76 ? 0.07
0.81 ? 0.2
0.83 ? 0.03
0.25 ? 0.2
0.16 ? 0.08
0.16 ? 0.04
0.2 ? 0.1
0.033 ? 0.004
0.30 ? 0.02
0.28 ? 0.06
0.280 ? 0.004
4.2.3 Tumor Diagnosis using Microarrays
Our next benchmark is a dataset of 102 microarrays from prostate cancer patients [13]. Each of these
microarrays measures the expression levels of 12,600 genes. The dataset comprises 50 samples
from normal tissues (positive label) and 52 from tumor tissues (negative label). We simulate the
realisitc scenario that two sets of microarrays A and B are given with dissimilar proportions of tumor
samples, and we want to perform cancer diagnosis via classification, training on A and predicting
1
Regression data from http://www.liacc.up.pt/?ltorgo/Regression/DataSets.html;
classification data from UCI. Sets with numbers in brackets are examined by different sampling schemes.
on B. We select training examples via the biased selection scheme P (s = 1|y = 1) = 0.85 and
P (s = 1|y = ?1) = 0.15. The remaining data points form the test set. We then perform SVM
classification for the unweighted, KMM, and importance sampling approaches. The experiment
was repeated over 500 independent draws from the dataset according to our biased scheme; the 500
resulting test errors are plotted in [7]. The KMM achieves much higher accuracy levels than the
unweighted approach, and is very close to the importance sampling approach.
We study a very similar scenario on two breast cancer microarray datasets from [4] and [19], measuring the expression levels of 2,166 common genes for normal and cancer patients [18]. We train
an SVM on one of them and test on the other. Our reweighting method achieves significant improvement in classification accuracy over the unweighted SVM (see [7]). Hence our method promises to
be a valuable tool for cross-platform microarray classification.
Acknowledgements: The authors thank Patrick Warnat (DKFZ, Heidelberg) for providing the microarray datasets, and Olivier Chapelle and Matthias Hein for helpful discussions. The work is
partially supported by by the BMBF under grant 031U112F within the BFAM project, which is part
of the German Genome Analysis Network. NICTA is funded through the Australian Government?s
Backing Australia?s Ability initiative, in part through the ARC. This work was supported in part by
the IST Programme of the EC, under the PASCAL Network of Excellence, IST-2002-506778.
References
[1] G. Casella and R. Berger. Statistical Inference. Duxbury, Pacific Grove, CA, 2nd edition, 2002.
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[3] A. Gretton, K. Borgwardt, M. Rasch, B. Sch?olkopf, and A. Smola. A kernel method for the two-sampleproblem. In NIPS. MIT Press, 2006.
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[11] B. Sch?olkopf, J. Platt, J. Shawe-Taylor, A. J. Smola, and R. C. Williamson. Estimating the support of a
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[12] H. Shimodaira. Improving predictive inference under convariance shift by weighting the log-likelihood
function. Journal of Statistical Planning and Inference, 90, 2000.
[13] D. Singh, P. Febbo, K. Ross, D. Jackson, J. Manola, C. Ladd, P. Tamayo, A. Renshaw, A. DAmico, and
J. Richie. Gene expression correlates of clinical prostate cancer behavior. Cancer Cell, 1(2), 2002.
[14] I. Steinwart. On the influence of the kernel on the consistency of support vector machines. Journal of
Machine Learning Research, 2:67?93, 2002.
[15] I. Steinwart. Support vector machines are universally consistent. J. Compl., 18:768?791, 2002.
[16] M. Sugiyama and K.-R. M?uller. Input-dependent estimation of generalization error under covariate shift.
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[17] I. Tsochantaridis, T. Joachims, T. Hofmann, and Y. Altun. Large margin methods for structured and
interdependent output variables. Journal of Machine Learning Research, 2005.
[18] P. Warnat, R. Eils, and B. Brors. Cross-platform analysis of cancer microarray data improves gene expression based classification of phenotypes. BMC Bioinformatics, 6:265, Nov 2005.
[19] M. West, C. Blanchette, H. Dressman, E. Huang, S. Ishida, R. Spang, H Zuzan, J.A. Olson Jr, J.R.Marks,
and J.R.Nevins. Predicting the clinical status of human breast cancer by using gene expression profiles.
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[20] B. Zadrozny. Learning and evaluating classifiers under sample selection bias. In International Conference
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2,287 | 3,076 | A Complexity-Distortion Approach to
Joint Pattern Alignment
Andrea Vedaldi
Stefano Soatto
Department of Computer Science
University of California at Los Angeles
Los Angeles, CA 90035
{vedaldi,soatto}@cs.ucla.edu
Abstract
Image Congealing (IC) is a non-parametric method for the joint alignment of a collection of images affected by systematic and unwanted deformations. The method
attempts to undo the deformations by minimizing a measure of complexity of the
image ensemble, such as the averaged per-pixel entropy. This enables alignment
without an explicit model of the aligned dataset as required by other methods (e.g.
transformed component analysis). While IC is simple and general, it may introduce degenerate solutions when the transformations allow minimizing the complexity of the data by collapsing them to a constant. Such solutions need to be
explicitly removed by regularization.
In this paper we propose an alternative formulation which solves this regularization issue on a more principled ground. We make the simple observation that
alignment should simplify the data while preserving the useful information carried by them. Therefore we trade off fidelity and complexity of the aligned ensemble rather than minimizing the complexity alone. This eliminates the need
for an explicit regularization of the transformations, and has a number of other
useful properties such as noise suppression. We show the modeling and computational benefits of the approach to the some of the problems on which IC has been
demonstrated.
1
Introduction
Joint pattern alignment attempts to remove from an ensemble of patterns the effect of nuisance
transformations of a systematic nature. The aligned patterns have then a simpler structure and can
be processed more easily. Joint pattern alignment is not the same problem as aligning a pattern to
another; instead all the patterns are projected to a common ?reference? (usually a subspace) which
is unknown and needs to be discovered in the process.
Joint pattern alignment is useful in many applications and has been addressed by several authors.
Here we only review the methods that are most related the present work.
Transform Component Analysis [7] (TCA) explicitly models the aligned ensemble as a Gaussian
linear subspace of patterns. In fact, TCA is a direct extension of Probabilistic Principal Component
Analysis (PPCA) [10]: Patterns are generated as in standard PPCA and additional hidden layers
model the nuisance deformations. Expectation-maximization is used to learn the model from data
which result in their alignment. Unfortunately the method requires the space of transformations to
be quantized and it is not clear how well the approach could scale to complex scenarios.
Image Congealing (IC) [9] takes a different perspective. The idea is that, as the nuisance deformations should increase the complexity of the data, one should be able to identify and undo them by
contrasting this effect. Thus IC transforms the data to minimize an appropriate measure of the ?complexity? of the ensemble. With respect to TCA, IC results in a lighter formulation which enables
addressing more complex transformations and makes fewer assumptions on the aligned ensemble.
An issue with the standard formulation of IC is that it does not require the aligned data to be a faithful
representation of the original data. Thus simplifying the data might not only remove the nuisance
factors, but also the useful information carried by the patterns. For example, if entropy is used to
measure complexity, a typical degenerate solution is obtained by mapping all the data to a constant,
which results in minimum (null) entropy. Such solutions are avoided by explicitly regularizing the
transformations, in ways that are however rather arbitrary [9].
One should instead search for an optimal compromise between complexity of the simplified data
and preservation of the useful information (Sect. 2). This approach is not only more direct, but also
conceptually more straightforward as no ad hoc regularization needs to be introduced. We illustrate some of its relationship with rate-distortion theory (Sect. 2.1) and information bottleneck [2]
(Sect. 2.2) and we contrast it to IC (Sect. 2.4).
In Sect. 3 we specialize our model to the problem of image alignment as done in [9]. For this case,
we show that the new model has the same computational complexity of IC (Sect. 3.1). We also show
that a Gauss-Newton based algorithm is possible, which is useful to converge quickly during the final
stage of the optimization (Sect. 3.2; in a similar context a descent based algorithm was introduced
in [1]). In Sect. 4 we illustrate the practical behavior of the algorithm, showing how the complexitydistortion compromise affects the final solution. In particular, our results compare favorably with
the ones of [9], with the added simplicity and other benefits, such as noise suppression.
2
Problem formulation
We formulate joint pattern alignment as the problem of finding a deformed pattern ensemble which
is simpler but faithful to the original data. This is similar to a lossy compression problem [5, 4, 3]
and is in fact equivalent to it in some cases (Sect. 2.1).
A pattern (or data) ensemble x ? X is a random variable with density p(x). Similarly, an aligned
ensemble or alignment y ? X of the ensemble x is another variable y that has conditional statistic
p(y|x). We seek for an alignment that is ?simpler? than x but ?faithful? to x. The complexity R
of the alignment y is measured by an operator R = H(y) such as, for example, the entropy of the
random variable y (but we will see other options). The cost of representing x by y is expressed by a
distortion function d(x, y) ? R+ and the faithfulness of the alignment y is quantified as the expected
distortion D = E[d(x, y)].
Consider a class W of deformations w : X ? X acting on the patterns X . In order for the
alignment y to factor out W we consider a distortion function which is invariant to the action of W;
in particular, given a base distortion d0 (x, y), we consider the deformation invariant distortion
d(x, y) = min d0 (x, w(y))
w?W
Thus an aligned pattern y is faithful to a deformed pattern x if it is possible to map y to x by a
nuisance deformation w.
Figuring out the best alignment y boils down in optimizing p(y|x) for complexity and distortion.
However, this require trading off complexity and distortion and there is no unique way of doing
so. The distortion-complexity function D(R) gives the best distortion D that can be achieved by
alignments of complexity R. All such distortion-optimal alignments are equally good in principle,
and it is the application that poses an upper bound on the acceptable distortion.
D(R) can be computed by optimizing the distortion D w.r.t. p(y|x) while keeping constant the
complexity R. However it is usually easier optimize the Lagrangian
min D + ?R
p(y|x)
(1)
whose optimum is attained where the derivative of D(R) is equal to ??. Then by varying ? one
spans the graph of D(R) and finds all the optimal alignments for given complexities.
2.1
Relation to rate-distortion and entropy constrained vector quantization
If one chooses the mutual information I(x, y) as complexity measure H(y) in eq. (1), then (1)
becomes a rate-distortion problem and the function D(R) a rate-distortion function [5]. The formulation is valid both for discrete and continuous spaces X , but yields to a mapping p(y|x) that
is genuinely stochastic. Therefore the alignment y of a pattern x is in general not unique. This
is because in rate-distortion y is an auxiliary variable used to derive a deterministic code for long
sequences (x1 , . . . , xn ) of data, not for data x in isolation.
In contrast, entropy constrained vector quantization [4, 3] assumes that y is finite (i.e. that it spans
a finite subset of X ) and that it is functionally determined by x (i.e. y = y(x)). Then it measures
the complexity of y as the (discrete) entropy H(y). This is analogous to a rate-distortion problem,
except that one searches for a ?single letter? optimal coding y of x rather than an optimal coding
for long sequences (x1 , . . . , xn ). Unlike rate-distortion, however, the aligned ensemble y is discrete
even if the ensemble x is continuous.
2.2
Relation to information bottleneck
Information Bottleneck (IB) [2] is a special rate-distortion problem in which one compresses a
variable x while preserving the information carried by x about another variable z, representing the
task of interest. In this sense IB is similar to the idea proposed here. By designing an appropriate
distribution p(x, z) it may also be possible to obtain an alignment effect similar to the one we seek
here. For example, if W is a group of transformations, one may define z = z(x) = {w(x) : w ?
W}, for which z is indifferent exactly to the deformations w of x.
2.3
Alternative measures of complexity
Instead of the entropy H(y) or the mutual information I(x, y) we can use alternative measures
of complexity that yield to more convenient computations. An example is the averaged-per-pixel
entropy introduced by IC [9] and discussed in Sect. 3. Generalizing this idea, we assume that the
aligned data y depend functionally on the patterns x (i.e. y = y(x)) and we express the complexity
of y as the total entropy of lower dimensional projections ?1 (y), . . . , ?M (y), ?i : X ? Rk of the
ensemble.
Distortion and entropies are estimated empirically and non-parametrically. Concretely, given an
ensemble x1 , . . . , xK ? X of patterns, we recover transformations w1 , . . . , wK ? W and aligned
patterns y1 , . . . , yK ? X that minimize
M
K
K
X
1 X
1 X
d(xi , wi (yi )) ? ?
log pj (?j (yi )),
K i=1
K i=1
j=1
where the densities pj (?j (y)) are estimated from the samples ?j (y1 ), . . . , ?j (yK ) by histogramming (discrete case) or by a Parzen estimator [6] with Gaussian kernel g? (y) of variance ? (continuous case1 ), i.e.
N
1 X
pj (?j (y)) =
g? (?j (y) ? ?j (yi )).
N i=1
2.4
Comparison to image congealing
In IC [9], given data x1 , . . . , xK ? X , one looks for transformations v : X ? X , x 7? y such that
the density p(y) estimated from samples y1 = v1 (x1 ), . . . , yK = vK (xK ) has minimum entropy. If
the transformations enable to do so, one can minimize the entropy by mapping all the patterns to a
constant; to avoid this one considers the regularized cost function
X
H(y) + ?
R(vi )
(2)
i
1
The Parzen estimator implies that the differential entropy of the distributions pj is always lower bounded
by the entropy of the kernel g? . This prevents the differential entropy to have arbitrary small negative values.
where R(v) is a term penalizing unacceptable deformations. Compared to IC, in our formulation:
I The distortion term E[d(x, y)] substitutes the arbitrary regularization R(v).
I The aligned patterns y are not obtained by deforming the patterns x; instead y is obtained as a
simplification of x within an acceptable level of distortion. This fact induces a noise-cancellation
effect (Sect. 4).
I The transformations w can be rather general, even non-invertible. IC can use complex trans-
formations too, but most likely these would need to be heavily regularized as they would tend to
annihilate the patterns.
3
Application to joint image alignment
We apply our model to the problem of removing a family of geometric distortions from images. This
is the same application for which IC [7] was proposed in the first place.
We are given a set I1 (x), . . . , IK (x) of digital images (pattern ensemble) defined on a regular lattice
x ? ? ? R2 and with range in [0, 1]. The images may be affected by parametric transformations
wi (?) = w(?; qi ) : R2 ? R2 , so that
Ii (x) = Ti (wx) + ni (x),
x??
for templates (aligned ensemble2 ) Ti (y), y ? ? and residuals ni (x). Here qi is the vector of parameters of the transformation wi (for example, wi might be a 2-D affine transformation y = Lx + l
and qi the vector q = [L11 L21 L12 L22 l1 l2 ]).
The templates Ti (y), y ? ? are digital images themselves. In order to define Ti (wx) when wx 6? ?,
bilinear interpolation and zero-padding are used. Therefore the symbol Ti (wi x) really denotes the
quantity
T (wi x) = A(x; wi )Ti ,
x??
where A(x; wi ) is a row vector of mixing coefficients determined by wi and and the interpolation
method being used and Ti is the vector obtained by stacking the pixels of the template Ti (y), y ? ?.
We will also use the notation wi ? Ti = A(wi )Ti where the left hand side is the stacking of the
warped template T (wi x), x ? ? and A(wi ) is the matrix whose rows are the vectors A(x; wi ) for
x ? ?.
P
The distortion is defined to be the squared l2 norm of the residual d(Ii , w ? Ti ) = x?? (Ii (x) ?
Ti (wi x))2 . The complexity of the aligned ensemble T (y), y ? ? is computed as in Sect. 2.3 by
projecting on the image pixels and averaging their entropies (this is equivalent to assuming that the
pixels are statistically independent). For each pixel y ? ? a density p(T (y) = t), t ? [0, 1] is estimated non parametrically from the data {T1 (y), . . . , TK (y)} (we use Parzen window as explained
in Sect. 2.3). The complexity of a pixel is thus
H(T (y)) = ?
K
1 X
log p(Ti (y)).
K i=1
Finally the overall cost function is obtained by summing over all pixels and averaging over all
images:
L(w1 , . . . , wK , T1 , . . . , TK ) =
K
K
1 XX
1 XX
(Ii (x) ? Ti (wi x))2 ? ?
log p(Ti (y)). (3)
K i=1
K i=1
x??
3.1
y??
Basic search
In this section we show how the optimization algorithm from [7] can be adapted to work with the
new formulation. This algorithm is a simple coordinate maximization in the dimensions of the
search space:
2
With respect to Sect. 2 the patterns xi are now the images Ii and the alignment y are the templates Ti .
1: Estimate the probabilities p(T (y)), y ? ? from the templates {Ti (y) : i =
1, . . . , K}
2: For each pattern i = 1, . . . , K and for each component qji of the parameter
vector qi , try a few values of qji . For each value re-compute the cost function (3) and keep the best.
3: Repeat, refining the sampling step of the parameters.
This algorithm is appropriate if the dimensionality of the parameter vector q is reasonably small.
Here we consider affine transformations for the sake of the illustration, so that q is six-dimensional.
In (1.)
and (2.)
estimating the probabilities p(Ti (y)) and the cost function
L(w1 , . . . , wK , T1 , . . . , TK ) requires to know Ti (y). As a first order approximation (as the
final result will be refined by Gauss-Newton as explained in the next Section), we bypass this
problem and we simply set Ti = wi?1 ? Ii , exploiting the fact that the affine
P transformations wi are
invertible3 . Eventually all we do is substituting the regularization term i R(vi ) of [9] with the
expected distortion
K
K
1 XX
1 XX
?1
2
(Ii (x) ? wi ? (wi ? Ii (x))) =
(Ii (x) ? A(x; wi )A(wi?1 )Ii )2
K i=1
K i=1
x??
x??
Note that warping and un-warping the image Ii is a lossy operation even if wi is bijective because
the transformation, applied to digital images, introduces aliasing. Thus the new algorithm is simply
avoiding those transformations wi that would introduce excessive loss of fidelity.
3.2
Gauss-Newton search
With respect to IC, where only the transformations w1 , . . . , wK are estimated, here we compute the
templates T1 , . . . , Tk as well. While this might be not so important when a coarse approximation to
the solution has to be found (for which the algorithm of Sect. 3.1 can be used), it must be taken into
account to get refined results. This can be done (with a bit of numeric care) by Gauss-Newton (GN).
Applying Gauss-Newton requires to take derivatives with respect to the pixel values Ti (y). We
exploit the fact that the variables T (y) are continuous, as opposed to [9].
We still process a single image per time, reiterating several times across the whole ensemble
{I1 (x), . . . , IK (x)}. For a given image Ii we update the warp parameters qi and the template
Ti simultaneously. We exploit the fact that, as the number K of images is usually big, the density
p(T (y)) does not change significantly when only one of the templates Ti is being changed. Therefore p(T (y)) can be assumed constant in the computation of the gradient and the Hessian of the cost
function (3). The gradient is given by
X
X
X p(T
?wi
?L
? i (y))
?L
=
2?i (x)?Ti (wi x) > (x),
=
2?i (x)(A(x; wi )?y ) ?
>
?Ti (y)
p(Ti (y))
?qi
?qi
x??
x??
y??
where ?i (x) = Ti (wi x)?Ii (x) is the reconstruction residual, A(x; wi ) is the linear map introduced
in Sect. 3 and ?y = ?(z ? y) is the 2-D discrete delta function centered on y, encoded as a vector.
The approximated Hessian of the cost function (3) can be obtained as follows. First, we use the
Gauss-Newton approximation for the derivative w.r.t. the transformation parameters qi
X ?w>
?2L
?wi
?
2 i (x)?> Ti (wi x)?Ti (wi x) > (x)
?qi
?qi ?qi>
?q
i
x??
We then have
X
X p?(Ti (y))p(Ti (y)) ? p(T
? i (y))2
?2L
2
=
2(A(x;
w
)?
)
?
i
y
?Ti (y)2
p(Ti (y))2
x??
y??
X
?2L
=
2(A(x; wi )?y )(A(x; wi )?z )
?Ti (y)?Ti (z)
x??
X
X
?
?2L
?wi
=
2(A(x; wi )?y )?Ti (wi x) > +
2?i (x)A(x; wi ) D1 ?y
?Ti (y)?q >
?q
i
x??
x??
3
D2 ?y
? ?wi
?qi>
Our criterion avoids implicitly non-invertible affine transformations as they yield highly distorted codes.
Figure 1: Toy example. Top left. We distort the patterns by applying translations drawn uniformly
from the 8-shaped region (the center corresponds to the null translation). Top. We show the gradient
based algorithm while it gradually aligns the patterns by reducing the complexity of the alignment
y. Dark areas correspond to high values of the density of the alignment; we also superimpose the
trajectory of one of the patterns. Unfortunately the gradient based algorithm, being a local technique, gets trapped in two local modes (the modes can however be fused in a post-processing stage).
Bottom. The basic algorithm completely eliminates the effect of the nuisance transformations doing
a better job of avoiding local minima. Although for this simple problem the basic search is more
effective, on more difficult scenarios the extra complexity of the Gauss-Newton search pays off (see
Sect. 4).
where D1 is the discrete linear operator used to compute the derivative of Ti (y) along its first dimension and D2 the analogous operator for the second dimension. The second term of the last equation
gives a very small contribution and can be dropped.
The equations are all straightforward and result in la linear system
??>
?2L
????>
=?
?L
??>
where the vector ?> = q > T (y1 ) . . . T (yn ) has size in the order of the number of pixels of the
template T (y), y ? ?. While this system is large, it is also extremely sparse an can be solved rather
efficiently by standard methods [8].
4
Experiments
The first experiment (Fig.1) is a toy problem illustrating our method. We collect K patterns xi ,
i = 1, . . . , K which are arrays of M 2D points xi = (x1i , . . . , xM i ). Such points are generated
by drawing M i.i.d. samples from a 2-D Gaussian distribution and adding a random translation
wi ? R2 to them. The distribution of the translations wi is generic (in the example wi is drawn
uniformly from an 8-shaped region of the plane): This is not a problem as we do not need to make
any particular assumptions on wPbesides that it is a translation. The distortion d(xi , yi ) is simply the
m
sum of the Euclidean distances j=1 kyji + wi ? xji k2 between the patterns xi and the transformed
codes wi (yiQ
) = (y1i + wi , . . . , ymi +wi ). The distribution p(yi ) of the codes is assumed to factorize
as p(yi ) = j=1 p(yji ) where the p(yji ) are identical densities estimated by Parzen window from
all the available samples {yji , j = 1, . . . , M, i = 1, . . . , K}.
In the second experiment (Fig. 2) we align hand-written digits extracted from the NIST Special
Database 19. The results (Fig. 3) should be compared to the ones from [9]: They are of analogous
quality, but they were achieved without regularizing the class of admissible transformations. Despite
this, we did not observe any of the aligned patterns to collapse. In Fig. 4 we show the effect of
choosing different values of the parameter ? in the cost function (3). As ? is increased, the alignment
complexity is reduced and the fidelity of the alignment is degraded. By an appropriate choice of ?,
the alignment can be regarded as a ?restoration? or ?canonization? of the pattern which abstracts
from details of the specific instance.
Expected value per pixel
Entropy per pixel
Entropy per pixel
5
?1
?1.5
5
?1
5
10
10
?1.5
10
10
15
15
?2
15
15
?2
20
20
?2.5
40
20
?2.5
40
25
25
30
5 10 15 20 25
?4
0
?2.2
Rate
?1.8
0
5 10 15 20 25
20
?4
p(T(y)) along middle scanline
5 10 15 20 25
0
?2.2
10
0.2
Rate
?1.8
0
Distortion per pixel
15
0.4
?2
40
20
0.6
1
5
20
25
0.8
2
0 0
5 10 15 20 25
Distortion?rate diagram
x 10
3
10
5 10 15 20 25
25
30
4
15
0.2
?2
40
Distortion per pixel
20
0.4
0.5
20
25
0.6
1
0 0
p(T(y)) along middle scanline
0.8
1.5
25
20
5 10 15 20 25
Distortion?rate diagram
x 10
20
Distortion
5
Distortion
Expected value per pixel
Entropy per pixel
Entropy per pixel
0
0
5
5 10 15 20 25
5 10 15 20 25
Figure 2: Basic vs GN image alignment algorithms. Left. We show the results of applying the basic
image alignment algorithm of Sect. 3.1. The patterns are zeroes from the NIST Special Database
19. We show in writing order: The expected value E[T (y)]; the per-pixel entropy H(T (y)) (it can
be negative as it is differential); a 3-D plot of the same function H(T (y)); the distortion-complexity
diagram as the algorithm minimizes the function D + ?R (in green we show some lines of constant
cost); the probability p(T (y) = l) as l ? [0, 1] and y varies along the middle scan-line; and the perpixel distortion D(x) = E[(I(x)?T (wx))2 ]. Right. We demonstrate the GN algorithm of Sect. 3.2.
The algorithm achieves a significantly better solution in term of the cost function (3). Moreover GN
converges in only two sweeps of the dataset, while the basic algorithm after 10 sweeps is still slowly
moving. This is due to the fact that GN selects both the best search direction and step size, resulting
in a more efficient search strategy.
Figure 3: Aligned patterns. Left. A few patterns from NIST Special Database 19. Middle. Basic
algorithm: Results are very similar to [9], except that no regularization on the transformations is
used. Right. GN algorithm: Patterns achieve a better alignment due to the more efficient search
strategy; they also appear to be much more ?regular? due to the noise cancellation effect discussed
in Fig. 4. Bottom. More examples of patterns before and after GN alignment.
5
Conclusions
IC is a useful algorithm for joint pattern alignment, both robust and flexible. In this paper we showed
that the original formulation can be improved by realizing that alignment should result in a simplified
representation of the useful information carried by the patterns rather than a simplification of the
patterns. This results in a formulation that does not require inventing regularization terms in order
to prevent degenerate solutions. We also showed that Gauss-Newton can be successfully applied to
this problem for the case of image alignment and that this is in some regards more effective than the
original IC algorithm.
5
x 10
?4
Distortion
4
3
2
1
0
?2.3
?2.2
?2.1
Rate
?2
?1.9
(a) Distortion-Complexity
?1.8
(b) Not aligned
(c) Aligned
Figure 4: Distortion-complexity balance. We illustrate the effect of varying the parameter ? in (3).
(a) Estimated distortion-complexity function D(R). The green (dashed) lines have slope equal to
? and should be tangent to D(R) (Sect. 2). (b) We show the alignment T (wi x) of eight patterns
(rows) as ? is increased (columns). In order to reduce the entropy of the alignment, the algorithm
?forgets? about specific details of each glyph. (c) The same as (b), but aligned.
Acknowledgments
We would like to acknowledge the support of AFOSR FA9550-06-1-0138 and ONR N00014-03-10850.
References
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[2] K. Branson. The information bottleneck method. Lecture Slides, 2003.
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[5] T. M. Cover and J. A. Thomson. Elements of Information Theory. Wiley, 2006.
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2,288 | 3,077 | Learning annotated hierarchies from relational data
Daniel M. Roy, Charles Kemp, Vikash K. Mansinghka, and Joshua B. Tenenbaum
CSAIL, Dept. of Brain & Cognitive Sciences, MIT, Cambridge, MA 02139
{droy, ckemp, vkm, jbt}@mit.edu
Abstract
The objects in many real-world domains can be organized into hierarchies, where
each internal node picks out a category of objects. Given a collection of features and relations defined over a set of objects, an annotated hierarchy includes a
specification of the categories that are most useful for describing each individual
feature and relation. We define a generative model for annotated hierarchies and
the features and relations that they describe, and develop a Markov chain Monte
Carlo scheme for learning annotated hierarchies. We show that our model discovers interpretable structure in several real-world data sets.
1 Introduction
Researchers in AI and cognitive science [1, 7] have proposed that hierarchies are useful for representing and reasoning about the objects in many real-world domains. One of the reasons that
hierarchies are valuable is that they compactly specify categories at many levels of resolution, each
node representing the category of objects at the leaves below the node. Consider, for example, the
simple hierarchy shown in Figure 1a, which picks out five categories relevant to a typical university
department: employees, staff, faculty, professors, and assistant professors.
Suppose that we are given a large data set describing the features of these employees and the interactions among these employees. Each of the five categories will account for some aspects of
the data, but different categories will be needed for understanding different features and relations.
?Faculty,? for example, is the single most useful category for describing the employees that publish
papers (Figure 1b), but three categories may be needed to describe the social interactions among the
employees (Figure 1c). In order to understand the structure of the department, it is important not
only to understand the hierarchical organization of the employees, but to understand which levels in
the hierarchy are appropriate for describing each feature and each relation. Suppose, then, that an
annotated hierarchy is a hierarchy along with a specification of the categories in the hierarchy that
are relevant to each feature and relation.
The idea of an annotated hierarchy is one of the oldest proposals in cognitive science, and researchers
including Collins and Quillian [1] and Keil [7] have argued that semantic knowledge is organized
into representations with this form. Previous treatments of annotated hierarchies, however, usually
suffer from two limitations. First, annotated hierarchies are usually hand-engineered, and there
are few proposals describing how they might be learned from data. Second, annotated hierarchies
typically capture knowledge only about the features of objects: relations between objects are rarely
considered. We address both problems by defining a generative model for objects, features, relations,
and hierarchies, and showing how it can be used to recover an annotated hierarchy from raw data.
Our generative model for feature data assumes that the objects are located at the leaves of a rooted
tree, and that each feature is generated from a partition of the objects ?consistent? with the hierarchy.
A tree-consistent partition (henceforth, t-c partition) of the objects is a partition of the objects into
disjoint categories, i.e. each class in the partition is exactly the set of leaves descending from some
node in the tree. Therefore, a t-c partition can be uniquely encoded as the set of these nodes whose
leaf descendants comprise the classes (Figure 1a,b). The simplest t-c partition is the singleton set
(a)
(b)
False
Staff (S)
Pu
bl
Ha ishes
s
Dir Tenu
ec
t?D re
ep
os
it
True
S S
Employees (E)
Professors (P)
Faculty (F)
P E
Assistant Profs (A)
E
S
S
P
F
E
F
P
S,S
E
F
A
E
(c)
A
S,F
F
S
P
A
S,S
S,F
F,S
F,F
P,P P,A
F,S
friends with
S
P
A
tA
wA
S,E
P,S P,P P,A
A,P A,A
A
F
A,E
works with
orders around
Figure 1: (a) A hierarchy over 15 members of a university department: 5 staff members, 5 professors and
5 assistant professors. (b) Three binary features, each of which is associated with a different t-c partition of
the objects. Each class in each partition is labeled with the corresponding node in the tree. (c) Three binary
relations, each of which is associated with a different t-c partition of the set of object pairs. Each class in each
partition is labeled with the corresponding pair of nodes.
containing the root node, which places all objects into a single class. The most complex t-c partition
is the set of all leaves, which assigns each object to its own class. We assume that the features of
objects in different classes are independent, but that objects in the same class tend to have similar
features. Therefore, finding the categories in the tree most relevant to a feature can be formalized
as finding the simplest t-c partition that best accounts for the distribution of the feature (Figure 1b).
We define an annotated hierarchy as a hierarchy together with a t-c partition for each feature.
Although most discussions of annotated hierarchies focus on features, much of the data available
to human learners comes in the form of relations. Understanding the structure of social groups, for
instance, involves inferences about relations like admires(?, ?), friend-of (?, ?) and brother-of (?, ?).
Like the feature case, our generative model for relational data assumes that each (binary) relation
is generated from a t-c partition of the set of all pairs of objects. Each class in a t-c partition now
corresponds to a pair of categories (i.e. pair of nodes) (Figure 1c), and we assume that all pairs in a
given class tend to take similar values. As in the feature case, finding the categories in the tree most
relevant to a relation can be formalized as finding the t-c partition that best accounts for the distribution of the relation. The t-c partition for each relation can be viewed as an additional annotation of
the tree. The final piece of our generative model is a prior over rooted trees representing hierarchies.
Roughly speaking, the best hierarchy will then be the one that provides the best categories with
which to summarize all the features and relations.
Like other methods for discovering structure in data, our approach may be useful both as a tool
for data analysis and as a model of human learning. After describing our approach, we apply it to
several data sets inspired by problems faced by human learners. Our first analysis suggests that the
model recovers coherent domains given objects and features from several domains (animals, foods,
tools and vehicles). Next we show that the model discovers interpretable structure in kinship data,
and in data representing relationships between ontological kinds.
2 A generative model for features and relations
Our approach is organized around a generative model for feature data and relational data. For simplicity, we present our model for feature and relational data separately, focusing on the case where
we have a single binary feature or a single binary relation. After presenting our generative model,
we describe how it can be used to recover annotated hierarchies from data.
We begin with the case of a single binary feature and define a joint distribution over three entities:
a rooted, weighted, binary tree T with O objects at the leaves; a t-c partition of the objects; and
feature observations, d. For a feature, a t-c partition ? is a set of nodes {n1 , n2 , . . . , nk }, such that
each object is a descendant of exactly one node in ?. We will identify each node with the category
of objects descending from it. We denote the data for all objects in the category n as dn . If o is a
leaf (single object category), then do is the value of the feature for object o. In Figure 1b, three t-c
partitions associated with the hierarchy are represented and each class in each partition is labeled
with the corresponding category.
The joint distribution P (T, w, ?, d|?, ?f ) is induced by the following generative process:
i. Sample a tree T from a uniform distribution over rooted binary trees with O leaves (each
leaf will represent an object and there are O objects). Each node n represents a category.
ii. For each category n, sample its weight, wn , according to an exponential distribution with
parameter ?, i.e. p(wn |?) = ?e??wn .
iii. Sample a t-c partition ?f = {n1 , n2 , . . . , nk } ? ?(root-of(T )), where ?(n) is a
stochastic, set-valued function:
{n}
n is a leaf, or w.p. ?(wn )
?(n) =
(1)
?i ?(ni ) otherwise
where ?(x) = 1?e?x and ni are the children of n. Intuitively, categories with large weight
are more likely to be classes in the partition. For the publishes feature in Figure 1b, the t-c
partition is {F, S}.
iv. For each category n ? ?f , sample ?n ? Beta(?f , ?f ), where ?n is the probability that
objects in category n exhibit the feature f . Returning to the publishes example in Figure 1b,
two parameters, ?F and ?S , would be drawn for this feature.
v. For each object o, sample its feature value do ? Bernoulli(?n ), where n ? ?f is the
category containing o.
Consider now the case where we have a single binary relation defined over all ordered pairs of
objects {(oi , oj )}. In the relational case, our joint distribution is defined over a rooted, weighted,
binary tree; a t-c partition of ordered pairs of objects; and observed, relational data represented as a
matrix D where Di,j = 1 if the relation holds between oi and oj .
Given a pair of categories (ni , mj ), let ni ? mj be the set of all pairs of objects (oi , oj ) such that oi
is an object in the category ni and oj is an object in the category mj . With respect to pairs of trees, a
t-c partition, ?, is a set of pairs of categories {(n1 , m1 ), (n2 , m2 ), . . . , (nk , mk )} such that, for every
pair of objects (oi , oj ), there exists exactly one pair (nk , mk ) ? ? such that (oi , oj ) ? nk ? mk .
To help visualize these 2D t-c partitions, we can reorder the columns and rows of the matrix D
according to an in-order traversal of the binary tree T . Each t-c partition now splits the matrix
into contiguous, rectangular blocks (see Figure 1c, where each rectangular block is labeled with its
category pair). Assuming we have already generated a rooted, weighted binary tree, we now specify
the generative process for a single binary relation (c.f. steps iii through v in the feature case):
iii. Sample a t-c partition ?r = {(n1 , m1 ), . . . , (nk , mk )} ? ?(root-of(T ), root-of(T )),
where ?(n, m) is a stochastic, set-valued function:
?
w.p. ?(wn ) ? ?(wm )
?{(n, m)}
(2)
?(n, m) = ?i ?(ni , m) otherwise, w.p. 21
?
