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1,200 | 2,094 | Kernel Feature Spaces and
Nonlinear Blind Source Separation
Stefan Harmeling1?, Andreas Ziehe1 , Motoaki Kawanabe1, Klaus-Robert M?ller1,2
1
Fraunhofer FIRST.IDA, Kekul?str. 7, 12489 Berlin, Germany
2
University of Potsdam, Department of Computer Science,
August-Bebel-Strasse 89, 14482 Potsdam, Germany
{harmeli,ziehe,kawanabe,klaus}@first.fhg.de
Abstract
In kernel based learning the data is mapped to a kernel feature space of
a dimension that corresponds to the number of training data points. In
practice, however, the data forms a smaller submanifold in feature space,
a fact that has been used e.g. by reduced set techniques for SVMs. We
propose a new mathematical construction that permits to adapt to the intrinsic dimension and to find an orthonormal basis of this submanifold.
In doing so, computations get much simpler and more important our
theoretical framework allows to derive elegant kernelized blind source
separation (BSS) algorithms for arbitrary invertible nonlinear mixings.
Experiments demonstrate the good performance and high computational
efficiency of our kTDSEP algorithm for the problem of nonlinear BSS.
1 Introduction
In a widespread area of applications kernel based learning machines, e.g. Support Vector
Machines (e.g. [19, 6]) give excellent solutions. This holds both for problems of supervised
and unsupervised learning (e.g. [3, 16, 12]). The general idea is to map the data x i (i =
1, . . . , T ) into some kernel feature space F by some mapping ? : <n ? F. Performing
a simple linear algorithm in F, then corresponds to a nonlinear algorithm in input space.
Essential ingredients to kernel based learning are (a) VC theory that can provide a relation
between the complexity of the function class in use and the generalization error and (b) the
famous kernel trick
k(x, y) = ?(x) ? ?(y),
(1)
which allows to efficiently compute scalar products. This trick is essential if e.g. F is
an infinite dimensional space. Note that even though F might be infinite dimensional the
subspace where the data lies is maximally T -dimensional. However, the data typically
forms an even smaller subspace in F (cf. also reduced set methods [15]). In this work
we therefore propose a new mathematical construction that allows us to adapt to the intrinsic dimension and to provide an orthonormal basis of this submanifold. Furthermore,
this makes computations much simpler and provides the basis for a new set of kernelized
learning algorithms.
?
To whom correspondence should be addressed.
To demonstrate the power of our new framework we will focus on the problem of nonlinear
BSS [2, 18, 9, 10, 20, 11, 13, 14, 7, 17, 8] and provide an elegant kernel based algorithm
for arbitrary invertible nonlinearities. In nonlinear BSS we observe a mixed signal of the
following structure
xt = f (st ),
(2)
where xt and st are n ? 1 column vectors and f is a possibly nonlinear function from < n to
<n . In the special case where f is an n?n matrix we retrieve standard linear BSS (e.g. [8, 4]
and references therein). Nonlinear BSS has so far been only applied to industrial pulp data
[8], but a large class of applications where nonlinearities can occur in the mixing process
are conceivable, e.g. in the fields of telecommunications, array processing, biomedical data
analysis (EEG, MEG, EMG, . . .) and acoustic source separation. Most research has so far
centered on post-nonlinear models, i.e.
xt = f (Ast ),
(3)
where A is a linear mixing matrix and f is a post-nonlinearity that operates componentwise.
Algorithmic solutions of eq.(3) have used e.g. self-organizing maps [13, 10], extensions of
GTM [14], neural networks [2, 11] or ensemble learning [18] to unfold the nonlinearity
f . Also a kernel based method was tried on very simple toy signals; however with some
stability problems [7]. Note, that all existing methods are of high computational cost and
depending on the algorithm are prone to run into local minima. In our contribution to the
general invertable nonlinear BSS case we apply a standard BSS technique [21, 1] (that
relies on temporal correlations) to mapped signals in feature space (cf. section 3). This is
not only mathematically elegant (cf. section 2), but proves to be a remarkably stable and
efficient algorithm with high performance, as we will see in the experiments on nonlinear
mixtures of toy and speech data (cf. section 4). Finally, a conclusion is given in section 5.
2 Theory
An orthonormal basis for a subspace in F
In order to establish a linear problem in feature space that corresponds to some nonlinear problem in input space we need to specify how to map inputs x 1 , . . . , xT ? <n into
the feature space F and how to handle its possibly high dimensionality. In addition to
the inputs, consider some further points v1 , . . . , vd ? <n from the same space, that will
later generate a basis in F. Alternatively, we could use kernel PCA [16]. However, in
this paper we concentrate on a different method. Let us denote the mapped points by
?x := [?(x1 ) ? ? ? ?(xT )] and ?v := [?(v1 ) ? ? ? ?(vd )]. We assume that the columns of
?v constitute a basis of the column space1 of ?x , which we note by
span(?v ) = span(?x ) and rank(?v ) = d.
(4)
Moreover, ?v being a basis implies that the matrix ?>
v ?v has full rank and its inverse
exists. So, now we can define an orthonormal basis
1
?2
? := ?v (?>
v ?v )
(5)
the column space of which is identical to the column space of ?v . Consequently this basis
? enables us to parameterize all vectors that lie in the column space of ? x by some vectors
PT
in <d . For instance for vectors i=1 ??i ?(xi ), which we write more compactly as ?x ?? ,
and ?x ?? in the column space of ?x with ?? and ?? in <T there exist ?? and ?? in <d
such that ?x ?? = ??? and ?x ?? = ??? . The orthonormality implies
>
> >
>
?>
? ?x ?x ?? = ?? ? ??? = ?? ??
(6)
input space
<
n
feature space
span(?)
F
parameter space
<d
Figure 1: Input data are mapped to some submanifold of F which is in the span of some ddimensional orthonormal basis ?. Therefore these mapped points can be parametrized in <d . The
linear directions in parameter space correspond to nonlinear directions in input space.
which states the remarkable property that the dot product of two linear combinations of the
columns of ?x in F coincides with the dot product in <d . By construction of ? (cf. eq.(5))
the column space of ?x is naturally isomorphic (as vector spaces) to <d . Moreover, this
isomorphism is compatible with the two involved dot products as was shown in eq.(6). This
implies that all properties regarding angles and lengths can be taken back and forth between
the column space of ?x and <d . The space that is spanned by ? is called parameter space.
Figure 1 pictures our intuition: Usually kernel methods parameterize the column space of
?x in terms of the mapped patterns {?(xi )} which effectively corresponds to vectors in
<T . The orthonormal basis from eq.(5), however enables us to work in < d i.e. in the span
of ?, which is extremely valuable since d depends solely on the kernel function and the
dimensionality of the input space. So d is independent of T .
Mapping inputs
Having established the machinery above, we will now show how to map the input data to
the right space. The expressions
>
(?>
v ?v )ij = ?(vi ) ?(vj ) = k(vi , vj )
with i, j = 1 . . . d
are the entries of a real valued d ? d matrix ?>
v ?v that can be effectively calculated using
the kernel trick and by construction of v1 , . . . , vd , it has full rank and is thus invertible.
Similarly we get
>
(?>
v ?x )ij = ?(vi ) ?(xj ) = k(vi , xj ) with i = 1 . . . d,
j = 1...T,
which are the entries of the real valued d ? T matrix ?>
v ?x . Using both matrices we
compute finally the parameter matrix
1
?2 >
?v ?x
?x := ?> ?x = (?>
v ?v )
(7)
1
The column space of ?x is the space that is spanned by the column vectors of ?x , written
span(?x ).
1
?2
which is also a real valued d ? T matrix; note that (?>
is symmetric. Regarding
v ?v )
computational costs, we have to evaluate the kernel function O(d 2 ) + O(dT ) times and
eq.(7) requires O(d3 ) multiplications; again note that d is much smaller than T . Furthermore storage requirements are cheaper as we do not have to hold the full T ? T kernel
matrix but only a d ? T matrix. Also, kernel based algorithms often require centering in
F, which in our setting is equivalent to centering in <d . Fortunately the latter can be done
quite cheaply.
Choosing vectors for the basis in F
So far we have assumed to have points v1 , . . . , vd that fulfill eq.(4) and we presented
the beneficial properties of our construction. In fact, v1 , . . . , vd are roughly analogous
to a reduced set in the support vector world [15]. Note however that often we can only
approximately fulfill eq.(4), i.e.
span(?v ) ? span(?x ).
(8)
In this case we strive for points that provide the best approximation.
Obviously d is finite since it is bounded by T , the number of inputs, and by the dimensionality of the feature space. Before formulating the algorithm we define the function rk(n)
for numbers n by the following process: randomly pick n points v 1 , . . . , vn from the inputs
and compute the rank of the corresponding n ? n matrix ?>
v ?v . Repeating this random
sampling process several times (e.g. 100 times) stabilizes this process in practice. Then we
denote by rk(n) the largest achieved rank; note that rk(n) ? n. Using this definition we
can formulate a recipe to find d (the dimension of the subspace of F): (1) start with a large
d with rk(d) < d. (2) Decrement d by one as long as rk(d) < d holds. As soon as we
have rk(d) = d we found the d. Choose v1 , . . . , vd as the vectors that achieve rank d. As
an alternative to random sampling we have also employed k-means clustering with similar
results.
3 Nonlinear blind source separation
To demonstrate the use of the orthonormal basis in F, we formulate a new nonlinear BSS
algorithm based on TDSEP [21]. We start from a set of points v1 , . . . , vd , that are provided
by the algorithm from the last section such that eq.(4) holds. Next, we use eq.(7) to compute
1
?2 >
?x [t] := ?> ?(x[t]) = (?>
?v ?(x[t])
v ?v )
? <d .
Hereby we have transformed the time signals x[t] from input space to parameter space signals ?x [t] (cf. Fig.1). Now we apply the usual TDSEP ([21]) that relies on simultaneous
diagonalisation techniques [5] to perform linear blind source separation on ? x [t] to obtain
d linear directions of separated nonlinear components in input space. This new algorithm is
denoted as kTDSEP (kernel TDSEP); in short, kTDSEP is TDSEP on the parameter space
defined in Fig.1. A key to the success of our algorithm are the time correlations exploited
by TDSEP; intuitively they provide the ?glue? that yields the coherence for the separated
signals. Note that for a linear kernel functions the new algorithm performs linear BSS.
Therefore linear BSS can be seen as a special case of our algorithm.
Note that common kernel based algorithms which do not use the d-dimensional orthonormal basis will run into computational problems. They need to hold and compute with a
kernel matrix that is T ? T instead of d ? T with T d in BSS problems. A further
problem is that manipulating such a T ? T matrix can easily become unstable. Moreover
BSS methods typically become unfeasible for separation problems of dimension T .
!
Figure 2: Scatterplot of x1 vs x2 for nonlinear mixing and demixing (upper left and right) and linear
demixing and true source signals (lower left and right). Note, that the nonlinear unmixing agrees very
nicely with the scatterplot of the true source signal.
4 Experiments
In the first experiment the source signals s[t] = [s1 [t] s2 [t]]> are a sinusoidal and a sawtooth signal with 2000 samples each. The nonlinearly mixed signals are defined as (cf. Fig.2
upper left panel)
x1 [t]
x2 [t]
= exp(s1 [t]) ? exp(s2 [t])
= exp(?s1 [t]) + exp(?s2 [t]).
A dimension d = 22 of the manifold in feature space was obtained by kTDSEP using
a polynomial kernel k(x, y) = (x> y + 1)6 by sampling from the inputs. The basisgenerating vectors v1 , . . . , v22 are shown as big dots in the upper left panel of Figure
2. Applying TDSEP to the 22 dimensional mapped signals ?x [t] we get 22 components
in parameter space. A scatter plot with the two components that best match the source
signals are shown in the right upper panel of Figure 2. The left lower panel also shows for
comparison the two components that we obtained by applying linear TDSEP directly to the
mixed signals x[t]. The plots clearly indicate that kTDSEP has unfolded the nonlinearity
successfully while the linear demixing algorithm failed.
In a second experiment two speech signals (with 20000 samples, sampling rate 8 kHz) that
are nonlinearly mixed by
x1 [t]
= s1 [t] + s32 [t]
x2 [t]
= s31 [t] + tanh(s2 [t]).
This time we used a Gaussian RBF kernel k(x, y) = exp(?|x ? y|2 ). kTDSEP identified
d = 41 and used k-means clustering to obtain v1 , . . . , v41 . These points are marked as
?+? in the left panel of figure 4. An application of TDSEP to the 41 dimensional parameter
mixture
s1
s2
kTDSEP
TDSEP
x1
x2
u1
u2
u1
u2
0.56
0.63
0.72
0.46
0.89
0.04
0.07
0.86
0.09
0.31
0.72
0.55
Table 3: Correlation coefficients for the signals shown in Fig.4.
space yields nonlinear components whose projections to the input space are depicted in the
right lower panel. We can see that linear TDSEP (right middle panel) failed and that the
directions of best matching kTDSEP components closely resemble the sources.
To confirm this visual impression we calculated the correlation coefficients of the kTDSEP
and TDSEP solution to the source signals (cf. table 3). Clearly, kTDSEP outperforms the
linear TDSEP algorithm, which is of course what one expects.
5 Conclusion
Our work has two main contributions. First, we propose a new formulation in the field of
kernel based learning methods that allows to construct an orthonormal basis of the subspace
of kernel feature space F where the data lies. This technique establishes a highly useful
(scalar product preserving) isomorphism between the image of the data points in F and a
d-dimensional space <d . Several interesting things follow: we can construct a new set of
efficient kernel-based algorithms e.g. a new and eventually more stable variant of kernel
PCA [16]. Moreover, we can acquire knowledge about the intrinsic dimension of the data
manifold in F from the learning process.
Second, using our new formulation we tackle the problem of nonlinear BSS from the viewpoint of kernel based learning. The proposed kTDSEP algorithm allows to unmix arbitrary
invertible nonlinear mixtures with low computational costs. Note, that the important ingredients are the temporal correlations of the source signals used by TDSEP. Experiments on
toy and speech signals underline that an elegant solution has been found to a challenging
problem.
Applications where nonlinearly mixed signals can occur, are found e.g. in the fields of
telecommunications, array processing, biomedical data analysis (EEG, MEG, EMG, . . .)
and acoustic source separation. In fact, our algorithm would allow to provide a softwarebased correction of sensors that have a nonlinear characteristics e.g. due to manufacturing
errors. Clearly kTDSEP is only one algorithm that can perform nonlinear BSS; kernelizing
other ICA algorithms can be done following our reasoning.
Acknowledgements The authors thank Benjamin Blankertz, Gunnar R?tsch, Sebastian
Mika for valuable discussions. This work was partly supported by the EU project (IST1999-14190 ? BLISS) and DFG (JA 379/9-1, MU 987/1-1).
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Figure 4: A highly nonlinear mixture of two speech signals: Scatterplot of x1 vs x2 and the waveforms of the true source signals (upper panel) in comparison to the
best matching linear and nonlinear separation results are shown in the middle and lower panel, respectively.
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1,201 | 2,095 | Prodding the ROC Curve: Constrained
Optimization of Classifier Performance
Michael C. Mozer*+, Robert Dodier*, Michael D. Colagrosso*+,
C?sar Guerra-Salcedo*, Richard Wolniewicz*
* Advanced Technology Group + Department of Computer Science
Athene Software
University of Colorado
2060 Broadway
Campus Box 430
Boulder, CO 80302
Boulder, CO 80309
Abstract
When designing a two-alternative classifier, one ordinarily aims to maximize the
classifier?s ability to discriminate between members of the two classes. We
describe a situation in a real-world business application of machine-learning
prediction in which an additional constraint is placed on the nature of the solution: that the classifier achieve a specified correct acceptance or correct rejection
rate (i.e., that it achieve a fixed accuracy on members of one class or the other).
Our domain is predicting churn in the telecommunications industry. Churn
refers to customers who switch from one service provider to another. We propose four algorithms for training a classifier subject to this domain constraint,
and present results showing that each algorithm yields a reliable improvement in
performance. Although the improvement is modest in magnitude, it is nonetheless impressive given the difficulty of the problem and the financial return that it
achieves to the service provider.
When designing a classifier, one must specify an objective measure by which the classifier?s performance is to be evaluated. One simple objective measure is to minimize the
number of misclassifications. If the cost of a classification error depends on the target and/
or response class, one might utilize a risk-minimization framework to reduce the expected
loss. A more general approach is to maximize the classifier?s ability to discriminate one
class from another class (e.g., Chang & Lippmann, 1994).
An ROC curve (Green & Swets, 1966) can be used to visualize the discriminative
performance of a two-alternative classifier that outputs class posteriors. To explain the
ROC curve, a classifier can be thought of as making a positive/negative judgement as to
whether an input is a member of some class. Two different accuracy measures can be
obtained from the classifier: the accuracy of correctly identifying an input as a member of
the class (a correct acceptance or CA), and the accuracy of correctly identifying an input
as a nonmember of the class (a correct rejection or CR). To evaluate the CA and CR rates,
it is necessary to pick a threshold above which the classifier?s probability estimate is interpreted as an ?accept,? and below which is interpreted as a ?reject??call this the criterion.
The ROC curve plots CA against CR rates for various criteria (Figure 1a). Note that as the
threshold is lowered, the CA rate increases and the CR rate decreases. For a criterion of 1,
the CA rate approaches 0 and the CR rate 1; for a criterion of 0, the CA rate approaches 1
0
0
correct rejection rate
20
40
60
80
100
100
(b)
correct rejection rate
20
40
60
80
(a)
0
20
40
60
80
100
correct acceptance rate
0
20
40
60
80
100
correct acceptance rate
FIGURE 1. (a) two ROC curves reflecting discrimination performance; the dashed curve
indicates better performance. (b) two plausible ROC curves, neither of which is clearly
superior to the other.
and the CR rate 0. Thus, the ROC curve is anchored at (0,1) and (1,0), and is monotonically nonincreasing. The degree to which the curve is bowed reflects the discriminative
ability of the classifier. The dashed curve in Figure 1a is therefore a better classifier than
the solid curve.
The degree to which the curve is bowed can be quantified by various measures such
as the area under the ROC curve or d?, the distance between the positive and negative distributions. However, training a classifier to maximize either the ROC area or d? often
yields the same result as training a classifier to estimate posterior class probabilities, or
equivalently, to minimize the mean squared error (e.g., Frederick & Floyd, 1998). The
ROC area and d? scores are useful, however, because they reflect a classifier?s intrinsic
ability to discriminate between two classes, regardless of how the decision criterion is set.
That is, each point on an ROC curve indicates one possible CA/CR trade off the classifier
can achieve, and that trade off is determined by the criterion. But changing the criterion
does not change the classifier?s intrinsic ability to discriminate.
Generally, one seeks to optimize the discrimination performance of a classifier. However, we are working in a domain where overall discrimination performance is not as critical as performance at a particular point on the ROC curve, and we are not interested in the
remainder of the ROC curve. To gain an intuition as to why this goal should be feasible,
consider Figure 1b. Both the solid and dashed curves are valid ROC curves, because they
satisfy the monotonicity constraint: as the criterion is lowered, the CA rate does not
decrease and the CR rate does not increase. Although the bow shape of the solid curve is
typical, it is not mandatory; the precise shape of the curve depends on the nature of the
classifier and the nature of the domain. Thus, it is conceivable that a classifier could produce a curve like the dashed one. The dashed curve indicates better performance when the
CA rate is around 50%, but worse performance when the CA rate is much lower or higher
than 50%. Consequently, if our goal is to maximize the CR rate subject to the constraint
that the CA rate is around 50%, or to maximize the CA rate subject to the constraint that
the CR rate is around 90%, the dashed curve is superior to the solid curve. One can imagine that better performance can be obtained along some stretches of the curve by sacrificing performance along other stretches of the curve. Note that obtaining a result such as the
dashed curve requires a nonstandard training algorithm, as the discrimination performance
as measured by the ROC area is worse for the dashed curve than for the solid curve.
In this paper, we propose and evaluate four algorithms for optimizing performance in
a certain region of the ROC curve. To begin, we explain the domain we are concerned with
and why focusing on a certain region of the ROC curve is important in this domain.
1 OUR DOMAIN
Athene Software focuses on predicting and managing subscriber churn in the telecommunications industry (Mozer, Wolniewicz, Grimes, Johnson, & Kaushansky, 2000). ?Churn?
refers to the loss of subscribers who switch from one company to the other. Churn is a significant problem for wireless, long distance, and internet service providers. For example,
in the wireless industry, domestic monthly churn rates are 2?3% of the customer base.
Consequently, service providers are highly motivated to identify subscribers who are dissatisfied with their service and offer them incentives to prevent churn.
We use techniques from statistical machine learning?primarily neural networks and
ensemble methods?to estimate the probability that an individual subscriber will churn in
the near future. The prediction of churn is based on various sources of information about a
subscriber, including: call detail records (date, time, duration, and location of each call,
and whether call was dropped due to lack of coverage or available bandwidth), financial
information appearing on a subscriber?s bill (monthly base fee, additional charges for
roaming and usage beyond monthly prepaid limit), complaints to the customer service
department and their resolution, information from the initial application for service (contract details, rate plan, handset type, credit report), market information (e.g., rate plans
offered by the service provider and its competitors), and demographic data.
Churn prediction is an extremely difficult problem for several reasons. First, the business environment is highly nonstationary; models trained on data from a certain time
period perform far better with hold-out examples from that same time period than examples drawn from successive time periods. Second, features available for prediction are
only weakly related to churn; when computing mutual information between individual
features and churn, the greatest value we typically encounter is .01 bits. Third, information
critical to predicting subscriber behavior, such as quality of service, is often unavailable.
Obtaining accurate churn predictions is only part of the challenge of subscriber
retention. Subscribers who are likely to churn must be contacted by a call center and
offered some incentive to remain with the service provider. In a mathematically principled
business scenario, one would frame the challenge as maximizing profitability to a service
provider, and making the decision about whether to contact a subscriber and what incentive to offer would be based on the expected utility of offering versus not offering an
incentive. However, business practices complicate the scenario and place some unique
constraints on predictive models. First, call centers are operated by a staff of customer service representatives who can contact subscribers at a fixed rate; consequently, our models
cannot advise contacting 50,000 subscribers one week, and 50 the next. Second, internal
business strategies at the service providers constrain the minimum acceptable CA or CR
rates (above and beyond the goal of maximizing profitability). Third, contracts that Athene
makes with service providers will occasionally call for achieving a specific target CA and
CR rate. These three practical issues pose formal problems which, to the best of our
knowledge, have not been addressed by the machine learning community.
The formal problems can be stated in various ways, including: (1) maximize the CA
rate, subject to the constraint that a fixed percentage of the subscriber base is identified as
potential churners, (2) optimize the CR rate, subject to the constraint that the CA rate
should be ?CA, (3) optimize the CA rate, subject to the constraint that the CR rate should
be ?CR, and finally?what marketing executives really want?(4) design a classifier that
has a CA rate of ?CA and a CR rate of ?CR. Problem (1) sounds somewhat different than
problems (2) or (3), but it can be expressed in terms of a lift curve, which plots the CA rate
as a function of the total fraction of subscribers identified by the model. Problem (1) thus
imposes the constraint that the solution lies at one coordinate of the lift curve, just as problems (2) and (3) place the constraint that the solution lies at one coordinate of the ROC
curve. Thus, a solution to problems (2) or (3) will also serve as a solution to (1). Although
addressing problem (4) seems most fanciful, it encompasses problems (2) and (3), and
thus we focus on it. Our goal is not altogether unreasonable, because a solution to problem
(4) has the property we characterized in Figure 1b: the ROC curve can suffer everywhere
except in the region near CA ?CA and CR ?CR. Hence, the approaches we consider will
trade off performance in some regions of the ROC curve against performance in other
regions. We call this prodding the ROC curve.
2 FOUR ALGORITHMS TO PROD THE ROC CURVE
In this section, we describe four algorithms for prodding the ROC curve toward a target
CA rate of ?CA and a target CR rate of ?CR.
2.1 EMPHASIZING CRITICAL TRAINING EXAMPLES
Suppose we train a classifier on a set of positive and negative examples from a class?
churners and nonchurners in our domain. Following training, the classifier will assign a
posterior probability of class membership to each example. The examples can be sorted by
the posterior and arranged on a continuum anchored by probabilities 0 and 1 (Figure 2).
We can identify the thresholds, ?CA and ?CR, which yield CA and CR rates of ?CA and
?CR, respectively. If the classifier?s discrimination performance fails to achieve the target
CA and CR rates, then ?CA will be lower than ?CR, as depicted in the Figure. If we can
bring these two thresholds together, we will achieve the target CA and CR rates. Thus, the
first algorithm we propose involves training a series of classifiers, attempting to make classifier n+1 achieve better CA and CR rates by focusing its effort on examples from classifier n that lie between ?CA and ?CR; the positive examples must be pushed above ?CR and
the negative examples must be pushed below ?CA. (Of course, the thresholds are specific
to a classifier, and hence should be indexed by n.) We call this the emphasis algorithm,
because it involves placing greater weight on the examples that lie between the two thresholds. In the Figure, the emphasis for classifier n+1 would be on examples e5 through e8.
This retraining procedure can be iterated until the classifier?s training set performance
reaches asymptote.
In our implementation, we define a weighting of each example i for training classifier
n, ? in . For classifier 1, ? i1 = 1 . For subsequent classifiers, ? in + 1 = ? in if example i is
not in the region of emphasis, or ? in + 1 = ? e ? in otherwise, where ?e is a constant, ?e > 1.
2.2 DEEMPHASIZING IRRELEVANT TRAINING EXAMPLES
The second algorithm we propose is related to the first, but takes a slightly different perspective on the continuum depicted in Figure 2. Positive examples below ?CA?such as
e2?are clearly the most dif ficult positive examples to classify correctly. Not only are they
the most difficult positive examples, but they do not in fact need to be classified correctly
to achieve the target CA and CR rates. Threshold ?CR does not depend on examples such
as e2, and threshold ?CA allows a fraction (1??CA) of the positive examples to be classified
incorrectly. Likewise, one can argue that negative examples above ?CR?such as e10 and
e11?need not be of concern. Essentially , the second algorithm, which we term thedeemphasis algorithm, is like the emphasis algorithm in that a series of classifiers are trained,
but when training classifier n+1, less weight is placed on the examples whose correct clas?CA
e1 e2 e3
0
e4
?CR
e5
e6
e7
e8
churn probability
e9
e10 e11 e12 e13
1
FIGURE 2. A schematic depiction of all training examples arranged by the classifier?s
posterior. Each solid bar corresponds to a positive example (e.g., a churner) and each grey bar
corresponds to a negative example (e.g., a nonchurner).
sification is unnecessary to achieve the target CA and CR rates for classifier n. As with the
emphasis algorithm, the retraining procedure can be iterated until no further performance
improvements are obtained on the training set. Note that the set of examples given emphasis by the previous algorithm is not the complement of the set of examples deemphasized
by the current algorithm; the algorithms are not identical.
In our implementation, we assign a weight to each example i for training classifier n,
? in . For classifier 1, ? i1 = 1 . For subsequent classifiers, ? in + 1 = ? in if example i is not
in the region of deemphasis, or ? in + 1 = ? d ? in otherwise, where ?d is a constant, ?d <1.
2.3 CONSTRAINED OPTIMIZATION
The third algorithm we propose is formulated as maximizing the CR rate while maintaining the CA rate equal to ?CA. (We do not attempt to simultaneously maximize the CA rate
while maintaining the CR rate equal to ?CR.) Gradient methods cannot be applied directly
because the CA and CR rates are nondifferentiable, but we can approximate the CA and
CR rates with smooth differentiable functions:
1
1
CA ( w, t ) = ------ ? ? ? ( f ( x i, w ) ? t )
CR ( w, t ) = ------- ? ? ? ( t ? f ( x i, w ) ) ,
P i?P
N i?N
where P and N are the set of positive and negative examples, respectively, f(x,w) is the
model posterior for input x, w is the parameterization of the model, t is a threshold, and ??
?1
is a sigmoid function with scaling parameter ?: ? ? ( y ) = ( 1 + exp ( ? ?y ) ) . The larger ?
is, the more nearly step-like the sigmoid is and the more nearly equal the approximations
are to the model CR and CA rates. We consider the problem formulation in which CA is a
constraint and CR is a figure of merit. We convert the constrained optimization problem
into an unconstrained problem by the augmented Lagrangian method (Bertsekas, 1982),
which involves iteratively maximizing an objective function
2
?
A ( w, t ) = CR ( w, t ) + ? CA ( w, t ) ? ? CA + --- CA ( w, t ) ? ? CA
2
with a fixed Lagrangian multiplier, ?, and then updating ? following the optimization step:
? ? ? + ? CA ( w *, t * ) ? ? CA , where w * and t * are the values found by the optimization
step. We initialize ? = 1 and fix ? = 1 and ? = 10 and iterate until ? converges.
2.4 GENETIC ALGORITHM
The fourth algorithm we explore is a steady-state genetic search over a space defined by
the continuous parameters of a classifier (Whitley, 1989). The fitness of a classifier is the
reciprocal of the number of training examples falling between the ?CA and ?CR thresholds.
Much like the emphasis algorithm, this fitness function encourages the two thresholds to
come together. The genetic search permits direct optimization over a nondifferentiable criterion, and therefore seems sensible for the present task.
3 METHODOLOGY
For our tests, we studied two large data bases made available to Athene by two telecommunications providers. Data set 1 had 50,000 subscribers described by 35 input features
and a churn rate of 4.86%. Data set 2 had 169,727 subscribers described by 51 input features and a churn rate of 6.42%. For each data base, the features input to the classifier were
obtained by proprietary transformations of the raw data (see Mozer et al., 2000). We chose
these two large, real world data sets because achieving gains with these data sets should be
more difficult than with smaller, less noisy data sets. Plus, with our real-world data, we
can evaluate the cost savings achieved by an improvement in prediction accuracy. We performed 10-fold cross-validation on each data set, preserving the overall churn/nonchurn
ratio in each split.
In all tests, we chose ? CR = 0.90 and ? CA = 0.50 , values which, based on our
past experience in this domain, are ambitious yet realizable targets for data sets such as
these. We used a logistic regression model (i.e., a no hidden unit neural network) for our
studies, believing that it would be more difficult to obtain improvements with such a
model than with a more flexible multilayer perceptron. For the emphasis and deemphasis
algorithms, models were trained to minimize mean-squared error on the training set. We
chose ?e = 1.3 and ?d = .75 by quick exploration. Because the weightings are cumulative
over training restarts, the choice of ? is not critical for either algorithm; rather, the magnitude of ? controls how many restarts are necessary to reach asymptotic performance, but
the results we obtained were robust to the choice of ?. The emphasis and deemphasis algorithms were run for 100 iterations, which was the number of iterations required to reach
asymptotic performance on the training set.
4 RESULTS
Figure 3 illustrates training set performance for the emphasis algorithm on data set 1. The
graph on the left shows the CA rate when the CR rate is .9, and the graph on the right show
the CR rate when the CA rate is .5. Clearly, the algorithm appears to be stable, and the
ROC curve is improving in the region around (?CA, ?CR).
Figure 4 shows cross-validation performance on the two data sets for the four prodding algorithms as well as for a traditional least-squares training procedure. The emphasis
and deemphasis algorithms yield reliable improvements in performance in the critical
region of the ROC curve over the traditional training procedure. The constrained-optimization and genetic algorithms perform well on achieving a high CR rate for a fixed CA
rate, but neither does as well on achieving a high CA rate for a fixed CR rate. For the constrained-optimization algorithm, this result is not surprising as it was trained asymmetrically, with the CA rate as the constraint. However, for the genetic algorithm, we have little
explanation for its poor performance, other than the difficulty faced in searching a continuous space without gradient information.
5 DISCUSSION
0.4
0.845
0.395
0.84
0.39
0.835
0.385
CR rate
CA rate
In this paper, we have identified an interesting, novel problem in classifier design which is
motivated by our domain of churn prediction and real-world business considerations.
Rather than seeking a classifier that maximizes discriminability between two classes, as
measured by area under the ROC curve, we are concerned with optimizing performance at
certain points along the ROC curve. We presented four alternative approaches to prodding
the ROC curve, and found that all four have promise, depending on the specific goal.
Although the magnitude of the gain is small?an increase of about .01 in the CR rate
given a target CA rate of .50?the impro vement results in significant dollar savings. Using
a framework for evaluating dollar savings to a service provider, based on estimates of subscriber retention and costs of intervention obtained in real world data collection (Mozer et
0.38
0.83
0.825
0.375
0.82
0.37
0.815
0.365
0.81
0
5
10 15 20 25 30 35 40 45 50
Iteration
0
5
10 15 20 25 30 35 40 45 50
Iteration
FIGURE 3. Training set performance for the emphasis algorithm on data set 1. (a) CA rate as
a function of iteration for a CR rate of .9; (b) CR rate as a function of iteration for a CA rate of
.5. Error bars indicate +/?1 standard error of the mean.
0.840
0.385
0.835
0.380
0.830
0.375
0.825
CR rate
Data set 1
CA rate
ISP
Test Set
0.390
0.370
0.820
0.365
0.815
0.360
0.810
0.355
0.805
0.350
0.800
std
emph
deemph
constr
GA
std
emph
deemph
constr
GA
std
emph
deemph
constr
GA
0.900
0.375
0.350
CR rate
Data set 2
0.875
CA rate
Wireless
Test Set
0.850
0.325
0.825
0.300
0.800
std
emph
deemph
constr
GA
FIGURE 4. Cross-validation performance on the two data sets for the standard training
procedure (STD), as well as the emphasis (EMPH), deemphasis (DEEMPH), constrained
optimization (CONSTR), and genetic (GEN) algorithms. The left column shows the CA rate for
CR rate .9; the right column shows the CR rate for CA rate .5. The error bar indicates one
standard error of the mean over the 10 data splits.
al., 2000), we obtain a savings of $11 per churnable subscriber when the (CA, CR) rates
go from (.50, .80) to (.50, .81), which amounts to an 8% increase in profitability of the
subscriber intervention effort.
These figures are clearly promising. However, based on the data sets we have studied, it is difficult to know whether another algorithm might exist that achieves even greater
gains. Interestingly, all algorithms we proposed yielded roughly the same gains when successful, suggesting that we may have milked the data for whatever gain could be had,
given the model class evaluated. Our work clearly illustrate the difficulty of the problem,
and we hope that others in the NIPS community will be motivated by the problem to suggest even more powerful, theoretically grounded approaches.
6 ACKNOWLEDGEMENTS
No white males were angered in the course of conducting this research. We thank Lian Yan and
David Grimes for comments and assistance on this research. This research was supported in part by
McDonnell-Pew grant 97-18, NSF award IBN-9873492, and NIH/IFOPAL R01 MH61549?01A1.
7 REFERENCES
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Frederick, E. D., & Floyd, C. E. (1998). Analysis of mammographic findings and patient history
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Green, D. M., & Swets, J. A. (1966). Signal detection theory and psychophysics. New York: Wiley.
Mozer, M. C., Wolniewicz, R., Grimes, D., Johnson, E., & Kaushansky, H. (2000). Maximizing revenue by predicting and addressing customer dissatisfaction. IEEE Transactions on Neural Networks, 11, 690?696.
Whitley, D. (1989). The GENITOR algorithm and selective pressure: Why rank-based allocation of
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1,202 | 2,096 | Bayesian time series classification
Peter Sykacek
Department of Engineering Science
University of Oxford
Oxford, OX1 3PJ, UK
[email protected]
Stephen Roberts
Department of Engineering Science
University of Oxford
Oxford, OX1 3PJ, UK
[email protected]
Abstract
This paper proposes an approach to classification of adjacent segments
of a time series as being either of classes. We use a hierarchical model
that consists of a feature extraction stage and a generative classifier which
is built on top of these features. Such two stage approaches are often used
in signal and image processing. The novel part of our work is that we link
these stages probabilistically by using a latent feature space. To use one
joint model is a Bayesian requirement, which has the advantage to fuse
information according to its certainty.
The classifier is implemented as hidden Markov model with Gaussian
and Multinomial observation distributions defined on a suitably chosen
representation of autoregressive models. The Markov dependency is motivated by the assumption that successive classifications will be correlated. Inference is done with Markov chain Monte Carlo (MCMC) techniques. We apply the proposed approach to synthetic data and to classification of EEG that was recorded while the subjects performed different
cognitive tasks. All experiments show that using a latent feature space
results in a significant improvement in generalization accuracy. Hence
we expect that this idea generalizes well to other hierarchical models.
1 Introduction
Many applications in signal or image processing are hierarchical in the sense that a probabilistic model is built on top of variables that are the coefficients of some feature extraction
technique. In this paper we consider a particular problem of that kind, where a Gaussian
and Multinomial observation hidden Markov model (GMOHMM) is used to discriminate
coefficients of an Auto Regressive (AR) process as being either of
classes. Bayesian
inference is known to give reasonable results when applied to AR models ([RF95]). The
situation with classification is similar, see for example the seminal work by [Nea96] and
[Mac92]. Hence we may expect to get good results if we apply Bayesian techniques to
both stages of the decision process separately. However this is suboptimal since it meant
to establish a no probabilistic link between feature extraction and classification. Two arguments suggest the building of one probabilistic model which combines feature extraction
and classification:
Since there is a probabilistic link, the generative classifier acts as a prior for fea-
ture extraction. The advantage of using this prior is that it naturally encodes our
knowledge about features as obtained from training data and other sensors. Obviously this is the only setup that is consistent with Bayesian theory ([BS94]).
Since all inferences are obtained from marginal distributions, information is combined according to its certainty. Hence we expect to improve results since information from different sensors is fused in an optimal manner.
2 Methods
2.1 A Gaussian and Multinomial observation hidden Markov model
As we attempt to classify adjacent segments of a time series, it is very likely that we find
correlations between successive class labels. Hence our model has a hidden Markov model
([RJ86]) like architecture, with diagonal Gaussian observation models for continuous variables and Multinomial observation models for discrete variables. We call the architecture a Gaussian and Multinomial observation hidden Markov model or GMOHMM for
short. Contrary to the classical approach, where each class is represented by its own trained
HMM, our model has class labels which are child nodes of the hidden state variables. Figure 1 shows the directed acyclic graph (DAG) of our model. We use here the convention
found in [RG97], where circular nodes are latent and square nodes are observed variables.
2.1.1 Quantities of interest
We regard all variables in the DAG that represent the probabilistic model of the time series
as quantities of interest. These are the hidden states, , the variables of the latent feature
space, , and , the class labels, , the sufficient statistics of the AR process,
,
and the segments of the time series, . The DAG shows the observation model only for
the
-th state. We have
latent feature variables, , which represent the coefficients of
the preprocessing model for of the
-th segment at sensor
. The state conditional distributions, , are modeled by diagonal Gaussians. Variable is the latent model
indicator which represents the model order of the preprocessing model and hence the dimension of . The corresponding observation model is a Multinomial-one
distribution. The third observation, , represents the class label of the
-th segment. The
observation model for is again a Multinomial-one distribution. Note that depending on whether we know the class label or not, can be a latent variable or observed. The
child node of and is the observed variable
, which represents a sufficient statistics of the corresponding time series segment. The proposed approach requires to calculate
the likelihoods !
repeatedly. Hence using the sufficient statistics is a
computational necessity. Finally we use " to represent the precision of the residual noise
model. The noise level is a nuisance parameter which is integrated over.
2.1.2 Model coefficients
Since we integrate over all unknown quantities, there is no conceptual difference between
model coefficients and the variables described above. However there is a qualitative difference. Model parameters exist only once for the entire GMOHMM, whereas there is an
individual quantity of interest for every segment
. Furthermore the model coefficients are
only updated during model inference whereas all quantities of interest are updated during
model inference and for prediction. We have three different prior counts, #%$ , # & and #' ,
which define the Dirichlet priors of the corresponding probabilities. Variable ( denotes
the transition probabilities, that is )* ,+- /. (1032 . The model assumes a stationary hidden state sequence. This allows us to obtain the unconditional prior probability of states 4
from the recurrence relation 56$78 /. (1598:;- . The prior probability of the first hidden
state, 5 $ - , is therefore the normalized eigenvector of the transition probability matrix
( that corresponds to the eigenvalue . Variable
represents the probabilities of class ,
)* .
032 , which are conditional on as well. The prior probabilities for observing the model indicator are represented by 5 . The probability )* 8 . 5 0 2 is
again conditional on the state . As was mentioned above, represents the model order
of the time series model. Hence another interpretation of 5 is that of state dependent prior
probabilities for observing particular model orders. The observation models for are
dynamic mixtures of Gaussians, with one model for each sensor
. Variables and 1
represent the coefficients of all Gaussian kernels. Hence 1 8 is a variate Gaussian distribution. Another interpretation is that the discrete indicator variables
8 and determine together with and 1 a Gaussian prior over . The nodes ,
, , , and / define a hierarchical prior setting which is discussed below.
?
T
T
d
t
d
i?1
?
W
i
d
i
W
i+1
?
?
1
?
?
s
P
?
1
?1
?
?1
i,1
I
i,1
P
1
?
1
h
?
?
i,s
I
i,s
P
s
?
?
i,s
?s
?s
1
X
s
s
i,1
g
P
s
?i,1
?
s
?
?
?s
?1
?1
g
s
h
s
X
i,1
i,s
i,s
Figure 1: This figure illustrates the details of the proposed model as a directed acyclic
graph. The graph shows the model parameters and all quantities of interest: denotes the
hidden states of the HMM; are the class labels of the corresponding time series segments;
are the latent coefficients of the time series model and the corresponding model
indicator variables; is the precision of the residual noise. For tractable inference, we
extract from the time series the sufficient statistics
. All other variables denote
model coefficients: ( are the transition probabilities;
are the probabilities for class 3 ;
and 1 are mean vectors and covariance matrices of the Gaussian observation model
for sensor
; and ) are the probabilities for observing .
2.2 Likelihood and priors for the GMOHMM
Suppose that we are provided with
segments of training data, .
. The
likelihood function of the GMOHMM parameters is then obtained by summation over all
possible sequences, , of latent states, . The sums and integrals under the product make
the likelihood function of Equation (1) highly nonlinear. This may be resolved by using
Gibbs sampling [GG84], which uses tricks similar to those of the expectation maximization
algorithm.
)*
/. )* - -
-
)* : -
- 2
"!
#$
- -3 "- -3 - -3 "-
7
(1)
%& '
(
Gibbs sampling requires that we obtain full conditional distributions1 we can sample from.
The conjugate priors are adopted from [RG97]. Below square brackets and index are
used to denote a particular component of a vector or matrix. Each component mean, 032 ,
- : - , with denoting the mean and : - the
is given a Gaussian prior: 0 2
inverse covariance matrix. As we use diagonal covariance matrices, we may give each
diagonal element an independent Gamma prior: * 0 2 :; , where
denotes the shape parameter and
denotes the inverse scale parameter. The hyperparameter, ! , gets a component wise Gamma hyper prior:
/ . The state
032 , get a Dirichlet prior:
032
# & # & . The
conditional class probabilities,
transition probabilities, ( 0 2 , get a Dirichlet prior: ( 0 2
#$ #$ . The probabilities
for observing different model orders, 5 3 0 2 , depend on the state . Their prior is Dirichlet
5 032
# ' # '/ . The precision gets a Jeffreys? prior, i.e. the scale parameter
is set to 0.
2
*),+
- (0/
.- ( (0/ )1 - (0/
- (0/ )31
4- (0' /'
' ' )65
)75
:<;
)85 '9'
and = , B 8 is set between > ' = and and
/?- (0/ is typically beValues Afor
are between
A
@
B
A
@
B
!
tween
- (?/ and > - (?/ ! , with - (0/ denoting the input range of maximum likelihood
estimates for - (0/
. The mean, , is the midpoint of the maximum likelihood esti4@?B C- (0/ ! , where B A- (0/ is again
mates - (0/
. The inverse covariance matrix A- (?/ .
the range of the estimates at sensor
. We set the prior counts #&
and #$ and #' to .
2.3 Sampling from the posterior
During model inference we need to update all unobserved variables of the DAG, whereas
for predictions we update only the variables summarized in section 2.1.1. Most of the
updates are done using the corresponding full conditional distributions, which have the
same functional forms as the corresponding priors. These full conditionals follow closely
from what was published previously in [Syk00], with some modifications necessary (see
e.g. [Rob96]), because we need to consider the Markov dependency between successive
hidden states. As the derivations of the full conditionals do not differ much from previous
work, we will omit them here and instead concentrate on an illustration how to update the
latent feature space,
.
2.3.1 A representation of the latent feature space
:<D
The AR model in Equation (2) is a linear regression model. We use 2 to denote the
AR coefficients, to denote the model order and to denote a sample from the noise
process, which we assume to be Gaussian with precision .
E4- /
F - / .HG 2 : D 2 F - GJI L/ KE4- /
(2)
D -
As is indicated by the subscript , the value of the I -th AR coefficient depends on the
model order. Hence AR coefficients are not a convenient representation of the latent feature
1
These are the distributions obtained when we condition on all other variables of the DAG.
space. A much more convenient representation is provided by using reflection coefficients,
, (statistically speaking they are partial correlation coefficients), which relate to AR coefficients via
the order recursive Levinson algorithm. Below we use vector notation and the
symbol 2 to denote the upside down version of the AR coefficient vector.
2 +-
K
2
(3)
2 +- .
2 +-
from dynamically
We expect to observe only such data that was generated
pro 2 stable AR
2 . This
cesses. For such processes, the latent density is defined on
- G /
as probais in contrast with the proposed DAG, where we use a finite Gaussian mixture
2
bilistic model for the latent variable, which is is defined on
. In order to avoid
applying
! ,
this mismatch, we reparameterise the space of reflection coefficients by
2
to obtain a more convenient representation of the latent features.
. !;"
(4)
2.3.2 Within dimensional updates
The within dimensional updates can be done with a conventional Metropolis Hastings step.
Integrating out , we obtain a Student t distributed likelihood function of the AR coefficients. In order to obtain likelihood ratio 1, we propose from the multivariate Student-t
distribution shown below, reparameterise in terms of reflection coefficients and apply the
! transformation.
$#
.
!;"
/%&#
where
)
' (*)
.
+
.
+
#
with
)
-
:;-
G
.
- B ' '9' B
:;-,
B *G
(5)
, $ :;-,
+
-
=
B
The proposal uses + to denote the -dimensional sample auto-covariance matrix,
is
,
the sample variance, .
- 2 +- $ is a vector of sample autocorrelations at lags
to
and N denotes the number of samples of the time series . The proposal in
Equation (5) gives a likelihood ratio of . The corresponding acceptance probability is
K
/
92
# 6
5 92
5
587 2
5 7 2
5
:<;
5
(6)
555 = '
5 7 5
The determinant of the Jacobian arises because we7 transform the AR coefficients using
:
..0/ 2134
5
Equations (3) and (4).
2.3.3 Updating model orders
Updating model orders requires us to sample across different dimensional parameter
spaces. One way of doing this is by using the reversible jump MCMC which was recently
proposed in [Gre95]. > We implement the reversible jump move from parameter space
>
3 2 to parameter space 2 +- as partial proposal. That is we propose a reflection
coefficient from a distribution that is conditional on the AR coefficient / . Integrating
out the precision of the noise model ! we obtain again a Student-t distributed likelihood.
This suggests the following proposal:
#
.
where
-
)
' (
(7)
G !-
.
.
G !
= G
B G $ , $
B K = , $ K2 + $
-
/
with
!;
.
.
"! K = K2 + * '
statistics of the K -dimensional BAR process,
Equation B (7) makes
use of the sufficient
B
. ' ' +"! . We use to denote the number
and
to denote
B +- / and
, $ ofB observations
the estimated auto covariance at time lag to obtain . - - ' '
+ as dimensional sample covariance matrix. Assuming
that> the probability of
proposing this move
>
is independent of , the proposal from 3 2 to 2 +- has acceptance probability
:;-
:
!
: . . / G !-!
= 1 1 !! # G ! K 8' (8)
>
>
If we attempt an update from 2 +- to 2 , we have to invert the second argument
of the .0/ operation in Equation (8).
!
.
+
3 Experiments
Convergence of all experiments is analysed by applying the method suggested in [RL96]
to the sequence of observed data likelihoods (equation (1), when filling in all variables).
3.1 Synthetic data
Our first evaluation uses synthetic data. We generate a first order Markov sequence as target
labels (2 state values) with 200 samples used for training and 600 used for testing. Each
sample is used as label of a segment with 200 samples from an auto regressive process. If
. If the label is
the label is , we generate data using reflection coefficients
. The driving noise has variance . Due to sampling
, we use the model
effects we obtain a data set with Bayes error
. In order to make the problem more
realistic, we use a second state sequence to replace of the segments with white noise.
These ?artifacts? are not correlated with the class labels.
=
>' G >' >'
>
> ' G > ' > '
=0>
In order to assess the effect of using a latent feature space, we perform three different
tests: In the first run we use conventional feature extraction with a third order model and
estimates found with maximum likelihood; In a second run we use again a third order
model but integrate over feature values; Finally the third test uses the proposed architecture
with a prior over model order which is ?flat? between and .
>
When compared with conditioning on feature estimates, the latent features show increased
likelihood. The likelihood gets even larger when we regard both the feature values and the
model orders of the preprocessing stage as random variables. As can be seen in figure 2,
this effect is also evident when we look at the generalization probabilities which become
larger as well. We explain this by sharper ?priors? over feature values and model orders,
which are due to the information provided by temporal context 2 of every segment. This
reduces the variance of the observation models which in turn increases likelihoods and
target probabilities. Table 1 shows that these higher probabilities correspond to a significant
improvement in generalization accuracy.
Probabilities from conditioning
1
0.5
0
50
100
50
100
50
100
150
200
250
300
350
400
Probabilities from integrating over features
450
500
550
600
150
200
250
300
350
400
450
500
Probabilities from integrating over model orders and features
550
600
550
600
1
0.5
0
1
0.5
0
150
200
250
300
350
400
450
500
Figure 2: This figure shows the generalization probabilities obtained with different settings. We see that the class probabilities get larger when we regard features as random
variables. This effect is even stronger when both the features and the model orders are
random variables.
3.2 Classification of cognitive tasks
The data used in these experiments is EEG recorded from 5 young, healthy and untrained
subjects while they perform different cognitive tasks. We classify 2 task pairings: auditorynavigation and left motor-right motor imagination. The recordings were taken from 3 electrode sites: T4, P4 (right tempero-parietal for spatial and auditory tasks), C3? , C3? (left
motor area for right motor imagination) and C4? , C4? (right motor area for left motor
imagination). The ground electrode was placed just lateral to the left mastoid process. The
data were recorded using an ISO-DAM system (gain of and fourth order band pass
filter with pass band between Hz and Hz). These signals were sampled with 384
Hz and 12 bit resolution. Each cognitive experiment was performed times for
seconds.
Classification uses again the same settings as with the synthetic problem. The summary
in table 1 shows results obtained from fold cross validation, where one experiment is
used for testing whereas all remaining data is used for training. We observe again significantly improved results when we regard features and model orders as latent variables.
The values in brackets are the significance levels for comparing integration of features with
conditioning and full integration with integration over feature values only.
4 Discussion
We propose in this paper a novel approach to hierarchical time series processing which
makes use of a latent feature representation. This understanding of features and model
orders as random variables is a direct consequence of applying Bayesian theory. Empirical
2
In a multi sensor setting there is spatial context as well.
Table 1: Generalization accuracies of different experiments
experiment
synthetic
left vs. right motor
auditory vs. navigation
L '
'
'=
conditioning
L' = ( ' > : :- ) ' (> ' >0> = )
< ' ( ' = > ) ' (> ' > :)
' > '> =
' (= ' > )
marginalize features
full integration
evaluations show that theoretical arguments are confirmed by significant improvements in
generalization accuracy. The only disadvantage of having a latent feature space is that
all computations get more involved, since there are additional variables that have to be
integrated over. However this additional complexity does not render the method intractable
since the algorithm remains polynomial in the number of segments to be classified. Finally
we want to point out that the improvements observed in our results can only be attributed
to the idea of using a latent feature space. This idea is certainly not limited to time series
classification and should generalize well to other hierarchical architectures.
Acknowledgments
We want to express gratitude to Dr. Rezek, who made several valuable suggestions in the
early stages of this work. We also want to thank Prof. Stokes, who provided us with the
EEG recordings that were used in the experiments section. Finally we are also grateful for
the valuable comments provided by the reviewers of this paper. Peter Sykacek is currently
funded by grant Nr. F46/399 kindly provided by the BUPA foundation.
References
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[GG84] S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions and the Bayesian
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[Mac92] D. J. C. MacKay. The evidence framework applied to classification networks. Neural
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[Nea96] R. M. Neal. Bayesian Learning for Neural Networks. Springer, New York, 1996.
? Ruanaidh and W. J. Fitzgerald. Numerical Bayesian Methods Applied to Signal
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[RL96] A. E. Raftery and S. M. Lewis. Implementing MCMC. In W.R. Gilks, S. Richardson, and
D.J. Spiegelhalter, editors, Markov Chain Monte Carlo in practice, chapter 7, pages 115?
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In S.A. Solla, T.K. Leen, and K.-R. M?uller, editors, Advances in Neural Information Processing Systems 12, pages 638?644, Boston, MA, 2000. MIT Press.
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1,203 | 2,097 | Active Learning
in the Drug Discovery Process
Manfred K. Warmuth , Gunnar
R?atsch , Michael
Mathieson ,
Jun Liao , Christian Lemmen
Computer
Science Dep., Univ. of Calif. at Santa Cruz
FHG
FIRST,
Kekul?estr. 7, Berlin, Germany
DuPont Pharmaceuticals,150 California St. San Francisco.
manfred,mathiesm,liaojun @cse.ucsc.edu, [email protected],
[email protected]
Abstract
We investigate the following data mining problem from Computational
Chemistry: From a large data set of compounds, find those that bind to
a target molecule in as few iterations of biological testing as possible. In
each iteration a comparatively small batch of compounds is screened for
binding to the target. We apply active learning techniques for selecting
the successive batches.
One selection strategy picks unlabeled examples closest to the maximum
margin hyperplane. Another produces many weight vectors by running
perceptrons over multiple permutations of the data. Each weight vector
votes with its prediction and we pick the unlabeled examples for which
the prediction is most evenly split between and . For a third selection strategy note that each unlabeled example bisects the version space
of consistent weight vectors. We estimate the volume on both sides of
the split by bouncing a billiard through the version space and select unlabeled examples that cause the most even split of the version space.
We demonstrate that on two data sets provided by DuPont Pharmaceuticals that all three selection strategies perform comparably well and are
much better than selecting random batches for testing.
1 Introduction
Two of the most important goals in Computational Drug Design are to find active compounds in large databases quickly and (usually along the way) to obtain an interpretable
model for what makes a specific subset of compounds active. Activity is typically defined
All but last author received partial support from NSF grant CCR 9821087
Current address: Austrialian National University, Canberra, Austrialia. Partially supported by
DFG
(JA 379/9-1, MU 987/1-1) and travel grants from EU (Neurocolt II).
Current address: BioSolveIT GmbH, An der Ziegelei 75, Sankt Augustin, Germany
as binding to a target molecule. Most of the time an iterative approach to the problem is
employed. That is in each iteration a batch of unlabeled compounds is screened against the
target using some sort of biological assay[MGST97]. The desired goal is that many active
hits show up in the assays of the selected batches.
From the Machine Learning point of view all examples (compounds) are initially unlabeled. In each iteration the learner selects a batch of un-labeled examples for being labeled
as positive (active) or negative (inactive). In Machine Learning this type of problem has
been called ?query learning? [Ang88] ?selective sampling? [CAL90] or ?active learning?
[TK00]. A Round0 data set contains 1,316 chemically diverse examples, only 39 of which
are positive. A second Round1 data set has 634 examples with 150 positives. 1 This data
set is preselected on the basis of medicinal chemistry intuition. Note that our classification problem is fundamentally asymmetric in that the data sets have typically many more
negative examples and the Chemists are more interested in the positive hits because these
compounds might lead to new drugs. What makes this problem challenging is that each
compound is described by a vector of 139,351 binary shape features. The vectors are
sparse (on the average 1378 features are set per Round0 compound and 7613 per Round1
compound).
We are working with retrospective data sets for which we know all the labels. However, we
simulate the real-life situation by initially hiding all labels and only giving to the algorithm
the labels for the requested batches of examples (virtual screening). The long-term goal of
this type of research is to provide a computer program to the Chemists which will do the
following interactive job: At any point new unlabeled examples may be added. Whenever
a test is completed, the labels are given to the program. Whenever a new test needs to be set up, the
Chemist asks the program to suggest a batch of unlabeled compounds. The suggested batch might be
?edited? and augmented using the invaluable knowledge and intuition of the medicinal Chemist. The
hope is that the computer assisted approach allows
for mining larger data sets more quickly. Note that
compounds are often generated with virtual Combinatorial Chemistry. Even though compound descriptors can be computed, the compounds have not been
Figure 1: Three types of comsynthesized yet. In other words it is comparatively pounds/points:
are active,
are
easy to generate lots of unlabeled data.
inactive and
are yet unlabeled. The
Maximum Margin Hyperplane is used as
In our case the Round0 data set consists of compounds from Vendor catalog and corporate collec- the internal classifier.
tions. Much more design effort went into the harder Round1 data set. Our initial results are
very encouraging. Our selection strategies do much better than choosing random batches
indicating that the long-term goal outlined above may be feasible.
Thus from the Machine Learning point of view we have a fixed set of points in
that
are either unlabeled or labeled positive or negative. (See Figure 1). The binary descriptors
of the compounds are rather ?complete? and the data is always linearly separable. Thus
we concentrate on simple linear classifiers in this paper. 2 We analyzed a large number
of different ways to produce hyperplanes and combine hyperplanes. In the next section we
describe different selection strategies on the basis of these hyperplanes in detail and provide
an experimental comparison. Finally in Section 3 we give some theoretical justification for
why the strategies are so effective.
1
2
Data provided by DuPont Pharmaceuticals.
On the current data sets kernels did not improve the results (not shown).
2 Different Selection Criteria and their Performance
A selection algorithm is specified in three parts: a batch size, an initialization and a selection strategy. In practice it is not cost effective to test single examples at a time. We
always chose 5% of the total data set as our batch size, which matches reasonably with
typical experimental constraints. The initial batches are chosen at random until at least one
positive and one negative example are found. Typically this is achieved with the first batch.
All further batches are chosen using the selection strategy.
As we mentioned in the introduction, all our selection strategies are based on linear classifiers of the data labeled so far.
examples
All
are normalized to unit-length and we consider
homogeneous hyperplanes
is again unit
where the normal direction
length. A plane predicts with sign
on the example/compound .
Once we specify how the weight vector is found then the next batch is found by selecting the
unlabeled examples closest to this hyperplane. The simplest way to obtain such a weight
vector is to run a perceptron over the labeled data until it produces a consistent weight
vector (Perc). Our second selection strategy (called SVM) uses the maximum margin hyperplane [BGV92] produced by a Support Vector Machine. When using the perceptron
to predict for example handwritten characters, it has been shown that ?voting? the predictions of many hyperplanes improves the predictive performance [FS98]. So we always
start from the weight vector zero and do multiple passes over the data until the perceptron
is consistent. After processing each example we store the weight vector. We remember
all weight vectors for each pass 3 and do this for 100 random permutations of the labeled
examples. Each weight vector gets one vote. The prediction on an example is positive if
the total vote is larger than zero and we select the unlabeled examples whose total vote is
closest to zero4 . We call this selection strategy VoPerc.
The dot product is commutative. So when
then the point lies on the positive side
of the hyperplane . In a dual view the point lies on the positive side of the hyperplane
(Recall all instances and weight vectors
that
have
unit-length). A
weight vector
is
must lie on the -side of the plane
for
consistent with all -labeled examples
all . The set of all consistent weight vectors is called the version space which is a section
of the unit hypersphere bounded by the planes corresponding to the labeled examples. An
unlabeled hyperplane
bisects the version space. For our third selection strategy
(VolEst)
of
bounce
a billiard is bounced 1000 times
inside
the
version
space
and
the
fraction
!
"
#
points on the positive side of
is computed. The prediction for
is positive if ! is
larger than half and the strategy selects unlabeled points whose fraction is closest to half.
In Figure 2 (left) we plot the true positives and false positives w.r.t. the whole data set
for Perc and VoPerc showing that VoPerc performs slightly better. Also VoPerc has lower
variance (Figure 2 (right)). Figure 3 (left) shows the averaged true positives and false
positives of VoPerc, SVM, and VolEst. We note that all three perform similarly. We also
plotted ROC curves after each batch has been added (not shown). These plots also show
that all three strategies are comparable.
The three strategies VoPerc, SVM, and VolEst all perform much better than the corresponding strategies where the selection criterion is to select random unlabeled examples instead
of using a ?closest? criterion. For example we show in Figure 4 that SVM is significantly
better than SVM-Rand. Surprisingly the improvement is larger on the easier Round0 data
set. The reason is that the Round0 has a smaller fraction of positive examples (3%). Recall
3
Surprisingly with some smart bookkeeping this can all be done with essentially no computational
overhead. [FS98]
4
Instead of voting the predictions of all weight vectors one can also average all the weight vectors
after normalizing them and select unlabeled examples closest to the resulting single weight vector.
This way of averaging leads to slightly worse results (not shown).
30
25
standard deviation
number of examples
150
100
50
0
0
Perc true pos
Perc false pos
VoPerc true pos
VoPerc false pos
0.2
0.4
0.6
0.8
fraction of examples selected
Perc true pos
Perc false pos
VoPerc true pos
VoPerc false pos
20
15
10
5
0
0
1
0.2
0.4
0.6
0.8
fraction of examples selected
1
Figure 2: (left) Average (over 10 runs) of true positives and false positives on the entire Round1 data
set after each 5% batch for Perc and VoPerc. (right) Standard deviation over 10 runs.
150
total number of hits
number of examples
150
100
50
0
0
VoPerc true pos
VoPerc false pos
SVM true pos
SVM false pos
VolEst true pos
VolEst false pos
0.2
0.4
0.6
0.8
fraction of examples selected
100
50
5% batch size
1 example batch size
1
0
0
0.2
0.4
0.6
0.8
fraction of examples selected
1
Figure 3: (left) Average (over 10 runs) of true and false positives on entire Round1 data set after each
5% batch for VoPerc, SVM, and VolEst. (right) Comparison of 5% batch size and 1 example batch
size for VoPerc on Round1 data.
that the Round1 data was preselected by the Chemists for actives and the fraction was raised
to about 25%. This suggest that our methods are particularly suitable when few positive
examples are hidden in a large set of negative examples.
The simple strategy SVM of choosing unlabeled examples closest to the maximum margin
hyperplane has been investigated by other authors (in [CCS00] for character recognition
and in [TK00] for text categorization). The labeled points that are closest to the hyperplane
are called the support vectors because if all other points are removed then the maximum
margin hyperplane remains unchanged. In Figure 5 we visualize the location of the points
in relation to the center of the hyperplane. We show the location of the points projected
onto the normal direction of the hyperplane. For each 5% batch the location of the points
is scattered onto a thin stripe. The hyperplane crosses the stripe in the middle. In the left
plot the distances are scaled so that the support vectors are at distance 1. In the right
plot the geometric distance to the hyperplane is plotted. Recall that we pick unlabeled
points closest to the hyperplane (center of the stripe). As soon as the ?window? between
the support vectors is cleaned most positive examples have been found (compare with the
SVM curves given in Figure 3 (left)). Also shrinking the width of the geometric window
corresponds to improved generalization.
So far our three selection strategies VoPerc, SVM and VolEst have shown similar performance. The question is whether the performance criterion considered so far is suitable
for the drug design application. Here the goal is to label/verify many positive compounds
quickly. We therefore think that the total number of positives (hits) among all examples
tested so far is the best performance criterion. Note that the total number of hits of the random selection strategy grows linearly with the number of batches (In each random batch
40
150
30
number of examples
number of examples
35
25
20
random true pos
random false pos
closest true pos
closest false pos
15
10
100
random true pos
random false pos
closest true pos
closest false pos
50
5
0
0
0.2
0.4
0.6
0.8
fraction of examples selected
0
0
1
0.2
0.4
0.6
0.8
fraction of examples selected
1
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
?2
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
?0.3
fraction of examples selected
fraction of examples selected
Figure 4: Comparisons of SVM using random batch selection and closest batch selection. (left)
Round0 data. (right) Round1 data.
?1.5
?1
?0.5
0
0.5
1
1.5
2
?0.2
normalized distance to hyperplane
?0.1
0
0.1
0.2
0.3
geometric distance to hyperplane
Figure 5: (left) Scatter plot of the distance of examples to the maximum margin hyperplane normalized so support vectors are at 1. (right) Scatter plot of the geometric distance of examples to the
hyperplane. Each stripe shows location of a random sub-sample of points (Round1 data) after an
additional 5% batch has been labeled by SVM. Selected examples are black x, unselected positives
are red plus, unselected negatives are blue square.
we expect 5% hits). In contrast the total number of hits of VoPerc, SVM and VolEst is
5% in the first batch (since it is random) but much faster thereafter (See Figure 6). VoPerc
performs the best.
Since the positive examples are much more valuable in our application, we also changed
5
the
examples of largest positive distance
selection strategy SVM to selecting unlabeled
to the maximum margin hyperplane (SVM ) rather than smallest distance
.
Correspondingly VoPerc picks the unlabeled example with the highest vote and VolEst
picks the unlabeled example with the largest fraction ! . The total hit plots of the resulting
modified strategies SVM , VoPerc and VolEst are improved ( see Figure 7 versus Figure
6 ). However the generalization plots of the modified strategies (i.e. curves like Figure
3(left)) are slightly worse for the new versions. Thus in some sense the original strategies
are better at ?exploration? (giving better generalization on the entire data set) while the
modified strategies are better at ?exploitation? (higher number of total hits). We show this
trade-off in Figure
8 for SVM and SVM . The same trade-off occurs for the VoPerc
and VolEst
pairs of strategies(not shown).
Finally we investigate the effect of batch size on performance. For simplicity we only show
total hit plots for VoPerc( Figure 3 (right) ). Note that for our data a batch size of 5% (31
examples for Round1) is performing not much worse than the experimentally unrealistic
batch size of only 1 example. Only when the results for batch size 1 are much better than
5
In Figure 5 this means we are selecting from right to left
40
150
30
total number of hits
total number of hits
35
25
20
15
10
5
0
0
VoPerc
SVM
VolEst
0.2
0.4
0.6
0.8
fraction of examples selected
50
0
0
1
VoPerc
SVM
VolEst
100
0.2
0.4
0.6
0.8
fraction of examples selected
Figure 6: Total hit performance on Round0 (left) and Round1 (right) data of
VolEst with 5% batch size.
40
1
, VoPerc and
150
30
total number of hits
total number of hits
35
25
20
15
10
5
0
0
VoPerc+
SVM+
VolEst+
0.2
0.4
0.6
0.8
fraction of examples selected
1
100
50
VoPerc+
SVM+
VolEst+
0
0
0.2
0.4
0.6
0.8
fraction of examples selected
Figure 7: Total hit performance on Round0 (left) and Round1 (right) data of
VolEst with 5% batch size.
1
, VoPerc and
the results for larger batch sizes, more sophisticated selection strategies are worth exploring
that pick say a batch that is ?close? and at the same time ?diverse?.
At this point our data sets are still small enough that we were able to precompute all dot
products
(the kernel matrix). After this preprocessing, one pass of a perceptron is at most
, where is the number of labeled examples and
number of mistakes.
the
Finding the maximum margin hyperplane
is estimated at
time. For the computa
tion of VolEst we need to spend
per bounce of the billiard. In our implementations
we used SVM Light [Joa99] and the billiard algorithm of [Ruj97, RM00, HGC99].
If we have the internal hypothesis of the algorithm then for applying the selection criterion
we need to evaluate the hypothesis for each unlabeled point. This cost is proportional to
the number of support vectors for the SVM-based methods and proportional to the number
of mistakes for the perceptron-based methods. In the case of VolEst we again need
time per bounce, where is the number of labeled points.
Overall VolEst was clearly the slowest. For much larger data sets VoPerc seems to be the
simplest and the most adaptable.
3 Theoretical Justifications
As we see in Figure 5(right) the geometric margin of the support vectors (half the width
of the window) is shrinking as more examples are labeled. Thus the following goal is reasonable for designing selection strategies: pick unlabeled examples that cause the margin
to shrink the most. The simplest such strategy is to pick examples closest to the maximum margin hyperplane since these example are expected to change the maximum margin
150
number of examples
total number of hits
150
100
50
0
0
SVM+
SVM
0.2
0.4
0.6
0.8
1
fraction of examples selected
100
SVM+ true pos
SVM true pos
SVM+ false pos
SVM false pos
50
0
0
0.2
0.4
0.6
0.8
1
fraction of examples selected
Figure 8: Exploitation versus Exploration: (left) Total hit performance and (right) True and False
positives performance (right) of SVM and on Round 1 data
hyperplane the most [TK00, CCS00].
An alternative goal is to reduce the volume of the version space. This volume is a rough
measure of the remaining uncertainty
and
in the data. Recall that both the weight vectors
instances have unit length. Thus
is the distance
of the point to the plane as well
to the plane . The maximum margin
as (in the dual
view) the distance
of the point
hyperplane
is the point
in version space with the largest sphere that is completely
contained in the version space [Ruj97, RM00]. After labeling only one side of the plane
remains. So if passes close to the point
then about half of the largest sphere is
eliminated from the version space. So this is a second justification for selecting unlabeled
examples closest to the maximum margin hyperplane.
Our selection strategy VolEst starts from any point inside the version space and then
bounces a billiard 1000 times.
The billiard is almost always ergodic (See discussion in
[Ruj97]).
Thus the fraction ! of bounces on the positive side of an unlabeled hyperplane
is an
. Since it is unknown
estimate of the fraction of volume on the positive side of
how
will be labeled, the best example are those
that
split
the
version
space in half. Thus
in VolEst we select unlabeled points for which ! is closest to half. The thinking underlying our strategy VolEst is most closely related to the Committee Machine where random
concepts in the version space are asked to vote on the next random example and the label
of that example is requested only if the vote is close to an even split [SOS92].
We tried to improve our estimate of the volume
by replacing ! by the fraction of the total
#
trajectory located on the positive side of . On our two data sets this did not improve the
performance
(not shown). We also averaged the 1000 bounce points. The resulting weight
vector (an approximation to the center of mass of the version space) approximates the
so called Bayes point [Ruj97] which
has the following property: Any unlabeled hyperplane
passing through the Bayes point cuts the version space roughly 6 in half. We thus tested
a selection strategy which picks unlabeled points closest to the estimated center of mass.
This strategy was again indistinguishable from the other two strategies based on bouncing
the billiard.
We have no rigorous justification for the
variants of our algorithms.
4 Conclusion
We showed how the active learning paradigm ideally fits the drug design cycle. After some
deliberations we concluded that the total number of positive examples (hits) among the
tested examples is the best performance criterion for the drug design application. We found
6
Even in dimension two there is no point that does this exactly [Ruj97].
that a number of different selection strategies with comparable performance. The variants
that select the unlabeled examples with the highest score (i.e. the variants) perform better.
Overall the selection strategies based on the Voted Perceptron were the most versatile and
showed slightly better performance.
References
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| 2097 |@word bounced:1 exploitation:2 middle:1 version:17 seems:1 cal90:2 tried:1 pick:9 asks:1 versatile:1 harder:1 initial:2 contains:1 score:1 selecting:6 current:3 yet:2 scatter:2 must:1 fs98:3 cruz:1 shape:1 christian:1 dupont:3 plot:9 interpretable:1 atlas:1 half:7 selected:16 warmuth:1 plane:6 manfred:2 hypersphere:1 cse:1 location:4 successive:1 billiard:8 hyperplanes:5 herbrich:1 along:1 ucsc:1 consists:1 combine:1 overhead:1 inside:2 expected:1 rapid:1 roughly:1 encouraging:1 window:3 hiding:1 provided:2 estimating:1 bounded:1 underlying:1 mass:2 what:2 sankt:1 finding:1 remember:1 computa:1 voting:2 interactive:1 exactly:1 classifier:7 hit:19 scaled:1 unit:5 grant:2 positive:31 bind:1 mistake:2 might:2 chose:1 black:1 au:1 initialization:1 plus:1 challenging:1 averaged:2 seventeenth:1 practical:1 testing:2 practice:1 drug:7 significantly:1 word:1 suggest:2 get:1 onto:2 unlabeled:30 selection:26 close:3 applying:1 center:4 ergodic:1 simplicity:1 haussler:1 ralf:1 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1,204 | 2,098 | Transform-invariant image decomposition
with similarity templates
Chris Stauffer, Erik Miller, and Kinh Tieu
MIT Artificial Intelligence Lab
Massachusetts Institute of Technology
Cambridge, MA 02139
{stauffer,emiller,tieu}@ai.mit.edu
Abstract
Recent work has shown impressive transform-invariant modeling
and clustering for sets of images of objects with similar appearance.
We seek to expand these capabilities to sets of images of an object
class that show considerable variation across individual instances
(e.g. pedestrian images) using a representation based on pixel-wise
similarities, similarity templates. Because of its invariance to the
colors of particular components of an object, this representation enables detection of instances of an object class and enables alignment
of those instances. Further, this model implicitly represents the regions of color regularity in the class-specific image set enabling a
decomposition of that object class into component regions.
1
Introduction
Images of a class of objects are often not effectively characterized by a Gaussian
distribution or even a mixture of Gaussians. In particular, we are interested in
modeling classes of objects that are characterized by similarities and differences
between image pixels rather than by the values of those pixels. For instance, images
of pedestrians (at a certain scale and pose) can be characterized by a few regions
of regularity (RORs) such as shirt, pants, background, and head, that have fixed
properties such as constant color or constant texture within the region, but tend to
be different from each other. The particular color (or texture) of those regions is
largely irrelevant. We shall refer to sets of images that fit this general description
as images characterized by regions of regularity, or ICRORs.
Jojic and Frey [1] and others [2] have investigated transform-invariant modeling
and clustering for images of a particular object (e.g., an individual?s face). Their
method can simultaneously converge on a model and align the data to that model.
This method has shown positive results for many types of objects that are effectively
modeled by a Gaussian or a mixture of Gaussians. Their work with transformed
component analysis (TCA) shows promise for handling considerable variation within
the images resulting from lighting or slight misalignments. However, because these
models rely on an image set with a fixed mean or mixture of means, they are not
directly applicable to ICRORs.
We would also like to address transform-invariant modeling, but use a model which
is invariant to the particular color of component regions. One simple way to achieve
this is to use edge templates to model local differences in image color. In contrast,
we have chosen to model global similarities in color using a similarity template (ST).
While representations of pixel similarity have previously been exploited for segmentation of single images [3, 4], we have chosen to use them for aggregate modeling
of image sets. Similarity templates enable alignment of image sets and decomposition of images into class-specific pixel regions. We note also that registration of
two ICRORs can be accomplished by minimizing the mutual information between
corresponding pixels [5]. But, there is no obvious way of extending this method to
large sets of images without a combinatorial explosion.
Section 2 briefly introduces similarity templates. We investigate their uses for modeling and detection. Section 3 discusses dataset alignment. Section 4 covers their
application to decomposing a class-specific set of images into component regions.
Future avenues of research and conclusions are discussed Section 5.
2
Similarity templates
This section begins with a brief explanation of the similarity template followed
by the mechanics of computing and comparing similarity templates. A similarity
template S for an N -pixel image is an N xN matrix. The element Si,j represents
the probability that pixel locations pi and pj would result from choosing a region
and drawing (iid) two samples (pixel locations) from it. More formally,
X
Si,j =
p(r)p(pi |r)p(pj |r),
(1)
r
where p(r) is the probability of choosing region r and p(pi |r) is the probability of
choosing pixel location pi from region r.
2.1
The ?ideal? similarity template
Consider sampling pixel pairs as described above from an N -pixel image of a particular object (e.g., a pedestrian) segmented by an oracle into disjoint regions (e.g.,
shirt, pants, head, feet, background). Assuming each region is equally likely to be
sampled and that the pixels in the region are selected with uniform probability,
then
1 1 2
( R )( Sr ) if ri = rj
Si,j =
(2)
0
otherwise,
where R is the number of regions, Sr is the number of pixels in region r, and ri
is the region label of pi . If two pixels are from the same region, the corresponding
value is the product of the probability R1 of choosing a particular region and the
probability ( S1r )2 of drawing that pixel pair. This can be interpreted as a block
diagonal co-occurrence matrix of sampled pixel pairs.
In this ideal case, two images of different pedestrians with the same body size and
shape would result in the same similarity template regardless of the colors of their
clothes, since the ST is a function only of the segmentation. An ST of an image
without a pedestrian would exhibit different statistics. Note that even the ST of
an image of a blank wall (segmented as a single region) would be different because
pixels that are in different regions under the ideal pedestrian ST would be in the
same region.
Unfortunately, images do not typically come with labeled regions, and so computation of a similarity template is impossible. However, in this paper, we take
advantage of the observation that properties within a region, such as color, are
often approximately constant. Using this observation, we can approximate true
similarity templates from unsegmented images.
2.2
Computing similarity templates
For the purposes of this paper, our model for similarity is based solely on color.
Since there is a correlation between color similarity and two pixels being in the
same region, we approximate the corresponding value S?i,j with a measure of color
similarity:
1
?||Ii ? Ij ||2
?
Si,j =
,
(3)
exp
N Zi
?i2
where Ii and Ij are pixel color values, ?i2 is a parameter that adjusts the color
similarity measure as a function of the pixel color distribution in the image, and Zi
is the sum of the ith row. This normalization is required because large regions have
a disproportionate effect on the ST estimate. The choice of ?i2 had little effect on
the resulting ST.
If each latent region had a constant but unique color and the regions were of equal
size, then as ?i2 approaches zero this process reconstructs the ?ideal? similarity
template defined in Equation 1. Although region colors are neither constant nor
unique, this approximation has proven to work well in practice.
It is possible to add a spatial prior based on the relative pixel location to model the
fact that similarities tend to local, but we will rely on the statistics of the images
in our data set to determine whether (and to what extent) this is the case. Also, it
may be possible to achieve better results using a more complex color model (e.g.,
hsv with full covariance) or broadening the measure of similarity to include other
modalities (e.g., texture, motion, depth, etc.).
Figure 1 shows two views of the same similarity template. The first view represents
each pixel?s similarity to every other pixel. The second view contains a sub-image
for each pixel which highlights the pixels that are most likely produced by the same
region. Pixels in the shirt tend to highlight the entire shirt and the pants (to a
lesser amount). Pixels in the background tend to be very dissimilar to all pixels in
the foreground.
2.3
Aggregate similarity templates (AST)
We assume each estimated ST is a noisy measurement of the true underlying joint
distribution. Hence we compute an aggregate similarity template (AST) as the
mean S? of the ST estimates over an entire class-specific set of K images:
K
1 X ?k
Si,j .
S?i,j =
K
(4)
k=1
For this quantity to be meaningful, the RORs must be in at least partial correspondence across the training set. Note that this is a less restrictive assumption than
assuming edges of regions are in correspondence across an image set, since regions
have greater support. Being the mean of a set of probability distributions, the AST
is also a valid joint probability distribution.
(a)
(b)
Figure 1: (a) The N xN aggregate similarity template for pedestrian data set. (b)
An alternate view of (a). This view is a width2 xheight2 version of (a). Each subimage represents the row of the original AST that corresponds to that pixel. Each
sub-image highlights the pixels that are most similar to the pixel it represents.
2.4
Comparing similarity templates
To compare an estimated similarity template S? to an aggregate similarity template
S? we evaluate their dot product1 :
XX
? S)
? =
s(S,
S?i,j S?i,j .
(5)
i
j
We are currently investigating other measures for comparison. By thresholding the
ratio of the dot product of a particular image patch under and AST trained on
pedestrian image patches versus an AST trained on random image patches, we can
determine whether a person is present in the image. In previous work [6], we have
illustrated encouraging detection performance.
3
Data set alignment
In this paper, we investigate a more difficult problem: alignment of a set of images.
To explore this problem, we created a set of 128x64 images of simulated pedestrians.
These pedestrians were generated by creating four independently-colored regions
corresponding to shirts, pants, head, and background. Each region was given a
random color. The RGB components were chosen from a uniform distribution
[0, 1]. Then, independent Gaussian noise was added to each pixel (? = .1). Finally
the images were translated uniformly up to 25% of the size of the object. Figure 2
shows examples of these images.
1
In our experimentation KL-divergence, typically used to compare estimates of distributions, proved less robust.
Figure 2: A set of randomly generated ?pedestrian? images used in alignment experimetns.
Using the congealing procedure of Miller et al. [2], we iteratively estimated the
latent variables (translations) that maximized the probability of the image STs to
the AST and re-estimated the AST. We were able to align the images to within .5
pixels on average.
4
Decomposing the similarity template
This section explains how to derive a factorized representation from the AST that
will be useful for recognition of particular instances of a class and for further refinement of detection. This representation is also useful in approximating the template
to avoid the O(N 2 ) storage requirements.
An AST represents the similarity of pixels within an image across an entire classspecific data set. Pairwise statistics have been used for segmentation previously
[3]. Recently, work centered on factoring joint distributions has gained increasing
attention [7, 8, 9, 10]. Rather than estimating two sets of marginals (conditioned on
a latent variable) that explain co-occurrence data (e.g. word-document pairs), we
seek a single set of marginals conditioned on a latent variable (the ROR) that explain
our co-occurrence data (pixel position pairs). Hence, it is a density factorization
in which the two conditional factors are identical (Equation 1). We refer to this as
symmetric factorization of a joint density.
Also, rather than treating pixel brightness (darkness, redness, blueness, or hue) as
a value to be reconstructed in the decomposition, we chose to represent pixel similarity. In contrast to simply treating images as additive mixtures of basis functions
[9], our decomposition will get the same results on a database of images of digits
written in black on white paper or in white on a black board and color images
introduce no difficulties for our methods.
We would like to estimate the factors from Equation 1 that best reconstruct our
? Let S? be the estimate of S? constructed from these factors. Given
measured AST, S.
the number of regions R, it is possible to estimate the priors for each region p(r)
and the probability of each region producing each pixel p(pi |r). The error function
we minimize is the KL-divergence between the empirically measured S? and our
?
parameterized estimate S,
!
XX
?i,j
S
E=
(6)
S?i,j log
S?i,j
i
j
as in [8]. Because our model S? is symmetric, this case can be updated with only
two rules:
X
? i , pj )
S(p
pnew (pi |r) ? p(pi |r)
p(r)p(pj |r) ?
, and
(7)
S(pi , pj )
p
j
pnew (r) ? p(r)
XX
pi
pj
? i , pj )
S(p
p(pj |r)p(pi |r) ?
.
S(pi , pj )
(8)
50
100
150
200
250
300
350
400
450
500
50
100
150
200
250
300
350
400
450
500
Figure 3: The similarity template and the corresponding automatically generated
binary decomposition of the images in the pedestrian data set. The root node represents every pixel in the image. The first branch splits foreground vs. background
pixels. Other nodes correspond to shirt, legs, head, and background regions.
The more underlying regions we allow our model, the closer our estimate will approximate the true joint distribution. These region models tend to represent parts
of the object class. p(pi |r) will tend to have high probabilities for a set of pixels
belonging to the same region. We take advantage of the fact that aligned pedestrian
images are symmetric about the vertical axis by adding a ?reflected? aggregate similarity template to the aggregate similarity template. The resulting representation
provides a compact approximation of the AST (O(RN ) rather than O(N 2 )).
Rather than performing a straight R-way decomposition of the AST to obtain R
pixel region models, we extracted a hierarchical segmentation in the form of a binary
tree. Given the initial region-conditioned marginals p(pi |r0 ) and p(pi |r1 ), each pixel
was assigned to the region with higher likelihood. This was iteratively applied to
the ASTs defined for each sub-region. Region priors were set to 0.5 and not adapted
in order to encourage a balanced cut.
The probabilistic segmentation can be employed to accumulate robust estimates of
statistics of the region. For instance, the mean pixel value can be calculated as a
weighted mean where the pixels are weighted by p(pi |r).
4.1
Decomposing pedestrians
Because the data collected at our lab showed limited variability in lighting, background composition, and clothing, we used the MIT CBCL pedestrian data set
which contains images of 924 unique, roughly aligned pedestrians in a wide variety of environments to estimate the AST. Figure 3 shows the resulting hierarchical
segmentation for the pedestrian AST. Since this intuitive representation was derived automatically with absolutely no knowledge about pedestrians, we hope other
classes of objects can be similarly decomposed into RORs.
In our experience, a color histogram of all the pixels within a pedestrian is not
useful for recognition and was almost useless for data mining applications. Here
we propose a class-conditional color model. It determines a color model over each
region that our algorithm has determined contain similar color information within
this class of objects. This allows us to obtain robust estimates of color in the regions
Figure 4: Results of automatic clustering on three components: shirt, pants, and
the background. Each shows the feature, the most unusual examples of that region,
followed by the 12 most likely examples for the eight prototypical colors of that
region.
of regularity. Further, as a result of our probabilistic segmentation, the values of
p(pi |r) indicate which pixels are most regular in a region which enables us to weight
the contribution of each pixel to the color model.
For the case of pedestrian-conditional color models, the regions roughly correspond
to shirt color, pant color, feet color, head color, and some background color regions. The colors in a region of a single image can be modeled by color histograms,
Gaussians, or mixtures of Gaussians. These region models can be clustered across
images to determine a density of shirt colors, pant colors, and other region colors
within a particular environment. This enables not only an efficient factored color
component codebook, but anomaly detection based on particular regions and higher
order models of co-occurrences between particular types of regions. To illustrate
the effectiveness of our representation we chose the simplest model for the colors in
each region?a single Gaussian in RGB space. The mean and variance of each Gaussian was computed by weighting the pixels represented by the corresponding node
by p(pi |r). This biases the estimate towards the ?most similar? pixels in the region
(e.g., the center of the shirt or the center of the legs). This allows us to represent
the colors of each pedestrian image with 31 means and variances corresponding to
the (2treeheight ? 1) nodes.
We investigated unsupervised clustering on components of the conditional color
model. We fit a mixture of eight Gaussians to the 924 color means for each region.
Figure 4 shows the 12 pedestrians with the highest probability under each of the
eight models and the 12 most unusual pedestrians with respect to that region for
three of the nodes of the tree: shirt color, pant color, and color of the background.
Red, white, blue, and black shirts represent a significant portion of the database.
Blue jeans are also very common in the Boston area (where the CBCL database
was collected). Indoor scenes tended to be very dark, and cement is much more
common than grass.
5
Conclusions
While this representation shows promise, it is not ideal for many problems. First,
it is expensive in both memory and computation. Here, we are only using a simple
measure of pairwise similarity?color similarity. In the future, similarity templates
could be applied to different modalities including texture similarity, depth similarity,
or motion similarity.
While computationally intensive, we believe that similarity templates can provide
a unified approach to the extraction of possible class-specific targets from an image
database, alignment of the candidate images, and precomputation of meaningful features of that class. For the case of pedestrians, it could detect potential pedestrians
in a database, align them, derive a model of pedestrians, and extract the parameters
for each pedestrian. Once the features are computed, query and retrieval can be
done efficiently.
We have introduced a new image representation based on pixel-wise similarity. We
have shown its application in both alignment and decomposition of pedestrian images.
References
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[4] Boykov, Y., O. Veksler and R. Zabih. Fast Approximate Energy Minimization via
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[5] Viola, P. Alignment by Maximization of Mutual Information. MIT Artificial Intelligence Lab, Ph.D. Thesis AI-TR #1548, June, 1995.
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In 30th Annual Meeting of the Association for Computational Linguistics, Columbus,
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1,205 | 2,099 | A theory of neural integration in the
head-direction system
Richard H.R. Hahnloser , Xiaohui Xie and H. Sebastian Seung
Howard Hughes Medical Institute
Dept. of Brain and Cognitive Sciences
Massachusetts Institute of Technology
Cambridge, MA 02139
rhahnloser|xhxie|seung @mit.edu
Abstract
Integration in the head-direction system is a computation by which horizontal angular head velocity signals from the vestibular nuclei are integrated to yield a neural representation of head direction. In the thalamus, the postsubiculum and the mammillary nuclei, the head-direction
representation has the form of a place code: neurons have a preferred
head direction in which their firing is maximal [Blair and Sharp, 1995,
Blair et al., 1998, ?].
Integration is a difficult computation, given that head-velocities can vary
over a large range. Previous models of the head-direction system relied
on the assumption that the integration is achieved in a firing-rate-based
attractor network with a ring structure. In order to correctly integrate
head-velocity signals during high-speed head rotations, very fast synaptic
dynamics had to be assumed.
Here we address the question whether integration in the head-direction
system is possible with slow synapses, for example excitatory NMDA
and inhibitory GABA(B) type synapses. For neural networks with such
slow synapses, rate-based dynamics are a good approximation of spiking neurons [Ermentrout, 1994]. We find that correct integration during
high-speed head rotations imposes strong constraints on possible network architectures.
1 Introduction
Several network models have been designed to emulate the properties of head-direction
neurons (HDNs) [Zhang, 1996, Redish et al., 1996, Goodridge and Touretzky, 2000]. The
model by Zhang reproduces persistent activity during stationary head positions. Persistent
neural activity is generated in a ring-attractor network with symmetric excitatory and inhibitory synaptic connections. Independently, he and Redish et al. showed that integration
is possible by adding asymmetrical connections to the attractor network. They assumed
that the strength of these asymmetrical connections is modulated by head-velocity. When
the rat moves its head to the right, the asymmetrical connections induce a rightward shift
of the activity in the attractor network. A more plausible model without multiplicative
modulation of connections has been studied recently by Goodridge and Touretzky. There,
the head-velocity input has a modulatory influence on firing rates of intermittent neurons
rather than on connection strengths. The intermittent neurons are divided into two groups
that make spatially offset connections, one group to the right, the other to the left. The different types of neurons in the Goodridge and Touretzky model have firing properties that
are comparable to neurons in the various nuclei of the head-direction system.
What all these previous models have in common is that the integration is performed in an inherent double-ring network with very fast synapses (less than ms for
[Goodridge and Touretzky, 2000]). The connections made by one ring are responsible for
rightward turns and the connections made by the other ring are responsible for leftward
turns. In order to derive a network theory of integration valid for fast and slow synapses,
here we solve a simple double-ring network in the linear and in the saturated regimes.
An important property of the head-direction system is that the integration be linear over a
large range of head-velocities. We are interested in finding those type of synaptic connections that yield a large linear range and pose our findings as predictions on optimal network
architectures. Although our network is conceptually simpler than previous models, we
show that using two simple read-out methods, averaging and extracting the maximum, it is
possible to approximate head-velocity independent tuning curves as observed in the Postsubiculum (PoS) and anticipatory responses in the anterior dorsal thalamus (ADN).
2 Definition of the model
We assume that the number of neurons in the double-ring network is large and write its
dynamics as a continuous neural field
where
(1)
(2)
.-/
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;
102.3 4
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*
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$ %
('&
! " $ ,
('&
'
0 @?BADC
E
> denotes a rectification nonlinearity.
and
are the firing rates of
"> :
neurons in the left and right ring, respectively. The quantities D and F represent
synaptic
). is a
and
activations (amount of neurotransmitter release3G4 caused3 by the
firing
rates
354H
3
'
6
6
synaptic
time
constant.
The
vestibular
inputs
and
are
purely
excitatory,
354JI
3 IK354
3
'
6
. For simplicity, we assume that 6
is proportional to angular headvelocity. The synaptic connection profiles $ % between neurons on the same ring and $ ,
between neurons on different rings are given by:
LNM4OJM QPSRT
UWVB4L V XPGRT S<
(3)
$)%
$),
M4 M
VB4
V
*
, ,
and
define
the
intra
and
inter-ring
connection
strengths.
is
the
intra-ring
connection offset and the inter-ring offset.
3 Integration
3O
E . In
When the animal is not moving, the vestibular inputs to the two rings are equal, 6
this case, within a certain range of synaptic connections, steady bumps of activities appear
on the two rings. When
the head of the animal rotates, the activity bumps travel
3
3 at a velocity
determined by 6 . For perfect integration, should be proportional to 6
over the full
range of possible head-velocities. This is a difficult computational problem, in particular
for slow synapses.
4 Small head-velocity approximation
3
E ), the two stationary bumps of synaptic activation
When the head is not rotating ( 6
are of the form
0
L
0
4
4
'
'
and
(4)
" PGRT
(')
" PSR T
(')
:
:
4
where
is the current head direction and is the offset between the two bumps. How to
calculate , and is shown in the Appendix. The half width of these bumps is given by
WA PPSRT
S<
(5)
3 3S4
When the angular head velocity is small ( 6
), we linearize the dynamics around
the stationary solution Eq. (4), see Appendix. We find that
PGRT
0
4
G
('
('&
'
' '
(6)
" '
:
PGRT
;
10
4
F
('
('&
'
'J
(7)
"
:
M4
M
'
VB4F
'
V
PGRT * '
PGRT
T
where the velocity is given by
M XT
1*
3
"
6
and
'
M
'J
'
M4
T *
!
T
0
VB4 T
'
(8)
<
(9)
(10)
Equation (8) is the desired
result, relating the velocity of the two bumps to the differential
3
vestibular input 6 . In Fig. 1 we show simulation results using slow synapses (
E ms). The integration is V linear
Mover almost the entire range of head-velocities (up to
more than EE ! ) when
, i.e., when the amplitudes
V
NM of inter-ring and intra-ring
connections are equal. We point out that the condition
V cannot
4 directly be deduced
E ) was necessary to
from the above formulas, some empirical tuning (for
3 example
achieve this large range of linearity (large both in 6 and ).
"
# $%
'&%(
When the bumps move, their amplitudes tend to decrease. Fig. 1d shows the peak firing
rates of neurons in the two rings as a function of vestibular input. As can be seen, the firing
rates are a linear function of vestibular input, in agreement with equations 17 and 18 of the
Appendix. However, a linear firing-rate modulation by head velocity is not universal, for
some parameters we have seen asymmetrically head-velocity tuning, with a preference for
small head velocities (not shown).
a.
b.
800
600
800
Simulation
Theory
600
400
v (degrees/sec)
v (degrees/sec)
400
Simulation
Theory
200
0
?200
200
0
?200
?400
?400
?600
?600
?800
?1
?0.5
0
? b/b0
0.5
?800
?1
1
c.
?0.5
0.5
1
0.5
1
d.
75
600
Simulation
Theory
Left
Right
70
Firing rate (Hz)
400
v (degrees/sec)
0
? b/b0
200
0
?200
65
60
55
?400
?600
?1
?0.5
0
? b/b0
0.5
1
50
?1
?0.5
0
? b/b0
3 3G4
Figure 1: Velocity
V of
M activity
VB4 as a function of vestibular input 6 V . M a. Sublinear
bumps
E .V b. Supralinear
E ,
,*
,*
,
integration.
V 4
M
integration.
V 4
E . c. Linear (perfect) integration.
E . d. Head-velocity
,*
,
dependent modulation of firing rates
- (on the right and on the left ring). Same parameters
E ms. *
as in c.
, and
.
"
$
" $
" $
$
$
5 Saturating velocity
3
When 6 is very large, at some point, the left ring becomes inactive. Because inactivating
the left ring means that the push-pull competition between the two rings is minimized, we
are able to determine the saturating velocity of the double-ring network. The saturating
velocity is given by the on-ring connections $ % . Define
M4 @M XPGRT
('+*
$
M4 @M XPGRT
9A
* PSR T
* T
9A
* $ %
$&%
'
M4 M XPGRT
* PGRT
. Now, let
be the steady solution of a ring
where $)%
'
network with symmetric connections $&%
. By differentiating, it follows that
is the solution of a ring network with connections $
. Hence, the saturating
velocity
is given by
9A
* <
(11)
Notice that a traveling solution may not always exist if one ring is inactive (this is the case
when there are no intra-ring excitatory connections). However, even without a traveling
solution, equation (11) remains valid. In Figs. 1a and b, the saturating velocity is indicated
E !
by
dotted lines, in Fig. 1a we find
and in Fig. 1b
E the horizontal
.
" $%
'&%(
$'
'&'(
6 ADN and POs neurons
Goodridge and Touretzky?s integrator model was designed to emulate details of neuronal
tuning as observed in the different areas of the head-direction system. Wondering whether
the simple double ring studied here can also reproduce
multiple tuning curves, we analyze
simple read-out methods of the firing rates and . What we find is that two readout methods can indeed approximate response behavior resembling that of ADN and POs
neurons.
ADN
?BADC neurons:
By reading out firing rates using a maximum operation,
!
, anticipatory head-direction tuning arises due to the fact that there is
an activity offset between the two rings, equation (13). When the head turns to the right,
the activity on the right ring is larger than on the
left ring and so the tuning of
is
biased to the right. Similarly, for left turns,
is biased to the left. Thus, the activity
offset between the two rings leads to an anticipation time for ADN neurons, see Figure
2. Because, by assumption is head-velocity independent, it follows
that
is inversely
proportional to head-velocity (assuming perfect integration),
. In other words,
the anticipation time tends to be smaller for fast head rotations and larger for slow head
rotations.
D
POs
By reading out the double ring activity as an average,
neurons:
, neurons in POs do not have any anticipation time: because averaging is a symmetric
operation, all information about the direction of head rotations is lost.
Right turn
Left turn
Left ring
Right ring
Firing Rate
Max
Average
0
90
180
270
Head?direction (degs)
360
Figure 2: Snapshots of the activities on the two rings (top). Reading out the activities by
averaging and by a maximum operation (bottom).
7 Discussion
Here we discuss how the various connection parameters contribute to the double-ring network to function as an integrator. In particular we
3 discuss how parameters have to be tuned
in order to yield an integration that is large in 6 and in .
: By assumption the synaptic time constant is large. has the simplest effect
of all parameters on the integrator properties. According to equation (8), scales
the range of . Notice that if were small, a large range of could be trivially
achieved. The art here is to achieve this with large .
* : The connection offset * between neurons receiving similar vestibular input
is the sole parameter besides determing the saturating head-velocity, beyond
which integration is impossible. According to equation (11), the saturating velocity is large if * is close to E (we want the saturating velocity to be large). In
other words, for good integration, excitatory connections should be strongest (or
inhibitory connections weakest) for neuron pairs with preferred head-directions
differing by close to E .
: The connection offset between neurons receiving different
- vestibular input
determines the anticipation time of thalamic neurons. If
is large, then ,
$
$
the activity offset in equation (13) is large. And,- because is proportional to
(assuming perfect integration), we conclude that should preferentially be large
(close to E - ) if is to be large. Notice that by equation (8), the range of is not
affected by .
VB4
V
and VB:4 The inter-ring connections should
V=4 be mainly excitatory, which imE was found to be optimal). The
plies that
should not be too negative (
3
intuitive reason is the following. We want the integration to be as linear in 6 as
possible, which means that we want our linear expansions (6) and (7) to deviate
as little as possible from (4). Hence, the differential gain between the two rings
should be small, which is the case when the two rings excite
each other. The inter3
ring excitation makes sure, even for large values of 6 , that there are comparable
activity levels on the two rings. This is one of the main points of this study.
M4
M
and
: The intra-ring connections should be mainly inhibitory, which implies
M4
that
should be strongly negative. The reason for this is that inhibition is necessary to result in proper and stable integration. Since
M 4 inhibition cannot come
from the inter-ring connections,
it
has
to
come
from
M
V . Notice also that according to equation (15),
cannot be much larger than
. If this were the case,
the persistent activity in the no head-movement caseV would
M become unstable. For
linear integration we have found that the condition
is necessary; small
deviations from this condition cause the integrator to become sub- or supralinear.
$
8 Conclusion
We have presented a theory for integration in the head-direction system with slow synapses.
We have found that in order to achieve a large range of linear integration, there should be
strong excitatory connections between neurons with dissimilar head-velocity tuning and
inhibitory connections between neurons with similar head-velocity tuning (see the discussion). Similar to models of the occulomotor integrator [Seung, 1996], we have found that
linear
can only be achieved by precise tuning of synaptic weights (for example
V
Nintegration
M
).
Appendix
, it is convenient to go into a moving
To study the traveling pulse solution with velocity
B'
coordinate frame by the change of variables
. The stationary solution in the
moving frame reads
W
'
and '
(12)
E . In order to find the fixed points of equation
(12),
we use the ansatz (4) and
Set
equate the coefficients of the 3 Fourier modes T , PGRT
and the -independent mode.
This leads to
A PT
M
V
QT 1*
'
(13)
3 4
(14)
M4L.VB4F 4
'
' PGRT
M XPGRT
V
M T
8
*
'
1*
(15)
where the functions
4
U
D4
and
D " T
are given by
'&
PGRT
10
D "
'
T
0<
E . Eq. (13) determines the
The above set of equations fully characterize the solution for
offset between the two rings, eq. (15) determines the threshold
, eq. (14) the amplitude
and eq. (5) the bias .
3
When the vestibular input 6
is small, we assume that the perturbed solution around
and takes the form:
4
PGRT
(')
'
PGRT
4F
S<
')
'
We linearize the dynamics (12) (to first order in
) and equate the Fourier coefficients.
This leads to
M T
T
0
*
T
'
(16)
"
!
' and ' . We determine and by solving the
where
'
PSR T
'
4F ' .
'
linearized dynamics of the differential mode
Comparing once more the Fourier coefficients leads to
6 3 " 3
' ' T
0 0
(17)
6 "
' ' T
(18)
M 4
V 4
'
. By substituting and into Eq. (16), we find equation (8).
where
References
[Blair et al., 1998] Blair, H., Cho, J., and Sharp, P. (1998). Role of the lateral mammillary
nucleus in the rat head direction circuit: A combined single unit recording and lesion
study. Neuron, 21:1387?1397.
[Blair and Sharp, 1995] Blair, H. and Sharp, P. (1995). Anticipatory head diirection signals
in anterior thalamus: evidence for a thalamocortical circuit that integrates angular head
motion to compute head direction. The Journal of Neuroscience, 15(9):6260?6270.
[Ermentrout, 1994] Ermentrout, B. (1994). Reduction of conductance-based models with
slow synapses to neural nets. Neural Computation, 6:679?695.
[Goodridge and Touretzky, 2000] Goodridge, J. and Touretzky, D. (2000). Modeling attractor deformation in the rodent head-direction system. The Journal of Neurophysiology, 83:3402?3410.
[Redish et al., 1996] Redish, A., Elga, A. N., and Touretzky, D. (1996). A coupled attractor
model of the rodent head direction system. Network: Computation in Neural Systems,
7:671?685.
[Seung, 1996] Seung, H. S. (1996). How the brain keeps the eyes still. Proc. Natl. Acad.
Sci. USA, 93:13339?13344.
[Zhang, 1996] Zhang, K. (1996). Representation of spatial orientation by the intrinsic
dynamics of the head-direction cell ensemble: A theory. J. Neurosci., 16(6):2112?2126.
| 2099 |@word neurophysiology:1 pulse:1 linearized:1 simulation:4 reduction:1 tuned:1 current:1 comparing:1 anterior:2 activation:2 designed:2 stationary:4 half:1 contribute:1 preference:1 simpler:1 zhang:4 differential:3 become:2 persistent:3 g4:1 inter:5 indeed:1 behavior:1 brain:2 integrator:5 little:1 jm:1 becomes:1 linearity:1 circuit:2 what:2 differing:1 finding:2 unit:1 medical:1 appear:1 inactivating:1 tends:1 acad:1 wondering:1 firing:14 modulation:3 studied:2 goodridge:7 range:11 responsible:2 hughes:1 lost:1 area:1 universal:1 empirical:1 convenient:1 word:2 induce:1 elga:1 anticipation:4 cannot:3 close:3 influence:1 impossible:1 xiaohui:1 resembling:1 go:1 independently:1 simplicity:1 pull:1 coordinate:1 agreement:1 velocity:32 observed:2 bottom:1 role:1 calculate:1 readout:1 decrease:1 movement:1 seung:5 ermentrout:3 dynamic:7 solving:1 purely:1 rightward:2 po:5 emulate:2 various:2 neurotransmitter:1 fast:4 larger:3 plausible:1 solve:1 net:1 maximal:1 achieve:3 intuitive:1 competition:1 double:7 perfect:4 ring:46 derive:1 linearize:2 pose:1 qt:1 sole:1 b0:4 eq:6 strong:2 implies:1 blair:6 come:2 direction:22 correct:1 around:2 bump:8 substituting:1 vary:1 proc:1 travel:1 integrates:1 mit:1 always:1 rather:1 mainly:2 dependent:1 integrated:1 entire:1 reproduce:1 interested:1 orientation:1 animal:2 art:1 integration:26 spatial:1 field:1 equal:2 once:1 minimized:1 adn:5 richard:1 inherent:1 ime:1 mover:1 attractor:6 conductance:1 intra:5 saturated:1 natl:1 necessary:3 rotating:1 desired:1 xhxie:1 deformation:1 modeling:1 deviation:1 too:1 characterize:1 perturbed:1 cho:1 combined:1 deduced:1 peak:1 receiving:2 ansatz:1 nm:2 cognitive:1 redish:4 sec:3 coefficient:3 multiplicative:1 performed:1 analyze:1 inter3:1 relied:1 thalamic:1 equate:2 ensemble:1 yield:3 conceptually:1 synapsis:9 strongest:1 touretzky:8 sebastian:1 synaptic:10 definition:1 gain:1 massachusetts:1 nmda:1 amplitude:3 xie:1 response:2 anticipatory:3 strongly:1 angular:4 traveling:3 horizontal:2 mode:3 indicated:1 usa:1 effect:1 asymmetrical:3 hence:2 spatially:1 symmetric:3 read:3 during:3 width:1 steady:2 excitation:1 rat:2 m:3 d4:1 motion:1 mammillary:2 recently:1 common:1 rotation:5 spiking:1 ji:1 he:1 relating:1 cambridge:1 tuning:10 postsubiculum:2 trivially:1 similarly:1 nonlinearity:1 had:1 moving:3 stable:1 inhibition:2 showed:1 leftward:1 certain:1 seen:2 determine:2 signal:3 full:1 multiple:1 thalamus:3 divided:1 qpsr:1 prediction:1 represent:1 achieved:3 cell:1 want:3 biased:2 sure:1 hz:1 tend:1 recording:1 extracting:1 architecture:2 shift:1 inactive:2 whether:2 cause:1 modulatory:1 amount:1 s4:1 simplest:1 exist:1 inhibitory:5 notice:4 dotted:1 neuroscience:1 correctly:1 write:1 affected:1 group:2 threshold:1 place:1 almost:1 appendix:4 vb:1 comparable:2 activity:15 strength:3 constraint:1 fourier:3 speed:2 according:3 gaba:1 smaller:1 rectification:1 equation:12 remains:1 turn:6 discus:2 operation:3 denotes:1 top:1 move:2 question:1 quantity:1 rotates:1 lateral:1 sci:1 unstable:1 reason:2 assuming:2 code:1 besides:1 preferentially:1 difficult:2 negative:2 proper:1 neuron:25 snapshot:1 howard:1 precise:1 head:54 frame:2 intermittent:2 sharp:4 pair:1 connection:29 vestibular:10 address:1 able:1 beyond:1 regime:1 reading:3 max:1 technology:1 inversely:1 eye:1 coupled:1 deviate:1 fully:1 sublinear:1 proportional:4 nucleus:4 integrate:1 degree:3 imposes:1 excitatory:7 thalamocortical:1 bias:1 institute:2 differentiating:1 curve:2 valid:2 made:2 approximate:2 supralinear:2 preferred:2 keep:1 reproduces:1 assumed:2 conclude:1 excite:1 continuous:1 expansion:1 main:1 neurosci:1 profile:1 lesion:1 determing:1 neuronal:1 fig:5 slow:8 sub:1 position:1 ply:1 formula:1 xt:1 badc:2 offset:10 weakest:1 evidence:1 intrinsic:1 adding:1 push:1 rodent:2 psr:4 saturating:8 determines:3 ma:1 hahnloser:1 change:1 determined:1 averaging:3 asymmetrically:1 modulated:1 dorsal:1 arises:1 dissimilar:1 dept:1 |
1,206 | 21 | 674
PA'ITERN CLASS DEGENERACY IN AN UNRESTRICfED STORAGE
DENSITY MEMORY
Christopher L. Scofield, Douglas L. Reilly,
Charles Elbaum, Leon N. Cooper
Nestor, Inc., 1 Richmond Square, Providence, Rhode Island,
02906.
ABSTRACT
The study of distributed memory systems has produced a
number of models which work well in limited domains.
However, until recently, the application of such systems to realworld problems has been difficult because of storage limitations,
and their inherent architectural (and for serial simulation,
computational) complexity.
Recent development of memories
with unrestricted storage capacity and economical feedforward
architectures has opened the way to the application of such
systems to complex pattern recognition problems.
However,
such problems are sometimes underspecified by the features
which describe the environment, and thus a significant portion
of the pattern environment is often non-separable.
We will
review current work on high density memory systems and their
network implementations.
We will discuss a general learning
algorithm for such high density memories and review its
application to separable point sets. Finally, we will introduce an
extension of this method for learning the probability
distributions of non-separable point sets.
INTRODUcnON
Information storage in distributed content addressable
memories has long been the topic of intense study.
Early
research focused on the development of correlation matrix
memories 1, 2, 3, 4. Workers in the field found that memories of
this sort allowed storage of a number of distinct memories no
larger than the number of dimensions of the input space.
Further storage beyond this number caused the system to give
an incorrect output for a memorized input.
@ American Institute of Physics 1988
675
Recent work on distributed memory systems has focused on
single layer, recurrent networks.
Hopfield 5, 6 introduced a
method for the analysis of settling of activity in recurrent
networks.
This method defined the network as a dynamical
system for which a global function called the 'energy' (actually a
Liapunov function for the autonomous system describing the
Hopfield showed that
state of the network) could be defined.
flow in state space is always toward the fixed points of the
dynamical system if the matrix of recurrent connections satisfies
certain conditions.
With this property, Hopfield was able to
define the fixed points as the sites of memories of network
acti vity.
Like its forerunners, the Hopfield network is limited in
storage capacity. Empirical study of the system found that for
randomly chosen memories, storage capacity was limited to m ~
O.lSN, where m is the number of memories that could be
accurately recalled, and N is the dimensionality of the network
(this has since been improved to m ~ N, 7, 8). The degradation of
memory recall with increased storage density is directly related
to the proliferation in the state space of unwanted local minima
which serve as basins of flow.
UNRESTRICIEn STORAGE DENSITY MEMORIES
Bachman et al. 9 have studied another relaxation system
similar in some respects to the Hopfield network. However, in
contrast to Hopfield, they have focused on defining a dynamical
system in which the locations of the minima are explicitly
known.
In particular, they have chosen a system with a Liapunov
function given by
E = -IlL ~
Qj I Il- Xj I - L,
(1)
J
where E is the total 'energy' of the
describing the initial network activity
and Xj' the site of the jth memory, for
parameter related to the network size.
= Xj for some memory j according to
network, Il (0) is a vector
caused by a test pattern,
m memories in RN. L is a
Then 1l(0) relaxes to Il(T)
676
(2)
This system is isomorphic to the classical electrostatic potential
between a positive (unit) test charge, and negative charges Qj at
the sites Xj (for a 3-dimensional input space, and L = 1). The Ndimensional Coulomb energy function then defines exactly m
basins of attraction to the fixed points located at the charge sites
Xj.
It can been shown that convergence to the closest distinct
memory is guaranteed, independent of the number of stored
memories m, for proper choice of Nand L 9, to.
Equation 1 shows that each cell receives feedback from the
network in the form of a scalar
~ Q-I Jl- x-I- L
J J
J
?
(3)
Importantly, this quantity is the same for all cells; it is as if a
single virtual cell was computing the distance in activity space
between the current state and stored states. The result of the
computation is then broadcast to all of the cells in the network.
A 2-layer feedforward network implementing such a system has
been described elsewhere 10 .
The connectivity for this architecture is of order m?N, where
m is the number of stored memories and N is the dimensionality
of layer 1.
This is significant since the addition of a new
memory m' = m + 1 will change the connectivity by the addition
of N + 1 connections, whereas in the Hopfield network, addition
of a new memory requires the addition of 2N + 1 connections.
An equilibrium feedforward network with similar properties
has been under investigation for some time 11. This model does
not employ a relaxation procedure, and thus was not originally
framed in the language of Liapunov functions.
However, it is
possible to define a similar system if we identify the locations of
the 'prototypes' of this model as? the locations in state space of
potentials which satisfy the following conditions
Ej
= -Qj lRo for I j.t - Xj I < Aj
=0
for I fl - Xj I > A].
(4)
677
where Ro is a constant.
This form of potential is often referred to as the 'square-well'
potential. This potential may be viewed as a limit of the Ndimensional Coulomb potential, in which the l/R (L = l) well is
replaced with a square well (for which L ? l). Equation 4
describes an energy landscape which consists of plateaus of zero
potential outside of wells with flat, zero slope basins. Since the
landscape has only flat regions separated by discontinuous
boundaries, the state of the network is always at equilibrium,
and relaxation does not occur. For this reason, this system has
been called an equilibrium model. This model, also referred to
as the Restricted Coulomb Energy (RCE)14 model, shares the
property of unrestricted storage density.
LEARNING IN HIGH DENSITY MEMORIES
A simple learning algorithm for the placement of the wells has
been described in detail elsewhere 11, 12.
Figurel: 3-layer feedforward network. Cell i
computes the quantity IJl - xii and compares
to internal threshold Ai.
678
Reilly et. al. have employed a three layer feedforward
network (figure 1) which allows the generalization of a content
addressable memory to a pattern classification memory.
Because the locations of the minima are explicitly known in the
equilibrium model, it is possible to dynamically program the
energy function for an arbitrary energy landscape. This allows
the construction of geographies of basins associated with the
classes constituting the pattern environment. Rapid learning of
complex, non-linear, disjoint, class regions is possible by this
method 12, 13.
LEARNING NON-SEPARABLE CLASS REGIONS
Previous studies have focused on the acquisition of the
geography and boundaries of non-linearly separable point sets.
However, a method by which such high density models can
acquire the probability distributions of non-separable sets has
not been described.
Non-separable sets are defined as point sets in the state
space of a system which are labelled with multiple class
affiliations.
This can occur because the input space has not
carried all of the features in the pattern environment, or because
the pattern set itself is not separable. Points may be degenerate
with respect to the explicit features of the space, however they
may have different probability distributions within the
environment.
This structure in the environment is important
information for the identification of patterns by such memories
10 the presence of feature space degeneracies.
We now describe one possible mechanism for the acquisition
of the probability distribution of non-separable points.
It is
assumed that all points in some region R of the state space of the
network are the site of events Jl (0, Ci ) which are examples of
pattern classes C = {C 1 , ... , CM }. A basin of attraction, xk( C i ),
defined by equation 4, is placed at each site fl(O, Ci ) unless
(5)
that is, unless a memory at Xj (of the class Ci ) already contains
fl(O, Ci )? The initial values of Qo and Ro at xk(Ci) are a constant for
all sites Xj. Thus as events of the classes C 1 , ... , C M occur at a
particular site in R, multiple wells are placed at this location.
679
If a well x/ C i) correctly covers an event Jl (0, C i ), then the
charge at that site (which defines the depth of the well) is
incremented by a constant amount ~ Q o. In this manner, the
region R is covered with wells of all classes {C 1 , ... , C M }, with the
depth of well XiCi) proportional to the frequency of occurence of
C i at Xj.
The architecture of this network is exactly the same as that
already described. As before, this network acquires a new cell
for each well placed in the energy landscape. Thus we are able
to describe the meaning of wells that overlap as the competition
by multiple cells in layer 2 in firing for the pattern of activity in
the input layer.
APPLICATIONS
This system has been applied to a problem in the area of risk
assessment in mortgage lending. The input space consisted of
feature detectors with continuous firing rates proportional to the
values of 23 variables in the application for a mortgage. For this
set of features, a significant portion of the space was nonseparable.
Figures 2a and 2b illustrate the probability distributions of
high and low risk applications for two of the features. It is clear
that in this 2-dimensional subspace, the regions of high and low
risk are non-separable but have different distributions.
t-----------#llir----- Prob. = 1.0.
1000 Patterns
Prob.
0.0
Feature 1
= 0.5
1.0
Figure 2a: Probability distribution for High
and Low risk patterns for feature 1.
680
=
Prob.
1.0.
t 000 Patterns
1-----1----\---------
Prob.
0.0
= 0.5
1.0
Feature 2
Figure 2b: Probability distribution for High
and Low risk patterns for feature 2.
Figure 3 depicts the probability distributions acquired by
the system for this 2-dimensional subspace.
In this image,
circle radius is proportional to the degree of risk: Small circles
are regions of low risk, and large circles are regions of high
risk.
00
o
o
V
0 0
0
0
o
0:>0
t?.
0
0
00
0
o
o
00
0
0
0
00
0
o
Feature 1
Figure 3: Probability distribition for Low and
High risk.
Small circles indicate low risk
regIons and large circles indicate high risk
regions.
681
Of particular interest is the clear clustering of high and low risk
regions in the 2-d map. Note that the regions are in fact nonlinearly separable.
DISCUSSION
We have presented a simple method for the acquisition of
probability distributions in non-separable point sets.
This
method generates an energy landscape of potential wells with
depths that are proportional to the local probability density of
the classes of patterns in the environment. These well depths
set the probability of firing of class cells In a 3-layer
feedforward network.
Application of this method to a problem in risk assessment
has shown that even completely non-separable subspaces may
be modeled with surprising accuracy.
This method improves
pattern classification in such problems with little additional
computational burden.
This algorithm has been run in conjunction with the method
described by Reilly et. al. II for separable regions. This combined
system is able to generate non-linear decision surfaces between
the separable zones, and approximate the probability
distributions of the non-separable zones in a seemless manner.
Further discussion of this system will appear in future reports.
Current work is focused on the development of a more
general method for modelling the scale of variations in the
distributions.
Sensitivity to this scale suggests that the
transition from separable to non-separable regions is smooth
and should not be handled with a 'hard' threshold.
ACKNOWLEDGEMENTS
We would like to thank Ed Collins and Sushmito Ghosh for their
significant contributions to this work through the development
of the mortgage risk assessment application.
REFERENCES
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interactive memory. Math. Biosci. 14, 197-220 (1972).
682
[2] Cooper, L.N.: A possible organization of animal memory and
learning. In: Proceedings of the Nobel Symposium on Collective
Properties of Physical Systems, Lundquist, B., Lundquist, S.
(eds.). (24), 252-264 London, New York: Academic Press 1973.
[3] Kohonen, T.: Correlation matrix memories.
IEEE Trans.
Comput. 21, 353-359 (1972).
[4] Kohonen, T.: Associative memory - a system-theoretical
approach. Berlin, Heidelberg, New York: Springer 1977.
[5] Hopfield, J.J.: Neural networks and physical systems with
emergent collective computational abilities. Proc. Natl. Acad. Sci.
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[6] Hopfield, J.J.: Neurons with graded response have collective
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[8] Potter, T.W.: Ph.D. Dissertation in advanced technology,
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published in Proc. Nati. Acad. Sci. USA.
[10] Dembo, A., Zeitouni, 0.: ARO Technical Report, Brown
University, Center for Neural Science, Pr0vidence, R.I., (1987),
also submitted to Phys. Rev. A.
[11] Reilly, D.L., Cooper, L.N., Elbaum, C.: A neural model for
category learning. BioI. Cybern. 45, 35 -41 (1982).
[12] Reilly, D.L., Scofield, C., Elbaum, C., Cooper, L.N.: Learning
system architectures composed of multiple learning modules. to
appear in Proc. First In1'1. Conf. on Neural Networks (1987).
[13] Rimey, R., Gouin, P., Scofield, C., Reilly, D.L.: Real-time 3-D
object classification using a learning system. Intelligent Robots
and Computer Vision, Proc. SPIE 726 (1986).
[14] Reilly, D.L., Scofield, C. L., Elbaum, C., Cooper, L.N: Neural
Networks with low connectivity and unrestricted memory
storage density. To be published.
| 21 |@word simulation:1 bachman:1 initial:2 contains:1 current:3 surprising:1 liapunov:3 xk:2 dembo:2 dissertation:1 math:1 lending:1 location:5 symposium:1 incorrect:1 consists:1 acti:1 unlearning:1 introduce:1 manner:2 acquired:1 rapid:1 proliferation:1 nonseparable:1 little:1 cm:1 elbaum:4 ghosh:1 charge:4 interactive:1 unwanted:1 exactly:2 ro:2 unit:1 appear:2 positive:1 before:1 local:2 limit:1 acad:3 firing:3 rhode:1 mortgage:3 studied:1 dynamically:1 suggests:1 limited:3 palmer:1 procedure:1 addressable:2 area:1 empirical:1 reilly:7 storage:13 risk:14 cybern:1 map:1 center:1 focused:5 stabilizing:1 rimey:1 attraction:2 importantly:1 autonomous:1 variation:1 construction:1 pa:1 recognition:1 located:1 underspecified:1 module:1 region:14 incremented:1 environment:7 complexity:1 serve:1 completely:1 hopfield:10 emergent:1 separated:1 distinct:2 describe:3 london:1 outside:1 larger:1 ability:1 itself:1 associative:1 aro:1 kohonen:2 degenerate:1 competition:1 convergence:1 generating:1 object:1 illustrate:1 recurrent:3 indicate:2 radius:1 discontinuous:1 opened:1 memorized:1 virtual:1 implementing:1 generalization:1 geography:2 investigation:1 extension:1 equilibrium:4 early:1 proc:5 always:2 ej:1 conjunction:1 modelling:1 richmond:1 contrast:1 nand:1 classification:3 ill:1 development:4 animal:1 field:1 future:1 report:2 intelligent:1 inherent:1 employ:1 randomly:1 composed:1 nestor:1 replaced:1 organization:1 interest:1 natl:2 worker:1 intense:1 unless:2 circle:5 theoretical:1 increased:1 cover:1 stored:3 providence:1 combined:1 density:11 sensitivity:1 physic:1 connectivity:3 broadcast:1 conf:1 american:1 potential:8 inc:1 satisfy:1 caused:2 explicitly:2 portion:2 sort:1 slope:1 contribution:1 square:3 il:3 accuracy:1 identify:1 landscape:5 identification:1 accurately:1 produced:1 economical:1 published:2 submitted:1 plateau:1 detector:1 phys:1 ed:2 energy:9 acquisition:3 frequency:1 associated:1 spie:1 degeneracy:2 recall:1 dimensionality:2 improves:1 actually:1 originally:1 response:1 improved:1 april:1 anderson:1 until:1 correlation:2 receives:1 christopher:1 qo:1 assessment:3 defines:2 aj:1 usa:3 effect:1 consisted:1 brown:1 acquires:1 ijl:1 meaning:1 image:1 recently:1 charles:1 physical:2 jl:3 significant:4 biosci:1 ai:1 framed:1 language:1 robot:1 surface:1 electrostatic:1 closest:1 recent:2 showed:1 introducnon:1 certain:1 vity:1 affiliation:1 minimum:3 unrestricted:3 additional:1 employed:1 july:1 ii:1 multiple:4 smooth:1 technical:1 academic:1 long:1 serial:1 vision:1 sometimes:1 cell:8 addition:4 whereas:1 flow:2 presence:1 feedforward:6 relaxes:1 xj:10 architecture:4 prototype:1 qj:3 handled:1 lsn:1 york:2 covered:1 clear:2 amount:1 ph:1 category:1 generate:1 lro:1 disjoint:1 correctly:1 xii:1 threshold:2 douglas:1 relaxation:4 realworld:1 prob:4 run:1 architectural:1 decision:1 layer:8 fl:3 guaranteed:1 activity:4 occur:3 placement:1 flat:2 generates:1 leon:1 separable:18 according:1 describes:1 island:1 rev:1 restricted:1 equation:3 discus:1 describing:2 mechanism:1 feinstein:1 coulomb:3 clustering:1 rce:1 zeitouni:2 graded:1 classical:1 already:2 quantity:2 in1:1 subspace:3 distance:1 thank:1 berlin:1 capacity:3 sci:3 topic:1 toward:1 reason:1 nobel:1 potter:1 modeled:1 acquire:1 difficult:1 negative:1 implementation:1 proper:1 collective:4 neuron:2 defining:1 rn:1 arbitrary:1 introduced:1 nonlinearly:1 unpublished:1 connection:3 recalled:1 trans:1 beyond:1 able:3 dynamical:3 pattern:16 program:1 memory:35 event:3 overlap:1 settling:1 ndimensional:2 advanced:1 technology:1 carried:1 occurence:1 review:2 acknowledgement:1 nati:1 limitation:1 proportional:4 degree:1 basin:5 share:1 elsewhere:2 placed:3 jth:1 scofield:4 institute:1 distributed:3 feedback:1 dimension:1 boundary:2 depth:4 transition:1 computes:1 constituting:1 approximate:1 global:1 assumed:1 continuous:1 nature:1 heidelberg:1 complex:2 domain:1 linearly:1 allowed:1 site:9 referred:2 depicts:1 cooper:6 explicit:1 comput:1 bachmann:1 burden:1 ci:5 scalar:1 springer:1 satisfies:1 bioi:1 viewed:1 labelled:1 content:2 change:1 hard:1 degradation:1 called:2 total:1 isomorphic:1 zone:2 internal:1 collins:1 |
1,207 | 210 | 100
Servan-Schreiber, Printz and Cohen
The Effect of Catecholamines on Performance:
From Unit to System Behavior
David Servan-Schreiber, Harry Printz and Jonathan D. Cohen
School of Computer Science and Department of Psychology
Carnegie Mellon University
Pittsburgh. PA 15213
ABSTRACT
At the level of individual neurons. catecholamine release increases the
responsivity of cells to excitatory and inhibitory inputs. We present a
model of catecholamine effects in a network of neural-like elements.
We argue that changes in the responsivity of individual elements do
not affect their ability to detect a signal and ignore noise. However.
the same changes in cell responsivity in a network of such elements do
improve the signal detection performance of the network as a whole. We
show how this result can be used in a computer simulation of behavior
to account for the effect of eNS stimulants on the signal detection
performance of human subjects.
1 Introduction
The catecholamines-norepinephrine and dopamine-are neuroactive substances that are
presumed to modulate information processing in the brain, rather than to convey discrete
sensory or motor signals. Release of norepinephrine and dopamine occurs over wide
areas of the central nervous system. and their post-synaptic effects are long lasting.
These effects consist primarily of an enhancement of the response of target cells to other
afferent inputs, inhibitory as well as excitatory (see [4] for a review).
Increases or decreases in catecholaminergic tone have many behavioral consequences
including effects on motivated behaviors. attention, learning and memory. and motor
The Effect of Catecholamines on Performance: From Unit to System Behavior
behavior. At the information processing level, catecholamines appear to affect the ability
to detect a signal when it is imbedded in noise (see review in [3]).
In terms of signal detection theory, this is described as a change in the performance of the
system. However, there is no adequate account of how these changes at the system level
relate to the effect of catecholamines on individual cells. Several investigators [5,12,2]
have suggested that catecholamine-mediated increases in a cell's responsivity can be
interpreted as a change in the cell's signal-to-noise ratio. By analogy, they proposed that
this change at the unit level may account for changes in signal detection performance at
the behavioral level.
In the first part of this paper we analyze the relation between unit responsivity, signal-tonoise ratio and signal detection performance in a network of neural elements. We start
by showing that the changes in unit responsivity induced by catecholamines do not result
in changes in signal detection performance of a single unit. We then explain how, in
spite of this fact, the aggregrate effect of such changes in a chain of units can lead to
improvements in the signal detection performance of the entire network.
In the second part, we show how changes in gain - which lead to an increase in the
signal detection performance of the network - can account for a behavioral phenomenon.
We describe a computer simulation of a network performing a signal detection task that
has been applied extensively to behavioral research: the continuous performance test.
In this simulation, increasing the responsivity of individual units leads to improvements
in performance that closely approximate the improvement observed in human subjects
under conditions of increased catecholaminergic tone.
2 Effect of Gain on a Single Element
We assume that the response of a typical neuron can be described by a strictly increasing
function !G(x) from real-valued inputs to the interval (0, 1). This function relates the
strength of a neuron's net afferent input x to its probability of firing, or activation. We
do not require that!G is either continuous or differentiable.
For instance, the family of logistics, given by
1
!G(x)
= 1 + e-(G%+B)
has been proposed as a model of neural activation functions [7,1]. These functions are
all strictly increasing, for each value of the gain G> 0, and all values of the bias B.
The potentiating effect of catecholamines on responsivity can be modelled as a change
in the shape of its activation function. In the case of the logistic, this is achieved by
increasing the value of G, as illustrated in Figure 1. However, our analysis applies to
any suitable family of functions, {fG}. We require only that each member function!G
is strictly increasing, and that as G -;. 00, the family {fG} converges monotonically to
101
102
Servan-Schreiber, Printz and Cohen
0.0
b==::::L::==-._:::::::::=-----I'---__
-6.0
....L...-_ _ _ _ _ _- - - '
-<lJ)
-2.0
OJ)
2J)
-IJ)
6J)
" (Nell""..,)
Figure 1: Logistic Activation Function, Used to Model the Response Function of Neurons. Positive
net inputs correspond to excitatory stimuli, negative net inputs correspond to inhibitory stimuli.
For the graphs drawn here, we set the bias B to -1. The asymmetry arising from a negative bias
is often found in the response function of actual neurons [6].
the unit step function Uo almost everywhere. 1 Here. Uo is defined as
0 for x < 0
uo(x) -- { 1 for x -> 0
This means that as G increases. the value !G(x) gets steadily closer to 1 if x > O. and
steadily closer to 0 if x < O.
2.1
Gain Does Not Affect Signal Detection Performance
Consider the signal detection performance of a network in which the response of a single
unit is compared with a threshold to determine the presence or absence of a signal. We
assume that in the presence of the signal. this unit receives a positive (excitatory) net
afferent input Xs. and in the absence of the signal it receives a null or negative (inhibitory)
input XA. When zero-mean noise is added to this quantity. in the presence as well as the
absent:e of the signal, the unit's net input in each case is distributed around Xs or XA
respectively. Therefore its response is distributed around !G(xs) or !G(XA) respectively
(see Figure 2).
In other words, the input in the case where the signal is present is a random variable
Xs ? with probability density function (pdt) PXs and mean Xs, and in the absence of the
signal it is the random variable XA? with pdf PXA and mean XA. These then determine
the random variables YGS =!G(Xs) and YGA =!G(XA). with pdfs PYas and PYGA' which
represent the response in the presence or absence of the signal for a given value of the
gain. Figure 2 shows examples of PYas and PYGA for two different values of G. in the case
where!G is the biased logistic.
If the input pdfs PXs and PXA overlap. the output pdfs PYas and PYGA will also overlap.
Thus for any given threshold () on the y-axis used to categorize the output as "signal
present" or "signal absent," there will be some misses and some false alarms. The best
1 A sequence of functions {gil} converges almost everywhere
diverges, or converges to the wrong value, is of measure zero.
to
the function g if the set of points where it
The Effect of Catecholamines on Performance: From Unit to System Behavior
01)
?21)
41)
21)
%
01)
?21)
21)
41)
61)
(Nelb'plll)
61)
% (Nelillplll)
---------p-~----
----~p~----------
Figure 2: Input and Output Probability Density Functions. The curves at the bottom are the pdfs
of the net input in the signal absent (left) and signal present (right) cases. The curves along the
y-axis are the response pdfs for each case; they are functions of the activation y, and represent the
distribution of outputs. The top graph shows the logistic and response pdfs for G = 0.5, B = -1;
the bottom graph shows them for G = 1. 0, B -1.
=
the system can do is to select a threshold that optimizes performance. More precisely,
the expected payoff or performance of the unit is given by
E(O) = A + a:.
Pr(YGS ~
0) - (3. Pr(YGA
~
0)
where A, a:, and (3 are constants that together reflect the prior probability of the signal,
and the payoffs associated with correct detections or hits, correct ignores, false alarms
and misses. Note that Pr(YGS ~ 0) and Pr(YGA ~ 0) are the probabilities of a hit and a
false alarm, respectively.
By solving the equation dE/dB = 0 we can determine the value 0* that maximizes E. We
call 0* the optimal threshold. Our first result is that for any activation function f that
satisfies our assumptions, and any fixed input pdfs PXs and PXA the unit's performance at
optimal threshold is the same. We call this the Constant Optimal Performance Theorem,
which is stated and proved in [10]. In particular, for the logistic, increasing the gain
G does not induce better performance. It may change the value of the threshold that
yields optimal performance, but it does not change the actual performance at optimum.
This is because a strictly increasing activation function produces a point-to-point mapping
between the distributions of input and output values. Since the amount of overlap between
103
104
Servan-Schreiber, Printz and Cohen
the two input pdfs PXs and PXA does not change as the gain varies, the amount of overlap
in the response pdfs does not change either, even though the shape of the response pdfs
does change when gain increases (see Figure 2). 2
3
Effect of Gain on a Chain of Elements
Although increasing the gain does not affect the signal detection performance of a single
element, it does improve the perfonnance of a chain of such elements. By a chain, we
mean an arrangement in which the output of the firs t unit provides the input to another
unit (see Figure 3). Let us call this second element the response unit We monitor the
output of this second unit to detennine the presence or absence of a signal.
Input Unit
x
Response Unit
z
y
v
Figure 3: A Chain of Units. The output of the unit receiving the signal is combined with noise
to provide input to a second unit, called the response unit. The activation of the response unit is
compared to a threshold to determine the presence or absence of the signal.
As in the previous discussion. noise is added to the net input to each unit in the chain
in the presence as well as in the absence of a signal. We represent noise as a random
variable V. with pdf PV that we assume to be independent of gain. As in the single-unit
case, the input to the first unit is a random variable Xs. with pdf PXs in the presence
of the signal and a random variable X A ? with pdf PXA in the absence of the signal. The
output of the first unit is described by the random variables YGS and YGA with pdfs PYas
and PYGA ? Now. because noise is added to the net input of the response unit as well. the
input of the response unit is the random variable Zas =YGS + V or ZGA = YGA + V. again
depending on whetber the signal is present or absent We write PZas and pz.ru for the
pdfs of these random variables. fJZos is the convolution of pyos and PV, and pz.ru is the
convolution of PYGA and Pv. The effect of convolving the output pdfs of the input unit
with the noise distribution is to increase the overlap between the resulting distributions
(PZas and pz.ru). and therefore decrease the discriminability of the input to the response
unit.
How are these distributions affected by an increase in gain on the input unit? By the
Constant Optimal Perfonnance Theorem. we already know that the overlap between PYGS
and PYGA remains constant as gain increases. Furthermore. as stated above, we have
assumed that the noise distribution is independent of gain. It would therefore seem that
a change in gain should not affect the overlap between PZos and pz.ru. However. it is
2We present the intuitions underlying our results in tenns of the overlap between the pdfs. However, the
proofs themselves are analytical.
The Effect of Catecholamines on Performance: From Unit to System Behavior
possible to show that. under very general conditions, the overlap between PZos and pz.a..
decreases when the gain of the input unit increases, thereby improving perfonnance of the
two-layered system. We call this the chain effect; the Chain Performance Theorem [10]
gives sufficient conditions for its appearance. 3
Paradoxically. the chain effect arises because the noise added to the net input of the
response unit is not affected by variations in the gain. As we mentioned before, increasing
the gain separates the means of the output pdfs of the input unit. I-'(YGS) and I-'(YGA)
(eventhough this does not affect the performance of the first unit). Suppose all the
probability mass were concentrated at these means. Then PZos would be a copy of Pv
centered at I-'(YGS). and pz.a.. would be a copy of pv centered at I-'(YGS). Thus in this case,
increasing the gain does correspond to rigidly translating PZos and PZat. apart, thereby
reducing their overlap and improving performance.
1
10
.
]
..
~
-4/J
?2/J
O/J
2/J
4/J
6/J
x(Ne' rnplll)
-4/J
?2/J
O/J
2/J
4/J
6/J
x (Nell""Ul)
---------p-~-------
-------p~------------
Figure 4: Dependence of Chain Output Pdfs Upon Gain. These graphs use the same conventions
and input pdfs as Figure 2. They depict the output pdfs, in the presence of additive Gaussian noise,
for G =0.5 (top) and G = 1.0 (bottom),
A similar effect arises in more general circumstances, when PYas and PY(JA are not concentrated at their means. Figure 4 provides an example. illustrating PZas and PZat. for
three different values of the gain. The first unit outputs are the same as in Figure 2, but
3In this discussion, we have assumed that the same noise was added to the net input into each unit of a
chain. However, the improvement in performance of a chain of units with increasing gain does not depend on
this particular assumption.
105
106
Servan-Schreiber, Printz and Cohen
these have been convolved with the pdf PV of a Gaussian random variable to obtain the
curves shown. Careful inspection of the figure will reveal that the overlap between PZa
and PZaA decreases as the gain rises.
4 Simulation of the Continuous Performance Test
The above analysis has shown that increasing the gain of the response function of individual units in a very simple network can improve signal detection performance. We
now present computer simulation results showing that this phenomenon may account for
improvements of performance with catecholamine agonists in a common behavioral test
of signal detection.
The continous performance test (CPT) has been used extensively to study attention and
vigilance in behavioral and clinical research. Performance on this task has been shown to
be sensitive to drugs or pathological conditions affecting catecholamine systems [11.8.9].
In this task, individual letters are displayed tachystoscopically in a sequence on a computer
monitor. In one common version of the task, a target event is to be reported when two
consecutive letters are identical. During baseline performance. subjects typically fail to
report 10 to 20% of targets ("misses") and inappropriately report a target during 0.5 to
1% of the remaining events ("false alarms"). Following the administration of agents
that directly release catecholamines from synaptic terminals and block re-uptake from
the synaptic cleft (i.e., CNS stimulants such as amphetamines or methylphenidate) the
number of misses decreases. while the number of false alarms remains approximately the
same. Using standard signal detection theory measures, investigators have claimed that
this pattern of results reflects an improvement in the discrimination between signal and
non-signal events (d'), while the response criterion (f3) does not vary significantly [11.8,9].
We used the backpropagation learning algorithm to train a recurrent three layer network
to perform the CPT (see Figure 5). In this model, several simplifyng assumptions made
in the preceding section are removed: in contrast to the simple two-unit assembly. the
network contains three layers of units (input layer, intermediate - or hidden - layer, and
output layer) with some recurrent connections; connection weights between these layers
are developed entirely by the training procedure; as a result, the activation patterns on
the intermediate layer that are evoked by the presence or absence of a signal are also
determined solely by the training procedure; finally. the representation of the signal is
distributed over an ensemble of units rather than determined by a single unit
Following training, Gaussian noise with zero mean was added to the net input of each unit
in the intermediate and output layers as each letter was presented. The overall standard
deviation of the noise distribution and the threshold of the response unit were adjusted
to produce a performance equivalent to that of subjects under baseline conditions (13.0%
misses and 0.75% false alarms). We then increased the gain of all the intermediate
and output units from 1.0 to 1.1 to simulate the effect of catecholamine release in the
network. This manipulation resulted in rates of 6.6% misses and 0.78% false alarms.
The correspondence between the network's behavior and empirical data is illustrated in
Figure 5.
The EfTect of Catecholamines on Performance: From Unit to System Behavior
Letter Identification Module
c~5
16 ......-
~~~
"r. .??~J
I \
~ .y
_
_ _ _ _ _ _ _ _ _ _......,
......
~F. . ........
--0-
Sim. ... _
_
6om.F._
;
J
t
ol-____.........
~O::::::::::~I~__--J
a.s
&lmoAonI
Feature Input Module
Figure 5: Simulation of the Continuous Performance Task. Len panel: The recurrent three-layer
network (12 input units, 30 intermediate units, 10 output units and 1 response unit). Each unit
projects to all units in the subsequent layer. In addition, each output unit also projects to each unit
in the intermediate layer. The gain parameter G is the same for all intermediate and output units.
In the simulation of the placebo condition, G = 1; in the simulation of the drug condition, G = 1.1.
The bias B = -1 in both conditions. Right panel: Performance of human subjects [9], and of the
simulation, on the CPT. With methylphenidate misses dropped from 11.7% to 5.5%, false alarms
decreased from 0.6% to 0.5% (non-significant).
The enhancement of signal detection performance in the simulation is a robust effect. It
appears when gain is increased in the intermediate layer only, in the output layer only,
or in both layers. Because of the recurrent connections between the output layer and
the intermediate layer, a chain effect occurs between these two layers when the gain is
increased over anyone of them, or both of them. The impact of the chain effect is to
reduce the distortion, due to internal noise, of the distributed representation on the layer
receiving inputs from the layer where gain is increased. Note also that the improvement
takes place even though there is no noise added to the input of the response unit. The
response unit in this network acts only as an indicator of the strength of the signal in
the intermediate layer. Finally, as the Constant Optimal Performance Theorem predicts,
increasing the gain only on the response unit does not affect the performance of the
network.
5 Conclusion
Fluctuations in catecholaminergic tone accompany psychological states such as arousal,
motivation and stress. Furthermore, dysfunctions of catecholamine systems are implicated
in several of the major psychiatric disorders. However, in the absence of models relating
changes in cell function to changes in system performance, the relation of catecholamines
to behavior has remained obscure. The findings reported in this paper suggest that the
behavioral impact of catecholamines depend on their effects on an ensemble of units
operating in the presence of noise, and not just on changes in individual unit responses.
107
108
Servan-Schreiber, Printz and Cohen
Furthermore. they indicate how neuromodulatory effects can be incorporated in parallel
distributed processing models of behavior.
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1,208 | 2,100 | Efficiency versus Convergence of Boolean
Kernels for On-Line Learning Algorithms
Roni Khardon
Tufts University
Medford, MA 02155
[email protected]
Dan Roth
University of Illinois
Urbana, IL 61801
[email protected]
Rocco Servedio
Harvard University
Cambridge, MA 02138
[email protected]
Abstract
We study online learning in Boolean domains using kernels which capture feature expansions equivalent to using conjunctions over basic features. We demonstrate a tradeoff between the computational efficiency
with which these kernels can be computed and the generalization ability of the resulting classifier. We first describe several kernel functions
which capture either limited forms of conjunctions or all conjunctions.
We show that these kernels can be used to efficiently run the Perceptron algorithm over an exponential number of conjunctions; however we
also prove that using such kernels the Perceptron algorithm can make
an exponential number of mistakes even when learning simple functions. We also consider an analogous use of kernel functions to run the
multiplicative-update Winnow algorithm over an expanded feature space
of exponentially many conjunctions. While known upper bounds imply
that Winnow can learn DNF formulae with a polynomial mistake bound
in this setting, we prove that it is computationally hard to simulate Winnow?s behavior for learning DNF over such a feature set, and thus that
such kernel functions for Winnow are not efficiently computable.
1 Introduction
The Perceptron and Winnow algorithms are well known learning algorithms that make predictions using a linear function in their feature space. Despite their limited expressiveness,
they have been applied successfully in recent years to several large scale real world classification problems. The SNoW system [7, 2], for example, has successfully applied variations
of Perceptron [6] and Winnow [4] to problems in natural language processing. The system
first extracts Boolean features from examples (given as text) and then runs learning algorithms over restricted conjunctions of these basic features.
There are several ways to enhance the set of features after the initial extraction. One idea is
to expand the set of basic features using conjunctions such as
and use
these expanded higher-dimensional examples, in which each conjunction plays the role of
a basic feature, for learning. This approach clearly leads to an increase in expressiveness
and thus may improve
performance. However, it also dramatically increases the number of
features (from
to
if all conjunctions are used) and thus may adversely affect both the
computation time and convergence rate of learning.
This paper studies the computational efficiency and convergence of the Perceptron and
Winnow algorithms over such expanded feature spaces of conjunctions. Specifically, we
study the use of kernel functions to expand the feature space and thus enhance the learning abilities of Perceptron and Winnow; we refer to these enhanced algorithms as kernel
Perceptron and kernel Winnow.
1.1
Background: Perceptron and Winnow
Throughout its execution Perceptron maintains a weight vector
which is initially
Upon receiving an example
the algorithm predicts according to the
If the prediction is and the label is (false positive
linear threshold function
prediction) then the vector is set to
, while if the prediction is
and the label is
No change is made if the prediction is correct.
(false negative) then is set to
The famous Perceptron Convergence Theorem [6] bounds the number of mistakes which
the Perceptron algorithm can make:
( ') +* , -. /10
-1 "! / "for#all
,
2436; 583795:3
The Winnow algorithm [4] has a very similar structure.
Winnow maintains a hypothesis
vector <# which is initially >= BA Winnow is parameterized by a promotion
factor ? and a threshold @ 0
upon receiving an example C' +*D Winnow
predicts according to the threshold function ( @ If the prediction is and the label is
demotion step. If
then for all , such that =E the value of is set to GF ? ; =Hthis isthea value
of
is set
the prediction is and the label is then for all , such that
to ?I ; this is a promotion step. No change is made if the prediction is correct.
For our purposes the following mistake bound, implicit in [4], is of interest:
=
Theorem 2 Let the target function be a J -literal monotone disjunction K
in ' +*D labeled according
to
K the number
8L M NN M PO For any sequence of examples
of prediction mistakes made by Winnow ? @ is at most Q S J ?TU VXWZY\[ @
Q
QBR
$ $%&
be a sequence of labeled examples with
Theorem 1 Let
and
for all . Let
be such that
Then Perceptron makes at most
mistakes on this example sequence.
1.2
Our Results
Our first result in Section 2 shows that it is possible to efficiently run the kernel Perceptron
algorithm over an exponential number of conjunctive features:
]
]
Theorem 3 There is an algorithm that simulates Perceptron over the -dimensional feature space
of labeled examples
of all conjunctions of
basic features. Given a sequence
the prediction and update for each example take poly
time steps.
in
' ^*
This result is closely related to one of the main open problems in learning theory: efficient
learnability of disjunctions of conjunctions, or DNF (Disjunctive Normal Form) expres
is true
sions.1 Since linear threshold elements can represent disjunctions (e.g.
iff
), Theorems 1 and 3 imply that kernel Perceptron can be used to learn
DNF. However, in this framework the values of
and in Theorem 1 can be exponentially large, and hence the mistake bound given by Theorem 1 is exponential rather than
polynomial in
The question thus arises whether, for kernel Perceptron, the exponential
!_
`
&
M !_ M
1
Angluin [1] proved that DNF expressions cannot be learned efficiently using hypotheses which
are themselves DNF expressions from equivalence queries and thus also in the mistake bound model
which we are considering here. However this result does not preclude the efficient learnability of DNF
using a different class of hypotheses such as those generated by the kernel Perceptron algorithm.
upper bound implied by Theorem 1 is essentially tight. We give an affirmative answer, thus
showing that kernel Perceptron cannot efficiently learn DNF:
K
Theorem 4 There is a monotone DNF over and a sequence ofexamples
la
beled according to which causes the kernel Perceptron algorithm to make
mistakes.
K
@ =H` ? @
Turning to Winnow, an attractive feature of Theorem 2 is that for suitable the bound
is logarithmic in the total number of features
(e.g.
). Therefore, as
and
noted by several researchers [5], if a Winnow analogue of Theorem 3 could be obtained
this would imply efficient learnability of DNF. We show that no such analogue can exist:
`
?=
Theorem 5 There is no polynomial time algorithm which simulates Winnow over exponentially many monotone conjunctive features for learning monotone DNF, unless every
problem in #P can be solved in polynomial time.
We observe that, in contrast to Theorem 5, Maass and Warmuth have shown that the Winnow algorithm can be simulated efficiently over exponentially many conjunctive features
for learning some simple geometric concept classes [5].
While several of our results are negative, in practice one can achieve good performance
by using kernel Perceptron (if
is small) or the limited-conjunction kernel described in
Section 2 (if
is large). This is similar to common practice with polynomial kernels 2 where
typically a small degree is used to aid convergence. These observations are supported by
our preliminary experiments in an NLP domain which are not reported here.
2 Theorem 3: Kernel Perceptron with Exponentially Many Features
It is easily observed, and well known, that the hypothesis of the Perceptron algorithm
is
made. If we let
a sum of the previous examples on which prediction mistakes were
denote the label of example , then
where is the
set of examples on which the algorithm
made
a
mistake.
Thus
the
prediction
of Perceptron
on is 1 iff
.
"' ^ *
=
U =
B
=
For an example ' + * let
denote its transformation into an enhanced feature
space such as the space of all conjunctions.
To run
the Perceptron algorithm over the
where
is the
enhanced space we must predict iff
B
weight
B
vector
in the .
enhanced space;
from
the
above
discussion
this
holds
iff
Denoting
= B
this holds iff
.
Thus we never need to construct the enhanced
feature space explicitly; we need only be
able to compute the kernel function
efficiently. This is the idea behind all so-called
kernel methods, which can be applied to any algorithm (such as support vector machines)
whose prediction is a function of inner products of examples; see e.g. [3] for a discussion.
The result in Theorem 3 is simply obtained by presenting a kernel function capturing all
conjunctions. We also describe kernels for all monotone conjunctions which allow no
negative literals, and kernels capturing all (monotone) conjunctions of up to literals.
J
The general
includes all conjunctions (with positive and negative
case:
mustWhen
literals)
compute the number of conjunctions which are true in both and
. Clearly, any literal in such a conjunction must satisfy both and and thus the corresponding
must have the
= bit "!$in# &%' ( where
same
value. Counting all such conjunctions gives
)+*-,/. is the number of original features that have the
same value in and . This kernel has been obtained independently by [8].
2
Our Boolean kernels are different than standard polynomial kernels in that all the conjunctions
are weighted equally. While expressive power does not change, convergence and behavior, do.
Monotone Monomials: In some applications the total number
of basic features may
be very large but in any one example only a small number of features take value 1. In
other applications the number of features
may not be known in advance (e.g. due to
unseen words in text domains). In these cases it may be useful to consider only&%monotone
'(
+! #
monomials. To express all monotone monomials we take
where
)+*,/. ) is the number of active features common to both and .
=
!Y
A parameterized kernel: In general, one may want to trade off expressivity against
number of examples and convergence time. Thus we consider a parameterized kernel
which captures all conjunctions of size at most for some
&%The
' ( number of
. This kersuch conjunctions that satisfy both and is
"!$#
nel is reported also in [10].
For
monotone
conjunctions
of
size
at
most
we have
&%' (
.
+! #
=
J
=
J
J
3 Theorem 4: Kernel Perceptron with Exponentially Many Mistakes
We describe a monotone DNF target function and a sequence of labeled examples which
cause the monotone kernel Perceptron algorithm to make exponentially many mistakes.
C'
+* we write to denote the number of 1?s in and . to denote
)+*,/.!Y)
We use the following lemma (constants have not been optimized):
-bit
*
% ' +* % with ]
=
strings =E' %
Lemma 6 There is a set of
%
%
such that \=
F
for
, ] and ! #"
F%$ for ,&(' ]
Proof: The proof uses the probabilistic method. For each ,V= ] let ! X' +* be
probability 1/10.
chosen by independently
A setting each bit to with
- For
% Rany 2 , itandis clear
that )+*, - =
F a Chernoff bound implies
that
thus
.0/%*1 !
2 F
Similarly, for any ,=6
the probability that any satisfies
is
at
most
3
4
]
5 %'
R
F
7 #" -#=
DBA a Chernoff bound implies that .8/%*1 9 #" 0
+
)
1
*
we have
F
F
:
$
2
and thus the probability that any " with ;
R
,
=
6
5 2 ' satisfies
" 0
%
F
$ is at
<
2
3
<
2
4] R % is less than 1. Thus
For ] =
most _ R
the value of _ R
=
F and # #"
:F $ For any
we have each
for some choice of
0
which has
F we can set ^
F of the 1s to 0s, and the lemma is proved.
The target DNF is very simple: it is the single conjunction !_ While
the original
Perceptron algorithm over the
features makes at most poly
mistakes for
this target function,
we now show that the monotone kernel
Perceptron algorithm which
runs over all
monotone monomials can make >
mistakes.
Recall that at the beginning of the Perceptron algorithm?s
execution all
A since
= coordinates
Perceptronof
are 0. The first example is the negative example
incorrectly predicts 1 on this example. The resulting update causes the coefficient ?
corresponding
to the empty monomial (satisfied by any example ) to become but all
other coordinates of remain
0. The next example is the positive example
For this
=
so Perceptron incorrectly predicts Since
example we have
all
monotone conjunctions
are satisfied by this example the resulting update causes
? to become 0 and all
other
coordinates of to become 1. The next
described in Lemma 6. Since each such example has
examples are
the vectors
)=
F each
example is negative; however as we now show the Perceptron algorithm
will predict on each of these examples.
% , % and consider the hypothesis vector just before
Fix any value
example
is received. Since !=
F the value of
is a sum of the _ different
For
!
coordinates which correspond to the monomials satisfied by More precisely we
where contains the monomials which are
have
and #" for some '65
and contains the monomials which are satisfied
satisfied by
by but no #" with '5 We lower bound the two sums separately.
Let
be any monomial in By
contains at most
%$ variables
2 Lemma
6 any
monomials in Using the well known bound
and thus there can be
at
most
"
; "
and
is the binary entropy function there can be
where
34
at most
terms in
Moreover the value of each must be at least34 since
decreases by at most 1 for each example, and hence
;
By Lemma 6 for
On the other hand, for any
we clearly have
:$ every
any
satisfied by
must belong to and hence
3 -variable
monomial
Combining these inequalities we have
< 2
and hence the Perceptron prediction on is 1.
=
%
Q
Q
=,
)
=E,
_
? % F
F
H
?
=
_
0 F _
_
0
_
0
0
^*
.
'
are all nonempty monomials
conjunctions) over A sequence of la (monotone
consistent if it is consistent with some
beled examples
% " for isallmonotone
implies % " If is monotone
monotone function, i.e.
J
=
consistent and has ] labeled examples then clearly there is a monotone DNF formula consistent with which contains at most ] conjunctions. We consider the following problem:
KERNEL WINNOW PREDICTION ? @ (KWP)
of labeled examples
Instance: Monotone
consistent sequence C= A example
with each .' ^ *
and each .') +* unlabeled
' +*
Question: Is
vector
@ where
is the ` = . -dimensional
hypothesis
?
generated by running Winnow ? @ on the example sequence
In order to run Winnow over all
. nonempty monomials to learn monotone DNF, one
must be able to solve KWP efficiently. The main result of this section is proved by showing
4 Theorem 5: Learning DNF with Kernel Winnow is Hard
In this section, for
denotes the
-element vector whose coordinates
that KWP is computationally hard for any parameter settings which yield a polynomial
mistake bound for Winnow via Theorem 2.
Q S ? >
`
=
U and ? 0 @ be such that , *
QBR
WZY\[ Q @
=
Then KWP ? @
is #P-hard.
Proof of
it can easily be verified that
Theorem 7: For
! `
? and_#" @ as described above
? poly and @ The proof of the theorem is a
Theorem 7 Let
!
poly
poly
poly
reduction from the following #P-hard problem [9]: (See [9] also for details on #P.)
poly
= 8L4M 3
' +* ?
%_
= %
MONOTONE 2-SAT (M2SAT)
Instance: Monotone 2-CNF Boolean formula $ &% (' % )' *' % with %
integer such that
and each ,+
Question: Is $
i.e. does $ have at least satisfying assignments in
'
* A
R
4.1
High-Level Idea of the Proof
The high level idea of the proof is simple: let $
be an instance of M2SAT where
$ is defined over variables The Winnow algorithm maintains a weight
for each monomial
over variables We define
a 1-1 correspondence between
for $ and we give a sequence of
these monomials
and truth assignments
examples for Winnow which causes .if $
and
if $
' +*
=
=
=
A
some additional work ensures that
R
@
R
W YD[ ? =
=
_
In more detail, let =
_ WZYD[
Q
Q
Q
and
=
C
We describe a polynomial time transformation
which maps
an
-variable instance $
of M2SAT to an -variable instance of
KWP
consistent, each and belong
is monotone
? @ where X=E
to ' ^* and @ if and only if $ R
The Winnow variables
are divided into three sets and where
=
* = ' * and = ' * The unlabeled ex'ample
in and
are set to 1 and all variables in
is
R R
i.e. all variables
are set to 0. We thus have
= ( where = ?
= ?
and (
= '
? '
? We refer to monomials
= 5
as typemonomials,
= 5
as type- monomials, and
!
= 5 as type- monomials.
= 5
monomials
monomials
A
The example sequence is divided into four stages. Stage 1 results in - $ R as
described below the
variables in
correspond to the
variables in the CNF formula $
Stage 2 results in -
? " $ R for some final
positive integer # Stages
3 and 4 together
result in
(
- @ ? " Thus the
value
of
is
@# ? " $ R ^ so we have
@ if and only if $ R
approximately
Since all variables in
are 0 in if
includes a variable in then the value of
does not affect The variables in are ?slack variables? which (i) make Winnow
perform the correct promotions/demotions and (ii) ensure that is monotone consistent.
The value of is thus related to $
if and only if $
4.2
Details of the Proof
$& %('* ),+.-0/214365)
' +*
7 &8 =
= LM 3
8L = 3 =
=
= < % ? R_9
=
= R ;:
!:
? R9
We now show how the Stage 1 examples cause Winnow
to make a false positive predic and
for all other , in
tion on negative examples which have 8L =
=
=
as described above. For each such negative example
new slack variables
@ F 1
six repeated
=< =< ! are used as follows: Stage 13 hasinPW YDStage
instances of
[
the positive example which has >< % = =< _ = and all Q other bits 0. These examples
%@ ?BA L %@?BA %@?BA L %@?BA ? @ and hence
cause promotions which result in @
@
%
B
?
A
3 first with ><3
= =< =
groups of similar examples (the
L
@ F Two=< other
=<
C = = ) cause %@?BAD @ F and %@?BAE @ F The
the second with
next example in is the negative example which has 8L =
=
= example
for
=<
=<
=<
C
3
all other
in
and
all
other
bits
0.
For
this
=
=
=
0 %@ ?BA L %@ ?BAD (%@?BAE @ so Winnow makes a false positive prediction.
_
Since $ has at most
clauses and there
are negative examples per clause, this con_
struction can be carried out using
slack variables
9 3
F IHJ ) +. -0 /6 1K365) . The
Stage 2: Setting $&%'G
first Stage 2 example is a positive example
with
= for all ,
9 3 = and all other bits 0. Since each of the
Stage 1: Setting
. We define the following correspondence between
and monomials
truth assignments
if and only if
is not
present in
For each clause
in $ Stage 1 contains negative examples such
Assuming that (1) Winnow makes a
that
and
for all other
false positive prediction on each of these examples and (2) in Stage 1 Winnow never does a
promotion on any example which has any variable in set to 1, then after Stage 1 we will
Thus we will have
have that
if $
and
if $
for some
$
=
= R
9
6 ;3 :
= ? @ 0
: F
U ?
0
R
%
#=PW YD[ Q @ F
? " U@ ? "
#
= ?" R
N; :
"?
" @
N;:
R
?"
=>? " R N : U? " U@ F
$ '
=
=
=
? "
6 :
R
.
.
=>@) ? " R
@B
? @ " :
9
R R
: _
%
@
.? "
U@
( .? " _
@
R
For ease of notation let denote @9
the examples
? " We now
% describe
E _ ? "
in Stages 3 and 4 and show that they will cause
to satisfy
so ? " R % ? " and hence there is a unique smallest integer such that
Let %
= PW YD[
% ? " R Q ? " The Stage% 3 examples
will result in
% Using the
definition
it can be verified
we? " R have
the % fact that
? that ? " R
Hence
H % ? " of% @ and
?
?
?
H
=
?
"
"
"
"
R
R
? %
Q
=
We use the following lemma:
% % there is a monotone CNF $ ' over
Lemma 8 For all for all
' Boolean
variables which has at most clauses, has exactly satisfying assignments in
+* and can be constructed from and in poly
time.
Proof: The proof is by induction on . For the base case (= we have = and
Assuming the lemma is true for =< J we now prove it for = J 8
$ ' =
% % then the desired CNF is $ ' = ' $ -' Since $ -' has at most J
If
% % then the desired CNF
"
clauses $ ' has at most J " clauses. If
M $ -' _ O By distributing over each clause of $ -' _ O we can write
is $ ' =
R
'
as a CNF with at most J clauses. If T
$
= then $ ' = R
and are satisfied by this example have
monomials which contain
!
$
poly
we have -
after
Since
$
the resulting promotion we have
$
so
Let
Stage 2 consists of repeated
instances of the
described
above. After these promotions
we have
positive
example
Since $
we also
$
have
$
Stage 3: Setting
. At the start of Stage 3 each type- and type- monomial
There are
variables in and variables in so at the start of Stage 2
has
Since no example
we have
and (
in Stages 3 or 4
and (
satisfies any in
at the end of Stage 4
will still be
$
will still
Thus ideally at the end of Stage 4 the value of would
be
be
since this would imply that
$
for
which is at least if and only if $
However it is not necessary
as long as
must be an integer and
to assume
this exact value; since $
we will have
(
that
if and only if $
Let $ ' be an -clause monotone CNF formula over the variables in which has
satisfying assignments. Similar to Stage 1, for each clause of $ ' , Stage 3 has
negative
examples corresponding to that clause, and as in Stage 1 slack variables in are used to
ensure that Winnow makes a false positive prediction on each such negative example.
Thus
the examples in Stage 3 cause
where
Since six
slack variables in are used for each negative example and there are
negative
examples, the slack variables
are sufficient for Stage 3.
Stage 4: Setting
. All that remains is to perform &%
promotions on examples which have each in set to 1. This will cause
which is as desired.
:_
R_
=
:_
_
9
3
$ $7% ' FIH
;: _
? " R = C ? " : _ ? " R E _ ? "
? R %
% ? " #1R
#
R Q
=
.: _
R Q
% The first
and % that
It can be verified from the definitions of
examples in are all the same positive example which has each
in set to 1 and
The first time this example is received
It can
R =<
performs a promotion; after #
R
occurrences
" @
so Winnow
Q%
_
"
R
of this example -
=
<
?
:
? " R
?
>@
"
_
and
=>? " R
;:
The remaining examples in Stage 4 are
R
% repetitions of the positive example
Q
which has each in set to 1 and
=
If promotions
on each repetition of
R occurred
" R
: _
so
this example then we would have
=
?
?
R
we need
show that this is less than @ We reexpress this quantity as ?
? " R Ionly
: _
We have ? " R I : _
? " R % ? _ " R % @
? " ? " U@9 _ ? "
Some
R
Q_
R _ ? " so indeed U@
easy manipulations show that ?
be verified that
? @=
Finally, we observe that by construction the example sequence is monotone consistent.
Since
poly
and contains poly
examples the transformation from M2SAT to
KWP
is polynomial-time computable and the theorem is proved.
(Theorem 7)
5 Conclusion
It is necessary to expand the feature space if linear learning algorithms are to learn expressive functions. This work explores the tradeoff between computational efficiency and
convergence (i.e. generalization ability) when using expanded feature spaces. We have
shown that additive and multiplicative update algorithms differ significantly in this respect;
we believe that this fact could have significant practical implications. Future directions
include the utilization of the kernels developed here and studying convergence issues of
Boolean-kernel Perceptron and Support Vector Machines in the PAC model.
Acknowledgements: R. Khardon was supported by NSF grant IIS-0099446. D. Roth
was supported by NSF grants ITR-IIS-00-85836 and IIS-9984168 and by EPSRC grant
GR/N03167 while visiting University of Edinburgh. R. Servedio was supported by NSF
grant CCR-98-77049 and by a NSF Mathematical Sciences Postdoctoral Fellowship.
References
[1] D. Angluin. Negative results for equivalence queries. Machine Learning, 2:121?150, 1990.
[2] A. Carlson, C. Cumby, J. Rosen, and D. Roth. The SNoW learning architecture. Technical
Report UIUCDCS-R-99-2101, UIUC Computer Science Department, May 1999.
[3] N. Cristianini and J. Shaw-Taylor. An Introduction to Support Vector Machines. Cambridge
Press, 2000.
[4] N. Littlestone. Learning quickly when irrelevant attributes abound: A new linear-threshold
algorithm. Machine Learning, 2:285?318, 1988.
[5] W. Maass and M. K. Warmuth. Efficient learning with virtual threshold gates. Information and
Computation, 141(1):378?386, 1998.
[6] A. Novikoff. On convergence proofs for perceptrons. In Proceeding of the Symposium on the
Mathematical Theory of Automata, volume 12, pages 615?622, 1963.
[7] D. Roth. Learning to resolve natural language ambiguities: A unified approach. In Proc. of the
American Association of Artificial Intelligence, pages 806?813, 1998.
[8] K. Sadohara. Learning of boolean functions using support vector machines. In Proc. of the
Conference on Algorithmic Learning Theory, pages 106?118. Springer, 2001. LNAI 2225.
[9] L. G. Valiant. The complexity of enumeration and reliability problems. SIAM Journal of Computing, 8:410?421, 1979.
[10] C. Watkins. Kernels from matching operations. Technical Report CSD-TR-98-07, Computer
Science Department, Royal Holloway, University of London, 1999.
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1,209 | 2,101 | An Efficient, Exact Algorithm for Solving
Tree-Structured Graphical Games
Michael L. Littman
AT&T Labs- Research
Florham Park, NJ 07932-0971
mlittman?research.att.com
Michael Kearns
Department of Computer & Information Science
University of Pennsylvania
Philadelphia, PA 19104-6389
mkearns?cis.upenn.edu
Satinder Singh
Syntek Capital
New York, NY 10019-4460
baveja?cs. colorado. edu
Abstract
We describe a new algorithm for computing a Nash equilibrium in
graphical games, a compact representation for multi-agent systems
that we introduced in previous work. The algorithm is the first
to compute equilibria both efficiently and exactly for a non-trivial
class of graphical games.
1
Introduction
Seeking to replicate the representational and computational benefits that graphical models have provided to probabilistic inference, several recent works
have introduced graph-theoretic frameworks for the study of multi-agent systems (La Mura 2000; Koller and Milch 2001; Kearns et al. 2001). In the simplest
of these formalisms, each vertex represents a single agent, and the edges represent
pairwise interaction between agents. As with many familiar network models, the
macroscopic behavior of a large system is thus implicitly described by its local interactions, and the computational challenge is to extract the global states of interest.
Classical game theory is typically used to model multi-agent interactions, and the
global states of interest are thus the so-called Nash equilibria, in which no agent
has a unilateral incentive to deviate.
In a recent paper (Kearns et al. 2001), we introduced such a graphical formalism for
multi-agent game theory, and provided two algorithms for computing Nash equilibria when the underlying graph is a tree (or is sufficiently sparse). The first algorithm
computes approximations to all Nash equilibria, in time polynomial in the size of
the representation and the quality of the desired approximation. A second and
related algorithm computes all Nash equilibria exactly, but in time exponential in
the number of agents. We thus left open the problem of efficiently computing exact
equilibria in sparse graphs.
In this paper, we describe a new algorithm that solves this problem. Given as input
a graphical game that is a tree, the algorithm computes in polynomial time an exact Nash equilibrium for the global multi-agent system. The main advances involve
the definition of a new data structure for representing "upstream" or partial Nash
equilibria, and a proof that this data structure can always be extended to a global
equilibrium. The new algorithm can also be extended to efficiently accommodate
parametric representations of the local game matrices, which are analogous to parametric conditional probability tables (such as noisy-OR and sigmoids) in Bayesian
networks.
The analogy between graphical models for multi-agent systems and probabilistic
inference is tempting and useful to an extent. The problem of computing Nash
equilibria in a graphical game, however, appears to be considerably more difficult
than computing conditional probabilities in Bayesian networks. Nevertheless, the
analogy and the work presented here suggest a number of interesting avenues for
further work in the intersection of game theory, network models, probabilistic inference, statistical physics, and other fields.
The paper is organized as follows. Section 2 introduces graphical games and other
necessary notation and definitions. Section 3 presents our algorithm and its analysis ,
and Section 4 gives a brief conclusion.
2
Preliminaries
An n-player, two-action 1 game is defined by a set of n matrices Mi (1 ~ i ~ n),
each with n indices. The entry Mi(Xl, ... ,xn ) = Mi(X) specifies the payoff to player
i when the joint action of the n players is x E {O, I} n. Thus, each Mi has 2n entries.
If a game is given by simply listing the 2n entries of each of the n matrices, we will
say that it is represented in tabular form.
?
The actions and 1 are the pure strategies of each player, while a mixed strategy
for player i is given by the probability Pi E [0, 1] that the player will play 1. For
any joint mixed strategy, given by a product distribution p, we define the expected
payoff to player i as Mi(i/) = Ex~p[Mi(X)], where x'" pindicates that each Xj is 1
with probability Pj and with probability 1 - Pj.
?
We use p[i : P:] to denote the vector that is the same as p except in the ith
component, where the value has been changed to P:. A Nash equilibrium for the
game is a mixed strategy p such that for any player i, and for any value
E
[0,1], Mi(i/) ::::: Mi(p[i :
(We say that Pi is a best response to jJ.) In other
words, no player can improve its expected payoff by deviating unilaterally from a
Nash equilibrium. The classic theorem of Nash (1951) states that for any game,
there exists a Nash equilibrium in the space of joint mixed strategies (product
distri butions).
pm.
P:
An n-player graphical game is a pair (G, M), where G is an undirected graph 2 on n
1 At present , no polynomial-time algorithm is known for finding Nash equilibria even in
2-player games with more than two actions, so we leave the extension of our work to the
multi-action setting for future work.
2The directed tree-structured case is trivial and is not addressed in this paper.
vertices and M is a set of n matrices Mi (1 ::; i ::; n), called the local game matrices .
Player i is represented by a vertex labeled i in G. We use N G (i) ~ {I, ... , n}
to denote the set of neighbors of player i in G- those vertices j such that the
undirected edge (i , j) appears in G. By convention, NG(i) always includes i itself.
The interpretation is that each player is in a game with only his neighbors in G.
Thus, if ING(i) I = k, the matrix Mi has k indices, one for each player in NG(i) , and
if x E [0, Ilk, Mi(X) denotes the payoff to i when his k neighbors (which include
himself) play
The expected payoff under a mixed strategy jJ E [0, Ilk is defined
analogously. Note that in the two-action case, Mi has 2k entries, which may be
considerably smaller than 2n.
x.
Since we identify players with vertices in G, it will be easier to treat vertices symbolically (such as U, V and W) rather than by integer indices. We thus use Mv to
denote the local game matrix for the player identified with vertex V.
Note that our definitions are entirely representational, and alter nothing about the
underlying game theory. Thus, every graphical game has a Nash equilibrium. Furthermore, every game can be trivially represented as a graphical game by choosing
G to be the complete graph and letting the local game matrices be the original
tabular form matrices. Indeed, in some cases, this may be the most compact graphical representation of the tabular game. However, exactly as for Bayesian networks
and other graphical models for probabilistic inference, any game in which the local
neighborhoods in G can be bounded by k ? n, exponential space savings accrue.
The algorithm presented here demonstrates that for trees, exponential computational benefits may also be realized.
3
The Algorithm
If (G , M) is a graphical game in which G is a tree, then we can always designate
some vertex Z as the root. For any vertex V, the single neighbor of Von the path
from V to Z shall be called the child of V, and the (possibly many) neighbors of V
on paths towards the leaves shall be called the parents of V. Our algorithm consists
of two passes: a downstream pass in which local data structures are passed from the
leaves towards the root, and an upstream pass progressing from the root towards
the leaves.
Throughout the ensuing discussion, we consider a fixed vertex V with parents
U I , ... , Uk and child W. On the downstream pass of our algorithm, vertex V will
compute and pass to its child W a breakpoint policy, which we now define.
?
Definition 1 A breakpoint policy for V consists of an ordered set of W -breakpoints
=
< WI < W2 < ... < Wt-I < Wt = 1 and an associated set of V-values
VI , . .. ,Vt? The interpretation is that for any W E [0,1], if Wi-I < W < Wi for some
index i and W plays w, then V shall play Vii and if W = Wi for some index i , then
V shall play any value between Vi and Vi+I. We say such a breakpoint policy has
t - 1 breakpoints.
Wo
A breakpoint policy for V can thus be seen as assigning a value (or range of values)
to the mixed strategy played by V in response to the play of its child W. In a slight
abuse of notation, we will denote this breakpoint policy as a function Fv(w), with
the understanding that the assignment V = Fv(w) means that V plays either the
fixed value determined by the breakpoint policy (in the case that W falls between
breakpoints), or plays any value in the interval determined by the breakpoint policy
(in the case that W equals some breakpoint).
Let G V denote the subtree of G with root V, and let M~=w denote the subset
of the set of local game matrices M corresponding to the vertices in G V , except
that the matrix M v is collapsed one index by setting W = w, thus marginalizing
W out. On its downstream pass, our algorithm shall maintain the invariant that if
we set the child W = w, then there is a Nash equilibrium for the graphical game
(G v , M~=w) (an upstream Nash) in which V = Fv(w). If this property is satisfied
by Fv(w), we shall say that Fv(w) is a Nash breakpoint policy for V. Note that
since (Gv, M~=w) is just another graphical game, it of course has (perhaps many)
Nash equilibria, and V is assigned some value in each. The trick is to commit
to one of these values (as specified by Fv (w)) that can be extended to a Nash
equilibrium for the entire tree G, before we have even processed the tree below V .
Accomplishing this efficiently and exactly is one of the main advances in this work
over our previous algorithm (Kearns et al. 2001).
The algorithm and analysis are inductive: V computes a Nash breakpoint policy
Fv(w) from Nash breakpoint policies FUl (v), ... , FUk (v) passed down from its parents (and from the local game matrix Mv). The complexity analysis bounds the
number of breakpoints for any vertex in the tree. We now describe the inductive
step and its analysis.
3.1
Downstream Pass
For any setting it E [0, l]k for
-0 and w
~v(i1,w)
E [0,1] for W, let us define
== Mv(l,it,w) - Mv(O,it,w).
The sign of ~v(it, w) tells us V's best response to the setting of the local neighborhood -0 = it, W = w; positive sign means V = 1 is the best response, negative that
V = 0 is the best response, and 0 that V is indifferent and may play any mixed
strategy. Note also that we can express ~v(it,w) as a linear function of w:
~v(it,w) = ~v(it, O)
+ w(~v(it, 1) -
~v(it,
0)).
For the base case, suppose V is a leaf with child W; we want to describe the Nash
breakpoint policy for V. If for all w E [0,1], the function ~v(w) is non-negative
(non-positive, respectively), V can choose 1 (0, respectively) as a best response
(which in this base case is an upstream Nash) to all values W = w. Otherwise,
~ v (w) crosses the w-axis, separating the values of w for which V should choose
1, 0, or be indifferent (at the crossing point). Thus, this crossing point becomes
the single breakpoint in Fv(w). Note that if V is indifferent for all values of w, we
assume without loss of generality that V plays l.
The following theorem is the centerpiece of the analysis.
Theorem 2 Let vertex V have parents UI , ... ,Uk and child W, and assume V has
received Nash breakpoint policies FUi (v) from each parent Ui . Then V can efficiently
compute a Nash breakpoint policy Fv (w). The number of breakpoints is no more
than two plus the total number of breakpoints in the FUi (v) policies.
Proof: Recall that for any fixed value of v, the breakpoint policy FUi (v) specifies
either a specific value for Ui (if v falls between two breakpoints of FUi (v)) , or a range
of allowed values for Ui (if v is equal to a breakpoint). Let us assume without loss of
generality that no two FUi (v) share a breakpoint, and let Vo = 0 < VI < ... < Vs = 1
be the ordered union of the breakpoints of the FUi (v). Thus for any breakpoint Vi,
there is at most one distinguished parent Uj (that we shall call the free parent) for
which Fu; (Vi) specifies an allowed interval of play for Uj . All other Ui are assigned
fixed values by Fu; (ve). For each breakpoint Ve, we now define the set of values for
the child W that, as we let the free parent range across its allowed interval, permit
V to play any mixed strategy as a best response.
< VI < ... < Vs = 1 be the ordered union of the breakpoints of the parent policies Fu; (v). Fix any breakpoint Ve, and assume without loss
of generality that U I is the free parent of V for V = Ve. Let [a, b] be the allowed
interval ofUI specified by FUI (ve), and letui = Fu;(ve) for all 2 :::; i:::; k. We define
Definition 3 Let Vo = 0
W e = {w E [0,1]: (:lUI E [a,b])6.v(UI,U2, ... ,Uk,W) = O}.
In words, We is the set of values that W can play that allow V to play any mixed
strategy, preserving the existence of an upstream Nash from V given W = w.
The next lemma, which we state without proof and is a special case of Lemma 6 in
Kearns et al. (2001), limits the complexity of the sets We. It also follows from the
earlier work that We can be computed in time proportional to the size of V's local
game matrix - O(2k) for a vertex with k parents.
We say that an interval [a, b]
~
[0, 1] is floating if both a -I- 0 and b -I- 1.
Lemma 4 For any breakpoint Ve, the set We is either empty, a single interval, or
the union of two intervals that are not floating.
We wish to create the (inductive) Nash breakpoint policy Fv(w) from the sets W e
and the Fu; policies. The idea is that if w E We for some breakpoint index e,
then by definition of W e, if W plays wand the Uis play according to the setting
determined by the Fu; policies (including a fixed setting for the free parent of V),
any play by V is a best response-so in particular, V may play the breakpoint value
Ve, and thus extend the Nash solution constructed, as the UiS can also all be best
responses. For b E {O, I}, we define W b as the set of values w such that if W = w
and the Uis are set according to their breakpoint policies for V = b, V = b is a
best response. To create Fv (w) as a total function, we must first show that every
w E [0, 1] is contained in some We or W O or WI.
Lemma 5 Let Vo = 0 < VI < ... < Vs = 1 be the ordered union of the breakpoints
of the Fu; (v) policies. Then for any value w E [0, 1], either w E w b for some
bE {O, I} , or there exists an index e such that wE W e.
Proof: Consider any fixed value of w, and for each open interval (vi> vj+d determined by adjacent breakpoints, label this interval by V 's best response (0 or
1) to W = wand 0 set according to the Fu; policies for this interval. If either
the leftmost interval [O ,vd is labeled with 0 or the rightmost interval [v s -I , I] is
labeled with 1, then w is included in W O or WI , respectively (V playing 0 or 1
is a best response to what the Uis will play in response to a 0 or 1). Otherwise,
since the labeling starts at 1 on the left and ends at 0 on the right, there must be
a breakpoint Ve such that V's best response changes over this breakpoint. Let Ui
be the free parent for this breakpoint. By continuity, there must be a value of Ui
in its allowed interval for which V is indifferent to playing 0 or 1, so w E W e. This
completes the proof of Lemma 5.
Armed with Lemmas 4 and 5, we can now describe the construction of Fv(w). Since
every w is contained in some W e (Lemma 5), and since every W e is the union of at
most two intervals (Lemma 4), we can uniquely identify the set WeI that covers the
largest (leftmost) interval containing w = 0; let [0, a] be this interval. Continuing
in the same manner to the right, we can identify the unique set We2 that contains
v7r - - - - - --- ----- --- ----- --- ----- --- ----- -------- - r - - - - - -
v6 -
V
------~
--------
v5 ------------------------------ -
v4 -
-
--
-
--
-
--
-
--
-
-----
------------ - --
, - - - - - - - ' ---------------
------------- ------------ ---------------------
v3f------.- ---------------------- --------------------------- - -
v2 _______--'-_ _ _ _ _ _----L _________________________________ _
vI ----------------------------------------------------
w
Figure 1: Example of the inductive construction of Fv(w). The dashed horizontal
lines show the vrbreakpoints determined by the parent policies Fu; (v). The solid
intervals along these breakpoints are the sets We. As shown in Lemma 4, each of
these sets consists of either a single (possibly floating) interval, or two non-floating
intervals. As shown in Lemma 5, each value of w is covered by some We. The
construction of Fv(w) (represented by a thick line) begins on the left, and always
next "jumps" to the interval allowing greatest progress to the right.
w = a and extends farthest to the right of a. Any overlap between We 1 and We 2
can be arbitrarily assigned coverage by We 1 , and We2 "trimmed" accordingly; see
Figure 1. This process results in a Nash breakpoint policy Fv(w).
Finally, we bound the number of breakpoints in the Fv (w) policy. By construction,
each of its breakpoints must be the rightmost portion of some interval in WO, WI, or
some We. After the first breakpoint, each of these sets contributes at most one new
breakpoint (Lemma 4). The final breakpoint is at w = 1 and does not contribute
to the count (Definition 1). There is at most one We for each breakpoint in each
Fu; (v) policy, plus WO and WI, plus the initial leftmost interval and minus the
final breakpoint, so the total breakpoints in Fv(w) can be no more than two plus
the total number of breakpoints in the Fu; (v) policies. Therefore, the root of a size
n tree will have a Nash breakpoint policy with no more than 2n breakpoints.
This completes the proof of Theorem 2.
3.2
Upstream Pass
The downstream pass completes when each vertex in the tree has had its Nash
breakpoint policy computed. For simplicity of description, imagine that the root of
t he tree includes a dummy child with constant payoffs and no influence on t he root,
so the root's breakpoint policy has t he same form as the others in the tree.
To produce a Nash equilibrium, our algorithm performs an upstream pass over
the tree, starting from the root. Each vertex is told by its child what value to
play, as well as the value the child itself will play. The algorithm ensures that all
downstream vertices are Nash (playing best response to their neighbors). Given
this information, each vertex computes a value for each of its parents so that its
own assigned action is a best response. This process can be initiated by the dummy
vertex picking an arbitrary value for itself, and selecting the root's value according
to its Nash breakpoint policy.
Inductively, we have a vertex V connected to parents U1 , ... , Uk (or no parents if
V is a leaf) and child W. The child of V has informed V to chose V = v and that
it will play W = w. To decide on values for V's parents to enforce V playing a best
response, we can look at the Nash breakpoint policies FUi (v), which provide a value
(or range of values) for Ui as a function of v that guarantee an upstream Nash. The
value v can be a breakpoint for at most one Ui . For each Ui , if v is not a breakpoint
in FUi (v) , then Ui should be told to select Ui = FUi (v). If v is a breakpoint in
FUi (v), then Ui's value can be computed by solving ~V(Ul "'" Ui,"" Uk, w) = 0;
this is the value of Ui that makes V indifferent. The equation is linear in Ui and has
a solution by the construction of the Nash breakpoint policies on the downstream
pass. Parents are passed their assigned values as well as the fact that V = v.
When the upstream pass completes, each vertex has a concrete choice of action such
that jointly they have formed a Nash equilibrium.
The total running time of the algorithm can be bounded as follows. Each vertex
is involved in a computation in the downstream pass and in the upstream pass.
Let t be the total number of breakpoints in the breakpoint policy for a vertex V
with k parents. Sorting the breakpoints and computing the W? sets and computing
the new breakpoint policy can be completed in 0 (t log t + t2 k ). In the upstream
pass, only one breakpoint is considered, so 0 (log t + 2k) is sufficient for passing
breakpoints to the parents. By Theorem 2, t :S 2n , so the entire algorithm executes
in time O(n 2 10g n + n22k), where k is the largest number of neighbors of any vertex
in the network.
The algorithm can be implemented to take advantage of local game matrices provided in a parameterized form. For example, if each vertex's payoff is solely a
function of the number of 1s played by the vertex's neighbors, the algorithm takes
O(n 2 10gn + n 2 k), eliminating the exponential dependence on k.
4
Conclusion
The algorithm presented in this paper finds a single Nash equilibrium for a game
represented by a tree-structured network. By building representations of all equilibria, our earlier algorithm (Kearns et al. 2001) was able to select equilibria efficiently
according to criteria like maximizing the total expected payoff for all players. The
polynomial-time algorithm described in this paper throws out potential equilibria
at many stages, most significantly during the construction of the Nash breakpoint
policies. An interesting area for future work is to manipulate this process to produce
equilibria with particular properties.
References
Michael Kearns, Michael L. Littman, and Satinder Singh. Graphical models for game
theory. In Proceedings of the 17th Conference on Uncertainty in Artificial Int elligence
(UAI), pages 253- 260, 200l.
Daphne Koller and Brian Milch. Multi-agent influence diagrams for representing and
solving games. Submitted, 2001.
Pierfrancesco La Mura. Game networks. In Proceedings of the 16th Conference on Uncertainty in Artificial Intelligence (UAI), pages 335- 342, 2000.
J . F. Nash. Non-cooperative games. Annals of Math ematics, 54:286- 295, 1951.
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1,210 | 2,102 | Products of Gaussians
Christopher K. I. Williams
Division of Informatics
University of Edinburgh
Edinburgh EH1 2QL, UK
c. k. i. [email protected]
http://anc.ed.ac.uk
Felix V. Agakov
System Engineering Research Group
Chair of Manufacturing Technology
Universitiit Erlangen-Niirnberg
91058 Erlangen, Germany
F.Agakov@lft?uni-erlangen.de
Stephen N. Felderhof
Division of Informatics
University of Edinburgh
Edinburgh EH1 2QL, UK
[email protected]
Abstract
Recently Hinton (1999) has introduced the Products of Experts
(PoE) model in which several individual probabilistic models for
data are combined to provide an overall model of the data. Below we consider PoE models in which each expert is a Gaussian.
Although the product of Gaussians is also a Gaussian, if each Gaussian has a simple structure the product can have a richer structure.
We examine (1) Products of Gaussian pancakes which give rise to
probabilistic Minor Components Analysis, (2) products of I-factor
PPCA models and (3) a products of experts construction for an
AR(l) process.
Recently Hinton (1999) has introduced the Products of Experts (PoE) model in
which several individual probabilistic models for data are combined to provide an
overall model of the data. In this paper we consider PoE models in which each
expert is a Gaussian. It is easy to see that in this case the product model will
also be Gaussian. However, if each Gaussian has a simple structure, the product
can have a richer structure. Using Gaussian experts is attractive as it permits a
thorough analysis of the product architecture, which can be difficult with other
models , e.g. models defined over discrete random variables.
Below we examine three cases of the products of Gaussians construction: (1) Products of Gaussian pancakes (PoGP) which give rise to probabilistic Minor Components Analysis (MCA), providing a complementary result to probabilistic Principal
Components Analysis (PPCA) obtained by Tipping and Bishop (1999); (2) Products of I-factor PPCA models; (3) A products of experts construction for an AR(l)
process.
Products of Gaussians
If each expert is a Gaussian pi(xI8 i ) '" N(J1i' ( i), the resulting distribution of the
product of m Gaussians may be expressed as
By completing the square in the exponent it may be easily shown that p(xI8)
N(/1;E, (2:), where (E l = 2::1 (i l . To simplify the following derivations we will
assume that pi(xI8 i ) '" N(O, (i) and thus that p(xI8) '" N(O, (2:). J12: i
can be
obtained by translation of the coordinate system.
?
1
Products of Gaussian Pancakes
A Gaussian "pancake" (GP) is a d-dimensional Gaussian, contracted in one dimension and elongated in the other d - 1 dimensions. In this section we show that the
maximum likelihood solution for a product of Gaussian pancakes (PoGP) yields a
probabilistic formulation of Minor Components Analysis (MCA).
1.1
Covariance Structure of a GP Expert
Consider a d-dimensional Gaussian whose probability contours are contracted
in the direction w and equally elongated in mutually orthogonal directions
VI , ... , vd-l.We call this a Gaussian pancake or GP. Its inverse covariance may be
written as
d- l
( -1=
L ViV; /30 + ww
T
/3,;;,
(1)
i= l
where VI, ... ,V d - l, W form a d x d matrix of normalized eigenvectors of the covariance C. /30 = 0"0 2 , /3,;; = 0";;2 define inverse variances in the directions of elongation
and contraction respectively, so that 0"5 2 0"1. Expression (1) can be re-written in
a more compact form as
(2)
where w = wJ/3,;; - /30 and Id C jRdxd is the identity matrix. Notice that according
to the constraint considerations /30 < /3,;;, and all elements of ware real-valued.
Note the similarity of (2) with expression for the covariance of the data of a 1factor probabilistic principal component analysis model ( = 0"21d + ww T (Tipping
and Bishop, 1999) , where 0"2 is the variance of the factor-independent spherical
Gaussian noise. The only difference is that it is the inverse covariance matrix for
the constrained Gaussian model rather than the covariance matrix which has the
structure of a rank-1 update to a multiple of Id .
1.2
Covariance of the PoGP Model
We now consider a product of m GP experts, each of which is contracted in a single
dimension. We will refer to the model as a (I,m) PoGP, where 1 represents the
number of directions of contraction of each expert. We also assume that all experts
have identical means.
From (1), the inverse covariance of the the resulting (I,m) PoGP model can be
expressed as
m
C;;l
=L
Ci l
(3)
i=l
where columns of We Rdxm correspond to weight vectors of the m PoGP experts,
and (3E = 2::1 (3~i) > o.
1.3
Maximum-Likelihood Solution for PoGP
Comparing (3) with m-factor PPCA we can make a conjecture that in contrast
with the PPCA model where ML weights correspond to principal components of
the data covariance (Tipping and Bishop, 1999), weights W of the PoGP model
define projection onto m minor eigenvectors of the sample covariance in the visible
d-dimensional space, while the distortion term (3E Id explains larger variations l . This
is indeed the case.
In Williams and Agakov (2001) it is shown that stationarity of the log-likelihood
with respect to the weight matrix Wand the noise parameter (3E results in three
classes of solutions for the experts' weight matrix, namely
W
5
5W
0;
CE ;
CEW, W:j:. 0, 5:j:. CE ,
(4)
where 5 is the covariance matrix of the data (with an assumed mean of zero). The
first two conditions in (4) are the same as in Tipping and Bishop (1999), but for
PPCA the third condition is replaced by C-l W = 5- l W (assuming that 5 - 1 exists).
In Appendix A and Williams and Agakov (2001) it is shown that the maximum
likelihood solution for WML is given by:
(5)
where R c Rmxm is an arbitrary rotation matrix, A is a m x m matrix containing
the m smallest eigenvalues of 5 and U = [Ul , ... ,u m ] c Rdxm is a matrix of the
corresponding eigenvectors of 5. Thus, the maximum likelihood solution for the
weights of the (1, m) PoG P model corresponds to m scaled and rotated minor
eigenvectors of the sample covariance 5 and leads to a probabilistic model of minor
component analysis. As in the PPCA model, the number of experts m is assumed
to be lower than the dimension of the data space d.
The correctness of this derivation has been confirmed experimentally by using a
scaled conjugate gradient search to optimize the log likelihood as a function of W
and (3E.
1.4
Discussion of PoGP model
An intuitive interpretation of the PoGP model is as follows: Each Gaussian pancake
imposes an approximate linear constraint in x space. Such a linear constraint is that
x should lie close to a particular hyperplane. The conjunction of these constraints
is given by the product of the Gaussian pancakes. If m ? d it will make sense to
lBecause equation 3 has the form of a factor analysis decomposition, but for the inverse
covariance matrix, we sometimes refer to PoGP as the rotcaf model.
define the resulting Gaussian distribution in terms of the constraints. However, if
there are many constraints (m > d/2) then it can be more efficient to describe the
directions of large variability using a PPCA model, rather than the directions of
small variability using a PoGP model. This issue is discussed by Xu et al. (1991) in
what they call the "Dual Subspace Pattern Recognition Method" where both PCA
and MCA models are used (although their work does not use explicit probabilistic
models such as PPCA and PoGP).
MCA can be used , for example, for signal extraction in digital signal processing
(Oja, 1992), dimensionality reduction, and data visualization. Extraction of the
minor component is also used in the Pisarenko Harmonic Decomposition method
for detecting sinusoids in white noise (see, e.g. Proakis and Manolakis (1992), p .
911). Formulating minor component analysis as a probabilistic model simplifies
comparison of the technique with other dimensionality reduction procedures, permits extending MCA to a mixture of MCA models (which will be modeled as a
mixture of products of Gaussian pancakes) , permits using PoGP in classification
tasks (if each PoGP model defines a class-conditional density) , and leads to a number of other advantages over non-probabilistic MCA models (see the discussion of
advantages of PPCA over PCA in Tipping and Bishop (1999)).
2
Products of PPCA
In this section we analyze a product of m I-factor PPCA models, and compare it
to am-factor PPCA model.
2.1
I-factor PPCA model
Consider a I-factor PPCA model, having a latent variable Si and visible variables x.
The joint distribution is given by P(Si, x) = P(si) P(xlsi). We set P(Si) '" N(O, 1)
and P(XI Si) '" N(WiSi' (]"2) . Integrating out Si we find that Pi(x) '" N(O, Ci ) where
C = wiwT + (]"21d and
(6)
where (3 = (]"-2 and "(i = (3/(1 + (3 llwi W). (3 and "(i are the inverse variances in the
directions of contraction and elongation respectively.
The joint distribution of Si and x is given by
(7)
[s;
exp - -(3 - - 2x T WiSi
2 "(i
+ XT X]
.
(8)
Tipping and Bishop (1999) showed that the general m-factor PPCA model (mPPCA) has covariance C = (]"21d + WW T , where W is the d x m matrix of factor
loadings. When fitting this model to data, the maximum likelihood solution is to
choose W proportional to the principal components of the data covariance matrix.
2.2
Products of I-factor PPCA models
We now consider the product of m I-factor PPCA models, which we denote a
(1, m)-PoPPCA model. The joint distribution over 5 = (Sl' ... ,Srn)T and x is
13
P(x,s) ex: exp -"2
L [s;- -:- - 2x
m
i=l
T
W iSi
+ XT X ]
(9)
?
,,(,
Let zT d~f (xT , ST). Thus we see that the distribution of z is Gaussian with inverse
covariance matrix 13M, where
-W)
r -1
(10)
,
and r = diag("(l , ... ,"(m)' Using the inversion equations for partitioned matrices
(Press et al., 1992, p. 77) we can show that
(11)
where ~xx is the covariance of the x variables under this model. It is easy to confirm
that this is also the result obtained from summing (6) over i = 1, ... ,m.