?j ?(n, mj ) otherwise
where ni /mj are the children of n/m. To handle special cases, if both n, m are leaves,
?(n, m) = {n, m}; if only one of the nodes is a leaf, we default to the feature case on the
remaining tree, halting with probability ?(wn ) ? ?(wm ). Intuitively, if a pair of categories
(n, m) both have large weight, the process is more likely to group all pairs of objects in
n ? m into a single class. In Figure 1c, the t-c partition for the works with relation is
{(S, S), (S, F ), (F, S), (F, F )}.
iv. For each pair of categories (n, m) ? ?r , sample ?n,m ? Beta(?r , ?r ), where ?n,m is the
probability that the relation holds between any pair of objects in n ? m. For the works
with relation in Figure 1c, parameters would be drawn for each of the four classes in the t-c
partition.
v. For each pair of objects (oi , oj ), sample the relation Di,j ? Bernoulli(?n,m ), where
(n, m) ? ?r and (oi , oj ) ? (n, m). That is, the probability that the relation holds is
the same for all pairs in a given class.
For data sets with multiple relations and features, we assume that all relations and features are
conditionally independent given the weighted tree T .
2.1 Inference
Given observations of features and relations, we can use the generative model to ask various questions about the latent hierarchy and its annotations. We start by determining the posterior distribution
(r) R
on the weighted tree topologies, (T, w), given data D = ({d(f ) }F
}r=1 ) over O objects,
f =1 , {D
F
F features and R relations and hyperparameters ? and ? = ({?f }f =1 , {?r }R
r=1 ). By Bayes? rule,
P (T, w|D, ?, ?) ? P (T )
? 1
P (w|T, ?)
Y
?e??wn
n
P (D|T, w, ?)
YF
YR
P (d(f ) |T, w, ?f )
P (D(r) |T, w, ?r ) .
f =1
r=1
P
(f )
But P (d(f ) |T, w, ?f ) =
|?, ?f ), where P (?|T, w) is the distribution over
? P (?|T, w) P (d
t-c partitions induced by the stochastic function ? and P (d(f ) |?, ?f ) is the likelihood given
the partition, marginalizing over the feature probabilities, ?n . Because the classes are indepenQ
(f )
(f )
dent, P (d(f ) |?, ?f ) = n?? P (dn |n ? ?, ?f ), where Mf (n) = P (dn |n ? ?, ?f ) is the
(f )
marginal likelihood for dn , the features for objects in category n. For our binary-valued data
sets, Mf (n) is the standard marginal likelihood for the beta-binomial model. Because there are
an exponential number of t-c partitions, we present an efficient dynamic program for calculating
(f )
Tf (n) = P (dn |T, w, ?f ). Then, Tf (root-of(T )) = P (d(f ) |T, w, ?f ) is the desired quantity.
First observe that, for all objects (i.e. leaf nodes) o, Tf (o) = Mf (o). Let n be a node and assume no
ancestor of n is in ?. With probability ?(wn ) = 1 ? e?wn , category n will be a single class and the
contribution to Tf will be Mf (n). Otherwise, ?(n) splits category n into its children, n1 and n2 .
Now the possible partitions of the objects in category n are every t-c partition of the objects below n1
paired with every t-c partition below n2 . By independence, this contributes Tf (n1 )Tf (n2 ). Hence,
?(wn )Mf (n) + (1 ? ?(wn )) Tf (n1 )Tf (n2 ) if n is an internal node
Tf (n) =
Mf (n)
otherwise.
(r)
For the relational case, we describe a dynamic program Tr (n, m) that calculates P (Dn,m |T, w, ?r ),
the probability of all relations between objects in n?m, conditioned on the tree, having marginalized
(r)
out the t-c partitions and relation probabilities. Let Mr (n, m) = P (Dn,m |(n, m) ? ?, ?r ) be the
marginal likelihood of the relations in n ? m. For relations, Mf (n, m) is also the beta-binomial. If
n and m are both leaves, then Tr (n, m) = Mr (n, m). Otherwise,
Tr (n, m) = ?(wn ) ?(wm )Mr (n, m)
?
n is a leaf
?Tr (n, m1 )Tr (n, m2 )
+ (1 ? ?(wn ) ?(wm )) (Tr (n1 , m)Tr (n2 , m)
m is a leaf
?1
2 ? (Tr (n, m1 )Tr (n, m2 ) + Tr (n1 , m)Tr (n2 , m)) otherwise
The above dynamic programs have linear and quadratic complexity in the number of objects, respectively. Because we can efficiently compute the posterior density of a weighted tree, we can
search for the maximum a posteriori (MAP) weighted tree. Conditioned on the MAP tree, we can
efficiently compute the MAP t-c partition for each feature and relation. We find the MAP tree first,
rather than jointly optimizing for both the topology and partitions, because marginalizing over the
t-c partitions produces more robust trees; marginalization has a (Bayesian) ?Occam?s razor? effect
and helps avoid overfitting. MAP t-c partitions can be computed by a straightforward modification
of the above dynamic programs, replacing sums with max operations and maintaining a list of nodes
representing the MAP t-c partition at each node in the tree.
We chose to implement global search by building a Markov chain Monte Carlo (MCMC) algorithm
with the posterior as the stationary distribution and keeping track of the best tree as the chain mixes.
For all the results in this paper, we fixed the hyperparameters of all beta distributions to ? = 0.5
(i.e. the asymptotically least informative prior) and report the (empirical) MAP tree and MAP t-c
partitions conditioned on the tree. The MCMC algorithm searches for the MAP tree by cycling
through three Metropolis-Hastings (MH) moves adapted from [14]:
i. Subtree Pruning and Regrafting: Choose a node n uniformly at random (except the root).
Choose a non-descendant node m. Detach n from its parent and collapse the parent (remove
node, attaching the remaining child to the parent?s parent and adding the parent?s weight to
the child?s). Sample u ? Uniform(0, 1) and then insert a new node m? between m and its
parent. Attach n to m? , set wm? := (1 ? u)wm and set wm := uwm .
ii. Edge Weight Adjustment: Choose a node n uniformly at random (including the root) and
propose a new weight wn (e.g. let x be Normal(log(wt ), 1) and let new weight be ex ).
iii. Subtree Swapping: Choose a node n uniformly at random (except the root). Choose another
node n? such that neither n nor n? is a descendant of the other, and swap n and n? .
The first two moves suffice to make the chain ergodic; subtree swapping is included to improve
mixing. The first and last moves are symmetric. We initialized the chain on a random tree with
weights set to one, ran the chain for approximately one million iterations and assessed convergence
by comparing separate chains started from multiple random initial states.
2.2 Related Work
There are several methods that discover hierarchical structure in feature data. Hierarchical clustering
[4] has been successfully used for analyzing both biological data [18] and psychological data, but
cannot learn the annotated hierarchies that we consider. Bayesian hierarchical clustering (BHC) [6]
is a recent alternative which constructs a tree as a byproduct of approximate inference in a flat
clustering model, but lacks any notion of annotations. It is possible that a BHC-inspired algorithm
could be derived to find approximate MAP annotated hierarchies. Our model for feature data is most
closely related to methods for Bayesian phylogenetics [14]. These methods typically assume that
features are generated directly by a stochastic process over a tree. Our model adds an intervening
layer of abstraction by assuming that partitions are generated by a stochastic process over a tree, and
that features are generated from these partitions. By introducing a partition for each feature, we gain
the ability to annotate a hierarchy with the levels most relevant to each feature.
There are several methods for discovering hierarchical structure in relational data [5, 13], but none
of these methods provides a general purpose solution to the problem we consider. Most of these
methods take a single relation as input, and assume that the hierarchy captures an underlying community structure: in other words, objects that are often paired in the input are assumed to lie nearby
in the tree. Our approach handles multiple relations simultaneously, and allows a more flexible mapping between each relation and the underlying hierarchy. Different relations may depend on very
different regions of the hierarchy, and some relations may establish connections between categories
that are quite distant in the tree (see Figure 4).
Many non-hierarchical methods for relational clustering have also been developed [10, 16, 17]. One
family of approaches is based on the stochastic blockmodel [15], of which the Infinite Relational
Model (IRM) [9] is perhaps the most flexible. The IRM handles multiple relations simultaneously,
and does not assume that each relation has underlying community structure. The IRM, however,
does not discover hierarchical structure; instead it partitions the objects into a set of non-overlapping
categories. Our relational model is an extension of the blockmodel that discovers a nested set of
categories as well as which categories are useful for understanding each relation in the data set.
3 Results
We applied our model to three problems inspired by tasks that human learners are required to solve.
Our first application used data collected in a feature-listing task by Cree and McRae [2]. Participants
in this task listed the features that came to mind when they thought about a given object: when asked
to think about a lemon, for example, subjects listed features like ?yellow,? ?sour,? and ?grows on
trees.?1 We analyzed a subset of the full data set including 60 common objects and the 100 features
most commonly listed for these objects. The 60 objects are shown in Figure 2, and were chosen to
represent four domains: animals, food, vehicles and tools.
Figure 2 shows the MAP tree identified by our algorithm. The model discovers the four domains
as well as superordinate categories (e.g. ?living things?, including fruits, vegetables, and animals)
and subordinate categories (e.g. ?wheeled vehicles?). Figure 2 also shows MAP partitions for 10
1
Note that some of the features are noisy ? according to these data, onions are not edible, since none of the
participants chose to list this feature for onion.
strawberry
pineapple
grapefruit
apple
tangerine
nectarine
lemon
grape
orange
cucumber
carrot
radish
onions
lettuce
potato
clamp
drill
pliers
wrench
shovel
chisel
tomahawk
sledgehammer
axe
hammer
hoe
scissors
screwdriver
rake
crowbar
van
car
truck
bus
jeep
ship
bike
helicopter
train
motorcycle
tricycle
wheelbarrow
submarine
yacht
jet
seal
lion
dolphin
mouse
duck
tiger
rat
chicken
cat
deer
squirrel
sheep
pig
horse
cow
Food
Tools
Vehicles
Animals
a tool
an animal
made of metal
is edible
has a handle
is juicy
is fast
eaten in salads
is white
has 4 wheels
has 4 legs
analyzes
Antibiotic
Enzyme
Hormone
Lipid
Amino Acid
Carbohydrate
Steroid
Finding
Diseases
Sign or Symptom
Laboratory Result
Laboratory Procedure
Diagnostic Procedure
Therapeutic Procedure
Natural Process
Human?caused Process
Plant
Mammal
Bird
Animal
Anatomical Structure
Cell
Tissue
Cell Dysfunction
Pathologic Function
Disease or Syndrome
Cell Function
Physiologic Function
Organ Function
Acquired Abnormality
Congenital Abnormality
Figure 2: MAP tree recovered from a data set including 60 objects from four domains. MAP partitions for
several features are shown: the model discovers, for example, that ?is juicy? is associated with only one part of
the tree. The weight of each edge in the tree is proportional to its vertical extent.
Chemicals
affects
process of
causes
causes (IRM)
Figure 3: MAP tree recovered from 49 relations between entities in a biomedical data set. Four relations are
shown (rows and columns permuted to match in-order traversal of the MAP tree). Consider the circled subset of
the t-c partition for causes. This block captures the knowledge that ?chemicals? cause ?diseases.? The Infinite
Relational Model (IRM) does not capture the appropriate structure in the relation cause because it does not
model the latent hierarchy, instead choosing a single partition to describe the structure across all relations.
representative features. The model discovers that some features are associated only with certain
parts of the tree: ?is juicy? is associated with the fruits, and ?is metal? is associated with the manmade items. Discovering domains is a fundamental cognitive problem that may be solved early
in development [11], but that is ignored by many cognitive models, which consider only carefully
chosen data from a single domain (e.g. data including only animals and only biological features). By
organizing the 60 objects into domains and identifying a subset of features that are associated with
each domain, our model begins to suggest how infants may parse their environment into coherent
domains of objects and features.
Our second application explores the acquisition of ontological knowledge, a problem that has been
previously discussed by Keil [7]. We demonstrate that our model discovers a simple biomedical
ontology given data from the Unified Medical Language System (UMLS) [12]. The full data set includes 135 entities and 49 binary relations, where the entities are ontological categories like ?Sign or
Symptom?, ?Cell?, and ?Disease or Syndrome,? and the relations include verbs like causes, analyzes
and affects. We applied our model to a subset of the data including the 30 entities shown in Figure 3.
OM1
OM1
OM1
OM1
OF1
OF1
OF1
OF1
YF1
YF1
YF1
YF1
YM1
YM1
YM1
YM1
YF3
YF3
OF3
OF3
YF3
YF3
OF3
OF3
YM3
OM3
OM3
YM3
YM3
YM3
OM3
OM3
OF4
OF4
OF4
YF4
YF4
OF4
YF4
YF4
OM4
OM4
YM4
YM4
YM4
YM4
OM4
OM4
YM2
YM2
OM2
OM2
OM2
YM2
YM2
OM2
OF2
OF2
OF2
OF2
YF2
YF2
YF2
YF2
Section 1
Agngiya
Section 3
Adiaya
Section 4
Umbaidya
Section 2
Anowadya
Anowadya (IRM)
Figure 4: MAP tree recovered from kinship relations between 64 members of the Alyawarra tribe. Individuals
have been labelled with their age, gender and kinship section (e.g. ?YF1? is a young female from section 1).
MAP partitions are shown for four representative relations: the model discovers that different relations depend
on the tree in very different ways; hierarchical structure allows for a compact representation (c.f. IRM).
The MAP tree is an ontology that captures several natural groupings, including a category for ?living
things? (plant, bird, animal and mammal), a category for ?chemical substances? (amino acid, lipid,
antibiotic, enzyme etc.) and a category for abnormalities. The MAP partitions for each relation
identify the relevant categories in the tree relatively cleanly: the model discovers, for example, that
the distinction between ?living things? and ?abnormalities? is irrelevant to the first place of the
relation causes, since neither of these categories can cause anything (according to the data set). This
distinction, however, is relevant to the second place of causes: substances can cause abnormalities
and dysfunctions, but cannot cause ?living things?. Note that the MAP partitions for causes and
analyzes are rather different: one of the reasons why discovering separate t-c partitions for each
relation is important is that different relations can depend on very different parts of an ontology.
Our third application is inspired by the problem children face when learning the kinship structure
of their social group. This problem is especially acute for children growing up in Australian tribes,
which have kinship systems that are more complicated in many ways than Western kinship systems,
but which nevertheless display some striking regularities. We focus here on data from the Alyawarra
tribe [3]. Denham [3] collected a large data set by asking 104 tribe members to provide kinship terms
for each other. Twenty-six different terms were mentioned in total, and four of them are represented
in Figure 4. More than one kinship term may describe the relationship between a pair of individuals
? since the data set includes only one term per pair, some of the zeros in each matrix represent
missing data rather than relationships that do not hold. For simplicity, however, we assume that
relationships that were never mentioned do not exist.
The Alyawarra tribe is divided into four kinship sections, and these sections are fundamental to
the social structure of the tribe. Each individual, for instance, is permitted only to marry individuals
from one of the other sections. Whether a kinship term applies between a pair of individuals depends
on their sections, ages and genders [3, 8]. We analyzed a subset of the full data set including 64
individuals chosen to equally represent all four sections, both genders, and people young and old.
The MAP tree divides the individuals perfectly according to kinship section, and discovers additional
structure within each section. Group three, for example, is split by age and then by gender. The MAP
partitions for each relation indicate that different relations depend very differently on the structure
of the tree. Adiadya refers to a younger member of one?s own kinship section. The MAP partition
for this relation contains fine-level structure only along the diagonal, indicating that the model has
discovered that the term only applies between individuals from the same kinship section. Umbaidya
can be used only between members of sections 1 and 3, and members of sections 2 and 4. Again
the MAP partition indicates that the model has discovered this structure. In some places the MAP
partitions appears to overfit the data: the partition for Umbaidya, for example, appears to capture
some of the noise in this relation. This result may reflect the fact that our generative process is not
quite right for these data: in particular, it does not capture the idea that some of the zeroes in each
relation represent missing data.
4 Conclusions
We developed a probabilistic model that assumes that features and relations are generated over an
annotated hierarchy, and showed how this model can be used to recover annotated hierarchies from
raw data. Three applications of the model suggested that it is able to recover interpretable structure
in real-world data, and may help to explain the computational principles which allow human learners
to acquire hierarchical representations of real-world domains.
Our approach opens up several avenues for future work. A hierarchy specifies a set of categories,
and annotations indicate which of these categories are important for understanding specific features
and relations. A natural extension is to learn sets of categories that possess other kinds of structure,
such as factorial structure [17]. For example, the kinship data we analyzed may be well described
by three sets of overlapping categories where each individual belongs to a kinship section, a gender,
and an age group. We have already extended the model to handle continuous data and can imagine other extensions, including higher-order relations, multiple trees, and relations between distinct
sets of objects (e.g. given information, say, about the book-buying habits of a set of customers, this
extension of our model could discover a hierarchical representation of the customers and a hierarchical representation of the books, and discover the categories of books that tend to be preferred by
different kinds of customers). We are also actively exploring variants of our model that permit accurate online approximations for inference; e.g., by placing an exchangeable prior over tree structures
based on a Polya-urn scheme, we can derive an efficient particle filter.
We have shown that formalizing the intuition behind annotated hierarchies in terms of a prior on
trees and partitions and a noise-robust likelihood enabled us to discover interesting structure in realworld data. We expect a fruitful area of research going forward will involve similar marriages between intuitions about structured representation from classical AI and cognitive science and modern
inferential machinery from Bayesian statistics and machine learning.
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2,289 | 3,078 | Modeling Human Motion
Using Binary Latent Variables
Graham W. Taylor, Geoffrey E. Hinton and Sam Roweis
Dept. of Computer Science
University of Toronto
Toronto, M5S 2Z9 Canada
{gwtaylor,hinton,roweis}@cs.toronto.edu
Abstract
We propose a non-linear generative model for human motion data that uses an
undirected model with binary latent variables and real-valued ?visible? variables
that represent joint angles. The latent and visible variables at each time step receive directed connections from the visible variables at the last few time-steps.
Such an architecture makes on-line inference efficient and allows us to use a simple approximate learning procedure. After training, the model finds a single set
of parameters that simultaneously capture several different kinds of motion. We
demonstrate the power of our approach by synthesizing various motion sequences
and by performing on-line filling in of data lost during motion capture.
Website: http://www.cs.toronto.edu/?gwtaylor/publications/nips2006mhmublv/
1 Introduction
Recent advances in motion capture technology have fueled interest in the analysis and synthesis
of complex human motion for animation and tracking. Models based on the physics of masses
and springs have produced some impressive results by using sophisticated ?energy-based? learning
methods[1] to estimate physical parameters from motion capture data[2]. But if we want to generate
realistic human motion, we need to model all the complexities of the real dynamics and this is so
difficult to do analytically that learning is likely to be essential. The simplest way to generate new
motion sequences based on data is to concatenate parts of training sequences [3]. Another method is
to transform motion in the training data to new sequences by learning to adjusting its style or other
characteristics[4, 5, 6]. In this paper we focus on model driven analysis and synthesis but avoid the
complexities involved in imposing physics-based constraints, relying instead on a ?pure? learning
approach in which all the knowledge in the model comes from the data.
Data from modern motion capture systems is high-dimensional and contains complex non-linear
relationships between the components of the observation vector, which usually represent joint angles
with respect to some skeletal structure. Hidden Markov models cannot model such data efficiently
because they rely on a single, discrete K-state multinomial to represent the history of the time series.
To model N bits of information about the past history they require 2N hidden states. To avoid this
exponential explosion, we need a model with distributed (i.e. componential) hidden state that has a
representational capacity which is linear in the number of components. Linear dynamical systems
satisfy this requirement, but they cannot model the complex non-linear dynamics created by the
non-linear properties of muscles, contact forces of the foot on the ground and myriad other factors.
2 An energy-based model for vectors of real-values
In general, using distributed binary representations for hidden state in directed models of time series
makes inference difficult. If, however, we use a Restricted Boltzmann Machine (RBM) to model
the probability distribution of the observation vector at each time frame, the posterior over latent
variables factorizes completely, making inference easy. Typically, RBMs use binary logistic units
for both the visible data and hidden variables, but in our application the data (comprised of joint
angles) is continuous. We thus use a modified RBM in which the ?visible units? are linear, realvalued variables that have Gaussian noise[7, 8]. The graphical model has a layer of visible units v
and a layer of hidden units h; there are undirected connections between layers but no connections
within a layer. For any setting of the hidden units, the distribution of each visible unit is defined by
a parabolic log likelihood function that makes extreme values very improbable:1
? log p(v, h) =
X (vi ? ci )2
i
2?i2
?
X
b j hj ?
j
X vi
hj wij + const,
?i
i,j
(1)
where ?i is the standard deviation of the Gaussian noise for visible unit i. (In practice, we rescale our
data to have zero mean and unit variance. We have found that fixing ?i at 1 makes the learning work
well even though we would expect a good model to predict the data with much higher precision).
The main advantage of using this undirected, ?energy-based? model rather than a directed ?belief
net? is that inference is very easy because the hidden units become conditionally independent when
the states of the visible units are observed. The conditional distributions (assuming ?i = 1) are:
p(hj = 1|v) = f (bj +
X
vi wij ),
(2)
hj wij , 1),
(3)
i
p(vi |h) = N (ci +
X
j
where f (?) is the logistic function, N (?, V ) is a Gaussian, bj and ci are the ?biases? of hidden unit
j and visible unit i respectively, and wij is the symmetric weight between them.
Maximum likelihood learning is slow in an RBM but learning still works well if we approximately
follow the gradient of another function called the contrastive divergence[9]. The learning rule is:
?wij ? hvi hj idata ? hvi hj irecon ,
(4)
where the first expectation (over hidden unit activations) is with respect to the data distribution and
the second expectation is with respect to the distribution of ?reconstructed? data. The reconstructions
are generated by starting a Markov chain at the data distribution, updating all the hidden units in
parallel by sampling (Eq. 2) and then updating all the visible units in parallel by sampling (Eq. 3).
For both expectations, the states of the hidden units are conditional on the states of the visible units,
not vice versa. The learning rule for the hidden biases is just a simplified version of Eq. 4:
?bj ? hhj idata ? hhj irecon .
(5)
2.1 The conditional RBM model
The RBM we have described above models static frames of data, but does not incorporate any temporal information. We can model temporal dependencies by treating the visible variables in the
previous time slice as additional fixed inputs [10]. Fortunately, this does not complicate inference.
We add two types of directed connections (Figure 2): autoregressive connections from the past n
configurations (time steps) of the visible units to the current visible configuration, and connections
from the past m visibles to the current hidden configuration. The addition of these directed connections turns the RBM into a conditional RBM (CRBM). In our experiments, we have chosen
n = m = 3. These are, however, tunable parameters and need not be the same for both types of
directed connections. To simplify discussion, we will assume n = m and refer to n as the order of
the model.
1
For any setting of the parameters, the gradient of the quadratic log likelihood will always overwhelm the
gradient due to the weighted input from the binary hidden units provided the value vi of a visible unit is far
enough from its bias, ci .
Figure 1: In a trained model, probabilities of each feature being ?on? conditional on the data at the
visible units. Shown is a 100-hidden unit model, and a sequence which contains (in order) walking,
sitting/standing (three times), walking, crouching, and running. Rows represent features, columns
represent sequential frames.
Inference in the CRBM is no more difficult than in the standard
RBM. Given the data at time t, t ? 1, . . . , t ? n, the hidden units
at time t are conditionally independent. We can still use contrastive
divergence for training the CRBM. The only change is that when we
update the visible and hidden units, we implement the directed connections by treating data from previous time steps as a dynamically
changing bias. The contrastive divergence learning rule for hidden
biases is given in Eq. 5 and the equivalent learning rule for the temporal connections that determine the dynamically changing hidden
unit biases is:
(t?q)
(6)
?dij
? vit?q hhtj idata ? hhtj irecon .
Hidden layer
j
i
(t?q)
where dij
is the log-linear parameter (weight) connecting visible
unit i at time t ? q to hidden unit j for q = 1..n. Similarly, the
learning rule for the autoregressive connections that determine the
dynamically changing visible unit biases is:
(t?q)
?aki
? vkt?q vit ? hvit irecon .
(7)
(t?q)
where aki
unit i.
is the weight from visible unit k at time t ? q to visible
The autoregressive weights can model short-term temporal structure
very well, leaving the hidden units to model longer-term, higher level
structure. During training, the states of the hidden units are determined by both the input they receive from the observed data and the
input they receive from the previous time slices. The learning rule
for W remains the same as a standard RBM, but has a different effect because the states of the hidden units are now influenced by the
previous visible units. We do not attempt to model the first n frames
of each sequence.
Visible layer
t-2
t-1
t
Figure 2: Architecture of
our model (in our experiments, we use three previous
time steps)
While learning a model of motion, we do not need to proceed sequentially through the training data
sequences. The updates are only conditional on the past n time steps, not the entire sequence. As
long as we isolate ?chunks? of frames (the size depending on the order of the directed connections),
these can be mixed and formed into mini-batches. To speed up the learning, we assemble these
chunks of frames into ?balanced? mini-batches of size 100.
We randomly assign chunks to different mini-batches so that the chunks in each mini-batch are
as uncorrelated as possible. To save computer memory, time frames are not actually replicated in
mini-batches; we simply use indexing to simulate the ?chunking? of frames.
2.2 Approximations
Our training procedure relies on several approximations, most of which are chosen based on experience training similar networks. While training the CRBM, we replaced vi in Eq. 4 and Eq. 7
by its expected value and we also used the expected value of vi when computing the probability
of activation of the hidden units. However, to compute the one-step reconstructions of the data, we
used stochastically chosen binary values of the hidden units. This prevents the hidden activities from
transmitting an unbounded amount of information from the data to the reconstruction [11].
While updating the directed visible-to-hidden connections (Eq. 6) and the symmetric undirected
connections (Eq. 4) we used the stochastically chosen binary values of the hidden units in the first
term (under the data), but replaced hj by its expected value in the second term (under the reconstruction). We took this approach because the reconstruction of the data depends on the binary choices
made when selecting hidden state. Thus when we infer the hiddens from the reconstructed data,
the probabilities are highly correlated with the binary hidden states inferred from the data. On the
other hand, we stop after one reconstruction, so the binary choice of hiddens from the reconstruction
doesn?t correlate with any other terms, and there is no point including this extra noise.
Lastly, we note that the fine-tuning procedure as a whole is making a crude approximation in addition
to the one made by contrastive divergence. The inference step, conditional on past visible states,
is approximate because it ignores the future (it does not do smoothing). Because of the directed
connections, exact inference within the model should include both a forward and backward pass
through each sequence (we currently perform only a forward pass). We have avoided a backward
pass because missing values create problems in undirected models, so it is hard to perform learning
efficiently using the full posterior. Compared with an HMM, the lack of smoothing is a loss, but this
is more than offset by the exponential gain in representational power.
3 Data gathering and preprocessing
We used data from the CMU Graphics Lab Motion Capture Database as well as from [12] (see
acknowledgments). The processed data consists of 3D joint angles derived from 30 (CMU) or 17
(MIT) markers plus a root (coccyx, near the base of the back) orientation and displacement. For
both datasets, the original data was captured at 120Hz; we have downsampled it to 30Hz.
Six of the joint angle dimensions in the original CMU data had constant values, so they were eliminated. Each of the remaining joint angles had between one and three degrees of freedom. All of
the joint angles and the root orientation were converted from Euler angles to the ?exponential map?
parameterization [13]. This was done to avoid ?gimbal lock? and discontinuities. (The MIT data
was already expressed in exponential map form and did not need to be converted.)
We treated the root specially because it encodes a transformation with respect to a fixed global
coordinate system. In order to respect physics, we wanted our final representation to be invariant
to ground-plane translation and to rotation about the gravitational vertical. We represented each
ground-plane translation by an incremental ?forwards? vector and an incremental ?sideways? vector
relative to the direction the person was currently facing, but we represented height non-incrementally
by the distance above the ground plane. We represented orientation around the gravitational vertical
by the incremental change, but we represented the other two rotational degrees of freedom by the
absolute pitch and roll relative to the direction the person was currently facing.
The final dimensionality of our data vectors was 62 (for the CMU data) and 49 (for the MIT data).
Note that we eliminated exponential map dimensions that were constant zero (corresponding to
joints with a single degree of freedom). As mentioned in Sec. 2, each component of the data was
normalized to have zero mean and unit variance.
One advantage of our model is the fact that the data does not need to be heavily preprocessed or
dimensionality reduced. Brand and Hertzmann [4] apply PCA to reduce noise and dimensionality.
The autoregressive connections in our model can be thought of as doing a kind of ?whitening? of the
data. Urtasun et al. [6] manually segment data into cycles and sample at regular time intervals using
quaternion spherical interpolation. Dimensionality reduction becomes problematic when a wider
range of motions is to be modeled.
4 Experiments
After training our model using the updates described above, we can demonstrate in several ways
what it has learned about the structure of human motion. Perhaps the most direct demonstration,
which exploits the fact that it is a probability density model of sequences, is to use the model to
generate de-novo a number of synthetic motion sequences. Video files of these sequences are available on the website mentioned in the abstract; these motions have not been retouched by hand in
any motion editing software. Note that we also do not have to keep a reservoir of training data
sequences around for generation - we only need the weights of the model and a few valid frames for
initialization.
Causal generation from a learned model can be done on-line with no smoothing, just like the learning
procedure. The visible units at the last few time steps determine the effective biases of the visible
and hidden units at the current time step. We always keep the previous visible states fixed and
perform alternating Gibbs sampling to obtain a joint sample from the conditional RBM. This picks
new hidden and visible states that are compatible with each other and with the recent (visible)
history. Generation requires initialization with n time steps of the visible units, which implicitly
determine the ?mode? of motion in which the synthetic sequence will start. We used randomly
drawn consecutive frames from the training data as an initial configuration.
4.1 Generation of walking and running sequences from a single model
In our first demonstration, we train a single model on data containing both walking and running
motions; we then use the learned model to generate both types of motion, depending on how it is
initialized. We trained2 on 23 sequences of walking and 10 sequences of jogging (from subject 35
in the CMU database). After downsampling to 30Hz, the training data consisted of 2813 frames.
Figure 3: After training, the same
model can generate walking (top)
and running (bottom) motion (see
videos on the website). Each skeleton is 4 frames apart.
Figure 3 shows a walking sequence and a running sequence generated by the same model, using alternating Gibbs sampling (with the probability of hidden units being ?on? conditional on the current
and previous three visible vectors). Since the training data does not contain any transitions between
walking and running (and vice-versa), the model will continue to generate walking or running motions depending on where it is initialized.
4.2 Learning transitions between various styles
In our second demonstration, we show that our model is capable of learning not only several homogeneous motion styles but also the transitions between them, when the training data itself contains
2
A 200 hidden-unit CRBM was trained for 4000 passes through the training data, using a third-order model
(for directed connections). Weight updates were made after each mini-batch of size 100. The order of the
sequences was randomly permuted such that walking and running sequences were distributed throughout the
training data.
examples of such transitions. We trained on 9 sequences (from the MIT database, file Jog1 M) containing long examples of running and jogging, as well as a few transitions between the two styles.
After downsampling to 30Hz, this provided us with 2515 frames. Training was done as before, except that after the model was trained, an identical 200 hidden-unit model was trained on top of the
first model (see Sec. 5). The resulting two-level model was used to generate data. A video available
on the website demonstrates our model?s ability to stochastically transition between various motion
styles during a single generated sequence.
4.3 Introducing transitions using noise
In our third demonstration, we show how transitions between motion styles can be generated even
when such transitions are absent in the data. We use the same model and data as described in Sec.
4.1, where we have learned on separate sequences of walking and running. To generate, we use the
same sampling procedure as before, except that at each time we stochastically choose the hidden
states (given the current and previous three visible vectors) we add a small amount of Gaussian
noise to the hidden state biases. This encourages the model to explore more of the hidden state
space without deviating too far the current motion. Applying this ?noisy? sampling approach, we
see that the generated motion occasionally transitions between learned styles. These transitions
appear natural (see the video on the website).
4.4 Filling in missing data
Due to the nature of the motion capture process, which can be adversely affected by lighting and
environmental effects, as well as noise during recording, motion capture data often contains missing
or unusable data. Some markers may disappear (?dropout?) for long periods of time due to sensor failure or occlusion. The majority of motion editing software packages contain interpolation
methods to fill in missing data, but this leaves the data unnaturally smooth. These methods also
rely on the starting and end points of the missing data, so if a marker goes missing until the end
of a sequence, na??ve interpolation will not work. Such methods often only use the past and future
data from the single missing marker to fill in that marker?s missing values, but since joint angles are
highly correlated, substantial information about the placement of one marker could be gained from
the others. Our trained model has the ability to easily fill in such missing data, regardless of where
the dropouts occur in a sequence. Due to its approximate inference method which does not rely on
a backward pass through the sequence, it also has the ability to fill in such missing data on-line.
Filling in missing data with our model is very similar to generation. We simply clamp the known
data to the visible units, initialize the missing data to something reasonable (for example, the value
at the previous frame), and alternate between stochastically updating the hidden and visible units,
with the known visible states held fixed.
To demonstrate filling in, we trained a model exactly as described in Sec. 4.1 except that one walking
and one running sequence were left out of the training data to be used as test data. For each of these
walking and running test sequences, we erased two different sets of joint angles, starting halfway
through the test sequence. These sets were the joints in (1) the left leg, and (2) the entire upper
body. As seen in the video files on the website, the quality of the filled-in data is excellent and is
hardly distinguishable from the original ground truth of the test sequence. Figure 4 demonstrates the
model?s ability to predict the three angles of rotation of the left hip.
For the walking sequence (of length 124 frames), we compared our model?s performance to nearest
neighbor interpolation, a simple method where for each frame, the values on known dimensions are
compared to each example in the training set to find the closest match (measured by Euclidean distance in the normalized angle space). The unknown dimensions are then filled in using the matched
example. As reconstruction from our model is stochastic, we repeated the experiment 100 times and
report the mean. For the missing leg, mean squared reconstruction error per joint using our model
was 8.78, measured in normalized joint angle space, and summed over the 62 frames of interest.
Using nearest neighbor interpolation, the error was greater: 11.68. For the missing upper body,
mean squared reconstruction error per joint using our model was 20.52. Using nearest neighbor
interpolation, again the error was greater: 22.20.
2
1.5
1.5
Normalized joint angle
Normalized joint angle
2
1
0.5
0
?0.5
?1
?1.5
0
1
0.5
0
?0.5
?1
?1.5
20
40
60
80
Frame
100
120
140
?2
0
20
40
60
80
100
120
140
Frame
Figure 4: The model successfully fills in missing data using only the previous values of the joint
angles (through the temporal connections) and the current angles of other joints (through the RBM
connections). Shown are two of the three angles of rotation for the left hip joint (the plot of the third
is similar to the first). The original data is shown on a solid line, the model?s prediction is shown
on a dashed line, and the results of nearest neighbor interpolation are shown on a dotted line (see a
video on the website).
5 Higher level models
Once we have trained the model, we can add layers like in a Deep Belief
Network [14]. The previous layer CRBM is kept, and the sequence of hidden
state vectors, while driven by the data, is treated as a new kind of ?fully
observed? data. The next level CRBM has the same architecture as the first
(though we can alter the number of its units) and is trained in the exact same
way. Upper levels of the network can then model higher-order structure.
This greedy procedure is justified using a variational bound [14]. A twolevel model is shown in Figure 5.
k
j
We can also consider two special cases of the higher-level model. If we
i
keep only the visible layer, and its n-th order directed connections, we have
a standard AR(n) model with Gaussian noise. If we take the two-hidden
layer model and delete the first-level autoregressive connections, as well as
both sets of visible-to-hidden directed connections, we have a simplified
t-2
t-1
t
model that can be trained in 2 stages: first learning a static (iid) model of
pairs or triples of time frames, then using the inferred hidden states to train
a ?fully-observed? sigmoid belief net that captures the temporal structure of Figure 5: Higherlevel models
the hidden states.
6 Discussion
We have introduced a generative model for human motion based on the idea that local constraints
and global dynamics can be learned efficiently by a conditional Restricted Boltzmann Machine.
Once trained, our models are able to efficiently capture complex non-linearities in the data without
sophisticated pre-processing or dimensionality reduction. The model has been designed with human
motion in mind, but should lend itself well to other high-dimensional time series.