2.3
Maximum Likelihood solution for PoPPCA
Am-factor PPCA model has covariance a21d + WW T and thus, by the Woodbury
formula, it has inverse covariance j3 ld - j3W(a2 lm + WT W) - lW T . The maximum
likelihood solution for a m-PPCA model is similar to (5), i.e. W = U(A _a2Im)1/2 RT,
but now A is a diagonal matrix of the m principal eigenvalues, and U is a matrix
of the corresponding eigenvectors. If we choose RT = I then the columns of W are
orthogonal and the inverse covariance of the maximum likelihood m-PPCA model
has the form j3 ld - j3WrwT. Comparing this to (11) (with W = W) we see that the
difference is that the first term of the RHS of (11) is j3m1d , while for m-PPCA it is
j3 ld.
In section 3.4 and Appendix C.3 of Agakov (2000) it is shown that (for m
obtain the m-factor PPCA solution when
-
m
-
A<A' < - - A
- '
m -I '
i = 1, ...
,m,
:::=:
2) we
(12)
where Ais the mean of the d - m discarded eigenvalues, and Ai is a retained eigenvalue; it is the smaller eigenvalues that are discarded. We see that the covariance
must be nearly spherical for this condition to hold. For covariance matrices satisfying (12) , this solution was confirmed by numerical experiments as detailed in
(Agakov, 2000, section 3.5).
To see why this is true intuitively, observe that Ci 1 for each I-factor PPCA expert
will be large (with value 13) in all directions except one. If the directions of contraction for each Ci 1 are orthogonal, we see that the sum of the inverse covariances
will be at least (m - 1)13 in a contracted direction and m j3 in a direction in which
no contraction occurs. The above shows that for certain types of sample covariance matrix the (1 , m) PoPPCA solution is not equivalent to the m-factor PPCA
solution. However, it is interesting to note that by relaxing the constraint on the
isotropy of each expert's noise the product of m one-factor factor analysis models
can be shown to be equivalent to an m-factor factor analyser (Marks and Movellan,
2001).
???
?? ?
?
(c)
(b)
?
(d)
Figure 1: (a) Two experts. The upper one depicts 8 filled circles (visible units) and
4 latent variables (open circles), with connectivity as shown. The lower expert also
has 8 visible and 4 latent variables, but shifted by one unit (with wraparound). (b)
Covariance matrix for a single expert. (c) Inverse covariance matrix for a single
expert. (d) Inverse covariance for product of experts.
3
A Product of Experts Representation for an AR(l)
Process
For the PoPPCA case above we have considered models where the latent variables
have unrestricted connectivity to the visible variables. We now consider a product
of experts model with two experts as shown in Figure l(a). The upper figure depicts
8 filled circles (visible units) and 4 latent variables (open circles), with connectivity
as shown. The lower expert also has 8 visible and 4 latent variables, but shifted
by one unit (with wraparound) with respect to the first expert. The 8 units are,
of course, only for illustration- the construction is valid for any even number of
visible units.
Consider one hidden unit and its two visible children. Denote the hidden unit by s
the visible units as Xl and Xr (l, r for left and right). Set s '" N(O, 1) and
Xl
= as
+ bWI
Xr
= ?as + bwr
(13)
,
where WI and Wr are independent N(O , 1) random variables, and a , b are constants.
(This is a simple example of a Gaussian tree-structured process, as studied by a
number of groups including that led by Prof. Willsky at MIT; see e.g. Luettgen
et al. (1993).) Then (xf) = (x;) = a2 + b2 and (XIX r ) = ?a2 ? The corresponding
2 x 2 inverse covariance matrix has diagonal entries of (a 2 + b2 )j ~ and off-diagonal
entries of =t=a 2 j~ , where ~ = b2 (b 2 + 2a 2 ).
Graphically, the covariance matrix of a single expert has the form shown in Figure
l(b) (where we have used the + rather than - choice from (13) for all variables).
Figure l(c) shows the corresponding inverse covariance for the single expert, and
Figure 1(d) shows the resulting inverse covariance for the product of the two experts,
with diagonal elements 2(a 2 + b2 )j ~ and off-diagonal entries of =t=a 2 j~.
An AR(l) process of the circle with d nodes has the form Xi = aXi -
1 (mod d)
+ Zi,
where Zi ~ N(O,v). Thusp(X) <X exp-21v L:i(Xi-aXi - 1 (mod d))2 and the inverse
covariance matrix has a circulant tridiagonal structure with diagonal entries of
(1 + ( 2 )/v and off-diagonal entries of -a/v. The product of experts model defined
above can be made equivalent to the circular AR(I) process by setting
(14)
The ? is needed in (13) as when a is negative we require
the inverse covariances.
Xr
= -as + bWr
to match
We have shown that there is an exact construction to represent a stationary circular AR(I) process as a product of two Gaussian experts. The approximation
of other Gaussian processes by products of tree-structured Gaussian processes is
further studied in (Williams and Felderhof, 2001). Such constructions are interesting because they may allow fast approximate inference in the case that d is large
(and the target process may be 2 or higher dimensional) and exact inference is not
tractable. Such methods have been developed by Willsky and coauthors, but not
for products of Gaussians constructions.
Acknowledgements
This work is partially supported by EPSRC grant GR/L78161 Probabilistic Models
for Sequences. Much of the work on PoGP was carried out as part of the MSc
project of FVA at the Division of Informatics, University of Edinburgh. CW thanks
Sam Roweis, Geoff Hinton and Zoubin Ghahramani for helpful conversations on the
rotcaf model during visits to the Gatsby Computational Neuroscience Unit. FVA
gratefully acknowledges the support of the Royal Dutch Shell Group of Companies
for his MSc studies in Edinburgh through a Centenary Scholarship. SNF gratefully
acknowledges additional support from BAE Systems.
References
Agakov, F. (2000). Investigations of Gaussian Products-of-Experts Models. Master's
thesis, Division of Informatics, The University of Edinburgh. Available at http://'iI'iI'iI .
dai.ed.ac.uk/homes/felixa/all.ps.gz.
Hinton, G . E . (1999) . Products of experts. In Proceedings of the Ninth International
Conference on Artificial N eural N etworks (ICANN gg), pages 1- 6.
Luettgen , M. , Karl, W. , and Willsky, A. (1993). Multiscale Representations of Markov
Random Fields. IEEE Trans. Signal Processing, 41(12):3377- 3395.
Marks , T. and Movellan, J. (2001). Diffusion Networks, Products of Experts, and Factor
Analysis. In Proceedings of the 3rd International Conference on Independent Component
Analysis and Blind Source Separation.
OJ a, E. (1992). Principal Components, Minor Components, and Linear Neural Networks.
Neural N etworks, 5:927 - 935.
Press, W. H. , Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P. (1992). Num erical
Recipes in C. Cambridge University Press, Second edition .
Proakis, J. G. and Manolakis, D. G. (1992). Digital Signal Processing: Principles, Algorithms and Applications. Macmillan.
Tipping, M. E. and Bishop, C. M. (1999). Probabilistic principal components analysis. J.
Roy. Statistical Society B, 61(3) :611- 622.
Williams, C. K. I. and Agakov, F. V. (2001). Products of Gaussians and Probabilistic
Minor Components Analysis. Technical Report EDI-INF-RR-0043 , Division of Informatics, University of Edinburgh. Available at http://'iI'iI'iI. informatics. ed. ac. ukl
publications/report/0043.html.
Williams, C. K. I. and Felderhof, S. N. (2001). Products and Sums of Tree-Structured
Gaussian Processes. In Proceedings of the ICSC Symposium on Soft Computing 2001
(SOCO 2001).
Xu, L. and Krzyzak, A. and Oja, E. (1991). Neural Nets for Dual Subspace Pattern
Recogntion Method. International Journal of Neural Systems, 2(3):169- 184.
A
ML Solutions for PoGP
Here we analyze the three classes of solutions for the model covariance matrix which
result from equation (4) of section 1.3.
The first case W
= 0 corresponds to a
minimum of the log-likelihood.
In the second case, the model covariance e~ is equal to the sample covariance 5.
From expression (3) for e i;l we find WWT = 5 - 1 - ;3~ ld. This has the known
solution W = Um(A - 1 - ;3~ lm)1 /2 RT , where Um is the matrix of the m eigenvectors
of 5 with the smallest eigenvalues and A is the corresponding diagonal matrix of the
eigenvalues. The sample covariance must be such that the largest d - m eigenvalues
are all equal to ;3~; the other m eigenvalues are matched explicitly.
Finally, for the case of approximate model covariance (5W = e~w, 5 =f. e~) we, by
analogy with Tipping and Bishop (1999), consider the singular value decomposition
of the weight matrix, and establish dependencies between left singular vectors of
W = ULR T and eigenvectors of the sample covariance 5. U = [U1 , U2 , ... , um] C
lRdxm is a matrix of left singular vectors of W with columns constituting an orthonormal basis, L = diag(h,l2, ... ,lm) C lRmxm is a diagonal matrix of the singular values of Wand R C lRmxm defines an arbitrary rigid rotation of W. For this
case equation (4) can be written as 5UL = e ~ UL , where e~ is obtained from (3) by
applying the matrix inversion lemma [see e.g. Press et al. (1992)]. This leads to
5UL = e~UL
(;3i;lld - ;3i;l W(;3~ + WTW) -l WT )UL
U(;3i;l lm - ;3i;l LRT(;3~ lm + RL2RT) -l RL)L
U(;3i; l lm - ;3i;l(;3~ L -2 + Im) -l) L.
(15)
Notice that the term ;3;1 1m - ;3;l(;3~ L -2 + Im)-l in the r.h.s. of equation (15) is
just a scaling factor of U. Equation (15) defines the matrix form of the eigenvector
equation, with both sides post-multiplied by the diagonal matrix L.
If li
=f. 0 then (15) implies that
e ~ U i = 5Ui = AiUi, Ai = ;3i;1(1 - (;3~li2 + 1) - 1),
(16)
where Ui is an eigenvector of 5, and Ai is its corresponding eigenvalue. The scaling
factor li of the ith retained expert can be expressed as li = (Ail - ;3~)1/2 .
Obviously, if li = 0 then Ui is arbitrary. If li = 0 we say that the direction corresponding to Ui is discarded, i.e. the variance in that direction is explained merely
by noise. Otherwise we say that Ui is retained. All potential solutions of W may
then be expressed as
W = Um(D - ;3~ lm)1 /2 RT ,
(17)
where R C lRmxm is a rotation matrix, Um = [U1U2 ... um] C lRdxm is a matrix whose
columns correspond to m eigenvectors of 5, and D = diag( d 1 , d 2 , ... , dm ) C lRm xm
such that di = Ail if Ui is retained and d i = ;3~ if Ui is discarded.
It may further be shown (Williams and Agakov (2001)) that the optimal solution
for the likelihood is reached when W corresponds to the minor eigenvectors of the
sample covariance 5.
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1,211 | 2,103 | ACh, Uncertainty, and Cortical Inference
Peter Dayan
Angela Yu
Gatsby Computational Neuroscience Unit
17 Queen Square, London, England, WC1N 3AR.
[email protected]
[email protected]
Abstract
Acetylcholine (ACh) has been implicated in a wide variety of
tasks involving attentional processes and plasticity. Following
extensive animal studies, it has previously been suggested that
ACh reports on uncertainty and controls hippocampal, cortical and
cortico-amygdalar plasticity. We extend this view and consider
its effects on cortical representational inference, arguing that ACh
controls the balance between bottom-up inference, influenced by
input stimuli, and top-down inference, influenced by contextual
information. We illustrate our proposal using a hierarchical hidden Markov model.
1 Introduction
The individual and joint computational roles of neuromodulators such as
dopamine, serotonin, norepinephrine and acetylcholine are currently the focus of
intensive study.5, 7, 9?11, 16, 27 A rich understanding of the effects of neuromodulators
on the dynamics of networks has come about through work in invertebrate systems.21 Further, some general computational ideas have been advanced, such as
that they change the signal to noise ratios of cells. However, more recent studies,
particularly those focusing on dopamine,26 have concentrated on specific computational tasks.
ACh was one of the first neuromodulators to be attributed a specific role. Hasselmo and colleagues,10, 11 in their seminal work, proposed that cholinergic (and,
in their later work, also GABAergic12 ) modulation controls read-in to and readout from recurrent, attractor-like memories, such as area CA3 of the hippocampus.
Such memories fail in a characteristic manner if the recurrent connections are operational during storage, thus forcing new input patterns to be mapped to existing
memories. Not only would these new patterns lose their specific identity, but,
worse, through standard synaptic plasticity, the size of the basin of attraction of
the offending memory would actually be increased, making similar problems more
likely. Hasselmo et al thus suggested, and collected theoretical and experimental
evidence in favor of, the notion that ACh (from the septum) should control the suppression and plasticity of specific sets of inputs to CA3 neurons. During read-in,
high levels of ACh would suppress the recurrent synapses, but make them readily
plastic, so that new memories would be stored without being pattern-completed.
Then, during read-out, low levels of ACh would boost the impact of the recurrent
weights (and reduce their plasticity), allowing auto-association to occur.
The ACh signal to the hippocampus can be characterized as reporting the unfamiliarity of the input with which its release is associated. This is analogous to its
characterization as reporting the uncertainty associated with predictions in theories
of attentional influences over learning in classical conditioning.4 In an extensive
series of investigations in rats, Holland and his colleagues14, 15 have shown that
a cholinergic projection from the nucleus basalis to the (parietal) cortex is important when animals have to devote more learning (which, in conditioning, is essentially synonymous with paying incremental attention) to stimuli about whose
consequences the animal is uncertain.20 We have4 interpreted this in the statistical terms of a Kalman filter, arguing that the ACh signal reported this uncertainty,
thus changing plasticity appropriately. Note, however, that unlike the case of the
hippocampus, the mechanism of action of ACh in conditioning is not well understood.
In this paper, we take the idea that ACh reports on uncertainty one step farther.
There is a wealth of analysis-by-synthesis unsupervised learning models of cortical processing.1, 3, 8, 13, 17, 19, 23 In these, top-down connections instantiate a generative
model of sensory input; and bottom-up connections instantiate a recognition model,
which is the statistical inverse of the generative model, and maps inputs into categories established in the generative model. These models, at least in principle, permit stimuli to be processed according both to bottom-up input and top-down expectations, the latter being formed based on temporal context or information from
other modalities. Top-down expectations can resolve bottom-up ambiguities, permitting better processing. However, in the face of contextual uncertainty, top-down
information is useless. We propose that ACh reports on top-down uncertainty,
and, as in the case of area CA3, differentially modulates the strength of synaptic
connections: comparatively weakening those associated with the top-down generative model, and enhancing those associated with bottom-up, stimulus-bound
information.2 Note that this interpretation is broadly consistent with existent electrophysiology data, and documented effects on stimulus processing of drugs that
either enhance (eg cholinesterase inhibitors) or suppress (eg scopolamine) the action of ACh.6, 25, 28
There is one further wrinkle. In exact bottom-up, top-down, inference using a
generative model, top-down contextual uncertainty does not play a simple role.
Rather, all possible contexts are treated simultaneously according to the individual posterior probabilities that they currently pertain. Given the neurobiologically
likely scenario in which one set of units has to be used to represent all possible
contexts, this exact inferential solution is not possible. Rather, we propose that a
single context is represented in the activities of high level (presumably pre-frontal)
cortical units, and uncertainty associated with this context is represented by ACh.
This cholinergic signal then controls the balance between bottom-up and top-down
influences over inference.
In the next section, we describe the simple hierarchical generative model that we
use to illustrate our proposal. The ACh-based recognition model is introduced in
section 3 and discussed in section 4.
2 Generative and Recognition Models
Figure 1A shows a very simple case of a hierarchical generative model. The generative model is a form of hidden Markov model (HMM), with a discrete hidden
state , which will capture the idea of a persistent temporal context, and a twodimensional, real-valued, output . Crucially, there is an extra layer, between
and . The state is stochastically determined from , and controls which of a
set of 2d Gaussians (centered at the corners of the unit square) is used to generate
. In this austere case, is the model?s representation of , and the key inference
!
4
3
2
1
4
3
2
1
0
2
2
1
1
0
?1 ?1
0
200
400
2
1
0
?1
?1 0 1 2
Figure 1: Generative model. A) Three-layer model "$#&%('*),+.-/102#3%4'*)5+4-6/178#,9;: with
dynamics (< ) in the " layer ( =8> "?A@B"C?ED [email protected] K(L ), a probabilistic mapping ( M ) from ";NO0
( =8> 0(? @P"C?RQ "[email protected] L V ), and a Gaussian model WX> 7YQ 04G with means at the corners of the unit
square and standard deviation IZJ V in each direction. The model is rotationally invariant;
only some of the links are shown for convenience. B) Sample sequence showing the slow
dynamics in " ; the stochastic mapping into 0 and the substantial overlap in 7 (different
symbols show samples from the different Gaussians shown in A).
problem will be to determine the distribution over which generated , given
the past experience [ ]\^`_ba ^4cdd d c ]\ ^fe and itself.
Figure 1B shows an example of a sequence of gihih steps generated from the model.
The state in the layer stays the same for an average of about jZh timesteps; and
then switches to one of the other states, chosen equally at random. The transition
matrix is kTlnmSoqpRlRm . The state in the layer is more weakly determined by the state
in the layer, with a probability of only j*r(g that !_ . The stochastic transition
from to is governed by the transition matrix s l mHtum . Finally, is generated as a
Gaussian about a mean specified by . The standard deviation of these Gaussians
( h dv in each direction) is sufficiently large that the densities overlap substantially.
The naive solution to inferring is to use only the likelihood term (ie only the
probabilities wyx Cz H{ ). The performance of this is likely to be poor, since the Gaussians in for the different values of overlap so much. However, and this is why
it is a paradigmatic case for our proposal, contextual information, in this case past
experience, can help to determine . We show how the putative effect of ACh in
controlling the balance between bottom-up and top-down inference in this model
can be used to build a good approximate inference model.
In order to evaluate our approximate model, we need to understand optimal inference in this case. Figure 2A shows the standard HMM inference model, which
calculates the exact posteriors wyx Cz [ H{ and wyx Cz [ H{ . This is equivalent to just the
forward part of the forwards-backwards algorithm22 (since we are not presently
interested in learning the parameters of the model). The adaptation to include the
layer is straightforward. Figures 3A;D;E show various aspects of exact inference
for a particular run. The histograms in figure 3A show that wyx z [ { captures quite
well the actual states ! | that generated the data. The upper plot shows the posterior probabilities of the actual states in the sequence ? these should be, and are,
usually high; the lower histogram the posterior probability of the other possible
states; these should be, and are, usually low. Figure 3D shows the actual state sequence | ; figure 3E shows the states that are individually most likely at each time
step (note that this is not the maximum likelihood state sequence, as found by the
Viterbi algorithm, for instance).
(
)
*
% $
&
'
"!#
$
Figure 2: Recognition models. A) Exact recognition model. =8> " ?ED F Q + ?ED F G is propagated to
provide the prior =8> "? Q +`?ED F G (shown by the lengths of the thick vertical bars) and thus the
prior =8> 0(? Q +`?ED F G . This is combined with the likelihood term from the data 7? to give the true
=8> 0 ? Q + ? G . B) Bottom-recognition model uses only a generic prior over 0 ? (which conveys no
information), and so the likelihood term dominates. C) ACh model. A single estimated
state ", ?ED F is used, in conjunction with its certainty -6? D F , reported by cholinergic activity, to
, >."C, ?nQ/"C, ?ED F]G over "C? (which is a mixture of a delta function and
produce an approximate prior =8
a uniform), and thus an approximate prior over 0 ? . This is combined with the likelihood to
, > 0f? Q +`? G , and a new cholinergic signal -? is calculated.
give an approximate =8
V
021436587 9;:
1X
0.25
0 0
0.8
00
4 W
> 3
2
1
0
1Y
R
021=3 < 5 7 9;:
? @ ABC?ED=> 5
A'\]H_^
R
PQ
1 NO
M
PQ
NM O
0
0
1
4Z
3
2
1
400 0
0UTE13S7 9K:
0 1 > 7 9K:
?EF&G
HI?J 2
A \]HI^
1
0
0
4[
3
2
1
400 0
02L 13S7 9K:
>L
A'\]H_^
1
400
Figure 3: Exact and approximation recognition. A) Histograms of the exact posterior distribution =8> 0!Q +`G over the actual state 0a? ` (upper) and the other possible states 0K@ b 0c? ` (lower,
written =8>'0 d ` G ). This shows the quality of exact representational inference. B;C) Comparison
, > 0(?nQ + G (C) with
of the purely bottom up =e4> 0f? Q 7 ?SG (B) and the ACh-based approximation =8
the true =8> 0(? Q +`G across all values of 0 . The ACh-based approximation is substantially more
accurate. D) Actual " ? . E) Highest probability " state from the exact posterior distribution.
F) Single " , state in the ACh model.
Figure 2B shows a purely bottom up model that only uses the likelihood terms
to infer the distribution over . This has wKfix z ]{ _hg x Cz H{ rji where i is a
normalization factor. Figure 3B shows the representational performance of this
model, through a scatter-plot of wkfZx z { against the exact posterior wyx z [ { . If
bottom-up inference was correct, then all the points would lie on the line of equality ? the bow-shape shows that purely bottom-up inference is relatively poor. Figure 4C shows this in a different way, indicating the difference between the average
summed log probabilities of the actual states under the bottom up model and those
under the true posterior. The larger and more negative the difference, the worse
the approximate inference. Averaging over l hZhih runs, the difference is m_4n h log
units (compared with a total log likelihood under the exact model of mpoq l h ).
3 ACh Inference Model
Figure 2C shows the ACh-based approximate inference model. The information
about the context comes in the form of two quantities: ]\^ , the approximated contextual state having seen [ ]\^ , and ]\ ^ , which is the measure of uncertainty in that
contextual state. The idea is that ]\ ^ is reported by ACh, and is used to control
(indicated by the filled-in ellipse) the extent to which top-down information based
on ]\^ is used to influence inference about . If we were given the full exact
posterior distribution wyx ]\^ c ]\^ z [ ]\ ^ { , then one natural definition for this ACh
signal would be the uncertainty in the most likely contextual state
]\^,_
l
lm
wyx ] \ ^`_ Xz [ ] \ ^ {
(1)
Figure 4A shows the resulting ACh signal for the case of figure 3. As expected,
ACh is generally high at times when the true state | is changing, and decreases
during the periods that | is constant. During times of change, top-down information is confusing or potentially incorrect, and so bottom-up information should
dominate. This is just the putative inferential effect of ACh.
However, the ACh signal of figure 4A was calculated assuming knowledge of the
true posterior, which is unreasonable. The model of figure 2C includes the key
approximation that the only other information from [ ]\ ^ about the state of is in
the single choice of context variable ]\ ^ . The full approximate inference algorithm
becomes
wyx]\ ^
w xCz ]\ ^
wyx uc z ]\
^
wyx c
]\^u{_
r
tlm ]\ ^l m oqp [ ]\^ approximation (2)
prior over
(3)
l wyx]\^!_ ]\^ { k lul m
]\^u{_ wyxCz ]\
^ ]\^u{ s l m tum
propagation to
(4)
z [ { wyx c z ]\^ ]\ ^ { wyx z { conditioning
(5)
wyx z [ {_ l wyx c _ z [ {
marginalization
(6)
wyx z [ {_ t w x _ c z [ {
marginalization
(7)
_ argmaxl wyx _ Xz [ {
contextual inference (8)
_ lm !"Al wyx _ Xz [ {
ACh level
(9)
u
c
]
\
4
^
c
]
\
^
where
are used as approximate sufficient statistics for [ , the number of states is
t (here
t _ g ), $#&% is the Kronecker delta, and the constant of
]\^u{_
]\ ^
proportionality in equation 5 normalizes the full conditional distribution. The last
two lines show the information that is propagated to the next time step; equation 6
shows the representational answer from the model, the distribution over given
[ . These computations are all local and straightforward, except for the representation and normalization of the joint distribution over and , a point to which
we return later. Crucially, ACh exerts its influence through equation 2. If ]\ ^ is
high, then the input stimulus controlled, likelihood term dominates in the conditioning process (equation 5); if ]\^ is low, then temporal context ( ]\^ ) and the
likelihood terms balance.
One potentially dangerous aspect of this inference procedure is that it might get
unreasonably committed to a single state ]\^ _ _Uddd because it does not represent explicitly the probability accorded to the other possible values of ]\^ given
[ ]\^ . A natural way to avoid this is to bound the ACh level from below by a constant, , making approximate inference slightly more stimulus-bound than exact
inference. This approximation should add robustness. In practice, rather than use
equation 9, we use
'
$_
'( lm' lm l wyx ._
z [ ){
(10)
1
!#"%$'&( )+*
0.5
0
1
-,.,./10 23 45
67 89 &
0.5
0
0
?10
?30
?50
?70
0
50
100
150
200
250
300
0.1
0.2
350
400
@BA 0CE#"
D GFIH.*KJ
@BA 0CL M-"GFNH'*KJ
0.3
0.4
:23;'
23 8<= >0.5
: 0.6
?
0.7
0.8
0.9
1
Figure 4: ACh model. A) ACh level from the exact posterior for one run. B) ACh level
- ? in the approximate model in the same run. Note the coarse similarity between A and
B. C) Solid: the mean extra representational cost for the true state 0 ? ` over that in the exact
posterior using the ACh model as a function of the minimum allowed ACh level O . Dashed:
the same quantity for the pure bottom-up model (which is equivalent to the approximate
model for O`@ ' ). Errorbars (which are almost invisible) show standard errors of the means
over ' I IfI trials.
Figure 4B shows the approximate ACh level for the same case as figure 4A, using
_ h d l . Although the detailed value of this signal is clearly different from that
arising from an exact knowledge of the posterior probabilities (in figure 4A), the
gross movements are quite similar. Note the effect of in preventing the ACh level
from dropping to h . Figure 3C shows that the ACh-based approximate posterior
values w x z [ { are much closer to the true values than for the purely bottom-up
model, particularly for values of wyx z [ ]{ near h and l , where most data lie. Figure 3F shows that inference about is noisy, but the pattern of true values | is
certainly visible. Figure 4C shows the effect of changing on the quality of inference about the true states | . This shows differences between approximate and
exact log probabilities of the true states | , averaged over l hihZh cases. If _ l , then
inference is completely stimulus-bound, just like the purely bottom-up model; values of less than h d o appear to do well for this and other settings of the parameters
of the problem. An upper bound on the performance of approximate inference can
be calculated in three steps by: i) using the exact posterior to work out and ,
ii) using these values to approximate wyx { as in equation 2, and iii) using this
approximate distribution in equation 4 and the remaining equations. The average
resulting cost (ie the average resulting difference from the log probability under
exact inference) is m j d v log units. Thus, the ACh-based approximation performs
well, and much better than purely bottom-up inference.
'
'
'
'
'
4 Discussion
We have suggested that one of the roles of ACh in cortical processing is to report
contextual uncertainty in order to control the balance between stimulus-bound,
bottom-up, processing, and contextually-bound, top-down processing. We used
the example of a hierarchical HMM in which representational inference for a middle layer should correctly reflect such a balance, and showed that a simple model
of the drive and effects of ACh leads to competent inference.
This model is clearly overly simple. In particular, it uses a localist representation
for the state , and so exact inference would be feasible. In a more realistic case,
distributed representations would be used at multiple levels in the hierarchy, and
so only one single context could be entertained at once. Then, it would also not be
possible to represent the degree of uncertainty using the level of activities of the
units representing the context, at least given a population-coded representation. It
would also be necessary to modify the steps in equations 4 and 5, since it would
be hard to represent the joint uncertainty over representations at multiple levels
in the hierarchy. Nevertheless, our model shows the feasibility of using an ACh
signal in helping propagate and use approximate information over time.
Since it is straightforward to administer cholinergic agonists and antagonists, there
are many ways to test aspects of this proposal. We plan to start by using the
paradigm of Ress et al,24 which uses fMRI techniques to study bottom-up and
top-down influences on the detection of simple visual targets. Preliminary simulation studies indicate that a hidden Markov model under controllable cholinergic
modulation can capture several aspects of existent data on animal signal detection
tasks.18
Acknowledgements
We are very grateful to Michael Hasselmo, David Heeger, Sham Kakade and Szabolcs K?ali for helpful discussions. Funding was from the Gatsby Charitable Foundation and the NSF. Reference [28] is an extended version of this paper.
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1,212 | 2,104 | Fast Parameter Estimation
Using Green's Functions
K. Y. Michael Wong
Department of Physics
Hong Kong University
of Science and Technology
Clear Water Bay, Hong Kong
[email protected]
FuIi Li
Department of Applied Physics
Xian Jiaotong University
Xian , China 710049
flli @xjtu. edu. en
Abstract
We propose a method for the fast estimation of hyperparameters
in large networks, based on the linear response relation in the cavity method, and an empirical measurement of the Green's function. Simulation results show that it is efficient and precise, when
compared with cross-validation and other techniques which require
matrix inversion.
1
Introduction
It is well known that correct choices of hyperparameters in classification and regression tasks can optimize the complexity of the data model , and hence achieve the
best generalization [1]. In recent years various methods have been proposed to estimate the optimal hyperparameters in different contexts, such as neural networks [2],
support vector machines [3, 4, 5] and Gaussian processes [5]. Most of these methods are inspired by the technique of cross-validation or its variant, leave-one-out
validation. While the leave-one-out procedure gives an almost unbiased estimate
of the generalization error, it is nevertheless very tedious. Many of the mentioned
attempts aimed at approximating this tedious procedure without really having to
sweat through it. They often rely on theoretical bounds, inverses to large matrices,
or iterative optimizations.
In this paper, we propose a new approach to hyperparameter estimation in large
systems. It is known that large networks are mean-field systems, so that when one
example is removed by the leave-one-out procedure, the background adjustment
can be analyzed by a self-consistent perturbation approach. Similar techniques
have been applied to the neural network [6], Bayesian learning [7] and the support
vector machine [5]. They usually involve a macroscopic number of unknown variables, whose solution is obtained through the inversion of a matrix of macroscopic
size, or iteration. Here we take a further step to replace it by a direct measurement
of the Green's function via a small number of learning processes. The proposed
procedure is fast since it does not require repetitive cross-validations, matrix inversions, nor iterative optimizations for each set of hyperparaemters. We will also
present simulation results which show that it is an excellent approximation.
The proposed technique is based on the cavity method, which was adapted from
disordered systems in many-body physics. The basis of the cavity method is a
self-consistent argument addressing the situation of removing an example from the
system. The change on removing an example is described by the Green's function,
which is an extremely general technique used in a wide range of quantum and
classical problems in many-body physics [8]. This provides an excellent framework
for the leave-one-out procedure. In this paper, we consider two applications of
the cavity method to hyperparameter estimation, namely the optimal weight decay
and the optimal learning time in feedforward networks. In the latter application,
the cavity method provides, as far as we are aware of, the only estimate of the
hyperparameter beyond empirical stopping criteria and brute force cross-validation.
2
Steady-State Hyperparameter Estimation
Consider the network with adjustable parameters w. An energy function E is
defined with respect to a set of p examples with inputs and outputs respectively
given by {IL and y'", JL = 1, ... ,p, where (IL is an N-dimensional input vector with
components
j = 1,? ?? ,N, and N ? 1 is macroscopic. We will first focus on
the dynamics of a single-layer feedforward network and generalize the results to
multilayer networks later. In single-layer networks, E has the form
e;,
E =
L f(X'",y'") + R(w).
(1)
'"
Here f( x'" , y'") represents the error function with respect to example JL. It is expressed in terms of the activation x'" == w? (IL. R( w) represents a regularization
term which is introduced to limit the complexity of the network and hence enhance
the generalization ability. Learning is achieved by the gradient descent dynamics
dWj(t) _ _ ~_oE_
dt
(2)
The time-dependent Green's function Gjk(t, s) is defined as the response of the
weight Wj at time t due to a unit stimulus added at time s to the gradient term with
respect to weight Wk, in the limit of a vanishing magnitude of the stimulus. Hence
if we compare the evolution of Wj(t) with another system Wj(t) with a continuous
perturbative stimulus Jhj(t), we would have
dWj(t) = _~ oE
dt
Now.
Jh()
J t ,
(3)
dsGjk(t,s)Jhk(s).
(4)
J
+
and the linear response relation
Wj(t) = Wj(t)
+L
J
k
Now we consider the evolution ofthe network w;'"(t) in which example JL is omitted
from the training set. For a system learning macroscopic number of examples, the
changes induced by the omission of an example are perturbative, and we can assume
that the system has a linear response. Compared with the original network Wj(t),
the gradient of the error of example JL now plays the role of the stimulus in (3).
Hence we have
(5)
Multiplying both sides by ~f and summing over j, we obtain
1-'( ) -
h t - x
I-'()
t
+
J
ds
[1
' " I-'G ( ) I-']OE(XI-'(S)'YI-')
N "7:~j jk t ,s ~k
oxl-'(s)'
(6)
Here hl-'(t) == V;\I-'(t) . ~ is called the cavity activation of example ft. When the
dynamics has reached the steady state, we arrive at
I-'
hI-'
= x
where, = limt--+oo
+,
OE(XI-' , yl-')
oxl-'
'
JdS[Ljk ~fGjk (t , s)~r]jN
(7)
is the susceptibility.
At time t , the generalization error is defined as the error function averaged over the
distribution of input (, and their corresponding output y, i.e. ,
(8)
where x == V; . (is the network activation. The leave-one-out generalization error is
an estimate of 109 given in terms ofthe cavity activations hI-' by fg = LI-' 10 (hI-' ,yl-')jp.
Hence if we can estimate the Green's function, the cavity activation in (7) provides
a convenient way to estimate the leave-one-out generalization error without really
having to undergo the validation process.
While self-consistent equations for the Green's function have been derived using
diagrammatic methods [9], their solutions cannot be computed except for the specific case of time-translational invariant Green's functions , such as those in Adaline
learning or linear regression. However, the linear response relation (4) provides a
convenient way to measure the Green's function in the general case. The basic idea
is to perform two learning processes in parallel, one following the original process
(2) and the other having a constant stimulus as in (3) with 6hj (t) = TJ6jk, where
8j k is the Kronecka delta. When the dynamics has reached the steady state, the
measurement Wj - Wj yields the quantity TJ Lk dsGjk(t, s).
J
A simple averaging procedure, replacing all the pairwise measurements between the
stimulation node k and observation node j, can be applied in the limit of large
N. We first consider the case in which the inputs are independent and normalized,
i.e., (~j) = 0, (~j~k) = 8j k. In this case, it has been shown that the off-diagonal
Green's functions can be neglected, and the diagonal Green's functions become selfaveraging, i.e. , Gjk(t , s) = G(t, s)8jk , independent of the node labels [9], rendering
, = limt--+oo J dsG(t, s).
In the case that the inputs are correlated and not normalized, we can apply standard
procedures of whitening transformation to make them independent and normalized
[1]. In large networks, one can use the diagrammatic analysis in [9] to show that
the (unknown) distribution of inputs does not change the self-averaging property of
the Green's functions after the whitening transformation. Thereafter, the measurement of Green's functions proceeds as described in the simpler case of independent
and normalized inputs. Since hyperparameter estimation usually involves a series
of computing fg at various hyperparameters, the one-time preprocessing does not
increase the computational load significantly.
Thus the susceptibility, can be measured by comparing the evolution of two processes: one following the original process (2), and the other having a constant
stimulus as in (3) with 8h j (t) = TJ for all j. When the dynamics has reached the
steady state, the measurement (Wj - Wj) yields the quantity TJ,.
We illustrate the extension to two-layer networks by considering the committee machine, in which the errorfunction takes the form E(2:: a !(x a), y) , where a = 1,? ??, nh
is the label of a hidden node, Xa == wa . [is the activation at the hidden node a,
and! represents the activation function. The generalization error is thus a function
of the cavity activations of the hidden nodes, namely, E9 = 2::JL E(2::a !(h~), yJL) /p,
where h~ = w~JL . (IL . When the inputs are independent and normalized, they are
related to the generic activations by
hJL- JL+'"
aE(2::c !(X~) , yJL)
a - Xa ~ "lab
a JL
'
Xb
b
(9)
where "lab = limt~ oo J dsGab(t, s) is the susceptibility tensor. The Green's function
Gab(t, s) represents the response of a weight feeding hidden node a due to a stimulus
applied at the gradient with respect to a weight feeding node b. It is obtained by
monitoring nh + 1 learning processes, one being original and each of the other nh
processes having constant stimuli at the gradients with respect to one of the hidden
nodes, viz.,
dw~~) (t) _
dt
1
aE
- - N ------=:(b)
aW aj
+ 'f)rSab ,
(10)
b = 1, ... ,nh?
When the dynamics has reached the steady state, the measurement (w~7
yields the quantity 'f)'Yab.
-
Waj)
We will also compare the results with those obtained by extending the analysis of
linear unlearning leave-one-out (LULOO) validation [6]. Consider the case that the
regularization R(w) takes the form of a weight decay term, R(w) = N 2::ab AabWa .
Wb/2. The cavity activations will be given by
hJL = JL + '"
a Xa ~
b
(
1-
,"
11
iJ 2:: j k ~'j(A + Q)~}bk~r
) aE(2:: c !(xn, yJL))
a JL
'
2::cjdk ~'j !'(xn(A + Q)~, dd'(x~)~r
Xb
1
(11)
where E~ represents the second derivative of E with respect to the student output
for example /1, and the matrix Aaj,bk = AabrSjk and Q is given by
Qaj,bk
= ~ 2: ~'j f'(x~)f'(x~)~r?
(12)
JL
The LULOO result of (11) differs from the cavity result of (9) in that the susceptibility "lab now depends on the example label /1, and needs to be computed by
inverting the matrix A + Q. Note also that second derivatives of the error term
have been neglected.
To verify the proposed method by simulations, we generate examples from a noisy
teacher network which is a committee machine
yJL = ~ erf
nh
(1yf2Ba ? f ) +
(Jzw
(13)
Here Ba is the teacher vector at the hidden node a. (J is the noise level. ~'j and
zJL are Gaussian variables with zero means and unit variances. Learning is done by
the gradient descent of the energy function
(14)
and the weight decay parameter ,X is the hyperparameter to be optimized. The
generalization error fg is given by
where the averaging is performed over the distribution of input { and noise z. It can
be computed analytically in terms of the inner products Qab = wa . Wb, Tab = Ba . Bb
and Rab = Ba . Wb [10]. However, this target result is only known by the teacher ,
since Tab and Rab are not accessible by the student.
Figure 1 shows the simulation results of 4 randomly generated samples. Four results
are compared: the target generalization error observed by the teacher, and those
estimated by the cavity method, cross-validation and extended LULOO. It can
be seen that the cavity method yields estimates of the optimal weight decay with
comparable precision as the other methods.
For a more systematic comparison, we search for the optimal weight decay in 10 samples using golden section search [11] for the same parameters as in Fig. 1. Compared
with the target results, the standard deviations of the estimated optimal weight decays are 0.3, 0.25 and 0.24 for the cavity method, sevenfold cross-validation and
extended LULOO respectively. In another simulation of 80 samples of the singlelayer perceptron, the estimated optimal weight decays have standard deviations of
1.2, 1.5 and 1.6 for the cavity method, tenfold cross-validation and extended LULOO respectively (the parameters in the simulations are N = 500, p = 400 and a
ranging from 0.98 to 2.56).
To put these results in perspective, we mention that the computational resources
needed by the cavity method is much less than the other estimations. For example,
in the single-layer perceptrons, the CPU time needed to estimate the optimal weight
decay using the golden section search by the teacher, the cavity method, tenfold
cross-validation and extended LULOO are in the ratio of 1 : 1.5 : 3.0 : 4.6.
Before concluding this section, we mention that it is possible to derive an expression
of the gradient dEg I d,X of the estimated generalization error with respect to the
weight decay. This provides us an even more powerful tool for hyperparameter
estimation. In the case of the search for one hyperparameter, the gradient enables
us to use the binary search for the zero of the gradient, which converges faster
than the golden section search. In the single-layer experiment we mentioned, its
precision is comparable to fivefold cross-validations, and its CPU time is only 4%
more than the teacher's search. Details will be presented elsewhere. In the case of
more than one hyperparameters, the gradient information will save us the need for
an exhaustive search over a multidimensional hyperparameter space.
3
Dynamical Hyperparameter Estimation
The second example concerns the estimation of a dynamical hyperparameter,
namely the optimal early stopping time, in cases where overtraining may plague
the generalization ability at the steady state. In perceptrons, when the examples
are noisy and the weight decay is weak, the generalization error decreases in the
early stage of learning, reaches a minimum and then increases towards its asymptotic value [12, 9]. Since the early stopping point sets in before the system reaches
the steady state, most analyses based on the equilibrium state are not applicable.
Cross-validation stopping has been proposed as an empirical method to control
overtraining [13]. Here we propose the cavity method as a convenient alternative.
0.52
G----8 target
eQ)
(b)
G----EJ cavity
0 - 0 LULOO
<=
0
~
.!::!
0.46
m
Q)
<=
Q)
0>
0.40
(d)
(c)
eQ)
<=
0
~
.!::!
m
Q)
<=
Q)
0>
0.40
o
0
weight decay A
2
weight decay A
Figure 1: (a-d) The dependence ofthe generalization error of the multilayer perceptron on the weight decay for N = 200, p = 700, nh = 3, (J = 0.8 in 4 samples. The
solid symbols locate the optimal weight decays estimated by the teacher (circle), the
cavity method (square), extended LULOO (diamond) and sevenfold cross-validation
(triangle) .
In single-layer perceptrons, the cavity activations of the examples evolve according
to (6), enabling us to estimate the dynamical evolution of the estimated generalization error when learning proceeds. The remaining issue is the measurement of
the time-dependent Green's function. We propose to introduce an initial homogeneous stimulus, that is, Jhj (t) = 1]J(t) for all j. Again, assuming normalized and
independent inputs with (~j) = 0 and (~j~k) = Jjk , we can see from (4) that the
measurement (Wj(t) - Wj(t)) yields the quantity 1]G(t, 0).
We will first consider systems that are time-translational invariant, i.e., G(t, s) =
G(t - s, 0). Such are the cases for Adaline learning and linear regression [9], where
the cavity activation can be written as
h'"(t) = x'"(t) +
J
dsG(t - s, 0) OE(X'"(S), y'").
ox,"(s)
(16)
This allows us to estimate the generalization error Eg(t) via Eg(t)
2:.," E(h'"(t), y'")/p, whose minimum in time determines the early stopping point.
To verify the proposed method in linear regression, we randomly generate examples from a noisy teacher with y'" = iJ . f'" + (Jzw Here iJ is the teacher vector with B2 = 1.
and z'" are independently generated with zero means and
unit variances. Learning is done by the gradient descent of the energy function
E(t) = 2:.," (y'" - w(t) . f'")2/2 . The generalization error Eg(t) is the error av-
e;
eraged over the distribution of input [ and their corresponding output y, i.e.,
Eg(t) = ((iJ . [ + (JZ - w? [)2/2). As far as the teacher is concerned, Eg(t) can
be computed as Eg(t) = (1 - 2R(t) + Q(t) + (J2)/2. where R(t) = w(t) . iJ and
Q(t) = W(t)2.
Figure 2 shows the simulation results of 6 randomly generated samples. Three results are compared: the teacher's estimate, the cavity method and cross-validation.
Since LULOO is based on the equilibrium state, it cannot be used in the present
context. Again, we see that the cavity method yields estimates of the early stopping time with comparable precision as cross-validation. The ratio of the CPU time
between the cavity method and fivefold cross-validation is 1 : 1.4.
For nonlinear regression and multilayer networks, the Green 's functions are not
time-translational invariant. To estimate the Green 's functions in this case, we have
devised another scheme of stimuli. Preliminary results for the determination of the
early stopping point are satisfactory and final results will be presented elsewhere.
1 .1
eQ.i
c::
a
~
.!::!
0.9
~
<l>
c::
<l>
0>
0.7
eQ.i
c::
a
~
c;;
Q.i
.!::!
0.9
c::
<l>
0>
0.7
0
2
time t
0
2
time t
0
2
4
time t
Figure 2: (a-f) The evolution of the generalization error of linear regression for
N = 500, p = 600 and (J = 1. The solid symbols locate the early stopping points
estimated by the teacher (circle), the cavity method (square) and fivefold crossvalidation (diamond).
4
Conclusion
We have proposed a method for the fast estimation of hyperparameters in large
networks, based on the linear response relation in the cavity method, combined
with an empirical method of measuring the Green's function. Its efficiency depends
on the independent and identical distribution of the inputs, greatly reducing the
number of networks to be monitored. It does not require the validation process
or the inversion of matrices of macroscopic size, and hence its speed compares
favorably with cross-validation and other perturbative approaches such as extended
LULOO. For multilayer networks, we will explore further speedup of the Green 's
function measurement by multiplexing the stimuli to the different hidden units into
a single network, to be compared with a reference network. We will also extend the
technique to other benchmark data to study its applicability.
Our initial success indicates that it is possible to generalize the method to more
complicated systems in the future. The concept of Green's functions is very general,
and its measurement by comparing the states of a stimulated system with a reference
one can be adopted to general cases with suitable adaptation. Recently, much
attention is paid to the issue of model selection in support vector machines [3, 4, 5].
It would be interesting to consider how the proposed method can contribute to these
cases.
Acknowledgements
We thank C. Campbell for interesting discussions and H. Nishimori for encouragement. This work was supported by the grant HKUST6157/99P from the Research
Grant Council of Hong Kong.
References
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(1995).
[2] G. B. Orr and K-R. Muller, eds., Neural Networks: Tricks of th e Trad e, Springer,
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1,213 | 2,105 | Algorithmic Luckiness
Ralf Herbrich
Microsoft Research Ltd.
CB3 OFB Cambridge
United Kingdom
rherb@microsoft?com
Robert C. Williamson
Australian National University
Canberra 0200
Australia
Bob. Williamson @anu.edu.au
Abstract
In contrast to standard statistical learning theory which studies
uniform bounds on the expected error we present a framework that
exploits the specific learning algorithm used. Motivated by the
luckiness framework [8] we are also able to exploit the serendipity
of the training sample. The main difference to previous approaches
lies in the complexity measure; rather than covering all hypotheses in a given hypothesis space it is only necessary to cover the
functions which could have been learned using the fixed learning
algorithm. We show how the resulting framework relates to the
VC, luckiness and compression frameworks. Finally, we present an
application of this framework to the maximum margin algorithm
for linear classifiers which results in a bound that exploits both the
margin and the distribution of the data in feature space.
1
Introduction
Statistical learning theory is mainly concerned with the study of uniform bounds
on the expected error of hypotheses from a given hypothesis space [9, 1]. Such
bounds have the appealing feature that they provide performance guarantees for
classifiers found by any learning algorithm. However, it has been observed that
these bounds tend to be overly pessimistic. One explanation is that only in the
case of learning algorithms which minimise the training error it has been proven
that uniformity of the bounds is equivalent to studying the learning algorithm's
generalisation performance directly.
In this paper we present a theoretical framework which aims at directly studying the
generalisation error of a learning algorithm rather than taking the detour via the
uniform convergence of training errors to expected errors in a given hypothesis space.
In addition, our new model of learning allows the exploitation of the fact that we
serendipitously observe a training sample which is easy to learn by a given learning
algorithm. In that sense, our framework is a descendant of the luckiness framework
of Shawe-Taylor et al. [8]. In the present case, the luckiness is a function of a given
learning algorithm and a given training sample and characterises the diversity of
the algorithms solutions. The notion of luckiness allows us to study given learning
algorithms at many different perspectives. For example, the maximum margin
algorithm [9] can either been studied via the number of dimensions in feature space,
the margin of the classifier learned or the sparsity of the resulting classifier. Our
main results are two generalisation error bounds for learning algorithms: one for
the zero training error scenario and one agnostic bound (Section 2). We shall
demonstrate the usefulness of our new framework by studying its relation to the
VC framework, the original luckiness framework and the compression framework of
Littlestone and Warmuth [6] (Section 3). Finally, we present an application of the
new framework to the maximum margin algorithm for linear classifiers (Section 4).
The detailed proofs of our main results can be found in [5].
We denote vectors using bold face, e.g. x = (Xl, ... ,xm ) and the length of this vector
by lxi, i.e. Ixl = m. In order to unburden notation we use the shorthand notation
Z[i:jJ := (Zi,"" Zj) for i :::; j. Random variables are typeset in sans-serif font. The
symbols P x , Ex [f (X)] and IT denote a probability measure over X, the expectation of
f (.) over the random draw of its argument X and the indicator function, respectively.
The shorthand notation Z(oo) := U;;'=l zm denotes the union of all m- fold Cartesian
products of the set Z. For any mEN we define 1m C {I, ... , m }m as the set of all
permutations of the numbers 1, ... ,m,
1m := {(i l , ... ,i m) E {I, ... ,m}m I'v'j f:- k: ij f:- id .
Given a 2m- vector i E hm and a sample z E z2m we define Wi : {I, ... , 2m} -+
{I, ... , 2m} by Wi (j) := ij and IIdz) by IIi (z) := (Z7ri(l), ... , Z7ri(2m))'
2
Algorithmic Luckiness
Suppose we are given a training sample z = (x, y) E (X x y)m = zm of size
mEN independently drawn (iid) from some unknown but fixed distribution P Xy =
P z together with a learning algorithm A : Z( 00) -+ yX . For a predefined loss
l : y x y -+ [0,1] we would like to investigate the generalisation error Gl [A, z] :=
Rl [A (z)] - infhEYx Rl [h] of the algorithm where the expected error Rl [h] of his
defined by
Rl [h] := Exy [l (h (X) ,Y)] .
Since infhEYx Rl [h] (which is also known as the Bayes error) is independent of A
it suffices to bound Rl [A (z)]. Although we know that for any fixed hypothesis h
the training error
~
1
Rdh,z]:=~
L
l(h(xi),Yi)
(X i ,Yi) E z
is with high probability (over the random draw of the training sample z E Z(oo))
close to Rl [h], this might no longer be true for the random hypothesis A (z). Hence
we would like to state that with only small probability (at most 8) , the expected
error Rl [A (z)] is larger than the training error HI [A (z), z] plus some sample and
algorithm dependent complexity c (A, z, 8),
Pzm (Rl [A (Z)] > HI [A (Z), Z]
+ c (A, Z,8)) < 8.
(1)
In order to derive such a bound we utilise a modified version of the basic lemma of
Vapnik and Chervonenkis [10].
Lemma 1. For all loss functions l : y x y -+ [0,1], all probability measures P z , all
algorithms A and all measurable formulas Y : zm -+ {true, false}, if mc 2 > 2 then
Pzm (( RdA (Z)] > HdA (Z) , Z]
+ c)
2P Z 2m ((HI [A (Z[l:m]) ,Z[(m+l):2mJJ
/\ Y (Z)) <
> HI [A (Z[l:mJ) ,Z[l:mJJ + ~) /\ Y (Z[l:m])) .
,
.I
V
J(Z)
Proof (Sketch). The probability on the r.h.s. is lower bounded by the probability of the conjunction of event on the l.h.s. and Q (z)
Rl [A (Z[l:mj)] Rl [A (Z[l:mj) ,Z(m+1):2m] < ~. Note that this probability is over z E z2m. If
we now condition on the first m examples, A (Z[l:mj) is fixed and therefore by an
application of Hoeffding's inequality (see, e.g. [1]) and since m?2 > 2 the additional
event Q has probability of at least ~ over the random draw of (Zm+1, ... , Z2m). 0
=
Use of Lemma 1 - which is similar to the approach of classical VC analysis reduces the original problem (1) to the problem of studying the deviation of the
training errors on the first and second half of a double sample z E z2m of size
2m. It is of utmost importance that the hypothesis A (Z[l:mj) is always learned
from the first m examples. Now, in order to fully exploit our assumptions of the
mutual independence of the double sample Z E z2m we use a technique known
as symmetrisation by permutation: since PZ2~ is a product measure, it has the
property that PZ2?> (J (Z)) = PZ2~ (J (ITi (Z))) for any i E hm. Hence, it suffices
to bound the probability of permutations Jri such that J (ITi (z)) is true for a given
and fixed double sample z. As a consequence thereof, we only need to count the
number of different hypotheses that can be learned by A from the first m examples
when permuting the double sample.
Definition 1 (Algorithmic luckiness). Any function L that maps an algorithm
A : Z( oo ) -+ yX and a training sample z E Z( oo ) to a real value is called an algorithmic luckiness. For all mEN, for any z E z2m , the lucky set HA (L , z) ~ yX is the
set of all hypotheses that are learned from the first m examples (Z7ri(1),???, Z7ri(m))
when permuting the whole sample z whilst not decreasing the luckiness, i.e.
(2)
where
Given a fixed loss function 1 :
y
x
y -+ [0,1]
the induced loss function set
?1 (HA (L,z)) is defined by
?1 (HA (L,z)) := {(x,y) r-+ 1(h(x) ,y)
I h E HA (L,z)}
.
For any luckiness function L and any learning algorithm A , the complexity of the
double sample z is the minimal number N1 (T, ?1 (HA (L, z)) ,z) of hypotheses h E
yX needed to cover ?1 (HA (L , z)) at some predefined scale T, i.e. for any hypothesis
hE HA (L, z) there exists a h E yX such that
(4)
To see this note that whenever J (ITi (z)) is true (over the random draw of permutations) then there exists a function h which has a difference in the training errors
on the double sample of at least ~ + 2T. By an application of the union bound we
see that the number N 1 (T, ?1 (HA (L , z)) , z) is of central importance. Hence, if we
are able to bound this number over the random draw of the double sample z only
using the luckiness on the first m examples we can use this bound in place of the
worst case complexity SUPzEZ2~ N1 (T, ?1 (HA (L , z)) ,z) as usually done in the VC
framework (see [9]).
Definition 2 (w- smallness of L). Given an algorithm A : Z (00 ) -+ yX and a loss
l : y x y -+ [a, 1] the algorithmic luckiness function Lis w- small at scale T E jR+ if
for all mEN, all J E (a , 1] and all P z
PZ2~ (Nl (T, ?"1 (1iA (L, Z)), Z)
> w (L (A, Z[l:ml) ,l, m, J,T)) < J.
,
"
v
S(Z)
Note that if the range of l is {a, I} then N 1 (2~ ' ?"1 (1iA (L, z)) , z) equals the number of dichotomies on z incurred by ?"1 (1iA (L , z)).
Theorem 1 (Algorithmic luckiness bounds). Suppos e we have a learning
algorithm A : Z( oo ) -+ yX and an algorithmic luckiness L that is w-small at
scale T for a loss function l : y X Y -+ [a, 1]. For any probability measure P z ,
any dEN and any J E (a , 1], with probability at least 1 - J over the random
draw of the training sample z E zm of size m, if w (L (A, z) ,l, m, J/4, T) :::; 2d
then
!
Rz[A (z)] :::; Rz[A (z), z] +
(d + 10g2
(~) )
+ 4T.
(5)
Furthermore, under the above conditions if the algorithmic luckiness L is wsmall at scale 2~ for a binary loss function l (".) E {a, I} and Rl [A (z), z] = a
then
(6)
Proof (Compressed Sketch). We will only sketch the proof of equation (5) ; the proof
of (6) is similar and can be found in [5]. First, we apply Lemma 1 with Y (z) ==
w (L (A,z) ,l,m,J/4,T) :::; 2d. We now exploit the fact that
PZ2~
(J (Z))
:Z2~
(J (Z)
1\
S (Z) ), +PZ2~ (J (Z)
1\
...,S (Z))
v
J
< 4+
:::: P Z 2 ~
(S(Z))
PZ2~
(J (Z) I\...,S (Z)) ,
which follows from Definition 2. Following the above-mentioned argument it suffices to bound the probability of a random permutation III (z) that J (III (z)) 1\
...,S (III (z)) is true for a fixed double sample z. Noticing that Y (z) 1\ ...,S (z) =>
Nl (T,?"l (1iA (L , z)) ,z) :::; 2d we see that we only consider swappings Jri for which
Nl (T,?"l (1iA (L,IIi (z))) ,IIi (z)) :::; 2d. Thus let us consider such a cover of
size not more than 2. By (4) we know that whenever J (IIi (z)) 1\ ...,S (IIi (z))
is true for a swapping i then there exists a hypothesis h E yX in the cover
(III (z)) [(m+1) :2ml] - Rl
(III (z)) [l:ml] > ~ + 2T. Using the
such that Rl
union bound and Hoeffding's inequality for a particular choice of PI shows that
PI (J (III (z)) 1\ ...,S (III (z))) :::; ? which finalises the proof.
D
[h,
[h,
A closer look at (5) and (6) reveals that the essential difference to uniform bounds
on the expected error is within the definition of the covering number: rather than
covering all hypotheses h in a given hypothesis space 1i ~ yX for a given double
sample it suffices to cover all hypotheses that can be learned by a given learning
algorithm from the first half when permuting the double sample. Note that the
usage of permutations in the definition of (2) is not only a technical matter; it
fully exploits all the assumptions made for the training sample, namely the training
sample is drawn iid.
3
Relationship to Other Learning Frameworks
In this section we present the relationship of algorithmic luckiness to other learning
frameworks (see [9, 8, 6] for further details of these frameworks).
VC Framework If we consider a binary loss function l (".) E {a, I} and assume
that the algorithm A selects functions from a given hypothesis space H ~ yX then
L (A, z) = - VCDim (H) is a w- smallluckiness function where
w
(Lo,l,m,8, 1) :S (2em)
-Lo
2m
-Lo
.
(7)
This can easily be seen by noticing that the latter term is an upper bound on
maxz EZ 2 ", I{ (l (h (Xl) ,yI) , ... ,l (h (X2m), Y2m)) : h E H}I (see also [9]). Note that
this luckiness function neither exploits the particular training sample observed nor
the learning algorithm used.
Luckiness Framework Firstly, the luckiness framework of Shawe-Taylor et al. [8]
only considered binary loss functions l and the zero training error case. In this work,
the luckiness ? is a function of hypothesis and training samples and is called wsmall if the probability over the random draw of a 2m sample z that there exists a
hypothesis h with w(?(h, (Zl, ... ,zm )), 8) < J'--h (2;" {(X , y) t--+ l (g (x) ,y) 1? (g , z) :::::
? (h, Z)}, z), is smaller than 8. Although similar in spirit, the classical luckiness
framework does not allow exploitation of the learning algorithm used to the same
extent as our new luckiness. In fact, in this framework not only the covering number
must be estimable but also the variation of the luckiness ? itself. These differences
make it very difficult to formally relate the two frameworks.
Compression Framework In the compression framework of Littlestone and
Warmuth [6] one considers learning algorithms A which are compression schemes,
i.e. A (z) = :R (e (z)) where e (z) selects a subsample z ~ z and :R : Z(oo) -+ yX
is a permutation invariant reconstruction function. For this class of learning algorithms, the luckiness L(A,z) = -le(z)1 is w- small where w is given by (7). In
order to see this we note that (3) ensures that we only consider permutations 7ri
where e (IIi (z)) :S Ie (z)l, i.e. we use not more than -L training examples from
z E z2m. As there are exactly
distinct choices of d training examples from
2m examples the result follows by application of Sauer's lemma [9]. Disregarding
constants, Theorem 1 gives exactly the same bound as in [6].
e;;)
4
A New Margin Bound For Support Vector Machines
In this section we study the maximum margin algorithm for linear classifiers, i.e. A :
Z(oo) -+ Hcp where Hcp := {x t--+ (? (x), w) I wE }C} and ? : X -+ }C ~ ?~ is known
as the feature mapping. Let us assume that l (h (x) ,y) = lO - l (h (x) ,y) := lIyh(x)::;o,
Classical VC generalisation error bounds exploit the fact that VCDim (Hcp) = nand
(7). In the luckiness framework of Shawe-Taylor et al. [8] it has been shown that we
can use fat1i.p h'z (w)) :S h'z (W))-2 (at the price of an extra 10g2 (32m) factor) in
place of VCDim (Hcp) where "(z (w) = min(xi,Yi)Ez Yi (? (Xi) , w) / Ilwll is known as
the margin. Now, the maximum margin algorithm finds the weight vector WMM that
maximises "(z (w). It is known that WMM can be written as a linear combination of
the ? (Xi). For notational convenience, we shall assume that A: Z(oo) -+ 1R(00) maps
to the expansion coefficients 0: such that Ilwall = 1 where Wa := 2:1~1 (XicfJ(Xi).
Our new margin bound follows from the following theorem together with (6).
Theorem 2. Let fi (x) be the smallest 10 > 0 such that {cfJ (Xl) , ... , cfJ (Xm) }
can be covered by at most i balls of radius less than or equal f. Let f z (w) be
.
Yi (4)(X i), W)
D
th
l
l
d
defi ne d by f z (W ) .. - mm( Xi, Yi)Ez 1
14>(Xi) II.llwll. ror e zero-one oss 0-1 an
the maximum margin algorithm A , the luckiness function
L(A ,Z )
=_ mIn
. {.~ E
",,-T
1'1
.>
_
~
(f (X)2:7=1 IA
i
fz
(Z)jl) 2 }
( )
,
(8)
W A(z)
is w-small at scale 112m w.r.t. the function
(
1)
w L o,l,m,8, 2m
=
(2em)- 2L O
-Lo
(9)
Proof (Sketch). First we note that by a slight refinement of a theorem of Makovoz
[7] we know that for any Z E zm there exists a weight vector w = 2:: 1 iiicfJ (Xi)
such that
(10)
and
a
WA(z)
E ]Rm has no more than - L (A, z) non-zero components. Although only
is of unit length, one can show that (10) implies that
(WA(z),
wi IIwll) ~ )1- f; (WA(z?).
Using equation (10) of [4] this implies that w correctly classifies Z E zm. Consider
a fixed double sample Z E z2m and let ko := L (A, (Zl , ... , zm )). By virtue of (3)
and the aforementioned argument we only need to consider permutations tri such
that there exists a weight vector w = 2:;:1 iijcfJ (Xj) with no more than ko non-zero
iij. As there are exactly (2;;) distinct choices of dE {I, ... , ko} training examples
from the 2m examples Z there are no more than (2emlko)kO different subsamples
to be used in w. For each particular subsample z ~ Z the weight vector w is a
member of the class of linear classifiers in a ko (or less) dimensional space. Thus,
from (7) it follows that for the given subsample z there are no more (2emlko)kO
different dichotomies induced on the double sample Z E z2m. As this holds for any
D
double sample, the theorem is proven.
There are several interesting features about this margin bound. Firstly, observe
that 2:;:1 IA (Z)j I is a measure of sparsity of the solution found by the maximum
margin algorithm which, in the present case, is combined with margin. Note that
for normalised data, i.e. IlcfJ Oil = constant, the two notion of margins coincide,
i.e. f z (w) = I Z (w). Secondly, the quantity fi (x) can be considered as a measure
of the distribution of the mapped data points in feature space. Note that for all
i E N, fi (x) :S 101 (x) :S maxjE{l ,... ,m} IlcfJ (xj)ll. Supposing that the two classconditional probabilities PX 1Y=y are highly clustered, 102 (x) will be very small. An
extension of this reasoning is useful in the multi-class case; binary maximum margin
classifiers are often used to solve multi-class problems [9]. There appears to be also
a close relationship of fi (x) with the notion of kernel alignment recently introduced
in [3]. Finally, one can use standard entropy number techniques to bound fi (x) in
terms of eigenvalues of the inner product matrix or its centred variants. It is worth
mentioning that although our aim was to study the maximum margin algorithm the
above theorem actually holds for any algorithm whose solution can be represented
as a linear combination of the data points.
5
Conclusions
In this paper we have introduced a new theoretical framework to study the generalisation error of learning algorithms. In contrast to previous approaches, we
considered specific learning algorithms rather than specific hypothesis spaces. We
introduced the notion of algorithmic luckiness which allowed us to devise data dependent generalisation error bounds. Thus we were able to relate the compression
framework of Littlestone and Warmuth with the VC framework. Furthermore, we
presented a new bound for the maximum margin algorithm which not only exploits
the margin but also the distribution of the actual training data in feature space.
Perhaps the most appealing feature of our margin based bound is that it naturally combines the three factors considered important for generalisation with linear
classifiers: margin, sparsity and the distribution of the data. Further research is
concentrated on studying Bayesian algorithms and the relation of algorithmic luckiness to the recent findings for stable learning algorithms [2].
Acknowledgements This work was done while RCW was visiting Microsoft Research Cambridge. This work was also partly supported by the Australian Research
Council. RH would like to thank Olivier Bousquet for stimulating discussions.
References
[1) M. Anthony and P. Bartlett. A Theory of Learning in Artificial Neural Networks.
Cambridge University Press, 1999.
[2) O. Bousquet and A. Elisseeff. Algorithmic stability and generalization performance. In
T. K. Leen , T. G. Dietterich, and V. Tresp, editors, Advances in N eural Information
Processing Systems 13, pages 196- 202. MIT Press, 2001.
[3) N. Cristianini, A. Elisseeff, and J. Shawe-Taylor. On optimizing kernel alignment .
Technical Report NC2-TR-2001-087, NeuroCOLT, http: //www.neurocolt.com. 2001.
[4) R. Herbrich and T . Graepel. A PAC-Bayesian margin bound for linear classifiers:
Why SVMs work. In T. K. Leen, T. G. Dietterich, and V. Tresp, editors, Advances
in Neural Information Processing Systems 13, pages 224- 230 , Cambridge, MA , 2001.
MIT Press.
[5) R. Herbrich and R. C. Williamson. Algorithmic luckiness. Technical report, Microsoft
Research, 2002.
[6) N . Littlestone and M. Warmuth. Relating data compression and learnability. Technical report, University of California Santa Cruz, 1986.
[7) Y . Makovoz. Random approximants and neural networks. Journal of Approximation
Theory, 85:98- 109, 1996.
[8) J. Shawe-Taylor, P. L. Bartlett, R. C. Williamson, and M. Anthony. Structural risk
minimization over data-dependent hierarchies. IEEE Transactions on Information
Theory, 44(5):1926- 1940, 1998.
[9) V . Vapnik. Statistical Learning Theory. John Wiley and Sons, New York, 1998.
[10) V. N. Vapnik and A. Y. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications,
16(2):264- 281, 1971.
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1,214 | 2,106 | Unsupervised Learning of Human Motion
Models
Yang Song, Luis Goncalves, and Pietro Perona
California Institute of Technology, 136-93, Pasadena, CA 9112 5, USA
yangs,luis,perona @vision.caltech.edu
Abstract
This paper presents an unsupervised learning algorithm that can derive
the probabilistic dependence structure of parts of an object (a moving human body in our examples) automatically from unlabeled data. The distinguished part of this work is that it is based on unlabeled data, i.e., the
training features include both useful foreground parts and background
clutter and the correspondence between the parts and detected features
are unknown. We use decomposable triangulated graphs to depict the
probabilistic independence of parts, but the unsupervised technique is
not limited to this type of graph. In the new approach, labeling of the
data (part assignments) is taken as hidden variables and the EM algorithm is applied. A greedy algorithm is developed to select parts and to
search for the optimal structure based on the differential entropy of these
variables. The success of our algorithm is demonstrated by applying it
to generate models of human motion automatically from unlabeled real
image sequences.
1 Introduction
Human motion detection and labeling is a very important but difficult problem in computer
vision. Given a video sequence, we need to assign appropriate labels to the different regions
of the image (labeling) and decide whether a person is in the image (detection). In [8, 7],
a probabilistic approach was proposed by us to solve this problem. To detect and label a
moving human body, a feature detector/tracker (such as corner detector) is first run to obtain
the candidate features from a pair of frames. The combination of features is then selected
based on maximum likelihood by using the joint probability density function formed by the
position and motion of the body. Detection is performed by thresholding the likelihood.
The lower part of Figure 1 depicts the procedure.
One key factor in the method is the probabilistic model of human motion. In order to avoid
exponential combinatorial search, we use conditional independence property of body parts.
In the previous work[8, 7], the independence structures were hand-crafted. In this paper,
we focus on the the previously unresolved problem (upper part of Figure 1): how to learn
the probabilistic independence structure of human motion automatically from unlabeled
training data, meaning that the correspondence between the candidate features and the parts
of the object is unknown. For example when we run a feature detector (such as LucasTomasi-Kanade detector [10]) on real image sequences, the detected features can be from
Unsupervised
Learning
algorithm
Feature
detector/
tracker
Probabilistic
Model of
Human Motion
Unlabeled
Training Data
Feature
detector/
tracker
Detection
and
Labeling
Testing: two frames
Figure 1: Diagram of the system.
Presence of Human?
Localization of parts?
target objects and background clutter with no identity attached to each feature. This case
is interesting because the candidate features can be acquired automatically. Our algorithm
leads to systems able to learn models of human motion completely automatically from real
image sequences - unlabeled training features with clutter and occlusion.
We restrict our attention to triangulated models, since they both account for much correlation between the random variables that represent the position and motion of each body
part, and they yield efficient algorithms. Our goal is to learn the best triangulated model,
i.e., the one that reaches maximum likelihood with respect to the training data. Structure learning has been studied under the graphical model (Bayesian network) framework
([2, 4, 5, 6]). The distinguished part of this paper is that it is an unsupervised learning
method based on unlabeled data, i.e., the training features include both useful foreground
parts and background clutter and the correspondence between the parts and detected features are unknown. Although we work on triangulated models here, the unsupervised technique is not limited to this type of graph.
This paper is organized as follows. In section 2 we summarize the main facts about the
triangulated probability model. In section 3 we address the learning problem when the
training features are labeled, i.e., the parts of the model and the correspondence between
the parts and observed features are known. In section 4 we address the learning problem
when the training features are unlabeled. In section 5 we present some experimental results.
2 Decomposable triangulated graphs
Discovering the probability structure (conditional independence) among variables is important since it makes efficient learning and testing possible, hence some computationally
intractable problems become tractable. Trees are good examples of modeling conditional
(in)dependence [2, 6]. The decomposable triangulated graph is another type of graph which
has been demonstrated to be useful for biological motion detection and labeling [8, 1].
A decomposable triangulated graph [1] is a collection of cliques of size three, where there
is an elimination order of vertices such that when a vertex is deleted, it is only contained
in one triangle and the remaining subgraph is again a collection of triangles until only one
triangle left. Decomposable triangulated graphs are more powerful than trees since each
node can be thought of as having two parents. Similarly to trees, efficient algorithms allow
fast calculation of the maximum likelihood interpretation of a given set of data.
"!,.$-/"!# &0 %("!2')1 %* "!$3)4
+'
Conditional independence among random variables (parts) can be described by a decombe the set of
parts, and
posable triangulated graph. Let
,
, is the measurement for . If the joint probability density function
can be decomposed as a decomposable triangulated graph, it can
"!$%& # ')(* +,(.-/(0
' (01
+ ( 2
- ( 43 ')5+)5-65
')5 2
+ 5 7
-65
9 : <;
>=
@?
C9 :
8
8BA
A
A
where
, 8
9 8 :
C9 : ,
C9 :
A
A are the cliques. 8 8
8
8
8 A
8 A
be written as,
- 0 0 # 0 4 # - 1 # 1
1 & 4 % ' - % 4
0 1 - 0 1 *4
(1)
, and
gives
the elimination order for the decomposable graph.
-/
0 1 4 % %
!0 ! 3
., - 4
,.- 4
,.- 4 ,.- 4 ,.- 4 ,.- 4
., - 4
,.- - 0 4 0 0 4 - 1 1 1 4 - 4 ,.- 4
3 Optimization of the decomposable triangulated graph
FE
are i.i.d samples from a probability density function,
Suppose D
HG
G
G , JI LK , are labeled data. We want to find the
where
MON D is maximized.
MON D is the
decomposable triangulated graph M , such that
probability of graph M being the ?correct? one given the observed data D . Here we use M
to denote both the decomposable graph and the conditional
(in)dependence depicted by the
P
graph. By Bayes? rule, MON D
DHN M
M
D , therefore if we can assume the
priors M are equal for different decompositions, then our goal is to find the structure
triangulated
M which can maximize DHN M C.9 From
: the previous
9 : section, a decomposable
C9 :
A
A , then DHN M
graph M is represented by 8
8
8 A
can be computed as follows,
QSR T VU 1 W YX Z\[ 3 ] !
'(1
+,(
-( Z_[ 3
+ 5 7
- 5
(2)
^
%& ,^
7a
where `
is differential entropy or conditional differential entropy [3] (we consider
continuous random
here). Equation (2) is an approximation which converges
K b variables
c
-4
- 0 0 0 4 - 1 1 1 4 - 4
to equality for
due to the weak Law of Large numbers and definitions and
properties
9 : of differential
C9 : entropy [3,C9 2, 4,
: 5, 6]. We want to find the decomposition
8
8
8BA
A
A such that the above equation can be maximized.
3.1 Greedy search
Though for tree cases, the optimal structure can be obtained efficiently by the maximum
spanning tree algorithm [2, 6], for decomposable triangulated graphs, there is no existing
algorithm which runs in polynomial time and guarantees to the optimal solution [9]. We
develop a greedy algorithm
For
: to grow the graph by the property of decomposable graphs.
9
A
A
each possible choice
of
(the
last
vertex
of
the
last
triangle),
find
the
best
which
? ed 5 ef 5
9 :
A as 8 A , i.e., the
, then
get5 the d best
can maximize `
5 child
? <g
f 5 of edge A
N
vertex (part) that can maximize `
. The next vertex is added one by
one to the existing graph by choosing the best child of all the edges (legal parents)
of the
:
existing graph until all the vertices are added to the graph. For each choice of A , one such
graph can be grown, so there are
candidate graphs. The final result is the graph with the
DFN M among the graphs.
highest hji/k
- 4 /- 4
4
,.- 4
ml - n
? porqts - - l - =u?
rw
v4 o 4 l v44
The above algorithm is efficient. The total search cost is
, which is on the order of
. The algorithm is a greedy algorithm, with no
guarantee that the global optimal solution could be found. Its effectiveness will be explored
through experiments.
- <g ( d ( f ( 4
3.2 Computation of differential entropy - translation invariance
.% %
10 - 4
-/ d ( f ( 4
In thev greedy
and `
,
r= search algorithm, we need to compute `
. If we assume
they are jointly
Gaussian, then the differential entropy can
z|{ that
I
G N }N , where is the dimension and } is the covariance matrix.
be computed by hxiyk
In our applications, position and velocity are used as measurements for each body part, but
humans can be present at different locations of the scene. In order to make the Gaussian
assumption reasonable, translations
s 9 s : s need to be removed. Therefore, we use local coordinate
s
, i.e., we can take one body part (for example 8 ) as
system for each triangle 8
the origin, and use relative positions
body parts. More formally, let denote a
g ( d ( f ( A
g ( d ( for
f ( other
vector of positions
describe
positions
s
? g ( if we
g ( A
d ( ? g ( f ( ? g ( d . ( Then
f ( ?
relative to 8 , we obtain,
.
This can
be written as
, where [12]
- 4
- - 4
, with
4
Z
Z
.
In the greedy search algorithm, the differential entropy of all the possible triplets are
needed and different triplets are with different origins. To reduce computational cost, notice
that
]
[ &
]
[ &
and
3
]
[ &
(4)
!
(5)
G
From the above equations, we can first estimate the mean and covariance } of
(including all the body parts and without removing translation), then take the dimensions
corresponding
9 s
eg ( d ( to the
f ( triangle and use equations (4) and (5) to get the mean and covariance
for
procedure can be applied to pairs (for example,
can be
9 .s Similar
: s
taken as origin for (
)) to achieve translation invariant.
- 4
4 Unsupervised learning of the decomposable graph
0 1
% &%
In this section, we consider the case when only unlabeled data are available. Assume we
K
tI
K
HE
G
have a data set of samples D
. Each sample
,
, is
G
a group of detected features which contains the target object, but
is unlabeled, which
means the correspondence between the candidate features and the parts of the object is
unknown. For example when we run a feature detector (such as Lucas-Tomasi-Kanade
detector [10]) on real image sequences, the detected features can be from target objects and
background clutter with no identity attached to each feature. We want to select the useful
composite parts of the object and learn the probability structure from D .
4.1 All foreground parts observed
Here we first assume that all the foreground parts are observed for each sample. If the labelG
ing for each
is taken as a hidden variable, then the EM algorithm can be used to learn
the probability structure and parameters. Our method was developed from [11], but here
we learn the probabilistic independence structure and all the candidate features are with the
I
FG
FG
same
contains
I type. Let ` G denote the labeling for . If
!
9 features,
9 then ` G is an
M ( M is the back-dimensional vector with each element taken a value from
E
ground clutter label). The observations for the EM algorithm are D
,
` G EG$# , and the parameters to optimize are the probability
the hidden variables are "
(in)dependence structure (i.e. the decomposable triangulated graph) and parameters for its
associated probability density function. We use M to represent both the probability strucG
ture and the parameters. If we assume that
s are independent from each other and ` G
G
only depends on
, then the likelihood function to maximize is,
0
%
0 1
QSR T VU W QSRT VU 1 W & SQ R T W
]
SQ RT ('*] ),+.- '
^ ^ 0/ 1 W 1& Q RT W
&
(6)
'
#
FG
where ` G is the th possible labeling for
, and G is the set of all such labelings. Optimization directly over equation (6) is hard, and
v the EM algorithm solves the optimization
problem iteratively. In EM, for each iteration , we will optimize the function,
W % 1W %
#
#
Q RT
]
]
]
#
0
VU W % 1 U W %
#
QSRT
W % 1
W %
^
&
#
]
1
W % 3 QSR T
0
/
^
& ' ) +.- ' ^
#
]
QSRT
*/ W %
^
^
& ( ' ) +.- ' 0/
#
^
^
0/
W %
(7)
#
G
` G given
where
G is the probability
of ` G
and the decomposable
s
v the observation
probability structure M
. For each iteration ,
G is a fixed number for a hypothesis ` G .
G can be computed as,
#
W %
]
W %
(8)
'*) ^ */
(
#
#
s
HG
We will discuss the computation of ` G
M
below. Under the labeling hypothG
G
` G ,
is divided into the foreground features
, which are parts of the
esis ` G
G
G
object, and background (clutter)
. If the foreground features
are independent of
G
, then,
clutter
W
1 W 2 W
^ 0/
^ */
^ 0/
1 ^ 0/ W 2
1 ^ */ W 2 ^ 0/ 1 W 2 W
(9)
For simplicity, we will assume the priors
` G N M are the same for different ` G , and
M are the same for different graph structures.
If we assume uniform background denG
g G
sities [11, 8], then
, where 8 is the volume of the space a
N` G M
background feature lies in, is the same for different ` G . Under probability decomposition
HG
N ` G M can be computed as in equation (1). Therefore the maximization of
M ,
*/
^
1
*/
#
,.- 4
,.-
W % 7
^
#
,.- # 04
,.- # 4
# 4 - 0 4 3 #
,.-
# 4
*/
equation (7) is equivalent to maximizing,
W % 1 W % C
#
]
X
]
]
& ( '*)
]
& ( '*)
]!
0/ %&
0/
QSRT
1
^
*/
#
W %
QSR T
' 0/( 1
+ */( 7
- *(/ 1& QSRT
+)*/ 5
-60/5
#
#
I
3
(10)
),
For most problems, the number of possible labelings is very large (on the order of
so it is computationally prohibitive to sum over all the possible ` G as in equation (10).
However, if there is one hypothesis labeling ` G that is much better than other hypotheses,,
i.e.
G corresponding to ` G is much larger than other
G ?s, then
G can be taken as
and other
G ?s as . Hence equation (10) can be approximated as,
# #
W % 1W %
# ##
gG ( dG (
X
#
]
#
fG (
#
#
] ! QSR T
'0/!( 1
+*/"(
-*(/" 1& QSR T
+0/!5
-*/"5
& % &
#
(11)
where
and
are measurements corresponding to the best labeling `#G .
Comparing
with
equation
(2)
and also by the weak law of large numbers, we know for
v
iteration , if the best hypothesis
s ` G is used as the ?true? labeling, then the decomposable
triangulated graph structure M can be obtained through the algorithm described in section
G
3. One approximation we make here is that the best hypothesis labeling ` G for each
is
really dominant among all the possible labelings so that hard assignment for labelings can
be used. This is similar to the situation of K-means vs. mixture of Gaussian for clustering
problems. We evaluate this approximation in experiments.
#
#
The whole algorithm can be summarized as follows. Given some random initial
v v guess of
the decomposable graph structure M and its parameters, then for iteration , ( is from
until the algorithm converges),
s
FG
E step: for each
, use M
to find the best labeling ` G and then compute the differential
entropies;
M step: use the differential
entropies to run the greedy graph growing algorithm described
s
in section 3 and get M .
0
#
4.2 Dealing with missing parts (occlusion)
So far we assume that all the parts are observed. In the case of some parts missing, the
measurements for the missing parts can be taken as additional hidden variables [11], and
the EM algorithm can be modified to handle the missing parts.
For each hypothesis ` G , let G denote the measurements
of the observed parts, G be
A
G
G
G A A be the measurements of
the measurements for the missing parts, and
the whole object (to reduce clutter in the notation, we assume that the dimensions can be
sorted in this way). For each EM iteration,
we need to compute G
and } G
to obtain
s
the differential entropies and then M with its parameters. Taking ` G and G as hidden
variables, we can get,
2
[
]
Z
2
2
Z
[
]
2
]
! [
(12)
!
Z
.
!
2
(13)
! ! , and
!
.
! !
! ! .
G
All the expectations
are conditional
expectations with respect to
` G
` G and
s
G are the measurements of the observed
decomposable graph structure M
. Therefore,
s
foreground parts under ` G
` G . Since M
is Gaussian distributed, conditional expec G and G s G A can be computed from observed parts G and the mean
tation
Where
- 4
-4
!
a
0
#
- 04
M
and covariance matrix of
0
#
.
5 Experiments
We tested the greedy algorithm on labeled motion capture data (Johansson displays) as in
[8], and the EM-like algorithm on unlabeled detected features from real image sequences.
5.1 Motion capture data
Our motion capture data consist of the 3-D positions of 14 markers fixed rigidly on a
subject?s body. These positions were tracked with 1mm accuracy as the subject walked
back and forth, and projected to 2-D.
Under Gaussian assumption, we first estimated the joint probability density function (mean
and covariance) of the data. From the estimated mean and covariance, we can compute
differential entropies for all the possible triplets and pairs and further run the greedy search
algorithm to find the approximated best triangulated model. Figure 2(a) shows the expected
likelihood (differential entropy) of the estimated joint pdf, of the best triangulated model
from the greedy algorithm, of the hand-constructed model from [8], and of randomly generated models. The greedy model is clearly superior to the hand-constructed model and the
random models. The gap to the original joint pdf is partly due to the strong conditional independence assumptions of the triangulated model, which are an approximation of the true
data?s pdf. Figure 2(b) shows the expected likelihood using 50 synthetic datasets. Since
these datasets were generated from 50 triangulated models, the greedy algorithm (solid
curve) can match the true model (dashed curve) extremely well. The solid line with error
bars are the expected likelihoods of random triangulated models.
?110
?135
?140
expected log likelihood
expected log likelihood
estimated joint pdf
?120
best trangulated model from greedy search
?130
?145
triangulated model used in previous papers
?150
?140
?155
?150
?160
randomly generated triangulated models
?160
0
500
1000
1500
2000
2500
3000
index of randomly generated triangulated models
?165
0
10
20
30
40
50
60
index of randomly generated triangulated models
(a)
(b)
Figure 2: Evaluation of greedy search.
5.2 Real image sequences
There are three types of sequences used here: (I) a subject walks from left to right (Figure
3(a,b)); (II) a subject walks from right to left; (III) a subject rides a bike from left to right
(Figure3(c,d)). Left-to-right walking motion models were learned from type I sequences
and tested on all types of sequences to see if the learned model can detect left-to-right
walking and label the body parts correctly. The candidate features were obtained from a
Lucas-Tomasi-Kanade algorithm [10] on two frames. We used two frames to simulate the
difficult situation, where due to extreme body motion or to loose and textured clothing and
occlusion, tracking is extremely unreliable and each feature?s lifetime is short.
Evaluation of the EM-like algorithm. As described in section 4.1, one approximation
we made is taking the best hypothesis labeling instead of summing over all the possible
hypotheses (equation (11)). This approximation was evaluated by checking how the loglikelihoods evolve with EM iterations and if they converge. Figure 4(a) shows the results of
learning a 12-feature model. We used random initializations, and each curve of Figure 4(a)
corresponds to one such random initialization. From Figure 4(a) we can see that generally
the log-likelihoods grow and converge well with the iterations of EM.
Models obtained. Figure 4 (b) and (c) show the best model obtained after we ran the
EM algorithms for 11 times. Figure 4(b) gives the mean positions and mean velocities
(shown in arrows) of the parts. Figure 4(c) shows the learned decomposable triangulated
probabilistic structure. The letter labels show the body parts correspondence.
Figure 3 shows samples of the results. The red dots (with letter labels) are the maximum
likelihood configuration from the left-to-right walking model. The horizontal bar at the
bottom left of each frame shows the likelihood
of the
The short vers
? best
s configuration.
tical bar gives the threshold where
G for all the test data. If
* ,
,
G
B
F
J CD
#
G
F
J C
B
D
H
H
H
H
E
IE A L
I
(a)walking detected
K
A
L
(b)detected
I
F G B
C D
E A
L
(c)not detected
F
C
G
I
B
DE
A
L
(d)not detected
Figure 3: Sample frames from left-to-right walking (a-b) and biking sequences (c-d). The dots
(either filled or empty) are the features selected by Tomasi-Kanade algorithm [10] on two frames.
The filled dots (with letter labels) are the maximum likelihood configuration from the left-to-right
walking model. The horizontal bar at the bottom left of each frame shows the likelihood of the best
configuration. The short vertical bar gives the threshold for detection.
?100
G
40
log likelihood
?120
J
C
G
B
C
D
80
?140
100
?160
H
H
120
?180
140
160
?200
180
?220
0
F
B
D
F
J
60
5
10
iterations of EM
15
20
200
I
100
E
150
K A L
200
I
250
E
K
A
L
(a)
(b)
(c)
Figure 4: (a) Evaluation of the EM-like algorithm. Log-likelihood vs. iterations of EM for different
random initializations. (b) and (c) show the best model obtained after we ran the EM-like algorithms
for 11 times.
the likelihood is greater than the threshold, a left-to-right walking person is detected.The
detection rate is 100% for the left-to-right walking vs. right-to-left walking, and 87% for
the left-to-right walking vs. left-to-right biking.
6 Conclusions
In this paper we have described a method for learning the structure and parameters of a
decomposable triangulated graph in an unsupervised fashion from unlabeled data. We have
applied this method to learn models of biological motion that can be used to reliably detect
and label biological motion.
Acknowledgments
Funded by the NSF Engineering Research Center for Neuromorphic Systems Engineering
(CNSE) at Caltech (NSF9402726), and by an NSF National Young Investigator Award to
PP (NSF9457618). We thank Charless Fowlkes for bringing the Chow and Liu?s paper to
our attention. We thank Xiaolin Feng for providing the real image sequences.
References
[1] Y. Amit and A. Kong, ?Graphical templates for model registration?, IEEE Transactions on
Pattern Analysis and Machine Intelligence, 18:225?236, 1996.
[2] C.K. Chow and C.N. Liu, ?Approximating discrete probability distributions with dependence
trees?, IEEE Transactions on Information Theory, 14:462?467, 1968.
[3] T.M. Cover and J.A. Thomas, Elements of Information Theory, John Wiley and Sons, 1991.
[4] N. Friedman and M. Goldszmidt, ?Learning bayesian networks from data?, Technical report,
AAAI 1998 Tutorial, http://robotics.stanford.edu/people/nir/tutorial/, 1998.
[5] M.I. Jordan, editor, Learning in Graphical Models, MIT Press, 1999.
[6] M. Meila and M.I. Jordan, ?Learning with mixtures of trees?, Journal of Machine Learning
Rearch, 1:1?48, 2000.
[7] Y. Song, X. Feng, and P. Perona, ?Towards detection of human motion?, In Proc. IEEE CVPR
2000, volume 1, pages 810?817, June 2000.
[8] Y. Song, L. Goncalves, E. Di Bernardo, and P. Perona, ?Monocular perception of biological
motion in johansson displays?, Computer Vision and Image Understanding, 81:303?327, 2001.
[9] Nathan Srebro, ?Maximum likelihood bounded tree-width markov networks?, In UAI, pages
504?511, San Francisco, CA, 2001.
[10] C. Tomasi and T. Kanade, ?Detection and tracking of point features?, Tech. Rep. CMU-CS-91132,Carnegie Mellon University, 1991.
[11] M. Weber, M. Welling, and P. Perona, ?Unsupervised learning of models for recognition?, In
Proc. ECCV, volume 1, pages 18?32, June/July 2000.
[12] Markus Weber, Unsupervised Learning of Models for Object Recognition, Ph.d. thesis, Caltech,
May 2000.
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1,215 | 2,107 | Tree-based reparameterization for
approximate inference on loopy graphs
Martin J. Wainwright, Tommi Jaakkola, and Alan S. Will sky
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
Cambridge, MA 02139
[email protected]
[email protected]
[email protected]
Abstract
We develop a tree-based reparameterization framework that provides a new conceptual view of a large class of iterative algorithms
for computing approximate marginals in graphs with cycles. It
includes belief propagation (BP), which can be reformulated as a
very local form of reparameterization. More generally, we consider
algorithms that perform exact computations over spanning trees
of the full graph. On the practical side, we find that such tree
reparameterization (TRP) algorithms have convergence properties
superior to BP. The reparameterization perspective also provides
a number of theoretical insights into approximate inference, including a new characterization of fixed points; and an invariance
intrinsic to TRP /BP. These two properties enable us to analyze
and bound the error between the TRP /BP approximations and
the actual marginals. While our results arise naturally from the
TRP perspective, most of them apply in an algorithm-independent
manner to any local minimum of the Bethe free energy. Our results also have natural extensions to more structured approximations [e.g. , 1, 2].
1
Introduction
Given a graphical model, one important problem is the computation of marginal
distributions of variables at each node. Although highly efficient algorithms exist
for this task on trees, exact solutions are prohibitively complex for more general
graphs of any substantial size. This difficulty motivates the use of approximate
inference algorithms, of which one of the best-known and most widely studied is
belief propagation [3], also known as the sum-product algorithm in coding [e.g., 4].
Recent work has yielded some insight into belief propagation (BP). Several researchers [e.g., 5, 6] have analyzed the single loop case, where BP can be reformulated as a matrix powering method. For Gaussian processes on arbitrary graphs,
two groups [7, 8] have shown that the means are exact when BP converges. For
graphs corresponding to turbo codes, Richardson [9] established the existence of
fixed points, and gave conditions for their stability. More recently, Yedidia et al. [1]
showed that BP corresponds to constrained minimization of the Bethe free energy,
and proposed extensions based on Kikuchi expansions [10]. Related extensions to
BP were proposed in [2]. The paper [1] has inspired other researchers [e.g., 11, 12] to
develop more sophisticated algorithms for minimizing the Bethe free energy. These
advances notwithstanding, much remains to be understood about the behavior of
BP.
The framework of this paper provides a new conceptual view of various algorithms
for approximate inference, including BP. The basic idea is to seek a reparameterization of the distribution that yields factors which correspond, either exactly or
approximately, to the desired marginal distributions. If the graph is acyclic (i.e.,
a tree) , then there exists a unique reparameterization specified by exact marginal
distributions over cliques. For a graph with cycles, we consider the idea of iteratively reparameterizing different parts of the distribution, each corresponding to an
acyclic subgraph. As we will show, BP can be interpreted in exactly this manner ,
in which each reparameterization takes place over a pair of neighboring nodes. One
of the consequences of this interpretation is a more storage-efficient "message-free"
implementation of BP. More significantly, this interpretation leads to more general
updates in which reparameterization is performed over arbitrary acyclic subgraphs,
which we refer to as tree-based reparameterization (TRP) algorithms.
At a low level, the more global TRP updates can be viewed as a tree-based schedule
for message-passing. Indeed, a practical contribution of this paper is to demonstrate that TRP updates tend to have better convergence properties than local
BP updates. At a more abstract level, the reparameterization perspective provides
valuable conceptual insight, including a simple tree-consistency characterization of
fixed points, as well as an invariance intrinsic to TRP /BP. These properties allow
us to derive an exact expression for the error between the TRP /BP approximations
and the actual marginals. Based on this exact expression, we derive computable
bounds on the error. Most of these results, though they emerge very naturally in
the TRP framework , apply in an algorithm-independent manner to any constrained
local minimum of the Bethe free energy, whether obtained by TRP /BP or an alternative method [e.g. , 11, 12]. More details of our work can be found in [13, 14].
1.1
Basic notation
An undirected graph Q = (V, ?) consists of a set of nodes or vertices V = {l , ... ,N}
that are joined by a set of edges ?. Lying at each node s E V is a discrete
random variable Xs E {a, ... ,m - I}. The underlying sample space X N is the
set of all N vectors x = {x s I S E V} over m symbols, so that IXNI = m N .
We focus on stochastic processes that are Markov with respect to Q, so that the
Hammersley-Clifford theorem [ e.g., 3] guarantees that the distribution factorizes
as p(x) ex: [lc Ee 'l/Jc(xc) where 'l/Jc(xc) is a compatibility function depending only
on the subvector Xc = {xs I SEC} of nodes in a particular clique C. Note that
each individual node forms a singleton clique, so that some of the factors 'l/Jc may
involve functions of each individual variable. As a consequence, if we have independent measurements Ys of Xs at some (or all) of the nodes, then Bayes' rule
implies that the effect of including these measurements - i.e., the transformation
from the prior distribution p(x) to the conditional distribution p(x I y) - is simply
to modify the singleton factors. As a result, throughout this paper, we suppress
explicit mention of measurements, since the problem of computing marginals for
either p(x) or p(x Iy) are of identical structure and complexity. The analysis of
this paper is restricted to graphs with singleton ('l/Js) and pairwise ('l/Jst} cliques.
However, it is straightforward to extend reparameterization to larger cliques, as in
cluster variational methods [e.g., 10].
1.2
Exact tree inference as reparameterization
Algorithms for optimal inference on trees have appeared in the literature of various fields [e.g., 4, 3]. One important consequence of the junction tree representation [15] is that any exact algorithm for optimal inference on trees actually computes
marginal distributions for pairs (s, t) of neighboring nodes. In doing so, it produces
an alternative factorization p(x) = TI sEV P s TI(s,t)E? Pst/(PsPt ) where Ps and Pst
are the single-node and pairwise marginals respectively. This {Ps , Pst} representation can be deduced from a more general factorization result on junction trees [e.g.
15]. Thus, exact inference on trees can be viewed as computing a reparameterized factorization of the distribution p(x) that explicitly exposes the local marginal
distributions.
2
Tree-based reparameterization for graphs with cycles
The basic idea of a TRP algorithm is to perform successive reparameterization updates on trees embedded within the original graph. Although such updates are
applicable to arbitrary acyclic substructures, here we focus on a set T 1 , ... , TL
of embedded spanning trees. To describe TRP updates, let T be a pseudomarginal probability vector consisting of single-node marginals Ts(x s ) for 8 E V;
and pairwise joint distributions Tst (x s, Xt) for edges (s, t) E [. Aside from positivity and normalization (Lx s Ts = 1; L xs , xt Tst = 1) constraints, a given vector T is arbitraryl , and gives rises to a parameterization of the distribution as
p(x; T) ex: TI sEV Ts TI(S,t)E? Tst/ {(L x. Tst)(L Xt Tst )}, where the dependence of Ts
and Tst on x is omitted for notational simplicity. Ultimately, we shall seek vectors
T that are consistent - i.e. , that belong to <C = {T I Lx. Tst = Tt \;/ (8, t) E [}. In
the context of TRP, such consistent vectors represent approximations to the exact
marginals of the distribution defined by the graph with cycles.
We shall express TRP as a sequence of functional updates Tn I-t T n+1 , where
superscript n denotes iteration number. We initialize at TO via T~t = Ii 'l/Js'I/Jt'I/Jst
and T~ = Ii 'l/Js TItEN(S) [L X t 'l/Jst'I/Jt], where Ii denotes a normalization factor; and
N(8) is the set of neighbors of node 8. At iteration n, we choose some spanning
tree Ti(n) with edge set [i(n), and factor the distribution p(x; Tn) into a product
of two terms
ex:
(la)
ex:
(lb)
corresponding, respectively, to terms in the spanning tree; and residual terms over
edges in [/ [i(n) removed to form Ti(n). We then perform a reparameterization
update on pi(n) (x; Tn) - explicitly:
pi(n)
(x'; Tn)
for all (s,t) E
[i(n)
(2)
x, s.t( x ~ ,x;)=(x. ,xtl
with a similar update for the single-node marginals {Ts I s E V}. These marginal
computations can be performed efficiently by any exact tree algorithm applied to
Ti(n). Elements of T n+1 corresponding to terms in ri(n) (x; Tn) are left unchanged
lIn general, T need not be the actual marginals for any distribution.
(i.e., Ts~+l = Tst for all (8, t) E E/Ei(n)) . The only restriction placed on the spanning
tree set T 1, ... ,TL is that each edge (8, t) E E belong to at least one spanning tree.
For practical reasons, it is desirable to choose a set of spanning trees that leads to
rapid mixing throughout the graph. A natural choice for the spanning tree index
i(n) is the cyclic ordering, in which i(n) == n(modL) + 1.
2.1
BP as local reparameterization
Interestingly, BP can be reformulated in a "message-free" manner as a sequence
of local rather than global reparameterization operations. This message-free version of BP directly updates approximate marginals, Ts and Tst, with initial values determined from the initial messages M~t and the original compatibility functions of the graphical model as T~ = Ii 'l/Js ITu EN(S) M~s for all 8 E V and
T~t = Ii 'l/Jst'l/Js'l/Jt ITu EN(s)/t M~s ITuEN(t) /s M~t for all (8, t) E E, where Ii denotes a normalization factor. At iteration n, these quantities are updated according
to the following recursions:
(3a)
T;'t
(3b)
The reparameterization form of BP decomposes the graph into a set of two-node
trees (one for each edge (8, t)); performs exact inference on such tree via equation (3b); and merges the marginals from each tree via equation (3a). It can be
shown by induction [see 13] that this simple reparameterization algorithm is equivalent to the message-passing version of BP.
2.2
Practical advantages of TRP updates
Since a single TRP update suffices to transmit information globally throughout the
graph, it might be expected to have better convergence properties than the purely
local BP updates. Indeed, this has proven to be the case in various experiments that
we have performed on two graphs (a single loop of 15 nodes, and a 7 x 7 grid). We
find that TRP tends to converge 2 to 3 times faster than BP on average (rescaled
for equivalent computational cost); more importantly, TRP will converge for many
problems where BP fails [13]. Further research needs to address the optimal choice
of trees (not necessarily spanning) in implementing TRP.
3
Theoretical results
The TRP perspective leads to a number of theoretical insights into approximate
inference, including a new characterization of fixed points, an invariance property,
and error analysis.
3.1
Analysis of TRP updates
Our analysis of TRP updates uses a cost function that is an approximation to the
Kullback-Leibler divergence between p(x; T) and p(x; U) - namely, the quantity
Xs
Given an arbitrary U E C, we show that successive iterates {Tn} of TRP updates
satisfy the following "Pythagorean" identity:
G(U ; T n)
=
G(U ; T n+ l ) + G(T n+1; T n)
(4)
which can be used to show that TRP fixed points T * satisfy the necessary conditions
to be local minima of G subject to the constraint T * E C. The cost function G,
though distinct from the Bethe free energy [1] , coincides with it on the constraint
set C, thereby allowing us to establish the equivalence of TRP and BP fixed points.
3.2
Characterization of fixed points
From the reparameterization perspective arises an intuitive characterization of any
TRP /BP fixed point T *. Shown in Figure l(a) is a distribution on a graph with
T 1:
T~T;
T4 ;
T3~
T2~
T; T;
T; T ~
T5:
T 1:
TtT;
T2~
T2*T;
T3~
T; T ~
(a) Fixed point on full graph
(b) Tree consistency condition.
Figure 1. Illustration of fixed point consistency condition. (a) Fixed point T * =
{T;, T;t } on the full graph with cycles. (b) Illustration of consistency condition on
an embedded tree. The quantities {T;, T;t } must be exact marginal probabilities
for any tree embedded within the full graph.
cycles, parameterized according to the fixed point T * = {Ts*t, T;}. The consistency
condition implies that if edges are removed from the full graph to form a spanning
tree, as shown in panel (b) , then the quantities Ts*t and Ts* correspond to exact
marginal distributions over the tree. This statement holds for any acyclic substructure embedded within the full graph with cycles - not just the spanning trees
Tl , ... ,TL used to implement TRP. Thus, algorithms such as TRP /BP attempt
to reparameterize a distribution on a graph with cycles so that it is consistent with
respect to each embedded tree.
It is remarkable that the existence of such a parameterization (though obvious for
trees) should hold for a positive distribution on an arbitrary graph. Also noteworthy
is the parallel to the characterization of max-product 2 fixed points obtained by
Freeman and Weiss [16]. Finally, it can be shown [13, 14] that this characterization,
though it emerged very naturally from the TRP perspective, applies more generally
to any constrained local minimum of the Bethe free energy, whether obtained by
TRP /BP, or an alternative technique [e.g., 11, 12].
2Max-product is a related but different algorithm for computing approximate MAP
assignments in graphs with cycles.
3.3
Invariance of the distribution
A fundamental property of TRP updates is that they leave invariant the full distribution on the graph with cycles. This invariance follows from the decomposition of
equation (1): in particular, the distribution pi(n) (x; Tn) is left invariant by reparameterization; and TRP does not change terms in ri(n) (x; Tn). As a consequence, the
overall distribution remains invariant - i.e., p(x; Tn) == p(x; TO) for all n. By continuity of the map T f-7 p(x; T) , it follows that any fixed point T* of the algorithm
also satisfies p(x; T*) == p(x; TO). This fixed point invariance is also an algorithmindependent result - in particular, all constrained local minima of the Bethe free
energy, regardless of how they are obtained, are invariant in this manner [13, 14].
This invariance has a number of important consequences. For example, it places
severe restrictions on cases (other than trees) in which TRP /BP can be exact;
see [14] for examples. In application to the linear-Gaussian problem, it leads to an
elementary proof of a known result [7, 8] - namely, the means must be exact if the
BP updates converge.
3.4
Error analysis
Lastly, we can analyze the error arising from any TRP /BP fixed point T* on an
arbitrary graph. Of interest are the exact single-node marginals Ps of the original distribution p(x; TO) defined by the graph with cycles, which by invariance are
equivalent to those of p(x; T*). Now the quantities Ts* have two distinct interpretations: (a) as the TRP /BP approximations to the actual single-node marginals on
the full graph; and (b) as the exact marginals on any embedded tree (as in Figure 1).
This implies that the approximations T; are related to the actual marginals P s on
the full graph by a relatively simple perturbation - namely, removing edges from
the full graph to reveal an embedded tree. From this observation, we can derive the
following exact expression for the difference between the actual marginal PS;j and
the TRP /BP approximation 3 T;j:
ri(X; T * ) }
.J
lEpi (x;T * ) [{ Z(T*) - 1 J(x s = J)
(5)
where i E {1, ... ,L} is an arbitrary spanning tree index; pi and ri are defined in
equation (1a) and (1b) respectively; Z(T*) is the partition function of p(x; T*);
J(xs = j) is an indicator function for Xs to take the value j; and lEpi (x;T * ) denotes
expectation using the distribution pi(x; T*).
Unfortunately, while the tree distribution pi (x; T*) is tractable, the argument of the
expectation includes all terms r i (x ; T*) removed from the original graph to form
spanning tree Ti. Moreover, computing the partition function Z (T*) is intractable.
These difficulties motivate the development of bounds on the error.
In [14], we use convexity arguments to derive a particular set of bounds on the
approximation error. Such error bounds, in turn, can be used to compute upper
and lower bounds on the actual marginals Ps;l. Figure 2 illustrates the TRP /BP
approximation, as well as these bounds on the actual marginals for a binary process
on a 3 x 3 grid under two conditions. Note that the tightness of the bounds is closely
related to approximation accuracy. Although it is unlikely that these bounds will
remain quantitatively useful for general problems on large graphs, they may still
yield useful qualitative information.
3The notation T;;j denotes the
/h
element of the vector T; .
Bounds on single node marginals
Bounds on single node marginals
0.9
0.9
0.8
0.8
0.7
:;:::'0.6
"
:5-"b.5
e
"- o.
0.2
0.1
?1~~--~--~4~~5---6~~~~~~
Node number
(a) Weak potentials
4
5
6
Node number
(b) Strong mixed potentials
Figure 2. Behavior of bounds on 3 x 3 grid. Plotted are the actual marginals P s;l
versus the TRP approximations T;'l> as well as upper and lower bounds on the
actual marginals. (a) For weak potentials, TRP /BP approximation is excellent;
bounds on exact marginals are tight. (b) For strong mixed potentials, approximation is poor. Bounds are looser, and for certain nodes, the TRP /BP approximation
lies above the upper bounds on the actual marginal P 8 ;1 .
Much of the analysis of this paper -- including reparameterization, invariance,
and error analysis -- can be extended [see 14] to more structured approximation
algorithms [e.g., 1, 2]. Figure 3 illustrates the use of bounds in assessing when to
use a more structured approximation. For strong attractive potentials on the 3 x 3
grid, the TRP /BP approximation in panel (a) is very poor, as reflected by relatively
loose bounds on the actual marginals. In contrast, the Kikuchi approximation in
(b) is excellent, as revealed by the tightness of the bounds.
4
Discussion
The TRP framework of this paper provides a new view of approximate inference;
and makes both practical and conceptual contributions. On the practical side, we
find that more global TRP updates tend to have better convergence properties than
local BP updates. The freedom in tree choice leads to open problems of a graphtheoretic nature: e.g., how to choose trees so as to guarantee convergence, or to
optimize the rate of convergence?
Among the conceptual insights provided by the reparameterization perspective are
a new characterization of fixed points; an intrinsic invariance; and analysis of the
approximation error. Importantly, most of these results apply to any constrained
local minimum of the Bethe free energy, and have natural extensions [see 14] to
more structured approximations [e.g., 1, 2].
Acknowledgments
This work partially funded by ODDR&E MURI Grant DAAD19-00-1-0466; by ONR
Grant N00014-00-1-0089; and by AFOSR Grant F49620-00-1-0362; MJW also supported
by NSERC 1967 fellowship.
References
[1] J. Yedidia, W. T. Freeman, and Y. Weiss. Generalized belief propagation. In NIPS
13, pages 689- 695. MIT Press, 2001.
Bounds on single node marginals
-
- - - -0 - - - -0- -
Bounds on single node marginals
-
e- - - -
M
M
0.8
_ 0 - - - - 0 - - - -0- - - - ?l - - - -
;o: V
"
:5-"b.
"
:5-"b.5
?>
a.. 0.4
?>
a.. 0.4
0.3
0.3
:~ II=-:= ~~r~~lured
0.2 rr -+-:Ac-,tu--:
al----.
0.1
-+- 0 ?
TAP I BP
Bounds
?1~~==~~-4~~5~~-~-~~
Node number
(a) TRP /BP
approx. 1
~rl=-e~B=o=un=ds~==~~~~-~-~-~
?1
4
5
Node number
(b) Kikuchi
Figure 3. When to use a more structured approximation? (a) For strong attractive potentials on the 3 x 3 grid, BP approximation is poor, as reflected by loose
bounds on the actual marginal. (b) Kikuchi approximation [1] for same problem
is excellent; corresponding bounds are tight.
[2] T. P. Minka. A family of algorithms for approximate Bayesian inference. PhD thesis,
MIT Media Lab, 2001.
[3] J. Pearl. Probabilistic reasoning in intelligent systems. Morgan Kaufman, San Mateo,
1988.
[4] F. Kschischang and B. Frey. Iterative decoding of compound codes by probability
propagation in graphical models. IEEE Sel. Areas Comm., 16(2):219- 230, February
1998.
[5] J. B. Anderson and S. M. Hladnik. Tailbiting map decoders. IEEE Sel. Areas Comm.,
16:297- 302, February 1998.
[6] Y. Weiss. Correctness of local probability propagation in graphical models with loops.
Neural Computation, 12:1-41, 2000.
[7] Y. Weiss and W. T. Freeman. Correctness of belief propagation in Gaussian graphical
models of arbitrary topology. In NIPS 12, pages 673- 679 . MIT Press, 2000 .
[8] P. Rusmevichientong and B. Van Roy. An analysis of turbo decoding with Gaussian
densities. In NIPS 12, pages 575- 581. MIT Press, 2000.
[9] T. Richardson. The geometry of turbo-decoding dynamics. IEEE Trans. Info. Theory,
46(1):9- 23, January 2000.
[10] R. Kikuchi. The theory of cooperative phenomena. Physical Review, 81:988- 1003,
1951.
[11] M. Welling and Y. Teh. Belief optimization: A stable alternative to loopy belief
propagation. In Uncertainty in Artificial Intelligence, July 2001.
[12] A. Yuille. A double-loop algorithm to minimize the Bethe and Kikuchi free energies.
Neural Computation, To appear, 2001.
[13] M. J . Wainwright, T. Jaakkola, and A. S. Willsky. Tree-based reparameterization for
approximate estimation on graphs with cycles. LIDS Tech. report P-2510: available
at http://ssg.rnit.edu/group/rnjyain/rnjyain.shtrnl, May 2001.
[14] M. Wainwright . Stochastic processes on graphs with cycles: geometric and variational
approaches. PhD thesis, MIT, Laboratory for Information and Decision Systems,
January 2002.
[1 5] S. L. Lauritzen. Graphical models. Oxford University Press, Oxford, 1996.
[16] W. Freeman and Y. Weiss. On the optimality of solutions of the max-product belief
propagation algorithm in arbitrary graphs. IEEE Trans. Info. Theory, 47:736- 744,
2001.
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1,216 | 2,108 | Semi-Supervised MarginBoost
F. d'Alche-Buc
LIP6,UMR CNRS 7606,
Universite P. et M. Curie
75252 Paris Cedex, France
Yves Grandvalet
Heudiasyc, UMR CNRS 6599,
Universite de Technologie de Compiegne,
BP 20.529, 60205 Compiegne cedex, France
florence. [email protected]
Yves. [email protected]
Christophe Ambroise
Heudiasyc, UMR CNRS 6599,
Universite de Technologie de Compiegne,
BP 20.529, 60205 Compiegne cedex, France
Christophe A [email protected]
Abstract
In many discrimination problems a large amount of data is available but
only a few of them are labeled. This provides a strong motivation to
improve or develop methods for semi-supervised learning. In this paper,
boosting is generalized to this task within the optimization framework of
MarginBoost . We extend the margin definition to unlabeled data and
develop the gradient descent algorithm that corresponds to the resulting
margin cost function. This meta-learning scheme can be applied to any
base classifier able to benefit from unlabeled data. We propose here to
apply it to mixture models trained with an Expectation-Maximization
algorithm. Promising results are presented on benchmarks with different
rates of labeled data.
1
Introduction
In semi-supervised classification tasks, a concept is to be learnt using both labeled
and unlabeled examples. Such problems arise frequently in data-mining where the
cost of the labeling process can be prohibitive because it requires human help as in
video-indexing, text-categorization [12] and medical diagnosis. While some works
proposed different methods [16] to learn mixture models [12], [1], SVM [3], cotrained machines [5] to solve this task, no extension has been developed so far for
ensemble methods such as boosting [7, 6]. Boosting consists in building sequentially a linear combination of base classifiers that focus on the difficult examples.
For AdaBoost and extensions such as MarginBoost [10], this stage-wise procedure
corresponds to a gradient descent of a cost functional based on a decreasing function
of the margin, in the space of linear combinations of base classifiers.
We propose to generalize boosting to semi-supervised learning within the framework of optimization. We extend the margin notion to unlabeled data, derive the
corresponding criterion to be maximized, and propose the resulting algorithm called
Semi-Supervised MarginBoost (SSMBoost). This new method enhances our previ-
ous work [9] based on a direct plug-in extension of AdaBoost in the sense that all
the ingredients of the gradient algorithm such as the gradient direction and the
stopping rule are defined from the expression of the new cost function. Moreover,
while the algorithm has been tested using the mixtures of models [1], 55MBoost
is designed to combine any base classifiers that deals with both labeled and unlabeled data. The paper begins with a brief presentation of MarginBoost (section 2).
Then, in section 3, the 55MBoost algorithm is presented. Experimental results are
discussed in section 5 and we conclude in section 6.
2
Boosting with MarginBoost
Boosting [7, 6, 15] aims at improving the performance of any weak "base classifier" by linear combination. We focus here on normalized ensemble classifiers
gt E LinCH) whose normalized 1 coefficients are noted aT = I ~: I and each base
classifier with outputs in [-1, 1] is hT E 1{:
t
gt(x) =
L aThT(x)
(1)
T=l
Different contributions [13, 14],[8], [10] have described boosting within an optimization scheme, considering that it carries out a gradient descent in the space of linear
combinations of base functions. We have chosen the MarginBoost algorithm, a variant of a more general algorithm called Any Boost [10], that generalizes AdaBoost
and formally justifies the interpretation in terms of margin. If S is the training
sample {(Xi,Yi) , i = l..l}, MarginBoost, described in Fig. 1, minimizes the cost
functional C defined for any scalar decreasing function c of the margin p :
I
C(gt) =
L c(p(gt(Xi), Yi)))
(2)
i=l
Instead of taking exactly ht+l = - \1C(gt) which does not ensure that the resulting
function gt+! belongs to Lin(1{), ht+! is chosen such as the inner product 2 - <
\1C(gt), ht+l > is maximal. The equivalent weighted cost function to be maximized
can thus be expressed as :
JF = L Wt(i)Yiht+! (Xi)
(3)
iES
3
Generalizing MarginBoost to semi-supervised
classification
3.1
Margin Extension
For labeled data, the margin measures the quality of the classifier output. When no
label is observed, the usual margin cannot be calculated and has to be estimated.
A first estimation could be derived from the expected margin EypL(gt(X) , y). We
can use the output of the classifier (gt(x) + 1)/2 as an estimate of the posterior
probability P(Y = +llx). This leads to the following margin pi; which depends on
the input and is linked with the response of the classifier:
lOr>
0 and L1 norm is used for normalization:
2< f, 9 >=
LiE S
f(X;)g(Xi)
IOrl
= L~=l
Or
Let wo(i) = l/l , i = 1, ... ,l.
Let go(x) = 0
For t = 1 ... T (do the gradient descent):
1. Learn a gradient direction htH E 1i with a high value of
J{ = L,iEswt(i)YihtH(Xi)
2. Apply the stopping rule: if J{ ::::: L,iES Wt(i)Yigt(Xi) then return
gt else go on.
3. Choose a step-length for the obtained direction by a line-search or
by fixing it as a constant f
a ttlh t t')
4 . Add the new direction to obtain 9HI = (l a t I9t+
lattl l
5. Fix the weight distribution: Wt
+1
=
c'(p(9ttl(Xi),Yi))
2: jE S c'(p(9ttl(Xj),Yj))
Figure 1: MarginBoost algorithm (with L1 normalization of the combination coefficients)
Another way of defining the extended margin is to use directly the maximum a
posteriori estimate of the true margin. This MAP estimate depends on the sign of
the classifier output and provides the following margin definition pC; :
(5)
3.2
Semi-Supervised MarginBoost : generalization of marginBoost to
deal with unlabeled data
The generalization of the margin can be used to define an appropriate cost functional
for the semi-supervised learning task. Considering that the training sample S is now
divided into two disjoint subsets L for labeled data and U for unlabeled data, the
cost falls into two parts involving PL = P and PU:
(6)
iEL
iEU
The maximization of - < \lC(gt), htH > is equivalent to optimize the new quantity
JtS that falls now into two terms J{ = Jf + J? The first term one can be directly
obtained from equation (3) :
Jf = LWt(i).YihtH(Xi)
(7)
iEL
The second term, J? , can be expressed as following:
(8)
with the weight distribution
Wt
now defined as :
c'(pL(9t( Xi),Yi))
Wt
(z.) -_
{
if i E L
..
with
If z E U
IWt l
c'(PU(9t(Xi)))
IWt l
This expression of
product:
IWt
I=
2:=
Wt
(i)
(9)
iES
JP comes directly from differential calculus and the chosen inner
( )()
'VC gt Xi
if x = Xi and i E L
if x = x, and i E U
YiC'(Pd9t(Xi),Yi))
= { c'(p U (g t (x.))) apU(9t(Xi))
a9t( Xi)
t
(10)
0
Pu
Implementation of 55MBoost with margins pI[; and
requires their derivatives.
Let us notice that the "signed margin", pus, is not derivable at point O. However,
according to the results of convex analysis (see for instance [2]), it is possible to
define the "sub derivative' of Pus since it is a continuous and convex function. The
value of the sub derivative corresponds here to the average value of the right and
left derivatives.
apUS(gt(Xi)) = {sign(g(Xi))
agt (Xi)
0
if X :f": 0
if x = 0
(11)
And, for the "squared margin" Pu 9 , we have:
apu 9 (gt(Xi)) = 2g(Xi)
agt(Xi)
(12)
This completes the set of ingredients that must be incorporated into the algorithm
of Fig. 1 to obtain 55MBoost.
4
Base Classifier
The base classifier should be able to make use of the unlabeled data provided by the
boosting algorithm. Mixture models are well suited for this purpose, as shown by
their extensive use in clustering. Hierarchical mixtures provide flexible discrimination tools, where each conditional distribution f(xlY = k) is modelled by a mixture
of components [4]. At the high level, the distribution is described by
K
f(x; if? =
2:= Pk!k (x; Ok)
,
(13)
k=l
where K is the number of classes, Pk are the mixing proportions, Ok the conditional
distribution parameters, and if> denotes all parameters {Pk; 0df=l. The high-level
description can also be expressed as a low-level mixture of components, as shown
here for binary classification:
Kl
f(x;if? =
K2
2:= PkJkl(X;Okl) + 2:= Pk2!k2(X;Ok2)
(14)
With this setting, the EM algorithm is used to maximize the log-likelihood with
respect to if> considering the incomplete data is {Xi, Yi}~= l and the missing data
is the component label Cik, k = 1, ... , K 1 + K2 [11]. An original implementation
of EM based on the concept of possible labels [1] is considered here. It is well
adapted to hierarchical mixtures, where the class label Y provides a subset of possible
components. When Y = 1 the first Kl modes are possible, when Y = -1 the last
K2 modes are possible, and when an example is unlabeled, all modes are possible.
A binary vector Zi E {0,1}(Kl+ K2) indicates the components from which feature
vector Xi may have been generated, in agreement with the assumed mixture model
and the (absence of) label Yi. Assuming that the training sample {Xi, Zi }i=l is i.i.d ,
the weighted log-likelihood is given by
I
L(<I> ;{Xi,zdi=l
= LWt(i) log (j(Xi,zi;<I?) ,
(15)
i=l
where Wt(i) are provided by boosting at step t. L is maximized using the following
EM algorithm:
E-Step Compute the expectation of L( <I>; {Xi , zdi=l) conditionally to {Xi , zdi=l
and the current value of <I> (denoted <I>q):
Kl+K2
L
Wt(i)Uik log (Pk!k(Xi; Ok))
i=l k=l
ZikPk!k(Xi; Ok)
L? ZUP?!?( Xi; O?)
I
L
with Uik
(16)
M-Step Maximize Q(<I>I<I>q) with respect to <I>.
Assuming that each mode k follows a normal distribution with mean ILk'
and covariance ~k ' <I>q+l = {ILk+! ; ~k+!;Pk+l}f~iK2 is given by:
(17)
(18)
5
Experimental results
Tests of the algorithm are performed on three benchmarks of the boosting literature:
twonorm and ringnorm [6] and banana [13]. Information about these datasets and
the results obtained in discrimination are available at www.first.gmd.de/-raetsch/
10 different samples were used for each experiment.
We first study the behavior of 55MBoost according the evolution of the test error
with increasing rates of unlabeled data (table 1). We consider five different settings
where 0%, 50%, 75%, 90% and 95% of labels are missing. 55MB is tested for the
margins P~ and Pu with c(x) = exp( -x). It is compared to mixture models and
AdaBoost. 55MBoost and AdaBoost are trained identically, the only difference
being that AdaBoost is not provided with missing labels.
Both algorithms are run for T = 100 boosting steps, without special care of overfitting. The base classifier (called here base(EM)) is a hierarchical mixture model with
an arbitrary choice of 4 modes per class but the algorithm (which may be stalled
in local minima) is restarted 100 times from different initial solutions, and the best
final solution (regarding training error rate) is selected. We report mean error rates
together with the lower and upper quartiles in table 1. For sake of space, we did
not display the results obtained without missing labels: in this case, AdaBoost and
55MBoost behave nearly identically and better than EM only for Banana.
For rates of unlabeled data inferior to 95% , 55MBoost beats slightly AdaBoost
for Ringnorm and Twonorm (except for 75%) but is not able to do as well as
Table 1: Mean error rates (in %) and interquartiles obtained with 4 different percentages
of unlabeled data for mixture models base(EM), AdaBoost and 55MBoost.
Ringnorm
50%
75%
90%
95%
base(EM)
AdaBoost
55MBoost pS
55MBoost pg
Twonorm
2.1 [ 1.7,
1.8[ 1.6,
1. 7[ 1.5,
1. 7[ 1.6,
50%
2.1]
2.0]
1.8]
1.8]
4.3[ 1.9,
3.1[ 1.9,
2.0 [ 1.5,
2.O[ 1.4,
75%
5.7]
4.1]
2.4]
2.5]
9.5 [ 2.7,12.0]
11.5[ 4.2 ,12.1]
3.7[ 2.1, 4.8]
4.5 [ 2.2, 3.6]
90%
23.7 [1 4.5,27.0]
28.7[11.5,37.6]
6.9[ 5.6,10.7]
8.1 [ 4.2, 9.0]
95%
base(EM)
AdaBoost
55MBoost pS
55MBoost pg
Banana
3.2 [ 2.7,
3.2[ 2.9,
2.7[ 2.5,
2.7[ 2.5,
50%
3.1]
3.2]
2.9]
2.8]
6.5[ 3.0,
3.2[ 3.0,
3.4 [ 2.8,
3.4 [ 2.8,
75%
9.0] 20.6[10.3,22.5] 24.8[18.3,31.9]
3.5] 11.0[ 5.2,14.2] 38.9[29.4,50.0]
4.3] 10.1 [ 5.8,13.6] 20.4[11.9,32.3]
4.2] 11.0[ 5.6,16.2] 21.1 [1 2.5,30.8]
90%
95%
base(EM)
18.2[16.7,18.6] 21.8[18.0,25.0] 26.1[20.7,29.8] 31.7[23.8,35.8]
AdaBoost
12.6[11.7,13.1] 15.2 [13.0,16.8] 22.1 [18.0,24.3] 37.5 [32.2,42.2]
55MBoost pS 13.3 [1 2.7,14.3] 17.0[15.3,17.8] 22.2[18.0,28.0] 28.3 [20.2,35.2]
55MBoost pg 13.3[12.8,14.2] 16.9[15.6,17.8] 22.8[18.3,29.3] 28.6 [21.5,34.2]
AdaBoost on Banana data. One possible explanation is that the discrimination
frontiers involved in the banana problem are so complex that the labels really bring
crucial informations and thus adding unlabeled data does not help in such a case.
Pu
obtains
Nevertheless, at rate 95% which is the most realistic situation, the margin
the minimal error rate for each of the three problems. It shows that it is worth
boosting and using unlabeled data.
As there is no great difference between the two proposed margins, we conducted
further experiments using only the
Pu'
Second, in order to study the relation between the presence of noise in the dataset
and the ability of 55MBoost to enhance generalization performance, we draw in
Fig. 2, the test errors obtained for problems with different values of Bayes error
when varying the rate of labeled examples. We see that even for difficult tasks (very
noisy problems), the degradation in performance for large subsets of unlabeled data
is still low. This reflects some consistency in the behavior of our algorithm.
Third, we test the sensibility of 55MBoost to overfitting. Overfitting can usually
be avoided by techniques such as early stopping, softenizing of the margin ([1 3],
[14]) or using an adequate margin function such as 1 - tanh(p) instead of exp( -p)
[10]. Here we keep using c = exp and ran 55MBoost with a maximal number of
step T = 1000 with 95% of unlabeled data. Of course, this does not correspond to
a realistic use of boosting in practice but it allows to check if the algorithm behaves
consistently in terms of gradient steps number. It is remarkable that no overfitting
is observed and in the Twonorm case (see Fig. 3), the test error still decreases
! We also observe that the standard error deviation is reduced at the end of the
process. For the banana problem (see Fig. 3 b.), we observe a stabilization near
the step t = 100. A massive presence of unlabeled data implies thus a regularizing
effect.
Bayes error:;;; 2.3%
Bayes error:;;; 15 .7%
Bayes error:;;; 3 1 .2%
50
40
20
10
?0L---~,7
0 --~20
7---~~
7----4~0--~5~0--~6=0--~7=0----8=0----9=0~~
'00
Rate of missing labels (%)
Figure 2: Consistency of the 55MBoost behavior: evolution of test error versus the missing
labels rate with respect to various Bayes error (twonorm ).
Mean (Error Test) +/- 1 std
70
70
Mean (Error test)
60'
,
I~
Mean of Error Test +/- std
Mean of Error test
I
60
,
~ 50
I
i \!), \
\
0: 40"
"' \
"
~_~~
~",~~,.~.
'-.
-.I~"
__
-r~/_
~
~~--
10
oL-~
o
__
~
~
__ _ __
~
~
-'-",---.- - --/----_ L_ __ L_ _~_ _~_ _~_ _L_~
~
~
~
~
Steptofgradient descent(boosting process}
~
~
_
?OL-~'OO~-2~OO~~3~OO--~400~~500~-=~~~7~OO~~8=OO--~~~~'~
Step t of gradient descent
Figure 3: Evolution of Test error with respect to maximal number T of iterations with
95% of missing labels (Two norm and Banana).
6
Conclusion
MarginBoost algorithm has been extended to deal with both labeled and unlabeled
data. Results obtained on three classical benchmarks of boosting litterature show
that it is worth using additional information conveyed by the patterns alone. No
overfitting was observed during processing 55MBoost on the benchmarks when
95% of the labels are missing: this should mean that the unlabeled data should
playa regularizing role in the ensemble classifier during the boosting process. After
applying this method to a large real dataset such as those of text-categorization,
our future works on this theme will concern the use of the extended margin cost
function on the base classifiers itself like multilayered perceptrons or decision trees.
Another approach could also be conducted from the more general framework of
AnyBoost that optimize any differential cost function.
References
[1] C. Ambroise and G. Govaert. EM algorithm for partially known labels. In IFCS 2000,
july 2000.
[2] J.-P. Aubin. L 'analyse non lineaire et ses applications d l'economie. Masson , 1984.
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M. Kearns, and S. Solla, editors, Advances in Neural Information Processing Systems,
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In Proceedings of th e 1998 Conference on Computational Learning Th eory, July 1998.
[6] L. Breiman. Prediction games and arcing algorithms. Technical Report 504, Statistics
Department , University of California at Berkeley, 1997.
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Machin e Learning: Proceedings of th e Thirteenth International Conference, pages
148- 156. Morgan Kauffman, 1996.
[8] J. Friedman, T. Hastie, and R. Tibshirani. Additive logistic regression: a statistical
view of boosting. The Annals of Statistics, 28(2):337- 407, 2000.
[9] Y. Grandvalet, F. d'Alche Buc, and C. Ambroise. Boosting mixture models for semisupervised learning. In ICANN 2001 , august 200l.
[10] L. Mason , J. Baxter, P. L. Bartlett, and M. Frean. Functional gradient techniques
for combining hypotheses. In Advances in Large Margin Classifiers. MIT, 2000.
[11] G.J. McLachlan and T. Krishnan. Th e EM algorithm and extensions. Wiley, 1997.
[12] K Nigam, A. K McCallum, S. Thrun, and T. Mitchell. Text classification from
labeled and unlabeled documents using EM. Machine learning, 39(2/3):135- 167,
2000.
[13] G. Riitsch, T. Onoda, and K-R. Muller. Soft margins for AdaBoost. Technical report,
Department of Computer Science, Royal Holloway, London , 1998.
[14] G. Riitsch, T. Onoda, and K-R. Muller. Soft margins for AdaBoost. Machine Learning, 42(3):287- 320, 200l.
[15] R. E. Schapire, Y. Freund, P. Bartlett, and W. S. Lee. Boosting the margin: A
new explanation for the effectiveness of voting methods. Th e Annals of Statistics,
26(5):1651- 1686, 1998.
[16] Matthias
Seeger.
Learning
with
data,www.citeseer.nj.nec.com/seegerOllearning.html.
labeled
and
unlabeled
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1,217 | 2,109 | Categorization by Learning
and Combining Object Parts
Bernd Heisele Thomas Serre Massimiliano Pontil Thomas Vetter Tomaso Poggio
Center for Biological
and Computational Learning, M.I.T., Cambridge, MA, USA
Honda R&D Americas, Inc., Boston, MA, USA
Department of Information Engineering, University of Siena, Siena, Italy
Computer Graphics Research Group, University of Freiburg, Freiburg, Germany
heisele,serre,tp @ai.mit.edu [email protected] [email protected]
Abstract
We describe an algorithm for automatically learning discriminative components of objects with SVM classifiers. It is based on growing image
parts by minimizing theoretical bounds on the error probability of an
SVM. Component-based face classifiers are then combined in a second
stage to yield a hierarchical SVM classifier. Experimental results in face
classification show considerable robustness against rotations in depth and
suggest performance at significantly better level than other face detection
systems. Novel aspects of our approach are: a) an algorithm to learn
component-based classification experts and their combination, b) the use
of 3-D morphable models for training, and c) a maximum operation on
the output of each component classifier which may be relevant for biological models of visual recognition.
1 Introduction
We study the problem of automatically synthesizing hierarchical classifiers by learning discriminative object parts in images. Our motivation is that most object classes (e.g. faces,
cars) seem to be naturally described by a few characteristic parts or components and their
geometrical relation. Greater invariance to viewpoint changes and robustness against partial occlusions are the two main potential advantages of component-based approaches compared to a global approach.
The first challenge in developing component-based systems is how to choose automatically
a set of discriminative object components. Instead of manually selecting the components,
it is desirable to learn the components from a set of examples based on their discriminative
power and their robustness against pose and illumination changes. The second challenge is
to combine the component-based experts to perform the final classification.
2 Background
Global approaches in which the whole pattern of an object is used as input to a single
classifier were successfully applied to tasks where the pose of the object was fixed. In [6]
Haar wavelet features are used to detect frontal and back views of pedestrians with an SVM
classifier. Learning-based systems for detecting frontal faces based on a gray value features
are described in [14, 13, 10, 2].
Component-based techniques promise to provide more invariance since the individual components vary less under pose changes than the whole object. Variations induced by pose
changes occur mainly in the locations of the components. A component-based method for
detecting faces based on the empirical probabilities of overlapping rectangular image parts
is proposed in [11]. Another probabilistic approach which detects small parts of faces is
proposed in [4]. It uses local feature extractors to detect the eyes, the corner of the mouth,
and the tip of the nose. The geometrical configuration of these features is matched with
a model configuration by conditional search. A related method using statistical models is
published in [9]. Local features are extracted by applying multi-scale and multi-orientation
filters to the input image. The responses of the filters on the training set are modeled as
Gaussian distributions. In [5] pedestrian detection is performed by a set of SVM classifiers
each of which was trained to detect a specific part of the human body.
In this paper we present a technique for learning relevant object components. The technique
starts with a set of small seed regions which are gradually grown by minimizing a bound
on the expected error probability of an SVM. Once the components have been determined,
we train a system consisting of a two-level hierarchy of SVM classifiers. First, component
classifiers independently detect facial components. Second, a combination classifier learns
the geometrical relation between the components and performs the final detection of the
object.
3 Learning Components with Support Vector Machines
3.1 Linear Support Vector Machines
Linear SVMs [15] perform pattern recognition for two-class problems by determining the
separating hyperplane with maximum distance to the closest points in the training set.
These points are called support vectors. The decision function of the SVM has the form:
!
#"%$ $
(1)
where is the number of data points and
is the class label of the data point
& . The coefficients are the solution of a quadratic programming
problem. The margin
'
is the distance of the support vectors to the hyperplane, it is given by:
'
$
( ) *
(2)
The margin is an indicator of the separability of the data. In fact, the expected error probability of the SVM, +-,.0/1/ , satisfies the following bound [15]:
+2,3.4/5/%6
where
9
$
+87:' 9<;
;>=
(3)
is the diameter of the smallest sphere containing all data points in the training set.
3.2 Learning Components
Our method automatically determines rectangular components from a set of object images.
The algorithm starts with a small rectangular component located around a pre-selected
point in the object image (e.g. for faces this could be the center of the left eye). The component is extracted from each object image to build a training set of positive examples. We
also generate a training set of background patterns that have the same rectangular shape as
the component. After training an SVM on the component data we estimate the performance
of the SVM based on the upper bound on the error probability. According to Eq. (3) we
calculate:
' 9<;
;
*
(4)
As shown in [15] this quantity can be computed by solving a quadratic programming problem. After determining we enlarge the component by expanding the rectangle by one
pixel into one of the four directions (up, down, left, right). Again, we generate training
data, train an SVM and determine . We do this for expansions into all four directions
and finally keep the expansion which decreases the most. This process is continued until
the expansions into all four directions lead to an increase of . In order to learn a set of
components this process can be applied to different seed regions.
4 Learning Facial Components
Extracting face patterns is usually a tedious and time-consuming work that has to be done
manually. Taking the component-based approach we would have to manually extract each
single component from all images in the training set. This procedure would only be feasible
for a small number of components. For this reason we used textured 3-D head models [16]
to generate the training data. By rendering the 3-D head models we could automatically
generate large numbers of faces in arbitrary poses and with arbitrary illumination. In addition to the 3-D information we also knew the 3-D correspondences for a set of reference
points shown in Fig. 1a). These correspondences allowed us to automatically extract facial
components located around the reference points. Originally we had seven textured head
models acquired by a 3-D scanner. Additional head models were generated by 3-D morph"
ing between
all pairs of the original head models. The heads were rotated between
and
in depth. The faces were illuminated by ambient light and a single directional
" light
and
pointing
towards
the
center
of
the
face.
The
position
of
the
light
varied
between
in azimuth and between
and
in elevation. Overall, we generated 2,457 face
images of size 58 58. Some examples of synthetic face images used for training are shown
in Fig. 1b).
The negative training set initially consisted of 10,209 58 58 non-face patterns randomly
extracted from 502 non-face images. We then applied bootstrapping to enlarge the training
data by non-face patterns that look similar to faces. To do so we trained a single linear
SVM classifier and applied it to the previously used set of 502 non-face images. The false
positives (FPs) were added to the non-face training data to build the final training set of
size 13,654.
We started with fourteen manually selected seed regions of size 5 5. The resulting components were located around the eyes (17 17 pixels), the nose (15 20 pixels), the mouth
$
(31 15 pixels), the cheeks (21 20 pixels), the lip (13 16 pixels), the nostrils (
pixels), the corners of the mouth (18 11 pixels), the eyebrows (19 15 pixels), and the
bridge of the nose (18 16 pixels).
a)
b)
Figure 1: a) Reference points on the head models which were used for 3-D morphing and
automatic extraction of facial components. b) Examples of synthetic faces.
5 Combining Components
An overview of our two-level component-based classifier is shown in Fig. 2. On the first
level the component classifiers independently detect components of the face. Each classifier was trained on a set of facial components and on a set of non-face patterns generated
from the training set described in Section 4. On the second level the combination classifier performs the detection of the face based on the outputs of the component classifiers.
The maximum real-valued outputs of each component classifier within rectangular search
regions around the expected positions of the components are used as inputs to the combination classifier. The size of the search regions was estimated from the mean and the standard
deviation of the locations of the components in the training images. The maximum operation is performed both during training and at run-time. Interestingly it turns out to be
similar to the key pooling mechanism postulated in a recent model of object recognition in
the visual cortex [8]. We also provide the combination classifier with the precise positions
of the detected components relative to the upper left corner of the 58
58 window. Overall
we have three values per component classifier that are propagated to the combination classifier: the maximum output of the component classifier and the & - image coordinates of
the maximum.
Left Eye
Eye
Left
expert:
expert:
Linear SVM
SVM
Linear
*
*Outputs of component
experts: bright intensities
indicate high confidence.
(O1 , X 1 , Y1 )
.
.
.
Nose expert:
expert:
Nose
Linear SVM
SVM
Linear
(O1 , X 1 , Y1 ,..., O14 , X 14 , Y14 )
*
(Ok , X k , Yk )
.
.
.
Mouth
Mouth
expert:
expert:
Linear SVM
SVM
Linear
1. Shift 58x58 window
over input image
2. Shift component
experts over
58x58 window
Combination
Combination
classifier:
classifier:
Linear SVM
SVM
Linear
*
(O14 , X 14 , Y14 )
3. For each component k,
determine its maximum
output within a search
region and its location:
(Ok , X k , Yk )
4. Final decision:
face / background
Figure 2: System overview of the component-based classifier.
6 Experiments
In our experiments we compared the component-based system to global classifiers. The
component system consisted of fourteen linear SVM classifiers for detecting the components and a single linear SVM as combination classifier. The global classifiers were a single
linear SVM and a single second-degree polynomial SVM both trained on the gray values
of the whole face pattern. The training data for these three classifiers consisted of 2,457
synthetic gray face images and 13,654 non-face gray images
" of size
58 58. The positive
test set consisted of 1,834 faces rotated between about
and
in depth. The faces
were manually extracted from the CMU PIE database [12]. The negative test set consisted
of 24,464 difficult non-face patterns that were collected by a fast face detector [3] from
web images. The FP rate was calculated relative to the number of non-face test images.
Because of the resolution required by the component-based system, a direct comparison
with other published systems on the standard MIT-CMU test set [10] was impossible. For
an indirect comparison, we used a second-degree polynomial SVM [2] which was trained
on a large set of 19 19 real face images. This classifier performed amongst the best face
detection systems on the MIT-CMU test set. The ROC curves in Fig. 3 show that the
component-based classifier is significantly better than the three global classifiers. Some
detection results generated by the component system are shown in Fig. 4.
Figure 3: Comparison between global classifiers and the component-based classifier.
Figure 4: Faces detected by the component-based classifier.
A natural question that arises is about the role of geometrical information. To answer this
question?which has relevant implications for models of cortex?we tested another system in
which the combination classifier receives as inputs only the output of each component classifier but not the position of its maximum. As shown in Fig. 5 this system still outperforms
the whole face systems but it is worse than the system with position information.
Figure 5: Comparison between a component-based classifier trained with position information and a component-based classifier without position information.
7 Open Questions
An extension under way of the component-based approach to face identification is already
showing good performances [1]. Another natural generalization of the work described here
involves the application of our system to various classes of objects such as cars, animals,
and people. Still another extension regards the question of view-invariant object detection. As suggested by [7] in a biological context and demonstrated recently by [11] in
machine vision, full pose invariance in recognition tasks can be achieved by combining
view-dependent classifiers. It is interesting to ask whether the approach described here
could also be used to learn which views are most discriminative and how to combine them
optimally. Finally, the role of geometry and in particular how to compute and represent
position information in biologically plausible networks, is an important open question at
the interface between machine and biological vision.
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[16] T. Vetter. Synthesis of novel views from a single face. International Journal of
Computer Vision, 28(2):103?116, 1998.
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1,218 | 211 | A Large-Scale Neural Network
A LARGE-SCALE NEURAL NETWORK
WHICH RECOGNIZES HANDWRITTEN
KANJI CHARACTERS
Yoshihiro Mori
Kazuki Joe
ATR Auditory and Visual Perception Research Laboratories
Sanpeidani Inuidani Seika-cho Soraku-gun Kyoto 619-02 Japan
ABSTRACT
We propose a new way to construct a large-scale neural network for
3.000 handwritten Kanji characters recognition. This neural network
consists of 3 parts: a collection of small-scale networks which are
trained individually on a small number of Kanji characters; a network
which integrates the output from the small-scale networks, and a
process to facilitate the integration of these neworks. The recognition
rate of the total system is comparable with those of the small-scale
networks. Our results indicate that the proposed method is effective for
constructing a large-scale network without loss of recognition
performance.
1 INTRODUCTION
Neural networks have been applied to recognition tasks in many fields. with good results
[Denker, 1988][Mori,1988][Weideman, 1989]. They have performed better than
conventional methods. However these networks currently operate with only a few
categories, about 20 to 30. The Japanese writing system at present is composed of about
3,000 characters. For a network to recognize this many characters, it must be given a
large number of categories while maintaining its level of performance.
To train small-scale neural networks is not a difficult task. Therefore. exploring methods
for integrating these small-scale neural networks is important to construct a large-scale
network. If such methods could integrate small-scale networks without loss of the
performance, the scale of neural networks would be extended dramatically. In this paper,
we propose such a method for constructing a large-scale network whose object is to
recognize 3,000 handwritten Kanji characters, and report the result of a part of this
network. This method is not limited to systems for character recognition, and can be
applied to any system which recognizes many categories.
2 STRATEGIES FOR A LARGE-SCALE NETWORK
Knowing the current recognition and generalization capacity of a neural network. we
realized that constructing a large-scale monolithic network would not be efficient or
415
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Mori and Joe
effective. Instead, from the start we decided on a building blocks approach
[Mori,1988] [Waibel,1988]. There are two strategies to mix many small-scale networks.
2.1 Selective Neural Network (SNN)
In this strategy, a large-scale neural network is made from many small-scale networks
which are trained individually on a small number of categories, and a network (SNN)
which selects the appropriate small-scale network (Fig. I). The advantage of this strategy
is that the information passed to a selected small-scale networks is always appropriate for
that network. Therefore, training these small-scale networks is very easy. But on the
other hand, increasing the number of categories will substantially increase the training
time of the SNN, and may make it harder for the SNN to retain high perfonnance.
Furthennore, the error rate of the SNN will limit the perfonnance of the whole system.
2.2 Integrative Neural Network (INN)
In this strategy, a large-scale neural network is made from many small-scale networks
which are trained individually on a small number of categories. and a network (INN)
which integrates the output from these small-scale networks(Fig. 2). The advantage of
this strategy is that every small-scale network gets information and contributes to finding
the right answer. Therefore, it is possible to use the knowledge distributed among each
small-scale network. But in some respects. various devices are needed to make the
integration easier.
The common advantage with both strategies just mentioned is that the size of each neural
network is relatively small, and it does not take a long time to train these networks. Each
small-scale networks is considered an independent part of the whole system. Therefore,
retraining these networks (to improve the performance of the whole system) will not take
too long.
~__....
O,utput
Sub
Net
1
? ?
Neural Network
(Selection Type)
':1U:U/W:::::::/:::/E::::::::.
: : Suspending
/ Network
(:::::{::::::::::::::::.:::::::: ~
Fig. 1 SNN Strategy
A Large-Scale Neural Network
Output
Neural Network
(Integration Type)
? ?
Fig. 2 INN Strategy
3 STRUCTURE OF LARGE-SCALE NETWORK
The whole system is constructed using three kinds of neural networks. The ftrst one,
called a SubNet, is an ordinary three layered feed forward type neural network trained
using the Back Propagation learning algorithm. The second kind of network is called a
SuperNet. This neural network makes its decision by integrating the outputs from all the
SubNets. This network is also a 3-layered feed-forward net, but is larger than the Subnets.
The last network, which we call an OtherFilter, is devised to improve the integration of
the S uperNet. This OtherFilter network was designed using the L VQ algorithm
[Khonen,1988]. There are also some changes made in the BP learning algorithm
especially for pattern recognition [Joe,1989].
We decided that, based on the time it takes for learning, there should be 9 categories in
each small-scale network. The 3,000 characters are separated into these small groups
through the K-means clustering method, which allows similar characters to be grouped
together. The separation occurs in two stages. First, 11 groups of 270 characters each are
formed, then each group is separated into 30 smaller units. In this way, 330 groups of 9
characters each are obtained. We choose the INN strategy to use distributed knowledge to
full advantage. The 9-character units are SubNets, which are integrated in 2 stages. First
30 SubNets are integrated by a higher level network SuperNet. Altogether, 11 SuperNets
are needed to recognize all 3,000 characters. SuperNets are in turn integrated by a higher
level network, the HyperNet. More precisely, the role and structure of these kinds of
networks are as follows:
3.1 SubNet
A feature vector extracted from handwritten patterns is used as the input (described in
Section 4.1). The number of units in the output layer is the same as the number of
categories to be recognized by the SubNet. In short, the role of a SubNet is to output the
similarity between the input pattern and the categories allotted to the SubNet. (Fig. 3)
3.2 SuperNet
The outputs from each SubNet fIltered by the OtherFilter network are used as the input to
417
418
Mori and Joe
the SuperNet. The number of units in an output layer is the same as the number of
SubNets belonging to a SuperNet. In shortt the role of SuperNet is to select the SubNet
which covers the category corresponding to the input patterns. (Fig. 5)
Output
Horizontal
+45?diagonal
Vertical
Original Pattern
Fig. 3 S ubNet
3.3 OtherFIIter
45(9x5) reference vectors are assigned to each SubNet. LVQ is used to adapt these
reference vectors t so that each input vector has a reference of the correct SubNet as its
closest reference vector. The
OtherFilter method is to frrst measure
the distance between all the reference
d
vectors and one input vector. The
mean distance and normal deviation of
distance are calculated. The distance
between a S ubNet and an input vector
is defmed to be the smallest distance
of that SubNet's reference vectors to
? References
the input vector .
?
?
XInput Vector
Fig4. Shape of OtherFilter
?
f(xo}=l 1(1+ e (x n-M+2d)/Cd) (1)
Xn : The Distance of Nth SubNet
M : The Mean of Xn
d : The Variance of Xn
C : Constant
A Large-Scale Neural Network
This distance modified by equation (1) is multiplied by the outputs of the SubNet. and fed
into the SuperNet. The outputs of SubNets whose distance is greater than the mean
distance are suppressed. and the outputs of SubNets whose distance is smaller than the
mean distance are amplified. In this way. the outputs of SubNets are modified to improve
the integration of the higher level SuperNet. (Fig. 5)
HyperNet 1
SuperNet 11
SubNet 330
OtherFilter 12
Other-Filter
FigS. Outline of the Whole System
4 RECOGNITION EXPERIMENT
4.1 TRAINING PATTERN
The training samples for this network were chosen from a database of about 3000 Kanji
characters [Saito 1985]. For each character. there are 200 handwritten samples from
different writers. 100 are used as training samples. and the remaining 100 are used to test
recognition accuracy of the trained network. All samples in the database consist of 64 by
63 dots.
419
420
Mori and Joe
JlQ
~
~~
~
.J-~
~~lJ
~~
~ ~
~
,~
~~
~
~~
-V'#f)
~
~
.orfffi
~~
J..~
~i2
O~
~
DI~J
o/N{
Fig. 6 Examples of training pattern
4.2 LDCD FEATURE
If we were to use this pattern as the input to our neural net, the number of units required
in the input layer would be too large for the computational abilities of current computers.
Therefore, a feature vector extracted from the handwritten patterns is used as the input. In
the "LDCD feature" [Hagita 1983], there are 256 dimensions computing a line segment
length along four directions: horizontal, vertical, and two diagonals in the 8 by 8 squares
into which the handwritten samples are divided.
t"
:61
o
horizontal
component
Fig 7. LDCD Feature
4.3 RECOGNITION RESULTS
In the work reported here, one SuperNet, 30 SubNets and one OtherFilter were
constructed for recognition experiments. SubNets were trained until the recognition of
training samples reaches at least 99%. With these SubNets, the mean recognition rate of
test patterns was 92%. This recognition rate is higher than that of conventional methods.
A SuperNet which integrates the output modified by OtherFilter from 30 trained SubNets
A Large-Scale Neural Network
was then constructed. The number of units in the input layer of the SuperNet was 270.
This SuperNet was trained until the performance of training samples becomes at least
93%. With this SuperNet, the recognition rate of test patterns was 74%, though that of
OtherFilter was 72%. The recognition rate of a system without the OtherFilter of test
patterns was 55%.
5 CONCLUSION
We have here proposed a new way of constructing a large-scale neural network for the
recognition of 3,000 handwritten Kanji characters. With this method, a system
recognizing 270 Kanji characters was constructed. This system will become a part of a
system recognizing 3,000 Kanji characters. Only a modest training time was necessary
owing to the modular nature of the system. Moreover, this modularity means that only a
modest re-training time is necessary for retraining an erroneous neural network in the
whole system. The overall system performance can be improved by retraining just that
neural network, and there is no need to retrain the whole system. However, the
performance of the OtherFilter is not satisfactory. We intend to improve the OtherFilter,
and build a large-scale network for the recognition of 3,000 handwritten Kanji characters
by the method reported here.
Acknowledgments
We are grateful to Dr. Yodogawa for his support and encouragement. Special thanks to
Dr. Sei Miyake for the ideas he provided in our many discussions. The authors would like
to acknowledge, with thanks, the help of Erik McDermott for his valuable assistance in
writing this paper in English.
References
[Denier, 1988]
l.S.Denker, W.R.Gardner, H.P. Graf, D.Henderson, R.E. Howard,
W.Hubbard, L.DJackel. H.S.Baird, I.Guyon : "Neural Network Recognizer for HandWritten ZIP Code Digits", NEURAL INFORMATION PROCESSING SYSTEMS 1.
pp.323-331, Morgan Kaufmann. 1988
[Mori,1988]
Y.Mori. K.Yokosawa : "Neural Networks that Learn to Discriminate
Similar Kanji Characters". NEURAL INFORMATION PROCESSING SYSTEMS 1,
pp.332-339, Morgan Kaufmann. 1988
[Weideman.1989]W.E.Weideman. M.T.Manry. H.C.Yau ; tI A COMPARISON OF A
NEAREST NEIGHBOR CLASSIFIER AND A NEURAL NETWORK FOR NUMERIC
HANDPRINT CHARACTER RECOGNITION". UCNN89(Washington), VoLl, pp.117120, June 1989
421
422
Mori and Joe
Alex Waibel, "Consonant Recognition by Modular Construction of
[Waibel, 1988]
Large Phonemic Time-Delay Neural Networks", NEURAL INFORMATION
PROCESSING SYSTEMS 1, pp.215-223, Morgan Kaufmann, 1988
[Joo,1989]
KJoo, Y.Mori, S.Miyake : "Simulation of a Large-Scale Neural
Networks on a Parallel Computer", 4th Hypercube Concurrent Computers,1989
[Khonen,1988] T.Kohonen, G.Barna, R.Chrisley : "Statistical Pattern Recognition
with Neural Networks", IEEE, Proc.of ICNN, YoU, pp.61-68, July 1988
[Saito,1985]
T.Saito, H.Yamada, K.Yamamoto : "On the Data Base ETL9 of
Handprinted Characters in 1IS Chinese Characters and Its Analysis", J68-D, 4, 757-764,
1985
[Hagita,1983]
N.Hagita, S.Naito, I.Masuda : "Recognition of Handprinted Chinese
Characters by Global and Local Direction Contributivity Density-Feature", J66-D, 6,
722-729,1983
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1,219 | 2,110 | A Neural Oscillator Model of Auditory
Selective Attention
Stuart N. Wrigley and Guy J. Brown
Department of Computer Science, University of Sheffield, Regent Court,
211 Portobello Street, Sheffield S1 4DP, UK.
[email protected], [email protected]
Abstract
A model of auditory grouping is described in which auditory attention
plays a key role. The model is based upon an oscillatory correlation
framework, in which neural oscillators representing a single perceptual
stream are synchronised, and are desynchronised from oscillators
representing other streams. The model suggests a mechanism by which
attention can be directed to the high or low tones in a repeating sequence
of tones with alternating frequencies. In addition, it simulates the
perceptual segregation of a mistuned harmonic from a complex tone.
1
Introduction
In virtually all listening situations, we are exposed to a mixture of sound energy from
multiple sources. Hence, the auditory system must separate an acoustic mixture in order to
create a perceptual description of each sound source. It has been proposed that this process
of auditory scene analysis (ASA) [2] takes place in two conceptual stages: segmentation
in which the acoustic mixture is separated into its constituent ?atomic? units, followed by
grouping in which units that are likely to have arisen from the same source are
recombined. The perceptual ?object? produced by auditory grouping is called a stream.
Each stream describes a single sound source.
Few studies have investigated the role of attention in ASA; typically, ASA is seen as a
precursor to attentional mechanisms, which simply select one stream as the attentional
focus. Recently, however, it has been suggested that attention plays a much more
prominent role in ASA. Carlyon et al. [4] investigated how attention influences auditory
grouping with the use of a rapidly repeating sequence of high and low tones. It is known
that high frequency separations and/or high presentation rates encourage the high tones
and low tones to form separate streams, a phenomenon known as auditory streaming [2].
Carlyon et al. demonstrated that auditory streaming did not occur when listeners attended
to an alternative stimulus presented simultaneously. However, when they were instructed
to attend to the tone sequence, auditory streaming occurred as normal. From this, it was
concluded that attention is required for stream formation and not only for stream selection.
It has been proposed that attention can be divided into two different levels [9]: low-level
exogenous attention which groups acoustic elements to form streams, and a higher-level
endogenous mechanism which performs stream selection. Exogenous attention may overrule conscious (endogenous) selection (e.g. in response to a sudden loud bang). The work
presented here incorporates these two types of attention into a model of auditory grouping
(Figure 1). The model is based upon the oscillatory correlation theory [10], which
suggests that neural oscillations encode auditory grouping. Oscillators corresponding to
grouped auditory elements are synchronised, and are desynchronised from oscillators
encoding other groups. This theory is supported by neurobiological findings that report
ALI
Correlogram
Signal
Cochlear
Filtering
Hair
cell
Cross
Channel
Correlation
Attentional
Stream
Neural
Oscillator
Network
Figure 1: Schematic diagram of the model (the attentional leaky integrator is labelled ALI).
synchronised oscillations in the auditory system [6]. Within the oscillatory correlation
framework, attentional selection can be implemented by synchronising attentional activity
with the stream of interest.
2
The model
2.1 Auditory periphery
Cochlear filtering is modelled by a bank of 128 gammatone filters with centre frequencies
equally spaced on the equivalent rectangular bandwidth (ERB) scale between 50 Hz and 2.5
kHz [3]. Auditory nerve firing rate is approximated by half-wave rectifying and square root
compressing the output of each filter. Input to the model is sampled at a rate of 8 kHz.
2.2 Pitch and harmonicity analysis
It is known that a difference in fundamental frequency (F0) can assist the perceptual
segregation of complex sounds [2]. Accordingly, the second stage of the model extracts pitch
information from the simulated auditory nerve responses. This is achieved by computing the
autocorrelation of the activity in each channel to form a correlogram [3]. At time t, the
autocorrelation of channel i with lag ? is given by:
P ?1
A ( i, t , ? ) =
? r ( i, t ? k )r ( i, t ? k ? ? )w ( k )
(1)
k=0
Here, r is the auditory nerve activity. The autocorrelation for channel i is computed using a 25
ms rectangular window w (P = 200) with lag steps equal to the sampling period, up to a
maximum lag of 20 ms. When the correlogram is summed across frequency, the resulting
?summary correlogram? exhibits a large peak at the lag corresponding to the fundamental
period of the stimulus. An accurate estimate of the F0 is found by fitting a parabolic curve to
the three samples centred on the summary peak.
The correlogram may also be used to identify formant and harmonic regions due to their
similar patterns of periodicity [11]. This is achieved by computing the correlations between
adjacent channels of the correlogram as follows:
L ?1
1
C ( i ) = --L
? A? ( i, t, ? )A? ( i + 1, t, ? )
(2)
?=0
Here, A?( i, t, ? ) is the autocorrelation function of (1) which has been normalised to have zero
mean and unity variance; L is the maximum autocorrelation lag in samples (L = 160).
2.3 Neural oscillator network
The network consists of 128 oscillators and is based upon the two-dimensional locally
excitatory globally inhibitory oscillator network (LEGION) of Wang [10], [11]. Within
LEGION, oscillators are synchronised by placing local excitatory links between them.
Additionally, a global inhibitor receives excitation from each oscillator, and inhibits every
oscillator in the network. This ensures that only one block of synchronised oscillators can be
active at any one time. Hence, separate blocks of synchronised oscillators - which correspond
to the notion of a segment in ASA - arise through the action of local excitation and global
inhibition.
The model described here differs from Wang?s approach [10] in three respects. Firstly, the
network is one-dimensional rather than two-dimensional; we argue that this is more plausible.
Secondly, excitatory links can be global as well as local; this allows harmonically-related
segments to be grouped. Finally, we introduce an attentional leaky integrator (ALI), which
selects one block of oscillators to become the attentional stream (i.e., the stream which is in the
attentional foreground).
The building block of the network is a single oscillator, which consists of a reciprocally
connected excitatory unit and inhibitory unit whose activities are represented by x and y
respectively:
3
x? = 3x ? x + 2 ? y + I o
(3a)
x
y? = ? ? ? 1 + tanh ---? ? y
?
??
(3b)
Here, ?, ? and ? are parameters. Oscillations are stimulus dependent; they are only observed
when Io > 0, which corresponds to a periodic solution to (3) in which the oscillator cycles
between an ?active? phase and a ?silent? phase. The system may be regarded as a model for the
behaviour of a single neuron, or as a mean field approximation to a group of connected
neurons. The input Io to oscillator i is a combination of three factors: external input Ir ,
network activity and global inhibition as follows:
Io = I r ?W z S ( z, ? z ) +
? Wik S ( xk, ?x )
(4)
k?i
Here, Wik is the connection strength between oscillators i and k; xk is the activity of oscillator
k. The parameter ?x is a threshold above which an oscillator can affect others in the network
and Wz is the weight of inhibition from the global inhibitor z. Similar to ?x, ?z acts a threshold
above which the global inhibitor can affect an oscillator. S is a squashing function which
compresses oscillator activity to be within a certain range:
1
S ( n, ? ) = -----------------------------?K ( n ? ? )
1+e
(5)
Here, K determines the sharpness of the sigmoidal function. The activity of the global inhibitor
is defined as
?
?
z? = H ? ? S ( xk, ? x ) ? 0.1? ? z
?
?
(6)
k
where H is the Heaviside function (H(n) = 1 for n ? 0, zero otherwise).
2.3.1 Segmentation
A block of channels are deemed to constitute a segment if the cross-channel correlation (2) is
greater than 0.3 for every channel in the block. Cross-correlations are weighted by the energy
of each channel in order to increase the contrast between spectral peaks and spectral dips.
These segments are encoded by a binary mask, which is unity when a channel contributes to a
segment and zero otherwise. To improve the resolution and separation of adjacent segments,
the cross-frequency spread of a segment is restricted to 3 channels. Oscillators within a
segment are synchronized by excitatory connections. The external input (Ir) of an oscillator
whose channel is a member of a segment is set to Ihigh otherwise it is set to Ilow.
2.3.2 Harmonicity grouping
Excitatory connections are made between segments if they are consistent with the current F0
estimate. A segment is classed as consistent with the F0 if a majority of its corresponding
correlogram channels exhibit a significant peak at the fundamental period (ratio of peak height
to channel energy greater than 0.46). A single connection is made between the centres of
harmonically related segments subject to old-plus-new constraints.
The old-plus-new heuristic [2] refers to the auditory system?s preference to ?interpret any part
of a current group of acoustic components as a continuation of a sound that just occurred? .
This is incorporated into the model by attaching ?age trackers? to each channel of the network.
Excitatory links are placed between harmonically related segments only if the two segments
are of similar age. The age trackers are leaky integrators:
+
B? k = d ( g [ M k ? B k ] ? [ 1 ? H ( M k ? Bk ) ]cBk )
(7)
= 0 otherwise. Mk is the (binary) value of the segment mask at
Here, [n] = n if n ? 0 and
channel k; small values of c and d result in a slow rise (d) and slow decay (c) for the integrator.
g is a gain factor.
+
[n]+
Consider two segments that start at the same time; the age trackers for their constituent
channels receive the same input, so the values of Bk will be the same. However, if two
segments start at different times, the age trackers for the earlier segment will have already
increased to a non-zero value when the second segment starts. This ?age difference? will
dissipate over time, as the values of both sets of leaky integrators approach unity.
2.3.3 Attentional leaky integrator (ALI)
Each oscillator is connected to the attentional leaky integrator (ALI) by excitatory links; the
strength of these connections is modulated by endogenous attention. Input to the ALI is given
by:
?
?
?
ali = H ? ? S ( x k, ? x )T k ? ? ALI? ? ali
?
?
(8)
k
?ALI is a threshold above which network activity can influence the ALI. Tk is an attentional
weighting which is related to the endogenous interest at frequency k:
T k = 1 ? ( 1 ? A k )L
(9)
Here, Ak is the endogenous interest at frequency k and L is the leaky integrator defined as:
+
L? = a ( b [ R ? L ] ? [ 1 ? H ( R ? L ) ]fL )
(10)
Small values of f and a result in a slow rise (a) and slow decay (f) for the integrator. b is a gain
factor. R = H ( xmax ) where xmax is the largest output activity of the network. The build-up of
attentional interest is therefore stimulus dependent. The attentional interest itself is modelled
as a Gaussian according to the gradient model of attention [7]:
A k = max A e
k
k?p
?----------22?
(11)
Here, Ak is the normalised attentional interest at frequency channel k and maxAk is the
maximum value that Ak can attain. p is the channel at which the peak of attentional interest
occurs, and ? determines the width of the peak.
A segment or group of segments are said to be attended to if their oscillatory activity coincides
temporally with a peak in the ALI activity. Initially, the connection weights between the
oscillator array and the ALI are strong: all segments feed excitation to the ALI, so all segments
are attended to. During sustained activity, these weights relax toward the Ak interest vector
such that strong weights exist for channels of high attentional interest and low weights exist
for channels of low attentional interest. ALI activity will only coincide with activity of the
channels within the attentional interest peak and any harmonically related (synchronised)
activity outside the Ak peak. All other activity will occur within a trough of ALI activity. This
behaviour allows both individual tones and harmonic complexes to be attended to using only a
single Ak peak.
The parameters for all simulations reported here were ? = 0.4, ? = 6.0, ? = 0.1, Wz = 0.5, ?z =
0.1, ?x = -0.5 and K = 50, d = 0.001, c = 5, g = 3, a = 0.0005, f = 5, b = 3, maxAk = 1, ? = 3, ?ALI
= 1.5, Ilow = -5.0, Ihigh = 0.2.The inter- and intra- segment connections have equal weights of
1.1.
3
Evaluation
Throughout this section, output from the model is represented by a ?pseudospectrogram? with
time on the abscissa and frequency channel on the ordinate. Three types of information are
superimposed on each plot. A gray pixel indicates the presence of a segment at a particular
frequency channel, which is also equivalent to the external input to the corresponding
oscillator: gray signifies Ihigh (causing the oscillator to be stimulated) and white signifies Ilow
(causing the oscillator to be unstimulated). Black pixels represent active oscillators (i.e.
oscillators whose x value exceeds a threshold value). At the top of each figure, ALI activity is
shown. Any oscillators which are temporally synchronised with the ALI are considered to be
in the attentional foreground.
3.1 Segregation of a component from a harmonic complex
Darwin et al. [5] investigated the effect of a mistuned harmonic upon the pitch of a 12
component complex tone. As the degree of mistuning of the fourth harmonic increased
towards 4%, the shift in the perceived pitch of the complex also increased. This effect was less
pronounced for mistunings of more than 4%; beyond 8% mistuning, little pitch shift was
observed. Apparently, the pitch of a complex tone is calculated using only those channels
which belong to the corresponding stream. When the harmonic is subject to mistunings below
8%, it is grouped with the rest of the complex and so can affect the pitch percept. Mistunings
of greater than 8% cause the harmonic to be segregated into a second stream, and so it is
excluded from the pitch percept.
B
C
1.5
Pitch shift (Hz)
120
100
120
100
80
60
40
20
Channel
Channel
Channel
A
120
100
80
60
40
20
80
60
40
20
D
Darwin
Model
1
0.5
0
0
20
40 60
Time (ms)
80
0
20
40 60
Time (ms)
80
0
20
40 60
Time (ms)
80
0
2
4
6
8
Mistuning of 4th harmonic (%)
Figure 2: A,B,C: Network response to mistuning of the fourth harmonic of a 12 harmonic
complex (0%, 6% and 8% respectively). ALI activity is shown at the top of each plot.
Gray areas denote the presence of a segment and black areas denote oscillators in the
active phase. Arrows show the focus of attentional interest. D: Pitch shift versus degree of
mistuning. A Gaussian derivative is fitted to each data set.
120
Channel
100
80
60
40
20
0
100
200
300
Time (ms)
400
500
600
Figure 3: Captor tones preceding the complex capture the fourth harmonic into a separate
stream. ALI activity (top) shows that this harmonic is the focus of attention and would be
?heard out? . The attentional interest vector (Ak) is shown to the right of the figure.
This behaviour is reproduced by our model (Figure 2). All the oscillators at frequency
channels corresponding to harmonics are temporally synchronised for mistunings up to 8%
(plots A and B) signifying that the harmonics belong to the same perceptual group. Mistunings
beyond 8% cause the mistuned harmonic to become desychronised from the rest of the
complex (plot C) - two distinct perceptual groups are now present: one containing the fourth
harmonic and the other containing the remainder of the complex tone. A comparison of the
pitch shifts found by Darwin et al. and the shifts predicted by the model is shown in plot D.
The pitch of the complex was calculated by creating a summary correlogram (similar to that
used in section 2.2) using frequency channels contained within the complex tone group. Only
segment channels below 1.1 kHz were used for this summary since low frequency (resolved)
harmonics are known to dominate the pitch percept [8].
Darwin et al. also showed that the effect of mistuning was diminished when the fourth
harmonic was ?captured? from the complex by four preceding tones at the same frequency. In
this situation, no matter how small the mistuning, the harmonic is segregated from the
complex and does not influence the pitch percept. Figure 3 shows the capture of the harmonic
with no mistuning. Attentional interest is focused on the fourth harmonic: oscillator activity
for the captor tone segments is synchronised with the ALI activity. During the 550 ms before
the complex tone onset, the age tracker activities for the captor tone channels build up. When
the complex tone begins, there is a significant age difference between the frequency channels
stimulated by the fourth harmonic and those stimulated by the remainder of the complex. Such
a difference prevents excitatory harmonicity connections from being made between the fourth
harmonic and the remaining harmonics. This behaviour is consistent with the old-plus-new
heuristic; a current acoustic event is interpreted as a continuation of a previous stimulus.
The old-plus-new heuristic can be further demonstrated by starting the fourth harmonic before
the rest of the complex. Figure 4 shows the output of the model when the fourth harmonic is
subject to a 50 ms onset asynchrony. During this time, the age trackers of channels excited by
the fourth harmonic increase to a significantly higher value than those of the remaining
harmonics. Once again, this prevents excitatory connections being made between the fourth
harmonic and the other harmonically related segments. The early harmonic is desynchronised
from the rest of the complex: two streams are formed. However, after a period of time, the
importance of the onset asynchrony decreases as the channel ages approach their maximal
values. Once this occurs, there is no longer any evidence to prevent excitatory links from
being made between the fourth harmonic and the rest of the complex. Grouping by
harmonicity then occurs for all segments: the complex and the early harmonic synchronise to
form a single stream.
3.2 Auditory streaming
Within the framework presented here, auditory streaming is an emergent property; all events
which occur over time, and are subject to attentional interest, are implicitly grouped. Two
temporally separated events at different frequencies must both fall under the Ak peak to be
grouped. It is the width of the Ak peak that determines frequency separation-dependent
streaming, rather than local connections between oscillators as in [10]. The build-up of
streaming [1] is modelled by the leaky integrator in (9). Figure 5 shows the effect of two
different frequency separations on the ability of the network to perform auditory streaming and
shows a good match to experimental findings [1], [4]. At low frequency separations, both the
high and low frequency segments fall under the attentional interest peak; this allows the
oscillator activities of both frequency bands to influence the ALI and hence they are
considered to be in the attentional foreground. At higher frequency separations, one of the
frequency bands falls outside of the attentional peak (in this example, the high frequency tones
fall outside) and hence it cannot influence the ALI. Such behaviour is not seen immediately,
because the attentional interest vector is subject to a build up effect as described in (9).
Initially the attentional interest is maximal across all frequencies; as the leaky integrator value
increases, the interest peak begins to dominate and interest in other frequencies tends toward
zero.
4
Discussion
A model of auditory attention has been presented which is based on previous neural oscillator
work by Wang and colleagues [10], [11] but differs in two important respects. Firstly, our
network is unidimensional; in contrast, Wang?s approach employs a two-dimensional timefrequency grid for which there is weak physiological justification. Secondly, our model
regards attention as a key factor in the stream segregation process. In our model, attentional
interest may be consciously directed toward a particular stream, causing that stream to be
selected as the attentional foreground.
Few auditory models have incorporated attentional effects in a plausible manner. For example,
Wang?s ?shifting synchronisation? theory [3] suggests that attention is directed towards a
stream when its constituent oscillators reach the active phase. This contradicts experimental
findings, which suggest that attention selects a single stream whose salience is increased for a
sustained period of time [2]. Additionally, Wang?s model fails to account for exogenous
reorientation of attention to a sudden loud stimulus; the shifting synchronisation approach
would multiplex it as normal with no attentional emphasis. By ensuring that the minimum Ak
value for the attentional interest is always non-zero, it is possible to weight activity outside of
the attentional interest peak and force it to influence the ALI. Such weighting could be derived
from a measure of the sound intensity present in each frequency channel.
We have demonstrated the model? s ability to accurately simulate a number of perceptual
phenomena. The time course of perception is well simulated, showing how factors such as
mistuning and onset asynchrony can cause a harmonic to be segregated from a complex tone.
It is interesting to note that a good match to Darwin?s pitch shift data (Figure 2D) was only
found when harmonically related segments below 1.1 kHz were used. The dominance of lower
(resolved) harmonics on pitch is well known [8], and our findings suggest that the correlogram
does not accurately model this aspect of pitch perception.
120
Channel
100
80
60
40
20
0
50
100
150
200
250
300
350
Time (ms)
Figure 4: Asynchronous onset of the fourth harmonic causes it to segregate into a separate
stream. The attentional interest vector (Ak) is shown to the right of the figure.
100
Channel
Channel
100
90
80
90
80
0
200
400
Time (ms)
600
0
200
400
Time (ms)
600
Figure 5: Auditory streaming at frequency separations of 5 semitones (left) and 3
semitones (right). Streaming occurs at the higher separation. The timescale of adaptation
for the attentional interest has been reduced to aid the clarity of the figures.
The simulation of two tone streaming shows how the proposed attentional mechanism and its
cross-frequency spread accounts for grouping of sequential events according to their proximity
in frequency. A sequence of two tones will only stream if one set of tones fall outside of the
peak of attentional interest. Frequency separations for streaming to occur in the model (greater
than 3 to 4 semitones) are in agreement with experimental data, as is the timescale for the
build-up of the streaming effect [1].
In summary, we have proposed a physiologically plausible model in which auditory streams
are encoded by a unidimensional neural oscillator network. The network creates auditory
streams according to grouping factors such as harmonicity, frequency proximity and common
onset, and selects one stream as the attentional foreground. Current work is concentrating on
expanding the system to include binaural effects, such as inter-ear attentional competition [4].
References
[1] Anstis, S. & Saida, S. (1985) Adaptation to auditory streaming of frequency-modulated tones. J. Exp.
Psychol. Human 11 257-271.
[2] Bregman, A. S. (1990) Auditory Scene Analysis. Cambridge MA: MIT Press.
[3] Brown, G. J. & Cooke, M. (1994) Computational auditory scene analysis. Comput. Speech Lang. 8,
pp. 297-336.
[4] Carlyon, R. P., Cusack, R., Foxton, J. M. & Robertson, I. H. (2001) Effects of attention and unilateral
neglect on auditory stream segregation. J. Exp. Psychol. Human 27(1) 115-127.
[5] Darwin, C. J., Hukin, R. W. & Al-Khatib, B. Y. (1995) Grouping in pitch perception: Evidence for
sequential constraints. J. Acoust. Soc. Am. 98(2)Pt1 880-885.
[6] Joliot, M., Ribary, U. & Llin?s, R. (1994) Human oscillatory brain activity near 40 Hz coexists with
cognitive temporal binding. Proc. Natl. Acad. Sci. USA 91 11748-51.
[7] Mondor, T. A. & Bregman, A. S. (1994) Allocating attention to frequency regions. Percept.
Psychophys. 56(3) 268-276.
[8] Moore, B. C. J. (1997) An introduction to the psychology of hearing. Academic Press.
[9] Spence, C. J., Driver, J. (1994) Covert spatial orienting in audition: exogenous and endogenous
mechanisms. J. Exp. Psychol. Human 20(3) 555-574.
[10] Wang, D. L. (1996) Primitive auditory segregation based on oscillatory correlation. Cognitive Sci.
20 409-456.
[11] Wang, D. L. & Brown, G. J. (1999) Separation of speech from interfering sounds based on
oscillatory correlation. IEEE Trans. Neural Networks 10 684-697.
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1,220 | 2,111 | Computing Time Lower Bounds for
Recurrent Sigmoidal Neural Networks
Michael Schmitt
Lehrstuhl Mathematik und Informatik, Fakultat fUr Mathematik
Ruhr-Universitat Bochum, D- 44780 Bochum, Germany
[email protected]
Abstract
Recurrent neural networks of analog units are computers for realvalued functions. We study the time complexity of real computation in general recurrent neural networks. These have sigmoidal,
linear, and product units of unlimited order as nodes and no restrictions on the weights. For networks operating in discrete time,
we exhibit a family of functions with arbitrarily high complexity,
and we derive almost tight bounds on the time required to compute
these functions. Thus, evidence is given of the computational limitations that time-bounded analog recurrent neural networks are
subject to.
1
Introduction
Analog recurrent neural networks are known to have computational capabilities that
exceed those of classical Turing machines (see, e.g., Siegelmann and Sontag, 1995;
Kilian and Siegelmann, 1996; Siegelmann, 1999). Very little, however, is known
about their limitations. Among the rare results in this direction, for instance,
is the one of Sima and Orponen (2001) showing that continuous-time Hopfield
networks may require exponential time before converging to a stable state. This
bound, however, is expressed in terms of the size of the network and, hence, does
not apply to fixed-size networks with a given number of nodes. Other bounds
on the computational power of analog recurrent networks have been established by
Maass and Orponen (1998) and Maass and Sontag (1999). They show that discretetime recurrent neural networks recognize only a subset of the regular languages in
the presence of noise. This model of computation in recurrent networks, however,
receives its inputs as sequences. Therefore, computing time is not an issue since
the network halts when the input sequence terminates. Analog recurrent neural
networks, however, can also be run as "real" computers that get as input a vector
of real numbers and, after computing for a while, yield a real output value. No
results are available thus far regarding the time complexity of analog recurrent
neural networks with given size.
We investigate here the time complexity of discrete-time recurrent neural networks
that compute functions over the reals. As network nodes we allow sigmoidal units,
linear units, and product units- that is, monomials where the exponents are ad-
justable weights (Durbin and Rumelhart, 1989) . We study the complexity of real
computation in the sense of Blum et aI. (1998). That means, we consider real numbers as entities that are represented exactly and processed without restricting their
precision. Moreover, we do not assume that the information content of the network
weights is bounded (as done, e.g., in the works of Balcazar et aI. , 1997; Gavalda and
Siegelmann, 1999). With such a general type of network, the question arises which
functions can be computed with a given number of nodes and a limited amount of
time. In the following , we exhibit a family of real-valued functions ft, l 2: 1, in one
variable that is computed by some fixed size network in time O(l). Our main result
is, then, showing that every recurrent neural network computing the functions ft
requires at least time nW /4). Thus, we obtain almost tight time bounds for real
computation in recurrent neural networks.
2
Analog Computation in Recurrent Neural Networks
We study a very comprehensive type of discrete-time recurrent neural network that
we call general recurrent neural network (see Figure 1). For every k, n E N there is
a recurrent neural architecture consisting of k computation nodes YI , . . . , Yk and n
input nodes Xl , ... , x n . The size of a network is defined to be the number ofits computation nodes. The computation nodes form a fully connected recurrent network.
Every computation node also receives connections from every input node. The input
nodes play the role of the input variables of the system. All connections are parameterized by real-valued adjustable weights. There are three types of computation
nodes: product units, sigmoidal units, and linear units. Assume that computation
node i has connections from computation nodes weighted by Wil, ... ,Wi k and from
input nodes weighted by ViI, .. . ,Vi n. Let YI (t) , . . . ,Yk (t) and Xl (t), ... ,X n (t) be the
values of the computation nodes and input nodes at time t, respectively. If node i
is a product unit, it computes at time t + 1 the value
(1)
that is, after weighting them exponentially, the incoming values are multiplied.
Sigmoidal and linear units have an additional parameter associated with them, the
threshold or bias ()i . A sigmoidal unit computes the value
where (J is the standard sigmoid (J( z ) = 1/ (1
simply outputs the weighted sum
+ e- Z ).
If node i is a linear unit, it
We allow the networks to be heterogeneous, that is, they may contain all three types
of computation nodes simultaneously. Thus, this model encompasses a wide class of
network types considered in research and applications. For instance, architectures
have been proposed that include a second layer of linear computation nodes which
have no recurrent connections to computation nodes but serve as output nodes (see,
e.g. , Koiran and Sontag, 1998; Haykin, 1999; Siegelmann, 1999). It is clear that in
the definition given here, the linear units can function as these output nodes if the
weights of the outgoing connections are set to O. Also very common is the use
of sigmoidal units with higher-order as computation nodes in recurrent networks
(see, e.g., Omlin and Giles, 1996; Gavalda and Siegelmann, 1999; Carrasco et aI.,
2000). Obviously, the model here includes these higher-order networks as a special
case since the computation of a higher-order sigmoidal unit can be simulated by
first computing the higher-order terms using product units and then passing their
.
I
I
sigmoidal, product, and linear units
computation
nodes
.
Yl
Yk
t
input nodes
Xl
Xn
I
Figure 1: A general recurrent neural network of size k. Any computation node may
serve as output node.
outputs to a sigmoidal unit. Product units , however, are even more powerful than
higher-order terms since they allow to perform division operations using negative
weights. Moreover, if a negative input value is weighted by a non-integer weight,
the output of a product unit may be a complex number. We shall ensure here that
all computations are real-valued. Since we are mainly interested in lower bounds,
however, these bounds obviously remain valid if the computations of the networks
are extended to the complex domain.
We now define what it means that a recurrent neural network N computes a function
f : ~n --+ llt Assume that N has n input nodes and let x E ~n. Given tE N,
we say that N computes f(x) in t steps if after initializing at time 0 the input
nodes with x and the computation nodes with some fixed values, and performing t
computation steps as defined in Equations (1) , (2) , and (3) , one of the computation
nodes yields the value f(x). We assume that the input nodes remain unchanged
during the computation. We further say that N computes f in time t if for every
x E ~n , network N computes f in at most t steps. Note that t may depend
on f but must be independent of the input vector. We emphasize that this is
a very general definition of analog computation in recurrent neural networks. In
particular, we do not specify any definite output node but allow the output to occur
at any node. Moreover, it is not even required that the network reaches a stable
state, as with attractor or Hopfield networks. It is sufficient that the output value
appears at some point of the trajectory the network performs. A similar view of
computation in recurrent networks is captured in a model proposed by Maass et al.
(2001). Clearly, the lower bounds remain valid for more restrictive definitions of
analog computation that require output nodes or stable states. Moreover, they
hold for architectures that have no input nodes but receive their inputs as initial
values of the computation nodes. Thus, the bounds serve as lower bounds also for
the transition times between real-valued states of discrete-time dynamical systems
comprising the networks considered here.
Our main tool of investigation is the Vapnik-Chervonenkis dimension of neural
networks. It is defined as follows (see also Anthony and Bartlett, 1999): A dichotomy
of a set S ~ ~n is a partition of S into two disjoint subsets (So , Sd satisfying
So U S1 = S. A class :F of functions mapping ~n to {O, I} is said to shatter S if
for every dichotomy (So , Sd of S there is some f E :F that satisfies f(So) ~ {O}
and f(S1) ~ {I}. The Vapnik-Chervonenkis (VC) dimension of :F is defined as
4"'+4",IL
'I
-1---Y-2----Y-5~1
S~
output
Y5
Y4
Figure 2: A recurrent neural network computing the functions fl in time 2l
+ 1.
the largest number m such that there is a set of m elements shattered by F. A
neural network given in terms of an architecture represents a class of functions
obtained by assigning real numbers to all its adjustable parameters, that is, weights
and thresholds or a subset thereof. The output of the network is assumed to be
thresholded at some fixed constant so that the output values are binary. The VC
dimension of a neural network is then defined as the VC dimension of the class of
functions computed by this network.
In deriving lower bounds in the next section, we make use of the following result
on networks with product and sigmoidal units that has been previously established
(Schmitt, 2002). We emphasize that the only constraint on the parameters of the
product units is that they yield real-valued, that is, not complex-valued, functions.
This means further that the statement holds for networks of arbitrary order, that is,
it does not impose any restrictions on the magnitude of the weights of the product
units.
Proposition 1. (Schmitt, 2002, Theorem 2) Suppose N is a feedforward neural
network consisting of sigmoidal, product, and linear units. Let k be its size and W
the number of adjustable weights. The VC dimension of N restricted to real-valued
functions is at most 4(Wk)2 + 20Wk log(36Wk).
3
Bounds on Computing Time
We establish bounds on the time required by recurrent neural networks for computing a family of functions fl : JR -+ JR, l 2:: 1, where l can be considered as a measure
of the complexity of fl. Specifically, fl is defined in terms of a dynamical system as
the lth iterate of the logistic map ?>(x) = 4x(1 - x), that is,
fl(X)
{
= 1,
?>(x)
l
?>(fl- l (x))
l > 2.
We observe that there is a single recurrent network capable of computing every fl
in time O(l).
Lemma 2. There is a general recurrent neural network that computes fl in time
2l + 1 for every l.
Proof. The network is shown in Figure 2. It consists of linear and second-order
units. All computation nodes are initialized with 0, except Yl, which starts with 1
and outputs 0 during all following steps. The purpose of Yl is to let the input x
output
Figure 3: Network Nt.
enter node Y2 at time 1 and keep it away at later times. Clearly, the value fl (x)
results at node Y5 after 2l + 1 steps.
D
The network used for computing fl requires only linear and second-order units. The
following result shows that the established upper bound is asymptotically almost
tight, with a gap only of order four . Moreover, the lower bound holds for networks
of unrestricted order and with sigmoidal units.
Theorem 3. Every general recurrent neural network of size k requires at least time
cl l / 4 j k to compute function fl' where c> 0 is some constant.
Proof. The idea is to construct higher-order networks Nt of small size that have
comparatively large VC dimension. Such a network will consist of linear and product
units and hypothetical units that compute functions fJ for certain values of j. We
shall derive a lower bound on the VC dimension of these networks. Assuming that
the hypothetical units can be replaced by time-bounded general recurrent networks,
we determine an upper bound on the VC dimension of the resulting networks in
terms of size and computing time using an idea from Koiran and Sontag (1998) and
Proposition 1. The comparison of the lower and upper VC dimension bounds will
give an estimate of the time required for computing k
Network Nt, shown in Figure 3, is a feedforward network composed of three networks
? r(1) , JVI
? r(2) , JVI
.r(3) . E ach networ k JVI
? r(/1) ,J.L = 1, 2, 3 , h as l ?lnput no d es Xl'
(/1) .. . , x I(/1)
JVI
and 2l + 2 computation nodes yb/1), ... , Y~r~l (see Figure 4). There is only one
adjustable parameter in Nt, denoted w, all other weights are fixed. The computation
nodes are defined as follows (omitting time parameter t):
for J.L
= 3,
for J.L = 1,2,
y~/1)
fll'--1 (Y~~)l) for i = 1, ... ,l and J.L = 1,2,3,
y}~{
y~/1) . x~/1), for i = 1, .. . ,l and
(/1)
Y21+l
(/1)
YIH
+ ... + Y21(/1)
J.L = 1,2,3,
c
- 1 2 3
lor
J.L , , ?
The nodes Yb/1) can be considered as additional input nodes for N//1), where N;(3)
gets this input from w, and N;(/1) from N;(/1+l) for J.L = 1,2. Node Y~r~l is the
output node of N;(/1), and node Y~~~l is also the output node of Nt. Thus, the entire
network has 3l + 6 nodes that are linear or product units and 3l nodes that compute
functions h, fl' or f12.
output
8
r - - - - - - - - - - - - '.....L - - - - - - - - - - - ,
I
I
B
B
t
t
I x~p)1
~
----t
input:
w
or
output of N;(P+1)
Figure 4: Network N;(p).
We show that Ni shatters some set of cardinality [3, in particular, the set S = ({ ei :
i = 1, . .. , [})3, where ei E {O, 1}1 is the unit vector with a 1 in position i and
elsewhere. Every dichotomy of S can be programmed into the network parameter
w using the following fact about the logistic function ? (see Koiran and Sontag,
1998, Lemma 2): For every binary vector b E {O, l}m, b = b1 .?. bm , there is some
real number w E [0,1] such that for i = 1, ... , m
?
E
{
[0,1 /2)
if bi = 0,
(1/2,1]
if bi = 1.
Hence, for every dichotomy (So, Sd of S the parameter w can be chosen such that
every (ei1' ei2 , ei3) E S satisfies
1/2 if (eillei2,eis) E So,
1/2 if (eillei2,eiJ E S1.
Since h +i2 H i 3 .12 (w) = ?i1 (?i2'1 (?i3 .1 2(w))), this is the value computed by Ni on
input (eill ei2' ei3), where ei" is the input given to network N;(p). (Input ei" selects
the function li"'I,,-1 in N;(p).) Hence, S is shattered by Ni, implying that Ni has
VC dimension at least [3.
Assume now that Ii can be computed by a general recurrent neural network of size
at most kj in time tj. Using an idea of Koiran and Sontag (1998), we unfold the
network to obtain a feedforward network of size at most kjtj computing fj. Thus we
can replace the nodes computing ft, ft, fl2 in Nz by networks of size k1t1, kltl, k12t12,
respectively, such that we have a feedforward network
consisting of sigmoidal,
product, and linear units. Since there are 3l units in Nl computing ft, ft, or fl2
and at most 3l + 6 product and linear units, the size of Nt is at most c1lkl2tl2
for some constant C1 > O. Using that Nt has one adjustable weight, we get from
Proposition 1 that its VC dimension is at most c2l2kr2tr2 for some constant C2 > o.
On the other hand, since Nz and Nt both shatter S, the VC dimension of Nt is at
least l3. Hence, l3 ~ C2l2 kr2 tr2 holds, which implies that tl2 2: cl 1/ 2/ kl2 for some
c > 0, and hence tl 2: cl 1/ 4/ kl .
D
'!J
Lemma 2 shows that a single recurrent network is capable of computing every
function fl in time O(l). The following consequence of Theorem 3 establishes that
this bound cannot be much improved.
Corollary 4. Every general recurrent neural network requires at least time 0(ll /4 )
to compute the functions fl.
4
Conclusions and Perspectives
We have established bounds on the computing time of analog recurrent neural
networks. The result shows that for every network of given size there are functions
of arbitrarily high time complexity. This fact does not rely on a bound on the
magnitude of weights. We have derived upper and lower bounds that are rather
tight- with a polynomial gap of order four- and hold for the computation of a
specific family of real-valued functions in one variable. Interestingly, the upper
bound is shown using second-order networks without sigmoidal units, whereas the
lower bound is valid even for networks with sigmoidal units and arbitrary product
units. This indicates that adding these units might decrease the computing time
only marginally. The derivation made use of an upper bound on the VC dimension
of higher-order sigmoidal networks. This bound is not known to be optimal. Any
future improvement will therefore lead to a better lower bound on the computing
time.
We have focussed on product and sigmoidal units as nonlinear computing elements.
However, the construction presented here is generic. Thus, it is possible to derive
similar results for radial basis function units, models of spiking neurons, and other
unit types that are known to yield networks with bounded VC dimension. The
questions whether such results can be obtained for continuous-time networks and for
networks operating in the domain of complex numbers, are challenging. A further
assumption made here is that the networks compute the functions exactly. By a
more detailed analysis and using the fact that the shattering of sets requires the
outputs only to lie below or above some threshold, similar results can be obtained
for networks that approximate the functions more or less closely and for networks
that are subject to noise.
Acknowledgment
The author gratefully acknowledges funding from the Deutsche Forschungsgemeinschaft (DFG). This work was also supported in part by the ESPRIT Working Group
in Neural and Computational Learning II, NeuroCOLT2, No. 27150.
References
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Schmitt, M. (2002). On the complexity of computing and learning with multiplicative neural networks. Neural Computation, 14. In press.
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1,221 | 2,112 | Activity Driven Adaptive Stochastic
Resonance
Gregor Wenning and Klaus Oberrnayer
Department of Electrical Engineering and Computer Science
Technical University of Berlin
Franklinstr. 28/29 , 10587 Berlin
{grewe , oby}@cs.tu-berlin.de
Abstract
Cortical neurons might be considered as threshold elements integrating in parallel many excitatory and inhibitory inputs. Due to
the apparent variability of cortical spike trains this yields a strongly
fluctuating membrane potential, such that threshold crossings are
highly irregular. Here we study how a neuron could maximize its
sensitivity w.r.t. a relatively small subset of excitatory input. Weak
signals embedded in fluctuations is the natural realm of stochastic
resonance. The neuron's response is described in a hazard-function
approximation applied to an Ornstein-Uhlenbeck process. We analytically derive an optimality criterium and give a learning rule
for the adjustment of the membrane fluctuations, such that the
sensitivity is maximal exploiting stochastic resonance. We show
that adaptation depends only on quantities that could easily be
estimated locally (in space and time) by the neuron. The main
results are compared with simulations of a biophysically more realistic neuron model.
1
Introduction
Energetical considerations [1] and measurements [2] suggest , that sub-threshold
inputs, i.e. inputs which on their own are not capable of driving a neuron , play an
important role in information processing. This implies that measures must be taken,
such that the relevant information which is contained in the inputs is amplified in
order to be transmitted. One way to increase the sensitivity of a threshold device is
the addition of noise. This phenomenon is called stochastic resonance (see [3] for a
review) , and has already been investigated and experimentally demonstrated in the
context of neural systems (e.g. [3, 4]). The optimal noise level, however , depends
on the distribution of the input signals, hence neurons must adapt their internal
noise levels when the statistics of the input is changing. Here we derive and explore
an activity depend ent learning rule which is intuitive and which only depends on
quantities (input and output rates) which a neuron could - in principle - estimate.
The paper is structured as follows. In section 2 we describe the neuron model and we
introduce the m embrane potential dynamics in its hazard function approximation.
In section 3 we characterize stochastic resonance in this model system and we calculate the optimal noise level as a function of t he input and output rates. In section 4
we introduce an activity dependent learning rule for optimally adjusting the internal noise level, demonstrate its usefulness by applying it to t he Ornstein-Uhlenbeck
neuron and relate the phenomenon of stochastic resonance to its experimentally
accessible signature: the adaptation of the neuron 's transfer function . Section 5
contains a comparison to the results from a biophysically more realistic neuron
model. Section 6, finally, concludes with a brief discussion.
2
The abstract Neuron Model
Figure 1 a) shows the basic model setup. A leaky integrate-and-fire neuron receives
.",
a)
~;=".~
train with rate As
"0
>8 >
{5
-5"
'-<
O/l
b)
0.8
c
0.7
;::
0.6
.~
rateAo
8
0.9
"
0.5
E
~
0.4
"-'
0
0.2
.0
0.1
~
/'IWn
I
.S
2 N balanced Poisson
spike trains with rates
As
;::
0.
0.3
00
0.2
0.4
0.6
0.8
average membrane potential
Figure 1: a)The basic model setup. For explanation see text . b) A family of
Arrhenius type hazard functions for different noise levels. 1 corresponds to the
threshold and values below 1 are subthreshold .
e
a "signal" input , which we assume to be a Poisson distributed spike train with a rate
As. The rate As is low enough , so that the membrane potential V of the neuron
remains sub-threshold and no output spikes are generated . For the following we
assume that the information the input and output of the neuron convey is coded by
its input and output rates As and Ao only. Sensitivity is then increased by adding
2N balanced excitatory and inhibitory "noise" inputs (N inputs each) with rates
An and Poisson distributed spikes . Balanced inputs [5, 6] were chosen , because they
do not affect t he average membrane potential and allow to separate the effect of
decreasing the distance of the neuron's operating point to the threshold potential
from the effect of increasing the variance of the noise. Signal and noise inputs
are coupled to t he neuron via synaptic weights Ws and Wn for the signal and noise
inputs . The threshold of the neuron is denoted bye. Without loss of generality the
membrane time-constant, the neuron 's resting potential, and the neuron 's threshold
are set to one, zero , and one , respectively.
If the total rate 2N An of incoming spikes on t he "noise" channel is large and the
individual coupling constants Wn are small , the dynamics of the m embrane potential
can b e approximated by an Ornstein-Uhlenbeck process,
dV
=-V
dt
+ J.l
dt
+
(J"
dW,
(1)
where drift J.l and variance (J" are given by J.l = wsA s and (J"2 = w1A s + 2NwYvAN,
and where dW describes a Gaussian noise process with m ean zero and variance
one [8]. Spike initiation is included by inserting an absorbing boundary with reset.
Equation (1) can b e solved an alytically for special cases [8], but here we opt for
a more versatile approximation (cf. [7]). In this approximation, the probability of
crossing the threshold , which is proportional to the instantaneous output rate of
the neuron , is described by an effective transfer function. In [7] several transfer
functions were compared in their performance, from which we choose an Arrheniustype function ,
Ao(t) = c exp{ _ (e - ~(t))2},
(2)
cr
e-
where
x(t) is the distance in voltage between the noise free trajectory of the
membrane potential x(t) and the threshold x(t) is calculated from eq. (1) without
its diffusion term. Note that x(t) is a function of As, c is a constant. Figure 1 b)
shows a family of Arrhenius type transfer functions for different noise levels cr.
3
e,
Stochastic Resonance in an Ornstein- Uhlenbeck Neuron
Several measures can be used to quantify the impact of noise on the quality of signal
transmission through threshold devices . A natural choice is the mutual information
[9] between the distributions p( As) and p( Ao) of input and output rates, which we
will discuss in section 4, see also figure 3f. In order to keep the analysis and the
derivation of the learning rule simple , however, we first consider a scenario, in which
a neuron should distinguish between two sub-threshold input rates As and As + ~s.