In relatively low-dimensional or unstructured data (for example if we were to model a single isolated
joint) a single-layer model might be expected to have difficulty since such cyclic time series contain
several subsequences which are locally very similar but occur in different phases of the overall
cycle. It would be possible to preserve the global phase information by using a much higher order
model, but for higher dimensional data such as full body motion capture this is unnecessary because
the whole configuration of joint angles and angular velocities never has any phase ambiguity. So
the single-layer version of our model actually performs much better on higher-dimensional data.
Models with more hidden layers are able to implicitly model longer-term temporal information, and
thus will mitigate this effect.
We have demonstrated that our model can effectively learn different styles of motion, as well as
the transitions between these styles. This differentiates our approach from PCA-based approaches
which only accurately model cyclic motion, and additionally must build separate models for each
type of motion. The ability of the model to transition smoothly, however, is dependent on having
sufficient examples of such transitions in the training data. We plan to train on larger datasets encompassing such transitions between various styles of motion. If we augment the data with some
static skeletal and identity parameters (in essence mapping a person?s unique identity to a set of features), we should be able to use the same generative model for many different people, and generalize
individual characteristics from one type of motion to another. Finally, our model is not limited to a
single source of data. In the future, we hope to integrate low-level vision data captured at the same
time as motion; we could then learn the correlations between the vision stream and the joint angles.
Acknowledgments
The first data set used in this project was obtained from mocap.cs.cmu.edu. This database
was created with funding from NSF EIA-0196217. The second data set used in this project was
obtained from http://people.csail.mit.edu/ehsu/work/sig05stf/. For Matlab
playback of motion and generation of videos, we have used Neil Lawrence?s motion capture toolbox
(http://www.dcs.shef.ac.uk/?neil/mocap/).
References
[1] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner, ?Gradient-based learning applied to document recognition,? Proceedings of the IEEE, vol. 86, pp. 2278?2324, November 1998.
[2] C. K. Liu, A. Hertzmann, and Z. Popovic, ?Learning physics-based motion style with nonlinear inverse
optimization,? ACM Trans. Graph., vol. 24, no. 3, pp. 1071?1081, 2005.
[3] O. Arikan, D. A. Forsyth, and J. F. O?Brien, ?Motion synthesis from annotations,? in Proc. SIGGRAPH,
2002.
[4] M. Brand and A. Hertzmann, ?Style machines.,? in Proc. SIGGRAPH, pp. 183?192, 2000.
[5] Y. Li, T. Wang, and H.-Y. Shum, ?Motion texture: a two-level statistical model for character motion
synthesis,? in Proc. SIGGRAPH, pp. 465?472, 2002.
[6] R. Urtasun, P. Glardon, R. Boulic, D. Thalmann, and P. Fua, ?Style-based Motion Synthesis,? Computer
Graphics Forum, vol. 23, no. 4, pp. 1?14, 2004.
[7] M. Welling, M. Rosen-Zvi, and G. E. Hinton, ?Exponential family harmoniums with an application to
information retrieval.,? in Proc. NIPS 17, 2005.
[8] Y. Freund and D. Haussler, ?Unsupervised learning of distributions of binary vectors using 2-layer networks,? in Proc. NIPS 4, 1992.
[9] G. E. Hinton, ?Training products of experts by minimizing contrastive divergence.,? Neural Comput,
vol. 14, pp. 1771?1800, Aug 2002.
[10] I. Sutskever and G. E. Hinton, ?Learning multilevel distributed representations for high-dimensional sequences,? Tech. Rep. UTML TR 2006-003, University of Toronto, 2006.
[11] Y. W. Teh and G. E. Hinton, ?Rate-coded restricted Boltzmann machines for face recognition,? in Proc.
NIPS 13, 2001.
[12] E. Hsu, K. Pulli, and J. Popovi?c, ?Style translation for human motion,? ACM Trans. Graph., vol. 24, no. 3,
pp. 1082?1089, 2005.
[13] F. S. Grassia, ?Practical parameterization of rotations using the exponential map,? J. Graph. Tools, vol. 3,
no. 3, pp. 29?48, 1998.
[14] G. E. Hinton, S. Osindero, and Y.-W. Teh, ?A fast learning algorithm for deep belief nets,? Neural Comp.,
vol. 18, no. 7, pp. 1527?1554, 2006.
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2,290 | 3,079 | TrueSkill
TM :
A Bayesian Skill Rating System
Ralf Herbrich
Tom Minka
Microsoft Research Ltd.
Thore Graepel
Microsoft Research Ltd.
Cambridge, UK
Microsoft Research Ltd.
Cambridge, UK
[email protected]
Cambridge, UK
[email protected]
[email protected]
Abstract
We present a new Bayesian skill rating system which can be viewed as a
generalisation of the Elo system used in Chess.
The new system tracks
the uncertainty about player skills, explicitly models draws, can deal with
any number of competing entities and can infer individual skills from team
results. Inference is performed by approximate message passing on a factor
graph representation of the model. We present experimental evidence on
the increased accuracy and convergence speed of the system compared to
Elo and report on our experience with the new rating system running in a
large-scale commercial online gaming service under the name of
1
TrueSkill.
Introduction
Skill ratings in competitive games and sports serve three main functions. First, they allow
players to be matched with other players of similar skill leading to interesting, balanced
matches.
Second, the ratings can be made available to the players and to the interested
public and thus stimulate interest and competition. Thirdly, ratings can be used as criteria
of quali cation for tournaments. With the advent of online gaming, the interest in rating
systems has increased dramatically because the quality of the online experience of millions
of players each day are at stake.
In 1959, Arpad Elo developed a statistical rating system for Chess, which was adopted by
the World Chess Federation FIDE in 1970 [4]. The key idea behind the
Elo system
[2] is to
model the probability of the possible game outcomes as a function of the two players' skill
i exhibits performance pi N (pi ; si ; 2 ) normally
si with xed variance 2 . The probability that player 1 wins
is given by the probability that his performance p1 exceeds the opponent's performance p2 ,
s s
(1)
P (p1 > p2 js1 ; s2 ) = 1p 2 ;
2
ratings
s1 and s2 .
In a game each player
distributed around their skills
denotes the cumulative density of a zero-mean unit-variance Gaussian. After the
s and s are updated such that the observed game outcome becomes
more likely and s + s = const. is maintained. Let y = +1 if player 1 wins, y =
1 if player
2 wins and y = 0 if a draw occurs. Then the resulting (linearised) Elo update is given by
s
s + y, s
s y and
p
y+1
sp s
=
;
| {z }
2 2
where
game, the skill ratings
1
1
1
2
1
2
2
2
1
2
K Factor
where
0 < < 1 determines the weighting of the new evidence versus the old estimate.
Most currently used Elo variants use a logistic distribution instead of a Gaussian because
it is argued to provide a better t for Chess data. From the point of view of statistics the
Elo system addresses the problem of estimating from paired comparison data [1] with the
Gaussian variant corresponding to the
the
Bradley-Terry
Thurstone Case V
model and the logistic variant to
model.
In the Elo system, a player's rating is regarded as provisional as long as it is based on less
than a xed number of, say, 20 games. This problem was addressed by Mark Glickman's
Bayesian rating system
Glicko
[5] which introduces the idea of modeling the belief about a
player's skill as a Gaussian belief distribution characterised by a mean
and a variance 2 .
An important new application of skill rating systems are multiplayer online games that
greatly bene t from the ability to create online matches in which the participating players
have roughly even skills and hence enjoyable, fair and exciting game experiences. Multiplayer
online games provide the following challenges:
1. Game outcomes often refer to teams of players yet a skill rating for individual players
is needed for future matchmaking.
2. More than two players or teams compete such that the game outcome is a permutation of teams or players rather than just a winner and a loser.
In this paper we present a new rating system,
TrueSkill, that addresses both these challenges
in a principled Bayesian framework. We express the model as a factor graph (Section 2)
and use approximate message passing (Section 3) to infer the marginal belief distribution
over the skill of each player. In Section 4 we present experimental results on real-world data
generated by Bungie Studios during the beta testing of the Xbox title Halo 2 and we report
on our experience with the rating system running in the Xbox Live service.
2
Factor Graphs for Ranking
n players f1; : : : ; ng in a game let k teams compete in a match.
k non-overlapping subsets Aj f1; : : : ; ng of the
player population, Ai \ Aj = ; if i 6= j . The outcome r := (r1 ; : : : ; rk ) 2 f1; : : : ; k g is
speci ed by a rank rj for each team j , with r = 1 indicating the winner and with the
possibility of draws when ri = rj . The ranks are derived from the scoring rules of the game.
We model the probability P (rjs; A) of the game outcome r given the skills s of the participating players and the team assignments A := fA1 ; : : : ; Ak g. From Bayes' rule we obtain
From among a population of
The team assignments are speci ed by
the posterior distribution
P (rjs; A) p (s)
:
(2)
P (rjA)
Qn
2
We assume a factorising Gaussian prior distribution, p(s) :=
i=1 N (si ; i ; i ). Each player
2
i is assumed to exhibit a performance pi N (pi ; si ; ) in the game, centred around their
2
skill si with xed variance . The performance tj of team j is modeled as the sum of the
P
performances of its members, tj :=
i2Aj pi . Let us reorder the teams in ascending order
of rank, r(1) r(2) r(k) . Disregarding draws, the probability of a game outcome r
p (sjr; A) =
is modeled as
P (rj ft1 ; : : : ; tk g) = P tr(1) > tr(2) > > tr(k) ;
that is, the order of performances generates the order in the game outcome. If draws are
r(j) < r(j+1) requires tr(j) > tr(j+1) + " and the draw outcome
r(j) = r(j+1) requires jtr(j) tr(j+1) j ", where " > 0 is a draw margin that can be calculated
permitted the winning outcome
from the assumed probability of draw.
1
We need to be able to report skill estimates after each game and will therefore use an online
learning scheme referred to as
Gaussian density ltering
[8]. The posterior distribution is
approximated to be Gaussian and is used as the prior distribution for the next game. If the
1
The transitive relation 1 draws with 2 is not modelled exactly by the relation
j " and jt2
the three teams despite the possibility that jt1
which is non-transitive. If
jt1
t2
j
t3
"
t3 > ".
j
jt1
t2
j ",
then the model generates a draw among
N (s ; ; )
N (s ; ; )
N (s ; ; )
N (s ; ; )
s1
s2
s3
s4
1
1
2
1
2
N (p ; s ; )
1
3
2
p1
p2
3
1
3
t2
I(d1 = t1
t2 )
2
4
p4
I(t3 = p4 )
t3
I(d2 = t2
4
t3 )
d2
2
5
I(d1 > ")
skills
all teams
4
6
d1
the
N (p ; s ; )
I(t2 = p2 + p3 )
t1
2
4
4
2
3
p3
I(t1 = p1 )
Figure 1:
4
N (p ; s ; )
2
2
2
3
3
N (p ; s ; )
2
1
2
2
2
I(jd2j ")
An example TrueSkill factor graph.
si for
performances of
There are four types of variables:
pi for the performances of all
and dj for the team performance dierences.
of all players,
players,
ti
for the
The rst row of factors encode the
(product) prior; the product of the remaining factors characterises the likelihood for the
game outcome Team 1 > Team 2 = Team 3.
The arrows indicate the optimal message
passing schedule: First, all light arrow messages are updated from top to bottom. In the
following, the schedule over the team performance (dierence) nodes are iterated in the
order of the numbers. Finally, the posterior over the skills is computed by updating all the
dark arrow messages from bottom to top.
skills are expected to vary over time, a Gaussian dynamics factor
introduced which leads to an additive variance component of
2
N (si;t ; si;t ;
) can be
2
+1
in the subsequent prior.
k = 3 teams with team assignments A = f1g, A = f2; 3g
= f4g. Let us further assume that team 1 is the winner and that teams 2 and 3
draw, i.e., r := (1; 2; 2). We can represent the resulting joint distribution p (s; p; tjr; A) by
Let us consider a game with
and
1
A3
2
the factor graph depicted in Figure 1.
A factor graph is a bi-partite graph consisting of variable and factor nodes, shown in Figure 1
as gray circles and black squares, respectively. The function represented by a factor graph
in our case the joint distribution
p (s; p; tjr; A)is given by the product of all the (potential)
functions associated with each factor. The structure of the factor graph gives information
about the dependencies of the factors involved and is the basis of e cient inference algorithms. Returning to Bayes rule (2), the quantities of interest are the posterior distribution
p (si jr; A)
over skills given game outcome
r
and team associations
A.
The
p (si jr; A) are
fpi g and
calculated from the joint distribution integrating out the individual performances
the team performances
fti g,
p (sjr; A) =
Z
1
1
Z
1
1
p (s; p; tjr; A) dp dt :
Factor
f () = I( > ") (win)
12
Vf
2
v (t,?,?)
v (t,?,?)
? = 0.50
? = 1.00
? = 4.00
4
8
6
0
4
?2
2
?4
0
?6
f () = I(j j ") (draw)
6
? = 0.50
? = 1.00
? = 4.00
10
Function
Factor
?4
?2
0
2
4
?6
?6
6
?4
?2
t
? = 0.50
? = 1.00
? = 4.00
4
6
? = 0.50
? = 1.00
? = 4.00
1
0.8
w (t,?,?)
w (t,?,?)
Wf
0.8
Function
2
t
1
0.6
0.4
0.2
0
?6
0
0.6
0.4
0.2
?4
?2
0
2
4
0
?6
6
?4
?2
t
0
2
4
6
t
Figure 2: Update rules for the the approximate marginals for dierent values of the draw
margin
": For a two-team game,
the parameter
t
represents the dierence of team perfor-
mances between winner and loser. Hence, in the win column (left) negative values of
indicate a surprise outcome leading to a large update.
t
In the draw column (right) any
stark deviation of team performances is surprising and leads to a large update.
3
Approximate Message Passing
The sum-product algorithm in its formulation for factor graphs [7] exploits the sparse connection structure of the graph to perform e cient inference of single-variable marginals
by message passing. The message passing for continuous variables is characterised by the
following equations (these follow directly from the distributive law):
p (vk )
where
=
mf !vj (vj )
=
mvk !f (vk )
=
Y
2
f Fvk
Z
mf !vk (vk )
Z
f (v)
Y
2
f~ Fvk
nff g
Y
6
i=j
(3)
mvi !f (vi ) dvnj
mf~!vk (vk ) ;
(4)
(5)
Fvk denotes the set of factors connected to variable vk and vnj denotes the components
v except for its j th component. If the factor graph is acyclic and the messages
of the vector
can be calculated and represented exactly then each message needs to be calculated only
once and the marginals
p(vk ) can be calculated from the messages by virtue of (3).
As can be seen from Figure 1 the TrueSkill factor graph is in fact acyclic and the majority
of messages can be represented compactly as 1dimensional Gaussians. However, (4) shows
that messages 2 and 5 from the comparison factors
dierences
di
I( > ") or I(jj ") to the performance
in Figure 1 are non Gaussianin fact, the true message would be the (non-
Gaussian) factor itself.
Following the
Expectation Propagation
algorithm [8], we approximate these messages as
well as possible by approximating the marginal
a Gaussian
p^(di )
p(di )
with the same mean and variance as
via moment matching resulting in
p(di ).
For Gaussian distributions,
Factor
Update equation
x
mf !x
x
2
y
x
N (x; y; c )
2
y1
mf !y
yn
follows from
x
I(x > ")
1
0
n
X
new
@
f !x
aj
y1
j =1
yj
!yj
2
a=
1 66
bn
4
3
b1
.
.
.
bn
+1
1
7
7
5
c
p p
1 W
f (d= c; " c)
p
pc; "pc)
d + c Vf (d=
1 Wf (d=pc; "pc)
f !x ;
d := x f !x
xnew
xnew
c := x
I(jxj ")
f
yn
I(yn = a> [y1 ; ; yn 1 ; x])
jj
1
yj f !yj A
mf !yn
mf !x
1
a2j
A
f !yj
j =1 yj
x
x
mf> !x
N
0
n
X
@
mf !yn
I(x = b> y)
f !y ) 1
x; y; c2 = N y; x; c2 .
2
fnew
!x
yn
1 + c (y
f !x
I(x = a> y)
y1
a :=
new
mf !x
x
a (y f !y )
a ( y f !y )
fnew
!x
fnew
!x
mf !x
1
v2
m
x + 2
v
new
N (x; m; v )
x
x +
xnew
p(x) and the messages mf !x for
N (; ; ) in terms of
2
precision, :=
, and precision adjusted mean, := . The
Table 1: The update equations for the (cached) marginals
all factor types of a TrueSkill factor graph. We represent Gaussians
their canonical parameters:
missing update equation for the message or the marginal follow from (6).
moment matching is known to minimise the KullbackLeibler divergence. Then, we exploit
the fact that from (3) and (5) we have
p^ (di ) = m
^ f !di (di ) mdi!f (di )
m
^ f !di (di ) =
,
p^ (di )
:
mdi!f (di )
(6)
Table 1 gives all the update equations necessary for performing inference in the TrueSkill
factor graph. The top four rows result from standard Gaussian integrals. The bottom rule
is the result of the moment matching procedure described above. The four functions are
the additive and multiplicative correction term for the mean and variance of a (doubly)
truncated Gaussian and are given by (see also Figure 2):
VI(>") (t; ") :=
N (t ") ; W
I >" (t; ") := VI >" (t; ") VI >" (t; ) + t " ;
(t ")
(
)
(
)
(
)
:= N ((" " t)t) N( ("" tt)) ;
(t; ") := VI jj>" (t; ") + (" t) N("(" t)t) +((" +" t) Nt) (" + t) :
VI(jj>") (t; ")
WI(jj>")
2
(
)
Since the messages 2 and 5 are approximate, we need to iterate over all messages that are
on the shortest path between any two approximate marginals
p^(di ) until
the approximate
marginals do not change anymore. The resulting optimal message passing schedule can be
found in Figure 1 (arrows and caption).
4
Experiments and Online Service
4.1 Halo 2 Beta Test
In order to assess the performance of the TrueSkill algorithm we performed experiments
on the game outcome data set generated by Bungie Studios during the beta testing of the
2
Xbox title Halo 2 . The data set consists of thousands of game outcomes for four dierent
types of games: 8 players against each other (Free for All), 4 players vs. 4 players (Small
Teams), 1 player vs. 1 player (Head to Head), and 8 players vs. 8 players (Large Teams).
The draw margin
" for each factor node was set by counting the fraction of draws between
" to the chance of drawing
teams (empirical draw probability) and relate the draw margin
by
"
draw probability = pn + n
where
n1 and n2
I(j j ")
"
"
2
pn + n = 2 pn + n
1;
are the number of players in each of the two teams compared by a I( > ")
1
or
1
2
node (see Figure 1). The performance variance
2 were set to the standard values (see next section).
1
2
2
and the dynamics variance
We compared the TrueSkill algorithm
= 0:07; this corresponds to a
24 on the Elo scale which is considered a good and stable dynamics (see [4]).
When we had to process a team game or a game with more than two teams we used the
so-called duelling heuristic: For each player, compute the 's in comparison to all other
players based on the team outcome of the player and every other player and perform an
update with the average of the 's. The approximate message passing algorithm described
to Elo with a Gaussian performance distribution (1) and
K
factor of
in the last section is extremely e cient; in all our experiments the runtime of the ranking
algorithm was within twice the runtime of the simple Elo update.
Predictive Performance
The following table presents the prediction error (fraction of
teams that were predicted in the wrong order before the game) for both algorithms (column
2 and 3).
This measure is di cult to interpret because of the interplay of ranking and
matchmaking: Depending on the (unknown) true skills of all players, the smallest achievable
prediction error could be as big as 50%. In order to compensate for this latent, unknown
variable, we arranged a competition between ELO and TrueSkill: We let each system predict
which games it considered most tightly matched and presented them to the other algorithm.
The algorithm that predicts more game outcomes correctly has a better ability to identify
tight matches. For TrueSkill we used the matchmaking criterion (7) and for Elo we used
the dierence in Elo scores,
ELO full
Free for All
Small Teams
Head to Head
Large Teams
34.92%
32.14%
33.24%
39.49%
s 1 s2 .
TrueSkill full
30.82%
32.44%
38.15%
35.23%
ELO challenged
38.30%
42.55%
40.57%
44.12%
TrueSkill challenged
35.64%
37.17%
30.83%
29.94%
It can be seen from column 4 and 5 of this table that TrueSkill is signi cantly better at
predicting the tight matches (the challenge set was always 20% of the total number of
games in each game mode).
2
Available for download at
http://research.microsoft.com/mlp/apg/downloads.htm
Match Quality
One of the main applications of a rating system is to be
able to match players of similar skill. In order to com-
35
pare the ability of Elo and TrueSkill on this task, we
30
25
tight then it would be very likely to observe a draw.
Thus, we plot the fraction of draws (out of all possible
draws) accumulating over the match quality order assigned by each system. In the graph on the right we see
% of pairwise draws
sorted the games based on the match quality assigned
by both systems to each game. If the match was truly
that TrueSkill is signi cantly better than Elo for both
Small Team
Free for All
20
15
10
Head To Head
5
the Free for All and Head to Head game mode but
ELO
TrueSkill
0
0
fails in Small Teams. This is possibly due to the vio-
20
40
60
80
100
% of games
lation of the additive team performance model as most
games in this mode are Capture-the-Flag games.
Win Probability
The perceived quality of a rating system for players is in
terms of their winning ratio: if the winning ratio is high
then player was erroneously assigned too weak opposi-
30
ond experiment we processed the Halo 2 dataset but rejected games that did not meet a certain match quality
threshold. For the games thus selected, we computed
the winning ratio of each player and, depending on the
minimal number of games played by each player, measured the average deviation of the winning probability
from 50% (the optimal winning ratio). The resulting
plot on the right (for the Head to Head game mode)
shows that with TrueSkill ever players with very few
Avg. ? of Win. Prob. from 50%
tion by the ranking system (and vice versa). In a sec-
Halo 2
ELO
TrueSkill
25
20
15
10
5
0
0
5
10
15
20
Minimal Number of Games
games got mostly fair matches (with a winning probability within 35% to 65%).
Convergence Properties
Finally, we plotted two exemplary convergence trajectories for two of the highest rated players in the Free
for All game mode (Solid line: TrueSkill; Dashed line:
Elo). As can be seen, TrueSkill automatically chooses
the correct learning rate whereas Elo only slowly converges to the target skill. In fact, TrueSkill comes close
to the information theoretic limit of
code a ranking of
n
n log(n) bits to en-
players. For 8 player games, the
information theoretic limit is
log(n)= log(8) 5 games
per player on average and the observed convergence for
these two players is
10 games!
4.2 TrueSkill in Xbox 360 Live
Xbox Live is Microsoft's console online gaming service. It lets players play together across
the world in hundreds of dierent titles.
As of September 2005 Xbox Live had over 2
million subscribed users who had accrued over 1.3 billion hours on the service. The new and
improved Xbox 360 Live service oers automatic player rating and matchmaking using the
TrueSkill algorithm. The system processes hundreds of thousands of games per day making
it one of the largest applications of Bayesian inference to date.
In Xbox Live we use a scale given by a prior
0 = 25 and 02 = (25=3)2 corresponding to a
probability for positive skills of approximately
99%.
The variance of performance is given
by
2 = (0 =2)2
and the dynamics variance is chosen to be
2 = (0 =100)2.
The TrueSkill
1%
3i. This choice ensures that the top of the leaderboards (a listing of all
players according to
3) are only populated by players that are highly skilled with high
certainty, having worked up their way from 0 =
3 . Pairwise matchmaking of players
is performed using a match quality criterion derived as the draw probability relative to the
highest possible draw probability in the limit " ! 0,
s
!
2
(
i j )
q
; i ; j ; i ; j :=
2 + + exp 2 2 + + : (7)
skill of a player
lower quantile
i
i
is currently displayed as a conservative skill estimate given by the
0
draw
0
2
2
2
2
2
i
2
j
2
2
2
i
j
Note that the matchmaking process can be viewed as a process of sequential experimental
design [3]. Since the quality of a match is determined by the unpredictability of its outcome,
the goals of matchmaking and nding the most informative matches are aligned!
As a fascinating by-product we have the opportunity to study TrueSkill in action with player
populations of hundreds of thousands of players. While we are only just beginning to analyse
the vast amount of resulting data, we have already made some interesting observations.
1. Games dier in the number of eective skill levels. Games of chance (e.g., single
game Backgammon or UNO) have a narrow skill distribution while games of skill
(e.g., semi-realistic racing games) have a wide skill distribution.
2. Matchmaking and skill display result in a feedback loop back to the players, who
often view their skill estimate as a reward or punishment for performance. Some
players try to protect or boost their skill rating by either stopping to play, by
carefully choosing their opponents, or by cheating.
3. The total skill distribution is shifted to below the prior distribution if players new to
the system consistently lose their rst few games. When a skill reset was initiated,
we found that the eect disappeared with tighter matchmaking enforced.
5
Conclusion
TrueSkill is a globally deployed Bayesian skill rating algorithm based on approximate message passing in factor graphs. It has many theoretical and practical advantages over the Elo
system and has been demonstrated to work well in practice.
While we speci cally focused on the TrueSkill algorithm, many more interesting models
can be developed within the factor graph framework presented here.
In particular, the
factor graph formulation is applicable to the family of constraint classi cation models [6]
that encompass a wide range of multiclass and ranking problems. Also, instead of ranking
individual entities one can use feature vectors to build a ranking function, e.g., for web pages
represented as bags-of-words. Finally, we are planning to run a full time-independent EP
analysis across chess games to obtain TrueSkill ratings for chess masters of all times.
Acknowledgements
We would like to thank Patrick O'Kelley, David Shaw and Chris Butcher
for interesting discussions. We also thank Bungie Studios for providing the data.
References
[1] A. H. David. The Method of Paired Comparisons. Charles Gri n and Company, London, 1969.
[2] A. E. Elo. The rating of chess players: Past and present. Arco Publishing, New York, 1978.
[3] V. V. Fedorov. Theory of optimal experiments. Academic Press, New York, 1972.
[4] M. E. Glickman. A comprehensive guide to chess ratings. Amer. Chess Journal, 3:59102, 1995.
[5] M. E. Glickman. Parameter estimation in large dynamic paired comparison experiments. Applied
Statistics, 48:377394, 1999.
[6] S. Har-Peled, D. Roth, and D. Zimak. Constraint classi cation: A new approach to multiclass
classi cation and ranking. In NIPS 15, pages 785792, 2002.
[7] F. R. Kschischang, B. Frey, and H.-A. Loeliger. Factor graphs and the sum-product algorithm.
IEEE Trans. Inform. Theory, 47(2):498519, 2001.
[8] T. Minka. A family of algorithms for approximate Bayesian inference. PhD thesis, MIT, 2001.
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2,291 | 308 | A Framework for the Cooperation
of Learning Algorithms
Leon Bottou
Patrick Gallinari
Laboratoire de Recherche en Informatique
Universite de Paris XI
91405 Orsay Cedex
France
Abstract
We introduce a framework for training architectures composed of several
modules. This framework, which uses a statistical formulation of learning
systems, provides a unique formalism for describing many classical
connectionist algorithms as well as complex systems where several
algorithms interact. It allows to design hybrid systems which combine the
advantages of connectionist algorithms as well as other learning algorithms.
1 INTRODUCTION
Many recent achievements in the connectionist area have been carried out by designing
systems where different algorithms interact. For example (Bourlard & Morgan, 1991)
have mixed a Multi-Layer Perceptron (MLP) with a Dynamic Programming algorithm.
Another impressive application (Le Cun, Boser & al., 1990) uses a very complex multilayer architecture, followed by some statistical decision process. Also, in speech or image
recognition systems, input signals are sequentially processed through different modules.
Modular systems are the most promising way to achieve such complex tasks. They can
be built using simple components and therefore can be easily modified or extended, also
they allow to incorporate into their architecture some structural a priori knowledge about
the task decomposition. Of course, this is also true for connectionism, and important
781
782
Bottou and Gallinari
progresses in this field could be achieved if we were able to train multi-modules
architectures.
In this paper, we introduce a formal framework for designing and training such
cooperative systems. It provides a unique formalism for describing both the different
modules and the global system. We show that it is suitable for many connectionist
algorithms, which allows to make them cooperate in an optimal way according to the
goal of learning. It also allows to train hybrid systems where connectionist and classical
algorithms interact. Our formulation is based on a probabilistic approach to the problem
of learning which is described in section 2. One of the advantages of this approach is to
provide a formal definition of the goal of learning. In section 3, we introduce modular
architectures where each module can be described using this framework, and we derive
explicit formulas for training the global system through a stochastic gradient descent
algorithm. Section 4 is devoted to examples, including the case of hybrid algorithms
combining MLP and Learning Vector Quantization (Bollivier, Gallinari & Thiria, 1990).
2
LEARNING SYSTEMS
The probabilistic formulation of the problem of learning has been extensively studied for
three decades (Tsypkin 1971), and applied to control, pattern recognition and adaptive
signal processing. We recall here the main ideas and refer to (Tsypkin 1971) for a detailed
presentation.
2.1
EXPECTED COST
Let x be an instance of the concept to learn. In the case of a pattern recognition problem
for example, x would be a pair (pattern, class). The concept is mathematically defined by
an unknown probability density function p(x) which measures the likelihood of instance
x.
We shall use a system parameterized by w to perform some task that depends on p(x).
Given an example x, we can define a local cost, J(x,w), that measures how well our
system behaves on that example. For instance, for classification J would be zero if the
system puts a pattern in the correct class, or positive in case of misclassification.
Learning consists in finding a parameter w? that optimises some functional of the model
parameters. For instance, one would like to minimize the expected cost (1).
C(w)
=f J(x,w) p(x)dx
(1)
The expected cost cannot be explicitely computed, because the density p(x) is unknown.
Our only knowledge of the process comes from a series of observations {X1 ... xn} drawn
from the unknown density p(x). Therefore, the quality of our system can only be
measured through the realisations J(x,w) of the local cost function for the different
observations.
A Framework for the Cooperation of Learning Algorithms
2.2
STOCHASTIC GRADIENT DESCENT
Gradient descent algorithms are the simplest minimization algorithms. We cannot,
however, compute the gradient of the expected cost (1), because p(x) is unknown.
Estimating these derivatives on a training set {X1 ... xn}, gives the gradient algorithm (2),
where VJ denotes the gradient of J(x,w) with respect to w, and Et, a small positive
constant. the "learning rate".
1 n
Wt+ 1 =Wt - Et - 2. V J(Xj,wt}
n 1. 1
(2)
The stochastic gradient descent algorithm (3) is an alternative to algorithm (2). At each
iteration, an example Xt is drawn at random, and a new value of w is computed.
(3)
Algorithm (3) is faster and more reliable than (2), it is the only solution for training
adaptive systems like Neural networks (NN). Such stochastic approximations have been
extensively studied in adpative signal processing (Benveniste. Metiver & Priouret, 1987).
(Ljung & Soderstrom, 1983). Under certain conditions, algorithm (3) converges almost
surely (Bottou, 1991). (White, 1991) and allows to reach an optimal state of the system.
3
MODULAR LEARNING SYSTEMS
Most often, when the goal of learning is complex, it can be achieved more easily by
using a decomposition of the global task into several simpler subtasks which for instance
reflect some a priori knowledge about the structure of the problem. One can use this
decomposition to build modular architectures where each module will correspond to one of
the subtasks.
Within this framework, we will use the expected risk (1) as the goal of learning. The
problem now is to change the analytical formulation of the functional (1) so as to
introduce the modular decomposition of the global task. In (1), the analytic expression of
the local cost J(x,w) has two meanings: it describes a parametric relationship between the
inputs and the outputs of the system, and measures the quality of the system. To
introduce the decomposition, one may write this local cost J(x,w) as the composition of
several functions. One of them will take into account the local error and therefore measure
the quality of the system; the others will correspond to the decomposition of the
parametric relationship between the inputs and the outputs of the system (Figure 1). Each
of the modules will therefore receive some inputs from other modules or the external
world and produce some outputs which will be sent to other modules.
783
784
B ottou and Gallinari
a I-y-p
Figure 1: A modular system
In classical systems. these modules correspond to well defmed processing stages like e.g.
signal processing. filtering. feature extraction. classification. They are trained sequentially
and then linked together to build a complete processing system which takes some inputs
(e.g. raw signals) and produces some outputs (e.g. classes). Neither the assumed
decomposition. nor the behavior of the different modules is guaranteed to optimally
contribute to the global goal of learning. We will show in the following that it is
possible to optimally train such systems.
3.1
TRAINING MODULAR SYSTEMS
Each function in the above composition defmes a local processing stage or module whose
outputs are defined by a parametric function of its inputs (4).
V' je y-1 (n), Yj
=fj(
(Xk) ke X-1(n) , (Wi) ie W-1 (n) )
(4)
y-1 (n) ( resp. X-1 (n). and W-1 (n) ) denotes the set of subscripts associated to the outputs
Y ( resp. inputs x and parameters W ) of module n. Conversely. output Yj ( resp. input xk
and parameter Wi ) belongs to module Y(j) ( resp. X(k) and W(i) ).
Modules are linked so as to build a feed-forward topology which is expressed by a
function cj).
V'k, xk = Y~(k)
(5)
We shall consider that the first module only feeds the system with examples and that the
last module only computes Ylast = J(x,w).
Following (Le Cun. 1988). we can compute the derivatives of J with a Lagrangian
method. Let a and ~ be the Lagrange coefficients for constraints (4) and (5).
L=J-
L ~k(Xk-Y~(k)) - L aj (Yr!j( (Xk) ke X-1Y(j), (Wi) ie W-1Y(j) ))
(6)
k
j
By equating the derivatives of L with respect to x and Yto zero. we get recursive formulas
for computing a and ~ in a single backward pass along the acyclic graph cj).
A Framework for the Cooperation of Learning Algorithms
(7)
alast = 1,
Then, the derivatives of J with respect to the weights are:
dJ
dL
-(w) = -(aRw) =
dwi
dwi ,.."
~
?.J
d I:
a'J :LJ..
:l. ...
(8)
je y-1W{i) UYVI
Once we have computed the derivatives of the local cost J(x,w), we can apply the
stochastic gradient descent algorithm (3) for minimizing of the expected cost C(w).
We shall say that each module is defined by the equations in (7) and (8) that characterize
its behavior. These equations are:
? a forward equation (F)
Yj = fj(
(xl<) keX-1(n) ,(Wi) ieW1(n) )
? a backward equation (B)
~=
L
a
. Ell
J dXk
jeY-'X(k)
? a gradient equation (G)
dJ
~i=-.=
dwl
a/ :
Laj ~
je Y-'W(i) awl
The remaining equations do not depend on the nature of the modules. They describe how
modules interact during training. Like back-propagation, they address the credit
assignment problem between modules by globally minimizing a single cost function.
Training such a complex system actually consists in cooperatively training its
components.
4 EXAMPLES
Most learning algorithms, as well as new algorithms may be expressed as modular
learning systems. Here are some simple examples of modules and systems.
4.1
LINEAR AND QUASI-LINEAR SYSTEMS
MODULE
SYMBOL
FORWARD
BACKWARD
GRADIENT
Matrix product
Wx
Yi-tWikXk
~k'""L<Xjwik
~ik=aixk
i
Mean square error
MSE
Perceptron error
Perceptron
Sigmoid
sigmo'id
J. t{d k'Xk)2
~k=-2 (dk-xk)
J.- t(d k-1 9t +(Xk?Xk ~k-- (dk- 19t+(Xk?
Yk?f(Xk)
~k,""f'(Xk)ak
A few basic modules are defined in the above table. Figure 2 gives examples of linear and
quasi linear algorithms derived by combining these modules.
785
786
Bottou and Gallinari
(
Wx
L
(
r.r
Examples
H
Wx
MSE
J
~~ perceptroj'" J
L
--1
sigmo'(d
L
r
H
Wx
H
Examples ---'
sigmoId
H
MSE
r
J
ExalT1lles _ _ _ _ _ _ _ _ _ _ _ _ _..1
1
Figure 2: An Adaline, a Perceptron, and a 2-Layer Perceptron.
Some MLP architectures, Time Delay Networks for instance, use local connections and
shared weights. Such complex architectures may be constructed by defining either quasilinear unit modules or complex matrix operations modules like convolutions. The latter
solution leads to more efficient implementations. Figure 3 gives an example of
convolution module, composed of several matrix products modules.
w
I
I
I
I
Xk
Yk
-
&.. ( Convolve ) - .Yk
Figure 3: A convolution module, composed of several matrix product modules.
4 ?2
EUCLIDIAN DISTANCE BASED ALGORITHMS
A wide class of learning systems are based on the measure of euclidian distances. Again,
defining an euclidian distance module and some adequate cost functions allows for
handling most euclidian distance based algorithms. Here are some examples:
MODULE
SYMBOL
BACKWARD
GRADIENT
(x-w)2
~k=-2tUj(Wjk-Xk)
Ajk=2Uj(Wjk- Xk)
Minimum
Min
~ko-1, ~k,oko-O
LVQ 1 error
LVQ1
Euclidian distance
FORWARD
If the nearest reference Xk? is associated to the correct class
J - Xk o=Min{xiJ
~ko .1, ~k,ok?-O
else
J - -Xko=-Min{xiJ ~ko -1, ~k,ok?.O
Combining an euclidian distance module with a "minimum" error module gives a Kmeans algorithm; combining it with a LVQI error module gives the LVQI algorithm
(Figure 4).
A Framework for the Cooperation of Learning Algorithms
t~~J
.... J
Examples
Figure 4: K-Means and Learning Vector Quantization.
4.3
HYBRID ALGORITHMS
Hybrid algorithms which may combine classical and connectionist learning algorithms are
easily defined by chaining appropriate modules. Figure 5, for instance, depicts an
algorithm combining a MLP layer and LVQ1. This algorithm has been described and
empirically compared to other pattern recognition algorithms in (Bollivier, Gallinari &
Thiria, 1990).
(
Wx
1
H
sigmO?id)-.{ (w
Examples
-X)2]-1 LVa 1 ~ J
1
Figure 5: An hybrid algorithm combining a MLP and LVQ.
Cooperative training gives a framework and a possible implementation for such
algorithms. Nevertheless, there are still specific problems (e.g. convergence,
initialization) which require a careful study. More complex hybrid systems, including
combinations of Markov Models and Time Delay Networks, have been described within
this framework in (Bottou,1991).
5 CONCLUSION
Cooperative training of modular systems provides a unified view of many learning
algorithms, as well as hybrid systems which combine classical or connectionist
algorithms. Our formalism provides a way to define specific modules and to combine
them into a cooperative system. This allows to design and implement complex learning
systems which eventually incorporate structural a priori knowledge about the task.
Acknowledgements
During this work, L.B. was supported by DRET grant nO 87/808/19.
References
Benveniste A., Metivier M., Priouret P. (1987) Algorithmes adaptatifs et approximations
stochastiques, Masson
Bollivier M. de, Gallinari P. & Thiria S. (1990) Cooperation of neural nets for robust
classification, Procedings of I1CNN 90, San Diego, voll, 113-120.
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788
Boltou and Gallinari
Bottou L. (1991) Une approche thiorique de l' apprentissage connexionniste; applications
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Le Cun Y.: A theoretical framework for back-propagation (1988) in Touretzky D.?
Hinton G. & Sejnowsky T. (eds.) Proceedings of the 1988 Connectionist Models
Summer School. 21-28. Morgan Kaufmann (1988)
Le Cun Y.? Boser B .? & al.. (1990): Handwritten Digit Recognition with a BackPropagation Network- in D.Touretzky (ed.) Advances in Neural Information Processing
Systems 2. 396-404. Morgan Kaufmann
Ljung L. & SMerstriim T. (1983) Theory and Practice of Recursive Identification. MIT
Press
Tsypkin Ya. (1971) Adaptation and Learning in Automatic systems. Mathematics in
science and engineering. vol 73. Academic Press
White H. (1991) An Overview of Representation and Convergence results for Multilayer
feed-forward Networks. Touretzky D.S .? Lipmann R. (eds.) Advances in Neural
Information Processing Systems 3 (this volume). Morgan-Kaufman
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2,292 | 3,080 | PG-means: learning the number of clusters in data
Yu Feng
Greg Hamerly
Computer Science Department
Baylor University
Waco, Texas 76798
{yu feng, greg hamerly}@baylor.edu
Abstract
We present a novel algorithm called PG-means which is able to learn the number
of clusters in a classical Gaussian mixture model. Our method is robust and efficient; it uses statistical hypothesis tests on one-dimensional projections of the data
and model to determine if the examples are well represented by the model. In so
doing, we are applying a statistical test for the entire model at once, not just on a
per-cluster basis. We show that our method works well in difficult cases such as
non-Gaussian data, overlapping clusters, eccentric clusters, high dimension, and
many true clusters. Further, our new method provides a much more stable estimate
of the number of clusters than existing methods.
1
Introduction
The task of data clustering is important in many fields such as artificial intelligence, data mining,
data compression, computer vision, and others. Many different clustering algorithms have been developed. However, most of them require that the user know the number of clusters (k) beforehand,
while an appropriate value for k is not always clear. It is best to choose k based on prior knowledge
about the data, but this information is often not available. Without prior knowledge it can be especially difficult to choose k when the data have high dimension, making exploratory data analysis
difficult.
In this paper, we present an algorithm called PG-means (PG stands for projected Gaussian) which
is able to discover an appropriate number of Gaussian clusters and their locations and orientations.
Our method is a wrapper around the standard and widely used Gaussian mixture model. The paper?s
primary contribution is a novel method of determining if a whole mixture model fits its data well,
based on projections and statistical tests. We show that the new approach works well not only in
simple cases in which the clusters are well separated, but also in the situations where the clusters
are overlapped, eccentric, in high dimension, and even non-Gaussian. We show that where some
other methods tend to severely overfit, our method does not, and that our method is comparable to
but much faster than a recent variational Bayes-based approach for learning k.
2
Related work
Several algorithms have been proposed to determine k automatically. Most of these algorithms wrap
around either k-means or Expectation-Maximization for fixed k. As they proceed, they use splitting
or merging rules to increase or decrease k until a proper value is reached.
Pelleg and Moore [9] proposed the X-means algorithm, which is a regularization framework for
learning k with k-means. This algorithm tries many values for k and obtains a model for each k
value. Then X-means uses the Bayesian Information Criterion (BIC) to score each model [5, 12],
and chooses the model with the highest BIC score. Besides the BIC, other scoring criteria could also
be applied such as the Akaike Information Criterion [1], or Minimum Description Length [10]. One
drawback of the X-means algorithm is that the cluster covariances are all assumed to be spherical
and the same width. This can cause X-means to overfit when it encounters data that arise from
non-spherical clusters.
Hamerly and Elkan [4] proposed the G-means algorithm, a wrapper around the k-means algorithm.
G-means uses projection and a statistical test for the hypothesis that the data in a cluster come from
a Gaussian distribution. The algorithm grows k starting with a small number of centers. It applies
a statistical test to each cluster and those which are not accepted as Gaussians are split into two
clusters. Interleaved with k-means, this procedure repeats until every cluster?s data are accepted as
Gaussian. While this method does not assume spherical clusters and works well if true clusters is
well-separated, it has difficulties when true clusters overlap since the hard assignment of k-means
can clip data into subsets that look non-Gaussian.
Sand and Moore [11] proposed an approach based on repairing faults in a Gaussian mixture model.
Their approach modifies the learned model at the regions where the residual is large between the
model?s predicted density and the empirical density. Each modification adds or removes a cluster
center. They use a hill-climbing algorithm to seek a model which maximizes a model fitness scoring function. However, calculating the empirical density and comparing it to the model density is
difficult, especially in high dimension.
Tibshirani et al. [13] proposed the Gap statistic, which compares the likelihood of a learned model
with the distribution of the likelihood of models trained on data drawn from a null distribution. Our
experience has shown that this method works well for finding a small number of clusters, but has
difficulty as the true k increases.
Welling and Kurihara [15] proposed Bayesian k-means, which uses Maximization-Expectation
(ME) to learn a mixture model. ME maximizes over the hidden variables (assignment of examples
to clusters), and computes an expectation over model parameters (center locations and covariances).
It is a special case of variational Bayesian methods. Bayesian k-means works well but is slower than
our method.
None of these prior approaches perform well in all situations; they tend to overfit, underfit, or are
too computationally costly. These issues form the motivation for our new approach.
3
Methodology
Our approach is called PG-means, where PG stands for projected Gaussian and refers to the fact
that the method applies projections to the clustering model as well as the data, before performing
each hypothesis test for model fitness. PG-means uses the standard Gaussian mixture model with
Expectation-Maximization training, but any underlying algorithm for training a Gaussian mixture
might be used. Our algorithm starts with a simple model and increases k by one at each iteration
until it finds a model that fits the data well.
Each iteration of PG-means uses the EM algorithm to learn a model containing k centers. Each time
EM learning converges, PG-means projects both the dataset and the learned model to one dimension,
and then applies the Kolmogorov-Smirnov (KS) test to determine whether the projected model fits
the projected data. PG-means repeats this projection and test step several times for a single learned
model. If any test rejects the null hypothesis that the data follows the model?s distribution, then it
adds one cluster and starts again with EM learning. If every test accepts the null hypothesis for a
given model, then the algorithm terminates. Algorithm 1 describes the algorithm more formally.
When adding a new cluster PG-means preserves the k clusters it has learned and adds a new cluster.
This preservation helps EM converge more quickly on the new model. To find the best new model,
PG-means runs EM 10 times each time it adds a cluster with a different initial location for the new
cluster. The mean of each new cluster is chosen from a set of randomly chosen examples, and also
points with low model-assigned probability density. The initial covariance of the new cluster is based
on the average of the existing clusters? covariances, and the new cluster prior is assigned 1/k and
all priors are re-normalized. More than 10 EM applications could be used, as well as deterministic
annealing [14], to ensure finding the best new model. In our tests, deterministic annealing did not
improve the results of PG-means. As stated earlier, any training algorithm (not just EM) may be
Algorithm 1 PG-means (dataset X, confidence ?, number of projections p)
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
Let k ? 1. Initialize the cluster with the mean and covariance of X.
for i = 1 . . . p do
Project X and the model to one dimension with the same projection.
Use the KS test at significance level ? to test if the projected model fits the projected dataset.
If the test rejects the null hypothesis, then break out of the loop.
end for
if any test rejected the null hypothesis then
for i = 1 . . . 10 do
Initialize k + 1 clusters as the k previously learned plus one new cluster.
Run EM on the k + 1 clusters.
end for
Retain the model of k + 1 clusters with the best likelihood.
Let k ? k + 1, and go to step 2.
end if
Every test accepts the null hypothesis; stop and return the model.
used to fit a particular set of k Gaussian models. For example, one might use k-means if more speed
is desired.
3.1
Projection of the model and the dataset
PG-means is novel because it applies projection to the learned model as well as to the dataset prior to
testing for model fitness. There are several reasons to project both the examples and the model. First,
a mixture of Gaussians remains a mixture of Gaussians after being linearly projected. Second, there
are many effective and efficient tests for model fitness in one dimension, but in higher dimensions
such testing is more difficult.
Assume some data X is sampled from a single Gaussian cluster with distribution X ? N (?, ?) in
d dimensions. So ? = E[X] is the d ? 1 mean vector and ? = Cov[X] is the d ? d covariance
matrix. Given a d ? 1 projection vector P of unit length (||P || = 1), we can project X along P
as X 0 = P T X. Then X 0 ? N (?0 , ? 2 ), where ?0 = P T ? and ? 2 = P T ?P . We can project each
cluster model to obtain a one-dimensional projection of an entire mixture along P . Then we wish to
test whether the projected model fits the projected data.
The G-means and X-means algorithms both perform statistical tests for each cluster individually.
This makes sense because each algorithm is a wrapper around k-means, and k-means uses hard
assignment (each example has membership in only one cluster). However, this approach is problematic when clusters overlap, since the hard assignment results in ?clipped? clusters, making them
appear very non-Gaussian. PG-means tests all clusters and all data at once. Then if two true clusters
overlap, the additive probability of the learned Gaussians representing those clusters will correctly
model the increased density in the overlapping region.
3.2
The Kolmogorov-Smirnov test and critical values
After projection, PG-means uses the univariate Kolmogorov-Smirnov [7] test for model fitness. The
KS test statistic is D = maxX |F (X) ? S(X)| ? the maximum absolute difference between the true
CDF F (X) with the sample CDF S(X). The KS test is only applicable if F (X) is fully specified;
however, PG-means estimates the model with EM, so F (X) cannot be specified a priori. The best
we can do is use the parameter estimates, but this will cause us to accept the model too readily. In
other words, the probability of a Type I error will be too low and PG-means will tend to choose
models with too few clusters. Lilliefors [6] gave a table of smaller critical values for the KS test
which correct for estimated parameters of a single univariate Gaussian. These values come from
Monte Carlo calculations. Along this vein, we create our own test critical values for a mixture of
univariate Gaussians.
To generate the critical values for the KS test statistic, we use the projected one-dimensional model
that has been learned to generate many different datasets, and then measure the KS test statistic
for each dataset. Then we find the KS test statistic that is in the ? range we desire, which is the
critical value we want. Fortunately, this can be done efficiently and does not dominate the running
time of our algorithm. It is much more efficient than if we were to generate datasets from the
full dimensional data and project these to obtain the statistic distribution, yet they are equivalent
approaches. Further optimization is?
possible when we follow Lilliefors? observation that the critical
value decreases as approximately n, for sufficiently large n, which we have also observed in
our simulations with mixtures of Gaussians. Therefore,pwe can use Monte Carlo simulations with
n0 n points, and scale the chosen critical value by n0 /n. A more accurate scaling given by
Dallal and Wilkinson [2] did not offer additional benefit in our tests. We use at most n0 = 3/?,
which is 3000 points for ? = 0.001. The Monte Carlo simulations can be easily parallelized, and
our implementation uses two computational threads.
3.3
Number of projections
We wish to use a small but sufficient number of projections and tests to discover when a model does
not fit data well. Each projection provides a different view of model fitness along that projection?s
direction. However, a projection can cause the data from two or more true clusters to be collapsed
together, so that the test cannot see that there should be multiple densities used to model them.
Therefore multiple projections are necessary to see these model and data discrepancies.
We can choose the projections in several different ways. Random projection [3] provides a useful
framework, which is what we use in this paper. Other possible methods include using the leading
directions from principal components analysis, which gives a stable set of vectors which can be
re-used, or choosing k ? 1 vectors that span the same subspace spanned by the k cluster centers.
Consider two cluster centers ?1 and ?2 in d dimensions and the vector which connects them, m =
?2 ??1 . We assume for simplicity
p that the two clusters have the same spherical covariance ? and are
c-separated, that is, ||m|| ? c trace(?). We follow Dasgupta?s conclusion that c-separation is the
natural measure for Gaussians [3]. Now consider the projection of m along some randomly chosen
vector P ? N (0, 1/dI). We use this distribution because in high dimension P will be approximately
unit-length. The probability that P is a ?good? projection, i.e. that it maintains c-separation between
the cluster means when projected, is
s
!
p
T ?P
?
dP
T
1/2
P r |P m| ? c P T ?P > 1 ? Erf c
= 1 ? Erf
2
2c trace(?)
where Erf is the standard Gaussian error function. Here we have used the relation P T ?P =
trace(?)/d when ? is spherical and ||P || = 1. If ? is not spherical, then this is true in an expected sense, i.e. E[P T ?P ] = trace(?)/d when ||P || = 1. If we perform p random projections,
we wish that the probability that all p projections are ?bad? to be less than some ?:
p
p
P r(p bad projections) = Erf
1/2 < ?
p
Therefore we need approximately p < log(?)/ log(Erf( 1/2)) ? ?2.6198 log(?) projections to
find a projection that keeps the two cluster means c-separated. For ? = 0.01, this is only 12 projections, and for ? = 0.001, this is only 18 projections.
3.4
Algorithm complexity
PG-means converges as fast as EM on any given k, and it repeats EM every time it adds a cluster.
Let K be the final learned number of clusters on n data points. PG-means runs in O(K 2 nd2 l +
Kn log(n)) time, where l is the number of iterations required for EM convergence. The n log(n)
term comes from the sort required for each KS test, and the d2 comes from using full covariance
matrices. PG-means uses a fixed number of projections for each model and each projection is
linear in n, d, and k; therefore the projections do not increase the algorithm?s asymptotic run time.
Note also that EM starts with k learned centers and one new randomly initialized center, so EM
convergence is much faster in practice than if all k + 1 clusters were randomly initialized. We must
also factor in the cost of the Monte Carlo simulations for determining the KS test critical value,
which are O(Kd2 n log(n)/?) for each simulation. For fixed alpha, this does not increase the runtime significantly, and in practice the simulations are a minor part of the running time.
Eccentricity=4
Learned k
Eccentricity=1
Learned k
c=2
c=4
c=6
25
22
22
20
20
20
15
18
18
10
2 4
8
16
16
2 4
8
16
16
60
40
40
40
30
30
20
20
20
0
2 4
10
8
16
dimension
2 4
10
8
16
dimension
2 4
8
16
PG?means
G?means
X?means
BKM
2 4
8
16
dimension
Figure 1: Each point represents the average number of clusters learned for various types of synthetic datasets. The true number of clusters is 20. The error bars denote the standard errors for the
experiments (except for BKM, which was run once for each dataset type).
Eccentricity=4
VI metric score
Eccentricity=1
VI metric score
c=2
c=4
c=6
1
0.4
0.4
0.5
0.2
0.2
0
0
10
20
1.5
0
0
10
20
0
0
1
1
0.5
0.5
1
10
20
PG?means
G?means
X?means
BKM
0.5
0
0
10
dimension
20
0
0
10
dimension
20
0
0
10
dimension
20
Figure 2: Each point represents the average VI metric comparing the learned clustering to the correct
labels for various types of synthetic datasets. Lower values are better. For each algorithm except
BKM we provide standard error bars (BKM was run once for each dataset type).
4
Experimental evaluation
We perform several experiments on synthetic and real-world datasets to illustrate the utility of
PG-means and compare it with G-means, X-means, and Bayesian k-means (BKM). For synthetic
datasets, we experiment with Gaussian and non-Gaussian data. We use ? = 0.001 for both PGmeans and G-means. For each model, PG-means uses 12 projections and tests, corresponding to an
error rate of ? < 0.01 that it incorrectly accepts. All our experiments use MATLAB on Linux 2.4 on
a dual-processor dual-hyperthreaded Intel Xeon 3.06 GHz computer with 2 gigabytes of memory.
Figure 1 shows the number of clusters found by running PG-means, G-means, X-means and BKM
on many synthetic datasets. Each of these datasets has 4000 points in d = 2, 4, 8 and 16 dimensions.
PG?means
G?means
X?means
Figure 3: The leftmost dataset has 10 true clusters with significant overlap (c = 1). Though PGmeans finds only 4 clusters, the model is very reasonable. On the right are the results for PG-means,
G-means, and X-means on a dataset with 5 true eccentric and overlapping clusters. PG-means finds
the correct model, while the others overfit with 15 and 19 clusters.
All of the data are drawn from a mixture of 20 true Gaussians. The centers of the clusters in each
dataset are chosen randomly, and each cluster generates the same number of points. Each Gaussian
mixture dataset is specified by the average c-separation between each cluster center and its nearest
neighbor (either
2, 4 or 6) and each cluster?s eccentricity (either 1 or 4). The eccentricity of is defined
p
as Ecc = ?max /?min where ?max and ?min are the maximum and minimum eigenvalues of the
cluster covariance. An eccentricity of 1 indicates a spherical Gaussian. We generate 10 datasets
of each type and run PG-means, G-means and X-means on each, and we run BKM on only one of
them due to the running time of BKM. Each algorithm starts with one center, and we do not place
an upper-bound on the number of clusters.
It is clear that PG-means performs better than G-means and X-means when the data are eccentric
(Ecc=4), especially when the clusters overlap (c = 2). In this situation G-means and X-means
tend to overestimate the number of clusters. The rightmost plots in Figure 3 further illustrate this
overfitting. PG-means is much more stable in its estimate of the number of clusters, unlike Gmeans and X-means which can dramatically overfit depending on the type of data. BKM generally
does very well, but is less efficient than PG-means. For example, on a set of 24 different datasets
each having 4000 points from 10 clusters, 2-16 dimensions and varying separations/eccentricities,
PG-means was three times faster than BKM.
Figure 1 only gives the information regarding the learned number of clusters, which is not enough to
measure the true quality of learned models. In order to better evaluate the approaches, we use Meila?s
VI (Variation of Information) metric [8] to compare the induced clustering to the true labels. The
VI metric is non-negative and lower values are better. It is zero when the two compared clusterings
are identical (modulo clusters being relabeled). Figure 2 shows the average VI metric obtained by
running PG-means, G-means, X-means, and BKM on the same synthetic datasets as in Figure 1.
PG-means does about as well as the other algorithms when the data are spherical and well-separated
(see the top-right plot). However, the top-left plot shows that PG-means does not perform as well as
G-means, X-means and BKM for spherical and overlapping data. The reason is that two spherical
clusters overlap, they can look like a single eccentric cluster. Since PG-means can capture eccentric
clusters effectively, it will accept these two overlapped spherical clusters as one cluster. But for the
same case, G-means and X-means will probably recognize them as two different clusters. Therefore,
although PG-means gives fewer clusters for spherical and overlapping data, the models it learns are
reasonable. Figure 3 shows how 10 true overlapping clusters may look like far fewer clusters, and
that PG-means can find an appropriate model with only 4 clusters.
High dimensional data of any finite-variance distribution looks more Gaussian when linearly projected to a randomly chosen lower-dimensional space. Projection is a weighted sum of the original
dimensions, and the sum of many random variables with finite variance tends to be Gaussian, according to the central limit theorem. Thus PG-means should be useful for high-dimensional data
which are not Gaussian. To test this, we perform experiments on high-dimensional non-Gaussian
synthetic datasets. These datasets are generated in a similar way of generating our synthetic Gaussian datasets, except that each true cluster has a uniform distribution. Each cluster is not necessarily
axis-aligned or square; it is scaled for eccentricity and rotated. Each dataset has 4000 points in 8
dimensions equally distributed among 20 clusters. The eccentricity and c-separation values for the
datasets are both 4. We run PG-means, G-means and X-means on 10 different datasets and BKM
Table 1: Results for synthetic non-Gaussian data and the handwritten digits dataset. Each nonGaussian dataset contains 4000 points in 8 dimensions sampled from 20 true clusters each having
uniform distribution. The eccentricity and c-separation are both 4. We run each algorithm except
BKM on ten such datasets, and BKM on one. The digits dataset consists of 10 classes and 9298
examples.
Algorithm
PG-means
G-means
X-means
BKM
Non-Gaussian datasets (20 true clusters)
Learned k
VI metric
20 ? 0
0?0
42.2 ? 3.67 0.673 ? 0.071
27.7 ? 1.28 0.355 ? 0.059
20
0
Handwritten digits dataset (10 true classes)
Learned k VI metric
14
2.045
48
3.174
29
2.921
15
1.980
on one of them. The results are shown in the left part of Table 1. G-means and X-means overfit
the non-Gaussian datasets, while PG-means and BKM both perform excellently in the number of
clusters learned and in learning the true labels according to the VI metric.
We tested all of these algorithms on the U.S. Postal Service handwritten digits dataset (both the train
and test portions, obtained from http://www-stat.stanford.edu/?tibs/ElemStatLearn/data.html). Each
example is a grayscale image of a handwritten digit. There are 9298 examples in the dataset, and
each example has 256 pixels (16 pixels on a side). The dataset has 10 true classes (digits 0-9). Our
goal is to cluster the dataset without knowing the true labels and analyze the result to find out how
well PG-means captures the true classes.
We use random linear projection to project the dataset to 16 dimensions and run PG-means, Gmeans, X-means, and BKM on it. The results are shown in the right side of Table 1. PG-means
gives 14 centers, which is closest to the true value. It also obtains nearly the best VI metric score.
On the other hand, G-means and X-means find many more classes than the truth, which do not help
them score well on the VI metric, and BKM takes over twice as long as PG-means.
5
Conclusions and future work
We presented a new algorithm called PG-means for learning the number of Gaussian clusters k in
data. Starting with one center, it grows k gradually. For each k, it learns a model using ExpectationMaximization. Then it projects both the model and the dataset to one dimension and tests for model
fitness with the Kolmogorov-Smirnov test and its own critical values. It performs multiple projections and tests per model, to avoid being fooled by a poorly chosen projection. If the model does not
fit well, PG-means adds an additional cluster. This procedure repeats until one model is accepted
by all tests. We proved that only a small number of these fast tests are required to have good performance at finding model differences. In the future we will investigate methods of finding better
projections for our task. We also hope to develop approximations to the critical values of the KS test
on Gaussian mixtures, to avoid the cost of Monte Carlo simulations.
PG-means finds better models than G-means and X-means when the true clusters are eccentric or
overlap, especially in low-dimension. On high-dimensional data PG-means also performs very well.
PG-means gives far more stable estimates of the number of clusters than the other two methods
over many different types of data. Compared with Bayesian k-means, we showed that PG-means
performs comparably, though PG-means is several times faster in our tests and uses less memory.
Though PG-means looks for general Gaussian clusters, we showed that PG-means works well on
high-dimensional non-Gaussian data, due to the central limit theorem and our use of projection. Our
techniques would also be applicable as a wrapper around the k-means algorithm, which is really
just a mixture of spherical Gaussians, or any other mixture of Gaussians with limited covariance.
On the real-world handwritten digits dataset PG-means finds a very good clustering with nearly the
correct number of classes, and PG-means and BKM are equally close to identifying the original
labels among the algorithms we tested.
We believe that the project-and-test procedure that PG-means uses is a useful method for determining fitness of a given mixture of Gaussians. However, the underlying standard EM clustering
algorithm dominates the runtime and is difficult to initialize well, which are well-known problems.
The project-and-test framework of PG-means does not depend on EM in any way, and could be
wrapped around any other better method of finding a Gaussian mixture.
Acknowledgements: We thank Dennis Johnston, Sanjoy Dasgupta, Charles Elkan, and the anonymous reviewers for helpful suggestions. We also thank Dan Pelleg and Ken Kurihara for sending us
their source code.
References
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tested:2 |
2,293 | 3,081 | Efficient Methods for Privacy Preserving Face
Detection
Shai Avidan
Mitsubishi Electric Research Labs
201 Broadway
Cambridge, MA 02139
[email protected]
Moshe Butman
Department of Computer Science
Bar Ilan University
Ramat-Gan, Israel
[email protected]
Abstract
Bob offers a face-detection web service where clients can submit their images for
analysis. Alice would very much like to use the service, but is reluctant to reveal
the content of her images to Bob. Bob, for his part, is reluctant to release his face
detector, as he spent a lot of time, energy and money constructing it. Secure MultiParty computations use cryptographic tools to solve this problem without leaking
any information. Unfortunately, these methods are slow to compute and we introduce a couple of machine learning techniques that allow the parties to solve the
problem while leaking a controlled amount of information. The first method is an
information-bottleneck variant of AdaBoost that lets Bob find a subset of features
that are enough for classifying an image patch, but not enough to actually reconstruct it. The second machine learning technique is active learning that allows
Alice to construct an online classifier, based on a small number of calls to Bob?s
face detector. She can then use her online classifier as a fast rejector before using
a cryptographically secure classifier on the remaining image patches.
1
Introduction
The Internet triggered many opportunities for cooperative computing in which buyers and sellers
can meet to buy and sell goods, information or knowledge. Placing classifiers on the Internet allows
buyers to enjoy the power of a classifier without having to train it themselves. This benefit is hindered by the fact that the seller, that owns the classifier, learns a great deal about the buyers? data,
needs or goals. This raised the need for privacy in Internet transactions. While it is now common to
assume that the buyer and the seller can secure their data exchange from the rest of the world, we
are interested in a stronger level of security that allows the buyer to hide his data from the seller as
well. Of course, the same can be said about the seller, who would like to maintain the privacy of his
hard-earned classifier.
Secure Multi-Party Computation (SMC) are based on cryptographic tools that let two parties, Alice
and Bob, to engage in a protocol that will allow them to achieve a common goal, without revealing
the content of their input. For example, Alice might be interested in classifying her data using Bobs?
classifier without revealing anything to Bob, not even the classification result, and without learning
anything about Bobs? classifier, other than a binary answer to her query.
Recently, Avidan & Butman introduced Blind Vision [1] which is a method for securely evaluating a
Viola-Jones type face detector [12]. Blind Vision uses standard cryptographic tools and is painfully
slow to compute, taking a couple of hours to scan a single image. The purpose of this work is to
explore machine learning techniques that can speed up the process, at the cost of a controlled leakage
of information.
In our hypothetical scenario Bob has a face-detection web service where clients can submit their
images to be analyzed. Alice would very much like to use the service, but is reluctant to reveal the
content of the images to Bob. Bob, for his part, is reluctant to release his face detector, as he spent a
lot of time, energy and money constructing it.
In our face detection protocol Alice raster scans the image and sends every image patch to Bob
to be classified. We would like to replace cryptographically-based SMC methods with Machine
Learning algorithms that might leak some information but are much faster to execute. The challenge
is to design protocols that can explicitly control the amount of information leaked. To this end we
propose two, well known, machine learning techniques. One based on the information bottleneck
and the other on active learning.
The first method is a privacy-preserving feature selection which is a variant of the informationbottleneck principle to find features that are useful for classification but not for signal reconstruction. In this case, Bob can use his training data to construct different classifiers that offer different
trade-offs of information leakage versus classification accuracy. Alice can then choose the trade-off
that suits her best and send only those features to Bob for classification. This method can be used,
for example, as a filtering step that rejects a large number of the image patches as having no face included in them, followed by a SMC method that will securely classify the remaining image patches,
using the full classifier that is known only to Bob.
The second method is active learning and it helps Alice choose which image patches to send to Bob
for classification. This method can be used either with the previous method or directly with an SMC
protocol. The idea being that instead of sending all image patches to Bob for classification, Alice
might try to learn from the interaction as much as she can and use her online trained classifier to
reject some of the image patches herself. This can minimize the amount of information revealed to
Bob, if the parties use the privacy-preserving features or the computational load, if the parties are
using cryptographically-based SMC methods.
2
Background
Secure multi-party computation originated from the work of Yao [14] who gave a solution to the
millionaire problem: Two parties want to find which one has a larger number, without revealing
anything else about the numbers themselves. Later, Goldriech et al. [5] showed that any function
can be computed in such a secure manner. However, the theoretical construct was still too demanding
to be of practical use. An easy introduction to Cryptography is given in [9] and a more advanced
and theoretical treatment is given in [4]. Since then many secure protocols were reported for various
data mining applications [7, 13, 1]. A common assumption in SMC is that the parties are honest
but curious, meaning that they will follow the agreed-upon protocol but will try to learn as much as
possible from the data-flow between the two parties. We will follow this assumption here.
The information bottleneck principle [10] shows how to compress a signal while preserving its
information with respect to a target signal. We offer a variant of the self-consistent equations used
to solve this problem and offer a greedy feature selection algorithm that satisfy privacy constraints,
that are represented as the percentage of the power spectrum of the original signal.
Active learning methods assume that the student (Alice, in our case) has access to an Oracle (Bob)
for labeling. The usual motivation in active learning is that the Oracle is assumed to be a human operator and having him label data is a time consuming task that should be avoided. Our motivation is
similar, Alice would like to avoid using Bob because of the high computational cost involved in case
of cryptographically secure protocols, or for fear of leaking information in case non-cryptographic
methods are used. Typical active learning applications assume that the distribution of class size is
similar [2, 11]. A notable exception is the work of [8] that propose an active learning method for
anomaly detection. Our case is similar as image patches that contain faces are rare in an image.
3
Privacy-preserving Feature Selection
Feature selection aims at finding a subset of the features that optimize some objective function, typically a classification task [6]. However, feature selection does not concern itself with the correlation
of the feature subset with the original signal.
This is handled with the information bottleneck method [10], that takes a joint distribution p(x, y)
and finds a compressed representation of X, denoted by T , that is as informative about Y . This is
achieved by minimizing the following functional:
min L : L ? I(X; T ) ? ?I(T ; Y )
p(t|x)
(1)
where ? is a trade-off parameter that controls the trade off between compressing X and maintaining
information about Y . The functional L admits a set of self-consistent equations that allows one to
find a suitable solution.
We map the information bottleneck idea to a feature selection algorithm to obtain a Privacypreserving Feature Selection (PPFS) and describe how Bob can construct such a feature set. Let
Bob have a training set of image patches, their associated label and a weight associated with every
feature (pixel) denoted {xn , yn , sn }N
n=1 . Bob?s goal is to find a feature subset I ? {i1 , . . . , ik } s.t.
a classifier F (x(I)) will minimize the classification error, where x(I) denotes a sample x that uses
only the features in the set I. Formally, Bob needs to minimize:
min
F
N
X
(F (xn (I)) ? yn ))2
(2)
n=1
subject to
X
si < ?
i?I
where ? is a user defined threshold that defines the amount of information that can be leaked.
We found it useful to use the PCA spectrum to measure the amount of information. Specifically, Bob
computes the PCA space of all the face images in his database and maps all the data to that space,
without reducing dimensionality. The weights {sn }N
n=1 are now set to the eigenvalues associated
with each dimension in the PCA space. This avoids the need to compute the mutual information
between pixels by making the assumption that features do not carry mutual information with other
features beyond second order statistics.
Algorithm 1 Privacy-Preserving Feature Selection
Input: {xn , yn , sn }N
n=1
Threshold ?
Number of iterations T
Output: A privacy-preserving strong classifier F (x)
? Start with weights wn = 1/N n = 1, 2, . . . , N , F (x) = 0, I = ?
? Repeat for t = 1, 2, . . . , T
P
? Set working index set J = I ? {j|sj + i?I si < ?}
? Repeat for j ? J
? Fit a regression stump gj (x(j)) ? aj (x(j) > ?j ) + bj to the j-th feature, x(j)
PN
wn (yn ?(aj (xn (j)>?j +bj )2
PN
? Compute error ej = n=1
n=1
wn
? Set ft = gi where ei < ej ?j ? J
? Update:
F (x) ? F (x) + ft (x)
(3)
?yn ft (xn )
(4)
(5)
wn ? wn e
I ? I ? {i}
Boosting was used for feature selection before [12] and Bob takes a similar approach here. He uses
a variant of the gentleBoost algorithm [3] to find a greedy solution to (2). Specifically, Bob uses
gentleBoost with ?stumps? as the weak classifiers where each ?stump? works on only one feature.
The only difference from gentleBoost is in the choice of the features to be selected. In the original
algorithm all the features are evaluated in every iteration of the algorithm, but here Bob can only use
a subset of the features. In each iteration Bob can use features that were already selected or those
that adding them will not increase the total weight of selected features beyond the threshold ?.
Once Bob computed the privacy-preserving feature subset, the amount of information it leaks and
its classification accuracy he publishes this information on the web. Alice then needs to map her
image patches to this low-dimensional privacy-preserving feature space and send the data to Bob for
classification.
4
Privacy-Preserving Active Learning
In our face detection example Alice needs to submit many image patches to Bob for classification.
This is computationally expensive if SMC methods are used and reveals information, in case the
privacy-preserving feature selection method discussed earlier is used. Hence, it would be beneficial
if Alice could minimize the number of image patches she needs to send Bob for classification. This
is where she might use active learning. Instead of raster scanning the image and submitting every
image patch for classification she sends a small number of randomly selected image patches, and
based on their label, she determines the next group of image patches to be sent for classification. We
found that substantial gains can be made this way.
Specifically, Alice maintains an RBF network that is trained on-line, based on the list of labeled
prototypes. Let {cj , yj }M
j=1 be the list of M prototypes that were labeled so far. Then, Alice constructs a kernel matrix K where Kij = k(ci , cj ) and solves the least squares equation Ku = y,
where y = [y1 , . . . , yM ]T . The kernel Alice uses is a Gaussian kernel whose width is set to be the
range of the prototype coordinates, in each dimension. The score of each image patch x is given by
h(x) = [k(x, c1 ), . . . , k(x, cM )]u.