Optimal distinguishability is achieved if the difference ~o of the corresponding
output rates is maximal, i.e. if
~o =
/(As
+ ~ s) - /(As)
(3)
= max ,
where / is the transfer function given by eq. (2). Obviously there is a close connection between these two measures , because increasing both of them leads to an
increase in the entropy of p( Ao) .
Fig. 2 shows plots of the difference
0.16
~o
of output rates vs. the level of noise, cr , for
0.4
~s
0.14
~s
0.35
AS= 7
0.12
:5
0
<:::]
:5
0
0.1
<:::]
0.08
0.06
0.04
0.02
0.05
00
50
0 2
100
[per cent]
50
100
[per cent]
Figure 2: ~ o vs. cr 2 for two different base rates As = 2 (left) and 7 (right) and 10
different values of ~ s = 0.01 , 0.02 , ... , 0.1. cr 2 is given in per cent of the maximum
cr 2 = 2N W;An. The arrows above t he x-axis indicate the position of the maximum
according to eq. (3), the arrowh eads below the x-axis indicate the optimal value
computed using eq. (5) (67% and 25%). Parameters were: N = la , An = 7,
Ws = 0.1 , and Wn E [0, 0.1].
different rates As and different values of ~ s . All curves show a clear maximum at a
particular noise level. The optimal noise level increases wit h decreasing t he input
rate As, but is roughly independent of the difference ~ s as long as ~ s is small.
Therefore, one optimal noise level holds even if a neuron has to distinguish several
sub-threshold input rates - as long as these rates are clustered around a given base
rate As.
The optimal noise level for constant As (stationary states) is given by the condition
d
d(j2 (f(A s + ~ s) - f(As)) = 0 ,
(4)
where f is given by eq. (2). Equation (4) can be evaluated in the limit of small
values of ~ s using a Taylor expansion up to the second order. We obtain
(j;pt
=
2(1 - ws As)2
(5)
if the main part of the variance of the membrane potential is a result of the balanced
.
'" 2N WNAN
2 , (f
2 -- - (1log(Ao/C)
- W, A, )2 , eq. (2) , eq. (5)
mput
, l."
e.f
1 (j 2 '"
c . eq. (1)) . S'mce (jopt
is equivalent to 1 + 2 log( Ao (A; ;0"2)) = O. This shows that the optimal noise level
depends either only on As or on Ao(As; (j2), both are quantities which are locally
available at the cell.
4
Adaptive Stochastic Resonance
We now consider the case , that a neuron needs to adapt its internal noise level
because the base input rate As changes. A simple learning rule which converges to
the optimal noise level is given by
~(j2
=
- f
(j2
log( - 2-) ,
(j opt
(6)
where the learning parameter f determines the time-scale of adaptation . Inserting
the corresponding expressions for the actual and the optimal variance we obtain a
learning rule for the weights W n ,
~wn =
-f
I og ( ( 2NAnw; )2 ) .
2 1 - ws As
(7)
Note, t hat equivalent learning rules (in the sense of eq. (6)) can be formulat ed for
the number N of the noise inputs and for their rates An as well. The r.h. s. of eqs .
(6) and (7) depend only on quantities which are locally available at the neuron.
Fig. 3ab shows the stochastic adaptation of the noise level, using eq.
randomly distributed As which are clustered around a base rate.
(7) , to
Fig. 3c-f shows an application ofthe learning rule, eq. (7) to an Ornstein-Uhlenbeck
neuron whose noise level needs to adapt to three different base input rates. T he
figure shows t he base input rate As (Fig. 3a). In fig. 3b the adaptation of Wn
according to eq. (7) is shown (solid line), for comparison t he Wn which maximizes
eq. (3) is also displayed (dash ed dotted lin e). Mutual information was calculated
between a distribution of randomly chosen input rates which are clustered around
the base rate As. The Wn that maximizes mutual Information between input and
output rates is displayed in fig. 3d (dashed lin e). Fig. 3e shows the ratio ~ o / ~ s
computed by using eq. (3) and the Wn calculated with eq. (8) (dashed dotted line)
and the same ratio for the quadratic approximation. Fig. 3f shows the mutual
information between the input and output rates as a function of the changing w n .
I[n~[ :~--/
0
0
2500
3000
1500
2000
2500
3000
0~
500 1000 1500
0
2000
1000
1500
~I
I,rJ
2000
500
?
0.1 5
w
110
b)
n 0.1
I1S
0
0.05
00
10
AS
o:i,ek '
500
1000
1500
2000
2500
3000
W
O'h
n 0.1
a)
500
I
1000
d)
..
?
5
00
As
500
1000
1500
2000
2500
3000
':1C)
0
0
I
500
1000
?
1500
ri"
I I
2500
3000
2000 2500 3000
?
time[updatesteps1
time [update steps]
Figure 3: a) Input rates As are evenly distributed around a base rate with width
0.5, in each time step one As is presented . b) Application of the learning rule eq.
(7) to t he rates shown in a). Adaptation of the noise level to t hree different input
base rates As. c ) The three base rates As. d) Wn as a function of time according
to eq. (7) (solid line) , the optimal Wn that maximizes eq. (3) (dashed dotted line)
and the optimal Wn that maximizes the m ut ual information between t he input and
output rates (dashed). T he opt imal values of Wn as the quadratic approximation,
eq. (5) yield are indicated by the black arrows. e ) The ratio b.. o / b.. s computed
from eq. (3) (dashed dotted line) and t he quadratic approximation (solid line) . f)
Mut ual information between input and output rates as a function of base rate and
changing synaptic coupling constant W n . For calculating the mutual information
the input rates were chosen randomly from the interval [As - 0.25 , As + 0.25] in each
time step . Parameters as in fig . 2.
T he fig ure shows , that the learning rule, eq. (7) in t he quadratic approximation
leads to values for () which are near-optimal, and that optimizing the difference of
output rates leads to results similar to t he optimization of the m ut ual information .
5
Conductance based Model Neuron
To check if and how t he results from the abstract model carryover to a biophysically
mode realistic one we explore a modified Hodgkin-Huxley point neuron with an
additional A-Current (a slow potassium current) as in [11] . T he dynamics of the
membrane potential V is described by t he following equation
C~~
- gL(V(t ) - EL) -
!iNam~ h(t)(V -
ENa)
- !iKn(t)4(V - EK) - !iAa ~ b(t)(V - EK)
+ l syn
+ la pp,
(8)
the parameters can be found in the appendix. All parameters are kept fixed through-
a
a)
II~
"}
'N
?::l
10
c
tS
80 ,---------------------------------,
b)
70
60
50
::I
~
o
~ 5
~
peak
conductances
8
40
~
.~
30
.5
20
<t>
U
00'----- 0 ,""
2 =:::...0-.4""""'-=--,L---~--,~
,2---"1.4
~
~
'i3
10
00
2
4
6
B
10
noiselevel in multiples of peak conductances
Figure 4: a) Transfer function for the conductance based model neuron with additional balanced input , a = 1, 2, 3, 4 b ) Demonstration of SR for the conductance
based model neuron. The plot shows the resonance for two different base currents
lapp = 0.7 and lapp = 0.2 and a E [0, 10].
~
-I
~
&
'0
-~
E
90 ,------ - - - - - - - - -- -- - - - - - - - - - - ---,
7
a)
E3 -
-B
~
~
5
--&.,
4-
b
:2
.j
1
70 -
~ 60 -
~
3
=
0)
gp ao -
50 40 -
1G
_~30 ~
20 -
-~
10 -
:;::
0 0~----------~~------~~~1
0.5
I drift [n~]
?0~----------0
~
.5
~--~------~
s"U-othresl""1<:>ld p<:>te:n.t:ial ( 8 =
1
)
Figure 5: a) Optimal noise-level as a function ofthe base current in the conductance
based model. b) Optimal noise-level as a function of the noise-free membrane
potential in the abstract model.
out all shown data. As balanced input we choose an excitatory Poisson spike train
with rate Ane = 1750 Hz and an inhibitory spike train with rate Ani = 750 Hz .
These spike trains are coupled to the neuron via synapses resulting in a synaptic
current as in [12]
ls yn = ge(V(t) - E e) + gi(V(t) - Ei)).
(9)
Every time a spike arrives at the synapse the conductance is increased by its peak
conductance ge i and decreases afterwards exponentially like exp{ - _t_
, } . The corT e, t
I
responding parameters are ge = a * 0.02 * gL , gi = a * 0.0615 * gL. The common
factor a is varied in the simulations and adjusts the height of the peak conductances,
gL is the leak conductance given above . Excitatory and inhibitory input are called
balanced if the impact of a spike-train at threshold is the same for excitation and
inhibition
TegeAne(Ee - B)
with Te i
I
= - TigiAni(Ei -
B)
= ~ J!fooo ge,i(t)dt . Note that the factor a does cancel in eq .
ge,t
(10)
(10).
Fig. 4a displays transfer funct ions in the conductance b ased setting with balanced
input. A family of functions with varying p eak conduct ances for the balanced input
is drawn .
~
100 r-----~----~----~----~----~-----.
d)
rJJ
~
~
50
?OL'~--~--~----~--~~--~--~
50
'-J
100
150
200
250
300
,-...,
i':k"-:-..:~. :
'"
0
50
1~-:o
50
100
150
200
250
300
100
150
200
250
300
Figure 6: Adaptive SR in the conductance based model. a) Currents drawn from
a uniform distribution of width 0.2 nA centered around base currents of 3, 8, 1 nA
respectively. b) Optimal noise-level in terms of a. Optimality refers to a semi-linear
fit to the data of fig. 5a. c) adapting the peak conductances using a in a learning
rule like eg. (8). d) Difference in spike count , for base currents I ? 0.1 nA and
using a as specified in c) .
For studying SR in t he conductance based framework , we apply the same paradigm
as in the abstract model. Given a certain average membrane potential, which is
adjusted via injecting a current I (in nA), we calculate the difference in the output
rate given a certain difference in the average membrane potential (mediated via the
injected current) I ? t:.I. A demonstration of stochastic resonance in the conductance based neuron can be seen in fig. 4b. In fig. 5a the optimal noise-level, in
terms of multiples a of the peak conductances , is plotted versus all currents that
yield a sub-threshold membrane voltage. For comparison we give the corresponding
relationship for the abstract model in fig. 5b .
Fig. 6 shows the performance of the conductance based model using a learning
rule like eg. (7). Since we do not have an analytically derived expression for (J opt
in the conductance based case, the relation (Jopt (I) , necessary for using eg. (7),
corresponds to a semi-linear fit to the (a opt , I) relation in fig. 5a.
6
Conclusion and future directions
In our contribution we have shown , that a simple and activity driven learning rule
can be given for the adaptation of the optimal noise level in a stochastic resonance
setting. Th e results from the abstract framework are compared with results from a
conductance based model neuron. A biological plausible m echanism for implem enting adaptive stochastic resonance in conductance based neurons is currently under
investigation.
Acknowledgments
Supported by: Wellcome Trust (061113/Z/00)
App e ndix: Parame t ers for the conductance based mode l n e uron
=
:':2
=
somatic conductances/ion-channel properties: em
1.0
,gL 0.05 ~ ,gNa
100 ~,gJ( = 40 ~,gA = 20 ~,EL = -65 mV,ENa = 55 mV, EJ(
- 80 mV, TA
=
= 20 ms ,
synaptic coupling : Ee = 0 mV, Ei = -80 mV, Te = 5 ms , Ti = 10 ms ,
spike initiation: dh = ~ dn = ngo - n <jQ = ~
mCO = <>m<>-ti3m'
dt
Th
O:m
= -O.l(V
55)/18),
h oo = <>h":i3h'
n co = <>n+i3 n '
O:h
O:n
'
dt
Tn'
dt
TA
'
+ 30)/(exp( -O .l(V + 30)) -
1), f3m = 4exp( -(V
+
= 0.07exp(-(V + 44)/20), f3h = l/( exp(-O. l(V + 14)) + 1) ,
= -O.Ol(V + 34)/(exp( -O.l(V + 34)) -1) , f3n = 0.125exp( -(V +
44)/80)
a oo = l/(exp( -(V + 50)/20) + 1) , boo = l/(exp((V
Th = in/(O:h + f3h) , Tn = in/(O:n + f3n), in = 0.1
+ 80)/6) + 1),
References
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| 2112 |@word nd:1 simulation:2 t_:1 solid:3 versatile:1 ld:1 contains:1 series:1 cort:1 current:12 wilkens:1 must:2 realistic:3 plot:2 update:1 v:2 stationary:1 device:2 ial:1 inam:1 height:1 dn:1 behavioral:1 introduce:2 rapid:1 roughly:1 ol:2 f3h:2 decreasing:2 actual:1 increasing:2 maximizes:4 every:1 ti:1 f3m:1 yn:1 engineering:1 limit:1 switching:1 ure:1 fluctuation:2 firing:1 might:1 black:1 co:2 steveninck:1 acknowledgment:1 adapting:1 pre:1 integrating:1 refers:1 suggest:1 close:1 ga:1 context:1 applying:1 equivalent:2 demonstrated:1 neurona:1 zador:1 l:2 wit:1 rule:13 adjusts:1 gammaitoni:1 dw:2 discharge:1 pt:1 play:1 element:2 crossing:2 approximated:1 role:1 electrical:1 solved:1 calculate:2 tsodyks:1 sompolinsky:1 decrease:1 balanced:10 leak:1 dynamic:3 signature:1 depend:2 funct:1 easily:1 lapp:2 cat:1 w1a:1 train:8 derivation:1 describe:1 effective:1 sejnowski:2 oby:1 klaus:1 salina:1 apparent:1 whose:1 plausible:1 ikn:1 statistic:1 gi:2 gp:1 obviously:1 rr:1 maximal:2 reset:1 adaptation:7 tu:1 relevant:1 inserting:2 j2:4 amplified:1 formulat:1 intuitive:1 ent:1 exploiting:1 potassium:1 abstr:1 transmission:1 converges:1 derive:2 coupling:3 oo:2 eq:23 soc:1 c:1 implies:1 indicate:2 quantify:1 bulsara:1 direction:1 stochastic:16 centered:1 respons:1 feeding:1 ao:8 clustered:3 investigation:1 opt:5 biological:1 adjusted:1 hold:1 around:5 considered:1 exp:10 shriki:1 driving:1 injecting:1 currently:1 hansel:1 gaussian:1 modified:1 i3:2 cr:6 ej:1 og:1 voltage:2 varying:1 derived:1 check:1 contrast:1 sense:1 dependent:1 el:2 plesser:1 w:4 relation:2 jq:1 orientation:1 denoted:1 resonance:14 special:1 mutual:6 cancel:1 future:1 carryover:1 escape:1 modern:1 randomly:3 individual:1 mut:1 fire:2 ab:1 conductance:22 detection:1 highly:1 arrives:1 implication:1 capable:1 necessary:1 conduct:1 taylor:1 plotted:1 theoretical:1 increased:2 modeling:1 cover:1 newsome:1 cost:1 subset:1 i1s:1 usefulness:1 uniform:1 paddle:1 characterize:1 optimally:1 peak:6 sensitivity:4 accessible:1 physic:1 na:4 choose:2 ek:3 potential:14 de:2 coding:1 mv:5 ornstein:5 depends:4 parallel:1 contribution:2 variance:5 yield:3 subthreshold:1 ofthe:2 weak:1 biophysically:3 trajectory:1 app:1 synapsis:1 synaptic:5 ed:2 pp:1 frequency:1 adjusting:1 realm:1 ut:2 syn:1 mco:1 ta:2 dt:6 response:1 synapse:1 evaluated:1 strongly:1 generality:1 anderson:2 receives:1 ei:3 trust:1 nonlinear:1 mode:3 quality:1 indicated:1 effect:2 fooo:1 analytically:2 hence:1 tuckwell:1 eg:3 embrane:2 width:2 excitation:1 m:3 demonstrate:1 tn:2 consideration:1 instantaneous:1 common:1 absorbing:1 exponentially:1 volume:1 he:21 resting:1 measurement:1 cambridge:1 ena:2 tuning:1 cortex:2 operating:1 inhibition:1 gj:1 base:15 own:1 optimizing:1 driven:2 scenario:1 certain:2 initiation:2 criterium:1 transmitted:1 seen:1 additional:2 maximize:1 paradigm:1 signal:7 semi:2 dashed:5 ii:1 multiple:2 rj:1 afterwards:1 technical:1 adapt:3 long:2 hazard:3 lin:2 coded:1 impact:2 basic:2 poisson:4 ane:1 uhlenbeck:5 achieved:1 cell:2 irregular:1 ion:2 addition:1 interval:1 sr:3 hz:2 induced:1 ngo:1 ee:2 near:1 enough:1 wn:12 boo:1 affect:1 fit:2 ased:1 implem:1 expression:2 e3:1 clear:1 locally:3 band:1 inhibitory:4 fish:1 dotted:4 estimated:1 neuroscience:2 per:3 threshold:17 drawn:2 changing:3 ani:1 diffusion:1 kept:1 injected:1 franklinstr:1 hodgkin:1 telecommunication:1 family:3 appendix:1 ct:1 dash:1 distinguish:2 display:1 quadratic:4 activity:4 huxley:1 ri:1 optimality:2 relatively:1 department:1 structured:1 according:3 membrane:13 describes:1 jopt:2 em:1 dv:1 taken:1 wellcome:1 equation:3 remains:1 wsa:1 discus:1 count:1 ge:5 studying:1 available:2 apply:1 fluctuating:1 hat:1 cent:3 hree:1 responding:1 lampl:1 cf:1 thomas:1 calculating:1 gregor:1 already:1 quantity:4 spike:14 distance:2 separate:1 berlin:3 evenly:1 parame:1 gna:1 relationship:1 ratio:3 demonstration:2 setup:2 relate:1 neuron:38 t:1 displayed:2 neurobiology:1 variability:2 arrhenius:2 varied:1 somatic:1 drift:2 specified:1 connection:1 distinguishability:1 below:2 max:1 explanation:1 gillespie:1 natural:2 ual:3 imal:1 brief:1 axis:2 grewe:1 concludes:1 mediated:1 coupled:2 moss:1 text:1 review:1 embedded:1 loss:1 proportional:1 versus:1 integrate:2 shadlen:1 principle:1 metabolic:1 excitatory:5 jung:1 gl:5 bye:1 free:2 supported:1 visua:1 allow:1 laughlin:1 wide:1 leaky:1 distributed:4 van:1 boundary:1 calculated:3 cortical:5 curve:1 adaptive:4 eak:1 keep:1 ances:1 incoming:1 channel:2 transfer:7 ruyter:1 nature:2 ean:1 expansion:1 investigated:1 gerstner:1 main:2 neurosci:2 arrow:2 noise:36 edition:1 convey:1 neuronal:1 fig:17 slow:1 wiley:1 sub:5 position:1 rjj:1 uron:1 er:1 adding:1 i11:1 te:3 mce:1 entropy:1 explore:2 adjustment:1 contained:1 corresponds:2 determines:1 dh:1 russel:1 i3h:1 ferster:1 experimentally:2 change:1 included:1 total:1 called:2 f3n:2 invariance:1 la:2 internal:3 mput:1 phenomenon:2 |
1,222 | 2,113 | On the Generalization Ability
of On-line Learning Algorithms
Nicol`o Cesa-Bianchi
DTI, University of Milan
via Bramante 65
26013 Crema, Italy
[email protected]
Alex Conconi
DTI, University of Milan
via Bramante 65
26013 Crema, Italy
[email protected]
Claudio Gentile
DSI, University of Milan
via Comelico 39
20135 Milano, Italy
[email protected]
Abstract
In this paper we show that on-line algorithms for classification and regression can be naturally used to obtain hypotheses with good datadependent tail bounds on their risk. Our results are proven without requiring complicated concentration-of-measure arguments and they hold
for arbitrary on-line learning algorithms. Furthermore, when applied to
concrete on-line algorithms, our results yield tail bounds that in many
cases are comparable or better than the best known bounds.
1 Introduction
One of the main contributions of the recent statistical theories for regression and classification problems [21, 19] is the derivation of functionals of certain empirical quantities
(such as the sample error or the sample margin) that provide uniform risk bounds for all the
hypotheses in a certain class. This approach has some known weak points. First, obtaining
tight uniform risk bounds in terms of meaningful empirical quantities is generally a difficult task. Second, searching for the hypothesis minimizing a given empirical functional
is often computationally expensive and, furthermore, the minimizing algorithm is seldom
incremental (if new data is added to the training set then the algorithm needs be run again
from scratch).
On-line learning algorithms, such as the Perceptron algorithm [17], the Winnow algorithm [14], and their many variants [16, 6, 13, 10, 2, 9], are general methods for solving
classification and regression problems that can be used in a fully incremental fashion. That
is, they need (in most cases) a short time to process each new training example and adjust
their current hypothesis. While the behavior of these algorithms is well understood in the
so-called mistake bound model [14], where no assumptions are made on the way the training sequence is generated, there are fewer results concerning how to use these algorithms
to obtain hypotheses with small statistical risk.
Littlestone [15] proposed a method for obtaining small risk hypotheses from a run of an
arbitrary on-line algorithm by using a cross validation set to test each one of the hypotheses
generated during the run. This method does not require any convergence property of the online algorithm and provides risk tail bounds that are sharper than those obtainable choosing,
for instance, the hypothesis in the run that survived the longest. Helmbold, Warmuth,
and others [11, 6, 8] showed that, without using any cross-validation sets, one can obtain
expected risk bounds (as opposed to the more informative tail bounds) for a hypothesis
randomly drawn among those generated during the run.
In this paper we prove, via refinements and extensions of the previous analyses, that online algorithms naturally lead to good data-dependent tail bounds without employing the
complicated concentration-of-measure machinery needed by other frameworks [19]. In
particular we show how to obtain, from an arbitrary on-line algorithm, hypotheses whose
risk is close to
with high probability (Theorems 2 and 3), where is the amount of
training data and is a data-dependent quantity measuring the cumulative loss of the online algorithm on the actual training data. When applied to concrete algorithms, the loss
bound translates into a function of meaningful data-dependent quantities. For classification problems, the mistake bound for the -norm Perceptron algorithm yields a tail risk
bound in terms of the empirical distribution of the margins ? see (4). For regression problems, the square loss bound for ridge regression yields a tail risk bound in terms of the
eigenvalues of the Gram matrix ? see (5).
2 Preliminaries and notation
# %$&(')'('*+ !
be arbitrary sets and
. An example is a pair
, where is an
Let
instance belonging to and
is the label associated with . Random variables will
be denoted in upper case and their realizations will be in lower case. We let be the
pair of random variables
, where
and take values in
and , respectively.
Throughout the paper, we assume that data are generated i.i.d. according to an unknown
probability distribution over . All probabilities and expectations will be understood with
to denote the vectorrespect to this underlying distribution. We use the short-hand
valued random variable
.
1
1 ,
:DC 1 E GFIH
:
"!
-
,./0
2(3(#,45 7698;:<#, =>? @BA
A hypothesis is any (measurable) mapping from instances
to predictions
, where is a given decision space. The risk of is defined by
,
where
is a nonnegative loss function. Unless otherwise specified, we will
assume that takes values in
for some known
. The on-line algorithms we
investigate are defined within a well-known mathematical model, which is a generalization
of a learning model introduced by Littlestone [14] and Angluin [1]. Let a training sequence
be fixed. In this learning model, an on-line
algorithm processes the examples in
one at a time in trials, generating a sequence of
hypotheses
. At the beginning of the -th trial, the algorithm receives the
and uses its current hypothesis
to compute a prediction
instance
for the label associated with . Then, the true value of the label
is disclosed and
, measuring how bad is the prediction
the algorithm suffers a loss
for the label . Before the next trial begins, the algorithm generates a new hypothesis
which may or may not be equal to
. We measure the algorithm?s performance on by
its cumulative loss
8 JKLA
,
J MNKOM7P
Q ! RS$T
U$V>(')'('W(
X#YZ 0 !
! !
Q!
,\[]?, $ )'(')'*+, !
^
4_
,/_ `*$
,4_ `*$T/_Ba 1
]_
/_
b_
:<#,/_`W$&/_#V
<_B
,S_`W$&/_c
_
Q!,_
,/_ `*$
d
Q ! 5 ! :<g, _`W$ _ >
_ >'
_fe.$
In our analysis, we will write h and ij[<(')'('Wi when we want to stress the fact that the
!
cumulative loss and the hypotheses of the on-line algorithm are functions of the random
sample ! . In particular, throughout the paper i [ will denote the (deterministic) initial
hypothesis of an arbitrary on-line algorithm and, for each kml7^nl , io_ will be a random
variable denoting the ^ -th hypothesis of the on-line algorithm and such that the value of
ij_B 5$&)')'('*+ ! does not change upon changes in the values of p_fq.$(')'('W+ ! .
Our goal is to relate the risk of the hypotheses produced by an on-line algorithm running
on an i.i.d. sequence
to the cumulative loss
of the algorithm on that sequence.
r!
hs# r!t
hsB !t
The cumulative loss
will be our key empirical (data-dependent) quantity. Via our
analysis we will obtain bounds of the form
k k l
2)3 gi [ )'(')'*i ! hsB n!t
where gi [<(')'('Wi is a specific function of the sequence of hypotheses i9[])')'('Wi
!
!
produced by the algorithm, and is a suitable positive constant. We will see that for specific on-line algorithms the ratio hs# r!t & can be further bounded in terms of meaningful
empirical quantities.
Our method centers on the following simple concentration lemma about bounded losses.
:
J l:Zl K . Let an arbitrary
i [ (')'('Wi ! when it is run
Lemma 1 Let be an arbitrary bounded loss satisfying
on-line algorithm output (not necessarily distinct) hypotheses
on . Then for any
we have
n!
JjM l k
k l '
k d ! 2)3 i _ `*$ h K
_fe.$
Proof. For each ^ Gk<(')')'* , set /_ `*$ 2)3 ij_`W$) :Lij_ `*$< 9_#V /_# . We have
k d ! _ `*$ k d ! 2)3 i _ `*$ h '
_fe.$
_fe.$
Furthermore, KOl
_ `*$ lNK , since : takes values in 8 J?K A . Also,
698 \_ `*$ _`W$+AS 72)3g,4_`W$) =6 8 : g,\_ `*$< 9_gV /_# _`W$?Ab+ J
where _ `*$ denotes the -algebra generated by %$(')')'*> _ `*$ . A direct application of the
Hoeffding-Azuma inequality [3] to the bounded random variables S[b)'(')'* `W$ proves the
!
lemma.
!
"
3 Concentration for convex losses
In this section we investigate the risk of the average hypothesis
d
, k ! , _`W$
_ e.$
$#&%('
,\[<?, $ (')'('*?, !
1
1
1
Theorem 2 Let be convex and : C F 8 J\K A be convex in the first argument. Let an
arbitrary on-line algorithm for : output (not necessarily distinct) hypotheses i [ )')'('Wi
!
M k the following holds
when the algorithm is run on r! . Then for any J M
2)3( i h K k l '
where
are the hypotheses generated by some on-line algorithm run on
training examples.1 The average hypothesis generates valid predictions whenever the decision space is convex.
1
Notice that the last hypothesis
+-,
)
*
is not used in this average.
$ !_ e $ L: g,4_:`W$& />G'
$! !_ e $ )2 3 g, _`W$
!
: , /V
l
2(3 ,4 l
Proof. Since is convex in the first argument, by Jensen?s inequality we have
Taking expectation with respect to
yields
. Using the last inequality along with Lemma 1 yields the thesis.
? @
"
Q !c &&
!_fe.$ (2 3 gi _`W$
This theorem, which can be viewed as the tail bound version of the expected bound in [11],
implies that the risk of the average hypothesis is close to
for ?most? samples
. On the other hand, note that it is unlikely that
concentrates around
, at least without taking strong assumptions on the underlying on-line algorithm.
Q!
6L8 h A
An application of Theorem 2 will be shown is Section 5. Here we just note that by applying
this theorem to the Weighted Majority algorithm [16], we can prove a version of [5, Theorem 4] for the absolute loss without resorting to sophisticated concentration inequalities
(details in the full paper).
4 Penalized risk estimation for general losses
:
If the loss function is nonconvex (such as the 0-1 loss) then the risk of the average hypothesis cannot be bounded in the way shown in the previous section. However, the risk
of the best hypothesis, among those generated by the on-line algorithm, cannot be higher
than the average risk of the same hypotheses. Hence, Lemma 1 immediately tells us that,
at
under no conditions on the loss function other than boundedness, for most samples
least one of the hypotheses generated has risk close to
. In this section we give
a technique (Lemma 4) that, using a penalized risk estimate, finds with high probability
such a hypothesis. The argument used is a refinement of Littlestone?s method [15]. Unlike
Littlestone?s, our technique does not require a cross validation set. Therefore we are able
to obtain bounds on the risk whose main term is
, where is the size of the whole
set of examples available to the learning algorithm (i.e., training set plus validation set in
Littlestone?s paper). Similar observations are made in [4], though the analysis there does
actually refer only to randomized hypotheses with 0-1 loss (namely, to absolute loss).
Q!
Q !t
Q !c &
,S_ by
_
^ =^t5
where ^ is the length of the suffix Q _fq.$ (')')' Q of the training sequence that the on-line
!
algorithm had not seen yet when ,/_ was generated, _ is the cumulative loss of ,4_ on that
suffix, and
k
k
/
5
'
Our algorithm chooses the hypothesis ,o ,
_ , where
^
[
< 3 _ `W $ _ ^ ^t '
!
For the sake of simplicity, we will restrict to losses : with range 8 J(kVA . However, it should
Let us define the penalized risk estimate of hypothesis
be clear that losses taking values in arbitrary bounded real interval can be handled using
techniques similar to those shown in Section 3. We prove the following result.
Theorem 3 Let an arbitrary on-line algorithm output (not necessarily distinct) hypotheses
when it is run on . Then, for any
, the hypothesis chosen using
the penalized risk estimate based on satisfies
i [<(')')'i !
!
2)3 i h
JjM l k
k k l '
*
i
The proof of this theorem is based on the two following technical lemmas.
Lemma 4 Let an arbitrary on-line algorithm output (not necessarily distinct) hypotheses
when it is run on . Then for any
the following holds:
i [<(')')'i !
!
J M M k
2)3 i [ _ *` $ 2)3 i _ ^t
l '
!
Proof. Let
<3 [ _ `*$ 2(3( ij_B
^t
S' Let further i i , h
h , and set for brevity !
hZ_
_ ^
h
r '
For any fixed J we have
2)3 i= 2(3( i
d `W$
l ! _ =^t%l S2)3 ij_B 2(3( i < S' (1)
_fe [
Now, if
_ =^t%l (
_ lO2)3 ij_# =^t
holds then either
or
2)3 i (
)
2(3 gi _ 2)3 i %M
hold. Hence for any fixed ^ we can write
_ ^t%l )*2)3 i _ 2)3 i )
l _ l 2(3(gi _ ^tS 2(3( i _ 2(3(gi <
2(3( i *2) 3 ij_g 2)3 i <
2)3(gij_# 2)3 i )%M )S\2)3 ij_# O2)3 i ( <
l _ l 2(3(gi _ ^t
2)3 i ( )
2)3(gi _ 2)3 i %M S\2)3 i _ O2)3 i <9'
Probability (3) is zero if
. Hence, plugging (2) into (1) we can write
2(3 i O2)3(gi (
(
d! `*$
l
_ l 2(3 gi _B ^t
2(3(gi
_fe [
d `*$
l k ! _ O2)3(gi _ =^t
_fe [
l k k
or
(2)
(3)
where in the last two inequalities we applied Chernoff-Hoeffding bounds.
"
Lemma 5 Let an arbitrary on-line algorithm output (not necessarily distinct) hypotheses
when it is run on . Then for any
the following holds:
i [ (')')'i !
[ _ ! `*$ 2)3 ij_B
!
h
^t
J M M k
k
k
k
l '
Proof. We have
d! `*$
k
[ _ ! `W$ 2)3g,4_g =^t
%l f_ e [ g2)3 g,4_#
d `*$
k ! 2(3 #, _
_fe [
d `*$
M k ! 2(3 #,\_#
_fe [
`*$
d
!
k
l 2(3 #,\_#
_fe [
where the last inequality follows from !_fe.$ k ^ l
h
[ _ ! `*$ 2(3( ij_B ^t
d `W$
l k ! 2)3 ij_B h
_fe [
l
by Lemma 1 (with K Gk ).
=^t
k
d! `W$ k
_ e [ ^t
d! `W$ k k
_ e [ =^
k k
. Therefore
k
k
k
)
*
k
"
Proof (of Theorem 3). The proof follows by combining Lemma 4 and Lemma 5, and by
"
overapproximating the square root terms therein.
5 Applications
For the sake of concreteness we now sketch two generalization bounds which can be obtained through a direct application of our techniques.
_ `*$ H
_
N=H C
lGk$
,/_`W$T
_g5
S_ `*$V
_B% k] k
*L
, _ `*$
_ 7
_
S _ 4_ `*$ <_
_
! " #
h
k
r!
2(3( i
i
k %$'& (r> ! ) Ok )k * Ok $'& + (r> ! k k (4)
The -norm Perceptron algorithm [10, 9] is a linear threshold algorithm which keeps in the
-th trial a weight vector
. On instance
, the
algorithm predicts by
sign
,
where
and . If the algorithm?s prediction is wrong (i.e., if
) then the algorithm
performs the weight update
. Notice that
yields the classical
Perceptron algorithm [17]. On the other hand,
gets an algorithm which
performs like a multiplicative algorithm, such as the Normalized Winnow algorithm [10].
Applying Theorem 3 to the bound on the number of mistakes for the -norm Perceptron
algorithm shown in [9], we immediately obtain that, with probability at least
with
respect to the draw of the training sample , the risk
of the penalized estimator
is at most
^
*
$ &
) Y
J
and for any ( such that ( `*$ lRk . The margin-based quantity
!
in [20] and accounts for
_ e $
J\)k <_(
_ ) is called soft margin
the distribution of margin values achieved by the examples in Q ! with respect to hyperplane
( . Traditional data-dependent bounds using uniform convergence methods (e.g., [19]) are
typically expressed in terms of the sample margin (^aC5 _ (
_ l $ ) & & , i.e., in terms of
the fraction of training points whose margin is at most ) . The ratio
+ (" Q ! occurring
(r Q !tL
for any
*
-
in (4) has a similar flavor, though the two ratios are, in general, incomparable.
$ &
We remark that bound (4) does not have the extra log factors appearing in the analyses
based on uniform convergence. Furthermore, it is significantly better than the bound in [20]
whenever
is constant, which typically occurs when the data sequence is not linearly
separable.
H
As a second application, we consider the ridge regression algorithm [12] for square loss.
Assume
and
. This algorithm computes at the beginning of the
-th trial the vector
which minimizes *
, where
, where is
. On instance the algorithm predicts with
the ?clipping? function
if
,
if
and
are thus bounded by . We can apply
if
. The losses
for ridge regression
(see [22, 2]) and
Theorem 2 to the bound on the cumulative
loss
obtain that, with probability at least
with respect to the draw of the training sample
, the risk
of the average hypothesis estimator is at most
8 p 0 A
_ `*$
NJ
_
/, _ `*$<
r
L%$
r
L
Xl
Nl
_ , _ *` $
_
h
k
!
2)3 i
i
k ( hs+(r+ ! d ! _ _
_fe.$
!_ e $ $ _ (
for any ( H , where hs(r+ !c
^
_ `*e.$ $ $
_gp E_`W$
_g
Nl
L
,
#
#
k
(5)
_ , ^ Gk<(')')'* . Then simple
!
j
T
_
_fe.$ tk _ /
#
_
f
_
.
e
$
! ! #
where the _ ?s are the eigenvalues of
. The nonzero
eigenvalues of
!
!
!
! are the
. Risk bounds in terms of
same as the nonzero eigenvalues of the Gram matrix
! ! we defer to the full paper a
the eigenvalues of the Gram matrix were also derived in [23];
_ , denotes the determinant
-dimensional identity matrix and
is the transpose of .2 Let us
of matrix
is the
denote by
the matrix whose columns are the data vectors
linear algebra shows that
!
*
comparison between these results and ours. Finally, our bound applies also to kernel ridge
regression [18] by replacing the eigenvalues of
with the eigenvalues of the kernel
,
Gram matrix
, where is the kernel being considered.
_
k l ^+ l
! !
References
[1] Angluin, D. Queries and concept learning, Machine Learning, 2(4), 319-342, 1988.
[2] Azoury, K., and Warmuth, M. K. Relative loss bounds for on-line density estimation
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[3] K. Azuma. Weighted sum of certain dependend random variables. Tohoku Mathematical Journal, 68, 357?367, 1967.
2
Using a slightly different linear regression algorithm, Forster and Warmuth [7] have proven a
sharper bound on the expected relative loss. In particular, they have exhibited an algorithm computing
,#"
,
")(+*-,/.102354768 9
!
;:=<
hypothesis
such
>@?BA "#C
AF
EHGHI that in expectation (over ) the relative risk $&%'
is bounded by D
.
[4] A. Blum, A. Kalai, and J. Langford. Beating the hold-out: bounds for k-fold and
progressive cross-validation. In 12th COLT, pages 203?208, 1999.
[5] S. Boucheron, G. Lugosi, and P. Massart. A sharp concentration inequality with
applications. Random Structures and Algorithms, 16, 277?292, 2000.
[6] N. Cesa-Bianchi, Y. Freund, D. Haussler, D. P. Helmbold, R. E. Schapire, and M. K.
Warmuth. How to use expert advice. Journal of the ACM, 44(3), 427?485, 1997.
[7] J. Forster, and M. K. Warmuth. Relative expected instantaneous loss bounds. 13th
COLT, 90?99, 2000.
[8] Y. Freund and R. Schapire. Large margin classification using the perceptron algorithm. Machine Learning, 37(3), 277?296, 1999.
[9] C. Gentile The robustness of the -norm algorithms. Manuscript, 2001. An extended
abstract (co-authored with N. Littlestone) appeared in 12th COLT, 1?11, 1999.
[10] A. J. Grove, N. Littlestone, and D. Schuurmans. General convergence results for
linear discriminant updates, Machine Learning, 43(3), 173?210, 2001.
[11] D. Helmbold and M. K. Warmuth. On weak learning. JCSS, 50(3), 551?573, June
1995.
[12] A. Hoerl, and R. Kennard, Ridge regression: biased estimation for nonorthogonal
problems. Technometrics, 12, 55?67, 1970.
[13] J. Kivinen and M. K. Warmuth. Additive versus exponentiated gradient updates for
linear prediction. Information and Computation, 132(1), 1?64, 1997.
[14] N. Littlestone. Learning quickly when irrelevant attributes abound: A new linearthreshold algorithm. Machine Learning, 2, 285?318, 1988.
[15] N. Littlestone. From on-line to batch learning. In 2nd COLT, 269?284, 1989.
[16] N. Littlestone and M. K. Warmuth. The weighted majority algorithm. Information
and Computation, 108(2), 212?261, 1994.
[17] F. Rosenblatt. Principles of neurodynamics: Perceptrons and the theory of brain
mechanisms. Spartan Books, Washington, D.C., 1962.
[18] C. Saunders, A. Gammerman, and V. Vovk. Ridge Regression Learning Algorithm in
Dual Variables, In 15th ICML, 1998.
[19] J. Shawe-Taylor, P. Bartlett, R. Williamson, and M. Anthony, Structural Risk Minimization over Data-dependent Hierarchies. IEEE Trans. IT, 44, 1926?1940, 1998.
[20] J. Shawe-Taylor and N. Cristianini, On the generalization of soft margin algorithms,
2000. NeuroCOLT2 Tech. Rep. 2000-082, http://www.neurocolt.org.
[21] V.N. Vapnik, Statistical learning theory. J. Wiley and Sons, NY, 1998.
[22] V. Vovk, Competitive on-line linear regression. In NIPS*10, 1998. Also: Tech. Rep.
Department of Computer Science, Royal Holloway, University of London, CSD-TR97-13, 1997.
[23] R. C. Williamson, J. Shawe-Taylor, B. Sch?olkopf and A. J. Smola, Sample Based
Generalization Bounds, IEEE Trans. IT, to appear.
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1,223 | 2,114 | Analog Soft-Pattern-Matching Classifier
using Floating-Gate MOS Technology
Toshihiko YAMASAKI and Tadashi SHIBATA*
Department of Electronic Engineering, School of Engineering
*Department of Frontier Informatics, School of Frontier Science
The University of Tokyo
7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan
[email protected], [email protected]
Abstract
A flexible pattern-matching analog classifier is presented in conjunction with a robust image representation algorithm called Principal Axes Projection (PAP). In the circuit, the functional form of
matching is configurable in terms of the peak position, the peak height
and the sharpness of the similarity evaluation. The test chip was fabricated in a 0.6-?m CMOS technology and successfully applied to
hand-written pattern recognition and medical radiograph analysis using
PAP as a feature extraction pre-processing step for robust image coding.
The separation and classification of overlapping patterns is also experimentally demonstrated.
1
I ntr o du c ti o n
Pattern classification using template matching techniques is a powerful tool in implementing human-like intelligent systems. However, the processing is computationally very expensive, consuming a lot of CPU time when implemented as software running on general-purpose computers. Therefore, software approaches are not
practical for real-time applications. For systems working in mobile environment, in
particular, they are not realistic because the memory and computational resources
are severely limited. The development of analog VLSI chips having a fully parallel
template matching architecture [1,2] would be a promising solution in such applications because they offer an opportunity of low-power operation as well as very
compact implementation.
In order to build a real human-like intelligent system, however, not only the pattern
representation algorithm but also the matching hardware itself needs to be made
flexible and robust in carrying out the pattern matching task. First of all,
two-dimensional patterns need to be represented by feature vectors having substantially reduced dimensions, while at the same time preserving the human perception
of similarity among patterns in the vector space mapping. For this purpose, an image representation algorithm called Principal Axes Projection (PAP) has been de-
veloped [3] and its robust nature in pattern recognition has been demonstrated in the
applications to medical radiograph analysis [3] and hand-written digits recognition
[4]. However, the demonstration so far was only carried out by computer simulation.
Regarding the matching hardware, high-flexibility analog template matching circuits
have been developed for PAP vector representation. The circuits are flexible in a
sense that the matching criteria (the weight to elements, the strictness in matching)
are configurable. In Ref. [5], the fundamental characteristics of the building block
circuits were presented, and their application to simple hand-written digits was presented in Ref. [6]. The purpose of this paper is to demonstrate the robust nature of
the hardware matching system by experiments. The classification of simple
hand-written patterns and the cephalometric landmark identification in gray-scale
medical radiographs have been carried out and successful results are presented. In
addition, multiple overlapping patterns can be separated without utilizing a priori
knowledge, which is one of the most difficult problems at present in artificial intelligence.
2
I ma g e re pr es e n tati on by P AP
PAP is a feature extraction technique using the edge information. The input image
(64x64 pixels) is first subjected to pixel-by-pixel spatial filtering operations to detect edges in four directions: horizontal (HR); vertical (VR); +45 degrees (+45); and
?45 degrees (-45). Each detected edge is represented by a binary flag and four edge
maps are generated. The two-dimensional bit array in an edge map is reduced to a
one-dimensional array of numerals by projection. The horizontal edge flags are accumulated in the horizontal direction and projected onto vertical axis. The vertical,
+45-degree and ?45-degree edge flags are similarly projected onto horizontal,
-45-degree and +45-degree axes, respectively. Therefore the method is called ?Principal Axes Projection (PAP)? [3,4]. Then each projection data set is series connected
in the order of HR, +45, VR, -45 to form a feature vector. Neighboring four elements are averaged and merged to one element and a 64-dimensional vector is finally obtained. This vector representation very well preserves the human perception
of similarity in the vector space. In the experiments below, we have further reduced
the feature vector to 16 dimensions by merging each set of four neighboring elements into one, without any significant degradation in performance.
3
C i r cui t c o nf i g ura ti ons
A B C VGG
A B C VGG
IOUT
IOUT
1
1
2
2
4
4
1
VIN
13
VIN
RST
RST
Figure 1: Schematic of vector element matching circuit: (a) pyramid (gain reduction) type; (b) plateau (feedback) type. The capacitor area ratio is indicated
in the figure.
The basic functional form of the similarity evaluation is generated by the shortcut
current flowing in a CMOS inverter as in Refs. [7,8,9]. However, their circuits were
utilized to form radial basis functions and only the peak position was programmable.
In our circuits, not only the peak position but also the peak height and the sharpness
of the peak response shape are made configurable to realize flexible matching operations [5].
Two types of the element matching circuit are shown in Fig. 1. They evaluate the
similarity between two vector elements. The result of the evaluation is given as an
output current (IOUT ) from the pMOS current mirror. The peak position is temporarily memorized by auto-zeroing of the CMOS inverter. The common-gate transistor
with VGG stabilizes the voltage supply to the inverter. By controlling the gate bias
VGG, the peak height can be changed. This corresponds to multiplying a weight factor to the element. The sharpness of the functional form is taken as the strictness of
the similarity evaluation. In the pyramid type circuit (Fig. 1(a)), the sharpness is
controlled by the gain reduction in the input. In the plateau type (Fig. 1(b)), the
output voltage of the inverter is fed back to input nodes and the sharpness changes
in accordance with the amount of the feedback.
!"!
#$%#&"# #
#
#
'( ')"'
'
'
'
16-dimension
15-vector
matching circuit
Decoder
4.5mm
Time-domain Winner-Take-All
Figure 2: Schematic of n-dimensional vector matching circuit utilizing the
pyramid type vector element circuits.
Figure 3: Photomicrograph of
soft-pattern-matching classi fier
circuit.
The total matching score between input and template vectors is obtained by taking
the wired sum of all I OUT ?s from the element matching circuits as shown in Fig. 2. A
multiplier circuit as utilized in Ref. [8] was eliminated because the radial basis function is not suitable for the template matching using PAP vectors. I SUM, the sum of
IOUT ?s, is then sunk through the nMOS with the VRAMP input. This forms a current
comparator circuit, which compares I SUM and the sink current in the nMOS with
VRAMP . The VOUT nodes are connected to a time-domain Winner-Take-All circuit [9].
A common ramp down voltage is applied to the VRAMP nodes of all vector matching
circuits. When V RAMP is ramped down from V DD to 0V, the vector matching circuit
yielding the maximum ISUM firstly upsets and its output voltage (VOUT ) shows a
0-to-1 transition. The time-domain WTA circuit senses the first upsetting signal and
memorizes the location in the open-loop OR-tree architecture [10]. In this manner,
the maximum-likelihood template vector is easily identified.
The circuits were designed and fabricated in a 0.6-?m double-poly triple-metal
CMOS technology. Fig. 3 shows the photomicrograph of a pattern classifier circuit
for 16-dimensional vectors. It contains 15 vector matching circuits. One element
matching circuit occupies the area of 150?m x 110?m. In the latest design, however,
the area is reduced to 54 ?m x 68 ?m in the same technology by layout optimization.
Further area reduction is anticipated by employing high-K dielectric films for capacitors since the capacitors occupy a large area. The full functioning of the chip
was experimentally confirmed [6]. In the following experiments, the simple vector
matching circuit in Fig. 2 was utilized to investigate the response from each template vector instead of just detecting the winner using the full chip.
E x per i me n tal r e s u l ts a nd di s c us si o n
Vector-element matching circuit
?
& & &'1- ( . 0
((
!
!
!
$#%
"
"
?
) ) )/- *,+.+ 0
+
"
?
4.1
!
!
?
4
!
"
Figure 4: Measured characteristics: (a) pyramid type; (b) plateau type. V GG was
varied from 3.0V to 4.5V, and control signals A~C from 000 to 111 for sharpness control.
Fig. 4 shows the measured characteristics of vector-element matching circuits in
both linear and log plots. The peak position was set at 1.05V by auto-zeroing. The
peak height was altered by V GG. Also, the operation mode was altered from the
above-threshold region to the sub-threshold region by V GG. In the plateau type circuit (Fig. 4(b)), I OUT becomes constant around the peak position and the flat region
widens in proportion to the amount of feedback. This is because the inverter operates so as to keep the floating gate potential constant in the high-gain region of the
inverter as in the case of virtual ground of an operational amplifier.
4.2
Matching of simple hand-w ritten patterns
Fig. 5 demonstrates the matching results for the simple input patterns. 16 templates
were stored in the matching circuit and several hand-written pattern vectors were
presented to the circuit as inputs. A slight difference in the matching score is observed between the pyramid type and the plateau type, but the answers are correct
for both types. Fig. 6 shows the effect of sharpness variation. As the sharpness gets
steeper, all the scores decrease. However, the score ratios between the winner and
loosers are increased, thus enhancing the winner discrimination margins. The
matching results with varying operational regimes of the circuit are given in Fig. 7.
The circuit functions properly even in the sub-threshold regime, demonstrating the
opportunity of extremely low power operation.
Presented
Patterns
Template Patterns
Template Patterns
Best
Matched
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Template #
Best
Matched
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Template #
Figure 5: Result of simple pattern matching: (a) pyramid type (left) where gain
reduction level was set with ABC=010; (b) plateau type (right) where feedback
ratio was set with ABC=101.
&%$
?
# "
!
& $ $
% $
$
,.-0/102434:<; 50=?64>@ 7A B , ; 8 C, , , -9, /9, 19, 29, 3
&
$ ' +
$ ' *
$ ' )
$ ' (
$ ,D-0/14203454647E, 8, , , -9, /, 19, 29, 3
:; = >@ A B ; C
Figure 6: Effect of sharpness variation in the pyramid type with ABC=010.
Template #4
VGG=4.0V
VGG=3.5V
VGG=3.0V
VGG=2.5V
Input Pattern
Best Matched
1 2 3 4 5 6 7 8 9 10111213141516
Template#
Figure 7: Matching results as a function of V GG. Correct results are obtained in
the sub-threshold regime as well as in the above-threshold regime (the pyramid
type was utilized).
4.3
Application to gray-scale medical radiograph analysis
In Fig. 8, are presented the result of cephalometric landmark identification experiments, where the Sella (pituitary gland) pattern search was carried out using the
same matching circuit. Since the 64-dimension PAP representation is essential for
grayscale image recognition, the 64-dimension vector was divided into four
16-dimension vectors and the matching scores were measured separately and then
summed up by off-chip calculation. The correct position was successfully identified
both in the above-threshold (Fig. 8(b)) and the sub-threshold (Fig. 8(c)) regimes
using the 14 learned vectors as templates. In the previous work [3], successful
search was demonstrated by the computer simulation.
!
?"$#
!
?"$#
%'&)(+*-,/.* 01-2 3-4658792 3*+:
%'&)(+*-,/.* 01-2 3-4658792 3*+:
; <8=>@?BAC9DFE D
?G
; <8=>@?HDDFE I
?G
Figure 8: Matching results of Sella search using pyramid type with ABC=000:
(a) input image; (b) above-threshold regime; (c) sub-threshold regime.
4.4
Separation of overlapping patterns
Suppose an unknown pattern is presented to the matching circuit. The pattern might
consist of a single or multiple overlapping patterns. Let X represent the input vector
and W1st the winner (best matched) vector obtained by the matching circuit. Let the
first matching trial be expressed as follows:
? W1st
1st trial: X ????
matching
Then, the residue vector (X-W1st) is generated. The subtraction is perfomed in the
vector space. When an element in the residue vector becomes negative, the value is
set to 0. Such operation is easily implemented using the floating gate technique.
Here, the residue was obtained by off-line calculation. If the input pattern is single,
the residue vector is meaningless: only the leftover edge information remains in the
residue vector. If the input consists of overlapping patterns, the edge information of
other patterns remains. If the residue vector is very small, we can expect that the
input is single. But in many cases, the residue vector is not so small due to the distortion in hand-written patterns. Thus, it is almost impossible to judge which is the
case only from the magnitude of the residue vector. Therefore, we proceed to the
second trial to find the second winner:
2nd trial: X ? W1st
matching
????
? W2nd
With the same sequence, the second residue vector (X-W1st-W2nd), the third
(X-W1st -W2nd-W3rd) and so forth are generated by repeating the winner subtraction
after each trial. Then, new template vectors are generated such as W1st +W2nd,
W1st +W2nd+W3rd, and so forth. If the input vector is that of a single pattern, the
matching score is the highest at W1st and the scores are lower at W1st +W2nd and
W1st +W2nd+W3rd. On the other hand, if the input vector is that of two overlapping
patterns, the score is the highest at W1st+W2nd. This procedure can be terminated
automatically when the new template composed of n overlapping patterns yields
lower score than that of n-1 overlapping patterns. In this manner, we are able to
know how many patterns are overlapping and what patterns are overlapping without
a priori knowledge. An example of separating multiple overlapping patterns is illustrated in Fig. 9.
Template Patterns
Presented Patterns
1st try:
2nd try:
3rd try:
Final try:
Best Matched
?
?
?
?
+
+
+
+
Figure 9: Experimental result illustrating the algorithm for separating overlapping patterns. The solid black bars indicate the winner locations.
Template Patterns
Presented
Patterns
Best
Matched
A
#1
B C
#2
+
D E F
#3
+
+
+
(a)
A
+
B
+
C
+
+
(b)
D
+
E
+
+
F
+
+
+
Figure 10: Measured results demonstrating separation of multiple overlapping
patterns: (a) result of separation and classification (A~F are depicted in (b));
(b) newly created templates such as W1st+W2nd, W1st +W2nd+W3rd, and so on.
Several other examples are shown in Fig. 10. Pattern #1 is correctly classified as a
single rectangle by yielding the higher score for single template than that for
W1st +W2nd. Pattern #3 consists of three overlapping patterns, but is erroneously
recognized as four overlapping patterns. However, the result is not against human
perception. When we look at pattern #3, a triangle is visible in the pattern. This
mistake is quite similar to that made by humans.
5
C on cl us i o ns
A soft-pattern matching circuit has been demonstrated in conjunction with a robust
image representation algorithm called PAP. The circuit has been successfully applied to hand-written pattern recognition and medical radiograph analysis. The recognition of overlapping patterns similar to human perception has been also experimentally demonstrated.
Acknowledgments
Test circuits were fabricated in the VDEC program (The Univ. of Tokyo), in collaboration with Rohm Corp. and Toppan Printing Corp. The work is partially supported by the Ministry of Education, Science, Sports and Culture under the
Grant-in-Aid for Scientific Research (No. 11305024) and by JST in the program of
CREST.
References
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[6] T. Yamasaki, K. Yamamoto and T. Shibata. (2001) Analog Pattern Classifier with Flexible Matching Circuitry Based on Principal-Axis-Projection Vector Representation. In Proc.
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[8] L. Theogarajan and L. A. Akers. (1996) A Multi-Dimentional Analog Gaussian Radial
Basis Circuit. In Proc. IEEE Int. Symp. Circuits Syst. (ISCAS ?96), Vol. 3, pp. III-543 -546
Atlanta, GA, USA, May, 1996.
[9] L. Theogarajan and L. A. Akers. (1997) A scalable low voltage analog Gaussian radial
basis circuit. IEEE Trans. on Circuits and Systems II, Volume 44, No. 11, pp. 977 ?979,
1997.
[10] K. Ito, M. Ogawa and T. Shibata. (2001) A High-Performance Time-Domain Winner-Take-All Circuit Employing OR-Tree Architecture. In Proc. 2001 Int. Conf. on Solid
State Devices and Materials (SSDM2001), pp. 94-95, Tokyo, Japan, Sep. 26-28, 2001.
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1,224 | 2,115 | 3 state neurons for contextual processing
Adam Kepecs* and Sridhar Raghavachari
Volen Center for Complex Systems
Brandeis University
Waltham MA 02454
{kepecs,sraghava}@brandeis.edu
Abstract
Neurons receive excitatory inputs via both fast AMPA and slow
NMDA type receptors. We find that neurons receiving input via
NMDA receptors can have two stable membrane states which are
input dependent. Action potentials can only be initiated from the
higher voltage state. Similar observations have been made in several brain areas which might be explained by our model. The interactions between the two kinds of inputs lead us to suggest that
some neurons may operate in 3 states: disabled, enabled and firing. Such enabled, but non-firing modes can be used to introduce
context-dependent processing in neural networks. We provide a
simple example and discuss possible implications for neuronal processing and response variability.
1
Introduction
Excitatory interactions between neurons are mediated by two classes of synapses:
AMPA and NMDA. AMPA synapses act on a fast time scale (TAMPA'" 5ms) , and
their role in shaping network dynamics has been extensively studied. The NMDA
type receptors are slow ((TNMDA '" 150ms) and have been mostly investigated for
their critical role in the induction of long term potentiation, which is thought to be
the mechanism for storing long term memories. Crucial to this is the unique voltage
dependence of NMDA receptors [6] that requires both the presynaptic neuron to
be active and the post-synaptic neuron to be depolarized for the channel to open.
However, pharamacological studies which block the NMDA receptors impair a variety of brain processes, suggesting that NMDA receptors also playa role in shaping
the dynamic activity of neural networks [10, 3, 8, 11, 2].
Therefore, we wanted to examine the role of NMDA receptors in post-synaptic
integration. Harsch and Robinson [4] have observed that injection of NMDA conductance that simulates synchronous synaptic input regularized firing while lowering response reliability. Our initial observations using a minimal model with
'The authors contributed equally to this work.
large NMDA inputs in a leaky dendrite showed a large regenerative depolarization. Neurons however, also possess a variety of potassium currents that are able
to limit these large excursions in voltage. In particular, recent observations show
that A-type potassium currents are abundant in dendrites of a variety of neurons
[7] . Combining these potassium currents with random NMDA inputs showed that
the membrane voltage alternated between two distinct subthreshold states. Similar
observations of two-state fluctuations have been made in vivo in several cortical
areas and the striatum [17, 9, 1]. The origin and possible functional relevance of
these fluctuations have remained a puzzle. We suggest that the NMDA type inputs
combined with potassium currents are sufficient to produce such membrane dynamics. Our results lead us to suggest that the fluctuations could be used to represent
contextual modulation of neuronal firing.
2
2 .1
NMDA-type input causes 2 state membrane fluctuations
Model
To examine the role of NMDA type inputs, we built a simple model of a cortical
neuron receiving AMPA and NMDA type inputs. To capture the spatial extent
of neuronal morphology we use a two-compartment model of pyramidal neurons
[15]. We represent the soma, proximal dendrites and the axon lumped into one
compartment containing the channels necessary for spike generation (INa and IK).
The dendritic compartment includes two potassium currents, a fast activating IKA
and the slower IKS along with a persistent sodium current INaP. The dendrite also
receives synaptic input as INMDA and IAMPA .
The membrane voltage of the neuron obeys the current balance equations:
while the dendritic voltage,
"\lid
obeys:
em
is the specific membrane capacitance which is taken to be 1 I1F / cm 2 for
where
both the dendrite and the soma for all cells and p =0.2, gc =0.05 determining the
electrotonic structure of the neuron.
The passive leak current in both the soma and dendrites were modeled as h eak =
El eak ), where gl eak was the leak conductance which was taken to be
0.3 mS/cm 2 for the soma and dendrite. El eak = -80mV was the leak reversal
potential for both the compartments. The voltage-dependent currents were modeled
according to the Hodgkin-Huxley formalism, with the gating variables obeying the
equation:
gl eak(V -
dx
(x oo(V) - x)
dt = ?x(ax(V)(1 - x) - ,sx(V)x) = ?x
Tx(V)
,
(3)
where x represents the activation/inactivation gates for the voltage-dependent currents.
The sodium current, INa = gNam~ h(VS - E Na ), where gNa = 45 mS/cm 2 and
sodium reversal potential, ENa = 55 mV with m oo(V) = a=(~)~~~(V). The
activation variables, O::m(V) = -O.l(V + 32)/[exp( -(V + 32)/10) - 1], 'sm(V) =
4exp( -[V + 57]/18); O::h(V) = 0.07 exp( -[V + 48]/20) and 'sh (V) = l/[exp( -{V +
18}/10) + 1], with ?m = ?h = 2.5.
The delayed rectifier potassium current, IKDr = gKn4(VS - EK), where gK = 9
mS/cm 2 and potassium reversal potential, EK = -80 mV with O::n(V) = -O.Ol(V +
34)/[exp( -(V + 34)/10) - 1], 'sn(V) = 0.125 exp( -[V + 44]/80), with ?n = 2.5.
In the dendrite, the persistent sodium current, INaP = gNapr~(V)(V - VNa ), with
roo(V) = 1/(1 + exp( -(V + 57)/5)) and gNaP =0.25 mS/cm 2 ? The two potassium
currents were hs = gKsq(V - VK), with qoo (V) = 1/(1 + exp( -(V + 50)/2)) and
Tq(V) = 200/(exp( -(V + 60)/10) + exp((V + 60)/10)) and gKS = 0.1 mS/cm 2 ; and
hA = gKAa~ (V)b(V - VK), with aoo(V) = 1/(1 + exp(-(V + 45)/6)), boo (V) =
1/(1 + exp(-(V + 56)/15)) and Tb(V) = 2.5(1 + exp((V + 60)/30)) and gKA = 10
mS/cm 2 .
The NMDA current, INMDA = fgNMDAS(V - ENMDA)/(l + 0.3[Mg] exp( -0.08V)),
where S was the activation variable and f denoted the inactivation of NMDA channels due to calcium entry. AMPA and NMDA inputs were modeled as conductance
kicks that decayed with TAMPA = 5 ms and TNMDA = 150 ms. Calcium dependent inactivation of the NMDA conductance was modeled as a negative feedback
df /dt = (foo - f)/2 , where f oo was a shallow sigmoid function that was 1 below a
conductance threshold of 2 ms/cm 2 and was inversely proportional to the NMDA
conductance above threshold. The coupling conductance is gc =0.1 mS/cm 2. The
asymmetry between the areas of the two compartments is taken into account in the
parameter p = somatic area/total area = 0.2. The temperature scaling factors
are ?h = ?n = 3.33. Other parameter values are: gLeak =0.3, gNa =36, gK =6 ,
gNaP =0.15, gKS =1, gKA =50 in mS/cm 2 unless otherwise noted; ELeak = -75,
ENa = +55, EK = -90, EKA = -80 in m V. Synchronous inputs were modeled as
a compound Poisson process representing 100 inputs firing at a rate A each spiking
with a probability of 0.1. Numerical integration was performed with a fourth-order
Runge-Kutta method using a 0.01 ms time step.
2.2
NMDA induced two-state fluctuations
Figure 1A shows the firing produced by inputs with high AMPA/NMDA ratio.
Figure 1B shows that the same spike train input delivered via synapses with a high
NMDA content results in robust two-state membrane behavior. We term the lower
and higher voltage states as UP and DOWN states respectively. Spikes caused by
AMPA-type inputs only occur during the up-state. In general, the same AMPA
input can only elicit spikes in the postsynaptic neuron when the NMDA input
switches that neuron into the up-state.
Transitions from down to up-state occur when synchronous NMDA inputs depolarize the membrane enough to cause the opening of additional NMDA receptor
channels (due to the voltage-dependence of their opening). This results in a regeneretive depolarization event, which is limited by the fast opening of IKA-type
Time [s]
Figure 1: Inputs with high AMPA-NMDA ratio cause the cell to spike (top trace,
=0.05, g N MDA =0.01). Strong NMDA inputs combined with potassium currents
(for the same AMPA input) result in fluctuations of the membrane potential between
two subthreshold states, with occasional firing due to the AMPA inputs (bottom trace,
gAMPA =0.01, gNMDA =0.1)
gA M PA
potassium channels. This up-state is stable because the regenerative nature and
long lifetime of NMDA receptor opening keeps the membrane depolarized, while
the slower I Ks potassium current prevents further depolarization. When input
ceases, NMDA channels eventually (TNMDA ~ 150ms) close and the membrane
jumps to the down-state. Note that while this bistable mechanism is intrinsic to
the membrane, it is also conditional upon input. Since the voltage threshold for
spike generation in the somal axon compartment is above the up-state, it acts as a
barrier. Thus, synchronous AMPA input in the down-state has a low probability of
eliciting a spike.
A number of previous experimental studies have reported similar phenomena in
various brain regions [16, 9, 1] where the two states persist even with all intrinsic
inward currents blocked but the inputs left intact [17] . Pharmacological block of
the potassium currents resulted in prolonged up-states [17] . These experimental
results suggested a conceptual model in which two-state fluctuations are (i) input
driven, (ii) the membrane states are stabilized by potassium currents. Nevertheless,
there remained a puzzle that (iii) up-state transitions are abrupt and (iv) the the
up-state is prolonged and restricted to a relatively narrow range of voltages. Our
model suggests a plausible mechanism for this phenomenon consistent with all experimental constraints. Below, we examine the origins of the two-state fluctuations
in light of these findings.
2.3
Analysis of two state fluctuations
Figure 2A shows the histogram of membrane potential for a neuron driven by combined AMPA and NMDA input at 30 Hz. There are two clear modes corresponding
to the up and down-states. The variability of the up-state and down-state voltages
is very low (u = 1.4 mV and 2.4 mV respectively) as observed. Figure 2B shows
the distribution of the up-state duration. The distribution of the up-state durations depend on the maximal NMDA conductance and the decay time constant
of NMDA (not shown), as well as the mean rate of NMDA inputs (Figure 2C).
A
B
C
O.
40
400
>-
~O.
30
U)
300
:c
E
'""
Q)
E
i=
CQ
.c
?0.
200
500
Time (ms)
100
1000
20 30 40 50
NMDA Rate (Hz)
Figure 2: A. Histogram of the up and down states. B. Dwell times of the up states C.
Mean duration of the up states increases with rate of NMDA inputs. Each histogram was
calculated over a run of 120 seconds.
Additionally, larger maximal potassium conductances shorten the duration of the
up states. Thus, we predict that the NMDA receptors are intimately involved in
shaping the firing characteristics of these neurons. Furthermore, our mechanistic
explanation leads a strong prediction about the functional role for these fluctuations
in neuronal processing.
3
Contextual processing with NMDA and AMPA pathways
Since NMDA and AMPA pathways have distinct roles in respectively switching and
firing our model neuron, we suggest the following conceptual model shown on Fig
3A. Without any input the neuron is at the rest or disabled state. Contextual input
(via NMDA receptors) can bring the neuron into an enabled state. Informational
(for instance, cue or positional) input (via AMPA receptors) can fire a neuron only
from this enabled state.
Where might such an architecture be used? In the CAl region of the hippocampus, pyramidal cells receive two distinct , spatially segregated input pathways: the
perforant path from cortex and the Schaffer collaterals from the CA3 region. The
perforant path has a very large NMDA receptor content [14] which is, interestingly, co-localized with high densities of I KA conductances [5]. Experimental [13]
and theoretical [12] observations suggest that these two pathways carry distinct
information. Lisman has suggested that the perforant path carries contextual information and the Schaffer collaterals bring sequence information [12]. Thus our model
seems to apply biophysically as well as suggest a possible way for CAl neurons to
carry out contextual computations. It is known that these cell can fire at specific
places in specific contexts. How might these different signals interact? As shown on
Fig3B , our model neuron can only fire spikes due to positional input when the right
context enables it. We note that a requirement for contextual processing is that the
two inputs be anatomically segregated, as they are in the CAl region. However,
we stress that the phenomenon of 2-state fluctuations itself is independent of the
location of the two kinds of inputs.
Figure 4A shows a similar processing scheme adapted for higher-order language
a. .;.~,. .,
A
Firing state
B'5
Context off
.~
<;;
~
~
0
Q.
""~ ~
0
~~'-)
?
~A/
'5
.~
Contextual input
Down-state / Disabled
Context on
<;;
~
~
0
Q.
g>
~~
Jll
Figure 3: A. Contextual input (high NMDA) switches the neuron from a rest state to
an up state. Informational input (high AMPA) cause the neuron to spike only from the
up state. B. Weak informational input can cause the cell to fire in conjunction with
the contextual input, (left traces) while strong informational input will not fire the cell
in the absence of contextual input (right traces). In this simulation, the soma/proximal
dendrite compartment receives AMPA input, while the NMDA input targets the dendritic
compartment.
processing. We simulated 3 neurons each receiving the same AMPA, informational
input. This might represent the word "green". Each of these neurons also receives
distinct contextual input via NMDA type receptors. These might, for instance,
represent specific noun groups: objects, people and fruit. The word "green" may
have very different meanings in these different contexts such as the color green, a
person who is a novice or an unripe fruit. We simulated this simple scenario shown
in Figure 4C. Even though each neuron receives the same strong AMPA input, their
firing seems uncorrelated. To evaluate the performance of the network in processing
contextual conjunctions, we measured the correlations between the information and
each contextual input. The most correlated at each moment was designated to be
the correct meaning. We then measured the number of spikes emitted by each
neuron during each "meaning" . Figure 4B shows that the neurons performed well ,
each tuned to fire preferentially in its appropriate context.
This simple example illustrates the use of a plausible biophysical mechanism for
computing conjuctions or multiplying with neurons.
4
Discussion
Voltage fluctuations between two subthresold levels with similar properties are observed in vivo in a variety of brain regions. Our model is in accordance with these
data and lead us to a new picture of how might these neuron operate in a functional
manner. Figure 3A shows our model operating as a 3-state device. It has a stable
low membrane state from which it cannot fire spikes, which we called disabled. It
also has a stable depolarized state from which action potentials can be elicited,
which this we call enabled state. Additionally, it has a firing state which is only
reachable from the enabled state.
What might be the role of the two non-firing states? We suggest that if high and
low NMDA-content pathways carry separate information these neurons can compute
A
"objects"
"people"
C
B
Contextual input:
90
0
"fruit"
111
0)(2)(3)
objects
people
frun
0
0
!6
o
J
~50
???
o 40
o
~
? 30
0
Sensory input:
0
"green"
o
2
4
0
1
2
3
I
1
2
3
1
Time(s)
Figure 4: A. Illustrative task for contextual processing in semantic inference. 3 neurons
each receive independent contextual (NMDA) and common informational (AMPA) input.
B. Voltage traces showing differences in firing patterns depending upon context. C. Each
neuron is tuned to its defined context. Correlation was measured between the informational
spike train and each contextual spike train smoothed with a gaussian filter (a = 60ms).
The most correlated context was defined to be the right one and the spikes of all neurons
were counted.
conjuctions, a simple form of multiplication. If the high NMDA-content pathway
carries contextual information then it would be in position to enable or disable a
neuron. In the enabled state, AMPA-type informational input could then fire a
neuron (Fig 3B).
We have presented a biophysical model for two-state fluctuations that is strongly
supported by data. One concern might be that most observations of 2-state fluctuations in vivo have been when the animal is anesthetized, implying that this kind
of neuronal dynamics is an artifact of the anesthetized state. However, these fluctuations have been observed in several different kinds of anesthesia, including local
anesthesia [16]. Furthermore, it has been shown that the duration of the up-states
correlate with orientation selectivity in visual cortical neurons suggesting that these
fluctuations might playa role in information processing. These observations suggest
that this phenomenon may be more indicative of a natural state of the cortex rather
than a by-product of anesthesia.
When the inputs with different AMPA/NMDA content are anatomically segregated,
t he NMDA input alone generates voltage fluctuations between a resting and depolarized state, while the AMPA input causes the neuron to spike when in the up-state.
This mechanism naturally leads to the suggestion that such two-state fluctuations
could have a function in computing context/input conjuctions. In summary, we
suggest the known biophysical mechanisms of some neurons can enable them two
operate as 3-state devices. In this mode of operation, the neurons could be used for
contextual processing.
Acknowledgments
We acknowledge John Lisman and John Fitzpatrick for useful discussion and suggestions.
2
3
References
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1,225 | 2,116 | Batch Value Function Approximation via
Support Vectors
Thomas G Dietterich
Department of Computet Science
Oregon State University
Corvallis, OR, 97331
[email protected]
Xin W"ang
Department of Computer Science
Oregon State University
Corvallis, OR, 97331
wangxi@cs. orst. edu
Abstract
We present three ways of combining linear programming with the
kernel trick to find value function approximations for reinforcement
learning. One formulation is based on SVM regression; the second
is based on the Bellman equation; and the third seeks only to ensure
that good moves have an advantage over bad moves. All formulations attempt to minimize the number of support vectors while
fitting the data. Experiments in a difficult, synthetic maze problem
show that all three formulations give excellent performance, but the
advantage formulation is much easier to train. Unlike policy gradient methods, the kernel methods described here can easily 'adjust
the complexity of the function approximator to fit the complexity
of the value function.
1
Introduction
Virtually all existing work on value function approximation and policy-gradient
methods starts with a parameterized formula for the value function or policy and
then seeks to find the best policy that can be represented in that parameterized form.
This can give rise to very difficult search problems for which the Bellman equation
is of little or no use. In this paper, we take a different approach: rather than fixing
the form of the function approximator and searching for a representable policy, we
instead identify a good policy and then search for a function approximator that
can represent it. Our approach exploits the ability of mathematical programming
to represent a variety of constraints including those that derive from supervised
learning, from advantage learning (Baird, 1993), and from the Bellman equation. By
combining the kernel trick with mathematical programming, we obtain a function
approximator that seeks to find the smallest number of support vectors sufficient to
represent the desired policy. This side-steps the difficult problem of searching for
a good policy among those policies representable by a fixed function approximator.
Our method applies to any episodic MDP, but it works best in domains-such as
resource-constrained scheduling and other combinatorial optimization problemsthat are discrete and deterministic.
2
Preliminaries
There are two distinct reasons for studying value function approximation methods.
The primary reason is to be able to generalize from some set of training experiences
to produce a policy that can be applied in new states that were not visited during
training. For example, in Tesauro's (1995) work on backgammon, even after training
on 200,000 games, the TD-gammon system needed to be able to generalize to new
board positions that it had not previously visited. Similarly, in Zhang's (1995) work
on space shuttle scheduling, each individual scheduling problem visits only a finite
number of states, but the goal is to learn from a series of "training" problems and
generalize to new states that arise in "test" problems. Similar MDPs have been
studied by Moll, Barto, Perkins & Sutton (1999).
The second reason to study function approximation is to support learning in continuous state spaces. Consider a robot with sensors that return continuous values.
Even during training, it is unlikely that the same vector of sensor readings will ever
be experienced more thaIl once. Hence, generalization is critical during the learning
process as well as after learning.
The methods described in this paper address only the first of these reason. Specifically, we study the problem of generalizing from a partial policy to construct a
complete policy for a Markov Decision Problem (MDP). Formally, consider a discrete time MDP M with probability transition function P(s/ls, a) (probability that
state Sl will result from executing action a in state s) and expected reward function
R(s/ls, a) (expected reward received from executing action a in state s and entering
state Sl). We will assume that, as in backgammon and space shuttle scheduling,
P(s/ls, a) and R(s/ls, a) are known and available to the agent, but that the state
space is so large that it prevents methods such as value iteration or policy iteration from being applied. Let L be a set of "training" states for which we have an
approximation V(s) to the optimal value function V*(s), s E L. In some cases, we
will also assume the availability of a policy 'ff consistent with V(s). The goal is to
construct a parameterized approximation yes; 8) that can be applied to all states
in M to yield a good policy if via one-step lookahead search. In the experiments
reported below, the set L contains states that lie along trajectories from a small set
of "training" starting states So to terminal states. A successful learning method will
be able to generalize to give a good policy for new starting states not in So. This
was the situation that arose in space shuttle scheduling, where the set L contained
states that were visited while solving "training" problems and the learned value
function was applied to solve "test" problems.
To represent states for function approximation, let X (s) denote a vector of features
describing the state s. Let K(X 1 ,X2 ) be a kernel function (generalized inner product) of the two feature vectors Xl and X 2 ? In our experiments, we have employed
the gaussian kernel: K(X1,X2;U) == exp(-IIX1 - X 2 11 2 / ( 2 ) with parameter u.
3
Three LP Formulations of Function Approximation
We now introduce three linear programming formulations of the function approximationproblem. We first express each of these formulations in terms of a generic
fitted function approximator V. Then, we implement V(s) as the dot product of
a weight vector W with the feature vector X (s): V(s) == W . X (s). Finally, we
apply the "kernel trick" by first rewriting W as a weighted sum of the training
points Sj E L, W == ~j ajX(sj), (aj 2: 0), and then replacing all dot products
between data points by invocations of the kernel function K. We assume L con-
tains all states along the best paths from So to terminal states and also all states
that can be reached from these paths in one step and that have been visited during
exploration (so that V is known). In all three formulations we have employed linear
objective functions, but quadratic objectives like those employed in standard support vector machines could be used instead. All slack variables in these formulations
are constrained- to be non-negative.
Formulation 1: Supervised Learning. The first formulation treats the value
function approximation problem as a supervised learning problem and applies the
standard c-insensitive loss function (Vapnik, 2000) to fit the function approximator.
minimize
L
[u(s)
+ v(s)]
S
subject to V(s)
+ u(s) 2:: V(s) - c; V(s) - v(s) :::; V(s) + c
"Is E L
In this formulation, u(s) and v(s) are slack variables that are non-zero only if V(s)
has an absolute deviation from V(s) of more than c. The objective function seeks to
minimize these absolute deviation errors. A key idea of support vector methods is
to combine this objective function with a penalty on the norm of the weight vector.
We can write this as
minimize
IIWlll + C L[u(s) + v(s)]
S
subject to W? X(s)
+ u(s) 2:: V(s) -
c;
W? X(s) - v(s) :::; V(s)
+c
"Is E L
The parameter C expresses the tradeoff between fitting the data (by driving the
slack variables to zero) and minimizing the norm of the weight vector. We have
chosen to minimize the I-norm of the weight vector (11Wlll == Ei IWi!), because this
is easy to implement via linear programming. Of course, if the squared Euclidean
norm of W is preferred, then quadratic programming methods could be applied to
minimize this.
Next, we introduce the assumption that W can be written as a weighted sum of the
data points themselves. Substituting this into the constraint equations, we obtain
minimize
L
aj
+ C L[u(s) + v(s)]
j
subject to
Ej
Ej
8
ajX(sj) . X(s) + u(s) ~ V(s) - c
ajX(sj) . X(s) - v(s) :::; V(s) + c
"Is E L
"Is E L
Finally, we can apply the kernel trick by replacing each dot product by a call to a
kernel function:
minimize
Laj
+ CL[u(s) + v(s)]
j
subject to
Ej
Ej
s
ajK(X(sj),X(s)) + u(s) 2:: V(s) - c
ajK(X(sj), X(s)) - v(s) :::; V(s) + c
"Is E L
"Is E L
Formulation 2: Bellman Learning. The second formulation introduces constraints from the Bellman equation V(s) == maxa ESI P(s'ls, a)[R(s'ls, a) + V(s')].
The standard approach to solving MDPs via linear programming is the following.
For each state s and action a,
minimize
L u(s; a)
s,a
subject to V(s) == u(s,a)
+ LP(s'ls,a)[R(s'ls,a) + V(s')]
s'
The idea is' that for the optimal action a* in state s, the slack variable u(s, a*)
can be driven to zero, while for non-optimal actions a_, the slack u(s, a_) will
remain non-zero. Hence, the minimization of the slack variables implements the
maximization operation of the Bellman equation.
We attempted to apply this formulation with function approximation, but the errors
introduced by the approximation make the linear program infeasible, because V(s )
must sometimes be less than the backed-up value Ls' P(s'ls, a)[R(s'ls, a) + V(s')].
This led us to the following formulation in which we exploit the approximate value
function 11 to provide "advice" to the LP optimizer about which constraints should
be tight and which ones should be loose. Consider a state s in L. We can group
the actions available in s into three groups: (a) the "optimal" action a* == 1f(s)
chosen by the approximate policy it, (b) other actions that are tied for optimum
(denoted by ao), and (c) actions that are sub-optimal (denoted by a_). We have
three different constraint equations, one for each type of action:
minimize
L[u(s, a*)
+ v(s, a*)] + LY(s, ao) + L z(s, a_)
s
subject to 17(s)
s,ao
+ u(s, a*) - v(s, a*)
s,a_
== L
P(s'ls, a*)[R(s'ls, a*)
+ V(s')]
s'
17(8) + y(s, ao) ~ L P(s'ls, ao)[R(s'ls, ao)
+ V(s')]
s'
17(s) + z(s, a_) ~ L P(s'ls, a_)[R(s'ls, a_)
+ V(s')] + ?
s'
The first constraint requires V(s) to be approximately equal to the backed-up value
of the chosen optimal action a*. The second constraint requires V(s) to be at least
as large as the backed-up value of any alternative optimal actions ao. If V(s) is
too small, it will be penalized, because the slack variable y(s, ao) will be non-zero.
But there is no penalty if V (s) is too large. The main effect of this constraint is
to drive the value of V(s') downward as necessary to satisfy the first constraint on
a*. Finally, the third constraint requires that V(s) be at least ? larger than the
backed-up value of all inferior actions a_. If these constraints can be satisfied with
all slack variables u, v, y, and z set to zero, then V satisfies the Bellman equation.
After applying the kernel trick and introducing the regularization objective, we
obtain the following Bellman formulation:
minimize
~ aj + C (,~_ u(s, a*) + v(s, a*) + y(s, ao) + z(s, a_))
subject to
~a.j [K(X(Sj),X(S)) J
LP(s'ls,a*)K(X(Sj),X(S'))] +
s'
u(s, a*) - v(s, a*) == L P(s'ls, a*)R(s'ls, a*)
~aj [K(X(Sj),X(S)) J
8'
LP(s'ls,ao)K(X(Sj),X(S'))] +y(s,ao)
~
~ LP(s'ls,ao)R(s'ls,ao)
s'
~O:j [K(X(Sj),X(S)) -
LP(S'IS,a_)K(X(Sj),X(S'))] +z(s,a_)
~
3
~ LP(s'ls,a_)R(s'ls,a_)
+?
8/
Formulation 3: Advantage Learning. The third formulation focuses on the
minimal constraints that must be satisfied to ensure that the greedy policy computed from V will be identical to the greedy policy computed from V (cf. Utgoff
& Saxena, 1987). Specifically, we require that the backed up value of the optimal
action a* be greater than the backed up values of all other actions a.
minimize
L
u(s,a*,a)
s,a*,a
subject to L P(s'ls, a*)[R(s'ls, a*)
+ V(s')] + u(s, a*, a)
8/
~ LP(s!ls,a)[R(s!ls,a)
+ V(s/)] +?
s/
There is one constraint and one slack variable u(s, a*, a) for every action executable
in state s except for the chosen optimal action a* = i"(s). The backed-up value of
a* must have an advantage of at least ? over any other action a, even other actions
that, according to V, are just as good as a*. After applying the kernel trick and
incorporating the complexity penalty, this becomes
minimize
Laj+C L
u(s,a*,a)
s,a*,a
j
subject to Laj L[P(s'ls,a*) -P(s'ls,a)]K(X(sj),X(s')) +u(s,a*,a) ~
j
s/
L P(s'ls, a)R(s'ls, a) - L P(s'ls, a*)R(s'ls, a*) + ?
s/
s/
Of course each of these formulations can easily be modified to incorporate a discount
factor for discounted cumulative reward.
4
Experimental Results
To compare these three formulations, we generated a set of 10 random maze problems as follows. In a 100 by 100 maze, the agent starts in a randomly-chosen square
in the left column, (0, y). Three actions are available in every state, east, northeast,
and southeast, which deterministically move the agent one square in the indicated
direction. The maze is filled with 3000 rewards (each of value -5) generated randomly from a mixture of a uniform distribution (with probability 0.20) and five 2-D
gaussians (each with probability 0.16) centered at (80,20), (80,60), (40,20), (40,80),
and (20,50) with variance 10 in each dimension. Multiple rewards generated for
a single state are accumulated. In addition, in column 99, terminal rewards are
generated according to a distribution that varies from -5 to +15 with minima at
(99,0), (99,40), and (99,80) and maxima at (99,20) and (99,60).
Figure 1 shows one of the generated mazes. These maze problems are surprisingly hard because unlike "traditional" mazes, they contain no walls. In traditional
n;tazes, the walls tend to guide the agent to the goal states by reducing what would
be a 2-D random walk to a random walk of lower dimension (e.g., 1-D along narrow
halls).
100
Rewards
90
-5
-10
80
-15
)(
-20
3IE
0
+
70
CZl
CZl
(1)
~
~
Vi
60
Vi
.s
50
Vi
40
bJJ
~
~
~
.?
(1)
~
30
20
10
10
20
30
40
50
60
70
80
90
100
Figure 1: Example randomly-generated maze. Agent enters at left edge and exits
at right edge.
We applied the three LP formulations in an incremental-batch method as shown
in Table 1. The LPs were solved using the CPLEX package from ILOG. The V
giving the best performance on the starting states in So over the 20 iterations
was saved and evaluated over all 100 possible starting states to obtain a measure of
generalization. The values of C and a were determined by evaluating generalization
on a holdout set of 3 start states: (0,30), (0,50), and (0,70). Experimentation showed
that C = 100,000 worked well for all three methods. We tuned 0- 2 separately for
each problem using values of 5, 10, 20, 40, 60, 80, 120, and 160; larger values
were preferred in case of ties, since they give better generalization. The results are
summarized in Figure 2.
The figure shows that the three methods give essentially identical performance, and
that after 3 examples, all three methods have a regret per start state of about
2 units, which is less than the cost of a single -5 penalty. However, the three
formulations differ in their ease of training and in the information they require.
Table 2 compares training performance in terms of (a) the CPU time required for
training, (b) the number of support vectors constructed, (c) the number of states in
which V prefers a tied-optimal action over the action chosen by n-, (d) the number
of states in which V prefers an inferior action, and (e) the number of iterations
performed after the best-performing iteration on the training set. A high score
on this last measure indicates that the learning algorithm is not converging well,
even though it may momentarily attain a good fit to the data. By virtually every
measure, the advantage formulation scores better. It requires much less CPU time
to train, finds substantially fewer support vectors, finds function approximators that
give better fit to .the data, and tends to converge better. In addition, the advantage?
Table 1:__ Incremental Batch Reinforcement Learning
Repeat 20 times:
For each start state So E 80 do
Generate 16 f-greedy trajectories using V
Record all transitions and rewards to build MDP model
Solve M via value "iteration to obtain V and 7r
if
L=0
For each start state 80 E 80 do
Generate trajectory according to -IT
Add to L all states visited along this trajectory
Apply LP method to L, V, and 7r to find new V
Perform Monte Carlo rollouts using greedy policy for V to evaluate each possible start state
Report total value of all start states.
Table 2: Measures of the quality of the training process (average over 10 MDPs)
180 1=
Sup
Bel
Adv
CPU
37.5
30.4
11.7
#SV
29.5
40.9
17.2
Sup
Bel
Adv
CPU
433.2
208.0
74.5
#SV
105.5
82.4
58.6
180 1=
#tie
70.5
62.0
46.7
1801 =
1
#tie
22.4
18.8
19.4
#bad
0.7
0.9
0.2
#iter
5.6
5.9
1.6
CPU
190.7
92.7
38.4
#SV
54.3
51.1
39.6
#bad
3.0
2.2
0.6
#iter
10.5
3.3
4.0
CPU
789.1
379.1
122.4
#SV
117.2
145.7
74.0
2
#tie
49.8
47.9
29.1
#bad
1.9
0.4
1.4
#iter
7.3
8.2
2.0
#bad
3.3
1.8
3.2
#iter
9.6
7.3
2.8
1801 =4
3
#tie
90.5
75.2
51.9
and Bellman formulations do not require the value of V, but only -fr. This makes
them suitable for learning to imitate a human-supplied policy.
5
Conclusions
This paper has presented three formulations. of batch value function approximation
by exploiting the power of linear programming to express a variety of constraints
and borrowing the kernel trick from support vector machines. All three formulations
were able to learn and generalize well on difficult synthetic maze problems. The
advantage formulation is easier and more reliable to train, probably because it places
fewer constraints on the value function approximation. Hence, we are now applying
the advantage formulation to combinatorial optimization problems in scheduling
and protein structure determination~
Acknowledgments
The authors gratefully acknowledge the support of AFOSR under contract F4962098-1-0375, and the NSF under grants IRl-9626584, I1S-0083292, 1TR-5710001197,
and E1A-9818414. We thank Valentina Zubek and Adam Ashenfelter for their
careful reading of the paper.
1200
~
u
1000
~
0
0.
~
.sa
800
0
B
'"d
Q)
a
a
600
0
~
......
Q.)
l-I
b1)
400
Q)
l-I
~
(5
~
t
200
0
0
2
3
4
5
Number of Starting States
Figure 2: Comparison of the total regret (optimal total reward - attained total
reward) summed over all 100 starting states for the three formulations as a function
of the number of start states in So. The three error bars represent the performance
of the supervised, Bellman, and advantage formulations (left-to-right). The bars
plot the 25th, 50th, and 75th percentiles computed over 10 randomly generated
mazes. Average optimal total reward on these problems is 1306. The random
policy receives a total reward of -14,475.
References
Baird, L. C. (1993). Advantage updating. Tech. rep. 93-1146, Wright-Patterson
AFB.
Moll, R., Barto, A. G., Perkins, T. J., & Sutton, R. S. (1999). Learning instanceindependent value functions to enhance local search. NIPS-II, 1017-1023.
Tesauro, G. (1995). Temporal difference learning and TD-Gammon. CACM, 28(3),
58-68.
Utgoff, P. E., & Saxena, S. (1987). Learning a preference predicate. In ICML-87,
115-121.
Vapnik, V. (2000). The Nature of Statistical Learning Theory, 2nd Ed. Springer.
Zhang, W., & Dietterich, T. G. (1995). A reinforcement learning approach to jobshop scheduling. In IJCAI95, 1114-1120.
| 2116 |@word norm:4 nd:1 seek:4 tr:1 series:1 contains:1 score:2 tuned:1 existing:1 written:1 must:3 plot:1 greedy:4 fewer:2 imitate:1 record:1 preference:1 zhang:2 five:1 mathematical:2 along:4 constructed:1 fitting:2 combine:1 introduce:2 expected:2 themselves:1 terminal:3 bellman:10 discounted:1 td:2 little:1 cpu:6 becomes:1 zubek:1 what:1 substantially:1 maxa:1 temporal:1 every:3 saxena:2 tie:5 unit:1 ly:1 grant:1 local:1 treat:1 tends:1 sutton:2 path:2 approximately:1 studied:1 ease:1 acknowledgment:1 regret:2 implement:3 episodic:1 attain:1 gammon:2 protein:1 scheduling:7 applying:3 deterministic:1 backed:7 starting:6 l:35 searching:2 programming:8 trick:7 updating:1 enters:1 solved:1 momentarily:1 adv:2 complexity:3 utgoff:2 reward:12 esi:1 solving:2 tight:1 patterson:1 exit:1 easily:2 represented:1 train:3 distinct:1 monte:1 cacm:1 larger:2 solve:2 ability:1 advantage:11 product:4 fr:1 combining:2 lookahead:1 exploiting:1 optimum:1 produce:1 incremental:2 executing:2 adam:1 derive:1 fixing:1 received:1 sa:1 c:2 differ:1 direction:1 saved:1 exploration:1 centered:1 human:1 require:3 ao:13 generalization:4 wall:2 preliminary:1 hall:1 wright:1 exp:1 substituting:1 driving:1 optimizer:1 smallest:1 combinatorial:2 visited:5 southeast:1 weighted:2 minimization:1 sensor:2 gaussian:1 modified:1 rather:1 arose:1 ej:4 shuttle:3 barto:2 focus:1 backgammon:2 indicates:1 tech:1 accumulated:1 unlikely:1 borrowing:1 among:1 denoted:2 constrained:2 summed:1 equal:1 once:1 construct:2 identical:2 icml:1 report:1 randomly:4 individual:1 rollouts:1 cplex:1 attempt:1 adjust:1 introduces:1 mixture:1 tgd:1 edge:2 partial:1 necessary:1 experience:1 filled:1 euclidean:1 walk:2 desired:1 minimal:1 fitted:1 column:2 maximization:1 cost:1 introducing:1 deviation:2 uniform:1 i1s:1 successful:1 northeast:1 predicate:1 too:2 reported:1 varies:1 sv:4 synthetic:2 ie:1 contract:1 enhance:1 squared:1 satisfied:2 return:1 summarized:1 availability:1 baird:2 oregon:2 satisfy:1 vi:3 performed:1 sup:2 reached:1 start:9 iwi:1 minimize:13 square:2 variance:1 yield:1 identify:1 yes:1 generalize:5 carlo:1 trajectory:4 drive:1 ed:1 con:1 holdout:1 attained:1 supervised:4 afb:1 formulation:31 evaluated:1 though:1 just:1 receives:1 replacing:2 ei:1 irl:1 quality:1 aj:4 indicated:1 mdp:4 dietterich:2 effect:1 contain:1 hence:3 regularization:1 entering:1 during:4 game:1 inferior:2 percentile:1 generalized:1 complete:1 executable:1 insensitive:1 corvallis:2 similarly:1 gratefully:1 had:1 dot:3 robot:1 add:1 showed:1 driven:1 tesauro:2 bjj:1 rep:1 approximators:1 minimum:1 greater:1 employed:3 converge:1 ii:1 multiple:1 determination:1 visit:1 converging:1 regression:1 essentially:1 iteration:6 kernel:12 represent:5 sometimes:1 laj:3 addition:2 separately:1 unlike:2 probably:1 subject:9 tend:1 virtually:2 call:1 easy:1 variety:2 fit:4 moll:2 jobshop:1 inner:1 idea:2 valentina:1 tradeoff:1 penalty:4 action:23 prefers:2 discount:1 ang:1 generate:2 sl:2 supplied:1 nsf:1 per:1 discrete:2 write:1 express:3 group:2 key:1 iter:4 rewriting:1 sum:2 package:1 parameterized:3 place:1 decision:1 quadratic:2 constraint:15 perkins:2 worked:1 x2:2 performing:1 department:2 according:3 representable:2 remain:1 lp:12 equation:8 resource:1 previously:1 describing:1 slack:9 loose:1 needed:1 studying:1 available:3 operation:1 gaussians:1 experimentation:1 apply:4 generic:1 batch:4 alternative:1 thomas:1 ensure:2 cf:1 exploit:2 giving:1 build:1 move:3 objective:5 primary:1 traditional:2 gradient:2 thank:1 reason:4 minimizing:1 difficult:4 negative:1 rise:1 policy:22 perform:1 ilog:1 markov:1 finite:1 acknowledge:1 situation:1 ever:1 introduced:1 required:1 bel:2 orst:2 learned:1 narrow:1 nip:1 ajx:3 address:1 able:4 bar:2 below:1 reading:2 thail:1 program:1 including:1 reliable:1 power:1 critical:1 suitable:1 mdps:3 f4962098:1 afosr:1 loss:1 approximator:7 agent:5 sufficient:1 consistent:1 course:2 penalized:1 surprisingly:1 last:1 repeat:1 infeasible:1 side:1 guide:1 absolute:2 dimension:2 transition:2 cumulative:1 maze:10 evaluating:1 author:1 reinforcement:3 ashenfelter:1 sj:13 approximate:2 preferred:2 tains:1 b1:1 a_:14 search:4 continuous:2 table:4 learn:2 nature:1 excellent:1 cl:1 domain:1 main:1 arise:1 x1:1 advice:1 ff:1 board:1 experienced:1 position:1 sub:1 deterministically:1 xl:1 lie:1 invocation:1 tied:2 third:3 formula:1 bad:5 svm:1 incorporating:1 vapnik:2 downward:1 easier:2 generalizing:1 led:1 prevents:1 contained:1 applies:2 springer:1 satisfies:1 goal:3 careful:1 ajk:2 hard:1 specifically:2 except:1 reducing:1 determined:1 total:6 experimental:1 xin:1 attempted:1 east:1 formally:1 support:10 incorporate:1 evaluate:1 |
1,226 | 2,117 | Effective size of receptive fields of inferior
temporal visual cortex neurons in natural scenes
Thomas P. Trappenberg
Dalhousie University
Faculty of Computer Science
5060 University Avenue, Halifax B3H 1W5, Canada
[email protected]
Edmund T. Rolls and Simon M. Stringer
University of Oxford,
Centre for Computational Neuroscience,
Department of Experimental Psychology,
South Parks Road, Oxford OX1 3UD, UK
edmund.rolls,[email protected]
Abstract
Inferior temporal cortex (IT) neurons have large receptive fields when a
single effective object stimulus is shown against a blank background, but
have much smaller receptive fields when the object is placed in a natural
scene. Thus, translation invariant object recognition is reduced in natural
scenes, and this may help object selection. We describe a model which
accounts for this by competition within an attractor in which the neurons
are tuned to different objects in the scene, and the fovea has a higher
cortical magnification factor than the peripheral visual field. Furthermore, we show that top-down object bias can increase the receptive field
size, facilitating object search in complex visual scenes, and providing a
model of object-based attention. The model leads to the prediction that
introduction of a second object into a scene with blank background will
reduce the receptive field size to values that depend on the closeness of
the second object to the target stimulus. We suggest that mechanisms of
this type enable the output of IT to be primarily about one object, so that
the areas that receive from IT can select the object as a potential target
for action.
1 Introduction
Neurons in the macaque inferior temporal visual cortex (IT) that respond to objects or faces
have large receptive fields when a single object or image is shown on an otherwise blank
screen [1, 2, 3]. The responsiveness of the neurons to their effective stimuli independent of
their position on the retina over many degrees is termed translation invariance. Translation
invariant object recognition is an important property of visual processing, for it potentially
enables the neurons that receive information from the inferior temporal visual cortex to
perform memory operations to determine whether for example the object has been seen
before or is associated with reward independently of where the image was on the retina.
This allows correct generalization over position, so that what is learned when an object is
shown at one position on the retina generalizes correctly to other positions [4].
If more than one object is present on the screen, then there is evidence that the neuron
responds more to the object at the fovea than in the parafoveal region [5, 6]. More recently,
it has been shown that if an object is presented in a natural background (cluttered scene),
and the monkey is searching for the object in order to touch it to obtain a reward, then the
receptive fields are smaller than when the monkey performs the same task with the object
against a blank background [7]. We define the size of a receptive field as twice the distance
from the fovea (the centre of the receptive field) to locations at which the response decreases
to half maximal. An analysis of IT neurons that responded to the target stimulus showed
that the average size of the receptive fields shrinks from approximately 56 degrees in a
blank background to approximately 12 degrees with a complex scene [8]. The responses of
an IT cell with a large receptive field are illustrated in Figure 1A. There the average firing
rates of the cell to an effective stimulus that the monkey had to touch on a touch-screen
to receive reward is shown as a function of the angular distance of the object from the
fovea. The solid line represents the results from experiments with the object placed in a
blank background. This demonstrates the large receptive fields of IT cells that have often
been reported in the literature [3]. In contrast, when the object is placed in a natural scene
(cluttered background), the size of the receptive field is markedly smaller (dashed line).
2 The model
We formalized our understanding of how the dependence of the receptive field size on various conditions could be implemented in the ventral visual processing pathway by developing a neural network model with the components sufficient to produce the above effects.
The model utilizes an attractor network representing the inferior temporal visual cortex,
and a neural input layer with several retinotopically organized modules representing the
visual scene in an earlier visual cortical area such as V4 (see Figure 1B). Each independent
module within ?V4? represents a small part of the visual field and receives input from earlier visual areas represented by an input vector for each possible location which is unique
for each object. Each module was 6 deg in width, matching the size of the objects presented
to the network. For the simulations we chose binary random input vectors representing objects with
components set to ones and the remaining
components
set to zeros.
is the number of nodes in each module and
is the sparseness
of the representation.
The structure labeled ?IT? represents areas of visual association cortex such as the inferior
temporal visual cortex and cortex in the anterior part of the superior temporal sulcus in
which neurons provide distributed representations of faces and objects [9, 3]. The activity
of nodes in this structure are governed by leaky integrator dynamics with time
constant
! "
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76
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0 of the ? th node is determined by a sigmoidal function from the activation
The firing
rate
0
!
"
CBD " E&FHGDIKJL
+M @A %"N
POQ1R! , where the parameters M and O
@A as
.
represent the gain and the offset, respectively. The constant represents the strength of
the activity-dependent global inhibition simulating the effects of inhibitory interneurons.
The external ?top-down? input vector
produces object-selective inputs, which are
used as the attentional drive when a visual search task is simulated. The strength of this
< =9S>
A.
B.
130
Average firing rate
120
Object bias
blank
background
110
IT
100
90
80
70
natural
background
60
50
0
10
20
30
40
50
60
Distance of gaze from target object
C.
1
V4
Weight factor
0.8
0.6
from cortical
magnification
factor
0.4
0.2
Gaussian
0
0
10
20
30
40
50
60
Visual Input
Eccentricity
Figure 1: A) Average activity of a macaque inferior temporal cortex neuron as a function
of the distance of the object from the fovea recorded in a visual search task when the object
was in a blank or a cluttered natural background. B) Outline of the model used in this study
with an attractor network labelled ?IT? that receives topgraphical organised inputs from
an input neural layer labeled ?V4?. Objects close to the fovea produce stronger inputs to
reflect the higher magnification factor of the visual representation close to the fovea. The
attractor network also receives top-down object-based inputs, to incorporate object-based
attention in a visual search task. C) The modulation factor used to weight inputs to IT from
V4 shown as a function of their distance from the fovea. The values on the solid line are
derived from cortical magnification factors, and were used in the simulations, whereas the
dotted line corresponds to a shifted Gaussian function.
8 9SA:;
. The recognition functionality of this
object bias is modulated by the value of
structure is modeled as an attractor neural network (ANN) with trained memories indexed
by representing particular objects. The memories are formed through Hebbian learning
on sparse patterns,
(2)
8
where A
*
- 8 (
P Q *
P N
S
*
- A 76
*
*
- 6 8 A 76 8S(
P
(set to 1in
the simulations below) is a normalization constant that depends
on
the learning rate,
is the sparseness of the training pattern in IT, and
are the
components of the pattern used to train the network. The weights
between the V4
nodes and IT nodes are trained by Hebbian learning of the form
(3)
to produce object representations in IT based on inputs in V4. The normalizing modulation
8 76 8
8
factor
allows the gain of inputs to be modulated as a function of their distance
from the fovea, and depends on the module to which the presynaptic node belongs. The
weight values between V4 and IT support translation invariant object recognition of a single
object in the visual field if the normalization factor is the same for each module and the
model is trained with the objects placed at every possible location in the visual field. The
translation invariance of the weight vectors between each V4 module and the IT nodes is
however explicitly modulated in our model by the module-dependent modulation factor
as indicated in Figure 1B by the width of the lines connecting V4 with IT.
The strength of the foveal module is strongest, and the strength decreases for modules
representing increasing eccentricity. The form of this modulation factor was derived from
the parameterization of the cortical magnification factors given by [10], 1 and is illustrated
in Figure 1C as a solid line. Similar results to the ones presented here can be achieved
with different forms of the modulation factor such as a shifted Gaussian as illustrated by
the dashed line in Figure 1C.
8 6 8S
3 Results
To study the ability of the model to recognize trained objects at various locations relative
to the fovea we tested the network with distorted versions of the objects, and measured
the ?correlation? between the target object and the final state of the attractor network. The
correlation was estimated from the normalized dot product between the target object vector
and the state of the IT network after a fixed amount of time sufficient for the network to
settle into a stable state. The objects were always presented on backgrounds with some
noise (introduced by flipping 2% of the bits in the scene) because the input to IT will
inevitably be noisy under normal conditions of operation. All results shown in the following
represent averages over 10 runs and over all patterns on which the network was trained.
3.1 Receptive fields are large in scenes with blank backgrounds
In the first experiments we placed only one object in the visual scene with different eccentricities relative to the fovea. The results of this simulation are shown in Figure 2A with
the line labeled ?blank background?. The value of the object bias
was set to 0 in
these simulations. Good object retrieval (indicated by large correlations) was found even
when the object was far from the fovea, indicating large IT receptive fields with a blank
background. The reason that any drop is seen in performance as a function of eccentricity is
because flipping 2% of the bits in the V4 modules introduces some noise into the recall process. The results demonstrate that the attractor dynamics can support translation invariant
object recognition even though the weight vectors between V4 and IT are not translation
invariant but are explicitly modulated by the modulation factor
derived from the
cortical magnification factor.
8 9 :;
8 76
3.2 The receptive field size is reduced in scenes with complex background
In a second experiment we placed individual objects at all possible locations in the visual
scene representing natural (cluttered) visual scenes. The resulting correlations between
the target pattern and asymptotic IT state are shown in Figure 2A with the line labeled
?natural background?. Many objects in the visual scene are now competing for recognition
by the attractor network, while the objects around the foveal position are enhanced through
the modulation factor derived by the cortical magnification factor. This results in a much
smaller size of the receptive field of IT neurons when measured with objects in natural
1
This parameterization is based on V1 data. However, it was shown that similar forms of the
magnification factor hold also in V4 [11]
A.
B.
Without object bias
blank
background
0.8
Correlation
Correlation
blank
background
1
1
0.6
0.4
0.8
0.6
0.4
natural
background
0.2
0.2
natural background
0
With object bias
0
10
20
30
40
Eccentricity
50
60
0
0
10
20
30
40
50
60
Eccentricity
Figure 2: Correlations as measured by the normalized dot product between the object vector
used to train IT and the state of the IT network after settling into a stable state with a single
object in the visual scene (blank background) or with other trained objects at all possible
locations in the visual scene (natural background). There is no object bias included in the
results shown in
graph A, whereas an object bias is included in the results shown in B with
in the experiments with a natural background and
in the
experiments with a blank background.
8 9SA:;
8 9SA:;
backgrounds.
3.3 Object-based attention increases the receptive field size, facilitating object
search in complex visual scenes
In addition to this major effect of the background on the size of the receptive field, which
parallels and we suggest may account for the physiological findings outlined in the introduction, there is also a dependence of the size of the receptive fields on the level of object
bias provided to the IT network. Examples are shown in Figure 2B where we used an object
bias. The object bias biasses the IT network towards the expected object with a strength determined by the value of
, and has the effect of increasing the size of the receptive
fields in both blank and natural backgrounds (see Figure 2B and compare to Figure 2A).
This models the effect found neurophysiologically [8].2
8 9SA:;
3.4 A second object in a blank background reduces the receptive field size
depending on the distance between the second object and the fovea
In the last set of experiments we placed two objects in an otherwise blank background.
The IT network was biased towards one of the objects designated as the target object (in
for example a visual search task), which was placed on one side of the fovea at different
eccentricities from the fovea. The second object, a distractor object, was placed on the
opposite side of the fovea at a fixed distance of degrees from the fovea. Results for
different values of are shown in 3A. The results indicate that the size of the receptive
field (for the target object) decreases with decreasing distance of the distractor object from
the fovea. The size of the receptive fields (the width at half maximal response) is shown
2
The larger values of
in the experiments with a natural background compared to the
experiments in a blank background reflects the fact that more attention may be needed to find objects
in natural cluttered environments.
in 3B. The size starts to increase linearly with increasing distance of the distractor object
from the fovea until the influence of the distractor on the size of the receptive field levels
off and approaches the value expected for the situation with one object in a visual scene
and a blank background.
A.
B.
70
Size of receptive field
Correlation
1
0.8
d=24
0.6
d=18
d=12
0.4
0.2
0
d=6
0
10
20
30
40
Eccentricity
50
60
60
50
40
30
20
10
5
10
15
20
25
30
35
40
d, distance of distractor from fovea
Figure 3: A) Correlations between the target object and the final state of the IT network
in experiments with two objects in a visual scene with a blank background. The different
curves correspond to different distances of the distractor object from the fovea. The
eccentricity refers to the distance between the target object and the fovea. B) The size of
the receptive field for the target as a function of the distance of the distractor object from
the fovea.
4 Discussion
When single objects are shown in a scene with a blank background, the attractor network
helps neurons to respond to an object with large eccentricities of this object relative to the
fovea of the agent. When the object is presented in a natural scene, other neurons in the
inferior temporal cortex become activated by the other effective stimuli present in the visual
field, and these forward inputs decrease the response of the network to the target stimulus
by a competitive process. The results found fit well with the neurophysiological data, in
that IT operates with almost complete translation invariance when there is only one object
in the scene, and reduces the receptive field size of its neurons when the object is presented
in a cluttered environment.
The model here provides an explanation of the real IT neuronal responses in natural scenes
and makes several predictions that can be explored experimentally. The model is compatible with the models developed by Gustavo Deco and colleagues (see, for example, [12, 13])
while specific simplifications and addition have been made to explore the variations in the
size of receptive fields in IT.
The model accounts for the larger receptive field sizes from the fovea of IT neurons in
natural backgrounds if the target is the object being selected compared to when it is not
selected [8]. The model accounts for this by an effect of top-down bias which simply
biasses the neurons towards particular objects compensating for their decreasing inputs
produced by the decreasing magnification factor modulation with increasing distance from
the fovea. Such object based attention signals could originate in the prefrontal cortex and
could provide the object bias for the inferotemporal cortex [14].
We proposed that the effective variation of the size of the receptive field in the inferior
temporal visual cortex enables the brain areas that receive from this area (including the
orbitofrontal cortex, amygdala, and hippocampal system) to read out the information
correctly from the inferior temporal visual cortex about individual objects, because the
neurons are responding effectively to the object close to the fovea, and respond very
much less to objects away from the fovea.3 This enables, for example, the correct reward
association of an object to be determined by pattern association in the orbitofrontal cortex
or amygdala, because they receive information essentially about the object at the fovea.
Without this shrinkage in the receptive field size, the areas that receive from the inferior
temporal visual cortex would respond to essentially all objects in a visual scene, and would
therefore provide an undecipherable babel of information about all objects present in the
visual scene. It appears that part of the solution to this potential binding problem that
is used by the brain is to limit the size of the receptive fields of inferior temporal cortex
neurons when natural environments are being viewed. The suggestion is that by providing
an output about what is at the fovea in complex scenes, the inferior temporal visual cortex
enables the correct reward association to be looked up in succeeding brain regions, and
then for the object to be selected for action if appropriate. Part of the hypothesis here is
that the coordinates of the object in the visual scene being selected for action are provided
by the position in space to which the gaze is directed [7].
Acknowledgments
This research was supported by the Medical Research Council, grant PG9826105, and by
the MRC Interdisciplinary Research Centre for Cognitive Neuroscience.
References
[1] C. G. Gross, R. Desimone, T. D. Albright, and Schwartz E. L. Inferior temporal cortex and
pattern recognition. Experimental Brain Research, 11:179?201, 1985.
[2] M. J. Tovee, E. T. Rolls, and P. Azzopardi. Translation invariance and the responses of neurons
in the temporal visual cortical areas of primates. Journal of Neurophysiology, 72:1049?1060,
1994.
[3] E. T. Rolls. Functions of the primate temporal lobe cortical visual areas in invariant visual
object and face recognition. Neuron, 27:205? 218, 2000.
[4] E. T. Rolls and A. Treves. Neural Networks and Brain Function. Oxford University Press,
Oxford, 1998.
[5] T. Sato. Interactions of visual stimuli in the receptive fields of inferior temporal neurons in
macaque. Experimental Brain Research, 77:23?30, 1989.
[6] E. T. Rolls and M. J. Tovee. The responses of single neurons in the temporal visual cortical
areas of the macaque when more than one stimulus is present in the visual field. Experimental
Brain Research, 103:409?420, 1995.
[7] E. T. Rolls, B. Webb, and M. C. A. Booth. Responses of inferior temporal cortex neurons to
objects in natural scenes. Society for Neuroscience Abstracts, 26:1331, 2000.
[8] E. T. Rolls, F. Zheng, and N. Aggelopoulos. Responses of inferior temporal cortex neurons to
objects in natural scenes. Society for Neuroscience Abstracts, 27, 2001.
[9] M. C. A. Booth and E. T. Rolls. View-invariant representations of familiar objects by neurons
in the inferior temporal visual cortex. Cerebral Cortex, 8:510?523, 1998.
3
Note that it is possible that a ?spotlight of attention? [15] can be moved away from the fovea,
but at least during normal visual search tasks, the neurons are sensitive to the object at which the
monkey is looking, that is which is on the fovea, as shown by [8]. Thus, spatial modulation of the
responsiveness of neurons at the V4 level can be influenced by location-specific attentional modulations originating, for example, in the posterior parietal cortex, which may be involved in directing
visual spatial attention [15].
[10] B.W. Dow, A.Z. Snyder, R.G. Vautin, and R. Bauer. Magnification factor and receptive field
size in foveal striate cortex of the monkey. Exp. Brain. Res., 44:213:228, 1981.
[11] R. Gattass, A.P.B. Sousa, and E. Covey. Cortical visual areas of the macaque: Possible substrates for pattern recognition mechanisms. Exp. Brain. Res., Supplement 11, 1985.
[12] G. Deco and J. Zihl. Top-down selective visual attention: A neurodynamical approach. Visual
Cognition, 8:119?140, 2001.
[13] E. T. Rolls and G. Deco. Computational neuroscience of vision. Oxford University Press,
Oxford, 2002.
[14] A. Renart, N. Parga, and E. T. Rolls. A recurrent model of the interaction between the prefrontal
cortex and inferior temporal cortex in delay memory tasks. In S.A. Solla, T.K. Leen, and K.-R.
Mueller, editors, Advances in Neural Information Processing Systems. MIT Press, Cambridge
Mass, 2000. in press.
[15] R. Desimone and J. Duncan. Neural mechanisms of selective visual attention. Annual Review
of Neuroscience, 18:193?222, 1995.
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1,227 | 2,118 | Cobot: A Social Reinforcement Learning Agent
Charles Lee Isbell, Jr.
Christian R. Shelton
AT&T Labs-Research
Stanford University
Michael Kearns
Satinder Singh
Peter Stone
University of Pennsylvania Syntek Capital AT&T Labs-Research
Abstract
We report on the use of reinforcement learning with Cobot, a software agent
residing in the well-known online community LambdaMOO. Our initial work on
Cobot (Isbell et al.2000) provided him with the ability to collect social statistics
and report them to users. Here we describe an application of RL allowing Cobot
to take proactive actions in this complex social environment, and adapt behavior
from multiple sources of human reward. After 5 months of training, and 3171
reward and punishment events from 254 different LambdaMOO users, Cobot
learned nontrivial preferences for a number of users, modifing his behavior based
on his current state. Here we describe LambdaMOO and the state and action
spaces of Cobot, and report the statistical results of the learning experiment.
1
Introduction
While most applications of reinforcement learning (RL) to date have been to problems
of control, game playing and optimization (Sutton and Barto1998), there has been a recent
handful of applications to human-computer interaction. Such applications present a number
of interesting challenges to RL methodology (such as data sparsity and inevitable violations
of the Markov property). These previous studies focus on systems that encounter human
users one at a time, such as spoken dialogue systems (Singh et al.2000).
In this paper, we report on an RL-based agent for LambdaMOO, a complex, open-ended,
multi-user chat environment, populated by a community of human users with rich and
often enduring social relationships. Our long-term goal is to build an agent who can learn
to perform useful, interesting and entertaining actions in LambdaMOO on the basis of user
feedback. While this is a deliberately ambitious and underspecified goal, we describe here
our implementation, the empirical experiences of our agent so far, and some of the lessons
we have learned about this challenging domain.
In previous work (Isbell et al.2000), we developed the software agent Cobot, who interacted
in various ways with LambdaMOO users. Cobot had two primary functions. First, Cobot
gathered ?social statistics? (e.g. how frequently and in what ways users interacted with
one another), and provided summaries of these statistics as a service. Second, Cobot had
rudimentary chatting abilities based on the application of information retrieval methods
to large documents. The original Cobot was entirely reactive , in that he never initiated
interaction with human users, but would only respond to their actions. As we documented
in our earlier paper, Cobot proved tremendously popular with LambdaMOO users, setting
the stage for our current efforts.
We modified Cobot to allow him to take certain actions (such as proposing conversation
topics, introducing users, or engaging in common word play routines) under his own initiative. The hope is to build an agent that will eventually take unprompted actions that are
meaningful, useful or amusing to users. Rather than hand-code complex rules specifying
Here we mean ?responding only to human-invoked interaction?, rather than ?non-deliberative?.
Characters in LambdaMOO have gender. Cobot?s description to users indicates that he is male.
when each action is appropriate (rules that would be inaccurate and quickly become stale),
we wanted Cobot to learn the individual and communal preferences of users. Thus, we provided a mechanism for users to reward or punish Cobot, and programmed Cobot to use RL
algorithms to alter his behavior on the basis of this feedback. The application of RL (or any
machine learning methodology) to such an environment presents a number of interesting
domain-specific challenges, including:
Choice of an appropriate state space. To learn how to act in a social environment such
as LambdaMOO, Cobot must represent the salient features. These should include social
information such as which users are present, how experienced they are in LambdaMOO,
how frequently they interact with one another, and so on.
Multiple reward sources. Cobot lives in an environment with multiple, often conflicting
sources of reward from different human users. How to integrate these sources reasonably
is a nontrivial empirical question.
Inconsistency and drift of user rewards and desires. Individual users may be inconsistent in the rewards they provide (even when they implicitly have a fixed set of preferences),
and their preferences may change over time (for example, due to becoming bored or irritated with an action). Even when their rewards are consistent, there can be great temporal
variation in their reward pattern.
Variability in user understanding. There is great variation in users? understanding of
Cobot?s functionality, and the effects of their rewards and punishments.
Data sparsity. Training data is scarce for many reasons, including user fickleness, and the
need to prevent Cobot from generating too much spam in the environment.
Irreproducibility of experiments. As LambdaMOO is a globally distributed community
of human users, it is virtually impossible to replicate experiments taking place there.
We do not have any simple answers (nor do we believe that simple answers exist), but here
we provide a case study of our choices and findings. Our primary findings are:
Inappropriateness of average reward. We found that the average reward that Cobot received over time, the standard measure of success for RL experiments, is an inadequate and
perhaps even inappropriate metric of performance in the LambdaMOO domain. Reasons
include that user preferences are not stationary, but drift as users become habituated or
bored with Cobot?s behavior; and the tendency for satisfied users to stop providing Cobot
with any feedback, positive or negative. Despite the inadequacy of average reward, we are
still able to establish several measures by which Cobot?s RL succeeds, discussed below.
A small set of dedicated ?parents?. While many users provided only a moderate or small
amount of RL training (rewards and punishments) to Cobot, a handful of users did invest
significant time in training him.
Some parents have strong opinions. While many of the users that trained Cobot did
not exhibit clear preferences for any of his actions over the others, some users clearly and
consistently rewarded and punished particular actions over the others.
Cobot learns matching policies. For those users who exhibited clear preferences through
their rewards and punishments, Cobot successfully learned corresponding policies of behavior.
Cobot responds to his dedicated parents. For those users who invested the most training time in Cobot, the observed distribution of his actions is significantly altered by their
presence.
Some preferences depend on state. Although some users for whom we have sufficient
data seem to have preferences that do not depend upon the social state features we constructed for the RL, others do in fact appear to change their preferences depending upon
prevailing social conditions.
The outline for the rest of the paper is as follows. In Section 2, we give brief background
on LambdaMOO. In Section 3, we describe our earlier (non-RL) work on Cobot. Section 4
provides some brief background on RL. In Sections 5, 6 and 7 we describe our implementation of Cobot?s RL action space, reward mechanisms and state features, respectively. Our
primary findings are presented in Section 8, and Section 9 offers conclusions.
2
LambdaMOO
LambdaMOO, founded in 1990 by Pavel Curtis at Xerox PARC, is the oldest continuously
operating MUD, a class of online worlds with roots in text-based multiplayer role-playing
games. MUDs (multi-user dungeons) differ from most chat and gaming systems in their
use of a persistent representation of a virtual world, often created by the participants, who
are represented as characters of their own choosing. LambdaMOO appears as a series of
interconnected rooms, populated by users and objects who may move between them. Each
room provides a shared chat channel, and typically has an elaborate text description that
imbues it with its own ?look and feel.? In addition to speech, users express themselves via
a large collection of verbs, allowing a rich set of simulated actions, and the expression of
emotional states:
(1)
(2)
(3)
(4)
(5)
(6)
Buster is overwhelmed by all these deadlines.
Buster begins to slowly tear his hair out, one strand at a time.
HFh comforts Buster.
HFh [to Buster]: Remember, the mighty oak was once a nut like you.
Buster [to HFh]: Right, but his personal growth was assured. Thanks anyway, though.
Buster feels better now.
Lines (1) and (2) are initiated by verb commands by user Buster, expressing his emotional
state, while lines (3) and (4) are examples of verbs and speech acts, respectively, by HFh.
Lines (5) and (6) are speech and verb acts by Buster. Though there are many standard verbs,
such as the use of the verb comfort in line (3) above, the variety is essentially unlimited,
as players have the ability to create their own verbs.
The rooms and objects in LambdaMOO are created by users themselves, who devise descriptions, and control access by other users. Users can also create objects with verbs that
can be invoked by other players. As last count, the database contains 118,154 objects,
including 4836 active user accounts. LambdaMOO?s long existence and its user-created
nature combine to give it one of the strongest senses of virtual community in the on-line
world. Many users have interacted extensively with each other over many years, and users
are widely acknowledged for their contribution of interesting objects. LambdaMOO is an
attractive environment for experiments in AI (Foner1997; Mauldin1994), including learning. The population is generally curious and technically savvy, and users are interested in
automated objects meant to display some form of intelligence.
3
Cobot
Cobot is a software agent residing in LambdaMOO. Like a human user, he connects via telnet, and from the point of view of the LambdaMOO server, is a user with all the rights and
responsibilities implied. Once actually connected, Cobot wanders into the Living Room,
where he spends most of his time. The Living Room is a central public place, frequented
both by many regulars, and by users new to LambdaMOO. There are several permanent
objects in the Living Room, including a couch with various features and a cuckoo clock.
The Living Room usually has between five and twenty users, and is perpetually busy. Over
a year, Cobot noted over 2.5 million separate events (about one event every eleven seconds)
Previously, we implemented a variety of functionality on Cobot centering around gathering and reporting social statistics. Cobot notes who takes what actions, and on whom.
Cobot can answer queries about these statistics, and describe the similarities and differences between users. He also has a rudimentary chatting ability based on the application
of information retrieval methods to large documents. He can also search the web to answer
specific questions posed to him. A more complete description of Cobot?s abilities, and his
early experiences as a social agent in LambdaMOO, can be found in (Isbell et al.2000).
Our focus here is to make Cobot proactive?i.e., let him take actions under his own
initiative?in a way that is useful, interesting, or pleasing to LambdaMOO users. It is
impossible to program rules anticipating when any given action is appropriate in such a
complex and dynamic environment, so we applied reinforcement learning to adapt directly
from user feedback. We emphasize that Cobot?s original reactive functionality remained
on during the RL experiment. Cobot?s persona is largely due to this original functionality,
and we felt it was most interesting, and even necessary, to add RL work in this context.
Null Action
Topic Change (4)
Roll Call (2)
Social
Commentary
Introductions
Choose to remain silent for this time period.
Introduce a conversational topic. Cobot declares that he wants to discuss sports
or politics, or he utters a sentence from either the sports section or political
section of the Boston Globe.
Initiate a ?roll call,? a common word play routine in LambdaMOO. For example, someone who is tired of Monica Lewinsky may emote ?TIRED OF
LEWINSKY ROLL CALL.? Sympathetic users agree with the roll call. Cobot
takes a recent utterance, and extracts either a single noun, or a verb phrase.
Make a comment describing the current social state of the Living Room, such
as ?It sure is quiet? or ?Everyone here is friendly.? These statements are based
on Cobot?s statistics from recent activity. Several different utterances possible,
but they are treated as a single action for RL purposes.
Introduce two users who have not yet interacted in front of Cobot.
Table 1: The 9 RL actions available to Cobot.
4
RL Background
In RL, problems of decision-making by agents interacting with uncertain environments are
usually modeled as Markov decision processes (MDPs). In the MDP framework, at each
time step the agent senses the state of the environment, and chooses and executes an action
from the set of actions available to it in that state. The agent?s action (and perhaps other
uncontrolled external events) cause a stochastic change in the state of the environment. The
agent receives a (possibly zero) scalar reward from the environment. The agent?s goal is to
choose actions so as to maximize the expected sum of rewards over some time horizon. An
optimal policy is a mapping from states to actions that achieves the agent?s goal.
Many RL algorithms have been developed for learning good approximations to an optimal
policy from the agent?s experience in its environment. At a high level, most algorithms
use this experience to learn value functions (or -values) that map state-action pairs to
the maximal expected sum of reward that can be achieved starting from that state-action
pair. The learned value function is used to choose actions stochastically, so that in each
state, actions with higher value are chosen with higher probability. In addition, many RL
algorithms use some form of function approximation (parametric representations of complex value functions) both to map state-action features to their values and to map states to
distributions over actions (i.e., the policy). See (Sutton and Barto1998) for an extensive
introduction to RL.
In the next sections, we describe the Cobot?s actions, our choice of state features, and how
we dealt with multiple sources of reward. The particular RL algorithm we use is a variant
of (Sutton et al.1999)?s policy gradient algorithm. Its details are beyond the scope of
this paper; however, see (Shelton2000) for details. One aspect of our RL algorithm that
is relevant to understanding our results is that we use a linear function approximator to
store our policy. In other words, for each state feature, we maintain a vector of real-valued
weights indexed by the possible actions. A positive weight for some action means that the
feature increases the probability of taking that action, while a negative weight decreases
the probability. The weight?s magnitude determines the strength of this contribution.
5
Cobot?s RL Actions
To have any hope of learning to behave in a way interesting to LambdaMOO users, Cobot?s
actions must ?make sense? to them, fit in with the social chat-based environment, and
minimize the risk of causing irritation. Conversation, word play, and emoting routines are
among the most common activity in LambdaMOO, so we designed a set of actions along
these lines, as detailed in Table 1. Many of these actions extract an utterance from the
recent conversations, or from a continually changing external source, such as the online
Boston Globe. Thus a single action may cause an infinite variety of behavior by Cobot.
At set time intervals (only every few minutes on average, to minimize spam), Cobot selects
an action to perform from this set according to a distribution determined by the Q-values
in his current state. Any rewards or punishments received before the next RL action are
attributed to the current action, and used to update Cobot?s value functions. It is worth
remembering that Cobot has two different categories of action: those actions taken proactively as a result of the RL, and those actions taken in response to a user?s action towards
Cobot. Some users are certainly aware of the distinction and can easily determine which
actions fall into which category, but other users may occasionally reward or punish Cobot
in response to a reactive action. Such ?erroneous? rewards and punishments act as a source
of noise in the training process.
6
The RL Reward Function
Cobot learns to behave directly from the feedback of LambdaMOO users, any of whom
can reward or punish him. There are both explicit and implicit feedback mechanisms. We
implemented explicit reward and punish verbs on Cobot that LambdaMOO users can
invoke at any time. These verbs give a numerical (positive and negative, respectively)
training signal to Cobot that is the basis of the RL. The signal is attributed as immediate
feedback for the current state and RL action, and ?backed up? to previous states and actions
in accordance with the standard RL algorithms.
There are several standard LambdaMOO verbs that are commonly used to express, sometimes playfully, approval or disapproval. Examples of the former include the verb hug, and
of the latter the verb spank. In the interest of allowing the RL process to integrate naturally
with the LambdaMOO environment, we chose to accept a number of such verbs as implicit
reward and punishment signals for Cobot; however, such implicit feedback is numerically
weaker than the feedback generated by the explicit mechanisms.
One fundamental design choice is whether to learn a single value function for the entire
community, or to learn separate value functions for each user based on individual feedback,
combining the value functions of those present to determine how to act at each moment.
We opted for the latter for three primary reasons.
First, it was clear that for learning to have any hope of success, ths system must represent
who is present at any given moment?different users simply have different personalities and
preferences. We felt that representing which users are present as additional state features
would throw away valuable domain information, as the RL would have to discover on its
own the primacy of user identity. Having separate reward functions for each user is thus a
way of asserting the importance of identity to the learning process.
Second, despite the extremely limited number of training examples available in this domain
per month), learning must be quick and significant. Without a clear sense that their
(
training has some impact on Cobot?s behavior, users will quickly lose interest in providing
feedback. A known challenge for RL is the ?curse of dimensionality,? (i.e. the size of the
state space increases exponentially with the number of state features). By avoiding the need
to represent the presence or absence of roughly 250 users, we are able to maintain a fairly
small state space and so speed up learning.
Third, we (correctly) anticipated the fact that certain users would provide an inordinate
amount of training to Cobot, and we did not want the overall policy followed by Cobot
to be dominated by the preferences of these individuals. By learning separate policies for
each user, and then combining these policies among those users present, we can limit the
impact any single user can have on Cobot?s actions.
7
Cobot?s RL State Features
The decision to maintain and learn separate value functions for each user means that we
can maintain separate state spaces as well, in the hopes of simplifying states and speeding
learning. Cobot can be viewed as running a large number of separate RL processes in
parallel, with each process having a different state space. The state space for a user contains
a number of features containing statistics about that particular user.
LambdaMOO is a social environment, and Cobot is learning to take social actions, so we
felt that his state features should contain information allowing him to gauge social activity
and relationships. Table 2 provides a description of the state features used for RL by Cobot
for each user. Even though we have simplified the state space by partitioning by user, the
state space for a single user remains sufficiently complex to preclude standard table-based
representation of value functions (also, each user?s state space is effectively infinite, as
there are real-valued state features). Thus, linear function approximation is used for each
user?s policy. Cobot?s RL actions are then chosen according to a mixture of the policies of
the users present. We refer the reader to (Shelton2000) for more details on the method by
which policies are learned and combined.
Social Summary
Vector
Mood Vector
Rates Vector
Current Room
Roll Call Vector
Bias
A vector of four numbers: the rate at which the user is producing events; the
rate at which events are being produced that are directed at the user; the percentage of the other users present who are among this user?s ten most frequently interacted-with users (?playmates?); and the percentage of the other
users present for whom this user is among their top ten playmates.
A vector measuring the recent use of eight groups of common verbs (e.g., one
group includes verbs grin and smile). Verbs were grouped according to how
well their usage was correlated.
A vector measuring the rate at which events are produced by those present.
The room where Cobot currently resides.
Indicates if Cobot?s currently saved roll call text has been used before, if someone has done a roll call since the last time Cobot did, and if there has been a
roll call since the last time Cobot grabbed new text.
Each user has one feature that is always ?on?; that is, this bias is always set to
a value of 1. Intuitively, it is the feature indicating the user?s ?presence.?
Table 2: State space of Cobot. Each user has his own state space and value function; the table thus
describes the state space maintained for a generic user.
8
Experimental Procedure and Findings
Cobot has been present in LambdaMOO more or less continuously since September, 1999.
The RL version of Cobot debuted May 10, 2000. Again, Cobot?s various reactive functionality was left intact for the duration of the RL experiment. Cobot is a working system
with real human users, and we wanted to perform the RL experiment in this context. Upon
launching the RL functionality publicly in the Living Room, Cobot logged all RL-related
data (states visited, actions taken, rewards received from each user, parameters of the value
functions, etc.) from May 10 until October 10, 2000. During this time, 63123 RL actions
were taken (in addition, of course, to many more reactive non-RL actions), and 3171 reward and punishment events were received from 254 different users. The findings we now
summarize are based on these extensive logs:
Inappropriateness of average reward. The most standard and obvious sign of successful RL would
be an increase in the average reward over time. Instead, as shown in Figure 1a, the average cumulative reward received by Cobot actually goes down. However, rather than indicating that users are
becoming more dissatisfied as Cobot learns, the decay in reward reveals some peculiarities of human
feedback in such an open-ended environment. There are at least two difficulties with average cumulative reward in an environment of human users. The first is that humans are fickle, and their tastes
and preferences may drift over time. Indeed, our experiences as users, and with the original reactive
functionality of Cobot, suggest that novelty is highly valued in LambdaMOO. Thus a feature that is
popular and exciting to users when it is introduced may eventually become an irritant (there are many
examples of this phenomenon). In RL terminology, we do not have a fixed, consistent reward function, and thus we are always learning a moving target. While difficult to quantify in such a complex
environment, this phenomenon is sufficiently prevalent in LambdaMOO to cast serious doubts on the
use of average cumulative reward as the primary measure of performance.
The second and related difficulty is that even when users do maintain relatively fixed preferences,
they tend to give Cobot less feedback of either type (reward or punishment) as he manages to learn
their preferences accurately. Simply put, once Cobot seems to be behaving as they wish, users feel
no need to continually provide reward for his ?correct? actions or to punish him for the occasional
?mistake.? This reward pattern is in contrast to typical RL applications, where there is an automated
and indefatigable reward source. Strong empirical evidence for this second phenomenon is provided
by User M and User S. These two users were among Cobot?s most dedicated trainers, each had strong
preferences for certain actions, and Cobot learned to strongly modify his behavior in their presence to
match their preferences. Nevertheless, both users tended to provide less frequent feedback to Cobot
as the experiment progressed, as shown in Figure 1a. We conclude that there are serious conceptual
difficulties with the use of average cumulative reward in such a human-centric application of RL, and
that alternative measures must be investigated, which we do below.
A small set of dedicated ?parents.? Among the 254 users who gave at least one reward or punishment event to Cobot, 218 gave 20 or fewer, while 15 gave 50 or more. Thus, we found that while
many users exhibited a passing interest in training Cobot, there was a small group that was willing to
invest nontrivial time and effort in teaching Cobot their preferences. In particular, User M and User
S, generated 594 and 69 rewards and punishments events, respectively.
By ?event?, we simply mean an RL action that received some feedback. The actual absolute
User O
User B
User C
User P
Roll Call. User O appears to especially dislike roll call actions when there have
been repeated roll calls and/or Cobot is repeating the same roll calls.
Rates. The overall rate of events being generating has slightly more relevance
than that of the rate of events being generated just by User O.
Social Summary. User B is effected by the presence of his friends. Not shown
here are other Social Summary features (deviating about 6 degrees). It appears
that User B is more likely to ignore Cobot when he is with many friends.
Roll Call. User C appears to have strong preferences about Cobot?s behavior
when a ?roll call party? is in progress (i.e., everyone is generating roll calls).
Room. User P would follow Cobot to his home, where he is generally alone, and
has trained him there. He appears to have different preferences for Cobot under
those circumstances.
Table 3: Relevant features for users with non-uniform policies. Several of our top users had some
features that deviated from their bias feature. The second column indicates the number of degrees
between the weight vectors for those features and the weight vectors for the bias feature. We have
only included features that deviated by more than 10 degrees. For the users above the double line,
we have included only features whose weights had a length greater than 0.2. Each of these users had
bias weights of length greater than 1. For those below the line, we have included only features with
a length greater than 0.1 (these all had bias weights of length much less than 1).
Some parents have strong opinions. For the vast majority of users who participated in the RL
training of Cobot, the policy learned was quite close to the uniform distribution. Quantification of
this statement is somewhat complex, since policies are dependent on state. However, we observed
that for most users the learned policy?s dependence on state was weak, and the resulting distribution
near uniform (though there are interesting and notable exceptions, as we shall see below). This
result is perhaps to be expected: most users provided too little feedback for Cobot to detect strong
preferences, and may not have been exhibiting strong and consistent preferences in the feedback
they did provide. However, there was again a small group of users for whom a highly non-uniform
policy was learned. In particular, for Users M and S mentioned above, the resulting policies were
relatively independent of state and their entropies were 0.03 and 1.93, respectively. (The entropy of
the uniform distribution over the actions is 2.2.) Several other users also exhibited less dramatic but
still non-uniform distributions. User M seemed to have a strong preference for roll call actions, with
the learned policy selecting these with probability 0.99, while User S preferred social commentary
actions, with the learned policy giving them probability 0.38. (Each action in the uniform distribution
is given weight 1/9 = 0.11.)
Cobot learns matching policies. In Figure 1b, we demonstrate that the policy learned by Cobot
for User M does in fact reflect the empirical pattern of rewards received over time. Similar results
obtain for User S, not shown here. Thus, repeated feedback given to Cobot for a non-uniform set of
preferences clearly pays off in a corresponding policy.
Cobot responds to his dedicated parents. The policies learned by Cobot for users can have strong
impact on the empirical distribution of actions he actually ends up taking in LambdaMOO. For User
M, we find that his presence causes a significant shift towards his preferences. In other words, Cobot
does his best to ?please? these dedicated trainers whenever they arrive in the Living Room, and
returns to a more uniform policy upon their departure.
Some preferences depend on state. Finally, we show that the policies learned by Cobot sometimes
depend upon the features Cobot maintains in his state. We use two facts about the RL weights
(described in Section 4) maintained by Cobot to determine which features are relevant for a given
user. First, we note that by construction, the RL weights learned for the bias feature described in
Table 2 represent the user?s preferences independent of state (since this feature is always on whenever
the user is present). Second, we note that because we initialized all weights to 0, only features with
non-zero weights will contribute to the policy that Cobot uses. Thus, we can determine that a feature
is relevant for a user if that feature?s weight vector is far from that user?s bias feature weight vector,
and from the all-zero vector. For our purposes, we have used (1) the normalized inner product (the
cosine of the angle between two vectors) as a measure of a feature?s distance from the bias feature,
and (2) a feature?s weight vector length to determine if it is away from zero. These measures show that
for most users, Cobot learned a state-independent policy (e.g., User M prefers roll calls); however, as
we can see in Table 3, Cobot has learned a policy for some users that depends upon state.
numerical reward received may be larger or smaller than 1 at any time, as implicit rewards provide
fractional reward, and the user may repeatedly reward or punish an action, with the feedback being
summed. For example, the total absolute value of rewards and punishments provided by User M was
607.63 over 594 feedback events.
Average Cumulative Reward per Timestep
Rewards / Policy / Empirical Distribution Comparision for User M
0.2
1
reward all users
abs reward all users
reward user M
abs reward user M
reward user S
abs reward user S
0.18
0.16
rewards
policy
empirical
0.8
0.14
0.6
0.4
change
Reward
0.12
0.1
0.2
0.08
0.06
0
0.04
?0.2
0.02
0
0
1
2
3
Time
4
5
?0.4
6
4
x 10
1
2
3
4
5
action
6
7
8
9
Figure 1: a) Average Cumulative Reward Over Time. b) Rewards received, policy learned, and
effect on actions for User M. Figure a) shows that average cumulative reward decreases over time,
for both total and absolute reward; however, Figure b shows that proper learning is taking place. For
each of the RL actions, three quantities are shown. The blue bars (left) show the average reward
given by User M for each action (the average reward given by User M across all actions has been
subtracted off to indicate relative preferences). The yellow bars (middle) show the policy learned by
Cobot for User M (the probability assigned to each action in the uniform distribution (1/9) has been
subtracted off). The red bars (right) show the empirical frequency with which each action was taken
in the presence of User M (minus the empirical frequency with which that action was taken by Cobot
over all time steps). These bars indicate the extent to which the presence of User M biases Cobot?s
behavior towards M?s preferences. We see that the policy learned by Cobot for User M aligns nicely
with the preferences expressed by M and that Cobot?s behavior shifts strongly towards the learned
policy for User M whenever M is present. To go beyond a qualitative visual analysis, we have defined
a metric that measures the extent to which two rankings of actions agree, while taking into account
that some actions are extremely close in the each ranking. The details are beyond the scope of the
paper, but the agreement between the action rankings shown here are in near-perfect agreement by
this measure. Similar results obtain for User S.
9
Conclusions
We have reported on our efforts to apply reinforcement learning in a complex human online
social environment where many of the standard assumptions (stationary rewards, Markovian behavior, appropriateness of average reward) are clearly violated. We feel that the
results obtained with Cobot so far are compelling, and offer promise for the application
of RL in such open-ended social settings. Cobot continues to take RL actions and receive
rewards and punishments from LambdaMOO users, and we plan to continue and embellish
this work as part of our overall efforts on Cobot.
References
Foner, L. (1997). Entertaining Agents: a Sociological Case Study. In Proceedings of the First
International Conference on Autonomous Agents.
Isbell, C. L., Kearns, M., Kormann, D., Singh, S., and Stone, P. (2000). Cobot in LambdaMOO: A
Social Statistics Agent. To appear in Proceedings of AAAI-2000.
Mauldin, M. (1994). Chatterbots, TinyMUDs, and the Turing Test: Entering the Loebner Prize
Competition. In Proceedings of the Twelfth National Conference on Artificial Intelligence.
Shelton, C. R. (2000). Balancing Multiple Sources of Reward in Reinforcement Learning. Submitted
for publication in Neural Information Processing Systems-2000.
Singh, S., Kearns, M., Littman, D., and Walker, M. (2000). Empirical Evaluation of a Reinforcement
Learning Dialogue System. To appear in Proceedings of AAAI-2000.
Sutton, R. S. and Barto, A. G. (1998). Reinforcement Learning: An Introduction. MIT Press, Cambridge, MA.
Sutton, R. S., McAllester, D., Singh, S., and Mansour, Y. (1999). Policy gradient methods for reinforcement learning with function approximation. In Neural Information Processing Systems1999.
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1,228 | 2,119 | Probabilistic principles in unsupervised learning
of visual structure: human data and a model
Shimon Edelman, Benjamin P. Hiles
& Hwajin Yang
Department of Psychology
Cornell University, Ithaca, NY 14853
se37,bph7,hy56 @cornell.edu
Nathan Intrator
Institute for Brain and Neural Systems
Box 1843, Brown University
Providence, RI 02912
Nathan [email protected]
Abstract
To find out how the representations of structured visual objects depend
on the co-occurrence statistics of their constituents, we exposed subjects
to a set of composite images with tight control exerted over (1) the conditional probabilities of the constituent fragments, and (2) the value of Barlow?s criterion of ?suspicious coincidence? (the ratio of joint probability
to the product of marginals). We then compared the part verification response times for various probe/target combinations before and after the
exposure. For composite probes, the speedup was much larger for targets that contained pairs of fragments perfectly predictive of each other,
compared to those that did not. This effect was modulated by the significance of their co-occurrence as estimated by Barlow?s criterion. For
lone-fragment probes, the speedup in all conditions was generally lower
than for composites. These results shed light on the brain?s strategies for
unsupervised acquisition of structural information in vision.
1 Motivation
How does the human visual system decide for which objects it should maintain distinct
and persistent internal representations of the kind typically postulated by theories of object
recognition? Consider, for example, the image shown in Figure 1, left. This image can be
represented as a monolithic hieroglyph, a pair of Chinese characters (which we shall refer
to as and ), a set of strokes, or, trivially, as a collection of pixels. Note that the second
option is only available to a system previously exposed to various combinations of Chinese
characters. Indeed, a principled decision whether to represent this image as ,
or otherwise can only be made on the basis of prior exposure to related images.
According to Barlow?s [1] insight, one useful principle is tallying suspicious coincidences:
two candidate fragments and should be combined into a composite object if the
probability of their joint appearance
is much higher than
, which is
the probability expected in the case of their statistical independence. This criterion may be
compared to the Minimum Description Length (MDL) principle, which has been previously
discussed in the context of object representation [2, 3]. In a simplified form [4], MDL calls
for representing explicitly as a whole if
, just as the principle
of suspicious coincidences does.
While the Barlow/MDL criterion
certainly indicates a suspicious coincidence, there are additional probabilistic considerations that may be used
in setting the degree of association between and . One example is the possi
ble perfect predictability of from and vice versa, as measured by
, then and are perfectly predictive of each
. If
other and should really be coded by a single symbol, whereas the MDL criterion may suggest merely that some association between the representation of and that of be estab
lished. In comparison, if and are not perfectly predictive of each other (
),
there is a case to be made in favor of coding them separately to allow for a maximally
expressive representation, whereas MDL may actually suggest a high degree of association
). In this study we investigated whether the human
(if
visual system uses a criterion based on
alongside MDL while learning (in an unsupervised manner) to represent composite objects.
AB
Figure 1: Left: how many objects are contained in image ? Without prior knowledge, a
reasonable answer, which embodies a holistic bias, should be ?one? (Gestalt effects, which
would suggest two convex ?blobs? [5], are beyond the scope of the present discussion).
Right: in this set of ten images, appears five times as a whole; the other five times
a fragment wholly contained in appears in isolation. This statistical fact provides
grounds for considering to be composite, consisting of two fragments (call the upper
one and the lower one ), because
, but
.
To date, psychophysical explorations of the sensitivity of human subjects to stimulus statistics tended to concentrate on means (and sometimes variances) of the frequency of various
stimuli (e.g., [6]. One recent and notable exception is the work of Saffran et al. [7], who
showed that infants (and adults) can distinguish between ?words? (stable pairs of syllables
that recur in a continuous auditory stimulus stream) and non-words (syllables accidentally
paired with each other, the first of which comes from one ?word? and the second ? from
the following one). Thus, subjects can sense (and act upon) differences in transition probabilities between successive auditory stimuli. This finding has been recently replicated, with
infants as young as 2 months, in the visual sequence domain, using successive presentation
of simple geometric shapes with controlled transition probabilities [8]. Also in the visual
domain, Fiser and Aslin [9] presented subjects with geometrical shapes in various spatial
configurations, and found effects of conditional probabilities of shape co-occurrences, in a
task that required the subjects to decide in each trial which of two simultaneously presented
shapes was more familiar.
The present study was undertaken to investigate the relevance of the various notions of
statistical independence to the unsupervised learning of complex visual stimuli by human
subjects. Our experimental approach differs from that of [9] in several respects. First,
instead of explicitly judging shape familiarity, our subjects had to verify the presence of a
probe shape embedded in a target. This objective task, which produces a pattern of response
times, is arguably better suited to the investigation of internal representations involved in
object recognition than subjective judgment. Second, the estimation of familiarity requires
the subject to access in each trial the representations of all the objects seen in the experi-
ment; in our task, each trial involved just two objects (the probe and the target), potentially
sharpening the focus of the experimental approach. Third, our experiments tested the pre
, and MDL, or Barlow?s
dictions of two distinct notions of stimulus independence:
ratio.
2 The psychophysical experiments
In two experiments, we presented stimuli composed of characters such as those in Figure 1
to nearly 100 subjects unfamiliar with the Chinese script. The conditional probabilities
of the appearance of individual characters were controlled. The experiments involved two
types of probe conditions: P TYPE=Fragment, or
(with
as the
(with
as
reference condition), and P TYPE=Composite, or
reference). In this notation (see Figure 2, left), and are ?familiar? fragments with controlled minimum conditional probability
, and are novel (low-probability)
fragments.
Each of the two experiments consisted of a baseline phase, followed by training exposure
(unsupervised learning), followed in turn by the test phase (Figure 2, right). In the baseline
and test phases, the subjects had to indicate whether or not the probe was contained in the
target (a task previously used by Palmer [5]). In the intervening training phase, the subjects
merely watched the character triplets presented on the screen; to ensure their attention, the
subjects were asked to note the order in which the characters appeared.
V
ABZ
VW
baseline/test
ABZ
target
reference
mask
probe
A
ABZ
AB
ABZ
4
test
3
2
1
probe
target
probe
Fragment
target
unsupervised training
Composite
Figure 2: Left: illustration of the probe and target composition for the two levels of P TYPE
(Fragment and Composite). For convenience, the various categories of characters that
appeared in the experiment are annotated here by Latin letters: , stand for characters
with controlled
, and stand for characters that
appeared only once throughout an experiment. In experiment 1, the training set was con
structed with
for some pairs, and
for others; in experiment 2,
Barlow?s suspicious coincidence ratio was also controlled. Right top: the structure of a
part verification trial (same for baseline and test phases). The probe stimulus was followed
by the target (each presented for
; a mask was shown before and after the target).
The subject had to indicate whether or not the former was contained in the latter (in this
example, the correct answer is yes). A sequence consisting of 64 trials like this one was
presented twice: before training (baseline phase) and after training (test phase). For ?positive? trials (i.e., probe contained in target), we looked at the S PEEDUP following training,
; negative trials were discarded. Right bottom: the
defined as
structure of a training trial (the training phase, placed between baseline and test, consisted
of 80 such trials). The three components of the stimulus appeared one by one for
to make sure that the subject attended to each, then together for
. The subject was
required to note whether the sequence unfolded in a clockwise or counterclockwise order.
!
The logic behind the psychophysical experiments rested on two premises. First, we knew
from earlier work [5] that a probe is detected faster if it is represented monolithically (that
is, considered to be a good ?object? in the Gestalt sense). Second, we hypothesized that a
composite stimulus would be treated as a monolithic object to the extent that its constituent
characters are predictable from each other, as measured by a high conditional probability,
, and/or by a high suspicious coincidence ratio, . The main prediction following
from these premises is that the S PEEDUP (the difference in response time between baseline
and test phases) for a composite probe should reflect the mutual predictability of the probe?s
constituents in the training set. Thus, our hypothesis ? that statistics of co-occurrence
determine the constituents in terms of which structured objects are represented ? would
be supported if the S PEEDUP turns out to be larger for those composite probes whose
constituents tend to appear together in the training set. The experiments, therefore, hinged
on a comparison of the patterns of response times in the ?positive? trials (in which the
probe actually is embedded in the target; see Figure 2, left) before and after exposure to the
training set.
400
Composite
Fragment
analog of speedup
0.3
speedup, ms
300
200
100
Composite
Fragment
0.2
0.1
0
?0.1
0
0.4
minCP
0.6
0.8
?0.2
0.4
1
0.6
0.8
minCP
1
Figure 3: Left: unsupervised learning of statistically defined structure by human subjects,
). The dependent variable S PEED - UP is defined as the difference in
experiment 1 (
between baseline and test phases (least-squares estimates of means and standard errors,
computed by the LSMEANS option of SAS procedure MIXED [10]). The S PEED - UP for
composite probes (solid line) with
exceeded that in the other conditions by
. Right: the results of a simulation of experiment 1 by a model derived from
about
the one described in [4]. The model was exposed to the same 80 training images as the
human subjects. The difference of reconstruction errors for probe and target served as the
analog of RT; baseline measurements were conducted on half-trained networks.
2.1 Experiment 1
Fourteen subjects, none of them familiar with the Chinese writing system, participated in
this experiment in exchange for course credit. Among the stimuli, two characters
. Alternatively, could
could be paired, in which case we had
be unpaired, with
,
(in this experiment, we held the suspicious
coincidence ratio
constant at ). For the paired
the minimum conditional probability
and the
two characters were perfectly predictable from each other, whereas for the unpaired
, and they were not. In the latter case probably should not be represented
as a whole.
!
As expected, we found the value of S PEED - UP to be strikingly different for composite
probes with
(
) compared to the other three conditions (about
);
see Figure 3, left. A mixed-effects repeated measures analysis of variance (SAS procedure
MIXED [10]) for S PEED - UP revealed a marginal effect of P TYPE (
) and a significant interaction P TYPE
interaction (
).
!
principle: S PEEDUP was generThis behavior conforms to the predictions of the
ally higher for composite probes, and disproportionately higher for composite probes with
. The subjects in experiment 1 proved to be sensitive to the
measure
of independence in learning to associate object fragments together. Note that the suspi
.
cious coincidence ratio was the same in both cases,
over and above the (constant-valued) MDLThus, the visual system is sensitive to
related criterion, according to which the propensity to form a unified representation of two
fragments, and , should be determined by [1, 4].
200
200
150
100
50
0.8
minCP
minCP=0.5
150
100
50
0
0.4
1
250
250
200
200
speedup, ms
speedup, ms
0.6
150
100
50
0
0
r=8.33
250
speedup, ms
speedup, ms
r=1.13
250
0
0.4
5
r
0.6
0.8
minCP
minCP=1.0
1
150
100
50
0
0
10
5
r
10
Figure 4: Human subjects, experiment 2 (
). The effect of
found in experiment 1 was modulated in a complicated fashion by the effect of the suspicious coincidence
ratio (see text for discussion).
2.2 Experiment 2
together.
In the second experiment, we studied the effects of varying both and
Because these two quantities are related (through the Bayes theorem), they cannot be manipulated independently. To accommodate this constraint, some subjects saw two sets of
, in the first sesstimuli, with
and with
and with
sion and other two sets, with
,
in the second session; for other subjects, the complementary combinations were used in
each session. Eighty one subjects unfamiliar with the Chinese script participated in this
experiment for course credit.
The results (Figure 4) showed that S PEEDUP was consistently higher for composite probes.
Thus, the association between probe constituents was strengthened by training in each of
the four conditions. S PEEDUP was also generally higher for the high suspicious coinci , and disproportionately higher for composite probes in the
dence ratio case,
case, indicating a complicated synergy between the two mea,
sures of dependence,
and . A mixed-effects repeated measures analysis of variance (SAS procedure MIXED [10]) for S PEED - UP revealed significant main effects of
) and (
P TYPE (
), as well as
) and
two significant two-way interactions,
(
). There was also a marginal three-way interaction,
P TYPE (
P TYPE (
).
!
!
!
The findings of these two psychophysical experiments can be summarized as follows: (1)
an individual complex visual shape (a Chinese character) is detected faster than a composite stimulus (a pair of such characters) when embedded in a 3-character scene, but this
advantage is narrowed with practice; (2) a composite attains an ?objecthood? status to the
extent that its constituents are predictable from each other, as measured either by the conditional probability,
, or by the suspicious coincidence ratio, ; (3) for composites,
the strongest boost towards objecthood (measured by response speedup following unsuper
is high and is low, or vice versa. The nature of
vised learning) is obtained when
this latter interaction is unclear, and needs further study.
3 An unsupervised learning model and a simulated experiment
The ability of our subjects to construct representations that reflect the probability of cooccurrence of complex shapes has been replicated by a pilot version of an unsupervised
learning model, derived from the work of [4]. The model (Figure 5) is based on the following observation: an auto-association network fed with a sequence of composite images
in which some fragment/location combinations are more likely than others develops a nonuniform spatial distribution of reconstruction errors. Specifically, smaller errors appear in
those locations where the image fragments recur. This information can be used to form a
spatial receptive field for the learning module, while the reconstruction error can signal its
relevance to the current input [11, 12].
In the simplified pilot model, the spatial receptive field (labeled in Figure 5, left, as ?relevance mask?) consists of four weights, one per quadrant:
,
. During
the
,
unsupervised training, the weights are updated by setting
where is the reconstruction error in trial , and and are learning constants. In a
simulation of experiment 1, a separate module with its own four-weight ?receptive field?
was trained for each of the composite stimuli shown to the human subjects. 1 The Euclidean
distance between probe and target representations at the output of the model served as the
analog of response time, allowing us to compare the model?s performance with that of the
humans. We found the same differential effects of
for Fragment and Composite probes in the real and simulated experiments; compare Figure 3, left (humans) with
Figure 3, right (model).
1
The full-fledged model, currently under development, will have a more flexible receptive field
structure, and will incorporate competitive learning among the modules.
input
error
input
adapt
?
erri
relevance
mask (RF)
ensemble of modules
auto?
associator
reconstructed
Figure 5: Left: the functional architecture of a fragment module. The module consists of
two adaptive components: a reconstruction network, and a relevance mask, which assigns
different weights to different input pixels. The mask modulates the input multiplicatively,
determining the module?s receptive field. Given a sequence of images, several such modules working in parallel learn to represent different categories of spatially localized patterns
(fragments) that recur in those images. The reconstruction error serves as an estimate of
the module?s ability to deal with the input ([11, 12]; in the error image, shown on the right,
white corresponds to high values). Right: the Chorus of Fragments (CoF) is a bank of
such fragment modules, each tuned to a particular shape category, appearing in a particular
location [13, 4].
4 Discussion
Human subjects have been previously shown to be able to acquire, through unsupervised
learning, sensitivity to transition probabilities between syllables of nonsense words [7] and
between digits [14], and to co-occurrence statistics of simple geometrical figures [9]. Our
results demonstrate that subjects can also learn (presumably without awareness; cf. [14]) to
treat combinations of complex visual patterns differentially, depending on the conditional
probabilities of the various combinations, accumulated during a short unsupervised training
session.
In our first experiment, the criterion of suspicious coincidence between the occurrences
and
conditions: in each case, we
of and was met in both
had
. Yet, the subjects? behavior indicated a significant
holistic bias: the representation they form tends to be monolithic ( ), unless imperfect
mutual predictability of the potential fragments ( and ) provides support for representing them separately. We note that a similar holistic bias, operating in a setting where a
single encounter with a stimulus can make a difference, is found in language acquisition:
an infant faced with an unfamiliar word will assume it refers to the entire shape of the most
salient object [15]. In our second experiment, both the conditional probabilities as such,
and the suspicious coincidence ratio were found to have the predicted effects, yet these
two factors interacted in a complicated manner, which requires a further investigation.
Our current research focuses on (1) the elucidation of the manner in which subjects process
statistically structured data, (2) the development of the model of structure learning outlined
in the preceding section, and (3) an exploration of the implications of this body of work for
wider issues in vision, such as the computational phenomenology of scene perception [16].
References
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[2] R. S. Zemel and G. E. Hinton. Developing population codes by minimizing description length. Neural Computation, 7:549?564, 1995.
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[4] S. Edelman and N. Intrator. A productive, systematic framework for the representation
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Neural Information Processing Systems 13, pages 10?16. MIT Press, 2001.
[5] S. E. Palmer. Hierarchical structure in perceptual representation. Cognitive Psychology, 9:441?474, 1977.
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the induction of category structure. Journal of Experimental Psychology: Learning,
Memory and Cognition, 12:241?256, 1986.
[7] J. R. Saffran, R. N. Aslin, and E. L. Newport. Statistical learning by 8-month-old
infants. Science, 274:1926?1928, 1996.
[8] N. Z. Kirkham, J. A. Slemmer, and S. P. Johnson. Visual statistical learning in infancy:
Evidence for a domain general learning mechanism. Cognition, -:?, 2002. in press.
[9] J. Fiser and R. N. Aslin. Unsupervised statistical learning of higher-order spatial
structures from visual scenes. Psychological Science, 6:499?504, 2001.
[10] SAS. User?s Guide, Version 8. SAS Institute Inc., Cary, NC, 1999.
[11] D. Pomerleau. Input reconstruction reliability estimation. In C. L. Giles, S. J. Hanson, and J. D. Cowan, editors, Advances in Neural Information Processing Systems,
volume 5, pages 279?286. Morgan Kaufmann Publishers, 1993.
[12] I. Stainvas and N. Intrator. Blurred face recognition via a hybrid network architecture.
In Proc. ICPR, volume 2, pages 809?812, 2000.
[13] S. Edelman and N. Intrator. (Coarse Coding of Shape Fragments) + (Retinotopy)
Representation of Structure. Spatial Vision, 13:255?264, 2000.
[14] G. S. Berns, J. D. Cohen, and M. A. Mintun. Brain regions responsive to novelty in
the absence of awareness. Science, 276:1272?1276, 1997.
[15] B. Landau, L. B. Smith, and S. Jones. The importance of shape in early lexical
learning. Cognitive Development, 3:299?321, 1988.
[16] S. Edelman. Constraints on the nature of the neural representation of the visual world.
Trends in Cognitive Sciences, 6:?, 2002. in press.
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Pineda
Time DependentAdaptive Neural Networks
Fernando J. Pineda
Center for Microelectronics Technology
Jet Propulsion Laboratory
California Institute of Technology
Pasadena, CA 91109
ABSTRACT
A comparison of algorithms that minimize error functions to train the
trajectories of recurrent networks, reveals how complexity is traded off for
causality. These algorithms are also related to time-independent
fonnalisms. It is suggested that causal and scalable algorithms are
possible when the activation dynamics of adaptive neurons is fast
compared to the behavior to be learned. Standard continuous-time
recurrent backpropagation is used in an example.
1 INTRODUCTION
Training the time dependent behavior of a neural network model involves the minimization
of a function that measures the difference between an actual trajectory and a desired
trajectory. The standard method of accomplishing this minimization is to calculate the
gradient of an error function with respect to the weights of the system and then to use the
gradient in a minimization algorithm (e.g. gradient descent or conjugate gradient).
Techniques for evaluating gradients and performing minimizations are well developed in the
field of optimal control and system identification, but are only now being introduced to the
neural network community. Not all algorithms that are useful or efficient in control problems
are realizable as physical neural networks. In particular, physical neural network algorithms
must satisfy locality, scaling and causality constraints. Locality simply is the constraint that
one should be able to update each connection using only presynaptic and postsynaptic
infonnation. There should be no need to use infonnation from neurons or connections that
are not in physical contact with a given connection. Scaling, for this paper, refers to the
Time Dependent Adaptive Neural Networks
scaling law that governs the amount of computation or hardware that is required to perform
the weight updates. For neural networks, where the number of weights can become very
large, the amount of hardware or computation required to calculate the gradient must scale
linearly with the number of weights. Otherwise, large networks are not possible. Finally,
learning algorithms must be causal since physical neural networks must evolve forwards in
time. Many algorithms for learning time-dependent behavior, although they are seductively
elegant and computationally efficient, cannot be implemented as physical systems because
the gradient evaluation requires time evolution in two directions. In this paper networks that
violate the causality constraint will be referred to as unphysical.
It is useful to understand how scalability and causality trade off in various gradient evaluation
algorithms. In the next section three related gradient evaluation algorithms are derived and
their scaling and causality properties are compared. The three algorithms demonstrate a
natural progression from a causal algorithm that scales poorly to an a causal algorithm that
scales linearly.
The difficulties that these exact algorithms exhibit appear to be inescapable. This suggests
that approximation schemes that do not calculate exact gradients or that exploit special
properties of the tasks to-be-Ieamed may lead to physically realizable neural networks. The
final section of this paper suggests an approach that could be exploited in systems where the
time scale of the to-be-Ieamed task is much slower than the relaxation time scale of the
adaptive neurons.
2 ANALYSIS OF ALGORITHMS
We will begin by reviewing the learning algorithms that apply to time-dependent recurrent
networks. The control literature generally derives these algorithms by taking a variational
approach (e.g. Bryson and Ho, 1975). Here we will take a somewhat unconventional
approach and restrict oursel yes to the domain of differential equations and their solutions. To
begin with, let us take a concrete example. Consider the neural system given by the equation
,
dx?
(it
=X i+
n
~w
I(x) + I
,=1
I
(1)
Where f(.) is a sigmoid shaped function (e.g. tanh(.)) and ~is an external input This system
is a well studied neural model (e.g. Aplevich, 1968; Cowan, 1967; Hopfield, 1984; Malsburg,
1973; Sejnowski, 1977). The goal is to find the weight matrix w that causes the states x(t)
of the output units to follow a specified trajectory x(t). The actually trajectory depends not
only on the weight matrix but also on the external input vector I. To find the weights one
minimizes a measure of the difference between the actual trajectory x(t) and the desired
trajectory ~(t). This measure is a functional of the trajectories and a function of the weights.
It is given by
tI
1
2
E(w ,t I,t ) =2
dt
,{t) - ~,{t))
(2)
f
.L
,e 0
(x
t
o
where 0 is the set of output units. We shall, only for the purpose of algorithm comparison,
711
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Pineda
make the following assumptions: (1) That the networks are fully connected (2) That all the
interval [tD,tr] is divided into q segments with numerical integrations performed using the
Euler method and (3) That all the operations are performed with the same precision. This will
allow us to easily estimate the amount of computation and memory required for each
algorithm relative to the others.
2.1 ALGORITHM A
If the objective function E is differentiated with respect to wn one obtains
-
aE
aw
=rs
Lnft!d t J i(t) P irit )
t
i=1
where
and where
o
gi(t)- x i(t)
J.= {
(3a)
'0
if i E 0
ififl.O
ax ,?
Pirs=-a-
(3b)
(3c)
Wrs
To evaluate Pirs' differentiate equation (1) with respect to wn and observe that the time
derivative and the partial derivative with respect to wn commute. The resulting equation is
dp irs ~L ( )
- d = ~ ij'X j Pjrs+Sir.
where
t
.1
(4a)
J=
(4b)
and where
(4c)
=
The initial condition for eqn. (4a) is p(t) O. Equations (1), (3) and (4) can be used to
calculate the gradient for a learning rule. This is the approach taken by Williams and Zipser
(1989) and also discussed by Pearlmutter(1988). Williams and Zipser further observe that
one can use the instantaneous value of p(t) and J(t) to update the weights continually
provided the weights change slowly. The computationally intensive part of this algorithm
occurs in the integration of equation (4a). There are n3 components to p hence there are Ji3
equations . Accordingly the amount of hardware or memory required to perform the
calculation will scale like n 3? Each of these equations requires a summation over all the
neurons, hence the amount of computation (measured in multiply-accumulates) goes like It
per time step, and there are q time steps, hence the total number of multiply-accumulates
scales like n4q Clearly, the scaling properties of this approach are very poor and it cannot
be practically applied to very large networks.
2.2 ALGORITHM B
Rather than numerically integrate the system of equations (4a) to obtain p(t), suppose we
write down the formal solution. This solution is
Time Dependent Adaptive Neural Networks
11
Pirs(t)='LKij(t,to)PjrsCt
0)+
j=1
'L"f'drKjj(t,f)Sjrs(i)
j=1
'0
(Sa)
The matrix K is defined by the expression
p(.r.. '~T
,) = ex
K (' 2'
L (x (T)))
(5b)
This matrix is known as the propagator or transition matrix. The expression for Pit. consists
of a homogeneous solution and a particular solution. The choice of initial condition Pirs(to)
0 leaves only the particular solution. If the particular solution is substituted back into eqn.
(3a), one eventually obtains the following expression for the gradient
=
aE
11
- = - 'Lf
'f
f '
d-r J;Ct)K irU ,-r)f(x s(-r))
(6)
rs
j=1
'0
'0
To obtain this expression one must observe that s.In can be expressed in terms of x? , i.e. use
eqn. (4c). This allows the summation over j to be performed trivially, thus resulting in
eqn.(6). The familiar outer product form of backpropagation is not yet manifest in this
expression. To uncover it, change the order of the integrations. This requires some care
because the limits of the integration are not the same. The result is
aw
aE
'L
11
-=-
aw
dt
f
If
d-rf
If
dt Jj(t)K irU ,-r)f(x sC-r))
(7)
rs
i=1
'0
l'
Inspection ofthis expression reveals that neither the summation over i nor the integration over
't includes x.(t), thus it is useful to factor it out. Consequently equation (7) takes on the
familiar outer product form of backpropagation
aE
If
-= -f
aw
rs
dt Y r(t)f(x sU))
(8)
l'
Where yr(t) is defined to be
If
11
Y r(-r) =-
'L f
i= 1
dt Jj(t)K irU ,-r)
(9)
t'
Equation (8), defines an expression for the gradient, provided we can calculate Yr(t) from eqn.
(9). In principle, this can be done since the propagator K and the vector J are both completely
determined by x(t). The computationally intensive part of this algorithm is the calculation
of K(t, 't) for all values of t and't. The calculation requires the integration of equations of the
form
(10)
dK ,-r) - L (x U) K (t ,-r)
i:
for q different values of't. There are n2different equations to integrate for each value of't
Consequently there are n2q integrations to be performed where the interval from to to tf is
divided into q intervals. The calculation of all the components ofK(t,'t), from tr to t ,scales
like n3q2, since each integration requires n multiply-accumulates per time step and there are
q time steps. Similarly, the memory requirements scale like n2q2. This is because K has n2
components for each (t,'t) pair and there are q2 such pairs.
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Equation (10) must be integrated forwards in time from t= 't to t = trand backwards in time
from t= 't to t = to. This is because K must satisfy K( 't?'t) = 1 (the identity matrix) for all
'to This condition follows from the definition of K eqn. (5b). Finally, we observe that
expression (9) is the time-dependent analog of the expression used by Rohwer and Forrest
(1987) to calculate the gradient in recurrent networks. The analogy can be made somewhat
more explicit by writingK(t,'t) as the inverse K-l('t,t). Thus we see that y( t) can be expressed
in terms of a matrix inverse just as in the Rohwer and Forrest algorithm.
2.3
ALGORITHM C
The final algorithm is familiar from continuous time optimal control and identification. The
algorithm is usually derived by performing a variation on the functional given by eqn. (2).
This results in a two-point boundary value problem. On the other hand, we know that y is
given by eqn. (9). So we simply observe that this is the particular solution of the differential
equation
dy
T
- ([t= L (x (t))y +J
(11)
Where LT is the transpose of the matrix defined in eqn. (4b). To see this simply substitute
the form for y into eqn. (11) and verify that it is indeed the solution to the equation.
The particular solution to eqn. (11) vanishes only if y(1r) = O. In other words: to obtain yet)
we need only integrate eqn. (11) backwards from the final condition y(t~ = O. This is just
the algorithm introduced to the neural network community by Pearlmutter (1988). This also
corresponds to the unfolding in time approach discussed by Rumelhart et al. (1986), provided
that all the equations are discretized and one takes At = 1.
The two point boundary value problem is rather straight forward to solve because the
equation for x(t) is independent of yet). Both x(t) and yet) can be obtained with n multiplyaccumulates per time step. There are q time steps from to to tfand bothx(t) and yet) have n
components, hence the calculation of x(t) and yet) scales like 02q. The weight update
equation also requires n2q mUltiply- accumulates. Thus the computational requirements of
the algorithm as a whole scale like n2q The memory required also scales like n2q, since it
is necessary to save each value of x(t) along the trajectory to compute yet).
2.4
SCALING VS CAUSALITY
The results of the previous sections are summarized in table 1 below. We see that we have
a progression of tradeoffs between scaling and causality. That is, we must choose between
a causal algorithm with exploding computational and storage requirements and an a causal
algorithm with modest storage requirements. There is no q dependence in the memory
requirments because the integral given in eqn. (3a) can be accumulated at each time step.
Algorithm B has some of the worst features of both algorithms.
Time Dependent Adaptive Neural Networks
Table 1: Comparison of three algorithms
Algorithm
A
B
C
Memory
Multiply
-accumulates
diirection of integations
x and p are both forward in time
x is forward, K is forward and backward
x is forward, y is backward in time.
Digital hardware has no difficulties (at least over finite time intervals) with a causal
algorithms provided a stack is available to act as a memory that can recall states in reverse
order. To the extent that the gradient calculations are carried out on digital machines, it makes
sense to use algorithm C because it is the most efficient. In analog VLSI however, it is
difficult to imagine how to build a continually running network that uses an a causal
algorithm. Algorithm A is attractive for physical implementation because it could be run
continually and in real time (Williams and Zipser, 1989). However, its scaling properties
preclude the possibility of building very large networks based on the algorithm. Recently,
Zipser (1990) has suggested that a divide and conquer approach may reduce the
computational and spatial complexity of the algorithm. This approach, although promising,
does not always work and there is as yet no convergence proof. How then, is it possible to
learn trajectories using local, scalable and causal algorithms? In the next section a possible
avenue of attack is suggested.
3 EXPLOITING DISPARATE TIME SCALES
I assert that for some classes of problems there are scalable and causal algorithms that
approximate the gradient and that these algorithms can be found by exploiting the disparity
in time scales found in these classes of problems. In particular, I assert that when the time
scale of the adaptive units is fast compared to the time scale of the behavior to be learned, it
is possible to find scalable and causal adaptive algorithms. A general formalism for doing
this will not be presented here, instead a simple, perhaps artificial, example will be presented.
This example minimizes an error function for a time dependent problem.
It is likely that trajectory generation in motor control problems are of this type. The
characteristic time scales of the trajectories that need to be generated are determined by
inertia and friction. These mechanical time scales are considerably longer than the electronic
time scales that occur in VLSI. Thus it seems that for robotic problems, there may be no need
to use the completely general algorithms discussed in section 2. Instead, algorithms that take
advantage of the disparity between the mechanical and the electronic time scales are likely
to be more useful for learning to generate trajectories.
he task is to map from a periodic input I(t) to a periodic output ~(t). The basic idea is to use
the continuous-time recurrent-backpropagation approach with slowly varying timedependent inputs rather than with static inputs. The learning is done in real-time and in a
continuous fashion. Consider a set of n "fast" neurons (i= 1,.. ,n) each of which satisfies the
715
716
Pineda
additive activation dynamics determined by eqn (1). Assume that the initial weights are
sufficientl y small that the dynamics of the network would be convergent if the inputs I were
constant. The external input vector ~ is applied to the network through the vector I. It has
been previously shown (pineda, 1988) that the ij-th component of the gradient ofE is equal
to yfjf(xf) where Xfj is the steady state solution of eqn. (1) and where yfjis a component of
the steady state solution of
dy
T
f
(12)
- = L (x )y +1
dt
where the components ofLT are given by eqn. (4.b). Note that the relative sign between
equations (11) and (12) is what enables this algorithm to be causal. Now suppose that instead
of a fixed input vector I, we use a slowly varying input I(t/'t ) where't is the characteristic
time scale over which the input changes significantly. If w~ take as lite gradient descent
algorithm, the dynamics defined by
dw
rs
't'w([t=Y i(t)X /t)
(13)
where't.. is the time constant that defines the (slow) time scale over which w changes and
where Xj is the instantaneous solution of eqn. (1) and Yj is the instantaneous solution of
eqn.(12) . Then in the adiabatic limit the Cartesian product yl(x) in eqn. (13) approximates
the negative gradient of the objective function E, that is
(14)
This approach can map one continuous trajectory into another continuous trajectory,
provided the trajectories change slowly enough. Furthermore, learning occurs causally and
scalably. There is no memory in the model, i.e. the output of the adaptive neurons depends
only on their input and not on their internal state. Thus, this network can never learn to
perform tasks that require memory unless the learning algorithm is modified to learn the
appropriate transitions. This is the major drawback of the adiabatic approach. Some state
information can be incorporated into this model by using recurrent connections - in which
case the network can have multiple basins and the final state will depend on the initial state
of the net as well as on the inputs, but this will not be pursued here.
Simple simulations were performed to verify that the approach did indeed perform gradient
descent. One simulation is presented here for the benefit of investigators who may wish to
verify the results. A feedforward network topology consisting of two input units, five hidden
units and two output units was used for the adaptive network. Units were numbered
sequentially, 1 through 9, beginning with the input layer and ending in the output layer. Time
dependent external inputs for the two input neurons were generated with time dependence
II = sin(27tt) and ~ = cos(2m). The targets for the output neurons were ~ = R sin(27tt) and
~9 =R cos(2m) where R = 1.0 + 0.lsin(6m). All the equations were simultaneously integrated
using 4th order Runge-Kutta with a time step of 0.1. A relaxation time scale was introduced
into the forward and backward propagation equations by multiplying the time derivatives in
eqns. (1) and (12) by't" and 'tyrespectively. These time scales were set to't" ='ty= 0.5. The
adaptive time scale of the weights was 't.. = 1.0. The error in the network was initially, E =
Time Dependent Adaptive Neural Networks
10 and the integration was cut off when the error reached a plateau at E = 0.12. The learning
curve is shown in Fig. 1. The trained trajectory did not exactly reach the desired solution. In
particular the network did not learn the odd order hannonic that modulates R. By way of
comparison, a conventional backpropagation approach that calculated a cumulative gradient
over the trajectory and used conjugate gradient for the descent, was able to converge to the
global minimum.
12,---------------------------------~
10-'
III
III
8 6 4-
m
m
m
m
m
m
ED
m
2 -
.
O+-__~---~~??~??E??~?.B..~..B..B. .~?.m??D?~???~~
o
I
I
10
20
I
30
40
50
Time
Figure 1: Learning curve. One time unit corresponds to a single oscillation
4 SUMMARY
The key points of this paper are: 1) Exact minimization algorithms for learning timedependent behavior either scale poorl y or else violate causality and 2) Approximate gradient
calculations will likely lead to causal and scalable learning algorithms. The adiabatic
approach should be useful for learning to generate trajectories of the kind encountered when
learning motor skills.
References herein to any specific commercial product, process, or service by trade name, trademark, manufacturer,
or otherwise, does not constitute or imply any endorsement by the Uoited States Government or the Jet Propulsion
Laboratory, California Institute of Technology. The work described in this paper was carried out at the
Center for Space Microelectonrics Technology, Jet Propulsion Laboratory, California Institute of
Technology. Support for the work came from the Air Force Office of Scientific Research through an
agreement with the National Aeronautics and Space Administration (AFOSR-ISSA-90-0027).
REFERENCES
Aplevich,J.D. (1968). Models of certain nonlinear systems. InE.R.Caianiello(Ed.),Neural
Networks, (pp. 110-115), Berlin: Springer Verlag.
Bryson, A. E. and Ho, Y. (1975). Applied Optimal Control: Optimization. Estimation. and
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Control. New York: Hemisphere Publishing Co.
Cowan, J. D. (1967). A mathematical theory of central nervous activity. Unpublished
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Hopfield, J. J. (1984). Neurons with graded response have collective computational
properties like those of two-state neurons. Proc. Nat. Acad. Sci. USA, Bio., ..8.l. 3088-3092.
Malsburg, C. van der (1973). Self-organization of orientation sensitive cells in striate cortex,
Kybernetic, 14,85-100.
Pearlmutter, B. A. (1988), Learning state space trajectories in recurrent neural networks: A
preliminary report, (Tech. Rep. AlP-54), Department of Computer Science , Carnegie Mellon
University, Pittsburgh, PA
Pineda, F. J. (1988). Dynamics and Architecture for Neural Computation. Journal of
Complexity,~, (pp.216-245)
Rowher R, R. and Forrest, B. (1987). Training time dependence in neural networks, In M.
CaudilandC.Butler,(Eds.),ProceedingsoftheIEEEFirstAnnuallnternationalConference
on Neural Networks, ~, (pp. 701-708). San Diego, California: IEEE.
Rumelhart, D. E., Hinton, G. E., and Willaims, R.J. (1986). Learning Internal
Representations by Error Propagation. In D. E. Rumelhart and J. L. McClelland, (Eds.),
Parallel Distributed Processing, (pp. 318-362). Cambridge: M.LT. Press.
Sejnowski, T. J. (1977). Storing covariance with nonlinearly interacting neurons. Journal
of Mathematical Biology, ~,303 .. 321.
Williams, R.I. and Zipser, D. (1989). A learning algorithm for continually running
fully recurrent neural networks. Neural Computation, 1, (pp. 270-280).
Zipser, D. (1990). Subgrouping reduces complexity and speeds up learning in recurrent
networks, (this volume).
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1,230 | 2,120 | Bayesian Predictive Profiles with
Applications to Retail Transaction Data
Igor V. Cadez
Information and Computer Science
University of California
Irvine, CA 92697-3425, U.S.A.
[email protected]
Padhraic Smyth
Information and Computer Science
University of California
Irvine, CA 92697-3425, U.S.A.
[email protected]
Abstract
Massive transaction data sets are recorded in a routine manner in
telecommunications, retail commerce, and Web site management.
In this paper we address the problem of inferring predictive individual profiles from such historical transaction data. We describe a generative mixture model for count data and use an an
approximate Bayesian estimation framework that effectively combines an individual?s specific history with more general population
patterns. We use a large real-world retail transaction data set to
illustrate how these profiles consistently outperform non-mixture
and non-Bayesian techniques in predicting customer behavior in
out-of-sample data.
1
Introduction
Transaction data sets consist of records of pairs of individuals and events, e.g., items
purchased (market basket data), telephone calls made (call records), or Web pages
visited (from Web logs). Of significant practical interest in many applications is
the ability to derive individual-specific (or personalized) models for each individual from the historical transaction data, e.g., for exploratory analysis, adaptive
personalization, and forecasting.
In this paper we propose a generative model based on mixture models and Bayesian
estimation for learning predictive profiles. The mixture model is used to address
the heterogeneity problem: different individuals purchase combinations of products
on different visits to the store. The Bayesian estimation framework is used to
address the fact that we have different amounts of data for different individuals.
For an individual with very few transactions (e.g., only one) we can ?shrink? our
predictive profile for that individual towards a general population profile. On the
other hand, for an individual with many transactions, their predictive model can be
more individualized. Our goal is an accurate and computationally efficient modeling
framework that smoothly adapts a profile to each individual based on both their own
historical data as well as general population patterns. Due to space limitations only
selected results are presented here; for a complete description of the methodology
and experiments see Cadez et al. (2001).
The idea of using mixture models as a flexible approach for modeling discrete and
categorical data has been known for many years, e.g., in the social sciences for latent
class analysis (Lazarsfeld and Henry, 1968). Traditionally these methods were only
applied to relatively small low-dimensional data sets. More recently there has been
a resurgence of interest in mixtures of multinomials and mixtures of conditionally
independent Bernoulli models for modeling high-dimensional document-term data
in text analysis (e.g., McCallum, 1999; Hoffman, 1999). The work of Heckerman
et al. (2000) on probabilistic model-based collaborative filtering is also similar in
spirit to the approach described in this paper except that we focus on explicitly
extracting individual-level profiles rather than global models (i.e., we have explicit
models for each individual in our framework). Our work can be viewed as being an
extension of this broad family of probabilistic modeling ideas to the specific case
of transaction data, where we deal directly with the problem of making inferences
about specific individuals and handling multiple transactions per individual. Other
approaches have also been proposed in the data mining literature for clustering
and exploratory analysis of transaction data, but typically in a non-probabilistic
framework (e.g., Agrawal, Imielinski, and Swami, 1993; Strehl and Ghosh, 2000;
Lawrence et al., 2001). The lack of a clear probabilistic semantics (e.g., for association rule techniques) can make it difficult for these models to fully leverage the
data for individual-level forecasting.
2
Mixture-Basis Models for Profiles
We have an observed data set D = {D1 , . . . , DN }, where Di is the observed data on
the ith customer, 1 ? i ? N . Each individual data set Di consists of one or more
transactions for that customer , i.e., Di = {yi1 , . . . , yij , . . . , yini }, where yij is the
jth transaction for customer i and ni is the total number of transactions observed
for customer i.
The jth transaction for individual i, yij , consists of a description of the set of
products (or a ?market basket?) that was purchased at a specific time by customer
i (and yi will be used to denote an arbitrary transaction from individual i). For
the purposes of the experiments described in this paper, each individual transaction
yij is represented as a vector of d counts yij = (nij1 , . . . nijc , . . . , nijC ), where nijc
indicates how many items of type c are in transaction yij , 1 ? c ? C.
We define a predictive profile as a probabilistic model p(yi ), i.e., a probability
distribution on the items that individual i will purchase during a store-visit. We
propose a simple generative mixture model for an individual?s purchasing behavior,
namely that a randomly selected transaction yi from individual i is generated by one
of K components in a K-component mixture model. The kth mixture component,
1 ? k ? K is a specific model for generating the counts and we can think of each of
the K models as ?basis functions? describing prototype transactions. For example,
in a clothing store, one might have a mixture component that acts as a prototype
for suit-buying behavior, where the expected counts for items such as suits, ties,
shirts, etc., given this component, would be relatively higher than for the other
items.
There are several modeling choices for the component transaction models for generating item counts. In this paper we choose a particularly simple memoryless
multinomial model that operates as follows. Conditioned on nij (the total number
of items in the basket) each of the individual items is selected in a memoryless
fashion by nij draws from a multinomial distribution Pk = (?k1 , . . . , ?kC ) on the C
possible items, ?kj = 1.
Probability
0.6
0.6
COMPONENT 1
0.4
0
0.2
0
10
20
30
40
50
Probability
0.6
COMPONENT 3
10
20
30
40
50
COMPONENT 4
0.2
0
10
20
30
40
50
0.6
Probability
0
0.4
0.2
0
0
10
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30
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50
0.6
COMPONENT 5
0.4
COMPONENT 6
0.4
0.2
0
0
0.6
0.4
0
COMPONENT 2
0.4
0.2
0.2
0
10
20
30
Department
40
50
0
0
10
20
30
Department
40
50
Figure 1: An example of 6 ?basis? mixture components fit to retail transaction
data.
Figure 1 shows an example of K = 6 such basis mixture components that have
been learned from a large retail transaction data (more details on learning will be
discussed below). Each window shows a different set of component probabilities
Pk , each modeling a different type of transaction. The components show a striking
bimodal pattern in that the multinomial models appear to involve departments
that are either above or below department 25, but there is very little probability
mass that crosses over. In fact the models are capturing the fact that departments
numbered lower than 25 correspond to men?s clothing and those above 25 correspond
to women?s clothing, and that baskets tend to be ?tuned? to one set or the other.
2.1
Individual-Specific Weights
We further assume that for each individual i there exists a set of K weights,
and in the general case
P these weights are individual-specific, denoted by ?i =
(?i1 , . . . , ?iK ), where k ?ik = 1. Weight ?ik represents the probability that when
individual i enters the store their transactions will be generated by component k.
Or, in other words, the ?ik ?s govern individual i?s propensity to engage in ?shopping
behavior? k (again, there are numerous possible generalizations, such as making the
?ik ?s have dependence over time, that we will not discuss here). The ?ik ?s are in
effect the profile coefficients for individual i, relative to the K component models.
This idea of individual-specific weights (or profiles) is a key component of our proposed approach. The mixture component models Pk are fixed and shared across
all individuals, providing a mechanism for borrowing of strength across individual
data. The individual weights are in principle allowed to freely vary for each individual within a K-dimensional simplex. In effect the K weights can be thought as
basis coefficients that represent the location of individual i within the space spanned
by the K basis functions (the component Pk multinomials). This approach is quite
similar in spirit to the recent probabilistic PCA work of Hofmann (1999) on mixture
models for text documents, where he proposes a general mixture model framework
that represents documents as existing within a K-dimensional simplex of multinomial component models.
The model for each individual is an individual-specific mixture model, where the
Number of items
8
TRAINING PURCHASES
6
4
2
0
0
5
10
15
20
25
30
35
40
45
50
40
45
50
Number of items
8
TEST PURCHASES
6
4
2
0
0
5
10
15
20
25
Department
30
35
Figure 2: Histograms indicating which products a particular individual purchased,
from both the training data and the test data.
0.2
Probability
PROFILE FROM GLOBAL WEIGHTS
0.15
0.1
0.05
0
0
5
10
15
20
25
30
35
40
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0.2
Probability
SMOOTHED HISTOGRAM PROFILE (MAP)
0.15
0.1
0.05
0
0
5
10
15
20
25
30
35
0.2
Probability
PROFILE FROM INDIVIDUAL WEIGHTS
0.15
0.1
0.05
0
0
5
10
15
20
25
30
35
Figure 3: Inferred ?effective? profiles from global weights, smoothed histograms, and
individual-specific weights for the individual whose data was shown in Figure 2.
weights are specific to individual i:
p(yij )
=
K
X
?ik p(yij |k)
k=1
=
K
X
k=1
?ik
C
Y
n
?kcijc .
c=1
where ?kc is the probability that the cth item is purchased given component k
and nijc is the number of items of category c purchased by individual i, during
transaction ij.
As an example of the application of these ideas, in Figure 2 the training data and
test data for a particular individual are displayed. Note that there is some predictability from training to test data, although the test data contains (for example)
a purchase in department 14 (which was not seen in the training data). Figure 3
plots the effective profiles1 for this particular individual as estimated by three different schemes in our modeling approach: (1) global weights that result in everyone
1
We call these ?effective profiles? since the predictive model under the mixture assump-
being assigned the same ?generic? profile, i.e., ?ik = ?k , (2) a maximum a posteriori (MAP) technique that smooths each individual?s training histogram with
a population-based histogram, and (3) individual weights estimated in a Bayesian
fashion that are ?tuned? to the individual?s specific behavior. (More details on
each of these methods are provided later in the paper; a complete description can
be found in Cadez et al. (2001)).
One can see in Figure 3 that the global weight profile reflects broad population-based
purchasing patterns and is not representative of this individual. The smoothed
histogram is somewhat better, but the smoothing parameter has ?blurred? the
individual?s focus on departments below 25. The individual-weight profile appears
to be a better representation of this individual?s behavior and indeed it does provide
the best predictive score (of the 3 methods) on the test data in Figure 2. Note that
the individual-weights profile in Figure 3 ?borrows strength? from the purchases of
other similar customers, i.e., it allows for small but non-zero probabilities of the
individual making purchases in departments (such as 6 through 9) if he or she has
not purchased there in the past. This particular individual?s weights, the ?ik ?s, are
(0.00, 0.47, 0.38, 0.00, 0.00.0.15) corresponding to the six component models shown
in Figure 1. The most weight is placed on components 2, 3 and 6, which agrees
with our intuition given the individual?s training data.
2.2
Learning the Model Parameters
The unknown parameters in our model consist of both the parameters of the K
multinomials, ?kc , 1 ? k ? K, 1 ? c ? C, and the vectors of individual-specific
profile weights ?i , 1 ? i ? N . We investigate two different approaches to learning
individual-specific weights:
? Mixture-Based Maximum Likelihood (ML) Weights: We treat the
weights ?i and component parameters ? as unknown and use expectationmaximization (EM) to learn both simultaneously. Of course we expect this
model to overfit given the number of parameters being estimated but we
include it nonetheless as a baseline.
? Mixture-Based Empirical Bayes (EB) Weights: We first use EM
to learn a mixture of K transaction models (ignoring individuals). We
then use a second EM algorithm in weight-space to estimate individualspecific weights ?i for each individual. The second EM phase uses a fixed
empirically-determined prior (a Dirichlet) for the weights. In effect, we are
learning how best to represent each individual within the K-dimensional
simplex of basis components. The empirical prior uses the marginal weights
(??s) from the first run for the mean of the Dirichlet, and an equivalent
sample size of n = 10 transactions is used in the results reported in the
paper. In effect, this can be viewed as an approximation to either a fully
Bayesian hierarchical estimation or an empirical Bayesian approach (see
Cadez et al. (2001) for more detailed discussion). We did not pursue the
fully Bayesian or empirical Bayesian approaches for computational reasons
since the necessary integrals cannot be evaluated in closed form for this
model and numerical methods (such as Markov Chain Monte Carlo) would
be impractical given the data sizes involved.
We also compare two other approaches for profiling for comparison: (1) Global
Mixture Weights: instead of individualized weights we set each individual?s
tion is not a multinomial that can be plotted as a bar chart: however, we can approximate
it and we are plotting one such approximation here
3.5
3.4
Negative LogP Score [bits/token]
3.3
Individualized MAP weights
3.2
3.1
Mixtures: Individualized ML weights
3
Mixtures: Global mixture weights
2.9
2.8
2.7
Mixtures: Individualized EB weights
2.6
2.5
0
10
20
30
40
50
60
70
Number of Mixture Components [k]
80
90
100
Figure 4: Plot of the negative log probability scores per item (predictive entropy) on
out-of-sample transactions, for various weight models as a function of the number
of mixture components K.
weight vector to the marginal weights (?i k = ?k ), and (2) Individualized MAP
weights: a non-mixture approach where we use an empirically-determined Dirichlet prior directly on the multinomials, and where the equivalent sample size of this
prior was ?tuned? on the test set to give optimal performance. This provides an (optimistic) baseline of using multinomial profiles directly, without use of any mixture
models.
3
Experimental Results
To evaluate our approach we used a real-world transaction data set. The data
consists of transactions collected at a chain of retail stores over a two-year period.
We analyze the transactions here at the store department level (50 categories of
items). We separate the data into two time periods (all transactions are timestamped), with approximately 70% of the data being in the first time period (the
training data) and the remainder in the test period data. We train our mixture and
weight models on the first period and evaluate our models in terms of their ability
to predict transactions that occur in the subsequent out-of-sample test period.
The training data contains data on 4339 individuals, 58,866 transactions, and
164,000 items purchased. The test data consists of 4040 individuals, 25,292 transactions, and 69,103 items purchased. Not all individuals in the test data set appear
in the training data set (and vice-versa): individuals in the test data set with no
training data are assigned a global population model for scoring purposes.
To evaluate the predictive power of each model, we calculate the log-probability
(?logp scores?) of the transactions as predicted by each model. Higher logp scores
mean that the model assigned higher probability to events that actually occurred.
Note that the mean negative logp score over a set of transactions, divided by the
total number of items, can be interpreted as a predictive entropy term in bits. The
lower this entropy term, the less uncertainty in our predictions (bounded below by
zero of course, corresponding to zero uncertainty).
Figure 4 compares the out-of-sample predictive entropy scores as a function of the
0
?50
?55
?50
?60
?100
logP, individual weights
logP, individual weights
?65
?150
?200
?250
?70
?75
?80
?85
?300
?90
?350
?95
?400
?400
?350
?300
?250
?200
?150
logP, global weights
?100
?50
0
?100
?100
?95
?90
?85
?80
?75
?70
logP, global weights
?65
?60
?55
?50
Figure 5: Scatter plots of the log probability scores for each individual on out-ofsample transactions, plotting log probability scores for individual weights versus log
probability scores for the global weights model. Left: all data, Right: close up.
number of mixture components K for the mixture-based ML weights, the mixturebased Global weights (where all individuals are assigned the same marginal mixture
weights), the mixture-based Empirical Bayes weights, and the non-mixture MAP
histogram method (as a baseline). The mixture-based approaches generally outperform the non-mixture MAP histogram approach (solid line). The ML-based mixture
weights start to overfit after about 6 mixture components (as expected). The Global
mixture weights and individualized mixture weights improve up to about K = 50
components and then show some evidence of overfitting. The mixture-based individual weights method is systematically the best predictor, providing a 15% decrease
in predictive entropy compared to the MAP histogram method, and a roughly 3%
decrease compared to non-individualized global mixture weights.
Figure 5 shows a more detailed comparison of the difference between individual
mixtures and the Global profiles, on a subset of individuals. We can see that the
Global profiles are systematically worse than the individual weights model (i.e., most
points are above the bisecting line). For individuals with the lowest likelihood (lower
left of the left plot) the individual weight model is consistently better: typically
lower weight total likelihood individuals are those with more transactions and items.
In Cadez et al. (2001) we report more detailed results on both this data set and a
second retail data set involving 15 million items and 300,000 individuals. On both
data sets the individual-level models were found to be consistently more accurate
out-of-sample compared to both non-mixture and non-Bayesian approaches. We
also found (empirically) that the time taken for EM to converge is roughly linear
as both a function of number of components and the number of transactions (plots
are omitted due to lack of space), allowing for example fitting of models with 100
mixture components to approximately 2 million baskets in a few hours.
4
Conclusions
In this paper we investigated the use of mixture models and approximate Bayesian
estimation for automatically inferring individual-level profiles from transaction data
records. On a real-world retail data set the proposed framework consistently outperformed alternative approaches in terms of accuracy of predictions on future unseen
customer behavior.
Acknowledgements
The research described in this paper was supported in part by NSF award IRI9703120. The work of Igor Cadez was supported by a Microsoft Graduate Research
Fellowship.
References
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of items in large databases, Proceedings of the ACM SIGMOD Conference on
Management of Data (SIGMOD?98), New York: ACM Press, pp. 207?216.
Cadez, I. V., Smyth, P., Ip, E., Mannila, H. (2001) Predictive profiles for transaction
data using finite mixture models, Technical Report UCI-ICS-01-67, Information and Computer Science, University of California, Irvine (available online at
www.datalab.uci.edu.