For the next round of classification Alice chooses the image patches with the highest h(x) score.
This is in line with [2, 11, 8] that consider choosing the examples of which one has the least amount
of information. In our case, Alice is interested in finding image patches that contain faces (which we
assume are labeled +1) but most of the prototypes will be labeled ?1, because faces are a rear event
in an image. As long as Alice does not sample a face image patch she will keep exploring the space
of image patches in her image, by sampling image patches that are farthest away from the current set
of prototypes. If an image patch that contains a face is sampled, then her online classifier h(x) will
label similar image patches with a high score, thus guiding the search towards other image patches
that might contain a face. To avoid large overlap between patches, we force a minimum distance, in
the image plane, between selected patches. The algorithm is given in algorithm 2.
Algorithm 2 Privacy-Preserving Active Learning
Input: {xi }N
i=1 unlabeled samples
Number M of classification calls allowed
Number s of samples to classify in each iteration
Output: Online classifier h(x)
? Choose s random samples {xi }si=1 , set C = [x1 , . . . , xs ] and obtain their labels y =
[y1 , . . . , ys ] from Bob.
? Repeat for m = 1, 2, ..., M times
? Construct the kernel matrix Kij = k(ci , cj ) and solve for the weight vector u through
least squares Ku = y.
? Evaluate h(xi ) = [k(xi , c1 ), . . . , k(xi , cm )]u ?i = 1, . . . , N .
? Choose top s samples with highest h(x) score, send them to Bob for classification and
add them, and their labels to C, y, respectively.
5
Experiments
We have conducted a couple of experiments to validate both methods.
Figure 1: Privacy preserving feature selection. We show the ROC curves of four strong classifiers,
each trained with 100 weak, ?stump? classifiers, but with different levels of information leakage.
The information leakage is defined as the amount of PCA spectrum captured by the features used
in each classifier. The number in parenthesis shows how much of the eigen spectrum is captured by
the features used in each classifier.
The first experiment evaluates the privacy-preserving feature selection method. The training set
consisted of 9666 image patches of size 24 ? 24 pixels each, split evenly to face/no-face images.
The test set was of similar size. We then run algorithm 1 with different levels of the threshold ? and
created a strong classifier with 100 weak, ?stump? based, classifiers. The ROC curves of several
such classifiers are shown in figure 1. We found that, for this particular dataset, setting ? = 0.1
gives identical results to a full classifier, without any privacy constraints. Reducing ? to 0.01 did
hurt the classification performance somewhat.
The second experiment tests the active learning approach. We assume that Alice and Bob use the
classifier with ? = 0.05 from the previous experiment, and measure how effective is the on-line
classifier that Alice constructs in rejecting no-face image patches.
Recall that there are three classifiers at play. One is the full classifier that Bob owns, the second is the
privacy-preserving classifier that Bob owns and the last is the on-line classifier that Alice constructs.
Alice uses the labels of Bobs? privacy-preserving classifier to construct her on-line classifier and the
questions is: how many image patches she can reject, without actually rejecting image patches that
will be classified as faces by the full classifier (that she knows nothing about)?
Before we performed the experiment, we conducted the following pre-processing operation: We
find, for each image, the scale at which the largest number of faces are detected using Bob?s full
classifier, and used only the image at that scale.
The experiment proceeds as follows. Alice chooses 5 image patches in each round, maps them to the
reduced PCA space and sends them to Bob for classification, using his privacy-preserving classifier.
Based on his labels, Alice then picks the next 5 image patches according to algorithm 2 and so on.
Alice repeats the process 10 times, resulting in 50 patches that are sent to Bob for classification. The
first 5 patches are chosen at random. Figure 2 shows the 50 patches selected by Alice, the online
classifier h and the corresponding rejection/recall curve, for several test images. The rejection/recall
curve shows how many image patches Alice can safely reject, based on h, without rejecting a face
that will be detected by Bobs? full classifier. For example, in the top row of figure 2 we see that
rejecting the bottom 40% of image patches based on the on-line classifier h will not reject any face
that can be detected with the full classifier. Thus 50 image patches that can be quickly labeled while
leaking very little information can help Alice reject thousands of image patches.
Next, we conducted the same experiment on a larger set of images, consisting of 65 of the
CMU+MIT database images1 . Figure 3 shows the results. We found that, on average (dashed line),
1
We used the 65 images in the newtest directory of the CMU+MIT dataset
(a-1)
(b-1)
(c-1)
(a-2)
(b-2)
(c-2)
(a-3)
(b-3)
(c-3)
Figure 2: Examples of privacy-preserving feature selection and active learning. (a) The input images
and the image patches (marked with white rectangles) selected by the active learning algorithm. (b)
The response image computed by the online classifier (the black spots correspond to the position of
the selected image patches). Brighter means a higher score. (c) The rejection/recall curve showing
how many image patches can be safely rejected. For example, panel (c-1) shows that Alice can reject
almost 50% of the image patches, based on her online classifier (i.e., response image), and not miss
a face that can be detected by the full classifier (that is known to Bob and not to Alice).
(a)
Figure 3: Privacy preserving active learning. Results on a dataset of 65 images. The figure shows
how many image patches can be rejected, based on the online classifier that Alice owns, without
rejecting a face. The horizontal axis shows how many image patches are rejected, based on the
on-line classifier, and the vertical axis shows how many faces are maintained. For example, the
figure shows (dashed line) that rejecting 20% of all image patches, based on the on-line classifier,
will maintain 80% of all faces. The solid line shows that rejecting 40% of all image patches, based
on the on-line classifier, will not miss a face in at least half (i.e. the median) of the images in the
dataset.
using only 50 labeled image patches Alice can reject up to about 20% of the image patches in an
image while keeping 80% of the faces in that image (i.e., Alice will reject 20% of the image patches
that Bob?s full classifier will classify as a face). If we look at the median (solid line), we see that for
at least half the images in the data set, Alice can reject a little more than 40% of the image patches
without erroneously rejecting a face.
We found that increasing the number of labeled examples from 50 to a few hundreds does not greatly
improve results, unless many thousands of samples are labeled, at which point too much information
might be leaked.
6
Conclusions
We described two machine learning methods to accelerate cryptographically secure classification
protocols. The methods greatly accelerate the performance of the system, while leaking a controlled
amount of information. The two methods are a privacy preserving feature selection that is similar to
the information bottleneck and an active learning technique that was found to be useful in learning
a rejector from an extremely small number of labeled data. We plan to keep investigating these
methods, apply them to classification tasks in other domains and develop new methods to make
secure classification faster to use.
References
[1] S. Avidan and M. Butman. Blind vision. In Proc. of European Conference on Computer Vision,
2006.
[2] Y. Baram, R. El-Yaniv, and K. Luz. Online choice of active learning algorithms. Journal of
Machine Learning Research, 5:255?291, March 2004.
[3] J. Friedman, T. Hastie, and R. Tibshirani. Additive logistic regression: a statistical view of
boosting, 1998.
[4] O. Goldreich. Foundations of Cryptography: Volume 1, Basic Tools. Cambridge University
Press, New York, 2001.
[5] O. Goldreich, S. Micali, and A. Wigderson. How to play any mental game or a completeness
theorem for protocols with honest majority. In ACM Symposium on Theory of Computing,
pages 218?229, 1987.
[6] I. Guyon and A. Elisseeff. An introduction to variable and feature selection. Journal of Machine Learning Research, 3:1157?1182, 2003.
[7] Y. Lindell and B. Pinkas. Privacy preserving data mining. In CRYPTO: Proceedings of Crypto,
2000.
[8] D. Pelleg and A. Moore. Active learning for anomaly and rare-category detection. In In
Advances in Neural Information Processing Systems 18, 2004.
[9] B. Schneier. Applied Cryptography. John Wiley & Sons, New York, 1996.
[10] N. Tishby, F. Pereira, and W. Bialek. The information bottleneck method. In In Proc. of 37th
Allerton Conference on communication and computation, 1999.
[11] S. Tong and D. Koller. Support vector machine active learning with applications to text classification. Journal of Machine Learning Research, 2:45?66, 2001.
[12] P. Viola and M. Jones. Rapid object detection using a boosted cascade of simple features. In
Coference on Computer Vision and Pattern Recognition (CVPR), 2001.
[13] R. N. Wright and Z. Yang. ?privacy-preserving bayesian network structure computataion on
distributed heterogeneous data?. In KDD ?04: Proceeding of the tenth ACM SIGKDD international conference on Knowledge discovery in data mining, pages 22?25, 2004.
[14] A. C. Yao. Protocols for secure computations. In Proc. 23rd IEEE Symp. on Foundations of
Comp. Science, pages 160?164, Chicago, 1982. IEEE.
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2,294 | 3,082 | No-regret Algorithms for Online Convex Programs
Geoffrey J. Gordon
Department of Machine Learning
Carnegie Mellon University
Pittsburgh, PA 15213
[email protected]
Abstract
Online convex programming has recently emerged as a powerful primitive for
designing machine learning algorithms. For example, OCP can be used for learning a linear classifier, dynamically rebalancing a binary search tree, finding the
shortest path in a graph with unknown edge lengths, solving a structured classification problem, or finding a good strategy in an extensive-form game. Several
researchers have designed no-regret algorithms for OCP. But, compared to algorithms for special cases of OCP such as learning from expert advice, these
algorithms are not very numerous or flexible. In learning from expert advice,
one tool which has proved particularly valuable is the correspondence between
no-regret algorithms and convex potential functions: by reasoning about these
potential functions, researchers have designed algorithms with a wide variety of
useful guarantees such as good performance when the target hypothesis is sparse.
Until now, there has been no such recipe for the more general OCP problem, and
therefore no ability to tune OCP algorithms to take advantage of properties of the
problem or data. In this paper we derive a new class of no-regret learning algorithms for OCP. These Lagrangian Hedging algorithms are based on a general
class of potential functions, and are a direct generalization of known learning rules
like weighted majority and external-regret matching. In addition to proving regret
bounds, we demonstrate our algorithms learning to play one-card poker.
1
Introduction
In a sequence of trials we must pick hypotheses y1 , y2 , . . . ? Y. After we choose yt , the correct
answer is revealed as a convex loss function ?t (yt ).1 Just before seeing the tth example, our total
Pt?1
loss is therefore Lt = i=1 ?i (yi ). If we had predicted using some fixed hypothesis y instead, then
Pt?1
our loss would have been i=1 ?i (y). Our total regret at time t is the difference between these two
losses, with positive regret meaning that we would have preferred y to our actual plays:
?t (y) = Lt ?
t?1
X
i=1
?i (y)
?t = sup ?t (y)
y?Y
We assume that Y is a compact convex subset of Rd that has at least two elements. In classical
no-regret algorithms such as weighted majority, Y is a simplex: the corners of Y represent pure
actions, the interior points of Y represent probability distributions over pure actions, and the number
of corners n is the same as the number of dimensions d. In a more general OCP, Y may have
1
Many problems use loss functions of the form ?t (yt ) = ?(yt , yttrue ), where ? is a fixed function such as
squared error and yttrue is a target output. The more general notation allows for problems where there may be
more than one correct prediction.
many more extreme points than dimensions, n ? d. For example, Y could be a convex set like
{y | Ay = b, y ? 0} for some matrix A and vector b, or it could even be a sphere.
The shape of Y captures the structure in our prediction problem. Each point in Y is a separate
hypothesis, but the losses of different hypotheses are related to each other because they are all embedded in the common representation space Rd . While we could run a standard no-regret algorithm
such as weighted majority on a structured Y by giving it hypotheses corresponding to the extreme
points c1 . . . cn of Y, this transformation would lose the connections among hypotheses (with a
corresponding loss in runtime and generalization ability).
Our algorithms below are stated in terms of linear loss functions, ?t (y) = ct ? y. If ?t is nonlinear
but convex, we can substitute the derivative at the current prediction, ??t (yt ), for ct , and our regret
bounds will still hold (see [1, p. 53]). We will write C for the set of possible gradient vectors ct .
2
Related Work
A large number of researchers have studied online prediction in general and OCP in particular. The
OCP problem dates back to Hannan in 1957 [2]. The name ?online convex programming? is due
to Zinkevich [3], who gave a clever gradient-descent algorithm. A similar algorithm and a weaker
bound were presented somewhat earlier in [1]: that paper?s GGD algorithm, using potential function
?0 (w) = kkwk22 , is equivalent to Zinkevich?s ?lazy projection? with a fixed learning rate. Another
clever algorithm for OCP was presented by Kalai and Vempala [4].
Compared to the above papers, the most important contribution of the current paper is its generality:
no previous family of OCP algorithms can use as flexible a class of potential functions. As an
illustration of the importance of this generality, consider the problem of learning from expert advice.
Well-known regret bounds for this problem are logarithmic in the number of experts (e.g., [5]); no
previous bounds for general OCP algorithms are sublinear in the number of experts, but logarithmic
bounds follow directly as a special case of our results [6, sec. 8.1.2]. Despite this generality, our
core result, Thm. 4 below, takes only half a dozen short equations to prove.
From the online prediction literature, the closest related work is that of Cesa-Bianchi and Lugosi [7],
which follows in the tradition of an algorithm and proof by Blackwell [8]. Cesa-Bianchi and Lugosi
consider choosing predictions from an essentially-arbitrary decision space and receiving outcomes
from an essentially-arbitrary outcome space. Together a decision and an outcome determine how
a marker Rt ? Rd will move. Given a potential function G, they present algorithms which keep
G(Rt ) from growing too quickly. This result is similar in flavor to our Thm. 4, and both Thm. 4
and the results of Cesa-Bianchi and Lugosi are based on Blackwell-like conditions. In fact, our
Thm. 4 can be thought of as the first generalization of well-known online learning results such as
Cesa-Bianchi and Lugosi?s to online convex programming.
The main differences between the Cesa-Bianchi?Lugosi results and ours are the restrictions on their
potential functions. They write
P their potential function as G(u) = f (?(u)); they require ? to
be additive (that is, ?(u) = i ?i (ui ) for one-dimensional functions ?i ), nonnegative, and twice
differentiable, and they require f : R+ 7? R+ to be increasing, concave, and twice differentiable.
These restrictions rule out many of the potential functions used here, and in fact they rule out most
online convex programming problems. The most restrictive requirement is additivity; for example,
when defining potentials for OCPs via Eq. (7) below, unless the set Y? can be factored as Y?1 ? Y?2 ?
. . . ? Y?N the potentials are generally not expressible as f (?(u)) for additive ?.
During the preparation of this manuscript, we became aware of the recent work of Shalev-Shwartz
and Singer [9]. This work generalizes some of the theorems in [6] and provides a very simple and
elegant proof technique for algorithms based on convex potential functions. However, it does not
consider the problem of defining appropriate potential functions for the feasible regions of OCPs
(as discussed in Sec. 5 below and in more detail in [6]); finding such functions is an important
requirement for applying potential-based algorithms to OCPs.
In addition to the general papers above, there are many no-regret algorithms for specific OCPs, such
as predicting as well as the best pruning of a decision tree [10], reorganizing a binary search tree so
that frequent items are near the root [4], and picking paths in a graph with unknown edge costs [11].
s1 ? 0
for t ? 1, 2, . . .
y?t ? f (st )
if y?t ? u > 0 then
yt ? y?t /(?
yt ? u)
else
yt ? arbitrary element of Y
fi
Observe ct , compute st+1 from (1)
end
1
0.8
0.6
0.4
0.2
0
?0.2
?0.4
?0.6
?0.8
?1
?1
?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
0.6
0.8
1
Figure 1: A set Y = {y1 + y2 = 1, y ? 0} (thick
dark line) and its safe set S (light shaded region).
3
(*)
Figure 2: The gradient form of the Lagrangian Hedging algorithm.
Regret Vectors
Lagrangian Hedging algorithms maintain their state in a regret vector, st , defined by the recursion
st+1 = st + (yt ? ct )u ? ct
(1)
with the base case s1 = 0. Here u is an arbitrary vector which satisfies y ? u = 1 for all y ? Y.
(If necessary we can append a constant element to each y so that such a u exists.) The regret vector
contains information about our actual losses and the gradients of our loss functions: from st we can
find our regret versus any y as follows. (This property justifies the name ?regret vector.?)
y ? st =
t?1
X
i=1
(yi ? ci )y ? u ?
t?1
X
i=1
y ? ci = Lt ?
t?1
X
i=1
y ? ci = ?t (y)
We can define a safe set, in which our regret is guaranteed to be nonpositive:
S = {s | (?y ? Y) y ? s ? 0}
(2)
The goal of the Lagrangian Hedging algorithm is to keep its regret vector st near the safe set S. S is
a convex cone: it is closed under positive linear combinations of its elements. And, it is polar [12]
to the cone of unnormalized hypotheses:
S ? = Y? ? {?y | y ? Y, ? ? 0}
(3)
4
The Main Algorithm
We will present the general LH algorithm first, then (in Sec. 5) a specialization which is often easier
to implement. The two versions are called the gradient form and the optimization form. The gradient
form is shown in Fig. 2. At each step it chooses its play based on the current regret vector st (Eq. (1))
and a closed convex potential function F (s) : Rd 7? R with subgradient f (s) : Rd 7? Rd . This
potential function is what distinguishes one instance of the LH algorithm from another. F (s) should
be small when s is in the safe set, and large when s is far from the safe set.
For example, suppose that Y is the probability simplex in Rd , so that S is the negative orthant in Rd .
(This choice of Y would be appropriate for playing a matrix game or predicting from expert advice.)
For this Y, two possible potential functions are
X
X
[si ]2+ /2
e?si ? ln d
F2 (s) =
F1 (s) = ln
i
i
where ? > 0 is a learning rate and [s]+ = max(s, 0). The potential F1 leads to the Hedge [5] and
weighted majority [13] algorithms, while the potential F2 results in external-regret matching [14,
Theorem B]. For more examples of useful potential functions, see [6].
To ensure the LH algorithm chooses legal hypotheses yt ? Y, we require the following (note the
constant 0 is arbitrary; any other k would work as well)
F (s) ? 0
?s ? S
(4)
Theorem 1 The LH algorithm is well-defined: define S as in (2) and fix a finite convex potential
function F . If F (s) ? 0 for all s ? S, then the LH algorithm picks hypotheses yt ? Y for all t.
(Omitted proofs are given in [6].) We can also define a version of the LH algorithm with an adjustable learning rate: replacing F (s) with F (?s) is equivalent to updating st with learning rate ?.
Adjustable learning rates will help us obtain regret bounds for some classes of potentials.
5
The Optimization Form
Even if we have a convenient representation of our hypothesis space Y, it may not be easy to work
directly with the safe set S. In particular, it may be difficult to define, evaluate, and differentiate a
potential function F which has the necessary properties. To avoid these difficulties, we can work
with an alternate form of the LH algorithm. This form, called the optimization form, defines F in
terms of a simpler function W which we will call the hedging function. It uses the same pseudocode
as the gradient form (Fig. 2), but on each step it computes F and ?F by solving an optimization
problem involving W and the hypothesis set Y (Eq. (8) below).
For example, two possible hedging functions are
P
P
?i ln y?i + ln d
if y? ? 0, i y?i = 1
iy
W1 (?
y) =
?
otherwise
X
2
W2 (?
y) =
y?i /2
(5)
(6)
i
If Y is the probability simplex in Rd , it will turn out that W1 (?
y /?) and W2 (?
y ) correspond to the
potentials F1 and F2 from Section 4 above. So, these hedging functions result in the weighted
majority and external-regret matching algorithms. For an example where the hedging function is
easy to write analytically but the potential function is much more complicated, see Sec. 8 or [6].
The optimization form of the LH algorithm using hedging function W is defined to be equivalent to
the gradient form using
F (s) = sup (s ? y? ? W (?
y ))
(7)
?
y??Y
Here Y? is defined as in (3).2 To implement the LH algorithm using the F of Eq. (7), we need an
efficient way to compute ?F . As Thm. 2 below shows, there is always a y? which satisfies
y? ? arg max (s ? y? ? W (?
y ))
(8)
?
y??Y
and any such y? is an element of ?F . So, the optimization form of the LH algorithm uses the same
pseudocode as the gradient form (Fig. 2), but uses Eq. (8) with s = st to compute y?t in line (?).
To gain an intuition for Eqs. (7?8), consider the example of external-regret matching. Since Y is
the unit simplex in Rd , Y? is the positive orthant in Rd . So, with W2 (?
y ) = k?
y k22 /2, the optimization
problem (8) will be equivalent to
1
y? = arg min ks ? y?k22
d
y??R+ 2
That is, y? is the projection of s onto Rd+ by minimum Euclidean distance. It is not hard to verify that
this projection replaces the negative elements of s with zeros, y? = [s]+ . Substituting this value for
y? back into (7) and using the fact that s ? [s]+ = [s]+ ? [s]+ , the resulting potential function is
X
X
F2 (s) = s ? [s]+ ?
[si ]2+ /2 =
[si ]2+ /2
i
i
as claimed above. This potential function is the standard one for analyzing external-regret matching.
Theorem 2 Let W be convex, dom W ? Y? be nonempty, and W (?
y ) ? 0 for all y?. Suppose the
sets {?
y | W (?
y ) + s ? y? ? k} are compact for all s and k. Define F as in (7). Then F is finite and
F (s) ? 0 for all s ? S. And, the optimization form of the LH algorithm using the hedging function
W is equivalent to the gradient form of the LH algorithm with potential function F .
2
Eq. (7) is similar to the definition of the convex dual W ? , but the supremum is over y? ? Y? instead of over
all y?. As a result, F and W ? can be very different functions. As discussed in [6], F can be expressed as the
dual of a function related to W .
6
Theoretical Results
Our main theoretical results are regret bounds for the LH algorithm. The bounds depend on the
curvature of our potential F , the size of the hypothesis set Y, and the possible slopes C of our loss
functions. Intuitively, F must be neither too curved nor too flat on the scale of the updates to st from
Eq. (1): if F is too curved then ?F will change too quickly and our hypothesis yt will jump around
a lot, while if F is too flat then we will not react quickly enough to changes in regret.
We will state our results for the gradient form of the LH algorithm. For the optimization form,
essentially the same results hold, but the constants are defined in terms of the hedging function
instead. Therefore, we never need to work with (or even be able to write down) the corresponding
potential function. For more details, see [6]. One result which is slightly tricky to carry over is
tuning learning rates. The choice of learning rate below and the resulting bound are the same as
for the gradient form, but the implementation is slightly different: to set a learning rate ? > 0, we
replace W (?
y ) with W (?
y /?).
We will need upper and lower bounds on F . We will assume
F (s + ?) ? F (s) + ? ? f (s) + Ck?k2
(9)
for all regret vectors s and increments ?, and
[F (s) + A]+ ? inf
Bks ? s? kp
?
s ?S
(10)
for all s. Here k ? k is an arbitrary finite norm, and A ? 0, B > 0, C > 0, and 1 ? p ? 2 are
constants. Eq. (9), together with the convexity of F , implies that F is differentiable and f is its
gradient; the LH algorithm is applicable if F is not differentiable, but its regret bounds are weaker.
We will bound the size of Y by assuming that
kyk? ? M
(11)
for all y in Y. Here, k ? k? is the dual of the norm used in Eq. (9) [12].
The size of our update to st (in Eq. (1)) depends on the hypothesis set Y, the cost vector set C, and
the vector u. We have already bounded Y; rather than bounding C and u separately, we will assume
that there is a constant D so that
E(kst+1 ? st k2 | st ) ? D
(12)
Here the expectation is taken with respect to our choice of hypothesis, so the inequality must hold
for all possible values of ct . (The expectation is only necessary if we randomize our choice of
hypothesis, as would happen if Y is the convex hull of some non-convex set. If interior points of Y
are valid plays, we need not randomize, so we can drop the expectation in (12) and below.)
Our theorem then bounds our regret in terms of the above constants. Since the bounds are sublinear
in t, they show that Lagrangian Hedging is a no-regret algorithm when we choose an appropriate
potential F .
Theorem 3 Suppose the potential function F is convex and satisfies Eqs. (4), (9) and (10). Suppose
that the problem definition is bounded according to (11) and (12). Then the LH algorithm (Fig. 2)
achieves expected regret
E(?t+1 (y)) ? M ((tCD + A)/B)1/p = O(t1/p )
versus any hypothesis y ? Y.
If p = 1 the above bound is O(t). But, suppose that we know ahead of time the number of trials t
we will see. Define G(s) = F (?s), where
p
? = A/(tCD)
Then the LH algorithm with potential G achieves regret
?
?
E(?t+1 (y)) ? (2M/B) tACD = O( t)
for any hypothesis y ? Y.
The full proof of Thm. 3 appears in [6]; here, we sketch the proof of one of the most important
intermediate results. Thm. 4 shows that, if we can guarantee E(st+1 ? st ) ? ?F (t) ? 0, then F (st )
cannot grow too quickly. This result is analogous to Blackwell?s approachability theorem: since the
level sets of F are related to S, we will be able to show st /t ? S, implying no regret.
Theorem 4 (Gradient descent) Let F (s) and f (s) satisfy Equation (9) with seminorm k ? k and
Pt?1
constant C. Let x0 , x1 , . . . be a sequence of random vectors. Write st = i=0 xi , and let D be a
constant so that E(kxt k2 | st ) ? D. Suppose that, for all t, E(xt ? f (st ) | st ) ? 0. Then for all t,
E(F (st+1 ) | s1 ) ? F (s1 ) ? tCD
P ROOF : The proof is by induction: for t ? 2, assume E(F (st ) | s1 ) ? F (s1 ) + (t ? 1)CD. (It is
obvious that the base case holds for t = 1.) Then:
F (st+1 )
= F (st + xt )
?
E(F (st+1 ) | st ) ?
E(F (st+1 ) | s1 ) ?
E(F (st+1 ) | s1 ) ?
which is the desired result.
7
F (st ) + xt ? f (st ) + Ckxt k2
F (st ) + CD
E(F (st ) | s1 ) + CD
F (s1 ) + (t ? 1)CD + CD
2
Examples
The classical applications of no-regret algorithms are learning from expert advice and learning to
play a repeated matrix game. TheseP
two tasks are essentially equivalent, since they both use the
probability simplex Y = {y | y ? 0, i yi = 1} for their hypothesis set. This choice of Y simplifies
the required algorithms greatly; with appropriate choices of potential functions, it can be shown that
standard no-regret algorithms such as Freund and Schapire?s Hedge [5], Littlestone and Warmuth?s
weighted majority [13], and Hart and Mas-Colell?s external-regret matching [14, Theorem B] are all
special cases of the LH algorithm.
A large variety of other online prediction problems can also be cast in our framework. These problems include path planning when costs are chosen by an adversary [11], planning in a Markov
decision process when costs are chosen by an adversary [15], online pruning of a decision tree [16],
and online balancing of a binary search tree [4]. More uses of online convex programming are
given in [1, 3, 4]. In each case the bounds for the LH algorithm will be polynomial or better in the
dimensionality of the appropriate hypothesis set and sublinear in the number of trials.
8
Experiments
To demonstrate that our theoretical bounds translate to good practical performance, we implemented
the LH algorithm with the potential function W2 from (6) and used it to learn policies for the game
of one-card poker. (The hypothesis space for this learning problem is the set of sequence weight
vectors, which is convex because one-card poker is an extensive-form game [17].)
In one-card poker, two players (called the gambler and the dealer) each ante $1 and receive one
card from a 13-card deck. The gambler bets first, adding either $0 or $1 to the pot. Then the dealer
gets a chance to bet, again either $0 or $1. Finally, if the gambler bet $0 and the dealer bet $1, the
gambler gets a second chance to bring her bet up to $1. If either player bets $0 when the other has
already bet $1, that player folds and loses her ante. If neither player folds, the higher card wins the
pot, resulting in a net gain of either $1 or $2 (equal to the other player?s ante plus the bet of $0 or
$1). In contrast to the usual practice in poker we assume that the payoff vector ct is observable after
each hand; the partially-observable extension is beyond the scope of this paper.
One-card poker is a simple game; nonetheless it has many of the elements of more complicated
games, including incomplete information, chance events, and multiple stages. And, optimal play
requires behaviors like randomization and bluffing. The biggest strategic difference between onecard poker and larger variants such as draw, stud, or hold-em is the idea of hand potential: while
0.6
0.6
Gambler bound
Dealer bound
Avg payoff
Minimax value
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
?0.1
?0.1
?0.2
?0.2
?0.3
?0.3
?0.4
0
50
100
150
200
250
Gambler bound
Dealer bound
Avg payoff
Minimax value
0.5
?0.4
0
50
100
150
200
250
Figure 3: Performance in self-play (left) and against a fixed opponent (right).
45679 and 24679 are almost equally strong hands in a showdown (they are both 9-high), holding
45679 early in the game is much more valuable because replacing the 9 with either a 3 or an 8 turns
it into a straight.
Fig. 3 shows the results of two typical runs: in both panels the dealer is using our no-regret algorithm.
In the left panel the gambler is also using our no-regret algorithm, while in the right panel the
gambler is playing a fixed policy. The x-axis shows number of hands played; the y-axis shows
the average payoff per hand from the dealer to the gambler. The value of the game, ?$0.064, is
indicated with a dotted line. The middle solid curve shows the actual performance of the dealer
(who is trying to minimize the payoff).
The upper curve measures the progress of the dealer?s learning: after every fifth hand we extracted
a strategy ytavg by taking the average of our algorithm?s predictions so far. We then plotted the
worst-case value of ytavg . That is, we plotted the payoff for playing ytavg against an opponent which
knows ytavg and is optimized to maximize the dealer?s losses. Similarly, the lower curve measures
the progress of the gambler?s learning.
In the right panel, the dealer quickly learns to win against the non-adaptive gambler. The dealer
never plays a minimax strategy, as shown by the fact that the upper curve does not approach the
value of the game. Instead, she plays to take advantage of the gambler?s weaknesses. In the left
panel, the gambler adapts and forces the dealer to play more conservatively; in this case, the limiting
strategies for both players are minimax.
The curves in the left panel of Fig. 3 show an interesting effect: the small, damping oscillations
result from the dealer and the gambler ?chasing? each other around a minimax strategy. One player
will learn to exploit a weakness in the other, but in doing so will open up a weakness in her own
play; then the second player will adapt to try to take advantage of the first, and the cycle will
repeat. Each weakness will be smaller than the last, so the sequence of strategies will converge to a
minimax equilibrium. This cycling behavior is a common phenomenon when two learning players
play against each other. Many learning algorithms will cycle so strongly that they fail to achieve the
value of the game, but our regret bounds eliminate this possibility.
9
Discussion
We have presented the Lagrangian Hedging algorithms, a family of no-regret algorithms for OCP
based on general potential functions. We have proved regret bounds for LH algorithms and demonstrated experimentally that these bounds lead to good predictive performance in practice. The regret
bounds for LH algorithms have low-order dependences on d, the number of dimensions in the hypothesis set Y. This low-order dependence means that the LH algorithms can learn well in prediction
problems with complicated hypothesis sets; these problems would otherwise require an impractical
amount of training data and computation time.
Our work builds on previous work in online learning and online convex programming. Our contributions include a new, deterministic algorithm; a simple, general proof; the ability to build algorithms
from a more general class of potential functions; and a new way of building good potential functions from simpler hedging functions, which allows us to construct potential functions for arbitrary
convex hypothesis sets. Future work includes a no-internal-regret version of the LH algorithm, as
well as a bandit-style version. The former will guarantee convergence to a correlated equilibrium in
nonzero-sum games, while the latter will allow us to work from incomplete observations of the cost
vector (e.g., as might happen in an extensive-form game such as poker).
Acknowledgments Thanks to Amy Greenwald, Martin Zinkevich, and Sebastian Thrun, as well
as Yoav Shoham and his research group. This work was supported by NSF grant EF-0331657 and
DARPA contracts F30602-01-C-0219, NBCH-1020014, and HR0011-06-0023. The opinions and
conclusions are the author?s and do not reflect those of the US government or its agencies.
References
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2,295 | 3,083 | Fast Discriminative Visual Codebooks
using Randomized Clustering Forests
Frank Moosmann?, Bill Triggs and Frederic Jurie
GRAVIR-CNRS-INRIA, 655 avenue de l?Europe, Montbonnot 38330, France
[email protected]
Abstract
Some of the most effective recent methods for content-based image classification work by extracting dense or sparse local image descriptors, quantizing them
according to a coding rule such as k-means vector quantization, accumulating histograms of the resulting ?visual word? codes over the image, and classifying these
with a conventional classifier such as an SVM. Large numbers of descriptors and
large codebooks are needed for good results and this becomes slow using k-means.
We introduce Extremely Randomized Clustering Forests ? ensembles of randomly
created clustering trees ? and show that these provide more accurate results, much
faster training and testing and good resistance to background clutter in several
state-of-the-art image classification tasks.
1
Introduction
Many of the most popular current methods for image classification represent images as collections of independent patches characterized by local visual descriptors. Patches can be sampled
densely [18, 24], randomly [15], or at selected salient points [14]. Various local descriptors exist
with different degrees of geometric and photometric invariance, but all encode the local patch appearance as a numerical vector and the more discriminant ones tend to be high-dimensional. The
usual way to handle the resulting set of descriptor vectors is to vector quantize them to produce socalled textons [12] or visual words [5, 22]. The introduction of such visual codebooks has allowed
significant advances in image classification, especially when combined with bag-of-words models
inspired by text analysis [5, 7, 22, 24, 25].
There are various methods for creating visual codebooks. K-means clustering is currently the most
common [5, 22] but mean-shift [9] and hierarchical k-means [17] clusterers have some advantages.
These methods are generative but some recent approaches focus on building more discriminative
codebooks [20, 24].
The above methods give impressive results but they are computationally expensive owing to the cost
of assigning visual descriptors to visual words during training and use. Tree based coders [11, 17, 23]
are quicker but (so far) somewhat less discriminative. It seems to be difficult to achieve both speed
and good discrimination.
This paper contributes two main ideas. One is that (small) ensembles of trees eliminate many of the
disadvantages of single tree based coders without losing the speed advantages of trees. The second
is that classification trees contain a lot of valuable information about locality in descriptor space that
is not apparent in the final class labels. One can exploit this by training them for classification then
ignoring the class labels and using them as ?clustering trees? ? simple spatial partitioners that assign
a distinct region label (visual word) to each leaf. Combining these ideas, we introduce Extremely
?
Current address: Institute of Measurement and Control, University of Karlsruhe, Germany. Contact: [email protected]
Figure 1: Using ERC-Forests as visual codebooks in bag-of-feature image classification.
Randomized Clustering Forests (ERC-Forests) ? ensembles of randomly created clustering trees.
We show that these have good resistance to background clutter and that they provide much faster
training and testing and more accurate results than conventional k-means in several state-of-the-art
image classification tasks.
In the rest of the paper, we first explain how decision trees can provide good visual vocabularies,
then we describe our approach and present experimental results and conclusions.
2
Tree Structured Visual Dictionaries
Our overall goal is to classify images according to the object classes that they contain (see figure 1).
We will do this by selecting or sampling patches from the image, characterizing them by vectors
of local visual descriptors and coding (quantizing) the vectors using a learned visual dictionary,
i.e. a process that assigns discrete labels to descriptors, with similar descriptors having a high
probability of being assigned the same label. As in text categorization, the occurrences of each label
(?visual word?) are then counted to build a global histogram (?bag of words?) summarizing the
image (?document?) contents. The histogram is fed to a classifier to estimate the image?s category
label. Unlike text, visual ?words? are not intrinsic entities and different quantization methods can
lead to very different performances.
Computational efficiency is important because a typical image yields 103 ?104 local descriptors and
data sets often contain thousands of images. Also, many of the descriptors generally lie on the background not the object being classified, so the coding method needs to be able to learn a discriminative
labelling despite considerable background ?noise?.
K-means and tree structured codes. Visual coding based on K-means vector quantization is effective but slow because it relies on nearest neighbor search, which remains hard to accelerate in
high dimensional descriptor spaces despite years of research on spatial data structures (e.g. [21]).
Nearest neighbour assignments can also be somewhat unstable: in high dimensions, concentration
of measure [2] tends to ensure that there are many centres with similar distances to any given point.
Component-wise decision trees offer logarithmic-time coding, but individual trees can rarely compete with a good K-means coding: each path through the tree typically accesses only a few of
the feature dimensions, so there is little scope for producing a consensus over many different dimensions. Nist?er et al.[17] introduced a tree coding based on hierarchical K-means. This uses all
components and gives a good compromise between speed and loss of accuracy.
Random forests. Despite their popularity, we believe that K-means codes are not the best compromise. No single data structure can capture the diversity and richness of high dimensional descriptors.
To do this an ensemble approach is needed. The theoretical and practical performance of ensemble
classifiers is well documented [1]. Ensembles of random trees [4] seem particularly suitable for
visual dictionaries owing to their simplicity, speed and performance [11]. Sufficiently diverse trees
can be constructed using randomized data splits or samples [4]. Extremely Randomized Trees (see
below) take this further by randomizing both attribute choices and quantization thresholds, obtaining even better results [8]. Compared to standard approaches such as C4.5, ER tree construction is
rapid, depends only weakly on the dimensionality and requires relatively little memory.
Clustering forests. Methods such as [11, 8] classify descriptors by majority voting over the treeassigned class labels. There are typically many leaves that assign a given class label. Our method
works differently after the trees are built. It uses the trees as spatial partitioning methods not classifiers, assigning each leaf of each tree a distinct region label (visual word). For the overall image
classification tasks studied here, histograms of these leaf labels are then accumulated over the whole
image and a global SVM classifier is applied. Our approach is thus related to clustering trees ?
decision trees whose leaves define a spatial partitioning or grouping [3, 13]. Such trees are able to
find natural clusters in high dimensional spaces. They can be built without external class labels,
but if labels are available they can be used to guide the tree construction. Ensemble methods and
particularly forests of extremely randomized trees again offer considerable performance advantages
here. The next section shows how such Extremely Randomized Clustering Forests can be used to
produce efficient visual vocabularies for image classification tasks.
3
Extremely Randomized Clustering Forests (ERC-Forests)
Our goal is to build a discriminative coding method. Our method starts by building randomized
decision trees that predict class labels y from visual descriptor vectors d = (f1 , . . . , fD ), where
fi , i = 1, . . . , D are elementary scalar features. For notational simplicity we assume that all of the
descriptors from a given image share the same label y. We train the trees using a labeled (for now)
training set L = {(dn , yn ), n = 1, . . . , N }. However we use the trees only for spatial coding, not
classification per se. During a query, for each descriptor tested, each tree is traversed from the root
down to a leaf and the returned label is the unique leaf index, not the (set of) descriptor label(s) y
associated with the leaf.
ERC-Trees. The trees are built recursively top down. At each node t corresponding to descriptor
space region Rt , two children l, r are created by choosing a boolean test Tt that divides Rt into two
disjoint regions, Rt = Rl ? Rr with Rl ? Rr = ?. Recursion continues until further subdivision
is impossible: either all surviving training examples belong to the same class or all have identical
values for all attributes. We use thresholds on elementary features as tests, Tt = {fi(t) ? ?t } for
some feature index i(t) and threshold ?t . The tests are selected randomly as follows. A feature index
i(t) is chosen randomly, a threshold ?t is sampled randomly from a uniform distribution, and the
2I(C,T )
,
resulting node is scored over the surviving points using Shannon entropy [8]: Sc(C, T ) = H
C +HT
where HC denotes entropy of the class label distribution, HT the entropy of the partition induced
by the test and I(C, T ) their mutual information. High scores indicate that the split separates the
classes well. This procedure is repeated until the score is higher than a fixed threshold Smin or until
a fixed maximum number Tmax of trials have been made. The test Tt that achieved the highest score
is adopted and the recursion continues.
The parameters (Smin , Tmax ) control the strength and randomness of the generated trees. High
values (e.g. (1, D) for normal ID3 decision tree learning) produce highly discriminant trees with
little diversity, while Smin = 0 or Tmax = 1 produce completely random trees.
ERC-Forests. Compared to standard decision tree learning, the trees built using random decisions
are larger and have higher variance. Class label variance can be reduced by voting over the ensemble
of trees (e.g. [15]), but here, instead of voting we treat each leaf in each tree as a separate visual word
and stack the leaf indices from each tree into an extended code vector for each input descriptor,
leaving the integration of votes to the final classifier. The resulting process is reminiscent of spatial
search algorithms based on random line projections (e.g. [10]), with each tree being responsible for
distributing the data across its own set of clusters.
Classifier forests are characterized by Breiman?s bound on the asymptotic generalization error [4],
P E? ? ? (1 ? s2 )/s2 where s measures the strength of the individual trees and ? measures the
correlation between them in terms of the raw margin. It would be interesting to optimize Smin and
Tmax to minimize the bound but we have not yet tried this. Experimentally, the trees appear to be
rather diverse while still remaining relatively strong, which should lead to good error bounds.
Application to visual vocabularies. In the experiments below, local features are extracted from the
training images by sampling sub-windows at random positions and scales1 and coding them using
a visual descriptor function. An ERC-Forest is then built using the given class labels. To control
the codebook size, we grow the trees fully then prune them back bottom up, recursively removing
the node with the lowest gain until either a specified threshold on the gain or a specified number of
1
For image classification, dense enough random sampling eventually outperforms keypoint based sampling
[18].
Figure 2: Left: example images from GRAZ-02. The rows are respectively Bikes (B), Cars (C) and
background (N). Right: some test patches that were assigned to a particular ?car? leaf (left) and a
particular ?bike? one (right)).
leaves is reached. One can also prune during construction, which is faster but does not allow the
number of leaf nodes to be controlled directly.
In use, the trees transform each descriptor into a set of leaf node indices with one element from each
tree. Votes for each index are accumulated into a global histogram and used for classification as
in any other bag of features approach. Independently of the codebook, the denser the sampling the
better the results, so typically we sample images more densely during testing than during codebook
training, c.f. [18].
Computational complexity. The worst-case complexity for building a tree is O(Tmax N k), where
N is the number of patches and k is the number of clusters/leaf nodes before pruning. With adversarial data the method cannot guarantee balanced trees so it can not do better than this, but in
our experiments on real data we always obtained well balanced trees at a practical complexity of
around O(Tmax N log k). The dependence on data dimensionality D is hidden in the constant Tmax ,
which needs to be set large enough to filter out irrelevant?feature dimensions, thus providing better
coding and more balanced
? trees. A value of Tmax ? O( D) has been suggested [8], leading to a
total complexity of O( DN log k). In contrast, k-means has a complexity of O(DN k) which is
more than 104 times larger for our 768-D wavelet descriptor with N = 20000 image patches and
k = 5000 clusters, not counting the number of iterations that k-means has to perform. Our method
is also faster in use ? a useful property given that reliable image classification requires large numbers
of subwindows to be labelled [18, 24]. Labeling a descriptor with a balanced tree requires O(log k)
operations whereas k-means costs O(kD).
4
Experiments
We present detailed results on the GRAZ-02 test set, http://www.emt.tugraz.at/?pinz/data/. Similar
conclusions hold for two other sets that we tested, so we comment only briefly on these. GRAZ-02
(figure 2-left) contains three object categories ? bicycles (B), cars (C), persons (P) ? and negatives
(N, meaning that none of B,C,P are present). It is challenging in the sense that the illumination is
highly variable and the objects appear at a wide range of different perspectives and scales and are
sometimes partially hidden. It is also neutral with respect to background, so it is not possible to
detect objects reliably based on context alone.
We tested various visual descriptors. The best choice turns out to depend on the database. Our
color descriptor uses raw HSL color pixels to produce a 768-D feature vector (16?16 pixels ? 3
colors). Our color wavelet descriptor transforms this into another 768-D vector using a 16?16 Haar
wavelet transform. Finally, we tested the popular grayscale SIFT descriptor [14], which returns
128-D vectors (4?4 histograms of 8 orientations).
We measure performance with ROC curves and classification rates at equal error rate (EER). The
method is randomized so we report means and variances over 10 learning runs. We use Smin = 0.5
but the exact value is not critical. In contrast Tmax has a significant influence on performance so it
Figure 3: ?Bike? visual words for 4 different images. The brightness denotes the posterior probability
for the visual word at the given image position to be labelled ?bike?.
is chosen using a validation set. For the 768-D Color Wavelet Descriptor on the GRAZ-02 dataset,
Tmax ? 50.
Our algorithm?s ability to produce meaningful visual words is illustrated in figure 3 (c.f. [16]).
Each white dot corresponds to the center of an image sub-window that reached an unmixed leaf
node for the given object category (i.e. all of the training vectors belonging to the leaf are labeled
with that category). Note that even though they have been learned on entire images without object
segmentation, the visual vocabulary is discriminative enough to detect local structures in the test
images that correspond well with representative object fragments, as illustrated in figure 2(right).
The tests here were for individual object categories versus negatives (N). We took 300 images from
each category, using images with even numbers for training and ones with odd numbers for testing.
For Setting 1 tests we trained on the whole image as in [19], while for Setting 2 ones we used the
segmentation masks provided with the images to train on the objects alone without background.
For the GRAZ-02 database the wavelet descriptors gave the best performance. We report results
for these on the two hardest categories, bikes and cars. For B vs. N we achieve 84.4% average EER
classification rate for setting 1 and 84.1% for setting 2, in comparison to 76.5% from Opelt et al.[19].
For C vs. N the respective figures are 79.9%, 79.8% and 70.7%. Remarkably, using segmentation
masks during training does not improve the image classification performance. This suggests that the
method is able to pick out the relevant information from a significant amount of clutter.
Comparing ERC-Forests with k-means and kd-clustering trees. Unless otherwise stated,
20 000 features (67 per image) were used to learn 1000 spatial bins per tree for 5 trees, and 8000
patches were sampled per image to build the resulting 5000-D histograms. The histograms are binarized using trivial thresholding at count 1 before being fed to the global linear SVM image classifier.
We also tested with histograms normalized to total sum 1, and with thresholding by maximizing the
mutual information of each dimension, but neither yielded better results for ERC-Forests.
Fig. 4 gives some quantitative results on the bikes category (B vs. N). Fig. 4(a) shows the clear
difference between our method and classical k-means for vocabulary construction. Note that we
were not able to extend the k-means curve beyond 20 000 windows per image owing to prohibitive
execution times. The figure also shows results for ?unsupervised trees? ? ERC-Forests built without using the class labels during tree construction. The algorithm remains the same, but the node
scoring function is defined as the ratio between the splits so as to encourage balanced trees similar
to randomized KD-trees. If only a few patches are sampled this is as good as k-means and much
faster. However the spatial partition is so bad that with additional test windows, the binarized histogram vectors become almost entirely filled with ones, so discrimination suffers. As the dotted line
shows, using binarization thresholds that maximize the mutual information can fix this problem but
the results are still far below ERC-Forests. This comparison clearly shows the advantages of using
supervision during clustering.
Fig. 4(b) shows that codebooks with around 5000 entries (1000 per tree) suffice for good results.
Fig. 4(c) shows that when the number of features used to build the codebooks is increased, the
Figure 4: Evaluation of the parameters for B vs. N in setting 2: classification rate at the EER, averaged over trials. The error bars indicate standard deviations. See the text for further explanations.
optimal codebook size also increases slightly. Also, if the trees are pruned too heavily they lose
discriminative power: it is better to grow them fully and do without pruning. Fig. 4(d) shows that
increasing the number of trees from 1 to 5 reduces the variance and increases the accuracy, with
little improvement beyond this. Here, the number of leaves per tree was kept constant at 1000, so
doubling the number of trees effectively doubles the vocabulary size.
We also tested our method on the 2005 Pascal Challenge dataset, http://www.pascalnetwork.org/challenges/VOC/voc2005. This contains four categories, motorbikes, bicycles, people
and cars. The goal is to distinguish each category from the others. Just 73 patches per image (50 000
in total over the 648 training images) were used to build the codebook. The maximum patch size
was 30% of the image size. SIFT descriptors gave the best results for coding. The chosen forest
contained four 7500-leaf trees, producing a 30 000-D histogram.
The results (M:95.8%, B:90.1%, P:94%, C:96%) were either similar to or up to 2% better than the
frontrunners in the 2005 Pascal Challenge [6], but used less information and had much faster processing times. A 2.8GHz P4 took around 20 minutes to build the codebook. Building the histograms
for the 684 training and 689 test images with 10 000 patches per image took only a few hours. All
times include both feature extraction and coding.
We also compared our results with those of Mar?ee et al. [15]. They use the same kind of tree
structures to classify images directly, without introducing the vocabulary layer that we propose. Our
EER error rates are consistently 5?10% better than theirs.
Finally, we tested the horse database from http://pascal.inrialpes.fr/data/horses. The task is difficult
because the images were taken randomly from the internet and are highly variable regarding subject
size, pose and visibility. Using SIFT descriptors we get an EER classification rate of 85.3%, which
is significantly better than the other methods that we are aware of. 100 patches per image were used
to build a codebook with 4 trees. 10 000 patches per image were used for testing.
5
Conclusions
Bag of local descriptor based image classifiers give state-of-the art results but require the quantization of large numbers of high-dimensional image descriptors into many label classes. Extremely
Randomized Clustering Forests provide a rapid and highly discriminative approach to this that outperforms k-means based coding in training time and memory, testing time, and classification accuracy. The method can use unlabelled data but it benefits significantly from labels when they are
available. It is also resistant to background clutter, giving relatively clean segmentation and ?popout? of foreground classes even when trained on images that contain significantly more background
features than foreground ones. Although trained as classifiers, the trees are used as descriptor-space
quantization rules with the final classification being handled by a separate SVM trained on the leaf
indices. This seems to be a promising approach for visual recognition, and may be beneficial in
other areas such as object detection and segmentation.
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2,296 | 3,084 | Bayesian Ensemble Learning
Hugh A. Chipman
Department of Mathematics and Statistics
Acadia University
Wolfville, NS, Canada
Edward I. George
Department of Statistics
The Wharton School
University of Pennsylvania
Philadelphia, PA 19104-6302
Robert E. McCulloch
Graduate School of Business
University of Chicago
Chicago, IL, 60637
Abstract
We develop a Bayesian ?sum-of-trees? model, named BART, where each tree is
constrained by a prior to be a weak learner. Fitting and inference are accomplished
via an iterative backfitting MCMC algorithm. This model is motivated by ensemble methods in general, and boosting algorithms in particular. Like boosting, each
weak learner (i.e., each weak tree) contributes a small amount to the overall model.
However, our procedure is defined by a statistical model: a prior and a likelihood,
while boosting is defined by an algorithm. This model-based approach enables a
full and accurate assessment of uncertainty in model predictions, while remaining
highly competitive in terms of predictive accuracy.
1
Introduction
We consider the fundamental problem of making inference about an unknown function f that predicts an output Y using a p dimensional vector of inputs x when Y = f (x) + , ? N (0, ? 2 ). To
do this, we consider modelling or at least approximating f (x) = E(Y | x), the mean of Y given
x, by a sum of m regression trees: f (x) ? g1 (x) + g2 (x) + . . . + gm (x) where each gi denotes a
binary regression tree.
The sum-of-trees model is fundamentally an additive model with multivariate components. It is
vastly more flexible than a single tree model which does not easily incorporate additive effects. Because multivariate components can easily account for high order interaction effects, a sum-of-trees
model is also much more flexible than typical additive models that use low dimensional smoothers
as components.
Our approach is fully model based and Bayesian. We specify a prior, and then obtain a sequence of
draws from the posterior using Markov chain Monte Carlo (MCMC). The prior plays two essential
roles. First, with m chosen large, it restrains the fit of each individual gi so that the overall fit is
made up of many small contributions in the spirit of boosting (Freund & Schapire (1997), Friedman
(2001)). Each gi is a ?weak learner?. Second, it ?regularizes? the model by restraining the overall
fit to achieve good bias-variance tradeoff. The prior specification is kept simple and a default choice
is shown to have good out of sample predictive performance.
Inferential uncertainty
is naturally quantified in the usual Bayesian way: variation in the MCMC
P
draws of f =
gi (evaluated at a set of x of interest) and ? indicates our beliefs about plausible
values given the data. Note that the depth of each tree is not fixed so that we infer the level of interaction. Our point estimate of f is the average of the draws. Thus, our procedure captures ensemble
learning (in which many trees are combined) both in the fundamental sum-of-trees specification and
in the model-averaging used to obtain the estimate.
2
The Model
The model consists of two parts: a sum-of-trees model, which we have named BART (Bayesian
Additive Regression Trees), and a regularization prior.
2.1
A Sum-of-Trees Model
To elaborate the form of a sum-of-trees model, we begin by establishing notation for a single tree
model. Let T denote a binary tree consisting of a set of interior node decision rules and a set
of terminal nodes, and let M = {?1 , ?2 , . . . , ?B } denote a set of parameter values associated with
each of the B terminal nodes of T . Prediction for a particular value of input vector x is accomplished
as follows: If x is associated with terminal node b of T by the sequence of decision rules from top
to bottom, it is then assigned the ?b value associated with this terminal node. We use g(x; T, M ) to
denote the function corresponding to (T, M ) which assigns a ?b ? M to x.
Using this notation, and letting gi (x) = g(x; Ti , Mi ), our sum-of-trees model can more explicitly
be expressed as
Y = g(x; T1 , M1 ) + g(x; T2 , M2 ) + ? ? ? + g(x; Tm , Mm ) + ,
(1)
? N (0, ? 2 ).
(2)
Unlike the single tree model, when m > 1 the terminal node parameter ?i given by g(x; Tj , Mj )
is merely part of the conditional mean of Y given x. Such terminal node parameters will represent
interaction effects when their assignment depends on more than one component of x (i.e., more
than one variable). Because (1) may be based on trees of varying sizes, the sum-of-trees model can
incorporate both direct effects and interaction effects of varying orders. In the special case where
every terminal node assignment depends on just a single component of x, the sum-of-trees model
reduces to a simple additive function.
With a large number of trees, a sum-of-trees model gains increased representation flexibility, which,
when coupled with our regularization prior, gives excellent out of sample predictive performance.
Indeed, in the examples in Section 4, we set m as large as 200. Note that with m large there
are hundreds of parameters of which only ? is identified. This is not a problem for our Bayesian
analysis. Indeed, this lack of identification is the reason our MCMC mixes well. Even when m is
much larger than needed to capture f (effectively, we have an ?overcomplete basis?) the procedure
still works well.
2.2
A Regularization Prior
The complexity of the prior specification is vastly simplified by letting the Ti be i.i.d, the ?i,b (node
b of tree i) be i.i.d given the set of T , and ? be independent of all T and ?. Given these independence
assumptions we need only choose priors for a single tree T , a single ?, and ?. Motivated by our
desire to make each g(x; Ti , Mi ) a small contribution to the overall fit, we put prior weight on small
trees and small ?i,b .
For the tree prior, we use the same specification as in Chipman, George & McCulloch (1998). In
this prior, the probability that a node is nonterminal is ?(1 + d)?? where d is the depth of the node.
In all examples we use the same prior corresponding to the choice ? = .95 and ? = 2. With this
choice, trees with 1, 2, 3, 4, and ? 5 terminal nodes receive prior probability of 0.05, 0.55, 0.28,
0.09, and 0.03, respectively. Note that even with this prior, trees with many terminal nodes can be
grown if the data demands it. At any non-terminal node, the prior on the associated decision rule
puts equal probability on each available variable and then equal probability on each available rule
given the variable.
For the prior on a ?, we start by simply shifting and rescaling Y so that we believe the prior probability that E(Y | x) ? (?.5, .5) is very high. We let ? ? N (0, ??2 ). Given the Ti and an x, E(Y | x)
?
is the sum of m independent ??s. The standard deviation of the sum is m ?? . We choose ?? so
2.5
conservative: df=10, quantile=.75
2.0
default: df=3, quantile=.9
0.0
0.5
1.0
1.5
aggressive: df=3, quantile=.99
0.0
0.5
1.0
1.5
2.0
2.5
3.0
sigma
Figure 1: Three priors on ? when ?
? = 2.
?
that .5 is within k standard deviations of zero: k m?? = .5. For example if k = 2 there is a 95%
(conditional) prior probability that the mean of Y is in (?.5, .5). k = 2 is our default choice and in
practice we typically rescale the response y so that its observed values range from -5. to .5. Note
that this prior increases the shrinkage of ?i,b (toward zero) as m increases.
For the prior on ? we start from the usual inverted-chi-squared prior: ? 2 ? ? ?/?2? . To choose the
hyperparameters ? and ?, we begin by obtaining a ?rough overestimate? ?
? of ?. We then pick a
degrees of freedom value ? between 3 and 10. Finally, we pick a value of q such as 0.75, 0.90 or
0.99, and set ? so that the qth quantile of the prior on ? is located at ?
? , that is P (? < ?
? ) = q. Figure
1 illustrates priors corresponding to three (?, q) settings when the rough overestimate is ?
? = 2. We
refer to these three settings, (?, q) = (10, 0.75), (3, 0.90), (3, 0.99), as conservative, default and
aggressive, respectively. For automatic use, we recommend the default setting (?, q) = (3, 0.90)
which tends to avoid extremes. Simple data-driven choices of ?
? we have used in practice are the
estimate from a linear regression or the sample standard deviation of Y . Note that this prior choice
can be influential. Strong prior beliefs that ? is very small could lead to over-fitting.
3
A Backfitting MCMC Algorithm
Given the observed data y, our Bayesian setup induces a posterior distribution
p((T1 , M1 ), . . . , (Tm , Mm ), ?| y) on all the unknowns that determine a sum-of-trees model.
Although the sheer size of this parameter space precludes exhaustive calculation, the following
backfitting MCMC algorithm can be used to sample from this posterior.
At a general level, our algorithm is a Gibbs sampler. For notational convenience, let T(i) be the
set of all trees in the sum except Ti , and similarly define M(i) . The Gibbs sampler here entails m
successive draws of (Ti , Mi ) conditionally on (T(i) , M(i) , ?):
(T1 , M1 )|T(1) , M(1) , ?, y
(T2 , M2 )|T(2) , M(2) , ?, y
..
.
(Tm , Mm )|T(m) , M(m) , ?, y,
(3)
followed by a draw of ? from the full conditional:
?|T1 , . . . Tm , M1 , . . . , Mm , y.
(4)
Hastie & Tibshirani (2000) considered a similar application of the Gibbs sampler for posterior sampling for additive and generalized additive models with ? fixed, and showed how it was a stochastic
generalization of the backfitting algorithm for such models. For this reason, we refer to our algorithm
as backfitting MCMC. In contrast with the stagewise nature of most boosting algorithms (Freund &
Schapire (1997), Friedman (2001), Meek, Thiesson & Heckerman (2002)), the backfitting MCMC
algorithm repeatedly resamples the parameters of each learner in the ensemble.
The idea is that given (T(i) , M(i) ) and ? we may subtract the fit from (T(i) , M(i) ) from both sides of
(1) leaving us with a single tree model with known error variance. This draw may be made following
the approach of Chipman et al. (1998) or the refinement of Wu, Tjelmeland & West (2007). These
methods draw (Ti , Mi ) | T(i) , M(i) , ?, y as Ti | T(i) , M(i) , ?, y followed by Mi | Ti , T(i) , M(i) , ?, y.
The first draw is done by the Metropolis-Hastings algorithm after integrating out Mi and the second
is a set of normal draws. The draw of ? is easily accomplished by subtracting all the fit from both
sides of (1) so the the are considered to be observed. The draw is then a standard inverted-chisquared.
The Metropolis-Hastings draw of Ti | T(i) , M(i) , ?, y is complex and lies at the heart of our method.
The algorithm of Chipman et al. (1998) proposes a new tree based on the current tree using one of
four moves. The moves and their associated proposal probabilities are: growing a terminal node
(0.25), pruning a pair of terminal nodes (0.25), changing a non-terminal rule (0.40), and swapping
a rule between parent and child (0.10). Although the grow and prune moves change the implicit
dimensionality of the proposed tree in terms of the number of terminal nodes, by integrating out Mi
from the posterior, we avoid the complexities associated with reversible jumps between continuous
spaces of varying dimensions (Green 1995).
We initialize the chain with m single node trees, and then iterations are repeated until satisfactory
convergence is obtained. At each iteration, each tree may increase or decrease the number of terminal nodes by one, or change one or two decision rules. Each ? will change (or cease to exist or be
born), and ? will change. It is not uncommon for a tree to grow large and then subsequently collapse
back down to a single node as the algorithm iterates. The sum-of-trees model, with its abundance
of unidentified parameters, allows for ?fit? to be freely reallocated from one tree to another. Because each move makes only small incremental changes to the fit, we can imagine the algorithm as
analogous to sculpting a complex figure by adding and subtracting small dabs of clay.
Compared to the single tree model MCMC approach of Chipman et al. (1998), our backfitting
MCMC algorithm mixes dramatically better. When only single tree models are considered, the
MCMC algorithm tends to quickly gravitate toward a single large tree and then gets stuck in a local
neighborhood of that tree. In sharp contrast, we have found that restarts of the backfitting MCMC
algorithm give remarkably similar results even in difficult problems. Consequently, we run one long
chain rather than multiple starts.
In some ways backfitting MCMC is a stochastic alternative to boosting algorithms for fitting linear
combinations of trees. It is distinguished by the ability to sample from a posterior distribution. At
each iteration, we get a new draw
f ? = g(x; T1 , M1 ) + g(x; T2 , M2 ) + . . . + g(x; Tm , Mm )
(5)
corresponding to the draw of Tj and Mj . These draws are a (dependent) sample from the posterior
distribution on the ?true? f . Rather than pick the ?best? f ? from these draws, the set of multiple
draws can be used to further enhance inference. We estimate f by the posterior mean of f which
is approximated by averaging the f ? over the draws. Further, we can gauge our uncertainty about
the actual underlying f by the variation across the draws. For example, we can use the 5% and 95%
quantiles of f ? (x) to obtain 90% posterior intervals for f (x).
4
Examples
In this section we illustrate the potential of our Bayesian ensemble procedure BART in a large experiment using 42 datasets. The data are a subset of 52 sets considered by Kim, Loh, Shih & Chaudhuri
Method
Lasso
Gradient
Boosting
Neural
Nets
Random
Forests
BART-cv
Parameter
shrinkage (in range 0-1)
# of trees
Shrinkage (multiplier of each tree added)
Max depth permitted for each tree
# hidden units
Weight decay
# of trees
% variables sampled to grow each node
Sigma prior: (?, q) combinations
# trees
? Prior: k value for ??
Values considered
0.1, 0.2, ..., 1.0
50, 100, 200
0.01, 0.05, 0.10, 0.25
1, 2, 3, 4
see text
.0001,.001, .01, .1, 1, 2, 3
500
10, 25, 50, 100
(3,0.90), (3,0.99), (10,0.75)
50, 200
2, 3, 5
Table 1: Operational parameters for the various competing models.
(2007). Ten datasets were excluded either because Random Forests was unable to use over 32 categorical predictors, or because a single train/test split was used in the original paper. All datasets
correspond to regression problems with between 3 and 28 numeric predictors and 0 to 6 categorical
predictors. Categorical predictors were converted into 0/1 indicator variables corresponding to each
level. Sample sizes vary from 96 to 6806 observations.
As competitors we considered linear regression with L1 regularization (the Lasso) (Efron, Hastie,
Johnstone & Tibshirani 2004) and four black-box models: Friedman?s (2001) gradient boosting,
random forests (Breiman 2001), and neural networks with one layer of hidden units. Implementation
details are given in Chipman, George & McCulloch (2006). Tree models were not considered, since
they tend to sacrifice predictive performance for interpretability.
We considered two versions of our Bayesian ensemble procedure BART. In BART-cv, the prior
hyperparameters (?, q, k, m) were treated as operational parameters to be tuned via cross-validation.
In BART-default, we set (?, q, k, m) = (3, 0.90, 2, 200). For both BART-cv and BART-default, all
specifications of the quantile q were made relative to the least squares linear regression estimate
?
? , and the number of burn-in steps and MCMC iterations used were determined by inspection of a
single long run. Typically 200 burn-in steps and 1000 iterations were used.
With the exception of BART-default (which has no tuning parameters), all free parameters in learners
were chosen via 5-fold cross-validation within the training set. The parameters considered and
potential levels are given in Table 1. The levels used were chosen with a sufficientlly wide range
that the optimal value was not at an extreme of the candidate values in most problems. Neural
networks are the only model whose operational parameters need additional explanation. In that
case, the number of hidden units was chosen in terms of the implied number of weights, rather
than the number of units. This design choice was made because of the widely varying number of
predictors across problems, which directly impacts the number of weights. A number of hidden units
was chosen so that there was a total of roughly u weights, with u = 50, 100, 200, 500 or 800. In all
cases, the number of hidden units was further constrained to fall between 3 and 30. For example,
with 20 predictors we used 3, 8 and 21 as candidate values for the number of hidden units.
The models were compared with 20 replications of the following experiment. For each replication,
we randomly chose 5/6 of the data as a training set and the remaining 1/6 was used for testing. As
mentioned above, 5-fold cv was used within each training set. In each of the 42 datasets, the response
was minimally preprocessed, applying a log or square root transformation if this made the histogram
of observed responses more bell-shaped. In about half the cases, a log transform was used to reduce
a right tail. In one case (Fishery) a square root transform was most appropriate. Finally, in order
to enable performance comparisons across all datasets, after possible nonlinear transformation, the
resultant response was scaled to have sample mean 0 and standard deviation 1 prior to any train/test
splitting.
A total of 42 ? 20 = 840 experiments were carried out. Results across these experiments are
summarized in Table 2, which gives mean RMSE values and Figure 2, which summarizes relative
performance using boxplots. In Figure 2, the relative performances are calculated as follows: In
Method
RMSE
BART-cv
0.5042
Boosting
0.5089
BART-default
0.5093
Random Forest
0.5097
Neural Net
0.5160
Lasso
0.5896
Table 2: Average test set RMSE values for each learner, combined across 20 train/test replicates of
42 datasets. The only statistically significant difference is Lasso versus the other methods.
BART?default
?
?
?
???
?
??
?
??
?? ???
?
?????? ??? ?
??
??
??
? ????
??
BART?cv
??
? ?
?
?
?
?
??
?????
??
???
? ? ? ? ?????
? ???? ???
Boosting
?
? ? ?? ?
???????
?
? ??
??
?
?
?
?
?????
??
?? ?
??
?
?????
?? ????
?
??? ????? ?
?
??
?
???
Neural Net
?
?
?
??
??
??
??? ??
?
?
???
?? ?
?? ?
?
?
?? ?
?? ??
?? ?
?? ?
? ?
Lasso
Random Forests
???
?
??
???
??
?
?
?
??
?
???
?? ??
?
? ?????
?
???
?
?
?? ?????
?? ??????
?
1.0
1.2
1.4
1.6
1.8
2.0
Figure 2: Test set RMSE performance relative to best (ratio of 1 means minimum RMSE test error).
Results are across 20 replicates in each of 42 datasets. Boxes indicate middle 50% of runs. Each
learner has the following percentage of ratios larger than 2.0, which are not plotted above: Neural
net: 5%, BART-cv: 6%, BART-default and Boosting: 7%, Random forests 10% and Lasso 21%.
each of the 840 experiments, the learner with smallest RMSE was identified. The relative ratio for
each learner is the raw RMSE divided by the smallest RMSE. Thus a relative RMSE of 1 means that
the learner had the best performance in a particular experiment. The central box gives the middle
50% of the data, with the median indicated by a vertical line. The ?whiskers? of the plot extend
to 1.5 times the box width, or the range of values, whichever comes first. Extremes outside the
whiskers are given by individual points. As noted in the caption, relative RMSE ratios larger than
2.0 are not plotted.
BART has the best performance, although all methods except the Lasso are not significantly different. The strong performance of our ?default? ensemble is especially noteworthy, since it requires no
selection of operational parameters. That is, cross-validation is not necessary. This results in a huge
computational savings, since under cross-validation, the number of times a learner must be trained
is equal to the number of settings times the number of folds. This can easily be 50 (e.g. 5 folds by
10 settings), and in this experiment it was 90!
BART-default is in some sense the ?clear winner? in this experiment. Although average predictive performance was indistinguishable from the other models, it does not require cross-validation.
Moreover, the use of cross-validation makes it impossible to interpret the MCMC output as valid uncertainty bounds. Not only is the default version of BART faster, but it also provides valid statistical
inference, a benefit not available to any of the other learners considered.
To further stress the benefit of uncertainty intervals, we report some more detailed results in the
analysis of one of the 42 datasets, the Boston Housing data. We applied BART to all 506 observations of the Boston Housing data using the default setting (?, q, k, m) = (3, 0.90, 2, 200) and the
linear regression estimate ?
? to anchor q. At each of the 506 predictor values x, we used 5% and
95% quantiles of the MCMC draws to obtain 90% posterior intervals for f (x). An appealing feature
of these posterior intervals is that they widen when there is less information about f (x). To roughly
illustrate this, we calculated Cook?s distance diagnostic Dx for each x (Cook 1977) based on a linear
least squares regression of y on x. Larger Dx indicate more uncertainty about predicting y with a
(b)
3.0
2.9
2.8
2.7
2.4
2.5
2.6
log median value
0.25
0.20
0.15
Posterior Interval Width
3.1
(a)
0.00
0.02
0.04
0.06
Cook?s distance
0.08
0
20
40
60
80
Crime
Figure 3: Plots from a single run of the Bayesian Ensemble model on the full Boston dataset. (a)
Comparison of uncertainty bound widths with Cook?s distance measure. (b) Partial dependence plot
for the effect of crime on the response (log median property value), with 90% uncertainty bounds.
linear regression at x. To see how the width of the 90% posterior intervals corresponded to Dx , we
plotted them together in Figure 3(a). Although the linear model may not be strictly appropriate, the
plot is suggestive: all points with large Dx values have wider uncertainty bounds.
Uncertainty bounds can also be used in graphical summaries such as a partial dependence plot
(Friedman 2001), which shows the effect of one (or more) predictor on the response, margining
out the effect of other predictors. Since BART provides posterior draws for f (x), calculation of a
posterior distribution for the partial dependence function is straightforward. Computational details
are provided in Chipman et al. (2006). For the Boston Housing data, Figure 3(b) shows the partial
dependence plot for crime, with 90% posterior intervals. The vast majority of data values occur for
crime < 5, causing the intervals to widen as crime increases and the data become more sparse.
5
Discussion
Our approach is a fully Bayesian approach to learning with ensembles of tree models. Because of the
nature of the underlying tree model, we are able to specify simple, effective priors and fully exploit
the benefits of Bayesian methodology. Our prior provides the regularization needed to obtain good
predictive performance. In particular, our default prior, which is minimially dependent on the data,
performs well compared to other methods which rely on cross-validation to pick model parameters.
We obtain inference in the natural Bayesian way from the variation in the posterior draws. While
predictive performance in always our first goal, many researchers want to interpret the results. In
this case, gauging the inferential uncertainty is essential. No other competitive methods do this in a
convenient way.
Chipman et al. (2006) and Abreveya & McCulloch (2006) provide further evidence of the predictive
performance of our approach. In addition Abreveya & McCulloch (2006) illustrate the ability of
our method to uncover interesting interaction effects in a real example. Chipman et al. (2006) and
and Hill & McCulloch (2006) illustrate the inferential capabilities. Posterior intervals are shown to
have good frequentist coverage. Chipman et al. (2006) also illustrates the method?s ability to obtain
inference in the very difficult ?big p, small n? problem, where there are few observations and many
potential predictors.
A common concern with Bayesian approaches is sensitivity to prior parameters. Chipman et al.
(2006) found that results were robust to a reasonably wide range of prior parameters, including
?, q, ?? , as well as the number of trees, m. m needs to be large enough to provide enough complexity
to capture f , but making m ?too large? does not appreciably degrade accuracy (although it does
make it slower to run). Chipman et al. (2006) provide guidelines for choosing m.