Heckerman, D., Chickering, D. M., Meek, C., Rounthwaite, R., and Kadie, C. (2000)
Dependency networks for inference, collaborative filtering, and data visualization.
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Hoffmann, T. (1999) Probabilistic latent sematic indexing, Proceedings of the ACM SIGIR
Conference 1999, New York: ACM Press, 50?57.
Lawrence, R.D., Almasi, G.S., Kotlyar, V., Viveros, M.S., Duri, S.S. (2001) Personalization of supermarket product recommendations, Data Mining and Knowledge
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Lazarsfeld, P. F. and Henry, N. W. (1968) Latent Structure Analysis, New York:
Houghton Mifflin.
McCallum, A. (1999) Multi-label text classification with a mixture model trained by EM,
in AAAI?99 Workshop on Text Learning.
Strehl, A. and J. Ghosh (2000) Value-based customer grouping from large retail datasets,
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1,231 | 2,121 | Analysis of Sparse Bayesian Learning
Anita C. Fanl
Michael E. Tipping
Microsoft Research
St George House, 1 Guildhall St
Cambridge CB2 3NH, U.K .
Abstract
The recent introduction of the 'relevance vector machine' has effectively demonstrated how sparsity may be obtained in generalised
linear models within a Bayesian framework. Using a particular
form of Gaussian parameter prior, 'learning' is the maximisation,
with respect to hyperparameters, of the marginal likelihood of the
data. This paper studies the properties of that objective function, and demonstrates that conditioned on an individual hyperparameter, the marginal likelihood has a unique maximum which
is computable in closed form. It is further shown that if a derived
'sparsity criterion' is satisfied, this maximum is exactly equivalent
to 'pruning' the corresponding parameter from the model.
1
Introduction
We consider the approximation, from a training sample, of real-valued functions,
a task variously referred to as prediction, regression, interpolation or function approximation. Given a set of data {xn' tn};;=l the 'target' samples tn = f(xn) + En
are conventionally considered to be realisations of a deterministic function f, potentially corrupted by some additive noise process. This function f will be modelled
by a linearly-weighted sum of M fixed basis functions {4>m (X)}~= l:
M
f(x) =
L
wm?>m(x),
(1)
m=l
and the objective is to infer values of the parameters/weights {Wm}~=l such that
is a 'good' approximation of f.
f
While accuracy in function approximation is generally universally valued, there has
been significant recent interest [2, 9, 3, 5]) in the notion of sparsity, a consequence
of learning algorithms which set significant numbers of the parameters Wm to zero.
A methodology which effectively combines both these measures of merit is that of
'sparse Bayesian learning', briefly reviewed in Section 2, and which was the basis for the recent introduction of the relevance vector machine (RVM) and related
models [6, 1, 7]. This model exhibits some very compelling properties, in particular
a dramatic degree of sparseness even in the case of highly overcomplete basis sets
(M ?N). The sparse Bayesian learning algorithm essentially involves the maximisation of a marginalised likelihood function with respect to hyperparameters in
the model prior. In the RVM , this was achieved through re-estimation equations,
the behaviour of which was not fully understood. In this paper we present further
relevant theoretical analysis of the properties of the marginal likelihood which gives
a much fuller picture of the nature of the model and its associated learning procedure. This is detailed in Section 3, and we close with a summary of our findings and
discussion of their implications in Section 4 (and which, to avoid repetition here,
the reader may wish to preview at this point).
2
Sparse Bayesian Learning
We now very briefly review the methodology of sparse Bayesian learning, more
comprehensively described elsewhere [6]. To simplify and generalise the exposition,
we omit to notate any functional dependence on the inputs x and combine quantities
defined over the training set and basis set within N- and M-vectors respectively.
Using this representation, we first write the generative model as:
t = f
+ ?,
(2)
where t = (t1"'" tN )T, f = (11, ... , fN)T and ? = (E1"'" EN)T. The approximator
is then written as:
f = <I>w,
(3)
where <I> = [?Pl'" ?PM] is a general N x M design matrix with column vectors ?Pm
and w = (W1, ... ,WM)T. Recall that in the context of (1), <I>nm = ?m(x n ) and
f
= {f(x1), .. .j(XN)P.
The sparse Bayesian framework assumes an independent zero-mean Gaussian noise
model, with variance u 2 , giving a multivariate Gaussian likelihood of the target
vector t:
p(tlw, ( 2) = (27r)-N/2 U -N exp { _lit
~:"2
}.
(4)
The prior over the parameters is mean-zero Gaussian:
M
p(wlo:) = (27r)-M/21I
a~,e exp
(
- a m2W2)
m
,
(5)
where the key to the model sparsity is the use of M independent hyperparameters
= (a1 " '" aM)T, one per weight (or basis vector), which moderate the strength
of the prior. Given 0:, the posterior parameter distribution is Gaussian and given
via Bayes' rule as p(wlt , 0:) = N(wIIL,~) with
0:
~ =
(A + u - 2<I> T<I? - 1
IL =
u - 2~<I>Tt,
(6)
and A defined as diag(a1, ... ,aM) . Sparse Bayesian learning can then be formulated as a type-II maximum likelihood procedure, in that objective is to maximise
the marginal likelihood, or equivalently, its logarithm ?(0:) with respect to the hyperparameters 0::
?(0:) = logp(tlo: , ( 2) = log
=
with C = u 2I
1
-"2
[Nlog27r
+ <I> A - l<I>T.
i:
p(tlw, ( 2) p(wlo:) dw,
+ log ICI + t C- 1t]
T
,
(7)
(8)
Once most-probable values aMP have been found 1 , in practice they can be plugged
into (6) to give a posterior mean (most probabletpoint estimate for the parameters
J.tMP and from that a mean final approximator: f MP = ()J.tMp?
Empirically, the local maximisation of the marginal likelihood (8) with respect to
a has been seen to work highly effectively [6, 1, 7]. Accurate predictors may be
realised, which are typically highly sparse as a result of the maximising values of
many hyperparameters being infinite. From (6) this leads to a parameter posterior
infinitely peaked at zero for many weights Wm with the consequence that J.tMP
correspondingly comprises very few non-zero elements.
However, the learning procedure in [6] relied upon heuristic re-estimation equations
for the hyperparameters, the behaviour of which was not well characterised. Also,
little was known regarding the properties of (8), the validity of the local maximisation thereof and importantly, and perhaps most interestingly, the conditions under
which a-values would become infinite. We now give, through a judicious re-writing
of (8), a more detailed analysis of the sparse Bayesian learning procedure.
3
3.1
Properties of the Marginal Likelihood ?(0:)
A convenient re-writing
We re-write C from (8) in a convenient form to analyse the dependence on a single
hyperparameter ai:
C = (]"21 +
2..: am?m?~' = (]"21 + 2..: a~1?m?~ + a-;1?i?T,
m
m# i
= C_ i + a-;1?i?T,
(9)
where we have defined C- i = (]"21+ Lm#i a;r/?m?~ as the covariance matrix with
the influence of basis vector ?i removed, equivalent also to ai = 00.
Using established matrix determinant and inverse identities, (9) allows us to write
the terms of interest in ?( a) as:
(10)
(11)
which gives
?(a) =
-~ [Nlog(2n) + log IC-il + tTC=;t
(?TC- 1 t)2
. ? T-' -1 ],
a. + ?i C_ i ?i
1
(?TC- 1 t)2
= ?(a-i) + -2 [logai -log(ai + ?TC=;?i) +
? T-' -1 ]
ai + ?i C_ i ?i
= ?( a-i) + ?( ai),
-logai + log(ai + ?TC=;?i) -
,
(12)
where ?(a-i) is the log marginal likelihood with ai (and thus Wi and ?i) removed
from the model and we have now isolated the terms in ai in the function ?(ai).
IThe most-probable noise variance (]"~p can also be directly and successfully estimated
from the data [6], but for clarity in this paper, we assume without prejudice to our results
that its value is fixed.
3.2
First derivatives of ?(0:)
Previous results. In [6], based on earlier results from [4], the gradient of the
marginal likelihood was computed as:
(13)
with fJi the i-th element of JL and ~ ii the i-th diagonal element of~. This then leads
to re-estimation updates for O::i in terms of fJi and ~ii where, disadvantageously,
these latter terms are themselves functions of O::i.
A new, simplified, expression.
In fact , by instead differentiating (12) directly,
(13) can be seen to be equivalent to:
(14)
where advantageously, O::i now occurs only explicitly since C - i is independent of O::i.
For convenience, we combine terms and re-write (14) as:
o::;lS;- (Qr - Si)
o?(o:) _
OO::i
2( O::i
+ Si)2
(15)
where, for simplification of this and forthcoming expressions, we have defined:
(16)
The term Qi can be interpreted as a 'quality' factor: a measure of how well c/>i
increases ?(0:) by helping to explain the data, while Si is a 'sparsity' factor which
measures how much the inclusion of c/>i serves to decrease ?(0:) through 'inflating'
C (i. e. adding to the normalising factor).
3.3
Stationary points of ?(0:)
Equating (15) to zero indicates that stationary points of the marginal likelihood
occur both at O::i = +00 (note that , being an inverse variance, O::i must be positive)
and for:
. _
O::t subject to Qr
S2t
Q; _Si'
(17)
> Si as a consequence again of O::i > o.
Since the right-hand-side of (17) is independent of O::i, we may find the stationary
points of ?(O::i) analytically without iterative re-estimation. To find the nature of
those stationary points, we consider the second derivatives.
3.4
3.4.1
Second derivatives of ?(0:)
With respect to O::i
Differentiating (15) a second time with respect to O::i gives:
-0::;2S;(O::i + Si)2 - 2(O::i
+ Si) [o::;lS;- (Qr - Si)]
2(O::i + Si)4
and we now consider (18) for both finite- and infinite-O::i stationary points.
(18)
Finite 0::. In this case, for stationary points given by (17), we note that the second
term in the numerator in (18) is zero, giving:
(19)
We see that (19) is always negative, and therefore ?(O::i) has a maximum, which
Si > and O::i given by (17).
must be unique, for
?
Q; -
Infinite 0::. For this case, (18) and indeed, all further derivatives, are uninformatively zero at O::i = 00 , but from (15) we can see that as O::i --+ 00, the sign of the
gradient is given by the sign of - (Q; - Si).
Q; -
If
Si > 0, then the gradient at O::i = 00 is negative so as O::i decreases ?(O::i)
must increase to its unique maximum given by (17). It follows that O::i = 00 is thus
a minimum. Conversely, if
Si < 0, O::i = 00 is the unique maximum of ?(O::i) .
If
Si = 0, then this maximum and that given by (17) coincide.
Q; -
Q; -
We now have a full characterisation of the marginal likelihood as a function of a
single hyperparameter, which is illustrated in Figure 1.
u.
10'
10?
I
Q;
Figure 1: Example plots of ?(ai) against a i (on a log scale) for
> Si (left) ,
showing the single maximum at finite ai, and
< Si (right), showing the
maximum at a i = 00.
Q;
3.4.2
With respect to
O::j,
j
i:- i
To obtain the off-diagonal terms of the second derivative (Hessian) matrix, it is
convenient to manipulate (15) to express it in terms of C. From (11) we see that
and
(20)
Utilising these identities in (15) gives:
(21)
We now write:
(22)
where 6ij is the Kronecker 'delta' function , allowing us to separate out the additional
(diagonal) term that appears only when i = j.
Writing, similarly to (9) earlier, C = C_ j
differentiating with respect to aj gives:
+ ajl?j?j,
substituting into (21) and
while we have
(24)
If all hyperparameters ai are individually set to their maximising values, i. e. a =
aMP such that alI8?(a)/8ai = 0, then even if all 8 2 ?(a)/8a; are negative, there
may still be a non-axial direction in which the likelihood could be increasing. We
now rule out this possibility by showing that the Hessian is negative semi-definite.
First, we note from (21) that if 8?(a)/8ai = 0, 'V;i = 0. Then, if v is a generic
nonzero direction vector:
<
(25)
where we use the Cauchy-Schwarz inequality. If the gradient vanishes, then for all
00, or from (21) , ?rC-1?i = (?rC- 1t)2. It follows
directly from (25) that the Hessian is negative semi-definite, with (25) only zero
where v is orthogonal to all finite a values.
i = 1, ... , M either ai =
4
Summary
Sparse Bayesian learning proposes the iterative maximisation of the marginal likelihood function ?(a) with respect to the hyperparameters a. Our analysis has
shown the following:
1. As a function of an individual hyperparameter ai, ?( a) has a unique maximum
computable in closed-form. (This maximum is, of course, dependent on the
values of all other hyperparameters.)
II. If the criterion
occurs at O:i =
model.
Qr - Si
00,
(defined in Section 3.2) is negative, this maximum
equivalent to the removal of basis function i from the
III. The point where all individual marginal likelihood functions ?(O:i) are maximised is a joint maximum (not necessarily unique) over all O:i.
These results imply the following consequences.
? From I, we see that if we update , in any arbitrary order, the O:i parameters
using (17), we are guaranteed to increase the marginal likelihood at each
step, unless already at a maximum. Furthermore, we would expect these
updates to be more efficient than those given in [6], which individually only
increase, not maximise, ? (O:i) .
? Result III indicates that sequential optimisation of individual O:i cannot
lead to a stationary point from which a joint maximisation over all 0: may
have escaped. (i.e. the stationary point is not a saddle point.)
? The result II confirms the qualitative argument and empirical observation
that many O:i -+ 00 as a result of the optimisation procedure in [6]. The
inevitable implication of finite numerical precision prevented the genuine
sparsity of the model being verified in those earlier simulations.
? We conclude by noting that the maximising hyperparameter solution (17)
remains valid if O:i is already infinite. This means that basis functions not
even in the model can be assessed and their corresponding hyperparameters
updated if desired. So as well as the facility to increase ?(0:) through
the 'pruning' of basis functions if
Si ::::: 0, new basis functions can
be introduced if O:i = 00 but
Si > O. This has highly desirable
computational consequences which we are exploiting to obtain a powerful
'constructive' approximation algorithm [8].
Qr Qr -
References
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and M. Goldszmidt , editors, Proceedings of th e 16th Conference on Uncertainty in
Artificial Intelligence, pages 46- 53. Morgan Kaufmann , 2000.
[2] S. Chen, D. L. Donoho, and M. A. Saunders. Atomic decomposition by basis pursuit.
Technical Report 479, Department of Statistics, Stanford University, 1995.
[3] Y. Grandvalet. Least absolute shrinkage is equivalent to quadratic penalisation. In
L. Niklasson, M. Boden, and T. Ziemske, editors, Proceedings of th e Eighth International Conference on Artificial N eural Networks (ICANN98) , pages 201- 206. Springer,
1998.
[4] D. J. C. MacKay. Bayesian interpolation. Neural Computation, 4(3):415- 447, 1992.
[5] A. J. Smola, B. Scholkopf, and G. Ratsch . Linear programs for automatic accuracy
control in regression. In Proceedings of th e Ninth Int ernational Conference on Artificial
N eural N etworks, pages 575- 580, 1999.
[6] M. E. Tipping. The Relevance Vector Machine. In S. A. Solla, T . K. Leen , and K.-R.
Muller, editors, Advances in N eural Information Processing Systems 12, pages 652- 658.
MIT Press, 2000.
[7] M. E. Tipping. Sparse kernel principal component analysis. In Advances in Neural
Information Processing Systems 13. MIT Press, 200l.
[8] M. E . Tipping and A. C. Faul. Bayesian pursuit. Submitted to NIPS*Ol.
[9] V. N. Vapnik. Statistical Learning Theory. Wiley, 1998.
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1,232 | 2,122 | Reducing multiclass to binary by
coupling probability estimates
Bianca Zadrozny
Department of Computer Science and Engineering
University of California, San Diego
La Jolla, CA 92093-0114
[email protected]
Abstract
This paper presents a method for obtaining class membership probability estimates for multiclass classification problems by coupling the probability estimates
produced by binary classifiers. This is an extension for arbitrary code matrices
of a method due to Hastie and Tibshirani for pairwise coupling of probability
estimates. Experimental results with Boosted Naive Bayes show that our method
produces calibrated class membership probability estimates, while having similar
classification accuracy as loss-based decoding, a method for obtaining the most
likely class that does not generate probability estimates.
1 Introduction
The two most well-known approaches for reducing a multiclass classification problem to
a set of binary classification problems are known as one-against-all and all-pairs. In the
one-against-all approach, we train a classifier for each of the classes using as positive examples the training examples that belong to that class, and as negatives all the other training
examples. In the all-pairs approach, we train a classifier for each possible pair of classes
ignoring the examples that do not belong to the classes in question.
Although these two approaches are the most obvious, Allwein et al. [Allwein et al., 2000]
have shown that there are many other ways in which a multiclass problem can be decomposed into a number of binary classification problems. We can represent each such decom1 0 1 k l , where k is the number of classes and l is
position by a code matrix M
the number of binary classification problems. If M c b
1 then the examples belonging to class c are considered to be positive examples for the binary classification problem
b. Similarly, if M c b
1 the examples belonging to c are considered to be negative
0 the examples belonging to c are not used in training
examples for b. Finally, if M c b
a classifier for b.
For example, in the 3-class case, the all-pairs code matrix is
c1
c2
c3
b1
1
1
0
b2
1
0
1
b3
0
1
1
This approach for representing the decomposition of a multiclass problem into binary prob-
lems is a generalization of the Error-Correcting Output Codes (ECOC) scheme proposed
by Dietterich and Bakiri [Dietterich and Bakiri, 1995]. The ECOC scheme does not allow
zeros in the code matrix, meaning that all examples are used in each binary classification
problem.
Orthogonal to the problem of choosing a code matrix for reducing multiclass to binary is
the problem of classifying an example given the labels assigned by each binary classifier.
Given an example x, Allwein et al. [Allwein et al., 2000] first create a vector v of length l
containing the -1,+1 labels assigned to x by each binary classifier. Then, they compute
the Hamming distance between v and each row of M, and find the row c that is closest to v
according to this metric. The label c is then assigned to x. This method is called Hamming
decoding.
For the case in which the binary classifiers output a score whose magnitude is a measure of confidence in the prediction, they use a loss-based decoding approach that takes
into account the scores to calculate the distance between v and each row of M, instead
of using the Hamming distance. This method is called loss-based decoding. Allwein et
al. [Allwein et al., 2000] present theoretical and experimental results indicating that this
method is better than Hamming decoding.
However, both of these methods simply assign a class label to each example. They do not
output class membership probability estimates P? C c X x for an example x. These
probability estimates are important when the classification outputs are not used in isolation
and must be combined with other sources of information, such as misclassification costs
[Zadrozny and Elkan, 2001a] or the outputs of another classifier.
Given a code matrix M and a binary classification learning algorithm that outputs probability estimates, we would like to couple the estimates given by each binary classifier in order
to obtain class probability membership estimates for the multiclass problem.
Hastie and Tibshirani [Hastie and Tibshirani, 1998] describe a solution for obtaining probability estimates P? C c X x in the all-pairs case by coupling the pairwise probability
estimates, which we describe in Section 2. In Section 3, we extend the method to arbitrary
code matrices. In Section 4 we discuss the loss-based decoding approach in more detail
and compare it mathematically to the method by Hastie and Tibshirani. In Section 5 we
present experimental results.
2 Coupling pairwise probability estimates
We are given pairwise probability estimates ri j x for every class i j, obtained by training
a classifier using the examples belonging to class i as positives and the examples belonging
to class j as negatives. We would like to couple these estimates to obtain a set of class
membership probabilities pi x
P C ci X x for each example x. The ri j are related
to the pi according to
ri j x P C i C i C
j X
x
pi x
pi x p j x
Since we additionally require that ?i pi x
1, there are k 1 free parameters and k k
1
2 constraints. This implies that there may not exist pi satisfying these constraints.
Let ni j be the number of training examples used to train the binary classifier that predicts
p?i x
p?i x
p? j x , Hastie and
ri j . In order to find the best approximation r?i j x
Tibshirani fit the Bradley-Terrey model for paired comparisons [Bradley and Terry, 1952]
by minimizing the average weighted Kullback-Leibler distance l x between r i j x and
r?i j x for each x, given by
l x
x
1
ij
ni j ri j x log
?
?ri j x
i j
r
ri j x log
1
1
ri j x
?ri j x
The algorithm is as follows:
1. Start with some guess for the ?pi x and corresponding ?ri j x .
2. Repeat until convergence:
(a) For each i 1 2 k
?pi x
? j i n i j ri j x
? j i ni j ?ri j x
?pi x
(b) Renormalize the ?pi x .
(c) Recompute the ?ri j x .
Hastie and Tibshirani [Hastie and Tibshirani, 1998] prove that the Kullback-Leibler distance between ri j x and r?i j x decreases at each step. Since this distance is bounded below
by zero, the algorithm converges. At convergence, the r?i j are consistent with the p? i . The
class predicted for each example x is c? x
argmax p? i x .
Hastie and Tibshirani also prove that the p? i x are in the same order as the non-iterative
estimates p?i x
? j i ri j x for each x. Thus, the p?i x are sufficient for predicting the
most likely class for each example. However, as shown by Hastie and Tibshirani, they
are not accurate probability estimates because they tend to underestimate the differences
between the p?i x values.
3 Extending the Hastie-Tibshirani method to arbitrary code matrices
For an arbitrary code matrix M, instead of having pairwise probability estimates, we have
an estimate rb x for each column b of M, such that
rb x P
C
c I
c
c I J
C c X
? c I pc x
? c I J pc x
x
1, respectively.
We would like to obtain a set of class membership probabilities pi
x for each example
x compatible with the rb
x and subject to ?i pi
x 1. In this case, the number of free
parameters is k 1 and the number of constraints is l 1, where l is the number of columns
of the code matrix.
Since for most code matrices l is greater than k 1, in general there is no exact solution to
where I and J are the set of classes for which M b
1 and M b
this problem. For this reason, we propose an algorithm analogous to the Hastie-Tibshirani
method presented in the previous section to find the best approximate probability estimates
p?i (x) such that
?rb x
?c I ?pc x
?c I J ?pc x
and the Kullback-Leibler distance between r?b x and rb x is minimized.
Let nb be the number of training examples used to train the binary classifier that corresponds to column b of the code matrix. The algorithm is as follows:
1. Start with some guess for the ?pi x and corresponding ?rb x .
2. Repeat until convergence:
(a) For each i 1 2 k
?pi x
?pi x
?b s t
?b s t
1 n b rb x
rb x
M i b 1 nb ?
M ib
?b s t
?b s t
1 nb 1 rb x
rb x
M i b 1 nb 1 ?
M ib
(b) Renormalize the ?pi x .
(c) Recompute the ?rb x .
If the code matrix is the all-pairs matrix, this algorithm reduces to the original method by
Hastie and Tibshirani.
1 and B i be the set of matrix
Let B i be the set of matrix columns for which M i
columns for which M c
1. By analogy with the non-iterative estimates suggested
by Hastie and Tibshirani, we can define non-iterative estimates p?i x
? b B
i rb x
?b B i 1 rb x . For the all-pairs code matrix, these estimates are the same as the ones
suggested by Hastie and Tibshirani. However, for arbitrary matrices, we cannot prove that
the non-iterative estimates predict the same class as the iterative estimates.
4 Loss-based decoding
In this section, we discuss how to apply the loss-based decoding method to classifiers that
output class membership probability estimates. We also study the conditions under which
this method predicts the same class as the Hastie-Tibshirani method, in the all-pairs case.
The loss-based decoding method [Allwein et al., 2000] requires that each binary classifier
output a margin score satisfying two requirements. First, the score should be positive if
the example is classified as positive, and negative if the example is classified as negative.
Second, the magnitude of the score should be a measure of confidence in the prediction.
The method works as follows. Let f x b be the margin score predicted by the classifier
corresponding to column b of the code matrix for example x. For each row c of the code
matrix M and for each example x, we compute the distance between f and M c as
dL x c
l
L M c b f x b
?
(1)
b 1
where L is a loss function that is dependent on the nature of the binary classifier and M c b
= 0, 1 or 1. We then label each example x with the label c for which dL is minimized.
If the binary classification learning algorithm outputs scores that are probability estimates,
they do not satisfy the first requirement because the probability estimates are all between 0
and 1. However, we can transform the probability estimates rb x output by each classifier
b into margin scores by subtracting 1
2 from the scores, so that we consider as positives
the examples x for which rb x is above 1/2, and as negatives the examples x for which
rb x is below 1/2.
We now prove a theorem that relates the loss-based decoding method to the HastieTibshirani method, for a particular class of loss functions.
Theorem 1 The loss-based decoding method for all-pairs code matrices predicts the same
class label as the iterative estimates p? i x given by Hastie and Tibshirani, if the loss function
is of the form L y
ay, for any a
0.
Proof: We first show that, if the loss function is of the form L y
ay, the loss-based
decoding method predicts the same class label as the non-iterative estimates p? i x , for the
all-pairs code matrix.
Dataset
satimage
pendigits
soybean
#Training Examples
4435
7494
307
#Test Examples
2000
3498
376
#Attributes
36
16
35
#Classes
7
10
9
Table 1: Characteristics of the datasets used in the experiments.
The non-iterative estimates p?i x are given by
?pc x
?
rb x
b B
c
? 1
b B
rb x
?
rb x
b B
c
c
?
b B
rb x
c
B c
B c and B c are the sets of matrix columns for which M
c 1 and M
c
where
1, respectively.
ay and f
x b rb
x 1
2, and eliminating the terms for which
Considering that L
y
M
c b 0, we can rewrite Equation 1 as
d x c
?
b B
c
a rb x
1
2
? a rb x
b B
c
1
2
a
?
b B
rb x
c
?
b B
rb x
c
For the all-pairs code matrix the following relationship holds: 1
2 B c
k 1
2, where k is the number of classes. So, the distance d x c is
d x c
a
?
b B
rb x
c
?
b B
rb x
c
B c
k
1
B
2
B
c
c
B
B c
c
1 2
It is now easy to see that the class c x which minimizes d x c for example x, also maximizes p?c x . Furthermore, if d x i d x j then p x i
p x j , which means that the
ranking of the classes for each example is the same.
Since the non-iterative estimates p? c x are in the same order as the iterative estimates p? c x ,
we can conclude that the Hastie-Tibshirani method is equivalent to the loss-based decoding
method if L y
ay, in terms of class prediction, for the all-pairs code matrix.
Allwein et al. do not consider loss functions of the form L y
ay, and uses non-linear
e y . In this case, the class predicted by loss-based decoding
loss functions such as L y
may differ from the one predicted by the method by Hastie and Tibshirani.
This theorem applies only to the all-pairs code matrix. For other matrices such that
B c
B c is a linear function of B c (such as the one-against-all matrix), we can prove
ay) predicts the same class as the non-iterative esthat loss-based decoding (with L y
timates. However, in this case, the non-iterative estimates do not necessarily predict the
same class as the iterative ones.
5 Experiments
We performed experiments using the following multiclass datasets from the UCI Machine
Learning Repository [Blake and Merz, 1998]: satimage, pendigits and soybean. Table
1 summarizes the characteristics of each dataset.
The binary learning algorithm used in the experiments is boosted naive Bayes
[Elkan, 1997], since this is a method that cannot be easily extended to handle multiclass
problems directly. For all the experiments, we ran 10 rounds of boosting.
Method
Loss-based (L y y)
Loss-based (L y e y )
Hastie-Tibshirani (non-iterative)
Hastie-Tibshirani (iterative)
Loss-based (L y y)
Loss-based (L y e y )
Extended Hastie-Tibshirani (non-iterative)
Extended Hastie-Tibshirani (iterative)
Loss-based (L y y)
Loss-based (L y e y )
Extended Hastie-Tibshirani (non-iterative)
Extended Hastie-Tibshirani (iterative)
Multiclass Naive Bayes
Code Matrix
All-pairs
All-pairs
All-pairs
All-pairs
One-against-all
One-against-all
One-against-all
One-against-all
Sparse
Sparse
Sparse
Sparse
-
Error Rate
0.1385
0.1385
0.1385
0.1385
0.1445
0.1425
0.1445
0.1670
0.1435
0.1425
0.1480
0.1330
0.2040
MSE
0.0999
0.0395
0.1212
0.0396
0.1085
0.0340
0.0651
Table 2: Test set results on the satimage dataset.
We use three different code matrices for each dataset: all-pairs, one-against-all and a sparse
random matrix. The sparse random matrices have 15 log2 k columns, and each element
is 0 with probability 1/2 and -1 or +1 with probability 1/4 each. This is the same type of
sparse random matrix used by Allwein et al.[Allwein et al., 2000]. In order to have good
error correcting properties, the Hamming distance ? between each pair of rows in the matrix
must be large. We select the matrix by generating 10,000 random matrices and selecting
the one for which ? is maximized, checking that each column has at least one 1 and one
1, and that the matrix does not have two identical columns.
We evaluate the performance of each method using two metrics. The first metric is the
error rate obtained when we assign each example to the most likely class predicted by
the method. This metric is sufficient if we are only interested in classifying the examples
correctly and do not need accurate probability estimates of class membership.
The second metric is squared error, defined for one example x as SE x
?j tj x
2
p j x , where p j x is the probability estimated by the method for example x and class
j, and t j x is the true probability of class j for x. Since for most real-world datasets true
labels are known, but not probabilities, t j x is defined to be 1 if the label of x is j and
0 otherwise. We calculate the squared error for each x to obtain the mean squared error (MSE). The mean squared error is an adequate metrics for assessing the accuracy of
probability estimates [Zadrozny and Elkan, 2001b]. This metric cannot be applied to the
loss-based decoding method, since it does not produce probability estimates.
Table 2 shows the results of the experiments on the satimage dataset for each type of code
matrix. As a baseline for comparison, we also show the results of applying multiclass Naive
Bayes to this dataset. We can see that the iterative Hastie-Tibshirani procedure (and its
extension to arbitrary code matrices) succeeds in lowering the MSE significantly compared
to the non-iterative estimates, which indicates that it produces probability estimates that
are more accurate. In terms of error rate, the differences between methods are small. For
one-against-all matrices, the iterative method performs consistently worse, while for sparse
random matrices, it performs consistently better. Figure 1 shows how the MSE is lowered
at each iteration of the Hastie-Tibshirani algorithm, for the three types of code matrices.
Table 3 shows the results of the same experiments on the datasets pendigits and soybean.
Again, the MSE is significantly lowered by the iterative procedure, in all cases. For the
soybean dataset, using the sparse random matrix, the iterative method again has a lower
error rate than the other methods, which is even lower than the error rate using the all-pairs
matrix. This is an interesting result, since in this case the all-pairs matrix has 171 columns
(corresponding to 171 classifiers), while the sparse matrix has only 64 columns.
Satimage
0.12
0.11
all?pairs
one?against?all
sparse
0.1
0.09
MSE
0.08
0.07
0.06
0.05
0.04
0.03
0
5
10
15
20
25
30
35
Iteration
Figure 1: Convergence of the MSE for the satimage dataset.
Method
Loss-based (L y y)
Loss-based (L y e y )
Hastie-Tibshirani (non-iterative)
Hastie-Tibshirani (iterative)
Loss-based (L y y)
Loss-based (L y e y )
Ext. Hastie-Tibshirani (non-it.)
Ext. Hastie-Tibshirani (it.)
Loss-based (L y y)
Loss-based (L y e y )
Ext. Hastie-Tibshirani (non-it.)
Ext. Hastie-Tibshirani (it.)
Multiclass Naive Bayes
Code Matrix
All-pairs
All-pairs
All-pairs
All-pairs
One-against-all
One-against-all
One-against-all
One-against-all
Sparse
Sparse
Sparse
Sparse
-
pendigits
Error Rate
MSE
0.0723
0.0715
0.0723
0.0747
0.0718
0.0129
0.0963
0.0963
0.0963
0.0862
0.1023
0.0160
0.1284
0.1266
0.1484
0.0789
0.1261
0.0216
0.2779
0.0509
soybean
Error Rate
MSE
0.0665
0.0665
0.0665
0.0454
0.0665
0.0066
0.0824
0.0931
0.0824
0.0493
0.0931
0.0073
0.0718
0.0718
0.0798
0.0463
0.0636
0.0062
0.0745
0.0996
Table 3: Test set results on the pendigits and soybean datasets.
6 Conclusions
We have presented a method for producing class membership probability estimates for
multiclass problems, given probability estimates for a series of binary problems determined
by an arbitrary code matrix.
Since research in designing optimal code matrices is still on-going
[Utschick and Weichselberger, 2001] [Crammer and Singer, 2000], it is important to
be able to obtain class membership probability estimates from arbitrary code matrices.
In current research, the effectiveness of a code matrix is determined primarily by the
classification accuracy. However, since many applications require accurate class membership probability estimates for each of the classes, it is important to also compare the
different types of code matrices according to their ability of producing such estimates. Our
extension of Hastie and Tibshirani?s method is useful for this purpose.
Our method relies on the probability estimates given by the binary classifiers to produce the
multiclass probability estimates. However, the probability estimates produced by Boosted
Naive Bayes are not calibrated probability estimates. An interesting direction for future
work is in determining whether the calibration of the probability estimates given by the
binary classifiers improves the calibration of the multiclass probabilities.
References
[Allwein et al., 2000] Allwein, E. L., Schapire, R. E., and Singer, Y. (2000). Reducing multiclass
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1,233 | 2,123 | Asymptotic Universality for Learning
Curves of Support Vector Machines
M.Opperl
R. Urbanczik 2
1 Neural Computing Research Group
School of Engineering and Applied Science
Aston University, Birmingham B4 7ET, UK.
[email protected]
2Institut Fur Theoretische Physik,
Universitiit Wurzburg Am Rubland, D-97074 Wurzburg, Germany
[email protected].
Abstract
Using methods of Statistical Physics, we investigate the rOle
of model complexity in learning with support vector machines
(SVMs). We show the advantages of using SVMs with kernels
of infinite complexity on noisy target rules, which, in contrast to
common theoretical beliefs, are found to achieve optimal generalization error although the training error does not converge to the
generalization error. Moreover, we find a universal asymptotics of
the learning curves which only depend on the target rule but not
on the SVM kernel.
1
Introduction
Powerful systems for data inference, like neural networks implement complex inputoutput relations by learning from example data. The price one has to pay for the
flexibility of these models is the need to choose the proper model complexity for a
given task, i.e. the system architecture which gives good generalization ability for
novel data. This has become an important problem also for support vector machines
[1]. The main advantage of SVMs is that the learning task is a convex optimization
problem which can be reliably solved even when the example data require the fitting
of a very complicated function. A common argument in computational learning
theory suggests that it is dangerous to utilize the full flexibility of the SVM to learn
the training data perfectly when these contain an amount of noise. By fitting more
and more noisy data, the machine may implement a rapidly oscillating function
rather than the smooth mapping which characterizes most practical learning tasks.
Its prediction ability could be no better than random guessing in that case. Rence,
modifications of SVM training [2] that allow for training errors were suggested to be
necessary for realistic noisy scenarios. This has the drawback of introducing extra
model parameters and spoils much of the original elegance of SVMs.
Surprisingly, the results of this paper show that the picture is rather different in
the important case of high dimensional data spaces. Using methods of Statistical
Physics, we show that asymptotically, the SVM achieves optimal generalization
ability for noisy data already for zero training error. Moreover, the asymptotic rate
of decay of the generalization error is universal, i.e. independent of the kernel that
defines the SVM. These results have been published previously only in a physics
journal [3].
As is well known, SVMs classify inputs y using a nonlinear mapping into a feature
vector w(y) which is an element of a Hilbert space. Based on a training set of m
inputs xl' and their desired classifications 71' , SVMs construct the maximum margin
hyperplane P in the feature space. P can be expressed as a linear combination of
the feature vectors w(xl'), and to classify an input y, that is to decide on which side
of P the image W(y) lies, one basically has to evaluate inner products W(xl') . W(y).
For carefully chosen mappings wand Hilbert spaces, inner products w(x) . w(y)
can be evaluated efficiently using a kernel function k(x, y) = w(x) . w(y), without
having to individually calculate the feature vectors w(x) and w(y). In this manner
it becomes computationally feasible to use very high and even infinite dimensional
feature vectors.
This raises the intriguing question whether the use of a very high dimensional
feature space may typically be helpful. So far, recent results [4, 5] obtained by
using methods of Statistical Mechanics (which are naturally well suited for analysing
stochastic models in high dimensional spaces), have been largely negative in this
respect. They suggest (as one might perhaps expect) that it is rather important to
match the complexity of the kernel to the target rule. The analysis in [4] considers
the case of N-dimensional inputs with binary components and assumes that the
target rule giving the correct classification 7 of an input x is obtained as the sign
of a function t(x) which is polynomial in the input components and of degree L.
The SVM uses a kernel which is a polynomial of the inner product x . y in input
space of degree K ;::: L, and the feature space dimension is thus O(N K ). In this
scenario it is shown, under mild regularity condition on the kernel and for large N,
that the SVM generalizes well when the number of training examples m is on the
order of N L . So the scale of the learning curve is determined by the complexity
of the target rule and not by the kernel. However, considering the rate with which
the generalization error approaches zero one finds the optimal N L 1m decay only
when K is equal to L and the convergence is substantially slower when K > L. So
it is important to match the complexity of the kernel to the target rule and using
a large value of K is only justified if L is assumed large and if one can use on the
order of N L examples for training.
In this paper we show that the situation is very different when one considers the
arguably more realistic scenario of a target rule corrupted by noise. Now one can no
longer use K = L since no separating hyperplane P will exist when m is sufficiently
large compared to N L. However when K > L, this plane will exist and we will show
that it achieves optimal generalization performance in the limit that N L 1m is small.
Remarkably, the asymptotic rate of decay of the generalization error is independent
of the kernel in this case and a general characterization of the asymptote in terms of
properties of the target rule is possible. In a second step we show that under mild
regularity conditions these findings also hold when k(x, y) is an arbitrary function
of x . y or when the kernel is a function of the Euclidean distance Ix - YI. The latter
type of kernels is widely used in practical applications of SVMs.
2
Learning with Noise: Polynomial Kernels
We begin by assuming a polynomial kernel k(x, y) = J(x? y) where J(z) =
Lf=oCk zk is of degree K. Denoting by P a multi-index P = (PI , ... ,PN) with Pi E
No, we set xp =
JTPTfnf:l %.r and the degree of xp is Ipi = Lf:l Pi?
The kernel
can then be described by features wp(x) = JCiPTxp since k(x,y) = Lp wp(x)wp(y),
where the summation runs over all multi-indices of degree up to K. To assure that
the features are real, we assume that the coefficients Ck in the kernel are nonnegative. A hyperplane in feature space is parameterized by a weight vector w with
components w p, and if 0 < TI'W . W(xl'), a point (xl', TI') of the training set lies on
the correct side of the plane. To express that the plane P has maximal distance to
the points of the training set, we choose an arbitrary positive stability parameter /'i,
and require that the weight vector w* of P minimize w . w subject to the constraints
/'i, < TI'W' w(xl'), for f.l = 1, ... ,m.
2.1
The Statistical Mechanics Formulation
Statistical Mechanics allows to analyze a variety of learning scenarios exactly in the
"thermodynamic limit" of high input dimensionality, when the data distributions
are simple enough. In this approach, one computes a partition function which
serves as a generating function for important averages such as the generalization
error. To define the partition problem for SVMs one first analyzes a soft version of
the optimization problem characterized by an inverse temperature f3. One considers
the partition function
z=
f dwe- ~f3w.w IT
8(TI'W'
w(xl') -
/'i,),
(1)
1'=1
where the SVM constraints are enforced strictly using the Heaviside step function
8. Properties of w * can be computed from In Z and taking the limit f3 -t 00.
To model the training data, we assume that the random and independent input
components have zero mean and variance liN . This scaling assures that the variance of w . w(xl') stays finite in the large N limit. For the target rule we assume
that its deterministic part is given by the polynomial t(x) = Lp JCiPTBpxp with
real parameters Bp. The magnitude of the contribution of each degree k to the
value of t(x) is measured by the quantities
1
Tk = Ck Nk
'"'
~ Bp2
(2)
p,lpl =k
where Nk = (N+; - I) is the number of terms in the sum. The degree of t(x) is L
and lower than K, so TL > 0 and TL+l = ... = TK = O. Note, that this definition
of t(x) ensures that any feature necessary for computing t(x) is available to the
SVM. For brevity we assume that the constant term in t(x) vanishes (To = 0) and
the normalization is Lk Tk = 1.
2.2
The Noise Model
In the deterministic case the label of a point x would simply be the sign of t(x).
Here we consider a nondeterministic rule and the output label is obtained using a
random variable Tu E {-1, 1} parameterized by a scalar u. The observable instances
of the rule, and in particular the elements of the training set, are then obtained by
independently sampling the random variable (x, Tt(x))' Simple examples are additive
noise, Tt(x) = sgn(t(x) + 77), or multiplicative noise, Tt(x) = sgn(t(x)77), where 77 is
a noise term independent of x. In general, we will assume that the noise does not
systematically corrupt the deterministic component, formally
1
(3)
1> Prob(Tu = sgn(u)) >"2 for all u.
So sgn( t( x)) is the best possible prediction of the output label of x, and the minimal
achievable generalization error is fmin = (8( -t(X)Tt(x)))x. In the limit of many
input dimensions N, a central limit argument yields that for a typical target rule
fmin = 2(8( -u)0(u))u , where u is zero mean and unit variance Gaussian. The
function 0 will play a considerable role in the sequel. It is a symmetrized form of
the probability p(u) that Tu is equal to 1, 0(u) = ~(p(u) + 1 - p( -u)).
2.3
Order Parameter Equations
One now evaluates the average of In Z (Eq. 1) over random drawings of training
data for large N in terms of t he order parameters
Q
r
(((W.1]i(X))2)Jw'
q=((w)w?1]i(X))2)x
and
Q-! ?w ?1]i(x))w B ?1]i(x))x .
(4)
Here the thermal average over w refers to the Gibbs distribution (1). For the large
N limit, a scaling of the training set size m must be specified, for which we make
t he generic Ansatz m = aNt, where I = 1, ... ,L. Focusing on the limit of large j3,
where the density on the weight vectors converges to a delta peak and q -+ Q, we
introduce the rescaled order parameter X = j3( Q - q) / St, with
t
St = i (1) -
L
Ci .
(5)
i=O
Note that this scaling with St is only possible since the degree K of the kernel
i(x, y) is greater than I, and thus St ?- O. Finally, we obtain an expression for it =
lim,B--+oo limN --+00 ?In Z)) St / (j3Nt ), where the double brackets denote averaging over
all training sets of size m . The value of it results from extremizing, with respect to
r, q and X, the function
it(r,q,X) =
-aq /0(-u)G (ru
X
\
+ ~v -~))
v0 u,v
~ (~: - X ~ 1) (1- -(X -1)TzS~;Ct + L~=l TJ
(6)
where G(z) = 8(z)z2, and u, v are independent Gaussian random variables with
zero mean and unit variance.
?
Since the stationary value of it is finite , w* . w*)) is of the order Nt. So the
higher order components of w* are small, (W;)2 ? 1 for Ipi > I, although these
components playa crucial role in ensuring that a hyperplane separating the training
points exists even for large a. But the key quantity obtained from Eq. (6) is the
stationary value of r which determines the generalization error of the SVM via
fg = (0(-u)8(ru + ~v))u,v, and in particular fg = fmin for r = 1.
2.4
Universal Asymptotics
We now specialize to the case that l equals L, the degree of the polynomial t(x) in
the target rule. So m = aNL and for large a, after some algebra, Eq. (6) yields
r = 1where
B(q)
A(q*) ~
4B(q*)2 a
(e(Y)8(-Y+Ii/yrj))}y
(e(Y)8(-Y+Ii/y7i) (_Y+Ii/y7i)2}y.
(7)
and
A(q)
Further q* = argminqqA(q), and con-
sidering the derivatives of qA(q) for q --+ 0 and q --+
condition (3) assures that qA(q) does have a minimum.
00,
one may show that
Equation (7) shows that optimal generalization performance is achieved on this scale
in the limit of large a. Remarkably, as long as K > L, the asymptote is invariant
to the choice of the kernel since A(q) and B(q) are defined solely in terms of the
target rule.
3
Extension to other Kernels
Our next goal is to understand cases where the kernel is a general function of the
inner product or of the distance between the vectors. We still assume that the
target rule is of finite complexity, i.e. defined by a polynomial and corrupted by
noise and that the number of examples is polynomial in N. Remarkably, the more
general kernels then reduce to the polynomial case in the thermodynamic limit.
Since it is difficult to find a description of the Hilbert space for k( x, y) which is useful
for a Statistical Physics calculation, our starting point is the dual representation:
The weight vector w* defining the maximal margin hyperplane P can be written
as a linear combination of the feature vectors w(x M ) and hence w* . w(y) = IJ(Y),
where
m
(8)
M=l
By standard results of convex optimization theory the AM are uniquely defined by the
Kuhn-Tucker conditions AM ::::: 0, TMIJ(X M) ::::: Ii (for all patterns), further requiring
that for positive AM the second of the two inequalities holds as an equality. One also
finds that w* . w* = 2:;=1 AM and for a polynomial kernel we thus obtain a bound
on 2:;=1AM since w* . w* is O(m).
We first consider kernels ?(x? y), with a general continuous function ? of the inner
product, and assume that ? can be approximated by a polynomial J in the sense
that ?(1) = J(l) and ?(z) - J(z) = O(ZK) for z --+ O. Now, with a probability
approaching 1 with increasing N, the magnitude of xM?xl/ is smaller than, say, N- 1/3
for all different indices {t and v as long as m is polynomial in N. So, considering Eq.
(8), for large N the functions ?(z) and J(z) will only be evaluated in a small region
around zero and at z = 1 when used as kernels of a SVM trained on m = aNL
examples. Using the fact that 2:;=1AM = O(m) we conclude that for large Nand
K > 3L the solution of the K uhn-Tucker conditions for J converges to the one for
?. So Eqs. (6,7) can be used to calculate the generalization error for ? by setting
ttl = ?(l) (O)/l! for l = 1, ... , L, when ? is an analytic function. Note that results in
[4] assure that ttl ::::: 0 if the kernel ?( X? y) is positive definite for all input dimensions
N. Further, the same reduction to the polynomial case will hold in many instances
where ? is not analytical but just sufficiently smooth close to O.
3.1
RBF Kernels
We next turn to radial basis function (RBF) kernels where k( x, y) depends only
on the Euclidean distance between two inputs, k(x,y) = <I>(lx - YI2). For binary
input components (Xi = ?N- 1 / 2 ) this is just the inner product case since <I>(lx Y12) = <I>(2 - 2x? y). However, for more general input distributions, e.g. Gaussian
input components, the fluctuations of Ixl around its mean value 1 have the same
magnitude as x . y even for large N, and an equivalence with inner product kernels
is not evident.
Our starting point is the observation that any kernel <I>(lx - Y12) which is positive
definite for all input dimensions N is a positive mixture of Gaussians [6]. More
precisely <I>(z) = fooo e-ez da(k) where the transform a(k) is nondecreasing. For
the special case of a single Gaussian one easily obtains features 'IT p by rewriting
<I>(lx - Y12) = e-lx-v I2/2 = e- 1x12/2ex've-lvI2 /2. Expanding the kernel e X ' v into
polynomial features, yields the features 'IT p(x) = e- 1x12 /2x pl
for <I>(lx _ YI2).
But, for a generic weight vector w in feature space,
M
w? 'IT(x) = ~Wp'ITp(x) = e-~lxI2 ~wp
M
(9)
is of order 1, and thus for large N the fluctuations of Ixl can be neglected.
This line of argument can be extended to the case that the kernel is a finite mixture
of Gaussians, <I>(z) = L~=l aie-'Y7z /2 with positive coefficients ai. Applying the
reasoning for a single Gaussian to each term in the sum, one obtains a doubly
indexed feature vector with components 'lTp,i(X) = e-'Y7IxI2/2(ai/';lpl/lpll)1/2xp. It
is then straightforward to adapt the calculation of the partition function (Eq. 16) to the doubly indexed features, showing that the kernel <I>(lx - Y12) yields the
same generalization behavior as the inner product kernel <I> (2 - 2x . y). Based on the
calculation, we expect the same equivalence to hold for general radial basis function
kernels, i.e. an infinite mixture of Gaussians, even if it would be involved to prove
that the limit of many Gaussians commutes with the large N limit.
4
Simulations
To illustrate the general results we first consider a scenario where a linear target rule,
corrupted by additive Gaussian noise, is learned using different transcendental RBF
kernels (Fig. 1) . While Eq. (7) predicts that the asymptote of the generalization
error does not depend on the kernel, remarkably, the dependence on the kernel
is very weak for all values of a. In contrast, the generalization error depends
substantially on the nature of the noise process. Figure 2 shows the finding for
a quadratic rule with additive noise for the cases that the noise is Gaussian and
binary. In the Gaussian case a 1/a decay of Eg to Emin is found, whereas for binary
noise the decay is exponential in a. Note that in both cases the order parameter r
approaches 1 as 1/a.
5
Summary
The general characterization of learning curves obtained in this paper demonstrates
that support vector machines with high order or even transcendental kernels have
definitive advantages when the training data is noisy. Further the calculations leading to Eq. (6) show that maximizing the margin is an essential ingredient of the
approach: If one just chooses any hyperplane which classifies the training data correctly, the scale of the learning curve is not determined by the target rule and far
more examples are needed to achieve good generalization. Nevertheless our findings
are at odds with many of the current t heoretical motivations for maximizing the
margin which argue that this minimizes the effective Vapnik Chervonenkis dimension of the classifier and thus ensures fast convergence of the training error to the
generalization error [1 , 2]. Since we have considered hard margins, in contrast to t he
generalization error, the training error is zero, and we find no convergence between
the two quantities. But close to optimal generalization is achieved since maximizing
the margin biases the SVM to explain as much as possible of the data in terms of a
low order polynomial. While the Statistical Physics approach used in this paper is
only exactly valid in the thermodynamic limit, the numerical simulations indicate
that the theory is already a good approximation for a realistic number of input
dimensions.
We thank Rainer Dietrich for useful discussion and for giving us his code for the simulations. The work of M.O. was supported by the EPSRC (grant no. GR/M81601)
and the British Council (ARC project 1037); R.U. was supported by the DFG and
the DAAD.
References
[1] C. Cortes and V. Vapnik. , Machine Learning, 20:273-297, 1995.
[2] N. Cristianini and J . Shawe-Taylor. Support Vector Machines. Cambridge U niversity Press , 2000.
[3] M. Opper and R. Urbanczik. Phys. Rev. Lett., 86:4410- 4413, 200l.
[4] R. Dietrich, M. Opper, and H. Sompolinsky. Phys. Rev. Lett., 82:2975 - 2978,
1999.
[5] S. Risau-Gusman and M. Gordon. Phys. Rev. E, 62:7092- 7099,2000.
[6] I. Schoenberg. Anal. Math, 39:811-841, 1938.
0.3
,-----r -- - - - - - - - - - - - - - - - - ,
(A)
(8)
(C)
(D)
0.2
D
6.
<>
0
(E)
tOg
0.1
- trllin -
o
-
-
-
-
-
10
5
20
15
a=P/N
Figure 1: Linear target rule corrupted by additive Gaussian noise rJ ((rJ) = 0, \rJ 2 ) =
1/16) and learned using different kernels. The curves are the theoretical prediction;
symbols show simulation results for N = 600 with Gaussian inputs and error bars
are approximately the size of the symbols. (A) Gaussian kernel, <I>(z) = e- kz with
k = 2/3. (B) Wiener kernel given by the non analytic function <I>(z) = e - e..jZ. We
chose c ~ 0.065 so that the theoretical prediction for this case coincides with (A).
(C) Gaussian kernel with k = 1/20, the influence of the parameter change on t he
learning curve is minimal. (D) Perceptron, ?(z) = z . Above a e ~ 7.5 vanishing
training error cannot be achieved and the SVM is undefined. (E) Kernel invariant
asymptote for (A,B,C).
0.1
-
-E~in-
-
o w-______
o
2
-
-
-
-
-
-
-
-
-
-
-
~______~_ _ _ _ _~_____ _ w
4
6
8
a = P/N2
Figure 2: A noisy quadratic rule (Tl = 0, T2 = 1) learned using the Gaussian kernel
with k = 1/20. The upper curve (simulations.) is for additive Gaussian noise as
in Fig. 1. The lower curve (simulations .) is for binary noise, rJ ? s with equal
probability. We chose s ~ 0.20 so that the value of fmin is the same for the two
noise processes. The inset shows that f9 decays as l/a for Gaussian noise, whereas
an exponential decay is found in the binary case. The dashed curves are the kernel
invariant asymptotes. The simulations are for N = 90 with Gaussian inputs, and
standard errors are approximately the size of the symbols.
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1,234 | 2,124 | Circuits for VLSI Implementation of
Temporally-Asymmetric Hebbian
Learning
Adria Bofill
Alan F. Murray
DanlOn P. Thompson
Dept. of Electrical Engineering
The University of Edinburgh
Edinburgh , EH93JL , UK
adria. [email protected]. uk
alan. murray @ee.ed.ac.uk
damon. thompson @ee.ed.ac. uk
Abstract
Experimental data has shown that synaptic strength modification
in some types of biological neurons depends upon precise spike timing differences between presynaptic and postsynaptic spikes. Several temporally-asymmetric Hebbian learning rules motivated by
this data have been proposed. We argue that such learning rules
are suitable to analog VLSI implementation. We describe an easily tunable circuit to modify the weight of a silicon spiking neuron
according to those learning rules. Test results from the fabrication
of the circuit using a O.6J.lm CMOS process are given.
1
Introduction
Hebbian learning rules modify weights of synapses according to correlations between
activity at the input and the output of neurons. Most artificial neural networks
using Hebbian learning are based on pulse-rate correlations between continuousvalued signals; they reduce the neural spike trains to mean firing rates and thus
precise timing does not carry information. With this approach the spiking nature
of biological neurons is just an efficient solution that evolution has produced to
transmit analog information over an unreliable medium.
In recent years, recorded data have indicated that synaptic strength modifications
are also induced by timing differences between pairs of presynaptic and postsynaptic
spikes [1][2]. A class of learning rules derived from these experimental dat a is illustrated in Figure 1 [2]-[4]. The "causal/non-causal" basis of these Hebbian learning
algorithms is present in all variants of this spike-timing dependent weight modification rule. When the presynaptic spike arrives at the synapse a few milliseconds
presynaptic spike
presynaptic spike
,tpre
tpre'
postsynaptic spike
postsynaptic spike
!post
tpost '
!'.w
tpre - tpost
tpre - tpost
(a)
(b)
Figure 1: Two temporally-asymmetric Hebbian learning rules drawing on
experimental data. The curves show the shape of the weight change (~W) for
differences between the firing times of the presynaptic (tpr e) and the postsynaptic
(tpost) neurons. When the presynaptic spike arrives at the synapse a few ms before the postsynaptic neuron fires , the weight of the synapse is increased. If the
postsynaptic neuron fires first, the weight is decreased.
before an output spike is generated, the synaptic efficiency increases. In contrast,
when the postsynaptic neuron fires first , the efficiency of the synapse is weakened.
Hence, only those synapses that receive spikes that appear to contribute to the
generation of the postsynaptic spike are reinforced. In [5] a similar spike-timing
difference based learning rule has been used to learn input sequence prediction in a
recurrent network. Studies reported in [4] indicate that the positive (potentiation)
element of the learning curve must be smaller than the negative (depression) to
obtain stable competitive weight modification.
Pulse signal representation has been used extensively in hardware implementations
of artificial neural networks [6] [7]. Such systems use pulses as a mere technological
solution to benefit from the robustness of binary signal transmission while making
use of analog circuitry for the elementary computation units. However , they do not
exploit the relative timing differences between individual pulses to compute. Also ,
analog hardware is not well-suited to the complexity of most artificial neural network
algorithms. The learning rules presented in Figure 1 are suitable for analog VLSI
because: (a) the signals involved in the weight modification are local to the neuron ,
(b) no temporal averaging of the presynaptic or postsynaptic activity is needed and
(c) they are remarkably simple compared to complex neural algorithms that impose
mathematical constraints in terms of accuracy and precision. An analog VLSI
implementation of a similar, but more complex, spike-timing dependent learning
rule can be found in [8].
We describe a circuit that implements the spike-timing dependent weight change
described above along with the t est results from a fabricated chip. We have fo cused
on the implem entation of the weight modification circuits, as VLSI spiking neurons
with tunable m embrane time constant and refractory p eriod have already b een
proposed in [9] and [10].
2
Learning circuit description
Figure 2 shows the weight change circuit and Figure 3 the form of signals required
to drive learning. These driving signals are generated by the circuits described in
Figure 4. The voltage across the weight capacitor , Cw in Figure 2, is modified
according to t he spike-timing dependent weight change rule discussed above. The
weight change, ~ W, is defined as -~ Vw so that the leakage of t he capacitor leads
Vw in the direction of weight decay. The circuits presented allow the control of:
(a) the abruptness of the transition between potentiation and depression at the
origin, (b) the difference between the areas under the curve in the potentiation and
depression regions, (c) the absolute value of the area under each side of the curve
and (d) the time constant of t he curve decay.
PI
Figure 2: W e ight change circuit
postsynaptic spike
up
n '- - -__
down
(a)
(b)
Figure 3: Stimulus for the w e ight change circuit
The weight change circuit of Figure 2 works as follows. When a falling edge of
either a postsynaptic or a presynaptic spike occurs , a short activation pulse is
generated which causes Cd ec to be charged to V pea k through transistor Nl. The
charge accumulated in Cd ec will leak to ground with a rate set by Vd ec ay ' The
resulting voltage at the gate of N3 produces a current flowing through P2-P3-N4. If
a presynaptic spike is active after the falling edge of a postsynaptic spike an activelow up pulse is applied to the gate of transistor P5. Thus, the current flowing
through N3 is mirrored to transistor P4 causing an increase in the voltage across
Cw that corresponds to a decrease in the weight. In contrast, when a presynaptic
spike precedes a postsynaptic spike an active-high down pulse is generated and the
current in N3 is mirrored to N5-N6 resulting in a discharge of Cw .
As the current in N2 is constant, the current integrated by Cw displays an exponential decay, if Vpeak is such that N3 is in sub-threshold mode. Hence, the rate
of decay of the learning curve is fixed by the ratio hlCdec. The abruptness of the
transition zone between potentiation and depression is set by the duration of both
the presynaptic and postsynaptic spike. Finally, an imbalance between the areas
under the positive and negative side of the curve can be introduced via Vdep and
Vpot . The effect of all these circuit parameters is exemplified by the test results
shown in the following section.
act
down
post_spike
(a)
(b)
Figure 4: Learning drivers. (a) Delayed act pulse generator. (b) Asynchronous controller for up and down signals
The circuit of Figure 4(a) , present in both the presynaptic and postsynaptic neurons,
generates a short act pulse with the falling edge of the output spike. The act pulses
are ORed at each synapse to produce the activation pulse applied to the weight
change circuit of Figure 2.
The other two driving signals , up and down, are produced by a small asynchronous
controller using standard and asymmetric C-elements [11] shown in Figure 4(b).
The internal signal q indicates if the last falling edge to occur corresponds to a
pre (q = 1) or a postsynaptic spike (q = 0). This ensures that an up signal that
decreases the weight is only generated when a presynaptic spike is active after the
falling edge of a postsynaptic spike. Similarly down is activated only when the
postsynaptic spike is active following a presynaptic spike falling edge.
Using the current flowing through N3 (Figure 2) to both increase and decrease the
weight allows us to match the curve at the potentiation and depression regions at
the exp ense of having to introduce the driving circuits of Figure 4.
3
Results from the temporally-asymmetric Hebbian chip
The circuit in Figure 2 has b een fabricated in a O.6J.lm standard CMOS process.
The driving signals (down, up and activation) are currently generated off-chip.
The circuit can be operated in t he p,s timescale, however , here we only present test
results with time constants similar to those suggested by experimental data and
studied using software models in [3]- [5].
3.5"==;;;r----;;.==---~--~==~
Vdecay = 515mV
t
2.5
t
pre
-I
- t
Vpeak = 694m V
= 1ms
T
'p
T act = 50j.ts
=2ms
V
\
pre
post
pot
=OV
Vdd - Vdep = OV
post
=5
t
Vdecay = 515mV
post
>'
V peak = 694mV
= 1ms
-I pre =5m
/
1.5
t
T
'p
= SOj.tS
T act
t
pre
-I
po st
V
=7.5ms
pot
=OV
tpost - tpre
Vdd - Vdep = OV
0.5
0.5
1.5
= 7.5ms
ms
/
00
2.5
=
i
- t
po"
0.5
1.5
Time ( s )
Time(s)
(a)
2.5
(b)
Figure 5: Test result s . Linearity. (a ) The voltage across Cw is initially set to
OV and increased by a sequence of consecutive pairs of pre and postsynaptic spikes.
The delays between presynaptic and postsynaptic firing times were set to 2ms , 5ms
and 7.5ms (b) The order of pre and postsynaptic spikes is reversed to decrease Vw .
In both plots the duration of the spikes, T sp , and the activation pulse, Ta ct , is set
to 1ms and and 50p,s respectively.
The learning window plots shown in Figures 6-8 were constructed with test data
from a sequence of consecutive presynaptic and postsynaptic spikes with different
delays . Before every new pair of presynaptic and postsynaptic spikes, the voltage
in Cw was reset to Vw = 2V . The weight change curves are similar for other initial
"reset" weight voltages owing to the linearity of the learning circuit for different Vw
values as shown in Figure 5. A power supply voltage of Vdd = 5V is used in all test
results shown.
80 , - - . - - , - - , - - , - - , - - , - - , - - - ,
V decay =516mV
100
Tsp =1ms
,,'k
=
Tact 501-15
V
=OV
>
50
""
pot
Vdd - Vdep = OV
-
Tsp= 1ms
Vpeak =716mV
--- V
=711mV
-- V
=701mV
60
Tact = 5O~IS
40
Vdd - Vdep = OV
Vpot = OV
>
E
E 20
0
~
-50
- - - v decay = 499mV
"
""
, ",
Vpeak = 701mV
- -
- - . . . . .~--.=.::.---.:.-
>~
>'
<l
Vdecay =517mV
Vpeak = 702mV
- ';",-;"--' ::"= '- -"'- -
l'
r
Vdecay =482mV
Vpeak = 699mV
,I
,
; -20
<l
-40
-60
- 100
-80~-~-~-~-~-~-~-~-----'
-25 -20
-15 - 10
-5
t p ,.
0
-
5
tpo,t (ms)
(a)
10
15
20
25
-40
-30
-20
-10
0
10
20
30
40
t pre - \ OSI (m s)
(b)
Figure 6: Test result s . (a) M aximum w e ig ht ch ange . (b) Learning window
decay. The decay of both tails of the learning window is set by Vdecay. A wid e
range of time constants can b e set. Note, however, that Vpea k needs to b e increased
slightly for faster decay rates to maintain exactly the same p eak value.
The maximum weight change is easily tuned with Vp e ak as shown in Figure 6( a).
Changing the value of Vp e ak modifies by the same amount the absolute value of the
peaks at both sides of the curve. The decay of the learning window is controlled
by Vd ec ay' An increase in Vd ecay causes both tails of the learning window to decay
faster as seen in Figure 6 (b). As m entioned above, matching between both sides of
the learning window is possible because the same source of current is used to both
increase and decrease the weight.
100
80
-
>
~-~-~~~~-~-~-~-~
Vdecay = SOOmV
Vpeak = 705mV
60
= 1ms
'p
Tact =50l1s
40
Vpot =OV
T
80
=9.3mV
60
_ ,_
dd
dep
Vdd - Vdep
>
E
>'0
I'
-20
<l
-40
;:
Vdecay = SOOmV
Vpeak = 705mV
Tsp= 1ms
Tact =
.:. 20
-
100
Vdd - Vdep = OV
- - - V - V = 3.8mV
40
,,_
50~lS
pot
,, ; '
Vdd -Vdep=OV
20
V =OV
pot
- - - V =47mV
- -- VPOt = 9~V
//
.-."';::: ,-;//
>' 0
,
<l
;:
<l
-60
-20
-40
-60
-80
-80
-1~go'---_--c1~
5 ---C10~~-5~-0~-~5-~-~
15:--~
-100
-20
20
- 15
- 10
-5
'pre - 'post (ms)
0
5
10
15
20
'pre - 'post (ms)
(a)
(b)
Figure 7: Test results. Imbalance between potentiation and depression.
The imbalance between the areas under the potentiation and depression regions of
the learning window is a critical parameter of this class of learning rules [3] [4]. The
circuit proposed can adjust the peak of the curve for potentiation and depression
independently (Figure 7). Vp ot can be used to reduce the area under the potentiation
region while keeping unchanged the depression part of the curve , thus setting the
overall area under the curve to a negative value (Figure 7(b)). Similarly, with
Vdd - Vd ep the area of the depression region can also be reduced (Figure 7 (a) ) .
100 ,---,--~--,,---,--~--,
100
50
_ _ Tsp = 100llS
Vpeak = 790mV
Vdecay = 499mV
Tact = 50flS
Vdd - Vdep =OV
Vpot =OV
---- T
"
>
E
50
i
100
=1ms
Vpeak = 699mV
Vdecay = 482mV
;: 100
<l
50
50
100 '---~--~--L--~--~-~
-15
-10
-5
0
5
10
15
'pre - 'post (ms)
Figure 8: Test results. Abruptness at the origin
The abruptness of the learning window at the origin (short delays between pre and
postsynaptic spikes) is set by th e duration of the spikes. Dat a in Figure 8 show
that th e two peaks of the learning window are separat ed by 2 times the durations
of the spikes (Tsp ).
4
Discussion and future work
Drawn from experimental data, several temporally-asymmetric Hebbian learning
rules have been proposed recently. These learning rules only strengthen the weights
when there is a causal relation between presynaptic and postsynaptic activity.
Purely random time coincidences between spikes will tend to decrease the weights.
Synaptic weight normalization is thus achieved via competition to drive postsynaptic spikes [4]. Predictive sequence learning has been achieved using a similar
time-difference learning rule based on the same data [5]. Other pulse-based learning rules have also been used to study how delay tuning could be achieved in the
sound source localization system of the barn owl [12].
A simple circuit to implement a general weight change block based on such learning
rules has been designed and partially fabricated. The main characteristics of the
learning rule, namely the abruptness at the origin, the rate of the decay of the
learning window, the imbalance between the potentiation and depression regions
and the rate of learning , can be tuned easily. The design also ensures that the
circuit can operate at different timescales. As shown, the fabricated circuits have
good linearity over a wide range of weight voltage values.
We are currently developing a second chip with a small network of temporally
asymmetric Hebbian spiking neurons using the circuit described in this paper. The
structure of the network will be reconfigurable. The small network will be used to
carry out movement planning exp eriments by learning of temporal sequences. We
envisage the application of networks of temporally-asymmetric Hebbian learning
silicon neurons as higher level processing stages for the integration of sensor and
motor activities in neuromorphic system. We will concentrate on auditory applications and adaptive, spike-based motion estimation. In both types of application,
naturally-occurring correlations in data can b e exploited to drive the pulse timingbased learning process.
Acknowledgelllents
We thank Robin Woodburn , Patrice Fleury and Martin Reekie for fruitful discussions during the design and tape out of the chip. We also acknowledge that the
circuits presented incorporate some of the insights into neuromorphic engineering
that one of the authors gained at the Telluride Workshop on Neuromorphic Engineering 2000 (http://www.ini.unizh.ch/telluride2000 /).
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1,235 | 2,125 | The Concave-Convex Procedure (CCCP)
A. L. Yuille and Anand Rangarajan *
Smith-Kettlewell Eye Research Institute,
2318 Fillmore Street,
San Francisco, CA 94115, USA.
Tel. (415) 345-2144. Fax. (415) 345-8455.
Email [email protected]
* Prof. Anand Rangarajan. Dept. of CISE, Univ. of Florida Room 301, CSE
Building Gainesville, FL 32611-6120 Phone: (352) 392 1507 Fax: (352) 392 1220
e-mail: [email protected]
Abstract
We introduce the Concave-Convex procedure (CCCP) which constructs discrete time iterative dynamical systems which are guaranteed to monotonically decrease global optimization/energy functions. It can be applied to (almost) any optimization problem and
many existing algorithms can be interpreted in terms of CCCP. In
particular, we prove relationships to some applications of Legendre
transform techniques. We then illustrate CCCP by applications to
Potts models, linear assignment, EM algorithms, and Generalized
Iterative Scaling (GIS). CCCP can be used both as a new way to
understand existing optimization algorithms and as a procedure for
generating new algorithms.
1
Introduction
There is a lot of interest in designing discrete time dynamical systems for inference
and learning (see, for example, [10], [3], [7], [13]).
This paper describes a simple geometrical Concave-Convex procedure (CCCP) for
constructing discrete time dynamical systems which can be guaranteed to decrease
almost any global optimization/energy function (see technical conditions in section (2)).
We prove that there is a relationship between CCCP and optimization techniques
based on introducing auxiliary variables using Legendre transforms. We distinguish
between Legendre min-max and Legendre minimization. In the former, see [6], the
introduction of auxiliary variables converts the problem to a min-max problem
where the goal is to find a saddle point. By contrast, in Legendre minimization, see
[8], the problem remains a minimization one (and so it becomes easier to analyze
convergence). CCCP relates to Legendre minimization only and gives a geometrical
perspective which complements the algebraic manipulations presented in [8].
CCCP can be used both as a new way to understand existing optimization algorithms and as a procedure for generating new algorithms. We illustrate this by
giving examples from Potts models, EM, linear assignment, and Generalized Iterative Scaling. Recently, CCCP has also been used to construct algorithms to
minimize the Bethe/Kikuchi free energy [13].
We introduce CCCP in section (2) and relate it to Legendre transforms in section (3). Then we give examples in section (4).
2
The Concave-Convex Procedure (CCCP)
The key results of CCCP are summarized by Theorems 1,2, and 3.
Theorem 1 shows that any function , subject to weak conditions, can be expressed
as the sum of a convex and concave part (this decomposition is not unique). This
implies that CCCP can be applied to (almost) any optimization problem.
Theorem 1. Let E(x) be an energy function with bounded Hessian [J2 E(x)/8x8x.
Then we can always decompose it into the sum of a convex function and a concave
function.
Proof. Select any convex function F(x) with positive definite Hessian with eigenvalues bounded below by f > o. Then there exists a positive constant A such that
the Hessian of E(x) + AF(x) is positive definite and hence E(x) + AF(x) is convex. Hence we can express E(x) as the sum of a convex part, E(x) + AF(x) , and a
concave part -AF(x).
Figure 1: Decomposing a function into convex and concave parts. The original function (Left Panel) can be expressed as the sum of a convex function (Centre Panel)
and a concave function (Right Panel). (Figure courtesy of James M. Coughlan).
Our main result is given by Theorem 2 which defines the CCCP procedure and
proves that it converges to a minimum or saddle point of the energy.
Theorem 2 . Consider an energy function E(x) (bounded below) of form E(x) =
Evex (x) + E cave (x) where Evex (x), E cave (x) are convex and concave functions of x
respectively. Then the discrete iterative CCCP algorithm ;zt f-7 ;zt+1 given by:
\1Evex
(x-t+l ) _- -\1Ecave
(x-t ),
(1)
is guaranteed to monotonically decrease the energy E(x) as a function of time and
hence to converge to a minimum or saddle point of E(x).
Proof. The convexity and concavity of Evex (.) and Ecave (.) means that Evex (X2) 2:
Evex (xd + (X2 -xd? ~ Evex (xd and Ecave (X4) :S Ecave (X3) + (X4 -X3)? ~ Ecave (X3 ),
for all X1 ,X2,X3,X4. Now set Xl = xt+l,X2 = xt,X3 = xt,X4 = xt+1. Using the
algorithm definition (i.e. ~Evex (xt+1) = -~Ecave (xt)) we find that Evex (xt+ 1) +
Ecave (xt+1) :S Evex (xt) + Ecave (xt), which proves the claim.
We can get a graphical illustration of this algorithm by the reformulation shown in
figure (2) (suggested by James M. Coughlan). Think of decomposing the energy
function E(x) into E 1(x) - E 2(x) where both E 1(x) and E 2(x) are convex. (This
is equivalent to decomposing E(x) into a a convex term E 1(x) plus a concave term
-E2(X)) . The algorithm proceeds by matching points on the two terms which have
the same tangents. For an input Xo we calculate the gradient ~ E2 (xo) and find the
point Xl such that ~ E1 (xd = ~ E2 (xo). We next determine the point X2 such that
~E1(X2) = ~E2 (X1)' and repeat.
7~------~--------~------,
60 50 40 -
30 -
20 -
o
10
O L---~=-~O-=~~~~----~
10
XO
Figure 2: A CCCP algorithm illustrated for Convex minus Convex. We want to
minimize the function in the Left Panel. We decompose it (Right Panel) into
a convex part (top curve) minus a convex term (bottom curve). The algorithm
iterates by matching points on the two curves which have the same tangent vectors,
see text for more details. The algorithm rapidly converges to the solution at x = 5.0.
We can extend Theorem 2 to allow for linear constraints on the variables X, for
example Li et Xi = aM where the {en, {aM} are constants. This follows directly
because properties such as convexity and concavity are preserved when linear constraints are imposed. We can change to new coordinates defined on the hyperplane
defined by the linear constraints. Then we apply Theorem 1 in this coordinate
system.
Observe that Theorem 2 defines the update as an implicit function of xt+ 1. In many
cases, as we will show, it is possible to solve for xt+1 directly. In other cases we may
need an algorithm, or inner loop , to determine xt+1 from ~Evex (xt+1). In these
cases we will need the following theorem where we re-express CCCP in terms of
minimizing a time sequence of convex update energy functions Et+1 (xt+1) to obtain
the updates xt+1 (i.e. at the tth iteration of CCCP we need to minimize the energy
Et+1 (xt+1 )). We include linear constraints in Theorem 3.
Theorem 3. Let E(x) = Evex (x) + E cave (x) where X is required to satisfy the linear
constraints Li et Xi = aM, where the {et}, {aM} are constants. Then the update rule
for xt+1 can be formulated as minimizing a time sequence of convex update energy
functions Et+1 (;rt+1):
(2)
where the lagrange parameters P'J1} impose linear comnstraints.
Proof. Direct calculation.
The convexity of EH1 (;rt+1) implies that there is a unique minimum corresponding
to ;rt+1. This means that if an inner loop is needed to calculate ;rt+1 then we can
use standard techniques such as conjugate gradient descent (or even CCCP).
3
Legendre Transformations
The Legendre transform can be used to reformulate optimization problems by introducing auxiliary variables [6]. The idea is that some of the formulations may
be more effective (and computationally cheaper) than others. We will concentrate
on Legendre minimization, see [7] and [8], instead of Legendre min-max emphasized
in [6]. An advantage of Legendre minimization is that mathematical convergence
proofs can be given. (For example, [8] proved convergence results for the algorithm
implemented in [7].)
In Theorem 4 we show that Legendre minimization algorithms are equivalent to
CCCP. The CCCP viewpoint emphasizes the geometry of the approach and complements the algebraic manipulations given in [8]. (Moreover, our results of the
previous section show the generality of CCCP while, by contrast, the Legendre
transform methods have been applied only on a case by case basis).
Definition 1. Let F(x) be a convex function. For each value y let F*(ff) =
minx{F(x) +y?x.}. Then F*(Y) is concave and is the Legendre transform of F(x).
Moreover, F (x) = max y { F* (y) - y. x} .
Property 1. F(.) and F*(.) are related by
a:; (fJ)
= {~~} - 1(_Y), -~~(x) =
{a{y* } -1 (x). (By { a{y* } -1 (x) we mean the value y such that a{y* (y) = x.)
Theorem 4. Let E1 (x) = f(x) + g(x) and E 2(x, Y) = f(x) + x? Y + h(i/), where
f(.), h(.) are convex functions and g(.) is concave. Then applying CCCP to E1 (x) is
equivalent to minimizing E2 (x, Y) with respect to x and y alternatively (for suitable
choices of g(.) and h(.).
Proof. We can write E1(X) = f(x) +miny{g*(Y) +x?y} where g*(.) is the Legendre
transform of g( .) (identify g(.) with F*( .) and g*(.) with F(.) in definition 1). Thus
minimizing E1 (x) with respect to x is equivalent to minimizing E1 (x, Y) = f(x) +
x . y + g* (Y) with respect to x and y. (Alternatively, we can set g* (Y) = h(Y)
in the expression for E 2(x,i/) and obtain a cost function E 2(x) = f(x) + g(x).)
Alternatively minimization over x and y gives: (i) of/ax = y to determine Xt+1 in
terms of Yt, and (ii) ag* / ay = x to determine Yt in terms of Xt which, by Property
1 of the Legendre transform is equivalent to setting y = -ag / ax. Combining these
two stages gives CCCP:
f (_)
a
ag (_)
ax Xt+1 = - ax Xt .
4
Examples of CCCP
We now illustrate CCCP by giving four examples: (i) discrete time dynamical
systems for the mean field Potts model, (ii) an EM algorithm for the elastic net,
(iii) a discrete (Sinkhorn) algorithm for solving the linear assignment problem, and
(iv) the Generalized Iterative Scaling (GIS) algorithm for parameter estimation.
Example 1.
Discrete Time Dynamical Systems for the Mean Field Potts
Model. These attempt to minimize discrete energy functions of form E[V] =
2: i ,j,a,b Tij ab Via V)b + 2: ia Bia Vi a, where the {Via} take discrete values {a, I} with
linear constraints 2:i Via = 1, Va.
Discussion. Mean field algorithms minimize a continuous effective energy E ett [S; T]
to obtain a minimum of the discrete energy E[V] in the limit as T f-7 a. The
{Sial are continuous variables in the range [0 ,1] and correspond to (approximate)
estimates of the mean states of the {Via}. As described in [12}, to ensure that the
minima of E[V] and E ett [S; T] all coincide (as T f-7 0) it is sufficient that T ijab
be negative definite. Moreover, this can be attained by adding a term -K 2: ia
to E[V] (for sufficiently large K) without altering the structure of the minima of
E[V] . Hence, without loss of generality we can consider 2: i ,j,a,b Tijab Via V)b to be a
concave function .
Vi!
We impose the linear constraints by adding a Lagrange multiplier term
2: a Pa {2: i Via - I} to the energy where the {Pa} are the Lagrange multipliers. The
effective energy becomes:
ia
i,j,a ,b
ia
a
We can then incorporate the Lagrange multiplier term into the convex part.
This gives: Evex [S] = T2: ia SialogSia + 2:aPa{2:iSia -I} and Ecave[S] =
2: i jab TijabSiaSjb + 2: ia BiaS ia ? Taking derivatives yields:
Evex [S] =
TI~~Sia + Pa
&g
&:s::~ (StH) = Bia?
&t E cave [S] = 2 2: j ,b TijabSjb + Bia? Applying eeep by setting
&:5;:e (st) gives T{l + log Sia (t + I)} + Pa = -2 2: j ,b TijabSjb(t)-
and
We solve for the Lagrange multipliers {Pal by imposing the constraints
1, Va. This gives a discrete update rule:
2:i Sia(t + 1) =
Sia (t + 1) =
e(-1/T){2 2:.J, b TijabSjb(t)+Oia}
'
.
2: c e( -1/T){2 2: j,b TijcbSjb(tl+Oi
c}
(4)
Algorithms of this type were derived in [lO}, [3} using different design principles.
Our second example relates to the ubiquitous EM algorithm. In general EM and
CCCP give different algorithms but in some cases they are identical. The EM algorithm seeks to estimate a variable f* = argmaxt log 2:{I} P(f, l), where {f}, {l} are
variables that depend on the specific problem formulation. It was shown in [4] that
this is equivalent to minimizing the following effective energy with respect to the
variables f and P(l): E ett [! , P(l)] = - ~ 2:1 P(l) log P(f, l) + ~ 2:{I} P(l) log P(l).
To apply CCCP to an effective energy like this we need either: (a) to decompose
E ett [!, P(l)] into convex and concave functions of f, P(l), or (b) to eliminate either
variable and obtain a convex concave decomposition in the remaining variable (d.
Theorem 4). We illustrate (b) for the elastic net [2]. (See Yuille and Rangarajan,
in preparation, for an illustration of (a)).
Example 2. The elastic net attempts to solve the Travelling Salesman Problem
(TSP) by finding the shortest tour through a set of cities at positions {Xi }' The
elastic net is represented by a set of nodes at positions {Ya} with variables {Sial
that determine the correspondence between the cities and the nodes of the net. Let
E el I [S, 171 be the effective energy for the elastic net, then the {y} variables can be
eliminated and the resulting Es[S] can be minimized using GGGP. (Note that the
standard elastic net only enforces the second set of linear constraints).
Discussion. The elastic net energy function can be expressed as [11]:
ia
a,b
where we impose the conditions L: a Sia = 1, V i and
i,a
L:i Sia =
1, V a.
The EM algorithm can be applied to estimate the {Ya}. Alternatively we can solve
for the {Ya} variables to obtain Yb = L:i a PabSiaXi where {Pab } = {Jab + 2')'Aab} -1.
We substitute this back into E ell [S, 171 to get a new energy Es[S] given by:
(6)
i ,j,a,b
i,a
Once again this is a sum of a concave and a convex part (the first term is concave
because of the minus sign and the fact that {Pba } and Xi . Xj are both positive semidefinite.) We can now apply GGGP and obtain the standard EM algorithm for this
problem. (See Yuille and Rangarajan, in preparation, for more details).
Our final example is a discrete iterative algorithm to solve the linear assignment
problem. This algorithm was reported by Kosowsky and Yuille in [5] where it was
also shown to correspond to the well-known Sinkhorn algorithm [9]. We now show
that both Kosowsky and Yuille's linear assignment algorithm, and hence Sinkhorn's
algorithm are examples of CCCP (after a change of variables).
Example 3. The linear assignment problem seeks to find the permutation matrix
{TI ia } which minimizes the energy E[m = L: ia TI ia A ia , where {Aia} is a set of
assignment values. As shown in [5} this is equivalent to minimizing the (convex)
Ep[P] energy given by Ep[P] = L: aPa + ~ L:i log L:a e-,B(Aia+Pa) , where the solution is given by TI;a = e-,B(Aia+Pa) / L:b e-,B(Aib+Pb) rounded off to the nearest
integer (for sufficiently large fJ). The iterative algorithm to minimize Ep[P] (which
can be re-expressed as Sinkhorn's algorithm, see [5}) is of form:
(7)
and can be re-expressed as GGGP.
Discussion. By performing the change of coordinates fJPa = - log r a V a (for r a
>
0, Va) we can re-express the Ep[P] energy as:
(8)
Observe that the first term of Er [r] is convex and the second term is concave (this
can be verified by calculating the Hessian). Applying CCCP gives the update rule:
1
rt+l =
a
2:= 2:::
i
e-,BAia
e-,BAibrt'
b
(9)
b
which corresponds to equation (7).
Example 4. The Generalized Iterative Scaling (GIS) Algorithm [ll for estimating
parameters in parallel.
Discussion. The GIS algorithm is designed to estimate the parameter
Xof a distri-
bution P(x : X) = eX.?(x) IZ[X] so that 2:::x P(x; X)?(x) = h, where h are observation data (with components indexed by j.t). It is assumed that ?fJ,(x) ::::: 0, V j.t,x,
hfJ, ::::: 0, V j.t, and 2:::fJ, ?fJ, (x) = 1, V x and 2:::fJ, hfJ, = 1. (All estimation problems of
this type can be transformed into this form [lj).
Darroch and Ratcliff [ll prove that the following GIS algorithm is guaranteed to
converge to value X* that minimizes the (convex) cost function E(X) = log Z[X]-X.h
and hence satisfies 2::: x P(x; X*)?(x) = h. The GIS algorithms is given by:
Xt+! =
Xt -
log ht
+ log h,
(10)
where ht = 2::: x P(x; Xt )?(x) {evaluate log h componentwise: (log h)fJ, = log hf),')
To show that GIS can be reformulated as CCCP, we introduce a new variable
iJ = eX (componentwise). We reformulate the problem in terms of minimizing
the cost function E,B [iJ] = log Z[log(iJ)] - h . (log iJ). A straightforward calculation shows that -h . (log iJ) is a convex function of iJ with first derivative being
-hi iJ (where the division is componentwise). The first derivative of log Z[log(iJ)] is
(II iJ) 2::: x ?(x)P(x: log ,8) (evaluated componentwise). To show that log Z[log(iJ)] is
concave requires computing its Hessian and applying the Cauchy-Schwarz inequality
using the fact that 2:::fJ, ?fJ,(x) = 1, V x and that ?fJ,(x) ::::: 0, V j.t,x. We can therefore apply CCCP to E,B [iJ] which yields l/iJH1 = l/iJt x Ilh x ht (componentwise) ,
which is GIS (by taking logs and using log ,8 = X).
5
Conclusion
CCCP is a general principle which can be used to construct discrete time iterative
dynamical systems for almost any energy minimization problem. It gives a geometric perspective on Legendre minimization (though not on Legendre min-max).
We have illustrated that several existing discrete time iterative algorithms can be reinterpreted in terms of CCCP (see Yuille and Rangarajan, in preparation, for other
examples). Therefore CCCP gives a novel way ofthinking about and classifying existing algorithms. Moreover, CCCP can also be used to construct novel algorithms.
See, for example, recent work [13] where CCCP was used to construct a double loop
algorithm to minimize the Bethe/Kikuchi free energy (which are generalizations of
the mean field free energy).
There are interesting connections between our results and those known to mathematicians. After this work was completed we found that a result similar to Theorem
2 had appeared in an unpublished technical report by D. Geman. There also are
similarities to the work of Hoang Tuy who has shown that any arbitrary closed
set is the projection of a difference of two convex sets in a space with one more
dimension. (See http://www.mai.liu.se/Opt/MPS/News/tuy.html).
Acknowledgements
We thank James Coughlan and Yair Weiss for helpful conversations. Max Welling
gave useful feedback on this manuscript. We thank the National Institute of Health
(NEI) for grant number R01-EY 12691-01.
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1,236 | 2,126 | Very loopy belief propagation for
unwrapping phase images
Brendan J . Freyl, Ralf Koetter2, Nemanja Petrovic 1 ,2
Probabilistic and Statistical Inference Group, University of Toronto
http://www.psi.toronto.edu
Electrical and Computer Engineering, University of Illinois at Urbana
1
2
Abstract
Since the discovery that the best error-correcting decoding algorithm can be viewed as belief propagation in a cycle-bound graph,
researchers have been trying to determine under what circumstances "loopy belief propagation" is effective for probabilistic inference. Despite several theoretical advances in our understanding of
loopy belief propagation, to our knowledge, the only problem that
has been solved using loopy belief propagation is error-correcting
decoding on Gaussian channels. We propose a new representation
for the two-dimensional phase unwrapping problem, and we show
that loopy belief propagation produces results that are superior to
existing techniques. This is an important result, since many imaging techniques, including magnetic resonance imaging and interferometric synthetic aperture radar, produce phase-wrapped images.
Interestingly, the graph that we use has a very large number of
very short cycles, supporting evidence that a large minimum cycle
length is not needed for excellent results using belief propagation.
1
Introduction
Phase unwrapping is an easily stated, fundamental problem in image processing
(Ghiglia and Pritt 1998). Each real-valued observation on a 1- or 2-dimensional
grid is measured modulus a known wavelength, which we take to be 1 without loss
of generality. Fig. Ib shows the wrapped, I-dimensional waveform obtained from the
original waveform shown in Fig. la. Every time the original waveform goes above 1
or below 0, it is wrapped to 0 or 1, respectively. The goal of phase unwrapping is to
infer the original, unwrapped curve from the wrapped measurements, using using
knowledge about which signals are more probable a priori.
In two dimensions, exact phase unwrapping is exponentially more difficult than 1dimensional phase unwrapping and has been shown to be NP-hard in general (Chen
and Zebker 2000). Fig. lc shows the wrapped output of a magnetic resonance
imaging device, courtesy of Z.-P. Liang. Notice the "fringe lines" - boundaries
across which wrappings have occurred. Fig. Id shows the wrapped terrain height
measurements from an interferometric synthetic aperture radar, courtesy of Sandia
National Laboratories, New Mexico.
(a)
(b)
(d)
Figure 1: (a) A waveform measured on a 1-dimensional grid. (b) The phase-wrapped version
ofthe waveform in (a), where the wavelength is 1. (c) A wrapped intensity ma p from a magnetic
resonance imaging device, measured on a 2-dimensional grid (courtesy of Z .-P. Liang). (d)
A wrapped topographic map measured on a 2-dimensional grid (courtesy of Sandia National
Laboratories, New Mexico) .
A sensible goal in phase unwrapping is to infer the gradient field of the original
surface. The surface can then be reconstructed by integration. Equivalently, the
goal is to infer the number of relative wrappings, or integer "shifts", between every
pair of neighboring measurements. Positive shifts correspond to an increase in the
number of wrappings in the direction of the x or y coordinate, whereas negative
shifts correspond to a decrease in the number of wrappings in the direction of the
x or y coordinate. After arbitrarily assigning an absolute number of wrappings to
one point, the absolute number of wrappings at any other point can be determined
by summing the shifts along a path connecting the two points. To account for
direction, when taking a step against the direction of the coordinate, the shift
should be subtracted.
When neighboring measurements are more likely closer together than farther apart
a priori, I-dimensional waveforms can be unwrapped optimally in time that is linear
in the waveform length. For every pair of neighboring measurements, the shift that
makes the unwrapped values as close together as possible is chosen. For example,
the shift between 0.4 and 0.5 would be 0, whereas the shift between 0.9 and 0.0
would be -1.
For 2-dimensional surfaces and images, there are many possible I-dimensional paths
between any two points. These paths should be examined in combination, since the
sum of the shifts along every such path should be equal. Viewing the shifts as state
variables, the cut-set between any two points is exponential in the size of the grid,
making exact inference for general priors NP-hard (Chen and Zebker 2000).
The two leading fully-automated techniques for phase unwrapping are the least
squares method and the branch cut technique (Ghiglia and Pritt 1998). (Some other
techniques perform better in some circumstances, but need additional information
or require hand-tweaking.) The least squares method begins by making a greedy
guess at the gradient between every pair of neighboring points. The resulting vector
field is not the gradient field of a surface, since in a valid gradient field, the sum of
the gradients around every closed loop must be zero (that is, the curl must be 0).
For example, the 2 x 2 loop of measurements 0.0, 0.3, 0.6, 0.9 will lead to gradients of
0.3,0.3,0.3, 0.1 around the loop, which do not sum to O. The least squares method
proceeds by projecting the vector field onto the linear subspace of gradient fields.
The result is integrated to produce the surface. The branch cut technique also
begins with greedy decisions for the gradients and then identifies untrustworthy
regions of the image whose gradients should not be used during integration. As
shown in our results section, both of these techniques are suboptimal.
Previously, we attempted to use a relaxed mean field technique to solve this problem
(Achan, Frey and Koetter 2001). Here, we take a new approach that works better
and is motivated by the impressive results of belief propagation in cycle-bound
graphs for error-correcting decoding (Wiberg, Loeliger and Koetter 1995; MacKay
and Neal 1995; Frey and Kschischang 1996; Kschischang and Frey 1998; McEliece,
MacKay and Cheng 1998). In contrast to other work (Ghiglia and Pritt 1998;
Chen and Zebker 2000; Koetter et al. 2001), we introduce a new framework for
quantitative evaluation, which impressively places belief propagation much closer
to the theoretical limit than other leading methods.
It is well-known that belief propagation (a.k.a. the sum-product algorithm, probability propagation) is exact in graphs that are trees (Pearl 1988), but it has been
discovered only recently that it can produce excellent results in graphs with many
cycles. Impressive results have been obtained using loopy belief propagation for
super-resolution (Freeman and Pasztor 1999) and for infering layered representations of scenes (Frey 2000) . However, despite several theoretical advances in our
understanding of loopy belief propagation (c.f. (Weiss and Freeman 2001)) and proposals for modifications to the algorithm (c.f. (Yedidia, Freeman and Weiss 2001)) ,
to our knowledge, the only problem that has been solved by loopy belief propagation
is error-correcting decoding on Gaussian channels.
We conjecture that although phase unwrapping is generally NP-hard, there exists a
near-optimal phase unwrapping algorithm for Gaussian process priors. Further, we
believe that algorithm to be loopy belief propagation.
2
Loopy Belief Propagation for Phase Unwrapping
As described above, the goal is to infer the number of relative wrappings , or integer
"shifts" , between every pair of neighboring measurements. Denote the x-direction
shift at (x,y) by a(x , y) and the y-direction shift at (x , y) by b(x , y), as shown in
Fig.2a. If the sum of the shifts around every short loop of 4 shifts (e.g., a(x,y) +
b(x + l,y) - a(x , y + 1) - b(x,y) in Fig. 2a) is zero, then perturbing a path will
not change the sum of the shifts along the path. So, a valid set of shifts S =
{a(x,y) , b(x , y) : x = 1, ... , N -1;y = 1, .. . , M -I} in an N x M image must
satisfy the constraint
a(x,y)
+ b(x + l,y)
- a(x,y
+ 1) -
b(x,y) = 0,
(1)
for x = 1, ... , N -1, Y = 1, ... , M -1. Since a(x, y) +b(x+ 1, y) -a(x, y+ 1) -b(x, y)
is a measure of curl at (x, y), we refer to (1) as a "zero-curl constraint", reflecting
the fact that the curl of a gradient field is O. In this way, phase unwrapping is
formulated as the problem of inferring the most probable set of shifts subject to
satisfying all zero-curl constraints.
We assume that given the set of shifts, the unwrapped surface is described by a loworder Gaussian process. The joint distribution over the shifts S = {a(x, y), b(x, y) :
x = 1, ... , N - 1; Y = 1, ... , M - I} and the wrapped measurements <I> = {?(x, y) :
(b)
(a)
x-direction shifts (' a's)
t.: ~-r1
X
X
X
X
'E
X [f X
X
X
X
X
t5
X
X
X
X
X
X
'6
X
X ) X
X
X
X
X
X ) X
X
X
X
X
X
X
X
X
Vi'
(x, y + l )
b(x, y )
X
a(x,y+ l )
-7
X (x+ l ,y+ l )
.<:::
I
I
<fl
<fl
c
0
b(x+ l ,y)
~
>.
(x,y)
X
-7
a(x,y)
X (x+ l ,y)
(d)
X
it2
t~
a(x, y)
it l
t
Figure 2: (a) Positive x-direction shifts (arrows labeled a) and positive y-direction shifts
(arrows labeled b) between neighboring measurements in a 2 X 2 patch of points (marked by
X 's) , (b) A graphical model that describes the zero-curl constraints (black discs) between
neighboring shift variables (white discs), 3-element probability vectors (J-L's) on the relative
shifts between neighboring variables (-1, 0, or +1) are propagated across the network: (c)
Constraint-to-shift vectors are computed from incoming shift-to-constraint vectors; (d) Shiftto-constraint vectors are computed from incoming constraint-to-shift vectors; (d) Estimates of
the marginal probabilities of the shifts given the data are computed by combining incoming
constra int-to-sh ift vectors,
0 :::; r/J(x, y)
< 1, x = 1, .. . , N; y
= 1, . . . , M } can be expressed in t he form
N- l M- l
P(S , <I? ex:
II II 5(a(x,y) +b(x +1 ,y) -
a(x,y +1 ) -b(x,y))
x=l y=l
N-l M
. II II
x= l y=l
N
e-(c/>(x+l,y)-c/>(x,y)-a(x,y))2/ 2u 2
M-l
II II
e-(C/>(x,y+1)-c/>(x,y)-b(x,y))2/ 2u 2 .
(2)
x= l y=l
The zero-curl constraints are enforced by 5 (.), which evaluates to 1 if its argument is
oand evaluates to 0 otherwise. We assume t he slope of t he surface is limited so t hat
t he unknown shifts take on t he values -1 , 0 and 1. a 2 is t he variance between two
neighboring measurements in t he unwrapped image, but we find t hat in practice it
can be estimated directly from t he wrapped image.
Phase unwrapping consists of making inferences about t he a's and b's in t he above
probability model. For example, t he marginal probability t hat t he x-direction shift
at (x,y) is k given an observed wrapped image <I> , is
P (a (x,y) = kl<I? ex:
L
P(S , <I? .
(3)
S:a(x ,y)=k
For an N x M grid, t he above sum has roughly 32N M terms and so exact inference
is intractable.
The factorization of t he joint distribution in (2) can be described by a graphical
model, as shown in Fig. 2b. In t his graph , each white disc sits on t he border between
two measurements (marked by x's), and corresponds to eit her an x-direction shift
(a's ) or a y-direction shift (b's) . Each black disc corresponds to a zero-curl constraint
(5(?) in (2)), and is connected to t he 4 shifts t hat it constrains to sum to O.
P robability propagation computes messages (3-vectors denoted by J-L) which are
passed in both directions on every edge in t he network. The elements of each 3vector correspond to t he allowed values of t he neighboring shift, -1 , 0 and 1. Each
of t hese 3-vectors can be t hought of as a probability distribut ion over t he 3 possible
values t hat t he shift can take on.
Each element in a constraint-to-shift message summarizes the evidence from the
other 3 shifts involved in the constraint, and is computed by averaging the allowed
configurations of evidence from the other 3 shifts in the constraint. For example, if
ILl ' IL2 and IL3 are 3-vectors entering a constraint as shown in Fig. 2c, the outgoing
3-vector, IL4' is computed using
1
f.t4i = L
j=-l
1
L
L
J(k
k=-ll=-l
+ l- i
- j)f.tljf.t2kf.t31,
(4)
and then normalized for numerical stability. The other 3 messages produced at the
constraint are computed in a similar fashion .
Shift-to-constraint messages are computed by weighting incoming constraint-to-shift
messages with the likelihood for the shift. For example, if ILl is a 3-vector entering
an x-direction shift as shown in Fig. 2d, the outgoing 3-vector, IL2 is computed using
(5)
and then normalized. Messages produced by y-direction shifts are computed in a
similar fashion.
At any step in the message-passing process, the messages on the edges connected
to a shift variable can be combined to produce an approximation to the marginal
probability for that shift, given the observations. For example, if ILl and IL2 are the
3-vectors entering an x-direction shift as shown in Fig. 2e, the approximation is
P(a(x , y) = il<I? = (f.tlif.t2i)/(Lf.tljf.t2j).
(6)
j
Given a wrapped image, the variance a 2 is estimated directly from the wrapped
image, the probability vectors are initialized to uniform distributions, and then
probability vectors are propagated across the graph in an iterative fashion. Different
message-passing schedules are possible, ranging from fully parallel, to a "forwardbackward-up-down" -type schedule, in which messages are passed across the network
to the right, then to the left, then up and then down. For an N x M grid, each
iteration takes O(N M) scalar computations.
After probability propagation converges (or, after a fixed number of iterations),
estimates of the marginal probabilities of the shifts given the data are computed, and
the most probable value of each shift variable is selected. The resulting configuration
of the shifts can then be integrated to obtain the unwrapped surface. If some
zero-curl constraints remain violated, a robust integration technique, such as least
squares integration (Ghiglia and Pritt 1998), should be used.
3
Experimental results
Generally, belief propagation in cycle-bound graphs is not guaranteed to converge.
Even if it does converge, the approximate marginals may not be close to the true
marginals. So, the algorithm must be verified by experiments.
On surfaces drawn from Gaussian process priors, we find that the belief propagation
algorithm produces significantly lower reconstruction errors than the least squares
method and the branch cut technique.
Here, we focus on the performances of the algorithms for real data recorded from a
synthetic aperture radar device (Fig. 1d) . Since our algorithm assumes the surface
is Gaussian given the shifts, a valid concern is that it will not perform well when
the Gaussian process prior is incorrect.
(a)
(b)
en
c
Q)
'6
CO
.....
-
0)
o
o
.....
..... 10- 1
.....
Q)
"0
~
::J
g
? 10- 2
Minimum wavelength
required for error-free
unwrapping using algs
that infer relative
~
shifts of -1 , 0 and +1
Q)
~
6
8
10
12
14
16 18 20 22 24 26
Wavelength, "-
Figure 3: (a) After 10 iterations of belief propagation using the phase-wrapped surface from
Fig. Id, hard decisions were made for the shift variables and the resulting shifts were integrated
to produce this unwrapped surface. (b) Reconstruction error versus wrapping wavelength for
our technique, the least squares method and the branch cuts technique .
Fig. 3a shows the surface that is obtained by setting (/2 to the mean squared difference between neighboring wrapped values, applying 10 iterations of belief propagation, making hard decisions for the integer shifts, and integrating the resulting
gradients. Since this is real data, we do not know the "ground truth" . However,
compared to the least squares method, our algorithm preserves more detail. The
branch cut technique is not able to unwrap the entire surface.
To obtain quantitative results on reconstruction error, we use the surface produced
by the least squares method as the "ground truth" . To determine the effect of wrapping wavelength on algorithm performance, we rewrap this surface using different
wavelengths. For each wavelength, we compute the reconstruction error for belief
propagation, least squares and branch cuts. Note that by using least squares to
obtain the ground truth, we may be biasing our results in favor of least squares.
Fig. 3b shows the logarithm of t he mean squared error in the gradient field of the
reconstructed surface as a function of the wrapping wavelength, >., on a log-scale.
(The plot for the mean squared error in t he surface heights looks similar.) As >. -+ 0,
unwrapping becomes impossible and as >. -+ 00, unwrapping becomes trivial (since
no wrappings occur), so algorithms have waterfall-shaped curves.
The belief propagation algorithm clearly obtains significantly lower reconstruction
errors. Viewed another way, belief propagation can tolerate much lower wrapping
wavelengths for a given reconstruction error. Also, it turns out that for this surface,
it is impossible for an algorithm that infers relative shifts of -1,0 and 1 to obtain a
reconstruction error of 0, unless A ::::: 12.97. Belief propagation obtains a zero-error
wavelength that is significantly closer to this limit than the least squares method
and the branch cuts technique.
4
Conclusions
Phase unwrapping is a fundamental problem in image processing and although it
has been shown to be NP-hard for general priors (Chen and Zebker 2000), we
conjecture there exists a near-optimal phase unwrapping algorithm for Gaussian
process priors. Further, we believe that algorithm to be loopy belief propagation.
Our experimental results show that loopy belief propagation obtains significantly
lower reconstruction errors compared to the least squares method and the branch
cuts technique (Ghiglia and Pritt 1998) , and performs close to the theoretical limit
for techniques that infer relative wrappings of -1, 0 and + 1. The belief propagation
algorithm runs in about the same time as the other techniques.
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Yedidia, J. , Freeman , W . T., and Weiss, Y. 2001. Generalized belief propagation. In
Advances in Neural Information Processing Systems 13. MIT Press, Cambridge MA.
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1,237 | 2,127 | On the Concentration of Spectral
Properties
John Shawe-Taylor
Royal Holloway, University of London
N ella Cristianini
BIOwulf Technologies
[email protected]
nello@support-vector. net
Jaz Kandola
Royal Holloway, University of London
[email protected]
Abstract
We consider the problem of measuring the eigenvalues of a randomly drawn sample of points. We show that these values can be
reliably estimated as can the sum of the tail of eigenvalues. Furthermore, the residuals when data is projected into a subspace is
shown to be reliably estimated on a random sample. Experiments
are presented that confirm the theoretical results.
1
Introduction
A number of learning algorithms rely on estimating spectral data on a sample of
training points and using this data as input to further analyses. For example in
Principal Component Analysis (PCA) the subspace spanned by the first k eigenvectors is used to give a k dimensional model of the data with minimal residual,
hence forming a low dimensional representation of the data for analysis or clustering. Recently the approach has been applied in kernel defined feature spaces
in what has become known as kernel-PCA [5]. This representation has also been
related to an Information Retrieval algorithm known as latent semantic indexing,
again with kernel defined feature spaces [2].
Furthermore eigenvectors have been used in the HITS [3] and Google's PageRank [1]
algorithms. In both cases the entries in the eigenvector corresponding to the maximal eigenvalue are interpreted as authority weightings for individual articles or web
pages.
The use of these techniques raises the question of how reliably these quantities can
be estimated from a random sample of data, or phrased differently, how much data is
required to obtain an accurate empirical estimate with high confidence. Ng et al. [6]
have undertaken a study of the sensitivity of the estimate of the first eigenvector to
perturbations of the connection matrix. They have also highlighted the potential
instability that can arise when two eigenvalues are very close in value, so that their
eigenspaces become very difficult to distinguish empirically.
The aim of this paper is to study the error in estimation that can arise from the
random sampling rather than from perturbations of the connectivity. We address
this question using concentration inequalities. We will show that eigenvalues estimated from a sample of size m are indeed concentrated, and furthermore the sum
of the last m - k eigenvalues is subject to a similar concentration effect, both results of independent mathematical interest. The sum of the last m - k eigenvalues
is related to the error in forming a k dimensional PCA approximation, and hence
will be shown to justify using empirical projection subspaces in such algorithms as
kernel-PCA and latent semantic kernels.
The paper is organised as follows. In section 2 we give the background results and
develop the basic techniques that are required to derive the main results in section
3. We provide experimental verification of the theoretical findings in section 4,
before drawing our conclusions.
2
Background and Techniques
We will make use of the following results due to McDiarmid. Note that lEs is the
expectation operator under the selection of the sample.
TheoreIll 1 (McDiarmid!4}) Let Xl, ... ,Xn be independent random variables taking values in a set A, and assume that f : An -+~, and fi : An- l -+ ~ satisfy for
l:::;i:::;n
Xl,??? , Xn
TheoreIll 2 (McDiarmid!4}) Let Xl, ... ,Xn be independent random variables taking values in a set A, and assume that f : An -+ ~, for 1 :::; i :::; n
sup
If(xI, ... , xn) - f(XI, ... , Xi -
I,
Xi, Xi+!,???, xn)1 :::; Ci,
We will also make use of the following theorem characterising the eigenvectors of a
symmetric matrix.
TheoreIll 3 (Courant-Fischer MiniIllax TheoreIll) If M E
ric, then for k = 1, ... , m,
Ak(M) =
max
v'Mv
min - - =
vlv
dim(T) = k O#v ET
min
max
dim(T) = m - k+IO#v E T
~mxm
is symmet-
v'Mv
vlv '
with the extrama achieved by the corresponding eigenvector.
The approach adopted in the proofs of the next section is to view the eigenvalues as
the sums of squares of residuals. This is applicable when the matrix is positive semidefinite and hence can be written as an inner product matrix M = XI X, where XI is
the transpose of the matrix X containing the m vectors Xl, . . . , Xm as columns. This
is the finite dimensional version of Mercer's theorem, and follows immediately if we
take X = V VA, where M = VA VI is the eigenvalue decomposition of M. There
may be more succinct ways of representing X, but we will assume for simplicity (but
without loss of generality) that X is a square matrix with the same dimensions as
M. To set the scene, we now present a short description of the residuals viewpoint.
The starting point is the singular value decomposition of X = U~V', where U and
V are orthonormal matrices and ~ is a diagonal matrix containing the singular
values (in descending order). We can now reconstruct the eigenvalue decomposition
of M = X' X = V~U'U~V' = V AV', where A = ~2. But equally we can construct
a matrix N = XX' = U~V'V~U' = UAU' , with the same eigenvalues as M.
As a simple example consider now the first eigenvalue, which by Theorem 3 and the
above observations is given by
v'Nv
max - O,t:vEIR = v'v
A1(M)
= max
O,t:vEIR=
m
max
O,t:vEIR =
v'XX'v
v'v
max
O,t:vE IR =
m
L
IIPv(xj)11 2 =
j=l
v'v
m
L
IIxjl12 -
j=l
min
O,t:vEIR=
L
IIP;-(xj)11 2
j=l
where Pv(x) (Pv..l (x)) is the projection of x onto the space spanned by v (space
perpendicular to v), since IIxI1 2 = IIPv(x)11 2+ IIPv
..l(x)112. It follows that the first
eigenvector is characterised as the direction for which sum of the squares of the
residuals is minimal.
Applying the same line of reasoning to the first equality of Theorem 3, delivers the
following equality
m
Ak =
max
L
min
dim(V) = k O,t:vEV .
IlPv(xj)112.
(1)
J=l
Notice that this characterisation implies that if v k is the k-th eigenvector of N, then
m
L
(2)
IlPv k (xj)112,
j=l
which in turn implies that if Vk is the space spanned by the first k eigenvectors,
then
Ak =
k
L
m
Ai =
L
m
IIPVk (Xj) 112 =
j=l
i=l
L
j=l
m
IIXj W-
L
IIP'* (Xj) 11 2,
(3)
j=l
where Pv(x) (PV(x)) is the projection of x into the space V (space perpendicular
to V). It readily follows by induction over the dimension of V that we can equally
characterise the sum of the first k and last m - k eigenvalues by
m
max
i= l
m
L
dim(V) = k .
m
IIPv(xj)11 2
=
J=l
m
L
L
.
m
IIxjl12 -
)= 1
k
L
min
L
dim(V) = k .
IIPv(xj)11 2,
)= 1
m
L
(4)
IIXjl12 - . Ai = dim(V)=k
min
IlPv(xj)112.
.
.
J=l
.=1
J=l
Hence, as for the case when k = 1, the subspace spanned by the first k eigenvalues
is characterised as that for which the sum of the squares of the residuals is minimal.
Frequently, we consider all of the above as occurring in a kernel defined feature
space, so that wherever we have written Xj we should have put ?>(Xj), where ?> is
the corresponding projection.
3
Concentration of eigenvalues
The previous section outlined the relatively well-known perspective that we now
apply to obtain the concentration results for the eigenvalues of positive semi-definite
matrices. The key to the results is the characterisation in terms of the sums of
residuals given in equations (1) and (4).
Theorem 4 Let K(x,z) be a positive semi-definite kernel function on a space X,
and let J-t be a distribution on X. Fix natural numbers m and 1 :::; k < m and let
S = (Xl"'" x m) E xm be a sample of m points drawn according to J-t. Th en for
all f > 0,
P{I~ )..k(S)
-lEs[~ )..k(S)ll 2: f} :::;
2exp (
-~:m)
,
where )..k (S) is the k-th eigenvalue of the matrix K(S) with entries K(S)ij
K(Xi,Xj) and R2 = maxx Ex K(x,x).
Proof: The result follows from an application of Theorem 1 provided
1
1
2
sup 1- )..k(S) - - )..k(S \ {xd)1 :::; Rim.
s m
m
Let S = S \ {Xi} and let V (11) be the k dimensional subspace spanned by the first
k eigenvectors of K(S) (K(S)). Using equation (1) we have
m
m
D
Surprisingly a very similar result holds when we consider the sum of the last m - k
eigenvalues.
Theorem 5 Let K(x, z) be a positive semi-definite kernel function on a space X,
and let J-t be a distribution on X. Fix natural numbers m and 1 :::; k < m and let
S = (Xl, ... , Xm) E xm be a sample of m points drawn according to J-t. Then for
all f > 0,
P{I~ )..>k(S)
-lEs [~ )..>k(S)ll 2: f} :::;
2 exp (
-~:m)
,
where )..>k(S) is the sum of all but the largest k eigenvalues of the matrix K(S) with
entries K(S)ij = K(Xi,Xj) and R2 = maxxEX K(x,x).
Proof: The result follows from an application of Theorem 1 provided
sup
1~)..>k(S)
s m
-
~)..>k(S \ {xd)1 :::; R2/m.
m
Let S = S \ {xd and let V (11) be the k dimensional subspace spanned by the first
k eigenvectors of K(S) (K(S)). Using equation (4) we have
m
j=l
#i
m
#i
D
j=l
Our next result concerns the concentration of the residuals with respect to a fixed
subspace. For a subspace V and training set S , we introduce the notation
1 m
Fv(S) = -
L
IIPV(Xi )112 .
m i=l
TheoreIll 6 Let J-t be a distribution on X. Fix natural numbers m and a subspace
V and let S = (Xl, .. . , Xm) E xm be a sample of m points drawn according to J-t.
Then for all t > 0,
P{IFv(S) -lEs [Fv(S)ll
~ t} ::::: 2exp (~~r;) .
Proof: The result follows from an application of Theorem 2 provided
sup IFv(S) - F(S \ {xd U {xi)1 ::::: R2/m.
S,X i
Clearly the largest change will occur if one of the points Xi and Xi is lies in the
subspace V and the other does not. In this case the change will be at most R2/m.
D
4
Experiments
In order to test the concentration results we p erformed experiments with the Breast
cancer data using a cubic polynomial kernel. The kernel was chosen to ensure that
the spectrum did not decay too fast.
We randomly selected 50% of the data as a 'training' set and kept the remaining
50% as a 'test' set. We centered the whole data set so that the origin of the feature
space is placed at the centre of gravity of the training set. We then performed an
eigenvalue decomposition of the training set. The sum of the eigenvalues greater
than the k-th gives the sum of the residual squared norms of the training points
when we project onto the space spanned by the first k eigenvectors. Dividing this by
the average of all the eigenvalues (which measures the average square norm of the
training points in the transformed space) gives a fraction residual not captured in
the k dimensional projection. This quantity was averaged over 5 random splits and
plotted against dimension in Figure 1 as the continuous line. The error bars give
one standard deviation. The Figure la shows the full spectrum, while Figure 1b
shows a zoomed in subwindow. The very tight error bars show clearly the very tight
concentration of the sums of tail of eigenvalues as predicted by Theorem 5.
In order to test the concentration results for subsets we measured the residuals of
the test points when they are projected into the subspace spanned by the first k
eigenvectors generated above for the training set. The dashed lines in Figure 1 show
the ratio of the average squares of these residuals to the average squared norm of the
test points. We see the two curves tracking each other very closely, indicating that
the subspace identified as optimal for the training set is indeed capturing almost
the same amount of information in the test points.
5
Conclusions
The paper has shown that the eigenvalues of a positive semi-definite matrix generated from a random sample is concentrated. Furthermore the sum of the last m - k
eigenvalues is similarly concentrated as is the residual when the data is projected
into a fixed subspace.
0.7,------,-------,-------,------,-------,-------,------,,------,
0.6
0.5
0.2
0.1
Projection Dimensionality
(a)
0.14,-----,-----,-----,-----,-----,-----,-----,-----,-----,-----,
0.12
0.1
e0.08
W
1
Cii
OJ
:g
en
\
&! 0.06
0.04
0.02
'1- -
I
- -:E-- -I- --:1:- _ '.[ __
O~--L---~--~--~---L-=~~~~======~~
o
10
20
30
40
50
60
Projection Dimensionality
70
80
90
100
(b)
Figure 1: Plots ofthe fraction of the average squared norm captured in the subspace
spanned by the first k eigenvectors for different values of k. Continuous line is
fraction for training set, while the dashed line is for the test set. (a) shows the full
spectrum, while (b) zooms in on an interesting portion.
Experiments are presented that confirm the theoretical predictions on a real world
dataset. The results provide a basis for performing PCA or kernel-PCA from a
randomly generated sample, as they confirm that the subset identified by the sample
will indeed 'generalise' in the sense that it will capture most of the information in
a test sample.
Further research should look at the question of how the space identified by a subsample relates to the eigenspace of the underlying kernel operator.
References
[1] S. Brin and L. Page. The anatomy of a large-scale hypertextual (web) search engine. In Proceedings of the Seventh International World Wide Web Conference,
1998.
[2] Nello Cristianini, Huma Lodhi, and John Shawe-Taylor. Latent semantic kernels
for feature selection. Technical Report NC-TR-00-080, NeuroCOLT Working
Group, http://www.neurocolt.org, 2000.
[3] J. Kleinberg. Authoritative sources in a hyperlinked environment. In Proceedings
of 9th ACM-SIAM Symposium on Discrete Algorithms, 1998.
[4] C. McDiarmid. On the method of bounded differences. In Surveys in Combinatorics 1989, pages 148- 188. Cambridge University Press , 1989.
[5] S. Mika, B. SchCilkopf, A. Smola, K.-R. MUller, M. Scholz, and G. Ratsch.
Kernel PCA and de-noising in feature spaces. In Advances in Neural Information
Processing Systems 11, 1998.
[6] Andrew Y. Ng, Alice X. Zheng, and Michael 1. Jordan. Link analysis, eigenvectors and stability. In To appear in the Seventeenth International Joint Conference on Artificial Intelligence (UCAI-Ol), 2001.
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1,238 | 2,128 | Online Learning with Kernels
Jyrki Kivinen
Alex J. Smola
Robert C. Williamson
Research School of Information Sciences and Engineering
Australian National University
Canberra, ACT 0200
Abstract
We consider online learning in a Reproducing Kernel Hilbert Space. Our
method is computationally efficient and leads to simple algorithms. In
particular we derive update equations for classification, regression, and
novelty detection. The inclusion of the -trick allows us to give a robust
parameterization. Moreover, unlike in batch learning where the -trick
only applies to the -insensitive loss function we are able to derive general trimmed-mean types of estimators such as for Huber?s robust loss.
1 Introduction
While kernel methods have proven to be successful in many batch settings (Support Vector
Machines, Gaussian Processes, Regularization Networks) the extension to online methods
has proven to provide some unsolved challenges. Firstly, the standard online settings for
linear methods are in danger of overfitting, when applied to an estimator using a feature
space method. This calls for regularization (or prior probabilities in function space if the
Gaussian Process view is taken).
Secondly, the functional representation of the estimator becomes more complex as the number of observations increases. More specifically, the Representer Theorem [10] implies
that the number of kernel functions can grow up to linearly with the number of observations. Depending on the loss function used [15], this will happen in practice in most cases.
Thereby the complexity of the estimator used in prediction increases linearly over time (in
some restricted situations this can be reduced to logarithmic cost [8]).
Finally, training time of batch and/or incremental update algorithms typically increases superlinearly with the number of observations. Incremental update algorithms [2] attempt to
overcome this problem but cannot guarantee a bound on the number of operations required
per iteration. Projection methods [3] on the other hand, will ensure a limited number of
updates per iteration. However they can be computationally expensive since they require
one matrix multiplication at each step. The size of the matrix is given by the number of
kernel functions required at each step.
Recently several algorithms have been proposed [5, 8, 6, 12] performing perceptron-like
updates for classification at each step. Some algorithms work only in the noise free case,
others not for moving targets, and yet again others assume an upper bound on the complexity of the estimators. In the present paper we present a simple method which will allows
the use of kernel estimators for classification, regression, and novelty detection and which
copes with a large number of kernel functions efficiently.
2 Stochastic Gradient Descent in Feature Space
"! #
%$&
&$
to be studied in
Reproducing Kernel Hilbert Space The class of functions
this paper are elements of an RKHS . This means that there exists a kernel
and a dot product
such that 1)
(reproducing property); 2) is
the closure of the span of all
with
. In other words, all
are linear
combinations of kernel functions.
' ' ()!*
Typically
is used as a regularization functional. It is the ?length of the weight
vector in feature space? as commonly used in SV algorithms. To state our algorithm we
need to compute derivatives of functionals defined on .
/.
0
/.
!21 ' '( we obtain 3546+-, ! . More general versions of
For the regularizer +-,
+-, /. !87 ' ' lead to 3 4 +-, /.( !97: ' ';' '=< 1 .
/.B ! we compute the derivative by using the reproFor the evaluation functional >@?A,
ducing property
of G
and
obtain
3 4 > ? , /. !%3 4
)!C
# . Consequently for
E
F
H
G
I
a function
which is differentiable in its third argument we obtain
D
3 4 D # KJL # !MD : # JL K
# . Below D will be the loss function.
Regularized Risk Functionals and
standard learning setting we are
# LLearning
N J N $OP QInG the
drawn
according to some underlying
supplied with pairs
of
observations
# J . Our aim is to predict the likely outcome
distribution R
at location . Several
# J may change over time, (ii) theJ training
# N KJ N
variants are possible: (i) R
sample
may be the next observation on which to predict which leads to a true online setting, or
(iii) we may want to find an algorithm which approximately minimizes a regularized risk
functional on a given training set.
D
SG% TGU
K J V;VV # /W KJ W J R # J
1 1
W
XZY\[^] , /. ! ` _ a D # cN KJ N # cN K
(1)
Nb
1
or, in order to avoid overly complex
hypotheses,
minimize the empirical risk plus an addi/.
tional regularization term +-, . This sum is known as the regularized risk
W
_
a
X"d#Y\e , /.f ! X"Y\[^] , /.5gih +-, /. ! ` D /N J N /N gkh +-, /. for hQl9m V (2)
Njb
1
We assume that we want to minimize a loss function
which penalizes
the deviation between an observation at location and the prediction
, based on
. Since
is unknown, a standard approach is to
observations
instead minimize the empirical risk
Common loss functions are the soft margin loss function [1] or the logistic loss for classification and novelty detection [14], the quadratic loss, absolute loss, Huber?s robust loss [9],
or the -insensitive loss [16] for regression. We discuss these in Section 3.
n
In some cases the loss function depends on an additional parameter such as the width of the
margin or the size of the -insensitive zone. One may make these variables themselves
parameters of the optimization problem [15] in order to make the loss function adaptive to
the amount or type of noise present in the data. This typically results in a term
or
added to
.
D # KJL #
X #d Y\e , /.
o n
Stochastic Approximation In order to find a good estimator we would like to minimize
. This can be costly if the number of observations is large. Recently several gradient
descent algorithms for minimizing such functionals efficiently have been proposed [13, 7].
Below we extend these methods to stochastic gradient descent by approximating
Xpd#Y\e , /.
X ,
f. ! D # K J #
gkh +-, /.
(3)
.
X
and then performing
with respect to
, . Here is
either randomly
_ ;VVV `gradient
or it isdescent
chosen from X
the
new
training
instance
observed
at time . Consequently
,
. with respect to is
the gradient of
3 4 X ,
. ! D : # J # K
; gSh 3 4 +-, /. ! D : J # K
; gSh/ V (4)
/. ! 1 ' ' ( . Analogous results hold for general +-, /. !
The
equality holds if +-,
7 ' last
' . The the update equations are( hence straightforward:
S
o 354 X ,
. V
(5)
$
Here
is the learning rate controlling h the size of updates undertaken at each iteration. We will return to the issue of adjusting at a later stage.
/.
Descent Algorithm For simplicity, assume that +-, !H1 ' ' ( . In this case (5) becomes
o D : KJ #
K
#
gihc E! _ o h o ( D : J #
K
; V (6)
by
While (6) is convenient to use for a theoretical analysis, it is not directly amenable to
computation. For this purpose we have to express as a kernel expansion
E! a N
# cN
N
where the
N are (previously seen) training patterns. Then (6) becomes
_ o h
J # K
:
o
D
m
#
for !
N ! o _ o D : h KJ N
for ! V
(7)
(8)
(9)
(10)
Eq. (8) means that at each iteration the kernel expansion may grow by one term. Furthermore, the cost
for training
at each step is not larger than the prediction cost:
have
once
we
computed
, is obtained by the value of the derivative of at
.
#
D # J # K
N
_ _ o
Instead
all coefficients we may simply cache the power series
h _ o ofh updating
h
_
( o "!@V;VV and pick suitable terms as needed. This is particularly
_ m useful
_
if the derivatives of the loss function D will only assume discrete values, say o as
is the case when using the soft-margin type loss functions (see Section 3).
Truncation The problem with (8) and (10) is that without any further measures, the number of basis functions # will grow without bound. This is not desirable since # determines
the amount of computation needed for prediction.
The regularization term helps us here. At
are shrunk
each iteration the coefficients with
by
. Thus after $ iterations
the coefficient will be reduced to
"% . Hence:
_ o h
# K with its first derivaProposition 1 (Truncation Error) For a loss function D JL
tive bounded by & and a kernel
with
norm '
; ;(' '*) , the truncation error
N frombounded
in incurred by
dropping
terms
the
kernel
expansion
of after $ update steps is
_ o h % &) . Furthermore,
bounded by
the total truncation error by dropping all terms
which are at least $ steps old is bounded by
a <.% _ h < N
d
+
,
(11)
' o
' ' Nb o &)0/ h < 1 _ o h % &)
1
N
N
_ o! h N
N
/N
< 1
h
requirements for the
The regularization parameter can thus be used to control the
storage
R
K
5
J
expansion. In addition, it naturally allows for distributions
that
change over time in
# N J N that are much older
which cases it is desirable to forget instances
than the average
d+ ,
!
Nb
Here
. Obviously the approximation quality increases expo.% of terms retained.
nentially with the number
time scale of the distribution change [11].
3 Applications
# ! g
We now proceed to applications of (8) and (10) to specific learning situations. We utilize
the standard addition of the constant offset to the function expansion,
i.e.
.
where
and
. Hence we also update into
O$
D JL !
$
,.
o 3 X
m _
o J
N
KK __
o
hh
o
Classification A typical loss function in SVMs is the soft margin, given by
. In this situation the update equations become
n
N N g N
N Jm Z J
J #
_
/
if
otherwise.
(12)
n
In classification with the -trick we avoid having to fix the margin by treating it as a
variable [15]. The value of is found automatically by using the loss function
D # JL # !
m n oJ
K o n
(13)
m ' _ is another parameter. Since has a much clearer intuitive meaning than
where '
h
n , it is easier to tune. On the other hand, one can show [15] that the specific choice
of
h
_
has no influence on the estimate in -SV classification. Therefore we may set !
and
obtain
N = n5 __ o NN KJm N " g J N n g _ o if J
/kn (14)
o = np
o
otherwise.
# #
Finally, if we choose the hinge-loss, D JL
!
m ;o
J
N KK __
o
hh NN Jm N Z g J N if J # / m
(15)
o
otherwise.
h ! m recovers the kernel-perceptron algorithm. For nonzero h we obtain the
Setting
kernel-perceptron with regularization.
Novelty Detection The results for novelty detection [14] are similar in spirit. The setting is most useful here particularly where the estimator acts as a warning device (e.g.
network intrusion detection) and we would like to specify an upper limit on the frequency
of alerts
(/ . The relevant loss function is
where
and usually [14] one uses
rather than
in order to avoid trivial
solutions. The update equations are
# Un
#
m n o # K o n
8$
Tg D KJc Z$i!
N n5 __ o NN Zm Kn g _ o K if /8n
(16)
o Kn o
otherwise.
Considering the update of n we can see that on average only a fraction of observations
fN
will be considered for updates. Thus we only have to store a small fraction of the .
Regression We consider the following four settings: squared loss, the -insensitive loss
using the -trick, Huber?s robust loss function, and trimmed mean estimators. For con where
venience we will only use estimates
rather than
. The
extension to the latter case is straightforward. We begin with squared loss where is given
by
the update equation is
Consequently
(17)
This means that we have to store every observation we make, or more precisely, the
prediction error we made on the observation. The -insensitive loss
avoids this problem but introduces a new parameter in turn ?
the width of the insensitivity zone . By making a variable of the optimization problem
we have
The update equations now have to be
stated in terms of , and which is allowed to change during the optimization process.
This leads to
if
otherwise.
(18)
This means that every time the prediction error exceeds , we increase the insensitivity
zone by . Likewise, if it is smaller than , the insensitive zone is decreased by
.
Next let us analyze the case of regression with Huber?s robust loss. The loss is given by
*$
8!
D # JL P
! (1 J o N K ( V _ h N #
o J o
K V
m QJ o # /o @
m J o #
D JL # N K!
N __ o h h NN m J
o "o
g
$%
D
D KJc # !
\o g 5V
o g _ o
J o l
_ o
J o #
#@o (1 if J o #
(19)
1( J o K ( otherwise.
# .
As before we obtain update equations by computing the derivative of D with respect to
N __ o NN J o # K if J o # l
(20)
o < 1 J o
K otherwise.
D JL # K!
Comparing (20) with (18) leads to the question whether might not also be adjusted
adaptively. This is a desirable goal since we may not know the amount of noise present in
the data. While the -setting allowed us to form such adaptive estimators for batch learning
with the -insensitive loss, this goal has proven elusive for other estimators in the standard
batch setting. In the online situation, however, such an extension is quite natural (see also
[4]). All we need to do is make a variable of the optimization problem and set
N
K _
o
K _ o
N g _
N < 1 J J o o # o o
4 Theoretical Analysis
m n o8J # K
Wb X
.
,
1
J o l
if
otherwise.
(21)
D KJc # !
Consider now the classification problem with the soft margin
loss
denote
; here is a fixed margin
parameter.
Let
the
hypothesis of
the online algorithm
after
seeing
the
first
observations.
Thus,
at
time
, the algorithm
.
, receives the correct outcome
, and upreceives an input , makes
its
prediction
dates
into according to (5). We now wish to bound the cumulative risk
its hypothesis
. The
motivation for such bounds is roughly
follows. Assume there is
areasdrawn,
some fixed distribution from which the examples
and define
4 X , /.
n
o _
1
# J
RX /.f
, ! ?
! , D # JL K .5gk h +-, /. V
J
` X
"
`
+ E ! (1 ' ' (
!
,
.Kg
Then it would be desirable for the online hypothesis
to converge towards
"
arg min
. If we can show that the cumulative risk is asymptotically
"
#
, we see that at least in some sense does converge to .
Hence, as a first step in our convergence analysis, we obtain an upper bound for the cumulative risk. In all the bounds of this section we assume
.
K# K J W b
lCm
1
' '
W
W
a X
. a X
, ' b
b
1
1
#
*! ` 1 (;
, L
. g ) ` 1 ( g _ V
(
Theorem 1 Let
be an example sequence such that
' ) for all
. Fix
)
, and choose the learning rate
. Then for any such that
'
we have
(22)
Notice that the bound does not depend on any probabilistic assumptions. If the example
sequence is such that some fixed predictor has a small cumulative risk, then the cumulative risk of the online algorithm will also be small. There is a slight catch here in that the
learning rate must be chosen a priori, and the optimal setting depends on . The longer
the sequence of examples, the smaller learning rate we want. We can avoid this by using a
learning rate that starts from a fairly large value and decreases as learning progresses. This
leads to a bound similar to Theorem 1 but with somewhat worse constant coefficients.
`
K# K J K W b
1
# ' ) ( for all
.
l*m
h
! _ 1 ( . Then for any such
' ' W
W
a X
. a X
.5g hf g h ( ` 1 ( g _
, ' b
, L
)
V (23)
b
1
1
Let us now consider
of Theorem 2 to a situation in which we assume that
J theareimplications
the examples
i.i.d. according to some fixed distribution R .
kG , such
' ) ( holds with
W
Theorem 3 Let R be a distribution over
that
#
W
_
_
b
probability for KJ
R . Let ! ` W b < 1 1 where is the
-th online
J 1 that
hypothesis based on an example sequence
is drawn i.i.d. according to R .
l*m , and use at update
the learning rate
Fix
! _
h
1 ( . Then for any such
' ' we have
that '
X , W .. ' X , . g hB g )
h ( ` < 1 ( g ` < 1 V
(24)
0,
Theorem 2 Let
be an example sequence such that
, and use at update the learning rate
Fix
that '
we have
If we know in advance how many examples we are going to draw, we can use a fixed
learning rate as in Theorem 1 and obtain somewhat better constants.
5 Experiments and Discussion
In our experiments we studied the performance of online -SVM algorithms in various
settings. They always yielded competitive performance. Due to space constraints we only
report the findings in novelty detection as given in Figure 1 (where the training algorithm
was fed the patterns sans class labels).
_ _
! m Vm _
Already after one pass through the USPS database (5000 training patterns, 2000 test patterns, each of them of size
pixels), which took in MATLAB less than 15s on a
433MHz Celeron, the results can be used for weeding out badly written digits. The ) to allow for a fixed fraction of detected ?outliers.? Based
setting was used (with
.
on the theoretical analysis of Section 4 we used a decreasing learning rate with
h
<
Conclusion We have presented a range of simple online kernel-based algorithms for a
variety of standard machine learning tasks. The algorithms have constant memory requirements and are computationally cheap at each update step. They allow the ready application
of powerful kernel based methods such as novelty detection to online and time-varying
problems.
Results after one pass through the USPS
database. We used Gaussian RBF kernels
with width
. The learn
ing rate was adjusted to
where
is
the number of iterations. Top: the first 50
patterns which incurred a margin error; bottom left: the 50 worst patterns according to
on the training set, bottom right:
the 50 worst patterns on an unseen test set.
(0! m V ! _
1W
`
o8n
Figure 1: Online novelty detection on the USPS dataset with
! m Vm _ .
Acknowledgments A.S. was supported by the DFG under grant Sm 62/1-1, J.K. &
R.C.W. were supported by the ARC. The authors thank Paul Wankadia for help with the
implementation.
References
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1,239 | 2,129 | Approximate Dynamic Programming
via Linear Programming
Daniela P. de Farias
Department of Management Science and Engineering
Stanford University
Stanford, CA 94305
pucci @stanford.edu
Benjamin Van Roy
Department of Management Science and Engineering
Stanford University
Stanford, CA 94305
bvr@stanford. edu
Abstract
The curse of dimensionality gives rise to prohibitive computational
requirements that render infeasible the exact solution of large- scale
stochastic control problems. We study an efficient method based
on linear programming for approximating solutions to such problems. The approach "fits" a linear combination of pre- selected
basis functions to the dynamic programming cost- to- go function.
We develop bounds on the approximation error and present experimental results in the domain of queueing network control, providing
empirical support for the methodology.
1
Introduction
Dynamic programming offers a unified approach to solving problems of stochastic
control. Central to the methodology is the cost- to- go function, which can obtained
via solving Bellman's equation. The domain of the cost- to- go function is the state
space of the system to be controlled, and dynamic programming algorithms compute and store a table consisting of one cost- to- go value per state. Unfortunately,
t he size of a state space typically grows exponentially in the number of state variables. Known as the curse of dimensionality, this phenomenon renders dynamic
programming intractable in the face of problems of practical scale.
One approach to dealing with this difficulty is to generate an approximation within
a parameterized class of functions , in a spirit similar to that of statistical regression. The focus of this paper is on linearly parameterized functions: one tries to
approximate the cost- to- go function J* by a linear combination of prespecified basis functions. Note that this scheme depends on two important preconditions for the
development of an effective approximation. First, we need to choose basis functions
that can closely approximate the desired cost-to-go function. In this respect, a suitable choice requires some practical experience or theoretical analysis that provides
rough information on the shape of the function to be approximated. "Regularities"
associated with the function, for example, can guide the choice of representation.
Second, we need an efficient algorithm that computes an appropriate linear combination.
The algorithm we study is based on a linear programming formulation, originally
proposed by Schweitzer and Seidman [5], that generalizes the linear programming
approach to exact dynamic programming, originally introduced by Manne [4]. We
present an error bound that characterizes the quality of approximations produced
by the linear programming approach. The error is characterized in relative terms,
compared against the "best possible" approximation of the optimal cost-to-go function given the selection of basis functions. This is the first such error bound for
any algorithm that approximates cost- to- go functions of general stochastic control
problems by computing weights for arbitrary collections of basis functions.
2
Stochastic control and linear programming
We consider discrete- time stochastic control problems involving a finite state space
lSI = N. For each state XES, there is a finite set of available
actions A x. Taking action a E A x when the current state is x incurs cost 9a(X) .
State transition probabilities Pa(x,y) represent, for each pair (x,y) of states and
each action a E A x, the probability that the next state will be y given that the
current state is x and the current action is a E Ax.
S of cardinality
A policy u is a mapping from states to actions. Given a policy u, the dynamics of
the system follow a Markov chain with transition probabilities Pu( x)(x, y). For each
policy u, we define a transition matrix Pu whose (x,y)th entry is Pu(x)(x,y).
The problem of stochastic control amounts to selection of a policy that optimizes
a given criterion. In this paper, we will employ as an optimality criterion infinitehorizon discounted cost of the form
Ju(x) =E
[~(i9U(Xd lxo =x] ,
where 9u(X) is used as shorthand for 9u(x)(X) and the discount factor a E (0,1)
reflects inter- temporal preferences. Optimality is attained by any policy that is
greedy with respect to the optimal cost-to-go function J*(x) = minu Ju(x) (a policy
u is called greedy with respect to J if TuJ = T J).
Let us define operators Tu and T by TuJ = 9u +aPuJ and T J = minu (9u + aPuJ).
The optimal cost-to-go function solves uniquely Bellman's equation J = T J. Dynamic programming offers a number of approaches to solving this equation; one of
particular relevance to our paper makes use of linear programming, as we will now
discuss. Consider the problem
max clJ
(1)
S.t.
T J;::: J,
where c is a vector with positive components, which we will refer to as staterelevance wei9hts. It can be shown that any feasible J satisfies J :::; J*. It follows
that, for any set of positive weights c, J* is the unique solution to (1).
Note that each constraint (T J)(x) ;::: J(x) is equivalent to a set of constraints
+ a L.YEs Pa(X ,y) J(y) ;::: J(x), Va E A x, so that the optimization problem
(1) can be represented as an LP, which we refer to as the exact LP.
9a(X)
As mentioned in the introduction, state spaces for practical problems are enormous
due to the curse of dimensionality. Consequently, the linear program of interest involves prohibitively large numbers of variables and constraints. The approximation
algorithm we study reduces dramatically the number of variables.
Let us now introduce the linear programming approach to approximate dynamic
programming. Given pre-selected basis functions (Pl, .. . , cPK, define a matrix If> =
[ cPl
cPK ]. With an aim of computing a weight vector f E ~K such that If>f
is a close approximation to J*, one might pose the following optimization problem:
max
s.t.
(2)
c'lf>r
Tlf>r
2::
If>r.
Given a solution f, one might then hope to generate near- optimal decisions by using
a policy that is greedy with respect to If>f.
As with the case of exact dynamic programming, the optimization problem (2) can
be recast as a linear program. We will refer to this problem as the approximate
LP. Note that, though the number of variables is reduced to K, the number of
constraints remains as large as in the exact LP. Fortunately, we expect that most
of the constraints will become irrelevant, and solutions to the linear program can
be approximated efficiently, as demonstrated in [3] .
3
Error Bounds for the Approximate LP
When the optimal cost- to- go function lies within the span of the basis functions,
solution of the approximate LP yields the exact optimal cost-to-go function. Unfortunately, it is difficult in practice to select a set of basis functions that contains
the optimal cost- to- go function within its span. Instead, basis functions must be
based on heuristics and simplified analyses. One can only hope that the span comes
close to the desired cost- to- go function.
For the approximate LP to be useful , it should deliver good approximations when
the cost- to- go function is near the span of selected basis functions. In this section,
we present a bound that ensure desirable results of this kind.
To set the stage for development of an error bound, let us establish some notation.
First, we introduce the weighted norms, defined by
1IJ111 "~ =
'"' ')'(x) IJ(x)l ,
~
xES
IIJlloo "~ =
max ')'(x) IJ(x)l,
xES
for any ')' : S f-t ~+. Note that both norms allow for uneven weighting of errors
across the state space.
We also introduce an operator H, defined by
(HV)(x) = max
aEAz
L Pa(x, y)V(y),
y
for all V : S f-t R For any V , (HV)(x) represents the maximum expected value
of V (y) if the current state is x and y is a random variable representing the next
state. Based on this operator, we define a scalar
kv =
for each V : S
f-t ~.
m,:x V(x) -
V(x)
a(HV)(x) ,
(3)
We interpret the argument V of H as a "Lyapunov function," while we view kv as
a "Lyapunov stability factor," in a sense that we will now explain. In the upcoming
theorem, we will only be concerned with functions V that are positive and that
make kv nonnegative. Also, our error bound for the approximate LP will grow
proportionately with kv, and we therefore want kv to be small. At a minimum, kv
should be finite , which translates to a condition
a(HV)(x) < V(x) ,
"Ix ES.
(4)
If a were equal to 1, this would look like a Lyapunov stability condition: the
maximum expected value (HV)(x) at the next time step must be less than the
current value V(x). In general, a is less than 1, and this introduces some slack in
the condition. Note also that kv becomes smaller as the (HV)(x)'s become small
relative to the V(x)'s. Hence, kv conveys a degree of "stability," with smaller values
representing stronger stability.
We are now ready to state our main result. For any given function V mapping S
to positive reals, we use l/V as shorthand for a function x I-t l/V(x).
Theorem 3.1 {2} Let f be a solution of the approximate LP. Then, for any v E 3rK
such that (<T>v) (x) > 0 for all xES and aH <T>v < <T>v ,
IIJ* - <T>flkc :::; 2k<I>v(c'<T>v) min
IIJ* - <T>rll oo,l/<I>v?
r
(5)
A proof of Theorem 3.1 can be found in the long version of this paper [2].
We highlight some implications of Theorem 3.1. First, the error bound (5) tells
that the the approximation error yielded by the approximate LP is proportional to
the error associated with the best possible approximation relative to a certain norm
11?lll,l/<I>v. Hence we expect that the approximate LP will have reasonable behavior
- if the choice of basis functions is appropriate, the approximate LP should yield a
relatively good approximation to the cost-to-go function , as long as the constants
k<I>v and c' <T>v remain small.
Note that on the left-hand side of (5), we measure the approximation error with the
weighted norm 11?lkc. Recall that the weight vector c appears in objective function
of the approximate LP (2) and must be chosen. In approximating the solution to a
given stochastic control problem, it seems sensible to weight more heavily portions
of the state space that are visited frequently, so that accuracy will be emphasized
in such regions. As discussed in [2], it seems reasonable that the weight vector c
should be chosen to reflect the relative importance of each state.
Finally, note that the Lyapunov function <T>v plays a central role in the bound of
Theorem 3.1. Its choice influences three terms on the right-hand-side of the bound:
1. the error minr
IIJ* - <T>rll oo,l/<I>v;
2. the Lyapunov stability factor k<I>v;
3. the inner product c' <T>v with the state- relevance weights.
An appropriately chosen Lyapunov function should make all three of these terms
relatively small. Furthermore, for the bound to be useful in practical contexts,
these terms should not grow much with problem size. We now illustrate with an
application in queueing problems how a suitable Lyapunov function could be found
and show how these terms scale with problem size.
3.1
Example: A Queueing Network
Consider a single reentrant line with d queues and finite buffers of size B. We assume
that exogenous arrivals occur at queue 1 with probability p < 1/2. The state x E ~d
indicates the number of jobs in each queue. The cost per stage incurred at state x
is given by
the average number of jobs per queue .
As discussed in [2] , under certain stability assumptions we expect that the optimal
cost-to-go function should satisfy
P2 I
Pl I
O J * ()
::::;
x::::; dX x
+ de x + Po,
for some positive scalars Po, Pl and P2 independent of d. We consider a Lyapunov
function V(x) = ~XIX + C for some constant C > 0, which implies
m}n IIJ* -lJ>rll oo,l/V
< IIJ*lloo,l /V
<
P2XlX + Plelx + dpo
max '-----'---..,-----'-x2: O
XiX + dC
Po
< P2 + Pl + C'
and the above bound is independent of the number of queues in the system.
Now let us study kv. We have
a(HV)(x)
C) +
< a [p
(~XIX + 2X 1/
< V(x)
(a+ap:;~:~),
1
+
(1- p)
(~XIX + C) ]
and it is clear that, for C sufficiently large and independent of d, there is a j3
independent of d such that aHV ::::; j3V, and therefore kv ::::; 1 ~ ,6 .
<
1
Finally, let us consider ciV. Discussion presented in [2] suggests that one might want
to choose c so as to reflect the stationary state distribution. We expect that under
some stability assumptions, the tail of the stationary state distribution will have an
(l!;l+l)d
upper bound with geometric decay [1]. Therefore we let c(x) =
plxl, for
some 0 < P < 1. In this case, c is equivalent to the conditional joint distribution of
d independent and identically distributed geometric random variables conditioned
on the event that they are less than B + 1, and we have
clV = E
[~t, xl + C I Xi < B + 1, i = 1, ... , d]
< 2 (1
~2p)2 + 1 ~ P + C,
where Xi, i = 1, .. . , d are identically distributed geometric random variables with
parameter 1 - p. It follows that clV is uniformly bounded over the number of
queues.
This example shows that the terms involved in the error bound (5) are uniformly
bounded both in the number of states in the system and in the number of state
variables, hence the behavior of the approximate LP does not deteriorate as the
problem size increases.
We finally present a numerical experiment to further illustrate the performance of
the approximate LP.
L
=-r
", - 3 / 11.5
Al - 1/11.5
)
~
~
- 4 / 11.5
1A8 - 2 .5/ 11.5
-----':7
IJ.z - 2 / 11.5
IC>I"'-" 'U
J
)
-
"" - 3/ 11.5
-
~5 - 3 / 11.5
-
A 2 - 1/11.5
) 1"4- 3 .1/11.5
l
Figure 1: System for Example 3.2.
Policy
Average Cost
Table 1: Average number of jobs after 50,000,000 simulation steps
3.2
An Eight-Dimensional Queueing Network
We consider a queueing network with eight queues. The system is depicted in Figure
1, with arrival P'i, i = 1,2) and departure (J.Li, i = 1, ... ,8) probabilities indicated.
The state x E ~8 represents the number of jobs in each queue. The cost-per-state
is g(x) = lxi, and the discount factor 0:: is 0.995. Actions a E {O, 1}8 indicate which
queues are being served; ai = 1 iff a job from queue i is being processed. We
consider only non-iddling policies and, at each t ime step, a server processes jobs
from one of its queues exclusively.
We choose c of the form c(x) = (1 - p)8 plxl. The basis functions are chosen to span
all polynomials in x of degree 2; therefore, the approximate LP has 47 variables.
Constraints (T<I>r)(x) 2: (<I>r)(x) for the approximate LP are generated by sampling
5000 states according to the distribution associated with c. Experiments were performed for p = 0.85,0.9 and 0.95, and p = 0.9 yielded the policy with smallest
average cost.
We compared the performance of the policy yielded by the approximate LP (ALP)
with that of first-in-first-out (FIFO), last-buffer-first-serve (LBFS)l and a policy
that serves the longest queue in each server (LONG). The average number of jobs
in the system for each policy was estimated by simulation. Results are shown in
Table 1. The policy generated by the approximate LP performs significantly better
than each of the heuristics, yielding more than 10% improvement over LBFS, the
second best policy. We expect that even better results could be obtained by refining
the choice of basis functions and state-relevance weights.
4
Closing Remarks and Open Issues
In t his paper we studied the linear programming approach to approximate dynamic
programming for stochastic control problems as a means of alleviating the curse of
1 LBFS serves the job that is closest to leaving the system; for example, if there are jobs
in queue 2 and in queue 6, a job from queue 2 is processed since it will leave the system
after going through only one more queue, whereas the job from queue 6 will still have to
go through two more queues. We also choose to assign higher priority to queue 8 than to
queue 3 since queue 8 has higher departure probability.
dimensionality. We provided an error bound based on certain assumptions on the
basis functions. The bounds were shown to be uniformly bounded in the number
of states and state variables in certain queueing problems.
Several questions remain open and are the object of future investigation: Can the
state-relevance weights in the objective function be chosen in some adaptive way?
Can we add robustness to the approximate LP algorithm to account for errors in the
estimation of costs and transition probabilities, i.e., design an alternative LP with
meaningful performance bounds when problem parameters are just known to be in
a certain range? How do our results extend to the average cost case? How do our
results extend to the infinite-state case? How does the quality of the approximate
value function, measure by the weighted L1 norm , translate into actual performance
of the associated greedy policy?
Acknowledgements
This research was supported by NSF CAREER Grant ECS-9985229, by the ONR
under Grant MURI N00014-00-1-0637, and by an IBM Research Fellowship.
References
[1] Bertsimas, D. , Gamarnik, D. & Tsitsiklis, J. , "Performance of Multiclass Markovian
Queueing Networks via Piecewise Linear Lyapunov Functions," submitted to Annals of
Applied Probability, 2000.
[2] de Farias, D.P. & Van Roy, B. , "The Linear Programming Approach to Approximate
Dynamic Programming," submitted to publication, 200l.
[3] de Farias, D.P. & Van Roy, B., "On Constraint Sampling for Approximate Linear
Programming," , submitted to publication , 200l.
[4] Manne, A.S., "Linear Programming and Sequential Decisions," Management Science
6, No.3, pp. 259-267, 1960.
[5] Schweitzer, P. & Seidmann, A. , "Generalized Polynomial Approximations in Markovian
Decision Processes," Journal of Mathematical Analysis and Applications 110, pp. 568582, 1985.
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1,240 | 213 | Connectionist Architectures for Multi-Speaker Phoneme Recognition
Connectionist Architectures/or Multi-Speaker
Phoneme Recognition
John B. Hampshire n and Alex Waibel
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213-3890
ABSTRACT
We present a number of Time-Delay Neural Network (TDNN) based
architectures for multi-speaker phoneme recognition (/b,d,g/ task). We
use speech of two females and four males to compare the performance
of the various architectures against a baseline recognition rate of 95.9%
for a single IDNN on the six-speaker /b,d,g/ task. This series of modular designs leads to a highly modular multi-network architecture capable
of performing the six-speaker recognition task at the speaker dependent
rate of 98.4%. In addition to its high recognition rate, the so-called
"Meta-Pi" architecture learns - without direct supervision - to recognize the speech of one particular male speaker using internal models
of other male speakers exclusively.
1 INTRODUCTION
References [1,2] have show the Tune-Delay Neural Network to be an effective classifier
of acoustic phonetic speech from individual speakers. The objective of this research has
been to extend the TDNN paradigm to the multi-speaker phoneme recognition task, with
the eventual goal of producing connectionist structures capable of speaker independent
phoneme recognition. In making the transition from single to multi-speaker tasks, we have
focused on modular architectures that perform the over-all recognition task by integrating
a number of smaller task-specific networks.
203
204
Hampshire and Waibel
Table 1: A synopsis of multi-speaker /b,d,g/ recognition results for six TDNN-based
architectures.
Architecture
Type
TDNN
baseline
single net
single net
PSTDNN
Multiple
TDNNs
Modular
TDNN
SID
multi net
Meta-Pi
multi net
1.1
multi net
multi net
Features
.Frequency shift
invariance
? arbitrated
classification
.2-stage training
.2-stage training
.Multiple TDNN
modules
.2-stage training
.Multiple TDNN
modules
.Bayesian MAP learning
.no explicit speaker LD.
Size
(connections)
Recognition Rate
3-speakers 6-speakers
6,233
97.3%
(I-ply) 5,357
(2-ply) 6,947
18,700
96.8%
97.2%
98.6%
18,650
37,400
144,000
97.3%
144,000
95.9%
97.1 %
-
-
96.3%
98.3%
-
98.4%
DATA
The experimental conditions for this research are detailed in [1]. Japanese speech data
from six professional announcers (2 female, 4 male) was sampled for the /b, d, g/
phonemes (approximately 250 training and 250 testing tokens per phoneme, per speaker).
Training for all of the modular architectures followed a general two-stage process: in
the first stage, speaker-dependent modules were trained on speech tokens from specific
individuals; in the second stage, the over-all modular structure was trained with speech
tokens from all speakers.
1.2
RESULTS
Owing to the number of architectures investigated, we present only brief descriptions of
each structure. Additional references are provided for readers interested in more detailed
descriptions of particular architectures. Table 1 summarizes our recognition results for
all of the network architectures described below. We list the type of architecture (single
or multi network), the important features of the design, its over-all size (in terms of
total connections), and its recognition performance on the specified multi-speaker task.
There are two principal multi-speaker tasks: a three male task, and a four male/two
female task: the six speaker task is considerably more difficult than its three speaker
counterpart, owing to the higher acoustic variance of combined male/female speech.
Connectionist Architectures for Multi-Speaker Phoneme Recognition
F
oot]
o-;J
F2
F1
Figure 1: The Frequency Shifting TDNN (FSTDNN) architecture.
2 ARCHITECTURE DESCRIPTIONS
TDNN: The TDNN [1,2] serves as our baseline multi-speaker experiment. Its recognition performance on single speaker speech is typically 98.5% [1,3]. The high acoustic
variance of speech drawn from six speakers - two of whom are female - reduces
the TDNN's performance significantly (95.9%). This indicates that architectures capable
of adjusting to markedly different speakers are necessary for robust multi-speaker and
speake~independentrecognJtion.
FSTDNN: In this design, a frequency shift invariant feature is added to the original
TDNN paradigm. The resulting architecture maps input speech into a first hidden layer
with three frequency ranges roughly corresponding to the three formants Fl - F3 (see
figure 1). Two variations of the basic design have been tested [4]: the first is a "one-ply"
architecture (depicted in the figure), while the second is a ''two-ply'' structure that uses
two plies of input to first hidden layer connections. While the frequency shift invariance
of this architecture has intuitive appeal, the resulting network has a very small number
of unique connections from the input to the first hidden layer (- 30, I-ply). This paucity
of connections has two ramifications. First, it creates a crude replica of the input layer
state in the first hidden layer, as a result, feature detectors that form in the connections
between the input and first hidden layers of the standard TDNN are now formed in the
connections between the first and second hidden layers of the FSTDNN. Second, the
crude input to first hidden layer replication results in some loss of information; thus, the
feature detectors of the FSTDNN operate on what can be viewed as a degraded version of
205
206
Hampsllire and Waibel
3?Way niIrMecI output
-----
W.:h;l,~
;-;-;?? .,..,..:;:
. .? ~
? ??~
?.
?
.. .,.. ............
.
,
.
.
..
......
iI
........
.
...
... .11;. I........
.
.:::.
.~
"I.
Inputla,.
~
........... , ?
.~..........
.r ~~:= .... ~~~:
~~
.~ :~:;~;;;;~~~
!
i: ~~. :~:;~~~;; j ~
.: ':-: ~:::::i;= i
Figure 2: The Multiple TDNN architecture: three identical networks trained with three
different objective functions.
the original input. The resulting over-all structure's recognition performance is typically
worse (-- 97%) than the baseline TDNN for the multi-speaker fb,d,g/ task.
Multiple TDNN: This design employs three TDNNs trained with the MSE, Cross Entropy
[5], and CFM [3] objective functions (see figure 2). The different objective functions
used to train the TDNNs form consistently different internal representations of the speech
signal. We exploit these differing representations by using the (potentially) conflicting
outputs of the three networks to form a global arbitrated classification decision. Taking
the normalized sum of the three networks' outputs constitutes a simple arbitration scheme
that typically reduces the single IDNN error rate by 30%.
[Modular TDNN: In this design, we use the connection strengths ofTDNNs fully trained
on individual speakers to form the initial connections of a larger multi-speaker network.
This resulting network's higher layer connections are retrained [6] to produce the final
multi-speaker network. This technique allows us to integrate speaker-dependent networks
into a larger structure, limiting the over-all training time and network complexity of the
final multi-speaker architecture. The 3-speaker modular TDNN (shown in figures 3 and
4) shows selective response to different tokens of speech. In figure 3, the network responds to a Idl phone with only one sub-network (associated with speaker "MNM"). In
fact, this Idl is spoken by "MNM". In figure 4, the same network responds to a fbI phone
spoken by "MHT' with all sub-networks. This selective response to utterances indicates
that the network is sensitive to utterances that are prototypical for all speakers as well
Connectionist Architectures for Multi-Speaker Phoneme Recognition
----
. . . . - ... .01
Figure 3: 3-speaker Modular TDNN responding to input with one module.
Figure 4: 3-speaker Modular TDNN responding to input with three modules.
as those that are unique to an individual. The recognition rate for the 3-speaker modular
TDNN is comparable to the baseline TDNN rate (97.3%); however, the 6-speaker modular TDNN (not shown) yields a substantially lower recognition rate (96.3%). We attribute
this degraded performance to the manner in which this modular structure integrates its
sub-networks. In particular, the sub-networks are integrated by the connections from the
second hidden to output layers. This scheme uses a very small number of connections to
perform the integrating function. As the number of speakers increases and the acoustic
variance of their speech becomes significant, the connection topology becomes inadequate
for the increasingly complex integration function. Interconnecting the sub-networks between the first and second hidden layers would probably improve performance, but the
improvement would be at the expense of modularity. We tried using a "Connectionist
Glue" enhancement to the 6-speaker network [4], but found that it did not result in a
significant recognition improvement.
Stimulus Identification (SID) network: This network architecture is conceptually very
similar to the Integrated Neural Network (INN) [7]. Figure 5 illustrates the network
in block diagram form. Stimulus specific networks (in this case, multiple TDNNs) are
trained to recognize the speech of an individual. Each of these multiple TDNNs forms
a module in the over-all network. The modules are integrated by a superstructure (itself
a multiple TDNN) trained to recognize the identity of the input stimulus (speaker). The
output activations of the integrating superstructure constitute multiplicative connections
that gate the outputs of the modules in order to form a global classification decision.
207
208
Hampshire and Waibel
Output_.
SdlDuI ?? LD. Not
Figure 5: A block diagram of the Stimulus identification (SID) network, which is very
similar to the Integrated Neural Network (INN) [7].
Reference [8] details the SID network's performance. The major advantages of this
architecture are its high degree of modularity (all modules and the integrating superstructure can be trained independently) and it's high recognition rate (98.3%). It's major
disadvantage is that it has no explicit mechanism for handling new speakers (see [8]).
The Meta-Pi Network: This network architecture is very similar to the SID network.
Figure 6 illustrates the network in action. Stimulus specific networks (in this case, multiple TDNNs) are trained to recognize the speech of an individual. Each of these multiple
TDNNs forms a module in the over-all network. The modules are integrated by a superstructure (itself a multiple TDNN) trained in Bayesian MAP fashion to maximize the
phoneme recognition rate of the over-all structure: the equations governing the error backpropagation through the Meta-Pi superstructure link the global objective function with
the output states of the network's speaker-dependent modules [8]. As with the the SID
network, the output activations of the integrating superstructure constitute multiplicative
connections that gate the outputs of the modules in order to form a global classification
decision. However, as mentioned above, the integrating superstructure is not trained independently from the modules it integrates. While this Bayesian MAP training procedure
is not as modularized as the SID network's training procedure, the resulting recognition
rate is comparable. Additionally, the Meta-Pi network forms very broad representations
of speaker types in order to perform its integration task. Reference [8] shows that the
Meta-Pi superstructure learns - without direct supervision - to perform its integra-
Connectionist Architectures for Multi-Speaker Phoneme Recognition
??
. . '.11""?'" ?...., ,. ...
~
~
??
:.::::::
... :.~' ~
..
...
tit
~:::: :
Input ....,..
.~.::::.~::~::: ~~
':.:=::
n
... ..:~:::::
.....
...........
... ??
"
?????????
_
I
..
Figure 6: The Meta-Pi network responding to the speech of one male (MHT) using
models of other males' speech exclusively.
tion function based on gross formant features of the speakers being processed; explicit
speaker identity is irrelevant. A by-product of this learning procedure and the general
representations that it fonns is that the Meta-Pi network learns to recognize the speech
of one male using modules trained for other males exclusively (see figure 6 and [8]).
3 CONCLUSION
We have presented a number ofTDNN-based connectionist architectures for multi-speaker
phoneme recognition. The Meta-Pi network combines the best features of a number of
these designs with a Bayesian MAP learning rule to fonn a connectionist classifier that
performs multi-speaker phoneme recognition at speaker-dependent rates. We believe that
the Meta-Pi network's ability to recognize the speech of one male using only models
of other male speakers is significant. It suggests speech recognition systems that can
maintain their own database of speaker models, adapting to new speakers when possible,
spawning new speaker-dependent learning processes when necessary, and eliminating
redundant or obsolete speaker-dependent modules when appropriate. The one major
disadvantage of the Meta-Pi network is its size. We are presently attempting to reduce the
network's size by 67% (target size: 48,000 connections) without a statistically significant
loss in recognition performance.
209
210
Hampshire and Waibel
Acknowledgements
We wish to thank Bell Communications Research, ATR Interpreting Telephony Research
Laboratories, and the National Science Foundation (EET-8716324) for their support of
this research. We thank Bellcore's David Burr, Daniel Kahn, and Candace Kamm and
Seimens' Stephen Hanson for their comments and suggestions, all of which served to
improve this work. We also thank CMU's Warp/iWarpl group for their support of our
computational requirements. Finally, we thank Barak Pearlmutter, Dean Pomerleau, and
Roni Rosenfeld for their stimulating conversations, insight, and constructive criticism.
References
[1] Waibel, A., Hanazawa, T., Hinton, G., Shikano, K., and Lang, K., "Phoneme
Recognition Using Time-Delay Neural Networks," IEEE Transactions on Acoustics. Speech and Signal Processing, vol. ASSP-37, March, 1989, pp. 328-339.
[2] Lang, K. "A Time-Delay Neural Network Architecture for Speech Recognition,"
Ph.D. Dissertation, Carnegie Mellon University technical report CMU-CS-89-185,
July, 31, 1989.
[3] Hampshire, J., Waibel, A., "A Novel Objective Function for Improved Phoneme
Recognition Using Time-Delay Neural Networks," Carnegie Mellon University
Technical Report CMU-CS-89-118, March, 1989. A shorter version of this technical report is published in the IEEE Proceedings of the 1989 International Joint
Conference on Neural Networks. vol. 1. pp. 235-241.
[4] Hampshire, J., Waibel, A., "Connectionist Architectures for Multi-Speaker Phoneme
Recognition," Carnegie Mellon University Technical Report CMU-CS-89-167, August, 1989.
[5] Hinton, G. E., "Connectionist Learning Procedures," Carnegie Mellon University
Technical Report CMU-CS-87-115 (version 2), December, 1987, pg. 14.
[6] Waibel, A., Sawai, H., and Shikano, K., "Modularity and Scaling in Large Phonemic
Neural Networks", IEEE Transactions on Acoustics. Speech and Signal Processing,
vol. ASSP-37, December, 1989, pp. 1888-1898.
[7] Matsuoka, T., Hamada, H., and Nakatsu, R., "Syllable Recognition Using Integrated
Neural Networks," IEEE Proceedings of the 1989 International Joint Conference
on Neural Networks, Washington, D.C., June 18-22, 1989, vol. 1, pp. 251-258.
[8] Hampshire, J., Waibel, A., ''The Meta-Pi Network: Building Distributed Knowledge Representations for Robust Pattern Recognition," Carnegie Mellon University
Technical Report CMU-CS-89-166, August, 1989.
liWarp is a registered trademark of Intel Corporation.
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1,241 | 2,130 | A Bayesian Model Predicts Human Parse
Preference and Reading Times in Sentence
Processing
Srini Narayanan
SRI International and ICSI Berkeley
[email protected]
Daniel Jurafsky
University of Colorado, Boulder
[email protected]
Abstract
Narayanan and Jurafsky (1998) proposed that human language comprehension can be modeled by treating human comprehenders as Bayesian
reasoners, and modeling the comprehension process with Bayesian decision trees. In this paper we extend the Narayanan and Jurafsky model
to make further predictions about reading time given the probability of
difference parses or interpretations, and test the model against reading
time data from a psycholinguistic experiment.
1
Introduction
Narayanan and Jurafsky (1998) proposed that human language comprehension can be modeled by treating human comprehenders as Bayesian reasoners, and modeling the comprehension process with Bayesian decision trees. In this paper, we show that the model accounts for parse-preference and reading time data from a psycholinguistic experiment on
reading time in ambiguous sentences.
Parsing, (generally called ?sentence processing? when we are referring to human parsing),
is the process of building up syntactic interpretations for a sentence from an input sequence
of written or spoken words. Ambiguity is extremely common in parsing problems, and
previous research on human parsing has focused on showing that many factors play a role
in choosing among the possible interpretations of an ambiguous sentence.
We will focus in this paper on a syntactic ambiguity phenomenon which has been repeatedly investigated: the main-verb (MV), reduced relative (RR) local ambiguity (Frazier &
Rayner, 1987; MacDonald, Pearlmutter, & Seidenberg, 1994; McRae, Spivey-Knowlton,
& Tanenhaus, 1998, inter alia) In this ambiguity, a prefix beginning with a noun-phrase and
an ambiguous verb-form could either be continued as a main clause (as in 1a), or turn out
to be a relative clause modifier of the first noun phrase (as in 1b).
1. a. The cop arrested the forger.
b. The cop arrested by the detective was guilty of taking bribes.
Many factors are known to influence human parse preferences. One such factor is the different lexical/morphological frequencies of the simple past and participial forms of the ambiguous verbform (arrested, in this case). Trueswell (1996) found that verbs like searched,
with a frequency-based preference for the simple past form, caused readers to prefer the
main clause interpretation, while verbs like selected, had a preference for a participle reading, and supported the reduced relative interpretation.
The transitivity preference of the verb also plays a role in human syntactic disambiguation.
Some verbs are preferably transitive, where others are preferably intransitive. The reduced
relative interpretation, since it involves a passive structure, requires that the verb be transitive. MacDonald, Pearlmutter, and Seidenberg (1994), Trueswell, Tanenhaus, and Kello
(1994) and other have shown that verbs which are biased toward an intransitive interpretation also bias readers toward a main clause interpretation.
Previous work has shown that a competition-integration model developed by SpiveyKnowlton (1996) could model human parse preference in reading ambiguous sentences
(McRae et al., 1998). While this model does a nice job of accounting for the reading-time
data, it and similar ?constraint-based? models rely on a complex set of feature values and
factor weights which must be set by hand. Narayanan and Jurafsky (1998) proposed an
alternative Bayesian approach for this constraint-combination problem. A Bayesian approach offers a well-understood formalism for defining probabilistic weights, as well as for
combining those weights. Their Bayesian model is based on the probabilistic beam-search
of Jurafsky (1996), in which each interpretation receives a probability, and interpretations
were pruned if they were much worse than the best interpretation. The model predicted
large increases in reading time when unexpected words appeared which were only compatible with a previously-pruned interpretation. The model was thus only able to characterize
very gross timing effects caused by pruning of interpretations.
In this paper we extend this model?s predictions about reading time to other cases where
the best interpretation turns out to be incompatible with incoming words. In particular, we
suggest that any evidence which causes the probability of the best interpretation to drop
below its next competitor will also cause increases in reading time.
2
The Experimental Data
We test our model on the reading time data from McRae et al. (1998), an experiment focusing on the effect of thematic fit on syntactic ambiguity resolution. The thematic role of
noun phrase ?the cop? in the prefix ?The cop arrested? is ambiguous. In the continuation
?The cop arrested the crook?, the cop is the agent. In the continuation ?The cop arrested by
the FBI agent was convicted for smuggling drugs?, the cop is the theme. The probabilistic
relationship between the noun and the head verb (?arrested?) biases the thematic disambiguation decision. For example, ?cop? is a more likely agent for ?arrest?, while ?crook? is
a more likely theme. McRae et al. (1998) showed that this ?thematic fit? between the noun
and verb affected phrase-by-phrase reading times in sentences like the following:
2. a. The cop / arrested by / the detective / was guilty / of taking / bribes.
b. The crook / arrested by / the detective / was guilty / of taking / bribes.
In a series of experiment on 40 verbs, they found that sentences with good agents (like cop
in 2a) caused longer reading times for the phrase the detective than sentences with good
themes (like crook in 2b). Figure 1 shows that at the initial noun phrase, reading time is
lower for good-agent sentences than good-patient sentences. But at the NP after the word
?by?, reading time is lower for good-patient sentences than good-agent sentences. 1
1
In order to control for other influences on timing, McRae et al. (1998) actually report reading
time deltas between a reduced relative and non-reduced relative for. It is these deltas, rather than raw
reading times, that our model attempts to predict.
Increased Reading Times (compared to
control)
70
60
50
40
30
20
10
0
The cop/crook arrested by
Good Agent
the detective
Good Patient
Figure 1: Self-paced reading times (from Figure 6 of McRae et al. (1998))
After introducing our model in the next section, we show that it predicts this cross-over in
reading time; longer reading time for the initial NP in good-patient sentences, but shorter
reading time for the post-?by? NP in good-patient sentences.
3
The Model and the Input Probabilities
In the Narayanan and Jurafsky (1998) model of sentence processing, each interpretation of
an ambiguous sentence is maintained in parallel, and associated with a probability which
can be computed via a Bayesian belief net. The model pruned low-probability parses, and
hence predicted increases in reading time when reading a word which did not fit into any
available parse. The current paper extends the Narayanan and Jurafsky (1998) model?s predictions about reading time. The model now also predicts extended reading time whenever
an input word causes the best interpretation to drop in probability enough to switch in rank
with another interpretation.
The model consists of a set of probabilities expressing constraints on sentence processing,
and a network that represents their independence relations:
Data
P(Agent verb, initial NP)
P(Patient verb, initial NP)
P(Participle verb)
P(SimplePast verb)
P(transitive verb)
P(intransitive verb)
P(RR initial NP, verb-ed, by)
P(RR initial NP, verb-ed, by,the)
P(Agent initial NP, verb-ed, by, the, NP)
P(MC SCFG prefix)
P(RR SCFG prefix)
Source
McRae et al. (1998)
McRae et al. (1998)
British National Corpus counts
British National Corpus counts
TASA corpus counts
TASA corpus counts
McRae et al. (1998) (.8, .2)
McRae et al. (1998) (.875. .125)
McRae et al. (1998) (4.6 average)
SCFG counts from Penn Treebank
SCFG counts from Penn Treebank
The first constraint expresses the probability that the word ?cop?, for example, is an agent,
given that the verb is ?arrested?. The second constraint expresses the probability that it is
a patient. The third and fourth constraints express the probability that the ?-ed? form of
the verb is a participle versus a simple past form (for example P(Participle ?arrest?)=.81).
These were computed from the POS-tagged British National Corpus. Verb transitivity
probabilities were computed by hand-labeling subcategorization of 100 examples of each
verb in the TASA corpus. (for example P(transitive ?entertain?)=.86). Main clause prior
probabilities were computed by using an SCFG with rule probabilities trained on the Penn
Treebank version of the Brown corpus. See Narayanan and Jurafsky (1998) and Jurafsky
(1996) for more details on probability computation.
4
Construction Processing via Bayes nets
Using Belief nets to model human sentence processing allows us to a) quantitatively evaluate the impact of different independence assumptions in a uniform framework, b) directly
model the impact of highly structured linguistic knowledge sources with local conditional
probability tables, while well known algorithms for updating the Belief net (Jensen (1995))
can compute the global impact of new evidence, and c) develop an on-line interpretation
algorithm, where partial input corresponds to partial evidence on the network, and the update algorithm appropriately marginalizes over unobserved nodes. So as evidence comes
in incrementally, different nodes are instantiated and the posterior probability of different
interpretations changes appropriately.
The crucial insight of our Belief net model
is to view specific interpretations as values of
latent variables that render top-down (
) and bottom-up evidence ( ) conditionally
independent (d-separate them (Pearl, 1988)). Thus syntactic, lexical, argument structure,
and other contextual information acts as prior or causal support for an interpretation, while
bottom-up phonological or graphological and other perceptual information acts as likelihood, evidential, or diagnostic support.
To applyour
to on-line disambiguation, we assume that there are a set of interpre
model
tations (
) that are consistent with the input data. At different stages of the
input, we compute the posterior probabilities of the different interpretations given the top
down and bottom-up evidence seen so far. 2
V = examine-ed type_of(Subj) = witness
P(A | v, ty(Subj))
P(Arg|v)
P(T | v, ty(Subj))
P(Tense|v)
Arg
Tense
AND
MV
thm
Tense = past
Sem_fit = Agent
Semantic_fit
AND
Arg = trans
Tense = pp
Sem_fit = Theme
RR
thm
Figure 2: The Belief net that represents lexical and thematic support for the two interpretations.
Figure 2 reintroduces the basic structure of our belief net model from Narayanan and Jurafsky (1998). Our model requires conditional probability distributions specifying the preference of every verb for different argument structures, as well its preference for different
tenses. We also compute the semantic fit between possible fillers in the input and different
conceptual roles of a given predicate. As shown in Figure 2, the and interpretations require the conjunction of specific values corresponding to tense, semantic fit and
argument structure features. Note that only the interpretation requires the transitive
argument structure.
2
In this paper, we will focus on the support from thematic, and syntactic features for the Reduced Relative (RR) and Main Verb (MV) interpretations at different stages of the input for the
examples
we saw earlier. So we will have two interpretations "!$# where %&')( * +, -*).,/10
2$3
4%& )( * + -* . /,06575 .
S
[.14] NP?> NP XP
S
[.48] S?> NP [V ...
[.92] S?> NP ...
VP
NP
#1[]
NP
NP VP
VP
V
the witness examined
The witness
MAIN VERB
Figure 3: The partial syntactic parse trees for the
ing an generating grammar.
MAIN CLAUSE
and the
S
NP
XP
VP
NP
Det
interpretations assum-
REDUCED RELATIVE
S
NP
examined
REDUCED RELATIVE
N
V
XP
Det
The witness examined
VP
N
The witness examined
Figure 4: The Bayes nets for the partial syntactic parse trees
The conditional probability of a construction given top-down syntactic evidence
is relatively simple to compute in an augmented-stochastic-context-free formalism (partial
parse trees shown in Figure 3 and the corresponding bayes net in Figure 4). Recall that
the prior probability gives the conditional probability of the right hand side of a
rule given the left hand side. The Inside/Outside algorithm applied to a fixed parse tree
structure is obtained exactly by casting parsing as a special instance of belief propagation.
The correspondences are straightforward a) the parse tree is interpreted as a belief network.
b)the non-terminal nodes correspond to random variables, the range of the variables being
the non-terminal alphabet, c) the grammar rules define the conditional probabilities linking
parent and child nodes, d) the
nonterminal at the root, as well as the terminals at the
leaves represent conditioning evidence to the network, and e) Conditioning on this evidence
produces exactly the conditional probabilities for each nonterminal node in the parse tree
and the joint probability distribution of the parse. 3
The overall posterior ratio requires propagating the conjunctive impact of syntactic and
lexical/thematic sources on our model. Furthermore, in computing the conjunctive impact of the lexical/thematic and syntactic support to compute and , we use the
NOISY- AND (assumes exception independence) model (Pearl, 1988) for combining conjunctive sources. In the case of the and interpretations. At various points, we
compute the
posterior
support
for
the
different
interpretations
using
the
following equa
tion.
. The first term is
3
One complication is that the the conditional distribution in a parse tree %&!, #"( $ / is not
the product distribution %&! ( $ /-%&!"( $ / (it is the conjunctive distribution). However, it is possible to generalize
the belief propagation equations to admit conjunctive distributions %&!, #"( $ /
3
and %&$" ( % / . The diagnostic (inside) support becomes & &' / 0)(+*-, ./& &0 /1& &2 /-%&0 12 ( ' /
and the causal support becomes 3&' / 0546(879, :;3&!< /1& &!=)/-%&' #= ( </ (details can be found at
http://www.icsi.berkeley.edu/ snarayan/scfg.ps).
the syntactic support
while
the second is the lexical and thematic support for a particular
interpretation (
).
5
Model results
9
8
7
6
5
MV/RR
4
3
2
1
0
NP verbed
Model Good Agent
by
Human Good Agent
the
Model Good Patient
NP
Human Good Patient
Figure 5: Completion data
We tested our model on sentences with the different verbs in McRae et al. (1998). For
each verb, we ran our model on sentences with Good Agents (GA) and Good Patients (GP)
for the initial NP. Our model results are consistent with the on-line disambiguation studies
with human subjects (human performance data from McRae et al. (1998)) and show that a
Bayesian implementation of probabilistic evidence combination accounts for garden-path
disambiguation effects.
Figure 5 shows the first result that pertains to the model predictions of how thematic fit
might influence sentence completion times. Our model shows close correspondence to the
human judgements about whether a specific ambiguous verb was used in the Main Clause
(MV) or reduced relative (RR) constructions. The human and model predictions were conducted at the verb (The crook arrested), by (the crook arrested by), the (the crook arrested
by the) and Agent NP (the crook arrested by the detective). As in McRae et al. (1998)
the data shows that thematic fit clearly influenced the gated sentence completion task. The
probabilistic account further captured the fact that at the by phrase, the posterior probability of producing an RR interpretation increased sharply, thematic fit and other factors
2.5
0.9
0.8
2.1
0.7
0.6
1.5
P(X)
P(MC)/P(RR)
2
1
0.541
0.13
0
The crook/detective
arrested by
(a)
Good Agent initial NP
0.4
0.3
0.7
0.5
0.5
the
0.2
0.1
0.04
detective
0.1
0
The cop arrested by
Good Patient Initial NP
Good Agent Main Clause
the detective
Good Agent RR
Figure 6: a) MV/RR for the ambiguous region showing a flip for the Good Agent (ga) case.
b) P(MV) and P(RR) for the Good Patient and Good Agent cases.
(b)
influenced both the sharpness and the magnitude of the increase.
The second result pertains to on-line reading times. Figure 6 shows how the human reading
time reduction effects (reduced compared to unreduced interpretations) increase for Good
Agents (GA) but decrease for Good Patients in the ambiguous region. This explains the
reading time data in Figure 1. Our model predicts this larger effect from the fact that the
most probable interpretation for the Good Agent case flips from the MV to the RR interpretation in this region. No such flip occurs for the Good Patient (GP) case. In Figure 6(a), we
see that the GP results already have the MV/RR ratio less than one (the RR interpretation
is superior) while a flip occurs for the GA sentences (from the initial state where MV/RR
to the final state where MV/RR
). Figure 6 (b) shows a more detailed view of
the GA sentences showing the crossing point where the flip occurs. This finding is fairly
robust (
of GA examples) and directly predicts reading time difficulties.
6
Conclusion
We have shown that a Bayesian model of human sentence processing is capable of modeling
reading time data from a syntactic disambiguation task. A Bayesian model extends current
constraint-satisfaction models of sentence processing with a principled way to weight and
combine evidence. Bayesian models have not been widely applied in psycholinguistics. To
our knowledge, this is the first study showing a direct correspondence between the time
course of maintaining the best a posteriori interpretation and reading time difficulty.
We are currently exploring how our results on flipping of preferred interpretation could
be combined with Hale (2001)?s proposal that reading time correlates with surprise (a surprising (low probability) word leads to large amounts of probability mass to be pruned) to
arrive at a structured probabilistic account of a wide variety of psycholinguistic data.
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1,242 | 2,131 | Dynamic Time-Alignment Kernel in
Support Vector Machine
Hiroshi Shimodaira
School of Information Science,
Japan Advanced Institute of
Science and Technology
[email protected]
Mitsuru Nakai
School of Information Science,
Japan Advanced Institute of
Science and Technology
[email protected]
Ken-ichi Noma
School of Information Science,
Japan Advanced Institute of
Science and Technology
[email protected]
Shigeki Sagayama
Graduate School of Information Science
and Technology,
The University of Tokyo
[email protected]
Abstract
A new class of Support Vector Machine (SVM) that is applicable to sequential-pattern recognition such as speech recognition is
developed by incorporating an idea of non-linear time alignment
into the kernel function. Since the time-alignment operation of
sequential pattern is embedded in the new kernel function, standard SVM training and classification algorithms can be employed
without further modifications. The proposed SVM (DTAK-SVM)
is evaluated in speaker-dependent speech recognition experiments
of hand-segmented phoneme recognition. Preliminary experimental results show comparable recognition performance with hidden
Markov models (HMMs).
1
Introduction
Support Vector Machine (SVM) [1] is one of the latest and most successful statistical
pattern classifier that utilizes a kernel technique [2, 3]. The basic form of SVM
classifier which classifies an input vector x ? Rn is expressed as
g(x) =
N
X
i=1
?i yi ?(xi ) ? ?(x) + b =
N
X
?i yi K(xi , x) + b,
(1)
i=1
0
where ? is a non-linear mapping function ?(x) : R n 7? Rn , (n n0 ), ??? denotes
the inner product operator, xi , yi and ?i are the i-th training sample, its class label,
and its Lagrange multiplier, respectively, K is a kernel function, and b is a bias.
Despite the successful applications of SVM in the field of pattern recognition such
as character recognition and text classification, SVM has not been applied to speech
recognition that much. This is because SVM assumes that each sample is a vector
of fixed dimension, and hence it can not deal with the variable length sequences
directly. Because of this, most of the efforts that have been made so far to apply
SVM to speech recognition employ linear time normalization, where input feature
vector sequences with different lengths are aligned to same length [4]. A variant
of this approach is a hybrid of SVM and HMM (hidden Markov model), in which
HMM works as a pre-processor to feed time-aligned fixed-dimensional vectors to
SVM [5]. Another approach is to utilize probabilistic generative models as a SVM
kernel function. This includes the Fisher kernels [6, 7], and conditional symmetric
independence (CSI) kernels [8], both of which employ HMMs as the generative models. Since HMMs can treat sequential patterns, SVM that employs the generative
models based on HMMs can handle sequential patterns as well.
In contrast to those approaches, our approach is a direct extension of the original
SVM to the case of variable length sequence. The idea is to incorporate the operation of dynamic time alignment into the kernel function itself. Because of this,
the proposed new SVM is called ?Dynamic Time-Alignment Kernel SVM (DTAKSVM)?. Unlike the SVM with Fisher kernel that requires two training stages with
different training criteria, one is for training the generative models and the second
is for training the SVM, the DTAK-SVM uses one training criterion as well as the
original SVM.
2
Dynamic Time-Alignment Kernel
We consider a sequence of vectors X = (x1 , x2 , ? ? ? , x L ), where xi ? Rn , L is the
length of the sequence, and the notation |X| is sometimes used to represent the
length of the sequence instead. For simplification, we at first assume the so-called
linear SVM that does not employ non-linear mapping function ?. In such case, the
kernel operation in (1) is identical to the inner product operation.
2.1
Formulation for linear kernel
Assume that we have two vector sequences X and V . If these two patterns are
equal in length, i.e. |X| = |V | = L, then the inner product between X and V can
be obtained easily as a summation of each inner product between xk and v k for
k = 1, ? ? ? , L:
X ?V
=
L
X
xk ? v k ,
(2)
k=1
and therefore an SVM classifier can be defined as given in (1). On the other hand
in case where the two sequences are different in length, the inner product can not
be calculated directly. Even in such case, however, some sort of inner product like
operation can be defined if we align the lengths of the patterns. To that end, let
?(k), ?(k) be the time-warping functions of normalized time frame k for the pattern
X and V , respectively, and let ??? be the new inner product operator instead of
the original inner product ???. Then the new inner product between the two vector
sequences X and V can be given by
X ?V
=
L
1X
x?(k) ? v ?(k) ,
L
(3)
k=1
where L is a normalized length that can be either |X|, |V | or arbitrary positive
integer.
There would be two possible types of time-warping functions. One is a linear timewarping function and the other is a non-linear time-warping function. The linear
time-warping function takes the form as
?(k) = d(|X|/L)ke,
?(k) = d(|V |/L)ke,
where dxe is the ceiling function which gives the smallest integer that is greater than
or equal to x. As it can be seen from the definition given above, the linear warping
function is not suitable for continuous speech recognition, i.e. frame-synchronous
processing, because the sequence lengths, |X| and |V |, should be known beforehand.
On the other hand, non-linear time warping, or dynamic time warping (DTW) [9] in
other word, enables frame-synchronous processing. Furthermore, the past research
on speech recognition has shown that the recognition performance by the non-linear
time normalization outperforms the one by the linear time normalization. Because
of these reasons, we focus on the non-linear time warping based on DTW.
Though the original DTW uses a distance/distortion measure and finds the optimal path that minimizes the accumulated distance/distortion, the DTW that is
employed for SVM uses inner product or kernel function instead and finds the optimal path that maximizes the accumulated similarity:
X ?V
subject to
=
max
?,?
L
1 X
m(k)x?(k) ? v ?(k) ,
M??
(4)
k=1
1 ? ?(k) ? ?(k + 1) ? |X|, ?(k + 1) ? ?(k) ? Q,
1 ? ?(k) ? ?(k + 1) ? |V |, ?(k + 1) ? ?(k) ? Q,
(5)
where m(k) is a nonnegative (path) weighting coefficient, M?? is a (path) normalizing factor, and Q is a constant constraining the local continuity. In the standard
PL
DTW, the normalizing factor M ?? is given as k=1 m(k), and the weighting coefficients m(k) are chosen so that M?? is independent of the warping functions.
The above optimization problem can be solved efficiently by dynamic programming.
The recursive formula in the dynamic programming employed in the present study
is as follows
(
)
G(i ? 1, j) + Inp(i, j),
G(i, j) = max G(i ? 1, j ? 1) + 2 Inp(i, j),
(6)
G(i, j ? 1) + Inp(i, j),
where Inp(i, j) is the standard inner product between the two vectors corresponding
to point i and j. As a result, we have
X ? V = G(|X|, |V |)/(|X| + |V |).
2.2
(7)
Formulation for non-linear kernel
In the last subsection, a linear kernel, i.e. the inner product, for two vector sequences with different lengths has been formulated in the framework of dynamic
time-warping. With a little constraint, similar formulation is possible for the case
where SVM?s non-linear mapping function ? is applied to the vector sequences. To
that end, ? is restricted to the one having the following form:
?(X) = (?(x1 ), ?(x2 ), ? ? ? , ?(x L )),
(8)
where ? is a non-linear mapping function that is applied to each frame vector x i ,
as given in (1). It should be noted that under the above restriction ? preserves the
original length of sequence at the cost of losing long-term correlations such as the
one between x1 and xL . As a result, a new class of kernel can be defined by using
the extended inner product introduced in the previous section;
Ks (X, V )
=
?(X) ? ?(V )
=
max
?,?
=
max
?,?
1
M??
L
X
(9)
m(k)?(x?(k) ) ? ?(v ?(k) )
(10)
k=1
L
1 X
m(k)K(x?(k) , v?(k) ).
M??
(11)
k=1
We call this new kernel ?dynamic time-alignment kernel (DTAK)?.
2.3
Properties of the dynamic time-alignment kernel
It has not been proven that the proposed function Ks (, ) is really an SVM?s admissible kernel which guarantees the existence of a feature space. This is because that
the mapping function to a feature space is not independent but dependent on the
given vector sequences. Although a class of data-dependent asymmetric kernel for
SVM has been developed in [10], our proposed function is more complicated and
difficult to analyze because the input data is a vector sequence with variable length
and non-linear time normalization is embedded in the function. Instead, what have
been known about the proposed function so far are (1) Ks is symmetric, (2) Ks
satisfies the Cauchy-Schwartz like inequality described bellow:
Proposition 1
Ks (X, V )2 ? Ks (X, X)Ks (V, V )
(12)
Proof For simplification, we assume that normalized length L is fixed, and omit
m(k) and M?? in (11). Using the standard Cauchy-Schwartz inequality, the following inequality holds:
Ks (X, V )
=
?
max
?,?
L
X
L
X
?(x?(k) ) ? ?(v ?(k) ) =
k=1
L
X
?(x?? (k) ) ? ?(v ?? (k) )
(13)
k=1
k ?(x?? (k) ) kk ?(v ?? (k) ) k,
(14)
k=1
where ?? (k), ?? (k) represent the optimal warping functions that maximize the RHS
of (13). On the other hand,
Ks (X, X)
=
max
?,?
L
X
?(x?(k) ) ? ?(x?(k) ) =
k=1
L
X
?(x?+ (k) ) ? ?(x?+ (k) ). (15)
k=1
Because here we assume that ?+ (k), ?+ (k) are the optimal warping functions that
maximize (15), for any warping functions including ? ? (k), the following inequality
holds:
L
L
X
X
Ks (X, X) ?
?(x?? (k) ) ? ?(x?? (k) ) =
k ?(x?? (k) ) k2 .
(16)
k=1
k=1
In the same manner, the following holds:
Ks (V, V )
?
L
X
k=1
?(v ?? (k) ) ? ?(v ?? (k) ) =
L
X
k=1
k ?(v ?? (k) ) k2 .
(17)
Therefore,
Ks (X, X)Ks (V, V ) ? Ks (X, V )2
L
X
?
k ?(x?? (k) ) k
2
k=1
=
L X
L
X
!
L
X
k ?(v ?? (k) ) k
k=1
2
!
L
X
?
k ?(x?? (k) ) kk ?(v ?? (k) ) k
k=1
k ?(x?? (i) ) kk ?(v ?? (j) ) k ? k ?(x?? (j) ) kk ?(v ?? (i) ) k
i=1 j=i+1
2
?0
!2
(18)
3
DTAK-SVM
Using the dynamic time-alignment kernel (DTAK) introduced in the previous section, the discriminant function of SVM for a sequential pattern is expressed as
g(X)
=
N
X
?i yi ?(X (i) ) ? ?(X) + b
(19)
?i yi Ks (X (i) , X) + b,
(20)
i=1
=
N
X
i=1
where X (i) represents the i-th training pattern. As it can be seen from these
expressions, the SVM discriminant function for time sequence has the same form
with the original SVM except for the difference in kernels. It is straightforward to
deduce the learning problem which is given as
N
min
W,b,?i
subject to
X
1
W ?W +C
?i ,
2
i=1
(i)
yi (W ? ?(X ) + b) ? 1 ? ?i ,
?i ? 0, i = 1, ? ? ? , N.
(21)
(22)
Again, since the formulation of learning problem defined above is almost the same
with that for the original SVM, same training algorithms for the original SVM can
be used to solve the problem.
4
Experiments
Speech recognition experiments were carried out to evaluated the classification performance of DTAK-SVM. As our objective is to evaluate the basic performance
of the proposed method, very limited task, hand-segmented phoneme recognition
task in which positions of target patterns in the utterance are known, was chosen.
Continuous speech recognition task that does not require phoneme labeling would
be our next step.
4.1
Experimental conditions
The details of the experimental conditions are given in Table 1. The training
and evaluation samples were collected from the ATR speech database: A-set (5240
Table 1: Experimental conditions
Speaker dependency
Phoneme classes
Speakers
Training samples
Evaluation samples
Signal sampling
Feature values
Kernel type
Experiment-1
Experiment-2
dependent
dependent
6 voiced consonants
5 vowels
5 males
5 males and 5 females
200 samples per phoneme
500 samples per phoneme
2,035 samples in all per 2500 samples in all per
speaker
speaker
12kHz, 10ms frame-shift
13-MFCCs and 13-?MFCCs
kx ?x k2
RBF (radial basis function): K(xi , xj ) = exp(? i ? 2 j )
100
C=0.1
C=1.0
C=10
95
90
# SVs / # training samples [%]
Correct classification rate [%]
100
85
80
75
70
65
60
55
50
C=0.1
C=1.0
C=10.0
80
60
40
20
0
0
2
4
6
8
RBF-sigma
(a) Recognition performance
10
1
2
3
4
5
6
7
8
9
10
RBF-sigma
(b) Number of SVs
Figure 1: Experimental results for Experiment-1 (6 voiced-consonants recognition)
showing (a) correct classification rate and (b) the number of SVs as a function of ?
(the parameter of RBF kernel).
Japanese words in vocabulary). In consonant-recognition task (Experiment-1), only
six voiced-consonants /b,d,g,m,n,N/ were used to save time. The classification task
of those 6 phonemes without using contextual information is considered as a relatively difficult task, whereas the classification of 5 vowels /a,i,u,e,o/ (Experiment-2)
is considered as an easier task.
To apply SVM that is basically formulated as a two-class classifier to the multiclass problem, ?one against the others? type of strategy was chosen. The proposed
DTAK-SVM has been implemented with the publicly available toolkit, SVMTorch
[11].
4.2
Experimental results
Fig. 1 depicts the experimental results for Experiment-1, where average values over
5 speakers are shown. It can be seen in Fig. 1 that the best performance of 95.8%
was achieved at ? = 2.0 and C = 10. Similar results were obtained for Experiment-2
as given in Fig. 2.
100
95
# SVs / # training samples [%]
Correct classification rate [%]
100
90
85
80
75
70
65
60
55
50
80
60
40
20
0
0
2
4
6
8
RBF-sigma
10
1
2
3
4
5
6
7
8
9
10
RBF-sigma
(a) Recognition performance
(b) Number of SVs
Figure 2: Experimental results for Experiment-2 (5 vowels recognition) showing
(a) correct classification rate and (b) the number of SVs as a function of ? (the
parameter of RBF kernel).
Table 2: Recognition performance comparison of DTAK-SVM with HMM. Results
of Experiment-1 for 1 male and 1 female speakers are shown. (numbers represent
correct classification rate [%])
Model
HMM (1 mix.)
HMM (4 mix.)
HMM (8 mix.)
HMM (16 mix.)
DTAK-SVM
# training samples/phoneme
male
female
50
100 200
50
100 200
75.0 69.1 77.1 72.2 65.5 76.6
83.3 84.7 90.9 77.3 76.4 86.4
82.8 87.0 92.4 74.6 79.3 88.5
79.9 85.0 93.2 72.9 78.7 89.8
83.8 85.9 92.1 83.5 81.8 87.7
Next, the classification performance of DTAK-SVM was compared with that of the
state-of-the-art HMM. In order to see the effect of generalization performance on
the size of training data set and model complexity, experiments were carried out
by varying the number of training samples (50, 100, 200), and mixtures (1,4,8,16)
for each state of HMM. The HMM used in this experiment was a 3-states, continuous density, Gaussian-distribution mixtures with diagonal covariances, contextindependent model. HTK [12] was employed for this purpose. The parameters of
DTAK-SVM were fixed to C = 10, ? = 2.0. The results for Experiment-1 with
respect to 1 male and 1 female speakers are given in Table 2.
It can be said from the experimental results that DTAK-SVM shows better classification performance when the number of training samples is 50, while comparable
performance when the number of samples is 200. One might argue that the number
of training samples used in this experiment is not enough at all for HMM to achieve
best performance. But such shortage of training samples occurs often in HMMbased real-world speech recognition, especially when context-dependent models are
employed, which prevents HMM from improving the generalization performance.
5
Conclusions
A novel approach to extend the SVM framework for the sequential-pattern classification problem has been proposed by embedding a dynamic time-alignment operation
into the kernel. Though long-term correlations between the feature vectors are omitted at the cost of achieving frame-synchronous processing for speech recognition, the
proposed DTAK-SVMs demonstrated comparable performance in hand-segmented
phoneme recognition with HMMs. The DTAK-SVM is potentially applicable to
continuous speech recognition with some extension of One-pass search algorithm
[9].
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http://www.idiap.ch/learning/SVMTorch.html.
[12] ?The Hidden Markov Model Toolkit (HTK).? http://htk.eng.cam.ac.uk/.
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1,243 | 2,132 | Duality, Geometry, and Support Vector
Regression
Jinbo Bi and Kristin P. Bennett
Department of Mathematical Sciences
Rensselaer Polytechnic Institute
Troy, NY 12180
[email protected], [email protected]
Abstract
We develop an intuitive geometric framework for support vector
regression (SVR). By examining when -tubes exist, we show that
SVR can be regarded as a classification problem in the dual space.