In practice, the stability of the MCMC makes the method easy to use. Typcially, it burns-in rapidly.
If the method is run twice with different seeds the same results are obtained both for fit and inference.
Code is publicly available in the R-package BayesTree.
Acknowledgments
The authors would like to thank three anonymous referees, whose comments improved an earlier
draft, and Wei-Yin Loh who generously provided the datasets used in the experiment. This research
was supported by the Natural Sciences and Engineering Research Council of Canada, the Canada
Research Chairs program, the Acadia Centre for Mathematical Modelling and Computation, the
University of Chicago Graduate School of Business, NSF grant DMS 0605102 and by NIH/NIAID
award AI056983.
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2,297 | 3,085 | Blind Motion Deblurring Using Image Statistics
Anat Levin?
School of Computer Science and Engineering
The Hebrew University of Jerusalem
Abstract
We address the problem of blind motion deblurring from a single image, caused
by a few moving objects. In such situations only part of the image may be blurred,
and the scene consists of layers blurred in different degrees. Most of of existing
blind deconvolution research concentrates at recovering a single blurring kernel
for the entire image. However, in the case of different motions, the blur cannot
be modeled with a single kernel, and trying to deconvolve the entire image with
the same kernel will cause serious artifacts. Thus, the task of deblurring needs to
involve segmentation of the image into regions with different blurs.
Our approach relies on the observation that the statistics of derivative filters in
images are significantly changed by blur. Assuming the blur results from a constant velocity motion, we can limit the search to one dimensional box filter blurs.
This enables us to model the expected derivatives distributions as a function of
the width of the blur kernel. Those distributions are surprisingly powerful in
discriminating regions with different blurs. The approach produces convincing
deconvolution results on real world images with rich texture.
1 Introduction
Motion blur is the result of the relative motion between the camera and the scene during image
exposure time. This includes both camera and scene objects motion. As blurring can significantly
degrade the visual quality of images, photographers and camera manufactures are frequently searching for methods to limit the phenomenon. One solution that reduces the degree of blur is to capture
images using shorter exposure intervals. This, however, increases the amount of noise in the image,
especially in dark scenes. An alternative approach is to try to remove the blur off-line. Blur is usually
modeled as a linear convolution of an image with a blurring kernel, also known as the point spread
function (or PSF). Image deconvolution is the process of recovering the unknown image from its
blurred version, given a blurring kernel. In most situations, however, the blurring kernel is unknown
as well, and the task also requires the estimation of the underlying blurring kernel. Such a process
is usually referred to as blind deconvolution.
Most of the existing blind deconvolution research concentrates at recovering a single blurring kernel for the entire image. While the uniform blur assumption is valid for a restricted set of camera
motions, it?s usually far from being satisfying when the scene contains several objects moving independently. Existing deblurring methods which handle different motions usually rely on multiple
frames. In this work, however, we would like to address blind multiple motions deblurring using a
single frame.
The suggested approach is fully automatic, under the following two assumptions. The first assumption is that the image consists of a small number of blurring layers with the same blurring kernel
within each layer. Most of the examples in this paper include a single blurred object and an unblurred background. Our second simplifying assumption is that the motion is in a single direction
?
Current address: MIT CSAIL, [email protected]
and that the motion velocity is constant, such as in the case of a moving vehicle captured by a static
camera. As a result, within each blurred layer, the blurring kernel is a simple one dimensional box
filter, so that the only unknown parameters are the blur direction and the width of the blur kernel.
Deblurring different motions requires the segmentation of the image into layers with different blurs
as well as the reconstruction of the blurring kernel in each layer. While image segmentation is an
active and challenging research area which utilizes various low level and high level cues, the only
segmentation cue used in this work is the degree of blur. In order to discriminate different degrees of
blur we use the statistics of natural images. Our observation is that statistics of derivatives responses
in images are significantly changed as a result of blur, and that the expected statistics under different
blurring kernels can be modeled. Given a model of the derivatives statistics under different blurring
kernels our algorithm searches for a mixture model that will best describe the distribution observed
in the input image. This results in a set of 2 (or some other small number) blurring kernels that were
used in the image. In order to segment the image into blurring layers we measure the likelihood
of the derivatives in small image windows, under each model. We then look for a smooth layers
assignment that will maximize the likelihood in each local window.
1.1 Related work
Blind deconvolution is an extensive research area. Research about blind deconvolution given a single image, usually concentrate at cases in which the image is uniformly blurred. A summary and
analysis of many deconvolution algorithms can be found in [14]. Early deblurring methods treated
blurs that can be characterized by a regular pattern of zeros in the frequency domain such as box
filter blurs [26]. This method is known to be very sensitive to noise. Even in the noise free case,
box filter blurs can not be identified in the frequency domain if different blurs are present. More
recent methods are making other assumptions about the image model. This includes an autoregressive process [22], spatial isotropy [28], power low distributions [8, 20], and piecewise-smoothness
edges modeling [3]. In a creative recent research which inspired our approach, Fergus et al [12]
use the statistics of natural images to estimate the blurring kernel (again, assuming a uniform blur).
Their approach searches for the max-marginal blurring kernel and a deblurred image, using a prior
on derivatives distribution in an unblurred image. They address more than box filters, and present
impressing reconstructions of complex blurring kernels. Our approach also relies on natural images
statistics, but it takes the opposite direction: search for a kernel that will bring the unblurred distribution close to the observed distribution. Thus, in addition to handling non uniform blurs, our
approach avoids the need to estimate the unblurred image in every step.
In [10], Elder and Zucker propose a scale space approach for estimating the scale of an edge. As
the edge?s scale provides some measure of blur this is used for segmenting an image into a focus
and out of focus layers. The approach was demonstrated on a rather picewise constant image, unlike
the rich texture patterns considered in this paper. In [4], blind restoration of spatially-varying blur
was studied in the case of astronomical images, which have statistics quite different from the natural
scenes addressed in this paper.
Other approaches to motion deblurring include hardware approaches [6, 17, 7], and using multiple
frames to estimate blur, e.g. [5, 21, 29].
Another related subject is the research on depth from focus or depth from defocus (see [9, 11] to
name a few), in which a scene is captured using multiple focus settings. As a scene point focus is
a function of its depth, the relative blur is used to estimate depth information. Again, most of this
research relies on more than a single frame.
Recent work in computer vision applied natural images priors for a variety of applications like
denoising [25, 24], super resolution [27], video matting [2], inpainting [16] and reflections decomposition [15].
2 Image statistics and blurring
Figure 1(a) presents an image of an outdoor scene, with a passing bus. The bus is blurred horizontally
as a result of the bus motion. In fig 1(b) we plot the log histogram of the vertical derivatives of this
image, and the horizontal derivatives within the blurred area (marked with a rectangle). As can be
0
0
0
vertical
horizontal
?1
Input
5 taps blur
21 taps blur
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horizontal
blurred vertical
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Figure 1: Blurred versus unblurred derivatives histograms. (a) Input image. (b) Horizontal derivatives within the blurred region versus vertical derivatives in the entire image. (c) Simulating different blurs in the vertical direction. (d) Horizontal derivatives within the blurred region matched with
blurred verticals (4 tap blur).
seen, the blur changes the shape of the histogram significantly. This suggests that the statistics of
derivative filters responses can be used for detecting blurred image areas.
How does the degree of blur affects the derivatives histogram? To answer this question we simulate
histograms of different blurs. Let f k denote the horizontal box kernel of size 1 ? k (that is, all
entries of fk equal 1/k). We convolve the image with the kernels f kT (where k runs from 1 to 30)
and compute the vertical derivatives distributions:
pk ? hist(dy ? fkT ? I)
(1)
T
where dy = [1 ? 1] . Some of those log histograms are plotted in fig 1(c). As the size of the blurring
kernel changes the derivatives distribution, we would also like to use the histograms for determining
the degree of blur. For example, as illustrated in fig 1(d), we can match the distribution of vertical
derivatives in the blurred area, and p 4 , the distribution of horizontal derivatives after blurring with a
4 tap kernel.
2.1 Identifying blur using image statistics
Given an image, the direction of motion blur can be selected as the direction with minimal derivatives variation, as in [28]. For the simplicity of the derivation we will assume here that the motion
direction is horizontal, and that the image contains a single blurred object plus an unblurred background. Our goal is to determine the size of the blur kernel. That is, to recover the filter f k which
is responsible for the blur observed in the image. For that we compute the histogram of horizontal
derivatives in the image. However, not all the image is blurred. Therefore, without segmenting the
blurred areas there is no single blurring model p k that will describe the observed histogram. Instead,
we try to describe the observed histogram with a mixture model. We define the log-likelihood of the
derivatives in a window with respect to each of the blurring models as:
log pk (Ix (j))
(2)
k (i) =
j?Wi
Where Ix (j) is the horizontal derivative in pixel j, and W i is a window around pixel i. Thus, k (i)
measures how well the i?th window is explained by a k-tap blur.
For an input image I and a given pair of kernels, we can measure the data log-likelihood by associating each window with the maximum likelihood kernel:
L(I|fk1 , fk2 ) =
max(k1 (i), k2 (i))
(3)
i?I
We search for a blurring model p k0 such that, when combined with the model p 1 (derivatives of the
unblurred image), will maximize the log-likelihood of the observed derivatives:
k0 = arg max L(I|f1 , fk )
k
(4)
One problem we need to address in defining the likelihoods is the fact that uniform areas, or areas
with pure horizontal edges (the aperture problem) don?t contain any information about the blur.
On the other hand, uniform areas receive the highest likelihoods from wide blur kernels (since the
derivatives distribution for wide kernels is more concentrated around zero, as can be observed in
figure 1(c)). When the image consists of large uniform areas, this bias the likelihood toward wider
blur kernels. To overcome this, we start by scanning the image with a simple edge detector and keep
only windows with significant vertical edges. In order to make our model consistent, when building
the blurred distribution models p k (eq 1), we also take into account only pixels within a window
around a vertical edge.
Note that since we deal here with one dimensional kernels, we can estimate the expected blurred
histogram pk (eq 1) from the perpendicular direction of the same image.
2.2 Segmenting blur layers
Once the blurring kernel f k has been found, we can use it to deconvolve the image, as in fig 2(b).
While this significantly improves the image in the blurred areas, serious artifacts are observed in
the background. Therefore, in addition to recovering the blurring kernel, we need to segment the
image into blurred and unblurred layers. We look for a smooth segmentation that will maximize the
likelihood of the derivatives in each region. We define the energy of a segmentation as:
E(x) =
?(x(i), i) +
eij |x(i) ? x(j)|
(5)
i
<ij>
where (x(i), i) = 1 (i) for x(i) = 0 and (x(i), i) = k (i) for x(i) = 1, < i, j > are neighboring
image pixels, and e ij is a smoothness term:
eij = ? + ?(|I(i) ? I ?fk (i)| + |I(j) ? I ?fk (j)|)
(6)
?fk
Here I
denotes the deconvolved image. The smoothness term is combined from two parts. The
first is just a constant penalty for assigning different labels to neighboring pixels, thus preferring
smooth segmentations. The second part encodes the fact that it is cheaper to cut the image in places
where there is no visual seam between the original and the deconvolved images (e.g. [1]).
Given the local likelihood scores and the energy definition, we would like to find the minimal energy
segmentation. This reduces to finding a min-cut in a graph. Given the segmentation mask x we
convolve it with a Gaussian filter to obtain a smoother seam. The final restorated image is computed
as:
R(i) = x(i)I ?fk (i) + (1 ? x(i))I(i)
(7)
3 Results
To compute a deconvolved image I ?fk given the blurring kernel, we follow [12] in using the matlab
implementation (deconvlucy) of the Richardson-Lucy deconvolution algorithm [23, 18].
Figure 2 presents results for several example images. For the doll example the image was segmented into 3 blurring layers. The examples of figure 2 and additional results are available in a
high resolution in the supplementary material. The supplementary file also includes examples with
non horizontal blurs. To determine the blur direction in those images we select the direction with
minimal derivatives variation, as in [28]. This approach wasn?t always robust enough.
For each image we show what happens if the segmentation is ignored and the entire image is deconvolved with the selected kernel (for the doll case the wider kernel is shown). While this improves the
result in the blurred area, strong artifacts are observed in the rest of the image. In comparison, the
third row presents the restorated images computed from eq 7 using the blurring layers segmentation.
We also show the local MAP labeling of the edges. White pixels are ones for which an unblurred
model receives a higher likelihood, that is 1 (i) > k (i), and for gray pixels 1 (i) < k (i) (for the
doll case there are 3 groups, defined in a similar way). The last row presents the segmentation contour. The output contour does not perfectly align with image edges. This is because our goal in the
segmentation selection is to produce visually plausible results. The smoothness term of our energy
(eq 6) does not aim to output an accurate segmentation, and it does not prefer to align segmentation
edges with image edges. Instead it searches for a cut that will make the seam between the layers
unobservable.
(a)
(b)
(c)
(d)
(e)
Figure 2: Deblurring Results. (a)Input image. (b)Applying the recovered kernel on the entire image.
(c)Our result. (d)Local classification of windows. (e)Segmentation contour
The recovered blur sizes for those examples were 12 pixels for the bicycles image and 4 pixels for
the bus. For the doll image a 9 pixels blur was identified in the skirt segment and a 2 pixels blur in the
doll head. We note that while recovering big degrees of blur as in the bicycles example is visually
more impressing, discriminating small degrees of blur as in the bus example is more challenging
from the statistical aspect. This is because the derivatives distributions in the case of small blurs are
much more similar to the distributions of unblurred images.
For the bus image the size of the blur kernel found by our algorithm was 4 pixels. To demonstrate
the fact that this is actually the true kernel size, we show in figure 3 the deconvolution results with a
3-tap filter and with a 5-tap filter. Stronger artifacts are observed in each of those cases.
(input)
(3-taps)
(4-taps)
(5-taps)
(matlab)
Figure 3: Deconvolving the bus image using different filters. The 4-tap filter selected by our algorithm yields best results
Next, we consider several simple alternatives to some of the algorithm parts. We start by investigating the need in segmentation and then discuss the usage of the image statistics.
Segmentation: As demonstrated in fig 2(b) deconvolving the entire image with the same kernel
damages the unblurred parts. One obvious solution is to divide the image into regions and match
a separate blur kernel to each region. As demonstrated by fig 2(d), even if we limit the kernel
choice in each local window to a small set of 2-3 kernels, the local decision could be wrong. For all
the examples in this paper we used 15 ? 35 windows. There is some tradeoff in selecting a good
window size. While likelihood measure based on a big window is more reliable, such a window
might cover regions from different blurring layers. Another alternative is to brake the image into
segments using an unsupervised segmentation algorithm, and match a kernel to each segment. The
fact that blur changes the derivatives distributions also suggests that it might be captured as a kind
of texture cue. Therefore, it?s particularly interesting to try segmenting the image using texture
affinities (e.g. [13, 19]). However, as this is an unsupervised segmentation process which does not
take into account the grouping goal, it?s hard to expect it to yield exactly the blurred layers. Fig 4(b)
presents segmentation results using the Ncuts framework of [19]. The output over-segments blur
layers, while merging parts of blurred and unblurred objects. Unsurprisingly, the recovered kernels
are wrong.
1
18
3
1
13
(a)
(b)
(c)
Figure 4: Deblurring using unsupervised segmentation. (a) Input. (b) Unsupervised segmentation
and the width of the kernel matched to each segment. (c) Result from deblurring each segment
independently.
Image statistics: We move to evaluating the contribution of the image statistics. To do that independently of the segmentation, we manually segmented the bus and applied the matlab blind
deconvolution function (deconvblind), initialized with a 1 ? 7 box kernel. Strong artifacts were
introduced as shown in the last column of fig 3.
The algorithm results also depend on the actual histograms used. Derivatives histograms of different
natural images usually have common characteristics such as the heavy tail structure. Yet, the histogram structure of different images is not identical, and we found that trying to deblur one image
using the statistics of a different image doesn?t work that well. For example, figure 5 shows the result
of deblurring the bus image using the bicycles image statistics. The selected blur in this case was
a 6-tap kernel, but deblurring the image with this kernel introduces artifacts. The classification of
pixels into layers using this model is wrong as well. Our solution was to work on each image using
the vertical derivatives histograms from the same image. This isn?t an optimal solution as when the
image is blurred horizontally some of the vertical derivatives are degraded as well. Yet, it provided
better results than using histograms obtained from different images.
(a)
(b)
(c)
(d)
Figure 5: Deblurring the bus image using the bicycles image statistics. (a) Applying the recovered kernel on the entire image. (b) Deblurring result. (c) Local classification of windows. (d)
Segmentation contour.
Limitations: Our algorithm uses simple derivatives statistics and the power of such statistics is
somewhat surprising. Yet, the algorithm might fail. One failure source is blurs which can?t be
described as a box filter, or failures in identifying the blur direction. Even when this isn?t the case,
the algorithm may fail to identify the correct blur size or it may not infer the correct segmentation.
Figure 6 demonstrate a failure. In this case the algorithm preferred a model explaining the bushes
texture instead of a model explaining the car blur. The bushes area consists of many small derivatives
which are explained better by a small blur model than by a no-blur model. On the other hand, the
car consists of very few vertical edges. As a result the algorithm selected a 6-pixels blur model. This
model might increase the likelihood of the bushes texture and the noise on the road, but it doesn?t
remove the blur of the car.
(a)
(b)
(c)
(d)
(e)
Figure 6: Deblurring failure. (a) Input. (b) Applying the recovered kernel (6-taps) on the entire
image. (c) Deblurring result. (d) Local classification. (e) Segmentation contour.
4 Discussion
This paper addresses the problem of blind motion deconvolution without assuming that the entire
image undergone the same blur. Thus, in addition to recovering an unknown blur kernel, we segment
the image into layers with different blurs. We treat this highly challenging task using a surprisingly
simple approach, relying on the derivatives distribution in blurred images. We model the expected
derivatives distributions under different degrees of blur, and those distributions are used for detecting
different blurs in image windows.
The box filters model used in this work is definitely limiting, and as pointed out by [12, 6], many
blurring patterns observed in real images are more complex.
A possible future research direction is to try to develop stronger statistical models which can include
stronger features in addition to the simple first order derivatives. Stronger models might enable us to
identify a wider class of blurring kernels rather than just box filters. Particularly, they could provide
a better strategy for identifying the blur direction. A better model might also avoid the need to detect
vertical edges and artificially limit the model to windows around edges.
In future work, it will also be interesting to try to detect different blurs without assuming a small
number of blurring layers. This will require estimating the blurs in the image in a continues way,
and might also provide a depth from focus algorithm that will work on a single image.
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2,298 | 3,086 | Support Vector Machines on a Budget
Ofer Dekel and Yoram Singer
School of Computer Science and Engineering
The Hebrew University
Jerusalem 91904, Israel
{oferd,singer}@cs.huji.ac.il
Abstract
The standard Support Vector Machine formulation does not provide its user with
the ability to explicitly control the number of support vectors used to define the
generated classifier. We present a modified version of SVM that allows the user
to set a budget parameter B and focuses on minimizing the loss attained by the B
worst-classified examples while ignoring the remaining examples. This idea can
be used to derive sparse versions of both L1-SVM and L2-SVM. Technically, we
obtain these new SVM variants by replacing the 1-norm in the standard SVM formulation with various interpolation-norms. We also adapt the SMO optimization
algorithm to our setting and report on some preliminary experimental results.
1
Introduction
The L1 Support Vector Machine (L1-SVM or SVM for short) [1, 2, 3] is a powerful technique for learning binary classifiers from examples. Given a training set {(xi , yi )}m
i=1 and
a positive
semi-definite
kernel
K,
the
SVM
solution
is
a
hypothesis
of
the
form
h(x)
=
sign
i?S ?i yi K(xi , x) + b , where S is a subset of {1, . . . , m}, {?i }i?S are real valued weights,
and b is a bias term. The set S defines the support of the classifier, namely, the set of examples that
actively participate in the classifier?s definition. The examples in this set are called support vectors,
and we say that the SVM solution is sparse if the fraction of support vectors (|S|/m) is reasonably
small.
Our first concern is usually with the accuracy of the classifier. However, in some applications, the
size of the support is equally important. Assuming that the kernel operator K can be evaluated in
constant time, the time-complexity of evaluating the classifier on a new instance is linear in the size
of S. Therefore, a large support defines a slow classifier. Classification speed is often important
and plays an especially critical role in real-time systems. For example, a classifier that drives a
phoneme detector in a speech recognition system is evaluated hundreds of times a second. If this
classifier does not manage to keep up with the rate at which the speech signal is acquired then
its classifications are useless, regardless of their accuracy. The size of the support also naturally
determines the amount of memory required to store the classifier. If a classifier is intended to run
in a device with a limited memory, such as a mobile telephone, there may be a physical limit on
the amount of memory available to store support vectors. The size of S may also effect the time
required to train an SVM classifier. Most modern SVM learning algorithms are active set methods,
namely, on every step of the training process, only a small set of active training examples are taken
into account. Knowing the size of S ahead of time would enable us to optimize the size of the active
set and possibly gain a significant speed-up in the training process.
The SVM mechanism does not give us explicit control over the size of the support. The user-defined
parameters of SVM have some influence on the size of S, but we often require more than this.
Specifically, we would like the ability to specify a budget parameter, B, which directly controls the
number of support vectors used to define the SVM solution. In this paper, we address this issue
and present budget-SVM, a minor modification to the standard L1-SVM formulation that allows
the user to set a budget parameter. The budget-SVM optimization problem focuses only on the B
worst-classified examples in the training set, ignoring all other examples.
The problem of sparsity becomes even more critical when it comes to L2-SVM [3], a variant of the
SVM problem that tends to have dense solutions. L2-SVM is sometimes preferred over L1-SVM
because it exhibits good generalization properties, as well as other desirable statistical characteristics [4]. We derive the budget-L2-SVM formulation by following the same technique used to derive
budget-L1-SVM.
The technique used to derive these SVM variants is as follows. We begin by generalizing the L1SVM formulation by replacing the 1-norm with an arbitrary norm. We obtain a general framework
for SVM-type problems, which we nickname Any-Norm-SVM. Next, we turn to the K-method of
norm interpolation to obtain the 1 ? ? interpolation-norm and the 2 ? ? interpolation-norm, and
use these norms in the Any-Norm-SVM framework. These norms have the property that they depend
only on the absolutely-largest elements of the vector. We rely on this property and show that our
SVM variants construct sparse solutions. For each of these norms, we present a simple modification
of the SMO algorithm [5], which efficiently solves the respective optimization problem.
Related Work The problem of approximating the SVM solution using a reduced set of examples
has received much previous attention [6, 7, 8, 9]. This technique takes a two-step approach: begin
by training a standard SVM classifier, perhaps obtaining a dense solution. Then, try to find a sparse
classifier which minimizes the L2 distance to the SVM solution. A potential drawback of this approach is that once the SVM solution has been found, the distribution from which the training set was
sampled no longer plays a role in the learning process. This ignores the fact that shifting the SVM
classifier by a fixed amount in different directions may have dramatically different consequences on
classification accuracy. We overcome this problem by taking the approach of [10] and reformulating
the SVM optimization problem itself in a way that promotes sparsity. Another technique used to
obtain a sparse kernel-machine takes advantage of the inherent sparsity of linear programming solutions, and formalizes the kernel-machine learning problem as a linear program [11]. This approach,
often called LP-SVM or Sparse-SVM, has been shown to generally construct sparse solutions, but
still lacks the ability to introduce an explicit budget parameter. Yet another approach involves randomly selecting a subset of the training set to serve as support vectors [12]. The problem of learning
a kernel-machine on a budget also appears in the online-learning mistake-bound framework, and it
is there where the term ?learning on a budget? was coined [13]. Two recent papers [14, 15] propose
online kernel-methods on a budget with an accompanying theoretical mistake-bound.
This paper is organized as follows. We present the generalized Any-Norm-SVM framework in
Sec. 2. We discuss the K-method of norm interpolation in Sec. 3 and put various interpolation
norms to use within the Any-Norm-SVM framework in Sec. 4. Then, in Sec. 5, we present some
preliminary experiments that demonstrate how the theoretical properties of our approach translate
into practice. We conclude with a discussion in Sec. 6. Due to the lack of space, some of the proofs
are omitted from this paper.
2
Any-Norm SVM
Let {(xi , yi )}m
i=1 be a training set, where every xi belongs to an instance space X and every yi ?
{?1, +1}. Let K : X ? X ? R be a positive semi-definite kernel, and let H be its corresponding
Reproducing Kernel Hilbert Space (RKHS) [16], with inner product ?, ?H . The L1 Support Vector
Machine is defined as the solution to the following convex optimization problem:
1
min
s.t. ? 1 ? i ? m yi f (xi ) + b ? 1 ? ?i ,
(1)
2 f, f H + C?1
f ?H, b?R, ??0
where ? is a vector of m slack variables, and C is a positive constant that controls the tradeoff
between the complexity of the learned classifier and how well it fits the training data. The value
of ?i is sometimes referred toas the hinge-loss attained by the SVM classifier on example i. The
m
1-norm, defined by ?1 = i=1 |?i |, is used to combine the individual hinge-loss values into a
single number.
L2-SVM is a variant of the optimization problem defined above, defined as follows:
2
1
min
s.t. ? 1 ? i ? m yi f (xi ) + b ? 1 ? ?i .
2 f, f H + C?2
f ?H, b?R, ??0
This formulation differs from
mthe L1 formulation in that the 1-norm is replaced by the squared 2norm, defined by ?22 = i=1 ?i2 . In this section, we take this idea even farther, and allow the
1-norm of L1-SVM to be replaced by any norm. Formally, let ? be an arbitrary norm defined on
Rm . Recall that a norm is a real valued operator such that for every v ? Rm and ? ? R it holds
that ?v = |?|v (positive homogeneity), v ? 0 and v = 0 if and only if v = 0 (positive
definiteness), and that satisfies the triangle inequality. Now consider the following optimization
problem:
1
min
s.t. ? 1 ? i ? m yi f (xi ) + b ? 1 ? ?i .
(2)
2 f, f H + C?
f ?H, b?R, ??0
L1-SVM is recovered by setting ? to be the 1-norm. Setting ? to be the 2-norm induces an
optimization problem which is close in nature to L2-SVM, but not identical to it since the 2-norm is
not squared. Combining the positive homogeneity property of ? with the fact that it satisfies the
triangle inequality ensures that the objective function of Eq. (2) is convex.
An important class of norms
mused extensively in our derivation is the family of p-norms, defined for
every p ? 1 by vp = ( j=1 |vj |p )1/p . A special member of this family is the ?-norm, which is
defined by v? = limp?? vp and can be shown to be equivalent to maxj |vj |. We also use the
notion of norm duality. Every norm on Rm has a dual norm which is also defined on Rm . The dual
norm of ? is denoted by ? and given by
u?v
=
max
u?v .
(3)
u = maxm
v?R
v
v?Rm : v=1
As its name implies, ? also satisfies the requirements of a norm. For example, H?older?s inequality [17] states that the dual of ? p is the norm ? q , where q = p/(p ? 1). The dual of the 1-norm
is the ?-norm and vice versa.
Using the definition of the dual norm, we now state the dual optimization problem of Eq. (2):
m
m
m
m
max
?i ? 12
?i ?j yi yj K(xi , xj ) s.t.
yi ?i = 0 and ? ? C .
??0
i=1
i=1 j=1
(4)
i=1
As a first sanity check, note that if ? in Eq. (2) is chosen to be the 1-norm, then ? is the ?norm, and the constraint ? ? C reduces to the familiar box-constraint of L1-SVM [3]. The proof
that Eq. (2) and Eq. (4) are indeed dual optimization problems relies on basic techniques in convex
analysis [18], and is omitted due
to the lack of space. Moreover, it can be shown that the solution
m
to Eq. (2) takes the form f (?) = i=1 ?i yi K(xi , ?), and that strong duality holds regardless of the
norm used. This allows us to forget about the primal problem in Eq. (2) and to focus on solving the
dual problem in Eq. (4). As with L1-SVM, the bias term, b, cannot be directly extracted from the
solution of the dual. The standard techniques used to find b in L1-SVM apply here as well [3].
We note that the Any-Norm-SVM formulation is not fundamentally different from the original L1SVM formulation. Both optimization problems have convex objective functions and linear constraints. More importantly, the only difference between their respective duals is in the dual-norm
constraint. Specifically, the objective function in Eq. (4) is a concave quadratic function for any
choice of ? . These facts enable us to efficiently solve the problem in Eq. (4) for any kernel K and
any norm using techniques similar to those used to solve the standard L1-SVM problem.
3
Interpolation Norms
In the previous section, we acquired the ability to replace the 1-norm in the definition of L1-SVM
with an arbitrary norm. We now use Peetre?s K-method of norm interpolation [19] to obtain norms
that promote the sparsity of the generated classifier. The K-method is a technique for smoothly
interpolating between a pair of norms. Let ? p1 : Rm ? R+ and ? p2 : Rm ? R+ be two
p-norms, and let ? q1 and ? q2 be their respective duals. Peetre?s K-functional with respect to
p1 and p2 , and with respect to the constant t > 0, is defined to be
wp1 + tzp2 .
(5)
vK(p1 ,p2 ,t) =
min
w,z : w+z=v
Peetre?s J-functional with respect to q1 , q2 , and with respect to the constant s > 0, is given by
(6)
uJ(q1 ,q2 ,s) = max uq1 , s uq2 .
The J-functional is obviously a norm: the properties of a norm all follow immediately from the
fact that ? q1 and ? q2 posses these properties. ? K(p1 ,p2 ,t) is also a norm, and moreover,
? K(p1 ,p2 ,t) and ? J(q1 ,q2 ,s) are dual to each other when t = 1/s. This fact can be proven using
elementary calculus, and this proof is omitted due to the lack of space.
We use the K-method to interpolate between the 1-norm and the ?-norm, and to interpolate between the 2-norm and the ?-norm. To gain some intuition on the behavior of these
interpolationm
p
norms, first note that for any p ? 1 and any v ? Rm it holds that maxi |vi |p ?
i=1 |vi | ?
p
1/p
m maxi |vi | , and therefore v? ? vp ? m v? . An immediate consequence of this is
that ? K(p,?,t) ? ? ? when 0 < t ? 1 and that ? K(p,?,t) ? ? p when m1/p ? t. In
other words, the interesting
? range of t for the 1 ? ? interpolation-norm is [1, m], and for the 2 ? ?
interpolation-norm is [1, m].
Next, we prove a theorem which states that interpolating a p-norm with the ?-norm is approximately
equivalent to restricting that p-norm to the absolutely-largest components of the vector. Specifically,
the 1 ? ? interpolation norm with parameter t (with t chosen to be an integer in [1, m]) is precisely
equivalent to taking the sum of the absolute values of the t absolutely-greatest elements of the vector.
Theorem 1. Let v be an arbitrary vector in Rm and let ? be a permutation on {1, . . . , m} such
that |v?(1) | ? . . . ? |v?(m) |. Then for any integer B in {1, . . . , m} it holds that vK(1,?,B) =
B
1/p
then it holds that
i=1 |v?(i) |, and for any 1 ? p < ?, if t = B
B
B
p 1/p
p 1/p
? vK(p,?,t) ?
+ B 1/p |v?(B) | .
i=1 |v?(i) |
i=1 |v?(i) |
Proof. Beginning with the lower bound, let w and z be such that w + z = v. Then
B
B
p 1/p
p 1/p
=
i=1 |v?(i) |
i=1 |w?(i) + z?(i) |
B
B
p 1/p
p 1/p
?
+
i=1 |w?(i) |
i=1 |z?(i) |
B
p 1/p
?
+ (B maxi |zi |p )1/p
i=1 |w?(i) |
m
p 1/p
?
+ tz? ,
i=1 |wi |
where the first inequality is the triangle inequality for the p-norm. Since
the above holds for any w
m
and z such that w + z = v, it also holds for the pair which minimizes ( i=1 |wi |p )1/p + tz? ,
and which defines vK(p,?,t) . Therefore, we have that,
B
p 1/p
? vK(p,?,t) .
(7)
i=1 |v?(i) |
?i =
Turning to the upper bound, let ? = |v?(B) |, and define for all 1 ? i ? m, w
? +z
? = v, and that
sign(vi ) max{0, |vi | ? ?} and z?i = sign(vi ) min{|vi |, ?}. Note that w
B
? 1 + B?
|v?(i) | = w
z? .
i=1
B
This proves that vK(1,?,B) ? i=1 |v?(i) | and together with Eq. (7) we have proven our claim
for p = 1. Moving on to the case of an arbitrary p, we have that
vK(p,?,t) =
? p + t?
min (wp + tz? ) ? w
z? .
w+z=v
? is at most as large as the absolute value of the
Since the absolute value of each element in w
? p ?
??(m) = 0, we have that w
corresponding element of v, and since w
??(r+1) = . . . = w
B
( i=1 |v?(i) |p )1/p . By definition, ?
z? = ? = |v?(B) |. This proves that vK(p,?,t) ?
B
( i=1 |v?(i) |p )1/p + t|v?(B) | and together with Eq. (7) this concludes our proof for arbitrary p.
4
Deriving Concrete Algorithms from the General Framework
Our first concrete algorithm is budget-L1-SVM, obtained by plugging the 1 ? ? interpolation-norm
with parameter B into the general Any-Norm-SVM framework. Relying on Thm. 1, we know that
this norm takes into account only the B largest values in ?. Since ? measures how badly each
example is misclassified, the budget-L1-SVM problem essentially optimizes the soft-margin with
respect to the B worst-classified examples. We now show that this property promotes the sparsity
of the budget-L1-SVM solution.
If there are less than B examples for which yi (f (xi )+b) < 1, then the KKT conditions of optimality
immediately imply that the number of support vectors is less than B. This holds true for every
instance of the Any-Norm-SVM framework, and is proven for L1-SVM in [3]. Therefore, we focus
on the more interesting case, where yi (f (xi ) + b) < 1 for at least B examples.
Theorem 2. Let B be an integer in {1, . . . , m}. Let (f, b, ?, ?) be an optimal primal-dual solution
of the primal problem in Eq. (2) and the dual problem in Eq. (4), where ? is chosen to be the 1??
interpolation-norm with parameter B. Define ?i = yi (f (xi ) + b) and let ? be a permutation of
{1, . . . , m} such that ??(1) ? . . . ? ??(m) . Assume that ??(B) < 1. Then, ?k = 0 if ??(B) < ?k .
Proof. We begin the proof by redefining ?i = max{1 ? ?i , 0} for all 1 ? i ? m and noting
that (f, b, ?, ?) remains a primal-dual solution to our problem. The benefit of starting with this
specific solution is that ??(1) ? . . . ? ??(m) . Let k be an index such that ??(B) < ?k and define
?k = 12 (?k + ??(B) ). Moreover, let ? be the vector obtained by replacing the k?th coordinate in
? with ?k , or in other words, ? = (?1 , . . . , ?k , . . . , ?m ). Using the assumption that ??(B) < 1,
we know that ??(B) > 0, and since ?k > ??(B) we get that ?k < ??(B) . We can now draw two
conclusions. First, ??(1) ? . . . ? ??(B) > ?k and therefore ? K(1,?,B) = ?K(1,?,B) . Second,
?k < ?k and therefore ? satisfies the constraints of Eq. (2). Overall, we obtain that (f, b, ? , ?) is
also a primal-dual solution to our problem. Moreover, we know that 1 ? ?k < ?k . Using the KKT
complementary slackness condition, it follows that ?k , the Lagrange multiplier corresponding to
this constraint, must equal 0.
Defining ?i and ? as above, a simple corollary of Thm. 2 is that the number of support vectors is
upper bounded by B in the case that ??(B) = ??(B+1) .
From our discussion in Sec. 3, we know that the dual of the 1 ? ? interpolation-norm is the function max{u? , (1/B)u1 }. Plugging this definition into Eq. (4) gives us the dual optimization problem of budget-L1-SVM. The constraint ? ? C simplifies to ?i ? C for all i and
m
i=1 ?i ? BC. To numerically solve this optimization problem, we turn to the Sequential Minimal Optimization (SMO) [5] technique. We briefly describe the SMO technique, and then discuss
its adaptation to our setting. SMO is an iterative process, which on every iteration selects a pair of
dual variables, ?k and ?l , and optimizes the dual problem with respect to them, leaving all other
variables fixed. The choice of the two variables is determined by a heuristic [5], and their optimal
values are calculated analytically. Assume that we start with a vector ? which is a feasible point
of the optimization
problem in Eq. (4). When restricted to the two active variables, ?k and ?l , the
constraint i=k,l ?i yi = 0 simplifies to ?knew yk + ?lnew yl = ?kold yk + ?lold yl . Put another way, we
can slightly overload our notation and define the linear functions
?k (?) = ?k + ?yk
and
?l (?) = ?l ? ?yl ,
(8)
and find the single variable ? which maximizes our constrained optimization problem. Since the
constraints in Eq. (4) define a convex and bounded feasible set, the intersection of the linear equalities in Eq. (8) with this feasible set restricts ? to an interval. The objective function, as a function of
the single variable ?, takes the form O(?) = P ?2 + Q? + c, where c is a constant,
P = K(xk , xl ) ? 12 K(xk , xk ) ? 12 K(xl , xl ) , Q = yk ? f (xk ) ? yl ? f (xl ) ,
m
and f is the current function in the RKHS (f ? i=1 ?i yi K(xi , ?)). Maximizing the objective
function in Eq. (4) with respect to ?k and ?l is equivalent to maximizing O(?) with respect to
? over an interval. P equals minus the Euclidean distance between the functions K(xk , ?) and
K(xl , ?) in the RKHS, and is therefore a negative number. Therefore, O(?) is a concave function
which attains a single (unconstrained) maximum. This maximum can be found analytically by
0 =
?O(?)
= 2P ? + Q
??
?
? =
?Q
.
2P
(9)
If this unconstrained optimum falls inside the feasible interval, then it is equivalent to the constrained
optimum. Otherwise, the constrained optimum falls on one of the two end-points of the interval.
Thus, we are left with the task of finding these end-points. To do so, we consider the remaining
constraints:
? (?) ? 0
? (?) ? C
(I) k
?i .
(II) k
(III) ?k (?) + ?l (?) ? BC ?
?l (?) ? 0
?l (?) ? C
i=k,l
The constraints in (I) translate to
yk = ?1 ? ? ? ?k
yl = ?1 ? ? ? ??l
yk = +1 ? ? ? ??k
yl = +1 ? ? ? ?l .
(10)
yk = +1 ? ? ? C ? ?k
yl = +1 ? ? ? ?l ? C .
(11)
m
1
i=1 ?i ? BC
2
m
1
i=1 ?i
2 BC ?
(12)
The constraints in (II) translate to
yk = ?1 ? ? ? ?k ? C
yl = ?1 ? ? ? C ? ?l
Constraint (III) translates to
yk = ?1 ? yl = +1
?
? ?
yk = +1 ? yl = ?1
?
? ?
.
Finding the end-points of the interval that confines ? amounts to finding the smallest upper bound
and the greatest lower bound in Eqs. (10,11,12). This concludes the analytic derivation of the SMO
update for budget-L1-SVM.
?
L2-SVM on a budget Next, we use the 2 ? ? interpolation-norm with parameter t = B in
the Any-Norm-SVM
framework, and obtain the budget-L2-SVM problem. Thm. 1 hints that setting
?
t = B makes the 2 ? ? interpolation-norm almost equivalent to restricting the 2-norm to the top
B elements in the vector ?. The support size of the budget-L2-SVM solution is strongly correlated
with the parameter B although the exact relation between the two is not as clear as before. Again
we begin with the
? dual formulation defined in Eq. (4), where the constraint ? ? C becomes
max{?2 , (1/ B)?1 } ? C. The intersection of this constraint with the other constraints defines
a convex and bounded feasible set, and its intersection with the linear equalities in Eq. (8) defines
an interval. The objective function in Eq. (4) is the same as before, so the unconstrained maximum
is once again given be Eq. (9). To obtain the constrained maximum, we must find the end-points of
the interval that confines ?. The dual-norm constraint can be written more explicitly as
?
(I) ?k (?) + ?l (?) ? BC ?
?i
(II) ?k2 (?) + ?l2 (?) ? C 2 ?
?i2 .
i=k,l
i=k,l
the constraint we had in the budget-L1-SVM case, and is given in terms
Constrain (I) is similar to?
of ? by replacing B with B in Eq. (12). Constraint (II
) is new, and can be written in terms of ? as
m
?2 + ?? + ? ? 0, where ? = ?k yk ? ?l yl and ? = 12 ( i=1 ?i2 ? C 2 ). It can be written even more
explicitly as
? ? 12 (?? + ? 2 ? 4?) and ? ? 12 (?? ? ? 2 ? 4?) .
(13)
In addition, we still have the constraint ? ? 0, which is common to every instance of the AnyNorm-SVM framework. This constraint is given in terms of ? in Eq. (10). Overall, the end-points
of the interval we are searching for are found by taking the smallest
upper bound and the greatest
?
lower bound in Eqs. (10,13) and Eq. (12) with B replaced by B.
700
500
450
600
400
500
350
300
s
s
400
300
250
200
150
200
100
100
50
200
400
600
B
800
1000
200
400
600
800
1000
B
Figure 1: Average test error of budget-L1-SVM (left) and budget-L2-SVM (right) for different
values of the budget parameter B and the pruning parameter s (all but s weights in ? are set to
zero). The test error in the darkest region is roughly 50%, and in the lightest region is roughly 5%.
5
Experiments
Many existing solvers for the standard L1-SVM problem define a positive threshold value close to
zero and replace every weight that falls below this threshold with zero. This heuristic significantly
reduces the time required for the algorithm to converge. In our setting, a more natural way to speed
up the learning process is to run the iterative SMO optimization algorithm for a fixed number of
iterations and then to keep only the B largest weights, setting the m ? B remaining weights to
zero. This pruning heuristic enforces the budget constraint in a brute-force way, and can be equally
applied to any kernel-machine. However, the natural question is how much will the pruning heuristic
affect the classification accuracy of the kernel-machine it is applied to. If our technique indeed lives
up to its theoretical promise, we expect the pruning heuristic to have little impact on classification
accuracy. On the other hand, if we train an L1-SVM and it so happens that the number of large
weights exceeds B, then applying the pruning heuristic should have a dramatic negative effect on
classification accuracy. The goal of our experiments is to demonstrate that this behavior indeed
occurs in practice.
We conducted our experiments using the MNIST dataset, which contains handwritten digits from
the 10 digit classes. We randomly generated 50 binary classification problems by first randomly
partitioning the 10 classes into two equally sized sets, and then randomly choosing a training set of
1000 examples and a test set of 4000 examples. The results reported below are averaged over these
50 problems. Although MNIST is generally thought to induce easy learning problems, the method
described above generates moderately difficult learning tasks.
For each binary problem, we trained both the L1 and the L2 budget SVMs with B =
20, 40, . . . , 1000. Note that ?K(1,?,B) grows roughly linearly with B, and that ?K(2,?,?B)
grows roughly
?like the square root of B. To compensate for this, we set C = 10/B in the L1 case
and C = 10/ B in the L2 case. This heuristic choice of C attempts to preserve the relative weight
of the regularization term with respect to the norm term in Eq. (2), across the various values of B.
In all of our experiments, we used a Gaussian kernel with ? = 1 (after scaling the data to have an
average unit norm). For each classifier trained, we pruned away all but the s largest weights, with
s = 20, 40, . . . , 1000, and calculated the test error. The average test error for every choice of B
(the budget parameter in the optimization problem) and s (the number of non-zero weights kept) is
summarized in Fig. 1. In practice, s and B should be equal, however we let s take different values
in our experiment to illustrate the characteristics of our approach. Note that the test-error attained
by L1-SVM (without a budget parameter) and L2-SVM are represented by the top-right corners of
the respective plots.
As expected, classification accuracy for any value of B deteriorates as s becomes small. However,
the accuracy attained by L1-SVM and L2-SVM can be equally attained using significantly less
support vectors.
6
Discussion
Using the Any-Norm-SVM framework with interesting norms enabled us to introduce a budget
parameter to the SVM formulation. However, the Any-Norm framework can be used for other
tasks as well. For example, we can interpolate between L1-SVM and L2-SVM by using the 1 ? 2
interpolation-norm. This gives the user the explicit ability to balance the trade-off between the pros
and cons of these two SVM variants. In [20] it is shown that there exists a constant c such that,
1/2
r
? m
2
cvK(1,2,?r) ?
|v
|
+
r
v
? vK(1,2,?r) .
j
j=1
j=r+1 j
These bounds give some insight into how such an interpolation would behave. Another possible
norm that can be used in our framework is the Mahalanobis norm (v = (v M v)1/2 , where M
is a positive definite matrix), which would define a loss function that takes into account pair-wise
relationships between examples.
Regarding our experiments, the rule-of-thumb we used to choose the parameter C is not always
optimal. It seems preferable to tune C individually for each B using cross-validation.
We are currently exploring extensions to our SMO variant that would quickly converge to the sparse
solution without the help of the pruning heuristic. We are also considering multiplicative update
optimization algorithms as an alternative to SMO.
References
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Press, 2000.
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2,299 | 3,087 | Natural Actor-Critic for Road Traffic Optimisation
Silvia Richter
Albert-Ludwigs-Universit?at
Freiburg, Germany
Douglas Aberdeen
National ICT Australia
Canberra, Australia
Jin Yu
National ICT Australia
Canberra, Australia.
[email protected]
[email protected]
[email protected]
Abstract
Current road-traffic optimisation practice around the world is a combination of
hand tuned policies with a small degree of automatic adaption. Even state-ofthe-art research controllers need good models of the road traffic, which cannot
be obtained directly from existing sensors. We use a policy-gradient reinforcement learning approach to directly optimise the traffic signals, mapping currently
deployed sensor observations to control signals. Our trained controllers are (theoretically) compatible with the traffic system used in Sydney and many other cities
around the world. We apply two policy-gradient methods: (1) the recent natural
actor-critic algorithm, and (2) a vanilla policy-gradient algorithm for comparison.
Along the way we extend natural-actor critic approaches to work for distributed
and online infinite-horizon problems.
1
Introduction
Optimising the performance of existing road networks is a cheap way to reduce the environmental,
social, and financial impact of ever increasing volumes of traffic. Road traffic optimisation can be
naturally cast as a reinforcement learning (RL) problem. Unfortunately it is in the hardest class of
RL problems, having a continuous state space and infinite horizon, and being partially observable
and difficult to model. We focus on the use of the natural actor-critic (NAC) algorithm [1] to solve
this problem through online interaction with the traffic system.
The NAC algorithm is an elegant combination of 4 RL methods: (1) policy-gradient actors to allow
local convergence under function approximation and partial observability; (2) natural gradients to
incorporate curvature statistics into the gradient-ascent; (3) temporal-difference critics to reduce
the variance of gradient estimates; (4) least-squares temporal difference methods to avoid wasting
information from costly environment interactions.
One contribution of this work is an efficient online version of NAC that avoids large gradient steps,
which cannot be guaranteed to be an improvement in the presence of stochastic gradients [2]. We
compare this online NAC to a simple online policy-gradient (PG) approach, demonstrating that
NAC converges in orders of magnitude less environment interactions. We also compare wall-clock
convergence time, and suggest that environments which can be simulated quickly and accurately can
be optimised faster with simple PG approaches rather than NAC.
This work has grown out of an interaction with the Sydney Road Traffic Authority. Our choice
of controls, observations, and algorithms all aim for practical large-scale traffic control. Although
our results are based on a simplified traffic simulation system, we could theoretically attach our
learning system into real-world traffic networks. Our simple simulator results demonstrate better
performance than the automatic adaption schemes used in current proprietary systems.
2
Traffic Control
The optimisation problem consists of finding signalling schedules for all intersections in the system
that minimise the average travel time, or similar objectives. This is complicated by the fact that
many of the influencing state variables cannot be readily measured. Most signal controllers in use
today rely only on state information gained from inductive loops in the streets.
A stream is a sequence of cars making the same turn (or going straight) through an intersection. A
phase is an interval where a subset of the lights at an intersection are green such that a set of streams
that will not collide have right of way. A signal cycle is completed when each phase has been on
once, the cycle time being the sum of phase times. Traditionally, control algorithms optimise traffic
flow via the phase scheme, the split, and the offset. The phase scheme groups the signal lights into
phases and determines their order. A split gives a distribution of the cycle time to the individual
phases. Offsets can be introduced to coordinate neighbouring intersections, creating a ?green wave?
for the vehicles travelling along a main road.
Approaches to Traffic Control can be grouped into three categories. Fixed time control strategies are
calculated off-line, based on historical data. TRANSYT, for example, uses evolutionary algorithms
and hill-climbing optimisation [3]. Traffic responsive strategies are real-time, calculating their policies from car counts determined from inductive-loop detectors. SCOOT and SCATS are examples in
use around the world [4, 5]. Third generation methods employ sophisticated dynamic traffic models
and try to find optimal lengths for all phases, given a fixed phase scheme, e.g., by dynamic programming [4]. The exponential complexity of the problem, however, limits these approaches to local
view of a few intersections. Reinforcement learning has also been applied [6], but in a way that uses
a value function for each car, which is unrealistic in today?s world. Common to most approaches is
that they deal with the insufficient state information by maintaining a model of the traffic situation,
derived from available sensor counts. However, imperfections in the model is a further source of
errors and performance may consequently suffer. Our methods avoid modelling.
We focus on the system that motivated this research, the Sydney Coordinated Adaptive Traffic System (SCATS) [5]. It is used in major cities throughout Australasia and North America. It provides
pre-specified plans that are computed from historical data. There is an additional layer of online
adaption based on a saturation-balancing algorithm, i.e., SCATS calculates phase timings so that all
phases are utilised by vehicles for a fixed percentage of the phase time. Small incremental updates
are performed once per cycle. Due to this slow update, rapidly fluctuating traffic conditions pose a
problem to SCATS. Furthermore, it does not optimise the global network performance, and its base
plans involve hand-tuning to encode offset values.
3
Partially Observable MDP Formulation
We cast the traffic problem as a partially observable Markov decision process (POMDP) with states
s in a continuous space S. The system is controlled by stochastic actions at ? A drawn from a
random variable (RV) conditioned on the current policy parameters ?t , and an observation of the
current state o(st ), according to Pr(at |o(st ), ?t ). In general POMDPs the observation function
o(st ) is stochastic. NAC, which has only been presented in literature for the fully observable case
so far, can be extended to POMDPs given deterministic observations, as this ensures compatibility
of the function approximation and avoids injecting noise into the least squares solution. Our traffic
POMDP fulfils this requirement. To simplify notation we set ot := o(st ). The state is an RV
evolving as a function of the previous state and action according to Pr(st+1 |st , at ). In the case of
road traffic these distributions are continuous and complex, making even approximate methods for
model based planning in POMDPs difficult to apply. The loop sensors only count cars that pass over
them, or are stationary on them. The controller needs access to the history to attempt to resolve some
hidden state. We later describe observations constructed from the sensor readings that incorporate
some important elements of history for intersection control.
3.1
Local Rewards
A common objective function in traffic control is the average vehicle travel time. However, it is
impossible to identify a particular car in the system, let alone what its travel time was. Sophisticated
modelling approaches are needed to estimate these quantities. We prefer a direct approach that
has the benefits of being trivial to measure from existing sensors and easing the temporal credit
assignment problem. For the majority of our experiments we treat each intersection as a local MDP.
It is trivial to count all cars that enter the intersection with loop detectors. Therefore, we chose the
instant reward rt,i to be the number of cars that entered intersection i over the last time step. The
objective for each intersection i is to maximise the normalised discounted throughput:
o
n
X?
Ri (?) = E (1 ? ?)
? t rt,i ? .
t=0
Discounting is important because it ensures that the controller prefers to pass cars through as early
as possible. While suboptimal policies (in terms of travel time) may achieve the optimal average
throughput over a time window, the discounted throughput criterion effectively minimises the total
waiting time at an intersection in the finite-horizon case [7]. Ignoring effects such as road saturation
and driver adaption (which we explore in our experiments), this would result in minimisation of the
system wide travel time. The use of local rewards speeds up learning, especially as the number of
intersections grows. Bagnell and Ng [8] demonstrate that local rewards alter the sample complexity
?
from worst case ?(I),
where I is the number of intersections, down to O(log I). Unfortunately,
the value of Ri (?) depends directly on the local steady state distribution Pr(si |?). Thus changes
to the policy of neighbouring intersections can adversely impact intersection i, by influencing the
distribution of si . A sufficiently small learning rate allows controllers to adapt to this effectively
non-stationary component of the local MDP. We may fail to find the globally optimal cooperative
policy without some communication of rewards, but it has proven very effective empirically.
4
Natural Actor-Critic Algorithms
Policy-gradient (PG) methods for RL are of interest because it can be easier to learn policies directly than to estimate the exact value of every state of the underlying MDP. While they offer only
local convergence guarantees, they do not suffer from the convergence problems exhibited by pure
value based methods under function approximation or partial observability [9]. On the other hand,
PG methods have suffered from slow convergence compared to value methods due to high variance
in the gradient estimates. The natural actor-critic method (NAC) [1] improves this with a combination of PG methods, natural gradients, value estimation, and least-squares temporal-difference
Q-learning (LSTD-Q). NAC computes gradient estimates in a batch fashion, followed by a search
for the best step size. We introduce an online stochastic gradient ascent using NAC estimates.
Stochastic gradient ascent methods often outperform stochastic batch methods [2]. We begin with
the Bellman equation for fixed parameters ? where the value of action a in state s is Q(s, a). This
can also be written as the value V(s) plus the advantage of action a in state s, or A(s, a):
Z
Q(s, a) = V(s) + A(s, a) = r(s, a) + ? Pr(s0 |s, a)V(s0 ) ds0 .
(1)
S
We substitute linear approximators for the value and advantage functions, with parameter vectors v
?
? a) := (?? log Pr(a|o(s), ?))| w, leading to
and w respectively: V(s)
:= o(s)| v, A(s,
Z
o(s)| v + (?? log Pr(a|o(s), ?t ))| w = r(s, at ) + ? Pr(s0 |s, a)o(s0 )| v ds0 .
(2)
S
? a) has the nice property that
The surprising choice of ?? log Pr(a|o(s), ?) as features for A(s,
the parameters w turn out to be the naturalised gradient of the long-term average reward (and it
is compatible with the policy parameterisation [9]). To see this we write out the policy-gradient
algorithm [9], where Pr(s|?) is the steady state probability of state s and b(s) is a baseline to reduce
the variance of gradient estimates
Z
Z
?? R(?) = Pr(s|?) ?? Pr(a|s)(Q(s, a) ? b(s)) da ds.
(3)
S
A
The obvious baseline for making Q(s, a) zero mean is b(s) = V(s), which gives Q(s, a) ? V(s) =
? a) for A(s, a) and make use of the fact
A(s, a). Again, we substitute the linear approximation A(s,
that our policy is actually a function of o := o(s) and ?:
Z
Z
?? R(?) = Pr(s|?) ?? Pr(a|o, ?)(?? log Pr(a|o, ?))| w da ds.
S
A
Further substituting ?? Pr(a|o, ?) by Pr(a|o, ?)?? log Pr(a|o, ?) gives
Z
Z
?? R(?) = Pr(s|?) Pr(a|o, ?)?? log Pr(a|o, ?)(?? log Pr(a|o, ?))| da ds w =: F? w
S
A
A key observation is that the matrix F? is the outer product of the log action gradient, integrated
over all states and actions. This is exactly the Fisher information matrix [1]. On the other hand,
the naturalisation of gradients consists of pre-multiplying the normal gradient by the inverse of the
Fisher matrix [10], leading to cancellation of the two Fisher matrices F??1 ?? R(?) = w.
We return to (2) and reformulate it as a temporal-difference estimate of Q(st , at ), noting in particular
that the integral is replaced by an approximation ?ot+1 vt of the discounted value of the observed
next state. This approximation introduces a zero-mean error ?. Rewriting as a linear system yields
(ot ? ?ot+1 )| vt + ?(ot , st , st+1 ) + (?? log Pr(at |ot , ?t ))| wt = r(st , at ).
[(?? log Pr(at |ot , ?t )| , (ot ? ?ot+1 )| ][wt| , vt| ]| + ?(ot , st , st+1 ) = r(st , at )
zt [(?? log Pr(at |ot , ?t )| , (ot ? ?ot+1 )| ][wt| , vt| ]| + ?(ot , st , st+1 ) = zt r(st , at ) =: gt (4)
In (4) we pre-multiply both sides by an eligibility trace zt [11], which gives us the LSTDQ algorithm for a single sample at step t. The NAC algorithm approximates w by averaging both sides over many time steps T and solving AT [w| , v| ]| = bT , where AT =
PT
1/T t=1 zt [(?? log Pr(at |ot , ?t )| , (ot ? ?ot+1 )| ] (averaging out the zero-mean noise), and
PT
bT = 1/T t=0 gt . By analogy with other second-order gradient methods we can view A as
containing curvature information about the optimisation manifold, accelerating learning. In the case
of NAC it is an elegant combination of the Fisher information matrix and critic information.
4.1
Online Infinite-Horizon Natural Actor Critic
We cannot perform a line search on a real world traffic system because during the line search we may
try arbitrarily poor step size values. Furthermore, the gradient estimates are noisy, disadvantaging
batch methods [2]. E.g., if wt is not accurate a line search can step a long way toward a suboptimal policy and get stuck because the soft-max function used to generate action distributions
has a minima for all large parameter values. Methods for preventing line search from going too far
typically counteract the advantages of using one at all. Thus, we propose the online version of NAC
in Algorithm 1, making a small parameter update at every time step which potentially accelerates
convergence because the policy can improve at every step. The main difference between the batch
and online versions is the avoidance of the O(d3 ) matrix inversion (although Cholesky factorisation
can help) for solving AT [w| , v| ]| = bT , where d = |?| + |o|. Instead, lines 12 to 15 implement a
trick used for Kalman filters: the Sherman-Morrison update of a matrix inverse [2]:
A?1 zy| A?1
.
(A + zy| )?1 = A?1 ?
1 + y| A?1 z
In other words, we always work in the inverse space. The update is O(d2 ), which is still expensive
compared to vanilla PG approaches. Faster methods would be possible if the rank one update of A
were of the restricted form A + zz| . NAC makes up for expensive computations by requiring orders
of magnitude fewer steps to converge to a good policy. We retain the aggregation of A,1 using
a rolling average implemented by the ? weighting (line 12); however we only use instantaneous
estimates gt to avoid multiple parameter updates based on the same rewards.
The O LPOMDP, or vanilla, algorithm [12] produces per step gradient estimates from the discounted
sum of ?? log Pr(at |ot , ?t ), multiplied by the instant reward rt . This is exactly wt if we set At := I
for all t. Other PG approaches [9, 10] are also specialisations of NAC. As the simplest and fastest
infinite-horizon algorithm we used O LPOMDP for comparison.
5
Policy-Gradient for Traffic Control
PG methods are particularly appealing for traffic control for several reasons. The local search,
Monte-Carlo gradient estimates, and local rewards improve scalability. The (almost) direct mapping
1
This means that At is a mixture of the Fisher matrices for many parameter values. This is unappealing
and we expected a discounted average to yield an At that better represents ?t . However, this performed poorly,
perhaps because decaying ? mitigates ill-conditioning in the Fisher matrix as parameter values grow [10].
Alg. 1: An Online Natural Actor-Critic
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
t = 1, A?1
1 = I, ?1 = [0], z1 = [0]
=step size, ?=Critic discount, ?=Actor discount
Get observation o1
while not converged do
Sample action at ? Pr(?|ot , ?t )
zt = ?zt?1 + [?? log Pr(at |ot , ?t )| , o|t ]|
Do action at
Get reward rt
gt = rt zt
Get observation ot+1
yt = [?? log Pr(?|ot , ?t )| , o|t ]| ? ?[0| , o|t+1 ]|
?t = 1 ? 1t
ut = (1 ? ?t )A?1
t?1 zt
q|t = yt| A?1
t?1
|
ut qt
1
A?1 ? 1.0+q
|
?t t?1
t zt
g
[wt| , vt| ]| = A?1
t
t
?t+1 = ?t + wt
=
A?1
t
16:
17:
18:
t?t+1
19: end while
Fig. 2: Convergence properties of NAC
(top) compared to O LPOMDP (bottom)
over 30 runs in the Offset scenario.
of raw sensor information to controls means that we avoid modelling, and also creates a controller
that can react immediately to fluctuations in traffic density. We emphasise that, while SCATS has
to adapt a single behaviour slowly, our policy is a rich mapping of sensor data to many different
behaviours. Neighbouring intersection controllers cooperate through common observations.
5.1
Our Simulator
We implemented a simple traffic simulation system, aiming at maximum simulation speed rather
than at an accurate model of traffic flow. We did, however, implement a realistic control interface.
We modelled the phase control protocol and sensor system based on information from the Sydney
traffic authority. Given that the learning algorithm does not depend directly on a model of the system,
just the ability to interact with it, our controller can be plugged into a more accurate simulation
without modification. Our simplifying assumptions include: all vehicles move at uniform speed;
road length is a multiple of the distance cars travel in one step; we ignore interactions between cars
or the relative positions of cars within one road segment except in intersection queues.
Roads are placed in a sparse grid, and intersections may be defined at grid nodes. Every intersection
has two queues per incoming road: one queue for right turns and one for going straight or left.
Vehicles drive on the left. Every vehicle has a destination intersection that it navigates to via a
shortest path. If a driver can select either intersection queue without changing route distance, they
choose the one that is currently green, or has the fewest cars in the queue. This models the adaption
of drivers to control policies. To account for the gap between a phase ending and the next starting
(inter-green time), and the fact that cars start up slowly, we restrict the number of cars that pass
through an intersection in the first green step. This factor deters strategies that flip rapidly between
phases. We represent saturated traffic conditions through a road capacity parameter. Cars are only
allowed to move into the next road segment if the number of cars there does not exceed 20.
5.2
The Control Architecture
Commonly intersections have 2 to 6 phases. Ours have 4 phases: for traffic coming from east or west
(EW) straight and left turns, EW right turns, north/south (NS) straight and left turns, and NS right
turns. At each time step (corresponding to about 5 seconds real-time) the controller decides which
phase to activate in the next step. We do not restrict the order of phases, but to ensure a reasonable
policy we enforce the constraint that all phases must be activated at least once within 16 time steps.
The controller input for intersection i is ot,i , constructed as follows: Cycle duration: 16 bits, where the nth bit is on in the nth
step of the cycle, supporting time based decisions like offsets.
Current phase: 4 bits, indicating the previous phase. Current
phase duration: 5 bits, indicating that we have spent no more
than 1, 2, 4, 8 or 13 continuous time steps in the current phase.
Phase durations: 5 bits per phase, in the same format as cur- Fig. 3: The intersection model,
rent duration, counting the total time spent in each phase in the showing 2 phases and detectors.
current cycle. Detector active: 8 bits for the 8 loop sensors indicating whether a car is waiting. Detector history: 3 bits per loop sensor, indicating a saturation level
of more than 0, more than half capacity, or capacity, in the current cycle. Neighbour information:
2 bits, giving a delayed comparison of the flows from neighbouring intersections, indicating where
traffic is expected from.
?t,i over the
The controller maps observations ot,i for intersection i to a probability distribution a
P phases using a linear approximator with outputs xt,i and the soft-max function. Let ?i be the
P ? |ot,i | matrix of parameters for intersection i. We additionally define U(p) as the unit vector
with a 1 in row p. Thus, assuming exp(xt,i ) is element-wise exponentiation
xt,i = ?i ot,i ,
?t,i = PP
a
exp(xt,i )
p=1
exp(xt,i (p))
?t,i )o|t,i ,
??i log Pr(at,i = p|ot,i , ?i ) = (U(p) ? a
,
?t,i (p);
Pr(at,i = p|ot,i , ?i ) = a
?? log Pr(at |ot , ?) = [?|?1 , . . . , ?|?i ]| .
We implemented two baseline controllers for comparison: (1) a uniform controller giving equal
length to all phases; (2) A SCATS inspired adaptive controller called S AT that tries to achieve a
saturation of 90% (thought to be used by SCATS) for all phases. The exact details of SCATS are
not available. We aimed to recreate just the adaptive parts of SCATS. It updates the policy once
per cycle depending on the current flows [7]. It does not implement the hand-tuned elements of the
SCATS controller, such as offsets between neighbouring intersections.
6
Experiments
Our first 4 experiments demonstrate scenarios where we expect PG learning to outperform the adaptive S AT controller. The 5th experiment is a large scale experiment where we had no particular prior
reason to expect PG to outperform S AT.
Fluctuating. This scenario focuses on an intersection in the centre of a crossroads. The traffic
volume entering the system on the NS and EW traffic axes is proportional to a sine and cosine
function of the time, respectively. Thus the optimal policy at the centre intersection also oscillates
with the traffic volume. This scenario is realistic because upstream intersections release periodic
bursts of traffic, which then disperse as they travel along the road. SCATS is known to adapt too
slowly to deal well with such situations. The road length is 3 units, and vehicles travel through 3
intersections and along 4 roads, leading to an optimal travel time of 12. On average 3 cars enter the
system per time step from each direction.
Our results quote the average travel time (TT). Tab. 1 shows that NAC and O LPOMDP both improve
upon the uniform controller and S AT. The two PG algorithms get similar results across all scenarios.
However, Tab. 2 shows that NAC does so in up to 3 orders of magnitude fewer learning steps, but
sometimes requires more CPU time. In a real deployment NAC would be able to keep up with realtime, thus we are much more concerned about reducing learning steps. The tables quote a single
run with tuned parameters. To check the reliability of convergence and compare the properties of
the two algorithms, Fig. 2 displays the results of 30 runs for both algorithms in one of our scenarios.
We also ran the original NAC algorithm, using batch estimates of the direction in a line search to
test whether an online NAC is advantageous. We were able to construct a line search that converged
faster than online NAC, but always to a significantly worse policy (TT 23 instead of 14.3); or a
line search that reached the same policy, but no faster than online. We analysed the sensitivity of
training to the removal of observation features. Performance was degraded after removing any set
of observations. Removing multiple observations caused a smooth degradation of the policy [7].
Burst. Intersection controllers can learn to ?cooperate? by using common observations. In this
scenario, we make use of only the neighbours feature in the observations, so the controller must use
the detector counts of its neighbours to anticipate traffic. The road network is the same as in the
Fluctuating scenario. A steady stream of 1 car per step travels EW, so that it is usually good for
the centre controller to give maximum time to the EW-straight phase. With the small probability of
0.02 per step, we input a group of 15 cars from the north, travelling south. When this happens, the
controller should interrupt its normal policy and give more time to the NS-straight phase.
Table 1 shows that both algorithms learn a good policy in terms of travel time. When examining the
learned policies, we noted that the centre intersection controller had indeed learned to switch to the
NS-straight phase just in time before the cars arrive, something that S AT and SCATS cannot do.
Offset. Many drivers have been frustrated by driving along a main street, to be constantly interrupted
by red lights. This scenario demonstrates learning an offset between neighbouring intersections, a
feature that needs to be hand-tuned in SCATS. We model one arterial with 3 controlled intersections,
neglecting any traffic flowing in from side roads. The road length is two units for 4 roads, resulting
in an optimal travel time of 8. We restricted the observations to the cycle duration, meaning that our
controllers learned from time information only.
NAC learned an optimal policy in this scenario. S AT performed badly because it had no means of
implementing an offset. We discovered, however, that learning an optimal policy is difficult, e.g. we
failed for a road length of 3 (given limited time). Learning is hard because intersection n + 1 can
only begin to learn when intersection n has already converged to an optimal policy.
Adaptive Driver. In this scenario, local reward optimisation fails to find the global optimum, so we
use the number of cars in the system as the global reward (which minimises the average travel time
assuming constant input of cars). This reward is hard to estimate in the real world, but we want to
demonstrate the ability of the system to learn cooperative policies using a global reward. Like the
previous crossroads scenarios, we have NS and EW streams that interact only at a central intersection
(D, in Fig. 4). The streams have the same volume, so the optimal policy splits time uniformly
between the NW-straight and EW-straight phases. An additional stream of cars is generated in the
south-west corner, at Intersection H, and travels diagonally east to Intersection E. Two equally short
routes are available by going straight, or turning east at Intersection F. However, cars that turn east
to join the northbound traffic flow must then turn east again at Intersection D, forcing the controller
of that intersection to devote time to a third NS-right phase and forcing the main volume of traffic
to pause. The optimal strategy is actually to route all cars north from Intersection F, so they join the
main eastbound traffic flow. This scenario relies on a driver model that prefers routes with shorter
waiting times at intersections among routes of equal distance. Our observations for this scenario
consist only of the phase durations, informing the controller how much time it has spent in each
phase during the current cycle.
Although the average TT of the PG algorithms was only slightly better than S AT?s, their policies
were radically different. S AT routed cars equally north and east at the critical intersection F, whilst
the PG algorithms routed cars north. In this scenario, a slightly larger volume of of vehicles made
S AT cause permanent traffic jams, while the PG algorithms still found the correct policy. In this
scenario PG even beat our hand-coded optimal deterministic policy because it used stochasticity to
find a ?mixed? policy, giving phases a fractional number of steps on average.
Large Scale Optimisation. In a 10 ? 10 intersection network, perhaps modelling a central business district, each node potentially produces two cars at each time step according to a randomly
initialised probability between 0 and 0.25. The two destinations for the cars from each source are
initialised randomly, but stay fixed during the experiment to generate some consistent patterns within
the network. Driver adaption and stochastic route choices also create some realistic variance.
We used local rewards and all observations. O LPOMDP gave an average travel time improvement of
20% over S AT even though this scenario was not tailored for our controller. Such savings in a real
city would be more than significant. NAC required around 52,000 iterations to improve on S AT.
This would be 3 days of experience in the real world to achieve equivalent performance.
Tab. 1: Comparison of best travel times (TT)for all methods
and all scenarios. Evaluation for the PG algorithms was over
100,000 steps, after reaching steady state.
Scen.
Random Unif. S AT NAC O LPOMDP
Fluct
250.0
102.0 21.5
14.3 13.4
197.0
35.0 18.4
13.4 13.5
Burst
Offset
17.9
15.0 12.0
8.0
8.0
A.D.
251.0
74.2 17.2
15.8 16.0
60.5
54.7 35.1
29.8 27.9
100 int.
Fig. 4: Adaptive driver scenario.
Tab. 2: Optimisation parameters and run times for all scenarios for the PG algorithms. Optimisation was
performed for t steps. ?Secs? is wall-clock time.
Scen.
NAC
TT
t
secs
Fluct.
14.3 4.5 ? 106
860,549
10?5
13.4 4.4 ? 106
25,454
10?4
Burst
Offset
8.0 2.1 ? 106
1,973 5 ? 10?5
A.D.
15.8 9.3 ? 107
867,267
10?7
5
100 int. 29.8 2.9 ? 10 1,077,151
10?4
7
?
0.9
0.9
0.98
0.98
0.9
?
0.95
0.95
0.9
0.95
0.95
TT
13.4
13.5
8.0
16.0
27.9
t
1.1 ? 109
9.7 ? 108
6.3 ? 108
2.2 ? 109
3.0 ? 108
O LPOMDP
secs
491,298
10?3
35,572
10?4
8,546 5 ? 10?6
807,496
10?6
1,029,428
10?5
?
0.9
0.9
0.98
0.98
0.9
Conclusion
We described an online stochastic ascent policy-gradient procedure based on the natural actor-critic
algorithm. We used it in a distributed road traffic problem to demonstrate where machine learning
could improve upon existing proprietary traffic controllers. Our future work will apply this approach to realistic and approved simulators. Improved algorithms will be developed to cope with
the increased noise and temporal credit assignment problem inherent in realistic systems.
Acknowledgments
National ICT Australia is funded by the Australian Government?s Backing Australia?s Ability program and the Centre of Excellence program.
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