Hard and soft -tubes are constructed by separating the convex
or reduced convex hulls respectively of the training data with the
response variable shifted up and down by . A novel SVR model is
proposed based on choosing the max-margin plane between the two
shifted datasets. Maximizing the margin corresponds to shrinking
the effective -tube. In the proposed approach the effects of the
choices of all parameters become clear geometrically.
1
Introduction
Support Vector Machines (SVMs) [6] are a very robust methodology for inference
with minimal parameter choices. Intuitive geometric formulations exist for the
classification case addressing both the error metric and capacity control [1, 2]. For
linearly separable classification, the primal SVM finds the separating plane with
maximum hard margin between two sets. The equivalent dual SVM computes the
closest points in the convex hulls of the data from each class. For the inseparable
case, the primal SVM optimizes the soft margin of separation between the two
classes. The corresponding dual SVM finds the closest points in the reduced convex
hulls. In this paper, we derive analogous arguments for SVM regression (SVR).
We provide a geometric explanation for SVR with the -insensitive loss function.
From the primal perspective, a linear function with no residuals greater than corresponds to an -tube constructed about the data in the space of the data attributes
and the response variable [6] (see e.g. Figure 1(a)). The primary contribution of this
work is a novel geometric interpretation of SVR from the dual perspective along
with a mathematically rigorous derivation of the geometric concepts. In Section
2, for a fixed > 0 we examine the question ?When does a ?perfect? or ?hard?
-tube exist??. With duality analysis, the existence of a hard -tube depends on
the separability of two sets. The two sets consist of the training data augmented
with the response variable shifted up and down by . In the dual space, regression
becomes the classification problem of distinguishing between these two sets. The
geometric formulations developed for the classification case [1] become applicable to
the regression case. We call the resulting formulation convex SVR (C-SVR) since it
is based on convex hulls of the augmented training data. Much like in SVM classification, to compute a hard -tube, C-SVR computes the nearest points in the convex
hulls of the augmented classes. The corresponding maximum margin (max-margin)
planes define the effective -tube. The size of margin determines how much the
effective -tube shrinks. Similarly, to compute a soft -tube, reduced-convex SVR
(RC-SVR) finds the closest points in the reduced convex hulls of the two augmented
sets.
This paper introduces the geometrically intuitive RC-SVR formulation which is a
variation of the classic -SVR [6] and ?-SVR models [5]. If parameters are properly
tuned, the methods perform similarly although not necessarily identically. RCSVR eliminates the pesky parameter C used in -SVR and ?-SVR. The geometric
role or interpretation of C is not known for these formulations. The geometric
roles of the two parameters of RC-SVR, ? and , are very clear, facilitating model
selection, especially for nonexperts. Like ?-SVR, RC-SVR shrinks the -tube and
has a parameter ? controlling the robustness of the solution. The parameter
acts as an upper bound on the size of the allowable -insensitive error function. In
addition, RC-SVR can be solved by fast and scalable nearest-point algorithms such
as those used in [3] for SVM classification.
2
When does a hard -tube exist?
y
y
y
D
?
y
+
D
+
D
?
D
-
x
D-
D-
(b)
(a)
(d)
(c)
x
+
x
x
Figure 1: The (a) primal hard 0-tube, and dual cases: (b) dual strictly separable > 0 ,
(c) dual separable = 0 , and (d) dual inseparable < 0.
SVR constructs a regression model that minimizes some empirical risk measure
regularized to control capacity. Let x be the n predictor variables and y the dependent response variable. In [6], Vapnik proposed using the -insensitive loss function
L (x, y, f) = |y ? f(x)| = max (0, |y ? f(x)| ? ), in which an example is in error if
its residual |y ? f(x)| is greater than . Plotting the points in (x, y) space as in Figure 1(a), we see that for a ?perfect? regression model the data fall in a hard -tube
about the regression line. Let (Xi , yi) be an example where i = 1, 2, ? ? ? , m, Xi is the
ith predictor vector, and yi is its response. The training data are then (X, y) where
Xi is a row of the matrix X ? Rm?n and y ? Rm is the response. A hard -tube
for a fixed > 0 is defined as a plane y = w 0x + b satisfying ?e ? y ? Xw ? be ? e
where e is an m-dimensional vector of ones.
When does a hard -tube exist? Clearly, for large enough such a tube always
exists for finite data. The smallest tube, the 0-tube, can be found by optimizing:
min
w,b,
s.t. ? e ? y ? Xw ? be ? e
(1)
Note that the smallest tube is typically not the -SVR solution. Let D + and D? be
formed by augmenting the data with the response variable respectively increased
and decreased by , i.e. D + = {(Xi , yi + ), i = 1, ? ? ? , m} and D ? = {(Xi , yi ?
), i = 1, ? ? ? , m}. Consider the simple problem in Figure 1(a). For any fixed > 0,
there are three possible cases: > 0 in which strict hard -tubes exist, = 0
in which only 0-tubes exist, and < 0 in which no hard -tubes exist. A strict
hard -tube with no points on the edges of the tube only exists for > 0 . Figure
1(b-d) illustrates what happens in the dual space for each case. The convex hulls of
D+ and D? are drawn along with the max-margin plane in (b) and the supporting
plane in (c) for separating the convex hulls.
Clearly, the existence of the tube is directly related to the separability of D + and
D? . If > 0 then a strict tube exists and the convex hulls of D + and D? are strictly
separable1 . There are infinitely many possible -tubes when > 0 . One can see
that the max-margin plane separating D + and D? corresponds to one such . In
fact this plane forms an ?
tube where > ? ? 0 . If = 0, then the convex hulls
of D+ and D? are separable but not strictly separable. The plane that separates
the two convex hulls forms the 0 tube. In the last case, where < 0 , the two sets
D+ and D? intersect. No -tubes or max-margin planes exist.
It is easy to show by construction that if a hard -tube exists for a given > 0 then
the convex hulls of D + and D? will be separable. If a hard -tube exists, then there
exists (w, b) such that
(y + e) ? Xw ? be ? 0,
(y ? e) ? Xw ? be ? 0.
(2)
X
0
For any convex combination of D + , (y+e)
0 u where e u = 1, u ? 0 of points
(Xi , yi + ), i = 1, 2, ? ? ? , m, we have (y + e)0 u ? w0(X0 u) ? b ? 0. Similarly for
X0
0
D? , (y?e)
0 v where e v = 1, v ? 0 of points (Xi , yi ? ), i = 1, 2, ? ? ? , m, we have
(y ? e)0 v ? w0 (X0 v) ? b ? 0. Then the plane y = w 0x + b in the -tube separates the
two convex hulls. Note the separating plane and the -tube plane are the same. If
no separating plane exists, then there is no tube. Gale?s Theorem2 of the alternative
can be used to precisely characterize the -tube.
0
Theorem 2.1 (Conditions for existence of hard -tube) A hard -tube exists
for a given > 0 if and only if the following system in (u, v) has no solution:
X0 u = X0 v, e0 u = e0 v = 1, (y + e)0 u ? (y ? e)0 v < 0, u ? 0, v ? 0.
(3)
Proof A hard -tube exists if and only if System (2) has a solution. By Gale?s
Theorem of the alternative [4], system (2) has a solution if and only if the following
alternative system has no solution: X0 u = X0 v, e0 u = e0 v, (y + e)0 u ? (y ? e)0 v =
?1, u ? 0, v ? 0. Rescaling by ?1 where ? = e0 u = e0 v > 0 yields the result.
1
We use the following definitions of separation of convex sets. Let D + and D? be
nonempty convex sets. A plane H = {x : w 0 x = ?} is said to separate D + and D? if
w0x ? ?, ?x ? D + and w0 x ? ?, ?x ? D ? . H is said to strictly separate D + and D? if
w0x ? ? + ? for x ? D + , and w0x ? ? ? ? for each x ? D ? where ? is a positive scalar.
2
The system Ax ? c has a (or has no) solution if and only if the alternative system
A0 y = 0, c0 y = ?1, y ? 0 has no (or has a) solution.
Note that if ? 0 then (y + e)0 u ? (y ? e)0 v ? 0. for any (u, v) such that
X0 u = X0 v, e0 u = e0 v = 1, u, v ? 0. So as a consequence of this theorem, if
D+ and D? are separable, then a hard -tube exists.
3
Constructing the -tube
For any > 0 infinitely many possible -tubes exist. Which -tube should be used?
The linear program (1) can be solved to find the smallest 0-tube. But this corresponds to just doing empirical risk minimization and may result in poor generalization due to overfitting. We know capacity control or structural risk minimization is
fundamental to the success of SVM classification and regression.
We take our inspiration from SVM classification. In hard-margin SVM classification,
the dual SVM formulation constructs the max-margin plane by finding the two
nearest points in the convex hulls of the two classes. The max-margin plane is
the plane bisecting these two points. We know that the existence of the tube is
linked to the separability of the shifted sets, D + and D? . The key insight is that
the regression problem can be regarded as a classification problem between D + and
D? . The two sets D + and D? defined as in Section 2 both contain the same number
of data points. The only significant difference occurs along the y dimension as the
response variable y is shifted up by in D + and down by in D? . For > 0,
the max-margin separating plane corresponds to a hard ?-tube where > ? ? 0.
The resulting tube is smaller than but not necessarily the smallest tube. Figure
1(b) shows the max-margin plane found for > 0 . Figure 1(a) shows that the
corresponding linear regression function for this simple example turns out to be the
0 tube. As in classification, we will have a hard and soft -tube case. The soft
-tube with ? 0 is used to obtain good generalization when there are outliers.
3.1
The hard -tube case
We now apply the dual convex hull method to constructing the max-margin plane
for our augmented sets D + and D? assuming they are strictly separable, i.e. > 0.
The problem is illustrated in detail in Figure 2. The closest points of D + and D? can
be found by solving the following dual C-SVR quadratic program:
min
u,v
s.t.
1
2
X0
(y+e)0
0
u?
X0
(y?e)0
2
v
(4)
e0 u = 1, e v = 1, u ? 0, v ? 0.
X0
Let the closest points in the convex hulls of D + and D? be c = (y+e)
? and
0 u
X0
d = (y?e)0 v
? respectively. The max-margin separating plane bisects these two
? of the plane is the difference between them, i.e., w
points. The normal (w,
? ?)
? =
0
0
0
?
Xu
??Xv
?, ? = (y + e) u
? ? (y ? e)0 v
?. The threshold, ?b, is the distance from the
?
origin
to the
point
halfway
between the two closest points along the normal: b =
y0 u
? +y0 v
?
? +X0 v
?
0 X0 u
0
?
?
?
w
?
+?
. The separating plane has the equation w
? x+ ?y? b = 0.
2
2
Rescaling this plane yields the regression function.
Dual C-SVR (4) is in the dual space. The corresponding Primal C-SVR is:
Figure 2: The solution ?-tube found by C-SVR can have ? < . Squares are original data.
Dots are in D + . Triangles are in D ? . Support Vectors are circled.
1
2
min
w,?,?,?
2
kwk + 12 ? 2 ? (? ? ?)
(5)
Xw + ?(y + e) ? ?e ? 0
Xw + ?(y ? e) ? ?e ? 0.
s.t.
Dual C-SVR (4) can be derived by taking the Wolfe or Lagrangian dual [4] of primal
C-SVR (5) and simplifying.
We prove that the optimal plane from C-SVR bisects the ? tube. The supporting
planes for class D + and class D? determines the lower and upper edges of the ?-tube
respectively. The support vectors from D + and D? correspond to the points along
the lower and upper edges of the ?-tube. See Figure 2.
Theorem 3.1 (C-SVR constructs ?-tube) Let the max-margin plane obtained
? ?b = 0 where w
by C-SVR
? 0x+
?y?
? = X0 u
? ?X0 v
?, ?? = (y+e)0 u
? ?(y?e)0 v
?, and
0(4) be
w
0
0
0
y
u
?
+y
v
?
v
?
0
?b = w
?
? 0 X u?+X
+?
. If > 0, then the plane y = w x + b corresponds
2
2
to an ?-tube of training data (Xi , yi), i = 1, 2, ? ? ? , m where w = ? w??? , b =
? = ?
??
? ??
2??
?
b
??
and
< .
Proof First, we show ?? > 0. By the Wolfe duality theorem [4], ?
? ? ?? > 0,
since the objective values of (5) and the negative objective value of (4) are equal at
optimality. By complementarity, the closest points are right on the margin planes
? ??
? ? ?? = 0 respectively, so ?
? + e)0 u
w
? 0x + ?y
? = 0 and w
? 0x + ?y
?=w
? 0 X0 u
? + ?(y
? and
?
?+
?
?
0 0
0
?
?
?
?
?
?=w
? Xv
? + ?(y?e) v
?. Hence b = 2 , and w,
? ?, ?
? , and ? satisfy the constraints of
?
?
? ? 0. Then subtract the
problem (5), i.e., Xw+
? ?(y+e)?
?
? e ? 0, Xw+
? ?(y?e)?
?e
? ??
? ??
second inequality from the first inequality: 2?
? + ?? ? 0, that is, ?? ? ??
2 > 0
because > 0 ? 0. Rescale constraints by ??? < 0, and reverse the signs. Let
?
w = ? w??? , then the inequalities become Xw ? y ? e ? ???? e, Xw ? y ? ?e ? ??? e.
Let b =
?
b
,
??
then
?
?
??
= b+
??
? ??
2??
??
? ??
= b ? ??
. Substituting into the previous
??
2??
??
? ??
??
? ??
e?be,
Xw?y
?
?
?
e?be. Denote
2??
2??
and
inequalities yields Xw?y ? ?
?
? ?
? = ? ??
< . These inequalities become Xw + be ? y ? ?e, Xw + be ? y ? ??
e.
2??
Hence the plane y = w 0x + b is in the middle of the ?
< tube.
3.2
The soft -tube case
For < 0 , a hard -tube does not exist. Making large to fit outliers may result
in poor overall accuracy. In soft-margin classification, outliers were handled in the
y
2?^
x
Figure 3: Soft ?-tube found by RC-SVR: left: dual, right: primal space.
dual space by using reduced convex hulls. The same strategy works for soft -tubes,
see Figure 3. Instead of taking the full convex hulls of D + and D? , we reduce the
convex hulls away from the difficult boundary cases. RC-SVR computes the closest
points in the reduced convex hulls
2
X0
X0
1
min
u
?
v
0
0
2
(y+e)
(y?e)
u,v
(6)
0
0
s.t.
e u = 1, e v = 1, 0 ? u ? De, 0 ? v ? De.
Parameter D determines the robustness of the solution by reducing the convex hull.
D limits the influence of any single point. As in ?-SVM, we can parameterize D
1
where m is the number of points. Figure 3 illustrates the case
by ?. Let D = ?m
for m = 6 points, ? = 2/6, and D = 1/2. In this example, every
Pmpoint in the
reduced convex hull must depend on at least two data points since i=1 ui = 1 and
0 ? ui ? 1/2. In general, every point in the reduced convex hull can be written as
the convex combination of at least d1/De = d? ? me. Since these points are exactly
the support vectors and there are two reduced convex hulls, 2 ? d?me is a lower
bound on the number of support vectors in RC-SVR. By choosing ? sufficiently
large, the inseparable case with ? 0 is transformed into a separable case where
once again our nearest-points-in-the-convex-hull-problem is well defined.
As in classification, the dual reduced convex hull problem corresponds to computing
a soft -tube in the primal space. Consider the following soft tube version of the
primal C-SVR (7) which has its Wolfe Dual RC-SVR (6):
2
min
1
2
s.t.
Xw + ?(y + e) ? ?e + ? ? 0, ? ? 0
Xw + ?(y ? e) ? ?e ? ? ? 0, ? ? 0
w,?,?,?,? ,?
kwk + 12 ? 2 ? (? ? ?) + C(e0 ? + e0 ?)
(7)
The results of Theorem 3.1 can be easily extended to soft -tubes.
Theorem 3.2 (RC-SVR constructs soft ?-tube) Let the soft max-margin
? ? ?b = 0 where w
plane obtained by RC-SVR (6) be w
? 0x + ?y
? = X0 u
? ? X0 v
?,
0
0
0
0
0
y
u
?
+y
v
?
X
u
?
+X
v
?
0
0
? If 0 < ? 0, then
?? = (y + e) u
? ? (y ? e) v
?, and ?b =
w
?+
?.
2
2
?
? ?
the plane y = w 0x + b corresponds to a soft ? = ? ??
< -tube of training data
2??
(Xi , yi ), i = 1, 2, ? ? ? , m, i.e., a ?-tube of reduced convex hull of training data where
?
?
? + e)0 u
? ? e)0 v
w = ?w
, b = ?b? and ?
?=w
? 0 X0 u
? + ?(y
?, ?? = w
? 0 X0 v
? + ?(y
?.
??
Notice that the ?
? and ?? determine the planes parallel to the regression plane and
through the closest points in each reduced convex hull of shifted data. In the
inseparable case, these planes are parallel but not necessarily identical to the planes
obtained by the primal RC-SVR (7).
Nonlinear C-SVR and RC-SVR can be achieved by using the usual kernel trick. Let
? by a nonlinear mapping of x such that k(Xi , Xj ) = ?(Xi ) ? ?(Xj ). The objective
function of C-SVR (4) and RC-SVR (6) applied to the mapped data becomes
Pm
Pm Pm
1
((ui ? vi )(uj ? vj )(?(Xi ) ? ?(Xj ) + yi yj )) + 2 i=1 (yi (ui ? vi))
j=1
i=1
2
P
P
P
m
m
m
= 21 i=1 j=1 ((ui ? vi )(uj ? vj )(k(Xi , Xj ) + yi yj )) + 2 i=1 (yi (ui ? vi ))
(8)
The final regression P
model after optimizing C-SVR or RC-SVR with kernels takes
m
ui ? v?i ) k(Xi , x) + ?b, where u
?i = u???i , v?i = v???i , ?? = (?
the form of f(x) = i=1 (?
u?
0
0
(?
u
+?
v
)
y
(?
u
+?
v
)
K(?
u
??
v
)
+
where Kij = k(Xi , Xj ).
v
?)0 y + 2, and the intercept term ?b =
2??
4
2
Computational Results
We illustrate the difference between RC-SVR and -SVR on a toy linear problem3.
Figure 4 depicts the functions constructed by RC-SVR and -SVR for different
values of . For large , -SVR produces undesirable results. RC-SVR constructs the
same function for sufficiently large. Too small can result in poor generalization.
2.5
2.5
2
2
1.5
1.5
? = 0.75
1
0.5
0
? = 0.45
0.5
? = 0.25
0
(a)
? = 0.15
?0.5
?1
? = 0.75, 0.45, 0.25
1
(b)
?0.5
0
1
2
3
4
5
6
?1
0
1
2
3
4
5
6
Figure 4: Regression lines from (a) -SVR and (b) RC-SVR with distinct .
In Table 1, we compare RC-SVR, -SVR and ?-SVR on the Boston Housing problem.
Following the experimental design in [5] we used RBF kernel with 2? 2 = 3.9, C =
500?m for -SVR and ?-SVR, and = 3.0 for RC-SVR. RC-SVR, -SVR, and ?-SVR
are computationally similar for good parameter choices. In -SVR, is fixed. In
RC-SVR, is the maximum allowable tube width. Choosing is critical for -SVR
but less so for RC-SVR. Both RC-SVR and ?-SVR can shrink or grow the tube
according to desired robustness. But ?-SVR has no upper bound.
5
Conclusion and Discussion
By examining when -tubes exist, we showed that in the dual space SVR can be
regarded as a classification problem. Hard and soft -tubes are constructed by separating the convex or reduced convex hulls respectively of the training data with
the response variable shifted up and down by . We proposed RC-SVR based on
choosing the soft max-margin plane between the two shifted datasets. Like ?-SVM,
RC-SVR shrinks the -tube. The max-margin determines how much the tube can
shrink. Domain knowledge can be incorporated into the RC-SVR parameters
3
The data consist of (x, y): (0 0), (1 0.1), (2 0.7), (2.5 0.9), (3 1.1) and (5 2). The
CPLEX 6.6 optimization package was used.
Table 1: Testing Results for Boston Housing, MSE= average of mean squared errors of
25 testing points over 100 trials, STD: standard deviation
RC-SVR
-SVR
?-SVR
2?
MSE
STD
MSE
STD
?
MSE
STD
0.1
37.3
72.3
0
11.2
8.3
0.1
9.6
5.8
0.2
11.2
7.6
1
10.8
8.2
0.2
8.9
7.9
0.3
10.7
7.3
2
9.5
8.2
0.3
9.5
8.3
0.4
9.6
7.4
3
10.3
7.3
0.4
10.8
8.2
0.5
8.9
8.4
4
11.6
5.8
0.5
10.9
8.3
0.6
10.6
9.1
5
13.6
5.8
0.6
11.0
8.4
0.7
11.5
9.3
6
15.6
5.9
0.7
11.2
8.5
0.8
12.5
9.8
7
17.2
5.8
0.8
11.1
8.4
and ?. The parameter C in ?-SVM and -SVR has been eliminated. Computationally, no one method is superior for good parameter choices. RC-SVR alone
has a geometrically intuitive framework that allows users to easily grasp the model
and its parameters. Also, RC-SVR can be solved by fast nearest point algorithms.
Considering regression as a classification problem suggests other interesting SVR
formulations. We can show -SVR is equivalent to finding closest points in a reduced
convex hull problem for certain C, but the equivalent problem utilizes a different
metric in the objective function than RC-SVR. Perhaps other variations would yield
even better formulations.
Acknowledgments
Thanks to referees and Bernhard Sch?
olkopf for suggestions to improve this work.
This work was supported by NSF IRI-9702306, NSF IIS-9979860.
References
[1] K. Bennett and E. Bredensteiner. Duality and Geometry in SVM Classifiers. In
P. Langley, eds., Proc. of Seventeenth Intl. Conf. on Machine Learning, p 57?64,
Morgan Kaufmann, San Francisco, 2000.
[2] D. Crisp and C. Burges. A Geometric Interpretation of ?-SVM Classifiers. In
S. Solla, T. Leen, and K. Muller, eds., Advances in Neural Info. Proc. Sys., Vol
12. p 244?251, MIT Press, Cambridge, MA, 1999.
[3] S.S. Keerthi, S.K. Shevade, C. Bhattacharyya and K.R.K. Murthy, A Fast Iterative Nearest Point Algorithm for Support Vector Machine Classifier Design,
IEEE Transactions on Neural Networks, Vol. 11, pp.124-136, 2000.
[4] O. Mangasarian. Nonlinear Programming. SIAM, Philadelphia, 1994.
[5] B. Sch?
olkopf, P. Bartlett, A. Smola and R. Williamson. Shrinking the Tube:
A New Support Vector Regression Algorithm. In M. Kearns, S. Solla, and D.
Cohn eds., Advances in Neural Info. Proc. Sys., Vol 12, MIT Press, Cambridge,
MA, 1999.
[6] V. Vapnik. The Nature of Statistical Learning Theory. Wiley, New York, 1995.
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1,244 | 2,133 | Covariance Kernels from Bayesian
Generative Models
Matthias Seeger
Institute for Adaptive and Neural Computation
University of Edinburgh
5 Forrest Hill, Edinburgh EH1 2QL
[email protected]
Abstract
We propose the framework of mutual information kernels for
learning covariance kernels, as used in Support Vector machines
and Gaussian process classifiers, from unlabeled task data using
Bayesian techniques. We describe an implementation of this framework which uses variational Bayesian mixtures of factor analyzers
in order to attack classification problems in high-dimensional spaces
where labeled data is sparse, but unlabeled data is abundant.
1
Introduction
Kernel machines, such as Support Vector machines or Gaussian processes, are powerful and frequently used tools for solving statistical learning problems. They are
based on the use of a kernel function which encodes task prior knowledge in a
Bayesian manner. In this paper, we propose the framework of mutual information (MI) kernels for learning covariance kernels from unlabeled task data using
Bayesian techniques. This section introduces terms and concepts. We also discuss
some general ideas for discriminative semi-supervised learning and kernel design
in this context. In section 2, we define the general framework and give examples.
We note that the Fisher kernel [4] is a special case of a MI kernel. MI kernels for
mixture models are discussed in detail. In section 3, we describe an implementation
for a MI kernel for variational Bayesian mixtures of factor analyzers models and
show results of preliminary experiments.
the semi-supervised classification problem, a labeled dataset Dl
{(Xl,tl), ... ,(Xm ,tm)} as well as an unlabeled set Du = {x m+1 ,""Xm+n} are
In
given for training, both i.i.d. drawn from the same unknown distribution, but the
labels for Du cannot be observed. Here, Xi E I~.P and ti E {-1, +1}.1 Typically,
m = IDll is rather small, and n = IDul ?m. Our aim is to fit models to Du in a
Bayesian way, thereby extracting (posterior) information, then use this information
to build a covariance kernel K. Afterwards, K will be plugged into a supervised
kernel machine, which is trained on the labeled data Dl to perform the classification
task.
1
For simplicity, we only discuss binary labels here.
It is important to distinguish very clearly between these two learning scenarios.
For fitting D u , we use Bayesian density estimation. After having chosen a model
family {p(xIOn and a prior distribution P(O) over parameters 0, the posterior
distribution P(OIDu) ex P(DuIO)P(O), where P(DuIO) = rr::~'~l P(xiIO), encodes
all information that Du contains about the latent (i.e. unobserved) parameters 0. 2
The other learning scenario is supervised classification, using a kernel machine.
Such architectures model a smooth latent function y (x) E ~ as a random process,
together with a classification noise model P(tly).3 The covariance kernel K specifies
the prior distribution for this process: namely, a-priori, y(x) is assumed to be a
Gaussian process with zero mean and covariance function K , i.e. K(x(1) , X(2 )) =
E[y(x(1))Y(X(2))]; see e.g. [10] for details. In the following, we use the notation
a = (ai)i = (al' ... ,aI)' for vectors, and A = (ai ,j )i,j for matrices respectively. The
prime denotes transposition. diag a is the matrix with diagonal a and 0 elsewhere.
N(xlJ.t,~) denotes the Gaussian density with mean J.t and covariance matrix ~.
Within the standard discriminative Bayesian classification scenario, unlabeled data
cannot be used. However, it is rather straightforward to modify this scenario by
introducing the concept of conditional priors (see [6]). If we have a discriminant
model family {P(tlx; w
a conditional prior P(w 10) allows to encode prior knowledge and assumptions about how information about P(x) (i.e. about 0) influences
our assumptions about a-priori probabilities over discriminants w. For example,
the P(wIO) could be Occam priors, expressing the intuitive fact that for many
problems, the notion of "simplicity" of a discriminant function depends strongly on
what is known about the input distribution P(x). For a given problem, it is in general not easy to come up with a useful conditional prior. However, once such a prior
is specified, we can in principle use the same powerful techniques for approximate
Bayesian inference that have been developed for supervised discriminative settings.
Semi-supervised techniques that can be seen as employing conditional priors include
co-training [1], feature selection based on clustering [7] and the Fisher kernel [4]. For
a probabilistic kernel technique, P( w 10) is fully specified by a covariance function
K(x(1) , X(2) 10) depending on O. The problem is therefore to find covariance kernels
which (as GP priors) favour discriminants in some sense compatible with what we
have learned about the input distribution P(x).
n,
Kernel techniques can be seen as nonparametric smoothers, based on the (prior)
assumption that if two input points are "similar" (e.g. "close" under some distance),
their labels (and latent outputs y) should be highly correlated. Thus, one generic
way of learning kernels from unlabeled data is to learn a distance between input
points from the information about P( x). A frequently used assumption about
how classification labels may depend on P(x) is the cluster hypothesis: we assume
discriminants whose decision boundaries lie between clusters in P(x) to be a-priori
more likely than such that label clusters inconsistently. A general way of encoding
this hypothesis is to learn a distance from P(x) which is consistent with clusters in
P(x) , i.e. points within the same cluster are closer under this distance than points
from different clusters. We can then try to embed the learned distance d(x(1), X(2))
approximately in an Euclidean space, i.e. learn a mapping ? : X r-+ ?( x) E ~l
such that d(x(1) , X(2)) :=;::j 11 ?(x(1)) - ?(X(2)) II for all pairs from Du. Then, a natural
kernel function would be K(x(1) , X(2)) = exp( - ,BII?(x(1)) - ?(x(2))11 2). In this
paper, however, we follow a simpler approach, by considering a similarity measure
2In practice, computation of P(OIDu) is hardly ever feasible , but powerful approximation techniques can be used.
3 A natural choice for binary classification is to represent the log odds log(P(t =
+1Ix)/P(t = -1Ix)) by y(x) .
which immediately gives rise to a covariance kernel, without having to compute an
approximate Euclidean embedding.
Remark: Our main aim in this paper is to construct kernels that can be learned
from unlabeled data only. In contrast to this, the task of learning a kernel from
labeled data is somewhat simpler and can be approached in the following generic
way: start with a parametric model family {y(x; w)} , with the interpretation that
y(x;w) models the log odds log(P(t = +llx)/P(t = -llx)). Fitting these models
to labeled data D[ , we obtain a posterior P(wIDI) . Now , a natural covariance
kernel for our problem is simply K(x(1),X(2)) = Jy(x(1);w)y(x(2 );w)Q(w)dw,
where (say) Q(w) <X P(wID[)AP(W)l - A (or an approximation thereof). For A = 0,
we obtain the prior covariance kernel for our model, while for larger A the kernel
incorporates more and more posterior information. The kernel proposed in [8] can
be seen as approximation to this approach.
2
Mutual Information Kernels
In this section, we begin by introducing the framework of mutual information kernels. Given a mediator distribution Pm e d (()) over parameters (), we define the joint
distribution Q(x(1) , X(2)) mediated by Pm e d (()) as
Q(x(1) , X(2))
=
J
Pmed (())P(x(1)I())P(x(2)1())d().
(1)
The sample mutual information between x(1) and X(2) under this distribution is
(1)
I(x
(2)
,x
_
Q(X(l) , X(2))
) - log Q(x(1))Q(X(2)) '
(2)
where Q(x) = JQ(x , x)dx. I(x(1) , x(2)) is called the mutual information (MI)
score. In a very concrete sense, it measures the similarity between x(1 ) and X(2) with
respect to the generative process represented by the mediator distribution Pm e d (()):
it is the amount of information they share via the mediator variable () ~ Pm e d (()) .
Note that Q(x(1), X(2)) can be seen as inner product in a space of functions () f-t R,
the features of X(k) being (P(x(k)I()))o, weighted by the distribution Pm e d . 4 X(k) is
represented by its likelihood under all possible models.
Covariance kernels have to satisfy the property of positive definiteness 5 , and the MI
score I does not. However, applying a standard transformation (called exponential
embedding (EE) here), we arrive at
K(x(1) X(2)) =
,
e - (I(x (l) ,x(1))+I(x(2),x(2))) /2+I(x(1),x(2))
=
Q(x(1), X(2))
vQ(x(1) , x(1))Q(X(2), X(2))
(3)
EE becomes familiar if we note that it transforms the standard inner product
x(1)' X(2) into the well-known Radial Basis Function (RBF) kernel 6
(4)
4When comparing X ( l) , X ( 2) via the inner product, we are not interested in correlating
their features uniformly, but rather focus on regions of high volume under Pm e d .
5 K is positive definite if the matrix (K(X(k ll , X(k2?)hl ,k2 is positive definite for every
set {x(1 ), ... , X (K ) } of distinct points.
60ne can show that if j is itself a kernel , and j -+ I< under EE, then 1<(3 is also a
kernel for all (3 > 0 (see e.g. [3]) .
or the weighted inner product x(1)'VX(2) into the squared-exponential kernel (e.g.
[10]). It is easy to show that K in (3) is a valid covariance kernel 7 , and we refer to
it as mutual information (MI) kernel.
Example: Let P(xIO) = N(xIO, (p/2)I) (spherical Gaussian with mean 0),
Prned(O) = N(OIO, aI). Then, the MI kernel K is the RBF kernel (4) with
(3 = 4/(p(4 + pia)). Thus, the RBF kernel is a special case of a MI kernel.
2.1
Mediator distribution. Model-trust scaling.
The mediator distribution Prned(O), motivated earlier in this section, should ideally
encode information about the x generation process, just as the Bayesian posterior
P(OIDu). On the other hand , we need to be able to control the influence that
information from sources such as unlabeled data Du can have on the kernel (relying
too much on such sources results in lack of robustness, see e.g. [6] for details). Here,
we propose model-trust scaling (MTS) , by setting
(5)
Prned varies with A from the (usually vague) prior P(O) (A = 0) towards the sharp
posterior P(OID u) (A = n), rendering the Du information (via the model) more
and more influence upon the kernel K. The concrete effect of MTS on the kernel
depends on the model family.
Example (continued): Again, P(xIO) = N(xIO , (p/2)I) , with a flat prior P(O) = 1
on the mean. Then, P(OIDu) = N(Olx , (p/2n)I), where x = n - 1 L:;~;>~l Xi, and
Prned(O) = N(Olx, (p/2A)I) (after (5)). Thus, the MI kernel is again the RBF kernel
(4) with (3 = 2/(p(2 + A)) . For the more flexible model P(xIO) = N(xIJL , ~),
=
(JL,~) and the conjugate Jeffreys prior, the MI kernel is computed in [5].
?
If the Bayesian analysis is done with conjugate prior-model pairs, the corresponding
MI kernel can be computed easily, and for many of these cases, MTS has a very
simple, analytic form (see [5]). In general, approximation techniques developed for
Bayesian analysis have to be applied. For example, applying the Laplace approximation to the computations on a model with flat prior P(O) = 1 results in the Fisher
kernel [4]8, see e.g. [5]. However, in this paper we favour another approximation
technique (see section 3).
2.2
Mutual Information Kernels for Mixture Models
If we apply the MI kernel framework to mixture models P(x 10, 7T") = Ls 7f sP(x lOs),
we run into a problem. As mentioned in section 1, we would like our kernel at least
partly to encode the cluster hypothesis, i.e. K(x(1), X(2)) should be small if x(1), X(2)
come from different clusters in P(x ),9 but the opposite is true (for not too small
7 Q(x(1 ), X ( 2)) is an inner product (therefore a kernel), for the rest of the argument see
[3], section 5.
8This was essentially observed by the authors of [4] on workshop talks, but has not
been published to our knowledge. The fascinating idea of the Fisher kernel has indeed
been the main motivation and inspiration for this paper.
9This does not mean that we (a-priori) believe they should have different labels, but
only that the label (or better: the latent yO) at one of them should not depend strongly
on yO at the other.
A). To overcome this problem, we generalize Q(x(1), X(2)):
S
Q(X(1),X(2)) =
L
WS1 ,S2
J
P(x(1) IOsJP(X(2) IOs2)Prn ed(O) dO,
(6)
8} , 82=1
where W = (W S1 ,S2)Sl ,S2 is symmetric with nonnegative entries and positive elements
on the diagonal. The MI kernel K is defined as before by (3) , based on the new
Q. If Prned(O,rr) = ITsPrn ed(Os)Prn ed(rr) (which is true for the cases we will be
interested in), we see that the original MI kernel arises as special case WS1,S2 =
EPm ed[7fS17fs2]' Now, by choosing W = diag(Epm ed[7fs])s, we arrive at a MI kernel
K which (typically) behaves as expected w.r.t. cluster separation (see figure 1), but
does not exhibit long-range correlations between joined components. In the present
work, we restrict ourselves to this diagonal mixture kernel. Note that this kernel
can be seen as (normalized) mixture of MI kernels over the component models.
Figure 1: Kernel contours on 2-cluster dataset (A = 5,100,30)
Figure 1 shows contour plots 10 of the diagonal mixture kernel for VB-MoFA (see
section 3), learned on a 500 cases dataset sampled from two Gaussians with equal
covariance (see subsection 3.1). We plot K(a , x) for fixed a (marked by a cross)
against all x , the height between contour lines is 0.1. The left and middle plot
have the lower cluster's centre as a, with A = 5, A = 100 respectively, the right
plot's a lies between the cluster centres, A = 30. The effect of MTS can be seen by
comparing left and middle, note the different sharpness of the slopes towards the
other cluster and the different sizes and shapes of the "high correlation" regions. As
seen on the right, points between clusters have highest correlation with other such
inter-cluster points, a feature that may be very useful for successful discrimination.
3
Experiments with Mixtures of Factor Analyzers
In this section, we describe an implementation of a MI kernel , using variational
Bayesian mixtures of factor analyzers (VB-MoFA) [2] as density models. These are
able to combine local dimensionality reduction (using noisy linear transformations
u -+ x from low-dimensional latent spaces) with good global data fit using mixtures.
VB-MoFA is a variational approximation to Bayesian analysis on these models, able
to deliver the posterior approximations we require for an MI kernel.
We employ the diagonal mixture kernel (see subsection 2.2). Instead of implementing MTS analytically, we compute the VB approximation to the true posterior (i.e.
A = n), then simply apply the scaling to this distribution. Prned(0, rr) factorizes as
required in subsection 2.2. The integrals P(x(1) IOs)p(X(2) IOs)Prn ed(Os) dOs in (6)
J
lOProduced using the first-order approximation (see 3) to the MI kernel. Plots using the
one-step variational approximation (see 3) have a somewhat richer structure.
are not analytically tractable. Our first idea was to approximate them by applying
the VB technique once more, ending up with what we called one-step variational
approximations. Unfortunately, the MI kernel approximation based on these terms
cannot be shown to be positive definite anymore l l ! Thus, in the moment we use a
less elegant and, we feel , less accurate approximation (details can be found in [5])
based on first-order Taylor expansions.
In the remainder of this section we compare the VB-MoFA kernel with the RBF
kernel (4) on two datasets, using a Laplace GP classifier (see [10]). In each case
we sample a training pool, a kernel dataset Du and a test set (mutually exclusive).
The VB-MoFA diagonal mixture kernel is learned on Du. For a given training set
size m, a run consists of sampling a training set Dl and a holdout set Dh (both of
size m) from the training pool, tuning kernel parameters by validation on D h , then
testing on the test set. We use the same Dl, Du for both kernels. For each training
set size, we do L = 30 runs. Results are presented by plotting means and 95% t-test
confidence intervals of test errors over runs.
3.1
Two Gaussian clusters
The dataset is sampled from two 2-d Gaussians with same non-spherical covariance
(see figure 1) , one for each class (the Bayes error is 2.64%) . We use n = 500 points
for D u , a training pool of 100 and a test set of 500 points. The learning curves in
figure 2 show that on this simple toy problem, on which the fitted VB-MoFA model
represents the cluster structure in P(x) almost perfectly, the VB-MoFA MI kernel
outperforms the RBF kernel for samples sizes n :::; 40.
,
_ 0 .225
~
~
02
~
0.15
02
~O. 175
~ 0.175
~
IL
0.15
~O.125
rI
~~~~~---7---~
~~~~~~--~
Training
set size n
I
I
',~~~~---7---.~
, --~--~--~~
Trair>ngsets;zen
Figure 2: Learning curves on 2-cluster dataset. Left: RBF kernel; right: MI kernel
3.2
Handwritten Digits (MNIST): Twos against threes
We report results of preliminary experiments using the subset of twos and threes
of the MNIST Handwritten Digits database 12 . Here, n = IDul = 2000, the training
pool contains 8089, the test set 2000 cases. We employ a VB-MoFA model with 20
components, fitted to Du. We use a very simple baseline (BL) algorithm (see [6],
section 2.3) based on the component densities from the VB-MoFA model13 , which
llThanks to an anonymous reviewer for pointing out this flaw.
12The 28 x 28 images were downsampled to size 8 x 8.
13The estimates P(xls) are obtained by integrating out the parameters (}s using the
variational posterior approximation. The integral is not analytic, and we use a one-step
variational approximation to it .
allows us to assess the "purity" of the component clusters w.r.t. the labels 1 \ this
algorithm is the only one not based on a kernel. Furthermore, we show results for
the one-step variational approximation to the MI kernel 15 (MIOLD ). The learning
curves are shown in figure 3.
to.>
...
1
~
r
II
.....
, II
~ 0.'
?
t? .
! ??
T.-." ,",_ ,
,
II I
j"
i
I ..
!
,,_
...
_,
=?
II I I
T,_ ... _,
j"
iI ..
!
[
Figure 3: Learning curves on MNIST twos/threes. Upper left: RBF kernel; upper
middle: Baseline method; upper right: VB-MoFA MI kernel (first-order approx.) ;
lower left: VB-MoFA MI "kernel" (one-step var. approx.)
The results are disappointing. The fact that the first-order approximation to the
MI kernel performs worse than the one-step variational approximation (although
the latter may fail to be positive definite) , indicates that the former is a poorer
approximation. The latter renders results close to the baseline method, while the
smoothing RBF kernel makes much better use of a growing number of labeled examples 16 This indicates that the conditional prior, as represented by the VB-MoFA
MI kernel, behaves nonsmooth and overrides label information in regions where it
should not. We suspect this problem to be related to the high dimensionality of
the input space, in which case probability densities tend to have a large dynamic
range, and mixture component responsibility estimates tend to behave very nonsmooth. Thus, it seems to be necessary to extend the basic MI kernel framework
by new scaling mechanisms in order to produce a smoother encoding of the prior
assumptions.
14The baseline algorithm is based on the assumption that, given the component index
s, the input point x and the label t are independent. Only the conditional probabilities
P(t ls) are learned, while P(xls) and pes) is obtained from the VB-MoFA model fitted to
unlabeled data only. Thus, success/failure of this method should be closely related to the
degree of purity of the component clusters w.r.t . the labels.
15This is somewhat inconsistent, since we use a kernel function which might not be
positive definite in a context (GP classification) which requires a covariance function.
16Note also that RBF kernel matrices can be evaluated significantly faster than such
using the VB-MoFA MI kernel.
4
Related work. Discussion
The present work is probably most closely related to the Fisher kernel (see subsection 2.1). The arguments concerning mixture models (see subsection 2.2) apply
there as well. Haussler [3] contains a wealth of material about kernel design for discrete objects x. Watkins [9] mentions that expressions like Q in (1) are valid kernels
for discrete x and countable parameter spaces. Very recently we came across [11],
which essentially describes a special case of the diagonal mixture kernel (see subsection 2.2) for Gaussian components with diagonal covariances 17 . The author calls
Q a stochastic equivalence predicate. He is interested in distance learning, does not
apply his method to kernel machines and does not give a Bayesian interpretation.
We have presented a general framework for kernel learning from unlabeled data and
described an approximate implementation using VB-MoFA models. A straightforward application of this technique to high-dimensional real-world data did not prove
successful, and in future work we will explore new ideas for extending the basic MI
kernel framework in order to be able to deal with high-dimensional input spaces.
Acknowledgments
We thank Chris Williams for many inspiring discussions , furthermore Ralf Herbrich, Amos Storkey, Hugo Zaragoza and Neil Lawrence. Matt Beal helped us a
lot with VB-MoFA. The author gratefully acknowledges support through a research
studentship from Microsoft Research Ltd.
References
[1] Avrim Blum and Tom Mitchell. Combining labeled and unlabeled data with CoTraining. In Proceedings of COLT, 1998.
[2] Z. Ghahramani and M. Beal. Variational inference for Bayesian mixtures of factor
analysers. In Advances in NIPS 12. MIT Press, 1999.
[3] David Haussler. Convolution kernels on discrete structures. Technical Report UCSCCRL-99-10 , University of California, Santa Cruz, July 1999.
[4] Tommi S. Jaakkola and David Haussler. Exploiting generative models in discriminative classifiers. In Advances in N eural Information Processing Systems 11, 1998.
[5] Matthias Seeger. Covariance kernels from Bayesian generative models. Technical
report , 2000. Available at http : //yyy . dai . ed. ac . ukr seeger /papers . html.
[6] Matthias Seeger. Learning with labeled and unlabeled data. Technical report, 2000.
Available at http://yyy .dai. ed. ac. ukrseeger/papers .html.
[7] Martin Szummer and Tommi Jaakkola. Partially labeled classification with Markov
random walks. In Advances in NIPS
14. MIT Press, 200l.
[8] Koji Tsuda, Motoaki Kawanabe, Gunnar Ratsch, Soeren Sonnenburg, and KlausRobert Muller. A new discriminative kernel from probabilistic models . In Advances
in NIPS 14. MIT Press, 200l.
[9] Chris Watkins. Dynamic alignment kernels. Technical Report CSD-TR-98-11 , Royal
Holloway, University of London, 1999.
[10] Christopher K.1. Williams and David Barber. Bayesian classification with Gaussian
processes. IEEE Trans. PAMI, 20(12):1342- 1351, 1998.
[11] Peter Yianilos. Metric learning via normal mixtures. Technical report , NEC Research ,
Princeton, 1995.
17The a parameter in this work is related to MTS in this case.
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1,245 | 2,134 | Model-Free Least Squares Policy Iteration
Michail G. Lagoudakis
Department of Computer Science
Duke University
Durham, NC 27708
[email protected]
Ronald Parr
Department of Computer Science
Duke University
Durham, NC 27708
[email protected]
Abstract
We propose a new approach to reinforcement learning which combines
least squares function approximation with policy iteration. Our method
is model-free and completely off policy. We are motivated by the least
squares temporal difference learning algorithm (LSTD), which is known
for its efficient use of sample experiences compared to pure temporal
difference algorithms. LSTD is ideal for prediction problems, however it
heretofore has not had a straightforward application to control problems.
Moreover, approximations learned by LSTD are strongly influenced by
the visitation distribution over states. Our new algorithm, Least Squares
Policy Iteration (LSPI) addresses these issues. The result is an off-policy
method which can use (or reuse) data collected from any source. We have
tested LSPI on several problems, including a bicycle simulator in which
it learns to guide the bicycle to a goal efficiently by merely observing a
relatively small number of completely random trials.
1 Introduction
Linear least squares function approximators offer many advantages in the context of reinforcement learning. While their ability to generalize is less powerful than black box
methods such as neural networks, they have their virtues: They are easy to implement and
use, and their behavior is fairly transparent, both from an analysis standpoint and from
a debugging and feature engineering standpoint. When linear methods fail, it is usually
relatively easy to get some insight into why the failure has occurred.
Our enthusiasm for this approach is inspired by the least squares temporal difference learning algorithm (LSTD) [4]. LSTD makes efficient use of data and converges faster than
other conventional temporal difference learning methods. Although it is initially appealing
to attempt to use LSTD in the evaluation step of a policy iteration algorithm, this combination can be problematic. Koller and Parr [5] present an example where the combination of
LSTD style function approximation and policy iteration oscillates between two bad policies
in an MDP with just 4 states. This behavior is explained by the fact that linear approximation methods such as LSTD compute an approximation that is weighted by the state
visitation frequencies of the policy under evaluation. Further, even if this problem is overcome, a more serious difficulty is that the state value function that LSTD learns is of no use
for policy improvement when a model of the process is not available.
This paper introduces the Least Squares Policy Iteration (LSPI) algorithm, which extends
the benefits of LSTD to control problems. First, we introduce LSQ, an algorithm that
learns least squares approximations of the state-action ( ) value function, thus permitting
action selection and policy improvement without a model. Next we introduce LSPI which
uses the results of LSQ to form an approximate policy iteration algorithm. This algorithm
combines the policy search efficiency of policy iteration with the data efficiency of LSTD.
It is completely off policy and can, in principle, use data collected from any reasonable
sampling distribution. We have evaluated this method on several problems, including a
simulated bicycle control problem in which LSPI learns to guide the bicycle to the goal by
observing a relatively small number of completely random trials.
2 Markov Decision Processes
We assume that the underlying control problem is a Markov Decision Process (MDP). An
MDP is defined as a 4-tuple
where: is a finite set of states; is a finite set of
actions; is a Markovian transition model where
represents the probability of
going from state to state with action ; and is a reward function IR,
such that !
represents the reward obtained when taking action in state and
ending up in state " .
We will be assuming that the MDP has an infinite horizon and that future rewards are
discounted exponentially with a discount factor #%$'& ()+*"
. (If we assume that all policies
are proper, our results generalize to the undiscounted case.) A stationary policy , for an
MDP is a mapping ,-./43
10 , where ,2!"
is the action the agent takes at state . The
state-action value function
5
is defined over all possible combinations of states and
actions and indicates the expected, discounted total reward when taking action in state
and following policy , thereafter. The exact
-values for all state-action pairs can be
found by solving the linear system of the Bellman equations :
3
!6
7
28'9:!6
7
<;=#>@?BA !
3
! ,2
C
? A
M3
where 9:3R37
D8FE S3 GH
I J!6
KL
. In matrix format, the
system becomes
8
3
9N;O#QP
9
, where
and
are vectors of size T DTUT 0MT and P
is a stochastic matrix of
3
describes the transitions from pairs 7
to pairs !@,2!H
@
.
size T DTLT 0 T7OT DTLT 0MT
. P
For every MDP, there exists an optimal policy, ,V , which maximizes the expected, discounted return of every state. Policy iteration is a method of discovering this policy by
iterating through a sequence of monotonically improving policies. Each iteration consists of two phases. Value determination computes the state-action values for a policy ,.WYX[Z by solving the above system. Policy improvement defines the next policy as
3fhgLi
,.WYX[\^]@ZC"
J8`_a
bc_d)e
7
. These steps are repeated until convergence to an optimal policy, often in a surprisingly small number of steps.
3 Least Squares Approximation of Q Functions
Policy iteration relies upon the solution of a system of linear equations to find the Q values
for the current policy. This43
is impractical for large state and action spaces. In such cases we
may wish to approximate
with a parametric function approximator and do some form
of approximate policy iteration. We now address the problem of finding a set of parameters
that maximizes the accuracy of our approximator. A common class of approximators is
the so called linear architectures, where the value function is approximated as a linear
weighted combination of
basis functions (features):
3
!
>
8
!6
7
8
!7
S
]
where is a set of weights (parameters). In general,
-T DTLT 0 T and so, the linear system
above now becomes an overconstrained system over the parameters :
# P
3
9F;O#QP
9
3
3
where is a
T DTLT 0MT!)
matrix. We are interested in a set3 of weights
that yields a fixed
3
point in value function space, that is a value function
that is invariant under
8
one step of value determination followed by orthogonal
projection to the space spanned by
the basis functions. Assuming that the columns of are linearly independent this is
]
3
9N;O#QP
3
3
8
8
3
8
3
# P
]
9
We note that this is the standard fixed point approximation method for linear value functions
with the exception3 that the problem is formulated in terms of Q values instead of state
values. For any P , the solution is guaranteed to exist for all but finitely many # [5].
4 LSQ: Learning the State-Action Value Function
3
In the previous section we assumed that a model M
P
of the underlying MDP is available. In many practical applications, such a model is not available and the value function
or, more precisely, its parameters have to be learned from sampled data. These sampled
data are tuples of the form:
L
, meaning that in state , action was taken, a reward was received, and the resulting state was . These data can be collected from actual
(sequential) episodes or from random queries to a generative model of the MDP. In the
extreme case, they can be experiences of other agents on the same MDP. We
know that
3
8! , where
the desired
set
of
weights
can
be
found
as
the
solution
of
the
system,
3
9 .
8
# P
and
8
3
The matrix P
and the vector 9 are unknown and so, and cannot be
a
3
determined
priori. However, and can be approximated using samples. Recall that , P
, and 9
are of the form:
FHG A:I
(*),+.-0/213- 4 5
%&
&
&
%&
A
&
A
6 606
0
(*),+./7184 5
"$#
'
),+.-0/ 1J-0/ +0KL42(?),+0KM/ N),+K4 4 5
@ A
&
B
60606
(*),+:9 ;<9=/ 139 >?9=4 5
CED
G ATI
F
%&
&
'
&
'
F
B
@ A
A
A
606O6
),+:/71J/+ K 42UV),+./ 1J/ + K 4
B
606O6
),+T9 ;J9=/ 139 >?9W/+0KL42UV),+:9 ;<9X/ 139 >?9Y/+0K4
G A I
A
6O606
),+ 9 ;<9 /21 9 >*9 /+ K 42(?),+ K / NQ),+ K 4 4 5
),+ - /21 - /+ K 42UV),+ - /21 - / + K 4
FHG A I
RS#
FPG A I
A
6O606
),+./21J/+ K 42(?),+ K / NQ),+ K 4 4 5
FPG A I
"$#
@ A
Given a set of samples, Z 8\["^]0_GJ]0_
K]0_ :]0_@
+T<`M8 *6OaRbcb dfe , where the :]0_G
J]0_I
are sampled from according to distribution g and the 6] _ are sampled according to
3
K]0_ T :]0_GJ]0_I
, we can construct approximate versions of
,P
, and 9 as follows :
hi
i
8
i
j
bc
!:]m
<]m
i
o
p
P
q
3
8
i
j
nbo
!K]lk @,sr K
l
] k
t
i
o
bc
] _C
] _I
hi
nbo
]lk
]lk
!H]m @,
] _
t
r K
cb
r K
] m
t
i
o
o
cb
K] _ @,
hi
o
p
i
9
8
j
] k
l
bb
] _
bb
:]m
nbo
o
p
o
These approximations can be thought of as first sampling rows from
according to g
and then, conditioned on 3 these
samples,
as
sampling
terms
from
the
summations
in the
corresponding rows of P
and 9 . The sampling distribution from the summations is
governed by the underlying dynamics (
U
) of the process as the samples in Z are
taken directly from the MDP.
Given
,P
q
3
, and 9 ,
and can be approximated as
8
3
# P
q
and
9
8
With d uniformly
distributed samples over pairs of states and actions
mations and are consistent approximations of the true and :
d
8
T
TUT
T
and
The Markov property ensures that
the solution
3
sufficiently large d whenever
exists:
3
] C
8
8
d
T DTUT
T
3
!7
, the approxi-
d
28
T
TUT
T
will converge to the true solution
]
d
T DTUT
T
]
28
3
for
3
8$
In the more general case, where g is not uniform, we will compute a weighted projection,
which minimizes the g weighted distance in the projection step. Thus, state is implicitly
assigned weight g!K
and the projection minimizes the weighted sum of squared
errors
3
with respect to g . In LSTD, for example, g is the stationary distribution of P , giving high
weight to frequently visited states, and low weight to infrequently visited states.
As with LSTD, it is easy to see that approximations ( 6 ) derived from different
]
]
sets of samples (Z Z ) can be combined additively to yield a better approximation that
]
corresponds to the combined set of samples:
8
;
]
and
8
]
;
This observation
leads to an incremental update rule for and . Assume that initially 8
( and
8
( . For a fixed policy, a new sample !6
contributes to the approximation
according to the following update equation :
;
5
+
5
#
! ,2
@
and
.;
7
We call this new algorithm LSQ due to its similarity to LSTD. However, unlike LSTD, it
computes Q functions and does not expect the data to come from any particular Markov
chain. It is a feature of this algorithm that it can use the same set of samples to compute Q
values for any policy representation that offers an action choice for
each in the set. The
policy merely determines which !,2K!@,2!H
@
is added to for each sample. Thus,
LSQ can use every single sample available to it no matter what policy is under evaluation.
We note that if a particular set of projection
weights are desired, it is straightforward to
reweight the samples as they are added to .
Notice that apart from storing the samples, LSQ requires only J7
space independently
of the size of the state and the action
space. For each sample in Z , LSQ incurs a cost of
and and a one time cost of J7 "
to solve the system and
J
to update the matrices
find
the weights. Singular value decomposition (SVD) can be used for robust inversion of
as it is not always a full rank matrix.
LSQ includes LSTD as a special case where there is only one action available. It is also
possible to extend LSQ to LSQ( ) in a way that closely resembles LSTD( ) [3], but in
that case the sample set must consist of complete episodes generated using the policy under evaluation, which again raises the question of bias due to sampling distribution, and
prevents the reusability of samples. LSQ is also applicable in the case of infinite and continuous state and/or action spaces with no modification. States and actions are reflected
only through the basis functions of the linear approximation and the resulting value function can cover the entire state-action space with the appropriate set of continuous basis
functions.
5 LSPI: Least Squares Policy Iteration
The LSQ
algorithm provides a means of learning an approximate state-action value funcS3
tion,
!6
7
, for any fixed policy , . We now integrate LSQ into an approximate policy
iteration algorithm. Clearly, LSQ is a candidate for the value determination step. The key
insight is that we can achieve the policy improvement step without ever explicitly representing our policy and without any sort
of model. Recall that in policy improvement, , WYX[\^]@Z
43
!7
. Since LSQ computes Q functions directly,
will pick the action that maximizes
we do not need a model to determine our improved policy; all the information we need is
contained implicitly in the weights parameterizing our Q functions 1:
, WUX[\^]IZ !
8
_a
bc_d
e
7
8
_a
b2c_d
e
!7
We close the loop simply by requiring that LSQ performs this maximization for each
when constructing the matrix for a policy. For very large or continuous action spaces,
explicit maximization over may be impractical. In such cases, some sort of global nonlinear optimization may be required to determine the optimal action.
Since LSPI uses LSQ to compute approximate Q functions, it can use any data source for
samples. A single set of samples may be used for the entire optimization, or additional samples may be acquired, either through trajectories or some other scheme, for each iteration
of policy iteration. We summarize the LSPI algorithm in Figure 1. As with any approximate policy iteration algorithm, the convergence of LSPI is not guaranteed. Approximate
policy iteration variants are typically analyzed in terms of a value function approximation
error and an action selection error [2]. LSPI does not require an approximate policy representation, e.g., a policy function or ?actor? architecture, removing one source of error.
Moreover, the direct computation of linear Q functions from any data source, including
stored data, allows the use of all available data to evaluate every policy, making the problem of minimizing value function approximation error more manageable.
6 Results
We initially tested LSPI on variants of the problematic MDP from Koller and Parr [5],
essentially simple chains of varying length. LSPI easily found the optimal policy within
a few iterations using actual trajectories. We also tested LSPI on the inverted pendulum
problem, where the task is to balance a pendulum in the upright position by moving the cart
to which it is attached. Using a simple set of basis functions and samples collected from
random episodes (starting in the upright position and following a purely random policy),
LSPI was able to find excellent policies using a few hundred such episodes [7].
Finally, we tried a bicycle balancing problem [12] in which the goal is to learn to balance
and ride a bicycle to a target position located 1 km away from the starting location. Initially,
the bicycle?s orientation is at an angle of 90 to the goal. The state description is a six
dimensional vector D
, where is the angle of the handlebar, is the vertical
1
This is the same principle that allows action selection without a model in Q-learning. To our
knowledge, this is the first application of this principle in an approximate policy iteration algorithm.
LSPI (
/( / //*N./
)
// ( : Number of basis functions
// : Basis functions
// : Discount factor
// N : Stopping criterion
N # N),+./ 4
(default:
// : Initial policy, given as ,
//
: Initial set of samples, possibly empty
#
N K #
N
repeat
Update
N # N K
/
/( / /*N
// Add/remove
samples, or leave unchanged
K
#
// K
/ /( /
/
// = LSQ ( K
)
// that is, ( )
)
)
until (
)
(optional)
N K
(
N=
PLSQ
N K
return
K #
// In essence,
#
N
// return
Figure 1: The LSPI algorithm.
angle of the bicycle, and is the angle of the
bicycle to the goal. The actions are the torque
applied to the handlebar (discretized to [ a7
(RC; a<e ) and the displacement of the rider
(discretized to [ (3 ( aR()G;(3 ( aJe ). In our experiments, actions are restricted to be either
or (or nothing) giving a total of 5 actions2 . The noise in the system is a uniformly
distributed term in & ( (8a7G;( (8a added to the displacement component of the action. The
dynamics of the bicycle are based on the model described by Randl?v and Alstr?m [12]
and the time step of the simulation is set to (3 (R* seconds.
The state-action value function !7
for a fixed action
combination of 20 basis functions:
<* D
D
is approximated by a linear
C
8 ,
8
where
for ( and
,
for ( . Note that the state variable is
completely ignored. This block of basis functions is repeated for each of the 5 actions, giving a total of 100 basis functions and weights. Training data were collected by initializing
the bicycle to a random state around the equilibrium position and running small episodes
of 20 steps each using a purely random policy. LSPI was applied on training sets of different sizes and the average performance is shown in Figure 2(a). We used the same data
set for each run of policy iteration and usually obtained convergence in 6 or 7 iterations.
Successful policies usually reached the goal in approximately 1 km total, near optimal performance. We also show an annotated set of trajectories to demonstrate the performance
improvement over multiple steps of policy iteration in Figure 2(b).
The following design decisions influenced the performance of LSPI on this problem: As is
typical with this problem, we used a shaping reward [10] for the distance to the goal. In this
case, we used (3 (R* of the net change (in meters) in the distance to the goal. We found that
when using full random trajectories, most of our sample points were not very useful; they
occurred after the bicycle had already entered into a ?death spiral? from which recovery
was impossible. This complicated our learning efforts by biasing the samples towards
hopeless parts of the space, so we decided to cut off trajectories after 20 steps. This created
an additional problem because there was no terminating reward signal to indicate failure.
We approximated this with an additional shaping reward, which was proportional to the
2
Results are similar for the full 9-action case, but required more training data.
100
200
6th iteration
(crash)
Starting
Position
90
80
0
Percentage of trials reaching the goal
3rd iteration
70
2nd iteration (crash)
Goal
60
?200
5th and 7th
iteration
50
40
4th and 8th
iteration
?400
30
20
?600
10
0
1st iteration
0
500
1000
1500
2000
Number of training episodes
2500
3000
?800
?200
0
200
400
600
800
1000
1200
(a)
(b)
Figure 2: The bicycle problem: (a) Percentage of final policies that reach the goal, averaged
over 200 runs of LSPI for each training set size; (b) A sample run of LSPI based on 2500
training trials. This run converged in 8 iterations. Note that iterations 5 and 7 had different
Q-values but very similar policies. This was true of iterations
4
and
8 as well. The weights
of the ninth differed from the eighth by less than *H( ] in
, indicate convergence.
The curves at the end of the trajectories indicating where the bicycle has looped back for a
second pass through the goal.
net change in the square of the vertical angle. This roughly approximated the likeliness
of falling at the end of a truncated trajectory. Finally, we used a discount of ( ( , which
seemed to yield more robust performance.
We admit to some slight unease about the amount of shaping and adjusting of parameters
that was required to obtain good results on this problem. To verify that we had not eliminated the learning problem entirely through shaping, we reran some experiments using a
discount of ( . In this case LSQ simply projects the immediate reward function into the
column space of the basis functions. If the problem were tweaked too much, acting to
maximize the projected immediate reward would be sufficient to obtain good performance.
On the contrary, these runs always produced immediate crashes in trials.
7 Discussion and Conclusions
We have presented a new, model-free approximate policy iteration algorithm called LSPI,
which is inspired by the LSTD algorithm. This algorithm is able to use either a stored
repository of samples or samples generated dynamically from trajectories. It performs
action selection and approximate policy iteration entirely in value function space, without
any need for model. In contrast to other approaches to approximate policy iteration, it does
not require any sort of approximate policy function.
In comparison to the memory based approach of Ormoneit and Sen [11], our method makes
stronger use of function approximation. Rather than using our samples to implicitly construct an approximate model using kernels, we operate entirely in value function space and
use our samples directly in the value function projection step. As noted by Boyan [3] the
matrix used by LSTD and LSPI can be viewed as an approximate, compressed model.
This is most compelling if the columns of are orthonormal. While this provides some
intuitions, a proper transition function cannot be reconstructed directly from , making a
possible interpretation of LSPI as a model based method an area for future research.
In comparison to direct policy search methods [9, 8, 1, 13, 6], we offer the strength of
policy iteration. Policy search methods typically make a large number of relatively small
steps of gradient-based policy updates to a parameterized policy function. Our use of policy
iteration generally results in a small number of very large steps directly in policy space.
Our experimental results demonstrate the potential of our method. We achieved good performance on the bicycle task using a very small number of randomly generated samples
that were reused across multiple steps of policy iteration. Achieving this level of performance with just a linear value function architecture did require some tweaking, but the
transparency of the linear architecture made the relevant issues much more salient than
would be the case with any ?black box? approach. We believe that the direct approach to
function approximation and data reuse taken by LSPI will make the algorithm an intuitive
and easy to use first choice for many reinforcement learning tasks. In future work, we plan
to investigate the application of our method to multi-agent systems and the use of density
estimation to control the projection weights in our function approximator.
Acknowledgments
We would like to thank J. Randl?v and P. Alstr?m for making their bicycle simulator available. We also thank C. Guestrin, D. Koller, U. Lerner and M. Littman for helpful discussions. The first author would like to thank the Lilian-Boudouri Foundation in Greece for
partial financial support.
References
[1] J. Baxter and P.Bartlett. Reinforcement learning in POMDP?s via direct gradient ascent. In
Proc. 17th International Conf. on Machine Learning, pages 41?48. Morgan Kaufmann, San
Francisco, CA, 2000.
[2] D. Bertsekas and J. Tsitsiklis. Neuro-Dynamic Programming. Athena Scientific, Belmont,
Massachusetts, 1996.
[3] Justin A. Boyan. Least-squares temporal difference learning. In I. Bratko and S. Dzeroski,
editors, Machine Learning: Proceedings of the Sixteenth International Conference, pages 49?
56. Morgan Kaufmann, San Francisco, CA, 1999.
[4] S. Bradtke and A. Barto. Linear least-squares algorithms for temporal difference learning.
Machine Learning, 22(1/2/3):33?57, 1996.
[5] D. Koller and R. Parr. Policy iteration for factored mdps. In Proceedings of the Sixteenth
Conference on Uncertainty in Artificial Intelligence (UAI-00). Morgan Kaufmann, 2000.
[6] V. Konda and J. Tsitsiklis. Actor-critic algorithms. In NIPS 2000 editors, editor, Advances in
Neural Information Processing Systems 12: Proceedings of the 1999 Conference. MIT Press,
2000.
[7] M. G. Lagoudakis and R. Parr. Model-Free Least-Squares policy iteration. Technical Report
CS-2001-05, Department of Computer Science, Duke University, December 2001.
[8] A. Ng and M. Jordan. PEGASUS: A policy search method for large MDPs and POMDPs.
In Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence (UAI-00).
Morgan Kaufmann, 2000.
[9] A. Ng, R. Parr, and D. Koller. Policy search via density estimation. In Advances in Neural
Information Processing Systems 12: Proceedings of the 1999 Conference. MIT Press, 2000.
[10] Andrew Y. Ng, Daishi Harada, and Stuart Russell. Policy invariance under reward transformations: theory and application to reward shaping. In Proc. 16th International Conf. on Machine
Learning, pages 278?287. Morgan Kaufmann, San Francisco, CA, 1999.
[11] D. Ormoneit and S. Sen. Kernel-based reinforcement learning. To appear, Machine Learning,
2001.
[12] J. Randl?v and P. Alstr?m. Learning to drive a bicycle using reinforcement learning and shaping. In The Fifteenth International Conference on Machine Learning, 1998. Morgan Kaufmann.
[13] R. Sutton, D. McAllester, S. Singh, and Y. Mansour. Policy gradient methods for reinforcement
learning with function approximation. In Advances in Neural Information Processing Systems
12: Proceedings of the 1999 Conference, 2000. MIT Press.
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1,246 | 2,135 | Novel iteration schemes for the Cluster
Variation Method
Hilbert J. Kappen
Department of Biophysics
Nijmegen University
Nijmegen, the Netherlands
bert?mbfys.kun.nl
Wim Wiegerinck
Department of Biophysics
Nijmegen University
Nijmegen, the Netherlands
wimw?mbfys.kun.nl
Abstract
The Cluster Variation method is a class of approximation methods containing the Bethe and Kikuchi approximations as special
cases. We derive two novel iteration schemes for the Cluster Variation Method. One is a fixed point iteration scheme which gives a
significant improvement over loopy BP, mean field and TAP methods on directed graphical models. The other is a gradient based
method, that is guaranteed to converge and is shown to give useful
results on random graphs with mild frustration. We conclude that
the methods are of significant practical value for large inference
problems.
1
Introduction
Belief Propagation (BP) is a message passing scheme, which is known to yield exact
inference in tree structured graphical models [1]. It has been noted by several
authors that Belief Propagation can can also give impressive results for graphs that
are not trees [2].
The Cluster Variation Method (CVM), is a method that has been developed in the
physics community for approximate inference in the Ising model [3]. The CVM approximates the joint probability distribution by a number of (overlapping) marginal
distributions (clusters). The quality of the approximation is determined by the size
and number of clusters. When the clusters consist of only two variables, the method
is known as the Bethe approximation. Recently, the method has been introduced
by Yedidia et a1.[4] into the machine learning community, showing that in the Bethe
approximation, the CVM solution coincides with the fixed points of the belief propagation algorithm. For clusters consisting of more than two variables, [4] present a
message passing scheme called generalized belief propagation (GBP). This approximation to the free energy is often referred to as the Kikuchi approximation. They
show, that GBP gives a significant improvement over the Bethe approximation for
a small two dimensional Ising lattice with random couplings. However, for larger
latices, both GBP and BP fail to converge [4, 5].
In [5] the CCCP method is proposed, which is a double loop iteration algorithm that
is guaranteed to converge for the general CVM problem. Intuitively, the method
consists of iteration a sequence of convex subproblem (outer loop) each of which is
solved using a fixed point iteration method (inner loop). In this sense, the method
is similar to the UPS algorithm of [6] which identifies trees as subproblems.
In this paper, we propose two algorithms, one is a fixed point iteration procedure, the other a gradient based method. We show that the fixed point iteration
method gives very fast convergence and accurate results for some classical directed
graphical models. However, for more challenging cases the fixed point method does
not converge and the gradient based approach, which is guaranteed to converge, is
preferable.
2
The Cluster Variation Method
In this section, we briefly present the cluster variation method. For a more complete
treatment see for instance [7]. Let x = (Xl, ... ,xn ) be a set of variables, where each
Xi can take a finite number of values. Consider a probability distribution on X of
the form
1_ -H(x)
( ) __
Z =
e-H(x)
PH X - Z(H)e
2:=
x
It is well known, that PH can be obtained as the minimum of the free energy, which
is a functional over probability distributions of the following form:
FH(P)
= (H) + (logp) ,
(1)
where the expectation value is taken with respect to the distribution p , i.e. (H) =
L x P(x)H(x). When one minimizes FH(P) with respect to P under the constraint
of normalization L x P(X) = 1, one obtains PH.
Computing marginals of PH such as PH(Xi) or PH(Xi, Xj) involves sums over all
states, which is intractable for large n. Therefore, one needs tractable approximations to PH. The cluster variation method replaces the probability distribution
PH(X) by a large number of (possibly overlapping) probability distributions , each
describing a sub set (cluster) of variables. Due to the one-to-one correspondence
between a probability distribution and the minima of a free energy we can define approximate probability distributions by constructing approximate free energies and
computing their minimum. This is achieved by approximating Eq. 1 in terms of the
cluster probabilities. The solution is obtained by minimizing this approximate free
energy subject to normalization and consistency constraints.
Define clusters as subsets of distinct variables: Xa = (XiI' ... ,Xik), with 1 ~ i j ~ n.
Consider the set of clusters P that describe the interactions in H and write H as a
sum of these interactions:
H(x) =
Hl(xoJ
2:=
a EP
We now define a set of clusters B, that will determine our approximation in the
cluster variation method. For each cluster a E B, we introduce a probability
distribution Pa(xa) which jointly must approximate p(x). B should at least contain
the interactions in p(x) in the following way: Va E P => 30:' E B,a c a'. In
addition, we demand that no two clusters in B contain each other: a, a' E B =>
a rt a', a' rt a. The minimal choice for B is to chose clusters from P itself. The
maximal choice for B is the cliques obtained when constructing the junction tree[8].
In this case, the clusters in B form a tree structure and the CVM method is exact.
Define a set of clusters M that consist of any intersection of a number of clusters
of B: M = {,BI,B = nkak, ak E B}, and define U = BuM. Once U is given, we
define numbers a/3 recursively by the Moebius formula
1=
L
ao;,
o;EU,o;"J/3
In particular, this shows that ao; = 1 for
0: E
V(3 E U
B.
The Moebius formula allows us to rewrite (H) in terms of the cluster probabilities
(H) = Lao; LPo;(xo;)Ho;(xo;),
o;EU
x"
(2)
with Ho;(xo;) = L./3EP,/3co; Hh(X/3) . Since interactions Hh may appear in more than
one Ho;, the constants ao; ensure that double counting is compensated for.
Whereas (H) can be written exactly in terms of Po;, this is not the case for the
entropy term in Eq. 1. The approach is to decompose the entropy of a cluster 0: in
terms of 'connected entropies' in the following way: 1
(3)
x"
/3Co;
where the sum over (3 contains all sub clusters of 0:. Such a decomposition can be
made for any cluster. In particular it can be made for the 'cluster' consisting of all
variables, so that we obtain
S = - LP(x) logp(x) = L Sh?
x
/3
(4)
The cluster variation method approximates the total entropy by restricting this
latter sum to only clusters in U and re-expressing Sh in terms of So;, using the
Moebius formula and the definition Eq. 3.
(5)
/3EU
/3EU 0;"J/3
o;EU
Since So; is a function of Po; (Eq. 3) , we have expressed the entropy in terms of
cluster probabilities Po; .
The quality of this approximation is illustrated in Fig. 1 for the SK model. Note,
that both the Bethe and Kikuchi approximation strongly deteriorate around J = 1,
which is where the spin-glass phase starts. For J < 1, the Kikuchi approximation is
superior to the Bethe approximation. Note, however, that this figure only illustrates
the quality of the truncations in Eq. 5 assuming that the exact marginals are known.
It does not say anything about the accuracy of the approximate marginals using
the approximate free energy.
Substituting Eqs. 2 and 5 into the free energy Eq. 1 we obtain the approximate
free energy of the Cluster Variation method. This free energy must be minimized
subject to normalization constraints L.x" Po; (x o; ) = 1 and consistency constraints
Po;(X/3) = P/3(X/3),
with Po; (X/3) = L. x
"\f3
0:,(3 E U,(3 C 0:.
(6)
Po; (xo;).
IThis decomposition is similar to writing a correlation in terms of means and covariance.
For instance when a = (i) , S(i) = SIi) is the usual mean field entropy and S(ij) = Sli) +
SIj)
+ Slij)
defines the two node correction Slij)"
12
10
8
>a.
e
c
6
lJ.J
4
"-
2
"-
"-
"-
0
0.5
1.5
2
J
Figure 1: Exact and approximate entropies for the fully connected Boltzmann-Gibbs
distribution on n = 10 variables with random couplings (SK model) as a function of mean
coupling strength. Couplings Wij are chosen from a normal Gaussian distribution with
mean zero and standard deviation J /..;n. External fields ()i are chosen from a normal
Gaussian distribution with mean zero and standard deviation 0.1. The exact entropy is
computed from Eq. 4. The Bethe and Kikuchi approximations are computed using the
approximate entropy expression Eq. 5 with exact marginals and by choosing B as the set
of all pairs and all triplets, respectively.
The set of consistency constraints can be significantly reduced because some constraints imply others. Let 0:,0:', .. . denote clusters in Band fJ, fJ', ... denote clusters
in M.
? If fJ
c fJ' Co: and Pa(x/3') = P/3' (x/3') and Pa(x/3 ) = P/3(x/3), then P/3' (x/3) =
P/3 (X/3)' This means that constraints between clusters in M can be removed .
fJ c fJ' c 0: , 0:' and Pa(x/3') = Pa' (x/3') and p,,,(x/3) = P/3 (x /3 ), then
Pa,(x/3) = P/3 (x/3)' This means that some constraints between clusters in B
? If
and M can be removed.
We denote the remaining necessary constraints by
0:
---t
fJ.
Adding Lagrange multipliers for the constraints we obtain the Cluster Variation
free energy:
aEU
- L Aa (LPa(Xa) aEU
x"
1) -
L
/3 EM
x"
L L Aa/3 (X/3) (Pa(x /3 ) - P/3 (x/3))
a-+/3 X f3
(7)
3
Iterating Lagrange multipliers
c(vm), a E U equal to zero, one can express the cluster probabilities in
By setting 88F
PO! X o:
terms of the Lagrange multipliers:
exp (-Ha(Xa)
;
+ L )..a(3 (X(3))
a
exp (-H(3 (X(3 ) - a1 L
;
(8)
(3 f-a
(3
)..a(3 (X(3 ))
(9)
(3 a-t (3
The remaining task is to solve for the Lagrange multipliers such that all constraints
(Eq. 6) are satisfied. We present two ways to do this.
When one substitutes Eqs. 8-9 into the constraint Eqs. 6 one obtains a system of
coupled non-linear equations. In Yedidia et al.[4] a message passing algorithm was
proposed to find a solution to this problem. Here, we will present an alternative
method, that solves directly in terms of the Lagrange multipliers.
3.1
Fixed point iteration
Consider the constraints Eq. 6 for some fixed cluster fJ and all clusters a -+ fJ and
define B(3 = {a E Bla -+ fJ }? We wish to solve for all constraints a -+ fJ, with
a E B(3 by adjusting )..a(3, a E B(3. This is a sub-problem with IB(3 IIX(3 I equations
and an equal number of unknowns, where IB(3 1 is the number of elements of B(3
and IX(3 1 is the number of values that x(3 can take. The probability distribution P(3
(Eq. 9) depends only on these Lagrange multipliers. Pa (Eq. 8) depends also on other
Lagrange multipliers. However, we consider only its dependence on )..a(3 , a E B(3 ,
and consider all other Lagrange multipliers as fixed . Thus,
(10)
with Pa independent of ).. a(3, a E B(3 .
Substituting, Eqs. 9 and 10 into Eq. 6, we obtain a set of linear equations for
)..a(3 (x(3 ) which we can solve in closed form:
)..a(3 (X(3 ) = -
a(3
alB IH(3 (X(3 ) - L A aa do gPa (X(3 )
l
+
(3
a'
with
Aaa =
l
1
/ja a l
-
--c=--:-
a(3
+ IB(3 1
We update the probabilities with the new values of the Lagrange multipliers using
Eqs. 9 and 10. We repeat the above procedure for all fJ E M until convergence.
3.2
Gradient descent
We define an auxiliary cost function
C = L LP(3 (X(3 ) log P(3 ((X(3 )) = L Ca(3
a(3 Xf3
Pa x(3
a(3
(11)
that is zero when all constraints are satisfied and positive otherwise and minimize
this cost function with respect to the Lagrange multipliers )..a(3 (X(3 ). The gradient
of C is given by:
8C
Pf3(Xf3
-Pf3(Xf3
- -) ""'
~ (log
( )) - Calf3 ) - ""'
~ (PIaf3' (
xf3) - Pa ())
xf3
af3
a/-tf3
Pa' Xf3
13 ' +--a
with
4
4.1
Numerical results
Directed Graphical models
We show the performance of the fixed point iteration procedure on several 'real
world' directed graphical models. In figure 2a, we plot the exact single node
marginals against the approximate marginals for the Asia problem [8]. Clusters
in B are defined according to the conditional probability tables. Convergence was
reached in 6 iterations using fixed point iteration. Maximal error on the marginals
is 0.0033. For comparison, we computed the mean field and TAP approximations,
as previously introduced by [9]. Although TAP is significantly better than MF, it
is far worse than the CVM method. This is not surprising, since both the MF and
TAP approximation are based on single node approximation, whereas the CVM
method uses potentials up to size 3.
In figure 2b, we plot the exact single node marginals against the approximate CVM
marginals for the alarm network [10]. The structure and CPTs were downloaded
from www.cs.huji.ac.il;-nir. Clusters in B are defined according to the conditional probability tables and maximally contain 5 variables. Convergence was
reached in 15 iterations using fixed point iteration. Maximal error on the marginals
is 0.029. Ordinary loopy BP gives an error in the marginals of approximately 0.25
[2]. Mean field and TAP methods did not give reproducible results on this problem.
0.5 , - - - - - - - - - ---.--f'l---+
..' .'
(fj
0.4
0.8
coc
(fj
co
.a, 0.6
?~0 . 3
C1l
x
x
E
x
ec. 0.2
0.1
E
:2 0.4
>
c.
<t:
Co
()
+
0.2
-F?
.'
.'
,...
OL-----------~
o
Exact marginals
(a) Asia problem (n
= 8).
0.5
Exact marginals
(b) Alarm problem (n
= 37).
Figure 2: Comparison of single node marginals on two real world problems.
Finally, we tested the cluster variation method on randomly generated directed
graphical models. Each node is randomly connected to k parents. The entries of
the probability tables are randomly generated between zero and one. Due to the
large number of loops in the graph, the exact method requires exponential time in
the maximum clique size, which can be seen from Table 1 to scale approximately
linear with the network size. Therefore exact computation is only feasible for small
graphs (up to size n = 40 in this case).
For the CVM, clusters in B are defined according to the conditional probability
tables. Therefore, maximal cluster size is k + 1. On these more challenging cases,
the fixed point iteration method does not converge. The results shown are obtained
with conjugate gradient descent on the auxiliary cost function Eq. 11. The results
are shown in Table 1.
n
10
20
30
40
50
Iter
16
189
157
148
132
IGI
8
12
16
21
26
Potential error
0.018
0.019
0.033
0.048
-
Margin error
0.004
0.029
0.130
0.144
-
G
9.7e-ll
2.4e-4
2.1e-3
3.6e-3
4.5e-3
Table 1: Comparison of CYM method for large directed graphical models. Each node
is connected to k = 5 parents. IGI is the tree width of the triangulated graph required
for the exact computation. Iter is the number of conjugate gradient descent iterations of
the CYM method. Potential error and margin error are the maximum absolute distance
(MAD) in any of the cluster probabilities and single variable marginals computed with
CYM, respectively. G is given by Eq. 11 after termination of CYM.
4.2
Markov networks
We compare the Bethe and Kikuchi approximations for the SK model with n = 5
neurons as defined in Fig. 1. We expect that for small J the CVM approximation
gives accurate results and deteriorates for larger J.
We compare the Bethe approximation, where we define clusters for all pairs of nodes
and a Kikuchi approximation where we define clusters for all sub sets of three nodes.
The results are given in Table 2. We see that for the Bethe approximation, the
results of the fixed point iteration method (FPI) and the gradient based approach
agree. For the Kikuchi approximation the fixed point iteration method does not
converge and results are omitted. As expected, the Kikuchi approximation gives
more accurate results than the Bethe approximation for small J.
5
Conclusion
We have presented two iteration schemes for finding the minimum of the constraint
problem Eq. 7. One method is a fixed point iteration method that is equivalent
to belief propagation for pairwise interactions. This method is very fast and gives
very accurate results for 'not too complex' graphical models , such as real world
directed graphical models and frustrated Boltzmann distributions in the Bethe approximation. However, for more complex graphs such as random directed graphs or
more complex approximations, such as the Kikuchi approximation, the fixed point
iteration method does not converge. Empirically, it is found that smoothing may
somewhat help , but certainly does not solve this problem. For these more complex problems we propose to minimize an auxiliary cost function using a gradient
Bethe
FPI
J
0.25
0.50
0.75
1.00
1.50
2.00
Iter
Error
7
9
13
17
38
75
0.000161
0.001297
0.004325
0.009765
0.027217
0.049955
gradient
Iter
Error
7
11
14
15
16
20
0.000548
0.001263
0.004392
0.009827
0.027323
0.049831
Kikuchi
gradient
Iter
Error
120
221
86
49
150
137
0.000012
0.000355
0.021176
0.036882
0.059977
0.088481
Table 2: Comparison of Bethe and Kikuchi approximation for Boltzmann distributions.
Iter is the number of iterations needed. Error is the MAD of single variable marginals.
based method. Clearly, this approach is guaranteed to converge. Empirically, we
have found no problems with local minima. However , we have found that obtaining
solut ion with C close to zero may require many iterations.
Acknowledgments
This research was supported in part by the Dutch Technology Foundation (STW).
I would like to thank Taylan Cemgil for providing his Matlab graphical models
toolkit and Sebino Stramaglia (Bari, Italy) for useful discussions.
References
[1] J. Pearl. Probabilistic reasoning in intelligent systems: Networks of Plausible Inference. Morgan Kaufmann, San Francisco, California, 1988.
[2] Kevin P. Murphy, Yair Weiss, and Michael I. Jordan . Loopy belief propagation for
approximate inference: An empirical study. In Proceedings of Uncertainty in AI,
pages 467- 475, 1999.
[3] R. Kikuchi. Physical R eview, 81:988, 1951.
[4] J.S. Yedidia, W.T. Freeman, and Y. Weiss. Generalized belief propagation. In T.K.
Leen , T.G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems 13 (Proceedings of th e 2000 Conference), 2001. In press.
[5] A.L. Yuille and A. Rangarajan. The convex-concave principle. In T.G. Dieterich,
S. Becker, and Z. Ghahramani, editors, Advances in Neural Information Processing
Systems, volume 14, 2002. In press.
[6] Y. Teh and M. Welling. The unified propagation and scaling algorithm. In T.G.
Dieterich, S. Becker, and Z. Ghahramani, editors, Advances in Neural Information
Processing Systems, volume 14, 2002. In press.
[7] H.J. Kappen. The cluster variation method for approximate reasoning in medical
diagnosis. In G. Nardulli and S. Stramaglia, editors, Modeling Bio-medical signals.
World-Scientific, 2002. In press.
[8] S.L. Lauritzen and D.J. Spiegelhalter. Local computations with probabilties on graphical structures and their application to expert systems. J. Royal Statistical society B ,
50:154- 227, 1988.
[9] H.J . Kappen and W .A.J.J. Wiegerinck. Second order approximations for probability
models. In Todd Leen, Tom Dietterich, Rich Caruana, and Virginia de Sa, editors,
Advances in Neural Information Processing Systems 13, pages 238- 244. MIT Press,
2001.
[10] 1. Beinlich, G. Suermondt, R. Chaves, and G. Cooper. The alarm monitoring system:
A case study with two probabilistic inference techniques for belief networks. In 2'nd
European Conference on AI in Medicin e, 1989.
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1,247 | 2,136 | Generalization Performance of Some Learning
Problems in Hilbert Functional Spaces
Tong Zhang
IBM T.J. Watson Research Center
Yorktown Heights, NY 10598
[email protected]
Abstract
We investigate the generalization performance of some learning problems in Hilbert functional Spaces. We introduce a notion of convergence
of the estimated functional predictor to the best underlying predictor, and
obtain an estimate on the rate of the convergence. This estimate allows
us to derive generalization bounds on some learning formulations.
1 Introduction
!"$#&%' (
)
In order to estimate a good predictor from a set of training data
randomly drawn
from , it is necessary to start with a model of the functional relationship. In this paper,
we consider models that are subsets in some Hilbert functional spaces * . Denote by + +-,
the norm in * , we consider models in the set ./10 324*657+ 8+ ,:9;=< , where
; is a parameter that can be used to control the size of the underlying model family. We
would like to find the best model in . which is given by:
> ?@ ACB7DFIKJMEHL G NH !8"O
()
(1)
By introducing a non-negative Lagrangian multiplier PQSR , we may rewrite the above
problem as:
> T@ ACB7IKDFJ E , G U NH !8"V
(
W PX + Y+Z,\[ )
(2)
We shall only consider this equivalent formulation in this paper. In addition, for technical
reasons, we also assume that ^]
C_ is a convex function of ] .
In machine learning, our goal is often to predict an unobserved output value based on an
observed input vector . This requires us to estimate a functional relationship
from a set of example pairs of
. Usually the quality of the predictor
can be
that is problem dependent. In machine learning,
measured by a loss function
we assume that the data
are drawn from an underlying distribution which is not
so that the expected true loss of given below is as small
known. Our goal is to find
as possible:
K
8)8)Y)&
>
, we consider the following estimation
'2 . :
> ?@ ACB7DFEHG
W PX + 8+ Z, [ )
(3)
IKJ , U
> >
The goal of this paper is to show that as
,
in probability under appropri-
Given training examples
method to approximate the optimal predictor
ate regularity conditions. Furthermore, we obtain an estimate on the rate of convergence.
Consequences of this result in some specific learning formulations are examined.
2 Convergence of the estimated predictor
+ + ,
2 EHD I I ?
+ +8, R
I
W 2
2 *
=
*
*
*
2 *
2 *
*
It is clear that * can be regarded as a representing feature vector of in * . This representation can be computed as follows. Let + @MA B7D @-, I " , then it is not difficult to
/ + * +, and + ?+ * +, . It follows that * ?1
see that + .* 0
+ 2 + . Note that this
method of computing * is not important for the purpose of this paper.
Since * can now be considered as a linear functional using the feature space
representation
* of , we can use the idea from [6] to analyze the convergence behavior
>
of in * . Following
[6], using the linear representation of , we differentiate (2) at
>
the optimal solution , which leads to the following first order condition:
M ! 4 3 > *
5 * W P > TR
(4)
^
]
^
]
]
where 3
C_8 is the derivative of
C_ with respect to if is smooth; it denotes a
subgradient (see [4]) otherwise. Since we have assumed that ^]
_ is a convex function
of ] , we know that ^] -
_W ^] ] 8 3 ^] K
C_ 9 ^]
_ . This implies the following
Z
Z
inequality:
>
>
>
>
>
W
3
9
(
Assume that input belongs to a set . We make the reasonable assumption that is point
wise continuous under the
topology: ,
where
is in the sense that
. This assumption is equivalent to ! #"%$
'&
' ( ) . The condition implies that each data point can be regarded as a bounded
linear functional * on
such that
: *
. Since a Hilbert space
is
self-dual, we can represent * by an element in . For notational simplicity, we shall
defined as *
for all
, where denotes the inner product of .
let *
which is equivalent to:
> (
& W PX > Z W
>
>
U 3 (
9 > (
W PX > Z )
Note that we have used Z to denote
we have
>
(
W PX > Z Q
> W4P > M > > [ W PX 6> > Z
+ + Z,
. Also note that by the definition of
> (
W PX > Z )
> ,
Therefore by comparing the above two inequalities, we obtain:
>
>
U 43 (
*
9 + 3 > (
5* W4P > + ,
PX > > Z 9
X
>+ 6 > &+8, 9 +
PX
P+
This implies that
> W P > M > > [
+ 6> > + , )
3 >
* W P > +,
>
3
* ! 3 > (
5* + , )
(5)
> > >
(
Note that the last equality follows from the first order condition (4). This is the only place
the condition is used. In (5), we have already bounded the convergence of to in terms
of the convergence of the empirical expectation of a random vector 3
* to its
mean. The latter is often easier to estimate. For example, if its variance can be bounded,
then we may use the Chebyshev inequality to obtain a probability bound. In this paper,
we are interested in obtaining an exponential probability bound. In order to do so, similar
to the analysis in [6], we use the following form of concentration inequality which can be
found in [5], page 95:
R R + +,Q 9 X Z Q _ Z W + + , 9 Z
*
Z
3 > (
* MX ! 3 > (
5*
X
9 W 9 W4 7 X )
X
R R
Q
M ! 3 > (
Y+* +, 9 Z
X
+ > > +8, Q 9 , 5 P Z Z / ! _ Z W4P" )
_
>
X
Theorem 2.1 ([5]) Let be zero-mean
independent random vectors
in a Hilbert space
If there exists
such that for all natural numbers
:
/
Then for all
:
,
.
.
.
We may now use the following form of Jensen?s inequality to bound the moments of the
zero-mean random vector
:
From inequality (5) and Theorem 2.1, we immediately obtain the following bound:
Theorem 2.2 If there exists
such that for all natural numbers
. Then for all
:
:
Although Theorem 2.2 is quite general, the quantity and
on the right hand side of
the bound depend on the optimal predictor which requires to be estimated. In order to
obtain a bound that does not require any knowledge of the true distribution , we may
impose the following assumptions: both 3
and *
are bounded. Observe that
*
, we obtain the following result:
"
IKJ ,$# I
^ ]
_
+ +8,
Corollary 2.1 Assume that 2 , 5 IKJ ,%# I " 9'& . Also assume that the loss
function ^]
C_ satisfies the condition that 3 ^]
C_( 9 , then )* R :
+ > > + , Q+ 9 X, , P Z Z / ! & Z Z W & P" ()
+ +,
3 Generalization performance
We study some consequences of Corollary 2.1, which bounds the convergence rate of the
estimated predictor to the best predictor.
3.1 Regression
6
Z 6 Z ifif 6 9Q )
Z
It is clear that is continuous differentiable and 3
( 9 for all and
not hard to check that
and :
Z
K
3 -
9 X 8ZM)
Z
Z
Z
We consider the following type of Huber?s robust loss function:
7
(6)
. It is also
Using this inequality and (4), we obtain:
>
PX >
PX >
>
U M ! (
W + +Z, [ U M ! (
W + +8Z, [
\ ! > (
> (
3 > (
> > W
9 ! X > > Z W PX + > > +Z, )
> >
If we assume that + +8, 9 and IKJ ,$# I " 9 & , then
M ! >
9 M ! >
W PX + > + Z, + > + Z, W ZX
PX + > > + Z,
& Z W4P()
(7)
M ! > (
! > (
9 P" + > &+, W Z & X Z W P()
This gives the following inequality:
>
It is clear that the right-hand side of the above inequality does not depend on the unobserved
function . Using Corollary 2.1, we obtain the following bound:
+
X IKJ ,%! # I "
P Z Z / R & Z
9 & P
M ! > (
9 M ! > (
W4"P + > &+, W Z & X Z W
Theorem 3.1 Using loss function (6) in (3). Assume that
/ , with probability of at least
, 5
&
R
> 2 . *
9 & , then
Z , we have
P()
Theorem 3.1 compares the performance of the computed function with that of the optimal
predictor
in (1). This style of analysis has been extensively used in the
literature. For example, see [3] and references therein. In order to compare with
their
results, we can rewrite Theorem 3.1 in another form as: with probability of at least ,
M ! >
9 M ! >
W
HG
)
In [3], the authors employed a covering number analysis which led to a bound of the form
(for squared loss)
M ! > (
9 M ! > (
W
HG
HG
for finite dimensional problems. Note that the constant in their
depends on the pseudodimension, which can be infinity for problems considered in this paper. It is possible to employ their analysis using some covering number bounds for general Hilbert spaces. However, such an analysis would have led to a result of the following form for our problems:
M ! > (
9 M ! > (
W
G
G
()
It is also interesting to compare Theorem 3.1 with the leave-one-out analysis in [7]. The
generalization error averaged over all training examples for squared loss can be bounded
as
! >
9 W
P M ! > (
W4P + > +Z, )
This result is not directly
> comparable with Theorem 3.1 since the right hand side includes
an extra term of P+ + Z, . Using the analysis in this paper, we may obtain a similar result
from (7) which leads to an average bound of the form:
! > (
9 M ! >
W P + > + Z, &W
P Z ()
It is clear that the term resulted in our paper is not as good as
from [7].
However analysis in this paper leads to probability bounds while the leave-one-out analysis
in [7] only gives average bounds. It is also worth mentioning that it is possible to refine
the analysis presented in this section to obtain a probability bound which when averaged,
, rather than
in the current analysis.
gives a bound with the correct term of
However due to the space limitation, we shall skip this more elaborated derivation.
In addition to the above style bounds, it is also interesting to compare the generalization
performance of the computed function to the empirical error of the computed function.
Such results have occurred, for example, in [1]. In order to obtain a comparable result, we
may use a derivation similar to that of (7), together with the first order condition of (3) as
follows:
3 > *
5* W P >
R)
This leads to a bound
of the form:
> (
9
> (
W PX + > &+ Z,
+ > + Z, W X Z & Z W P )
X IKJ ,%! # I " 9 & , then
P
P Z Z / R & Z Z , we have
! >
> (
>
>
9 Z & Z W P W U M ! (
(
[ )
Combining the above inequality and (7), we obtain the following theorem:
+ 9 &
Theorem 3.2 Using loss function (6) in (3). Assume that
/ , with probability of at least
&
, 5
R
> (
M
!
U
>
>
Unlike Theorem 3.1, the bound given in Theorem 3.2 contains a term
which relies on the unknown optimal predictor . From Theorem 3.1,
we know that this term does not affect the performance of the estimated function when
>
[
>
compared with the performance of . In order for us to compare with the bound in [1]
obtained from an algorithmic stability point of view, we make the additional assumption
for all
. Note that this assumption is also required in [1]. Using
that
/
Hoeffding?s inequality, we obtain that with probability of at most , 5
,
> 9
! >
P Z Z ! R & Z Z
/
& )
> (
% P
M ! >
,
HG
> (
9 & Z W P ! R & Z Z HG
W Z
)
PZ
Together with Theorem 3.2, we have with probability of at least
This compares very favorably to the following bound in [1]:1
M ! >
> (
9
X
X
X
& P Z Z W Z P & W ! P& Z W X HG Z )
Z
3.2 Binary classification
20 <
QTR
1R
In binary classification, the output value
is a discrete variable. Given a continu, we consider the following prediction rule: predict
if
, and
ous model
otherwise. The classification error (we shall ignore the point
predict
,
which is assumed to occur rarely) is
(
R
if
if
9 R
R )
Unfortunately, this classification error function is not convex, which cannot be handled in
our formulation. In fact, even in many other popular methods, such as logistic regression
and support vector machines, some kind of convex formulations have to be employed. We
shall thus consider the following soft-margin SVM style loss as an illustration:
7DF@-,
R )
(8)
Note that the separable case of this loss was investigated in [6]. In this case, 3
denotes a subgradient
rather than gradient
since
is non-smooth: at
,
3
2 U
CR [ ; 3
when & and 3
7R when .
> >
Since , 9 and ,
9
, we know that if + +, 9 ,
Z
Z
then
M ! > (
9 ! > &
(
X
>
>
&
9
&
()
P Z Z ! R & Z
M ! >
Using the standard Hoeffding?s inequality, we have with probability of at most
/
, 5
,
1
&
>
&
W P / !
& )
In [1], there was a small error after equation (11). As a result, the original bound in their paper
was in a form equivalent to the one we cite here with
replaced by
.
>
When &
R , it is usually better to use a different (multi , 5 P Z form
plicative)
inequality, which implies that with probability of at most
Z / ! Rof& Z Hoeffding?s
,
X
>
>
M ! &
D @ , &
(
P Z Z / & Z ()
Together with Corollary 2.1, we obtain the following margin-percentile result:
IKJ ,%# I " 9
R + 9 & P
P Z Z / ! R & Z , we have
X
>
>
!
M ! (
9 &
W P / & )
We also have with probability of at least
, P Z Z / ! R & Z ,
X
X
M ! >
9 DF@-, > &
(
CP Z KZ / & Z8 ()
Theorem 3.3 Using loss function (8) in (3). Assume that
/ , with probability of at least
&
, 5
&
, then
We may obtain from Theorem 3.3 the following result: with probability of at least
! >
9
>
!
R G
PZ
&Z
&W
GX
,
)
It is interesting to compare this result with margin percentile style bounds from VC analysis. For example, Theorem
4.19 in [2] implies that there exists a constant such that with
probability of at least : for all we have
! > (
9
P
>
W
& PZ HG Z W HG
Z
!
R
& Z
)
We can see that if we assume that is small and the margin
is also small,
then the above bound with this choice of is inferior to the bound in Theorem 3.3. Clearly,
this implies that our analysis has some advantages over VC analysis due to the fact that we
directly analyze the numerical formulation of support vector classification.
4 Conclusion
In this paper, we have introduced a notion of the convergence of the estimated predictor
to the best underlying predictor for some learning problems in Hilbert spaces. This generalizes an earlier study in [6]. We derived generalization bounds for some regression and
classification problems. We have shown that results from our analysis compare favorably
with a number of earlier studies. This indicates that the concept introduced in this paper
can lead to valuable insights into certain numerical formulations of learning problems.
References
[1] Olivier Bousquet and Andr?e Elisseeff. Algorithmic stability and generalization performance. In Todd K. Leen, Thomas G. Dietterich, and Volker Tresp, editors, Advances
in Neural Information Processing Systems 13, pages 196?202. MIT Press, 2001.
[2] Nello Cristianini and John Shawe-Taylor. An Introduction to Support Vector Machines
and other Kernel-based Learning Methods. Cambridge University Press, 2000.
[3] Wee Sun Lee, Peter L. Bartlett, and Robert C. Williamson. The importance of convexity in learning with squared loss. IEEE Trans. Inform. Theory, 44(5):1974?1980,
1998.
[4] R. Tyrrell Rockafellar. Convex analysis. Princeton University Press, Princeton, NJ,
1970.
[5] Vadim Yurinsky. Sums and Gaussian vectors. Springer-Verlag, Berlin, 1995.
[6] Tong Zhang. Convergence of large margin separable linear classification. In Advances
in Neural Information Processing Systems 13, pages 357?363, 2001.
[7] Tong Zhang. A leave-one-out cross validation bound for kernel methods with applications in learning. In 14th Annual Conference on Computational Learning Theory,
pages 427?443, 2001.
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1,248 | 2,137 | Relative Density Nets: A New Way to
Combine Backpropagation with HMM's
Andrew D. Brown
Department of Computer Science
University of Toronto
Toronto, Canada M5S 3G4
[email protected]
Geoffrey E. Hinton
Gatsby Unit, UCL
London, UK WCIN 3AR
[email protected]
Abstract
Logistic units in the first hidden layer of a feedforward neural network compute the relative probability of a data point under two
Gaussians. This leads us to consider substituting other density
models. We present an architecture for performing discriminative
learning of Hidden Markov Models using a network of many small
HMM's. Experiments on speech data show it to be superior to the
standard method of discriminatively training HMM's.
1
Introduction
A standard way of performing classification using a generative model is to divide the
training cases into their respective classes and t hen train a set of class conditional
models. This unsupervised approach to classification is appealing for two reasons. It
is possible to reduce overfitting, because t he model learns the class-conditional input
densities P(xlc) rather t han the input -conditional class probabilities P(clx). Also,
provided that the model density is a good match to the underlying data density
then the decision provided by a probabilistic model is Bayes optimal. The problem
with this unsupervised approach to using probabilistic models for classification is
that, for reasons of computational efficiency and analytical convenience, very simple
generative models are typically used and the optimality of the procedure no longer
holds. For this reason it is usually advantageous to train a classifier discriminatively.
In this paper we will look specifically at the problem of learning HMM 's for classifying speech sequences. It is an application area where the assumption that the HMM
is the correct generative model for the data is inaccurate and discriminative methods
of training have been successful. The first section will give an overview of current
methods of discriminatively training HMM classifiers. We will then introduce a new
type of multi-layer backpropagation network which takes better advantage of the
HMM 's for discrimination. Finally, we present some simulations comparing the two
methods.
19 ' S1
c1
1
1
1
="
[tn] [tn][t n] HMM 's
\V
Sequence
Figure 1: An Alphanet with one HMM per class. Each computes a score for the
sequence and this feeds into a softmax output layer.
2
Alphanets and Discriminative Learning
The unsupervised way of using an HMM for classifying a collection of sequences is to
use the Baum-Welch algorithm [1] to fit one HMM per class. Then new sequences
are classified by computing the probability of a sequence under each model and
assigning it to the one with the highest probability. Speech recognition is one of the
commonest applications of HMM 's, but unfortunately an HMM is a poor model of
the speech production process. For this reason speech researchers have looked at the
possibility of improving the performance of an HMM classifier by using information
from negative examples - examples drawn from classes other than the one which
the HMM was meant to model. One way of doing this is to compute the mutual
information between the class label and the data under the HMM density, and
maximize that objective function [2].
It was later shown that this procedure could be viewed as a type of neural network
(see Figure 1) in which the inputs to the network are the log-probability scores
C(Xl:TIH) of the sequence under hidden Markov model H [3]. In such a model
there is one HMM per class, and the output is a softmax non-linearity:
(1)
Training this model by maximizing the log probability of correct classification leads
to a classifier which will perform better than an equivalent HMM model trained
solely in a unsupervised manner. Such an architecture has been termed an "AIphanet" because it may be implemented as a recurrent neural network which mimics
the forward pass of the forward-backward algorithm.l
3
Backpropagation Networks as Density Comparators
A multi-layer feedforward network is usually thought of as a flexible non-linear
regression model, but if it uses the logistic function non-linearity in the hidden
layer, there is an interesting interpretation of the operation performed by each
hidden unit. Given a mixture of two Gaussians where we know the component
priors P(9) and the component densities P(xl9) then the posterior probability that
Gaussian, 90 , generated an observation x , is a logistic function whose argument is
the negative log-odds of the two classes [4] . This can clearly be seen by rearranging
lThe results of the forward pass are the probabilities of the hidden states conditioned
on the past observations, or "alphas" in standard HMM terminology.
the expression for the posterior:
P(xI9o)P(Qo)
P(xI9o)P(Qo) + P(xI9d P (Qd
P(Qolx)
1
1 + exp {-log
P(x IQo) P(x lQd
log
(2)
P(Qo) }
P(Ql)
If the class conditional densities in question are multivariate Gaussians
P(xI9k) =
121f~1-~ exp {-~(x -
with equal covariance matrices,
written in this familiar form:
~,
Pk)T ~-l(X - Pk)}
(3)
then the posterior class probability may be
1
P(Qo Ix) = -l-+-e-xp-{-=---(:-x=Tw-+-b---:-)
(4)
where,
w
(5)
b
(6)
Thus, the multi-layer perceptron can be viewed as computing pairwise posteriors
between Gaussians in the input space, and then combining these in the output layer
to compute a decision.
4
A New Kind of Discriminative Net
This view of a feedforward network suggests variations in which other kinds of
density models are used in place of Gaussians in the input space. In particular,
instead of performing pairwise comparisons between Gaussians, the units in the
first hidden layer can perform pairwise comparisons between the densities of an
input sequence under M different HMM's. For a given sequence the log-probability
of a sequence under each HMM is computed and the difference in log-probability
is used as input to the logistic hidden unit. 2 This is equivalent to computing the
posterior responsibilities of a mixture of two HMM's with equal prior probabilities.
In order to maximally leverage the information captured by the HMM's we use (~)
hidden units so that all possible pairs are included. The output of a hidden unit h
is given by
(7)
where we have used (mn) as an index over the set, (~) , of all unordered pairs of
the HMM's. The results of this hidden layer computation are then combined using
a fully connected layer of free weights, W, and finally passed through a soft max
function to make the final decision.
ak
=
L
W(m ,n)kh(mn)
(8)
(mn) E (~)
(9)
2We take the time averaged log-probability so that the scale of the inputs is independent
of the length of the sequence.
Density
Comparator
Units
Figure 2: A multi-layer density net with HMM's in the input layer. The hidden
layer units perform all pairwise comparisons between the HMM 's.
where we have used u(?) as shorthand for the logistic function, and Pk is the value
of the kth output unit. The resulting architecture is shown in figure 2. Because
each unit in the hidden layer takes as input the difference in log-probability of two
HMM 's, this can be thought of as a fixed layer of weights connecting each hidden
unit to a pair of HMM's with weights of ?l.
In contrast to the Alphanet , which allocates one HMM to model each class, this network does not require a one-to-one alignment between models and classes and it gets
maximum discriminative benefit from the HMM's by comparing all pairs. Another
benefit of this architecture is that it allows us to use more HMM's than there are
classes. The unsupervised approach to training HMM classifiers is problematic because it depends on the assumption that a single HMM is a good model of the data
and, in the case of speech, this is a poor assumption. Training the classifier discriminatively alleviated this drawback and the multi-layer classifier goes even further in
this direction by allowing many HMM's to be used to learn the decision boundaries
between the classes. The intuition here is that many small HMM's can be a far
more efficient way to characterize sequences than one big HMM. When many small
HMM's cooperate to generate sequences, the mutual information between different
parts of generated sequences scales linearly with the number of HMM's and only
logarithmically with the number of hidden nodes in each HMM [5].
5
Derivative Updates for a Relative Density Network
The learning algorithm for an RDN is just the backpropagation algorithm applied
to the network architecture as defined in equations 7,8 and 9. The output layer is
a distribution over class memberships of data point Xl:T, and this is parameterized
as a softmax function. We minimize the cross-entropy loss function:
K
f =
2: tk logpk
(10)
k= l
where Pk is the value of the kth output unit and tk is an indicator variable which is
equal to 1 if k is the true class. Taking derivatives of this expression with respect
to the inputs of the output units yields
of
- = t k - Pk
oak
(11)
O?
o?
Oak
(12)
- , - - - - = (tk - Pk)h(mn)
OW(mn) ,k
oak OW(mn) ,k
The derivative of the output of the (mn)th hidden unit with respect to the output
of ith HMM, ?i, is
oh(mn)
(13)
~ = U(?m - ?n)(l - U(?m - ?n))(bim - bin)
where (bim - bin) is an indicator which equals +1 if i = m, -1 if i = n and zero
otherwise. This derivative can be chained with the the derivatives backpropagated
from the output to the hidden layer.
For the final step of the backpropagation procedure we need the derivative of the
log-likelihood of each HMM with respect to its parameters. In the experiments we
use HMM 's with a single, axis-aligned, Gaussian output density per state. We use
the following notation for the parameters:
?
?
?
?
?
A: aij is the transition probability from state i to state j
II: 7ri is the initial state prior
f./,i: mean vector for state i
Vi: vector of variances for state i
1-l: set of HMM parameters {A , II, f./" v}
We also use the variable St to represent the state of the HMM at time t. We make
use of the property of all latent variable density models that the derivative of the
log-likelihood is equal to the expected derivative of the joint log-likelihood under
the posterior distribution. For an HMM this means that:
O?(Xl:TI1-l)
'"
0
o1-l i
= ~ P(Sl:Tlxl:T' 1-l) o1-l i log P(Xl:T' Sl:TI1-l)
(14)
Sl:T
The joint likelihood of an HMM is:
(logP(Xl:T ' Sl:TI1-l)) =
T
L(b81 ,i)log 7ri
+ LL(b "jb
8
8
,_1 ,i)log aij
+
i,j
t=2
~ ~(b8" i) [-~ ~IOgVi'd ~ ~(Xt'd -
f./,i,d) 2 /Vi,d]
-
+ canst
(15)
where (-) denotes expectations under the posterior distribution and (b 8 , ,i) and
(b 8 , ,jb8 '_1 ,i) are the expected state occupancies and transitions under this distribution. All the necessary expectations are computed by the forward backward algorithm. We could take derivatives with respect to this functional directly, but that would require doing constrained gradient descent on the probInstead, we reparameterize the model using a
abilities and the variances.
softmax basis for probability vectors and an exponential basis for the variance parameters.
This choice of basis allows us to do unconstrained optimization in the new basis.
The new parameters are defined as follows:
. _
a' J -
exp(e;; ?)
(e (a? ) ,
2:
JI
exp
1JI
. _
7r, -
exp(e; ~?)
2:
if
exp
(e i(~?)'
.
_
(v)
V"d - exp(Oi,d )
This results in the following derivatives:
O?(Xl :T 11-l)
oO(a)
'J
T
L
t= 2
[(b 8 , ,jb 8 '_1 ,i) - (b 8 '_1 ,i)aij ]
(16)
8?(Xl:T 11?)
80(7r)
?
8?(Xl:T 11?)
8f..li,d
8?(Xl:T 11?)
80(v)
.,d
(8
S1
,i) -
(17)
1fi
T
l)8 st ,i)(Xt,d -
(18)
f..li ,d)/Vi ,d
t= l
1 T
2"l)8st ,i)
[(Xt ,d - f..li ,d)2/Vi ,d -
IJ
(19)
t= l
When chained with the error signal backpropagated from the output, these derivatives give us the direction in which to move the parameters of each HMM in order
to increase the log probability of the correct classification of the sequence.
6
Experiments
To evaluate the relative merits of the RDN, we compared it against an Alphanet
on a speaker identification task. The data was taken from the CSLU 'Speaker
Recognition' corpus. It consisted of 12 speakers uttering phrases consisting of 6
different sequences of connected digits recorded multiple times (48) over the course
of 12 recording sessions. The data was pre-emphasized and Fourier transformed
in 32ms frames at a frame rate of lOms. It was then filtered using 24 bandpass,
mel-frequency scaled filters. The log magnitude filter response was then used as the
feature vector for the HMM's. This pre-processing reduced the data dimensionality
while retaining its spectral structure.
While mel-cepstral coefficients are typically recommended for use with axis-aligned
Gaussians, they destroy the spectral structure of the data, and we would like to
allow for the possibility that of the many HMM's some of them will specialize on
particular sub-bands of the frequency domain. They can do this by treating the
variance as a measure of the importance of a particular frequency band - using
large variances for unimportant bands, and small ones for bands to which they pay
particular attention.
We compared the RDN with an Alphanet and three other models which were implemented as controls. The first of these was a network with a similar architecture
to the RDN (as shown in figure 2), except that instead of fixed connections of ?1,
the hidden units have a set of adaptable weights to all M of the HMM's. We refer
to this network as a comparative density net (CDN). A second control experiment
used an architecture similar to a CDN without the hidden layer, i.e. there is a single
layer of adaptable weights directly connecting the HMM's with the softmax output
units. We label this architecture a CDN-l. The CDN-l differs from the Alphanet
in that each softmax output unit has adaptable connections to the HMM's and we
can vary the number of HMM's, whereas the Alphanet has just one HMM per class
directly connected to each softmax output unit. Finally, we implemented a version
of a network similar to an Alphanet, but using a mixture of Gaussians as the input density model. The point of this comparison was to see if the HMM actually
achieves a benefit from modelling the temporal aspects of the speaker recognition
task.
In each experiment an RDN constructed out of a set of, M, 4-state HMM's was
compared to the four other networks all matched to have the same number of free
parameters, except for the MoGnet. In the case of the MoGnet, we used the same
number of Gaussian mixture models as HMM's in the Alphanet, each with the
same number of hidden states. Thus, it has fewer parameters, because it is lacking
the transition probabilities of the HMM. We ran the experiment four times with
a)
0.95
~
~
0.9
b)
~
0.95
e
E=:l
~
0.9
e =
0.85
0.85
0.8
0.8
0.75
0.75
8
0.7
0.65
0
0.7
0.6
0.6
0.55
Alphanet
MaGnet
CDN
EJ
0.65
0.55
RDN
B
CDN-1
RDN
Alphanet
Architecture
C)
$
D
~
d)
e
~
0.9
8
*
c
0
CDN-1
~
*0.8
?in
gj
Ci
gj
CiO.5
B
MeG net
Architecture
CDN
U
gO.7
~
~0.6
?in
Alphanet
~
a:
~
~O.8
RDN
CDN
~
a:
0.6
MaGnet
Architecture
CDN-1
8
0.4
0.3
RDN
Alphanet
MeGnet
CDN
CDN-1
Architecture
Figure 3: Results of the experiments for an RDN with (a) 12, (b) 16, (c) 20 and
(d) 24 HMM's.
values of M of 12, 16, 20 and 24. For the Alphanet and MoGnet we varied the
number of states in the HMM's and the Gaussian mixtures, respectively. For the
CDN model we used the same number of 4-state HMM's as the RDN and varied
the number of units in the hidden layer of the network. Since the CDN-1 network
has no hidden units, we used the same number of HMM's as the RDN and varied
the number of states in the HMM. The experiments were repeated 10 times with
different training-test set splits. All the models were trained using 90 iterations of
a conjugate gradient optimization procedure [6] .
7
Results
The boxplot in figure 3 shows the results of the classification performance on the
10 runs in each of the 4 experiments. Comparing the Alphanet and the RDN we
see that the RDN consistently outperforms the Alphanet. In all four experiments
the difference in their performance under a paired t-test was significant at the level
p < 0.01. This indicates that given a classification network with a fixed number of
parameters, there is an advantage to using many small HMM 's and using all the
pairwise information about an observed sequence, as opposed to using a network
with a single large HMM per class.
In the third experiment involving the MoGnet we see that its performance is comparable to that of the Alphanet. This suggests that the HMM's ability to model the
temporal structure of the data is not really necessary for the speaker classification
task as we have set it Up.3 Nevertheless, the performance of both the Alphanet and
3If we had done text-dependent speaker identification, instead of multiple digit phrases
the MoGnet is less than the RDN.
Unfortunately the CDN and CDN-l networks perform much worse than we expected. While we expected these models to perform similarly to the RDN, it seems
that the optimization procedure takes much longer with these models. This is probably because the small initial weights from the HMM's to the next layer severely
attenuate the backpropagated error derivatives that are used to train the HMM's.
As a result the CDN networks do not converge properly in the time allowed.
8
Conclusions
We have introduced relative density networks, and shown that this method of discriminatively learning many small density models in place of a single density model
per class has benefits in classification performance. In addition, there may be a
small speed benefit to using many smaller HMM 's compared to a few big ones.
Computing the probability of a sequence under an HMM is order O(TK 2 ), where T
is the length of the sequence and K is the number of hidden states in the network.
Thus, smaller HMM 's can be evaluated faster. However, this is somewhat counterbalanced by the quadratic growth in the size of the hidden layer as M increases.
Acknowledgments
We would like to thank John Bridle, Chris Williams, Radford Neal, Sam Roweis ,
Zoubin Ghahramani, and the anonymous reviewers for helpful comments.
References
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Signal Processing, pp. 49- 53, 1986.
[3] J. Bridle, "Training stochastic model recognition algorithms as networks can
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gradient code available from http ://www .gatsby.ucl.ac.uk/~edward/code/.
then this might have made a difference.
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1,249 | 2,138 | Scaling of Probability-Based Optimization
Algorithms
J. L. Shapiro
Department of Computer Science University of Manchester
Manchester, M13 9PL U.K. [email protected]
Abstract
Population-based Incremental Learning is shown require very sensitive scaling of its learning rate. The learning rate must scale with
the system size in a problem-dependent way. This is shown in two
problems: the needle-in-a haystack, in which the learning rate must
vanish exponentially in the system size, and in a smooth function
in which the learning rate must vanish like the square root of the
system size. Two methods are proposed for removing this sensitivity. A learning dynamics which obeys detailed balance is shown to
give consistent performance over the entire range of learning rates.
An analog of mutation is shown to require a learning rate which
scales as the inverse system size, but is problem independent.
1
Introduction
There has been much recent work using probability models to search in optimization
problems. The probability model generates candidate solutions to the optimization
problem. It is updated so that the solutions generated should improve over time.
Usually, the probability model is a parameterized graphical model , and updating
the model involves changing the parameters and possibly the structure of the model.
The general scheme works as follows,
? Initialize the model to some prior (e.g. a uniform distribution);
? Repeat
- Sampling step: generate a data set by sampling from the probability
model;
- Testing step: test the data as solutions to the problem;
- Selection step: create a improved data set by selecting the better
solutions and removing the worse ones;
- Learning step: create a new probability model from the old model
and the improved data set (e.g. as a mixture of the old model and the
most likely model given the improved data set);
? until (stopping criterion met)
Different algorithms are largely distinguished by the class of probability models
used. For reviews of the approach including the different graphical models which
have been used, see [3, 6]. These algorithms have been called Estimation of Distribution Algorithms (EDA); I will use that term here.
EDAs are related to genetic algorithms; instead of evolving a population, a generative model which produces the population at each generation is evolved. A
motivation for using EDAs instead of GAs is that is that in EDAs the structure of
the graphical model corresponds to the form of the crossover operator in GAs (in
the sense that a given graph will produce data whose probability will not change
much under a particular crossover operator). If the EDA can learn the structure of
the graph, it removes the need to set the crossover operator by hand (but see [2]
for evidence against this).
In this paper, a very simple EDA is considered on very simple problems. It is shown
that the algorithm is extremely sensitive to the value of learning rate. The learning
rate must vanish with the system size in a problem dependent way, and for some
problems it has to vanish exponentially fast. Two correctives measures are considered: a new learning rule which obeys detailed balance in the space of parameters,
and an operator analogous to mutation which has been proposed previously.
2
The Standard PBIL Algorithm
The simplest example of a EDA is Population-based Incremental Learning (PBIL)
which was introduced by Baluja [1]. PBIL uses a probability model which is a
product of independent probabilities for each component of the binary search space.
Let Xi denote the ith component of X, an L-component binary vector which is a
state of the search space. The probability model is defined by the L-component
vector of parameters 'Y ~), where 'Yi(t ) denotes the probability that Xi = 1 at time
t.
The algorithm works as follows,
? Initialize 'Yi(O) = 1/2 for all i;
? Repeat
- Generate a population of N strings by sampling from the binomial
distribution defined by 1(t).
- Find the best string in the population x*.
- Update the parameters 'Yi(t + 1) = 'Yi(t) + a[xi - 'Yi (t)] for all i.
? until (stopping criterion met)
The algorithm has only two parameters, the size of the population N and the
learning parameter a.
3
3.1
The sensitivity of PBIL to the learning rate
PBIL on a flat landscape
The source of sensitivity of PBIL to the learning rate lies in its behavior on a flat
landscape. In this case all vectors are equally fit , so the "best" vector x* is a random
vector and its expected value is
(1)
(where (-) denotes the expectation operator) Thus, the parameters remain unchanged on average. In any individual run, however, the parameters converge
rapidly to one of the corners of the hypercube. As the parameters deviate from
1/2 they will move towards a corner of the hypercube. Then the population generated will be biased towards that corner, which will move the parameters closer
yet to that corner, etc. All of the corners of the hypercube are attractors which,
although never reached, are increasingly attractive with increasing proximity. Let
us call this phenomenon drift. (In population genetics, the term drift refers to the
loss of genetic diversity due to finite population sampling. It is in analogy to this
that the term is used here.)
Consider the average distance between the parameters and 1/2,
D(t) ==
1 (1
L 2: "2 - 'Yi (t)
?
)2
(2)
Solving this reveals that on average this converges to 1/4 with a characteristic time
T
= -1/ 10g(1 - 0: 2) ~ 1/0: 2 for 0: ~ O.
(3)
The rate of search on any other search space will have to compete with drift.
3.2
PBIL and the needle-in-the haystack problem
As a simple example of the interplay between drift and directed search, consider the
so-called needle-in-a-haystack problem. Here the fitness of all strings is 0 except for
one special string (the "needle") which has a fitness of 1. Assume it is the string
of all 1 'so It is shown here that PBIL will only find the needle if 0: is exponentially
small, and is inefficient at finding the needle when compared to random search.
rrf=1
'Yi(t).
Consider the probability of finding the needle at time t, denoted O(t) =
Consider times shorter than T where T is long enough that the needle may be
found multiple times, but 0:2T -+ 0 as L -+ 00. It will be shown for small 0: that
when the needle is not found (during drift), 0 decreases by an amount 0: 2LO/2,
whereas when the needle is found, 0 increases by the amount o:LO. Since initially,
the former happens at a rate 2L times greater than the latter, 0: must be less than
2 - (L - 1) for the system to move towards the hypercube corner near the optimum,
rather than towards a random corner.
When the needle is not found, the mean of O(t) is invariant, (O(t + 1)) = O(t).
However, this is misleading, because 0 is not a self-averaging quantity; its mean
is affected by exponentially unlikely events which have an exponentially big effect.
A more robust measure of the size of O(t) is the exponentiated mean of the log of
O(t) . This will be denoted by [0] == exp (log 0). This is the appropriate measure of
the central tendency of a distribution which is approximately log-normal [4], as is
expected of O(t) early in the dynamics, since the log of 0 is the sum of approximately
independent quantities.
The recursion for 0 expanded to second order in 0: obeys
[O(t
+ 1)] =
{
[O(t)] [1
[O(t)] [1
- 10:2 L] .
+ ~L + ~'0:2 L(L -
needle not found
1)] ;
needle found.
(4)
In these equations, 'Yi(t) has also been expanded around 1/2.
Since the needle will be found with probability O(t) and not found with probability
1 - O(t), the recursion averages to,
[O(t + 1)] = [O(t)] (1 -
~0:2 L) + [0(t)]2
[O:L -
~0:2 L(L + 1)] .
(5)
The second term actually averages to [D(t)] (D(t)) , but the difference between (D)
and [D] is of order 0:, and can be ignored.
Equation (5) has a stable fixed point at 0 and an unstable fixed point at 0:/2 +
O( 0: 2 L). If the initial value of D(O) is less than the unstable fixed point, D will
decay to zero. If D(O) is greater than the unstable fixed point, D will grow. The
initial value is D(O) = 2- ?, so the condition for the likelihood of finding the needle
to increase rather than decrease is 0: < 2-(?-1).
1.1 ,-----~-~--~-~--,_________,
120
a
Figure 1: Simulations on PBIL on needle-in-a-haystack problem for L = 8,10,11,12
(respectively 0, +, *, 6). The algorithm is run until no parameters are between 0.05
and 0.95, and averaged over 1000 runs. Left: Fitness of best population member at
convergence versus 0:. The non-robustness of the algorithm is clear; as L increases,
0: must be very finely set to a very small value to find the optimum. Right: As
previous, but with 0: scaled by 2?. The data approximately collapses, which shows
that as L increases, 0: must decrease like 2-? to get the same performance.
Figure 1 shows simulations of PBIL on the needle-in-a-haystack problem. These
confirm the predictions made above, the optimum is found only if 0: is smaller than
a constant times 2?. The algorithm is inefficient because it requires such small 0:;
convergence to the optimum scales like 4?. This is because the rate of convergence
to the optimum goes like Do:, both of which are 0(2-?).
3.3
PBIL and functions of unitation
One might think that the needle-in-the-haystack problem is hard in a special way,
and results on this problem are not relevant to other problems. This is not be true,
because even smooth functions have fiat subspaces in high dimensions. To see this,
consider any continuous, monotonic function of unit at ion u, where u =
L~ Xi , the
number of 1 's in the vector. Assume the the optimum occurs when all components
are l.
t
The parameters 1 can be decomposed into components parallel and perpendicular
to the optimum. Movement along the perpendicular direction is neutral, Only
movement towards or away from the optimum changes the fitness. The random
strings generated at the start of the algorithm are almost entirely perpendicular to
the global optimum, projecting only an amount of order 1/..JL towards the optimum.
Thus, the situation is like that of the needle-in-a-haystack problem. The perpendicular direction is fiat, so there is convergence towards an arbitrary hypercube corner
with a drift rate,
TJ.. '"
a?
(6)
from equation (3). Movement towards the global optimum occurs at a rate,
a
Til '" VL?
(7)
Thus, a must be small compared to l/VL for movement towards the global optimum
to win.
A rough argument can be used to show how the fitness in the final population
depends on a. Making use of the fact that when N random variables are drawn
from a Gaussian distribution with mean m and variance u 2 , the expected largest
value drawn is m + J2u 2 10g(N) for large N (see, for example, [7]) , the Gaussian
approximation to the binomial distribution, and approximating the expectation of
the square root as the square root of the expectation yields,
(u(t
+ 1)) =
(u(t))
+ aJ2 (v(t)) 10g(N),
(8)
-b
where v(t) is the variance in probability distribution, v(t) =
L i Ii (t)[l - li(t)].
Assuming that the convergence of the variance is primarily due to the convergence
on the flat subspace, this can be solved as,
(u(oo))
~
Jlog(N)
1
"2 + aV'iL .
(9)
The equation must break down when the fitness approaches one, which is where the
Gaussian approximation to the binomial breaks down.
0.9
0 .9
0 .8
0 .8
0.7
0 .7
0 .6
~
~ 0.6
~ 0.5
u..
I
NO .5
'" 0.4
0.4
0.3
0 .3
0 .2
0 .2
0.1
0
0
0.2
0.4
0.6
0.8
20
a
Figure 2: Simulations on PBIL on the unitation function for L = 16,32,64,128,256
(respectively D , 0, +, *, 6) . The algorithm is run until all parameters are closer to
1 or 0 than 0.05, and averaged over 100 runs. Left: Fitness of best population
member at convergence versus a. The fitness is scaled so that the global optimum
has fitness 1 and the expected fitness of a random string is O. As L increases, a
must be set to a decreasing value to find the optimum. Right: As previous, but
with a scaled by VL. The data approximately collapses, which shows t hat as L
increases, a must decrease like VL to get the same performance. The smooth curve
shows equation (9).
Simulations of PBIL on the unitation function confirm these predictions. PBIL fails
to converge to the global optimum unless a is small compared to l/VL. Figure 2
shows the scaling of fitness at convergence with aVL, and compares simulations
with equation (9).
4
Corrective 1 -
Detailed Balance PBIL
One view of the problem is that it is due to the fact that the learning dynamics
does not obey detailed balance. Even on a flat space, the rate of movement of
the parameters "Yi away from 1/2 is greater than the movement back. It is wellknown that a Markov process on variables x will converge to a desired equilibrium
distribution 7r(x) if the transition probabilities obey the detailed balance conditions,
w(x'lx)7r(x) = w(xlx')7r(x'),
(10)
where w(x'lx) is the probability of generating x' from x. Thus, any search algorithm
searching on a flat space should have dynamics which obeys,
w(x'lx)
= w(xlx'),
(11)
and PEIL does not obey this. Perhaps the sensitive dependence on a would be
removed if it did.
There is a difficulty in modifying the dynamics of PBIL to satisfy detailed balance,
however. PEIL visits a set of points which varies from run to run, and (almost)
never revisits points. This can be fixed by constraining the parameters to lie on a
lattice. Then the dynamics can be altered to enforce detailed balance.
Define the allowed parameters in terms of a set of integers ni. The relationship
between them is.
I - ~(1 - a)ni, ni > 0;
{
(12)
"Yi =
!(1- a) lni l,
ni < 0;
ni = O.
2'
Learning dynamics now consists of incrementing and decrementing the n/s by 1;
when xi = 1(0) ni is incremented (decremented).
Transforming variables via equation (12), the uniform distribution in "Y becomes in
n,
P (n) = _a_(I_ a) lnl.
(13)
2-a
4.0.1
Detailed balance by rejection sampling
One of the easiest methods for sampling from a distribution is to use the rejection
method. In this , one has g(x'lx) as a proposal distribution; it is the probability of
proposing the value x' from x. Then, A(x'lx) is the probability of accepting this
change. Detailed balance condition becomes
g(x'lx)A(x'lx)7r(x) = g(xlx')A(xlx')7r(x') .
(14)
For example, the well-known Metropolis-Hasting algorithm has
A(x'lx) = min
(1, :~~}:(~}I~})'
(15)
The analogous equations for PEIL on the lattice are,
A(n
+ lin)
A(n-lln)
=
. [1- "Y (n+l)
]
mm
"Y(n)
(1 - a), 1
(16)
min[{~;(~~(1-a),I].
(17)
In applying the acceptance formula, each component is treated independently. Thus,
moves can be accepted on some components and not on others.
4.0.2
Results
Detailed Balance PBIL requires no special tuning of parameters, at least when
applied to the two problems of the opening sections. For the needle-in-a-haystack,
simulations were performed for 100 values of (): between 0 and 0.4 equally spaced for
L = 8,9,10,11,12; 1000 trials of each, population size 20, with the same convergence
criterion as before, simulation halts when all "Ii'S are less than 0.05 or greater than
0.95. On none of those simulations did the algorithm fail to contain the global
optimum in the final population.
For the function of unitation, Detailed Balance PBIL appears to always find the
optimum if run long enough. Stopping it when all parameters fell outside the range
(0.05,0.95), the algorithm did not always find the global optimum. It produced
an average fitness within 1% of the optimum for (): between 0.1 and 0.4 and L =
32, 64,128,256 over a 100 trials, but for learning rates below 0.1 and L = 256 the
average fitness fell as low as 4% below optimum. However, this is much improved
over standard PBIL (see figure 2) where the average fitness fell to 60% below the
optimum in that range.
5
Corrective 2 -
Probabilistic mutation
Another approach to control drift is to add an operator analogous to mutation in
GAs. Mutation has the property that when repeatedly applied, it converges to a
random data set. Muhlenbein [5] has proposed that the analogous operator ED As
estimates frequencies biased towards a random guess. Suppose ii is the fraction of
l's at site i. Then, the appropriate estimate of the probability of a 1 at site i is
ii + m
(18)
"Ii = 1 + 2m'
where m is a mutation-like parameter. This will be recognized as the maximum
posterior estimate of the binomial distribution using as the prior a ,a-distribution
with both parameters equal to mN + 1; the prior biases the estimate towards 1/2.
This can be applied to PBIL by using the following learning rule,
(
"Ii t
+
1)
"Ii(t)
=
+ (): [x; - "Ii (t)] + m
1 + 2m
.
(19)
With m = 0 it gives the usual PBIL rule; when repeatedly applied on a flat space
it converges to 1/2.
Unlike Detailed Balance PBIL, this approach does required special scaling of the
learning rate, but the scaling is more benign than in standard PBIL and is problem
independent. It is determined from three considerations. First, mutation must
be large enough to counteract the effects of drift towards random corners of the
hypercube. Thus, the fixed point of the average distance to 1/2, (D(t + 1)) defined
in equation (2) , must be sufficiently close to zero. Second, mutation must be small
enough that it does not interfere with movement towards the parameters near the
optimum when the optimum is found. Thus, the fixed point of equation (19) must be
sufficiently close to 0 or 1. Finally, a sample of size N sampled from the fixed point
distribution near the hypercube corner containing the optimum should contain the
optimum with a reasonable probability (say greater than 1 - e- 1 ). Putting these
considerations together yields,
logN
m
():
-L- ? -(): ? -.
4
(20)
5.1
Results
To satisfy the conditions in equation 20, the mutation rate was set to m ex: a 2 ,
and a was constrained to be smaller than log (N)/L. For the needle-in-a-haystack,
the algorithm behaved like Detailed Balance PElL. It never failed to find the optimum for the needle-in-a-haystack problems for the sizes given previously. For the
functions of unitation, no improvement over standard PBIL is expected, since the
scaling using mutation is worse, requiring a < 1/ L rather than a < 1/..fL. However, with tuning of the mutation rate, the range of a's with which the optimum
was always found could be increased over standard PBIL.
6
Conclusions
The learning rate of PBIL has to be very small for the algorithm to work, and
unpredictably so as it depends upon the problem size in a problem dependent way.
This was shown in two very simple examples. Detailed balance fixed the problem
dramatically in the two cases studied. Using detailed balance, the algorithm consistently finds the optimum over the entire range of learning rates. Mutation also fixed
the problem when the parameters were chosen to satisfy a problem-independent set
of inequalities.
The phenomenon studied here could hold in any EDA, because for any type of
model, the probability is high of generating a population which reinforces the move
just made. On the other hand, more complex models have many more parameters, and also have more sources of variability, so the issue may be less important.
It would be interesting to learn how important this sensitivity is in EDAs using
complex graphical models.
Of the proposed correctives, detailed balance will be more difficult to generalize to
models in which the structure is learned. It requires an understanding of algorithm's
dynamics on a flat space, which may be very difficult to find in those cases. The
mutation-type operator will easier to generalize, because it only requires a bias
towards a random distribution. However, the appropriate setting of the parameters
may be difficult to ascertain.
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[7] Jonathan L. Shapiro and Adam Priigel-Bennett. Maximum entropy analysis of genetic
algorithm operators. Lecture Notes in Computer Science, 993:14- 24, 1995.
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1,250 | 2,139 | Stability-Based Model Selection
Tilman Lange, Mikio L. Braun, Volker Roth, Joachim M. Buhmann
(lange,braunm,roth,jb)@cs.uni-bonn.de
Institute of Computer Science, Dept. III,
University of Bonn
R?omerstra?e 164, 53117 Bonn, Germany
Abstract
Model selection is linked to model assessment, which is the problem of
comparing different models, or model parameters, for a specific learning
task. For supervised learning, the standard practical technique is crossvalidation, which is not applicable for semi-supervised and unsupervised
settings. In this paper, a new model assessment scheme is introduced
which is based on a notion of stability. The stability measure yields an
upper bound to cross-validation in the supervised case, but extends to
semi-supervised and unsupervised problems. In the experimental part,
the performance of the stability measure is studied for model order selection in comparison to standard techniques in this area.
1 Introduction
One of the fundamental problems of learning theory is model assessment: Given a specific
data set, how can one practically measure the generalization performance of a model trained
to the data. In supervised learning, the standard technique is cross-validation. It consists in
using only a subset of the data for training, and then testing on the remaining data in order to
estimate the expected risk of the predictor. For semi-supervised and unsupervised learning,
there exist no standard techniques for estimating the generalization of an algorithm, since
there is no expected risk. Furthermore, in unsupervised learning, the problem of model
order selection arises, i.e. estimating the ?correct? number of clusters. This number is part
of the input data for supervised and semi-supervised problems, but it is not available for
unsupervised problems.
We present a common point of view, which provides a unified framework for model assessment in these seemingly unrelated areas of machine learning. The main idea is that an
algorithm generalizes well, if the solution on one data set has small disagreement with the
solution on another data set. This idea is independent of the amount of label information
which is supplied to the problem, and the challenge is to define disagreement in a meaningful way, without relying on additional assumptions, e.g. mixture densities. The main
emphasis lies on developing model assessment procedures for semi-supervised and unsupervised clustering, because a definitive answer to the question of model assessment has
not been given in these areas.
In section 3, we derive a stability measure for solutions to learning problems, which allows us to characterize the generalization in terms of the stability of solutions on different
sets. For supervised learning, this stability measure is an upper bound to the 2-fold cross-
validation error, and can thus be understood as a natural extension of cross-validation to
semi-supervised and unsupervised problems.
For experiments (section 4), we have chosen the model order selection problem in the
unsupervised setting, which is one of the relevant areas of application as argued above. We
compare the stability measure to other techniques from the literature.
2 Related Work
For supervised learning problems, several notions of stability have been introduced ([10],
[3]). The focus of these works lies on deriving theoretical generalization bounds for supervised learning. In contrast, this work aims at developing practical procedures for model
assessment, which are also applicable in semi- and unsupervised settings. Furthermore, the
definition of stability developed in this paper does not build upon the cited works.
Several procedures have been proposed for inferring the number of clusters of which we
name a few here. Tibshirani et al. [14] propose the Gap Statistic that is applicable to Euclidian data only. Given a clustering solution, the total sum of within-cluster dissimilarities is
computed. This quantity computed on the original data is compared with the average over
data which was uniformly sampled from a hyper-rectangle containing the original data.
The which maximizes the gap between these two quantities is the estimated number of
clusters. Recently, resampling-based approaches for model order selection have been proposed that perform model assessment in the spirit of cross validation. These approaches
share the idea of prediction strength or replicability as a common trait. The methods exploit the idea that a clustering solution can be used to construct a predictor, in order to
compute a solution for a second data set and to compare the computed and predicted class
memberships for the second data set. In an early study, Breckenridge [4] investigated the
usefulness of this approach (called replication analysis there) for the purpose of cluster
validation. Although his work does not lead to a directly applicable procedure, in particular
not for model order selection, his study suggests the usefulness of such an approach for
the purpose of validation. Our method can be considered as a refinement of his approach.
Fridlyand and Dudoit [6] propose a model order selection procedure, called Clest, that also
builds upon Breckenridge?s work. Their method employs the replication analysis idea by
repeatedly splitting the available data into two parts. Free parameters of their method are
the predictor, the measure of agreement between a computed and a predicted solution and
a baseline distribution similar to the Gap Statistic. Because these three parameters largely
influence the assessment, we consider their proposal more as a conceptual framework than
as a concrete model order estimation procedure. In particular, the predictor can be chosen
independent of the clustering algorithm which can lead to unreliable results (see section
3). For the experiments in section 4, we used a linear discriminant analysis classifier, the
Fowlkes-Mellows index for solution comparison (c.f. [9, 6]) and the baseline distribution of
the Gap Statistic. Tibshirani et al. [13] formulated a similar method (Prediction Strength)
for inferring the number of clusters which is based on using nearest centroid predictors.
Roughly, their measure of agreement quantifies the similarity of two clusters in the computed and in the predicted solution. For inferring a number of clusters, the least similar
pair of clusters is taken into consideration. The estimated is the largest for which the
similarity is above some threshold value. Note that the similarity for
is always above
this threshold.
3 The Stability Measure
We begin by introducing a stability measure for supervised learning. Then, the stability
measure is generalized to semi-supervised and unsupervised settings. Necessary modifications for model order selection are discussed. Finally, a scheme for practical estimation of
the stability is proposed.
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D 8F= ;
;
8 8= 8F= =
@ B 8KLD 8F= = GM @ B 8ND 8F= GPO @ B 8Q= ND 8F= = G
8ND 8F= =
8= LD 8 8F= LD 8F= =
=
+FR
+R
+ RFS K = +
1PR S +)R TKUWV @ B +)R = 6ED = GXM 1PR S +)R S O T UWV @ B +)R = 6 D +)R S = G (1)
We call the second term the stability of the predictor + and denote its expectation by Y + :
(2)
Y + , KZ\[ T UWV @ B +)R = ] D +)R S = GC^
; means perfect stability
We call the value of Y + stability cost to stress the fact that Y
and
of Y mean
with respect to and
= onlargebothvalues
Z large 6`ainstability.
Z
1PR +)R Taking
9 M ZXY expectations
+ . If + 9 iscobtained
sides yields 1 +_R
risk
b , then 1&R +)R ZXd?efgih)byj 1Pempirical
R + 9 M
dWeFfgih)minimization
jkZ
1PR + 9 over
Ed?eFfsome
gih)j 1 hypothesis
+ , and onesetobtains
Z 1 + R l`mgidWeFh)fj 1 + M Z 1 + R l`nZ
1 R + R 9 M Y + 0
(3)
Stability and Supervised Learning The supervised learning problem is defined as follows. Let
be a sequence of random variables where
are drawn i.i.d. from some probability distribution
. The
are the objects and
are the labels. The task is to find a labeling function
which minimizes the expected risk, given by
, using
only a finite sample of data as input. Here is the so-called loss function. For classification, we take the - -loss defined by
iff
and
else.
A measure of the stability of the labeling function learned is derived as follows. Note that
, since
for three labels , and , it holds that
implies
or
. Now let and be two data sets drawn independently
from the same source, and denote the predictor trained on by . Then, the test risk of
can be bounded by introducing
:
By eq. (3), the stability defined in (2) yields an upper bound on the generalization error.
It can be shown that there exists a converse upper bound, if the minimum is unique and
well-separated, such that
implies
.
Note that the stability measures the disagreement between labels on training data and test
data, both assigned by . This asymmetry arises naturally and directly measures the generalization performance of . Furthermore, the stability can be interpreted as the expected
empirical risk of with respects to the labels computed by itself (compare (1) and (2)).
Therefore, stability measures the self-consistency of . This interpretation is also valid in
the semi-supervised and unsupervised settings. Practical evaluation of the stability amounts
to 2-fold cross-validation. No improvement can therefore be expected in this area. However,
unlike cross-validation, stability can also be defined in settings where no label information
is available. This property of the method will be discussed in the remainder of this section.
Z 1 _+ R o.JdWeFfgh)j 1 +
+
+ +
Y . ;
+
Semi-supervised Learning Semi-supervised learning problems are defined as follows.
of an object
might not be known. This fact is encoded by setting
The label
, since is not a valid label. At least one labeled point must be given for every class.
Furthermore, for the present discussion, we assume that we do not have a fully labeled data
set for testing purposes.
There exist two alternatives in defining the solution to a semi-supervised learning problem.
In the first alternative, the solution is a labeling function defined on the whole object
space as in supervised learning. Then, the stability (eq. (2)) can be readily computed and
measures the confidence for the (unknown) training error.
The second alternative is that the solution is not given by a labeling function on the whole
object space, but only by a labeling function on the training set . Labeling functions
;
$
;
+
R
which are defined on the training set only will be denoted by to stress the difference. The
labeling on will be denoted by
, which is only defined on . As mentioned above,
the stability compares labels given to the training data with predicted labels. In the current
setting, there are no predicted labels, because is defined on the
training set only. One
possibility to obtain predicted labels is to introduce a predictor , which is trained using
to predict labels on the new set . Leaving as a free parameter, we define the
stability for semi-supervised learning as
R
@ B
K
Z
[
Y
, T UW V
semi
=
= ] D RFS = GC^F
(4)
Of course, the choice of influences the value of the stability. We need a condition
on the
prediction step to select . First note that (4) is the expected empirical risk of with respect
to the data source
. Analogously to supervised learning, the minimal attainable
stability
measures the extent to which classes overlap, or how consistent
s
emi
the labels are. Therefore, should be chosen to minimize s emi . Unfortunately, the
construction of non-asymptotically Bayes optimal learning algorithms is extremely difficult
and, therefore, we should not expect that
there exists a universally applicable constructive
procedure for automatically building given an .
In practice, some has to be chosen. This choice will yield larger stability costs, i.e. worse
stability, and can therefore not fake stability. Furthermore, it is often possible to construct
good predictors in practice.
Note that (4) measures the mismatch between the label genera
tor
and
the
predictor
.
Intuitively,
can lead to good stability only if the strategy of and
are similar. For unsupervised learning, as discussed in the next paragraph, the choices for
various standard techniques are natural. For example, -means clustering suggests to use
nearest centroid classification. Minimum spanning tree type clustering algorithms suggest
nearest neighbor classifiers, and finally, clustering algorithms which fit a parametric density
model should use the class posteriors computed by the Bayes rule for prediction.
dWe Y K lR
Y
E
0
Unsupervised Learning The unsupervised learning setting
is given as the problem of
. The solution
is again a function only defined
labeling a finite data set
on . From this definition, it becomes clear that we again need a predictor as in the second
alternative of semi-supervised learning.
For unsupervised learning, another problem arises. Since no specific label values are prescribed for the classes, label indices might be permuted from one instance to another, even
when the partitioning is identical. For example, keeping the same classes, exchanging the
class labels and leads to a new partitioning, which is not structurally different. In other
words, label values are only known up to a permutation. In view of this non-uniqueness of
the representation of a partitioning, we define the permutation relating indices on the first
set to the second set by the one which maximizes the agreement between the classes. The
stability then reads
@B = ] D = G ^F
Z
?
d
e
[
Y ,
h T UWV
(5)
S
Note that the minimization
take place inside the expectation, becausethe permutation
K = .hasIn topractice,
depends on the data
it is not necessary to compute
all permutations,
un
because the problem is solvable by the Hungarian method in
[11].
Model Order Selection The problem of model order selection consists in determining
the number of clusters to be estimated, and exists only in unsupervised learning.
The range of the stability depends on , therefore stability values cannot be compared
for different values of . For unsupervised
learning, the stability minimized over is
bounded from above by
, since for a larger instability, there exists a relabeling
Y
`
Y
which has smaller stability costs. This stability value is asymptotically achieved by the
random predictor which assigns uniformly drawn labels to objects. Normalizing by
the stability of the random predictor yields values independent of . We thus define the
re-normalized stability as
(6)
un
un
un
Y Y Y
9
Resampling Estimate of the Stability In practice, a finite data set
is given,
and the best model should be estimated. The stability is defined in terms of an expectation,
which has to be estimated for practical applications. Estimation of over a hypothesis set
is feasible if has finite VC-dimension, since the VC-dimension for estimating is the
same as for the empirical risk, a fact which is not proved here. In order to estimate the
stability, we propose the following resampling scheme: Iteratively split the data set into
disjoint halves, and compare the solutions on these sets as defined above for the respective
cases. After the model having the smallest value of is determined, train this model again
on the whole data to obtain the result.
Note that it is necessary to split into disjoint subsets, because common points potentially increase the stability artificially. Furthermore, unlike in cross-validation, both sets must have
the same size, because both are used as inputs to training algorithms. For semi-supervised
and unsupervised learning, the comparison might entail predicting labels on a new set, and
for the latter also minimizing over permutation of labels.
b
Y
b
Y
Y
4 Stability for Model Order Selection in Clustering: Experimental
Results
We now provide experimental evidence for the usefulness of our approach to model order
selection, which is one of the hardest model assessment problems. First, the algorithms are
compared for toy data, in order to study the performance of the stability measure under
well-controlled conditions. However, for real-world applications, it does not suffice to be
better than competitors, but one has to provide solutions which are reasonable within the
framework of the application. Therefore, in a second experiment, the stability measure is
compared to the other methods for the problem of clustering gene expression data.
Experiments are conducted using a deterministic annealing variant of -means [12] and
Path-Based Clustering [5] optimized via an agglomerative heuristic. For all data sets, we
average over resamples for
. For the Gap Statistic and Clest1 random
samples are drawn from the baseline. For Clest and Prediction Strength, the number of
resamples
is chosen the same as for our method. The threshold for Prediction Strength is
set to
. As mentioned above, the nearest centroid classifier is employed for the purpose
of prediction when using -means, and a variant of the nearest neighbor classifier is used
for Path-Based Clustering which can be regarded as a combination of Minimum Spanning
Tree clustering and Pairwise Clustering [5, 8].
We compare the proposed stability index of section 3 with the Gap Statistic, Clest and with
Tibshirani?s Prediction Strength method using two toy data sets and a microarray data set
taken from [7]. Table 1 summarizes the estimated number of clusters of each method.
;
;
;
;
;
Toy Data Sets The first data set consists of three fairly well separated point clouds, generated from three Gaussian distributions ( points from the first and the second and
in figure 1(a),
points from the third were drawn). Note that for some , for example
the variance in the stability over different resamples is quite high. This effect is due to the
model mismatch, since for
, the clustering of the three classes depends highly on the
subset selected in the resampling. This means that besides the absolute value of the stability
1
See section 2 for a brief overview over these techniques.
Data Set
Stability
Method
3 Gaussians
3 Rings -means
3 Rings Path-Based
Golub et al. data
Gap
Statistic
;
Clest
Prediction
Strength
?true?
number
or
Table 1: The estimated model orders for the two toy and the microarray data set.
costs, additional information about the fit can be obtained from the distribution of the stability costs over the resampled subsets. For this data set, all methods under comparison are
able to infer the ?true? number of clusters
. Figures 1(d) and 1(a) show the clustered
data set and the proposed stability index. For
, the stability is relatively high, which
is due to the hierarchical structure of the data set, which enables stable merging of the two
smaller sub-clusters.
In the ring data set (depicted in figures 1(e) and 1(f)), one can naturally distinguish three
ring shaped clusters that violate the modeling assumptions of -means since clusters are not
spherically distributed. Here, -means is able to identify the inner circle as a cluster with
. Thus, the stability for this number of clusters is highest (figure 1(b)). All other
methods except Clest infer
for this data set with -means. Applying the proposed
stability estimator with Path-Based Clustering on the same data set yields highest stability
for
, the ?correct? number of clusters (figures 1(f) and 1(c)). Here, all other methods
. The Gap Statistic fails here because it directly incorporates the
fail and estimate
assumption of spherically distributed data. Similarly, the Prediction Strength measure and
Clest (in the form we use here) use classifiers that only support linear decision boundaries
which obviously cannot discriminate between the three ring-shaped clusters. In all these
cases, the basic requirement for a validation scheme is violated, namely that it must not
incorporate additional assumptions about the group structure in a data set that go beyond the
ones of the clustering principle employed. Apart from that, it is noteworthy that the stability
with -means is significantly worse than the one achieved with Path-Based Clustering,
which indicates that the latter is the better choice for this data set.
Application to Microarray Data Recently, several authors have investigated the possibility of identifying novel tumor classes based solely on gene expression data [7, 2, 1].
Golub et al. [7] studied in their analysis the problem of classifying and clustering acute
leukemias. The important question of inferring an appropriate model order remains unaddressed in their article and prior knowledge is used instead. In practice however, such
knowledge is often not available.
Acute leukemias can be roughly divided into two groups, acute myeloid leukemia (AML)
and acute lymphoblastic leukemia (ALL) where the latter can furthermore be subdivided
into B-cell ALL and T-cell ALL. Golub et al. used a data set of 72 leukemia samples (25
AML, 47 ALL of which 38 are B-cell ALL samples)2 . For each sample, gene expression
was monitored using Affymetrix expression arrays.
We apply the preprocessing steps as in Golub et al. resulting in a data set consisting of 3571
genes and 72 samples. For the purpose of cluster analysis, the feature set was additionally
reduced by only retaining the 100 genes with highest variance across samples. This step
is adopted from [6]. The final data set consists of 100 genes and 72 samples. We have
performed cluster analysis using -means and the nearest centroid rule. Figure 2 shows
2
Available at http://www-genome.wi.mit.edu/cancer/
0.8
1
0.7
0.5
0.9
0.8
0.6
0.4
0.7
0.5
0.4
index
index
index
0.6
0.5
0.4
0.3
0.3
0.2
0.3
0.2
0.2
0.1
0.1
0.1
0
0
2
3
4
5
6
7
number of clusters
8
9
0
2
10
(a) The stability index for
the Gaussians data set
with -means.
3
4
5
6
7
number of clusters
8
9
10
2
(b) The stability index for
the three-ring data set with
-means Clustering.
4
5
6
7
number of clusters
8
9
10
(c) The stability index for
the three-ring data set with
Path-Based Clustering.
8
3
5
5
4
4
3
3
2
2
1
1
6
4
2
0
0
?1
?1
?2
?2
?3
?3
0
?2
?4
?4
?6
?4
?2
0
2
4
6
(d) Clustering solution on
the full data set for
.
8
?5
?5
?4
?4
?3
?2
?1
0
1
2
3
4
(e) Clustering solution on
the full data set for
.
5
?5
?5
?4
?3
?2
?1
0
1
2
3
4
5
(f) Clustering solution on
the full data set for
.
Figure 1: Results of the stability index on the toy data (see section 4).
, we estimate the highest stability. We expect
the corresponding stability curve. For
separates AML, B-cell ALL
ALL samples from each
that clustering with
and
ofT-cell
other. With respect to the known ground-truth labels,
the samples (66 samples) are
correctly classified (the Hungarian method is used to map the clusters to the ground-truth).
Of the competitors, only Clest is able to infer the ?correct? number of cluster
while
the Gap Statistic largely overestimates the number of clusters. The Prediction strength does
not provide any reasonable result as it estimates
. Note, that for
similar stability
is achieved. We cluster the data set again
for
and
compare
the
result
with the ALL ?
of the samples (62 samples) are correctly
AML labeling of the data. Here,
identified.
We conclude that our method is able to infer biologically relevant model orders. At the
same time, a is suggested that leads to high accuracy w.r.t. the ground-truth. Hence, our
re-analysis demonstrates that we could have recovered a biologically meaningful grouping
in a completely unsupervised manner.
C
5 Conclusion
The problem of model assessment was addressed in this paper. The goal was to derive a
common framework for practical assessment of learning models. Starting with defining a
stability measure in the context of supervised learning, this measure was generalized to
semi-supervised and unsupervised learning. The experiments concentrated on model or-
0.7
0.6
index
0.5
0.4
0.3
0.2
0.1
0
2
3
4
5
6
7
number of clusters
8
9
10
Figure 2: Resampled stability for the leukemia dataset vs. number of classes (see sec. 4).
der selection for unsupervised learning, because this is the area where the need for widely
applicable model assessment strategies is highest. On toy data, the stability measure outperforms other techniques, when their respective modeling assumptions are violated. On
real-world data, the stability measure compares favorably to the best of the competitors.
Acknowledgments. This work has been supported by the German Research Foundation
(DFG), grants #Buh 914/4, #Buh 914/5.
References
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profiling. Nature, 406(3):536 ? 540, 2000.
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1,251 | 214 | A Neural Network to Detect Homologies in Proteins
A Neural Network to Detect
Homologies in Proteins
Yoshua Bengio
Yannick Pouliot
School of Computer Science
McGill University
Montreal, Canada H3A 2A7
Department of Biology
McGill University
Montreal Neurological Institute
Samy Bengio
Patrick Agin
Departement dlnformatique
Universite de Montreal
Departement d'Informatique
U niversite de Montreal
.A BSTRACT
In order to detect the presence and location of immunoglobulin (Ig) domains from amino acid sequences we built a system
based on a neural network with one hidden layer trained with
back propagation. The program was designed to efficiently
identify proteins exhibiting such domains, characterized by a
few localized conserved regions and a low overall homology.
When the National Biomedical Research Foundation (NBRF)
NEW protein sequence database was scanned to evaluate the
program's performance, we obtained very low rates of false
negatives coupled with a moderate rate of false positives.
1 INTRODUCTION
Two amino acid sequences from proteins are homologous if they can be
aligned so that many corresponding amino acids are identical or have similar
chemical properties. Such subsequences (domains) often exhibit similar three
dimensional structure. Furthemore, sequence similarity often results from
common ancestors. Immunoglobulin (Ig) domains are sets of ,a-sheets bound
423
424
Bengio, Bengio, Pouliot and Agin
by cysteine bonds and with a characteristic tertiary structure. Such domains
are found in many proteins involved in immune, cell adhesion and receptor
functions. These proteins collectively form the immunoglobulin superfamily
(for review, see Williams and Barclay, 1987). Members of the superfamily
often possess several Ig domains. These domains are characterized by wellconserved groups of amino acids localized to specific subregions. Other residues outside of these regions are often poorly conserved, such that there is
low overall homology between Ig domains, even though they are clearly
members of the same superfamily.
Current search programs incorporating algorithms such as the Wilbur-Lipman
algorithm (1983) or the Needleman-Wunsch algorithm (1970) and its modification by Smith and Waterman (1981) are ill-designed for detecting such
domains because they implicitly consider each amino acid to be equally important. This is not the case for residues within domains such as the Ig
domain, since only some amino acids are well conserved, while most are variable. One solution to this problem are search algorithms based upon the statistical occurrence of a residue at a particular position (Wang et al., 1989;
Gribskov et al., 1987). The Profile Analysis set of programs published by the
University of Wisconsin Genetics Computer Group (Devereux et al., 1984)
rely upon such an algorithm. Although Profile Analysis can be applied to
search for domains (c./. Blaschuk, Pouliot & Holland 1990), the output from
these programs often suffers from a high rate of false negatives and positives.
Variations in domain length are handled using the traditional method of penalties proportional to the nuinber of gaps introduced, their length and their position. This approach entails a significant amount of spurious recognition if
there is considerable variation in domain length to be accounted for.
We have chosen to address these problems by training a neural network to
recognize accepted Ig domains. Perceptrons and various types of neural networks have been used previously in biological research with various degrees of
success (cf. Stormo et al., 1982; Qian and Sejnowski, 1988). Our results suggest that they are well suited for detecting relatively cryptic sequence patterns
such as those which characterize Ig domains. Because the design and training
procedure described below is relatively simple, network-based search programs constitute a valid solution to problems such as searching for proteins
assembled from the duplication of a domain.
2 ALGORITHM, NETWORK DESIGN AND TRAINING
The network capitalizes upon data concerning the existence and localization
of highly conserved groups of amino acids characteristic of the Ig domain. Its
design is similar in several respects to neural networks we have used in the
study of speech recognition (Bengio et al., 1989). Four conserved subregions
(designated P1-P4) of the Ig domain homology were identified. These roughly
correspond to ,a-strands B, C, E and F, respectively, of the Ig domain (see
also Williams and Barclay, 1988). Amino acids in these four groups are not
necessarily all conserved, but for each subregion they show a distribution very
different from the distribution generally observed elsewhere in these proteins.
Hence the first and most important stage of the system learns about these
joint distributions. The program scans proteins using a window of 5 residues.
A Neural Network to Detect Homologies in Proteins
The first stage of the system consists of a 2-layer feedforward neural network
(5 X 20 inputs - 8 hidden - 4 outputs; see Figure 1) trained with back propagation (Rumelhart et al., 1986). Better results were obtained for the recognition
of these conserved regions with this architecture than without hidden layer
(similar to a perceptron). The second stage evaluates, based upon the stream
of outputs generated by the first stage, whether and where a region similar to
the Ig domain has been detected. This stage currently uses a simple dynamic
programming algorithm, in which constraints about order of subregions and
distance between them are explicitly programmed. We force the recognizer to
detect a sequence of high values (above a threshold) for the four conserved
regions, in the correct order and such that the sum of the values obtained at
the four recognized regions is greater than a certain threshold. Weak penalties
are applied for violations of distance constraints between conserved subregions (e.g., distance between P1 and P2, P2 and P3, etc) based upon simple
rules derived from our analysis of Ig domains. These rules have little impact if
strong homologies are detected, such that the program easily handles the large
variation in domain size exhibited by Ig domains. It was necessary to explicitly formulate these constraints given the low number of training examples as
well as the assumption that the distance between groups is not a critical
discriminating factor. We have assumed that inter-region subsequences probably do not significantly influence discrimination.
4 output units
representing
4 features of
the Ig domain
8 hidden
units
20
possible
amino
acids
window scanning 5 consecutive residues
Figure 1: Structure of the neural network
425
426
Bengio, Bengio, Pouliot and Agin
filename : A22771.NEW
input sequence name: 19 epsilon chain C region - Human
HOMOLOGY starting at 24
VTLGCLATGYFPEPVMVTWDTGSLNGTTMTLPATTLTLSGHYAT1SLLTVSGAWAKQMFTC
P1
P2
P3
P4
Ending at 84. Score = 3.581
HOMOLOGY starting at 130
1QLLC LVSGYTPGT1NITWLEDGQVMDVD LSTASTTQEGE LASTQSE LTLSQKHWLSDRTYTC
P1
P2
P3
P4
Ending at 192. Score = 3.825
HOMOLOGY starting at 234
PTITCLVVDLAPSKGTVNLTWSRASGKPVNHSTRKEEKQRNGTLTVTSTLPVGTRDW1EGETYQC
P1
P2
P3
P4
Ending at 298. Score = 3.351
HOMOLOGY starting at 340
RTLACLIQNFMPED1SVQWLHNEVQLPDARHSTTQPRKTKGSGFFVFSRLEVTRAEWEQKDEF1C
P1
P2
P3
P4
Ending at 404. Score - 3.402
Figure 2: Sample output from a search of NEW. Ig domains
present within the constant region of an epsilon Ig chain
(NBRF file number A22771) are listed with the position of
P1-P4 (see text). The overall score for each domain is also listed.
As a training set we used a group of 30 proteins comprising bona fide Ig
domains (Williams and Barclay, 1987). In order to increase the size of the
training set, additional sequences were stochastically generated by substituting
residues which are not in critical positions of the domain. These substitutions
were designed not to affect the local distribution of residues to minimize
changes in the overall chemical character of the region.
The program was evaluated and optimized by scanning the NBRF protein databases (PROTEIN and NEW) version 19. Results presented below are based
upon searches of the NEW database (except where otherwise noted) and were
generated with a cutoff value of 3.0. Only complete sequences from vertebrates, insects (including Drosophila melanogaster) and eUkaryotic viruses
were scanned. This corresponds to 2422 sequences out of the 4718 present in
the NEW database. Trial runs with the program indicated that a cutoff threshold of between 2.7 and 3.0 eliminates the vast majority of false positives with
little effect upon the rate of false negatives. A sample output is listed in Figure 2.
3 RESULTS
When the NEW protein sequence database of NBRF was searched as
described above, 191 proteins were identified to possess at least one Ig
domain. A scan of the 4718 proteins comprising the NEW database required
an average of 20 hours of CPU time on a VAX 11/780. This is comparable to
other computationally intensive programs (e.g., Profile Analysis). When run
on a SUN 4 computer, similar searches required 1.3 hours of CPU time. This
is sufficiently fast to allow the user to alter the cutoff threshold repeatedly
when searching for proteins with low homology.
A Neural Network to Detect Homologies in Proteins
Table 1: Output from a search of the NEW protein sequence database.
Domains are sorted according to overall score.
3.0017 ClAss II hlstocompatlb. ant'fen, Hl,A-OR bec:a- I chain precursor (REM) . Hu,.,.n 3.4295 " bPPII chain V region - Mouse H 37-10
3.014& NonsJMdf'k: cross?ructtng an,..,. precursor? Human
3.429519 bppa chlln V region - Moule H37-&4
3.0161 ,..teffl-dertylld growth factor receptor precursor ? Moun
3.4295 Ig kappa chlln V regions - Moun Hn-C6 and H22f>2S
3.0164 Til class I hlscocomp.alib. ,nUgen. Til-, alpha chain ? Mouse
3.4331 T-uU rectPtOr alpha chain precursor V '~'on IP71) . Mouse
3.0164 Ta. class I hlstocomPliUb. ant.n. Tj? b _Iphll chain? Moust
3.0223 Vttronectln recept:or alph_ ,h.n precursor ' HUman
3.0226 T-CtllsurfKe gtycoprotetn ly-3 precursor ' Moun
3.0244 Klnase-,". trlnstormlng ploteln (srd (EC 2.7. 1.? ) . AVI,an urcomil VirUS
),0350 It alptt.." chlln C region - Humin
J.OJ50 It alptt.., I chain C regIOn - Human
3.0J?0 It alph..,2 chlln C region. A2m( I) lilorype ' Human
3.0-409 Gr.nulocyte-macroph.ge colonv?sUmulaUng flCcor I precursor - Moust
J.04I' HLA dass 1 hlstocomparlb . ant~en. Ilph. chain precursor' Human
3.0492 HADH-ubtquktOne ox~or.uctase (EC 1.6.S.3). chlln 5 - Fruit fly (Drosophila)
3.0501 NAIlH?.biq.lnOn??? Ido.ed.cu.. (Ee I.6.S.31. chain I ? F.... IIy (O.os.pllilal
3.0511 HLA clas ? htstocomp.Mlb. ant'9tft. DP bet. chain precursor - clone
3.0511 HLA cia, ? hlstocompatlb. ant...,. DP4 bet. chain ptecunor - HUmin
3.0SISHLA cia, ? hlstocomplltlb . ? nt.... OPW4 bet. I chain ptecursor - Human
3.0520 Class n histocompaUb. ? nt'gen. HLA-OQ beta ch.ln precursor (REM) - Human
].0561 rroteln' ryroSlne kinase (Ee 2. 7.1.1 12). lymphocyte - Moun
].0669 H-2 clas. hJstoc:ompaUb. ant'9tft, A?beca?2 chain ptecursor - Mouse
3.072] T-cell ree.,.or pnvna cham precursor'll 'eglOn (MNCI) - Mouse
3.072J T~ reeepeor glfTV1'\a cham ptecursor 'II regAon IRAeII} - Mouse
3.072J T-cell ree_or glfTV1'\a cha... ptecursor 'II 'eglon IRAe4) . Mouse
3.072J T-eeR ree_or glfTV1'\a chain ptecursor 'II region 'RAC42) . Mouse
3.072J T-c:eII ree_or glfTV1'\a chatn ptecursor 'II region (RACSo) . Mouse
3.0750 T-ctl r?_or bet. cha'" V region (C.F~ ? Mouse
J.07&01g hefty Chain V retlon ? Moule 251 .3
3.0711 T-col(
bOlA ch .. n "'eglon (SUp?T 'I . Hum...
3.0711 H?Z cia. I hi>.ocompotlb. ....Igen. Q7 olpllo ch.ln " ..c.rs" . Mo...
3.0717 "?2 class I hlsrocompatlb . ? nr'left, OS IIlpha ch.ln precursor - Mouse
3.0912 MytiIn-assoclatld gtycoptoteln 11236 long form precurso' - Rilt
] .0912 MyefIIt-.socl.rld g~oproteln 1&2]6 shon form precursor - R.t
3.09&2 MyoIIrtoesoc ... ed 91\'<.pr ???ln precu .. or. b,aln . Ra.
3.09&2 ~soc".ed "'90 glyc.pr ....n prec ...... Rat
3.0991 Closs I hls.ocompotlb. ""'VOn. BolA .Iph. ch.ln prec ????? (BLI?51 . Bovl....
J.099aOass I htstocompatlb_ antigen, loLA alpha chilln precursor (BU ?]) - Bovtne
J. I 04& H-2 clas I hlstocompatlb. IIntM1en. K?" a6pha chilln precursor? Mouse
J . I0&61g h.vy chain precursor'll regIOn - Mous. VCAM3 2
J. I 128 T-cell rcepeor .Ipha ,haln precursor V region (MO I 3~ - Mouse
J. II29 T<ell ree_or detta chain V region ION?4) . Moule
3.1192 T<ell rcepeor bet. chain precursor'll region IVAk) - Mouse
3.126S T -c:eU ree_or glfTV1'\a cham ptecursor 'II regIOn IK20) ? Human
3.1 J47 T-c:eU ree_or alph. chilln precursor V region (HAPOS) - Human
3. 1623 T-cen surface gtycoprotetn COl ptecurs.or . Human
3.1623: T-c:eI surface gtvcoprolelft COl prottln precunor . Human
3.1776 .... e-nma-3 chain C reQ1on . C]mlb) allotype Humin
3.1931 HypothetICal proc",n HQlf 2 ? C..,tome.;JaloY1rLls Istraln AD 169)
3.2041 SodIum channel prottln II Rat
3.20441g huvy chain'll re.;Jlon Afr"An cla*fd "og
3.2141 SURF- I protein' Mouu
].2207 T-cell recepc:or alpha chain pr~ Uls.or \0' 1f"910n (HAP 10~ . Human
3.2300 1et.-2-mlCroglobulin pre<",r~Or Hun-,an
] .2300 Beu-2-mlCroglobulln. modtflfd Human
3.2106 rreonancy-spt:Clflc bela I Qlyc opror~tn E prf'(u rs or Human
3.2344lgE Fc receptor Ilpha ,haln prKufSor Hurnan
3.2420 T-c:ell surflCe Qtycoprotetn C02 pte<unor Pat
3.2422 H?2 class N htuocompaub .nIl9~n I A I~OOI bf'ta cham precursor - Moun
3.25;2 HLA elms II hlstocompaub .ntlgen . op."..e aloha I cham precursor? Human
3.2552 HLA class II hlsro(ompat lb ,ntlgf'n ~ 8 .Ipha O'laln precursor' Human
3.2654 T-c,1/ surface glvcoproteln CO&' JI K (hJlln pfKur sor . Rat
3.2726 Myelin PO ptoteln? Bovtne
J.21141t .Iptt.., 1 chain e regton Huma"
3.21141g .Iph.., I chaIR C recJlon HumM
3.2120 Thy-I membrane glycoprotein p,e<u,~or Mouse
3.2&40 5mh clas II hlstocompilub anogen prlKurSor Ehrenberg smote-rat
1.3039 X-lInkld chronk: granulomatous dlSeast plotem Human
3.3013 rregnartCy-speclflC bela I Vl lyc oprot<t:ln ( pr<t:(unor Human
) . 3013 Prt9n ....cy-speclfiC beta' I gtycoprotetn (J pr KurSOf . Humiln
3. 30 .... T-cell recepror bell chain precUlsor \I r~IOn t 16) Human
3.3251 It pnrna- I Ill) garnma? 2b fe receptor p,KU'lor Mouse
3.]414 HypodMlkll hvbr~ IQIT?cell receplOr prftuts.or \I ff:9lon (SUp?T 1~ - Humiln
3.]414. heavy chain precursor V II recJlon Human 71 2
3.]414 Ig heavy chain precursor V II reqton Human 71 ?
3.)417 Nellral celf IdhHk)n prot~ln pfftUnOr Mou se
3. 35 If Ig epsilon chatn C recJlon Human
1 35 I I Ig epsYon chain C recJlon . HUman
3.]S22 T-c:ell rectpc:or alpha chain V reqlon (80fl alpha I) Moun
3.J605 lIft.ry gtycoptoteln I . Humil"
3.3131 T-c:eII receptor garrvnil-I chain C 'ecJlon IMNGI and MNCn - Moust
3.]131 T-c:ell ,eceplor gamma I chain C IPglOn Mouu
3.3861 T?cell 9IftVTIa chain precursor V rf:'CJlon ('II j) Moun
3.4024 Ie ep51"n chilln C recJlon . Human
3.4024" epSolkJn chain C region ? Human
3.4110 Ig heavy chain V region ? Mouse Hl6-2
3.41 3J I, heavy chlln 'II region ? Mouse H]7 ?60
3.41521g heavy chain V rec)lon . Mouse H 18-S.1 S
3.41 S5 191 kappe chlln V region? Mouse HP9
3.4171191 heavy chain'll region? Mouse If6
3.4191 Ig kappa! chain'll region ? Mouse HieS .4l) I
3.4199lg heavy cha"l V region ? Mouse ]010
3.4199 ? heavy cha," V regIon? Mouse II CR kt I I
3.4211 191 heavy Chal" V r<t:9lon ? MOllse HPll and HP27
3.421] Prt9nancy,spt:Clflc b<t:ta? I glycoprotein ( prKursor . Human
3.4213 Prt9nancy?speclfK btla? 1 g tyC OPfO(tln 0 prKursor . HUman
3421 I T?celt receplor beta chain prKUls o r V ffglon (4 C3) . Mouse
].4211 T-cell receptor beta chain precursor 1/ region (810) Mouse
34212 Sodium channel prott'ln II Rat
3 429S Ig kappa ch.ln V rt'Qlon (HZ8-A.1) Mouse H28-A2
3429519 kilppa ch-lln V r~lon . Mous.e H I S& 89H4
3.429519 kappa chain V recJlon Mouse H 37 ] I I
3.4295 Ig kappa chain V region ? MouS<t: H]] 40
3.429S Ig kappa chain V ft:qlon Mouse H 3 7 ")
) 4295 ~ k.3ppa chain V rt:910n Mouse Hll 45
,_or
34572 T?ceU surface glycoprotein CO) epsilon chain - Human
3.4594 T~en sI,.Ia,. gtycDprote .... CO. precursor? Mouse
3.4594 T'ul) surrace gtycoproteln lyt?2 precursor? Mouse
3.4595 T-c:eII recePior .. ptta chain precursor V region (HAPO$ - Humin
3.4606T-c:ell rec_or gamma-2 chlln C region eMHC& Ind MN(9) ? Mouse
3.4614 T-c:eII receptor g.nwna ch.ln C region (PfER) ? Human
3.4614 T-c:ell receJKor gamrna-I chlln C region - Hu~n
3.4614 T-cell receptOr gamrna-2 chlln C region - Human
3.4620 It heevy chain V regkln - Mouse H 146-2413
3.4620. heavy chain V region - Mouse HI 5a-I9H4
3.4620 19 heavy chain" .eglon . M???? H3S,C&
l46lO I, heavy chain Pfecursor V region? Mouse M~J3
3.4690 T-c:eII rec_or beta- I ch.ln e regIOn' Human
3.4690T-c:eIf receptor beta-I chain C regIOn? Moyse
3.4690 T-c:en receptor bK~2 chain C regIOn - Hum.n
3.4690 T-cell receptor bK~2 chain C regtOn - Human
3.4769 ? ~3 chain e reg~on. G3m(b) allOrypa - Hum.n
] .479& It k.ppa ,haln V region - Mouse H 146-2483
3.479& It k.ppa Ch.ln V region - Mouse H36-2
3.479& It kappa ch.m V region - Mouse H37-62
3.479& It kappa ch.m V region - Mouse HH?12
3.4110 It kappa chain V-I retlon . HUman WII( I)
3.48-iO Peroxklase (Ee I.Il.l.n precursor - Human
3.4&&& PIa~tv. 9rowth IKIM reeeptor precursor - Mouse
3._5 N.t<h prot..... f ?? ,. fly
3._5 N.tch pr....... f ?? I. fly
3.4983: T<" recepror beta chain precursor V rt9lon (MT I-I) - Human
3.491J T<eII receptOr beta-2 cheln precursor V regkMt MOlT' 4' Human
].4991", kappII chain Pfecursor V region - Mouse Set-.
3.S035 Alkol... pII.. pII.... (EC 3. 1.3.11 p.ecU"Of ? H.man
3.5061 "heavy choln" 'eglo.. ? M.... H 37?&2
3.SO&2 Closs R hlsllOCompotlb.
HIA-DR botaoZ ch .... proc.rs.r (REMI . H.m...
3.5012 H-2 class. hlstocomPllttb. antigen. ?.a/k bet .. 2 chain PfKyrsor - Mouse
3.5012 H-2 class nhlstoCOlnpallb. .,It~. E1I beu-2 ch.ln precursor' Mouse
] .SOI2 HLA class II hlstocompallb. anll9en, OR I beta chain (cklne 69) - Humin
3.5012 HLA class. hlsbKompKIb. antigen. OR bela chain precursor
3.S012 HLA class II hlstocompatlb_anUgtft. Ollt beta chain precursor A5) - Hum.n
3.5012 HLA class I hlstocomp.lltib. antlgen, Ollt- I bet. ch .... precursor - Human
3.5012 HLA class I hlstocompilrlb. antlten. OR-4 betll chain' Human
3.S012 HLA class h hlstocompatlb. anrlttft, DR-5 kli chain precursor' Hum.n
3.509419 IiIm~S chlln C region - Mouse
3.S 144lg .lphl?2 ch.", e region. A2m( I) alforype - Human
3.5150 Ig heavy chain V region? Mouse H2a-A2
3.5180 Biliary gtycoprDtein I- Human
3.5193 Ig heavy chain V region - Mouse H37-45
3 5193 Ig heavy chain V regions - Mouse HJ7?80 and H]7-43
35211 Ig IMnbda chain ptecursor V region' Rat
15264 Ig huvy chain V region - Mouse H ]7-62
] S]161g heavy chain V region - Mouse H37? 311
] 533419 heavy chain V region ? Mouse HH?4O
] SJ72 T'cl'll receptor beta cha,n precyrsor V region (ATlI2'2) . Human
3 S435 Ig heavy chain V region - Mouse HleS? 401
3 SS79 Ig heavy chain V region - Mouse H]7-14
35603 Ig IMnbdl?2 chain e region - Ral
3.5666 J9 heavy chain V region - Moust 8 I? , henEallve slquence)
] 5709ll11ary glyCoprotein I- Human
] 5741 Nonspecific cross -reacting antigen precursor - Human
35115 Ig epsilon chain e region? Human
3.5115 Ig epsilon chain e rec,kln ' Human
3.5194 Neur.1 cell adheskln ptoteln precursor? Mouse
3.5912 Ig bppa chain V region - Mouse H]7-60
35971 Ig kilppa chain precursor'll region - Rat IR2
36020 Ig kappa chain V region? Mous. IF6
] 6020 fg kappa chilin V region? Mouse 3010
36027 T'cell receptor beell chain V region (K~ATU - Human
36071 19 heavy chlln V region ? Mouse HP20
36071 Ig heavy chain V regktn ? Mouse HP25
1.6120 T-cl'll receptor alptta ch.ln V regIOn (5c.en - Mouse
36'20 T-cell receptor alptta ch.'n V region (U~ ? Mouse
3.6120 T-cell receptor alph. ch.ln Pfecursor V region (214) ? Mouse
3.6120 T-cell receptor alptt. chain ptecursor V region (4.e]) . Mouse
] 6120 T?cell rec.pc:or Ilptt. chilin precursor'll reqlon (810) ? Mouse
].6302 HLA class 1/ hlSloCompatlb. antigen OX alpha chain prKursor - Human
] 6302 HLA class .. hlstocompatlb. anlAgen. OQ alph. chain precursor' Humiln
36461 T-ul! receptor alptta chilln precursor V region (HAPSIij - Human
] 646S Ig kappa chain precursor V ch.'n - Moys. s.e,-b
36539 Heur.1 un adhesion ptotetn precursor? Mouse
3.6636Ig huvy chain V region - Mouse BI -&'VI1V2 (untatlve slquence)
] 6771 Ig kappa chain precursor V-HI regAon - Human SU?OHl?6
36791 Ig kappa chain V region - Mouse H Ia-S415
36&)J Myelln-assoclatld gtvc:optoteln 11236 tong form ptecursor ? Rae:
3.6&)) Myelln'ilSsoc"tld g~op,oteln IB236 shon form prKursor . Rat
3.6&)3 Myelln-MsOClatld g~oproteln precursor. brain ~ Rar
].6&)J Myebn?assocl.r:. lar,. gtyc:oproteln precursor ? Rat
3.7102 It kappa chain V-III 'eglon - HUman C8
3.7170 Ig kappa chain V-I regIon ' HUman WII(2)
] 7341 Ig lambdl chain e region' Chicken
] 7505 Ig hppa chain precursor V?I region? Human Natm-6
] 75351g heavy chain precursor V regIOn - Mouse 129
] 7600 Ig lambda?5 chain C region - Mouse
3.7779 19 h~avy chain V reg60n - Mouse HP 12
] 790719 kappa chain V region 30S precursor - Humiln
] 790719 kappa chain precursor '1,111- Human Nalm-6
] 7909 19 heavy chain V region? Mouse HP21
] &017 Ntural cell adhHk)n proUtn precursor' Mouse
] 81 ao Ig mu chain e rtglon. b allele? Mouse
3824719 epSilon chain C region - Human
3 8247 ~ epsilon chilln e region - Human
3 &440 ~ kilppa chll" precursor V region? Mouse MAkH
3867119 klppa chain precursor II region? Rat IRI62
1In_
427
428
Bengio, Bengio, Pouliot and Agin
Table 2: Efficiency of detection for some Ig superfamily proteins present in NEW. Mean scores of recognized Ig domains
for each protein type are listed. Recognition efficiency is calculated by dividing the number of proteins correctly identified
(Le., bearing at least one Ig domain) by the total number of
proteins identified by their file description as containing an Ig
domain, multiplied by 100. Numbers in parentheses indicate
the number of complete protein sequences of each type for
each species. All complete sequences for light and heavy immunoglobulin chains of human and mouse origin were
scanned. The threshold was set at 3.0. ND: not done.
Protein
Immunoglobulins,
mouse,
all forms
Immunoglobulins,
human,
all forms
H-2 class II,
all forms
HLA class II,
all forms
T-cell receptor
chains,
mouse,
all forms
T-cell receptor
chains,
human,
all forms
Mean score of
detected domains
(max 4.00)
3.50
Recognition emciency for
Ig-bearing proteins
(see le2end)
98.2 % (55)
3.48
93.8 % (16)
3.33
ND
3.36
ND
3.32
ND
3.41
ND
The vast majority of proteins which scored above 3.0 were of human, mouse,
rat or rabbit origin. A few viral and insect proteins also scored above the
threshold. All proteins in the training set and present in either the NEW or
PROTEIN databases were detected. Proteins detected in the NEW database
are listed in Table I and sorted according to score. Even though only human
MHC class I and II were included in the training set, both mouse H-2 class I
and II were detected. Bovine and rat transplantation antigens were also
detected. These proteins are homologs of human MHC's. For proteins which
include more than one Ig domain contiguously arranged (e.g., carcinoembryonic antigen), all domains were detected if they were sufficiently well conserved. However, domains lacking a feature or possessing a degenerate
feature scored much lower (usually below 3.0) such that they are not recognized when using a threshold value of 3. Recognition of human and mouse immunoglobulin sequences was used to measure recognition efficiency. The rate
of false negatives for immunoglobulins was very low for both species (Table
II). Table III lists the 13 proteins categorized as false positives detected when
searching with a threshold of 3.0. Relative to the total number of domains
detected, this corresponds to a false positive rate of 6.8%. In the strict sense
some of these proteins are not false positives because they do exhibit the expected features of the Ig domain in the correct order. However, inter-feature
A Neural Network to Detect Homologies in Proteins
distances for these pseudo-domains are very different from those observed in
bona fide Ig domains. Proteins which are rich in ,B-sheets, such as rat sodium
channel II and fruit-fly NADH-ubiquinone oxidoreductase chain 1 are also
abundant among the set of false positives. This is not surprising since the Ig
domain is composed of ,B-strands. One solution to this problem lies in the use
of a larger training set as well as the addition of a more intelligent second
stage designed to evaluate inter-feature distances so as to increase the specificity of detection.
Table 3: False positives obtained when searching NEW with a threshold of
3.0. Proteins categorized as false positives are listed. See text for details.
3.0244 Kinase-related transforming protein (src) (Ee 2.7.1.-)
3.0409 Granulocyte-macrophage colony-stimulating
3.0492 NADH-ubiquinone oxidoreductase (Ee 1.6.5.3), chain 5
3.0508 NADH-ubiquinone oxidoreductase (Ee 1.6.5.3), chain 1
3.0561 Protein-tyrosine kinase (Ee 2.7.1.112), lymphocyte - Mouse
3.1931 Hypothetical protein HQLF2 - Cytomegalovirus (strain AD169)
3.2041 Sodium channel protein II - Rat
3.2147 SURF-1 protein - Mouse
3.3039 X-linked chronic granulomatous disease protein - Human
3.4840 Peroxidase (Ee 1.11.1.7) precursor - Human
3.4965 Notch protein - Fruit fly
3.4965 Notch protein - Fruit fly
3.5035 Alkaline phosphatase (EC 3.1.3.1) precursor - Human
5 DISCUSSION
The detection of specific protein domains is becoming increasingly important
since many proteins are constituted of a succession of domains. Unfortunately, domains (Ig or otherwise) are often only weakly homologous with each
other. We have designed a neural network to detect proteins which comprise
Ig domains to evaluate this approach in helping to solve this problem. Alternatives to neural network-based search programs exist. Search programs can
be designed to recognize the flanking Cys-termini regions to the exclusion of
other domain features since these flanks are the best conserved features of Ig
domains (c/. Wang et ai., 1989). However, even Cys-termini can exhibit poor
overall homology and therefore generate statistically insignificant homology
scores when analyzed with the ALIGN program (NBRF) (cf. Williams and
Barclay, 1987). Other search programs (such as Profile Analysis) cannot efficiently handle the large variations in domain size exhibited by the Ig domain
(mostly comprised between 45 and 70 residues). Search results become corrupted by high rates of false positives and negatives. Since the size of the
NBRF protein databases increases considerably each year, the problem of
false positives promises to become crippling if these rates are not substantially
decreased. In view of these problems we have found the application of a
neural network to the detection of Ig domains to be an advantageous solution.
As the state of biological knowledge advances, new Ig domains can be added
to the training set and training resumed. They can learn the statistical features
429
430
Bengio, Bengio, Pouliot and Agio
of the conserved subregions that permit detection of an Ig domain and generalize to new examples of this domain that have a similar distribution. Previously unrecognized and possibly degenerate homologous sequences are therefore likely to be detected.
Acknowledgments
This research was supported by a grant from the Canadian Natural Sciences
and Engineering Research Council to Y.B. We thank CISTI for graciously allowing us access to their experimental BIOMOLE' system.
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1,252 | 2,140 | Learning Attractor Landscapes for
Learning Motor Primitives
Auke Jan Ijspeert1,3?, Jun Nakanishi2 , and Stefan Schaal1,2
University of Southern California, Los Angeles, CA 90089-2520, USA
2
ATR Human Information Science Laboratories, Kyoto 619-0288, Japan
3
EPFL, Swiss Federal Institute of Technology, Lausanne, Switzerland
[email protected], [email protected], [email protected]
1
Abstract
Many control problems take place in continuous state-action spaces,
e.g., as in manipulator robotics, where the control objective is often defined as finding a desired trajectory that reaches a particular
goal state. While reinforcement learning offers a theoretical framework to learn such control policies from scratch, its applicability to
higher dimensional continuous state-action spaces remains rather
limited to date. Instead of learning from scratch, in this paper we
suggest to learn a desired complex control policy by transforming
an existing simple canonical control policy. For this purpose, we
represent canonical policies in terms of differential equations with
well-defined attractor properties. By nonlinearly transforming the
canonical attractor dynamics using techniques from nonparametric
regression, almost arbitrary new nonlinear policies can be generated without losing the stability properties of the canonical system. We demonstrate our techniques in the context of learning a
set of movement skills for a humanoid robot from demonstrations
of a human teacher. Policies are acquired rapidly, and, due to the
properties of well formulated differential equations, can be re-used
and modified on-line under dynamic changes of the environment.
The linear parameterization of nonparametric regression moreover
lends itself to recognize and classify previously learned movement
skills. Evaluations in simulations and on an actual 30 degree-offreedom humanoid robot exemplify the feasibility and robustness
of our approach.
1
Introduction
Learning control is formulated in one of the most general forms as learning a control
policy u = ?(x, t, w) that maps a state x, possibly in a time t dependent way, to an
action u; the vector w denotes the adjustable parameters that can be used to optimize the policy. Since learning control policies (CPs) based on atomic state-action
representations is rather time consuming and faces problems in higher dimensional
and/or continuous state-action spaces, a current topic in learning control is to use
?
http://lslwww.epfl.ch/?ijspeert/
higher level representations to achieve faster and more robust learning [1, 2]. In this
paper we suggest a novel encoding for such higher level representations based on the
analogy between CPs and differential equations: both formulations suggest a change
of state given the current state of the system, and both usually encode a desired
goal in form of an attractor state. Thus, instead of shaping the attractor landscape
of a policy tediously from scratch by traditional methods of reinforcement learning,
we suggest to start out with a differential equation that already encodes a rough
form of an attractor landscape and to only adapt this landscape to become more
suitable to the current movement goal. If such a representation can keep the policy
linear in the parameters w, rapid learning can be accomplished, and, moreover, the
parameter vector may serve to classify a particular policy.
In the following sections, we will first develop our learning approach of shaping attractor landscapes by means of statistical learning building on preliminary previous
work [3, 4].1 Second, we will present a particular form of canonical CPs suitable
for manipulator robotics, and finally, we will demonstrate how our methods can be
used to classify movement and equip an actual humanoid robot with a variety of
movement skills through imitation learning.
2
Learning Attractor Landscapes
We consider a learning scenario where the goal of control is to attain a particular
attractor state, either formulated as a point attractor (for discrete movements) or
as a limit cycle (for rhythmic movements). For point attractors, we require that the
CP will reach the goal state with a particular trajectory shape, irrespective of the
initial conditions ? a tennis swing toward a ball would be a typical example of such
a movement. For limit cycles, the goal is given as the trajectory shape of the limit
cycle and needs to be realized from any start state, as for example, in a complex
drumming beat hitting multiple drums during one period. We will assume that,
as the seed of learning, we obtain one or multiple example trajectories, defined by
positions and velocities over time. Using these samples, an asymptotically stable
CP is to be generated, prescribing a desired velocity given a particular state 2 .
Various methods have been suggested to solve such control problems in the literature. As the simplest approach, one could just use one of the demonstrated
trajectories and track it as a desired trajectory. While this would mimic this one
particular trajectory, and scaling laws could account for different start positions
[5], the resultant control policy would require time as an explicit variable and thus
become highly sensitive toward unforeseen perturbations in the environment that
would disrupt the normal time flow. Spline-based approaches [6] have a similar
problem. Recurrent neural networks were suggested as a possible alternative that
can avoid explicit time indexing ? the complexity of training these networks to obtain stable attractor landscapes, however, has prevented a widespread application
so far. Finally, it is also possible to prime a reinforcement learning system with
sample trajectories and pursue one of the established continuous state-action learning algorithms; investigations of such an approach, however, demonstrated rather
limited efficiency [7]. In the next sections, we present an alternative and surprisingly
simple solution to learning the control problem above.
1
Portions of the work presented in this paper have been published in [3, 4]. We here
extend these preliminary studies with an improvement and simplification of the rhythmic
system, an integrated view of the interpretation of both the discrete and rhythmic CPs, the
fitting of a complete alphabet of Grafitti characters, and an implementation of automatic
allocation of centers of kernel functions for locally weighted learning.
2
Note that we restrict our approach to purely kinematic CPs, assuming that the movement system is equipped with an appropriate feedback and feedforward controller that can
accurately track the kinematic plans generated by our policies.
Table 1: Discrete and Rhythmic control policies. ?z , ?z , ?v , ?v , ?z , ?z , ?, ?i and ci are
positive constants. x0 is the start state of the discrete system in order to allow nonzero initial conditions. The design parameters of the discrete system are ? , the temporal
scaling factor, and g, the goal position. The design parameters of the rhythmic system
are ym , the baseline of the oscillation, ? , the period divided by 2?, and ro , the amplitude
of oscillations. The parameters wi are fitted to a demonstrated trajectory using Locally
Weighted Learning.
Discrete
PN
?
?i wiT v
i=1
? y? = z + P
N
i=1
?i
? z? = ?z (?z (g ? y) ? z)
? = [v]
v
? v? = ?v (?v (g ? x) ? v)
? x? = v ?
?
2
0
?i = exp ?hi ( x?x
g?x0 ? ci )
ci ? [0, 1]
2.1
Rhythmic
PN
?
?i wiT v
i=1
? y? = z + P
N
i=1
?i
? z? = ?z (?z (ym ? y) ? z)
? = [r cos ?, r sin ?]T
v
? ?? = 1
? r? = ??(r ? r0 )
?
?
?i = exp ?hi (mod(?, 2?) ? ci )2
ci ? [0, 2?]
Dynamical systems for Discrete Movements
Assume we have a basic control policy (CP), for instance, instantiated by the second
order attractor dynamics
? z? = ?z (?z (g ? y) ? z)
? y? = z + f
(1)
where g is a known goal state, ?z , ?z are time constants, ? is a temporal scaling
factor (see below) and y, y? correspond to the desired position and velocity generated
by the policy as a movement plan. For appropriate parameter settings and f = 0,
these equations form a globally stable linear dynamical system with g as a unique
point attractor. Could we insert a nonlinear function f in Eq.1 to change the rather
trivial exponential convergence of y to allow more complex trajectories on the way
to the goal? As such a change of Eq.1 enters the domain of nonlinear dynamics,
an arbitrary complexity of the resulting equations can be expected. To the best
of our knowledge, this has prevented research from employing generic learning in
nonlinear dynamical systems so far. However, the introduction of an additional
canonical dynamical system (x, v)
? v? = ?v (?v (g ? x) ? v)
and the nonlinear function f
PN
i=1 ?i wi v
f (x, v, g) = P
N
i=1 ?i
? x? = v
(2)
?
?
?i = exp ?hi (x/g ? ci )2
(3)
can alleviate this problem. Eq.2 is a second order dynamical system similar to
Eq.1, however, it is linear and not modulated by a nonlinear function, and, thus,
its monotonic global convergence to g can be guaranteed with a proper choice of
?v and ?v . Assuming that all initial conditions of the state variables x, v, y, z are
initially zero, the quotient x/g ? [0, 1] can serve as a phase variable to anchor the
Gaussian basis functions ?i (characterized by a center ci and bandwidth hi ), and v
can act as a ?gating term? in the nonlinear function (3) such that the influence of
this function vanishes at the end of the movement. Assuming boundedness of the
weights wi in Eq.3, it can be shown that the combined dynamical system (Eqs.1?3)
asymptotically converges to the unique point attractor g.
Given that f is a normalized basis function representation with linear parameterization, it is obvious that this choice of a nonlinearity allows applying a variety of
?1
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Figure 1: Examples of time evolution of the discrete CPs (left) and rhythmic CPs (right).
The parameters wi have been adjusted to fit y? demo (t) = 10 sin(2?t) exp(?t2 ) for the discrete CPs and y? demo (t) = 2? cos(2?t) ? 6? sin(6?t) for the rhythmic CPs.
learning algorithms to find the wi . For learning from a given sample trajectory,
characterized by a trajectory ydemo (t), y? demo (t) and duration T , a supervised learning problem can be formulated with the target trajectory ftarget = ? y? demo ? zdemo
for Eq.1 (right), where zdemo is obtained by integrating Eq.1 (left) with ydemo instead of y. The corresponding goal state is g = ydemo (T ) ? ydemo (t = 0), i.e., the
sample trajectory was translated to start at y = 0. In order to make the nominal
(i.e., assuming f = 0) dynamics of Eqs.1 and 2 span the duration T of the sample
trajectory, the temporal scaling factor ? is adjusted such that the nominal dynamics
achieves 95% convergence at t = T . For solving the function approximation problem, we chose a nonparametric regression technique from locally weighted learning
(LWL) [8] as it allows us to determine the necessary number of basis functions, their
centers ci , and bandwidth hi automatically ? in essence, for every basis function
?i , LWL performs a locally weighted regression of the training data to obtain an
approximation of the tangent of the function to be approximated within the scope of
the kernel, and a prediction for a query point is achieved by a ?i -weighted average
of the predictions all local models. Moreover, as will be explained later, the parameters wi learned by LWL are also independent of the number of basis functions,
such that they can be used robustly for categorization of different learned CPs.
In summary, by anchoring a linear learning system with nonlinear basis functions in
the phase space of a canonical dynamical system with guaranteed attractor properties,
we are able to learn complex attractor landscapes of nonlinear differential equations
without losing the asymptotic convergence to the goal state.
2.2
Extension to Limit Cycle Dynamics
The system above can be extended to limit cycle dynamics by replacing the canonical system (x, v) with, for instance, the following rhythmic system which has a
stable limit cycle in terms of polar coordinates (?, r):
? ?? = 1
? r? = ??(r ? r0 )
(4)
Similar to the discrete system, the rhythmic canonical system serves to provide
? = [r cos ?, r sin ?]T and phase variable mod(?, 2?) to
both an amplitude signal v
the basis function ?i of the control policy (z, y):
PN
T?
i=1 ?i wi v
(5)
? z? = ?z (?z (ym ? y) ? z)
? y? = z + P
N
i=1 ?i
where ym is an anchor point for the oscillatory trajectory. Table 1 summarizes the
proposed discrete and rhythmic CPs, and Figure 1 shows exemplary time evolutions
of the complete systems.
2.3
Special Properties of Control Policies based on Dynamical Systems
Spatial and Temporal Invariance An interesting property of both discrete and
rhythmic CPs is that they are spatially and temporally invariant. Scaling of the goal
g for the discrete CP and of the amplitude r0 for the rhythmic CP does not affect the
topology of the attractor landscape. Similarly, the period (for the rhythmic system)
and duration (for the discrete system) of the trajectory y is directly determined
by the parameter ? . This means that the amplitude and durations/periods of
learned patterns can be independently modified without affecting the qualitative
shape of trajectory y. In section 3, we will exploit these properties to reuse a
learned movement (such as a tennis swing, for instance) in novel conditions (e.g
toward new ball positions).
Robustness against Perturbations When considering applications of our approach to physical systems, e.g., robots and humanoids, interactions with the environment may require an on-line modification of the policy. An obstacle can, for
instance, block the trajectory of the robot, in which case large discrepancies between
desired positions generated by the control policy and actual positions of the robot
will occur. As outlined in [3], the dynamical system formulation allows feeding back
an error term between actual and desired positions into the CPs, such that the time
evolution of the policy is smoothly paused during a perturbation, i.e., the desired
position y is modified to remain close to the actual position y?. As soon as the
perturbation stops, the CP rapidly resumes performing the (time-delayed) planned
trajectory. Note that other (task-specific) ways to cope with perturbations can be
designed. Such on-line adaptations are one of the most interesting properties of
using autonomous differential equations for CPs.
Movement Recognition Given the temporal and spatial invariance of our policy
representation, trajectories that are topologically similar tend to be fit by similar parameters wi , i.e., similar trajectories at different speeds and/or different amplitudes
will result in similar wi . In section 3.3, we will use this property to demonstrate
the potential of using the CPs for movement recognition.
3
Experimental Evaluations
3.1 Learning of Rhythmic Control Policies by Imitation
We tested the proposed CPs in a learning by demonstration task with a humanoid
robot. The robot is a 1.9-meter tall 30 DOFs hydraulic anthropomorphic robot
with legs, arms, a jointed torso, and a head [9]. We recorded trajectories performed
by a human subject using a joint-angle recording system, the Sarcos Sensuit (see
Figure 2, top). The joint-angle trajectories are fitted by the CPs, with one CP
per degree of freedom (DOF). The CPs are then used to replay the movement
in the humanoid robot, using an inverse dynamics controller to track the desired
trajectories generated by the CPs. The actual positions y? of each DOF are fed back
into the CPs in order to take perturbations into account.
Using the joint-angle recording system, we recorded a set of rhythmic movements
such as tracing a figure 8 in the air, or a drumming sequence on a bongo (i.e.
without drumming sticks). Six DOFs for both arms were recorded (three at the
shoulder, one at the elbow, and two at the wrist). An exemplary movement and its
replication by the robot is demonstrated in Figure 2 (top). Figure 2 (left) shows the
joint trajectories over one period of an exemplary drumming beat. Demonstrated
and learned trajectories are superposed. For the learning, the base frequency was
extracted by hand such as to provide the parameter ? to the rhythmic CP.
Once a rhythmic movement has been learned by the CP, it can be modulated
in several ways. Manipulating r0 and ? for all DOFs amounts to simultaneously
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1
0
Figure 2: Top: Humanoid robot learning a figure-8 movement from a human demonstration. Left: Recorded drumming movement performed with both arms (6 DOFs per
arm). The dotted lines and continuous lines correspond to one period of the demonstrated and learned trajectories, respectively. Right: Modification of the learned rhythmic pattern (flexion/extension of the right elbow, R EB). A: trajectory learned by the
rhythmic CP, B: temporary modification with r?0 = 2r0 , C: ?? = ? /2, D: y?m = ym + 1
(dotted line), where r?0 , ??, and y?m correspond to modified parameters between t=3s
and t=7s. Movies of the human subject and the humanoid robot can be found at
http://lslwww.epfl.ch/?ijspeert/humanoid.html.
modulate the amplitude and period of all DOFs, while keeping the same phase
relation between DOFs. This might be particularly useful for a drumming task
in order to replay the same beat pattern at different speeds and/or amplitudes.
Alternatively, the r0 and ? parameters can be modulated independently for the
DOFs each arm, in order to be able to change the beat pattern (doubling the
frequency of one arm, for instance). Figure 2 (right) illustrates different modulations
which can be generated by the rhythmic CPs. For reasons of clarity, only one DOF
is showed. The rhythmic CP can smoothly modulate the amplitude, frequency, and
baseline of the oscillations.
3.2 Learning of Discrete Control Policies by Imitation
In this experiment, the task for the robot was to learn tennis forehand and backhand
swings demonstrated by a human wearing the joint-angle recording system. Once
a particular swing has been learned, the robot is able to repeat the swing motion
to different cartesian targets, by providing new goal positions g to the CPs for the
different DOFs. Using a system of two-cameras, the position of the ball is given
to an inverse kinematic algorithm which computes these new goals in joint space.
When the new ball positions are not too distant from the original cartesian target,
the modified trajectories reach the ball with swing motions very similar to those
used for the demonstration.
3.3 Movement Recognition using the Discrete Control Policies
Our learning algorithm, Locally Weighted Learning [8], automatically sets the number of the kernel functions and their centers ci and widths hi depending on the complexity of the function to be approximated, with more kernel functions for highly
Figure 3: Humanoid robot learning a forehand swing from a human demonstration.
nonlinear details of the movement. An interesting aspect of locally weighted regression is that the regression parameters wi of each kernel i do not depend on the
other kernels, since regression is based on a separate cost function for each kernel.
This means that kernel functions can be added or removed without affecting the
parameters wi of the other kernels.
We here use this feature to perform movement recognition within a large variety
of trajectories, based on a small subset of kernels at fixed locations c i in phase
space. These fixed kernels are common for fitting all the trajectories, in addition
to the kernels automatically added by the LWL algorithm. The stability of their
parameters wi w.r.t. other kernels generated by LWL makes them well-suited for
comparing qualitative trajectory shapes.
To illustrate the possibility of using the CPs for movement recognition (i.e., recognition of spatiotemporal patterns, not just spatial patterns as in traditional character
recognition), we carried out a simple task of fitting trajectories performed by a human user when drawing two-dimensional single-stroke patterns. The 26 letters of
the Graffiti alphabet used in hand-held computers were chosen. These characters
are drawn in a single stroke, and are fed as a two-dimensional trajectory (x(t), y(t))
to be fitted by our system. Five examples of each character were presented (see
Figure 4 for four examples).
Fixed sets of five kernels per DOF were set aside for movement recognition. The
wT w
correlation |waa||wbb | between their parameter vectors wa and wb of character a and
b can be used to classify movements with similar velocity profiles (Figure 4, right).
For instance, for the 5 instances of the N, I, P, S, characters, the correlation is
systematically higher with the four other examples of the same character. These
similarities in weight space can therefore serve as basis for recognizing demonstrated
movements by fitting them and comparing the fitted parameters wi with those
of previously learned policies in memory. In this example, a simple one-nearestneighbor classifier in weight space would serve the purpose. Using such a classifier
within the whole alphabet (5 instances of each letter), we obtained a 84% recognition
rate (i.e. 110 out of the 130 instances had a highest correlation with an instance of
the same letter). Further studies are required to evaluate the quality of recognition
in larger training and test sets ? what we wanted to demonstrate is the ability
for recognition without any specific system tuning or sophisticated classification
algorithm.
4
Conclusion
Based on the analogy between autonomous differential equations and control policies, we presented a novel approach to learn control policies of basic movement skills
by shaping the attractor landscape of nonlinear differential equations with statistical learning techniques. To the best of our knowledge, the presented approach is the
first realization of a generic learning system for nonlinear dynamical systems that
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Figure 4: Left: Examples of two-dimensional trajectories fitted by the CPs. The demonstrated and fitted trajectories are shown with dotted and continuous lines, respectively.
Right: Correlation between the weight vectors of the 20 characters (5 of each letter) fitted
by the system. The gray scale is proportional to the correlation, with black corresponding
to a correlation of +1 (max. correlation) and white to a correlation of 0 or smaller.
can guarantee basic stability and convergence properties of the learned nonlinear
systems. We demonstrated the applicability of the suggested techniques by learning various movement skills for a complex humanoid robot by imitation learning,
and illustrated the usefulness of the learned parameterization for recognition and
classification of movement skills. Future work will consider (1) learning of multidimensional control policies without assuming independence between the individual
dimensions, and (2) the suitability of the linear parameterization of the control
policies for reinforcement learning.
Acknowledgments
This work was made possible by support from the US National Science Foundation (Awards
9710312 and 0082995), the ERATO Kawato Dynamic Brain Project funded by the Japan
Science and Technology Corporation, the ATR Human Information Science Laboratories,
and Communications Research Laboratory (CRL).
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1,253 | 2,141 | Global Versus Local Methods
in Nonlinear Dimensionality Reduction
Vin de Silva
Department of Mathematics,
Stanford University,
Stanford. CA 94305
[email protected]
Joshua B. Tenenbaum
Department of Brain and Cognitive Sciences,
Massachusetts Institute of Technology,
Cambridge. MA 02139
[email protected]
Abstract
Recently proposed algorithms for nonlinear dimensionality reduction fall
broadly into two categories which have different advantages and disadvantages: global (Isomap [1]), and local (Locally Linear Embedding [2],
Laplacian Eigenmaps [3]). We present two variants of Isomap which
combine the advantages of the global approach with what have previously been exclusive advantages of local methods: computational sparsity and the ability to invert conformal maps.
1 Introduction
In this paper we discuss the problem of nonlinear dimensionality reduction (NLDR): the
task of recovering meaningful low-dimensional structures hidden in high-dimensional data.
An example might be a set of pixel images of an individual?s face observed under different pose and lighting conditions; the task is to identify the underlying variables (pose angles, direction of light, etc.) given only the high-dimensional pixel image data. In many
cases of interest, the observed data are found to lie on an embedded submanifold of the
high-dimensional space. The degrees of freedom along this submanifold correspond to the
underlying variables. In this form, the NLDR problem is known as ?manifold learning?.
Classical techniques for manifold learning, such as principal components analysis (PCA)
or multidimensional scaling (MDS), are designed to operate when the submanifold is embedded linearly, or almost linearly, in the observation space. More generally there is a
wider class of techniques, involving iterative optimization procedures, by which unsatisfactory linear representations obtained by PCA or MDS may be ?improved? towards more
successful nonlinear representations of the data. These techniques include GTM [4], self
organising maps [5] and others [6,7]. However, such algorithms often fail when nonlinear
structure cannot simply be regarded as a perturbation from a linear approximation; as in
the Swiss roll of Figure 3. In such cases, iterative approaches tend to get stuck at locally
optimal solutions that may grossly misrepresent the true geometry of the situation.
Recently, several entirely new approaches have been devised to address this problem. These
methods combine the advantages of PCA and MDS?computational efficiency; few free
parameters; non-iterative global optimisation of a natural cost function?with the ability to
recover the intrinsic geometric structure of a broad class of nonlinear data manifolds.
These algorithms come in two flavors: local and global. Local approaches (LLE [2], Laplacian Eigenmaps [3]) attempt to preserve the local geometry of the data; essentially, they
seek to map nearby points on the manifold to nearby points in the low-dimensional representation. Global approaches (Isomap [1]) attempt to preserve geometry at all scales,
mapping nearby points on the manifold to nearby points in low-dimensional space, and
faraway points to faraway points.
The principal advantages of the global approach are that it tends to give a more faithful
representation of the data?s global structure, and that its metric-preserving properties are
better understood theoretically. The local approaches have two principal advantages: (1)
computational efficiency: they involve only sparse matrix computations which may yield
a polynomial speedup; (2) representational capacity: they may give useful results on a
broader range of manifolds, whose local geometry is close to Euclidean, but whose global
geometry may not be.
In this paper we show how the global geometric approach, as implemented in Isomap,
can be extended in both of these directions. The results are computational efficiency and
representational capacity equal to or in excess of existing local approaches (LLE, Laplacian
Eigenmaps), but with the greater stability and theoretical tractability of the global approach.
Conformal Isomap (or C-Isomap) is an extension of Isomap which is capable of learning the
structure of certain curved manifolds. This extension comes at the cost of making a uniform
sampling assumption about the data. Landmark Isomap (or L-Isomap) is a technique for
approximating a large global computation in Isomap by a much smaller set of calculations.
Most of the work focuses on a small subset of the data, called the landmark points.
The remainder of the paper is in two sections. In Section 2, we describe a perspective on
manifold learning in which C-Isomap appears as the natural generalisation of Isomap. In
Section 3 we derive L-Isomap from a landmark version of classical MDS.
2 Isomap for conformal embeddings
2.1 Manifold learning and geometric invariants
We can view the problem of manifold learning as an attempt to invert a generative model
for a set of observations. Let be a -dimensional domain contained in the Euclidean
, and let
be a smooth embedding, for some
. The object of
space
manifold learning is to recover and based on a given set
of observed data in
.
The observed data arise as follows. Hidden data
are generated randomly in , and are
then mapped by to become the observed data, so
.
The problem as stated is ill-posed: some restriction is needed on if we are to relate the
observed geometry of the data to the structure of the hidden variables
and itself. We
will discuss two possibilities. The first is that is an isometric embedding in the sense of
Riemannian geometry; so preserves infinitesmal lengths and angles. The second possibility is that is a conformal embedding; it preserves angles but not lengths. Equivalently,
at every point
there is a scalar
such that infinitesimal vectors at get magnified in length by a factor
. The class of conformal embeddings includes all isometric
embeddings as well as many other families of maps, including stereographic projections
such as the Mercator projection.
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)(
%*
!"
One approach to solving a manifold learning problem is to identify which aspects of the
geometry of are invariant under the mapping . For example, if is an isometric embedding then by definition infinitesimal distances are preserved. But more is true. The length
of a path in is defined by integrating the infinitesimal distance metric along the path.
The same is true in
, so preserves path lengths. If
are two points in , then
the shortest path between and lying inside is the same length as the shortest path
-
,+.-
'
-
between
and
along
. Thus geodesic distances are preserved. The conclusion is that is isometric with
, regarded as metric spaces under geodesic distance.
Isomap exploits this idea by constructing the geodesic metric for
approximately as a
matrix, using the observed data alone.
To solve the conformal embedding problem, we need to identify an observable geometric
invariant of conformal maps. Since conformal maps are locally isometric up to a scale
factor
, it is natural to try to estimate
at each point
in the observed data. By
rescaling, we can then restore the original metric structure of the data and proceed as in
Isomap. We can do this by noting that a conformal map rescales local volumes in by a
factor
. Hence if the
hidden data are sampled uniformly in , the local density of the
observed data will be
. It follows that the conformal factor
can be estimated
in terms of the observed local data density, provided that the original sampling is uniform.
C-Isomap implements a version of this idea which is independent of the dimension .
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This uniform sampling assumption may appear to be a severe restriction, but we believe
it reflects a necessary tradeoff in dealing with a larger class of maps. Moreover, as we
illustrate below, our algorithm appears in practice to be robust to moderate violations of
this assumption.
2.2 The Isomap and C-Isomap algorithms
There are three stages to Isomap [1]:
!
'
1. Determine a neighbourhood graph of the observed data
in a suitable way.
For example, might contain iff is one of the nearest neighbours of
(and vice versa). Alternatively, might contain the edge iff
,
for some .
,
,
2. Compute shortest paths in the graph for all pairs of data points. Each edge
in the graph is weighted by its Euclidean length
, or by some other useful
metric.
3. Apply MDS to the resulting shortest-path distance matrix
ding of the data in Euclidean space, approximating .
to find a new embed-
The premise is that local metric information (in this case, lengths of edges in the
neighbourhood graph) is regarded as a trustworthy guide to the local metric structure in the
original (latent) space. The shortest-paths computation then gives an estimate of the global
metric structure, which can be fed into MDS to produce the required embedding.
It is known that Step 2 converges on the true geodesic structure of the manifold given
sufficient data, and thus Isomap yields a faithful low-dimensional Euclidean embedding
whenever the function is an isometry. More precisely, we have (see [8]):
, with respect to a density
Theorem. Let be sampled from a bounded convex region in
function
.
Let
be
a
-smooth
isometric
embedding
of
that region in
. Given
, for a suitable choice of neighbourhood size parameter or , we have
+
)(
recovered distance
original distance
with probability at least
, provided that the sample size is sufficiently large.
formula is taken to hold for all pairs of points simultaneously.]
[The
C-Isomap is a simple variation on Isomap. Specifically, we use the -neighbours method
in Step 1, and replace Step 2 with the following:
2a. Compute shortest paths in the graph for
all pairs of data points. Each edge in
the graph is weighted by
is the mean distance
. Here
of to its nearest neighbours.
,
Using similar arguments to those in [8], one can prove a convergence theorem for CIsomap. The exact formula for the
weights
is not critical in the asymptotic analysis. The
point is that the rescaling factor
is an asymptotically accurate approximation
to the conformal scaling factor in the neighbourhood of and .
Theorem. Let be sampled uniformly from a bounded convex region
in
. Let be
a -smooth conformal embedding of that region in
. Given
, for a suitable
choice of neighbourhood size parameter , we have
recovered distance
original distance
with probability at least
, provided that the sample size is sufficiently large.
+
(
It is possible but unpleasant to find explicit lower bounds for the sample size. Qualitatively, we expect to require a larger sample size for C-Isomap since it depends on two
approximations?local data density and geodesic distance?rather than one. In the special
case where the conformal embedding is actually an isometry, it is therefore preferable to
use Isomap rather than C-Isomap. This is borne out in practice.
2.3 Examples
We ran C-Isomap, Isomap, MDS and LLE on three ?fishbowl? examples with different data
distributions, as well as a more realistic simulated data set. We refer to Figure 1.
Fishbowls: These three datasets differ only in the probability density used to generate
the points. For the conformal fishbowl (column 1), 2000 points were generated randomly
and then projected stereographically (hence conformally
uniformly in a circular disk
mapped) onto a sphere. Note the high concentration of points near the rim. There is no
metrically faithful way of embedding a curved fishbowl inside a Euclidean plane, so classical MDS and Isomap cannot succeed. As predicted, C-Isomap does recover the original
disk structure of (as does LLE). Contrast with the uniform fishbowl (column 2), with data
points sampled using a uniform measure on the fishbowl itself. In this situation C-Isomap
behaves like Isomap, since the rescaling factor is approximately constant; hence it is unable
to find a topologically faithful 2-dimensional representation. The offset fishbowl (column 3)
is a perturbed version of the conformal fishbowl; points are sampled in using a shallow
Gaussian offset from center, then stereographically projected onto a sphere. Although the
theoretical conditions for perfect recovery are not met, C-Isomap is robust enough to find a
topologically correct embedding. LLE, in contrast, produces topological errors and metric
distortion in both cases where the data are not uniformly sampled in (columns 2 and 3).
Face images: Artificial images of a face were rendered as
pixel images and rasterized into 16384-dimensional vectors. The images varied randomly and independently
in two parameters: left-right pose angle and distance from camera . There is a natural
family of conformal transformations for this
if we ignore perspective distor data manifold,
, for
, which has the effect of shrinking
tions in the closest images: namely
or magnifying the apparent size of images by a constant factor. Sampling uniformly in
and in gives a data set approximately satisfying the required conditions for C-Isomap.
We generated 2000 face images in this way, spanning the range indicated by Figure 2. All
four algorithms returned a two-dimensional embedding of the data. As expected, C-Isomap
returns the cleanest embedding, separating the two degrees of freedom reliably along the
horizontal and vertical axes. Isomap returns an embedding which narrows predictably as
the face gets further away. The LLE embedding is highly distorted.
(
conformal fishbowl
uniform fishbowl
offset fishbowl
face images
MDS
MDS
MDS
MDS
Isomap: k = 15
Isomap: k = 15
Isomap: k = 15
Isomap: k = 15
C?Isomap: k = 15
C?Isomap: k = 15
C?Isomap: k = 15
C?Isomap: k = 15
LLE: k = 15
LLE: k = 15
LLE: k = 15
LLE: k = 15
Figure 1: Four dimensionality reduction algorithms (MDS, Isomap, C-Isomap, and LLE)
are applied to three versions of a toy ?fishbowl? dataset, and to a more complex data manifold of face images.
Figure 2: A set of 2000 face images were randomly generated, varying independently in
two parameters: distance and left-right pose. The four extreme cases are shown.
3 Isomap with landmark points
The Isomap algorithm has two computational bottlenecks. The first is calculating the
shortest-paths distance matrix
. Using Floyd?s algorithm this is
; this can be
improved to
by implementing Dijkstra?s algorithm with Fibonacci heaps
( is the neighbourhood size). The second bottleneck is the MDS eigenvalue calculation,
which involves a full
matrix and has complexity
. In contrast, the eigenvalue
computations in LLE and Laplacian Eigenmaps are sparse (hence considerably cheaper).
L-Isomap addresses both of these inefficiencies at once. We designate of the data points to
be landmark points, where
. Instead of computing
, we compute the
matrix
of distances from each data point to the landmark points only. Using a new procedure LMDS (Landmark MDS), we find a Euclidean embedding of the data using
instead of
. This leads to an enormous savings when is much less than , since
can be computed using Dijkstra in
time, and LMDS runs in
.
LMDS is feasible precisely because we expect the data to have a low-dimensional embedding. The first step is to apply classical MDS to the landmark points only, embedding them
faithfully in . Each remaining point can now be located in
by using its known distances from the landmark points as constraints. This is analogous to the Global Positioning
System technique of using
a finite number of distance readings to identify a geographic
location. If
and the landmarks are in general position, then there are enough
constraints to locate uniquely. The landmark points
may be chosen randomly, with
taken to be sufficiently larger than the minimum
to ensure stability.
3.1 The Landmark MDS algorithm
LMDS begins by applying classical MDS [9,10] to the landmarks-only distance matrix
. We recall
the procedure. The first step is to construct an ?inner-product? matrix
; here
is the matrix of squared
distances and
is the ?centering? matrix
. Next find the eigenvalues
and eigenvectors
of
defined by the formula
. Write for the positive eigenvalues (labelled so that
), and for
the corresponding eigenvectors
(written as column vectors); non-positive eigenvalues are
ignored. Then for
the required optimal -dimensional embedding vectors are given
as the columns of the matrix:
..
.
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0/
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*))+) ,
.-
233 5 6'87 .:'9 ;=<<
3 5 7 . -- 9 <
1
4 5 7 >
.- 9
The embedded data are automatically mean-centered with principal components aligned
/
with the axes, most significant first. If ? has no negative eigenvalues, then the dimensional embedding is perfect; otherwise there is no exact Euclidean embedding.
@
"A
The second stage of LMDS is to embed the remaining points in
. Let
denote the
column vector of squared distances between a data point and the landmark points. The
embedding vector is related linearly to
by the formula:
-
where
D
is the column mean of
A
C
1
B
-
D "A&
1
1
and 23 B is the
pseudoinverse transpose of
33 .:- '9 55 6 ' ; <<<
1CB .:- 9 .
4 ..5 >
.- 9
:
Original points
L?Isomap: k=8
20 landmarks
L?Isomap: k=8
10 landmarks
L?Isomap: k=8
4 landmarks
L?Isomap: k=8
3 landmarks
Swiss roll embedding
LLE: k=18
LLE: k=14
LLE: k=10
LLE: k=6
Figure 3: L-Isomap is stable over a wide range of values for the sparseness parameter
(the number of landmarks). Results from LLE are shown for comparision.
The final (optional) stage is to use PCA to realign the data with the coordinate axes. A full
discussion of LMDS will appear in [11]. We note two results here:
1. If is a landmark point, then the embedding given by LMDS is consistent with
the original MDS embedding.
2. If the distance matrix
can be represented exactly by a Euclidean config, and if the landmarks are chosen so that their affine span in that
uration in
configuration is -dimensional (i.e. in general position), then LMDS will recover
the configuration exactly, up to rotation and translation.
A good way to satisfy the affine span condition is to pick
landmarks randomly, plus
a few extra for stability. This is important for Isomap, where the distances are inherently
slightly noisy. The robustness of LMDS to noise depends on the matrix norm
. If
is very small, then all the landmarks lie close to a hyperplane and LMDS
performs poorly with noisy data. In practice, choosing a few extra landmark points gives
satisfactory results.
5
1 B
3.2 Example
Figure 3, shows the results of testing L-Isomap on a Swiss roll data set. 2000 points were
generated uniformly in a rectangle (top left) and mapped into a Swiss roll configuration
in
. Ordinary Isomap recovers the rectangular structure correctly
provided that the
neighbourhood parameter is not too large (in this case
works). The tests show
that this peformance is not significantly degraded when L-Isomap is used. For each , we
chose landmark points at random; even down to 4 landmarks the embedding closely approximates the (non-landmark) Isomap embedding. The configuration of three landmarks
was chosen especially to illustrate the affine distortion that may arise if the landmarks lie
close to a subspace (in this case, a line). For three landmarks chosen at random, results are
generally much better.
In contrast, LLE is unstable under changes in its sparseness parameter (neighbourhood
size). To be fair, is principally a topological parameter and only incidentally a sparseness
parameter for LLE. In L-Isomap, these two roles are separately fulfilled by and .
4 Conclusion
Local approaches to nonlinear dimensionality reduction such as LLE or Laplacian Eigenmaps have two principal advantages over a global approach such as Isomap: they tolerate a
certain amount of curvature and they lead naturally to a sparse eigenvalue problem. However, neither curvature tolerance nor computational sparsity are explicitly part of the formulation of the local approaches; these features emerge as byproducts of the goal of trying
to preserve only the data?s local geometric structure. Because they are not explicit goals but
only convenient byproducts, they are not in fact reliable features of the local approach. The
conformal invariance of LLE can fail in sometimes surprising ways, and the computational
sparsity is not tunable independently of the topological sparsity of the manifold. In contrast, we have presented two extensions to Isomap that are explicitly designed to remove
a well-characterized form of curvature and to exploit the computational sparsity intrinsic
to low-dimensional manifolds. Both extensions are amenable to algorithmic analysis, with
provable conditions under which they return accurate results; and they have been tested
successfully on challenging data sets.
Acknowledgments
This work was supported in part by NSF grant DMS-0101364, and grants from Schlumberger, MERL and the DARPA Human ID program. The authors wish to thank Thomas
Vetter for providing the range and texture maps for the synthetic face; and Lauren Schmidt
for her help in rendering the actual images using Curious Labs? ?Poser? software.
References
[1] Tenenbaum, J.B., de Silva, V. & Langford, J.C (2000) A global geometric framework for nonlinear
dimensionality reduction. Science 290: 2319?2323.
[2] Roweis, S. & Saul, L. (2000) Nonlinear dimensionality reduction by locally linear embedding.
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[3] Belkin, M. & Niyogi, P. (2002) Laplacian eigenmaps and spectral techniques for embedding and
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Processing Systems 14. MIT Press.
[4] Bishop, C., Svensen, M. & Williams, C. (1998) GTM: The generative topographic mapping.
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[5] Kohonen, T. (1984) Self Organisation and Associative Memory. Springer-Verlag, Berlin.
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[8] Bernstein, M., de Silva, V., Langford, J.C. & Tenenbaum, J.B. (December 2000) Graph approximations to geodesics on embedded manifolds. Preprint may be downloaded at the URL:
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1,254 | 2,142 | Dopamine Induced Bistability Enhances
Signal Processing in Spiny Neurons
Aaron J. Gruber l ,2, Sara A. Solla2,3, and James C. Houk 2,l
Departments of Biomedical Engineeringl, Physiology2, and Physics and Astronomy3
Northwestern University, Chicago, IL 60201
{ a-gruberl, solla, j-houk }@northwestern.edu
Abstract
Single unit activity in the striatum of awake monkeys shows a
marked dependence on the expected reward that a behavior will
elicit. We present a computational model of spiny neurons, the
principal neurons of the striatum, to assess the hypothesis that direct neuromodulatory effects of dopamine through the activation
of D 1 receptors mediate the reward dependency of spiny neuron
activity. Dopamine release results in the amplification of key ion
currents, leading to the emergence of bistability, which not only
modulates the peak firing rate but also introduces a temporal and
state dependence of the model's response, thus improving the detectability of temporally correlated inputs.
1
Introduction
The classic notion of the basal ganglia as being involved in purely motor processing
has expanded over the years to include sensory and cognitive functions. A surprising new finding is that much of this activity shows a motivational component.
For instance, striatal activity related to visual stimuli is dependent on the type of
reinforcement (primary vs secondary) that a behavior will elicit [1]. Task-related
activity can be enhanced or suppressed when a reward is anticipated for correct
performance, relative to activity when no reward is expected. Although the origin
of this reward dependence has not been experimentally verified, dopamine modulation is likely to playa role. Spiny neurons in the striatum, the input to the basal
ganglia, receive a prominent neuromodulatory input from dopamine neurons in the
substantia nigra pars compacta. These dopamine neurons discharge in a rewarddependent manner [2]; they respond to the delivery of unexpected rewards and to
sensory cues that reliably precede the delivery of expected rewards.
Activation of dopamine receptors alters the response characteristics of spiny neurons
by modulating the properties of voltage-gated ion channels, as opposed to simple
excitatory or inhibitory effects [3]. Activation of the D1 type dopamine receptor
alone can either enhance or suppress neural responses depending on the prior state
of the spiny neuron [4]. Here, we use a computational approach to assess the
hypothesis that the modulation of specific ion channels through the activation of
D1 receptors is sufficient to explain both the enhanced and suppressed single unit
responses of medium spiny neurons to reward-predicting stimuli.
We have constructed a biophysically grounded model of a spiny neuron and used it to
investigate whether dopamine neuromodulation accounts for the observed rewarddependence of striatal single-unit responses to visual targets in the memory guided
saccade task described by [1]. These authors used an asymmetric reward schedule and compared the response to a given target in rewarded as opposed to unrewarded cases. They report a substantial reward-dependent difference; the majority
of these neurons showed a reward-related enhancement of the intensity and duration
of discharge, and a smaller number exhibited a reward-related depression. The authors speculated that D1 receptor activation might account for enhanced responses,
whereas D2 receptor activation might explain the depressed responses. The model
presented here demonstrates that neuromodulatory actions of dopamine through
D1 receptors suffice to account for both effects, with interesting consequences for
information processing.
2
Model description
The membrane properties of the model neuron result from an accurate representation of a minimal set of currents needed to reproduce the characteristic behavior of
spiny neurons. In low dopamine conditions, these cells exhibit quasi two-state behavior; they spend most of their time either in a hyperpolarized 'down' state around
-85 mV, or in a depolarized 'up' state around -55 mV [5]. This bimodal character
of the response to cortical input is attributed to a combination of inward rectifying (IRK) and outward rectifying (ORK) potassium currents [5]. IRK contributes
a small outward current at hyperpolarized membrane potentials, thus providing
resistance to depolarization and stabilizing the down state. ORK is a major hyperpolarizing current that becomes activated at depolarized potentials and opposes
the depolarizing influences of excitatory synaptic and inward ionic currents; it is
their balance that determines the membrane potential of the up state. In addition
to IRK and ORK currents, the model incorporates the L-type calcium (L-Ca) current that starts to provide an inward current at subthreshold membrane potentials,
thus determining the voltage range of the up state. This current has the ability to
increase the firing rate of spiny neurons and is critical to the enhancement of spiny
neuron responses in the presence of D1 agonists [4].
Our goal is to design a model that provides a consistent description of membrane
properties in the 100 - 1000 ms time range. This is the characteristic range of
duration for up and down state episodes; it also spans the time course of short
term modulatory effects of dopamine. The model is constructed according to the
principle of separation of time scales: processes that operate in the 100-1000 ms
range are modeled as accurately as possible, those that vary on a much shorter time
scale are assumed to instantaneously achieve their steady-state values, and those
that occur over longer time scales, such as slow inactivation, are assumed constant.
Thus, the model does not incorporate currents which inactivate on a short time
scale, and cannot provide a good description of rapid events such as the transitions
between up and down states or the generation of action potentials.
The membrane of a spiny neuron is modeled here as a single compartment with
steady-state voltage-gated ion currents. A first order differential equation relates
the temporal change in membrane potential (Vm ) to the membrane currents (Ii),
(1)
The right hand side of the equation includes active ionic, leakage, and synaptic
currents. The multiplicative factor 'Y models the modulatory effects of D1 receptor
activation by dopamine, to be described in more detail later. Ionic currents are
modeled using a standard formulation; the parameters are as reported in the biophysical literature, except for adjustments that compensate for specific experimental
conditions so as to more closely match in vivo realizations.
All currents except for L-Ca are modeled by the product of a voltage gated conductance and a linear driving force , Ii = gi (Vm - E i ), where Ei is the reversal potential
of ion species i and gi is the corresponding conductance. The leakage conductance
is constant; the conductances for IRK and ORK are voltage gated, gi = ?hLi (Vm ),
where 9i is the maximum conductance and Li (Vm ) is a logistic function of the
membrane potential. Calcium currents are not well represented by a linear driving
force model; extremely low intracellular calcium concentrations result in a nonlinear
driving force well accounted for by the Goldman-Hodgkin-Katz equation [6],
h -C a
=
PL -C aLL -C a (Vm )
2) ([Ca] [Ca]oe _
(z 2VmF
RT
1 _ e-?r
i -
zV",F
)
RT
'
(2)
where FL -C a is the maximum permeability. The resulting ionic currents are shown
in Fig 1A.
The synaptic current is modeled as the product of a conductance and a linear
driving force, 18 = g8(Vm - E 8), with E8 = O. The synaptic conductance includes
two types of cortical input: a phasic sensory-related component gp, and a tonic
context-related component gt, which are added to determine the total synaptic
input: g8 = ~(gp + gt). The factor ~ is a random variable that simulates the noisy
character of synaptic input.
Dopamine modulates the properties of ion currents though the activation of specific
receptors. Agonists for the D1 type receptor enhance the IRK and L-Ca currents
observed in spiny neurons [7, 8]. This effect is modeled by the factor 'Y in Eq 1.
An upper bound of'Y = 1.4 is derived from physiological experiments [7, 8]. The
lower bound at 'Y = 1.0 corresponds to low dopamine levels; this is the experimental
condition in which the ion currents have been characterized.
3
Static and dynamic properties
Stationary solutions to Eq 1 correspond to equilibrium values of the membrane
potential Vm consistent with specific values of the dopamine controlled conductance
gain parameter 'Y and the total synaptic conductance g8; fluctuations of g8 around
its mean value are ignored in this section: the noise parameter is set to ~ = 1.
Stationary solutions satisfy dVm/dt = 0; it follows from Eq 1 that they result from
(3)
Intersections between a curve representing the total ionic current (left hand side of
Eq 3) as a function of Vm and a straight line representing the negative of the synaptic
current (right hand side of Eq 3) determine the stationary values of the membrane
potential. Solutions to Eq 3 can be followed as a function of g8 for fixed 'Y by varying
the slope of the straight line. For 'Y = 1 there is only one such intersection for any
value of g8. At low dopamine levels, Vm is a single-valued monotonically increasing
function of g8, shown in Fig 1B (dotted line). This operational curve describes a
A
B
-30
2
>-50
N
E
()
::c 0 +--1----=::::::__
.sE
>
.6
-70
-90.j....i~=-.:::::;:...---~
-80
-60
Vm(mV)
o
10
9s (IlS/cm2)
20
Figure 1: Model characterization in low (-y = 1.0, dotted lines) and high (-y = 1.4,
solid lines) dopamine conditions. (A) Voltage-gated ion currents. (B) Operational
curves: stationary solutions to Eq 1.
gradual, smooth transition from hyperpolarized values of Vm corresponding to the
down state to depolarized values of Vm corresponding to the up state. At high
dopamine levels (-y = 1.4), the membrane potential is a single-valued monotonically
increasing function of the synaptic conductance for either g8 < 9.74 JLS/cm 2 or
g8 > 14.17 JLS/cm 2 . In the intermediate regime 9.74 JLS/cm 2 < g8 < 14.17 JLS/cm 2 ,
there are three solutions to Eq 3 for each value of g8. The resulting operational
curve, shown Fig 1B (solid line), consists of three branches: two stable and one
unstable. The two stable branches (dark solid lines) correspond to a hyperpolarized
down state (lower branch) and a depolarized up state (upper branch). The unstable
branch (solid gray line) corresponds to intermediate values of Vm that are not
spontaneously sustainable.
Bistability arises through a saddle node bifurcation with increasing 'Y and has a
drastic effect on the response properties of the model neuron in high dopamine
conditions. Consider an experiment in which 'Y is fixed at 1.4 and g8 changes slowly
so as to allow Vm to follow its equilibrium value on the operational curve for 'Y = 1.4
(see Fig 1B). As g8 increases, the hyperpolarized down state follows the lower stable
branch. As g8 reaches 14.17 JLS/cm 2 , the synaptic current suddenly overcomes the
mostly IRK hyperpolarizing current, and Vm depolarizes abruptly to reach an up
state stabilized by the activation of the hyperpolarizing aRK current. This is
the down to up (D-+U) state transition. As g8 is increased further, the up state
follows the upper stable branch, with a small amount of additional depolarization.
If g8 is now decreased, the depolarized up state follows the stable upper branch
in the downward direction. It is the inward L-Ca current which counteracts the
hyperpolarizing effect of the aRK current and stabilizes the up state until g8 reaches
9.74 JLS/cm 2 , where a net hyperpolarizing ionic current overtakes the system and
Vm hyperpolarizes abruptly to the down state. This is the up to down (U-+D) state
transition.
The emergence of bistability in high dopamine conditions results in a prominent
hysteresis effect. The state of the model, as described by the value of Vm , depends
not only on the current values of 'Y and g8' but also on the particular trajectory
followed by these parameters to reach their current values. The appearance of
bistability gives a well defined meaning to the notion of a down state and an up state:
in this case there is a gap between the two stable branches, while in low dopamine
conditions the transition is smooth, with no clear separation between states. We
generically refer to hyperpolarized potentials as the down state and depolarized
potentials as the up state, for consistency with the electrophysiological terminology.
A
Unrewarded Trial
Rewarded Trial
-30
-9p
-9p
-50
g> VI
:-Da
+/ .........
(
.. ~ .. '
E
> -70
;,...
.......;.;;.::.:.~p+Da
::'
.....
...
?????[j-+?~9p~
-90
0
B
Unrewarded Trial
Rewarded Trial
-30
-9p
>-50
VI
. .. ...
~ ... l
g
~
,
E
> -70
I'
.... ...
-90
0
.....
......
... ...
....
8
+9p
+Da
:
6.:+~p:
95 (J.tS/cm2)
......
,"
-Da
.....
16
0
8
95
16
95 (J.tS/cm2)
Figure 2: Response to a sensory related phasic input in rewarded and unrewarded
trials. (A) gt + gp > gD-+U? (B) gt + gp < g;.
An important feature of the model is that operational curves for all values of, intersect at a unique point, indicated by a circle in Fig 1B, for which V;' = - 55.1 m V
and g; = 13.2 J-tS / cm 2 . The appearance of this critical point is due to a perfect cancellation between the IRK and the L-Ca currents; it arises as a solut ion
to the equation I IRK + h-ca = O. When this condition is satisfied, solutions
to Eq 3 become independent of,. The existence of a critical point at a slightly
more depolarized membrane potential than the firing threshold at VI = - 58 m V
is an important aspect of our model; it plays a role in the mechanism that allows
dopamine to either enhance or depress the response of the model spiny neuron.
The dynamical evolution of Vm due to changes in both g8 and, follows from Eq 1.
Consider a scenario in which a tonic input gt maintains Vm below VI; the response to
an additional phasic input gp sufficient to drive Vm above VI depends on whether it
is associated with expected reward and thus triggers dopamine release. The response
of the model neuron depends on the combined synaptic input g8 in a manner that is
critically dependent on the expectation of reward. We consider two cases: whether
g8 exceeds gD-+U (Fig 2A) or remains below g; (Fig 2B).
If the phasic input is not associated with reward, the dopamine level does not increase (left panels in Fig 2). The square on the operational curve for , = 1 (dotted
line) indicates the equilibrium state corresponding to gt. A rapid increase from
g8 = gt to g8 = gt + gp (rightward solid arrow) is followed by an increase in Vm
towards its equilibrium value (upward dotted arrow). When the phasic input is
removed (leftward solid arrow), Vm decreases to its initial equilibrium value (down-
9D-U
9;
enhanced
amplitude
9t
N
E
c75
6
enhanced
amplitude
and
7.5
d)
No Response
O-l-----~-~___>,~"t_,
o
Figure 3: Modulation
of response in high
dopamine relative to
low dopamine conditions
as a function of the
strength of phasic and
tonic inputs.
ward dotted arrow). In unrewarded trials, the only difference between a larger and
a smaller phasic input is that the former results in a more depolarized membrane
potential and thus a higher firing rate. The firing activity, which ceases when the
phasic input disappears, encodes for the strength of the sensory-related stimulus.
Rewarded trials (right panels in Fig 2) elicit qualitatively different responses. The
phasic input is the conditioned stimulus that triggers dopamine release in the striatum, and the operational curve switches from the '"Y = 1 (dotted) curve to the
bistable '"Y = 1.4 (solid) curve. The consequences of this switch depend on the
strength of the phasic input. If g8 exceeds the value for the D-+ U transition (Fig
2A) , Vm depolarizes towards the upper branch of the bistable operational curve.
This additional depolarization results in a noticeably higher firing rate than the
one elicited by the same input in an unrewarded trial (Fig 2A, left panel). When
the phasic input is removed, the unit hyperpolarizes slightly as it reaches the upper
branch of the bistable operational curve. If gt exceeds gU--+D, the unit remains in
the up state until '"Y decreases towards its baseline level. If this condition is met
in a rewarded trial, the response is not only larger in amplitude but also longer in
duration. In contrast to these enhancements, if g8 is not sufficient to exceed g; (Fig
2B), Vm hyperpolarizes towards the lower branch of the bistable operational curve.
The unit remains in the down state until '"Y decreases towards its baseline level. In
this type of rewarded trial, dopamine suppresses the response of the unit.
The analysis presented above provides an explanatory mechanism for the observation of either enhanced or suppressed spiny neuron activity in the presence of
dopamine. It is the strength of the total synaptic input that selects between these
two effects; the generic features of their differentiation are summarized in Fig 3.
Enhancement occurs whenever the condition g8 > gD--+U is met, while activity is
suppressed if g8 < g;. The separatrix between enhancement and suppression always lies in a narrow band limited by g8 = gD--+U and g8 = g;. Its precise location
will depend on the details of the temporal evolution of '"Y as it rises and returns to
baseline. But whatever the shape of '"Y(t) might be, there will be a range of values
of g8 for which activity is suppressed, and a range of values of g8 for which activity
is enhanced.
4
Information processing
Dopamine induced bistability improves the ability of the model spiny neuron to detect time correlated sensory-related inputs relative to a context-related background.
To illustrate this effect, consider g8 = ~(gt + gp) as a random variable. The multiplicative noise is Gaussian, with <~>= 1 and <e >= 1.038. The total probability
density function (PDF) shown in Fig 4A for gt = 9.2 J-LS/cm 2 consists of two PDFs
corresponding to gp = 0 (left; black line) and gp = 5.8 J-LS/cm 2 (right; grey line).
These two values of gp occur with equal prior probability; time correlations are
introduced through a repeat probability Pr of retaining the current value of gp in
the subsequent time step. The total PDF shown in Fig 4A does not depend on the
value of Pr. Performance at the task of detecting the sensory-related input (gp -::f- 0)
is limited by the overlap of the corresponding PDFs [9]; optimal separation of the
two PDFs in Fig 4A results in a Bayesian error of 10.46%.
c
B
A
D
o
-60
Vm (mV)
Vm (mV)
-30
Vm (mV)
Figure 4: Probability density functions for (A) synaptic input, (B) membrane potential at "( = 1, (C) membrane potential at "( = 1.4 for un correlated inputs (Pr = 0.5) ,
and (D) membrane potential at "( = 1.4 for correlated inputs (Pr = 0.975).
The transformation of g8 into Vm through the "( = 1 operational curve results in
the PDFs shown in Fig 4B; here again, the total PDF does not depend on Pr.
An increase in the separation of the two peaks indicates an improved signal-tonoise ratio, but an extension in the tails of the PDFs counteracts this effect: the
Bayesian error stays at 10.46%, in agreement with theoretical predictions [9] that
hold for any strictly monotonic map from g8 into Vm . For the "( = 1.4 operational
curve, the PDFs that characterize Vm depend on Pr and are shown in Fig 4C
(Pr = 0.5, for which gp is independently drawn from its prior in each time step) and
4D (Pr = 0.975, which describes phasic input persistance for about 400 ms). The
implementation of Bayesian optimal detection of gp -::f- 0 for "( = 1.4 requires three
separating boundaries; the corresponding Bayesian errors stand at 10.46% for Fig
4C and 4.23% for Fig 4D. A single separating boundary in the gap between the two
stable branches is suboptimal, but is easily implement able by the bistable neuron.
This strategy leads to detection errors of 20.06% for Fig 4C and 4.38% for Fig 4D.
Note that the Bayesian error decreases only when time correlations are included,
and that in this case, detection based on a single separating boundary is very close
to optimal. The results for "( = 1.4 clearly indicate that ambiguities in the bistable
region make it harder to identify temporally uncorrelated instances of gp -::f- 0 on
the basis of a single separating boundary (Fig 4C), while performance improves if
instances with gp -::f- 0 are correlated over time (Fig 4D). Bistability thus provides a
mechanism for improved detection of time correlated input signals.
5
Conclusions
The model presented here incorporates the most relevant effects of dopamine neuromodulation of striatal medium spiny neurons via D1 receptor activation. In the
absence of dopamine the model reproduces the bimodal character of medium spiny
neurons [5]. In the presence of dopamine, the model undergoes a bifurcation and
becomes bistable. This qualitative change in character provides a mechanism to
account for both enhancement and depression of spiny neuron discharge in response
to inputs associated with expectation of reward. There is only limited direct experimental evidence of bistability in the membrane potential of spiny neurons: the
sustained depolarization observed in vitro following brief current injection in the
presence of D1 agonists [4] is a hallmark of bistable responsiveness.
The activity of single striatal spiny neurons recorded in a memory guided saccade
task [1] is strongly modulated by the expectation of reward as reinforcement for correct performance. In these experiments, most units show a more intense response of
longer duration to the presentation of visual stimuli indicative of upcoming reward;
a few units show instead suppressed activity. These observations are consistent
with properties of the model neuron, which is capable of both types of response to
such stimuli. The model identifies the strength of the total excitatory cortical input
as the experimental parameter that selects between these two response types, and
suggests that enhanced responses can have a range of amplitudes but attenuated
responses result in an almost complete suppression of activity, in agreement with
experimental data [1].
Bistability provides a gain mechanism that nonlinearly amplifies both the intensity
and duration of striatal activity. This amplification, exported through thalamocortical pathways, may provide a mechanism for the preferential cortical encoding of
salient information related to reward acquisition. The model indicates that through
the activation of D1 receptors, dopamine can temporarily desensitize spiny neurons
to weak inputs while simultaneously sensitizing spiny neurons to large inputs. A
computational advantage of this mechanism is the potential adaptability of signal
modulation: the brain may be able to utilize the demonstrated plasticity of corti costriatal synapses so that dopamine release preferentially enhances salient signals
related to reward. This selective enhancement of striatal activity would result in a
more informative efferent signal related to achieving reward.
At the systems level, dopamine plays a significant role in the normal operation
of the brain, as evident in the severe cognitive and motor deficits associated with
pathologies ofthe dopamine system (e.g. Parkinson's disease, schizophrenia). Yet at
the cellular level, the effect of dopamine on the physiology of neurons seems modest.
In our model , a small increase in the magnitude of both IRK and L-Ca currents
elicited by D1 receptor activation suffices to switch the character of spiny neurons
from bimodal to truly bistable, which not only modulates the frequency of neural
responses but also introduces a state dependence and a temporal effect. Other
models have suggested that dopamine modulates contrast [9], but the temporal
effect is a novel aspect that plays an important role in information processing.
6
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1,255 | 2,143 | A Convergent Form of Approximate Policy
Iteration
Theodore J. Perkins
Department of Computer Science
University of Massachusetts Amherst
Amherst, MA 01003
[email protected]
Doina Precup
School of Computer Science
McGill University
Montreal, Quebec, Canada H3A 2A7
[email protected]
Abstract
We study a new, model-free form of approximate policy iteration which
uses Sarsa updates with linear state-action value function approximation
for policy evaluation, and a ?policy improvement operator? to generate
a new policy based on the learned state-action values. We prove that if
the policy improvement operator produces -soft policies and is Lipschitz
continuous in the action values, with a constant that is not too large, then
the approximate policy iteration algorithm converges to a unique solution from any initial policy. To our knowledge, this is the first convergence result for any form of approximate policy iteration under similar
computational-resource assumptions.
1 Introduction
In recent years, methods for reinforcement learning control based on approximating value
functions have come under fire for their poor, or poorly-understood, convergence properties. With tabular storage of state or state-action values, algorithms such as Real-Time
Dynamic Programming, Q-Learning, and Sarsa [2, 13] are known to converge to optimal
values. Far fewer results exist for the case in which value functions are approximated using generalizing function approximators, such as state-aggregators, linear approximators,
or neural networks. Arguably, the best successes of the field were generated in this way
(e.g., [15]), and there are a few positive convergence results, particularly for the case of linear approximators [16, 7, 8]. However, simple examples demonstrate that many standard
reinforcement learning algorithms, such as Q-Learning, Sarsa, and approximate policy iteration, can diverge or cycle without converging when combined with generalizing function
approximators (e.g., [1, 6, 4]).
One classical explanation for this lack of convergence is that, even if one assumes that the
agent?s environment is Markovian, the problem is non-Markovian from the agent?s point
of view?the state features and/or the agent?s approximator architecture may conspire to
make some environment states indistinguishable. We focus on a more recent observation,
which faults the discontinuity of the action selection strategies usually employed by reinforcement learning agents [5, 10]. If an agent uses almost any kind of generalizing function
approximator to estimate state-values or state-action values, the values that are learned depend on the visitation frequencies of different states or state-action pairs. If the agent?s
behavior is discontinuous in its value estimates, as is the case with greedy and -greedy
behavior [14], then slight changes in value estimates may result in radical changes in the
agent?s behavior. This can dramatically change the relative frequencies of different states
or state-action pairs, causing entirely different value estimates to be learned.
One way to avoid this problem is to ensure that small changes in action values result in
small changes in the agent?s behavior?that is, to make the agent?s policy a continuous
function of its values. De Farias and Van Roy [5] showed that a form of approximate
value iteration which relies on linear value function approximations and softmax policy improvement is guaranteed to possess fixed points. For partially-observable Markov decision
processes, Perkins and Pendrith [10] showed that observation-action values that are fixed
points under Q-Learning or Sarsa update rules are guaranteed to exist if the agent uses
any continuous action selection strategy. Both of these papers demonstrate that continuity
of the agent?s action selection strategy leads to the existence of fixed points to which the
algorithms can converge. In neither case, however, was convergence established.
We take this line of reasoning on step further. We study a form of approximate policy iteration in which, at each iteration: (1) Sarsa updating is used to learn weights for a linear
approximation to the action value function of the current policy (policy evaluation), and
then (2) a ?policy improvement operator? determines a new policy based on the learned
action values (policy improvement).1 We show that if the policy improvement operator,
analogous to the action selection strategy of an on-line agent, is -soft and Lipschitz continuous in the action values, with a constant that is not too large, then the sequence of policies
generated is guaranteed to converge. This technical requirement formalizes the intuition
that the agent?s behavior should not change too dramatically when value estimates change.
2 Markov Decision Processes and Value Functions
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We consider infinite-horizon discounted Markov decision problems [3]. We assume that
the Markov decision process has a finite state set, , and a finite action set, , with sizes
and
. When the process is in state and the agent chooses action ,
the agent receives an immediate reward with expectation , and the process transitions to
next state with probability
. Let be the length
vector of expected immediate
rewards following each state-action pair (
).
A stochastic policy, , assigns a probability distribution over to each
. The prob. If
ability that the agent chooses action when the process is in state is denoted
is deterministic in state , i.e., if
for some and
for all
,
then we write
. For
let , , and denote, respectively, the state
of the process at time , the action chosen by the agent at time , and the reward received
by the agent at time . For policy , the state-value function,
, and state-action value
function (or just action-value function),
, are defined as:
where the expectation is with respect to the stochasticity of the process and the fact that the
agent chooses actions according to , and
is a discount factor. It is well-known
[11] that there exists at least one deterministic, optimal policy
for which
for all , , and .
Policy is called -soft if
for all and . For any
, let
denote the set
of -soft policies. Note that a policy, , can be viewed as an element of
, and
can
be viewed as a compact subset of
. We make the following assumption:
Assumption 1 Under any policy , the Markov decision process behaves as an irreducible, aperiodic Markov chain over the state set .
1
The algorithm can also be viewed as batch-mode Sarsa with linear action-value function approximation.
?
????????????????????????????????????
Inputs: initial policy , and policy improvement operator .
for i=0,1,2,. . . do
Policy evaluation: Sarsa updates under policy , with linear function approximation.
Initialize
arbitrarily.
With environment in state :
.
Choose according to
Observe ,
.
Repeat for
until converges:
.
Choose according to
T
? # ?
1? ?
-
2 % . 0/1/0/ 2
A2 2
2 0> 2 # 2 E
2
@
12 1 2
!" V #
Observe
.
# 2 E
2
Policy improvement:
.
end for
????????????????????????????????????
Figure 1: The version of approximate policy iteration that we study.
The approximate policy iteration algorithm we propose learns linear approximations to the
action value functions of policies. For this purpose, we assume that each state-action pair
is represented by a length $ feature vector
. (In this paper, all vectors are
columns
unless transposed.) For weights
, the approximate action-value for
%
is
denotes the transpose of
. Letting be the
& , where
-by- $ matrix whose rows correspond to the feature vectors of the state-action pairs, the
entire
approximate action-value function given by weights is represented by the vector
%
. We make the following assumption:
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6
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Assumption 2 The columns of are linearly independent.
?
3
4
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3 Approximate Policy Iteration
The standard, exact policy iteration algorithm [3] starts with an arbitrary policy
and
alternates between two steps: policy evaluation, in which (' is computed, and pol*' can be computed
icy improvement, in which a new policy, ) , is computed.
in various ways, including dynamic programming or solving a system of linear equations. !" is taken to be a greedy, deterministic policy with respect to (' . That is,
,+.- /0#+21
('
+2-3/0+
1
for all . Policy it)
54
(' . It is well-known that the sequence of policies
eration terminates when (')687
'"697
'
generated is monotonically improving in the sense that
for all , and
that the algorithm terminates after a finite number of iterations [3].
# ,
Bertsekas and Tsitsiklis [4] describe several versions of approximate policy iteration in
which the policy evaluation step is not exact. Instead,
is approximated by a weighted
linear combination of state features, with weights determined by Monte Carlo or TD( : )
learning rules. However, they assume that the policy improvement step is the same as
in the standard policy iteration algorithm?the next policy is greedy with respect to the
(approximate) action values of the previous policy. Bertsekas and Tsitsiklis show that if the
approximation error in the evaluation step is low, then such algorithms generate solutions
that are near optimal [4]. However, they also demonstrate by example that the sequence
of policies generated does not converge for some problems, and that poor performance can
result when the approximation error is high.
We study the version of approximate policy iteration shown in Figure 1. Like the versions
studied by Bertsekas and Tsitsiklis, we assume that policy evaluation is not performed
exactly. In particular, we assume that Sarsa updating is used to learn the weights of a linear
approximation to the action-value function. We use action-value functions instead of statevalue functions so that the algorithm can be performed based on interactive experience
with the environment, without knowledge of the state transition probabilities. The weights
learned in the policy evaluation step converge under conditions specified by Tsitsiklis and
Van Roy [17], one of which is Assumption 2.
6 T UXW
6 6 TJUXW V6 6 #TJ6 UXW # 6 6 6
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O )
C?
The key difference from previous work is that we assume a generic policy improvement operator, , which maps every
to a stochastic policy. This operator may produce,
for example, greedy policies, -greedy policies, or policies with action selection probabilities based on the softmax
[14]. is Lipschitz continuous with constant if, for
function
all
,
, where denotes the Euclidean
norm. is -soft if, for all
,
is -soft. The fact that we allow for a policy
improvement step that is not strictly greedy enables us to establish the following theorem.
Theorem 1 For any infinite-horizon Markov decision process satisfying Assumption 1, and
for any
, there exists
such that if is -soft and Lipschitz continuous with
constant , then the sequence of policies generated by the approximate policy iteration
, regardless of the
algorithm in Figure 1 converges to a unique limiting policy
choice of .
QR
In other words, if the behavior of the agent does not change too greatly in response to
changes in its action value estimates, then convergence is guaranteed. The remainder of
the paper is dedicated to proving this theorem. First, however, we briefly consider what the
theorem means and what some of its limitations are. The strength of the theorem is that it
states a simple condition under which a form of model-free reinforcement learning control
based on approximating value functions converges for a general class of problems. The
theorem does not specify a particular constant, , which ensures convergence; it merely
states that such a constant exists. The values of (and hence, range of policy improvement
operators) which ensure convergence depend on properties of the decision process, such
as its transition probabilities and rewards, which we assume to be unknown. The theorem
also offers no guarantee on the quality of the policy to which the algorithm converges.
Intuitively, if the policy improvement operator is Lipschitz continuous with a small constant
, then the agent is limited in the extent to which it can optimize its behavior. For example,
even if an agent correctly learns that the value of action is much higher than the value
of action , limits the frequency with which the agent can choose in favor of , and
this may limit performance. The practical importance of these considerations remains to
be seen, and is discussed further in the conclusions section.
$
4 Proof of Theorem 1
4.1 Probabilities Related to State-Action Pairs
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Because the approximate policy iteration algorithm in Figure 1 approximates action-values,
our analysis relies extensively on certain probabilities that are associated with state-action
pairs. First, we define
to be the
-bymatrix whose entries correspond to the
probabilities that one state-action pair follows another when the agent behaves according
row and
column of is
to . That is, the element on the
.
can be viewed as the stochastic transition matrix of a Markov chain over state-action
pairs.
.
Lemma 1 There exists
such that for all , 7
Proof: Let and be fixed, and let
and
. Then 7
4
. It is readily shown that for any two -by- matrices
K4 7 K 4
4 O ) # E
2
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:
7
:
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7 K 4
!!4 7 ! ! 4 4 4 7
4 A4
and
whose elements different in absolute value by at most ,
.
. Hence,
Under Assumption 1, fixing a policy, , induces an irreducible, aperiodic Markov chain
over . Let
denote the stationary probability of state . We define
to be
the length
vector whose
element is
. Note that the elements of
sum to one. If
for all and , then all elements of
are positive and it
is easily verified that
is the unique stationary distribution of the irreducible, aperiodic
Markov chain over state-action pairs with transition matrix
.
Lemma 2 For any
.
, there exists
such that for all
,
Proof: For any
, let
be the largest eigenvalue of
with modulus strictly less
than 1.
is well-defined since the transition matrix of any irreducible, aperiodic Markov
chain has precisely one eigenvalue equal to one [11]. Since the eigenvalues of a matrix
are continuous in the elements of the matrix [9], and since
is compact, there exists
for some
. Seneta [12], showed that for any
and
and stationary
two irreducible aperiodic Markov chains with transition matrices
distributions
and , on a state set with elements,
,
where
is the largest eigenvalue of
with modulus strictly less than one. Let
.
.
Lastly, we define
,
to be the matrix whose diagonal is
.
. It is easy to show that for any
4.2 The Weights Learned in the Policy Evaluation Step
4
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)
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4
4
! 24 7 ! 4 @ !
4 7 24 7 @ 4 4 ! 4 4
@
@
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O )
Consider the approximate policy evaluation step of the algorithm in Figure 1. Suppose
that the agent follows policy and uses Sarsa updates to learn weights , and suppose
that
for all and . Then
is the stochastic transition matrix of an irreducible, aperiodic Markov chain over state-action pairs, and
has the unique stationary
distribution of that chain on its diagonal. Under standard conditions on the learning rate
parameters for the updates, , Tsitsiklis and Van Roy [17] show that the weights converge
to the unique solution to the equation:
(1)
(Note that we have translated their result for TD( ) updating of approximate state-values
to Sarsa, or TD(0), updating of approximate state-action values.) In essence, this equation
says that the ?expected update? to the weights under the stationary distribution, , is zero.
Let
and
. Tsitsiklis and Van Roy [17] show that
is invertible, hence we can write
for the unique weights which satisfy
Equation 1.
Lemma 3 There exist
and
such that for all
.
Proof: For the first claim,
. For the second claim,
,
and
4 7 ! 4
@ ! 4 7 4 7 4
@ $! 4 7 ! 4
4
% 42 7 ! "4 @ ! 42 7 42 7 4 @ ! 427 ! 4 4
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4 %
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24 7
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24 7
4
4 7 24 7 4 4 4 4 %% 24 7 %% 4
4 7 4 7 4
# 4 47 7 4 7 4
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% 4 4 7 7 % 4
4 4 7
4
4
24 7 24 7 4 % % 24 7 % % 4 24 7 4 & 4
24 7 4 &
'
24 7 4 (& )'
!
!
where the last line follows from Lemmas 1 and 2 and the facts
for any
.
, there exists such that
Lemma 4 For any
and
for all
.
Proof: By Lemmas 1 and 2, and by the continuity of matrix inverses [11],
is a continuous function of . Thus,
is a continuous function of . Because
is a compact
subset of
, and because continuous functions map compact sets to compact sets, the
existence of the bound, , follows.
For any
-bymatrix , let
. That is,
measures
how
small a vector of length one can become under left-multiplication by matrix .
, there exists
Lemma 5 For any
,
such that for all
.
Proof: Lemma 7, in the Appendix, shows that is a continuous mapping and that is
,
. Since
positive for any non-singular matrix. For any
is continuous, and
compact, the infimum is attained by some
. Thus
, where the last inequality follows because is non-singular.
Lemma 6 For any
.
Proof: Let
Thus:
, there exists
such that for all
,
be arbitrary. From Equation 1,
and
.
(2)
The left hand side of Equation 2 follows from Lemmas 5 and 7; the right hand side follows
from Lemmas 3 and 4.
4.3 Contraction Argument
O )
V 6 4 ! 4 ! 4 6 4
Proof of Theorem 1: For a given infinite-horizon discounted Markov decision problem,
let
and be fixed. Suppose that is Lipschitz continuous with constant , where
is yet to be determined. Let
be arbitrary. The policies that result from
and% after one% iteration of the approximate policy% iteration % algorithm of% Figure% 1 are
7
respectively.
7
Observe that:
7 2and
from Lemma 6. If
, where the% last step follows
%
7 , then for some
7
! . Each
we have
!
iteration of the approximate policy iteration is a contraction. By the
% Contraction Mapping
Theorem [3], there is a unique fixed point of the mapping #"
, and the sequence of
policies generated according to that mapping from any initial policy converges to the fixed
point.
6K4
E Q R
4
4
4
4
V
6
6
6
6
G ) %
6 4 V 6 4
V 6 4$
K6 % 4
Note that since the sequence of policies, , converges, and since
is a continuous function
of
,
the
sequence
of
approximate
action-value
functions
computed
by the algorithm,
%
' , also converges.
5 Conclusions and Future Work
We described a model-free, approximate version of policy iteration for infinite-horizon discounted Markov decision problems. In this algorithm, the policy evaluation step of classical
policy iteration is replaced by learning a linear approximation to the action-value function
using on-line Sarsa updating. The policy improvement step is given by an arbitrary policy
improvement operator, which maps any possible action-value function to a new policy. The
main contribution of the paper is to show that if the policy improvement operator is -soft
and Lipschitz continuous in the action-values, with a constant that is not too large, then
the approximate policy iteration algorithm is guaranteed to converge to a unique, limiting
policy from any initial policy. We are hopeful that similar ideas can be used to establish the
convergence of other reinforcement learning algorithms, such as on-line Sarsa or Sarsa( : )
control with linear function approximation.
The magnitude of the constant that ensures convergence depends on the model of the
environment and on properties of the feature representation. If the model is not known, then
choosing a policy improvement operator that guarantees convergence is not immediate.
To be safe, an operator for which is small should be chosen. However, one generally
prefers to be large, so that the agent can exploit its knowledge by choosing actions with
higher estimated action-values as frequently as possible. One approach to determining
a proper value of would be to make an initial guess and begin the approximate policy
iteration procedure. If the contraction property fails on any iteration, one should choose a
new policy improvement operator that is Lipschitz continuous with a smaller constant. A
potential advantage of this approach is that one can begin with a high choice of , which
allows exploitation of action value differences, and switch to lower values of only as
necessary. It is possible that convergence could be obtained with much higher values of
than are suggested by the bound in the proof of Theorem 1.
Discontinuous improvement operators/action selection strategies can lead to nonconvergent behavior for many reinforcement learning algorithms, including Q-Learning,
Sarsa, and forms of approximate policy iteration and approximate value iteration. For
some of these algorithms, (non-unique) fixed points have been shown to exist when the
action selection strategy/improvement operator is continuous [5, 10]. Whether or not convergence also follows remains to be seen. For the algorithm studied in this paper, we have
constructed an example demonstrating non-convergence with improvement operators that
are Lipschitz continuous but with too large of a constant. In this case, it appears that the
Lipschitz continuity assumption we use cannot be weakened. One direction for future work
is determining minimal restrictions on action selection (if any) that ensure the convergence
of other reinforcement learning algorithms.
Ensuring convergence answers one standing objection to reinforcement learning control
methods based on approximating value functions. However, an important open issue for
our approach, and for other approaches advocating continuous action selection [5, 10], is
to characterize the solutions that they produce. We know of no theoretical guarantees on
the quality of solutions found, and there is little experimental work comparing algorithms
that use continuous action selection with those that do not.
Acknowledgments
Theodore Perkins was supported in part by National Science Foundation grants ECS0070102 and ECS-9980062. Doina Precup was supported in part by grants from NSERC
and FQNRT.
References
[1] L. C. Baird. Residual algorithms: Reinforcement learning with function approximation. In Proceedings of the Twelfth International Conference on Machine Learning, pages 30?37. Morgan
Kaufmann, 1995.
[2] A. G. Barto, S. J. Bradtke, and S. P. Singh. Learning to act using real-time dynamic programming. Artificial Intelligence, 72(1):81?138, 1995.
[3] D. P. Bertsekas. Dynamic Programming and Optimal Control, Volumes 1 and 2. Athena
Scientific, 2001.
[4] D. P. Bertsekas and J. N. Tsitsiklis. Neuro-Dynamic Programming. Athena Scientific, 1996.
[5] D. P. De Farias and B. Van Roy. On the existence of fixed points for approximate value iteration
and temporal-difference learning. Journal of Opt. Theory and Applications, 105(3), 2000.
[6] G. Gordon. Chattering in Sarsa( ).
www.cs.cmu.edu/ ggordon, 1996.
CMU Learning Lab Internal Report. Available at
[7] G. Gordon. Approximate Solutions to Markov Decision Processes. PhD thesis, Carnegie Mellon
University, 1999.
[8] G. J. Gordon. Reinforcement learning with function approximation converges to a region.
Advances in Neural Information Processing Systems 13, pages 1040?1046. MIT Press, 2001.
[9] C. D. Meyer. Matrix Analysis and Applied Linear Algebra. SIAM, 2000.
[10] T. J. Perkins and M. D. Pendrith. On the existence of fixed points for Q-learning and Sarsa in
partially observable domains. In Proceedings of the Nineteenth International Conference on
Machine Learning, 2002.
[11] M. L. Puterman. Markov Decision Processes: Disrete Stochastic Dynamic Programming. John
Wiley & Sons, Inc, New York, 1994.
[12] E. Seneta. Sensitivity analysis, ergodicity coefficients, and rank-one updates for finite markov
chains. In W. J. Stewart, editor, Numerical Solutions of Markov Chains. Dekker, NY, 1991.
[13] S. Singh, T. Jaakkola, M. L. Littman, and C. Szepesvari. Convergence results for single-step
on-policy reinforcement-learning algorithms. Machine Learning, 38(3):287?308, 2000.
[14] R. S. Sutton and A. G. Barto. Reinforcement Learning: An Introduction. MIT Press/Bradford
Books, Cambridge, Massachusetts, 1998.
[15] G. J. Tesauro. TD-Gammon, a self-teaching backgammon program, achieves master-level play.
Neural Computation, 6(2):215?219, 1994.
[16] J. N. Tsitsiklis and B. Van Roy. Optimal stopping of markov processes: Hilbert space theory,
approximation algorithms, and an application to pricing high-dimensional financial derivatives.
IEEE Transactions on Automatic Control, 44(10):1840?1851, 1999.
[17] J. N. Tsitsiklis and B. Van Roy. An analysis of temporal-difference learning with function
approximation. IEEE Transactions on Automatic Control, 42(5):674?690, 1997.
Appendix
MO ))
T
Lemma 7 For -by- matrix
1.
2.
3.
4.
0
M
, let
for all ,
iff is non-singular,
for any
,
is continuous.
=
. Then:
,
% .' '1/0/1/
I I = I0 I
:
I :
)
I
"0
: 0 = "0
: I )0
: I
all
Ilet 1 I +2-3/ )00
= : . Then
I )0 I : I I "0 I0 3. Now,
for
:
"0 : I I1 M )0 : I I1 M
"0 3 : I I I I1 . Thus, )0 : I0 .
Proof: The first three points readily follow from elementary arguments.
We focus on
the last point. We want to show that given a sequence
of
matrices
,
"
that converge
to
some
,
then
.
Note
that
"
means
that
"0
+.- / 0
. Let
. Then )0
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called:1 bradford:1 experimental:1 internal:1 |
1,256 | 2,144 | Generalized 2 Linear 2 Models
Geoffrey J. Gordon
[email protected]
Abstract
We introduce the Generalized 2 Linear 2 Model, a statistical estimator which combines features of nonlinear regression and factor analysis. A (GL)2M approximately decomposes a rectangular matrix
X into a simpler representation j(g(A)h(B)). Here A and Bare
low-rank matrices, while j, g, and h are link functions. (GL)2Ms
include many useful models as special cases, including principal
components analysis, exponential-family peA, the infomax formulation of independent components analysis, linear regression, and
generalized linear models. They also include new and interesting
special cases, one of which we describe below. We also present an
iterative procedure which optimizes the parameters of a (GL)2M.
This procedure reduces to well-known algorithms for some of the
special cases listed above; for other special cases, it is new.
1
Introduction
Let the m x n matrix X contain an independent sample from some unknown distribution. Each column of X represents a training example, and each row represents a
measured feature of the examples. It is often reasonable to assume that some of the
features are redundant , that is, that there exists a reduced set of I features which
contains all or most of the information in X.
If the reduced features are linear functions of the original features and the distributions of the elements of X are Gaussian, redundancy means we can write X as
the product of two smaller matrices U and V with small sum of squared errors.
This factorization is essentially a singular value decomposition: U must span the
first I dimensions of the left principal subspace of X, while V T must span the first
I dimensions of the right principal subspace. (Since the above requirements do not
uniquely determine U and V, the SVD traditionally imposes additional restrictions
which we will ignore in this paper.)
The SVD has a long list of successes in machine learning, including information
retrieval applications such as latent semantic analysis [1] and link analysis [2]; pattern recognition applications such as "eigenfaces" [3]; structure from motion algorithms [4]; and data compression tools [5]. Unfortunately, the SVD makes two
assumptions which can limit its accuracy as a learning tool.
The first assumption is the use of the sum of squared errors between X and UV as
a loss function. Squared error loss means that predicting 1000 when the answer is
1010 is as bad as saying -7 when the answer is 3. The second assumption is that
the reduced features are linear functions of the original features. Instead, X might
be a nonlinear function of UV, and U and V might be nonlinear functions of some
other matrices A and B. To address these shortcomings, we propose the model
x = f(g(A)h(B))
(1)
for the expected value of X. We also propose allowing non-quadratic loss functions
for the error (X - X) and the parameter matrices A and B. The fixed functions
are called link functions. By analogy to generalized linear models [6], we call equation (1) a Generalized 2 Linear 2 Model: generalized2 because it uses link functions
for the parameters A and B as well as the prediction X, and linear2 because like
the SVD it is bilinear.
As long as we choose link and loss functions that match each other (see below for the
definition of matching link and loss), there will exist efficient algorithms for finding
A and B given X, f, g, and h. Because (1) is a generalization of the SVD, (GL)2Ms
are drop-in replacements for SVDs in all of the applications mentioned above, with
better reconstruction performance when the SVD's error model is inaccurate. In
addition, they open up new applications (see section 7 for one) where an SVD would
have been unable to provide a sufficiently accurate reconstruction.
2
Matching link and loss functions
Whenever we try to optimize the predictions of a nonlinear model, we need to worry
about getting stuck in local minima. One example of this problem is when we try
to fit a single sigmoid unit with parameters (J E lRd to training inputs Xi E lRd and
target outputs Yi E lR under squared error loss:
Yi = 10git(Zi)
Zi
= Xi
. (J
Even for small training sets, the number of local minima of L can grow exponentially
with the dimension d [7]. On the other hand, if we optimize the same predictions
Yi under the logarithmic loss function ~i[Yi log Yi + (1 - Vi) 10g(1 - Yi)] instead of
squared error, our optimization problem is convex. Because the logistic link works
with the log loss to produce a convex optimization problem, we say they match each
other [7]. Matching link-loss pairs are important because minimizing a convex loss
function is usually far easier than minimizing a non convex one.
We can use any convex function F(z) to generate a matching pair of link and loss
functions. The loss function which corresponds to F is
(2)
where F*(y) is defined so that minz DF(Z I y) = O. (F* is the convex dual of F [8],
and D F is the generalized Bregman divergence from Z to Y [9].)
Expression (2) is nonnegative, and it is globally convex in all of the ZiS (and therefore
also in (J since each Zi is a linear function of (J). If we write f for the gradient of F,
the derivative of (2) with respect to Zi is f(Zi) - Vi. So, (2) will be zero if and only
if Yi = f(Zi) for all i; in other words, using the loss (2) implies that Yi = f(z;) is
our best prediction of Vi, and f is therefore our matching link function.
We will need two facts about convex duals below. The first is that F* is always
convex, and the second is that the gradient of F* is equal to f - l. (Also, convex
duality is defined even when F, G, and H aren't differentiable. If they are not,
replace derivatives by subgradients below.)
3
Loss functions for (G L )2Ms
In (GL)2Ms, matching loss functions will be particularly important because we need
to deal with three separate nonlinear link functions. We will usually not be able
to avoid local minima entirely; instead, the overall loss function will be convex in
some groups of parameters if we hold the remaining parameters fixed.
We will specify a (GL)2M by picking three link functions and their matching loss
functions. We can then combine these individual loss functions into an overall loss
function as described in section 4; fitting a (GL)2M will then reduce to minimizing
the overall loss function with respect to our parameters. Each choice of links results
in a different (G L)2M and therefore potentially a different decomposition of X.
The choice of link functions is where we should inject our domain knowledge about
what sort of noise there is in X and what parameter matrices A and B are a priori
most likely. Useful link functions include f (x) = x (corresponding to squared error
and Gaussian noise), f (x) = log x (unnormalized KL-di vergence and Poisson noise),
and f(x) = (1 + e- x) - l (log-loss and Bernoulli noise).
The loss functions themselves are only necessary for the analysis; all of our algorithms need only the link functions and (in some cases) their derivatives. So, we
can pick the loss functions and differentiate to get the matching link functions; or,
we can pick the link functions directly and not worry about the corresponding loss
functions. In order for our analysis to apply, the link functions must be derivatives
of some (possibly unknown) convex functions.
Our loss functions are D F , DG, and DH where
G : lRmxl H lR
are convex functions. We will abuse notation and call F, G, and H loss functions as
well: F is the prediction loss, and its derivative f is the prediction link; it provides
our model of the noise in X. G and H are the parameter losses, and their derivatives
g and h are the parameter links; they tell us which values of A and B are a priori
most likely. By convention, since F takes an m x n matrix argument , we will say
that the input and output to f are also m x n matrices (similarly for g and h).
4
The model and its fixed point equations
We will define a (GL)2M by specifying an overall loss function which relates the
parameter matrices A and B to the data matrix X. If we write U = g(A) and
V = h(B), the (GL)2M loss function is
L(U, V) = F(UV) - X
Here A
0
0
UV
+ G*(U) + H*(V)
(3)
B is the "matrix dot product," often written tr(AT B).
Expression (3) is a sum of three Bregman divergences, ignoring terms which don't
depend on U and V: it is DF(UV I X)+DG(O I U) +DH(O I V). The F-divergence
tends to pull UV towards X, while the G- and H-divergences favor small U and V.
To further justify (3), we can examine what happens when we compute its derivatives with respect to U and V and set them to O. The result is a set of fixed-point
equations that the optimal parameter settings must satisfy:
UT(X - f(UV))
B
(4)
(X - f(UV))VT
A
(5)
To understand these equations, we can consider two special cases. First, if we let
G* go to zero (so there is no pressure to keep U and V small) , (4) becomes
UT(X - f(UV)) = 0
(6)
which tells us that each column of the error matrix must be orthogonal to each
column of U. Second, if we set the prediction link to be f(UV) = UV, (6) becomes
UTUV = UTX
which tells us that for a given U, we must choose V so that UV reconstructs X
with the smallest possible sum of squared errors.
5
Algorithms for fitting (GL)2Ms
We could solve equations (4- 5) with any of several different algorithms. For example, we could use gradient descent on either U, V or A, B. Or, we could use the
generalized gradient descent [9] update rule (with learning rate a):
A +-", (X - f(UV))V T
B +-", UT(X - f(UV))
The advantage of these algorithms is that they are simple to implement and don't
require additional assumptions on F , G , and H. They can even work when F, G,
and Hare nondifferentiable by using subgradients.
In this paper, though, we will focus on a different algorithm. Our algorithm is based
on Newton's method , and it reduces to well-known algorithms for several special
cases of (GL)2Ms. Of course, since the end goal is solving (4-5), this algorithm will
not always be the method of choice; instead, any given implementation of a (GL)2M
should use the simplest algorithm that works.
For our Newton algorithm we will need to place some restrictions on the prediction
and parameter loss functions. (These restrictions are only necessary for the Newton
algorithm; more general loss functions still give valid (GL)2Ms, but require different
algorithms.) First, we will require (4-5) to be differentiable. Second, we will restrict
F(Z) = LFij (Zij)
H(B) = L Hj(B. j )
ij
j
These definitions fix most of the second derivatives of L(U, V) to be zero, simplifying
and speeding up computation. Write f ij , gi, and h j for the respective derivatives.
With these restrictions, we can linearize (4) and (5) around our current guess at
the parameters, then solve the resulting equations to find updated parameters. To
simplify notation, we can think of (4) as j separate equations, one for each column
of V. Linearizing with respect to Vj gives:
(U T DjU + Hj)(Vr w
-
Vj) = UT(X.j - f.j(UV j )) - B. j
where the l x l matrix H j is the Hessian of Hi at V j ' or equivalently the inverse of
the Hessian of Hj at B.j; and the m x m diagonal matrix Dj contains the second
derivatives of F with respect to the jth column of its argument. That is,
Now , collecting terms involving Vjew yields:
We can recognize (7) as a weighted least squares problem with weights
precision H j , prior mean Vj + H j1 B-j , and target outputs
UV j
+ Dj1(x.j
VJ5j,
prior
- f(UV j ))
Similarly, we can linearize with respect to rows of U to find the equation
UreW(VDiVT + G i ) = ((Xi. - j;.(Ui.V))Di1 + Ui.V)DiV T + Ui. G i - Ai. (8)
where G i is the Hessian of Gi and Di contains the second derivatives of F with
respect to the ith row of its argument.
We can solve one copy of (7) simultaneously for each column of V, then replace V
by vnew. Next we can solve one copy of (8) simultaneously for each row of U, then
replace U by unew. Alternating between these two updates will tend to reduce (3).1
6
Related models
There are many important special cases of (GL)2Ms. We derive two in this section;
others include principal components analysis, "sensible" PCA, linear regression,
generalized linear models, and the weighted majority algorithm. (Our Newton algorithm turns into power iteration for PCA and iteratively-reweighted least squares
for GLMs.) (GL)2Ms are related to generalized bilinear models; the latter include
some of the above special cases, but not ICA, weighted majority, or the example of
section 7. There are natural generalizations of (GL)2Ms to multilinear interactions.
Finally, some models such as non-negative matrix factorization [10] and generalized low-rank approximation [11] are cousins of (GL)2Ms: they use a loss function
which is convex in either factor with the other fixed but which is not a Bregman
divergence.
6.1
Independent components analysis
In ICA, we assume that there is a hidden matrix V (the same size as X) of independent random variables, and that X was generated from V by applying a square
matrix U. We seek to recover the mixing matrix U and the sources V; in other
words , we want to decompose X = UV so that the elements of V are as nearly
independent as possible.
The info max algorithm for ICA assumes that the elements of V follow distributions
with heavy tails (i.e. , high kurtosis). This assumption helps us find independent
components because the sum of two heavy-tailed random variables tends to have
lighter tails, so we can search for U by trying to make the elements of V follow a
heavy-tailed distribution.
In our notation, the fixed point of the info max algorithm for ICA is
_ U T = tanh(V)XT
(9)
(see, e.g., equation (11) or (13) of [12]). To reproduce (9) , we will let the prediction
link f be the identity, and we will let the duals of the parameter loss functions be
G*(U)
-dogdet U
H*(V)
E
L log cosh
Vij
ij
iTo guarantee convergence, we can check that (3) decreases and reduce our step size if
we encounter problems. (Since U T D j U, H j , V Di V T, and G i are all positive definite, the
Newton update directions are descent directions; so, there always exists a small enough
step size.) We have not found this check necessary in practice.
where
f
is a small positive real number. Then equations (4) and (5) become
UT(X - UV)
(X - UV)VT
ttanh(V)
-fU - T
(10)
(11)
since the derivative of log cosh v is tanh v and the derivative of log det U is U - T .
Right-multiplying (10) by (UV)T and substituting in (11) yields
_u T
Now since UV -+ X as
6.2
f
= tanh(V)(UV)T
(12)
-+ 0, (12) is equivalent to (9) in the limit of vanishing
f.
Exponential family peA
To duplicate exponential family PCA [13], we can set the prediction link f arbitrarily and let the parameter links 9 and h be large multiples of the identity. Our
Newton algorithm is applicable under the assumptions of [13], and (7) becomes
(13)
Equation (13) along with the corresponding modification of (8) should provide a
much faster algorithm than the one proposed in [13], which updates only part of U
or V at a time and keeps updating the same part until convergence before moving
on to the next one.
7
Example: robot belief states
Figure 1 shows a map of a corridor in the CMU CS building. A robot navigating
in this corridor can sense both side walls and compute an accurate estimate of its
lateral position. Unless it is near an observable feature such the lab door near the
middle of the corridor, however, it can't accurately resolve its position along the
corridor and it can't tell whether it is pointing left or right.
In order to plan to achieve a goal in this environment, the robot must maintain
a belief state (a probability distribution representing its best information about
the unobserved state variables). The map shows the robot's starting belief state:
it is at one end of the corridor facing in, but it doesn't know which end. We
collected a training set of 400 belief states by driving the robot along the corridor and
feeding its sensor readings to a belief tracker [14]. To simulate a larger environment
with greater uncertainty, we artificially reduced sensor range and increased error.
Figure 1 shows two of the collected beliefs.
Planning is difficult because belief states are high-dimensional: even in this simple
world there are 550 states (275 positions at lOcm intervals along the corridor x 2
orientations), so a belief is a vector in ]R550. Fortunately, the robot never encounters
most belief states. This regularity can make planning tractable: if we can identify
a few features which extract the important information from belief states, we can
plan in low-dimensional feature space instead of high-dimensional belief space.
We factored the matrix of belief states using feature space ranks l = 3,4, 5. For the
prediction link f(Z) we used exp(Z) (componentwise); this link ensures that the
predicted beliefs are positive, and treats errors in small probabilities as proportionally more important than errors in large ones. (The matching loss for f is a Poisson
log-likelihood or unnormalized KL-divergence.) For the parameter link h we used
10 12 I, corresponding to H* = lO - 12 11V11 2 /2 (a weak bias towards small V).
~,~I~,A
~""~
,_
~,~I- - - -cj~
:L \ .~
\_
~,1:- - - - - - c.- - - -: :,fc- \~"'~\_ ~
~t ,. A. 1 ~,~
I -------c:L,.
-----=-----'
-----=-----,1
-----=-----'
A-----"-----..t
, /____,________=_\
Figure 1: Belief states. Left panel: overhead map of corridor with initial belief b1 ;
belief state bso (just before robot finds out which direction it's pointing); belief bgo
(just after finding out). Right panel: reconstruction of bso with 3, 4, and 5 features.
We set G* = 1O- 1211U11 2j2 +6..(U), where 6.. is 0 when the first column of U contains
all Is and 00 otherwise. This loss function fixes the first column of U, representing
our knowledge that one feature should be a normalizing constant so that each belief
sums to 1. The subgradient of G* is 1O- 12U + [k, 0], so equation (5) becomes
(X - exp(UV))VT
= 1O- 12U + [k, 0]
Here [k,O] is a matrix with an arbitrary first column and all other elements 0; this
matrix has enough degrees of freedom to compensate for the constraints on U.
Our Newton algorithm handles this modified fixed point equation without difficulty.
So, this (GL)2M is a principled and efficient way to decompose a matrix of probability distributions. So far as we know this model and algorithm have not been
described in the literature.
Figure 1 shows our reconstructions of a representative belief state using I = 3, 4,5
features (one of which is a normalizing constant that can be discarded for planning).
The I = 5 reconstruction is consistently good across all 400 beliefs, while the I = 4
reconstruction has minor artifacts for some beliefs. A small number of restarts is
required to achieve good decompositions for I = 3 where the optimization problem
is most constrained. For comparison, a traditional SVD requires a matrix of rank
about 25 to achieve the same mean-squared reconstruction error as our rank-3
decomposition. It requires rank about 85 to match our rank-5 decomposition.
Examination of the learned U matrix (not shown) for I = 4 reveals that the corridor is mapped into two smooth curves in feature space, one for each orientation.
Corresponding states with opposite orientations are mapped into similar feature
vectors for one half of the corridor (where the training beliefs were sometimes confused about orientation) but not the other (where there were no training beliefs
that indicated any connection between orientations). Reconstruction artifacts occur
when a curve nearly self-intersects and causes confusion between states. This selfintersection happens because of local minima in the loss function; with more flexibility (I = 5) the optimizer is able to untangle the curves and avoid self-intersection.
Our success in compressing the belief state translates directly into success in planning; see [15] for details. By comparison, traditional SVD on either the beliefs or
the log beliefs produces feature sets which are unusable for planning because they
do not achieve sufficiently good reconstruction with few enough features.
8
Discussion
We have introduced a new general class of nonlinear regression and factor analysis
model, presenting a derivation and algorithm, reductions to previously-known special cases, and a practical example. The model is a drop-in replacement for PCA,
but can provide much better reconstruction performance in cases where the PCA
error model is inaccurate. Future research includes online algorithms for parameter
adjustment; extensions for missing data; and exploration of new link functions.
Acknowledgments
Thanks to Nick Roy for helpful comments and for providing the data analyzed
in section 7. This work was supported by AFRL contract F30602-01-C-0219,
DARPA's MICA program, and by AFRL contract F30602- 98- 2- 0137, DARPA's
CoABS program. The opinions and conclusions are the author's and do not reflect
those of the US government or its agencies.
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1,257 | 2,145 | Learning Sparse Multiscale Image
Representations
Phil Sallee
Department of Computer Science and
Center for Neuroscience, UC Davis
1544 Newton Ct.
Davis, CA 95616
[email protected]
Bruno A. Olshausen
Department of Psychology and
Center for Neuroscience, UC Davis
1544 Newton Ct.
Davis, CA 95616
[email protected]
Abstract
We describe a method for learning sparse multiscale image representations using a sparse prior distribution over the basis function
coefficients. The prior consists of a mixture of a Gaussian and a
Dirac delta function, and thus encourages coefficients to have exact
zero values. Coefficients for an image are computed by sampling
from the resulting posterior distribution with a Gibbs sampler. The
learned basis is similar to the Steerable Pyramid basis, and yields
slightly higher SNR for the same number of active coefficients. Denoising using the learned image model is demonstrated for some
standard test images, with results that compare favorably with
other denoising methods.
1
Introduction
Increasing interest has been given to the use of overcomplete representations for
natural scenes, where the number of basis functions exceeds the number of image
pixels. One reason for this is that overcompleteness allows for more stable, and thus
arguably more meaningful, representations in which common image features can be
well described by only a few coefficients, regardless of where they are located in
the image, how they are rotated, or how large they are [8, 6]. This may translate
into gains in coding efficiency for image compression, and improved accuracy for
tasks such as denoising. Overcomplete representations have been shown to reduce
Gibbs-like artifacts common to thresholding methods employing critically sampled
wavelets [4, 3, 9].
Common wavelet denoising approaches generally apply either a hard or softthresholding function to coefficients which have been obtained by filtering an image
with a the basis functions. One can view these thresholding methods as a means
of selecting coefficients for an image based on an assumed sparse prior on the coefficients [1, 2]. This statistical framework provides a principled means of selecting
an appropriate thresholding function. When such thresholding methods are applied
to overcomplete representations, however, problems arise due to the dependencies
between coefficients. Choosing optimal thresholds for a non-orthogonal basis is still
an unsolved problem. In one approach, orthogonal subgroups of an overcomplete
shift-invariant expansion are thresholded separately and then the results are combined by averaging [4, 3]. In addition, if the coefficients are obtained by filtering
the noisy image, there will be correlations in the noise that should be taken into
account.
Here we address two major issues regarding the use of overcomplete representations
for images. First, current methods make use of various overcomplete wavelet bases.
What is the optimal basis to use for a specific class of data? To help answer this
question, we describe how to adapt an overcomplete wavelet basis to the statistics of natural images. Secondly, we address the problem of properly inferring the
coefficients for an image when the basis is overcomplete. We avoid problems associated with thresholding by using the wavelet basis as part of a generative model,
rather than a simple filtering mechanism. We then sample the coefficients from
the resulting posterior distribution by simulating a Markov process known as a
Gibbs-sampler.
Our previous work in this area made use of a prior distribution peaked at zero and
tapering away smoothly to obtain sparse coefficients [7]. However, we encountered
a number of significant limitations with this method. First, the smooth priors
do not force inactive coefficients to have values exactly equal to zero, resulting in
decreased coding efficiency. Efficiency may be partially regained by thresholding
the near-zero coefficients, but due to the non-orthogonality of the representation
this will produce sub-optimal results as previously mentioned. The maximum a
posteriori (MAP) estimate also introduced biases in the learning process. These
effects can be partially compensated for by renormalizing the basis functions, but
other parameters of the model such as those of the prior could not be learned.
Finally, the gradient ascent method has convergence problems due to the power
spectrum of natural images and the overcompleteness of the representation. Here we
resolve these problems by using a prior distribution which is composed of a mixture
of a Gaussian and a Dirac delta function, so that inactive coefficients are encouraged
to have exact zero values. Similar models employing a mixture of two Gaussians
have been used for classifying wavelet coefficients into active (high variance) and
inactive (low variance) states [2, 5]. Such a classification should be even more
advantageous if the basis is overcomplete. A method for performing Gibbs-sampling
for the Delta-plus-Gaussian prior in the context of an image pyramid is derived, and
demonstrated to be effective at obtaining very sparse representations which match
the form of the imposed prior. Biases in the learning are overcome by sampling
instead of using a MAP estimate.
2
Wavelet image model
Each observed image I is assumed to be generated by a linear superposition of basis
functions which are columns of an N by M weight matrix W, with the addition of
Gaussian noise ?:
I = W a + ?,
(1)
where I is an N -element vector of image pixels and a is an M -element vector of basis
coefficients. In order to achieve a practical implementation which can be seamlessly
scaled to any size image, we assume that the basis function matrix W is composed of
a small set of spatially localized mother wavelet functions ?i (x, y), which are shifted
to each position in the image and rescaled by factors of two. Unlike typical wavelet
transforms which use a single 1-D mother wavelet function to generate 2-D functions
by inner product, we do not constrain the functions ?i (x, y) to be 1-D separable.
The functions ?i (x, y) provide an efficient way to perform computations involving
W by means of convolutions. Basis functions of coarser scales are produced by
upsampling the ?i (x, y) functions and blurring with a low-pass filter ?(x, y), also
known as the scaling function. The image model above may be re-expressed to
make these parameters explicit:
I(x, y)
g l (x, y)
= g 0 (x, y) + ?(x, y)
l+1
P
g (x, y) ? 2 ? ?(x, y) + i ali (x, y) ? ?i (x, y)
=
al (x, y)
(2)
l <L?1
(3)
l =L?1
where the coefficients ali (x, y) are indexed by their position (x, y), band (i) and
level of resolution (l) within the pyramid (l = 0 is the highest resolution level). The
symbol ? denotes convolution, and ? 2 denotes upsampling by two and is defined as
f ( x2 , y2 ) x even & y even
f (x, y) ? 2 ?
(4)
0
otherwise
The probability of generating an image I, given coefficients a, parameters ?, assuming Gaussian i.i.d. noise ? (with variance 1/?N ), is
P (I|a, ?)
1 ? ?N |I?W a|2
.
e 2
Z ?N
=
(5)
The prior probability over each coefficient ai is modeled as a mixture of a Gaussian
distribution and a Dirac delta function ?(ai ). A binary state variable si for each
coefficient indicates whether the coefficient ai is active (any real value), or inactive
(zero). The probability of a coefficient vector a given a binary state vector s and
model parameters ? = {W, ?N , ?a , ?s } is defined as
Y
P (ai |si , ?)
(6)
P (a|s, ?) =
i
P (ai |si , ?)
=
(
?(ai )
1
Z?a
e
?
?a
i
2
a2i
if
si = 0,
if
si = 1
(7)
i
where ?a is a vector with elements ?ai . The probability of a binary state s is
P (s|?) =
1 ? 1 sT ? s s
.
e 2
Z ?s
(8)
Matrix ?s is assumed to be diagonal (for now), with nonzero elements ?si . The
form of the prior is shown graphically in figure 1. Note that the parameters W, ? a ,
and ?s are themselves parameterized by a much smaller set of parameters. Only
the mother wavelet function ?i (x, y) and a single ?si and ?ai parameter need to be
learned for each wavelet band, since we are assuming translation invariance.
The total image probability is obtained by marginalizing over the possible coefficient
and state values:
Z
X
P (I|?) =
P (s|?) P (I|a, ?)P (a|s, ?) da
(9)
s
3
Sampling and Inference
We show how to sample from the posterior distribution P (a, s|I, ?) for an image
I using a Gibbs sampler. For each coefficient and state variable pair (ai ,si ), we
10
0
-1
10
-2
10
-3
10
-4
10
-5
10
-6
10
0.5
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
Figure 1: Prior distribution (dashed), and histogram of samples taken from the
posterior distribution (solid) plotted for a single coefficient. The y-axis is plotted
on a log scale.
sample from the posterior distribution conditioned on the image and the remaining
coefficients a?i : P (ai , si |I, a?i , s?i , ?). After all coefficients (and state variables) have
been updated, this process is repeated until the system has reached equilibrium. To
infer an optimal representation a for an image I (for coding or denoising purposes),
we can either average a number of samples to estimate the posterior mean, or with
minor adjustment locate a posterior maximum by raising the posterior distribution
to a power (1/T ) and annealing T to zero. To sample from P (ai , si |I, a?i , s?i , ?), we
first draw a value for si from P (si |I, a?i , s?i , ?), then draw ai from P (ai |si , I, a?i , s?i , ?).
For P (si |I, a?i , s?i , ?) we have:
P (si |I, a?i , s?i , ?)
? P (si |s?i , ?)
where
P (si |s?i , ?)
=
P (I|ai , a?i , ?)
=
Z
P (I|ai , a?i , ?)P (ai |si , ?)dai
1
e?
si
,
Zsi |s?i
1 ? ?ni (ai ?bi )2
e 2
,
Z ?ni
and
?ni = ?N |Wi |2 ,
?s
i
2
bi =
Wi ? (I ? W ai=0 )
.
|Wi |2
(10)
(11)
(12)
(13)
The notation Wi denotes column i of matrix W, |Wi | is the length of vector Wi ,
and ai=0 denotes the current coefficient vector a except with ai set to zero. Thus,
bi denotes the value for ai which minimizes the reconstruction error (while holding
a?i constant). Since si can only take on two values, we can compute equation 10 for
si = 0 and si = 1, integrating over the possible coefficient values. This yields the
following sigmoidal activation rule as a function of bi :
P (si = 1|I, a?i , s?i , ?)
=
1
1+e
??i (b2i ?ti )
(14)
where
?2ni
1
?i =
,
2 ? n i + ? ai
? n i + ? ai
? ai
ti =
?si ? log
.
?2ni
? n i + ? ai
(15)
For P (ai |si , I, a?i , s?i , ?) we have:
P (ai |si , I, a?i , s?i , ?)
=
(
?(ai )
? i bi
, ?n
N ( ?n n+?
a
i
i
1
+?ai
i
)
if si = 0,
if si = 1
(16)
To perform this procedure on a wavelet pyramid, the inner product computations
necessary to compute bi can be performed efficiently by means of convolutions with
the mother wavelet functions ?i (x, y). The ?N , ?si and ?ai parameters may be
adapted to a specific image during the inference process by use of the update rules
described in the next section. This method was found to be particularly useful for
denoising, when the variance of the noise was assumed to be unknown.
4
Learning
Our objective for learning is to adjust the parameters, ?, to maximize the average
log-likelihood of images under the model:
?? = arg max hlog P (I|?)i
(17)
?
The parameters are updated by gradient ascent on this objective, which results in
the following update rules:
??si
??ai
??i (x, y)
?
?
1
2
*
1
2
*
1
1
1 + e 2 ? si
si
? si
1
? a2i
? ai
P (a,s|I,?)
P (a,s|I,?)
+
D
E
? ?N he(x, y) ? ai (x, y)iP (a,s|I,?)
+
(18)
(19)
(20)
where ? denotes cross correlation and e(x, y) is the reconstruction error computed
by e = I ? W a. Only a center portion of the cross correlation with the extent of
the ?i (x, y) functions is computed to update the parameters. The outer brackets
denotes averaging over many images. The notation hiP () denotes averaging the
quantity in brackets while sampling from the specified distribution.
5
Results
The image model was trained on 22 512x512 pixel grayscale natural images (not
whitened). These images were generated from color images taken from a larger
database of photographic images 1 . Smaller images (64x64 pixels) were selected
randomly for sampling during training. To simplify the learning procedure, sampling was performed on a single spatial frequency scale. Each image was bandpass
filtered for an octave range before sampling from the posterior for that scale. The
1
Images were downloaded from philip.greenspun.com with permission from Philip
Greenspun.
(a)
(b)
Figure 2: (a) Mother wavelet functions ?i (x, y) adapted for 2, 4 and 6 bands and
corresponding power spectra showing power as a function of spatial frequency in
the 2D Fourier plane. (b) Equivalent mother wavelets and spectra for the 4-band
Steerable Pyramid.
?ai and ?si parameters were constrained to be the same for all orientation bands and
were adapted over many images with ?N fixed. Shown in figure 2 are the learned
?i (x, y) which parameterize W , with their corresponding 2D spectra. Three different degrees of overcompleteness were tested. The results are shown for 2 band, 4
band and 6 band wavelet bases. As the degree of overcompleteness increases, the
resulting functions show tighter tuning to orientation. The basis filters for a 4 band
Steerable Pyramid [10] are also shown for comparison, to illustrate the similarity
to the learned functions.
27
learned
steer
26.5
26
SNR (dB)
25.5
25
24.5
24
23.5
23
22.5
1.0
2.0
3.0
4.0
5.0
% nonzeros
Figure 3: Sparsity comparison between the learned basis (top) and the steerable
basis (bottom). The y axis represents the signal-to-noise ratio (SNR) in dB achieved
for each method for a given percentage of nonzeros.
5.1
Sparsity
We evaluated the sparsity of the representations obtained with the four band learned
functions and the sampling method with those obtained using the same sampling
method and the four band Steerable Pyramid filters [10]. In order to explore the
SNR curves for each basis, we used a variety of values for ?s so as to obtain different
levels of sparsity. The same images were used for both bases. The results are given
in figure 3. Each dot on the line represents a different value of ?s . The results were
similar, with the learned basis yielding slightly higher SNR (about 0.5 dB) for the
same number of active coefficients.
5.2
Denoising
We evaluated our inference method and learned basis functions by denoising images
containing known amounts of additive i.i.d. Gaussian noise. Denoising was accomplished by averaging samples taken from the posterior distribution for each image
via Gibbs sampling to approximate the posterior mean. Gibbs sampling was performed on a four level pyramid using the 6 band learned wavelet basis, and also
using the 6 band Steerable basis. The ?N , ?si and ?ai parameters were adapted to
each noisy image during sampling for blind denoising in which the noise variance
was assumed to be unknown. We compared these results to the wiener2 function in
MATLAB, and also to BayesCore [9], a Bayesian method for computing an optimal
soft thresholding, or coring, function for a generalized Laplacian prior. For wiener2,
the best neighborhood size was used for each image. Table 1 gives the SNR results
for each method when applied to some standard test images for three different levels of i.i.d. Gaussian noise with standard deviation ?. Figure 4 shows a cropped
subregion of the results for the ?Einstein? image with ? = 10.
6
Summary and Conclusions
We have shown that a wavelet basis and a mixture prior composed of a Dirac delta
function and a Gaussian can be adapted to natural images resulting in very sparse
image representations. The resulting basis is very similar to a Steerable basis, both
in appearance and sparsity of the resulting image representations. It appears that
the Steerable basis may be nearly optimal for producing sparse representations of
natural scenes. Denoising results indicate that using a sparse prior and an inference
method to properly account for the non-orthogonality of the representation may
yield a significant improvement over wavelet coring methods that use filtered coefficients. More work needs to be done to determine whether the coding gains achieved
are due to the choice of prior versus the basis or inference/estimation method used.
Acknowledgments Supported by NIMH R29-MH057921. Phil Sallee?s work was
also supported in part by a United States Department of Education Government
Assistance in Areas of National Need (DOE-GAANN) grant #P200A980307.
Image
Einstein
Lena
Goldhill
Fruit
noise level
? = 10
? = 20
? = 30
? = 10
? = 20
? = 30
? = 10
? = 20
? = 30
? = 10
? = 20
? = 30
noisy
12.40
6.40
2.89
13.61
7.59
4.07
13.86
7.83
4.28
16.25
10.24
6.70
wiener2
15.80
12.61
10.95
19.05
15.51
13.25
17.56
14.32
12.64
21.87
18.15
15.97
BayesCore S6
16.36
13.44
11.81
19.91
16.88
14.99
18.14
15.18
13.61
22.09
18.97
17.21
D+G S6
16.47
13.80
12.28
20.37
17.46
15.48
18.10
15.41
13.92
22.78
19.61
17.72
D+G L6
16.19
13.79
12.29
20.21
17.54
15.55
17.90
15.41
13.95
22.38
19.42
17.66
Table 1: SNR values (in dB) for noisy and denoised images contaminated with
additive i.i.d. Gaussian noise of std.dev. ?. ?D+G? means Delta-plus-Gaussian
prior, ?S6? means 6-Band Steerable basis, and ?L6? means 6-Band Learned basis.
original
noisy (?=10) SNR=12.3983
wiener2 SNR=15.8033
BayesCore steer6 SNR=16.3591
D+G steer6 SNR=16.4714
D+G learned6 SNR=16.1939
Figure 4: Denoising example. A cropped subregion of the Einstein image and
denoised images for each noise reduction method for noise std.dev. ?=10.
References
[1] Abromovich F, Sapatinas T, Silverman B (1996), Wavelet Thresholding via a Bayesian
Approach, preprint.
[2] Chipman H, Kolaczyk E, McCulloch R (1997) Adaptive bayesian wavelet shrinkage,
J. Amer. Statist. Assoc. 92(440): 1413-1421.
[3] Chang SG, Yu B, Vetterli M (2000). Spatially Adaptive Wavelet Thresholding with
Context Modelling for Image Denoising. IEEE Trans. on Image Proc., 9(9): 1522-1531.
[4] Coifman RR, Donoho DL (1995). Translation-invariant de-noising, in Wavelets and
Statistics, A.Antoniadis and G. Oppenheim, Eds. Berlin, Germany: Springer-Varlag.
[5] Crouse MS, Nowak RD and Baraniuk RG (1998) Wavelet-based Statistical Signal
Processing using Hidden Markov Models, IEEE Trans. Signal Proc., 46(4): 886-902.
[6] Freeman WT, Adelson EH (1991) The Design and Use of Steerable Filters. IEEE
Trans. Patt. Anal. and Machine Intell., 13(9): 891-906.
[7] Olshausen BA, Sallee P, Lewicki MS (2001) Learning sparse image codes using a
wavelet pyramid architecture, Adv. in Neural Inf. Proc. Sys., 13: 887-893.
[8] Simoncelli EP, Freeman WT, Adelson EH, Heeger DJ (1992) Shiftable multiscale transforms, IEEE Transactions on Information Theory, 38(2): 587-607.
[9] Simoncelli EP, Adelson EH (1996) Noise removal via Bayesian wavelet coring, Presented at: 3rd IEEE International Conf. on Image Proc., Laussanne Switzerland.
[10] Simoncelli EP, Freeman WT (1995). The Steerable Pyramid: A Flexible Architecture
for Multi-scale Derivative Computation, IEEE Int. Conf. on Image Processing.
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1,258 | 2,146 | Conditional Models on the Ranking Poset
Guy Lebanon
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
John Lafferty
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
[email protected]
[email protected]
Abstract
A distance-based conditional model on the ranking poset is presented
for use in classification and ranking. The model is an extension of the
Mallows model, and generalizes the classifier combination methods
used by several ensemble learning algorithms, including error correcting
output codes, discrete AdaBoost, logistic regression and cranking. The
algebraic structure of the ranking poset leads to a simple Bayesian interpretation of the conditional model and its special cases. In addition to a
unifying view, the framework suggests a probabilistic interpretation for
error correcting output codes and an extension beyond the binary coding
scheme.
1 Introduction
Classification is the task of associating a single label
with a covariate . A generalization of this problem is conditional ranking, the task of assigning to a full or partial
ranking of the items in . This paper studies the algebraic structure of this problem, and
proposes a combinatorial structure called the ranking poset for building probability models
for conditional ranking.
In ensemble approaches to classification and ranking, several base models are combined to
produce a single ranker or classifier. An important distinction between different ensemble
methods is whether they use discrete inputs, ranked inputs, or confidence-rated predictions. In the case of discrete inputs, the base models provide a single item in , and no
preference for a second or third choice is given. In the case of ranked input, the base classifiers output a full or partial ranking over . Of course, discrete input is a special case
of ranked input, where the partial ranking consists of the single topmost item. In the case
of confidence-rated predictions, the base models again output full or partial rankings, but
in addition provide a confidence score, indicating how much one class should be preferred
to another. While confidence-rated predictions are sometimes preferable as input to an ensemble method, such confidence scores are often not available (as is typically the case in
metasearch), and even when they are available, the scores may not be well calibrated.
This paper investigates a unifying algebraic framework for ensemble methods for classification and conditional ranking, focusing on the cases of discrete and ranked inputs.
, which consists of
Our approach is based on the ranking poset on items, denoted
the collection of all full and partial rankings equipped with the partial order given by re-
finement of rankings. The structure of the poset of partial ranking over gives rise to
natural invariant distance functions that generalize Kendall?s Tau and the Hamming distance. Using these distance functions we define a conditional model
where
. This conditional model generalizes several existing models
for classification and ranking, and includes as a special case the Mallows model [11]. In
addition, the model represents algebraically the way in which input classifiers are combined
in certain ensemble methods, including error correcting output codes [4], several versions
of AdaBoost [7, 1], and cranking [10].
In Section 2 we review some basic algebraic concepts and in Section 3 we define the ranking poset. The new model and its Bayesian interpretation are described in Section 4. A
derivation of some special cases is given in Section 5, and we conclude with a summary in
Section 6.
2 Permutations and Cosets
We begin by reviewing some basic concepts from algebra, with some of the notation and
definitions borrowed from Critchlow [2].
Identifying the items to be ranked
with the numbers
, if denotes a
permutation of
, then !"
denotes the rank given to item " and $# "
denotes
the item assigned to rank " . The collection of all permutations of -items forms the nonabelian symmetric group of order , denoted % . The multiplicative notation '& )(*
is used to denote function composition.
#
The subgroup of
; thus,
%
%
consisting of all permutations that fix the top
%
,(- .
%
#
The right coset
#
%
+ positions is denoted
!"
!(/"0
" (1
+2
34(5 3 67 '
%
#
(1)
8
(2)
is equivalent to a partial ranking, where there is a full ordering of the + top-ranked items.
.
The set of all such partial rankings forms the quotient space % :9%
#
= of positive integers that sum
An ordered partition of is a sequence ;<(
to . Such an ordered partition corresponds to a partial ranking of type ; with items
in the first position, > items in the second position and so on. No further information
is conveyed about orderings within each position. A partial ranking of the top + items is
aL special case with ?@
L (A+CB*
D E(F ( @(F G
L IHD E( .JK+ . More formally, let
(* G G
G
>M(N BO G
B > G
&&&
= (* BO&&&PB = # BO
.
Then the subgroupLR%R[ QOS( LR
% [ GTVUW&&&2UX% ZY contains all permutations W% for which
!
\
(
the set equality
holds for each " ; that is, all permutations that only permute
LR[
within each . A partial ranking of type ; is equivalent to a coset % Q and the set of such
partial rankings forms the quotient space % :9%]Q .
We now describe a convenient notation for permutations and cosets. In the following, we
list items separated by vertical lines, indicating that the items on the left side of the line are
preferred to (ranked higher than) the items on the right side of the line. For example, the
permutation !7 3M(*^
7!_^G`(a
!cb d(*b is denoted by ^:e b . A partial ranking %gf
# h
where the top 3 items are b
^
is denoted by b2 ^i 8 j
k . A classification
may thus be
h
denoted by be
^
7j
k . A partial ranking % Q where ;l(mb:
^ with items
b
k ranked in
the first position is denoted by G
b:
k: ^n
j .
A distance function o on %
is a function oNpq% rUl% Nsut that satisfies the usual
properties: o2
7g(wv , o2
xyv when a(A
z , o2
C(wo2c
, and the triangle
T
PSfrag replacements
T
T
T
T
T
T
T
T
T
T
PSfrag replacements
T
Figure 1: The Hasse diagram of
h (left) and a partial Hasse diagram of (right). Some
of the lines are dotted for easier visualization.
o2
:B o2
for all
.% . In addition, since the indexing
inequality oc
of the items 8
is arbitrary, it is appropriate to require invariance to relabeling of .
Formally, this amounts to right invariance o2
(/o2:
: , for all
'%
A popular right invariant distance on %
]c
(
#
[
is Kendall?s Tau ]
, given by
[ # c "
JX # _7
(3)
where 2E( for ,xAv and 2E(4v otherwise [8]. Kendall?s Tau ]
can be
interpreted as the number of discordant pairs of items between and , or the minimum
number of adjacent transpositions needed to bring $# to D# . An adjacent transposition
flips a pair of items that have adjacent ranks. Critchlow [2] derives extensions of Kendall?s
Tau and other distances on %
to distances on partial rankings.
3 The Ranking Poset
We first define partially ordered sets and then proceed to define the ranking poset. Some of
the definitions below are taken from [12], where a thorough introduction to posets can be
found.
A partially ordered set or poset is a pair "!d
$#q , where ! is a set and # is a binary relation
that satisfies (1) %#
, (2) if %#
and &#
then )(
, and (3) if %#
and &#('
then )#*' for all
2
+' ,! . We write ,when ,#
and ,( z
. We say that
covers and write /.
when 0and there is no ' 1! such that 0-)' and '2.A
finite poset is completely described by the covering relation. The planar Hasse diagram of
"!d
3#q is the graph for which the elements of ! are the nodes and the edges are given by
the covering relation. In addition, we require that if 4.
then is drawn higher than .
The ranking poset
is the poset in which the elements are all possible cosets %65 ,
% . The partial order of
where 7 is an ordered partition of and
is defined by
refinement; that is, &-l if we can get from to by adding vertical lines. Note that
is different from the poset of all set partitions of G
2
ordered by partition refinement
since in
the order of the partition elements matters. Figure 1 shows the Hasse diagram
of
and a portion of the Hasse diagram of .
h
A subposet 8E
$#:9! of "!d
3#<;$ is defined by 8>=?! and &#:9
if and only if @#:; .
A chain is a poset in which every two elements are comparable. A saturated chain A of
.m&&&:. .
length + is a sequence of elements
! that satisfy .
A chain of ! is a maximal chain if there is no other saturated chain of ! that contains it. A
graded poset of rank is a poset in which every maximal chain has length . In a graded
poset, there is a rank or grade function )p !As 3v:
such that (mv if is a
minimal element and !(2 2B if 4. .
It is easy to see that
is a graded poset of rank XJ, and the rank of every element
is the number of vertical lines in its denotation. We use to denote the subposet of
\ . In particular, the elements in the + th grade,
consisting of
all of which are incomparable, are denoted by . Full orderings occupy the topmost
grade . Classifications "
" reside in . Other elements of are
#
multilabel classifications li G
2
where = G G
.
4 Conditional Models on the Ranking Poset
We now present a family of conditional models defined in terms of the ranking poset. To
. That is, o2
( oc
: for
begin, suppose that o is a right invariant function on
all
and
% . Here right invariance is defined with respect to the natural
action of %
on
, given by
[
[
[
T3
TZi I&&&i
[
[
[
&$y(5 3 T 0
T
0
&&&3i 3
I
(4)
The function o may or may not be a metric; its interpretation as a measure of dissimilarity,
however, remains.
We will examine several distances that are based on the covering relation . of
. Down
and up moves on the Hasse diagram will be denoted by and respectively. A distance
o defined in terms of and moves is easily shown to be right invariant because the
group action of %
does not change the covering relation between any two elements; that
is, the group action of %
on
commutes with and moves:
JJJJ0s
JJ JJ0s
JJ:JJ0s
(5)
JJ: JJ0s
While the metric properties of o are not[ required in our model, the right invariance property
is essential since we want to treat all in the same manner.
We are now ready to give the general form of a conditional model on
. Let o be an
invariant function, as above. The model takes as input + rankings
>
contained
in some subset =
of the ranking poset. For example, each ! could be an element of
, which will be the ?carrier density?
# . Let " be a probability mass function on
or default model. Then o and " specify an exponential model $#M given by
c6$#M (
:
9
where & 6587 t ,
normalizing constant
%
'&D
(#d (
%
/0
'&
(#M
)8+*",.-
, and .! ;
=
<.=?>
/0
+*",.-
! 21
! 1
=
!o2
+!$34
.
The term
! oc
!
43
%
(6)
'&
#M is the
(7)
, 7 &3 #M forms a probability distribution over 9(= .
Given a data set c
# , the parameters 5 will[ typically
[ be selected by maxi( 1 [
2 $# , a marginal likelihood
mizing the conditional loglikelihood
&
or posterior. Under mild regularity conditions,
will be convex and have a unique global
1
maximum.
Thus, conditional on #
[
[
4.1 A Bayesian interpretation
We now derive a Bayesian interpretation for the model given by (6). Our result parallels
the interpretation of multistage ranking models given by Fligner and Verducci [6]. The key
fact is that, under appropriate assumptions, the normalizing term does not depend on the
partial ordering in the one-dimensional case.
Proposition 4.1. Suppose that o is right invariant and that
of % . If %
acts transitively on 9 then
for all
7
9 and
1
=
< (
=
is invariant under the action
<
(8)
t .
Proof. First, note that since =
is invariant under the action of % , it follows that
(
% . [Indeed,
7
for each[
by the invariance
assumption,[ ! and :7
!
[
since for l(< T 8 &&&3
we have " !(< 3 # T0 nI&&&i 3 # 0
such
that " (/ .
acts transitively on 9 , for all
Now, since %
We thus have that
%
1
(
=
(
(
(
Thus, we can write
fact depend on .
%
7
M(
1
%
<
9 there is # .%
such that "#E(
$ .
(9)
< % %
=
< ' %
=
< '
=
(by right invariance of & )
(10)
(11)
(by invariance of ( )
(12)
since the normalizing constant for
1
9 does not in
The underlying generative model is given as follows. Assume that
9 is drawn from
the prior " c and that
are independently drawn from generalized Mallows
models
*) c .!
(
%
) < )
(13)
3!
1
where .! . Then under the conditions of Proposition
4.1, we have from Bayes? rule
that the posterior distribution over is given by
- ) ) < )
%
)8,+ ! * ) _ +!
)8
+ ! !30#
- ) 1 . )/ 0< * )
(
(14)
%
<+=?> )8, + ! * ) _ +!
+ ! ! # <+=?> c
1
$#M
(
(15)
We thus have the following characterization of
7&3$#M .
Proposition 4.2. If o is right invariant, is invariant under the action of % , and %
acts
transitively on 9 , then the model
&$#M defined in equation (6) is the posterior under
*)
& , with prior S " .
independent sampling of generalized Mallows models, 2! S
The
[ conditions of this proposition are satisfied, for example, when 91(*%
% :9G% 5 as is assumed in the special cases of the next section.
:9G%
5
and
(
5 Special Cases
This section derives several special cases of model (6), corresponding to existing ensemble methods. The special cases correspond to different choices of 9]
5 and o in the
definition of the model. In each case c is taken to be uniform, though the extension
to non-uniform is immediate. Following [9], the unnormalized versions of all the
models may be easily derived, corresponding to the exponential loss used in boosting.
5.1 Cranking and Mallows
model
Let 5N( t
(9 ( /( (N% , and let oc
be the minimum number of down#
up ( ) moves on the Hasse diagram of needed to bring to . Since adjacent
transpositions of permutations may be identified with a down move followed by an up
move over the Hasse diagram, oc
is equal to Kendall?s Tau ]c
. For example,
]7 ^: b
b2 ^i ( b and the corresponding path in Figure 1 is
^: b
^ b
^i 8 b
^i G
b
^: be
^n
be
b ^:e
In this case model (6) becomes the cranking model [10]
6&2 (
- ) T ) < )
%
#V
&2
&
+! '
%
t
(16)
The Bayesian interpretation in this case is well known, and is derived in [6]. The generative
model is independent sampling of ! from a Mallows model whose location parameter
is and whose scale parameter is ! . Other special cases that fall into this category are the
models of Feigin [5] and Critchlow1 and Verducci [3].
5.2 Logistic models
Let 5a( t , 9 (6*(a% 9%
number of up-down
# , and let oc
be the minimum
( ) moves in the Hasse diagram. Since 9 ( ( % :9%
o2
(/o2 %
#
#
:
0%
#!(
v
#
^
if #
!(
otherwise
#2#
(17)
In this case model (6) becomes equivalent to the multiclass generalization of logistic regression. If the normalization constraints in the corresponding convex primal problem are
removed, the model becomes discrete
[ AdaBoost.M2; that is, o2
!8 29Z^ becomes the
(discrete) multiclass weak learner
3v:
in the usual boosting notation. See [9]
for details on the correspondence between exponential models and the unnormalized models that correspond to AdaBoost.
5.3 Error correcting output codes
A more interesting special case of the algebraic structure described in Sections 3 and 4
is where the ensemble method is error correcting output coding (ECOC) [4]. Here we set
#
, l(
9 (,% :9G%
9
, and take the parameter space to be
[
(18)
55(- & Ot ( >q(*&&&G(
and
v G
1
1
1
1
As before, o2
is the minimal number of up-down ( ) moves in the Hasse diagram
needed to bring to .
T
Since ( %
# , the model computes probabilities of classifications . On input , the base rankers output !G 2
?9 , which corresponds to one of the binary
classifiers in ECOC for the appropriate column of the binary coding matrix.
For example,
consider a binary classifier trained on the coding column
v
v:
v:
vG . On an input ,
the classifier outputs 0 or 1, corresponding to the partial rankings ( ^n
7j
k
e
b and
@(* G
b ^n
j:
0kn
, respectively.
Since
@% 9G% #
and
9
o2
(
(
o2 %
^
[ [
[
= i [ =
if #
#
(20)
otherwise.
For example, if (*^:e
b
j:
k
and X(N^n
j:
k
2i G
b , then o2
`(
from the sequence of moves
^:e
b
j:
k
(19)
^: j:
0kn
2i G
b
, as can be seen
^n
j:
k
2i G
b`
(21)
If '(* 8 ^
b:
7j:
0kn
and '(
n^
7j
kn
e
b , then o2
( ^ , with the sequence of moves
^n
7j
k
b
^
7j
kn
b ^
7j:
0kn
e b ^
7j:
0kn
e
bM (22)
^n
b
7j
k
[
Since (
! , the exponent of the model becomes ! o2
+! . At test time, the model
1
1 the label corresponding to the partial ranking
1
( arg
, < n6$#M . Now,
thus selects
since is strictly negative, #M is a monotonically decreasing function in ! o2
+! .
1
Equivalence with the ECOC decision rule thus follows from the fact that ! oc
+!nJ+
is the Hamming distance between the appropriate row of the coding matrix and the concatenation of the bits returned from the binary classifiers.
Thus, with the appropriate definitions of 9\
and o , the conditional model on the ranking
poset is a probabilistic formulation of ECOC that yields the same classification decisions.
This suggests ways in which ECOC might be naturally extended. First, relaxing the constraint `(
> (*&&&8( results in a more general model that corresponds to ECOC with a
1 Hamming
1
1
weighted
distance,
or index sensitive ?channel,? where the learned weights may
adapt to the precision of the various base classifiers. Another simple generalization results
from using a nonuniform carrier density .
A further generalization is achieved by considering that
[ [ for a given
[ [ coding[ [ matrix,[ the
[
trained classifier for a given column outputs either = i =
or = =
depending on the input . Allowing the output of the classifier instead to belong to other
grades of
results in a model that corresponds to error correcting output codes with nonbinary codes. While this is somewhat antithetic to the original spirit of ECOC?reducing
multiclass to binary?the base classifiers in ECOC are often multiclass classifiers such as
decision trees in [4]. For such classifiers, the task instead can be viewed as reducing multiclass to partial ranking. Moreover, there need not be an explicit coding matrix. Instead,
the input rankers may output different partial rankings for different inputs, which are then
combined according to model (6). In this way, a different coding matrix is built for each
example in a dynamic manner. Such a scheme may be attractive in bypassing the problem
of designing the coding matrix.
6 Summary
An algebraic framework has been presented for classification and ranking, leading to conditional models on the ranking poset that are defined in terms of an invariant distance or
dissimilarity function. Using the invariance properties of the distances, we derived a generative interpretation of the probabilistic model, which may prove to be useful in model
selection and validation. Through different choices of the components
9]
and o , the
family of models was shown to include as special cases the Mallows model, and the
classifier combination methods used by logistic models, boosting, cranking, and error correcting output codes. In the case of ECOC, the poset framework shows how probabilities
may be assigned to partial rankings in a way that is consistent with the usual definitions of
ECOC , and suggests several natural extensions.
Acknowledgments
We thank D. Critchlow, G. Hulten and J. Verducci for helpful input on the paper. This work
was supported in part by NSF grant CCR-0122581.
References
[1] M. Collins, R. E. Schapire, and Y. Singer. Logistic regression, AdaBoost and Bregman distances. Machine Learning, 48, 2002.
[2] D. E. Critchlow. Metric Methods for Analyzing Partially Ranked Data. Lecture Notes
in Statistics, volume 34, Springer, 1985.
[3] D. E. Critchlow and J. S. Verducci. Detecting a trend in paired rankings. Journal of
the Royal Statistical Society C, 41(1):17?29, 1992.
[4] T. G. Dietterich and G. Bakiri. Solving multiclass learning problems via errorcorrecting codes. Journal of Artificial Intelligence Research, 2:263?286, 1995.
[5] P. D. Feigin. Modeling and analyzing paired ranking data. In M. A. Fligner and
J. S. Verducci, editors, Probability Models and Statistical Analyses for Ranking Data.
Springer, 1992.
[6] M. A. Fligner and J. S. Verducci. Posterior probabilities for a concensus ordering.
Psychometrika, 55:53?63, 1990.
[7] Y. Freund and R. E. Schapire. Experiments with a new boosting algorithm. In International Conference on Machine Learning, 1996.
[8] M. G. Kendall. A new measure of rank correlation. Biometrika, 30, 1938.
[9] G. Lebanon and J. Lafferty. Boosting and maximum likelihood for exponential models. In Advances in Neural Information Processing Systems, 15, 2001.
[10] G. Lebanon and J. Lafferty. Cranking: Combining rankings using conditional probability models on permutations. In International Conference on Machine Learning,
2002.
[11] C. L. Mallows. Non-null ranking models. Biometrika, 44:114?130, 1957.
[12] R. P. Stanley. Enumerative Combinatorics, volume 1. Wadsworth & Brooks/Cole
Mathematics Series, 1986.
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1,259 | 2,147 | Efficient Learning Equilibrium *
Ronen I. Brafman
Computer Science Department
Ben-Gurion University
Beer-Sheva, Israel
email: [email protected]
Moshe Tennenholtz
Computer Science Department
Stanford University
Stanford, CA 94305
e-mail: [email protected]
Abstract
We introduce efficient learning equilibrium (ELE), a normative approach to learning in non cooperative settings. In ELE, the learning algorithms themselves are required to be in equilibrium. In
addition, the learning algorithms arrive at a desired value after
polynomial time, and deviations from a prescribed ELE become irrational after polynomial time. We prove the existence of an ELE
in the perfect monitoring setting, where the desired value is the
expected payoff in a Nash equilibrium. We also show that an ELE
does not always exist in the imperfect monitoring case. Yet, it
exists in the special case of common-interest games. Finally, we
extend our results to general stochastic games.
1
Introduction
Reinforcement learning in the context of multi-agent interaction has attracted the
attention of researchers in cognitive psychology, experimental economics, machine
learning, artificial intelligence, and related fields for quite some time [8, 4]. Much
of this work uses repeated games [3, 5] and stochastic games [10, 9, 7, 1] as models
of such interactions.
The literature on learning in games in game theory [5] is mainly concerned with the
understanding of learning procedures that if adopted by the different agents will
converge at end to an equilibrium of the corresponding game. The game itself may
be known; the idea is to show that simple dynamics lead to rational behavior, as
prescribed by a Nash equilibrium. The learning algorithms themselves are not required to satisfy any rationality requirement; it is what they converge to, if adopted
by all agents that should be in equilibrium. This is quite different from the classical perspective on learning in Artificial Intelligence, where the main motivation
The second author permanent address is: Faculty of Industrial Engineering and Management , Technion, Haifa 32000, Israel. This work was supported in part by Israel Science
Foundation under Grant #91/02-1. The first author is partially supported by the Paul
Ivanier Center for Robotics and Production Management .
for learning stems from the fact that the model of the environment is unknown.
For example, consider a Markov Decision Process (MDP). If the rewards and transition probabilities are known then one can find an optimal policy using dynamic
programming. The major motivation for learning in this context stems from the
fact that the model (i.e. rewards and transition probabilities) is initially unknown.
When facing uncertainty about the game that is played, game-theorists appeal to
a Bayesian approach, which is completely different from a learning approach; the
typical assumption in that approach is that there exists a probability distribution
on the possible games, which is common-knowledge. The notion of equilibrium is
extended to this context of games with incomplete information, and is treated as
the appropriate solution concept. In this context, agents are assumed to be rational
agents adopting the corresponding (Bayes-) Nash equilibrium, and learning is not
an issue.
In this work we present an approach to learning in games, where there is no known
distribution on the possible games that may be played - an approach that appears to
be much more reflective of the setting studied in machine learning and AI and in the
spirit of work on on-line algorithms in computer science. Adopting the framework of
repeated games, we consider a situation where the learning algorithm is a strategy
for an agent in a repeated game. This strategy takes an action at each stage based
on its previous observations, and initially has no information about the identity of
the game being played. Given the above, the following are natural requirements for
the learning algorithms provided to the agents:
1. Individual Rationality: The learning algorithms themselves should be in
equilibrium. It should be irrational for each agent to deviate from its
learning algorithm, as long as the other agents stick to their algorithms,
regardless of the what the actual game is.
2. Efficiency:
(a) A deviation from the learning algorithm by a single agent (while the
other stick to their algorithms) will become irrational (i.e. will lead to
a situation where the deviator 's payoff is not improved) after polynomially many stages.
(b) If all agents stick to their prescribed learning algorithms then the expected payoff obtained by each agent within a polynomial number of
steps will be (close to) the value it could have obtained in a Nash
equilibrium, had the agents known the game from the outset.
A tuple of learning algorithms satisfying the above properties for a given class of
games is said to be an Efficient Learning Equilibrium[ELE]. Notice that the learning
algorithms should satisfy the desired properties for every game in a given class
despite the fact that the actual game played is initially unknown. Such assumptions
are typical to work in machine learning. What we borrow from the game theory
literature is the criterion for rational behavior in multi-agent systems. That is, we
take individual rationality to be associated with the notion of equilibrium. We also
take the equilibrium of the actual (initially unknown) game to be our benchmark
for success; we wish to obtain a corresponding value although we initially do not
know which game is played.
In the remaining sections we formalize the notion of efficient learning equilibrium,
and present it in a self-contained fashion. We also prove the existence of an ELE
(satisfying all of the above desired properties) for a general class of games (repeated
games with perfect monitoring) , and show that it does not exist for another. Our
results on ELE can be generalized to the context of Pareto-ELE (where we wish
to obtain maximal social surplus), and to general stochastic games. These will be
mentioned only very briefly, due to space limitations. The discussion of these and
other issues, as well as proofs of theorems, can be found in the full paper [2].
Technically speaking, the results we prove rely on a novel combination of the socalled folk theorems in economics, and a novel efficient algorithm for the punishment
of deviators (in games which are initially unknown).
2
ELE: Definition
In this section we develop a definition of efficient learning equilibrium. For ease of
exposition, our discussion will center on two-player repeated games in which the
agents have an identical set of actions A. The generalization to n-player repeated
games with different action sets is immediate, but requires a little more notation.
The extension to stochastic games is fully discussed in the full paper [2].
A game is a model of multi-agent interaction. In a game, we have a set of players,
each of whom performs some action from a given set of actions. As a result of the
players' combined choices, some outcome is obtained which is described numerically
in the form of a payoff vector, i.e., a vector of values, one for each of the players.
A common description of a (two-player) game is as a matrix. This is called a game
in strategic form. The rows of the matrix correspond to player 1 's actions and the
columns correspond to player 2's actions. The entry in row i and column j in the
game matrix contains the rewards obtained by the players if player 1 plays his ith
action and player 2 plays his jth action.
In a repeated game (RG) the players playa given game G repeatedly. We can view
a repeated game, with respect to a game G, as consisting of infinite number of
iterations, at each of which the players have to select an action of the game G .
After playing each iteration, the players receive the appropriate payoffs, as dictated
by that game's matrix, and move to a new iteration.
For ease of exposition we normalize both players' payoffs in the game G to be nonnegative reals between and some positive constant Rmax . We denote this interval
(or set) of possible payoffs by P = [0, Rmax].
?
In a perfect monitoring setting, the set of possible histories of length t is (A2 X p2)t,
and the set of possible histories, H, is the union of the sets of possible histories for all
t 2 0, where (A2 x p 2)O is the empty history. Namely, the history at time t consists
of the history of actions that have been carried out so far, and the corresponding
payoffs obtained by the players. Hence, in a perfect monitoring setting, a player
can observe the actions selected and the payoffs obtained in the past, but does not
know the game matrix to start with. In an imperfect monitoring setup, all that a
player can observe following the performance of its action is the payoff it obtained
and the action selected by the other player. The player cannot observe the other
player's payoff. The definition of the possible histories for an agent naturally follows.
Finally, in a strict imperfect monitoring setting, the agent cannot observe the other
agents' payoff or their actions.
Given an RG , a policy for a player is a mapping from H, the set of possible histories ,
to the set of possible probability distributions over A. Hence, a policy determines the
probability of choosing each particular action for each possible history. A learning
algorithm can be viewed as an instance of a policy.
We define the value for player 1 (resp. 2) of a policy profile (1f, p), where 1f is a
policy for player 1 and p is a policy for player 2, using the expected average reward
criterion as follows. Given an RG M and a natural number T, we denote the
expected T -step undiscounted average reward of player 1 (resp. 2) when the players
follow the policy profile (1f,p), by U1 (M,1f,p,T) (resp. U2 (M,1f,p,T)). We define
Ui(M, 1f, p) = liminfT--+oo Ui(M, 1f, p, T) for i = 1,2.
Let M denote a class of repeated games. A policy profile (1f, p) is a learning equilibrium w.r.t. M if'rh' , p',M E M, we have that U1 (M,1f',p) :::; U 1 (M,1f,p), and
U2 (M,1f,p') :::; U2 (M , 1f,p). In this paper we mainly treat the class M of all repeated games with some fixed action profile (i.e. , in which the set of actions available
to all agents is fixed). However, in Section 4 we consider the class of common-interest
repeated games. We shall stick to the assumption that both agents have a fixed ,
identical set A of k actions.
Our first requirement, then, is that learning algorithms will be treated as strategies.
In order to be individually rational they should be the best response for one another.
Our second requirement is that they rapidly obtain a desired value. The definition
of this desired value may be a parameter, the most natural candidate - though not
the only candidate - being the expected payoffs in a Nash equilibrium of the game.
Another appealing alternative will be discussed later.
Formally, let G be a (one-shot) game, let M be the corresponding repeated game,
and let n(G) be a Nash-equilibrium of G. Then, denote the expected payoff of agent
i in n(G) by Nl/i(n(G)).
A policy profile (1f, p) is an efficient learning equilibrium with respect to the class
of games M if for every E > 0, < 8 < 1, there exists some T > 0, where T is
polynomial in ~,~, and k , such that with probability of at least 1 - 8: (1) For
every t 2: T and for every repeated game M E M (and its corresponding one-shot
game, G), Ui(M, 1f , p, t) 2: Nl/i(n(G)) - E for i = 1,2, for some Nash equilibrium
n(G), and (2) If player 1 (resp. 2) deviates from 1f to 1f' (resp. from p to p') in
iteration l, then U1 (M, 1f', p, l + t) :::; U 1 (M, 1f, p, l + t) + E (resp. U2 (M , 1f, p', l + t) :::;
U2 (M, 1f, p, l + t) + E) for every t 2: T.
?
Notice that a deviation is considered irrational if it does not increase the expected
payoff by more than E. This is in the spirit of E-equilibrium in game theory. This
is done mainly for ease of mathematical exposition. One can replace this part of
the definition, while getting similar results, with the requirement of "standard"
equilibrium, where a deviation will not improve the expected payoff, and even with
the notion of strict equilibrium, where a deviation will lead to a decreased payoff.
This will require, however, that we restrict our attention to games where there
exist a Nash equilibrium in which the agents' expected payoffs are higher than their
probabilistic maximin values.
The definition of ELE captures the insight of a normative approach to learning in
non-cooperative settings. We assume that initially the game is unknown, but the
agents will have learning algorithms that will rapidly lead to the values the players
would have obtained in a Nash equilibrium had they known the game. Moreover ,
as mentioned earlier, the learning algorithms themselves should be in equilibrium.
Notice that each agent's behavior should be the best response against the other
agents' behaviors, and deviations should be irrational, regardless of what the actual
(one-shot) game is.
3
Efficient Learning Equilibrium: Existence
Let M be a repeated game in which G is played at each iteration. Let A =
{al' ... , ak} be the set of possible actions for both agents. Finally let there be
an agreed upon ordering over the actions. The basic idea behind the algorithm is
as follows. The agents collaborate in exploring the game. This requires k 2 moves.
Next, each agent computes a Nash equilibrium of the game and follows it. If more
than one equilibrium exists, then the first one according to the natural lexicographic
ordering is used. l If one of the agents does not collaborate in the initial exploration
phase, the other agent "punishes" this agent. We will show that efficient punishment is feasible. Otherwise, the agents have chosen a Nash-equilibrium, and it is
irrational for them to deviate from this equilibrium unilaterally.
This idea combines the so-called folk-theorems in economics [6], and a technique
for learning in zero-sum games introduced in [1]. Folk-theorems in economics deal
with a technique for obtaining some desired behavior by making a threat of employing a punishing strategy against a deviator from that behavior. When both
agents are equipped with corresponding punishing strategies, the desired behavior will be obtained in equilibrium (and the threat will not be materialized - as a
deviation becomes irrational). In our context however , when an agent deviates in
the exploration phase, then the game is not fully known, and hence punishment
is problematic; moreover, we wish the punishment strategy to be an efficient algorithm (both computationally, and in the time a punishment will materialize and
make deviations irrational). These are addressed by having an efficient punishment algorithm that guarantees that the other agent will not obtain more than
its maximin value, after polynomial time, although the game is initially unknown
to the punishing agent. The latter is based on the ideas of our R-max algorithm,
introduced in [1].
More precisely, consider the following algorithm, termed the ELE algorithm.
The ELE algorithm:
Player 1 performs action ai one time after the other for k times, for all i = 1,2, ... , k.
In parallel, player 2 performs the sequence of actions (al' ... ,ak) k times.
If both players behaved according to the above then a Nash equilibrium of the corresponding (revealed) game is computed, and the players behave according
to the corresponding strategies from that point on. If several Nash equilibria exist, one is selected based on a pre-determined lexicographic ordering.
lIn particular, the agents can choose the equilibrium selected by a fixed shared
algorithm.
If one of the players deviated from the above, we shall call this player the adversary
and the other player the agent. Let G be the Rmax-sum game in which the
adversary's payoff is identical to his payoff in the original game, and where
the agent's payoff is Rmax minus the adversary payoffs. Let M denote the
corresponding repeated game. Thus, G is a constant-sum game where the
agent's goal is to minimize the adversary's payoff. Notice that some of
these payoffs will be unknown (because the adversary did not cooperate
in the exploration phase). The agent now plays according to the following
algorithm:
Initialize: Construct the following model M' of the repeated game M, where the
game G is replaced by a game G' where all the entries in the game matrix
are assigned the rewards (R max , 0). 2
In addition, we associate a boolean valued variable with each joint-action
{assumed, known}. This variable is initialized to the value assumed.
Repeat:
Compute and Act: Compute the optimal probabilistic maximin of G'
and execute it.
Observe and update: Following each joint action do as follows : Let a be
the action the agent performed and let a' be the adversary's action.
If (a, a') is performed for the first time, update the reward associated
with (a,a') in G', as observed, and mark it known. Recall- the agent
takes its payoff to be complementary to the (observed) adversary's
payoff.
We can show that the policy profile in which both agents use the ELE algorithm is
indeed an ELE. Thus:
Theorem 1 Let M be a class of repeated games.
w.r.t. M given perfect monitoring.
Then, there exists an ELE
The proof of the above Theorem, contained in the full paper, is non-trivial. It rests
on the ability of the agent to "punish" the adversary quickly, making it irrational
for the adversary to deviate from the ELE algorithm.
4
Imperfect monitoring
In the previous section we discussed the existence of an ELE in the context of the
perfect monitoring setup. This result allows us to show that our concepts provide
not only a normative, but also a constructive approach to learning in general noncooperative environments. An interesting question is whether one can go beyond
that and show the existence of an ELE in the imperfect monitoring case as well.
Unfortunately, when considering the class M of all games, this is not possible.
Theorem 2 There exist classes of games for which an ELE does not exist given
imperfect monitoring.
2The value 0 given to the adversary does not play an important role here.
Proof (sketch): We will consider the class of all 2 x 2 games and we will show
that an ELE does not exist for this class under imperfect monitoring.
Consider the following games:
1. Gl:
6,
o
M= ( 5, -100
2. G2:
M =
0,100 )
1, 500
(6,5,119 0, 1)
1, 10
Notice that the payoffs obtained for a joint action in Gland G 2 are identical for
player 1 and are different for player 2.
The only equilibrium of G 1 is where both players play the second action, leading
to (1,500). The only equilibrium of G2 is where both players play the first action,
leading to (6,9). (These are unique equilibria since they are obtained by removal of
strictly dominated strategies.)
Now, assume that an ELE exists, and look at the corresponding policies of the
players in that equilibrium. Notice that in order to have an ELE, we must visit the
entry (6,9) most of the times if the game is G2 and visit the entry (1 ,500) most of
the times if the game is G 1; otherwise, player 1 (resp. player 2) will not obtain a
high enough value in G2 (resp. Gl), since its other payoffs in G2 (resp. Gl) are
lower than that.
Given the above, it is rational for player 2 to deviate and pretend that the game is
always Gland behave according to what the suggested equilibrium policy tells it
to do in that case. Since the game might be actually G 1, and player 1 cannot tell
the difference, player 2 will be able to lead to playing the second action by both
players for most times also when the game is G2, increasing its payoff from 9 to 10,
contradicting ELE. I
The above result demonstrates that without additional assumptions, one cannot
provide an ELE under imperfect monitoring. However, for certain restricted classes
of games, we can provide an ELE under imperfect monitoring, as we now show.
A game is called a common-interest game if for every joint-action, all agents receive
the same reward. We can show:
Theorem 3 Let M c - i be the class of common-interest repeated games in which the
number of actions each agent has is a. There exists an ELE for M c - i under strict
imperfect monitoring.
Proof (sketch): The agents use the following algorithm: for m rounds , each agent
randomly selects an action. Following this, each agent plays the action that yielded
the best reward. If multiple actions led to the best reward, the one that was used
first is selected. m is selected so that with probability 1 - J every joint-action will
be selected. Using Chernoff bound we can choose m that is polynomial in the size
of the game (which is a k , where k is the number of agents) and in 1/ J.
I
This result improves previous results in this area, such as the combination of Qlearning and fictitious play used in [3]. Not only does it provably converge in
polynomial time, it is also guaranteed, with probability of 1 - J to converge to the
optimal Nash-equilibrium of the game rather than to an arbitrary (and possibly
non-optimal) Nash-equilibrium.
5
Conclusion
We defined the concept of an efficient learning equilibria - a normative criterion for
learning algorithms. We showed that given perfect monitoring a learning algorithm
satisfying ELE exists, while this is not the case under imperfect monitoring. In
the full paper [2] we discuss related solution concepts, such as Pareto ELE. A
Pareto ELE is similar to a (Nash) ELE, except that the requirement of attaining
the expected payoffs of a Nash equilibrium is replaced by that of maximizing social
surplus. We show that there fexists a Pareto-ELE for any perfect monitoring setting,
and that a Pareto ELE does not always exist in an imperfect monitoring setting. In
the full paper we also extend our discussion from repeated games to infinite horizon
stochastic games under the average reward criterion. We show that under perfect
monitoring, there always exists a Pareto ELE in this setting. Please refer to [2] for
additional details and the full proofs.
References
[1] R. I. Brafman and M. Tennenholtz. R-max - a general polynomial time algorithm for near-optimal reinforcement learning. In IJCAI'Ol, 200l.
[2] R. I. Brafman and M. Tennenholtz. Efficient learning equilibrium. Technical
Report 02-06, Dept. of Computer Science, Ben-Gurion University, 2002.
[3] C. Claus and C. Boutilier. The dynamics of reinforcement learning in cooperative multi-agent systems. In Proc. Workshop on Multi-Agent Learning, pages
602- 608, 1997.
[4] I. Erev and A.E. Roth. Predicting how people play games: Reinforcement
learning in games with unique strategy equilibrium. American Economic Review, 88:848- 881, 1998.
[5] D. Fudenberg and D. Levine. The theory of learning in games. MIT Press,
1998.
[6] D. Fudenberg and J. Tirole. Game Theory. MIT Press, 1991.
[7] J. Hu and M.P. Wellman. Multi-agent reinforcement learning: Theoretical
framework and an algorithms. In Proc. 15th ICML , 1998.
[8] L. P. Kaelbling, M. L. Littman, and A. W. Moore. Reinforcement learning: A
survey. Journal of AI Research, 4:237- 285, 1996.
[9] M. L. Littman. Markov games as a framework for multi-agent reinforcement
learning. In Proc. 11th ICML, pages 157- 163, 1994.
[10] L.S. Shapley. Stochastic Games. In Proc. Nat. Acad. Scie. USA, volume 39,
pages 1095- 1100, 1953.
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1,260 | 2,148 | Coulomb Classifiers: Generalizing
Support Vector Machines via an Analogy
to Electrostatic Systems
Sepp Hochreiter? , Michael C. Mozer? , and Klaus Obermayer?
?
Department of Electrical Engineering and Computer Science
Technische Universit?at Berlin, 10587 Berlin, Germany
?
Department of Computer Science
University of Colorado, Boulder, CO 80309?0430, USA
{hochreit,oby}@cs.tu-berlin.de, [email protected]
Abstract
We introduce a family of classifiers based on a physical analogy to
an electrostatic system of charged conductors. The family, called
Coulomb classifiers, includes the two best-known support-vector
machines (SVMs), the ??SVM and the C?SVM. In the electrostatics analogy, a training example corresponds to a charged conductor
at a given location in space, the classification function corresponds
to the electrostatic potential function, and the training objective
function corresponds to the Coulomb energy. The electrostatic
framework provides not only a novel interpretation of existing algorithms and their interrelationships, but it suggests a variety of new
methods for SVMs including kernels that bridge the gap between
polynomial and radial-basis functions, objective functions that do
not require positive-definite kernels, regularization techniques that
allow for the construction of an optimal classifier in Minkowski
space. Based on the framework, we propose novel SVMs and perform simulation studies to show that they are comparable or superior to standard SVMs. The experiments include classification
tasks on data which are represented in terms of their pairwise proximities, where a Coulomb Classifier outperformed standard SVMs.
1
Introduction
Recently, Support Vector Machines (SVMs) [2, 11, 9] have attracted much interest in
the machine-learning community and are considered state of the art for classification
and regression problems. One appealing property of SVMs is that they are based
on a convex optimization problem, which means that a single minimum exists and
can be computed efficiently. In this paper, we present a new derivation of SVMs
by analogy to an electrostatic system of charged conductors. The electrostatic
framework not only provides a physical interpretation of SVMs, but it also gives
insight into some of the seemingly arbitrary aspects of SVMs (e.g., the diagonal of
the quadratic form), and it allows us to derive novel SVM approaches. Although we
are the first to make the analogy between SVMs and electrostatic systems, previous
researchers have used electrostatic nonlinearities in pattern recognition [1] and a
mechanical interpretation of SVMs was introduced in [9].
In this paper, we focus on the classification of an input vector x ? X into one of
two categories, labeled ?+? and ???. We assume a supervised learning paradigm
in which N training examples are available, each example i consisting of an input
xi and a label yi ? {?1, +1}. We will introduce three electrostatic models that
are directly analogous to existing machine-learning (ML) classifiers, each of which
builds on and generalizes the previous. For each model, we describe the physical
system upon which it is based and show its correspondence to an ML classifier.
1.1 Electrostatic model 1: Uncoupled point charges
Consider an electrostatic system of point charges populating a space X 0 homologous
to X . Each point charge corresponds to a particular training example; point charge
i is fixed at location?xi in X 0 , and
? has a charge
? of sign yi . ?We define two sets of
fixed charges: S + = xi | yi = +1 and S ? = xi | yi = ?1 . The charge of point
i is Qi ? yi ?i , where ?i ? 0 is the amount of charge, to be discussed below.
We briefly review some elementary physics. If a unit positive charge is at x in
X 0 , it will be attracted to all charges in S ? and repelled by all charges in S + . To
? the attractive and repelling forces
move the charge from x to some other location x,
must be overcome at every point along the trajectory; the path integral of the force
along the trajectory is called the work and does not depend on the trajectory. The
potential at x is the work that must be done to move a unit positive charge from a
reference point (usually infinity) to x.
? j ?
PN
The potential at x is ? (x) =
j=1 Qj G x , x , where G is a function of the
distance. In electrostatic systems with point charges, G (a, b) = 1/ ka ? bk 2 . From
this definition, one can see that the potential at x is negative (positive) if x is in
a neighborhood of many negative (positive) charges. Thus, the potential indicates
the sign and amount of charge in the local neighborhood.
Turning back to the ML classifier, one might propose a classification rule for some
input x that assigns the label ?+? if ?(x) > 0 or ??? otherwise. Abstracting
from the electrostatic system, if ?i = 1 and G is a function that decreases sufficiently steeply with distance, we obtain a nearest-neighbor classifier. This potential
classifier can be also interpreted as Parzen windows classifier [9].
1.2 Electrostatic model 2: Coupled point charges
Consider now an electrostatic model that extends the previous model in two respects. First, the point charges are replaced by conductors, e.g., metal spheres.
Each conductor i has a self?potential coefficient, denoted Pii , which is a measure
of how much charge it can easily hold; for a metal sphere, Pii is related to sphere?s
diameter. Second, the conductors in S + are coupled, as are the conductors in S ? .
?Coupling? means that charge is free to flow between the conductors. Technically,
S + and S ? can each be viewed as a single conductor.
In this model, we initially place the same charge ?/N on each conductor, and allow
charges within S + and S ? to flow freely (we assume no resistance in the coupling
and no polarization of the conductors). After the charges redistribute, charge will
tend to end up on the periphery of a homogeneous neighborhood of conductors,
because like charges repel. Charge will also tend to end up along the S + ?S ?
boundary because opposite charges attract. Figure 1 depicts the redistribution of
charges, where the shading is proportional to the magnitude ?i . An ML classifier
can be built based on this model, once again using ?(x) > 0 as the decision rule
for classifying an input x. In this model, however, the ?i are not uniform; the
conductors with large ?i will have the greatest influence on the potential function.
Consequently, one can think of ?i as the weight or importance of example i. As we
will show shortly, the examples with ?i > 0 are exactly support vectors of an SVM.
-
+
+
+
+
+
+
+
+
+
+
+
+
-
+ +
+
+
+
+
+
+
+
+
-
-
-
- - - - - - -
Figure 1: Coupled conductor system following charge redistribution. Shading reflects the charge magnitude, and the contour indicates a zero potential.
The redistribution of charges in the electrostatic system is achieved via minimization
of the Coulomb energy. Imagine placing the same total charge magnitude, m, on
S + and S ? by dividing it uniformly among the conductors, i.e., ?i = m/ |S yi |. The
free charge flow in S + and S ? yields a distribution of charges, the ?i , such that
Coulomb energy is minimized.
To introduce Coulomb energy, we begin with some preliminaries. The potential at
conductor i, ?(xi ), which we will denote more compactly as ?i , can be described
PN
in terms of the coefficients of potential Pij [10]: ?i = j=1 Pij Qj , where Pij is the
potential induced on conductor i by charge Qj on conductor j; Pii ? Pij ? 0 and
Pij = Pji . If each conductor i is a metal sphere centered at xi and has radius ri
(radii are enforced to be small enough so that the spheres do not touch? each other),
?
the system can be modeled by a point charge Qi at xi , and Pij = G xi , xj as in
the previous section [10]. The self-potential, Pii , is defined as a function of ri . The
Coulomb energy is defined in terms of the potential on the conductors, ?i :
E =
N
N
1 X
1 T
1X
?i Q i =
Q P Q =
Pij yi yj ?i ?j .
2 i=1
2
2 i,j=1
When the energy minimum is reached, the potential ?i will be the same for all
connected i ? S + (i ? S ? ); we denote this potential ?S + (?S ? ).
Two additional constraints on the system of coupled conductors are necessary in
order to interpret the system in terms of existing machine learning models. First,
the positive and negative potentials must be balanced, i.e., ?S + = ??S ? . This
constraint is achieved by setting the reference point of the potentials ?through
? b,
PN
i
+
?
b = ?0.5 (?S + ?S ), into the potential function: ? (x) = i=1 Qi G x , x + b.
Second, the conductors must be prevented from reversing the sign of their charge,
i.e., ?i ? 0, and from holding more than a quantity C of charge, i.e., ?i ? C. These
requirements can be satisfied in the electrostatic model by disconnecting a conductor
i from the charge flow in S + or S ? when ?i reaches a bound, which will subsequently
freeze its charge. Mathematically, the requirements are satisfied by treating energy
minimization as a constrained optimization problem with 0 ? ?i ? C.
The electrostatic system corresponds to a ??support vector machine (??SVM)
[9]
P
?
with
kernel
G
if
we
set
C
=
1/N
.
The
electrostatic
system
assures
that
+
i =
i?S
P
?
=
0.5
?.
The
identity
holds
because
the
Coulomb
energy
is
exactly
the
?
i
i?S
??SVM quadratic objective function, and the thresholded electrostatic potential
evaluated at a location is exactly the SVM decision rule. The minimization of
potentials differences in the systems S + and S ? corresponds to the minimization
of slack variables in the SVM (slack variables express missing potential due to the
upper bound on ?i ). Mercer?s condition [6], the essence of the nonlinear SVM
theory,Ris equivalent to the fact that continuous electrostatic energy is positive, i.e.,
E =
G (x, z) h (x) h (z) dx dz ? 0. The self-potentials of the electrostatic
system provide an interpretation to the diagonal elements in the quadratic objective
function of the SVM. This interpretation of the diagonal elements allows us to
introduce novel kernels and novel SVM methods, as we discuss later.
1.3 Electrostatic model 3: Coupled point charges with battery
In electrostatic model 2, we control the magnitude of charge applied to S + and S ? .
Although we apply the same charge magnitude to each, we do not have to control
the resulting potentials ?S + and ?S ? , which may be imbalanced. We compensate
for this imbalance via the potential offset b. In electrostatic model 3, we control the
potentials ?S + and ?S + directly by adding a battery to the system. We connect
S + to the positive pole of the battery with potential +1 and S ? to the negative
pole with potential ?1. The battery ensures that ?S + = +1 and ?S ? = ?1 because
charges flow from the battery into or out of the system until the systems take on
the potential of the battery poles. The battery can then be removed. The potential
?i = yi is forced by the battery on conductor i. The total Coulomb energy is the
energy from
P model 2 minus
P the work done by the battery. The work done by the
battery is i?N yi Qi = i?N ?i . The Coulomb energy is
N
N
N
X
X
1 T
1 X
?i =
?i .
Q P Q ?
Pij yi yj ?i ?j ?
2
2 i,j=1
i=1
i=1
This physical system corresponds
to a C?support vector machine (C?SVM) [2, 11].
P
The C?SVM requires that i yi ?i = 0; although this constraint may not be fulfilled
in the system described here, it can be enforced by a slightly different system [4]. A
more straightforward relation to the C?SVM is given in [9] where the authors show
that every ??SVM has the same class boundaries as a C?SVM with appropriate C.
2
Comparison of existing and novel models
2.1 Novel Kernels
The electrostatic perspective makes it easy to understand why SVM algorithms can
break down in high-dimensional spaces: Kernels with rapid fall-off induce small potentials and consequently, almost every conductor retains charge. Because a charged
conductor corresponds to a support vector, the number of support vectors is large,
which leads to two disadvantages: (1) the classification procedure is slow, and (2) the
expected generalization error increases with the number of support vectors [11]. We
therefore should use kernels that do not drop off exponentially. The self?potential
permits the use of kernels that would otherwise be invalid, such as a generalization
?
??l
?
?
?
?
of the electric field: G xi , xj := ?xi ? xj ?2 and G xi , xi := ri?l = Pii , where
ri the radius of the ith sphere. The ri s are increased
to?their maximal values, i.e.
?
until they hit other conductors (ri = 0.5 minj ?xi ? xj ?2 ). These kernels, called
?Coulomb kernels?, are invariant to scaling of the input space in the sense that
scaling does not change the minimum of the objective function. Consequently, such
kernels are appropriate for input data with varying local densities. Figure 2 depicts
a classification task with input regions of varying density. The optimal class boundary is smooth in the low data density regions and has high curvature in regions,
where the data density is high. The classification boundary was constructed using
??
??l/2
?2
?
?
, which is an
a C-SVM with a Plummer kernel G xi , xj := ?xi ? xj ? + ?2
2
approximation to our novel Coulomb kernel but lacks its weak singularities.
Figure 2: Two class data with a dense region and trained with a SVM using the
new kernel. Gray-scales indicate the weights ? support vectors are dark. Boundary
curves are given for the novel kernel (solid), best RBF-kernel SVM which overfits
at high density regions where the resulting boundary goes through a dark circle
(dashed), and optimal boundary (dotted).
2.2 Novel SVM models
Our electrostatic framework can be used to derive novel SVM approaches [4], two
representative examples of which we illustrate here.
2.2.1 ??Support Vector Machine (??SVM):
We can exploit the physical interpretation of Pii as conductor i?s self?potential. The
Pii ?s determine the smoothness of the charge distribution at the energy minimum.
We can introduce a parameter ? to rescale the self potential ? Piinew = ? Piiold .
? controls the complexity of the corresponding SVM. With this modification, and
with C = ?, electrostatic model 3 becomes what we call the ??SVM.
2.2.2 p?Support Vector Machine (p?SVM):
At the Coulomb energy minimum the electrostatic potentials equalize: ? i ? yi =
0, ?i (y is the label vector). This motivates the introduction of potential difference,
2
1
1 T T
1 T
T T
2 kP Q + yk2 = 2 Q P P Q + Q P y + 2 y y as the objective. We obtain
1 T
min
? Y P T P Y ? ? 1T Y P Y ?
?
2
subject to
1T P Y ? = 0 , |?i | ? C,
where 1 is the vector of ones and Y := diag(y). We call this variant of the
optimization problem the potential-SVM (p-SVM). Note that the p-SVM is similar
to the ?empirical kernel map? [9]. However P appears in the objective?s linear term
and the constraints. We classify in a space where P is a dot product matrix. The
constraint 1T P Y ? = 0 ensures that the average potential for each class is equal.
By construction, P T P is positive definite; consequently, this formulation does not
require positive definite kernels. This characteristic is useful for problems in which
the properties of the objects to be classified are described by their pairwise proximities. That is, suppose that instead of representing each input object by an explicit
feature vector, the objects are represented by a matrix which contains a real number indicating the similarity of each object to each other object. We can interpret
the entries of the matrix as being produced by an unknown kernel operating on
unknown feature vectors. In such a matrix, however, positive definiteness cannot
be assured, and the optimal hyperplane must be constructed in Minkowski space.
3
Experiments
UCI Benchmark Repository. For the representative models we have introduced, we perform simulations and make comparisons to standard SVM variants.
All datasets (except ?banana? from [7]) are from the UCI Benchmark Repository
and were preprocessed in [7]. We did 100-fold validation on each data set, restricting
the training set to 200 examples, and using the remainder of examples for testing.
We compared two standard architectures, the C?SVM and the ??SVM, to our novel
architectures: to the ??SVM, to the p?SVM, and to a combination of them, the
??p?SVM. The ??p?SVM is a p?SVM regularized like a ??SVM. We explored the
use of radial basis function (RBF), polynomial (POL), and Plummer (PLU) kernels.
Hyperparameters were determined by 5?fold cross validation on the first 5 training
sets. The search for hyperparameter was not as intensive as in [7].
Table 1 shows the results of our comparisons on the UCI Benchmarks. Our two
novel architectures, the ??SVM and the p?SVM, performed well against the two
existing architectures (note that the differences between the C? and the ??SVM
are due to model selection). As anticipated, the p?SVM requires far fewer support vectors. Additionally, the Plummer kernel appears to be more robust against
hyperparameter and SVM choices than the RBF or polynomial kernels.
C
RBF
POL
PLU
6.4
22.8
6.1
RBF
POL
PLU
33.6
36.0
33.4
RBF
POL
PLU
28.7
33.7
28.8
?
?
p
thyroid
9.4
7.7
5.4
12.6
7.0
13.3
6.2
6.1
5.7
breast?cancer
31.6
33.8 32.4
25.7 29.6 27.1
33.1
33.4 30.6
german
29.3
29.0 27.8
29.6 26.2 31.8
28.5
33.3 27.1
?-p
C
?
8.6
6.9
6.1
21.4
20.4
16.3
19.1
20.4
16.3
33.7
29.1
33.4
13.2
35.3
15.7
36.7
35.0
15.7
?
heart
17.9
19.3
16.3
banana
13.2
11.5
15.7
p
?-p
22.4
23.0
17.4
17.8
19.3
16.3
11.6
22.4
21.9
13.4
11.5
15.7
28.8
26.2
33.3
Table 1: Mean % misclassification on 5 UCI Repository data sets. Each cell in
the table is obtained via 100 replications splitting the data into training and test
sets. The comparison is among five SVMs (the table columns) using three kernel
functions (the table rows). Cells in bold face are the best result for a given data set
and italicized the second and third best.
Pairwise Proximity Data. We applied our p?SVM and the generalized SVM
(G?SVM) [3] to two pairwise-proximity data sets. The first data set, the ?cat cortex? data, is a matrix of connection strengths between 65 cat cortical areas and was
provided by [8], where the available anatomical literature was used to determine
proximity values between cortical areas. These areas belong to four different coarse
brain regions: auditory (A), visual (V), somatosensory (SS), and frontolimbic (FL).
The goal was to classify a given cortical area as belonging to a given region or
not. The second data set, the ?protein? data, is the evolutionary distance of 226 sequences of amino acids of proteins obtained by a structural comparison [5] (provided
by M. Vingron). Most of the proteins are from four classes of globins: hemoglobin-ff
(H-ff), hemoglobin-fi (H-fi), myoglobin (M), and heterogenous globins (GH). The
goal was to classify a protein as belonging to a given globin class or not. As Table 2
shows, our novel architecture, the p?SVM, beats out an existing architecture in the
literature, the G?SVM, on 5 of 8 classification tasks, and ties the G?SVM on 2 of
8; it loses out on only 1 of 8.
Size
G-SVM
G-SVM
G-SVM
p-SVM
p-SVM
p-SVM
Reg.
?
0.05
0.1
0.2
0.6
0.7
0.8
cat
V
18
4.6
4.6
6.1
3.1
3.1
3.1
cortex
A
SS
10
18
3.1 3.1
3.1 6.1
1.5 3.1
1.5 6.1
3.1 4.6
3.1 4.6
FL
19
1.5
1.5
3.1
3.1
1.5
1.5
Reg.
?
0.05
0.1
0.2
300
400
500
protein data
H-? H-?
M
72
72
39
1.3
4.0
0.5
1.8
4.5
0.5
2.2
8.9
0.5
0.4
3.5 0.0
0.4 3.1 0.0
0.4
3.5 0.0
GH
30
0.5
0.9
0.9
0.4
0.9
1.3
Table 2: Mean % misclassifications for the cat-cortex and protein data sets using
the p?SVM and the G?SVM and a range of regularization parameters (indicated in
the column labeled ?Reg.?). The result obtained for the cat-cortex data is via leaveone-out cross validation, and for the protein data is via ten-fold cross validation.
The best result for a given classification problem is printed in bold face.
4
Conclusion
The electrostatic framework and its analogy to SVMs has led to several important
ideas. First, it suggests SVM methods for kernels that are not positive definite.
Second, it suggests novel approaches and kernels that perform as well as standard
methods (will undoubtably perform better on some problems). Third, we demonstrated a new classification technique working in Minkowski space which can be used
for data in form of pairwise proximities. The novel approach treats the proximity
matrix as an SVM Gram matrix which lead to excellent experimental results.
We argued that the electrostatic framework not only characterizes a family of
support-vector machines, but it also characterizes other techniques such as nearest
neighbor classification. Perhaps the most important contribution of the electrostatic framework is that, by interrelating and encompassing a variety of methods,
it lays out a broad space of possible algorithms. At present, the space is sparsely
populated and has barely been explored. But by making the dimensions of this
space explicit, the electrostatic framework allows one to easily explore the space
and discover novel algorithms. In the history of machine learning, such general
frameworks have led to important advances in the field.
Acknowledgments
We thank G. Hinton and J. Schmidhuber for stimulating conversations leading to
this research and an anonymous reviewer who provided helpful advice on the paper.
References
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1,261 | 2,149 | Identity Uncertainty and Citation Matching
Hanna Pasula, Bhaskara Marthi, Brian Milch, Stuart Russell, Ilya Shpitser
Computer Science Division, University Of California
387 Soda Hall, Berkeley, CA 94720-1776
pasula, marthi, milch, russell, [email protected]
Abstract
Identity uncertainty is a pervasive problem in real-world data analysis. It
arises whenever objects are not labeled with unique identifiers or when
those identifiers may not be perceived perfectly. In such cases, two observations may or may not correspond to the same object. In this paper,
we consider the problem in the context of citation matching?the problem of deciding which citations correspond to the same publication. Our
approach is based on the use of a relational probability model to define
a generative model for the domain, including models of author and title
corruption and a probabilistic citation grammar. Identity uncertainty is
handled by extending standard models to incorporate probabilities over
the possible mappings between terms in the language and objects in the
domain. Inference is based on Markov chain Monte Carlo, augmented
with specific methods for generating efficient proposals when the domain
contains many objects. Results on several citation data sets show that
the method outperforms current algorithms for citation matching. The
declarative, relational nature of the model also means that our algorithm
can determine object characteristics such as author names by combining
multiple citations of multiple papers.
1 INTRODUCTION
Citation matching is the problem currently handled by systems such as Citeseer [1]. 1 Such
systems process a large number of scientific publications to extract their citation lists. By
grouping together all co-referring citations (and, if possible, linking to the actual cited
paper), the system constructs a database of ?paper? entities linked by the ?cites(p 1 , p2 )?
relation. This is an example of the general problem of determining the existence of a set
of objects, and their properties and relations, given a collection of ?raw? perceptual data;
this problem is faced by intelligence analysts and intelligent agents as well as by citation
systems.
A key aspect of this problem is determining when two observations describe the same
object; only then can evidence be combined to develop a more complete description of the
object. Objects seldom carry unique identifiers around with them, so identity uncertainty
is ubiquitous. For example, Figure 1 shows two citations that probably refer to the same
paper, despite many superficial differences. Citations appear in many formats and are rife
with errors of all kinds. As a result, Citeseer?which is specifically designed to overcome
such problems?currently lists more than 100 distinct AI textbooks published by Russell
1
See citeseer.nj.nec.com. Citeseer is now known as ResearchIndex.
[Lashkari et al 94] Collaborative Interface Agents, Yezdi Lashkari, Max Metral, and Pattie
Maes, Proceedings of the Twelfth National Conference on Articial Intelligence, MIT Press,
Cambridge, MA, 1994.
Metral M. Lashkari, Y. and P. Maes. Collaborative interface agents. In Conference of the
American Association for Artificial Intelligence, Seattle, WA, August 1994.
Figure 1: Two citations that probably refer to the same paper.
and Norvig on or around 1995, from roughly 1000 citations. Identity uncertainty has been
studied independently in several fields. Record linkage [2] is a method for matching up
the records in two files, as might be required when merging two databases. For each pair
of records, a comparison vector is computed that encodes the ways in which the records
do and do not match up. EM is used to learn a naive-Bayes distribution over this vector
for both matched and unmatched record pairs, so that the pairwise match probability can
then be calculated using Bayes? rule. Linkage decisions are typically made in a greedy
fashion based on closest match and/or a probability threshold, so the overall process is
order-dependent and may be inconsistent. The model does not provide for a principled
way to combine matched records. A richer probability model is developed by Cohen et
al [3], who model the database as a combination of some ?original? records that are correct
and some number of erroneous versions. They give an efficient greedy algorithm for finding
a single locally optimal assignment of records into groups.
Data association [4] is the problem of assigning new observations to existing trajectories
when multiple objects are being tracked; it also arises in robot mapping when deciding if
an observed landmark is the same as one previously mapped. While early data association systems used greedy methods similar to record linkage, recent systems have tried to
find high-probability global solutions [5] or to approximate the true posterior over assignments [6]. The latter method has also been applied to the problem of stereo correspondence,
in which a computer vision system must determine how to match up features observed in
two or more cameras [7]. Data association systems usually have simple observation models (e.g., Gaussian noise) and assume that observations at each time step are all distinct.
More general patterns of identity occur in natural language text, where the problem of
anaphora resolution involves determining whether phrases (especially pronouns) co-refer;
some recent work [8] has used an early form of relational probability model, although with
a somewhat counterintuitive semantics.
Citeseer is the best-known example of work on citation matching [1]. The system groups
citations using a form of greedy agglomerative clustering based on a text similarity metric
(see Section 6). McCallum et al [9] use a similar technique, but also develop clustering
algorithms designed to work well with large numbers of small clusters (see Section 5).
With the exception of [8], all of the preceding systems have used domain-specific algorithms and data structures; the probabilistic approaches are based on a fixed probability
model. In previous work [10], we have suggested a declarative approach to identity uncertainty using a formal language?an extension of relational probability models [11]. Here,
we describe the first substantial application of the approach. Section 2 explains how to
specify a generative probability model of the domain. The key technical point (Section 3)
is that the possible worlds include not only objects and relations but also mappings from
terms in the language to objects in the domain, and the probability model must include a
prior over such mappings. Once the extended model has been defined, Section 4 details the
probability distributions used. A general-purpose inference method is applied to the model.
We have found Markov chain Monte Carlo (MCMC) to be effective for this and other applications (see Section 5); here, we include a method for generating effective proposals based
on ideas from [9]. The system also incorporates an EM algorithm for learning the local
probability models, such as the model of how author names are abbreviated, reordered, and
misspelt in citations. Section 6 evaluates the performance of four datasets originally used
to test the Citeseer algorithms [1]. As well as providing significantly better performance,
our system is able to reason simultaneously about papers, authors, titles, and publication
types, and does a good job of extracting this information from the grouped citations. For
example, an author?s name can be identified more accurately by combining information
from multiple citations of several different papers. The errors made by our system point to
some interesting unmodeled aspects of the citation process.
2 RPMs
Reasoning about identity requires reasoning about objects, which requires at least some of
the expressive power of a first-order logical language. Our approach builds on relational
probability models (RPMs) [11], which let us specify probability models over possible
worlds defined by objects, properties, classes, and relations.
2.1 Basic RPMs
At its most basic, an RPM, as defined by Koller et al [12], consists of
? A set C of classes denoting sets of objects, related by subclass/superclass relations.
? A set I of named instances denoting objects, each an instance of one class.
? A set A of complex attributes denoting functional relations. Each complex attribute A has a domain type Dom[A] ? C and a range type Range[A] ? C.
? A set B of simple attributes denoting functions. Each simple attribute B has a
domain type Dom[B] ? C and a range V al[B].
? A set of conditional probability models P (B|P a[B]) for the simple attributes.
P a[B] is the set of B?s parents, each of which is a nonempty chain of (appropriately typed) attributes ? = A1 . ? ? ? .An .B 0 , where B 0 is a simple attribute. Probability models may be attached to instances or inherited from classes. The parent
links should be such that no cyclic dependencies are formed.
? A set of instance statements, which set the value of a complex attribute to an
instance of the appropriate class.
We also use a slight variant of an additional concept from [11]: number uncertainty, which
allows for multi-valued complex attributes of uncertain cardinality. We define each such
attribute A as a relation rather than a function, and we associate with it a simple attribute #[A] (i.e., the number of values of A) with a domain type Dom[A] and a range
{0, 1, . . . , max #[A]}.
2.2 RPMs for citations
Figure 2 outlines an RPM for the example citations of Figure 1. There are four classes,
the self-explanatory Author, Paper, and Citation, as well as AuthorAsCited, which represents not actual authors, but author names as they appear when cited. Each citation we
wish to match leads to the creation of a Citation instance; instances of the remaining three
classes are then added as needed to fill all the complex attributes. E.g., for the first citation
of Figure 1, we would create a Citation instance C1 , set its text attribute to the string ?Metral M. ...August 1994.?, and set its paper attribute to a newly created Paper
instance, which we will call P1 . We would then introduce max(#[author]) (here only 3,
for simplicity) AuthorAsCited instances (D11 , D12 , and D13 ) to fill the P1 .obsAuthors (i.e.,
observed authors) attribute, and an equal number of Author instances (A 11 , A12 , and A13 )
to fill both the P1 .authors[i] and the D1i .author attributes. (The complex attributes would
be set using instance statements, which would then also constrain the cited authors to be
equal to the authors of the actual paper. 2 ) Assuming (for now) that the value of C1 .parse
2
Thus, uncertainty over whether the authors are ordered correctly can be modeled using probabilistic instance statements.
A11
Author
A12
surname
#(fnames)
fnames
A13
A21
D11
AuthorAsCited
surname
#(fnames)
fnames
author
A22
A23
D12
D13
D21
D22
Paper
D23
Citation
#(authors)
authors
title
publication type
P1
P2
#(obsAuthors)
obsAuthors
obsTitle
parse
C1
C2
text
paper
Figure 2: An RPM for our Citeseer example. The large rectangles represent classes: the
dark arrows indicate the ranges of their complex attributes, and the light arrows lay out
all the probabilistic dependencies of their basic attributes. The small rectangles represent
instances, linked to their classes with thick grey arrows. We omit the instance statements
which set many of the complex attributes.
is observed, we can set the values of all the basic attributes of the Citation and AuthorAsCited instances. (E.g., given the correct parse, D11 .surname would be set to Lashkari,
and D12 .fnames would be set to (Max)). The remaining basic attributes ? those of the
Paper and Author instances ? represent the ?true? attributes of those objects, and their
values are unobserved.
The standard semantics of RPMs includes the unique names assumption, which precludes
identity uncertainty. Under this assumption, any two papers are assumed to be different
unless we know for a fact that they are the same. In other words, although there are many
ways in which the terms of the language can map to the objects in a possible world, only
one of these identity mappings is legal: the one with the fewest co-referring terms. It is then
possible to express the RPM as an equivalent Bayesian network: each of the basic attributes
of each of the objects becomes a node, with the appropriate parents and probability model.
RPM inference usually involves the construction of such a network. The Bayesian network
equivalent to our RPM is shown in Figure 3.
3 IDENTITY UNCERTAINTY
In our application, any two citations may or may not refer to the same paper. Thus, for
citations C1 and C2 , there is uncertainty as to whether the corresponding papers P 1 and P2
are in fact the same object. If they are the same, they will share one set of basic attributes;
A11.
surname
D12.
#(fnames)
D12.
surname
A11.
fnames
D11.
#(fnames)
D12.
fnames
A21.
#(fnames)
A13.
surname
A12.
fnames
A21.
fnames
A13.
fnames
A13.
#(fnames)
D13.
surname
D11.
fnames
D11.
surname
D13.
#(fnames)
C1.
#(authors)
P1.
title
C1.
text
P1.
pubtype
C1.
obsTitle
A21.
surname
A23.
surname
A22.
fnames
D22.
#(fnames)
D12.
surname
D21.
#(fnames)
D22.
fnames
A23.
fnames
A23.
#(fnames)
D23.
surname
D21.
fnames
D13.
fnames
C1.
parse
A22.
#(fnames)
A22.
surname
A12.
#(fnames)
A12.
surname
A11.
#(fnames)
D23.
fnames
D21.
surname
D23.
#(fnames)
C2.
#(authors)
P2.
title
C2.
parse
C2.
text
C2.
obsTitle
P2.
pubtype
Figure 3: The Bayesian network equivalent to our RPM, assuming C 1 6= C2 .
if they are distinct, there will be two sets. Thus, the possible worlds of our probability
model may differ in the number of random variables, and there will be no single equivalent Bayesian network. The approach we have taken to this problem [10] is to extend the
representation of a possible world so that it includes not only the basic attributes of a set
of objects, but also the number of objects n and an identity clustering ?, that is, a mapping
from terms in the language (such as P1 ) to objects in the world. We are interested only
in whether terms co-refer or not, so ? can be represented by a set of equivalence classes of
terms. For example, if P1 and P2 are the only terms, and they co-refer, then ? is {{P1 , P2 }};
if they do not co-refer, then ? is {{P1 }, {P2 }}.
We define a probability model for the space of extended possible worlds by specifying the
prior P (n) and the conditional distribution P (?|n). As in standard RPMs, we assume that
the class of every instance is known. Hence, we Q
can simplify these distributions further
by factoring them by class, so that, e.g., P (?) = C?C P (?C ). We then distinguish two
cases:
? For some classes (such as the citations themselves), the unique names assumptions
remains appropriate. Thus, we define P (?Citation ) to assign a probability of 1.0
to the one assignment where each citation object is unique.
? For classes such as Paper and Author, whose elements are subject to identity uncertainty, we specify P (n) using a high-variance log-normal distribution. 3 Then
we make appropriate uniformity assumptions to construct P (?C ). Specifically, we
assume that each paper is a priori equally likely to be cited, and that each author is
a priori equally likely to write a paper. Here, ?a priori? means prior to obtaining
any information about the object in question, so the uniformity assumption is entirely reasonable. With these assumptions, the probability of an assignment ? C,k,m
that maps k named instances to m distinct objects, when C contains n objects, is
given by
1
n!
P (?C,k,m ) =
(n ? m)! nk
When n > m, the world contains objects unreferenced by any of the terms. However, these filler objects are obviously irrelevant (if they affected the attributes of
some named term, they would have been named as functions of that term.) Therefore, we never have to create them, or worry about their attribute values.
Our model assumes that the cardinalities and identity clusterings of the classes are independent of each other, as well as of the attribute values. We could remove these assumptions.
For one, it would be straightforward to specify a class-wise dependency model for n or ?
using standard Bayesian network semantics, where the network nodes correspond to the
cardinality attributes of the classes. E.g., it would be reasonable to let the total number of
papers depend on the total number of authors. Similarly, we could allow ? to depend on the
attribute values?e.g., the frequency of citations to a given paper might depend on the fame
of the authors?provided we did not introduce cyclic dependencies.
4 The Probability Model
We will now fill in the details of the conditional probability models. Our priors over the
?true? attributes are constructed off-line, using the following resources: the 1990 Census data on US names, a large A.I. BibTeX bibliography, and a hand-parsed collection of
500 citations. We learn several bigram models (actually, linear combinations of a bigram
model and a unigram model): letter-based models of first names, surnames, and title words,
as well as higher-level models of various parts of the citation string. More specifically, the
values of Author.fnames and Author.surname are modeled as having a a 0.9 chance of being
3
Other models are possible; for example, in situations where objects appear and disappear, P (?)
can be modeled implicitly by specifying the arrival, transition, and departure rates [6].
drawn from the relevant US census file, and a 0.1 chance of being generated using a bigram
model learned from that file. The prior over Paper.titles is defined using a two-tier bigram
model constructed using the bibliography, while the distributions over Author.#(fnames),
Paper.#(authors), and Paper.pubType 4 are derived from our hand-parsed file. The conditional distributions of the ?observed? variables given their true values (i.e., the corruption models of Citation.obsTitle, AuthorAsCited.surname, and AuthorAsCited.fnames) are
modeled as noisy channels where each letter, or word, has a small probability of being
deleted, or, alternatively, changed, and there is also a small probability of insertion. AuthorAsCited.fnames may also be abbreviated as an initial. The parameters of the corruption
models are learnt online, using stochastic EM.
Let us now return to Citation.parse, which cannot be an observed variable, since citation
parsing, or even citation subfield extraction, is an unsolved problem. It is therefore fortunate that our approach lets us handle uncertainty over parses so naturally. The state space
of Citation.parse has two different components. First of all, it keeps track of the citation
style, defined as the ordering of the author and title subfields, as well as the format in which
the author names are written. The prior over styles is learned using our hand-segmented
file. Secondly, it keeps track of the segmentation of Citation.text, which is divided into
an author segment, a title segment, and three filler segments (one before, one after, and
one in between.) We assume a uniform distribution over segmentations. Citation.parse
greatly constrains Citation.text: the title segment of Citation.text must match the value of
Citation.obsTitle, while its author segment must match the combined values of the simple
attributes of Citation.obsAuthors. The distributions over the remaining three segments of
Citation.text are defined using bigram models, with the model used for the final segment
chosen depending on the publication type. These models were, once more, learned using
our pre-segmented file.
5 INFERENCE
With the introduction of identity uncertainty, our model grows from a single Bayesian
network to a collection of networks, one for each possible value of ?. This collection can be
rather large, since the number of ways in which a set can be partitioned grows very quickly
with the size of the set. 5 Exact inference is, therefore, impractical. We use an approximate
method based on Markov chain Monte Carlo.
5.1 MARKOV CHAIN MONTE CARLO
MCMC [13] is a well-known method for approximating an expectation over some distribution ?(x), commonly used when the state space of x is too large to sum over. The weighted
sum over the values of x is replaced by a sum over samples from ?(x), which are generated
using a Markov chain constructed to have ?(x) as a stationary distribution.
There are several ways of building up an appropriate Markov chain. In the Metropolis?
Hastings method (M-H), transitions in the chain are constructed in two steps. First, a
candidate next state x0 is generated from the current state x, using the (more or less arbitrary) proposal distribution q(x0 |x). The probability that
to x0 is actually made is
the move
0
)q(x|x0 )
the acceptance probability, defined as ?(x0 |x) = min 1, ?(x
?(x)q(x0 |x) .
Such a Markov chain will have the right stationary distribution ?(x) as long as q is defined
in such a way that the chain is ergodic. It is even possible to factor q into separate proposals
for various subsets of variables. In those situations, the variables that are not changed by the
transition cancel in the ratio ?(x0 )/?(x), so the required calculation can be quite simple.
4
Publication types range over {article, conference paper, book, thesis, and tech report}
This sequence is described by the Bell numbers, whose asymptotic behaviour is more than exponential.
5
5.2 THE CITATION-MATCHING ALGORITHM
The state space of our MCMC algorithm is the space of all the possible worlds, where
each possible world contains an identity clustering ?, a set of class cardinalities n, and the
values of all the basic attributes of all the objects. Since the ? is given in each world, the
distribution over the attributes can be represented using a Bayesian network as described
in Section 3. Therefore, the probability of a state is simply the product pf P (n), P (?), and
the probability of the hidden attributes of the network.
Our algorithm uses a factored q function. One of our proposals attempts to change n using
a simple random walk. The other suggests, first, a change to ?, and then, values for all the
hidden attributes of all the objects (or clusters in ?) affected by that change. The algorithm
for proposing a change in ?C works as follows:
Select two clusters a1 , a2 ? ?C 6
Create two empty clusters b1 and b2
place each instance i ? a1 ? a2 u.a.r. into b1 or b2
Propose ?0C = ?C ? {a1, a2} ? {b1, b2}
Given a proposed ?0C , suggesting values for the hidden attributes boils down to recovering
their true values from (possibly) corrupt observations, e.g., guessing the true surname of
the author currently known both as ?Simth? and ?Smith?. Since our title and name noise
models are symmetric, our basic strategy is to apply these noise models to one of the
observed values. In the case of surnames, we have the additional resource of a dictionary
of common names, so, some of the time, we instead pick one of the set of dictionary entries
that are within a few corruptions of our observed names. (One must, of course, careful
to account for this hybrid approach in our acceptance probability calculations.) Parses are
handled differently: we preprocess each citation, organizing its plausible segmentations
into a list ordered in terms of descending probability. At runtime, we simply sample from
these discrete distributions. Since we assume that boundaries occur only at punctuation
marks, and discard segmentations of probability < 10?6 , the lists are usually quite short. 7
The publication type variables, meanwhile, are not sampled at all. Since their range is so
small, we sum them out.
5.3 SCALING UP
One of the acknowledged flaws of the MCMC algorithm is that it often fails to scale. In
this application, as the number of papers increases, the simplest approach ? one where
the two clusters a1 and a2 are picked u.a.r ? is likely to lead to many rejected proposals,
as most pairs of clusters will have little in common. The resulting Markov chain will mix
slowly. Clearly, we would prefer to focus our proposals on those pairs of clusters which are
actually likely to exchange their instances. We have implemented an approach based on the
efficient clustering algorithm of McCallum et al [9], where a cheap distance metric is used
to preprocess a large dataset and fragment it into many canopies, or smaller, overlapping
sets of elements that have a non-zero probability of matching. We do the same, using
word-matching as our metric, and setting the thresholds to 0.5 and 0.2. Then, at runtime,
our q(x0 |x) function proposes first a canopy c, and then a pair of clusters u.a.r. from c.
(q(x|x0 ) is calculated by summing over all the canopies which contain any of the elements
of the two clusters.)
6 EXPERIMENTAL RESULTS
We have applied the MCMC-based algorithm to the hand-matched datasets used in [1].
(Each of these datasets contains several hundred citations of machine learning papers, about
half of them in clusters ranging in size from two to twenty-one citations.) We have also
6
7
Note that if the same cluster is picked twice, it will probably be split.
It would also be possible to sample directly from a model such as a hierarchical HMM
Face
Reinforcement
Reasoning
Constraint
349 citations, 242 papers
406 citations, 148 papers
514 citations, 296 papers 295 citations, 199 papers
Phrase matching
94%
79%
86%
89%
RPM + MCMC
97%
94%
96%
93%
Table 1: Results on four Citeseer data sets, for the text matching and MCMC algorithms.
The metric used is the percentage of actual citation clusters recovered perfectly; for the
MCMC-based algorithm, this is an average over all the MCMC-generated samples.
implemented their phrase matching algorithm, a greedy agglomerative clustering method
based on a metric that measures the degrees to which the words and phrases of any two
citations overlap. (They obtain their ?phrases? by segmenting each citation at all punctuation marks, and then taking all the bigrams of all the segments longer than two words.)
The results of our comparison are displayed in Figure 1, in terms of the Citeseer error metric. Clearly, the algorithm we have developed easily beats our implementation of phrase
matching.
We have also applied our algorithm to a large set of citations referring to the textbook Artificial Intelligence: A Modern Approach. It clusters most of them correctly, but there are a
couple of notable exceptions. Whenever several citations share the same set of unlikely errors, they are placed together in a separate cluster. This occurs because we do not currently
model the fact that erroneous citations are often copied from reference list to reference
list, which could be handled by extending the model to include a copiedFrom attribute.
Another possible extension would be the addition of a topic attribute to both papers and authors: tracking the authors? research topics might enable the system to distinguish between
similarly-named authors working in different fields. Generally speaking, we expect that
relational probabilistic languages with identity uncertainty will be a useful tool for creating
knowledge from raw data.
References
[1]
[2]
[3]
[4]
[5]
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[7]
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[12]
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S. Lawrence, K. Bollacker, and C. Lee Giles. Autonomous citation matching. In Agents, 1999.
I. Fellegi and A. Sunter. A theory for record linkage. In JASA, 1969.
W. Cohen, H. Kautz, and D. McAllester. Hardening soft information sources. In KDD, 2000.
Y. Bar-Shalom and T. E. Fortman. Tracking and Data Association. Academic Press, 1988.
I. J. Cox and S. Hingorani. An efficient implementation and evaluation of Reid?s multiple
hypothesis tracking algorithm for visual tracking. In IAPR-94, 1994.
H. Pasula, S. Russell, M. Ostland, and Y. Ritov. Tracking many objects with many sensors. In
IJCAI-99, 1999.
F. Dellaert, S. Seitz, C. Thorpe, and S. Thrun. Feature correspondence: A markov chain monte
carlo approach. In NIPS-00, 2000.
E. Charniak and R. P. Goldman. A Bayesian model of plan recognition. AAAI, 1993.
A. McCallum, K. Nigam, and L. H. Ungar. Efficient clustering of high-dimensional data sets
with application to reference matching. In KDD-00, 2000.
H. Pasula and S. Russell. Approximate inference for first-order probabilistic languages. In
IJCAI-01, 2001.
A. Pfeffer. Probabilistic Reasoning for Complex Systems. PhD thesis, Stanford, 2000.
A. Pfeffer and D. Koller. Semantics and inference for recursive probability models. In
AAAI/IAAI, 2000.
W.R. Gilks, S. Richardson, and D.J. Spiegelhalter. Markov chain Monte Carlo in practice.
Chapman and Hall, London, 1996.
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1,262 | 215 | Pulse-Firing Neural Chips for Hundreds of Neurons
PULSE-FIRING NEURAL CIDPS
FOR HUNDREDS OF NEURONS
Michael Brownlow
Lionel Tarassenko
Dept. Eng. Science
Univ. of Oxford
Oxford OX1 3PJ
Alan F. Murray
Dept. Electrical Eng.
Univ. of Edinburgh
Mayfield Road
Edinburgh EH9 3JL
Alister Hamilton
II Song Han(l)
H. Martin Reekie
Dept. Electrical Eng.
U niv. of Edinburgh
ABSTRACT
We announce new CMOS synapse circuits using only three
and four MOSFETsisynapse. Neural states are asynchronous
pulse streams, upon which arithmetic is performed directly.
Chips implementing over 100 fully programmable synapses
are described and projections to networks of hundreds of
neurons are made.
1 OVERVIEW OF PULSE FIRING NEURAL VLSI
The inspiration for the use of pulse firing in silicon neural networks is
clearly the electrical/chemical pulse mechanism in "real" biological neurons.
Asynchronous, digital voltage pulses are used to signal states t Si ) through
synapse weights { Tij } to emulate neural dynamics. Neurons fire voltage
pulses of a frequency determined by their level of activity but of a constant
magnitude (usually 5 Volts) [Murray,1989a]. As indicated in Fig. 1,
synapses perform arithmetic directly on these asynchronous pulses, to
increment or decrement the receiving neuron's activity. The activity of a
receiving neuron i, Xi is altered at a frequency controlled by the sending
neuron j, with state Sj by an amount determined by the synapse weight
(here, T ij ).
1 On secondment from the Korean Telecommunications Authority
785
786
Brownlow, Tarassenko, Murray, Hamilton, Han and Reekie
Sj> 0
lij > 0
Sj> 0
lij < 0
Sj = 0
lij > 0
Sj > 0
Tij = 0
Sj > 0
lij < 0
I
S.I
x?
t=:=
I
.fL.fL.n-IL
veo
Figure 1 : Pulse stream synapse functionality
A silicon neural network based on this technique is therefore an
asynchronous, analog computational structure. It is a hybrid between
analog and digital techniques in that the individual neural pulses are digital
voltage spikes, with all the robustness to noise and ease of regeneration that
this implies. These and other characteristics of pulse stream networks will
be discussed in detail later in this paper. Pulse stream methods, developed
in Edinburgh, have since been investigated by other groups - see for
instance [EI-Leithy,1988, Daniell, 1989].
1.1. WHY PULSE STREAMS?
There are some advantages in the use of pulse streams, and pulse rate
encoding, in implementing neural networks. It should be admitted here that
the initial move towards pulse streams was motivated by the desire to
implement pseudo-analog circuits on an essentially digital CMOS process. It
was a decision based at the time on expediency rather than on great vision
on our part, as we did not initially appreciate the full benefits of this form
of pulse stream arithmetic [Murray,1987].
Pulse-Firing Neural Chips for Hundreds of Neurons
For example, the voltages on the terminals of a MOSFET, VGS and VDS
could clearly be used to code a neural synapse weight and state respectively,
doing away with the need for pulses. In the pulse stream form, however,
we can arrange that only VGS is an "unknown". The device equations are
therefore easily simplified, and furthermore the body effect is more
predictable. In an equivalent continuous - time circuit, VDS will also be a
variable, which codes information. Predicting the transistor's operating
regime becomes more difficult, and the equation cannot be simplified.
Aside of the transistor - level advantages, giving rise to extremely compact
synapse circuits, there may be architectural advantages. There are certainly
architectural consequences. Digital pulses are easier to regenerate, easier to
pass between chips, and generally far more noise - insensitive than analog
voltages, all of which are significant advantages in the VLSI context.
Furthermore, the relationship to the biological exemplar should not be
ignored. It is at least interesting - whether it is significant remains to be
seen.
2 FULLY ANALOG PULSE STREAM SYNAPSES
Our early pulse stream chips proved the viability of the pulse stream
technique [Murray,1988a]. However, the area occupied by the digital
weight storage memory was unacceptably large. Furthermore, the use of
pseudo-clocks in an analog circuit was both aesthetically unsatisfactory and
detrimental to smooth dynamical behaviour, and using separate signal paths
for excitation and inhibition was both clumsy and inefficient. Accordingly,
we have developed a family of fully programmable, fully analog synapses
using dynamic weight storage, and operating on individual pulses to perform
arithmetic. We have already reported time-modulation synapses based on
this technique, and a later section of this paper will present the associated
chips [Murray,1988b, Murray,1989b].
2.1. TRANSCONDUCTANCE MULTIPLIER SYNAPSES
The equation of interest is that for the drain-source current, IDs, for a
MOSFET in the linear or triode region:IDS =
j.l.C ox W [
-1:--
(VGS - V T
)
VDS -
VDs2]
2--
(1)
Here, Cox is the oxide capacitance/area, j.l. the carrier mobility, W the
transistor gate width, L the transistor gate length, and VGS, VT, VDS the
transistor gate-source, threshold and drain-source voltages respectively.
.
. f
.
ThIs
expressIon
or IDS contams
a use f ul prod uct term:- j.l.CLox W x VGS X V .
DS
However, it also contains two other terms in V DS x VT and VDs2.
One approach might be to ignore this imperfection in the multiplication, in
the hope that the neural parallelism renders it irrelevant. We have chosen,
rather, to remove the unwanted terms via a second M OSFET, as shown in
Fig. 2.
787
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Brownlow, Tarassenko, Murray, Hamilton, Han and Reekie
13 = 11-12
Figure 2 : Use of a second MOSFET to remove nonlinearities
(a transconductance multiplier).
The output current 13 is now given by:W1
13 = JJ.Cox [ L1 (VGSl - VT
W2
L 2 (VGS2 - V T
)
VDS1 -
)
VDS2
+
W 1 vDsl
L1
2
(2)
W 2 VDsL]
L2
2
The secret now is to select W 1, L 1, W 2, L 2, VGSb VGS2 , VDS1 and VDS2 to
cancel all terms except
W1
JJ.Cox L1 VGSl X VDS1
(3)
This is a fairly well-known circuit, and constitutes a Transconductance
Multiplier. It was reported initially for use in signal processing chips such
as filters [Denyer,1981 , Han,1984]. It would be feasible to use it directly in
a continuous time network, with analog voltages representing the {Sj}. We
choose to use it within a pulse-stream environment, to minimise the
uncertainty in determining the operating regime, and terminal voltages, of
the MOSFETs, as described above.
Fig. 3 shows two related pulse stream synapse based on this technique. The
presynaptic neural state Sj is represented by a stream of 0-5V digital,
asynchronous voltage pulses Vj ? These are used to switch a current sink and
source in and out of the synapse, either pouring current to a fixed voltage
node (excitation of the postsynaptic neuron), or removing it (inhibition).
The magnitude and direction of the resultant current pulses are determined
by the synapse weight, currently stored as a dynamic, analog voltage Tij.
Pulse-Firing Neural Chips for Hundreds of Neurons
(a)
(b)
State V J
Reference 1
r1
Reference V r
:r:
Tij
Vfixed
Vfixed
Referen~
Reference 3
I
Figure 3 : Use of a transconductance multiplier to
form fully programmable pulse-stream synapses.
The fixed voltage VJixed and the summation of the current pulses to give an
activity Xj = 'LTjjSj are both provided by an Operational Amplifier
integrator circuit, whose saturation characteristics incidentally apply a
sigmoid nonlinearity. The transistors Tl and T4 act as power supply
"on/off" switches in Fig. 3a, and in Fig 3b are replaced by a single
transistor, in the output "leg" of the synapse, Transistors T2 and T3 form the
transconductance multiplier. One of the transistors has the synapse voltage
Tij on its gate, the other a reference voltage, whose value determines the
crossover point between excitation and inhibition. The gate-source voltages
on T2 and T3 need to be substantially greater than the drain-source
voltages, to maintain linear operation. This is not a difficult constraint to
satisfy.
The attractions of these cells are that all the transistors are n-type, removing
the need for area-hungry isolation well structures, and In Fig. 3a, the
vertical line of drain-source connections is topologically attractive,
producing very compact layout, while Fig. 3b has fewer devices. It is not
yet clear which will prove optimal.
2.2. ASYNCHRONOUS "SWITCHED CAPACITOR" SYNAPSE
Fig. 4 shows a further variant, in the form of a "switched capacitor" pulse
stream synapse. Here the synapse voltage Tij is electrically buffered to
switched capacitor structure, clocked by the presynaptic neural pulse
waveforms. Packets of charge are therefore "metered out" to the current
integrator whose magnitude is controlled by Tij (positive or negative), and
789
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Brownlow, Tarassenko, 1\1urray, Hamilton, Han and I{eekie
whose frequency by the presynaptic pulse rate. The overall principle is
therefore the same as that described for the transconductance multiplier
synapses, although the circuit level details are different.
-
Vj
Buffer
Vj
Integrator
Tij
~
T
I
X
1
/
-
I
Figure 4 : Asynchronous, "switched capacitor" pulse stream synapse.
Conventional synchronous switched capacitor techniques have been used in
neural integration [Tsividis,1987], but nowhere as directly as in this
example.
2.3. CHIP DETAILS AND RESULTS
Both the time-modulation and switched capacitor synapses have been tested
fully in silicon, and Fig. 5 shows a section of the time-modulation test chip.
This synapse currently occupies 174x73jl.m.
Figure 5 : Section, and single synapse, from time-modulation chip.
Pulse-Firing Neural Chips for Hundreds of Neurons
Three distinct pulse-stream synapse types have been presented, with
different operating schemes and characteristics. None has yet been used to
configure a large network, but this is now being done. Current estimates
for the number of synapses implementable using the two techniques
described above are as shown in Table 1, using an 8mmx8mm die as an
example.
The lack of direct scaling between transistor count and synapse count (e.g.
why does the factor 4111 not manifest itself as a much larger increase in
synapse count) can be explained. The raw number of transistors is not the
only factor in determining circuit area. Routing of power supplies, synapse
weight address lines, as well as storage capacitor size all take their toll, and
are common to both of the above synapse circuits. Furthermore, in analog
circuitry, transistors are almost certainly larger than minimum geometry,
and generally significantly larger, to minimise noise problems. This all gives
rise to a larger area than might be expected from simple arguments.
Clearly, however, we are in position to implement serious sized networks,
firstly with the time-modulation synapse, which is fully tested in silicon, and
later with the transconductance type, which is still under detailed design and
layout.
Table 1 : Estimated synapse count on 8mm die
SYNAPSE
Time modulation
Transconductance
Switched Capacitor
NO. OF
TRANSISTORS
ESTIMATED
NETWORK SIZE
11
4
4
= 6400 synapses
=15000 synapses
= 14000 synapses
In addition, we are developing new oscillator forms, techniques to
counteract leakage from dynamic nodes, novel inter-chip signalling strategies
specifically for pulse-stream systems, and non-volatile (a-Si) pulse stream
synapses. These are to be used for applications in text-speech synthesis,
pattern analysis and robotics. Details will be published as the work
progresses.
Acknowledgements
The authors are grateful to the UK Science and Engineering Research
Council, and the European Community (ESPRIT BRA) for its support of
this work. Dr. Han is grateful to the Korean Telecommunications
Authority, from whence he is on secondment in Edinburgh, and
KOSEF(Korea) for partial financial support.
791
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Brownlow, Tarassenko, Murray, Hamilton, Han and Reekie
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P. M. Daniell, W. A. J. Waller, and D. A. Bisset, "An
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Denyer ,1981.
P. B. Denyer and J. Mavor, "MOST Transconductance Multipliers for
Array Applications," lEE Proc. Pt. 1, vol. 128, no. 3, pp. 81-86, June
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EI-Leithy,1988.
N. EI-Leithy, M. Zaghloul, and R. W. Newcomb, "Implementation of
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| 215 |@word cox:3 pulse:46 eng:3 solid:1 electronics:2 initial:1 contains:1 current:8 si:2 yet:2 remove:2 aside:1 fewer:1 device:2 unacceptably:1 signalling:1 accordingly:1 smith:2 authority:2 node:2 firstly:1 rc:1 direct:1 supply:2 symposium:1 prove:1 mayfield:1 inter:1 secret:1 expected:1 integrator:3 terminal:2 becomes:1 provided:1 circuit:12 substantially:1 developed:2 pseudo:2 act:1 charge:1 unwanted:1 esprit:1 uk:1 control:1 hamilton:7 producing:1 carrier:1 positive:1 engineering:1 waller:1 consequence:1 encoding:1 oxford:2 id:3 firing:8 path:1 modulation:6 might:2 ease:1 implement:2 area:5 crossover:1 significantly:1 projection:1 road:1 cannot:1 storage:3 context:1 equivalent:1 conventional:1 layout:2 attraction:1 array:1 financial:1 increment:1 pt:1 nowhere:1 tarassenko:7 electrical:3 region:1 predictable:1 environment:1 dynamic:4 grateful:2 upon:1 sink:1 easily:1 chip:13 emulate:1 represented:1 univ:2 mosfet:3 distinct:1 artificial:1 whose:4 larger:4 regeneration:1 itself:1 advantage:4 toll:1 transistor:15 product:1 lionel:1 r1:1 incidentally:1 cmos:2 exemplar:1 ij:1 progress:1 aesthetically:1 implies:1 direction:1 waveform:1 newcomb:1 functionality:1 filter:2 packet:1 occupies:1 routing:1 implementing:2 behaviour:1 niv:1 biological:2 summation:1 mm:1 great:1 circuitry:1 arrange:1 early:1 mosfets:1 proc:4 currently:2 council:1 hope:1 clearly:3 imperfection:1 rather:2 occupied:1 voltage:18 june:3 unsatisfactory:1 portland:1 whence:1 initially:2 vlsi:5 overall:1 integration:2 fairly:1 park:1 cancel:1 constitutes:1 t2:2 serious:1 micro:1 individual:2 replaced:1 geometry:1 fire:1 maintain:1 amplifier:1 interest:1 certainly:2 configure:1 partial:1 conj:1 korea:1 mobility:1 instance:1 hundred:6 reported:2 stored:1 lee:2 off:1 receiving:2 michael:1 vgs:5 synthesis:1 w1:2 choose:1 dr:1 conf:1 oxide:1 inefficient:1 nonlinearities:1 int:1 oregon:1 satisfy:1 stream:23 performed:1 later:3 doing:1 il:1 kaufmann:1 characteristic:3 t3:2 raw:1 none:1 brownlow:5 published:1 synapsis:14 frequency:3 pp:10 resultant:1 associated:1 proved:1 manifest:1 synapse:25 done:1 ox:1 furthermore:4 uct:1 clock:1 d:2 ei:3 ox1:1 lack:1 indicated:1 effect:1 multiplier:7 inspiration:1 chemical:1 volt:1 attractive:1 anastassiou:1 width:1 excitation:3 die:2 clocked:1 l1:3 hungry:1 novel:1 sigmoid:1 common:1 volatile:1 overview:1 leithy:3 pouring:1 insensitive:1 jl:1 discussed:1 analog:10 he:1 silicon:4 significant:2 buffered:1 nonlinearity:1 han:9 operating:4 inhibition:3 irrelevant:1 buffer:1 vt:3 seen:1 minimum:1 greater:1 morgan:1 bra:1 signal:3 ii:1 arithmetic:8 full:1 alan:1 smooth:1 coded:1 controlled:3 variant:1 essentially:1 vision:1 veo:1 robotics:1 cell:1 addition:1 source:7 w2:1 capacitor:9 viability:1 switch:2 xj:1 isolation:1 zaghloul:1 minimise:2 vgs2:2 whether:1 motivated:1 expression:1 synchronous:1 ul:1 song:2 render:1 speech:1 jj:2 programmable:5 ignored:1 tij:8 generally:2 clear:1 detailed:1 amount:1 estimated:2 vol:5 group:1 four:1 threshold:1 pj:1 sum:1 counteract:1 letter:2 uncertainty:1 telecommunication:2 topologically:1 family:1 announce:1 almost:1 architectural:2 decision:2 bisset:1 scaling:1 eh9:1 fl:2 expediency:1 alister:1 activity:4 constraint:1 argument:1 extremely:1 transconductance:9 martin:1 developing:1 electrically:1 postsynaptic:1 leg:1 explained:1 daniell:3 equation:3 remains:1 count:4 mechanism:1 sending:1 operation:1 apply:1 away:1 robustness:1 gate:5 triode:1 giving:1 murray:19 appreciate:1 leakage:1 move:1 capacitance:1 already:1 spike:1 strategy:1 detrimental:1 separate:1 vd:4 presynaptic:3 code:2 length:1 relationship:1 korean:2 difficult:2 negative:1 rise:2 design:1 implementation:2 unknown:1 perform:2 vertical:1 neuron:12 regenerate:1 implementable:1 august:1 community:1 connection:1 address:1 usually:1 dynamical:1 parallelism:1 pattern:1 regime:2 saturation:1 memory:1 analogue:2 power:2 hybrid:1 predicting:1 representing:1 scheme:1 altered:1 lij:4 text:1 l2:1 acknowledgement:1 drain:4 multiplication:1 determining:2 fully:8 interesting:1 digital:7 switched:8 reekie:5 principle:1 asynchronous:9 edinburgh:5 benefit:1 author:1 made:1 simplified:2 far:1 sj:8 nov:1 compact:2 ignore:1 active:1 xi:1 continuous:2 prod:1 why:2 table:2 operational:1 tsividis:3 investigated:1 european:1 vj:3 did:1 decrement:1 noise:3 body:1 fig:9 tl:1 clumsy:1 position:1 resistor:1 removing:2 denyer:3 magnitude:3 t4:1 easier:2 admitted:1 desire:1 determines:1 ma:2 sized:1 towards:1 oscillator:1 feasible:1 determined:3 except:1 specifically:1 pas:1 select:1 support:2 dept:3 tested:2 |
1,263 | 2,150 | Bayesian Monte Carlo
Carl Edward Rasmussen and Zoubin Ghahramani
Gatsby Computational Neuroscience Unit
University College London
17 Queen Square, London WC1N 3AR, England
edward,[email protected]
http://www.gatsby.ucl.ac.uk
Abstract
We investigate Bayesian alternatives to classical Monte Carlo methods
for evaluating integrals. Bayesian Monte Carlo (BMC) allows the incorporation of prior knowledge, such as smoothness of the integrand,
into the estimation. In a simple problem we show that this outperforms
any classical importance sampling method. We also attempt more challenging multidimensional integrals involved in computing marginal likelihoods of statistical models (a.k.a. partition functions and model evidences). We find that Bayesian Monte Carlo outperformed Annealed
Importance Sampling, although for very high dimensional problems or
problems with massive multimodality BMC may be less adequate. One
advantage of the Bayesian approach to Monte Carlo is that samples can
be drawn from any distribution. This allows for the possibility of active
design of sample points so as to maximise information gain.
1 Introduction
Inference in most interesting machine learning algorithms is not computationally tractable,
and is solved using approximations. This is particularly true for Bayesian models which
require evaluation of complex multidimensional integrals. Both analytical approximations,
such as the Laplace approximation and variational methods, and Monte Carlo methods
have recently been used widely for Bayesian machine learning problems. It is interesting
to note that Monte Carlo itself is a purely frequentist procedure [O?Hagan, 1987; MacKay,
1999]. This leads to several inconsistencies which we review below, outlined in a paper
by O?Hagan [1987] with the title ?Monte Carlo is Fundamentally Unsound?. We then
investigate Bayesian counterparts to the classical Monte Carlo.
Consider the evaluation of the integral:
(1)
where is
a probability (density), and
is the
we wish to integrate. For
function
example,
could
be
the
posterior
distribution
and
the
predictions
by a model
made
with parameters , or
could be the parameter prior and
the likelihood
so that equation (1) evaluates the marginal likelihood (evidence) for a model. Classical
Monte Carlo makes the approximation:
(2)
where
are random (not necessarily independent) draws
, which converges
to
. from
the right answer in the limit of large numbers
of
samples,
If
sampling
directly
from
is hard, or if high density regions in
do not match up with areas where
has large
magnitude,
it is also possible to draw samples from some importance
sampling distribution
to obtain the estimate:
(3)
As O?Hagan [1987] points out, there are two important
objections to these procedures.
not only depends on the values of
but also on the enFirst, the estimator
tirely
same set of samples
choice of the sampling distribution . Thus, if the
arbitrary
,
were obtained from
, conveying exactly the same information about
two different sampling distributions, two different estimates of would be obtained. This
dependence on irrelevant (ancillary)
information is unreasonable and violates the Likelihood Principle. The second
objection
is that classical Monte Carlo procedures entirely
ignore the values
of the
when forming the estimate. Consider the simple example of
three points
that are sampled from and the third happens to fall on the same point as the
second,
, conveying no extra information about the integrand. Simply aver-
aging the integrand at these three points, which is the classical Monte Carlo estimate, is
clearly inappropriate; it would make much more sense to average the first two (or the first
and third). In practice points are unlikely to fall on top of each other in continuous spaces,
however, a procedure that weights points equally regardless of their spatial distribution is
ignoring relevant information. To summarize the objections, classical Monte Carlo bases
its estimate on irrelevant information and throws away relevant information.
We seek to turn the problem of evaluating the integral (1) into a Bayesian inference problem
which, as we will see, avoids the inconsistencies of classical Monte Carlo and
can result
in better estimates. To do this, we think of the unknown desired quantity
as being
random. Although this interpretation is not the most usual one, it is entirely consistent with
the Bayesian view that all forms of uncertainty are represented
using
probabilities: in this
case uncertainty
arises because
we
afford to compute
at every location. Since
cannot
the desired
is a function
of
(which is unknown until we evaluate it) we proceed
to obtain the posterior over
by putting a prior on , combining it with the observations
which in turn implies a distribution over the desired .
A very convenient way of putting priors over functions is through Gaussian Processes (GP).
Under a GP prior
the joint distribution
of any (finite) number of function values (indexed
by the inputs, ) is Gaussian:
! # "
(4)
where here we take the mean to be zero. The covariance matrix is given by the covariance
function, a convenient choice being:1
"$ &
%('*)
+
$
:
,+.- /1032546 79
8
;
>: =
:
:
6
$
* < +
(5)
where the parameters are hyperparameters. Gaussian processes, including optimization
of hyperparameters, are discussed in detail in [Williams and Rasmussen, 1996].
1
Although the function values obtained are assumed to be noise-free, we added a tiny constant to
the diagonal of the covariance matrix to improve numerical conditioning.
2 The Bayesian Monte Carlo Method
The Bayesian Monte Carlo method starts with a prior
over the function,
and makes
inferences about from
giving the
a set
of samples
posterior distribution
. Under a GP prior the posterior is (an infinite dimensional
joint)
since the
Gaussian;
integral
eq. (1) is just a linear projection (on the direction defined
by
), the posterior
is also Gaussian, and fully characterized by its mean and
variance. The average over functions of eq. (1) is the expectation of the average function:
where
(6)
is the posterior mean function. Similarly, for the variance:
%('*)
6
6
6
%('*) 4
=
(7)
where
is the posterior covariance. The standard results for the GP model for the
=
posterior mean and covariance
are:
"! $#
#
"
&%
and
%('*) 4
6
'!
"
$#
(!
)%
!
*#
(8)
where and are the observed inputs and function values respectively. In general combining eq. (8) with eq. (6-7) may lead to expressions which are difficult to evaluate, but there
are several interesting special cases.
If the density
and thecovariance
eq. (5) are both Gaussian, we obtain ana
,+ .function
-
and the Gaussian kernels on the data points are
lytical results. In detail, if
*/
10 diag
+ +
then the expectation evaluates to:
'2
"
%
,+ - 8
32
0
%
%
-5476
/1032 6
.8
:9
*/
6
;+
*0<4<-
%
6
*/
;+
(9)
a result which has previously been derived under the name of Bayes-Hermite Quadrature
[O?Hagan, 1991]. For the variance, we get:
+.- 7
7=
=
0
%
->4&6?= %
=
6 "
$8
32
%
2
(10)
2
lead to analytical results include polynomial
with as defined in eq. (9). Other choices
that
kernels and mixtures of Gaussians for
.
2.1 A Simple Example
To illustrate the method we evaluated the integral of a one-dimensional function
under a
Gaussian
density
(figure
1,
left).
We
generated
samples
independently
from
, evalu
ated
at those points, and optimised the hyperparameters of our Gaussian process fit
to the function. Figure 1 (middle) compares the error in the Bayesian Monte Carlo (BMC)
estimate of the integral (1) to the Simple Monte Carlo (SMC) estimate using the same samples.
we would
expect the squared error in the Simple Monte Carlo estimate decreases
is the
< Aswhere
sample size. In contrast, for more than about 10 samples, the
as
BMC
estimate improves at a much higher rate. This is achieved because the prior on allows
the method to interpolate between sample points.
Moreover, whereas the SMC estimate is
invariant to permutations of the values on the axis, BMC makes use of the smoothness of
the function. Therefore, a point in a sparse region is far more informative about the shape
of the function for BMC than points in already densely sampled areas. In SMC if two samples happen to fall close to each other the function value there will be counted with double
weight.
This effect means that large numbers of samples are needed to adequately represent
.
BMC
circumvents this problem by analytically integrating its mean function w.r.t.
.
In figure 1 left, the negative log density of the true value of the integral under the predictive distributions are compared for BMC and SMC. For not too small sample sizes, BMC
outperforms SMC. Notice however, that for very small sample sizes BMC occasionally
has
very bad performance.
This
is
due
to
examples
where
the
random
draws
of
lead
to
func
tion values
that are consistent with much longer length scale than the true function;
the mean prediction becomes somewhat inaccurate, but worse still, the inferred variance
becomes very small (because a very slowly varying function is inferred), leading to very
poor performance compared to SMC. This problem is to a large extent caused by the optimization of the length scale hyperparameters of the covariance function; we ought instead
to have integrated over all possible length scales. This integration would effectively ?blend
in? distributions with much larger variance (since the data is also consistent with a shorter
length scale), thus alleviating the problem, but unfortunately this is not possible in closed
form. The problem disappears for sample sizes of around 16 or greater.
In the previous example, we chose
to be Gaussian. If you wish to use BMC to integrate
w.r.t. non-Gaussian densities then an importance re-weighting trick becomes necessary:
<
(11)
and is a Gaussian and
where the Gaussian process models
is
an arbitrary density
which
can
be
evaluated.
See
Kennedy [1998] for extension to non
Gaussian
.
2.2 Optimal Importance Sampler
For the simple example discussed above, it is also interesting to ask whether the efficiency
of SMC could be improved by generating
independent samples from more-cleverly designed distributions.
As
we
have
seen
in
equation
(3), importance sampling gives an unbi
ased estimate of by sampling
from
and computing:
where
wherever
(12)
. The variance of this estimator is given by:
56
(13)
Using calculus of variations it is simple to show that the optimal (minimum variance) importance sampling distribution is:
(14)
which we can substitute into equation (13) to get
the minimum variance, . If
is
always non-negative
or
non-positive
then
,
which
is
unsurprising
given
that
we
needed to know in advance to normalise . For functions that take on both positive and
?2
10
Bayesian inference
Simple Monte Carlo
Optimal importance
0.4
?3
10
average squared error
0.3
0.2
0.1
0
?0.1
?0.2
?4
10
?5
10
?6
10
?0.3
?0.4
?0.5
?4
Bayesian inference
Simple Monte Carlo
minus log density of correct value
function f(x)
measure p(x)
0.5
20
15
10
5
0
?5
?7
?2
0
2
10
4
1
10
2
10
sample size
1
10
2
10
sample size
Figure 1: Left: a simple one-dimensional function
(full) and Gaussian density (dashed)
with respect to which we wish to integrate . Middle: average squared error for simple Monte Carlo sampling from (dashed), the optimal achievable bound for importance
sampling (dot-dashed), and the Bayesian Monte Carlo estimates. The values plotted are
averages over up to 2048 repetitions. Right: Minus the log of the Gaussian predictive density with mean eq. (6) and variance eq. (7), evaluated at the true value of the integral (found
by numerical integration), ?x?. Similarly for the Simple Monte Carlo procedure, where the
mean and variance of the predictive distribution are computed from the samples, ?o?.
<
6
negative values
which is a constant times the variance of
a Bernoulli random variable (sign
). The lower bound from this optimal importance
sampler as a function of number of samples is shown in figure 1, middle. As we can
see, Bayesian Monte Carlo improves on the optimal importance sampler considerably. We
stress that the optimal importance sampler is not practically achievable since it requires
knowledge of the quantity we are trying to estimate.
3 Computing Marginal Likelihoods
We now consider the problem of estimating the marginal likelihood of a statistical model.
This problem is notoriously difficult and very important, since it allows for comparison of
different models. In the physics literature it is known as free-energy estimation. Here we
compare the Bayesian Monte Carlo method to two other techniques: Simple Monte Carlo
sampling (SMC) and Annealed Importance Sampling (AIS).
Simple Monte Carlo, sampling from the prior, is generally considered inadequate for this
problem, because the likelihood is typically sharply peaked and samples from the prior are
unlikely to fall in these confined areas, leading to huge variance in the estimates (although
they are unbiased). A family of promising ?thermodynamic integration? techniques for
computing marginal likelihoods are discussed under the name of Bridge and Path sampling
in [Gelman and Meng, 1998] and Annealed Importance Sampling (AIS) in [Neal, 2001].
The central idea is to divide one difficult integral into a series of easier ones, parameterised
by (inverse) temperature, . In detail:
-
-
%
where
and
is the inverse temperature of the annealing schedule and
where
To compute
each fraction we sample from equilibrium from the distribution
%
and compute importance weights:
,!
%
(15)
!"
!
%
%
;
%
%
%
.
(16)
In practice can be set to 1, to allow very slow reduction in temperature. Each of the
intermediate ratios are much easier to compute than the original ratio, since the likelihood
function to the power of a small number is much better behaved that the likelihood itself.
Often elaborate non-linear cooling schedules are used, but for simplicity we will just take
a linear schedule for the inverse temperature. The samples at each temperature are drawn
using a single Metropolis proposal, where the proposal width is chosen to get a fairly high
fraction of acceptances.
The model in question for which we attempt to compute the marginal likelihood was itself a Gaussian process regression fit to the an artificial dataset suggested
+ - by [Friedman,
1988].2 We had 9 length scale hyperparameters, a signal variance ( ) and an explicit
noise variance parameter. Thus the marginal likelihood is an integral
a 7 dimensional
over
priors.
hyperparameter space. The log of the hyperparameters are given
Figure 2 shows a comparison of the three methods. Perhaps surprisingly, AIS and SMC are
seen to be very comparable, which can be due to several reasons: 1) whereas the SMC samples are drawn independently, the AIS samples have considerable auto-correlation because
of the Metropolis generation mechanism, which hampers performance for low sample sizes,
2) the annealing schedule was not optimized nor the proposal width adjusted with temperature, which might possibly have sped up convergence. Further, the difference between
AIS and SMC would be more dramatic in higher dimensions and for more highly peaked
likelihood functions (i.e. more data).
The Bayesian Monte Carlo method was run on the same samples as were generate
by
the
AIS procedure. Note that BMC can use samples from any distribution, as long as
can
be evaluated. Another obvious choice for generating samples for BMC would be to use
an MCMC method to draw samples from the posterior. Because BMC needs to model the
integrand using a GP, we need to limit the number of samples
computation (for fitting
. Thussince
hyperparameters
and
computing
the
?s)
scales
as
for
sample
greater than
7 we limit the number of samples to 7 , chosen equally spaced fromsize
the AIS Markov
chain. Despite this thinning of the samples we see a generally superior performance of
BMC, especially for smaller sample sizes. In fact, BMC seems to perform equally well for
almost any of the investigated sample sizes. Even for this fairly large number of samples,
the generation of points from the AIS still dominates compute time.
4 Discussion
An important aspect which we have not explored in this paper is the idea that the GP model
used to fit the integrand gives errorbars (uncertainties) on the integrand. These error bars
2
was 100 samples generated from the 5-dimensional function "!$#&%('(''(%)!+*&,./&021)35The
4 "67! data
# !98&,;:=< 0 "!
>@? 0 ' AB, 8 : /&0 !9C:=A! * :=D , where D is zero mean unit variance Gaussian
noise and the inputs are sampled independently from a uniform [0, 1] distribution.
Log Marginal Likelihood
?45
?50
?55
?60
True
SMC
AIS
BMC
?65
?70
3
10
4
10
Number of Samples
5
10
Figure 2: Estimates of the marginal likelihood for different sample sizes using Simple
Monte Carlo sampling (SMC; circles, dotted line), Annealed Importance Sampling (AIS;
, dashed line), and Bayesian Monte Carlo (BMC;
triangles, solid line). The true value
sample
(solid straight line) is estimated 6
from
a
single
long run of AIS. For comparison,
(which is an upper bound on the true value).
the maximum log likelihood is
could be used to conduct an experimental design,
i.e. active learning. A simple approach
would
and
be to evaluate the function at points where the GP has large uncertainty
is not too small:
the
expected contribution to the uncertainty in the estimate of the
. For a fixed Gaussian Process covariance function these design
integral scales as
points can often be pre-computed, see e.g. [Minka, 2000]. However, as we are adapting the
covariance function depending on the observed function values, active learning would have
to be an integral part of the procedure. Classical Monte Carlo approaches cannot make use
of active learning since the samples need to be drawn from a given distribution.
When using BMC to compute marginal likelihoods, the Gaussian covariance function used
here (equation 5) is not ideally suited to modeling the likelihood. Firstly, likelihoods are
non-negative whereas the prior is not restricted in the values the function can take. Secondly, the likelihood tends to have some regions of high magnitude and variability and
other regions which are low and flat; this is not well-modelled by a stationary covariance
function. In practice this misfit between the GP prior and the function modelled has even
occasionally led to negative values for the estimate of the marginal likelihood! There could
be several approaches to improving the appropriateness of the prior. An importance distribution such as one computed from a Laplace approximation or a mixture of Gaussians
can be used to dampen the variability in the integrand [Kennedy, 1998]. The GP could be
used to model the log of the likelihood [Rasmussen, 2002]; however this makes integration
more difficult.
The BMC method outlined in this paper can be extended in several ways. Although the
choice of Gaussian process priors is computationally convenient in certain circumstances,
in general other function approximation priors can be used to model the integrand. For
discrete (or mixed) variables the GP model could still be used with appropriate choice of
covariance function. However, the resulting sum (analogous to equation 1) may be difficult
to evaluate. For discrete , GPs are not directly applicable.
Although BMC has proven successful on the problems presented here, there are several
limitations to the approach. High dimensional integrands can prove difficult to model. In
such cases a large number of samples may be required to obtain good estimates of the
function. Inference using a Gaussian Process prior is at present limited computationally
to a few thousand samples. Further, models such as neural networks and mixture models
exhibit an exponentially large number of symmetrical modes in the posterior. Again modelling this with a GPprior
would typically be difficult. Finally, the BMC method requires
that the distribution
can be evaluated. This contrasts with classical MC
many
where
methods only require that samples can be drawn from some distribution
, for which
the normalising constant is not necessarily known (such as in equation 16). Unfortunately,
this limitation makes it difficult, for example, to design a Bayesian analogue to Annealed
Importance Sampling.
We believe that the problem of computing an integral using a limited number of function
evaluations should be treated as an inference problem and that all prior knowledge about
the function being integrated should be incorporated into the inference. Despite the limitations outlined above, Bayesian Monte Carlo makes it possible to do this inference and
can achieve performance equivalent to state-of-the-art classical methods despite using a
fraction of sample evaluations, even sometimes exceeding the theoretically optimal performance of some classical methods.
Acknowledgments
We would like to thank Radford Neal for inspiring discussions.
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1,264 | 2,151 | Discriminative Binaural Sound Localization
Ehud Ben-Reuven and Yoram Singer
School of Computer Science & Engineering
The Hebrew University, Jerusalem 91904, Israel
[email protected], [email protected]
Abstract
Time difference of arrival (TDOA) is commonly used to estimate the azimuth of a source in a microphone array. The most common methods
to estimate TDOA are based on finding extrema in generalized crosscorrelation waveforms. In this paper we apply microphone array techniques to a manikin head. By considering the entire cross-correlation
waveform we achieve azimuth prediction accuracy that exceeds extrema
locating methods. We do so by quantizing the azimuthal angle and
treating the prediction problem as a multiclass categorization task. We
demonstrate the merits of our approach by evaluating the various approaches on Sony?s AIBO robot.
1 Introduction
In this paper we describe and evaluate several algorithms to perform sound localization in
a commercial entertainment robot. The physical system being investigated is composed of
a manikin head equipped with a two microphones and placed on a manikin body. This type
of systems is commonly used to model sound localization in biological systems and the
algorithms used to analyze the signal are usually inspired from neurology. In the case of an
entertainment robot there is no need to be limited to a neurologically inspired model and
we will use combination of techniques that are commonly used in microphone arrays and
statistical learning. The focus of the work is the task of localizing an unknown stationary
source (compact in location and broad in spectrum). The goal is to find the azimuth angle
of the source relative to the head.
A common paradigm to approximately find the location of a sound source employs a microphone array and estimates time differences of arrival (TDOA) between microphones
in the array (see for instance [1]). In a dual-microphone array it is usually assumed that
the difference in the two channels is limited to a small time delay (or linear phase in frequency domain) and therefore the cross-correlation is peaked at the the time corresponding
to the delay. Thus, methods that search for extrema in cross-correlation waveforms are
commonly used [2]. The time delay approach is based on the assumption that the sound
waves propagate along a single path from the source to the microphone and that the microphone response of the two channels for the given source location is approximately the
same. In order for this to hold, the microphones should be identical, co-aligned, and, near
each other relative to the source. In addition there should not be any obstructions between
or near the microphones. The time delay assumption fails in the case of a manikin head: the
microphone are antipodal and in addition the manikin head and body affect the response
in a complex way. In our system the distance to the supporting floor was also significant.
Our approach for overcoming these difficulties is composed of two stages. First, we perform signal processing based on the generalized cross correlation transform called Phase
Transform (PHAT) also called Cross Power Spectrum Phase (CPSP). This signal processing removes to a large extent variations due the sound source. Then, rather than proceeding
with peak-finding we employ discriminative learning methods by casting the azimuth estimation as a multiclass prediction problem. The results achieved by combining the two
stages gave improved results in our experimental setup.
This paper is organized as follows. In Sec. 2 we describe how the signal received in the
two microphones was processed to generate accurate features. In Sec. 3 we outline the
supervised learning algorithm we used. We then discuss in Sec. 4 approaches to combined predictions from multiple segments. We describe experimental results in Sec. 5 and
conclude with a brief discussion in Sec. 6.
2 Signal Processing
Throughout the paper we denote signals in the time domain by lower case letters and in the
frequency domain by upper case letters. We denote the convolution operator between two
signals by and the correlation operator by . The unknown source signal is denoted by
and thus its spectrum is . The source signal passes through different physical setup and is
received at the right and left microphones. We denote the received signals by and . We
model the different physical media, the signal passes through, as two linear systems whose
frequency response is denoted by and . In addition the signals are contaminated with
noise that may account for non-linear effects such as room reverberations (see for instance
[3] for more detailed noise models). Thus, the received signals can be written in the time
and frequency domain as,
(1)
(2)
Since the source signal is typically non-stationary we break each training and test signal into segments and perform the processing described in the sequel based on short-time
Fourier transform. Let be the number of segments a signal is divided into and the
number of samples in a single segment. Each is multiplied by a Hanning window and
padded with zeros to smooth the end-of-segment effects and increase the resolution of the
short-time Fourier transform (see for instance [8]). Denote by and the left and right
signal-segments after the above processing. Based on the properties of the Fourier transform, the local cross-correlation between the two signals can be computed efficiently by
the inverse Fourier transform, denoted , of the product of the spectrum of and the
complex conjugate of the spectrum of ,
! #" %
$
&
(3)
Had the difference between the two signals been a mere time delay due to the different
location of the microphones, the cross correlation would have obtained its maximal value
at a point which corresponds to the time-lag between the received signals. However, since
the source signal passes through different physical media the short-time cross-correlation
does not necessarily obtain a large value at the time-lag index. It is therefore common (see
for instance [1]) to multiply the spectrum of the cross-correlation by a weighting function
in order to compensate for the differences in the frequency responses obtained at the two
microphones. Denoting the spectral shaping function for the ' th segment by ( , the generalization cross-correlation from Eq. (3) is, !
)*
+
" ( $ & . For
?plain? cross-correlation, ( ,.-0/ is equal to 1 at each (discrete) frequency - . In our tests
we found that a globally-equalized cross-correlation gives better results. The transform is
obtained by setting, ( ,.-0/ 1325476 where 486 is the average over all measurements and both
channels of 9 ,:-0/ 9 ; . Finally, for PHAT the weight for the spectral point - is,
1
( ,.-0/
:
,
<
/
9
$ ,:-</ 9
To further motivate and explain the PHAT weighting scheme, we build on the derivation
in [5] and expand the PHAT assuming that the noise is zero. In PHAT the spectral value at
frequency point - (prior to the inverse Fourier transform) is,
( ,.-0/ ,.-0/ $ ,:-</
., -0/ $ ., -0/
9 ., -0/ $ ., -0/ 9
(4)
Inserting Eq. (1) and Eq. (2) into Eq. (4) without noise we get,
(
,:-0/ ,.-0/ $ ,.-0/
:, -</ ,:-</ $ ., -0/ $ ., -0/
- / ., -0/ $ ., -0/ $ ., -0/ 9
9 ,.0
Therefore, assuming the noise is zero, PHAT eliminates the
contribution of the unknown source and the entire waveform of PHAT is only a function of the physical setup. If all
other physical parameters are constant, the PHAT waveform
(as well as its peak location) is a function of the azimuth angle of the sound source relative to the manikin head. This
is of course an approximation and the presence of noise and
changes in the environment result in a waveform that deviates from the closed-form given in Eq. (5). In Fig. 1 we show
the empirical average of the waveform for PHAT and for the
equalized cross-correlation, the vertical bars represent an error of 1 . In both cases, the location of the maximal correlation is clearly at as expected. Nonetheless, the high variance, especially in the case of the equalized cross-correlation
imply that classification of individual segments may often be
rather difficult.
$
9 9:9 9
(5)
0.6
0.5
0.4
0.3
0.2
0.1
0
?0.1
?5
?4
?3
?2
?1
0
1
2
3
4
5
0
1
2
3
4
5
0.05
0.04
0.03
0.02
0.01
0
?0.01
?0.02
?0.03
?5
?4
?3
?2
?1
Figure 1: Average wave-
form with standard deviIn practice, we found that it suffices to take only the ener
afation for
getic portion of the generalized cross-correlation waveforms
ter performing PHAT (top)
by considering only time lags of
through
samples. In
and the equalized crosswhat follows we will take this part to be the waveform. Forcorrelation (bottom).
mally, the feature vector of the ' th segment is defined as,
, ! ,
/
! ,
/ /
(6)
87
were
was set to be bigger than the maximal lag in samples between the two channels,
2 where is the head diameter and is speed of sound.
Summarising, the signal processing we perform is based on short time Fourier transform
of the signals received at the two microphones. From the two spectrums we then compute
the generalized cross-correlation using one of the three weighting schemes described above
and taking only
*1 samples of the resulting waveforms as the feature vectors. We now
move our focus to classification of a single segment.
3 Single Segment Classification
Traditional approaches to sound localization search for the the position of the extreme value
in the generalized cross-correlation waveform that were derived in Sec. 2. While being
intuitive, this approach is prone to noise. Peak location can be considered as a reduction
in dimensionality, from
1 to 1 , of the feature vectors , however we have shown
in Eq. 5 that the entire waveform of PHAT can be used as a feature vector to localise the
source. Indeed, in Sec. 5 we report experimental results which show that peak-finding is
significantly inferior to methods that we now describe, that uses the entire waveform. In all
techniques, peak-location and waveform, we used supervised learning to build a model of
the data using a training set and then used a test set to evaluate the learned model.
In a supervised learning setting, we have access to labelled examples and the goal is to
find a mapping from the instance domain (the peak-location or waveforms in our setting)
to a response variable (the azimuth angle). Since the angle is a continuous variable the
first approach that comes to mind is using a linear or non-linear regressors. However,
we found that regression algorithms such as Widrow-Hoff [10] yielded inferior results.
Instead of treating the learning problem as a regression problem, we quantized the angle and
converted the sound localization problem into a multiclass decision problem. Formally, we
where
into non-overlapping intervals
bisected the interval
78
/
,
/
<
2
<
2 and
1 72 1 . We now can transform the
1
! where
real-valued angle of the ' th segment, , into a discrete variable
78
%& . After this quantization, the training set is composed of instance-label
" iff $
#
pairs '
)(+ * and the first task is to find a classification rule from the peak-location or
, . We will first describe the method used for peak-location
waveforms space into 1
77
and then we will describe two discriminative methods to classify the waveform. The first
is based on a multiclass version of the Fisher linear discriminant [7] and is very simple to
implement. The second employs recent advances in statistical learning and can be used in
an online fashion. It thus can cope, to some extent with changes, in the environment such
as moving elements that change the reverberation properties of the physical media.
Peak location classification: Due to the relative low sampling frequency (-/. 10 " 21 )
spline interpolation was used to improve the peak location. In microphone arrays it is common to translate the peak-location to an estimate of the source azimuth using a geometric
formula. However, this was found to be inappropriate due to the internal reverberations
generated by the manikin head. We thus used the classification method describe in [4].
The peak location data was modelled using a separate histogram for each direction " . For
a given direction
%& , all the training measurements for which
%& are used to
!
, ! *1 / /
3 6
build a single histogram: , ! 9 " / 4
where ;< is 1 if
5 798:*
%
; is true and otherwise, is the size of the bin in the histogram, = >?; =
, and 9 9
is the number of bins. An estimate of the probability density function was taken to be
A
! B !
, ! *1 / / 9 " / 25
the normalized histogram step function: @ , 9 " / , C
%
" .
where
is the number of training measurements for which #
%
In order to classify new test data we simply compute the likelihood of the observed measurement under each distribution and choose the class attaining the maximal likelihood
(ML) score with respect to the distribution defined by the histogram,
A
D+EFHGID:J @
@
%
, 9" /
(7)
Multiclass Fisher discriminant: Generalising the Fisher discriminant for binary classification problems to multiclass settings, each class is modelled as a multivariate normal
distribution. To do so we divide the training set into subsets where the " th subset corresponds to measurements from azimuth in
%& . The density function of the " th class is
A
, 9" /
K
, :;
1
/ 9L
%
9+M
JNPO
1
, R
Q
%
/ S L %
,
Q
% U/ T
where S is the transpose of , '
1 is its dimensionality, Q denotes the mean of
%
the normal distribution, and L
the covariance matrix. Each mean and covariance matrix
%
are set to be the maximum likelihood estimates,
Q@ %
1
%W5 V 7
8* %
YX @L %
1
%
1 65 7
V
8* %
,
Q @ % /
S,
New test waveforms were then classified using the ML formula, Eq. 7.
Q@ %
/
The advantage of Fisher linear discriminant is that it is simple and easy to implement.
However, it degenerates if the training data is non-stationary, as often is the case in sound
localization problems due to effects such as moving objects. We therefore also designed,
implemented and tested a second discriminative methods based on the Perceptron.
Online Learning using Multiclass Perceptron with Kernels: Despite, or because of,
its age the Perceptron algorithm [9] is a simple and effective algorithm for classification.
We chose the Perceptron algorithm for its simplicity, adaptability, and ease in incorporating
Mercer kernels described below. The Perceptron algorithm is a conservative online algorithm: it receives an instance, outputs a prediction for the instance, and only in case it made
a prediction mistake the Perceptron update its classification rule which is a hyperplane.
Since our setting requires building a multiclass rule, we use the version described in [6]
which generalises the Perceptron to multiclass settings. We first describe the general form
of the algorithm and then discuss the modifications we performed in order to adapt it to the
sound localization problem.
To extend the Perceptron algorithm to multiclass problem we maintain hyperplanes (one
per class) denoted
87
. The algorithm works in an online fashion working on one
example at a time. On the ' th round, the algorithm gets a new instance and set the
predicted class to be the index of the hyperplane attaining the largest inner-product with
the input instance, @ D+EF GID:J
If the algorithm made a prediction error,
%
%
, it updates the set of hyperplanes.
In [6] a family of possible update
that is @
schemes was given. In this work we have used the so called uniform update which is
very simple to implement and also attained very good results. The uniform update moves
the hyperplane corresponding to the correct label 78 in the direction of and all the
hyperplanes whose inner-products were larger than 78 away from . Formally, let
" 9 " X % 7 8 We update the hyperplanes as follows,
"
(8)
"
= 8 =
%
%
C
then we keep
and if "
% intact. This update of the hyperplanes is performed
only on rounds on which there was a prediction error. Furthermore, on such rounds only
a subset of the vectors is updated and thus the algorithm is called ultraconservative. The
multiclass Perceptron algorithm is guaranteed to converge to a perfect classification rule
if the data can be classified perfectly by an unknown set of hyperplanes. When the data
cannot be classified perfectly then an alternative competitive analysis can be applied.
The problem with above algorithm is that it allows only linear classification rules. However, linear classifiers may not suffice to obtain in many applications, including the sound
localization application. We therefore incorporate kernels into the multiclass Perceptron.
A kernel is an inner-product operator B
where is the instance space (for
instance, PHAT waveforms). An explicit way to describe is via a mapping B
,' /
, ' / . Common kernels
from to an inner-products space such that , '
' /
,
/ , / .
are RBF kernels and polynomial kernels which take the form
Any learning algorithm that is based on inner-products with a weighted sum of vectors can
be converted to a kernel-based version by explicitly keeping the weighted combination of
vectors. In the case of the multiclass Perceptron we replace the update from Eq. 8 with a
?kernelized? version,
, /
"
(9)
"
= 8 = , /
%
%
Since we cannot compute , / explicitly we instead perform bookkeeping of the weights
associated with each , / and compute a inner-products using the kernel functions. For
, / with a new instance is
instance, the inner-product of a vector 3
, /
, /
3
3
,
/.
Algorithm
PHAT + Poly Kernels, D=5
PHAT + Fisher
PHAT + Peak-finding
Equalized CrossCor + Peak-finding
Err
Table 1: Summary of results of sound localization methods for a single segment.
In our experiments we found that polynomial kernel of degree yielded the best results.
The results are summarised in Table 1. We defer the discussion of the results to Sec. 5.
4 Multi-segment Classification
The accuracy of a single segment classifier is too low to make our approach practical. However, if the source of sound does not move for a period of time, we can accumulate evidence
from multiple segments in order to increase the accuracy. Due to the lack of space we only
outline the multi-segment classification procedure for the Fisher discriminant and compare
it to smoothing and averaging techniques used in the signal processing community.
In multi-segment classification we are given waveforms for which we assume that
6
. Each
the source angle did not change in this period, i.e., , - 1
78
6
small window was processed independently to give a feature vector . We then converted the waveform feature vector into a probability estimate for each discrete angle
A 6
direction, , 9
%& / using the Fisher discriminant. We next assumed that the probability estimates for consecutive windows are independent. This is of course a false assumption. However, we found that methods which compensate for the dependencies did
not yield substantial improvements. The probability density function of the entire winA
A @ , 6 9
%& /
and the ML estimation for @ is
" 6! *
dow is therefore @ , ! 9
%& / #
! $ GID:J&% '(*) A @ , ! 9
%& / We compared
@
the Maximum Likelihood decision un
der the independence assumption with the following commonly used signal processing
technique. We averaged the power spectrum and cross power spectrum of the different
windows and only then we proceeded to compute the generalized cross correlation wave +.- $ 0
form, ! " ( ,
is the average over the measurements
/ &
where +
6
3 6! * 1
in the same window, + 1
The averaged weight function for the
!
PHAT waveform is now ( ,.-0/ 129 +2- ,:-</ $ ,:-</ / 9
When using averaged power
spectrum it is also possible to define a smoothed coherent transform (SCOT) [1]. The
weight vector in this case is identical to the PHAT weight in the single segment case,
( ,.-0/ 1243 + - ,:-0/ $ ,:-0/ / + - ,:-0/ $ ,:-0/ / . Finally, we applied the classification
techniques for the single segments on the resulting (smoothed or averaged) waveform.
5 Experimental Results
In this section we report and discuss results of experiments that we performed with the
various learning algorithms for single-segments and multiple segments. Measurements
where made using the Sony ERS-210 AIBO robot. The sampling frequency was fixed to
- . )1 0 " 2
1 and the robot?s uni-directional microphone without automatic level control
was used. The robot was laid on a concrete floor in a regular office room, the room reverberations was 50687:9 0<4 . A loudspeaker, playing speech data from multiple speakers,
was placed 1 ; in front of the robot and ; above its plane, the background noise
<>9@? ACB . A PC connected through a wireless link to the robot directed its head
was =
relative to the speaker. The location of the sound source was limited to be in front of the
head (
) at a fixed constant elevation and in jumps of 1 . Therefore,
78
the number of classes, , for training is 1 . An illustration of the system is given in Fig. 2.
Algorithm
Max. Likl. PHAT + Fisher
SCOT + Fisher
Smoothed PHAT + Fisher
Smoothed PHAT + Peak-finding
SCOT + Peak-finding
Err
Table 2: Summary of results of sound localization methods for multiple segments.
Further technical details can be obtained from http://udi.benreuven.com. (MATLAB is a
trademark of Mathworks, Inc. and AIBO is a trademark of Sony and its affiliates.) For each
head direction ? segments of data were collected. Each segment is )1 0; 84 long. The
segments were collected with a partial overlap of 1 ; 4 . For each direction, the measurements were divided into equal amounts of train and test measurements. The total number of
segments per class, , is . Therefore, altogether there were
%
%
segments for training and the same amount for evaluation. An FFT of size 1 was used to
generate un-normalized cross-correlations, equalized cross-correlations, and PHAT waveforms. From the transformed waveforms 1 1 samples where taken (
in Eq. 6). Extrema
locations in histograms were found using 9 9 ? 1 bins.
We used two evaluation measures for comparing the different algorithms. The first, denoted
+ !3! , is the empirical classification error that counts the number of times the predicted
(discretized) angle was different than the true angle, that is, + !3! 3 +( *
@ .
The second evaluation measure, denoted , is the average absolute( difference
between
3 (
9 @ 9 . It should be kept
the predicted angle and the true angle,
*
( same direction set as the training
in mind that the test data was obtained from the
data. Therefore,
is an appropriate evaluation measure of the errors in our experi
mental setting. However, alternative evaluation methods should be devised for general
recordings when the test signal is not confined to a finite set of possible directions.
The accuracy results with respect to both measures on the test data for the various representations and algorithms are summarized in Table 1.
It is clear from the results that traditional methods which search for extrema in the waveforms
are inferior to the discriminative methods. As a
by-product we confirmed that equalized crosscorrelations is inferior to PHAT modelling for
high SNR with strong reverberations, similar results were reported in [11]. The two discrimiFigure 2: Acquisition system overview.
native methods achieve about the same results.
Using the Perceptron algorithm with degree
achieves the best results but the difference between the Perceptron and the multiclass Fisher
discriminant is not statistically significant. It is worth noting again that we also tested linear regression algorithms. Their performance turns to be inferior to the discriminative
multiclass approaches. A possible explanation is that the multiclass methods employ multiple hyperplanes and project each class onto a different hyperplane while linear regression
methods seek a single hyperplane onto which example are projected.
Although Fisher?s discriminant and the Perceptron algorithm exhibit practically the same
performance, they have different merits. While Fisher?s discriminant is very simple
to implement and is space efficient the Perceptron is capable to adapt quickly and
achieves high accuracy even with small amounts of training data. In Fig 3 we compare the error rates of Fisher?s discriminant and the Perceptron on subsets of the training data. The Perceptron clearly outperforms Fisher?s discriminant when the number of training examples is less than but once about examples are pro
vided the two algorithms are indistinguishable. This suggests that online algorithms
may be more suitable when the sound source is stationary only for short periods.
Last we compared multi-segment results. Multisegment classification was performed by taking
? 1 consecutive measurements over a window of 0; 4 during which the source location remained fix. In Table 2 we report classification results for the various multi-segment
techniques. (Since the Perceptron algorithm
used a very large number of kernels we did not
implement a multi-segment classification using
the Perceptron. We are currently conducting research on space-efficient kernel-based methods
for multi-segment classification.) Here again,
the best performing method is Fisher?s discrimFigure 3: Error rates of Fisher?s disinant that combines the scores directly without
criminant and the Perceptron for variaveraging and smoothing leads the pack. The reous training sizes.
sulting prediction accuracy of Fisher?s discriminant is good enough to make the solution practical so long as the sound source is fixed and the recording conditions do not change.
48
Perceptron
Fisher
47
46
Error Rate
45
44
43
42
41
40
39
1000
2000
3000
4000
5000
6000
Number of Examples
6 Discussion
We have demonstrated that by using discriminative methods highly accurate sound localization is achievable on a small commercial robot equipped with a binaural hearing that
are placed inside a manikin head. We have confirmed that PHAT is superior to plain crosscorrelation. For classification using multiple segments classifying the entire PHAT waveform gave better results than various techniques that smooth the power spectrum over the
segments. Our current research is focused on efficient discriminative methods for sound
localization in changing environments.
References
[1] C. H. Knapp and G. C. Carter. The generalized correlation method for estimation of time delay.
IEEE Transactions on ASSP, 24(4):320-327,1976.
[2] M. Omologo and P. Svaizer. Acoustic event localization using a crosspowerspectrum phase based
technique. Proceedings of ICASSP1994, Adelaide, Australia, 1994.
[3] T. Gustafsson and B.D. Rao. Source Localization in Reverberant Environments: Statistical Analysis. Submitted to IEEE Trans. on Speech and Audio Processing, 2000.
[4] N. Strobel and R. Rabenstein. Classification of Time Delay Estimates for Robust Speaker Localization ICASSP, Phoenix, USA, March 1999.
[5] J. Benesty Adaptive eigenvalue decomposition algorithm for passive acoustic source localization
J. Acoust. Soc. Am. 107 (1), January 2000
[6] K. Crammer and Y. Singer. Ultraconservative online algorithms for multiclass problems. In Proc.
of the 14th Annual Conf. on Computational Learning Theory, 2001.
[7] R. O. Duda, P. E. Hart. Pattern Classification. Wiley, 1973.
[8] B. Porat. A course in Digital Signal Processing. Wiley, 1997.
[9] F. Rosenblatt. The Perceptron: A probabilistic model for information storage and organization
in the brain. Psychological Review, 65:386?407, 1958.
[10] B. Widrow and M. E. Hoff. Adaptive switching circuits. 1960 IRE WESCON Convention
Record, pages 96?104, 1960.
[11] P. Aarabi, A. Mahdavi. The Relation Between Speech Segment Selectivity and Time-Delay
Estimation Accuracy. In Proc. of IEEE Conf. on Acoustics Speech and Signal Processing, 2002.
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1,265 | 2,152 | Interpreting Neural Response Variability as
Monte Carlo Sampling of the Posterior
Patrik O. Hoyer and Aapo Hyv?arinen
Neural Networks Research Centre
Helsinki University of Technology
P.O. Box 9800, FIN-02015 HUT, Finland
http://www.cis.hut.fi/phoyer/
[email protected]
Abstract
The responses of cortical sensory neurons are notoriously variable, with
the number of spikes evoked by identical stimuli varying significantly
from trial to trial. This variability is most often interpreted as ?noise?,
purely detrimental to the sensory system. In this paper, we propose an alternative view in which the variability is related to the uncertainty, about
world parameters, which is inherent in the sensory stimulus. Specifically, the responses of a population of neurons are interpreted as stochastic samples from the posterior distribution in a latent variable model. In
addition to giving theoretical arguments supporting such a representational scheme, we provide simulations suggesting how some aspects of
response variability might be understood in this framework.
1 Introduction
During the past half century, a wealth of data has been collected on the response properties
of cortical sensory neurons. The majority of this research has focused on how the mean
firing rates of individual neurons depend on the sensory stimulus. Similarly, mathematical
models have mainly focused on describing how the mean firing rate could be computed
from the input. One aspect which this research does not address is the high variability of
cortical neural responses. The trial-to-trial variation in responses to identical stimuli are
significant [1, 2], and several trials are typically required to get an adequate estimate of the
mean firing rate.
The standard interpretation is that this variability reflects ?noise? which limits the accuracy
of the sensory system [2, 3]. In the standard model, the firing rate is given by
where
rate
stimulus
noise
(1)
is the ?tuning function? of the cell in question. Here, the magnitude of the noise
may depend on the stimulus. Experimental results [1, 2] seem to suggest that the amount of
variability depends only on the mean firing rate, i.e. stimulus , and not on the particular
Current address: 4 Washington Place, Rm 809, New York, NY 10003, USA
stimulus that evoked it. Specifically, spike count variances tend to grow in proportion to
spike count means [1, 2]. This has been taken as evidence for something like a Poisson
process for neural firing.
This standard view is not completely satisfactory. First, the exquisite sensitivity and the
reliability of many peripheral neurons (see, e.g. [3]) show that neurons in themselves need
not be very unreliable. In vitro experiments [4] also suggest that the large variability does
not have its origin in the neurons themselves, but is a property of intact cortical circuits.
One is thus tempted to point at synaptic ?background? activity as the culprit, attributing the
variability of individual neurons to variable inputs. This seems reasonable, but it is not
quite clear why such modulation of firing should be considered meaningless noise rather
than reflecting complex neural computations.
Second, the above model does a poor job of explaining neural responses in the phenomenon
known as ?visual competition?: When viewing ambiguous (bistable) figures, perception,
and the responses of many neurons with it, oscillates between two distinct states (for a
review, see [5]). In other words, a single stimulus can yield two very different firing rates
in a single neuron depending on how the stimulus is interpreted. In the above model, this
means that either (a) the noise term needs to have a bimodal distribution, or (b) we are
forced to accept the fact that neurons can be tuned to stimulus interpretations, rather than
stimuli themselves. The former solution is clearly unattractive. The latter seems sensible,
but we have then simply transformed the problem of oscillating firing rates into a problem
of oscillating interpretations: Why should there be variability (over time, and over trials) in
the interpretation of a stimulus?
What would be highly desirable is a theoretical framework in which the variability of responses could be shown to have a specific purpose. One suggestion [6] is that variability
could improve the signal to noise ratio through a phenomenon known as ?stochastic resonance?. Another recent suggestion is that variability contributes to the contrast invariance
of visual neurons [7].
In this paper, we will propose an alternative explanation for the variability of neural responses. This hypothesis attempts to account for both aspects of variability described
above: the Poisson-like ?noise? and the oscillatory responses to ambiguous stimuli. Our
suggestion is based on the idea that cortical circuits implement Bayesian inference in latent variable models [8, 9, 10]. Specifically, we propose that neural firing rates might be
viewed as representing Monte Carlo samples from the posterior distribution over the latent
variables, given the observed input. In this view, the response variability is related to the
uncertainty, about world parameters, which is inherent in any stimulus. This representation would allow not only the coding of parameter values but also of their uncertainties.
The latter could be accomplished by pooling responses over time, or over a population of
redundant cells.
Our proposal has a direct connection to Monte Carlo methods widely used in engineering.
These methods use built-in randomness to solve difficult problems that cannot be solved
analytically. In particular, such methods are one of the main options for performing approximate inference in Bayesian networks [11]. With that in mind, it is perhaps even a bit
surprising that Monte Carlo sampling has not, to our knowledge, previously been suggested
as an explanation for the randomness of neural responses.
Although the approach proposed is not specific to sensory modality, we will here, for concreteness, exclusively concentrate on vision. We shall start by, in the next section, reviewing the basic probabilistic approach to vision. Then we will move on to further explain the
proposal of this contribution.
2 The latent variable approach to vision
2.1 Bayesian models of high-level vision
Recently, a growing number of researchers have argued for a probabilistic approach to
vision, in which the functioning of the visual system is interpreted as performing Bayesian
inference in latent variable models, see e.g. [8, 9, 10]. The basic idea is that the visual
input is seen as the observed data in a probabilistic generative model. The goal of vision
is to estimate the latent (i.e. unobserved or hidden) variables that caused the given sensory
stimulation.
In this framework, there are a number of world parameters that contribute to the observed
data. These could be, for example, object identities, dimensions and locations, surface
properties, lighting direction, and so forth. These parameters are not directly available to
the sensory system, but must be estimated from the effects that they have on the images
projected onto the retinas. Collecting all the unknown world variables into the vector
and all sensory data into the vector , the probability that a given set of world parameters
caused a given sensory stimulus is
(2)
describes
is known
is the prior probability of the set of world parameters , and
where
how sensory data is generated from the world parameters. The distribution
as the posterior distribution.
A specific perceptual task then consists of estimating some subset of the world variables,
given the observed data [10]. In face recognition, for example, one wants to know the
identity of a person but one does not care about the specific viewpoint or the direction of
lighting. Note, however, that sometimes one might specifically want to estimate viewpoint
or lighting, disregarding identity, so one cannot just automatically throw out that information [10]. In a latent variable model, all relevant information is contained in the complete
posterior distribution identity viewpoint lighting sensory data . To estimate the identity
one must use the marginal posterior identity sensory data , obtained by integrating out
the viewpoint and lighting variables. Bayesian models of high-level vision model the visual
system as performing these types of computations, but typically do not specify how they
might be neurally implemented.
2.2 Neural network models of low-level vision
This probabilistic approach has not only been suggested as an abstract framework for vision, but in fact also as a model for interpreting actual neural firing patterns in the early
visual cortex [12, 13]. In this line of research, the hypothesis is that the activity of individual neurons can be associated with hidden state variables, and that the neural circuitry
implements probabilistic inference.1
The model of Olshausen and Field [12], known as sparse coding or independent component analysis (ICA) [14], depending on the viewpoint taken, is perhaps the most influential latent variable model of early visual processing to date. The hidden variables
are
independent and sparse, such as is given, for instance, by the double-sided exponential
distribution
. The observed data vector is then given by a
,
linear combination of the , plus additive isotropic Gaussian noise. That is,
1
Here, it must be stressed that in these low-level neural network models, the hidden variables that
the neurons represent are not what we would typically consider to be the ?causal? variables of a visual
scene. Rather, they are low-level visual features similar to the optimal stimuli of neurons in the early
visual cortex. The belief is that more complex hierarchical models will eventually change this.
where is a matrix of
model parameters (weights), and is Gaussian with zero mean and
covariance matrix
.
How does this abstract probabilistic model relate to neural processing? Olshausen and
Field showed that when the model parameters are estimated (learned) from natural image
data, the basis vectors (columns of ) come to resemble V1 simple cell receptive fields.
Moreover, the latent variables relate to the activities of the corresponding cells. Specifically, Olshausen and Field suggested [12] that the firing rates of the neurons correspond
to
the maximum a posteriori (MAP) estimate of the latent variables, given the image input:
.
An important problem with this kind of a MAP representation is that it attempts to represent a complex posterior distribution using only a single point (at the maximum). Such a
representation cannot adequately represent multimodal posterior distributions, nor does it
provide any way of coding the uncertainty of the value (the width of the peak). Many other
proposed neural representations of probabilities face similar problems [11] (however, see
[15] for a recent interesting approach to representing distributions). Indeed, it has been said
[10, 16] that how probabilities actually are represented in the brain is one of the most important unanswered questions in the probabilistic approach to perception. In the next section
we suggest an answer based on the idea that probability distributions might be represented
using response variability.
3 Neural responses as samples from the posterior distribution?
As discussed in the previous section, the distribution of primary interest to a sensory system is the posterior distribution over world parameters. In all but absolutely trivial models,
computing and representing such a distribution requires approximative methods, of which
one major option is Monte Carlo methods. These generate stochastic samples from a given
distribution, without explicitly calculating it, and such samples can then be used to approximately represent or perform computations on that distribution [11].
Could the brain use a Monte Carlo approach to perform Bayesian inference? If neural
firing rates are used (even indirectly) to represent continuous-valued latent variables, one
possibility would be for firing rate variability to represent a probability distribution over
these variables. Here, there are two main possibilities:
(a) Variability over time. A single neuron could represent a continuous distribution if
its firing rate fluctuated over time in accordance with the distribution to be represented. At each instant in time, the instantaneous firing rate would be a random
sample from the distribution to be represented.
(b) Variability over neurons. A distribution could be instantaneously represented if
the firing rate of each neuron in a pool of identical cells was independently and
randomly drawn from the distribution to be represented.
Note that these are not exclusive, both types of variability could potentially coexist. Also
note that both cases lead to trial-to-trial variability, as all samples are assumed independent.
Both possibilities have their advantages. The first option is much more efficient in terms
of the number of cells required, which is particularly important for representing highdimensional distributions. In this case, dependencies between variables can naturally be
represented as temporal correlations between neurons representing different parameters.
This is not nearly as straightforward for case (b). On the other hand, in terms of processing
speed, this latter option is clearly preferred to the former. Any decisions should optimally
be based on the whole posterior distribution, and in case (a) this would require collecting
samples over an extended period of time.
10
10
10
10
1
1
1
1
0.1
0.1
1
10
0.1
0.1
1
10
0.1
0.1
1
10
0.1
0.1
1
10
Figure 1: Variance of response versus mean response, on log-log axes, for 4 representative
model neurons. Each dot gives the mean (horizontal axis) and variance (vertical axis) of
the response of the model neuron in question to one particular stimulus. Note that the scale
of responses is completely arbitrary.
We will now explain how both aspects of response variability described in the introduction
can be understood in this framework. First, we will show how a simple mean-variance relationship can arise through sampling in the independent component analysis model. Then,
we will consider how the variability associated with the phenomenon of visual competion
can be interpreted using sampling.
3.1 Example 1: Posterior sampling in ICA
Here, we sample the posterior distribution in the ICA model of natural images, and show
how this might relate to the conspicious variance-mean relation of neural response variability. First, we used standard ICA methods [17] to estimate a complete basis for the
-pixel natural image patches. Motivated by
40-dimensional principal subspace of
the non-negativity of neural firing rates we modified the model to assume single-sided exponential priors
[18], and augmented the basis so that a pair of neurons
coded separately for the positive and negative parts of each original independent component. We then took 50 random natural image patches and sampled the posterior distributions for all 50 patches , taking a total of 1000 samples in each case. 2
From the 1000 collected samples, we calculated the mean and variance of the response of
each neuron to each stimulus separately. We then plotted the variance against the mean
independently for each neuron in log-log coordinates. Figure 1 shows the plots from 4
randomly selected neurons. The crucial thing to note is that, as for real neurons [1], the
variance of the response is systematically related to the mean response, and does not seem
to depend on the particular stimulus used to elicit a given mean response. This feature of
neural variability is perhaps the single most important reason to believe that the variability
is meaningless noise inherent in neural firing; yet we have shown that something like this
might arise through sampling in a simple probabilistic model.
Following [1, 2], we fitted lines to the plots, modeling the variance as var
mean .
Over the whole population (80 model neurons), the mean values of and were
and
, with population standard deviations
and (respectively). Although these
values do not actually match those obtained from physiology (most reports give values of
between 1 and 2, and close to 1, see [1, 2]), this is to be expected. First, the values of these
parameters probably depend on the specifics of the ICA model, such as its dimensionality
and the noise level; we did not optimize these to attempt to fit physiology. Second, and
more importantly, we do not believe that ICA is an exact model of V1 function. Rather, the
visual cortex would be expected to employ a much more complicated, hierarchical, image
2
This was accomplished using a Markov Chain Monte Carlo method, as described in the Appendix. However, the technical details of this method are not very relevant to this argument.
model. Thus, our main goal was not to show that the particular parameters of the variancemean relation could be explained in this framework, but rather the surprising fact that such
a simple relation might arise as a result of posterior sampling in a latent variable model.
3.2 Example 2: Visual competition as sampling
As described in the introduction, in addition to the mean-variance relationship observed
throughout the visual cortex, a second sort of variability is that observed in visual competition. This phenomenon arises when viewing a bistable figure, such as the famous Necker
cube or Rubin?s vase/face figure. These figures each have two interpretations (explanations) that both cannot reasonably explain the image simultaneously. In a latent variable
image model, this corresponds to the case of a bimodal posterior distribution.
When such figures are viewed, the perception oscillates between the two interpretations (for
a review of this phenomenon, see [5]). This corresponds to jumping from mode to mode
in the posterior distribution. This can directly be interpreted as sampling of the posterior.
When the stimulus is modified so that one interpretations is slightly more natural than the
other one, the former is dominant for a relatively longer period compared with the latter
(again, see [5]), just as proper sampling takes relatively more samples from the mode which
has larger probability mass. Although the above might be considered purely ?perceptual?
sampling, animal studies indicate that especially in higher-level visual areas many neurons
modulate their responses in sync with the animal?s perceptions [5, 19]. This link proves
that some form of sampling is clearly taking place on the level of neural firing rates as well.
Note that this phenomenon might be considered as evidence for sampling scheme (a) and
against (b). If we instantaneously could represent whole distributions, we should be able to
keep both interpretations in mind simultaneously. This is in fact (weak) evidence against
any scheme of representing whole distributions instantaneously, by the same logic.
4 Conclusions
One of the key unanswered questions in theoretical neuroscience seems to be: How are
probabilities represented by the brain? In this paper, we have proposed that probability distributions might be represented using response variability. If true, this would also present
a functional explanation for the significant amount of cortical neural ?noise? observed. Although it is clear that the variability degrades performance on many perceptual tasks of the
laboratory, it might well be that it plays an important function in everyday sensory tasks.
Our proposal would be one possible way in which it might do so.
Do actual neurons employ such a computational scheme? Although our arguments and
simulations suggest that it might be possible (and should be kept in mind), future research
will be needed to answer that question. As we see it, key experiments would compare
measured firing rate variability statistics (single unit variances, or perhaps two-unit covariances) to those predicted by latent variable models. Of particular interest are cases where
contextual information reduces the uncertainty inherent in a given stimulus; our hypothesis
predicts that in such cases neural variability is also reduced.
A final question concerns how neurons might actually implement Monte Carlo sampling
in practice. Because neurons cannot have global access to the activity of all other neurons
in the population, the only possibility seems to be something akin to Gibbs sampling [20].
Such a scheme might require only relatively local information and could thus conceivably
be implemented in actual neural networks.
Acknowledgements ? Thanks to Paul Hoyer, Jarmo Hurri, Bruno Olshausen, Liam Paninski, Phil Sallee, Eero Simoncelli, and Harri Valpola for discussions and comments.
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[16] H. B. Barlow. Redundancy reduction revisited. Network: Computation in Neural Systems,
12:241?253, 2001.
[17] A. Hyv?arinen. Fast and robust fixed-point algorithms for independent component analysis.
IEEE Trans. on Neural Networks, 10(3):626?634, 1999.
[18] P. O. Hoyer. Modeling receptive fields with non-negative sparse coding. In E. De Schutter,
editor, Computational Neuroscience: Trends in Research 2003. Elsevier, Amsterdam, 2003. In
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[19] N. K. Logothetis and J. D. Schall. Neuronal correlates of subjective visual perception. Science,
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of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6:721?741, 1984.
Appendix: MCMC sampling of the non-negative ICA posterior
The posterior probability of , upon observing , is given by
(3)
Taking the (natural) logarithm yields
(4)
where
is a vector of all ones. The crucial thing to note is that this function is quadratic in
. Thus, the posterior distribution has the form of a gaussian, except that of course it is only
defined for non-negative . Rejection sampling might look tempting, but unfortunately does
not work well in high dimensions. Thus, we will instead opt for a Markov Chain Monte
Carlo approach. Implementing Gibbs sampling [20] is quite straightforward. The posterior
distribution of
, given and all other hidden variables , is a one-dimensional density
that we will call cut-gaussian,
if
!"+$* #&%
%(
,
In this case, we have the following parameter values:
3
0
21
!
1
if .- if /
')(
!
1
(5)
and
54
(6)
Here, 1 denotes the 6 :th column of , and
3 denotes the current state vector but with
set to zero. Sampling from such a one-dimensional distribution is relatively simple. Just as
one can sample the corresponding (uncut) gaussian by taking uniformly distributed samples
and passing them through the inverse of the gaussian cumulative
on the interval
distribution function, the same can be done for a cut-gaussian distribution by constraining
the uniform sampling interval suitably.
Hence Gibbs sampling is feasible, but, as is well known, Gibbs sampling exhibits problems
when there are significant correlations between the sampled variables. Thus we choose to
use a sampling scheme based on a rotated co-ordinate system. The basic idea is to update
the state vector not in the directions of the component axes, as in standard Gibbs sampling,
but rather in the directions of the eigenvectors of . Thus we start by calculating
these eigenvectors, and cycle through them one at a time. Denoting the current unit-length
eigenvector to be updated 7 we have as a function of the step length ,
7
const
8
7
7
7
7
(7)
Again, note how this is a quadratic function of . Again, the non-negativity constraints
on require us to sample a cut-gaussian distribution. But this time there is an additional
complication: When the basis is overcomplete, some of the eigenvectors will be associated
with zero eigenvalues, and the logarithmic probability will be linear instead of quadratic.
Thus, in such a case we must sample a cut-exponential distribution,
:9
0=<
if
if
if
;
>/;
-
(8)
Like in the cut-gaussian case, this can be done by uniformly sampling the corresponding
interval and then applying the inverse of the exponential cumulative distribution function.
In summary: We start by calculating the eigensystem of the matrix , and set the
state vector to random non-negative values. Then we cycle through the eigenvectors
indefinitely, sampling from cut-gaussian or cut-exponential distributions depending on
the eigenvalue corresponding to the current eigenvector 7 , and updating the state vector
to
7 . MATLAB code performing and verifying this sampling is available at:
http://www.cis.hut.fi/phoyer/code/samplingpack.tar.gz
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1,266 | 2,153 | Prediction and Semantic Association
Thomas L. Griffiths & Mark Steyvers
Department of Psychology
Stanford University, Stanford, CA 94305-2130
{gruffydd,msteyver}@psych.stanford.edu
Abstract
We explore the consequences of viewing semantic association as
the result of attempting to predict the concepts likely to arise in a
particular context. We argue that the success of existing accounts
of semantic representation comes as a result of indirectly addressing
this problem, and show that a closer correspondence to human data
can be obtained by taking a probabilistic approach that explicitly
models the generative structure of language.
1
Introduction
Many cognitive capacities, such as memory and categorization, can be analyzed as
systems for efficiently predicting aspects of an organism's environment [1]. Previously, such analyses have been concerned with memory for facts or the properties
of objects, where the prediction task involves identifying when those facts might
be needed again, or what properties novel objects might possess. However, one of
the most challenging tasks people face is linguistic communication. Engaging in
conversation or reading a passage of text requires retrieval of a variety of concepts
from memory in response to a stream of information. This retrieval task can be
facilitated by predicting which concepts are likely to be needed from their context,
having efficiently abstracted and stored the cues that support these predictions.
In this paper, we examine how understanding the problem of predicting words
from their context can provide insight into human semantic association, exploring
the hypothesis that the association between words is at least partially affected
by their statistical relationships. Several researchers have argued that semantic
association can be captured using high-dimensional spatial representations , with
the most prominent such approach being Latent Semantic Analysis (LSA) [5]. We
will describe this procedure, which indirectly addresses the prediction problem. We
will then suggest an alternative approach which explicitly models the way language
is generated and show that this approach provides a better account of human word
association data than LSA, although the two approaches are closely related. The
great promise of this approach is that it illustrates how we might begin to relax some
of the strong assumptions about language made by many corpus-based methods.
We will provide an example of this, showing results from a generative model that
incorporates both sequential and contextual information.
2
Latent Semantic Analysis
Latent Semantic Analysis addresses the prediction problem by capturing similarity
in word usage: seeing a word suggests that we should expect to see other words
with similar usage patterns. Given a corpus containing W words and D documents,
the input to LSA is a W x D word-document co-occurrence matrix F in which fwd
corresponds to the frequency with which word w occurred in document d. This
matrix is transformed to a matrix G via some function involving the term frequency
fwd and its frequency across documents fw .. Many applications of LSA in cognitive
science use the transformation
gwd = IOg{fwd
+ 1}(1 -
Hw)
H
- _
2:D_
Wlog{W}
d-l f w
f w.
logD
w -
'
(1)
where Hw is the normalized entropy of the distribution over documents for each
word. Singular value decomposition (SVD) is applied to G to extract a lower
dimensional linear subspace that captures much of the variation in usage across
words. The output of LSA is a vector for each word, locating it in the derived
subspace. The association between two words is typically assessed using the cosine of
the angle between their vectors, a measure that appears to produce psychologically
accurate results on a variety of tasks [5] . For the tests presented in this paper,
we ran LSA on a subset of the TASA corpus, which contains excerpts from texts
encountered by children between first grade and the first year of college. Our subset
used all D = 37651 documents, and the W = 26414 words that occurred at least
ten times in the whole corpus, with stop words removed. From this we extracted a
500 dimensional representation, which we will use throughout the paper. 1
3
The topic model
Latent Semantic Analysis gives results that seem consistent with human judgments
and extracts information relevant to predicting words from their contexts, although
it was not explicitly designed with prediction in mind. This relationship suggests
that a closer correspondence to human data might be obtained by directly attempting to solve the prediction task. In this section, we outline an alternative approach
that involves learning a probabilistic model of the way language is generated. One
generative model that has been used to outperform LSA on information retrieval
tasks views documents as being composed of sets of topics [2,4]. If we assume that
the words that occur in different documents are drawn from T topics, where each
topic is a probability distribution over words, then we can model the distribution
over words in anyone document as a mixture of those topics
T
P(Wi)
= LP(Wil zi =j)P(Zi
=j)
(2)
j=l
where Zi is a latent variable indicating the topic from which the ith word was drawn
and P(wilzi = j) is the probability of the ith word under the jth topic. The words
likely to be used in a new context can be determined by estimating the distribution
over topics for that context, corresponding to P(Zi).
Intuitively, P(wlz = j) indicates which words are important to a topic, while P(z)
is the prevalence of those topics within a document. For example, imagine a world
where the only topics of conversation are love and research. We could then express
IThe dimensionality of the representation is an important parameter for both models in
this paper. LSA performed best on the word association task with around 500 dimensions,
so we used the same dimensionality for the topic model.
the probability distribution over words with two topics, one relating to love and the
other to research. The content of the topics would be reflected in P(wlz = j): the
love topic would give high probability to words like JOY, PLEASURE, or HEART, while
the research topic would give high probability to words like SCIENCE, MATHEMATICS,
or EXPERIMENT. Whether a particular conversation concerns love, research, or
the love of research would depend upon its distribution over topics, P(z), which
determines how these topics are mixed together in forming documents.
Having defined a generative model, learning topics becomes a statistical problem.
The data consist of words w = {Wl' ... , w n }, where each Wi belongs to some document di , as in a word-document co-occurrence matrix. For each document we
have a multinomial distribution over the T topics, with parameters ()(d), so for a
word in document d, P(Zi = j) = ();d;). The jth topic is represented by a multinomial distribution over the W words in the vocabulary, with parameters 1/i), so
P(wilzi = j) = 1>W. To make predictions about new documents, we need to assume
a prior distribution on the parameters (). Existing parameter estimation algorithms
make different assumptions about (), with varying results [2,4]. Here, we present a
novel approach to inference in this model, using Markov chain Monte Carlo with a
symmetric Dirichlet(a) prior on ()(di) for all documents and a symmetric Dirichlet(,B)
prior on 1>(j) for all topics. In this approach we do not need to explicitly represent
the model parameters: we can integrate out () and 1>, defining the model simply in
terms of the assignments of words to topics indicated by the Zi'
Markov chain Monte Carlo is a procedure for obtaining samples from complicated
probability distributions, allowing a Markov chain to converge to the taq~et distribution and then drawing samples from the states of that chain (see [3]). We
use Gibbs sampling, where each state is an assignment of values to the variables
being sampled, and the next state is reached by sequentially sampling all variables
from their distribution when conditioned on the current values of all other variables
and the data. We will sample only the assignments of words to topics, Zi. The
conditional posterior distribution for Zi is given by
'1
)
P( Zi=)Zi ,wex
where
Z- i
is the assignment of all
Zk
+ (3 n(di) + a
-',}
-',}
(d ' )
n_i,j + W (3 n_i,. + Ta
n eW;)
(.)
such that k
f:.
(3)
i, and n~~:j is the number
of words assigned to topic j that are the same as w, n~L is the total number of
words assigned to topic j, n~J,j is the number of words from document d assigned
to topic j, and n~J. is the total number of words in document d, all not counting
the assignment of the current word Wi. a,,B are free parameters that determine how
heavily these distributions are smoothed.
We applied this algorithm to our subset of the TASA corpus, which contains n =
5628867 word tokens. Setting a = 0.1,,B = 0.01 we obtained 100 samples of 500
topics, with 10 samples from each of 10 runs with a burn-in of 1000 iterations and
a lag of 100 iterations between samples. 2 Each sample consists of an assignment of
every word token to a topic, giving a value to each Zi. A subset of the 500 topics
found in a single sample are shown in Table 1. For each sample we can compute
2Random numbers were generated with the Mersenne Twister, which has an extremely
deep period [6]. For each run, the initial state of the Markov chain was found using an
on-line version of Equation 3.
FEEL
FEELINGS
FEELING
ANGRY
WAY
THINK
SHOW
FEELS
PEOPLE
FRIENDS
THINGS
MIGHT
HELP
HAPPY
FELT
LOVE
ANGER
BEING
WAYS
FEAR
MUSIC
BALL
GAME
TEAM
PLAY
DANCE
PLAYS
STAGE
PLAYED
BAND
AUDIENCE
MUSICAL
DANCING
RHYTHM
PLAYING
THEATER
DRUM
ACTORS
SHOW
BALLET
ACTOR
DRAMA
SONG
PLAY
BASEBALL
FOOTBALL
PLAYERS
GAMES
PLAYING
FIELD
PLAYED
PLAYER
COACH
BASKETBALL
SPORTS
HIT
BAT
TENNIS
TEAMS
SOCCER
SCIENCE
STUDY
SCIENTISTS
SCIENTIFIC
KNOWLEDGE
WORK
CHEMISTRY
RESEARCH
BIOLOGY
MATHEMATICS
LABORATORY
STUDYING
SCIENTIST
PHYSICS
FIELD
STUDIES
UNDERSTAND
STUDIED
SCIENCES
MANY
WORKERS
WORK
LABOR
JOBS
WORKING
WORKER
WAGES
FACTORY
JOB
WAGE
SKILLED
PAID
CONDITIONS
PAY
FORCE
MANY
HOURS
EMPLOYMENT
EMPLOYED
EMPLOYERS
FORCE
FORCES
MOT IO N
BODY
GRAVITY
MASS
PULL
NEWTON
OBJECT
LAW
DIRECTION
MOVING
REST
FALL
ACTING
MOMENTUM
DISTANCE
GRAVITATIONAL
PUSH
VELOCITY
Table 1: Each column shows the 20 most probable words in one of the 500 topics
obtained from a single sample. The organization of the columns and use of boldface
displays the way in which polysemy is captured by the model.
the posterior predictive distribution (and posterior mean for q/j)) :
P(wl z
4
= j, z, w) =
J
P(wl z
(.) ( Iz,
= j, ? J
0)
)P(? J
w) d?
( 0)
J
+ (3
+ W (3
n (W)
= _(;=,.J)_ _
nj
(4)
Predicting word association
We used both LSA and the topic model to predict the association between pairs
of words, comparing these results with human word association norms collected by
Nelson, McEvoy and Schreiber [7]. These word association norms were established
by presenting a large number of participants with a cue word and asking them to
name an associated word in response. A total of 4544 of the words in these norms
appear in the set of 26414 taken from the TASA corpus.
4.1
Latent Semantic Analysis
In LSA, the association between two words is usually measured using the cosine
of the angle between their vectors. We ordered the associates of each word in the
norms by their frequencies , making the first associate the word most commonly
given as a response to the cue. For example, the first associate of NEURON is BRAIN.
We evaluated the cosine between each word and the other 4543 words in the norms ,
and then computed the rank of the cosine of each of the first ten associates, or
all of the associates for words with less than ten. The results are shown in Figure
1. Small ranks indicate better performance, with a rank of one meaning that the
target word had the highest cosine. The median rank of the first associate was 32,
and LSA correctly predicted the first associate for 507 of the 4544 words.
4.2
The topic model
The probabilistic nature of the topic model makes it easy to predict the words likely
to occur in a particular context. If we have seen word WI in a document, then we
can determine the probability that word W2 occurs in that document by computing
P( w2IwI). The generative model allows documents to contain multiple topics, which
450
400
1_
LSA - cosine
LSA - inner product
Topi c model
D
1
350
300
II
250
200
150
l;r
100
50
o
lin
2
3
4
5
6
7
8
9
10
Associate number
Figure 1: Performance of different methods of prediction on the word association
task. Error bars show one standard error, estimated with 1000 bootstrap samples.
is extremely important to capturing the complexity of large collections of words
and computing the probability of complete documents. However, when comparing
individual words it is more effective to assume that they both come from a single
topic. This assumption gives us
(5)
z
where we use Equation 4 for P(wlz) and P(z) is uniform, consistent with the symmetric prior on e, and the subscript in Pi (w2lwd indicates the restriction to a single
topic. This estimate can be computed for each sample separately, and an overall
estimate obtained by averaging over samples. We computed Pi (w2Iwi) for the 4544
words in the norms, and then assessed the rank of the associates in the resulting
distribution using the same procedure as for LSA. The results are shown in Figure
1. The median rank for the first associate was 32, with 585 of the 4544 first associates exactly correct. The probabilistic model performed better than LSA, with
the improved performance becoming more apparent for the later associates .
4.3
Discussion
The central problem in modeling semantic association is capturing the interaction
between word frequency and similarity of word usage. Word frequency is an important factor in a variety of cognitive tasks, and one reason for its importance is its
predictive utility. A higher observed frequency means that a word should be predicted to occur more often. However, this effect of frequency should be tempered by
the relationship between a word and its semantic context . The success of the topic
model is a consequence of naturally combining frequency information with semantic
similarity: when a word is very diagnostic of a small number of topics, semantic
context is used in prediction. Otherwise, word frequency plays a larger role.
The effect of word frequency in the topic model can be seen in the rank-order
correlation of the predicted ranks of the first associates with the ranks predicted
by word frequency alone , which is p = 0.49. In contrast, the cosine is used in LSA
because it explicitly removes the effect of word frequency, with the corresponding
correlation being p = -0.01. The cosine is purely a measure of semantic similarity,
which is useful in situations where word frequency is misleading, such as in tests of
English fluency or other linguistic tasks, but not necessarily consistent with human
performance. This measure was based in the origins of LSA in information retrieval ,
but other measures that do incorporate word frequency have been used for modeling
psychological data. We consider one such measure in the next section.
5
Relating LSA and the topic model
The decomposition of a word-document co-occurrence matrix provided by the topic
model can be written in a matrix form similar to that of LSA. Given a worddocument co-occurrence matrix F, we can convert the columns into empirical estimates of the distribution over words in each document by dividing each column
by its sum. Calling this matrix P, the topic model approximates it with the nonnegative matrix factorization P ~ ?O, where column j of ? gives 4/j) , and column d
of 0 gives ()(d). The inner product matrix ppT is proportional to the empirical estimate of the joint distribution over words P(WI' W2)' We can write ppT ~ ?OOT ?T,
corresponding to P(WI ,W2) = L z"Z 2 P(wIl zdP(W2Iz2)P(ZI,Z2) , with OOT an empirical estimate of P(ZI , Z2)' The theoretical distribution for P(ZI, Z2) is proportional to 1+ 0::, where I is the identity matrix, so OOT should be close to diagonal.
The single topic assumption removes the off-diagonal elements, replacing OOT with
I to give PI (Wl ' W2) ex: ??T.
By comparison, LSA transforms F to a matrix G via Equation 1, then the SVD
gives G ~ UDV T for some low-rank diagonal D. The locations of the words along
the extracted dimensions are X = UD. If the column sums do not vary extensively,
the empirical estimate of the joint distribution over words specified by the entries in
G will be approximately P(WI,W2) ex: GG T . The properties of the SVD guarantee
that XX T , the matrix of inner products among the word vectors , is the best lowrank approximation to GG T in terms of squared error. The transformations in
Equation 1 are intended to reduce the effects of word frequency in the resulting
representation, making XX T more similar to ??T.
We used the inner product between word vectors to predict the word association
norms, exactly as for the cosine. The results are shown in Figure 1. The inner
product initially shows worse performance than the cosine, with a median rank
of 34 for the first associate and 500 exactly correct, but performs better for later
associates. The rank-order correlation with the predictions of word frequency for
the first associate was p = 0.46, similar to that for the topic model. The rankorder correlation between the ranks given by the inner product and the topic model
was p = 0.81, while the cosine and the topic model correlate at p = 0.69. The
inner product and PI (w2lwd in the topic model seem to give quite similar results,
despite being obtained by very different procedures. This similarity is emphasized
by choosing to assess the models with separate ranks for each cue word, since this
measure does not discriminate between joint and conditional probabilities. While
the inner product is related to the joint probability of WI and W2, PI (w2lwd is a
conditional probability and thus allows reasonable comparisons of the probability
of W2 across choices of WI , as well as having properties like asymmetry that are
exhibited by word association.
"syntax"
HE
YOU
THEY
I
SHE
WE
IT
PEOPLE
EVERYONE
OTHERS
SCIENTISTS
SOMEONE
WHO
NOBODY
ONE
SOMETHING
ANYONE
EVERYBODY
SOME
THEN
ON
AT
INTO
FROM
WITH
THROUGH
OVER
AROUND
AGAINST
ACROSS
UPON
TOWARD
UNDER
ALONG
NEAR
BEHIND
OFF
ABOVE
DOWN
BEFORE
BE
MAKE
GET
HAVE
GO
TAKE
DO
FIND
USE
SEE
HELP
KEEP
GIVE
LOOK
COME
WORK
MOVE
LIVE
EAT
BECOME
"semantics"
SAID
ASKED
THOUGHT
TOLD
SAYS
MEANS
CALLED
CRIED
S HOWS
ANSWERED
TELLS
REPLIED
SHOUTED
EXPLAINED
LAUGHED
MEANT
WROTE
SHOWED
BELIEVED
WHISPERED
MAP
NORTH
EARTH
SOUTH
POLE
MAPS
EQUATOR
WEST
LINES
EAST
AUSTRALIA
GLOBE
POLES
HEMISPHERE
LATITUDE
PLACES
LAND
WORLD
COMPASS
CONTINE NTS
DOCTOR
PATIENT
HEALTH
HOSPITAL
MEDICAL
CARE
PATIENTS
NURSE
DOCTORS
MEDICINE
NURSING
TREATMENT
NURSES
PHYSICIAN
HOSPITALS
DR
S ICK
ASSISTANT
EMERGENCY
PRACTICE
Table 2: Each column shows the 20 most probable words in one of the 48 "syntactic"
states of the hidden Markov model (four columns on the left) or one of the 150
"semantic" topics (two columns on the right) obtained from a single sample.
6
Exploring more complex generative models
The topic model, which explicitly addresses the problem of predicting words from
their contexts, seems to show a closer correspondence to human word association
than LSA. A major consequence of this analysis is the possibility that we may be
able to gain insight into some of the associative aspects of human semantic memory
by exploring statistical solutions to this prediction problem. In particular, it may
be possible to develop more sophisticated generative models of language that can
capture some of the important linguistic distinctions that influence our processing
of words. The close relationship between LSA and the topic model makes the latter
a good starting point for an exploration of semantic association, but perhaps the
greatest potential of the statistical approach is that it illustrates how we might go
about relaxing some of the strong assumptions made by both of these models.
One such assumption is the treatment of a document as a "bag of words" , in which
sequential information is irrelevant. Semantic information is likely to influence only
a small subset of the words used in a particular context, with the majority of the
words playing functional syntactic roles that are consistet across contexts. Syntax is
just as important as semantics for predicting words, and may be an effective means
of deciding if a word is context-dependent. In a preliminary exploration of the
consequences of combining syntax and semantics in a generative model for language,
we applied a simple model combining the syntactic structure of a hidden Markov
model (HMM) with the semantic structure of the topic model. Specifically, we used
a third-order HMM with 50 states in which one state marked the start or end of
a sentence, 48 states each emitted words from a different multinomial distribution,
and one state emitted words from a document-dependent multinomial distribution
corresponding to the topic model with T = 150. We estimated parameters for this
model using Gibbs sampling, integrating out the parameters for both the HMM and
the topic model and sampling a state and a topic for each of the 11821091 word
tokens in the corpus. 3 Some of the state and topic distributions from a single sample
after 1000 iterations are shown in Table 2. The states of the HMM accurately picked
out many of the functional classes of English syntax, while the state corresponding
to the topic model was used to capture the context-specific distributions over nouns.
3This larger number is a result of including low frequency and stop words.
Combining the topic model with the HMM seems to have advantages for both: no
function words are absorbed into the topics, and the HMM does not need to deal
with the context-specific variation in nouns. The model also seems to do a good job
of generating topic-specific text - we can clamp the distribution over topics to pick
out those of interest, and then use the model to generate phrases. For example, we
can generate phrases on the topics of research ( "the chief wicked selection of research
in the big months" , "astronomy peered upon your scientist's door", or "anatomy
established with principles expected in biology") , language ("he expressly wanted
that better vowel"), and the law ("but the crime had been severely polite and
confused" , or "custody on enforcement rights is plentiful"). While these phrases
are somewhat nonsensical , they are certainly topical.
7
Conclusion
Viewing memory and categorization as systems involved in the efficient prediction
of an organism's environment can provide insight into these cognitive capacities.
Likewise, it is possible to learn about human semantic association by considering
the problem of predicting words from their contexts. Latent Semantic Analysis
addresses this problem, and provides a good account of human semantic association.
Here, we have shown that a closer correspondence to human data can be obtained
by taking a probabilistic approach that explicitly models the generative structure
of language, consistent with the hypothesis that the association between words
reflects their probabilistic relationships. The great promise of this approach is the
potential to explore how more sophisticated statistical models of language, such as
those incorporating both syntax and semantics, might help us understand cognition.
Acknowledgments
This work was generously supported by the NTT Communications Sciences Laboratories.
We used Mersenne Twister code written by Shawn Cokus, and are grateful to Touchstone
Applied Science Associates for making available the TASA corpus, and to Josh Tenenbaum
for extensive discussions on this topic.
References
[1] J. R. Anderson. The Adaptive Character of Thought. Erlbaum, Hillsdale, NJ, 1990.
[2] D . M. Blei, A. Y. Ng, and M. 1. Jordan . Latent Dirichlet allocation. In T. G. Dietterich,
S. Becker, and Z. Ghahramani, eds, Advances in Neural Information Processing Systems
14, 2002.
[3] W . R. Gilks, S. Richardson, and D. J . Spiegelhalter, eds. Markov Chain Monte Carlo
in Practice. Chapman and Hall, Suffolk, 1996.
[4] T . Hofmann. Probabilistic Latent Semantic Indexing. In Proceedings of the TwentySecond Annual International SIGIR Conference, 1999.
[5] T. K. Landauer and S. T. Dumais. A solution to Plato's problem: The Latent Semantic
Analysis theory of acquisition, induction, and representation of knowledge. Psychological
Review, 104:211- 240, 1997.
[6] M. Matsumoto and T . Nishimura. Mersenne twister: A 623-dimensionally equidistributed uniform pseudorandom number generator. ACM Transactions on Modeling and
Computer Simulation, 8:3- 30, 1998.
[7] D. L. Nelson , C. L. McEvoy, and T. A. Schreiber. The University of South Florida
word association norms. http://www. usf. edu/FreeAssociation, 1999.
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1,267 | 2,154 | Margin-Based Algorithms
for Information Filtering
Nicol`o Cesa-Bianchi
DTI, University of Milan
via Bramante 65
26013 Crema, Italy
[email protected]
Alex Conconi
DTI, University of Milan
via Bramante 65
26013 Crema, Italy
[email protected]
Claudio Gentile
CRII, Universit`a dell?Insubria
Via Ravasi, 2
21100 Varese, Italy
[email protected]
Abstract
In this work, we study an information filtering model where the relevance
labels associated to a sequence of feature vectors are realizations of an
unknown probabilistic linear function. Building on the analysis of a restricted version of our model, we derive a general filtering rule based on
the margin of a ridge regression estimator. While our rule may observe
the label of a vector only by classfying the vector as relevant, experiments
on a real-world document filtering problem show that the performance
of our rule is close to that of the on-line classifier which is allowed to
observe all labels. These empirical results are complemented by a theoretical analysis where we consider a randomized variant of our rule and
prove that its expected number of mistakes is never much larger than that
of the optimal filtering rule which knows the hidden linear model.
1 Introduction
Systems able to filter out unwanted pieces of information are of crucial importance for
several applications. Consider a stream of discrete data that are individually labelled as
?relevant? or ?nonrelevant? according to some fixed relevance criterion; for instance, news
about a certain topic, emails that are not spam, or fraud cases from logged data of user
behavior. In all of these cases, a filter can be used to drop uninteresting parts of the stream,
forwarding to the user only those data which are likely to fulfil the relevance criterion.
From this point of view, the filter may be viewed as a simple on-line binary classifier.
However, unlike standard on-line pattern classification tasks, where the classifier observes
the correct label after each prediction, here the relevance of a data element is known only if
the filter decides to forward that data element to the user. This learning protocol with partial feedback is known as adaptive filtering in the Information Retrieval community (see,
e.g., [14]). We formalize the filtering problem as follows. Each element of an arbitrary
data sequence is characterized by a feature vector
and an associated relevance
label (say,
for relevant and
for nonrelevant). At each time
,
the filtering system observes the -th feature vector
and must decide whether or not to
forward it. If the data is forwarded, then its relevance label is revealed to the system,
The research was supported by the European Commission under the KerMIT Project No. IST2001-25431.
which may use this information to adapt the filtering criterion. If the data is not forwarded,
its relevance label remains hidden. We call
the -th instance of the data sequence and
the -th example. For simplicity, we assume
the pair
for all . There
are two kinds of errors the filtering system can make in judging the relevance of a feature
vector . We say that an example
and is classified
is a false positive if
as relevant by the system; similarly, we say that
is a false negative if
and
is classified as nonrelevant by the system. Although false negatives remain unknown,
the filtering system is scored according to the overall number of wrong relevance judgements it makes. That is, both false positives and false negatives are counted as mistakes.
In this paper, we study the filtering problem under the assumption that relevance judgements are generated using an unknown probabilistic linear function. We design filtering
rules that maintain a linear hypothesis and use the margin information to decide whether
to forward the next instance. Our performance measure is the regret; i.e., the number of
wrong judgements made by a filtering rule over and above those made by the rule knowing
the probabilistic function used to generate judgements. We show finite-time (nonasymptotical) bounds on the regret that hold for arbitrary sequences of instances. The only other
results of this kind we are aware of are those proven in [9] for the apple tasting model.
Since in the apple tasting model relevance judgements are chosen adversarially rather than
probabilistically, we cannot compare their bounds with ours. We report some preliminary
experimental results which might suggest the superiority of our methods as opposed to the
general transformations developed within the apple tasting framework. As a matter of fact,
these general transformations do not take margin information into account.
In Section 2, we introduce our probabilistic relevance model and make some preliminary
observations. In Section 3, we consider a restricted version of the model within which
we prove a regret bound for a simple filtering rule called SIMPLE - FIL. In Section 4, we
generalize this filtering rule and show its good performance on the Reuters Corpus Volume
1. The algorithm employed, which we call RIDGE - FIL, is a linear least squares algorithm
inspired by [2]. In that section we also prove, within the unrestricted probabilistic model,
a regret bound for the randomized variant P - RIDGE - FIL of the general filtering rule. Both
RIDGE - FIL and its randomized variant can be run with kernels [13] and adapted to the case
when the unknown linear function drifts with time.
2 Learning model, notational conventions and preliminaries
The relevance of
is given by a
(where
-valued random variable
means ?relevant?) such that there exists a fixed and unknown vector
,
,
. Hence
for which
for all
is relevant with probability
. The random variables
are assumed to be independent, whereas we do not make any assumption on the way the sequence !
! is
generated. In this model, we want to perform almost as well as the algorithm that knows
and forwards
if and only if "# . We consider linear-threshold filtering algorithms that predict the value of
through SGN $&%
is a
(' *) , where %
dynamically updated weight vector which might be intended as the current approximation
to , and ' is a suitable time-changing ?confidence? threshold. For any fixed sequence
of instances, we use +
to denote the margin
and ,+
to
!
!
denote the margin %
. We define the expected regret of the linear-threshold filtering algorithm at time as -.&
,+ /' 012 3-.4
+ 015 . We observe that in the conditional
-76 -probability space where ,+
8' is given we have
- 6 4
8'
;- 6 4
,+
;- 6 4
+
,+
('
092 8- 6 &
+
0:2 (- 6 4
92 8- 6 4
+
+
0:5
012<5 ,+
015?@ ,+
A'
8'
B+
=
+
C
=
ED +
>
D 5 ,+
8'
+
=
>
where we use to denote the Bernoulli random variable which is 1 if and only if predicate is true. Integrating over all possible values of ,+ 8' we obtain
-.&
('
,+
015 (-4
+
015
D+
=
D+
D-. ,+
D-.BD ,+
,+
=
=
8'
+
8'
A+
('
5
D@
+
(1)
D+
D
(2)
where the last inequality is Markov?s. These (in)equalities will be used in Sections 3 and 4.2
for the analysis of SIMPLE - FIL and P - RIDGE - FIL algorithms.
3 A simplified model
We start by analyzing a restricted model where each data element has the same unknown
probability of being relevant and we want to perform almost as well as the filtering rule
that consistently does the optimal action (i.e., always forwards if :
and never forwards otherwise). The analysis of this model is used in Section 4 to guide the design of
good filtering rules for the unrestricted model.
and let ,
be the sample average of
, where
is
Let
,
the number of forwarded data elements in the first time steps and , is the fraction of
true positives among the elements that have been forwarded. Obviously, the optimal
rule forwards if and only if
. Consider instead the empirical rule that forwards if
and only if ,
. This rule makes a mistake only when
!,
= . To make
the probability of this event go to zero with , it suffices that -BD ,
CD = D
CD
as , which can only happen if increases quickly enough with . Hence, data
should be forwarded (irrespective to the sign of the estimate ,
) also when the confidence
level for ,
gets too small with respect to . A problem in this argument is that large
deviation bounds require
for making -.D ,
CD D
CD small. But in
our case
is unknown. To fix this, we use the condition
,
. This looks
dangerous, as we use the empirical value of ,
to control the large deviations of ,
itself; however, we will show that this approach indeed works. An algorithm, which we
call SIMPLE - FIL, implementing the above line of reasoning takes the form of the following
'
, where '
"! # %$ . The
simple rule: forward if and only if ,
expected regret at time of SIMPLE - FIL is defined as the probability that SIMPLE - FIL makes
a mistake at time minus the probability that the optimal filtering rule makes a mistake,
that is -4
,
'
0 2
-.&
0 5 . The next result shows a logarithmic bound
on this regret.
of time
Theorem 1 The expected cumulative regret of SIMPLE - FIL after any number (
steps is at most & 2CD
CD '(! ) .
Proof sketch. We can bound the actual regret after
the definition of the filtering rule we have
time steps as follows. From (1) and
*
*
.-
,
,+
,+
, +
21 *,+ 435 % $
6"!
87 *, + 43
,
9 % $
6(!
875:
;6< =;>
*
-.&
4,
8'
/0:5
D CD
-
-
=
,
4,
8'
D CD
D CD
-4
=
015
A
-
!,
=
,
=
0/
Without loss of generality, assume
. We now bound =; < and 6
; >
, we have that ;6<
Since %$ = "! ,
implies that
%$
for some . Hence we can write
separately.
implies
% $
, +* 3 %$
6(!
9 %$ 7 ,+* * + 3 6
"! 7
* *% $
(! -
9/
, + +
* *%$ -
/ for
"!
,+ % $+
* * -
/
, + +
Applying Chernoff-Hoeffding [11] bounds to
, which is a sum of -valued
independent random variables, we obtain
=;<
,+ % $+
!
We now bound ;6> by adapting a technique from [8]. Let "#%$ '& "()$
*
+/.
& * + , )$ $ *- +. We have
*
*
, ,
;>
-
"
$
/
,+
,+ 10 " $ $ 3 2
*,+ 3 , ,
$ 9 % $
6(!
87
*
* % $
*
, + +54 -
" )$
/ , + 3 ,
$ %$
6(!
7
;76 ;78
Applying Chernoff-Hoeffding bounds again, we get =;96
, + (!
Finally, one can easily verify that ;:8
. Piecing everything together we get the desired
;
result.
; <
=
=
,
=
=
,
=
=
D =
D,
D CD
=
D,
D =
D CD
=
D,
CD>
D CD
,
=
!
-
2D ,
CD>
=
D CD
@
,
,
=
D,
CD>
D CD
,
,
,
=
D,
CD@
D CD
.
,
=
,
=
C
4 Linear least squares filtering
In order to generalize SIMPLE - FIL to the original (unrestricted) learning model described
in Section 2, we need a low-variance estimate of the target vector . Let
be the matrix
whose columns are the forwarded feature vectors after the first time steps and let
be the vector of corresponding observed relevance labels (the index will be momentarily
dropped). Note that
holds. Consider the least squares estimator
of , where
is the pseudo-inverse of
. For all belonging to the column
E
space of , this is an unbiased estimator of , that is B
/
To remove the assumption on , we make
full rank by adding
the identity . This also allows us to replace the pseudo-inverse with the standard inverse,
<
@<A< B%<A<
K
>= ?
<
E<A< B
<D<
<
GF @<D< !BC<H=JI
<D<
=
@<A< !BC<D=
E<A< B%< >=
RIDGE-FULL
RIDGE-FIL
FREQUENCY x 10
1
F-MEASURE
0.8
0.6
0.4
0.2
0
34 CATEGORIES
-measure for each one of the 34 filtering tasks. The -measure is defined by
, where is precision (fraction of relevant documents among the forwarded ones) and is recall
(fraction of forwarded documents among the relevant ones). In the plot, the filtering rule RIDGE - FIL
is compared with RIDGE - FULL which sees the correct label after each classification. While precision
and recall of RIDGE - FULL are balanced, RIDGE - FIL?s recall is higher than precision due to the need
of forwarding more documents than believed relevant. This in order to make the confidence of the
estimator converge to 1 fast enough. Note that, in some cases, this imbalance causes RIDGE - FIL to
achieve a slightly better -measure than RIDGE - FULL.
Figure 1:
@K
<D<
<D=
$
obtaining
, a ?sparse? version of the ridge regression estimator [12] (the
sparsity is due to the fact that we only store in the forwarded instances, i.e., those for
which we have a relevance labels). To estimate directly the margin , rather than ,
we further modify, along the lines of the techniques analyzed in [3, 6, 15], the sparse ridge
with the quantity %
, where
regression estimator. More precisely, we estimate
the % is defined by
<
K < %$ < % $ $ < %$ = %$
(3)
Using the Sherman-Morrison
can then write out the expectation of as
$
,formula,
we
F EI
,
which holds for all , , and all matrices
< %$ . Let % $ be the number of forwarded instances among % $ . In order to
generalize to the estimator (3) the analysis of
- , we need to find a large deviation
bound of the form
' , % $ ?
for all , , where
goes to zero ?sufficiently fast? as 4 . Though we have not been able to find such
%
4%
%
SIMPLE FIL
-8$BD %
A+
D
)
>
>
bounds, we report some experimental results showing that algorithms based on (3) and
inspired by the analysis of SIMPLE - FIL do exhibit a good empirical behavior on real-world
data. Moreover, in Section 4.2 we prove a bound (not based on the analysis of SIMPLE - FIL)
on the expected regret of a randomized variant of the algorithm used in the experiments.
For this variant we are able to prove a regret bound that scales essentially with the square
root of (to be contrasted with the logarithmic regret of SIMPLE - FIL).
4.1 Experimental results
We ran our experiments using the filtering rule that forwards
if SGN ;%
,
8'
where % is the estimator (3) and '
"! # %$ Note that this rule, which we call
RIDGE - FIL , is a natural generalization of SIMPLE - FIL to the unrestricted learning model; in
particular, SIMPLE - FIL uses a relevance threshold ' of the very same form as RIDGE - FIL,
although SIMPLE - FIL?s ?margin? ,
is defined differently. We tested our algorithm on a
, . " K< , .
.
1. Get and let
2. If
then forward , get label and update as follows:
$ ;
< <
$ < < ;
K , where if and is such
, otherwise;
that
.
3. Else forward with probability . If was forwarded then get label and do
the same updates as in 2; otherwise, do not make any update.
Algorithm: P - RIDGE - FIL.
Parameters: Real : ;
Initialization: % E4
Loop for
"
A
B5
9%
,+
,+
9
%
%
%
E
D D%
DD
@
,+
<%
D D%
3D D =
9
Figure 2: Pseudo-code for the filtering algorithm P - RIDGE - FIL. The performance of this
algorithm is analyzed in Theorem 3.
document filtering problem based on the first 70000 newswire stories from the Reuters
Corpus Volume 1. We selected the 34 Reuters topics whose frequency in the set of 70000
documents was between 1% and 5% (a plausible range for filtering applications). For
each topic, we defined a filtering task whose relevance judgements were assigned based
on whether the document was labelled with that topic or not. Documents were mapped
to real vectors using the bag-of-words representation. In particular, after tokenization we
lemmatized the tokens using a general-purpose finite-state morphological English analyzer
and then removed stopwords (we also replaced all digits with a single special character).
Document vectors were built removing all words which did not occur at least three times in
the corpus and using the TF-IDF encoding in the form
"! TF '(!7 DF , where TF is
the word frequency in the document, DF is the number of documents containing the word,
and is the total number of documents (if TF the TF-IDF coefficient was also set to
). Finally, all document vectors were normalized to length 1. To measure how the choice
of the threshold ' affects the filtering performance, we ran RIDGE - FIL with ' set to zero
on the same dataset as a standard on-line binary classifier (i.e., receiving the correct label
after every classification). We call this algorithm RIDGE - FULL. Figure 1 illustrates the
experimental results. The average -measure of RIDGE - FULL and RIDGE - FIL are, respectively,
and ; hence the threshold compensates pretty well the partial feedback in
the filtering setup. On the other hand, the standard Perceptron achieves here a -measure
in the classification task, hence inferior to that of RIDGE - FULL. Finally, we also
of
tested the apple-tasting filtering rule (see [9, STAP transformation]) based on the binary
classifier RIDGE - FULL. This transformation, which does not take into consideration the
margin, exhibited a poor performance and we did not include it in the plot.
4.2 Probabilistic ridge filtering
In this section we introduce a probabilistic filtering algorithm, derived from the (on-line)
ridge regression algorithm, for the class of linear probabilistic relevance functions. The
algorithm, called P - RIDGE - FIL, is sketched in Figure 2. The algorithm takes and a
probability value as input parameters and maintains a linear hypothesis % . If %
, then
is forwarded and %
gets updated according to the following two-steps ridge
regression-like rule. First, the intermediate vector %
is computed via the standard on-line
ridge regression algorithm using the inverse of matrix
. Then, the new vector %
is obtained by projecting %
onto the unit ball, where the projection is taken w.r.t. the
"
distance function
implies
; B%
4 (%
; (% . Note that D D % /D D =
%
%
. On the other hand, if %
0 then
is forwarded (and consequently
%
is updated) with some probability . The analysis of P - RIDGE - FIL is inspired by the
analysis in [1] for a related but different problem, and is based on relating the expected
regret in a given trial to a measure of the progress of % towards . The following lemma
will be useful.
Lemma 2 Using the notation of Figure 2, let be the trial when the -th update
< oc
<
= 8(!
curs. Then the following inequality holds:
,+
:
&+
:
; %
and
; B%
, where D /D denotes the determinant of matrix
; %
4 (%"
; 8% .
;
!
Theorem 3 Let !
. For all , if algorithm - ! ! , then its expected cumulative regret
of Figure 2 is run with
,+
, + is at most
!
!
" " " (! - #
Proof sketch. If is the trial when the -th forward takes place,
we define the random
& and ' 8 (! < < . If no update occurs
variables $ %
' . Let ) be the regret of - - in trial and ) be
in trial we set $ (
in trial . If
, then ) ) and
the regret of the update rule
$ can be lower bounded via Lemma 2. If
, then $ gets lower bounded via
Lemma 2 only with probability , while for the regret we can only use the trivial bound
)
. With probability 4 , instead, ) ) and $ . Let * be a
constant to be specified. We can write
) +
* $ , ) + * $
)
- * $
+
(4)
Now, it is easy to verify that in the conditional space where is given we have
and
. Thus, using Lemma
2 and Eq. (4) we can write
'
) +
* $ . ) + *
)
'
*6
Proof sketch. From Lemma 4.2 and Theorem 4.6 in [3] and the fact that D ,+ D = D D % D D =
<
= 8(!
4 B%
.
it follows that ,+
+
< ; B%
4
%
%" is a Bregman diverNow, the function
;
; B%
gence (e.g., [4, 10]), and it can be easily shown that %
in Figure 2 is the projection of %
onto the convex set
D D D D =
w.r.t.
; i.e., %
!
= =
%
.
By
a
projection
property
of
Bregman
divergences
(see,
e.g., the appendix in [10]) it follows that
4 B%
/
24 %
for all such
that D D D D = . Putting together gives the desired inequality.
D+
D
1
-.&
,+
012
-.&
+
%
D
D
$
;
B%
)
=
0
E
,+
,+
6
P RIDGE FIL
%
?
%
P RIDGE FIL
092
;
,+
!
,+
09>
6
6
:C
?!
,+
1>
!
,+
09>
! &+
,+
*
D ,+
?!
+
,+
D ,+
6 ?,+
1C
5
,+
6 ,+
+
+
,+
,+
A+
01C
(
+
C
D4,+
(
,+
; ?,+
D4,+
; ?,+
:C
0:C
01C
1
This parametrization requires the knowledge of / . It turns out one can remove this assumption
at the cost of a slightly more involved proof.
+ *
+*
* '
* '
+
(5)
where in the inequality we have dropped from factor and combined the resulting
and ) into ) . In turn, this term has been bounded
terms )
" by virtue of (2) with . At this
as )
point we work in the conditional space where
is given and distinguish the two cases
and . In the first case we have
- * * ' + * * '
whereas in the second case we have
* , * '
- ' # * * '
+ '
. We set *
where in both cases we used
sum over .
and
-
-
<
Notice that , + $
and that , + ' 8 "! < (! -
(e.g., [3], proof of Theorem 4.6 therein). After a few overapproximations (and taking the
and
) we obtain
worst between the two cases
#
#
, +
)
" "! - " #
"
This can be further upper bounded by
,+
A+
5D ,+
& ,+
,+
+
9>
?,+
!
1C
2D ,+
& ,+
,+
.01C
+
,+
,+
09>
01C
?
6 ,+
6 =
01C
6 ,+
,+
+
9>
=
6
,+
+
'
,+
,+
9
01
,+
1 ,+
2 9 ,+
+
A+
D ,+
D4,+
*
*
=
D D
?
C!
C
=
thereby concluding the proof.
1
=
=
=
&
?
>
?
D4,+
C
D ,+
=
=
$
)
01
,+
D4,+
D ,+
DD
,+
3+
,+
2
$
)
;
References
[1] Abe, N., and Long, P.M. (1999). Associative reinforcement learning using linear probabilistic
concepts. In Proc. ICML?99, Morgan Kaufmann.
[2] Auer, P. (2000). Using Upper Confidence Bounds for Online Learning. In Proc. FOCS?00,
IEEE, pages 270?279.
[3] Azoury, K., and Warmuth, M.K. (2001). Relative loss bounds for on-line density estimation
with the exponential family of distributions, Machine Learning, 43:211?246.
[4] Censor, Y., and Lent, A. (1981). An iterative row-action method for interval convex programming. Journal of Optimization Theory and Applications, 34(3), 321?353.
[5] Cesa-Bianchi, N. (1999). Analysis of two gradient-based algorithms for on-line regression.
Journal of Computer and System Sciences, 59(3):392?411.
[6] Cesa-Bianchi, N., Conconi, A., and Gentile, C. (2002). A second-order Perceptron algorithm.
In Proc. COLT?02, pages 121?137. LNAI 2375, Springer.
[7] Cesa-Bianchi, N., Long, P.M., and Warmuth, M.K. (1996). Worst-case quadratic loss bounds
for prediction using linear functions and gradient descent. IEEE Trans. NN, 7(3):604?619.
[8] Gavald`a, R., and Watanabe, O. (2001). Sequential sampling algorithms: Unified analysis and
lower bounds. In Proc. SAGA?01, pages 173?187. LNCS 2264, Springer.
[9] Helmbold, D.P., Littlestone, N., and Long, P.M. (2000). Apple tasting. Information and Computation, 161(2):85?139.
[10] Herbster, M. and Warmuth, M.K. (1998). Tracking the best regressor, in Proc. COLT?98, ACM,
pages 24?31.
[11] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. Journal
of the American Statistical Association, 58:13?30.
[12] Hoerl, A., and Kennard, R. (1970). Ridge regression: biased estimation for nonorthogonal problems. Technometrics, 12:55?67.
[13] Vapnik, V. (1998). Statistical learning theory. New York: J. Wiley & Sons.
[14] Voorhees, E., Harman, D. (2001). The tenth Text REtrieval Conference. TR 500-250, NIST.
[15] Vovk, V. (2001). Competitive on-line statistics. International Statistical Review, 69:213?248.
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1,268 | 2,155 | Independent Components Analysis
through Product Density Estimation
'frevor Hastie and Rob Tibshirani
Department of Statistics
Stanford University
Stanford, CA, 94305
{ hastie, tibs } @stat.stanford. edu
Abstract
We present a simple direct approach for solving the ICA problem,
using density estimation and maximum likelihood. Given a candidate orthogonal frame, we model each of the coordinates using a
semi-parametric density estimate based on cubic splines. Since our
estimates have two continuous derivatives , we can easily run a second order search for the frame parameters. Our method performs
very favorably when compared to state-of-the-art techniques.
1
Introduction
Independent component analysis (ICA) is a popular enhancement over principal
component analysis (PCA) and factor analysis. In its simplest form , we observe a
random vector X E IRP which is assumed to arise from a linear mixing of a latent
random source vector S E IRP,
(1)
X=AS;
the components Sj, j = 1, ... ,p of S are assumed to be independently distributed.
The classical example of such a system is known as the "cocktail party" problem.
Several people are speaking, music is playing, etc., and microphones around the
room record a mix of the sounds. The ICA model is used to extract the original
sources from these different mixtures.
Without loss of generality, we assume E(S) = 0 and Cov(S) = I , and hence
Cov(X) = AA T . Suppose S* = R S represents a transformed version of S, where R
is p x p and orthogonal. Then with A * = ART we have X* = A * S* = ARTR S =
X. Hence the second order moments Cov(X) = AAT = A * A *T do not contain
enough information to distinguish these two situations.
Model (1) is similar to the factor analysis model (Mardia, Kent & Bibby 1979),
where S and hence X are assumed to have a Gaussian density, and inference is
typically based on the likelihood of the observed data. The factor analysis model
typically has fewer than p components, and includes an error component for each
variable. While similar modifications are possible here as well, we focus on the
full-component model in this paper. Two facts are clear:
? Since a multivariate Gaussian distribution is completely determined by its
first and second moments, this model would not be able to distinguish A
and A * . Indeed, in factor analysis one chooses from a family of factor
rotations to select a suitably interpretable version.
? Multivariate Gaussian distributions are completely specified by their
second-order moments. If we hope to recover the original A, at least p - 1
of the components of S will have to be non-Gaussian.
Because of the lack of information in the second moments, the first step in an ICA
model is typically to transform X to have a scalar covariance, or to pre-whiten the
data. From now on we assume Cov(X) = I , which implies that A is orthogonal.
Suppose the density of Sj is Ij, j = 1, ... ,p, where at most one of the Ii are
Gaussian. Then the joint density of S is
p
(2)
Is(s) =
II Ii(Sj),
j= l
and since A is orthogonal, the joint density of X is
p
(3)
Ix(x) =
II Ii(aJ x),
j=l
where aj is the jth column of A . Equation (3) follows from S = AT X due to the
orthogonality of A , and the fact that the determinant in this multivariate transformation is 1.
In this paper we fit the model (3) directly using semi-parametric maximum likelihood. We represent each of the densities Ii by an exponentially tilted Gaussian
density (Efron & Tibshirani 1996).
(4)
where ? is the standard univariate Gaussian density, and gj is a smooth function,
restricted so that Ii integrates to 1. We represent each of the functions gj by a cubic
smoothing spline, a rich class of smooth functions whose roughness is controlled by a
penalty functional. These choices lead to an attractive and effective semi-parametric
implementation of ICA:
? Given A, each of the components Ii in (3) can be estimated separately by
maximum likelihood. Simple algorithms and standard software are available.
? The components gj represent departures from Gaussianity, and the expected log-likelihood ratio between model (3) and the gaussian density is
given by Ex 2: j gj(aJ X), a flexible contrast function.
? Since the first and second derivatives of each of the estimated gj are immediately available, second order methods are available for estimating the
orthogonal matrix A . We use the fixed point algorithms described in
(Hyvarinen & Oja 1999).
? Our representation of the gj as smoothing splines casts the estimation problem as density estimation in a reproducing kernel Hilbert space, an infinite
family of smooth functions. This makes it directly comparable with the
"Kernel ICA" approach of Bach & Jordan (2001), with the advantage that
we have O(N) algorithms available for the computation of our contrast
function, and its first two derivatives.
In the remainder of this article, we describe the model in more detail, and evaluate
its performance on some simulated data.
2
Fitting the Product Density leA model
Given a sample Xl, ... ,XN we fit the model (3),(4) by maximum penalized likelihood. The data are first transformed to have zero mean vector, and identity
covariance matrix using the singular value decomposition. We then maximize the
criterion
(5)
subject to
T
(6)
J
a j ak
bjk
?(s)e 9j (slds
(7)
't/j, k
1 't/j
For fixed aj and hence Sij = aT Xi the solutions for 9j are known to be cubic splines
with knots at each of the unique values of Sij (Silverman 1986). The p terms
decouple for fixed aj, leaving us p separate penalized density estimation problems.
We fit the functions 9j and directions aj by optimizing (5) in an alternating fashion ,
as described in Algorithm 1. In step (a), we find the optimal 9j for fixed 9j; in
Algorithm 1 Product Density leA algorithm
1. Initialize A (random Gaussian matrix followed by orthogonalization).
2. Alternate until convergence of A, using the Amari metric (16).
(a) Given A , optimize (5) w.r.t. 9j (separately for each j), using the
penalized density estimation algorithm 2.
(b) Given 9j , j = 1, ... ,p, perform one step of the fixed point algorithm 3
towards finding the optimal A.
step (b), we take a single fixed-point step towards the optimal A. In this sense
Algorithm 1 can be seen to be maximizing the profile penalized log-likelihood w.r.t.
A.
2.1
Penalized density estimation
We focus on a single coordinate, with N observations Si,
Si = Xi for some k). We wish to maximize
1, ... ,N (where
af
(8)
J
subject to ?(s)e 9 (slds = 1. Silverman (1982) shows that one can incorporate the
integration constraint by using the modified criterion (without a Lagrange multiplier)
N
(9)
~ l:= {lOg?(Si) + 9(Si )} >=1
J
?(s)e 9 (slds - A
J
91/ 2 (S)ds.
Since (9) involves an integral, we need an approximation. We construct a fine grid
of L values s; in increments ~ covering the observed values Si, and let
*
(10)
Y? =
#Si E (sf - ~/2, Sf
N
+ ~/2)
Typically we pick L to be 1000, which is more than adequate. We can then approximate (9) by
L
(11)
L
{Y; [log(?(s;))
+ g(s;)]- ~?(se)e9(sll}
J
- A
gI/2(s)ds.
?=1
This last expression can be seen to be proportional to a penalized Poisson loglikelihood with response Y;! ~ and penalty parameter A/~, and mean J-t(s) =
?(s)e 9(s). This is a generalized additive model (Hastie & Tibshirani 1990), with
an offset term log(?(s)), and can be fit using a Newton algorithm in O(L) operations. As with other GAMs, the Newton algorithm is conveniently re-expressed as
an iteratively reweighted penalized least squares regression problem, which we give
in Algorithm 2.
Algorithm 2 Iteratively reweighted penalized least squares algorithm for fitting
the tilted Gaussian spline density model.
1. Initialize 9 == O.
2. Repeat until convergence:
(a) Let J-t(s;) = ?(s;)e 9(sll, ? = 1, ... ,L, and w?
(b) Define the working response
(12)
z? = g(s*)
?
= J-t(s;).
+ Ye -
J-t(sf)
J-t( sf)
(c) Update g by solving the weighted penalized least squares problem
(13)
This amounts to fitting a weighted smoothing spline to the pairs (sf, ze)
with weights w? and tuning parameter 2A/~.
Although other semi-parametric regression procedures could be used in (13), the
cubic smoothing spline has several advantages:
? It has knots at all L of the pseudo observation sites sf' The values sf
can be fixed for all terms in the model (5), and so a certain amount of
pre-computation can be performed. Despite the large number of knots
and hence basis functions , the local support of the B-spline basis functions
allows the solution to (13) to be obtained in O(L) computations.
? The first and second derivatives of 9 are immediately available, and are
used in the second-order search for the direction aj in Algorithm 1.
? As an alternative to choosing a value for A, we can control the amount of
smoothing through the effective number of parameters, given by the trace
of the linear operator matrix implicit in (13) (Hastie & Tibshirani 1990).
? It can also be shown that because of the form of (9), the resulting density
inherits the mean and variance of the data (0 and 1); details will be given
in a longer version of this paper.
2.2
A fixed point method for finding the orthogonal frame
For fixed functions g1> the penalty term in (5) does not playa role in the search
for A. Since all of the columns aj of any A under consideration are mutually
orthogonal and unit norm, the Gaussian component
p
L log ?(aJ
Xi)
j=l
does not depend on A. Hence what remains to be optimized can be seen as the
log-likelihood ratio between the fitted model and the Gaussian model, which is
simply
C(A)
(14)
Since the choice of each gj improves the log-likelihood relative to the Gaussian, it is
easy to show that C(A) is positive and zero only if, for the particular value of A, the
log-likelihood cannot distinguish the tilted model from a Gaussian model. C(A) has
the form of a sum of contrast functions for detecting departures from Gaussianity.
Hyvarinen, Karhunen & Oja (2001) refer to the expected log-likelihood ratio as the
negentropy, and use simple contrast functions to approximate it in their FastICA
algorithm. Our regularized approach can be seen as a way to construct a flexible
contrast function adaptively using a large set of basis functions .
Algorithm 3 Fixed point update forA.
1. For j = 1, ... ,p:
(15)
where E represents expectation w.r.t. the sample
column of A.
Xi,
and
aj
is the jth
2. Orthogonalize A: Compute its SVD , A = UDV T , and replace A
f-
UV T .
Since we have first and second derivatives avaiable for each gj , we can mimic exactly
the fast fixed point algorithm developed in (Hyvarinen et al. 2001, page 189) ; see
algorithm 3. Figure 1 shows the optimization criterion C (14) above, as well as the
two criteria used to approximate negentropy in FastICA by Hyvarinen et al. (2001)
[page 184]. While the latter two agree with C quite well for the uniform example
(left panel), they both fail on the mixture-of-Gaussians example, while C is also
successful there.
Uniforms
x
Gaussian Mixtures
0
0
'",,;
'",,;
'",,;
x
"
'",,;
"
"0
C
"0
C
'",,;
'",,;
'",,;
'",,;
0
0
,,;
,,;
0.0
0.5
1.0
1.5
e
2.0
2.5
3.0
0.0
0.5
1.0
1.5
e
2.0
2.5
3.0
Figure 1: The optimization criteria and solutions found for two different examples in lR2
using FastICA and our ProDenICA . G 1 and G2 refer to the two functions used to define
negentropy in FastICA. In the left example the independent components are uniformly
distributed, in the right a mixture of Gaussians. In the left plot, all the procedures found
the correct frame; in the right plot, only the spline based approach was successful. The
vertical lines indicate the solutions found, and the two tick marks at the top of each plot
indicate the true angles.
3
Comparisons with fast ICA
In this section we evaluate the performance of the product density approach (ProDenICA) , by mimicking some of the simulations performed by Bach & Jordan (2001)
to demonstrate their Kernel ICA approach. Here we compare ProDenICA only with
FastICA; a future expanded version of this paper will include comparisons with other
leA procedures as well.
The left panel in Figure 2 shows the 18 distributions used as a basis of comparison.
These exactly or very closely approximate those used by Bach & Jordan (2001) . For
each distribution, we generated a pair of independent components (N=1024) , and
a random mixing matrix in ill? with condition number between 1 and 2. We used
our Splus implementation of the FastICA algorithm, using the negentropy criterion
based on the nonlinearity G 1 (s) = log cosh(s) , and the symmetric orthogonalization
scheme as in Algorithm 3 (Hyvarinen et al . 2001, Section 8.4.3). Our ProDenICA
method is also implemented in Splus. For both methods we used five random starts
(without iterations). Each of the algorithms delivers an orthogonal mixing matrix A
(the data were pre-whitenea) , which is available for comparison with the generating
orthogonalized mixing matrix A o. We used the Amari metric(Bach & Jordan 2001)
as a measure of the closeness of the two frames:
(16)
d(Ao,A) =
~ f.- (L~=1 Irijl
2p ~
i=1
max?lr?
?1
J"J
-1) + ~ f.- (Lf=1I rijl -1) ,
2p ~
max?lr??1
j=1"
"J
where rij = (AoA - 1 )ij . The right panel in Figure 2 shows boxplots of the pairwise
differences d(Ao, A F ) -d(Ao , Ap ) (x100), where the subscripts denote ProDenICA
or FastICA. ProDenICA is competitive with FastICA in all situations, and dominates in most of the mixture simulations. The average Amari error (x 100) for
FastICA was 13.4 (2.7), compared with 3.0 (0.4) for ProDenICA (Bach & Jordan
(2001) report averages of 6.2 for FastICA, and 3.8 and 2.9 for their two KernelICA
methods).
We also ran 300 simulations in
1R.4,
using N = 1000, and selecting four of the
,
b
,
0 ..,---_ _ _ _ _ _ _ _ _ _ _ _ _----,
JL ~
d
~
9
~
~- JJL
f
h
flL ~ ~
~~~
j
k
m
"
I
,
~~~
p
ro
ci
q
~~~
N
ci
o
ci
abcdefghijklmnopqr
distribution
Figure 2: The left panel shows eighteen distributions used for comparisons. These include
the "t", uniform, exponential, mixtures of exponentials, symmetric and asymmetric gaussian mixtures. The right panel shows boxplots of the improvement of ProDenICA over
FastICA in each case, using the Amari metric, based on 30 simulations in lR? for each
distribution.
18 distributions at random. The average Amari error (x 100) for FastICA was
26.1 (1.5), compared with 9.3 (0.6) for ProDenICA (Bach & Jordan (2001) report
averages of 19 for FastICA , and 13 and 9 for their two K ernelICA methods).
4
Discussion
The lCA model stipulates that after a suitable orthogonal transformation, the data
are independently distributed. We implement this specification directly using semiparametric product-density estimation. Our model delivers estimates of both the
mixing matrix A, and estimates of the densities of the independent components.
Many approaches to lCA, including FastICA, are based on minimizing approximations to entropy. The argument, given in detail in Hyvarinen et al. (2001) and
reproduced in Hastie, Tibshirani & Friedman (2001), starts with minimizing the
mutual information - the KL divergence between the full density and its independence version. FastICA uses very simple approximations based on single (or a small
number of) non-linear contrast functions , which work well for a variety of situations,
but not at all well for the more complex gaussian mixtures. The log-likelihood for
the spline-based product-density model can be seen as a direct estimate of the mutual information; it uses the empirical distribution of the observed data to represent
their joint density, and the product-density model to represent the independence
density. This approach works well in both the simple and complex situations automatically, at a very modest increase in computational effort. As a side benefit,
the form of our tilted Gaussian density estimate allows our log-likelihood criterion
to be interpreted as an estimate of negentropy, a measure of departure from the
Gaussian.
Bach & Jordan (2001) combine a nonparametric density approach (via reproducing
kernel Hilbert function spaces) with a complex measure of independence based on
the maximal correlation. Their procure requires O(N3) computations, compared to
our O(N). They motivate their independence measures as approximations to the
mutual independence. Since the smoothing splines are exactly function estimates
in a RKHS, our method shares this flexibility with their Kernel approach (and is
in fact a "Kernel" method). Our objective function, however, is a much simpler
estimate of the mutual information. In the simulations we have performed so far ,
it seems we achieve comparable accuracy.
References
Bach, F . & Jordan, M. (2001), Kernel independent component analysis, Technical
Report UCBjCSD-01-1166, Computer Science Division, University of California, Berkeley.
Efron, B. & Tibshirani, R. (1996), 'Using specially designed exponential families
for density estimation' , Annals of Statistics 24(6), 2431-246l.
Hastie, T. & Tibshirani, R. (1990), Generalized Additive Models, Chapman and
Hall.
Hastie, T., Tibshirani, R. & Friedman, J. (2001), The Elements of Statistical Learning; Data mining, Inference and Prediction, Springer Verlag, New York.
Hyvarinen, A., Karhunen, J. & Oja, E. (2001), Independent Component Analysis,
Wiley, New York.
Hyvarinen, A. & Oja, E. (1999), 'Independent component analysis: Algorithms and
applications' , Neural Networks .
Mardia, K., Kent, J. & Bibby, J. (1979), Multivariate Analysis, Academic Press.
Silverman, B. (1982), 'On the estimation of a probability density function by the
maximum penalized likelihood method', Annals of Statistics 10(3),795-810.
Silverman, B. (1986), Density Estimation for Statistics and Data Analysis, Chapman and Hall.
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1,269 | 2,156 | Adaptive Scaling for Feature Selection in SVMs
Yves Grandvalet
Heudiasyc, UMR CNRS 6599,
Universit?e de Technologie de Compi`egne,
Compi`egne, France
[email protected]
St?ephane Canu
PSI
INSA de Rouen,
St Etienne du Rouvray, France
[email protected]
Abstract
This paper introduces an algorithm for the automatic relevance determination of input variables in kernelized Support Vector Machines. Relevance is measured by scale factors defining the input space metric, and
feature selection is performed by assigning zero weights to irrelevant
variables. The metric is automatically tuned by the minimization of the
standard SVM empirical risk, where scale factors are added to the usual
set of parameters defining the classifier. Feature selection is achieved
by constraints encouraging the sparsity of scale factors. The resulting
algorithm compares favorably to state-of-the-art feature selection procedures and demonstrates its effectiveness on a demanding facial expression recognition problem.
1 Introduction
In pattern recognition, the problem of selecting relevant variables is difficult. Optimal
subset selection is attractive as it yields simple and interpretable models, but it is a combinatorial and acknowledged unstable procedure [2]. In some problems, it may be better
to resort to stable procedures penalizing irrelevant variables. This paper introduces such a
procedure applied to Support Vector Machines (SVM).
The relevance of input features may be measured by continuous weights or scale factors,
which define a diagonal metric in input space. Feature selection consists then in determining a sparse diagonal metric, and sparsity can be encouraged by constraining an appropriate
norm on scale factors. Our approach can be summarized by the setting of a global optimization problem pertaining to 1) the parameters of the SVM classifier, and 2) the parameters
of the feature space mapping defining the metric in input space. As in standard SVMs,
only two tunable hyper-parameters are to be set: the penalization of training errors, and
the magnitude of kernel bandwiths. In this formalism we derive an efficient algorithm to
monitor slack variables when optimizing the metric. The resulting algorithm is fast and
stable.
After presenting previous approaches to hard and soft feature selection procedures in the
context of SVMs, we present our algorithm. This exposure is followed by an experimental
section illustrating its performances and conclusive remarks.
2 Feature Selection via adaptive scaling
Scaling is a usual preprocessing step, which has important outcomes in many classification
methods including SVM classifiers [9, 3]. It is defined by a linear transformation within
the input space:
, where
diag
is a diagonal matrix
of scale
factors.
Adaptive scaling consists in letting to be adapted during the estimation process with the
explicit aim of achieving a better recognition rate. For kernel classifiers, is a set of hyperparameters of the learning process. According to the structural risk minimization principle
[8], can be tuned in two ways:
1. estimate the parameters of classifier by empirical risk minimization for several values of
to produce a structure of classifiers
multi-indexed by
. Select one element of the structure by finding the set
minimizing some estimate of generalization error.
2. estimate the parameters of classifier and the hyper-parameters
by empirical risk minimization, while a second level hyper-parameter, say , constrains
in order to avoid overfitting. This procedure produces a structure of classifiers indexed by , whose value is computed by minimizing some estimate of
generalization error.
!
"
!
"
%
#
$
!%
The usual paradigm consists in computing the estimate of generalization error for regularly
spaced hyper-parameter values and picking the best solution among all trials. Hence, the
first approach requires intensive computation, since the trials should be completed over a
-dimensional grid over
values.
&
'
Several authors suggested to address this problem by optimizing an estimate of generalization error with respect to the hyper-parameters. For SVM classifiers, Cristianini et al. [4]
first proposed to apply an iterative optimization scheme to estimate a single kernel width
hyper-parameter. Weston et al. [9] and Chapelle et al. [3] generalized this approach to
multiple hyper-parameters in order to perform adaptive scaling and variable selection.
The experimental results in [9, 3] show the benefits of this optimization. However, relying on the optimization of generalization error estimates over many hyper-parameters is
hazardous. Once optimized, the unbiased estimates become down-biased, and the bounds
provided by VC-theory usually hold for kernels defined a priori (see the proviso on the
radius/margin bound in [8]). Optimizing these criteria may thus result in overfitting.
%
#
!
In the second solution considered here, the estimate of generalization error is minimized
with respect to , a single (second level) hyper-parameter, which constrains
.
The role of this constraint is twofold: control the complexity of the classifier, and encourage variable selection in input space. This approach is related to some successful
soft-selection procedures, such as lasso and bridge [5] in the frequentist framework and
Automatic Relevance Determination (ARD) [7] in the Bayesian framework. Note that this
type of optimization procedure has been proposed for linear SVM in both frequentist [1]
and Bayesian frameworks [6]. Our method generalizes this approach to nonlinear SVM.
3 Algorithm
3.1 Support Vector Machines
(*),+.-$/ </ ? <A 0@ , where<CB functionB
214365 "798 ;:=<2> 798
The decision function provided by SVM is
is defined as:
(1)
B
1
where the parameters
8
are obtained by solving the following optimization problem:
<
<
4
1
3
1
:
7
<
) -
<
< BB
(2)
>
1
5
3
<
subject to
798 BB
with 5 defined as 5 . In this problem setting,
and the parameters of the
feature space mapping (typically a kernel bandwidth) are tunable hyper-parameters which
need to be determined by the user.
B
1
In [9, 3], adaptive scaling is performed by iteratively finding the parameters
8 of the
SVM classifier for a fixed value of
!
and minimizing aB bound on the estimate of generalization error with respect to hyper-parameters
. The algorithm
3.2 A global optimization problem
minimizes 1) the SVM empirical criterion with respect to parameters and 2) an estimate of
generalization error with respect to hyper-parameters.
!
In the present approach, we avoid the enlargement of the set of hyper-parameters by letting
to be standard parameters of the classifier. Complexity is controlled by
and
by constraining the magnitude of . The latter defines the single hyper-parameter of the
learning process related to scaling variables. The learning criterion is defined as follows:
<
1
1
<
3
:
)- )-
7
"
!
> < < 1 3 5 < "
subject to
7 8
$
&
: " &%
&%
<
B #B
BB#
'
(3)
BB &
B
%
In (3), the constraint on should favor sparse solutions. To allow
to go to zero, ( should
be
To encourage sparsity, zeroing a small #
should allow a high increase of &) ,
*, + positive.
' , hence ( should be small. In the limit of (-. , the constraint counts the number
Like in standard SVM classification, the minimization of an estimate of generalization error
is postponed to a later step, which consists in picking the best solution among all trials on
.
the two dimensional grid of hyper-parameters
of non-zero scale parameters, resulting in a hard selection procedure. This choice might
seem appropriate for our purpose, but it amounts to attempt to solve a highly non-convex
optimization problem, where the number of local minima grows exponentially with the
input dimension . To avoid this problem, we suggest to use (
, which is the smallest
value for which the problem is convex with the linear mapping
. Indeed, for
linear kernels, the constraint on amounts to minimize the standard SVM criterion where
the penalization
on the /10 norm is replaced by the penalization of the /3254 norm. Hence,
provides
47682
setting (
the solution of the / SVM classifier described in [1]. For non-linear
kernels however, the two solutions differ notably since the present algorithm modifies the
metric in input space, while the / SVM classifier modifies
the metric in feature space.
and Gaussian
Finally, note that unicity can be guaranteed for (
kernels with large
bandwidths ( -9 ).
&
5
%
3.3 An alternated optimization scheme
B
1 8
Problem (3) is complex; we propose to solve iteratively a series of simplier problems.
The function is first optimized with respect to parameters
for a fixed mapping
(standard SVM problem). Then, the parameters of the feature space mapping are
optimized
while some characteristics
of
are kept fixed: At step , starting
from a given
value, the optimal
are computed. Then
is determined by a
descent algorithm.
5
1 / 8
0
B
1
In this scheme, /
8!
0 are computed by solving the standard quadratic optiB
mization problem (2). Our implementation, based on an interior point method, will not
be detailed here. Several SVM retraining are necessary, but they are faster than the usual
training since the algorithm is initialized appropriately with the solutions of the preceding
round.
For solving the minimization problem with respect to , we use a reduced conjugate gradient technique. The optimization problem was simplified by assuming that some of the other
variables are fixed. We tried several versions: 1) fixed; 2) Lagrange multipliers
fixed;
3) set of support vectors fixed. For the three versions, the optimal value of , or at least the
optimal value of the slack variables
can be obtained by solving a linear program, whose
optimum is computed directly (in a single iteration). We do not detail our first version here,
since the two last ones performed much better. The main steps of the two last versions are
sketched below.
1
8
B
B
1
8/ 0
?
<
1
1 > 5
Regarding problem (3), 1 is sub-optimal when
3.4 Sclaling parameters update
Starting from an initial solution
, our goal is to update by solving a
simple intermediate problem providing an improved solution to the global problem (3). We
first assume that the Lagrange multipliers
defining are not affected by updates, so
that is defined as
.
1
1
varies; nevertheless is guaranteed to
be an admissible solution. Hence, we minimize an upper bound of the original primal
cost which guarantees that any admissible update (providing a decrease of the cost) of the
intermediate problem will provide a decrease of the cost of the original problem.
The intermediate optimization problem is stated as follows:
? <? <
> >
)
<
? @
<
>
:
>
subject to
<
&
: &%
&%
!
:<
@
<B
<
:
<
$ 7!
<CB
< BB
798 B B
'
BB &
(4)
Solving this problem is still difficult since the cost is a complex non-linear function of
scale factors. Hence, as stated above, will be updated by a descent algorithm. The latter
requires the evaluation of the cost and its gradient with respect to . In particular, this
means that we should be able to compute
and
for any value of .
<
<
<
<
'
For given values of
and
,
is the solution of the following problem:
<
<
:
)
-
<
? @ *< B
: > 798
subject to >
<
!
whose dual formulation is
>
:<
?
>
< <
:< >
<
subject to
<
<
?
@
<B
BB
B B B
(5)
: >
"
!
@
<
<
BB
(6)
<0B
" >
<
< <
and its derivative with respect to are easily computed. Parameters
With ,
by a conjugate reduced gradient technique, i.e. a conjugate gradient
are then updated
algorithm ensuring that the set of constraints on are always verified.
This linear problem is solved directly by the following algorithm: 1) sort
in descending order for all positive examples on the one
and for all negative examples on the other side; 2) compute the pairwise sum of sorted
side
values; 3) set
for all positive and negative examples whose sum is positive.
3.5 Updating Lagrange multipliers
B B
' 8 < <
>
1. for support vectors of the first category 2
? @ <*B
<
: > 7 8 >
<
Assume now that only the support vectors remain fixed while optimizing . This assump
tion is used to derive a rule to update at
reasonable computing cost the Lagrange multipliers
together with by computing
. At
, the following holds [3]:
2. for support vectors of the second category (such that
(7)
?"<
)
.
From these equations, and the assumption that support vectors remain support vectors (and
that their category do not change) one derives a system of linear equations defining the
derivatives of
and with respect to [3]:
8
?
1. for support vectors of the first category
@ <B
?
@ <B
> $ 7 : > 7 #8
:
?<
2. for support vectors of the second category
'
(8)
?
>
:
obey the constraint
3. Finally, the system is completed by stating that the Lagrange multipliers should
:
?
:
>
? <
(9)
1
The value of
is updated from these equations, and the step size is limited to ensure that
for support vectors of the first category. Hence, in this version, is also an
admissible sub-optimal solution regarding problem (3).
4 Experiments
In the experiments reported below, we used (
for the constraint on (3). The scale parameters were optimized with the last version, where the set of support vectors is assumed
were chosen using the span bound [3].
to be fixed. Finally, the hyper-parameters
Although the value of the bound itself was not a faithful estimate of test error, the average
loss induced by using the minimizer of these bounds was quite small.
B
%
4.1 Toy experiment
In [9], Weston et al. compared two versions of their feature selection algorithm, to standard
SVMs and filter methods (i.e. preprocessing methods selecting features either based on
Pearson correlation coefficients, Fisher criterion score, or the Kolmogorov-Smirnov statistic). Their artificial data benchmarks provide a basis for comparing our approach with
their, which is based on the minimization of error bounds. Two types of distributions are
provided, whose detailed characteristics are not given here. In the linear problem, 6 dimensions out of 202 are relevant. In the nonlinear problem, two features out of 52 are relevant.
For each distribution, 30 experiments are conducted, and the average test recognition rate
measures the performance of each method.
For both problems, standard SVM achieve a 50% error rate in the considered range of
training set sizes. Our results are shown in Figure 1.
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
10
20
30
40
50
75
100
0
10
20
30
40
50
75
100
Figure 1: Results obtained on the benchmarks of [9]. Left: linear problem; right nonlinear
problem. The number of training examples is represented on the -axis, and the average
test error rate on the -axis.
>
Our test performances are qualitatively similar to the ones obtained by gradient descent on
the radius/margin bound in [9], which are only improved by the forward selection algorithm
minimizing the span bound. Note however that Weston et al. results are obtained after a
correct number of features was specified by the user, whereas the present results were
obtained fully automatically. Knowing the number of features that should be selected by
the algorithm is somewhat similar
to select the optimal value of parameter ( for each .
'%
problem, for
In the non-linear
training examples, an average of 26.5 features are
selected; for
8 , an average of 6.6 features are selected. These figures show that
although our feature selection scheme is effective, it should be more stringent: a smaller
variables
value of ( would
of problem. The two relevant
more appropriate for this type
selected in be
are
of cases for
and
, in for n=50, and in for
. For these two sample sizes, they
and second.
are even always ranked first
Regarding
training times, the optimization of required an average of over 100 times
more computing time than standard SVM fitting for the linear problem and 40 times for the
nonlinear problem. These increases scale less than linearly with the number of variables,
and are certainly yet to be improved.
4.2 Expression recognition
We also tested our algorithm on a more demanding task to test its ability to handle a large
number of features. The considered problem consists in recognizing the happiness expression among the five other facial expressions corresponding to universal emotions (disgust,
sadness, fear, anger, and surprise). The data sets are made of
8 gray level images of
frontal faces, with standardized positions of eyes, nose and mouth. The training set comprises
8 positive images, and negative ones. The test set is made of positive images
and negative ones.
the raw pixel representation of images, resulting in 4200 highly correlated feaWe used
tures. For this task, the accuracy of standard SVMs is 92.6% (11 test errors). The recognition rate is not significantly affected by our feature selection scheme (10 errors), but more
than 1300 pixels are considered to be completely irrelevant at the end of the iterative procedure (estimating required about 80 times more computing time than standard SVM).
This selection brings some important clues for building relevant attributes for the facial
recognition expression task.
Figure 2 represents the scaling factors , where black is zero and white represents the
highest value. We see that, according to the classifier, the relevant areas for recognizing the
happiness expression are mainly in the mouth area, especially on the mouth wrinkles, and
to a lesser extent in the white of the eyes (which detects open eyes) and the outer eyebrows.
On the right hand side of this figure, we displayed masked support faces, i.e. support faces
scaled by the expression mask. Although we lost many important features regarding the
identity of people, the expression is still visible on these faces. Areas irrelevant for the
recognition task (forehead, nose, and upper cheeks) have been erased or softened by the
expression mask.
5 Conclusion
We have introduced a method to perform automatic relevance determination and feature
selection in nonlinear SVMs. Our approach considers that the metric in input space defines
a set of parameters of the SVM classifier. The update of the scale factors is performed
by iteratively minimizing an approximation of the SVM cost. The latter is efficiently minimized with respect to slack variables when the metric varies. The approximation of the cost
function is tight enough to allow large update of the metric when necessary. Furthermore,
because at each step our algorithm guaranties the global cost to decrease, it is stable.
Figure 2: Left: expression mask of happiness provided by the scaling factors ; Right,
top row: the two positive masked support face; Right, bottom row: four negative masked
support faces.
Preliminary experimental results show that the method provides sensible results in a reasonable time, even in very high dimensional spaces, as illustrated on a facial expression
recognition task. In terms of test recognition rates, our method is comparable with [9, 3].
Further comparisons are still needed to demonstrate the practical merits of each paradigm.
%
Finally, it may also be beneficial to mix the two approaches: the method of Cristianini et al.
[4] could be used to determine and
. The resulting algorithm would differ from [9, 3],
since the relative relevance of each feature (as measured by ) would be estimated by
empirical risk minimization, instead of being driven by an estimate of generalization error.
%
References
[1] P. S. Bradley and O. L. Mangasarian. Feature selection via concave minimization and
support vector machines. In Proc. 15th International Conf. on Machine Learning,
pages 82?90. Morgan Kaufmann, San Francisco, CA, 1998.
[2] L. Breiman. Heuristics of instability and stabilization in model selection. The Annals
of Statistics, 24(6):2350?2383, 1996.
[3] O. Chapelle, V. Vapnik, O. Bousquet, and S. Mukherjee. Choosing multiple parameters
for support vector machines. Machine Learning, 46(1):131?159, 2002.
[4] N. Cristianini, C. Campbell, and J. Shawe-Taylor. Dynamically adapting kernels in
support vector machines. In M. S. Kearns, S. A. Solla, and D. A. Cohn, editors, Advances in Neural Information Processing Systems 11. MIT Press, 1999.
[5] T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning: data
mining , inference, and prediction. Springer series in statistics. Springer, 2001.
[6] T. Jebara and T. Jaakkola. Feature selection and dualities in maximum entropy discrimination. In Uncertainity In Artificial Intellegence, 2000.
[7] R. M. Neal. Bayesian Learning for Neural Networks, volume 118 of Lecture Notes in
Statistics. Springer, 1996.
[8] V. N. Vapnik. The Nature of Statistical Learning Theory. Springer Series in Statistics.
Springer, 1995.
[9] J. Weston, S. Mukherjee, O. Chapelle, M. Pontil, T. Poggio, and V. Vapnik. Feature
selection for SVMs. In Advances in Neural Information Processing Systems 13. MIT
Press, 2000.
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1,270 | 2,157 | Learning to Perceive Transparency from
the Statistics of Natural Scenes
Anat Levin
Assaf Zomet
Yair Weiss
School of Computer Science and Engineering
The Hebrew University of Jerusalem
91904 Jerusalem, Israel
{alevin,zomet,yweiss}@cs.huji.ac.il
Abstract
Certain simple images are known to trigger a percept of transparency: the input image I is perceived as the sum of two images
I(x, y) = I1 (x, y) + I2 (x, y). This percept is puzzling. First, why
do we choose the ?more complicated? description with two images
rather than the ?simpler? explanation I(x, y) = I1 (x, y) + 0 ? Second, given the infinite number of ways to express I as a sum of two
images, how do we compute the ?best? decomposition ?
Here we suggest that transparency is the rational percept of a system that is adapted to the statistics of natural scenes. We present
a probabilistic model of images based on the qualitative statistics
of derivative filters and ?corner detectors? in natural scenes and
use this model to find the most probable decomposition of a novel
image. The optimization is performed using loopy belief propagation. We show that our model computes perceptually ?correct?
decompositions on synthetic images and discuss its application to
real images.
1
Introduction
Figure 1a shows a simple image that evokes the percept of transparency. The
image is typically perceived as a superposition of two layers: either a light square
with a dark semitransparent square in front of it or a dark square with a light
semitransparent square in front of it.
Mathematically, our visual system is taking a single image I(x, y) and representing
as the sum of two images:
I1 (x, y) + I2 (x, y) = I(x, y)
(1)
When phrased this way, the decomposition is surprising. There are obviously an
infinite number of solutions to equation 1, how does our visual system choose one?
Why doesn?t our visual system prefer the ?simplest? explanation I(x, y) = I 1 (x, y)+
0?
a
b
Figure 1: a. A simple image that evokes the percept of transparency. b. A simple
image that does not evoke the percept of transparency.
Figure 1b shows a similar image that does not evoke the percept of transparency.
Here again there are an infinite number of solutions to equation 1 but our visual
system prefers the single layer solution.
Studies of the conditions for the percept of transparency go back to the very first research on visual perception (see [1] and references within). Research of this type has
made great progress in understanding the types of junctions and their effects (e.g.
X junctions of a certain type trigger transparency, T junctions do not). However,
it is not clear how to apply these rules to an arbitrary image.
In this paper we take a simple Bayesian approach. While equation 1 has an infinite
number of possible solutions, if we have prior probabilities P (I1 (x, y)), P (I2 (x, y))
then some of these solutions will be more probable than others. We use the statistics
of natural images to define simple priors and finally use loopy belief propagation
to find the most probable decomposition. We show that while the model knows
nothing about ?T junctions? or ?X junctions?, it can generate perceptually correct
decompositions from a single image.
2
Statistics of natural images
A remarkably robust property of natural images that has received much attention
lately is the fact that when derivative filters are applied to natural images, the filter
outputs tend to be sparse [5, 7]. Figure 2 illustrates this fact: the histogram of the
horizontal derivative filter is peaked at zero and fall off much faster than a Gaussian.
Similar histograms are observed for vertical derivative filters and for the gradient
magnitude: |?I|.
There are many ways to describe the non Gaussian nature of this distribution
(e.g. high kurtosis, heavy tails). Figure 2b illustrates the observation made by
Mallat [4] and Simoncelli [8]: that the distribution is similar to an exponential
density with exponent
less than 1. We show the log probability for densities of the
?
form p(x) ? e?x . We assume x ? [0, 100] and plot the log probabilities so that
they agree on p(0), p(100). There is a qualitative difference between distributions
for which ? > 1 (when the log probability is convex) and those for which ? < 1
(when it becomes concave). As figure 2d shows, the natural statistics for derivative
deriv filter
corner operator
5
5
2.5
x 10
5
x 10
4.5
4
2
3.5
3
1.5
2.5
2
1
1.5
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0.5
0.5
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?150
0
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?100
?50
0
a
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200
0
0.5
1
1.5
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2.5
7
250
x 10
e
14
0
12
?2
10
?4
2
Gaussian:?x
Laplacian: ?x
?0.4
logprob
100
c
0
?0.2
50
1/2
?X
?6
8
1/4
?X
?8
?0.6
6
?10
4
?0.8
?12
2
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0
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x
b
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?14
0
0
50
100
150
200
0
0.5
1
1.5
d
2
2.5
7
250
x 10
f
Figure 2: a. A natural image. c Histogram of filter outputs. e Histogram of corner
detector outputs. d,e log histograms.
?
filters has the qualitative nature of a distribution e?x with ? < 1.
In [9] the sparsity of derivative filters was used to decompose an image sequence as
a sum of two image sequences. Will this prior be sufficient for a single frame ? Note
that decomposing the image in figure 1a into two layers does not change the output
of derivative filters: exactly the same derivatives exist in the single layer solution
as in the two layer solution. Thus we cannot appeal to the marginal histogram of
derivative filters to explain the percept of transparency in this image.
There are two ways to go beyond marginal histograms of derivative filters. We
can either look at joint statistics of derivative filters at different locations or orientations [6] or look at marginal statistics of more complicated feature detectors
(e.g. [11]).
We looked at the marginal statistics of a ?corner detector?. The output of the
?corner detector? at a given location x0 , y0 is defined as:
?
?
X
Ix2 (x, y)
Ix (x, y)Iy (x, y)
c(x0 , y0 ) = det(
w(x, y)
)
(2)
Ix (x, y)Iy (x, y)
Iy2 (x, y)
where w(x, y) is a small Gaussian window around x0 , y0 and Ix , Iy are the derivatives
of the image.
Figures 2e,f show the histogram of this corner operator on a typical natural
image.
?
Again, note that it has the qualitative statistic of a distribution e?x for ? < 1.
To get a more quantitative description of the statistics
we used maximum likelihood
?
to fit a distribution of the form P (x) = Z1 e?ax to gradient magnitudes and corner
detector histograms in a number of images. We found that the histograms shown
in figure 2 are typical: for both gradients and corner detectors the exponent was
less than 1 and the exponent for the corner detector was smaller than that of the
gradients. Typical exponents were 0.7 for the derivative filter and 0.25 for the corner
detector. The scaling parameter a of the corner detector was typically larger than
that of the gradient magnitude.
3
Simple prior predicts transparency
Motivated by the qualitative statistics observed in natural images we now define a
probability distribution over images. We define the log probability of an image by
means of a probability over its gradients:
X?
?
log P (Ix , Iy ) = ? log Z ?
|?I(x, y)|? + ?c(x, y)?
(3)
x,y
with ? = 0.7, ? = 0.25. The parameter ? was determined by the ratio of the scaling
parameters in the corner and gradient distributions.
Given a candidate decomposition of an image I into I1 and I2 = I ? I1 we define
the log probability of the decomposition as the sum of the log probabilities of the
gradients of I1 and I2 . Of course this is only an approximation: we are ignoring
dependencies between the gradients across space and orientation. Although this is
a weak prior, one can ask: is this enough to predict transparency? That is, is the
most probable interpretation of figure 1a one with two layers and the most probable
decomposition of figure 1b one with a single layer?
Answering this question requires finding the global maximum of equation 3. To
gain some intuition we calculated the log probability of a one dimensional family
of solutions. We defined s(x, y) the image of a single white square in the same
location as the bottom right square in figure 1a,b. We considered decompositions
of the form I1 = ?s(x, y),I2 = I ? I1 and evaluated the log probability for values of
? between ?1 and 2.
Figure 3a shows the result for figure 1a. The most probable decomposition is the
one that agrees with the percept: ? = 1 one layer for the white square and another
for the gray square. Figure 3b shows the result for figure 1b. The most probable
decomposition again agrees with the percept: ? = 0 so that one layer is zero and
the second contains the full image.
3.1
The importance of being non Gaussian
Equation 3 can be verbally described as preferring decompositions where the total
edge and corner detector magnitudes are minimal. Would any cost function that
has this preference give the same result?
Figure 3c shows the result with ? = ? = 2 for the transparency figure (figure 1a).
This would be the optimal interpretation if the marginal histograms of edge and
corner detectors were Gaussian. Now the optimal interpretation indeed contains
two layers but they are not the ones that humans perceive. Thus the non Gaussian
nature of the histograms is crucial for getting the transparency percept. Similar
?non perceptual? decompositions are obtained with other values of ?, ? > 1.
We can get some intuition for the importance of having exponents smaller than
1 from the following observation which considers the analog of the transparency
problem with scalars. We wish to solve the equation a + b = 1 and we have a prior
over positive scalars of the form P (x).
Observation: The MAP solution to the scalar transparency problem is obtained
with a = 1, b = 0 or a = 0, b = 1 if and only if log P (x) is concave.
The proof follows directly from the definition of concavity.
160
800
I1=?
I=
I1= ?
-log(prob)
600
-log(prob)
120
160
100
I1= ?
I=
700
180
-log(prob)
140
200
I=
500
400
140
300
80
120
60
-1
100
-1
0
?
1
2
200
?
0
a
1
2
100
-1
b
?
0
1
2
c
Figure 3: a-b. negative log probability (equation 3) for a sequence of decompositions of figure 1a,b respectively. The first layer is always a single square with
contrast ? and the second layer is shown in the insets. c. negative log probability
(equation 3) for a sequence of decompositions of figure 1a with ? = ? = 2.
4
Optimization using loopy BP
Finding the most likely decomposition requires a highly nonlinear optimization. We
chose to discretize the problem and use max-product loopy belief propagation to find
the optimum. We defined a graphical model in which every node gi corresponded to
a discretization of the gradient of one layer I1 at that location gi = (gix , giy )T . For
every value of gi we defined fi which represents the gradient of the second layer at
that location: fi = (Ix , Iy )T ? gi . Thus the two gradients fields {gi }, {fi } represent
a valid decomposition of the input image I.
The joint probability is given by:
Y
1 Y
P (g) =
?i (gi )
?ijkl (gi , gj , gk , gl )
Z i
(4)
<ijkl>
where < ijkl > refers to four adjacent pixels that form a 2x2 local square.
The local potential ?i (gi ) is based on the histograms of derivative filters:
?i (gi ) = e(?|g|
?
?|f |? )/T
(5)
where T is an arbitrary system ?temperature?.
The fourway potential: ?ijkl (gi , gj , gk , gl ) is based on the histogram of the corner
operator:
?ijkl (gi , gj , gk , gl ) = e??/T (det(gi gi
T
T
+gj gjT +gk gk
+gl glT )? +det(fi fiT +fj fjT +fk fkT +fl flT )? )
(6)
To enforce integrability of the gradient fields the fourway potential is set to zero
when gi , gj , gk , gl violate the integrability constraint (cf. [3]).
The graphical model defined by equation 4 has many loops. Nevertheless motivated
by the recent results on similar graphs [2, 3] we ran the max-product belief propagation algorithm on it. The max-product algorithm finds a gradient field {g i } that
is a local maximum of equation 4 with respect to a large neighbourhood [10]. This
gradient field also defines the complementary gradient field {fi } and finally we integrate the two gradient fields to find the two layers. Since equation 4 is completely
symmetric in {f } and {g} we break the symmetry by requiring that the gradient
in a single location gi0 belong to layer 1.
In order to run BP we need to somehow discretize the space of possible gradients
at each pixel. Similar to the approach taken in [2] we use the local potentials to
Input I
Output I1
Output I2
Figure 4: Output of the algorithm on synthetic images. The algorithm effectively
searches over an exponentially large number of possible decompositions and chooses
decompositions that agree with the percept.
sample a small number of candidate gradients at each pixel. Since the local potential
penalizes non zero gradients, the most probable candidates are gi = (Ix , Iy ) and
gi = (0, 0). We also added two more candidates at each pixel gi = (Ix , 0) and
gi = (0, Iy ). With this discretization there are still an exponential number of
possible decompositions of the image. We have found that the results are unchanged
when more candidates are introduced at each pixel.
Figure 4 shows the output of the algorithm on the two images in figure 1. An
animation that illustrates the dynamics of BP on these images is available at
www.cs.huji.ac.il/ ?yweiss. Note that the algorithm is essentially searching exponentially many decompositions of the input images and knows nothing about ?X
junctions? or ?T junctions? or squares. Yet it finds the decompositions that are
consistent with the human percept.
Will our simple prior also allow us to decompose a sum of two real images ? We
first tried a one dimensional family of solutions as in figure 3. We found that for
real images that have very little texture (e.g. figure 5b) the maximal probability
solution is indeed obtained at the perceptually correct solution. However, nearly
any other image that we tried had some texture and on such images the model failed
(e.g. 5a). When there is texture in both layers, the model always prefers a one layer
decomposition: the input image plus a zero image. To understand this failure,
recall that the model prefers decompositions that have few corners and few edges.
According to the simple ?edge? and ?corner? operators that we have used, real
images have edges and corners at nearly every pixel so the two layer decomposition
has twice as many edges and corners as the one layer decomposition. To decompose
general real images we need to use more sophisticated features to define our prior.
Even for images with little texture standard belief propagation with synchronous
a
b
c
d
Figure 5: When we sum two arbitrary images (e.g. in a.) the model usually prefers
the one layer solution. This is because of the texture that results in gradients and
corners at every pixel. For real images that are relatively texture free (e.g. in b.)
the model does prefer splitting into two layers (c. and d.)
updates did not converge. Significant manual tweaking was required to get BP to
converge. First, we manually divided the input image into smaller patches and ran
BP separately on each patch. Second, to minimize discretization artifacts we used
a different number of gradient candidates at each pixel and always included the
gradients of the original images in the list of candidates at that pixel. Third, to
avoid giving too much weight to corners and edges in textured regions, we increased
the temperature at pixels where the gradient magnitude was not a local maximum.
The results are shown at the bottom of 5. In preliminary experiments we have found
that similar results can be obtained with far less tweaking when we use generalized
belief propagation to do the optimization.
5
Discussion
The percept of transparency is a paradigmatic example of the ill-posedness of vision:
the number of equations is half the number of unknowns. Nevertheless our visual
systems reliably and effectively compute a decomposition of a single image into
two images. In this paper we have argued that this perceptual decomposition may
correspond to the most probable decomposition using a simple prior over images
derived from natural scene statistics.
We were surprised with the mileage we got out of the very simple prior we used: even
though it only looks at two operators (gradients, and cornerness) it can generate
surprisingly powerful predictions. However, our experiments with real images show
that this simple prior is not powerful enough. In future work we would like to
add additional features. One way to do this is by defining features that look for
?texture edges? and ?texture corners? and measuring their statistics in real images.
A second way to approach this is to use a full exponential family maximum likelihood
algorithm (e.g. [11]) that automatically learned which operators to look at as well
as the weights on the histograms.
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[11] Song Chun Zhu, Zing Nian Wu, and David Mumford. Minimax entropy principle and its application to texture modeling. Neural Computation, 9(8):1627?
1660, 1997.
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1,271 | 2,158 | ynamic
Causal Learning
Thomas L. Griffiths
David Danks
Institute for Human & Machine Cognition
Department of Psychology
University of West Florida
Stanford University
Stanford, CA 94305-2130
Pensacola, FL 32501
[email protected]
[email protected]
Joshua B. Tenenbaum
Department of Brain & Cognitive Sciences
~AIT
Cambridge, MA 02139
[email protected]
Abstract
Current psychological theories of human causal learning and
judgment focus primarily on long-run predictions: two by
estimating parameters of a causal Bayes nets (though for different
parameterizations), and a third through structural learning. This
paper focuses on people's short-run behavior by examining
dynamical versions of these three theories, and comparing their
predictions to a real-world dataset.
1
Introduction
Currently active quantitative models of human causal judgment for single (and
sometimes multiple) causes include conditional j}JJ [8], power PC [1], and Bayesian
network structure learning [4], [9]. All of these theories have some normative
justification, and all can be understood rationally in terms of learning causal Bayes
nets. The first two theories assume a parameterization for a Bayes net, and then
perform maximum likelihood parameter estimation. Each has been the target of
numerous psychological studies (both confirming and disconfirming) over the past
ten years. The third theory uses a Bayesian structural score, representing the log
likelihood ratio in favor of the existence of a connection between the potential cause
and effect pair. Recent work found that this structural score gave a generally good
account, and fit data that could be fit by neither of the other two models [9].
To date, all of these models have addressed only the static case, in which judgments
are made after observing all of the data (either sequentially or in summary format).
Learning in the real world, however, also involves dynamic tasks, in which
judgments are made after each trial (or small number). Experiments on dynamic
tasks, and theories that model human behavior in them, have received surprisingly
little attention in the psychological community. In this paper, we explore dynamical
variants of each of the above learning models, and compare their results to a real
data set (from [7]). We focus only on the case of one potential cause, due to space
and theoretical constraints, and a lack of experimental data for the multivariate case.
2
Real-World Data
In the experiment on which we focus in this paper [7], people's stepwise acquisition
curves were measured by asking people to determine whether camouflage makes a
tank more or less likely to be destroyed. Subjects observed a sequence of cases in
which the tank was either camouflaged or not, and destroyed or not. They were
asked after every five cases to judge the causal strength of the- camouflage on a
[-100, +100] scale, where -100 and +100 respectively correspond to the potential
cause always preventing or producing the effect. The learning curves, constructed
from average strength ratings, were:
50
Positive contingent
High P(E) non-contingent
Low P(E) non-contingent
Negative contingent
Me
an
jud
gm
ent
-50
10
15
20
25
30
35
40
Trials
Figure 1: Example of learning curves
In this paper, we focus on qualitative features of the learning curves. These learning
curves can be divided on the basis of the actual contingencies in the experimental
condition. There were two contingent conditions: a positive condition in which
peE I C) = .75 (the probability of the effect given the cause) and peE I -,C) = .25,
and a negative condition where the opposite was true. There were also two noncontingent conditions, one in which peE) = .75 and one in which peE) = .25,
irrespective of the presence or absence of the causal variable. We refer to the former
non-contingent condition as having a high peE), and the latter as having a low peE).
There are two salient, qualitative features of the acquisition curves:
3
3.1
1.
For contingent cases, the strength rating does not immediately reach the
final judgment, but rather converges to it slowly; and
2.
For non-contingent cases, there is an initial non-zero strength rating when
the probability of the effect, peE), is high, followed by convergence to zero.
Parameter Estimation Theories
Conditional
~p
The conditional f1P theory predicts that the causal strength rating for a particular
factor will be (proportional to) the conditional contrast for that factor [5], [8]. The
general form of the conditional contrast for a particular potential cause is given by:
f1P C.{X} = peE I C & X) - peE I -,C & X), where X ranges over the possible states of
the other potential causes. So, for example, if we have two potential causes, C 1 and
C2 , then there are two conditional contrasts for C 1 : f1P C l.{C2} = peE I C1 & C2 ) peE I -'C 1 & C2 ) and f1P C l.{-.C2} = peE I C1 & -,C2 ) - peE I-'C1 & -,C2 ). Depending
on the probability distribution, some conditional contrasts for a potential cause may
be undefined, and the defined contrasts for a particular variable may not agree. The
conditional I1P theory only makes predictions about a potential cause when the
underlying probability distribution is "well-behaved": at least one of the conditional
contrasts for the factor is defined, and all of the defined conditional contrasts for the
factor are equal. For a single cause-effect relationship, calculation of the J1P value is
a maximum likelihood parameter estimator assuming that the cause and the
background combine linearly to predict the effect [9J.
Any long-run learning model can model sequential data by being applied to all of
the data observed up to a particular point. That is, after observing n datapoints, one
simply applies the model, regardless of whether n is "the long-run." The behavior of
such a strategy for the conditional ~p theory is shown in Figure 2 (a), and clearly
fails to model accurately the above on-line learning curves. There is no gradual
convergence to asymptote in the contingent cases, nor is there differential behavior
in the non-contingent cases.
An alternative dynamical model is the Rescorla-Wagner model [6J, which has
essentially the same form as the well-known delta rule used for training simple
neural networks. The R-W model has been shown to converge to the conditionall1P
value in exactly the situations in which the I1P theory makes a prediction [2J. The
R-W model follows a similar statistical logic as the I1P theory: J1P gives the
maximum likelihood estimates in closed-form, and the R-W model essentially
implements gradient ascent on the log-likelihood surface, as the delta rule has been
shown to do. The R-W model produces' learning curves that qualitatively fit the
learning curves in Figure 1, but suffers from other serious flaws. For example,
suppose a subject is presented with trials of A, C, and E, followed by trials with only
A and E. In such a task, called backwards blocking, the R-W model predicts that C
should be viewed as moderately causal, but human subjects rate C as non-causal.
In the augmented R-W model [10J causal strength estimates (denoted by Vi, and
assumed to start at zero) change after each observed case. Assuming that b(.x) = 1 if
X occurs on a particular trial, and 0 otherwise, then strength estimates change by the
following equation:
aiO and ail are rate parameters (saliences) applied when Ci is present and absent,
respectively, and Po and PI are the rate parameters when E is present and absent,
respectively. By updating the causal strengths of absent potential causes, this model
is able to explain many of the phenomena that escape the normal R-W model, such
as backwards blocking.
Although the augmented R-W model does not always have the same asymptotic
behavior as the regular R-W model, it does have the same asymptotic behavior in
exactly those situations in which the conditional J1P theory makes a prediction
(under typical assumptions: aiO = -ail, Po = PI, and A = 1) [2]. To determine whether
the augmented R-W model also captures the qualitative features of people's
dynamic learning, we performed a simulation in which 1000 simulated individuals
were shown randomly ordered cases that matched the probability distributions used
in [7]. The model parameter values were A = 1.0, Q{)o = 0.4, alO = 0.7, au = -0.2, Po
= PI = 0.5, with two learned parameters: Vo for the always present background cause
Co, and VI for the potential cause C I . The mean values of VI, multiplied by 100 to
match scale with Figure 1, are shown in Figure 2 (b).
(b)
(a)
50
-50
5
10
15
20
25
30
35
40
10
15
20
25
30
35
40
5
10
15
20
25
30
35
40
5
10
15
20
25
30
35
40
50
50
-50
5
(e)
5
(d)
(c)
50~
,
=:.::=:=t
10
15
20
25
30
35
40
_5~~~~
5
10
15
20
25
30
35
40
-50
Figure 2: Modeling results. (a) is the maximum-likelihood estimate of fJJ , (b) is the
augmented R-W model, (c) is the maximum-likelihood estimate of causal power, (d)
is the analogue of augmented R-W model for causal power, (e) shows the Bayesian
strength estimate with a uniform prior on all parameters, and (f) does likewise with
a beta(I,5) prior on Va. The line-markers follow the conventions of Figure 1.
Variations in A only change the response scale. Higher values of lXoo (the salience of
the background) shift downward all early values of the learning curves, but do not
affect the asymptotic values. The initial non-zero values for the non-contingent
cases is proportional in size to (alO + al r), and so if the absence of the cause is more
salient than the presence, the initial non-zero value will actually be negative.
Raising the fJ values increases the speed of convergence to asymptote, and the
absolute values of the contingent asymptotes decrease in proportion to (fJo - fJI).
For the chosen parameter values, the learning curves for the contingent cases both
gradually curve towards an asymptote, and in the non-contingent, high peE) case,
there is an initial non-zero rating. Despite this qualitative fit and its computational
simplicity, the augmented R-W model does not have a strong rational motivation. Its
only rational justification is that it is a consistent estimator of fJJ: in the limit of
infinite data, it converges to fJJ under the same circumstances that the regular (and
well-motivated) R-W model does. But it does not seem to have any of the other
properties of a good statistical estimator: it is not unbiased, nor does it seem to be a
maximum likelihood or gradient-ascent-on-log-Iikelihood algorithm (indeed,
sometimes it appears to descend in likelihood). This raises the question of whether
there might be an alternative dynamical model of causal learning that produces the
appropriate learning curves but is also a principled, rational statistical estimator.
3.2
Power PC
In Cheng's power PC theory [1], causal strength estimates are predicted to be
(proportional to) perceived causal power: the (unobserved) probability that the
potential cause, in the absence of all other causes, will produce the effect. Although
causal power cannot be directly observed, it can be estimated from observed
statistics given some assumptions. The power PC theory predicts that, when the
assumptions are believed to be satisfied, causal power for (potentially) generative or
preventive 'causes will be estimated by the following equations:
?
p _
G eneratIve:
M
C
C-1-P(EI-,C)
Preventive: p =
e
- Me
p(EI-'C)
Because the power PC theory focuses on the long-run, one can easily d'etermine
which equation to use: simply wait until asymptote, determine J1P c , and then divide
by the appropriate factor. Similar equations can also be given for interactive causes.
Note that although the preventive causal power equation yields a positive number,
we should expect people to report a negative rating for preventive causes.
As with the t:JJ theory, the power PC theory can, in the case of a single cause-effect
pair, also be seen as a maximum li).<elihood estimator for the strength parameter of a
causal Bayes net, though one with a different parameterization than for conditional
t:JJ. Generative causes and the background interact to produce the effect as though
they were a noisy-OR gate. Preventive causes combine with them as a noisy-ANDNOT gate. Therefore, if the G/s are generative causes and lj's are preventive causes,
the theory predicts: P(E) =
I} (1- Ijl1- If (1- G,)]-
As for conditional J1P, simply applying the power PC equations to the sufficient
statistics for observed sequential data does not produce appropriate learning curves.
There is no gradual convergence in the contingent cases, and there is no initial
difference in the non-contingent cases. This behavior is shown in Figure 2 (c).
Instead, we suggest using an analogue of the augmented R-W model, which uses the
above noisy-ORlAND-NOT prediction instead of the linear prediction implicit in
the augmented R-W model. Specifically, we define the following algorithm (with all
parameters as defined before), using the notational device that the C/ s are
preventive and the Cj ' s are generative:
Unlike the R-W and augmented R-W models, there is no known characterization of
the long-run behavior of this iterative algorithm. However, we can readily determine
(using the equilibrium technique of [2]) the asymptotic Vi values for' one potential
cause (and a single, always present, generative background cause). If we make the
same simplifying assumptions as in Section 3.1, then this algorithm asymptotically
computes the causal power for C, regardless of whether C is generative or
preventive. We conjecture that this algorithm also computes the causal power for
multiple potential causes.
This iterative algorithm can only be applied if one knows whether each potential
cause is potentially generative or preventive. Furthermore, we cannot determine
directionality by the strategy of the power PC theory, as we do not necessarily have
the correct t:JJ sign during the short run. However, changing the classification of Ci
from generative to preventive (or vice versa) requires only removing from (adding
to) the estimate (i) the Vi term; and (ii) all terms in which Vi was the only generative
factor. Hence, we conjecture that this algorithm can be augmented to account for
reclassification of potential causes after learning has begun.
To simulate this dynamical version of the power PC theory, we used the same setup
as in Section 3 .1 (and multiplied preventive causal power ratings by -1 to properly
scale them). The parameters for this run were: A = 1.0, lXoo = 0.1, al0 = 0.5,
all = -0.4,/30 = /31 = 0.9, and the results are shown in Figure 2 (d). Parameter
variations have the same effects as for the augmented R-W model, except that
increasing lXoo reduces the size of the initial non-zero values in the non-contingent
conditions (instead of all conditions), and absolute values of the asymptotes in all
conditions are shifted by an amount proportional to (/30 - /31),
This dynamical theory produces the right sort of learning curves for these parameter
values, and is also a consistent estimator (converging to the power PC estimate in
the limit of infinite data). But as with the augmented R-W model, there is no
rational motivation for choosing this dynamic estimator: it is not unbiased, nor
maximum likelihood, nor an implementation of gradient ascent in log-likelihood.
The theory's main (and arguably only) advantage over the augmented R-W model is
that it converges to a quantity that is more typically what subj ects estimate in longrun experiments. But it is still not what we desire from a principled dynamic model.
4
Bayesian structure learning
The learning algorithms considered thus far are based upon the idea that human
causal judgments reflect the estimated value of a strength parameter in a particular
(assumed) causal structure. Simple maximum likelihood estimation of these strength
parameters does not capture the trends in the data, and so we have considered
estimation algorithms that do not have a strong rational justification. Weare thus
led to the question of whether human learning curves can be accounted for by a
rational process. In this section, we argue that the key to forming a rational,
statistical explanation of people's dynamical behavior is to take structural
uncertainty into account when forming parameter estimates.
Complete specification of the structure of a Bayesian network includes both the
underlying graph and choice of parameterization. For example, in the present task
there are three possible relationships between a potential cause C1 and an. effect E:
generative (h+), preventive (h_), or non-existent (h o). These three possibilities can
respectively be represented by a graph with a noisy-OR parameterization, one with a
noisy-AND-NOT parameterization, and one with no edge between the potential
cause and the effect. Each possibility is illustrated schematically in Figure 3.
ho
~@
~-~
?
?
+~N;_
Figure 3: Structural hypotheses for the Bayesian model. Co is an always present
background cause, C1 is the potential cause, and E the effect. The signs of arrows
indicate positive and negative influences on the outcome.
Previous work applying Bayesian structure learning to human causal judgment
focused on people .making the decision as to which of these structures best accounts
for the observed data [9]. That work showed that the likelihood of finding a causal
relationship rose with the base rate peE) in non-contingent cases, suggesting that
structural decisions are a relevant part of the present data. However, the rating scale
of the current task seems to encourage strength judgments rather, than purely
structural decisions, because it is anchored at the endpoints by two qualitatively
different causal strengths (strong generative, strong preventive). As a result,
subj ects' causal judgments appear to converge to causal power.
Real causal learning tasks often involve uncertainty about both structure and
parameters. Thus, even when a task demands ratings of causal strength, the
structural uncertainty should still be taken into account; we do this by considering a
hierarchy of causal models. The first level of this hierarchy involves structural
uncertainty, giving equal probability to the relationship between the variables being
generative, preventive, or non-existent. As mentioned in previous sections, the
parameterizations associated with the first two models lead to' a maximum
likelihood estimate of causal power. The second level of the hierarchy addresses
uncertainty over the parameters. With a constant background and a single cause,
there are two parameters for the noisy-OR and the noisy-AND-NOT models, Va and VI. If the cause and effect are unconnected, then only Va is required. Uncertainty in
all parameters can be expressed with distributions on the unit interval.
Using this set of m9dels, we can obtain a strength rating by taking the expectation
of the strength parameter Vi associated with a causal variable over the posterior
distribution on that parameter induced by the data. This expectation is taken over
both structure and parameters, allowing both factors to influence the result. In the
two-variable case, we can write this as
1
<11 >== L J11 "111 h,D) "hi D)dW
hEHO
where H = {h+, ha, h_}. The effective value of the strength parameter is a in the
model where there is no relationship between cause and effect, and should be
negative for preventive causes. We thus have:
<VI> = P(h+)f.l+ - P(h_)f.l-
where f.l+, f.l- are the posterior means of VI under h+ and h_ respectively.
While this theory is appealing from a rational and statistical point of view, it has
computational drawbacks. All four terms in the above expression are quite
computationally intensive to compute, and require an amount of information that
increases exponentially with the number of causes. Furthermore, the number of
different hypotheses we must consider grows exponentially with the number of
potential causes, limiting its applicability for multivariate cases.
We applied this model to the data of [7J, using a uniform prior over models, and
also over parameters. The results, averaged across 200 random orderings of trials,
are shown in Figure 2 (e). The predictions are somewhat symmetric with respect to
positive and negative contingencies and high and low peE). This symmetry is a
consequence of choosing a uniform (i.e., strongly uninformative) prior for the
parameters. If we instead take a uniform prior on VI and a beta(1,5) prior on Va,
consistent with a prior belief that effects occur only rarely without an observed
cause and similar to starting with zero weights in the algorithms presented above,
we obtain the results shown in Figure 2 (t). In both cases, the curvature of the
learning curves is a consequence of structural uncertainty, and the asymptotic values
reflect the strength of causal relationships. In the contingent cases, the probability
distribution over structures rapidly transfers all of its mass to the correct hypothesis,
and the result asymptotes at the posterior mean of' VI in that model, which will be
very close to causal power. The initial non-zero ratings in the non-contingent cases
result from h+ giving a slightly better account of the data than h_, essentially due to
the non-uniform prior on Va.
This structural account is only one means of understanding the rational basis for
these learning curves. Dayan and Kakade [3] provide a statistical theory of classical
conditioning based on Bayesian estimation of the parameters in a linear model
similar to that underlying 11P. Their theory accounts for phenomena that the
classical R-W theory does not, such as backwards blocking. They also give a neural
network learning model that approximates the Bayesian estimate, and that closely
resembles the augmented R-W model considered here. Their network model can
also produce the learning curves discussed in this paper. However, because it is
based on a linear model of causal interaction, it is not a good candidate for modeling
human causal judgments, which across various studies of asymptotic behavior seem
to be more closely approximated by parameter estimates' in noisy logic gates, as
instantiated in the power PC model [1] and our Bayesian model.
5
Conclusion
In this paper, we have outlined a range of dynamical models, from computationally
simple ones (such as simply applying conditional liP to the observed datapoints) to
rationally grounded ones (such as Bayesian structure/parameter estimation).
Moreover, there seems to be a tension in this domain in trying to develop a model
that is easily implemented in an individual and scales well with additional variables,
and one that has a rational statistical basis. Part of our effort here has been aimed at
providing a set of models that seem to equally well explain human behavior, but that
have different virtues besides their fit with the data. Human causal learning might
not scale up well, or it might not be rational; further discrimination among these
possible theories awaits additional data about causal learning curves.
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Judgments: The Role of Nonpresentation of Compound Stimulus Elements." Learning and
Motivation, 25: 127-151.
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1,272 | 2,159 | Rational Kernels
Corinna Cortes Patrick Haffner Mehryar Mohri
AT&T Labs ? Research
180 Park Avenue, Florham Park, NJ 07932, USA
corinna, haffner, mohri @research.att.com
Abstract
We introduce a general family of kernels based on weighted transducers or rational relations, rational kernels, that can be used for analysis of
variable-length sequences or more generally weighted automata, in applications such as computational biology or speech recognition. We show
that rational kernels can be computed efficiently using a general algorithm of composition of weighted transducers and a general single-source
shortest-distance algorithm. We also describe several general families of
positive definite symmetric rational kernels. These general kernels can
be combined with Support Vector Machines to form efficient and powerful techniques for spoken-dialog classification: highly complex kernels
become easy to design and implement and lead to substantial improvements in the classification accuracy. We also show that the string kernels
considered in applications to computational biology are all specific instances of rational kernels.
1 Introduction
In many applications such as speech recognition and computational biology, the objects
to study and classify are not just fixed-length vectors, but variable-length sequences, or
even large sets of alternative sequences and their probabilities. Consider for example the
problem that originally motivated the present work, that of classifying speech recognition
outputs in a large spoken-dialog application. For a given speech utterance, the output of a
large-vocabulary speech recognition system is a weighted automaton called a word lattice
compactly representing the possible sentences and their respective probabilities based on
the models used. Such lattices, while containing sometimes just a few thousand transitions,
may contain hundreds of millions of paths each labeled with a distinct sentence.
The application of discriminant classification algorithms to word lattices, or more generally
weighted automata, raises two issues: that of handling variable-length sequences, and that
of applying a classifier to a distribution of alternative sequences. We describe a general
technique that solves both of these problems.
Kernel methods are widely used in statistical learning techniques such as Support Vector
Machines (SVMs) [18] due to their computational efficiency in high-dimensional feature
spaces. This motivates the introduction and study of kernels for weighted automata. We
present a general family of kernels based on weighted transducers or rational relations,
rational kernels which apply to weighted automata. We show that rational kernels can be
computed efficiently using a general algorithm of composition of weighted transducers and
a general single-source shortest-distance algorithm.
We also briefly describe some specific rational kernels and their applications to spokendialog classification. These kernels are symmetric and positive definite and can thus be
combined with SVMs to form efficient and powerful classifiers. An important benefit of
S EMIRING
Boolean
Probability
Log
Tropical
S ET
('*),+-/.,021
.,02354
Table 1: Semiring examples. ! is defined by: "#$ % &
.
our approach is its generality and its simplicity: the same efficient algorithm can be used
to compute arbitrarily complex rational kernels. This makes highly complex kernels easy
to use and helps us achieve substantial improvements in classification accuracy.
2 Weighted automata and transducers
In this section, we present the algebraic definitions and notation necessary to introduce
rational kernels.
-/6
4
-96
4
Definition 1 ([7]) A system
-9677 84 is a semiring if:
7 is a commutative
monoid with identity element ; 7 is a monoid
with
identity
element ; distributes
6
over ; and is an annihilator for : for all :<;
: !& =>:#& .
Thus, a semiring is a ring that may lack negation. Table 2 lists some familiar examples
of semirings. In addition to the Boolean semiring and the probability semiring used to
combine probabilities, two semirings often used in
' )+ applications are the log semiring which
is isomorphic to the probability semiring via a
morphism, and the tropical semiring
which is derived from the log semiring using the Viterbi approximation.
6
Definition
2 A weighted
transducer ? over a semiring is an 8-tuple ?@&
-BA
4 finite-state
A
CDE FGHJIKLMJN where: is the finite input alphabet of the transducer; C is the
FPXWOQ4 E the
finite output alphabet; E is a finite
set
-VA
XW of
4 states;
6 set of initial states; GROSE the
set of final6 states; I@OTEU
CY
E a finite set of transitions;
6
the initial weight function; and N]ZGU[
the final weight function mapping
LZF\6 [
G to .
Weighted automata can be formally defined in a similar way by simply omitting the input
or the output labels.
.
.cb
.cb
Given a transition ;^I , we denote
.cb by _a` its origin or previous
.hgjiciiJ.lstate
k and d ` its destination state or next state, and e!.,` n g its
is an element of I m
b weight.
.5npb A path fQ&
d ` b 0
&o._agJb` , q$&srtcuucuvw . We extend d and _ to paths
with consecutive
b transitions:
.kXb
by setting: d ` f &xd `
and _a` f &o_a` . The weight function e can also be extended
to paths by defining
of its constituentz
b the. weight
i a path
. k bas the -product of-/the
z z|{}weights
4
g b iciof
!
e
`
f
s
&
!
e
`
Q
!
e
`
y
transitions:
.
We
denote
by
the
set
ofA paths from
z {
-9z
z {4
z
z {
to and by y
the set of paths from to with input label ";
~"a~%2
m { and output
label
case). These
- % (transducer
-/z definitions
z { 4 can be extended to subsets 7 O?E , by:
{4
y #J"8~%2
&?????j?V?~??????/y
~"a~%2 .
A transducer
is
regulated
if
the
output
weight associated by ? to any pair of input-output
?
4
string "a~% by:
`?
b V-
4
"8J%
6
&
?
b ?- 4
? ???? ?? 1 ? 3 ? ?M?
?b 4
b
bb
L a
_ `f ?
e!` f ^N2` d ` f
-
(1)
4
is well-defined and in . ` ?
" & when y FJ"8~%2G &x? . In the following, we will
assume that all the transducers considered are regulated. Weighted transducers are closed
under , and
g Kleene-closure. In particular, - the 4 -sum and -multiplications of two
transducers ? and ?M? are defined for each pair "a~% by:
g
b ?4
gb V4
b V4
(2)
` ? ???? "8J%
&
`?
"8J% S` ` ?2? "8~%
`?
g
????
b ?-
"8J%
4
&
?
`?
1X?M1l?V1X? ? v3 ?M3v??3?
gb ?- g vg 4
b ?4
" J% ?
` ??? ?
" ?J%?
(3)
3 Rational kernels
This section introduces rational kernels, presents a general algorithm for computing them
efficiently and describes several examples of rational kernels.
3.1 Definition
Definition
3 A kernel4
-BA
CDE FA GHJIKLMJN
m:
all "a~%\;
is rational if 6 there exist a weighted
&
6 transducer ?
Z
[ such that for
over the semiring
and a function
-
4
"8J%
&
-
6
`?
b ?-
4~4
"a~%
(4)
In general, is an arbitrary function mapping to . In some cases, it may be desirable
to assume
that it is a semiring morphism as in Section 3.6.6 It is often the identity function
6
when &o and 6may be a projection
when the semiring is the cross-product of and
6 {
another semiring ( &S ?
).
Rational kernels can be naturally extended to kernels over weighted automata. In the
following, to simplify the presentation, we will restrict ourselves to the case of acyclic
weighted automata which is the case of interest for our applications, but our results apply
similarly to arbitrary
Let and be two acyclic weighted automata
6 weighted
- automata.
4
over the semiring , then
is defined by:
4 &
!
?
1 ?3
`
b V- 4
b V4
b V- 4J4
" S` ? 8
" ~% ?`
%
(5)
More generally, the results mentioned in the following for strings apply all similarly
to
6
acyclic weighted automata. Since the set of weighted transducers over a semiring is also
closed
under -sum and -product [2, 3], it follows that
6
g rational kernels over ag semiring
are closed under sum and product.
We
denote
by
? the sum and by
? the
g
g
product of two rational kernels
and <? . Let ? and ?M? be the associated transducers of
these kernels, we have for example:
-
g
?
4-
"a~%
4
&
- - g
V4 b ?~4 4
` ? ?? ? "a~% &
g -
"8J%
4
4
" ~%
? a
(6)
In learning techniques such as those based on SVMs, we are particularly interested in
positive definite symmetric kernels, which guarantee the existence of a corresponding reproducing kernel Hilbert space. Not all rational kernels are positive definite symmetric but
in the following sections we will describe some general classes of rational kernels that have
this property.
Positive definite symmetric kernels can be used to construct other families of kernels
that also meet these
conditions
[17]. Polynomial kernels of degree _ are
from
4
-~ formed
? ?? 4 with
the- expression
,
and
Gaussian
kernels
can
be
formed
as
T
:
4
4
4
? "a~% 4 &
"8J"
%2~%
r
"8J% . Since the class of symmetric positive definite kernels is closed under sum [1], the sum of two positive definite rational kernels is also
a positive definite rational kernel.
In what follows, we will focus
for computing rational kernels. The al4 on the - algorithm
4
"a~% , or ! , for any two acyclic weighted automata, is
gorithm for computing
based on two general algorithms that we briefly present: composition of weighted trans6
ducers to combine , ? , and , and a general shortest-distance algorithm in a semiring
to compute the -sum of the weights of the successful paths of the combined machine.
3.2 Composition of weighted transducers
Composition is a fundamental operation on weighted transducers that can be used
in many
6
applications to create complex
g weighted transducers from simpler ones. Let 6 be a commutative semiring and let ? and ?M? be two weighted transducersg defined over such that
the input
alphabet of ? ? coincides withg the output alphabet of ? . Then, the composition
g
of ? and ?M? is a weighted transducer ?
??? which, when it is regulated, is defined for all
a:a/1.61
0
1
a:b/0
a:a/1.2
b:a/0.69
b:a/0.69
2
b:b/0.22
0
3/0
b:b/0.92
a:b/2.3
b:a/0.51
(a)
0
a:a/0.51
1
2/0
(b)
a:a/2.81
1
a:a/0.51
a:b/3.91
4
a:b/0.92
b:a/1.2
2
3/0
b:a/0.73
(c)
g
Figure 1: (a) Weighted transducer ? over the log semiring. (b) Weighted
transducer ? ?
g
over the log semiring. (c) Construction of the result of composition ?
? ? . Initial states
are represented by bold circles, final states by double circles. Inside each circle, the first
number indicates the state number, the second, at final states only, the value of the final
weight function N at that state. Arrows represent transitions and are labeled with symbols
followed by their corresponding weight.
"8J% by [2, 3, 15, 7]: 1
`?
g
???
b V-
"8J%
4
&??
gb V-
`?
"8
4
?` ???
b ?-
4
~%
(7)
A
Note that a transducer can be viewed as a matrix over a countable set m!
CKm and composition as the corresponding matrix-multiplication. There exists a general and efficient
composition algorithm for weighted transducers which takesg advantage of the sparsity of
the inputg transducers [14, 12]. States in the compositiong ?
??? of two weighted transducers ? and ?8W ? are identified with pairs of a state of ? and a state of ?8? . Leaving aside
transitions
with inputs or outputs, the following
rule specifies how to compute a transition
g
g
of ?
? ? from appropriate transitions of ? and ? ? :2
-9z g
J:? ~e
g z 4
?
-/z g{
z {4
-~-9z g z g{ 4
g
-/z
z { 4J4
and
(8)
~e ? ? &
:hl~e ?e ? ? ?
g
z g
In
zlg{ the worst case, all transitions of ? leaving a state match all those
-~-
ofg ?a
?
leaving
g
4-
state
E
I
E ?
, thus
4~4 the space and time complexity of composition is quadratic:
I ? . Fig.(1) (a)-(c) illustrate the algorithm when applied to the transducers of Fig.(1) (a)(b) defined over the log semiring. The intersection of two weighted automata is a special
case of composition. It corresponds to the case where the input and output label of each
transition are identical.
3.3 Single-source shortest distance algorithm over a semiring
Given a weighted automaton or transducer , the shortest-distance
from state
z
of final states G is defined as the -sum of all the paths from to G :
` zXb &
?
e `f
!
? ? ???}?? ?M?
6
b
?N2` d ` f
b*b
z
to the set
(9)
when this sum is well-defined and in , which is always the case when the semiring is w closed or when is acyclic [11], the case of interest
zXb in what follows.-
There
- exists a general
4
4
algorithm for computing the shortest-distance ` in linear time E ?
( ?? I ,
where ? denotes the maximum time to compute and ? the time to compute [11].
The algorithm
is a generalization of Lawler?s algorithm [8] to the case of an arbitrary
6
semiring . It is based on a generalized relaxation of the outgoing transitions of each
state of visited in reverse topological order [11].
1
We use a matrix notation for the definition of composition as opposed to a functional notation.
This is a deliberate choice motivated by an improved readability in many applications.
2
See [14, 12] for a detailed presentation of the algorithm including the use of a transducer filter
for dealing with -multiplicity in the case of non-idempotent semirings.
?:b/3
?:a/3
b:?/2
a:?/2
b:a/1
a:b/1
b:b/0
a:a/0
?:b
?:a
b:?
a:?
?:a
a:a b:?/? ?:a/? 3
a:?/?
?:b/?
a:a
2
b:b
1
?:b
b:b
a:a
b:b
0
0/0
?:b/?
?:a/?
(a)
a:a
b:b
?:b
?:a
b:?
a:?
4
?:a
?:b
5
(b)
Figure 2: Weighted transducers associated to two rational kernels. (a) Edit-distance kernel.
(b) Gappy -gram count kernel, with = 2.
3.4 Algorithm
Let
be a rational kernel and let ? be the associated weighted transducer. Let and
A
be two acyclic weighted automata.
and may represent just two strings "8J%^;
m or
may be any other complex weighted - acceptors.
By definition of rational kernels (Eq.(5))
4
can be computed by:
and the shortest-distance (Eq.(9)),
!
.
1. Constructing theb acyclic composed transducer &
?
2. Computing ` , the shortest-distance from the initial states of to its final states
using the shortest-distance algorithm described in the previous section.
b?4
.
3. Computing
`
-
4
Thus, the total complexity of the algorithm is ?
, where ? ,
, and
denote
the size of ? , and and the worst case complexity of computing
- 4 respectively
6
time as in many applica" , "?; . If we assume that can be computed- in constant
4
tions, then the
of the computation of
is quadratic with respect to
-
complexity
4
and is: ?
.
3.5 Edit-distance kernels
Recently, several kernels, string kernels, have been introduced in computational biology for
input vectors representing biological sequences [4, 19]. String kernels are specific instances
of rational kernels. Fig.(2) (a) shows the weighted transducer over the tropical semiring
associated to a classical type of string kernel. The kernel corresponds to an edit-distance
based on a symbol substitution with cost , deletion with cost r , and insertion of cost
. All classical edit-distances can be represented by weighted transducers over the tropical
semiring [13, 10]. The kernel computation algorithm just described can be used to compute
efficiently the edit-distance of two strings or two sets of strings represented by automata. 3
3.6 Rational kernels of the type ?
?
0 g
There exists a general method for constructing a 6positive definite and symmetric rational
kernel from a weighted transducer
? when Z
[ is
6
0 g a semiring morphism ? this
implies in particular that is commutative. Denote by ?
the inverse of ? , that is the
g
transducer obtained from ? by transposing
the
input
and
output
labels of each transition.
0
Then the composed transducer &S? ?
is symmetric and, when it is regulated, defines
3
We have proved and will present elsewhere a series of results related to kernels based on the
notion of edit-distance. In particular, we have shown that the classical edit-distance with equal
costs for insertion, deletion and substitution is not negative definite [1] and that the Gaussian kernel
is not positive definite.
a positive definite symmetric rational kernel
by definition of composition:
-
"8J%
- `
4
&
b ?-
"a~%
4~4
. Indeed, since
&
-
`?
b ?-
"8
4J4?i
-
is a semiring morphism,
`?
b ?-
%2
4J4
which shows that is symmetric. For any non-negative integer d and for all "a~% we define
a symmetric kernel
by:
- "8~% 4 & - ` ? b ?- "8 4J4?i - ` ? b V- %? 4J4
g
? ucuuv A
where the sum runs over all strings of length less or equal to d . Let
g
be an arbitrary ordering of nthese
For any
and any " uucuJ"
;
m,
& strings.
define
by:
- " n ~" X4 . Then, & with defined by n &
- b ?- then matrix
X 4J4
` ? " . Thus, the eigenvalues of are all non-negative,
- which
4
' implies that - is
4
"8J% &
"8J% ,
positive definite [1]. Since is a point-wise limit of ,
is also definite positive [1].
4 Application to spoken-dialog classification
Rational kernels can be used in a variety of applications ranging from computational biology to optical character recognition. This section singles out one specific application, that
of topic classification applied to the output of a speech recognizer. We will show how the
use of weighted transducers rationalizes the design and optimization of kernels. Simple
equations and graphs replace complex diagrams and intricate algorithms often used for the
definition and analysis of string kernels.
As mentioned in the introduction, the output of a speech recognition system associated
to a speech utterance is a weighted automaton called a word lattice representing a set of
alternative sentences and their respective probabilities based on the models used. Rational
kernels help address both the problem of handling variable-length sentences and that of
applying a classification algorithm to such distributions of alternatives.
The traditional solution to sentence classification is the ?bag-of-words? approach used in
information retrieval. Because of the very large dimension of the input space, the use of
large-margin classifiers such as SVMs [6] and AdaBoost [16] was found to be appropriate
in such applications.
One approach adopted in various recent studies to measure the topic-similarity of two sentences consists of counting their common non-contiguous -grams, i.e., their common
substrings of
words with possible insertions. These -grams can be extracted explicitly from each sentence [16] or matched implicitly through a string kernel [9]. We will
show that such kernels are rational and will describe how they can be easily constructed
and computed using the general algorithms given in the previous section. More generally,
we will show how rational kernels can be used to compute the expected counts of common
non-contiguous -grams of two weighted automata and thus define the topic-similarity of
two lattices. This will demonstrate the simplicity, power, and flexibility of our framework
for the design of kernels.
4.1 Application of ?
?
0 g
kernels
Consider a word lattice over the probability
semiring. can be viewed as a probability
A
distribution y
over all strings <;
m . The expected count or number of occurrences
- 4
of
1 ,
"
y
?
y
an -gram
sequence
in
a
string
for
the
probability
distribution
is:
where 1 denotes the number of occurrences of " in . It is easy to construct a weighted
transducer ?
that outputs the set of -grams of an input lattice with their corresponding
counts. Fig.(3) (a) shows that transducer, when the alphabet is reduced to
A expected
& :? and &?r . Similarly, the transducer ? H? of Fig.(3) (b) can be used to output
non-contiguous or gappy -grams with their expected counts. 4 Long gaps are penalized
!#" $
%
4
The transducers shown in the figures of this section are all defined over the probability semiring,
thus a transition corresponding to a gap in
is weighted by .
b:?
a:?
a:a
b:b
0
b:?
a:?
b:?
a:?
2
0
a:a
b:b
1
b:?
a:?
b:?/?
a:?/?
a:a
b:b
1
(a)
a:a
b:b
2
(b)
Figure 3: -gram transducers ( = 2) defined over the probability semiring. (a) Bigram
counter transducer ?a? . (b) Gappy bigram counter ? ?v? .
with a decay factor ?L Y : a gap of length reduces the count by L . A transducer
counting variable-length -grams is obtained by simply taking the sum of these transducers: ?
? &
? t? .
and L since our results are
In the remaining of this section, we will omit the subscript
independent of the choice of these parameters. Thus the topic-similarity of two strings or
lattices and based on the expected counts of theirs g common substrings is given by:
-
4
&
`
-
?
?
0 4
b
(10)
is of the type studied in section 3.6 and thus is symmetric and positive
The kernel
definite.
4.2 Computation
The specific form of the kernel
and the associativity of composition provide us with
several alternatives for computing .
General algorithm. We can use the general
0 g algorithm described in Section 3.4 to compute
by precomputing the transducer ? ?
. Fig.(2)(b) shows the result of that composition
in the case of - gappy4 bigrams. Using that algorithm, the complexity of the -computation
of
4
the kernel
as described in the previous section is quadratic
. This
particular example has been treated by ad hoc algorithms with a similar complexity, but
that only work with strings [9, 5] and not with weighted automata or lattices.
Other factoring. Thanks to the associativity of composition, we can consider a different
factoring of the composition cascade defining :
g
-
4
&
`-
?
0 g
4
- 0
?
4Vb
first and then composing the resulting
This factoring suggests computing ? and0 ?g
transducers rather than constructing ?
. The choice between the two methods does
?
(11)
not affect the overall time complexity of the algorithm, but in practice one method may be
preferable over the other. We are showing elsewhere that in the specific case of the counting
transducers such as those described in previous
the kernel computation can in fact
-
sections,
4
be performed in linear time, that is in
, in particular by using the notion of
failure functions.
4.3 Experimental results
0 g
We used the ?
-type kernel with SVMs for call-classification in the spoken language
?
understanding (SLU) component of the AT&T How May I Help You natural dialog system.
In this system, users ask questions about their bill or calling plans and the objective is to
assign a class to each question out of a finite set of 38 classes made of call-types and named
entities such as Billing Services, or Calling Plans.
In our experiments, we used 7,449 utterances as our training data and 2,228 utterances as
our test data. The feature space corresponding to our lattice kernel is that of all possible
trigrams over a vocabulary of 5,405 words. Training required just a few minutes on a single
processor of a 1GHz Intel Pentium processor Linux cluster with 2GB of memory and 256
KB cache. The implementation took only about a few hours and was entirely based on
the FSM library. Compared to the standard approach of using trigram counts over the
best recognized sentence, our experiments with a trigram rational kernel showed a
reduction in error rate at a rejection level.
5 Conclusion
In our classification experiments in spoken-dialog applications, we found rational kernels
to be a very powerful exploration tool for constructing and generalizing highly efficient
string and weighted automata kernels. In the design of learning machines such as SVMs,
rational kernels give us access to the existing set of efficient and general weighted automata
algorithms [13]. Prior knowledge about the task can be crafted into the kernel using graph
editing tools or weighted regular expressions, in a way that is often more intuitive and easy
to modify than complex matrices or formal algorithms.
References
[1] Christian Berg, Jens Peter Reus Christensen, and Paul Ressel. Harmonic Analysis on Semigroups. Springer-Verlag: Berlin-New York, 1984.
[2] Jean Berstel. Transductions and Context-Free Languages. Teubner Studienbucher: Stuttgart,
1979.
[3] Samuel Eilenberg. Automata, Languages and Machines, volume A-B. Academic Press, 1974.
[4] David Haussler. Convolution kernels on discrete structures. Technical Report UCSC-CRL-9910, University of California at Santa Cruz, 1999.
[5] Ralf Herbrich. Learning Kernel Classifiers. MIT Press, Cambridge, 2002.
[6] Thorsten Joachims. Text categorization with support vector machines: learning with many
relevant features. In Proc. of ECML-98. Springer Verlag, 1998.
[7] Werner Kuich and Arto Salomaa. Semirings, Automata, Languages. Number 5 in EATCS
Monographs on Theoretical Computer Science. Springer-Verlag, Berlin, Germany, 1986.
[8] Eugene L. Lawler. Combinatorial Optimization: Networks and Matroids. Holt, Rinehart, and
Winston, 1976.
[9] Huma Lodhi, John Shawe-Taylor, Nello Cristianini, and Christopher J. C. H. Watkins. Text
classification using string kernels. In NIPS, pages 563?569, 2000.
[10] Mehryar Mohri. Edit-Distance of Weighted Automata. In Jean-Marc Champarnaud and Denis
Maurel, editor, Seventh International Conference, CIAA 2002, volume to appear of Lecture
Notes in Computer Science, Tours, France, July 2002. Springer-Verlag, Berlin-NY.
[11] Mehryar Mohri. Semiring Frameworks and Algorithms for Shortest-Distance Problems. Journal of Automata, Languages and Combinatorics, 7(3):321?350, 2002.
[12] Mehryar Mohri, Fernando C. N. Pereira, and Michael Riley. Weighted automata in text and
speech processing. In ECAI-96 Workshop, Budapest, Hungary. ECAI, 1996.
[13] Mehryar Mohri, Fernando C. N. Pereira, and Michael Riley. The Design Principles of a
Weighted Finite-State Transducer Library. Theoretical Computer Science, 231:17?32, January
2000. http://www.research.att.com/sw/tools/fsm.
[14] Fernando C. N. Pereira and Michael D. Riley. Speech recognition by composition of weighted
finite automata. In Emmanuel Roche and Yves Schabes, editors, Finite-State Language Processing, pages 431?453. MIT Press, Cambridge, Massachusetts, 1997.
[15] Arto Salomaa and Matti Soittola. Automata-Theoretic Aspects of Formal Power Series.
Springer-Verlag: New York, 1978.
[16] Robert E. Schapire and Yoram Singer. Boostexter: A boosting-based system for text categorization. Machine Learning, 39(2/3):135?168, 2000.
[17] Bernhard Scholkopf and Alex Smola. Learning with Kernels. MIT Press: Cambridge, MA,
2002.
[18] Vladimir N. Vapnik. Statistical Learning Theory. John Wiley & Sons, New-York, 1998.
[19] Chris Watkins. Dynamic alignment kernels. Technical Report CSD-TR-98-11, Royal Holloway,
University of London, 1999.
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1,273 | 216 | Rule Representations in a Connectionist Chunker
Rule Representations in a Connectionist Chunker
David S. Touretzky
Gillette Elvgren
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
m
ABSTRACT
We present two connectionist architectures for chunking of symbolic
rewrite rules. One uses backpropagation learning, the other competitive
learning. Although they were developed for chunking the same sorts
of rules, the two differ in their representational abilities and learning
behaviors.
1 INTRODUCTION
Chunking is a process for generating, from a sequence of if-then rules, a more complex
rule that accomplishes the same task in a single step. It has been used to explain incremental human perfonnance improvement in a wide variety of cognitive, perceptual, and
motor tasks (Newell, 1987). The SOAR production system (Laird, Newell, & Rosenbloom, 1987) is a classical AI computer program that implements a "unified theory of
cognition" based on chunking.
SOAR's version of chunking is a symbolic process that examines the working memory
trace of rules contributing to the chunk. In this paper we present two connectionist
rule-following architectures that generate chunks a different way: they use incremental
learning procedures to infer the environment in which the chunk should fire. The first
connectionist architecture uses backpropagation learning, and has been described previously in (Touretzky, 1989a). The second architecture uses competitive learning. It
exhibits more robust behavior than the previous one, at the cost of some limitations on
the types of rules it can learn.
The knowledge to be chunked consists of context-sensitive rewrite rules on strings. For
example, given the two rules
431
432
Touretzky and Elvgren
RI:
R2:
"change D to B when followed by E"
"change A to C when followed by B"
the model would go through the following derivation: ADE - (Rule RI) ABE - (Rule
R2) CBE. Rule RI's firing is what enables rule R2 to fire. The model detects this and
formulates a chunked rule (RI-R2) that can accomplish the same task in a single step:
R I-R2:
AD - CB I _ E
Once this chunk becomes active, the derivation will be handled in a single step, this way:
ADE - (Chunk RI-R2) CBE. The chunk can also contribute to the formation of larger
chunks.
2
CHUNKING VIA BACKPROPAGATION
Our first experiment, a three-layer backpropagation chunker, is shown in Figure 1. The
input layer is a string buffer into which symbols are shifted one at a time, from the right.
The output layer is a "change buffer" that describes changes to be made to the string.
The changes supported are deletion of a segment, mutation of a segment, and insertion
of a new segment. Combinations of these changes are also permitted.
Rules are implemented by hidden layer units that read the input buffer and write changes
(via their a connections) into the change buffer. Then separate circuitry, not shown in
the figure, applies the specified changes to the input string to update the state of the input
buffer. The details of this string manipulation circuitry are given in (Touretzky, 1989b;
Touretzky & Wheeler, 1990).
We will now go through the ADE derivation in detail. The model starts with an empty
input buffer and two rules: R I and R2.1 After shifting the symbol A into the input buffer,
no rule fires-the change buffer is all zeros. After shifting in the D, the input buffer
contains AD, and again no rule fires. After shifting in the E the input buffer contains
ADE, and rule R I fires, writing a request in the change buffer to mutate input segment 2
(counting from the right edge of the buffer) to a B. The input buffer and change buffer
states are saved in temporary buffers, and the string manipulation circuitry derives a new
input buffer state, ABE. This now causes rule R2 to fire. 2 It writes a request into the
change buffer to mutate segment 3 to a C. Since it was RI's firing that triggered R2,
the conditions exist for chunk formation. The model combines RI's requested change
with that of R2, placing the result in the "chunked change buffer" shown on the right in
Figure I. Backpropagation is used to teach the hidden layer that when it sees the input
buffer pattern that triggered RI (ADE in this case) it should produce via its f3 connections
the combined change pattern shown in the chunked change buffer.
The model's training is "self-supervised:" its own behavior (its history of rule firings)
is the source of the chunks it acquires. It is therefore important that the chunking
1 The initial rule set is installed by an external teacher using backpropagation.
2Note that Rl applies to positions 1 and 2 of the buffer (counting from the right edge), while R2 applies to
positions 2 and 3. Rules are represented in a position-independent manner, allowing them to apply anywhere
in the buffer that their environment is satisfied. The mechanism for achieving this is explained in (Touretzky.
1989a).
Rule Representations in a Connectionist Chunker
Chunked Change:
Change Buffer:
cur: [change seg. 3 to "C"]
[ change seg. 2 to "B" and
change seg . 3 to "C" ]
prey: [change seg . 2 to "B"]
next:
cur:
prey:
tc
B
E
IA IB I E I
A D E
Figure 1: Architecture of the backpropagation chunker.
process not introduce any behavioral errors during the intennediate stages of learning,
since no external teacher is present to force the model back on track should its rule
representations become corrupted. The original rules are represented in the a connections
and the chunked rules are trained using the j3 connections, but the two rule sets share the
same hidden units and input connections, so interference can indeed occur. The model
must actively preserve its a rules by continuous rehearsal: after each input presentation,
backpropagation learning on a contrast-enhanced version of the a change pattern is used
to counteract any interference caused by training on the j3 patterns. Eventually, when the
j3 weights have been learned correctly, they can replace the a weights.
The parameters of the model were adjusted so that the initial rules had a distributed
representation in the hidden layer, Le., several units were responsible for implementing
each rule. Analysis of the hidden layer representations after chunking revealed that the
model had split off some of the RI units to represent the RI-R2 chunk; the remainder
were used to maintain the original RI rule.
The primary flaw of this model is fragility. Constant rehearsal of the original rule set, and
low learning rates, are required to prevent the a rules from being corrupted before the j3
rules have been completely learned. Furthermore, it is difficult to form long rule chains,
because each chunk further splits up the hidden unit population. Repeated splitting and
retraining of hidden units proved difficult, but the model did manage to learn an RI-R2R3 chunk that supersedes the RI-R2 chunk, so that ADE mutates directly to CFE. The
third rule was:
R3:
B~F/ C _E
"change B to F when between C and En
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Touretzky and Elvgren
Output Change Pattern
Competitive
Rule Units
Input String Buffer
Input Change Pattern
(Training Only)
Figure 2: Architecture of the competitive learning chunker.
3
CHUNKING VIA COMPETITIVE LEARNING
Our second chunker, shown in Figure 2, minimizes interference between rules by using
competitive learning to assign each rule a dedicated unit. As in the previous case, the
model is taught its initial rules by showing it input buffer states and desired change buffer
states. Chunks are then formed by running strings through the input buffer and watching
for pairs of rules that fire sequentially. The model recruits new units for the chunks
and teaches them to produce the new change buffer patterns (formed by composing the
changes of the two original rules) in appropriate environments.
A number of technical problems had to be resolved in order to make this scheme work.
First, we want to assign a separate unit to each rule, but not to each training example;
otherwise the model will use too many units and not generalize well. Second, the
encoding for letters we chose (see Table 1) is based on a Cartesian product, and so input
patterns are highly overlapping and close together in Hamming space. This makes the
job of the competitive learning algorithm more difficult. Third, there must be some way
for chunks to take priority over the component rules from which they were fonned, so
that an input sequence like ADE fires the chunk RI-R2 rather than the original rule Rl.
As we trace through the operation of the chunker we will describe our solutions to these
problems.
Rule units in the competitive layer are in one of three states: inactive (waiting to be
recruited), plastic (currently undergoing learning), and active (weights finalized; ready to
compete and fire.) They also contain a simple integrator (a counter) that is used to move
them from the plastic to the active state. Initially all units are inactive and the counter
Rule Representations in a Connectionist Chunker
Table 1: Input code for both chunking models.
A
B
C
D
E
F
1
1
1
0
0
0
0
0
0
1
1
1
1
0
0
0
1
1
0
0
0
0
0
0
1
1
0
0
0
1
is zero. As in any competitive learning scheme, the rule units' input weights are kept
normalized to unit vectors (Rumelhart & Zipser, 1986).
When the teacher presents a novel instance, we must determine if there is already some
partially-trained rule unit whose weights should be shaped by this instance. Due to our
choice of input code, it is not possible to reliably assign training instances to rule units
based solely on the input pattern, because "similar" inputs (close in Hamming space)
may invoke entirely different rules. Our solution is to use the desired change pattern as
the primary index for selecting a pool of plastic rule units; the input buffer pattern is
then used as a secondary cue to select the most strongly activated unit from this pool.
Let's consider what happens with the training example DE - BE. The desired change
pattern "mutate segment 2 to a B" is fed to the competitive layer, and the network looks
for plastic rule units whose change patterns exactly match the desired pattern. 3 If no such
unit is found, one is allocated from the inactive pool, its status is changed to "plastic,"
its input buffer weights are set to match the pattern in the input buffer, and its change
pattern input and and change pattern output weig.hts are set according to the desired
change pattern.
Otherwise, if a pool of suitable plastic units already exists, the input pattern DE is
presented to the competitive layer and the selected plsatic units compete to see which
most closely matches the input The winning unit's input buffer weights are then adjusted
by competitive learning to move the weight vector slightly closer to this input buffer
vector. The unit's counter is also bumped.
Several presentations are normally required before a rule unit's input weights settle into
their correct values, since the unit must determine from experience which input bit values
are significant and which should be ignored. For example, rule S 1 in Table 2 (the asterisk
indicates a wildcard) can be learned from the training instances ACF and ADF, since as
Table 1 shows, the letters C and D in the second segment have no bits in common.
Therefore the learning algorithm will concentrate virtually all of the weight vector's
magnitude in the connections that specify "A" as the first segment and "F' as the third.
Each time a rule unit's weights are adjusted by competitive learning, its counter is in3The units' thresholds are raised so that they can only become active if their weight vectors match the input
change buffer vector exactly.
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Touretzky and Elvgren
cremented. When the counter reaches a suitable value (currently 25), the unit switches
from the plastic to the active state. It is now ready to compete with other units for the
right to fire; its weights will not change further.
We now consider the formation of the model's first chunk. Assume that rules RI and
R2 have been acquired successfully. The model is trained by running random strings
through the input buffer and looking for sequences of rule firings. Suppose the model is
presented with the input string BFDADE. RI fires, producing BFDABE; this then causes
R2 to fire, producing BFDCBE. The model proceeds to form a chunk. The combined
change pattern specifies that the penultimate segment should be mutated to "B," and the
antepenultimate to "C." Since no plastic rule unit's change pattern weights match this
change, a fresh unit is allocated and its change buffer weights are set to reproduce this
pattern. The unit's input weights are set to detect the pattern BFDADE.
After several more examples of the RI-R2 firing sequence, the competitive learning
algorithm will discover that the first three input buffer positions can hold anything at all,
but the last three always hold ADE. Hence the weight vector will be concentrated on the
last three positions. When its counter reaches a value of 25, the rule unit will switch to
the active state.
Now consider the next time an input ending in ADE is presented. The network is in
performance mode now, so there is nothing in the input change buffer; the model is
looking only at the input string buffer. The RI unit will be fully satisfied by the input;
its normalized weight vector concentrates on just the last two positions, "DE," which
match exactly. The RI-R2 unit will also be fully satisfied; its normalized weight vector
looks for the sequence ADE. The latter unit is the one we want to win the competition.
We achieve this by scaling the activation function of competitive units by an additional
factor: the degree of distributedness of the weight vector. Units that distribute their input
weight over a larger number of connections likely represent complex chunks, and should
therefore have their activation boosted over rules with narrowly focused input vectors.
Once the unit encoding the RI-R2 chunk enters the active state, its more distributed input
weights assure that it will always win over the RI unit for an input like ADE. The RI
unit may still be useful to keep around, though, to handle a case like FDE -+ FBE that
does not trigger R2.
Sometimes a new chunk is learned that covers the same length input as the old, e.g.,
chunk RI-R2-R3 that maps ADE -+ CFE looks at exactly the same input positions as
chunk RI-R2. We therefore introduce one additional term into the activation function.
As part of the learning process, active units that contribute to the formation of a new
chunk are given a permanent, very small inhibitory bias. This ensures that RI-R2 will
always lose the competition to RI-R2-R3 once that chunk goes from plastic to active,
even though their weights are distributed to an equal degree.
Another special case that needs to be handled is when the competitive algorithm wrongly
splits a rule between two plastic units in the same pool, e.g., one unit might be assigned
the cases {A,B,C} ADE, and the other the cases {D.E,F} ADE. (In other words, one unit
looks for the bit pattern IOxxx in the first position, and the other unit looks for Olxxx.)
Rule Representations in a Connectionist Chunker
This is bad because it allows the weights of each unit to be more distributed than they need
to be. To correct the problem, whenever a plastic unit wins a competition our algorithm
makes sure that the nearest runner up is considerably less active than the winner. If its
activation is too high, the runner up is killed. This causes the survivor to readjust its
weights to describe the rule correctly, i.e., it will look for the input pattern ADE. If the
runner up was killed incorrectly (meaning it is really needed for some other rule), it will
be resurrected in response to future examples.
Finally, active units have a decay mechanism that is kept in check by the unit's firing
occasionally. If a unit does not fire for a long time (200 input presentations), its weights
decay to zero and it returns to the inactive state. This way. units representing chunks that
have been superseded will eventually be recycled.
4 DISCUSSION
Each of the two learning architectures has unique advantages. The backpropagation
learner can in principle learn arbitrarily complex rules. such as replacing a letter with
its successor. or reversing a subset of the input string. Its use of a distributed rule
representation allows knowledge of rule RI to participate in the forming of the RI-R2
chunk. However. this representation is also subject to interference effects. and as is often
the case with backprop. learning is slow.
The competitive architecture learns very quickly. It can form a greater number of chunks.
and can handle longer rule chains. since it avoids inteference by assigning a dedicated
unit to each new rule it learns.
Both learners are sensitive to changes in the distribution of input strings; new chunks
can form any time they are needed. Chunks that are no longer useful in the backprop
model will eventually fade away due to non-rehearsal; the hidden units that implement
these chunks will be recruited for other tasks. The competitive chunker uses a separate
decay mechanism to recycle chunks that have been superseded.
This work shows that connectionist techniques can yield novel and interesting solutions
to symbol processing problems. Our models are based on a sequence manipulation architecture that uses a symbolic description of the changes to be made (via the change
buffer), but the precise environments in which rules apply are never explicitly represented. Instead they are induced by the learning algorithm from examples of the models'
own behavior. Such self-supervised learning may play an important role in cognitive
development. Our work shows that it is possible to correctly chunk knowledge even
when one cannot predict the precise environment in which the chunks should apply.
Acknowledgements
This research was supported by a contract from Hughes Research Laboratories, by the
Office of Naval Research under contract number NOOOI4-86-K-0678. and by National
Science Foundation grant EET-8716324. We thank Allen Newell. Deirdre Wheeler. and
Akihiro Hirai for helpful discussions.
437
438
Touretzky and Elvgren
Table 2: Initial rule set for the competitive learning chunker.
SI:
S2:
S3:
S4:
S5:
A"'F
-+
BD
-+
{D,E,F}*E -+
{B,E}B
-+
{A,D}C -+
B*F
BF
{A,B,C}*A
CB
{C,F}C
Table 3: Chunks formed by the competitive learning chunker.
Chunk
EA*F -+ CB*F
ABD -+ CBF
AADF-+ CBFF
BE*E -+ CB*A
DEB -+ FEB
(Component Rules)
(SI,S4)
(SI,S2,S4)
(S I,S2,S I,S4)
(S3,S4)
(S4,S5)
Rererences
Laird, J. E., Newell, A., and Rosenbloom, P. S. (1987) Soar: An architecture for general
intelligence. Artificial Intelligence 33(1):1-64.
Newell, A. (1987) The 1987 William James Lectures: Unified Theories of Cognition.
Given at Harvard University.
Rurnelhart, D E., and Zipser, D. (1986) Feature discovery by competitive learning. In D.
E. Rumelhart and J. L. McClelland (eds.), Parallel Distributed Processing: Explorations
in the Microstructure oj Cognition. Cambridge, MA: MIT Press.
Touretzky. D. S. (1989a) Chunking in a connectionist network. Proceedings of the
Eleventh Annual Conference of the Cognitive Science Society, pp. 1-8. Hillsdale. NJ:
Erlbaum.
Touretzky, D. S. (1989b) Towards a connectionist phonology: the "many maps" approach to sequence manipulation. Proceedings of the Eleventh Annual Conference of the
Cognitive Science Society. pp. 188-195. Hillsdale. NJ: Erlbaurn.
Touretzky. D. S., and Wheeler. D. W. (1990) A computational basis for phonology. In D.
S. Touretzky (ed.), Advances in Neural Information Processing Systems 2. San Mateo.
CA: Morgan Kaufmann.
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1,274 | 2,160 | On the Dirichlet Prior and Bayesian
Regularization
Harald Steck
Artificial Intelligence Laboratory
Massachusetts Institute of Technology
Cambridge, MA 02139
[email protected]
Tommi S. Jaakkola
Artificial Intelligence Laboratory
Massachusetts Institute of Technology
Cambridge, MA 02139
[email protected]
Abstract
A common objective in learning a model from data is to recover
its network structure, while the model parameters are of minor interest. For example, we may wish to recover regulatory networks
from high-throughput data sources. In this paper we examine how
Bayesian regularization using a product of independent Dirichlet
priors over the model parameters affects the learned model structure in a domain with discrete variables. We show that a small
scale parameter - often interpreted as "equivalent sample size" or
"prior strength" - leads to a strong regularization of the model
structure (sparse graph) given a sufficiently large data set. In particular, the empty graph is obtained in the limit of a vanishing scale
parameter. This is diametrically opposite to what one may expect
in this limit, namely the complete graph from an (unregularized)
maximum likelihood estimate. Since the prior affects the parameters as expected, the scale parameter balances a trade-off between
regularizing the parameters vs. the structure of the model. We
demonstrate the benefits of optimizing this trade-off in the sense
of predictive accuracy.
1
Introduction
Regularization is essential when learning from finite data sets. In the Bayesian approach, regularization is achieved by specifying a prior distribution over the parameters and subsequently averaging over the posterior distribution. This regularization
provides not only smoother estimates of the parameters compared to maximum
likelihood but also guides the selection of model structures.
It was pointed out in [6] that a very large scale parameter of the Dirichlet prior can
degrade predictive accuracy due to severe regularization of the parameter estimates.
We complement this discussion here and show that a very small scale parameter
can lead to poor over-regularized structures when a product of (conjugate) Dirichlet
priors is used over multinomial conditional distributions (Section 3). Section 4
demonstrates the effect of the scale parameter and how it can be calibrated. We
focus on the class of Bayesian network models throughout this paper.
2
Regularization of Parameters
We briefly review Bayesian regularization of parameters. We follow the assumptions outlined in [6] : multinomial sample, complete data, parameter modularity,
parameter independence, and Dirichlet prior. Note that the Dirichlet prior over
the parameters is often used for two reasons: (1) the conjugate prior permits analytical calculations, and (2) the Dirichlet prior is intimately tied to the desirable
likelihood-equivalence property of network structures [6]. The Dirichlet prior over
the parameters 8' I11"i is given by
(1)
where 8Xi l11"i pertains to variable X i in state Xi given that its parents IIi are in joint
state 'Tri . The number of variables in the domain is denoted by n, and i = 1, ... , n.
The normalization terms in Eq. 1 involve the Gamma function r(?). There are
a number of approaches to specifying the positive hyper-parameters O:Xi ,11"i of the
Dirichlet prior [2, 1, 6] ; we adopt the common choice,
(2)
where p is a (marginal) prior distribution over the (joint) states, as this assignment
ensures likelihood equivalence of the network structures [6]. Due to lack of prior
knowledge, p is often chosen to be uniform, p(Xi,'Tri ) = 1/ (IXil?IIIil), where lXii ,
IIIi l denote the number of (joint) states [1]. The scale parameter 0: of the Dirichlet
prior is positive and independent of i, i.e. , 0: = L Xi ,11"i O:Xi ,11"i '
The average parameter value OXi l11" i ' which typically serves as the regularized parameter estimate given a network structure m , is given by
o
= E p(Ox i l ~i I D,m) [ 8
]=
Xi l11"i
Xi l11"i -
+
i, 11"i
+ O:X
'
Q
N Xi ,11"i
N
7ri
(3)
7ri
where N Xi ,11"i are the cell-counts from data D; E[?] is the expectation. Positive
hyper-parameters O:X i, 11"i lead to regularized parameter estimates, i.e., the estimated
parameters become "smoother" or " less extreme" when the prior distribution p is
close to uniform. An increasing scale parameter 0: leads to a stronger regularization, while in the limit 0: -+ 0, the (unregularized) maximum likelihood estimate is
obtained, as expected.
3
Regularization of Structure
In the remainder of this paper, we outline effects due to Bayesian regularization of
the Bayesian network structure when using a product of Dirichlet priors. Let us
briefly introduce relevant notation.
In the Bayesian approach to structure learning, the posterior probability of the
network structure m is given by p(mID) = p(Dlm)p(m)/p(D), where p(D) is the
(unknown) probability of given data D , and p(m) denotes the prior distribution over
the network structures; we assume p(m) > 0 for all m. Following the assumptions
outlined in [6], including the Dirichlet prior over the parameters 8, the marginal
likelihood p(Dlm) = Ep(O lm) [p(Dlm , 8)] can be calculated analytically. Pretending
that the (i.i.d.) data arrived in a sequential manner , it can be written as
N
p(Dlm) =
II II
n
N(k-l)
k = l i=l
N 7r ik
+ 0:
X:(':~l)
k
k
Xi ,11"i ,
+ O:11"k
i
(4)
where N(k-l) denotes the counts implied by data D(k-l) seen before step k along
the sequence (k = 1, ... , N). The (joint) state of variable Xi and its parents IIi
occurring in the kth data point is denoted by xf, 7rf. In Eq. 4, we also decomposed
the joint probability into a product of conditional probabilities according to the
Bayesian network structure m. Eq. 4 is independent of the sequential ordering of
the data points, and the ratio in Eq. 3 corresponds to the one in Eq. 4 when based
on data D(k - l) at each step k along the sequence.
3.1 Limit of Vanishing Scale-Parameter
This section is concerned with the limit of a vanishing scale parameter of the Dirichlet prior, a -+ O. In this limit Bayesian regularization depends crucially on the
number of zero-cell-counts in the contingency table implied by the data, or in other
words, on the number of different configurations (data points) contained in the
data. Let the Effective Number of Parameters (EP) be defined as
n
dk';) =
l: [l: I(Nxi,1rJ - l: I(N1rJ ],
(5)
where N Xi ,1ri' N1ri are the (marginal) cell counts in the contingency table implied
by data D; m refers to the Bayesian network structure, and 1(?) is an indicator
function such that I(z) = 0 if z = 0 and I(z) = 1 otherwise. When all cell
counts are positive, EP is identical to the well-known number of parameters (P),
dk';) = m ) = L:i(IXil - l)IIIil, which play an important role in regularizing the
learned network structure. The key difference is that EP accounts for zero-cellcounts implied by the data.
Let us now consider the behavior of the marginal likelihood (cf. Eq. 4) in the limit
of a small scale parameter a. We find
Proposition 1: Under the assumptions concerning the prior distribution outlined
in Section 2, the marginal likelihood of a Bayesian network structure m vanishes in
the limit a -+ 0 if the data D contain two or more different configurations. This
property is independent of the network structure. The leading polynomial order is
given by
d(=l
p(Dlm) "-' a EP
as a -+ 0,
(6)
4
which depends both on the network structure and the data. However, the dependence
on the data is through the number of different data points only. This holds independently of a particular choice of strictly positive prior distributions P(Xi ' IIi). If
the prior over the network structures is strictly positive, this limiting behavior also
holds for the posterior probability p( miD) .
In the following we give a derivation of Proposition 1 that also facilitates the intuitive
understanding of the result. First, let us consider the behavior of the Dirichlet
distribution in the limit a -+ O. The hyper-parameters a X i , 1r i vanish when a -+ 0,
and thus the Dirichlet prior converges to a discrete distribution over the parameter
simplex in the sense that the probability mass concentrates at a particular, randomly
chosen corner of the simplex containing B. I1ri (cf. [9]). Since the randomly chosen
points (for different 7ri, i) do not change when sampling (several) data points from
the distribution implied by the model , it follows immediately that the marginal
likelihood of any network structure vanishes whenever there are two or more different
configurations contained in the data.
This well-known fact also shows that the limit a -+ 0 actually corresponds to a very
strong prior belief [9, 12]. This is in contrast to many traditional interpretations
where the limit a -+ 0 is considered as "no prior information", often motivated
by Eq. 3. As pointed out in [9, 12], the interpretation of the scale parameter a
as "equivalent sample size" or as the" strength" of prior belief may be misleading,
particularly in the case where O:X i, 1ri < 1 for some configurations Xi, 7ri. A review
of different notions of "noninformative" priors (including their limitations) can be
found in [7]. Note that the noninformative prior in the sense of entropy is achieved
by setting O:Xi , 1ri = 1 for each Xi, 7ri and for all i = 1, ... , n. This is the assignment
originally proposed in [2]; however , this assignment generally is inconsistent with
Eq. 2, and hence with likelihood equivalence [6].
In order to explain the behavior of the marginal likelihood in leading order of the
scale parameter 0:, the properties of the Dirichlet distribution are not sufficient by
themselves. Additionally, it is essential that the probability distribution described
by a Bayesian network decomposes into a product of conditional probabilities, and
that there is a Dirichlet prior pertaining to each variable for each parent configuration. All these Dirichlet priors are independent of each other under the standard
assumption of parameter independence. Obviously, the ratio (for given k and i) in
Eq. 4 can only vanish in the limit 0: --+ 0 if N(~ - ~ = 0 while N(~- l) > 0; in other
Xi , 7r i
7ri
words, the parent-configuration 7rf must already have occurred previously along
the sequence (7rf is "old"), while the child-state xf occurs simultaneously with this
parent-state for the first time (xf, 7rf is "new"). In this case, the leading polynomial order of the ratio (for given k and i) is linear in 0:, assuming P(Xi' IIi) > 0;
otherwise the ratio (for given k and i) converges to a finite positive value in the
limit 0: --+ O. Consequently, the dependence of the marginal likelihood in leading
polynomial order on 0: is completely determined by the number of different configurations in the data. It follows immediately that the leading polynomial order in
0: is given by EP (d. Eq. 5). This is because the first term counts the number of
all the different joint configurations of Xi , IIi in the data, while the second term
ensures that EP counts only those configurations where (xf, 7rf) is " new" while 7rf
is "old".
Note that the behavior of the marginal likelihood in Proposition 1 is not entirely
determined by the network structure in the limit 0: --+ 0, as it still depends on the
data. This is illustrated in the following example. First, let us consider two binary
variables, Xo and Xl, and the data D containing only two data points, say (0,0)
and (1,1). Given data D, three Dirichlet priors are relevant regarding graph ml,
Xo --+ Xl, but only two Dirichlet priors pertain to the empty graph, mo. The
resulting additional "flexibility" due to an increased number of priors favours more
complex models: p(Dlmd ~ 0:, while p(Dlmo) ~ 0: 2 . Second, let us now assume
that all possible configurations occur in data D. Then we still have p(Dlmo) ~ 0: 2
for the empty graph. Concerning graph ml, however , the marginal likelihood now
also involves the vanishing terms due to the two priors pertaining to BX1 lxo =o and
BXl lxo=l, and it hence becomes p(Dlmd ~ 0: 3 .
This dependence on the data can be formalized as follows. Let us compare the
marginal likelihoods of two graphs, say m+ and m - . In particular, let us consider
two graphs that are identical except for a single edge, say A +- B between the
variables A and B. Let the edge be present in graph m+ and absent in m-. The
fact that the marginal likelihood decomposes into terms pertaining to each of the
variables (d. Eq. 4) entails that all the terms regarding the remaining variables
cancel out in the Bayes factor p(Dlm+)/p(Dlm-), which is the standard relative
Bayesian score. With the definition of the Effective Degrees of Freedom (EDF)l
(7)
we immediately obtain from Proposition 1 that p(Dlm+)/p(Dlm- ) ~
INote that EDF are not necessarily non-negative.
o:dEDF
in the
limit a -+ 0, and hence
Proposition 2: Let m+ and m- be the two network structures as defined above.
Let the prior belief be given according to Eq. 2. Then in the limit a -+ 0:
{-oo
I p(Dlm+)
if d EDF > 0,
( )
og p(Dlm- ) -+
+00 if dEDF < O.
8
The result holds independently of a particular choice of strictly positive prior distributions P(Xi' IIi). If the prior over the network structures is strictly positive, this
limiting behavior also holds for the posterior ratio.
A positive value of the log Bayes factor indicates that the presence of the edge
A f- B is favored , given the parents IIA ; conversely, a negative relative score
suggests that the absence of this edge is preferred. The divergence of this relative
Bayesian score implies that there exists a (small) positive threshold value ao > 0
such that, for any a < ao, the same graph(s) are favored as in the limit.
Since Proposition 2 applies to every edge in the network, it follows immediately
that the empty (complete) graph is assigned the highest relative Bayesian score
when EDF are positive (negative). Regularization of network structure in the case
of positive EDF is therefore extreme, permitting only the empty graph. This is
precisely the opposite of what one may have expected in this limit, namely the
complete graph corresponding to the unregularized maximum likelihood estimate
(MLE). In contrast, when EDF are negative, the complete graph is favored. This
agrees with MLE.
Roughly speaking, positive (negative) EDF correspond to large (small) data sets.
It is thus surprising that a small data set, where one might expect an increased
restriction on model complexity, actually gives rise to the complete graph, while
a large data set yields the - most regularized - empty graph in the limit a -+ O.
Moreover, it is conceivable that a "medium" sized data set may give rise to both
positive and negative EDF. This is because the marginal contingency tables implied
by the data with respect to a sparse (dense) graph may contain a small (large)
number of zero-cell-counts. The relative Bayesian score can hence become rather
unstable in this case, as completely different graph structures are optimal in the
limit a -+ 0, namely graphs where each variable has either the maximal number of
parents or none.
Note that there are two reasons for the hyper-parameters a Xi , i to take on small
values (cf. Eq. 2): (1) a small equivalent sample size a, or (2) a large number of
joint states, i.e. IXi l? IIIil ? a , due to a large number of parents (with a large
number of states). Thus, these hyper-parameters can also vanish in the limit of a
large number of configurations (x , 1f) even though the scale parameter a remains
fixed. This is precisely the limit defining Dirichlet processes [4], which, analogously,
produce discrete samples. With a finite data set and a large number of joint configurations, only the typical limit in Proposition 2 is possible. This follows from the
fact that a large number of zero-cell-counts forces EDF to be negative. The surprising behavior implied by Proposition 2 therefore does not carryover to Dirichlet
processes. As found in [8], however, the use of a product of Dirichlet process priors
in non parametric inference can also lead to surprising effects.
When dEDF = 0, it is indeed true that the value of the log Bayes factor can converge
to any (possibly finite) value as a -+ O. Its value is determined by the priors
P(Xi ' IIi), as well as by the counts implied by the data. The value of the Bayes
factor can be therefore easily set by adjusting the prior weights p(Xi' 1fi).
1f
3.2
Large Scale-Parameter
In the other limiting case, where a -+ 00 , the Bayes factor approaches a finite value,
which in general depends on the given data and on the prior distributions p(Xi' IIi).
lBF
2
1.5
1
0.5
-0.5
-1
z=3
z=o
~
100
150
200
250
300
a
Figure 1: The log Bayes factor (lBF) is depicted as a function of the scale parameter
It is assumed that the two variables A and B are binary and have no parents;
and that the "data" imply the contingency table: NA = O,B= O = NA = l,B = l = 10 + z
and NA=l,B=O = NA=O,B=l = 10 - z, where z is a free parameter determining the
statistical dependence between A and B. The prior p(Xi,II i ) was chosen to be
uniform.
0:.
This can be seen easily by applying the Stirling approximation in the limit 0: -+ 00
after rewriting Eq. 4 in terms of Gamma functions (cf. also [2, 6]). When the
popular choice of a uniform prior p(Xi,II i ) is used [1], then
p(Dlm+)
log p(Dlm-) -+ 0
as
0:-+00,
(9)
which is independent of the data. Hence, neither the presence nor the absence of
the edge between A and B is favored in this limit. Given a uniform prior over the
network structures, p(m) =const, the posterior distribution p(mID) over the graphs
thus becomes increasingly spread out as 0: grows, permitting more variable network
structures.
The behavior of the Bayes factor between the two limits 0: -+ 0 and 0: -+ 00 is exemplified for positive EDF in Figure 1: there are two qualitatively different behaviors,
depending on the degree of statistical dependence between A and B. A sufficiently
weak dependence results in a monotonically increasing Bayes factor which favors the
absence of the edge A +- B at any finite value of 0:. In contrast, given a sufficiently
strong dependence between A and B, the log Bayes factor takes on positive values
for all (finite) 0: exceeding a certain value 0:+ of the scale parameter. Moreover, 0:+
grows as the statistical dependence between A and B diminishes. Consequently,
given a domain with a range of degrees of statistical dependences, the number of
edges in the learned graph increases monotonically with growing scale parameter 0:
when each variable has at most one parent (i. e., in the class of trees or forests). This
is because increasingly weaker statistical dependencies between variables are recovered as 0: grows; the restriction to forests excludes possible "interactions" among
(several) parents of a variable. As suggested by our experiments, this increase in
the number of edges can also be expected to hold for general Bayesian network
structures (although not necessarily in a monotonic way).
This reveals that regularization of network structure tends to diminish with a growing scale parameter. Note that this is in the opposite direction to the regularization
of parameters (cf. Section 2). Hence, the scale parameter 0: of the Dirichlet prior
determines the trade-off between regularizing the parameters vs. the structure of the
Bayesian network model.
If a uniform prior over the network structures is chosen, p(m) = const, the above
discussion also holds for the posterior ratio (instead of the Bayes factor). The
behavior is more complicated, however, when a non-uniform prior is assumed. For
instance, when a prior is chosen that penalizes the presence of edges, the posterior
favours the absence of an edge not only when the scale parameter is sufficiently
small, but also when it is sufficiently large. This is apparent from Fig. 1, when the
log Bayes factor is compared to a positive threshold value (instead of zero).
4
Example
This section exemplifies that the entire model (parameters and structure) has to
be considered when learning from data. This is because regularization of model
structure diminishes, while regularization of parameters increases with a growing
scale parameter a of the Dirichlet prior, as discussed in the previous sections.
When the entire model is taken into account, one can use a sensitivity analysis in
order to determine the dependence of the learned model on the scale parameter
a, given the prior p(Xi' IIi) (cf. Eq. 2). The influence of the scale parameter a
on predictive accuracy of the model can be assessed by cross-validation or, in a
Bayesian approach, prequential validation [11, 3]. Another possibility is to treat
the scale parameter a as an additional parameter of the model to be learned from
data. Hence, prior belief regarding the parameters e can then enter only through
the (normalized) distributions P(Xi' IIi). Howeverl. note that this is sufficient to
determine the (average) prior parameter estimate (cf. Eq. 3) , i.e., when N = O.
Assuming an (improper) uniform prior distribution over a, its posterior distribution
is p(aID) ex: p(Dla), given data D. Then aD = argmaxaP(Dla), where p(Dla) =
I: m P(Dla,m)p(m)2 can be calculated exactly if the summation is feasible (like in
the example below). Alternatively, assuming that the posterior over a is strongly
peaked, the likelihood may also be approximated by summing over the k most likely
graphs m only (k = 1 in the most extreme case; empirical Bayes). Subsequently,
model structure m and parameters B can be learned with respect to the Bayesian
score employing aD.
e
In the following, the effect of various values assigned to the scale parameter a is
exemplified concerning the data set gathered from Wisconsin high-school students
by Sewell and Shah [10]. This domain comprises 5 discrete variables, each with
2 or 4 states; the sample size is 10,318. In this small domain, exhaustive search
in the space of Bayesian network structures is feasible (29,281 graphs). Both the
prior distributions p(m) for all m and P(Xi' IIi) are chosen to be uniform. Figure 2
shows that the number of edges in the graph with the highest posterior probability
grows with an increasing value of the scale parameter, as expected (cf. Section 3).
In addition, cross-validation indicates best predictive accuracy of the learned model
at a ~ 100, ... ,300, while the likelihood p(Dla) takes on its maximum at aD ~ 69.
Both approaches agree on the same network structure, which is depicted in Fig. 3.
This graph can easily be interpreted in a causal manner, as outlined in [5].3 We
note that this graph was also obtained in [5] due, however , to additional constraints
concerning network structure, as a rather small prior strength of a = 5 was used. For
comparison, Fig. 3 also shows the highest-scoring unconstraint graph due to a = 5,
which does not permit a causal interpretation, cf. also [5]. This illustrates that
the "right" choice of the scale parameter a of the Dirichlet prior, when accounting
for both model structure and parameters, can have a crucial impact on the learned
network structure and the resulting insight in the ("true") dependencies among the
variables in the domain.
2We assume that m and a are independent a priori, p(mla) = p(m).
3Since we did not impose any constraints on the network structure, unlike to [5] ,
Markov-equivalence leaves the orientation of the edge between the variables IQ and CP
unspecified.
a
5
50
100
200
300
500
1, 000
a.
6
7
7
7
7
7
8
XV 5
0.045
0.044
0.040
0.040
0.040
0.042
0.047
p(D la)
p(D laD)
10 ?w
0.13
0.05
10- 14
10- 30
10- 65
10- 151
Figure 2: As a function of a: number of
arcs (a.) in the highest-scoring graph;
average KL divergence in 5-fold crossvalidation (XV 5), std= 0.006; likelihood
of a when treated as an additional model
parameter (aD = 69).
SES: socioeconomic status SEX: gender of student
PE: parental encouragement CP: college plans
IQ: intelligence quotient
Figure 3: Highest-scoring (unconstraint)
graphs when a = 5 (left) , and when a =
46, ... ,522 (right). Note that the latter
graph can also be obtained at a = 5 when
additional constraints are imposed on the
structure, cf. [5].
Acknowledgments
We would like to thank Chen-Hsiang Yeang and the anonymous referees for valuable
comments. Harald Steck acknowledges support from the German Research Foundation (DFG) under grant STE 1045/1-1. Tommi Jaakkola acknowledges support
from Nippon Telegraph and Telephone Corporation, NSF ITR grant IIS-0085836,
and from the Sloan Foundation in the form of the Sloan Research Fellowship.
References
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[4] T. S. Ferguson. A Bayesian analysis of some nonparametric problems. Annals
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[6] D. Heckerman, D. Geiger, and D. M. Chickering. Learning Bayesian networks:
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[8] S. Petrone and A. E. Raftery. A note on the Dirichlet process prior in Bayesian
nonparametric inference with partial exchangeability. Technical Report 297,
University of Washington, Seattle, 1995.
[9] J. Sethuraman and R. C. Tiwari. Convergence of Dirichlet measures and the
interpretation of their parameter. In S. S. Gupta and J. O. Berger (Eds.),
Statistical Decision Theory and Related Topics III, pages 305- 15, 1982.
[10] W. Sewell and V. Shah. Social class, parental encouragement, and educational
aspirations. American Journal of Sociology, 73:559- 72, 1968.
[11] M. Stone. Cross-validatory choice and assessment of statistical predictions.
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[12] S. G. Walker and B. K. Mallick. A note on the scale parameter of the Dirichlet
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1,275 | 2,161 | Analysis of Information in Speech based
on MANOVA
Sachin s. Kajarekarl and Hynek Hermansky l ,2
1 Department of Electrical and Computer Engineering
OGI School of Science and Engineering at OHSU
Beaverton, OR
2International Computer Science Institute
Berkeley, CA
{ sachin,hynek} @asp.ogi.edu
Abstract
We propose analysis of information in speech using three sources
- language (phone), speaker and channeL Information in speech is
measured as mutual information between the source and the set of
features extracted from speech signaL We assume that distribution of features can be modeled using Gaussian distribution. The
mutual information is computed using the results of analysis of
variability in speech. We observe similarity in the results of phone
variability and phone information, and show that the results of the
proposed analysis have more meaningful interpretations than the
analysis of variability.
1
Introduction
Speech signal carries information about the linguistic message, the speaker, the
communication channeL In the previous work [1, 2], we proposed analysis of information in speech as analysis of variability in a set of features extracted from
the speech signal. The variability was measured as covariance of the features , and
analysis was performed using using multivariate analysis of variance (MANOVA).
Total variability was divided into three types of variabilities, namely, intra-phone
(or phone) variability, speaker variability, and channel variability. Effect of each
type was measured as its contribution to the total variability.
In this paper, we extend our previous work by proposing an information-theoretic
analysis of information in speech. Similar to MANOVA, we assume that speech
carries information from three main sources- language, speaker, and channeL We
measure information from a source as mutual information (MI) [3] between the
corresponding class labels and features. For example, linguistic information is measured as MI between phone labels and features. The effect of sources is measured
in nats (or bits). In this work, we show it is easier to interpret the results of this
analysis than the analysis of variability.
In general, MI between two random variables X and Y can be measured using
three different methods [4]. First, assuming that X and Y have a joint Gaussian
distribution. However, we cannot use this method because one of the variables - a set
of class labels - is discrete. Second, modeling distribution of X or Y using parametric
form , for example, mixture of Gaussians [4]. Third, using non-parametric techniques
to estimate distributions of X and Y [5]. The proposed analysis is based on the
second method, where distribution of features is modeled as a Gaussian distribution.
Although it is a strong assumption, we show that results of this analysis are similar
to the results obtained using the third method [5].
The paper is organized as follows. Section 2 describes the experimental setup.
Section 3 describes MAN OVA and presents results of MAN OVA. Section 4 proposes
information theoretic approach for analysis of information in speech and presents
the results. Section 5 compares these results with results from the previous study.
Section 6 describes the summary and conclusions from this work.
2
Experimental Setup
In the previous work [1 , 2], we have analyzed variability in the features using three
databases - HTIMIT, OGI Stories and TIMIT. In this work, we present results
of MAN OVA using OGI Stories database; mainly for the comparison with Yang's
results [5, 6]. English part of OGI Stories database consists of 207 speakers, speaking
for approximately 1 minute each. Each utterance is transcribed at phone level.
Therefore, phone is considered as a source of variability or source of information.
The utterances are not labeled separately by speakers and channels, so we cannot
measure speaker and channel as separate sources. Instead, we assume that different
speakers have used different channels and consider speaker+channel as a single
source of variability or a single source of information.
Figure 1 shows a commonly used time-frequency representation of energy in speech
signal. The y-axis represents frequency, x-axis represents time, and the darkness of
each element shows the energy at a given frequency and time. A spectral vector is
defined by the number of points on the y-axis, S(w, t m ). In this work, this vector
contains 15 points on Bark spectrum. The vector is estimated at every 10 ms using a
25 ms speech segment. It is labeled by the phone and the speaker and channel label
of the corresponding speech segment. A temporal vector is defined by a sequence
of points along time at a given frequency, S(wn, t). In this work, it consists of
50 points each in the past and the future with respect to the current observation
and the observation itself. As the spectral vectors are computed every 10 ms, the
temporal vector represents 1 sec of temporal information. The temporal vectors
are labeled by the phone and the speaker and channel label of the current speech
segment. In this work, the analysis is performed independently using spectral and
temporal vectors.
3
MANOVA
Multivariate analysis of variance (MANOVA) [7] is used to measure the variation
in the data, {X E R n }, with respect to two or more factors. In this work, we use
two factors - phone and speaker+channel. The underline model of MAN OVA is
(1)
where, i = 1"" ,p, represents phones, j = 1"" Be, represents speakers and channels. This equation shows that any feature vector, X ijk , can be approximated using
a sum of X.. , the mean of the data; Xi ., mean of the phone i; Xij., mean of the
speaker and channel j, and phone i; and Eij k, an error in this approximation. Using
~r- l0ms
Ii I
JJ I 1M
I
LL
I
I~I 11 IH~ifl~" III
I'I
~i
I
I I1I11
1
' 1!
I~
I ~ [I
1111
1;1, I
?
?
i
I III
1II1
IHII
I
n
I[l l~
'I
~
11111
I
Il:
I II ,UI
I
~IJ' n'I ~
il [ItJ:ii'
l!I I I'""I~ "'I 'I ~'I
u
1'1
ilJU JI II,I "i...i
"
uJlJ
111 1'111
1111
[III
r1111
III
II I
Temporal Vector
(Temporal Domain)
1111
---:JIIIII_ _ __
Spectral Vector
(Spectral Domain)
Figure 1: Time-frequency representation of logarithmic energies from speech signal
this model, the total covariance can be decomposed as follows
~total = ~p
+ ~s c + ~re sidual
(2)
where
N (X... _
"""' N i
~
~sc
LL -Nij
N (X, ZJ
i
~r esidual
X .. )t (X... - X .. )
-
t
-
-
X .. ) (X,
ZJ - X z. )
j
1""",,,"",,,"",
-
t
-
N ~ ~ ~(Xijk - Xij) (Xijk - Xij)
i
j
k
and, N is the data size and Nijk refers to the number of samples associated with
the particular combination of factors (indicated by the subscript).
The covariance terms are computed as follows. First, all the feature vectors (X)
belonging to each phone i are collected and their mean (Xi) is computed. The
covariance of these phone means, ~p, is the estimate of phone variability. Next, the
data for each speaker and channel j within each phone i is collected and the mean
of the data (X ij ) is computed. The covariance of the means of different speakers
averaged over all phones, ~s c, is the estimate of speaker variability. All the variability in the data is not explained using these sources. The unaccounted sources,
such as context and coarticulation, cause variability in the data collected from
one speaker speaking one phone through one channel. The covariance within each
phone, speaker, and channel is averaged over all phones, speakers, and channels,
and the resulting covariance, ~r e sidual, is the estimate of residual variability.
3.1
Results
Results of MAN OVA are interpreted at two levels - feature element and feature
vector. Results for each feature element are shown in Figure 2. Table 1 shows the
results using the complete feature vector. The contribution of different sources is
calculated as trace (~source )ltrace(~total). Note that this measure cannot be used
to compare variabilities across feature-sets with different number of features. Therefore, we cannot directly compare contribution of variabilities in time and frequency
domains. For comparison, contribution of sources in temporal domain is calculated
Table 1: Contribution of sources in spectral and temporal domains
o contribution
source
pectral Domain Temporal Domain
phone
35.3
4.0
speaker+channel
41.1
30.3
as trace(EtI',source E) /trace(EtI',totaIE) , where
eigenvectors of I',total.
ElOl x 15
is a matrix of 15 leading
In spectral domain, the highest phone variability is between 4-6 Barks. The highest
speaker and channel variability is between 1-2 Barks where phone variability is
the lowest. In temporal domain, phone variability spreads for approximately 250
ms around the current phone. Speaker and channel variability is almost constant
except around the current frame. This deviation is explained by the difference in
t he phonetic context among the phone instances across different speakers. Thus,
features for speakers within a phone differ not only because of different speaker
characteristics but also different phonetic contexts. This deviation is also seen in
the speaker and channel information in the proposed analysis. In the overall results
for each domain, spectral domain has higher variability due to different phones than
temporal domain. It also has higher speaker and channel variability than temporal
domain.
The disadvantage of this analysis is that it is difficult to interpret the results. For
example, how much phone variability is needed for perfect phone recognition? and
is 4% of phone variability in temporal domain significant? In order to answer these
questions, we propose an information theoretic analysis.
Phone
Speaker+Channel
7r,----------------------~
6
7 ,---~----~--------~
,
~
6
5
5 '
Q)
"'
g 4 ,.. ,. .. , _ ,_ , _ ,
..
<1l
_ , _ ,_ ' - ,- '
...
;
'?3
>
2
O L-----~----~------~
5
10
Frequency
(Critical Band Index)
15
-250
0
Time (ms)
Figure 2: Results of analysis of variability
250
4
Information-theoretic Analysis
Results of MAN OVA can not be directly converted to MI because the determinant
of source and residual covariances do not add to the determinant of total covariance.
Therefore, we propose a different formulation for the information theoretic analysis
as follows. Let {X E Rn} be a set of feature vectors, with probability distribution
p(X). Let h(X) be the entropy of X. Let Y = {Y1 , ... , Ym} be a set of different
factors and each Yi be a set of classes within each factor. For example, we can
assume that Y1 = {yf} represents phone factor and each yf represent a phone class.
Lets assume that X has two parts; one completely characterized by Y and another
part, Z , characterized by N(X) ""' N(O, Jn xn ), where J is the identity matrix. Let
J (X; Y) be the MI between X and Y. Assuming that we consider all the possible
factors for our analysis,
J(X;Y) = J(X;Y1, ... , Ym) = h(X)-h(X/Yl , ... ,Ym) = h(X)-h(Z) = D(PIIN) ,
where D() is the kullback-liebler distance [3] between distributions P and N. Using
the chain-rule, the left hand side can be expanded as follows ,
m
J(X; Y1 ,?.?, Yn ) = J(X; Yd
+ J(X; Y2 /Yd + l: J(X; Yi/Yi - l"'"
Y2 , Yd?
(3)
i=3
If we assume that there are only two factors Y1 and Y2 used for the analysis, then
this equation is similar to the decomposition performed using MAN OVA (Equation
2). The term on the left hand side is entropy of X which is the total information
in X that can be explained using Y . This is similar to the left-hand side term in
MANOVA that describes the total variability. On the right hand side, first term
is similar to the phone variability, second term is similar to the speaker variability,
and the last term which calculates the effect of unaccounted factors (Y3 , ... , Y m ) is
similar to the residual variability.
First and second terms on the right hand side of Equation 3 are computed as follows.
Yd = h(X) -
(4)
J(X; Y2 /Yd = h(X/Yd - h(X/Y1, Y2 ).
(5)
h () terms are estimated using parametric approximation to the total and conditional distribution It is assumed that the total distribution of features is a Gaussian
distribution with covariance ~. Therefore, h (X) = ~ log (2net I~I. Similarly, we
assume that the distribution of features of different phones (i) is a Gaussian distribution with covariances ~i' Therefore,
J(X;
h(X/Y1) =
~
h(X/Yd
l: p (y~)Iog (2net I~il
(6)
yiCYi
Finally, we assume that the distribution of features of different phones spoken by
different speakers is also a Gaussian distribution with covariances ~ij. Therefore,
h(X/Y1,Y2 )
=~
p(yLYOlog(2netl~ijl
l:
(7)
y;CY1,y~CY2
Substituting equations 6 and 7 in equations 4 and 5, we get
J(X ' Y;)=~lo
,
1
2
1
J(X;Y2 /Yd = -log
2
g
I~I
IT.Yi CY;
IT?
IT
.
(8)
1~ ,' IP(Yil
I~'IP(Y;)
YicY;'
i
j
Yl CY1 'Y2 CY2
.
j
I~i IP(Yi ' Y2)
(9)
Phone
Speaker +Channel
0.6 ,----~--~-~-------,
0.5
Ul
1.5
ca
-:2:c
,i
,
0.5
\
:2: 0.2 , .. ,_ , .. ,_ ,' ,; -
\
\
-' , - ,- ,- ,- ,_ ,_ ,_ 1- '"
.... ,. - .'..... , -
, _.,
"
0.1
5
10
Frequency
(Critical Band Index)
15
-250
0
250
Time (ms)
Figure 3: Results of information-theoretic analysis
Table 2: Mutual information between features and phone and speaker and channel
labels in spectral and temporal domains
source
phone
speaker+ channel
4 .1
Results
Figure 3 shows the results of information-theoretic analysis in spectral and temporal domain. These results are computed independently for each feature element.
In spectral domain, phone information is highest between 3-6 Barks. Speaker and
channel information is lowest in that range and highest between 1-2 Barks. Since
OGI Stories database was collected over different telephones, speaker+ channel information below 2 Barks ( :=::: 200 Hz ) is due to different telephone channels. In
temporal domain, the highest phone information is at the center (0 ms). It spreads
for approximately 200 ms around the center. Speaker and channel information is
almost constant across t ime except near the center.
Note that the nature of speaker and channel variability also deviates from the constant around the current frame. But, at the current frame , phone variability is
higher than speaker and channel variability. The results of analysis of informat ion show that, at the current frame, phone information is lower than speaker and
channel information. This difference is explained by comparing our MI results with
results from Yang et. al. [6] in the next section.
Table 2 shows the results for the complete feature vector. Note that there are some
practical issues in computing determinants in Equation 4 and 5. They are related
to data insufficiency, specifically, in temporal domain where the feature vector is
101 points and there are approximately 60 vectors per speaker per phone. We ob-
serve that without proper conditioning of covariances, the analysis overestimates
MI (l(X ; Yl , Y2 ) > H(Yl , Y2 )). This is addressed using the condition number to
limit the number of eigenvalues used in the calculation of determinants. Our hypothesis is that in presence of insufficient data, only few leading eigen vectors are
properly estimated. We have use condition number of 1000 to estimate determinant
of ~ and ~i, and condition number of 100 to estimate the determinant of ~ij. The
results show that phone information in spectral domain is 1.6 nats. Speaker and
channel information is 0.5 nats. In temporal domain, phone information is about
1.2 nats. Speaker and channel information is 5.9 nats. Comparison of results from
spectral and temporal domains shows that spectral domain has higher phone information than temporal domain. Temporal domain has higher speaker and channel
information than spectral domain.
Using these results, we can answer the questions raised in Section 3. First question
was how much phone variability is needed for perfect phone recognition? The answer to the question is H(Yd, because the maximum value of leX; Yd is H(Yd?
We compute H(Yl ) using phone priors. For this database, we get H(Yl ) = 3.42
nats, that means we need 3.42 nats of information for perfect phone recognition.
Question about significance of phone information in temporal domain is addressed
by comparing it with information-less MI level. The information-less MI is computed as MI between the current phone label and features at 500 ms in the past
or in the future . From our results, we get information-less MI equal to 0.0013 nats
considering feature at 500 ms in the past, and 0.0010 nats considering features at
500 ms in the future l . The phone information in temporal domain is 1.2 bits that
is greater than both the levels. Therefore it is significant.
5
Results in Perspective
In the proposed analysis, we estimated MI assuming Gaussian distribution for the
features. This assumption is validated by comparing our results with the results
from a study by Yang, et. al.,[6], where MI was computed without assuming any
parametric model for the distribution of features. Note that only entropies can
be directly compared for difference in the estimation technique [3]. However, MI
using Gaussian assumption can be equal to, less or more than the actual MI. In
the comparison of our results with Yang's results , we consider only the nature of
information observed in both studies. The difference in actual MI levels across the
two studies is related to the difference in the estimation techniques.
In spectral domain, Yang's study showed higher phone information between 3-8
Barks. The highest phone information was observed at 4 Barks. Higher speaker
and channel information was observed around 1-2 Barks. In temporal domain, their
study showed that phone information spreads for approximately 200 ms around the
current time frame. Comparison of results from this analysis and our analysis shows
that nature of phone information is similar in both studies. Nature of speaker and
channel information in spectral domain is also similar. We could not compare the
speaker and channel information in temporal domain because Yang's study did not
present these results.
In Section 4.1, we observed difference in the nature of speaker and channel variability, and speaker and channel information at Ii =5 Barks. Comparing MI levels
from our study to those from Yang's study, we observe that Yang's results show that
speaker and channel information at 5 Barks is less that the corresponding phone
information. This is consistent with results of analysis of variability, but not with
lInformation-less MI calculated using Yang et. al. is 0.019 bits
the proposed analysis of information. As mentioned before, this difference is due
to difference in the density estimation techniques used for computing MI. In the
future work, we plan to model the densities using more sophisticated techniques,
and improve the estimation of speaker and channel information.
6
Conclusions
We proposed analysis of information in speech using three sources of information
- language (phone), speaker and channel. Information in speech was measured as
MI between the class labels and the set of features extracted from speech signal.
For example, linguistic information was measured using phone labels and the features. We modeled distribution of features using Gaussian distribution. Thus we
related the analysis to previous proposed analysis of variability in speech. We observed similar results for phone variability and phone information. The speaker
and channel variability and speaker and channel information around the current
frame was different. This was shown to be related to the over-estimation of speaker
and channel information using unimodal Gaussian model. Note that the analysis of
information was proposed because its results have more meaningful interpretations
than results of analysis of variability. For addressing the over-estimation, we plan
to use more complex models ,such as mixture of Gaussians, for computing MI in
the future work.
Acknowledgments
Authors thank Prof. Andrew Fraser from Portland State University for numerous
discussions and helpful insights on this topic.
References
[1] S. S. Kajarekar , N. Malayath and H. Hermansky, "Analysis of sources of variability in speech," in Proc. of EUROSPEECH, Budapest, Hungary, 1999.
[2] S. S. Kajarekar, N. Malayath and H. Hermansky, "Analysis of speaker and
channel variability in speech," in Proc. of ASRU, Colorado, 1999.
[3] T. M. Cover and J. A. Thomas, Elements of Information theory, John Wiley &
Sons, Inc., 1991.
[4] J. A. Bilmes, "Maximum Mutual Information Based Reduction Strategies for
Cross-correlation Based Joint Distribution Modelling ," in Proc. of ICASSP,
Seattle, USA, 1998.
[5] H. Hermansky H. Yang, S. van Vuuren, "Relevancy of Time-Frequency Features
for Phonetic Classification Measured by Mutual Information," in ICASSP '99,
Phoenix, Arizona, USA, 1999.
[6] H. H. Yang, S. Sharma, S. van Vuuren and H. Hermansky, "Relevance of TimeFrequency Features for Phonetic and Speaker-Channel Classification," Speech
Communication, Aug. 2000.
[7] R. V. Hogg and E. A. Tannis, Statistical Analysis and Inference, PRANTICE
HALL, fifth edition, 1997.
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1,276 | 2,162 | Real-Time Monitoring of Complex Industrial
Processes with Particle Filters
Rub?en Morales-Men?endez
Dept. of Mechatronics and Automation
ITESM campus Monterrey
Monterrey, NL M?exico
[email protected]
Nando de Freitas and David Poole
Dept. of Computer Science
University of British Columbia
Vancouver, BC, V6T 1Z4, Canada
nando,poole @cs.ubc.ca
Abstract
This paper discusses the application of particle filtering algorithms to
fault diagnosis in complex industrial processes. We consider two ubiquitous processes: an industrial dryer and a level tank. For these applications, we compared three particle filtering variants: standard particle filtering, Rao-Blackwellised particle filtering and a version of RaoBlackwellised particle filtering that does one-step look-ahead to select
good sampling regions. We show that the overhead of the extra processing per particle of the more sophisticated methods is more than compensated by the decrease in error and variance.
1 Introduction
Real-time monitoring is important in many areas such as robot navigation or diagnosis
of complex systems [1, 2]. This paper considers online monitoring of complex industrial
processes. The processes have a number of discrete states, corresponding to different combinations of faults or regions of qualitatively different dynamics. The dynamics can be very
different based on the discrete states. Even if there are very few discrete states, exact monitoring is computationally unfeasible as the state of the system depends on the history of the
discrete states. However there is a need to monitor these systems in real time to determine
what faults could have occurred.
This paper investigates the feasibility of using particle filtering (PF) for online monitoring.
It also proposes some powerful variants of PF. These variants involve doing more computation per particle for each time step. We wanted to investigate whether we could do real-time
monitoring and whether the extra cost of the more sophisticated methods was worthwhile
in these real-world domains.
2 Classical approaches to fault diagnosis in dynamic systems
Most existing model-based fault diagnosis methods use a technique called analytical redundancy [3]. Real measurements of a process variable are compared to analytically calculated
Visiting Scholar (2000-2003) at The University of British Columbia.
values. The resulting differences, named residuals, are indicative of faults in the process.
Many of these methods rely on simplifications and heuristics [4, 5, 6, 7]. Here, we propose
a principled probabilistic approach to this problem.
3 Processes monitored
We analyzed two industrial processes: an industrial dryer and a level-tank. In each of these,
we physically inserted a sequence of faults into the system and made appropriate measurements. The nonlinear models that we used in the stochastic simulation were obtained
through open-loop step responses for each discrete state [8]. The parametric identification
procedure was guided by the minimum squares error algorithm [9] and validated with the
?Control Station? software [10]. The discrete-time state space representation was obtained
by a standard procedure in control engineering [8].
3.1 Industrial dryer
An industrial dryer is a thermal process that converts electricity to heat. As shown in
Figure 1, we measure the temperature of the output air-flow.
Figure 1: Industrial dryer.
Normal operation corresponds to low fan speed,
open air-flow grill and clean temperature
sensor (we denote this discrete state
). We induced 3 types of fault: faulty
fan,
faulty grill (the grill is closed), and
faulty fan and grill.
3.2 Level tank
Many continuous industrial processes need to control the amount of accumulated material
using level measurement, such as evaporators, distillation columns or boilers. We worked
with a level-tank system that exhibits the dynamic behaviour of these important processes,
see Figure 2. A by-pass pipe and two manual valves (
and ) where used to induce
typical faulty states.
4 Mathematical model
We adopted the following jump Markov linear Gaussian model:
,
!#"$ &%')( &*+
)./ &012)3 &*+54
-
Figure 2: Level-Tank
where ,
denotes the measurements,
denotes the unknown continuous
4 54
is a known control signal,
denotes the unknown discrete
states, *
states (normal operations and faulty conditions). The noise processes are i.i.d Gaussian:
%'
4 and 01
4 . The parameters 4 " 4 - 4 . 4 ('4 3 4
are
4
identified matrices with . ./
for any . The initial states are
. The important thing to notice is that for each realization of 1 , we have
and
a single linear-Gaussian model. If we knew , we could solve for exactly using the
Kalman filter algorithm.
'&
! #" %$
&
The aim of the analysis is to compute the marginal posterior distribution1 of the discrete states ,
. This distribution can be derived from the posterior distribution
4
,
by standard marginalisation. The posterior density satisfies the following recursion
( '& )&
*
&
'& 4 )& ,
&
*
'&
4 '& 2
,
& 2
*
, 54
*
*
4
4
, ,
2
&
(1)
This recursion involves intractable integrals. One, therefore, has to resort to some form of
numerical approximation scheme.
5 Particle filtering
'+-,/& . 4 )+/,/& . 4 % -+ ,/. 10/, 2
In the PF setting, we use a weighted set of samples (particles)
approximate the posterior with the following point-mass distribution
3
to
4 '& 54 '& ,
& 5 0 % 7
+/,-. 68:>@9<;-? =AB >@9<;-? A= ( 4 '& 54 '& 4
(
0
,-2
)
&
)
&
4
& denotes the Dirac-delta function.
where 6 8 >9C;/? =A B >9C;/? A= D(
Given E
parti
cles F )+/,/& .
4 '+/,-& . 2
10,/2
at time GIH
, approximately distributed according to
NOTATION: For a generic vector J , we adopt the notation JLKM NFOIP#J1K%Q@JRQ%SS%S%Q@J'NUTWV to denote all
the entries of this vector at time X . For simplicity, we use JN to denote both the random variable and its
realisation.
we express continuous probability distributions using YZP\[LJ N T instead of
]_^ P#J'Na`[LJ'NDConsequently,
T and discrete distributions using YbP#J'NUT instead of ]_^ P#J'N_cdJNWT . If these distributions
admit densities with respect to an underlying measure e (counting or Lebesgue), we denote these
densities by fgP#JNWT . For example, when considering the space hji , we will use the Lebesgue measure,
ekcl[LJ'N , so that YZP\[LJNWTFcmfgP#J'NUTL[LJN .
1
( '+-,/& .
4 '+/,-& . 2
,
& 2
, PF enables us to compute E particles '+/,-& . 4 '+/,-& . 0,-2
approximately distributed according to ( )+/,/& . 4 '+/,-& . ,
& , at time G . Since we cannot sample from
the posterior directly, the PF update is accomplished by introducing an appropriate importance proposal distribution ( )& 54 '& from which we can obtain samples. The basic
algorithm, Figure (3), consists of two steps: sequential importance sampling and selection
(see [11] for a detailed derivation). This algorithm uses the transition priors as proposal
distributions; 4 ,
4 2
. For the selection step, we
used a state-of-the-art minimum variance resampling algorithm [12].
'& )& &
Sequential importance sampling step
c Q)SCSCS Q , sample from the transition priors
N
Y P N N
K T and N
Y P\[ N N
K Q N
T
and set M N Q M N O N Q N Q M N K Q M N K S
For c Q)SCSCS Q , evaluate and normalize the importance weights
" f %$ N N
Q N
! N#
For
Selection step
&
M N Q
M N% ') ( * K with respect to high/low importance
weights ! N to obtain particles & M N Q M N ' ( * .
K
Multiply/Discard particles
G
Figure 3: PF algorithm at time .
6 Improved Rao-Blackwellised particle filtering
*
'& )&
*
&
& ' & * )& &
'& & '&
* ) &
)&
*
,
4
,
, it is
By considering the factorisation 4 ,
,
4
is Gaussian
possible to design more efficient algorithms. The density
5 ,
& .
and can be computed analytically if we know the marginal posterior density
This density satisfies the alternative recursion
*
* )& 5 ,
& * )&
,
&
*
&
)& *
&
, ,
14
&
, ,
2
*
&
(2)
If equation (1) does not admit a closed-form expression, then equation (2) does not admit one either and sampling-based methods are still required. (Also note that the term
, ,
14
& in equation (2) does not simplify to , & because there is a dependency on past values through .) Now assuming that we can use a weighted set of
samples
4 %
to represent the marginal posterior distribution
&
)&
*
'&
)+/,/& . /+ ,-. 0-, 2
the marginal density of
* 3 4 '& ,
&
0
)&
3
0
'& ,
&
50
,-2
%
+/,/. 6 >@9<;-? A= '& &4
is a Gaussian mixture
,+ * 4 '& '& 4 ,
& &@( '& ,
& 5 0
,-2
%
+/,-. * 4 '& ,
& 4 )+/,/& .
that can be computed efficiently with a stochastic bank of Kalman filters. That is, we use
PF to estimate the distribution of and exact computations (Kalman filter) to estimate the
mean and variance of . In particular, we sample and then propagate the mean
and covariance of with a Kalman filter:
$ +/,/.
e N
N
N N
+/,-.
K c
c
c
c
c
"
,
,
0
0
2
, !
where
&
$
P N
TDe N
K P N
T N
P N T N K P N T
P N T P N T
P N T N N K
P N T
P N T P N T
P N TDe N N K P N T N
e N
N K N
N K
P N
T
N K
P $ N $ N
N K T
N N K N N K
P N T
N K
P N T N N K Q
,
& , ,
, ,
&
, $
& 2
, "
,
& & and
,
,
&
0
.
c
N
K
$ N
N K
e N
N
" +/,-.
This is the basis of the RBPF algorithm that was adopted in [13]. Here, we introduce an
extra improvement. Let us expand the expression for the importance weights:
%
* ' & 5 ,
& & * '& 2
,
& * )& 2
4 ,
&
*
(3)
'& ,
&
) &
,
&
5 )& 2
4 ,
&
* , ,
& 2
4 '& * '&
4 ,
& 2
(4)
)&
4 ,
&
)& ,
& '& 2
4 ,
& * '& 2
,
& 2
, states that we are
The proposal choice,
not sampling past trajectories. Sampling past trajectories requires solving an intractable
integral [14].
'&
&
transition prior as proposal distribution: 2
4 ,
. Then, according to equation (4), the importance
weights simplify to the predictive density
the
*We could
)&
14 use
,
&
5
*
*
&
'& &
"! , $# ,
4 %
(5)
However, we can do better by noticing that according to equation (3), the optimal proposal distribution corresponds to the choice 1 )& 2
4 ,
& * )&
4 ,
& . This
%'
, 5 ,
2
4
distribution satisfies Bayes rule:
*
'& 2
4 ,
&
*
&
'& *
&
'&
&
, ,
2
4
4 ,
2
, ,
2
4
5
*
'&
(6)
and, hence, the importance weights simplify to
%'
*
&
, ,
2
4
'&
5
5 '& *
A2
&
, ,
2
4
'&
4
*
(7)
When the number of discrete states is small, say 10 or 100, we can compute the distributions
in equations (6) and (7) analytically. In addition to Rao-Blackwellisation, this leads to
substantial improvements over standard particle filters. Yet, a further improvement can still
be attained.
* '&
& * '&
&
Even when using the optimal importance distribution, there is a discrepancy arising from
the ratio
,
)( 2
,
2
in equation (3). This discrepancy is what causes
the well known problem of sample impoverishment in all particle filters [11, 15]. To circumvent it to a significant extent, we note that the importance weights do not depend on
(we are marginalising over this variable). It is therefore possible to select particles
be
fore the sampling step. That is, one chooses the fittest particles at time
using the
information at time . This observation leads to an efficient algorithm (look-ahead RBPF),
whose pseudocode is shown in Figure 4. Note that for standard PF, Figure 3, the importance weights depend on the sample , thus not permitting selection before sampling.
Selecting particles before sampling results in a richer sample set at the end of each time
step.
G H
G
Kalman prediction step
+/,-.
N_c Q%S%SS)Q compute
e N
N K P N T Q N
N K P N T Q $ N
N K P N T Q N
P N T
For i=1, . . . , N, and for
For i=1 , . . . , N , evaluate and normalize the importance weights
! N
mc f P $ N $ KM N K Q
M N K TFc i P $ N
N K P N T Q N
P N TT-f P N N
K T
A* K
Selection step
Multiply/Discard particles & e N
K Q N
K Q
M N K ' ( * K with respect to high/low
importance weights ! N to obtain particles & e N K Q N K Q M N K ' ( * .
K
Sequential importance sampling step
Kalman prediction. For i=1, . . . , N, and for N_c QS%S)S%Q using the resampled
information, re-compute
e N
N K P NWT Q N
N K P NUT Q $ N
N K P NWT Q N
P NUT
For N c Q)S%SS%Q compute
f P N
M N K Q $ KM NUT " P $ N
N K P NWT Q N
P NDTT-f P N N
K T
Sampling step
N
f P N
M N K Q $ KM NUT
Updating step
For i=1 , . . . , N, use one step of the Kalman recursion
the suffi
to
compute
cient statistics & e N Q N ' given & e N N K P N T Q N N K P N T ' .
&
G
GH
Figure 4: look-ahead RBPF algorithm at time . The
algorithm uses an optimal proposal
distribution. It also selects particles from time
using the information at time .
G
7 Results
The results are shown in Figures 5 and 6. The left graphs show error detection versus
computing time per time-step (the signal sampling time was 1 second). The right graphs
show the error detection versus number of particles. The error detection represents how
many discrete states were not identified properly, and was calculated for 25 independent
runs (1,000 time steps each). The graphs show that look-ahead RBPF works significantly
better (low error rate and very low variance). This is essential for real-time operation with
low error rates.
100
100
90
90
Real time
80
80
% Error Detection
% Error Detection
70
60
50
40
RBPF
30
PF
60
50
RBPF
PF
40
30
20
10
70
20
la?RBPF
10
la?RBPF
0 ?3
10
?2
10
?1
10
0
10
Computing time per timestep (=) sec
1
10
0
0
10
1
2
10
10
Number of particles
3
10
Figure 5: Industrial dryer: error detection vs computing time and number of particles.
The graphs also show that even for 1 particle, look-ahead RBPF is able to track the discrete
state. The reason for this is that the sensors are very accurate (variance=0.01). Consequently, the distributions are very peaked and we are simply tracking the mode. Note that
look-ahead RBPF is the only filter that uses the most recent information in the proposal
distribution. Since the measurements are very accurate, it finds the mode easily. We repeated the level-tank experiments with noisier sensors (variance=0.08) and obtained the
results shown in Figure 7. Noisier sensors, as expected, reduce the accuracy of look-ahead
RBPF with a small number of particles. However, it is still possible to achieve low error
rates in real-time. Since modern industrial and robotic sensors tend to be very accurate, we
conclude that look-ahead RBPF has great potential.
Acknowledgments
Ruben Morales-Men?endez was partly supported by the Government of Canada (ICCS) and
UBC CS department. David Poole and Nando de Freitas are supported by NSERC
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1,277 | 2,163 | One-Class LP Classifier for Dissimilarity
Representations
1
El?zbieta P?ekalska1 , David M.J.Tax2 and Robert P.W. Duin1
Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
2
Fraunhofer Institute FIRST.IDA, Kekul?str.7, D-12489 Berlin, Germany
[email protected],[email protected]
Abstract
Problems in which abnormal or novel situations should be detected can
be approached by describing the domain of the class of typical examples. These applications come from the areas of machine diagnostics,
fault detection, illness identification or, in principle, refer to any problem where little knowledge is available outside the typical class. In this
paper we explain why proximities are natural representations for domain
descriptors and we propose a simple one-class classifier for dissimilarity
representations. By the use of linear programming an efficient one-class
description can be found, based on a small number of prototype objects.
This classifier can be made (1) more robust by transforming the dissimilarities and (2) cheaper to compute by using a reduced representation set.
Finally, a comparison to a comparable one-class classifier by Campbell
and Bennett is given.
1 Introduction
The problem of describing a class or a domain has recently gained a lot of attention, since it
can be identified in many applications. The area of interest covers all the problems, where
the specified targets have to be recognized and the anomalies or outlier instances have to
be detected. Those might be examples of any type of fault detection, abnormal behavior,
rare illnesses, etc. One possible approach to class description problems is to construct oneclass classifiers (OCCs) [13]. Such classifiers are concept descriptors, i.e. they refer to all
possible knowledge that one has about the class.
An efficient OCC built in a feature space can be found by determining a minimal volume
hypersphere around the data [14, 13] or by determining a hyperplane such that it separates
the data from the origin as well as possible [11, 12]. By the use of kernels [15] the data is
implicitly mapped into a higher-dimensional inner product space and, as a result, an OCC in
the original space can yield a nonlinear and non-spherical boundary; see e.g. [15, 11, 12, 14].
Those approaches are convenient for data already represented in a feature space. In some
cases, there is, however, a lack of good or suitable features due to the difficulty of defining
them, as e.g. in case of strings, graphs or shapes. To avoid the definition of an explicit
feature space, we have already proposed to address kernels as general proximity measures
[10] and not only as symmetric, (conditionally) positive definite functions of two variables
[2]. Such a proximity should directly arise from an application; see e.g. [8, 7]. Therefore,
our reasoning starts not from a feature space, like in case of the other methods [15, 11, 12,
14], but from a given proximity representation. Here, we address general dissimilarities.
The basic assumption that an instance belongs to a class is that it is similar to examples
within this class. The identification procedure is realized by a proximity function equipped
with a threshold, determining whether an instance is a class member or not. This proximity
function can be e.g. a distance to an average representative, or a set of selected prototypes. The data represented by proximities is thus more natural for building the concept
descriptors, i.e. OCCs, since the proximity function can be directly built on them.
In this paper, we propose a simple and efficient OCC for general dissimilarity representations, discussed in Section 2, found by the use of linear programming (LP). Section 3
presents our method together with a dissimilarity transformation to make it more robust
against objects with large dissimilarities. Section 4 describes the experiments conducted,
and discusses the results. Conclusions are summarized in Section 5.
2 Dissimilarity representations
Although a dissimilarity measure D provides a flexible way to represent the data, there
are some constraints. Reflectivity and positivity conditions are essential to define a proper
measure; see also [10]. For our convenience, we also adopt the symmetry requirement.
We do not require that D is a strict metric, since non-metric dissimilarities may naturally
be found when shapes or objects in images are compared e.g. in computer vision [4, 7].
Let z and pi refer to objects to be compared. A dissimilarity representation can now be
seen as a dissimilarity kernel based on the representation set R = {p 1 , .., pN } and realized
by a mapping D(z, R) : F ? RN , defined as D(z, R) = [D(z, p1 ) . . . D(z, pN )]T . R
controls the dimensionality of a dissimilarity space D(?, R). Note also that F expresses a
conceptual space of objects, not necessarily a feature space. Therefore, to emphasize that
objects, like z or pi , might not be feature vectors, they will not be printed in bold.
The compactness hypothesis (CH) [5] is the basis for object recognition. It states that
similar objects are close in their representations. For a dissimilarity measure D, this means
that D(r, s) is small if objects r and s are similar.If we demand that D(r, s) = 0, if and only
if the objects r and s are identical, this implies that they belong to the same class. This can
be extended by assuming that all objects s such that D(r, s) < ?, for a sufficient small ?, are
so similar to r that they are members of the same class. Consequently, D(r, t) ? D(s, t) for
other objects t. Therefore, for dissimilarity representations satisfying the above continuity,
the reverse of the CH holds: objects similar in their representations are similar in reality
and belong, thereby, to the same class [6, 10].
Objects with large distances are assumed to be dissimilar. When the set R contains objects
from the class of interest, then objects z with large D(z, R) are outliers and should be
remote from the origin in this dissimilarity space. This characteristic will be used in our
OCC. If the dissimilarity measure D is a metric, then all vectors D(z, R), lie in an open
prism (unbounded from above1), bounded from below by a hyperplane on which the objects
from R are. In principle, z may be placed anywhere in the dissimilarity space D(?, R) only
if the triangle inequality is completely violated. This is, however, not possible from the
practical point of view, because then both the CH and its reverse will not be fulfilled.
Consequently, this would mean that D has lost its discriminating properties of being small
for similar objects. Therefore, the measure D, if not a metric, has to be only slightly nonmetric (i.e. the triangle inequalities are only somewhat violated) and, thereby, D(z, R) will
still lie either in the prism or in its close neigbourhood.
1
the prism is bounded if D is bounded
3 The linear programming dissimilarity data description
To describe a class in a non-negative dissimilarity space, one could minimize the volume of
the prism, cut by a hyperplane P : w T D(z, R) = ? that bounds the data from above2 (note
that non-negative dissimilarities impose both ? ? 0 and wi ? 0). However, this might be not
a feasible task. A natural extension is to minimize the volume of a simplex with the main
vertex being the origin and the other vertices v j resulting from the intersection of P and
the axes of the dissimilarity space (v j is a vector of all zero elements except for vji = ?/wi ,
given that wi 6= 0). Assume now that there are M non-zero weights of the hyperplane P , so
effectively, P is constructed in a RM . From geometry we know that the volume V of such
a simplex can be expressed as V = (VBase /M !) ? (?/||w||2 ), where VBase is the volume of
the base, defined by the vertices v j . The minimization of h = ?/||w||2 , i.e. the Euclidean
distance from the origin to P , is then related to the minimization of V .
Let {D(pi , R)}N
i=1 , N = |R| be a dissimilarity representation, bounded by a hyperplane P ,
i.e. wT D(pi , R) ? ? for i = 1, . . . , N , such that the Lq distance to the origin dq (0, P ) =
?/||w||p is the smallest (i.e. q satisfies 1/p + 1/q = 1 for p ? 1) [9]. This means that P can
be determined by minimizing ? ? ||w||p . However, when we require ||w||p = 1 (to avoid
any arbitrary scaling of w), the construction of P can be solved by the minimization of ?
only. The mathematical programming formulation of such a problem is [9, 1]:
min ?
(1)
s.t.
wT D(pi , R) ? ?,
i = 1, 2, .., N,
||w||p = 1, ? ? 0.
If p = 2, then P is found such that h is minimized, yielding a quadratic optimization problem. A much simpler
? LP formulation, realized for p = 1, is of our interest. Knowing that
||w||2 ? ||w||1 ? M||w||2 and by?the assumption of ||w||1 = 1, after simple calculations,
we find that ? ? h = ?/||w||2 ? M ?. Therefore, by minimizing d? (0, P ) = ?, (and
||w||1 = 1), h will be bounded and the volume of the simplex considered, as well.
By the above reasoning and (1), a class represented by dissimilarities can be characterized
by a linear proximity function with the weights w and the threshold ?. Our one-class
classifier CLPDD , Linear Programming Dissimilarity-data Description, is then defined as:
CLPDD (D(z, ?)) = I(
X
wj D(z, pj ) ? ?),
(2)
wj 6=0
where I is the indicator function. The proximity function is found as the solution to a soft
margin formulation (which is a straightforward extension of the hard margin case) with
? ? (0, 1] being the upper bound on the outlier fraction for the target class:
PN
min ? + ? 1N
i=1 ?i
(3)
s.t. wT D(pi , R) ? ? + ?i , i = 1, 2, .., N
P
j wj = 1, wj ? 0, ? ? 0, ?i ? 0.
In the LP formulations, sparse solutions are obtained, meaning that only some w j are positive. Objects corresponding to such non-zero weights, will be called support objects (SO).
The left plot of Fig. 1 is a 2D illustration of the LPDD. The data is represented in a metric
dissimilarity space, and by the triangle inequality the data can only be inside the prism
indicated by the dashed lines. The LPDD boundary is given by the hyperplane, as close to
the origin as possible (by minimizing ?), while still accepting (most) target objects. By the
discussion in Section 2, the outliers should be remote from the origin.
Proposition. In (3), ? ? (0, 1] is the upper bound on the outlier fraction for the target class,
i.e. the fraction of objects that lie outside the boundary; see also [11, 12]. This means that
1 PN
i=1 (1 ? CLPDD (D(pi , ?)) ? ?.
N
2
P is not expected to be parallel to the prism?s bottom hyperplane
D(. ,p j)
LPDD: min?
T
LPSD: min 1/N sum k(w K(xk ,S) + ? )
K(.,x j)
?
1
??
|| w ||1 = 1
w
0
Dissimilarity space
D(.,p i)
w
0
Similarity space
1
K(.,x i)
Figure 1: Illustrations of the LPDD in the dissimilarity space (left) and the LPSD in the similarity
space (right). The dashed lines indicate the boundary of the area which contains the genuine objects.
The LPDD tries to minimize the max-norm distance from the bounding hyperplane to the origin,
while the LPSD tries to attract the hyperplane towards the average of the distribution.
The proof goes analogously to the proofs given in [11, 12]. Intuitively, the proof
P follows
this: assume we have found a solution of (3). If ? is increased slightly, the term i ?i in the
objective function will change proportionally to the number of points that have non-zero ? i
(i.e. the outlier objects). At the optimum of (3) it has to hold that N ? ? #outliers.
Scaling dissimilarities. If D is unbounded, then some atypical objects of the target class
(i.e. with large dissimilarities) might badly influence the solution of (3). Therefore, we
propose a nonlinear, monotonous transformation of the distances to the interval [0, 1] such
that locally the distances are scaled linearly and globally, all large distances become close to
1. A function with such properties is the sigmoid function (the hyperbolical tangent can also
be used), i.e. Sigm(x) = 2/(1 + e?x/s ) ? 1, where s controls the ?slope? of the function,
i.e. the size of the local neighborhoods. Now, the transformation can be applied in an
element-wise way to the dissimilarity representation such that Ds (z, pi ) = Sigm(D(z, pi )).
Unless stated otherwise, the CLPDD will be trained on Ds .
A linear programming OCC on similarities. Recently, Campbell and Bennett have
proposed an LP formulation for novelty detection [3]. They start their reasoning from
a feature space in the spirit of positive definite kernels K(S, S) based on the set
S = {x1 , .., xN }. They restricted themselves to the (modified) RBF kernels, i.e. for
2
2
K(xi , xj ) = e?D(xi ,xj ) /2 s , where D is either Euclidean or L1 (city block) distance.
In principle, we will refer to RBFp , as to the ?Gaussian? kernel based on the Lp distance.
Here, to be consistent with our LPDD method, we rewrite their soft-margin LP formulation (a hard margin formulation is then obvious), to include a trade-off parameter ? (which
lacks, however, the interpretation as given in the LPDD), as follows:
PN
PN
min N1 i=1 (wT K(xi , S) + ?) + ? 1N
i=1 ?i
T
(4)
s.t.
w K(xi , S) + ? ? ??i , i = 1, 2, .., N
P
j wj = 1, wj ? 0, ?i ? 0.
Since K can be any similarity representation, for simplicity, we will call this method Linear
Programming Similarity-data Description (LPSD). The CLPSD is then defined as:
CLPSD (K(z, ?)) = I(
X
wj K(z, xj ) + ? ? 0).
(5)
wj 6=0
In the right plot of Fig. 1, a 2D illustration of the LPSD is shown. Here, the data is represented in a similarity space, such that all objects lie in a hypercube between 0 and 1.
Objects remote from the representation objects will be close to the origin. The hyperplane
is optimized to have minimal average output for the whole target set. This does not necessarily mean a good separation from the origin or a small volume of the OCC, possibly
resulting in an unnecessarily high outlier acceptance rate.
LPDD on the Euclidean representation
s = 0.3
1.5
s = 0.4
1.5
s = 0.5
1.5
s=1
1.5
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
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0
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1
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0
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0
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1
?0.5
0
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s=3
1.5
1
1
?0.5
0
0.5
1
LPSD based on RBF2
s = 0.3
1.5
s = 0.4
1.5
s = 0.5
1.5
s=1
1.5
1
1
1
1
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
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0
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1
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0
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1
?0.5
0
0.5
1
?0.5
0
0.5
s=3
1.5
1
1
?0.5
0
0.5
1
Figure 2: One-class hard margin LP classifiers for an artificial 2D data. From left to right, s takes
the values of 0.3d, 0.4d, 0.5d, d, 3d, where d is the average distance. Support objects are marked by
squares.
Extensions. Until now, the LPDD and LPSD were defined for square (dis)similarity matrices. If the computation of (dis)similarities is very costly, one can consider a reduced
representation set Rred ? R, consisting of n << N objects. Then, a dissimilarity or similarity representations are given as rectangular matrices D(R, Rred ) or K(S, Sred ), respectively.
Both formulations (3) and (4) remain the same with the only change that R/S is replaced by
Rred /Sred . An another reason to consider reduced representations is the robustness against
outliers. How to choose such a set is beyond the scope of this paper.
4 Experiments
Artificial datasets. First, we illustrate the LPDD and the LPSD methods on two artificial
datasets, both originally created in a 2D feature space. The first dataset contains two clusters with objects represented by Euclidean distances. The second dataset contains one uniform, square cluster and it is contaminated with three outliers. The
P objects are represented
by a slightly non-metric L0.95 dissimilarity (i.e. d0.95 (x, y) = [ i (xi ?yi )0.95 ]1/0.95 ). In
Fig. 2, the first dataset together with the decision boundaries of the LPDD and the LPSD
in the theoretical input space are shown. The parameter s used in all plots refers either to
the scaling parameter in the sigmoid function for the LPDD (based on D s ) or to the scaling
parameter in the RBF kernel. The pictures show similar behavior of both the LPDD and
the LPSD; the LPDD tends to be just slightly more tight around the target class.
LPDD on the Euclidean representation
1
? = 0.1; s = 0.2; e = 0
1
? = 0.1; s = 0.29; e = 0.04
1
? = 0.1; s = 0.46; e = 0.06
1
? = 0.1; s = 0.87; e = 0.06
1
0.5
0.5
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0
0
0
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0
LPSD based on RBF2
? = 0.1; s = 0.2; e = 0.04
1
? = 0.1; s = 0.29; e = 0.06
1
? = 0.1; s = 0.46; e = 0.06
1
0.5
1
? = 0.1; s = 0.87; e = 0.06
0
1
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0
0
0
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1
? = 0.1; s = 2.3; e = 0.08
0.5
1
? = 0.1; s = 2.3; e = 0.08
0
0.5
1
Figure 3: One-class LP classifiers, trained with ? = 0.1 for an artificial uniformly distributed 2D data
with 3 outliers. From left to right s takes the values of 0.7dm , dm , 1.6dm , 3dm , 8dm , where dm is
the median distance of all the distances. e refers to the error on the target set. Support objects are
marked by squares.
LPDD on the L0.95 representation
1
? = 0.1; s = 0.26; e = 0
1
? = 0.1; s = 0.37; e = 0.04
1
? = 0.1; s = 0.59; e = 0.06
1
? = 0.1; s = 1.1; e = 0.08
1
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LPSD based on RBF0.95
? = 0.1; s = 0.26; e = 0
1
? = 0.1; s = 0.37; e = 0.04
1
? = 0.1; s = 0.59; e = 0.06
1
? = 0.1; s = 1.1; e = 0.08
1
0.5
0.5
0.5
0.5
0.5
0
0
0
0
0
0
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1
0
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1
0
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1
0
? = 0.1; s = 3; e = 0.08
0.5
0.5
1
? = 0.1; s = 3; e = 0.06
1
0
0.5
1
Figure 4: One-class LP classifiers for an artificial 2D data. The same setting as in Fig.3 is used, only
for the L0.95 non-metric dissimilarities instead of the Euclidean ones. Note that the median distance
has changed, and consequently, the s values, as well.
1
LPDD
1
0.5
0.5
0
0
1
LPSD
? = 0.1; s = 0.41; e = 0.08
0
0.5
1
? = 0.1; s = 0.41; e = 0.06
1
? = 0.1; s = 1; e = 0.08
0
0.5
? = 0.1; s = 1; e = 0.08
1
? = 0.1; s = 0.4; e = 0.06
1
0.5
0.5
0
0
1
1
0
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1
? = 0.1; s = 0.4; e = 0.08
1
0.5
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0
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1
? = 0.1; s = 1; e = 0.08
0
0.5
? = 0.1; s = 1; e = 0.08
0
0.5
1
1
Figure 5: One-class LP classifiers, trained with ? = 0.1, for an artificial uniformly distributed 2D
data with 3 outliers, given by the L0.95 non-metric rectangular 50?6 dissimilarity representations.
The upper row shows the LPDD?s results and bottom row shows the LPSD?s results with the kernel
RBF0.95 . The objects of the reduced sets Rred and Sred are marked by triangles. Note that they differ
from left to right. e refers to the error on the target set. Support objects are marked by squares.
This becomes more clear in Fig. 3 and 4, where three outliers lying outside a single uniformly distributed cluster should be ignored when an OCC with a soft margin is trained.
From these figures, we can observe that the LPDD gives a tighter class description, which
is more robust against the scaling parameter and more robust against outliers, as well. The
same is observed when L0.95 dissimilarity is used instead of the Euclidean distances.
Fig. 5 presents some results for the reduced representations, in which just 6 objects are
randomly chosen for the set Rred . In the left four plots, Rred contains objects from the
uniform cluster only, and both methods perform equally well. In the right four plots, on
the other hand, Rred contains an outlier. It can be judged that for a suitable scaling s, no
outliers become support objects in the LPDD, which is often a case for the LPSD; see also
Fig. 4 and 3. Also, a crucial difference between the LPDD and LPSD can be observed w.r.t.
the support objects. In case of the LPSD (applied to a non-reduced representation), they lie
on the boundary, while in case of the LPDD, they tend to be ?inside? the class.
Condition monitoring. Fault detection is an important problem in the machine diagnostics: failure to detect faults can lead to machine damage, while false alarms can lead
to unnecessary expenses. As an example, we will consider a detection of four types of
fault in ball-bearing cages, a dataset [16] considered in [3]. Each data instance consists
of 2048 samples of acceleration taken with a Bruel and Kjaer vibration analyser. After
pre-processing with a discrete Fast Fourier Transform, each signal is characterized by 32
attributes. The dataset consists of five categories: normal behavior (NB), corresponding
Table 1: The errors of the first and second kind (in %) of the LPDD and LPSD on two dissimilarity
representations for the ball-bearing data. The reduced representations are based on 180 objects.
LPDD
LPDD-reduced
LPSD
LPSD-reduced
LPDD
LPDD-reduced
LPSD
LPSD-reduced
Euclidean representation
Method
Optimal s # of SO
NB T1
200.4
10
1.4 0.0
65.3
17
1.1 0.0
320.0
8
1.3 0.0
211.2
6
0.6 0.0
L1 dissimilarity representation
Method
Optimal s # of SO
NB T1
566.3
12
1.3 0.0
329.5
10
1.3 0.0
1019.3
8
0.9 0.0
965.7
5
0.3 0.0
Error
T2
45.0
20.2
46.7
39.9
T3
69.8
47.5
71.7
67.1
T4
70.0
50.9
74.5
69.5
Error
T2
1.6
2.3
2.2
3.5
T3
20.9
18.7
27.9
26.3
T4
19.8
16.9
27.2
27.5
to measurements made from new ball-bearings, and four types of anomalies, say, T 1 ? T4 ,
corresponding either to the damaged outer race or cages or a badly worn ball-bearing. To
compare our LPDD method with the LPSD method, we performed experiments in the same
way, as described in [3], making use of the same training set, and independent validation
and test sets; see Fig. 6.
The optimal values of s were found for both LPDD and
Train Valid. Test
LPSD methods by the use of the validation set on the EuNB
913
913
913
clidean and L1 dissimilarity representations. The results
T
747
747
1
are presented in Table 1. It can be concluded that the
T2
913
996
L1 representation is far more convenient for the fault deT3
996
tection, especially if we look at the fault type T3 and T4
T4
996
which were unseen in the validation process. The LPSD
performs better on normal instances (yields a smaller erFigure 6: Fault detection data.
ror) than the LPDD. This is to be expected, since the
boundary is less tight, by which less support objects (SO) are needed. On the contrary, the
LPSD method deteriorates w.r.t. the outlier detection. Note also that the reduced representation, based on randomly chosen 180 target objects (? 20%) mostly yields significantly
better performances in outlier detection for the LPDD, and in target acceptance for the
LPSD. Therefore, we can conclude that if a failure in the fault detection has higher costs
than the cost of misclassifying target objects, then our approach should be recommended.
5 Conclusions
We have proposed the Linear Programming Dissimilarity-data Description (LPDD) classifier, directly built on dissimilarity representations. This method is efficient, which means
that only some objects are needed for the computation of dissimilarities in a test phase.
The novelty of our approach lies in its reformulation for general dissimilarity measures,
which, we think, is more natural for class descriptors. Since dissimilarity measures might
be unbounded, we have also proposed to transform dissimilarities by the sigmoid function,
which makes the LPDD more robust against objects with large dissimilarities. We emphasized the possibility of using the LP procedures for rectangular dissimilarity/similarity
representations, which is especially useful when (dis)similarities are costly to compute.
The LPDD is applied to artificial and real-world datasets and compared to the LPSD detector as proposed in [3]. For all considered datasets, the LPDD yields a more compact target
description than the LPSD. The LPDD is more robust against outliers in the training set, in
particular, when only some objects are considered for a reduced representation. Moreover,
with a proper scaling parameter s of the sigmoid function, the support objects in the LPDD
do not contain outliers, while it seems difficult for the LPSD to achieve the same. In the
original formulation, the support objects of the LPSD tend to lie on the boundary, while
for the LPDD, they are mostly ?inside? the boundary. This means that a removal of such an
object will not impose a drastic change of the LPDD.
In summary, our LPDD method can be recommended when the failure to detect outliers is
more expensive than the costs of a false alarm. It is also possible to enlarge the description
of the LPDD by adding a small constant to ?. Such a constant should be related to the
dissimilarity values in the neighborhood of the boundary. How to choose it, remains an
open issue for further research.
Acknowledgements. This work is partly supported by the Dutch Organization for Scientific Research (NWO) and the European Community Marie Curie Fellowship. The authors are solely responsible for information communicated and the European Commission is not responsible for any views
or results expressed.
References
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support:14 dissimilar:1 violated:2 |
1,278 | 2,164 | Distance Metric Learning, with Application
to Clustering with Side-Information
Eric P. Xing, Andrew Y. Ng, Michael I. Jordan and Stuart Russell
University of California, Berkeley
Berkeley, CA 94720
epxing,ang,jordan,russell @cs.berkeley.edu
Abstract
Many algorithms rely critically on being given a good metric over their
inputs. For instance, data can often be clustered in many ?plausible?
ways, and if a clustering algorithm such as K-means initially fails to find
one that is meaningful to a user, the only recourse may be for the user to
manually tweak the metric until sufficiently good clusters are found. For
these and other applications requiring good metrics, it is desirable that
we provide a more systematic way for users to indicate what they consider ?similar.? For instance, we may ask them to provide examples. In
this paper, we present an algorithm that, given examples of similar (and,
if desired, dissimilar) pairs of points in , learns a distance metric over
that respects these relationships. Our method is based on posing metric learning as a convex optimization problem, which allows us to give
efficient, local-optima-free algorithms. We also demonstrate empirically
that the learned metrics can be used to significantly improve clustering
performance.
1 Introduction
The performance of many learning and datamining algorithms depend critically on their
being given a good metric over the input space. For instance, K-means, nearest-neighbors
classifiers and kernel algorithms such as SVMs all need to be given good metrics that reflect
reasonably well the important relationships between the data. This problem is particularly
acute in unsupervised settings such as clustering, and is related to the perennial problem of
there often being no ?right? answer for clustering: If three algorithms are used to cluster a
set of documents, and one clusters according to the authorship, another clusters according
to topic, and a third clusters according to writing style, who is to say which is the ?right?
answer? Worse, if an algorithm were to have clustered by topic, and if we instead wanted it
to cluster by writing style, there are relatively few systematic mechanisms for us to convey
this to a clustering algorithm, and we are often left tweaking distance metrics by hand.
In this paper, we are interested in the following problem: Suppose a user indicates that
certain points in an input space (say, ) are considered by them to be ?similar.? Can we
automatically learn a distance metric over that respects these relationships, i.e., one that
assigns small distances between the similar pairs? For instance, in the documents example,
we might hope that, by giving it pairs of documents judged to be written in similar styles,
it would learn to recognize the critical features for determining style.
One important family of algorithms that (implicitly) learn metrics are the unsupervised
ones that take an input dataset, and find an embedding of it in some space. This includes
algorithms such as Multidimensional Scaling (MDS) [2], and Locally Linear Embedding
(LLE) [9]. One feature distinguishing our work from these is that we will learn a full metric
over the input space, rather than focusing only on (finding an embedding for) the points in the training set. Our learned metric thus generalizes more easily to
previously unseen data. More importantly, methods such as LLE and MDS also suffer from
the ?no right answer? problem: For example, if MDS finds an embedding that fails to capture the structure important to a user, it is unclear what systematic corrective actions would
be available. (Similar comments also apply to Principal Components Analysis (PCA) [7].)
As in our motivating clustering example, the methods we propose can also be used in a
pre-processing step to help any of these unsupervised algorithms to find better solutions.
In the supervised learning setting, for instance nearest neighbor classification, numerous
attempts have been made to define or learn either local or global metrics for classification.
In these problems, a clear-cut, supervised criterion?classification error?is available and
can be optimized for. (See also [11], for a different way of supervising clustering.) This
literature is too wide to survey here, but some relevant examples include [10, 5, 3, 6],
and [1] also gives a good overview of some of this work. While these methods often
learn good metrics for classification, it is less clear whether they can be used to learn
good, general metrics for other algorithms such as K-means, particularly if the information
available is less structured than the traditional, homogeneous training sets expected by
them.
In the context of clustering, a promising approach was recently proposed by Wagstaff et
al. [12] for clustering with similarity information. If told that certain pairs are ?similar? or
?dissimilar,? they search for a clustering that puts the similar pairs into the same, and dissimilar pairs into different, clusters. This gives a way of using similarity side-information
to find clusters that reflect a user?s notion of meaningful clusters. But similar to MDS and
LLE, the (?instance-level?) constraints that they use do not generalize to previously unseen
data whose similarity/dissimilarity to the training set is not known. We will later discuss
this work in more detail, and also examine the effects of using the methods we propose in
conjunction with these methods.
2 Learning Distance Metrics
, and are given information that certain
!
and are similar
(1)
#
"#!if between
How can we learn a distance metric
points and that respects this;
Suppose we have some set of points
pairs of them are ?similar?:
specifically, so that ?similar? points end up close to each other?
Consider learning a distance metric of the form
"#! %$ '&( "# )$+*,* -.#/*,* & $+0 1-2# 4365 7-.# 98
5
5;:=<
(2)
5>$@?
To ensure that this be a metric?satisfying non-negativity and the triangle inequality?
we require that
be positive semi-definite,
.1 Setting
gives Euclidean
distance; if we restrict to be diagonal, this corresponds to learning a metric in which
the different axes are given different ?weights?; more generally, parameterizes a family
of Mahalanobis distances over .2 Learning such a distance metric is also equivalent
to finding a rescaling of a data that replaces each point with
and applying the
5
5
5 ACB
Technically, this also allows pseudometrics, where DFEHGJILKNMPO/QSR does not imply ITQUM .
that, but putting the original dataset through a non-linear basis function V and considering
W GXV/Note
GJI!O Y7VZGJMPONO\[L]^GXV/GJI!OY_VZGJMPONO , non-linear distance metrics can also be learned.
1
2
standard Euclidean metric to the rescaled data; this will later be useful in visualizing the
learned metrics.
A simple way of defining a criterion for the desired metric is to demand that
pairs of points
in
have, say, small squared distance between them:
. This is trivially solved with
, which is not
useful, and we add the constraint
to ensure that does not
collapse the dataset into a single point. Here, can be a set of pairs of points known to be
?dissimilar? if such information is explicitly available; otherwise, we may take it to be all
pairs not in . This gives the optimization problem:
B
&
,* *
- *,* &
,* *
-S!!** &
!
5 $ <
5
&
,* *
/-2!'*,* B&
" ,* * L
/-2!'*,* &# $
s.t.
(3)
5 : <8
5
(4)
B
% 5
(5)
The choice of the constant 1 in the right hand side of (4) is arbitrary but not important, and
changing it to any other positive constant results only in being replaced by
. Also,
this problem has an objective that is linear in the parameters , and both of the constraints
are also easily verified to be convex. Thus, the optimization problem is convex, which
enables us to derive efficient, local-minima-free algorithms to solve it.
We also note that, while one might consider various alternatives to (4), ?
? would not be a good choice despite its giving a simple linear constraint. It
would result in always being rank 1 (i.e., the data are always projected onto a line). 3
%
!'*,* B& '
5
& **
-
5
2.1 The case of diagonal
5
5 $)( *,+ 5 C C5 BCB 8 8 8ZC5
In the case that we want to learn a diagonal
an efficient algorithm using the Newton-Raphson method. Define
, we can derive
.
*,*
-2 *,* B& -0/1 +324 .
*,*
-2 *,* &657
5 : < ) is equivalent, up to a
It is straightforward
to show that minimizing - (subject to
5
multiplication of by a positive constant, to solving the original problem (3?5). We can
thus use Newton-Raphson to efficiently optimize - .4
- 5 %$ - 5 C 8 8 8Z5
2.2 The case of full
)$
5
9B
5
5 : <
8 :9<;
In the case of learning a full matrix , the constraint that
becomes slightly trickier
to enforce, and Newton?s method often becomes prohibitively expensive (requiring
time to invert the Hessian over
parameters). Using gradient descent and the idea of
iterative projections (e.g., [8]) we derive a different algorithm for this setting.
G>=@?BA C C A DFDFE EGIHBH IKJ"Y IL HBH EM ONO=@?BA C A PD:EQRHBH IKJ"Y IL HBH EM Q
STU VXW]ZY G N[STUVXWL]ZY Q
Y]\ Q'=^?_A A \ GJKI JHYSIL OCGJI`J YSIL O [G . Decomposing
] as ] Qa=cbJdfehg J g [J (always pos] i R ), this gives = Jg [J Y g J:NO= Jjg [J Y Q g J , which we recognize as a Rayleighsible since a
quotient
whose solution is given by (say) solving the generalized eigenvector problem
G like quantity
Y ge#
Q kKY Q g e for the principal eigenvector, and setting g M Q@lXlXl Q g b QSR .
To ensure that m
] iR , whicheqsisr true iff theeqsdiagonal
] JJ are non-negative, we actually
r , whereelements
t is a step-size parameter optimized via a
replace the Newton update npo
by tun]o
line-search to give the largest downhill step subject to Z
] JJuv.R .
3
The proof is reminiscent of the derivation of Fisher?s linear discriminant. Briefly, consider maximizing
, where
4
Iterate
Iterate
5 $ *P+ & !** 5 -25 ** 5"
5 $ *P+ & !** 5 -25 ** 5 B
5
until converges
5 $ 5
& - 5
until convergence
HBH Y BH H Q G:= = Y J M L O e M
J L
Figure 1: Gradient ascent + Iterative projection algorithm. Here,
).
matrices (
*&
HBHhHBH
is the Frobenius norm on
- 5 )$
,* * L
!'*,* &
5 )$
**
*,* B&
We pose the equivalent problem:
s.t.
(6)
5 : <8
5
We will use a gradient ascent step on -
(7)
(8)
to optimize (6), followed by the method of
iterative projections to ensure that the
constraints
(7) and (8) hold. Specifically, we will
repeatedly
take
a
gradient
step
, and then repeatedly project into
and
. This gives the
the sets
algorithm shown in Figure 1.5
The motivation
for the specific choice of the problem formulation (6?8) is that projecting
onto or can be done
Specifically, the first projection step
" #inexpensively.
!
involves minimizing a quadratic objective subject to
a single linear constraint; the solution to this is easily found by solving
time)
(in
of
a sparse system of linear equations. The second projection step onto , the space
%$ '& $ all
positive-semi
definite
matrices,
is
done
by
first
finding
the
diagonalization
,
&
(
(
is a diagonal matrix of ?s eigenvalues
and
the& columns
of
where
$
+
,
$
$
?s corresponding
eigenvectors, and taking
, where
*) contains
&
(
(
!
. (E.g., see [4].)
5 $ 5
$ 5
,* * L
"-U'** B&
5
B
* + & *,* 5 - 5 ** B 5 (
$ ( *,+ 8 8 8"
5
$ ( *,+ *
< 8 8 8Z
<
& - 5
B
5
5
B $ 5 5 : <
5 $
B
8 F9
5 $
3
5 $
3
3 Experiments and Examples
We begin by giving some examples of distance metrics learned on artificial data, and then
show how our methods can be used to improve clustering performance.
3.1 Examples of learned distance metrics
Consider the data shown in Figure 2(a), which is divided into two classes (shown by the
different symbols and, where available, colors). Suppose that points in each class are ?similar? to each other, and we are given reflecting this.6 Depending on whether we learn a
diagonal or a full , we obtain:
;<>=
;<
.0/ 12354016 879 1.036 :
?A@06 6 879 3.245 3.286 0.081
:
:
1.007 :
3.286 3.327 0.082
:
:
:
0.081 0.082 0.002
CB
To visualize this, we can use the fact discussed earlier that learning
is equivalent
to finding a rescaling of the data
, that hopefully ?moves? the similar pairs
&
5
&
*,* *,* &
5 ACB
q Er
q E
5
The algorithm shown in the figure includes a small refinement that the gradient step is taken the
direction of the projection of
onto the orthogonal subspace of 'D , so that it will ?minimally?
disrupt the constraint E . Empirically, this modification often significantly speeds up convergence.
6
In the experiments with synthetic data, F was a randomly sampled 1% of all pairs of similar
points.
e
2?class data (original)
2?class data projection (Newton)
5
5
0
0
5
z
z
z
2?class data projection (IP)
?5
?5
5
0
?5
y
?5
20
5
5
0
0
?5
0
y
x
?5
(a)
?5
5
0
20
0
0
y
x
?20
(b)
?20
x
(c)
]
Figure 2: (a) Original data, with the different classes indicated by the different symbols (and colors, where available). (b) Rescaling of data corresponding to learned diagonal . (c) Rescaling
corresponding to full .
]
3?class data (original)
3?class data projection (Newton)
2
?2
2
0
z
0
z
z
2
3?class data projection (IP)
?2
5
0
?5
y
?5
5
5
0
0
?2
0
y
x
?5
(a)
?5
5
0
2
y
(b)
]
Figure 3: (a) Original data. (b) Rescaling corresponding to learned diagonal
sponding to full .
5 ACB
2
0
x
]
0
?2
?2
x
(c)
. (c) Rescaling corre-
together. Figure 2(b,c) shows the result of plotting
. As we see, the algorithm has
successfully brought together the similar points, while keeping dissimilar ones apart.
Figure 3 shows a similar result for a case of three clusters whose centroids differ only
in the x and y directions. As we see in Figure 3(b), the learned diagonal metric correctly
ignores the z direction. Interestingly, in the case of a full , the algorithm finds a surprising
projection of the data onto a line that still maintains the separation of the clusters well.
5
3.2 Application to clustering
One application of our methods is ?clustering with side information,? in which we learn
a distance metric using similarity information, and cluster data using that metric. Specifically, suppose we are given , and told that each pair
means and belong
to the same cluster. We will consider four algorithms for clustering:
Z
C!
L
*,*
-*,* BB
U
2. Constrained K-means: K-means but subject to points
1. K-means using the default Euclidean metric
between points
to define distortion (and ignoring ).
cluster centroids
and
always being
assigned to the same cluster [12].7
**
"- *,* B&
3. K-means + metric: K-means but with distortion defined using the distance metric
learned from .
4. Constrained K-means + metric: Constrained K-means using the distance metric
learned from .
GJI J KNI L O
IKJ I L
U T
GJIKJ Y O M
This is implemented as the usual K-means, except if
F , then during the step in which
points are assigned to cluster centroids , we assign both and to cluster
. More generally, if we imagine drawing an edge between each pair of points in , then all
the points in each resulting connected component E are constrained to lie in the same cluster, which
we pick to be
.
7
GJI L Y O M
U T
O= A E GJIKJY O M
Porjected 2?class data
10
10
0
0
z
z
Original 2?class data
?10
?10
20
y
20
20
0
?20
?20
y
x
(a)
1.
2.
3.
4.
20
0
0
0
?20
?20
x
(b)
K-means: Accuracy = 0.4975
Constrained K-means: Accuracy = 0.5060
K-means + metric: Accuracy = 1
Constrained K-means + metric: Accuracy = 1
] ?s result is
]
$ 8 8 89 ) be the cluster to which point
is assigned by an automatic clustering
Let % (
algorithm, and let % be some ?correct? or desired clustering
of the data.
Following [?], in
the case of 2-cluster data, we will measure how well the % ?s match the % ?s according to
%
$ % $ %
$ %
$
.
<
8 -
Accuracy
B is the indicator function ( $ , * / $ < ). This is equivalent to
where
L
! drawn randomly from the dataset, our clustering
the
probability that for two points ,
Z
and ! belong to same or different
% agrees
with the ?true? clustering % on whether
Figure 4: (a) Original dataset (b) Data scaled according to learned metric. (
shown, but
gave visually indistinguishable results.)
clusters.8
As a simple example, consider Figure 4, which shows a clustering problem in which the
?true clusters? (indicated by the different symbols/colors in the plot) are distinguished by
their -coordinate, but where the data in its original space seems to cluster much better
according to their -coordinate. As shown by the accuracy scores given in the figure, both
K-means and constrained K-means failed to find good clusterings. But by first learning
a distance metric and then clustering according to that metric, we easily find the correct
clustering separating the true clusters from each other. Figure 5 gives another example
showing similar results.
We also applied our methods to 9 datasets from the UC Irvine repository. Here, the ?true
clustering? is given by the data?s class labels. In each, we ran one experiment using ?little? side-information , and one with ?much? side-information. The results are given in
Figure 6.9
We see that, in almost every problem, using a learned diagonal or full metric leads to
significantly improved performance over naive K-means. In most of the problems, using
a learned metric with constrained K-means (the 5th bar for diagonal , 6th bar for full )
also outperforms using constrained K-means alone (4th bar), sometimes by a very large
#
5
8
5
In the case of many (
) clusters, this evaluation metric tends to give inflated scores since
almost any clustering will correctly predict that most pairs are in different clusters. In this setting,
we therefore modified the measure averaging not only ,
drawn uniformly at random, but from
the same cluster (as determined by ) with chance 0.5, and from different clusters with chance 0.5, so
that ?matches? and ?mis-matches? are given the same weight. All results reported here used K-means
with multiple restarts, and are averages over at least 20 trials (except for wine, 10 trials).
9
F was generated by picking a random subset of all pairs of points sharing the same class . In
the case of ?little? side-information, the size of the subset was chosen so that the resulting number of
resulting connected components
(see footnote 7) would be very roughly 90% of the size of the
original dataset. In the case of ?much? side-information, this was changed to 70%.
!
IJ I L
!J
"$#
Projected data
50
50
0
0
z
z
Original data
?50
?50
50
y
50
50
0
?50
50
0
0
?50
y
x
(a)
1.
2.
3.
4.
0
?50
?50
K-means: Accuracy = 0.4993
Constrained K-means: Accuracy = 0.5701
K-means + metric: Accuracy = 1
Constrained K-means + metric: Accuracy = 1
]
Figure 5: (a) Original dataset (b) Data scaled according to learned metric. (
shown, but
gave visually indistinguishable results.)
Boston housing (N=506, C=3, d=13)
ionosphere (N=351, C=2, d=34)
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0
Kc=354
wine (N=168, C=3, d=12)
Kc=269
Kc=187
0
balance (N=625, C=3, d=4)
1
1
1
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0
Kc=127
soy bean (N=47, C=4, d=35)
Kc=548
Kc=400
0
protein (N=116, C=6, d=20)
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0
Kc=34
Kc=92
Kc=61
Kc=116
Kc=482
Kc=358
diabetes (N=768, C=2, d=8)
1
Kc=41
Kc=133
breast cancer (N=569, C=2, d=30)
0.8
Kc=153
] ?s result is
Iris plants (N=150, C=3, d=4)
1
Kc=447
x
(b)
0
Kc=694
Kc=611
Figure 6: Clustering accuracy on 9 UCI datasets. In each panel, the six bars on the left correspond to
an experiment with ?little? side-information F , and the six on the right to ?much? side-information.
From left to right, the six bars in each set are respectively K-means, K-means
diagonal metric, K-means full metric, Constrained K-means (C-Kmeans), C-Kmeans diagonal metric, and
C-Kmeans full metric. Also shown are : size of dataset; E : number of classes/clusters; : dimensionality of data;
: mean number of connected components (see footnotes 7, 9). 1 s.e. bars
are also shown.
"$#
D
Performance on Wine dataset
1
0.9
0.9
0.8
0.8
performance
performance
Performance on Protein dataset
1
0.7
0.6
0.6
kmeans
c?kmeans
kmeans + metric (diag A)
c?kmeans + metric (diag A)
kmeans + metric (full A)
c?kmeans + metric (full A)
0.5
0
0.1
kmeans
c?kmeans
kmeans + metric (diag A)
c?kmeans + metric (diag A)
kmeans + metric (full A)
c?kmeans + metric (full A)
0.7
0.5
0.2
0
0.1
ratio of constraints
(a)
0.2
I
ratio of constraints
(b)
Figure 7: Plots of accuracy vs. amount of side-information. Here, the -axis gives the fraction of all
pairs of points in the same class that are randomly sampled to be included in F .
margin. Not surprisingly, we also see that having more side-information in typically
leads to metrics giving better clusterings.
Figure 7 also shows two typical examples of how the quality of the clusterings found increases with the amount of side-information. For some problems (e.g., wine), our algorithm learns good diagonal and full metrics quickly with only a very small amount of
side-information; for some others (e.g., protein), the distance metric, particularly the full
metric, appears harder to learn and provides less benefit over constrained K-means.
4 Conclusions
We have presented an algorithm that, given examples of similar pairs of points in , learns
a distance metric that respects these relationships. Our method is based on posing metric
learning as a convex optimization problem, which allowed us to derive efficient, localoptima free algorithms. We also showed examples of diagonal and full metrics learned
from simple artificial examples, and demonstrated on artificial and on UCI datasets how
our methods can be used to improve clustering performance.
References
[1] C. Atkeson, A. Moore, and S. Schaal. Locally weighted learning. AI Review, 1996.
[2] T. Cox and M. Cox. Multidimensional Scaling. Chapman & Hall, London, 1994.
[3] C. Domeniconi and D. Gunopulos. Adaptive nearest neighbor classification using support vector machines. In Advances in Neural Information Processing Systems 14. MIT Press, 2002.
[4] G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins Univ. Press, 1996.
[5] T. Hastie and R. Tibshirani. Discriminant adaptive nearest neighbor classification. IEEE Transactions on Pattern Analysis and Machine Learning, 18:607?616, 1996.
[6] T.S. Jaakkola and D. Haussler. Exploiting generative models in discriminaive classifier. In Proc.
of Tenth Conference on Advances in Neural Information Processing Systems, 1999.
[7] I.T. Jolliffe. Principal Component Analysis. Springer-Verlag, New York, 1989.
[8] R. Rockafellar. Convex Analysis. Princeton Univ. Press, 1970.
[9] S.T. Roweis and L.K. Saul. Nonlinear dimensionality reduction by locally linear embedding.
Science 290: 2323-2326.
[10] B. Scholkopf and A. Smola. Learning with Kernels. In Press, 2001.
[11] N. Tishby, F. Pereira, and W. Bialek. The information bottleneck method. In Proc. of the 37th
Allerton Conference on Communication, Control and Computing, 1999.
[12] K. Wagstaff, C. Cardie, S. Rogers, and S. Schroedl. Constrained k-means clustering with background knowledge. In Proc. 18th International Conference on Machine Learning, 2001.
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1,279 | 2,165 | Charting a Manifold
Matthew Brand
Mitsubishi Electric Research Labs
201 Broadway, Cambridge MA 02139 USA
www.merl.com/people/brand/
Abstract
We construct a nonlinear mapping from a high-dimensional sample space
to a low-dimensional vector space, effectively recovering a Cartesian
coordinate system for the manifold from which the data is sampled.
The mapping preserves local geometric relations in the manifold and is
pseudo-invertible. We show how to estimate the intrinsic dimensionality
of the manifold from samples, decompose the sample data into locally
linear low-dimensional patches, merge these patches into a single lowdimensional coordinate system, and compute forward and reverse mappings between the sample and coordinate spaces. The objective functions
are convex and their solutions are given in closed form.
1
Nonlinear dimensionality reduction (NLDR) by charting
Charting is the problem of assigning a low-dimensional coordinate system to data points
in a high-dimensional sample space. It is presumed that the data lies on or near a lowdimensional manifold embedded in the sample space, and that there exists a 1-to-1 smooth
nonlinear transform between the manifold and a low-dimensional vector space. The datamodeler?s goal is to estimate smooth continuous mappings between the sample and coordinate spaces. Often this analysis will shed light on the intrinsic variables of the datagenerating phenomenon, for example, revealing perceptual or configuration spaces.
Our goal is to find a mapping?expressed as a kernel-based mixture of linear projections?
that minimizes information loss about the density and relative locations of sample points.
This constraint is expressed in a posterior that combines a standard gaussian mixture model
(GMM) likelihood function with a prior that penalizes uncertainty due to inconsistent projections in the mixture. Section 3 develops a special case where this posterior is unimodal
and maximizable in closed form, yielding a GMM whose covariances reveal a patchwork of
overlapping locally linear subspaces that cover the manifold. Section 4 shows that for this
(or any) GMM and a choice of reduced dimension d, there is a unique, closed-form solution
for a minimally distorting merger of the subspaces into a d-dimensional coordinate space,
as well as an reverse mapping defining the surface of the manifold in the sample space.
The intrinsic dimensionality d of the data manifold can be estimated from the growth process of point-to-point distances. In analogy to differential geometry, we call the subspaces
?charts? and their merger the ?connection.? Section 5 considers example problems where
these methods are used to untie knots, unroll and untwist sheets, and visualize video data.
1.1 Background
Topology-neutral
NLDR
algorithms can be divided into those that compute mappings, and
those that directly compute low-dimensional embeddings. The field has its roots in mapping algorithms: DeMers and Cottrell [3] proposed using auto-encoding neural networks
with a hidden layer ? bottleneck,? effectively casting dimensionality reduction as a compression problem. Hastie defined principal curves [5] as nonparametric 1 D curves that pass
through the center of ? nearby? data points. A rich literature has grown up around properly
regularizing this approach and extending it to surfaces. Smola and colleagues [10] analyzed
the NLDR problem in the broader framework of regularized quantization methods.
More recent advances aim for embeddings: Gomes and Mojsilovic [4] treat manifold completion as an anisotropic diffusion problem, iteratively expanding points until they connect
to their neighbors. The I SO M AP algorithm [12] represents remote distances as sums of a
trusted set of distances between immediate neighbors, then uses multidimensional scaling
to compute a low-dimensional embedding that minimally distorts all distances. The locally
linear embedding algorithm (LLE) [9] represents each point as a weighted combination of
a trusted set of nearest neighbors, then computes a minimally distorting low-dimensional
barycentric embedding. They have complementary strengths: I SO M AP handles holes well
but can fail if the data hull is nonconvex [12]; and vice versa for LLE [9]. Both offer embeddings without mappings. It has been noted that trusted-set methods are vulnerable to
noise because they consider the subset of point-to-point relationships that has the lowest
signal-to-noise ratio; small changes to the trusted set can induce large changes in the set of
constraints on the embedding, making solutions unstable [1].
In a return to mapping, Roweis and colleagues [8] proposed global coordination? learning
a mixture of locally linear projections from sample to coordinate space. They constructed
a posterior that penalizes distortions in the mapping, and gave a expectation-maximization
(EM) training rule. Innovative use of variational methods highlighted the difficulty of even
hill-climbing their multimodal posterior. Like [2, 7, 6, 8], the method we develop below is
a decomposition of the manifold into locally linear neighborhoods. It bears closest relation
to global coordination [8], although by a different construction of the problem, we avoid
hill-climbing a spiky posterior and instead develop a closed-form solution.
2
Estimating locally linear scale and intrinsic dimensionality
.
We begin with matrix of sample points Y = [y1 , ? ? ? , yN ], yn ? RD populating a Ddimensional sample space, and a conjecture that these points are samples from a manifold M of intrinsic dimensionality d < D. We seek a mapping onto a vector space
.
G(Y) ? X = [x1 , ? ? ? , xN ], xn ? Rd and 1-to-1 reverse mapping G?1 (X) ? Y such that
local relations between nearby points are preserved (this will be formalized below). The
map G should be non-catastrophic, that is, without folds: Parallel lines on the manifold in
RD should map to continuous smooth non-intersecting curves in Rd . This guarantees that
linear operations on X such as interpolation will have reasonable analogues on Y.
Smoothness means that at some scale r the mapping from a neighborhood on M to Rd is
effectively linear. Consider a ball of radius r centered on a data point and containing n(r)
data points. The count n(r) grows as rd , but only at the locally linear scale; the grow rate
is inflated by isotropic noise at smaller scales and by embedding curvature at larger scales.
.
To estimate r, we look at how the r-ball grows as points are added to it, tracking c(r) =
d
d log n(r) log r. At noise scales, c(r) ? 1/D < 1/d, because noise has distributed points in
all directions with equal probability. At the scale at which curvature becomes significant,
c(r) < 1/d, because the manifold is no longer perpendicular to the surface of the ball, so
the ball does not have to grow as fast to accommodate new points. At the locally linear
scale, the process peaks at c(r) = 1/d, because points are distributed only in the directions
of the manifold?s local tangent space. The maximum of c(r) therefore gives an estimate
of both the scale and the local dimensionality of the manifold (see figure 1), provided that
the ball hasn?t expanded to a manifold boundary? boundaries have lower dimension than
Scale behavior of a 1D manifold in 2-space
Point?count growth process on a 2D manifold in 3?space
1
10
radial growth process
1D hypothesis
2D hypothesis
3D hypothesis
radius (log scale)
samples
noise scale
locally linear scale
curvature scale
0
10
2
1
10
2
10
#points (log scale)
3
10
Figure 1: Point growth processes. L EFT: At the locally linear scale, the number of points
in an r-ball grows as rd ; at noise and curvature scales it grows faster. R IGHT: Using the
point-count growth process to find the intrinsic dimensionality of a 2D manifold nonlinearly
embedded in 3-space (see figure 2). Lines of slope 1/3 , 1/2 , and 1 are fitted to sections of the
log r/ log nr curve. For neighborhoods of radius r ? 1 with roughly n ? 10 points, the slope
peaks at 1/2 indicating a dimensionality of d = 2. Below that, the data appears 3 D because
it is dominated by noise (except for n ? D points); above, the data appears >2 D because of
manifold curvature. As the r-ball expands to cover the entire data-set the dimensionality
appears to drop to 1 as the process begins to track the 1D edges of the 2D sheet.
the manifold. For low-dimensional manifolds such as sheets, the boundary submanifolds
(edges and corners) are very small relative to the full manifold, so the boundary effect is
typically limited to a small rise in c(r) as r approaches the scale of the entire data set. In
practice, our code simply expands an r-ball at every point and looks for the first peak in
c(r), averaged over many nearby r-balls. One can estimate d and r globally or per-point.
3
Charting the data
In the charting step we find a soft partitioning of the data into locally linear low-dimensional
neighborhoods, as a prelude to computing the connection that gives the global lowdimensional embedding. To minimize information loss in the connection, we require that
the data points project into a subspace associated with each neighborhood with (1) minimal
loss of local variance and (2) maximal agreement of the projections of nearby points into
nearby neighborhoods. Criterion (1) is served by maximizing the likelihood function of a
Gaussian mixture model (GMM) density fitted to the data:
.
p(yi |?, ?) = ? j p(yi |? j , ? j ) p j = ? j N (yi ; ? j , ? j ) p j .
(1)
Each gaussian component defines a local neighborhood centered around ? j with axes defined by the eigenvectors of ? j . The amount of data variance along each axis is indicated
by the eigenvalues of ? j ; if the data manifold is locally linear in the vicinity of the ? j , all
but the d dominant eigenvalues will be near-zero, implying that the associated eigenvectors constitute the optimal variance-preserving local coordinate system. To some degree
likelihood maximization will naturally realize this property: It requires that the GMM components shrink in volume to fit the data as tightly as possible, which is best achieved by
positioning the components so that they ? pancake? onto locally flat collections of datapoints. However, this state of affairs is easily violated by degenerate (zero-variance) GMM
components or components fitted to overly small enough locales where the data density off
the manifold is comparable to density on the manifold (e.g., at the noise scale). Consequently a prior is needed.
Criterion (2) implies that neighboring partitions should have dominant axes that span similar subspaces, since disagreement (large subspace angles) would lead to inconsistent projections of a point and therefore uncertainty about its location in a low-dimensional coordinate space. The principal insight is that criterion (2) is exactly the cost of coding the
location of a point in one neighborhood when it is generated by another neighborhood? the
cross-entropy between the gaussian models defining the two neighborhoods:
D(N1 kN2 )
=
=
Z
dy N (y; ?1 ,?1 ) log
N (y; ?1 ,?1 )
N (y; ?2 ,?2 )
> ?1
?1
(log |??1
1 ?2 | + trace(?2 ?1 ) + (?2 ??1 ) ?2 (?2 ??1 ) ? D)/2. (2)
Roughly speaking, the terms in (2) measure differences in size, orientation, and position,
respectively, of two coordinate frames located at the means ?1 , ?2 with axes specified by
the eigenvectors of ?1 , ?2 . All three terms decline to zero as the overlap between the two
frames is maximized. To maximize consistency between adjacent neighborhoods, we form
.
the prior p(?, ?) = exp[? ?i6= j mi (? j )D(Ni kN j )], where mi (? j ) is a measure of co-locality.
Unlike global coordination [8], we are not asking that the dominant axes in neighboring
charts are aligned? only that they span nearly the same subspace. This is a much easier
objective to satisfy, and it contains a useful special case where the posterior p(?, ?|Y) ?
?i p(yi |?, ?)p(?, ?) is unimodal and can be maximized in closed form: Let us associate a
gaussian neighborhood with each data-point, setting ?i = yi ; take all neighborhoods to be
a priori equally probable, setting pi = 1/N; and let the co-locality measure be determined
from some local kernel. For example, in this paper we use mi (? j ) ? N (? j ; ?i , ?2 ), with
the scale parameter ? specifying the expected size of a neighborhood on the manifold in
sample space. A reasonable choice is ? = r/2, so that 2erf(2) > 99.5% of the density of
mi (? j ) is contained in the area around yi where the manifold is expected to be locally linear.
With uniform pi and ?i , mi (? j ) and fixed, the MAP estimates of the GMM covariances are
!,
?i =
? mi (? j ) (y j ? ?i )(y j ? ?i )> + (? j ? ?i )(? j ? ?i )> + ? j
? mi (? j ) (3).
j
j
Note that each covariance ?i is dependent on all other ? j . The MAP estimators for all
covariances can be arranged into a set of fully constrained linear equations and solved exactly for their mutually optimal values. This key step brings nonlocal information about
the manifold?s shape into the local description of each neighborhood, ensuring that adjoining neighborhoods have similar covariances and small angles between their respective
subspaces. Even if a local subset of data points are dense in a direction perpendicular to
the manifold, the prior encourages the local chart to orient parallel to the manifold as part
of a globally optimal solution, protecting against a pathology noted in [8]. Equation (3) is
easily adapted to give a reduced number of charts and/or charts centered on local centroids.
4
Connecting the charts
We now build a connection for set of charts specified as an arbitrary nondegenerate GMM. A
GMM gives a soft partitioning of the dataset into neighborhoods of mean ?k and covariance
?k . The optimal variance-preserving low-dimensional coordinate system for each neighborhood derives from its weighted principal component analysis, which is exactly specified
by the eigenvectors of its covariance matrix: Eigendecompose Vk ?k V>
k ? ?k with eigen.
values in descending order on the diagonal of ?k and let Wk = [Id , 0]V>
the operator
k be
.
projecting points into the kth local chart, such that local chart coordinate uki = Wk (yi ? ?k )
.
and Uk = [uk1 , ? ? ? , ukN ] holds the local coordinates of all points.
Our goal is to sew together all charts into a globally consistent low-dimensional coordinate
system. For each chart there will be a low-dimensional affine transform Gk ? R(d+1)?d
that projects Uk into the global coordinate space. Summing over all charts, the weighted
average of the projections of point yi into the low-dimensional vector space is
W j (y ? ? j )
u ji
.
.
d
c
p j|y (y)
?
xi |yi = ? G j
x|y = ? G j
p j|y (yi ),
(4)
1
1
j
j
where pk|y (y) ? pk N (y; ?k , ?k ), ?k pk|y (y) = 1 is the probability that chart k generates
point y. As pointed out in [8], if a point has nonzero probabilities in two charts, then there
should be affine transforms of those two charts that map the point to the same place in a
global coordinate space. We set this up as a weighted least-squares problem:
2
uki
u ji
.
.
G
G = [G1 , ? ? ? , GK ] = arg min ? pk|y (yi )p j|y (yi )
?
G
(5)
j
k
1
1
F
Gk ,G j i
Equation (5) generates a homogeneous set of equations that determines a solution up to an
affine transform of G. There are two solution methods. First, let us temporarily anchor one
neighborhood at the origin to fix this indeterminacy. This adds the constraint G1 = [I, 0]> .
.
To solve, define indicator matrix Fk = [0, ? ? ? , 0, I, 0, ? ? ? , 0]> with the identity ma.
trix occupying the kth block, such that Gk = GFk . Let the diagonal of Pk =
diag([pk|y (y1 ), ? ? ? , pk|y (yN )]) record the per-point posteriors of chart k. The squared error
of the connection is then a sum of of all patch-to-anchor and patch-to-patch inconsistencies:
"
#
2
2
U
U
.
.
1
E = ?
(GUk ? 0 )Pk P1
+ ?
(GU j ? GUk )P j Pk
F ; Uk = Fk 1k .
F
j6=k
k
(6)
Setting dE /dG = 0 and solving to minimize convex E gives
!
!?1
>
G =
?
Uk P2k
k
?
j6=k
P2j
U>
k ?
?
j6=k
Uk P2k P2j U>j
?
k
Uk P2k P21
U1
0
> !
.
(7)
We now remove the dependence on a reference neighborhood G1 by rewriting equation 5,
G = arg min ? j6=k k(GU j ? GUk )P j Pk k2F = kGQk2F = trace(GQQ> G> ) ,
(8)
G
.
where Q = ? j6=k U j ? Uk P j Pk . If we require that GG> = I to prevent degenerate
solutions, then equation (8) is solved (up to rotation in coordinate space) by setting G> to
the eigenvectors associated with the smallest eigenvalues of QQ> . The eigenvectors can be
computed efficiently without explicitly forming QQ> ; other numerical efficiencies obtain
by zeroing any vanishingly small probabilities in each Pk , yielding a sparse eigenproblem.
A more interesting strategy is to numerically condition the problem by calculating the
trailing eigenvectors of QQ> + 1. It can be shown that this maximizes the posterior
2
p(G|Q) ? p(Q|G)p(G) ? e?kGQkF e?kG1k , where the prior p(G) favors a mapping G
whose unit-norm rows are also zero-mean. This maximizes variance in each row of G
and thereby spreads the projected points broadly and evenly over coordinate space.
The solutions for MAP charts (equation (5)) and connection (equation (8)) can be applied
to any well-fitted mixture of gaussians/factors1 /PCAs density model; thus large eigenproblems can be avoided by connecting just a small number of charts that cover the data.
1 We
thank reviewers for calling our attention to Teh & Roweis ([11]? in this volume), which
shows how to connect a set of given local dimensionality reducers in a generalized eigenvalue problem that is related to equation (8).
charting
(projection onto coordinate space)
charting
best Isomap
LLE, n=5
LLE, n=6
LLE, n=7
LLE, n=8
LLE, n=9
LLE, n=10
XZ view
random subset
of local charts
XYZ view
data (linked)
embedding, XY view
XY view
original data
reconstruction
(back?projected coordinate grid)
best LLE
(regularized)
Figure 2: The twisted curl problem. L EFT: Comparison of charting, I SO M AP, & LLE.
400 points are randomly sampled from the manifold with noise. Charting is the only
method that recovers the original space without catastrophes (folding), albeit with some
shear. R IGHT: The manifold is regularly sampled (with noise) to illustrate the forward
and backward projections. Samples are shown linked into lines to help visualize the manifold structure. Coordinate axes of a random selection of charts are shown as bold lines.
Connecting subsets of charts such as this will also give good mappings. The upper right
quadrant shows various LLE results. At bottom we show the charting solution and the
reconstructed (back-projected) manifold, which smooths out the noise.
Once the connection is solved, equation (4) gives the forward projection of any point y
down into coordinate space. There are several numerically distinct candidates for the backprojection: posterior mean, mode, or exact inverse. In general, there may not be a unique
posterior mode and the exact inverse is not solvable in closed form (this is also true of [8]).
Note that chart-wise projection defines a complementary density in coordinate space
0
[Id , 0]?k [Id , 0]> 0
, Gk
px|k (x) = N (x; Gk
G>
(9)
k ).
1
0
0
Let p(y|x, k), used to map x into subspace k on the surface of the manifold, be a Dirac delta
function whose mean is a linear function of x. Then the posterior mean back-projection is
obtained by integrating out uncertainty over which chart generates x:
+
!
I
0
>
c
y|x = ? pk|x (x) ?k + Wk Gk
x ? Gk
,
(10)
0
1
k
(?)+
where
denotes pseudo-inverse. In general, a back-projecting map should not reconstruct the original points. Instead, equation (10) generates a surface that passes through the
weighted average of the ?i of all the neighborhoods in which yi has nonzero probability,
much like a principal curve passes through the center of each local group of points.
5
Experiments
Synthetic examples: 400 2 D points were randomly sampled from a 2 D square and embedded in 3 D via a curl and twist, then contaminated with gaussian noise. Even if noiselessly
sampled, this manifold cannot be ? unrolled? without distortion. In addition, the outer curl
is sampled much less densely than the inner curl. With an order of magnitude fewer points,
higher noise levels, no possibility of an isometric mapping, and uneven sampling, this is
arguably a much more challenging problem than the ? swiss roll? and ? s-curve? problems
featured in [12, 9, 8, 1]. Figure 2LEFT contrasts the (unique) output of charting and the
best outputs obtained from I SO M AP and LLE (considering all neighborhood sizes between
2 and 20 points). I SO M AP and LLE show catastrophic folding; we had to change LLE?s
b. data, yz view
c. local charts
d. 2D embedding
e. 1D embedding
1D ordinate
a. data, xy view
true manifold arc length
Figure 3: Untying a trefoil knot ( ) by charting. 900 noisy samples from a 3 D-embedded
1 D manifold are shown as connected dots in front (a) and side (b) views. A subset of charts
is shown in (c). Solving for the 2 D connection gives the ? unknot? in (d). After removing
some points to cut the knot, charting gives a 1 D embedding which we plot against true
manifold arc length in (e); monotonicity (modulo noise) indicates correctness.
Three principal degrees of freedom recovered from raw jittered images
pose
scale
expression
images synthesized via backprojection of straight lines in coordinate space
Figure 4: Modeling the manifold of facial images from raw video. Each row contains
images synthesized by back-projecting an axis-parallel straight line in coordinate space
onto the manifold in image space. Blurry images correspond to points on the manifold
whose neighborhoods contain few if any nearby data points.
regularization in order to coax out nondegenerate (>1 D) solutions. Although charting is
not designed for isometry, after affine transform the forward-projected points disagree with
the original points with an RMS error of only 1.0429, lower than the best LLE (3.1423) or
best I SO M AP (1.1424, not shown). Figure 2RIGHT shows the same problem where points
are sampled regularly from a grid, with noise added before and after embedding. Figure 3
shows a similar treatment of a 1 D line that was threaded into a 3 D trefoil knot, contaminated
with gaussian noise, and then ? untied? via charting.
Video: We obtained a 1965-frame video sequence (courtesy S. Roweis and B. Frey) of
20 ? 28-pixel images in which B.F. strikes a variety of poses and expressions. The video
is heavily contaminated with synthetic camera jitters. We used raw images, though image
processing could have removed this and other uninteresting sources of variation. We took a
500-frame subsequence and left-right mirrored it to obtain 1000 points in 20 ? 28 = 560D
image space. The point-growth process peaked just above d = 3 dimensions. We solved for
25 charts, each centered on a random point, and a 3D connection. The recovered degrees
of freedom? recognizable as pose, scale, and expression? are visualized in figure 4.
original data
stereographic map to 3D fishbowl
charting
Figure 5: Flattening a fishbowl. From the left: Original 2000?2D points; their stereographic mapping to a 3D fishbowl; its 2D embedding recovered using 500 charts; and the
stereographic map. Fewer charts lead to isometric mappings that fold the bowl (not shown).
Conformality: Some manifolds can be flattened conformally (preserving local angles) but
not isometrically. Figure 5 shows that if the data is finely charted, the connection behaves
more conformally than isometrically. This problem was suggested by J. Tenenbaum.
6
Discussion
Charting breaks kernel-based NLDR into two subproblems: (1) Finding a set of datacovering locally linear neighborhoods (? charts? ) such that adjoining neighborhoods span
maximally similar subspaces, and (2) computing a minimal-distortion merger (? connection? ) of all charts. The solution to (1) is optimal w.r.t. the estimated scale of local linearity
r; the solution to (2) is optimal w.r.t. the solution to (1) and the desired dimensionality d.
Both problems have Bayesian settings. By offloading the nonlinearity onto the kernels,
we obtain least-squares problems and closed form solutions. This scheme is also attractive
because large eigenproblems can be avoided by using a reduced set of charts.
The dependence on r, like trusted-set methods, is a potential source of solution instability. In practice the point-growth estimate seems fairly robust to data perturbations (to be
expected if the data density changes slowly over a manifold of integral Hausdorff dimension), while the use of a soft neighborhood partitioning appears to make charting solutions
reasonably stable to variations in r. Eigenvalue stability analyses may prove useful here.
Ultimately, we would prefer to integrate r out. In contrast, use of d appears to be a virtue:
Unlike other eigenvector-based methods, the best d-dimensional embedding is not merely
a linear projection of the best d + 1-dimensional embedding; a unique distortion is found
for each value of d that maximizes the information content of its embedding.
Why does charting performs well on datasets where the signal-to-noise ratio confounds
recent state-of-the-art methods? Two reasons may be adduced: (1) Nonlocal information
is used to construct both the system of local charts and their global connection. (2) The
mapping only preserves the component of local point-to-point distances that project onto
the manifold; relationships perpendicular to the manifold are discarded. Thus charting uses
global shape information to suppress noise in the constraints that determine the mapping.
Acknowledgments
Thanks to J. Buhmann, S. Makar, S. Roweis, J. Tenenbaum, and anonymous reviewers for
insightful comments and suggested ? challenge? problems.
References
[1] M. Balasubramanian and E. L. Schwartz. The IsoMap algorithm and topological stability. Science, 295(5552):7, January 2002.
[2] C. Bregler and S. Omohundro. Nonlinear image interpolation using manifold learning. In
NIPS?7, 1995.
[3] D. DeMers and G. Cottrell. Nonlinear dimensionality reduction. In NIPS?5, 1993.
[4] J. Gomes and A. Mojsilovic. A variational approach to recovering a manifold from sample
points. In ECCV, 2002.
[5] T. Hastie and W. Stuetzle. Principal curves. J. Am. Statistical Assoc, 84(406):502?516, 1989.
[6] G. Hinton, P. Dayan, and M. Revow. Modeling the manifolds of handwritten digits. IEEE
Trans. Neural Networks, 8, 1997.
[7] N. Kambhatla and T. Leen. Dimensionality reduction by local principal component analysis.
Neural Computation, 9, 1997.
[8] S. Roweis, L. Saul, and G. Hinton. Global coordination of linear models. In NIPS?13, 2002.
[9] S. T. Roweis and L. K. Saul. Nonlinear dimensionality reduction by locally linear embedding.
Science, 290:2323?2326, December 22 2000.
[10] A. Smola, S. Mika, B. Sch?lkopf, and R. Williamson. Regularized principal manifolds. Machine Learning, 1999.
[11] Y. W. Teh and S. T. Roweis. Automatic alignment of hidden representations. In NIPS?15, 2003.
[12] J. B. Tenenbaum, V. de Silva, and J. C. Langford. A global geometric framework for nonlinear
dimensionality reduction. Science, 290:2319?2323, December 22 2000.
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p21:1 regularizing:1 phenomenon:1 |
1,280 | 2,166 | Adapting Codes and Embeddings for Polychotomies
Gunnar R?atsch, Alexander J. Smola
RSISE, CSL, Machine Learning Group
The Australian National University
Canberra, 0200 ACT, Australia
Gunnar.Raetsch, Alex.Smola @anu.edu.au
Sebastian Mika
Fraunhofer FIRST
Kekulestr. 7
12489 Berlin, Germany
[email protected]
Abstract
In this paper we consider formulations of multi-class problems based on
a generalized notion of a margin and using output coding. This includes,
but is not restricted to, standard multi-class SVM formulations. Differently from many previous approaches we learn the code as well as the
embedding function. We illustrate how this can lead to a formulation
that allows for solving a wider range of problems with for instance many
classes or even ?missing classes?. To keep our optimization problems
tractable we propose an algorithm capable of solving them using twoclass classifiers, similar in spirit to Boosting.
1 Introduction
The theory of pattern recognition is primarily concerned with the case of binary classification, i.e. of assigning examples to one of two categories, such that the expected number of
misassignments is minimal. Whilst this scenario is rather well understood, theoretically as
well as empirically, it is not directly applicable to many practically relevant scenarios, the
most prominent being the case of more than two possible outcomes.
Several learning techniques naturally generalize to an arbitrary number of classes, such as
density estimation, or logistic regression. However, when comparing the reported performance of these systems with the de-facto standard of using two-class techniques in combination with simple, fixed output codes to solve multi-class problems, they often lack in
terms of performance, ease of optimization, and/or run-time behavior.
On the other hand, many methods have been proposed to apply binary classifiers to multiclass problems, such as Error Correcting Output Codes (ECOC) [6, 1], Pairwise Coupling
[9], or by simply reducing the problem of discriminating classes to ?one vs. the rest?
dichotomies. Unfortunately the optimality of such methods is not always clear (e.g., how
to choose the code, how to combine predictions, scalability to many classes).
Finally, there are other problems similar to multi-class classification which can not be
solved satisfactory by just combining simpler variants of other algorithms: multi-label
problems, where each instance should be assigned to a subset of possible categories, and
ranking problems, where each instance should be assigned a rank for all or a subset of possible outcomes. These problems can, in reverse order of their appearance, be understood as
more and more refined variants of a multi-variate regression, i.e.
two-class
multi-class
multi-label
ranking
multi-variate regression
Which framework and which algorithm in there one ever chooses, it is usually possible
to make out a single scheme common to all these: There is an encoding step in which
the input data are embedded into some ?code space? and in this space there is a code
book which allows to assign one or several labels or ranks respectively by measuring the
similarity between mapped samples and the code book entries. However, most previous
work is either focused on finding a good embedding given a fixed code or just optimizing
the code, given a fixed embedding (cf. Section 2.3).
The aim of this work is to propose (i) a multi-class formulation which optimizes the code
and the embedding of the training sample into the code space, and (ii) to develop a general
ranking technique which as well specializes to specific multi-class, multi-label and ranking
problems as it allows to solve more general problems. As an example of the latter consider
the following model problem: In chemistry people are interested in mapping sequences
to structures. It is not yet known if there is an one-to-one correspondence and hence the
problem is to find for each sequence the best matching structures. However, there are only
say a thousand sequences the chemists have good knowledge about. They are assigned,
with a certain rank, to a subset of say a thousand different structures. One could try to cast
this as a standard multi-class problem by assigning each training sequence to the structure
ranked highest. But then, there will be classes to which only very few or no sequences
are assigned and one can obviously hardly learn using traditional techniques. The machine
we propose is (at least in principle) able to solve problems like this by reflecting relations
between classes in the way the code book is constructed and at the same time trying to find
an embedding of the data space into the code space that allows for a good discrimination.
The remainder of this paper is organized as follows: In Section 2 we introduce some basic
notions of large margin, output coding and multi-class classification. Then we discuss the
approaches of [4] and [21] and propose to learn the code book. In Section 3 we propose
a rather general idea to solve resulting multi-class problems using two-class classifiers.
Section 4 presents some preliminary experiments before we conclude.
2 Large Margin Multi-Class Classification
! "
#
Denote by the sample space (not necessarily a metric space), by the space of possible
labels or ranks (e.g.
for multi-class problems where denotes the number
of classes, or
for a ranking problem), and let be a training sample of size ,
with
.
i.e.
$
Output Coding It is well known (see [6, 1] and references therein) that multi-class problems can be solved by decomposing a polychotomy into dichotomies and solving these
separately using a two-class technique. This can be understood as assigning to each class
a binary string
of length which is called a code word. This results
in an
binary code matrix. Now each of the columns of this matrix defines a partitioning of classes into two subsets, forming binary problems for which a classifier is
trained. Evaluation is done by computing the output of all learned functions, forming a
new bit-string, and then choosing the class such that some distance measure between this
string and the corresponding row of the code matrix is minimal, usually the Hamming distance. Ties can be broken by uniformly selecting a winning class, using prior information
or, where possible, using confidence outputs from the basic classifiers. 1
Since the codes for each class must be unique, there are
(for
)
possible code matrices to choose from. One possibility is to choose the codes to be errorcorrecting (ECOC) [6]. Here one uses a code book with e.g. large Hamming distance
between the code words, such that one still gets the correct decision even if a few of the
classifiers err. However, finding the code that minimizes the training error is NP-complete,
even for fixed binary classifiers [4]. Furthermore, errors committed by the binary classifiers are not necessarily independent, significantly reducing the effective number of wrong
bits that one can handle [18, 19]. Nonetheless ECOC has proven useful and algorithms
for finding a good code (and partly also finding the corresponding classifiers) have been
%
- .,
& % ' ( ) +*
,
%
,
,
1/ 0352 4 76 8 3 *
1
We could also use ternary codes, i.e.
<>=@?
ACB)AD? E , allowing for ?don?t care? classes.
9;: 8 *
proposed in e.g. [15, 7, 1, 19, 4]. Noticeably, most practical approaches suggest to drop the
requirement of binary codes, and instead propose to use continuous ones.
We now show how predictions with small (e.g. Hamming) distance to their appropriated
code words can be related to a large margin classifier, beginning with binary classification.
2.1 Large Margins
Dichotomies Here a large margin classifier is defined as a mapping
with the
property that
, or more specifically
with
, where is
some positive constant [20]. Since such a positive margin may not always be achievable,
one typically maximizes a penalized version of the maximum margin, such as
where ( ".
and (1)
! is a regularization constant and denotes the
Here is a regularization term,
( we could
( CNote
class
consideration.
that
for
the condition
(
#
(
rewrite
( (and
of" functions
( under
#
#
also
as
likewise
0
0
(
). In other words, we can express (the margin as the difference between the
from the target and the target .
for
distance of
$
& % @ %
* %
, &%(' *)
)
,+-
*
.) # !
(2)
/146
85 0327 $ & % )@ ( $ & ) #
)
This means that we
measure the minimal
relative difference in distance between , the
correct target & and any other target & % (cf. [4]). We obtain accordingly the following
Polychotomies While this insight by itself is not particularly useful, it paves the way for
an extension of the notion of the margin to multi-class problems: denote by a distance
measure and by
,
(
is the length of the code) target vectors
corresponding to class . Then we can define the margin
of an observation and
as
class with respect to
9 )
where $ & % ) #( $ & )@ # ( (3)
minimize
,
: < and ) = . For the time being we chose as a reference margin ?
for all %;
an adaptive means of choosing the reference margin can be implemented using the > -trick,
optimization problem:
which leads to an easier to control regularization parameter [16].
( ) ?0
$ &> ) -?D& @
0
Lemma 1 (Difference of Distance Measures) Denote
by
$ (& >& A ) ) )
*
*=
) a sym
)
&CB is convex in for all
metric distance measure.
Then the only case where $ >&
& &6B occurs if $ &> ) <$D & $E *) &GFIH ) , where H
*KJ * is symmetric.
) ( $ &MB1 ) is positive semidefinite. This is
)
Proof Convexity in implies that L E 0 $ &>
) is a function of ) only. The latter, however, implies that the only
only possible if L E 0 $ ) &>
)
joint terms in & and must be of linear nature in . Symmetry, on the other hand, implies
2.2 Distance Measures
Several choices of are possible. However, one can show that only
and related functions will lead to a convex constraint on :
$
)
&
that the term must be linear in , too, which proves the claim.
Lemma 1 implies that any distance functions other than the ones described above will lead
to optimization problems with potentially many local minima, which is not desirable. However, for quadratic we will get a convex optimization problem (assuming suitable
)
$
& % & % H means
$ &> ) ?D& ( ) ? 0
$ & % )@ ( $ & )@ ?D& % ? 0 ( ?D& ? 0 8 & % F )@ ( 8 & F )@ (4)
Note,
if the code words have the same length, the difference of the projections of
) ontothat
different code
We will indeed later consider a more
words
)@ determines
& % F ) the margin.
convenient case: $ & %
, which will lead to linear constraints only and
and then there are ways to efficiently solve (3). Finally, re-defining
that it is sufficient to consider only
. We obtain
allows us to use standard optimization packages. However, there are no principal limitations
about using the Euclidean distance.
If we choose
to be an error-correcting code, such as those in [6, 1], one will often have
. Hence, we use fewer dimensions than we have classes. This means that during
optimization we are trying to find
functions
,
,
from an dimensional subspace. In other words, we choose the subspace and perform
regularization by allowing only a smaller class of functions. By appropriately choosing the
subspace one may encode prior knowledge about the problem.
,
& %
4 $ & % ) # % )
,
( 8
8 7 ( 4 & % 4
8 & % F )@ ( 8 & F )@
2.3 Discussion and Relation to Previous Approaches
we have that (4) is equal
Note that for
and hence the problem of multi-class classification reverts to the problem
of solving binary classification problems of one vs. the remaining classes. Then our approach turns out to be very similar to the idea presented in [21] (except for some additional
slack-variables).
A different approach was taken in [4]. Here, the function is held fix and the code is
optimized. In their approach, the code is described as a vector in a kernel feature space and
one obtains in fact an optimization problem very similar to the one in [21] and (3) (again,
the slack-variables are defined slightly different).
Another idea which is quite similar to ours was also presented at the conference [5]. The
resulting optimization problem turns out to be convex, but with the drawback, that one can
either not fully optimize the code vectors or not guarantee that they are well separated.
Since these approaches were motivated by different ideas (one optimizing the code, the
other optimizing the embedding), this shows that the role of the code
and the embedding function is interchangeable if the function or the code, respectively, is fixed.
Our approach allows arbitrary codes for which a function is learned. This is illustrated in
Figure 1. The position of the code words (=?class centers?) determine the function . The
position of the centers relative to each other may reflect relationships between the classes
(e.g. classes ?black? & ?white? and ?white? & ?grey? are close).
)
)
)
&
& %
)
)
Figure 1: Illustration of embedding idea: The samples are mapped from the input space
into the code space via the embedding function , such that samples from the same class
are close to their respective code book vector (crosses on the right). The spatial organization
of the code book vectors reflects the organization of classes in the space.
)
2.4 Learning Code & Embedding
This leaves us with the question of how to determine a ?good? code and a suitable . As we
can see from (4), for fixed the constraints are linear in and vice versa, yet we have non-
)
&
)
&
)
&
convex constraints, if both and are variable. Finding the global optimum is therefore
computationally infeasible when optimizing and simultaneously (furthermore note that
any rotation applied to and will leave the margin invariant, which shows the presence
of local minima due to equivalent codes).
Instead, we propose the following method: for fixed code optimize over , and subsequently, for fixed , optimize over , possibly repeating the process. The first step follows
[4], i.e. to learn the code for a fixed function. Both steps separately can be performed fairly
efficient (since the optimization problems are convex; cf. Lemma 1).
This procedure is guaranteed to decrease the over all objective function at every step and
converges to a local minimum. We now show how a code maximizing the margin can be
found. To avoid a trivial solution (we can may virtually increase the margin by rescaling all
by some constant), we add
to the objective function. It can be shown that
one does not need an additional regularization constant in front of this term, if the distance
is linear on both arguments. If one prefers sparse codes, one may use the -norm instead.
In summary, we obtain the following convex quadratic program for finding the codes which
can be solved using standard optimization techniques:
&
)
)
)
&
&
4
?D& % ? 00
&
D 4
minimize
subject to
%
( 4#
) ?D& ?00 (
& & % F
"
and % :
for all .
(5)
The technique for finding the embedding will be discussed in more detail in Section 3.
Initialization To obtain a good initial code, we may either take recourse to readily available tables [17] or we may use a random code, e.g. by generating vectors uniformly
distributed on the -dimensional sphere. One can show that the probability that there
exists two such vectors (out of ) that have a smaller distance than is bounded by
(proof given in the full paper). Hence, with probability greater
than the random code vectors have distances greater than
from each other.2
(
0 9 9
0
,
0 2 * 8)
8
3 Column Generation for Finding the Embedding
There are several ways to setup and optimize the resulting optimization problem (3). For
instance in [21, 4] the class of functions is the set of hyperplanes in some kernel feature
space and the regularizer
is the sum of the -norms of the hyperplane
normal vectors.
! "
$# %
In this section we consider a different approach. Denote by
)(
" -, "./"
+*
a class of basis functions and let
'&
'01
2 . We choose
the regularizer
to be the -norm on the expansion coefficients. We are interested in
solving:
)
0
)
2
+
* , "
/ 03203/10435 "
687!9-:; <.7!9>=
?
(6)
(
(
& % & F 6
? ".
# : % )
subject to
To derive a column generation method
the dual optimization problem, or
A 4 @[12,
, " 2] we)
need
# and : % )
,
more specifically its constraints:
A 4 & ( & % # F A" CB )-% )
&
(7)
7 @
5 4
2
However, also note that this is quite a bit worse than the best packing, which scales with DCE
rather than D E . This is due a the union-bound argument in the proof, which requires us to sum
IKJ.L pairs have more than M distance.
over the probability that all DGFHD
.= ?
7
5 4
A 4 B
" )
#
and
. The idea of column generation is to start with a
,
restricted master problem, namely without the variables (i.e &
). Then one solves the
corresponding dual problem (7) and then finds the hypothesis that corresponds to a violated
constraint (and also one primal variable). This hypothesis is included in the optimization
problem, one resolves and finds the next violated constraint. If all constraints of the full
problem are satisfied, one has reached optimality.
"@
We now construct a hypothesis set from a scalar valued base-class
& , which has particularly nice properties for our purposes. The idea
where %
"
is to extend @ by multiplication with vectors
:
+
*
*
* ? ? )
+
Since there are infinitely many functions in this set , we have an infinite number of constraints in the dual optimization problem. By using the described column generation technique one can, however, find the solution of this semi-infinite programming problem [13].
We have to identify the constraint in" (7), which is maximally violated, i.e. one has to find a
?partitioning? and a hypothesis @ with maximal
A 4 "@ & ( & % F
F "@
(8)
7
5 4
"
"
for appropriate @ . Maximizing (8) with respect to ? ? is easy for a given @ :
8
"@
"@
"@
for , one chooses
2 ; if , then ? ? 0 and for
one chooses the minimizing" unit vector. However, finding and simultaneously is a
difficult problem, if not all @ are known in advance (see also [15]). We propose to test all
"
previously used hypotheses to find the best . As a second step one finds the hypothesis @
"@
that maximizes
. Only if one cannot find a hypothesis that violates a constraint, one
employs the more sophisticated techniques suggested in [15]. If there is no hypothesis
left that corresponds to a violated constraint, the dual optimization problem is optimal.
In this work we are mainly
interested in the case
, since then
and the
"
problem of finding @ simplifies greatly. Then we can use another learning algorithm that
minimizes or approximately minimizes the training error of a weighted training set (rewrite
(8)). This approach has indeed many similarities to Boosting. Following the ideas in [14]
one can show that there is a close relationship between our technique using the trivial code
and the multi-class boosting algorithms as e.g. proposed in [15].
F
>*
4 Extensions and Illustration
4.1 A first Experiment
In a preliminary set of experiments we use two benchmark data sets from the UCI benchmark repository: glass and iris. We used our column generation strategy as described
in Section 3 in conjunction with the code optimization problem to solve the combined optimization problem to find the code and the embedding. We used
. The algorithm has
only one model parameters ( ). We selected it by -fold cross validation on the training
data. The test error is determined by averaging over five splits of training and test data.
As base learning algorithm we chose decision trees (C4.5) which we only use as two-class
classifier in our column generation algorithm.
On the glass data set we obtained an error rate of
. In [1] an error of
was reported for SVMs using a polynomial kernel. We also computed the test error of multiclass decision trees and obtained
error. Hence, our hybrid algorithm could
relatively improve existing results by
. On the iris data we could achieve an error
rate of
and could slightly improve the result of decision trees (
).
" 8 #
!
,
8
However, SVMs beat our result with
error [1]. We conjecture that this is due to the
properties of decision trees which have problems generating smooth boundaries not aligned
with coordinate axes.
So far, we could only show a proof of concept and more experimental work is necessary.
It is in particular interesting to find practical examples, where a non-trivial choice of the
code (via optimization) helps simplifying the embedding and finally leads to additional
improvements. Such problems often appear in Computer Vision, where there are strong relationships between classes. Preliminary results indicate that one can achieve considerable
improvements when adapting codes and embeddings [3].
3
Figure 2: Toy example for learning missing classes. Shown is
the decision boundary and the confidence for assigning a sample
to the upper left class. The training set, however, did not contain
samples from this class. Instead, we used (9) with the information that each example besides belonging to its own class with
confidence two also belongs to the other classes with confidence
one iff its distance to the respective center is less than one.
2.5
2
1.5
1
0.5
0.5
1
1.5
2
2.5
3
4.2 Beyond Multi-Class
So far we only considered the case where there is only one class to which an example
belongs to. In a more general setting as for example the problem mentioned in the introduction, there can be several classes, which possibly have a ranking. We have the sets
, which contain all pairs of ?relations? between
#
contains all pairs of positive and negative classes of
the positive classes. The set
an example.
% % 0 % % 0
D#
* , "
% ? 0
B
?
&
minimize
?
?
"
0
B
' A
5& A %
& % ( & % # F ) 6 (
(9)
subject to
? %
0
%
& % for
( all& % ".# F ) 6
and
( B 0
?
0
for all ".
and % %
) "*
, ".A" and 7 !A" +
*#% )
0 & .
where 6
)
&
In this formulation one tries to find a code and an embedding , such that for each example the output wrt. each class this example has a relation with, reflects the order of this
relations (i.e. the examples get ranked appropriately). Furthermore, the program tries to
achieve a ?large margin? between relevant and irrelevant classes for each sample. Similar
formulations can be found in [8] (see also [11]).
Optimization of (9) is analogous to the column generation approach discussed in Section 3.
We omit details due to constraints on space. A small toy example, again as a limited proof
of concept, is given in Figure 2.
Connection to Ranking Techniques Ordinal regression through large margins [10] can
be seen as an extreme case of (9), where we have as many classes as" observations,
and each
"
pair of observations
has
to
satisfy
a
ranking
relation
,
if
is to
"
be preferred to . This formulation can of course also be understood as a special case of
multi-dimensional regression.
( (
5 Conclusion
We proposed an algorithm to simultaneously optimize output codes and the embedding of
the sample into the code book space building upon the notion of large margins. Further-
more, we have shown, that only quadratic and related distance measures in the code book
space will lead to convex constraints and hence convex optimization problems whenever
either the code or the embedding is held fixed. This is desirable since at least for these
sub-problems there exist fairly efficient techniques to compute these (of course the combined optimization problem of finding the code and the embedding is not convex and has
local minima). We proposed a column generation technique for solving the embedding optimization problems. It allows the use of a two-class algorithm, of which there exists many
efficient ones, and has connection to boosting. Finally we proposed a technique along the
same lines that should be favorable when dealing with many classes or even empty classes.
Future work will concentrate on finding more efficient algorithms to solve the optimization
problem and on more carefully evaluating their performance.
Acknowledgements We thank B. Williamson and A. Torda for interesting discussions.
References
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violated:4 |
1,281 | 2,167 | A Model for Learning Variance Components of
Natural Images
Michael S. Lewicki?
[email protected]
Yan Karklin
[email protected]
Computer Science Department &
Center for the Neural Basis of Cognition
Carnegie Mellon University
Abstract
We present a hierarchical Bayesian model for learning efficient codes of
higher-order structure in natural images. The model, a non-linear generalization of independent component analysis, replaces the standard assumption of independence for the joint distribution of coefficients with
a distribution that is adapted to the variance structure of the coefficients
of an efficient image basis. This offers a novel description of higherorder image structure and provides a way to learn coarse-coded, sparsedistributed representations of abstract image properties such as object
location, scale, and texture.
1 Introduction
One of the major challenges in vision is how to derive from the retinal representation
higher-order representations that describe properties of surfaces, objects, and scenes. Physiological studies of the visual system have characterized a wide range of response properties, beginning with, for example, simple cells and complex cells. These, however, offer
only limited insight into how higher-order properties of images might be represented or
even what the higher-order properties might be. Computational approaches to vision often derive algorithms by inverse graphics, i.e. by inverting models of the physics of light
propagation and surface reflectance properties to recover object and scene properties. A
drawback of this approach is that, because of the complexity of modeling, only the simplest and most approximate models are computationally feasible to invert and these often
break down for realistic images. A more fundamental limitation, however, is that this formulation of the problem does not explain the adaptive nature of the visual system or how it
can learn highly abstract and general representations of objects and surfaces.
An alternative approach is to derive representations from the statistics of the images themselves. This information theoretic view, called efficient coding, starts with the observation
that there is an equivalence between the degree of structure represented and the efficiency
of the code [1]. The hypothesis is that the primary goal of early sensory coding is to encode information efficiently. This theory has been applied to derive efficient codes for
? To
whom correspondence should be addressed
natural images and to explain a wide range of response properties of neurons in the visual
cortex [2?7].
Most algorithms for learning efficient representations assume either simply that the data
are generated by a linear superposition of basis functions, as in independent component
analysis (ICA), or, as in sparse coding, that the basis function coefficients are ?sparsified?
by lateral inhibition. Clearly, these simple models are insufficient to capture the rich structure of natural images, and although they capture higher-order statistics of natural images
(correlations beyond second order), it remains unclear how to go beyond this to discover
higher-order image structure.
One approach is to learn image classes by embedding the statistical density assumed by
ICA in a mixture model [8]. This provides a method for modeling classes of images and
for performing automatic scene segmentation, but it assumes a fundamentally local representation and therefore is not suitable for compactly describing the large degree of structure
variation across images. Another approach is to construct a specific model of non-linear
features, e.g. the responses of complex cells, and learn an efficient code of their outputs [9].
With this, one is limited by the choice of the non-linearity and the range of image regularities that can be modeled.
In this paper, we take as a starting point the observation by Schwartz and Simoncelli [10]
that, for natural images, there are significant statistical dependencies among the variances
of filter outputs. By factoring out these dependencies with divisive normalization, Schwartz
and Simoncelli showed that the model could account for a wide range of non-linearities
observed in neurons in the auditory nerve and primary visual cortex.
Here, we propose a statistical model for higher-order structure that learns a basis on the
variance regularities in natural images. This higher-order, non-orthogonal basis describes
how, for a particular visual image patch, image basis function coefficient variances deviate
from the default assumption of independence. This view offers a novel description of
higher-order image structure and provides a way to learn sparse distributed representations
of abstract image properties such as object location, scale, and surface texture.
Efficient coding of natural images
The computational goal of efficient coding is to derive from the statistics of the pattern
ensemble a compact code that maximally reduces the redundancy in the patterns with minimal loss of information. The standard model assumes that the data is generated using a set
of basis functions A and coefficients u:
x = Au ,
(1)
Because coding efficiency is being optimized, it is necessary, either implicitly or explicitly,
for the model to capture the probability distribution of the pattern ensemble. For the linear
model, the data likelihood is [11, 12]
p(x|A) = p(u)/| det A| .
(2)
The coefficients ui , are assumed to be statistically independent
p(u) = ? p(ui ) .
(3)
i
ICA learns efficient codes of natural scenes by adapting the basis vectors to maximize
the likelihood of the ensemble of image patterns, p(x1 , . . . , xN ) = ?n p(xn |A), which maximizes the independence of the coefficients and optimizes coding efficiency within the limits
of the linear model.
a
b
c
Figure 1: Statistical dependencies among natural image independent component basis coefficients. The scatter plots show for the two basis functions in the same row and column
the joint distributions of basis function coefficients. Each point represents the encoding of
a 20 ? 20 image patch centered at random locations in the image. (a) For complex natural
scenes, the joint distributions appear to be independent, because the joint distribution can
be approximated by the product of the marginals. (b) Closer inspection of particular image
regions (the image in (b) is contained in the lower middle part of the image in (a)) reveals
complex statistical dependencies for the same set of basis functions. (c) Images such as
texture can also show complex statistical dependencies.
Statistical dependencies among ?independent? components
A linear model can only achieve limited statistical independence among the basis function
coefficients and thus can only capture a limited degree of visual structure. Deviations from
independence among the coefficients reflect particular kinds of visual structure (fig. 1). If
the coefficients were independent it would be possible to describe the joint distribution as
the product of two marginal densities, p(ui , u j ) = p(ui )p(u j ). This is approximately true
for natural scenes (fig.1a), but for particular images, the joint distribution of coefficients
show complex statistical dependencies that reflect the higher-order structure (figs.1b and
1c). The challenge for developing more general models of efficient coding is formulating
a description of these higher-order correlations in a way that captures meaningful higherorder visual structure.
2 Modeling higher-order statistical structure
The basic model of standard efficient coding methods has two major limitations. First,
the transformation from the pattern to the coefficients is linear, so only a limited class
of computations can be achieved. Second, the model can capture statistical relationships
among the pixels, but does not provide any means to capture higher order relationships
that cannot be simply described at the pixel level. As a first step toward overcoming these
limitations, we extend the basic model by introducing a non-independent prior to model
higher-order statistical relationships among the basis function coefficients.
Given a representation of natural images in terms of a Gabor-wavelet-like representation
learned by ICA, one salient statistical regularity is the covariation of basis function coefficients in different visual contexts. Any specific type of image region, e.g. a particular kind
of texture, will tend to yield in large values for some coefficients and not others. Different
types of image regions will exhibit different statistical regularities among the variances of
the coefficients. For a large ensemble of images, the goal is to find a code that describes
these higher-order correlations efficiently.
In the standard efficient coding model, the coefficients are often assumed to follow a generalized Gaussian distribution
q
p(ui ) = ze?|ui /?i | ,
(4)
where z = q/(2?i ?[1/q]). The exponent q determines the distribution?s shape and weight
of the tails, and can be fixed or estimated from the data for each basis function coefficient.
The parameter ?i determines the scale of variation (usually fixed in linear models, since
basis vectors in A can absorb the scaling). ?i is a generalized notion of variance; for clarity,
we refer to it simply as variance below.
Because we want to capture regularities among the variance patterns of the coefficients,
we do not want to model the values of u themselves. Instead, we assume that the relative
variances in different visual contexts can be modeled with a linear basis as follows
?i = exp([Bv]i )
(5)
? = Bv .
? log?
(6)
where [Bv]i refers to the ith element of the product vector Bv. This formulation is useful
because it uses a basis to represent the deviation from the variance assumed by the standard model. If we assume that vi also follows a zero-centered, sparse distribution (e.g. a
generalized Gaussian), then Bv is peaked around zero which yields a variance of one, as
in standard ICA. Because the distribution is sparse, only a few of the basis vectors in B
are needed to describe how any particular image deviates from the default assumption of
independence. The joint distribution for the prior (eqn.3) becomes
L
ui q
? log p(u|B, v) ? ? [Bv] ,
(7)
e i
i
Having formulated the problem as a statistical model, the choice of the value of v for a
given u is determined by maximizing the posterior distribution
v? = argmax p(v|u, B) = argmax p(u|B, v)p(v)
(8)
v
v
Unfortunately, computing the most probable v is not straightforward. Because v specifies
the variance of u, there is a range of values that could account for a given pattern ? all that
changes is the probability of the first order representation, p(u|B, v). For the simulations
below, v? was estimated by gradient ascent.
By maximizing the posterior p(v|u, B), the algorithm is computing the best way to describe
how the distribution of vi ?s for the current image patch deviates from the default assumption
of independence, i.e. v = 0. This aspect of the algorithm makes the transformation from
the data to the internal representation fundamentally non-linear. The basis functions in
B represent an efficient, sparse, distributed code for commonly observed deviations. In
contrast to the first layer, where basis functions in A correspond to specific visual features,
higher-order basis functions in B describe the shapes of image distributions.
The parameters are adapted by performing gradient ascent on the data likelihood. Using the
generalized prior, the data likelihood is computed by marginalizing over the coefficients.
Assuming independence between B and v, the marginal likelihood is
p(x|A, B) =
Z
p(u|B, v)p(v)/| det A|dv .
(9)
This, however, is intractable to compute, so we approximate it by the maximum a posteriori
value v?
p(x|A, B) ? p(u|B, v? )p(?v)/| detA| .
(10)
We assume that p(v) = ?i p(vi ) and that p(vi ) ? exp(?|vi |). We adapt B by maximizing
the likelihood over the data ensemble
B = argmax ? log p(un |B, v? n ) + log p(B)
(11)
B
n
For reasons of space, we omit the (straightforward) derivations of the gradients.
Figure 2: A subset of the 400 image basis functions. Each basis function is 20x20 pixels.
3 Results
The algorithm described above was applied to a standard set of ten 512?512 natural images
used in [2]. For computational simplicity, prior to the adaptation of the higher-order basis
B, a 20 ? 20 ICA image basis was derived using standard methods (e.g. [3]). A subset of
these basis functions is shown in fig. 2.
Because of the computational complexity of the learning procedure, the number of basis
functions in B was limited to 30, although in principle a complete basis of 400 could be
learned. The basis B was initialized to small random values and gradient ascent was performed for 4000 iterations, with a fixed step size of 0.05. For each batch of 5000 randomly
sampled image patches, v? was derived using 50 steps of gradient ascent at a fixed step size
of 0.01.
Fig. 3 shows three different representations of the basis functions in the matrix B adapted to
natural images. The first 10 ? 3 block (fig.3a) shows the values of the 30 basis functions in
B in their original learned order. Each square represents 400 weights B i, j from a particular
v j to all the image basis functions ui ?s. Black dots represent negative weights; white,
positive weights. In this representation, the weights appear sparse, but otherwise show no
apparent structure, simply because basis functions in A are unordered.
Figs. 3b and 3c show the weights rearranged in two different ways. In fig. 3b, the dots representing the same weights are arranged according to the spatial location within an image
patch (as determined by fitting a 2D Gabor function) of the basis function which the weight
affects. Each weight is shown as a dot; white dots represent positive weights, black dots
negative weights. In fig. 3c, the same weights are arranged according to the orientation and
spatial scale of the Gaussian envelope of the fitted Gabor. Orientation ranges from 0 to ?
counter-clockwise from the horizontal axis, and spatial scale ranges radially from DC at the
bottom center to Nyquist. (Note that the learned basis functions can only be approximately
fit by Gabor functions, which limits the precision of the visualizations.)
In these arrangements, several types of higher-order regularities emerge. The predominant
one is that coefficient variances are spatially correlated, which reflects the fact that a common occurrence is an image patch with a small localized object against a relatively uniform
background. For example, the pattern in row 5, column 3 of fig. 3b shows that often the
coefficient variances in the top and bottom halves of the image patch are anti-correlated,
i.e. either the object or scene is primarily across the top or across the bottom. Because
vi can be positive or negative, the higher-order basis functions in B represent contrast in
the variance patterns. Other common regularities are variance-contrasts between two orientations for all spatial positions (e.g. row 7, column 1) and between low and high spatial
scales for all positions and orientations (e.g. row 9, column 3). Most higher-order basis
functions have simple structure in either position, orientation, or scale, but there are some
whose organization is less obvious.
a
b
c
Figure 3: The learned higher-order basis functions. The same weights shown in the original
order (a); rearranged according to the spatial location of the corresponding image basis
functions (b); rearranged according to frequency and orientation of image basis functions
(c). See text for details.
Figure 4: Image patches that yielded the largest coefficients for two basis functions in B.
The central block contains nine image patches corresponding to higher-order basis function
coefficients with values near zero, i.e. small deviations from independent variance patterns.
Positions of other nine-patch blocks correspond to the associated values of higher-order
coefficients, here v15 and v27 (whose weights to ui ?s are shown at the axes extrema). For
example, the upper-left block contains image patches for which v 15 was highly negative
(contrast localized to bottom half of patch) and v27 was highly positive (power predominantly at low spatial scales). This illustrates how different combinations of basis functions
in B define distributions of images (in this case, spatial frequency and location).
Another way to get insight into the code learned by the model is to display, for a large
ensemble of image patches, the patches that yield the largest values of particular v i ?s (and
their corresponding basis functions in B). This is shown in fig. 4.
As a check to see if any of the higher-order structure learned by the algorithm was simply
due to random variations in the dataset, we generated a dataset by drawing independent
samples un from a generalized Gaussian to produce the pattern xn = Aun . The resulting
basis B was composed only of small random values, indicating essentially no deviation
from the standard assumption of independence and unit variance. In addition, adapting
the model on a synthetic dataset generated from a hand-specified B recovers the original
higher-order basis functions.
It is also possible to adapt A and B simultaneously (although with considerably greater
computational expense). To check the validity of first deriving B for a fixed A, both matrices were adapted simultaneously for small 8 ? 8 patches on the same natural image data
set. The results for both the image basis matrix A and the higher-order basis B were qualitatively similar to those reported above.
4 Discussion
We have presented a model for learning higher-order statistical regularities in natural images by learning an efficient, sparse-distributed code for the basis function coefficient variances. The recognition algorithm is non-linear, but we have not tested yet whether it can
account for non-linearities similar to the types reported in [10].
A (cautious) neurobiological interpretation of the higher-order units is that they are analogous to complex cells which pool output over specific first-order feature dimensions.
Rather than achieving a simplistic invariance, however, the model presented here has the
specific goal of efficiently representing the higher-order structure by adapting to the statistics of natural images, and thus may predict a broader range of response properties than are
commonly tested physiologically.
One salient type of higher-order structure learned by the model is the position of image
structure within the patch. It is interesting that, rather than encoding specific locations, the
model learned a coarse code of position using broadly tuned spatial patterns. This could
offer novel insights into the function of the broad tuning of higher level visual neurons.
By learning higher-order basis functions for different classes of visual images, the model
could not only provide insights into other types of visual response properties, but could
provide a way to simplify some of the computations in perceptual organization and other
computations in mid-level vision.
References
[1] H. B. Barlow. Possible principles underlying the transformation of sensory messages. In W. A.
Rosenbluth, editor, Sensory Communication, pages 217?234. MIT Press, Cambridge, 1961.
[2] B. A. Olshausen and D. J. Field. Emergence of simple-cell receptive-field properties by learning
a sparse code for natural images. Nature, 381:607?609, 1996.
[3] A. J. Bell and T. J. Sejnowski. The ?independent components? of natural scenes are edge filters.
Vision Res., 37(23):3327?3338, 1997.
[4] J. H. van Hateren and A. van der Schaaf. Independent component filters of natural images
compared with simple cells in primary visual cortex. Proc. Royal Soc. Lond. B, 265:359?366,
1998.
[5] J. H. van Hateren and D. L. Ruderman. Independent component analysis of natural image
sequences yield spatiotemporal filters similar to simple cells in primary visual cortex. Proc.
Royal Soc. Lond. B, 265:2315?2320, 1998.
[6] P. O. Hoyer and A. Hyvarinen. Independent component analysis applied to feature extraction
from colour and stereo images. Network, 11(3):191?210, 2000.
[7] E. Simoncelli and B. Olshausen. Natural image statistics and neural representation. Ann. Rev.
Neurosci., 24:1193?1216, 2001.
[8] T-W. Lee and M. S. Lewicki. Unsupervised classification, segmentation and de-noising of
images using ICA mixture models. IEEE Trans. Image Proc., 11(3):270?279, 2002.
[9] P. O. Hoyer and A. Hyvarinen. A multi-layer sparse coding network learns contour coding from
natural images. Vision Research, 42(12):1593?1605, 2002.
[10] O. Schwartz and E. P. Simoncelli. Natural signal statistics and sensory gain control. Nat.
Neurosci., 4:819?825, 2001.
[11] B. A. Pearlmutter and L. C. Parra. A context-sensitive generalization of ICA. In International
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[12] J-F. Cardoso. Infomax and maximum likelihood for blind source separation. IEEE Signal
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1,282 | 2,168 | Rate Distortion Function in the Spin Glass State:
a Toy Model
Tatsuto Murayama and Masato Okada
Laboratory for Mathematical Neuroscience
RIKEN Brain Science Institute
Saitama, 351-0198, JAPAN
{murayama,okada}@brain.riken.go.jp
Abstract
We applied statistical mechanics to an inverse problem of linear mapping
to investigate the physics of optimal lossy compressions. We used the
replica symmetry breaking technique with a toy model to demonstrate
Shannon?s result. The rate distortion function, which is widely known
as the theoretical limit of the compression with a fidelity criterion, is
derived. Numerical study shows that sparse constructions of the model
provide suboptimal compressions.
1
Introduction
Many information-science studies are very similar to those of statistical physics. Statistical
physics and information science may have been expected to be directed towards common
objectives since Shannon formulated an information theory based on the concept of entropy. However, envisaging how this actually happened would have been difficult; that the
physics of disordered systems, and spin glass theory in particular, at its maturity naturally
includes some important aspects of information sciences, thus reuniting the two disciplines.
This cross-disciplinary field can thus be expected to develop much further beyond current
perspectives in the future [1].
The areas where these relations are particularly strong are Shannon?s coding theory [2] and
classical spin systems with quenched disorder, which is the replica theory of disordered
statistical systems [3]. Triggered by the work of Sourlas [4], these links have recently
been examined in the area of matrix-based error corrections [5, 6], network-based compressions [7], and turbo decoding [8]. Recent results of these topics are mostly based on
the replica technique. Without exception, their basic characteristics (such as channel capacity, entropy rate, or achievable rate region) are only captured by the concept of a phase
transition with a first-order jump between the optimal and the other solutions arising in the
scheme.
However, the research in the cross-disciplinary field so far can be categorized as a so-called
?zero-distortion? decoding scheme in terms of information theory: the system requires perfect reproduction of the input alphabets [2]. Here, the same spin glass techniques should
be useful to describe the physics of systems with a fidelity criterion; i.e., a certain degree
of information distortion is assumed when reproducing the alphabets. This framework is
called the rate distortion theory [9, 10]. Though processing information requires regarding the concept of distortions practically, where input alphabets are mostly represented by
continuous variables, statistical physics only employs a few approaches [11, 12].
In this paper, we introduce a prototype that is suitable for cross-disciplinary study. We
analyze how information distortion can be described by the concepts of statistical physics.
More specifically, we study the inverse problem of a Sourlas-type decoding problem by using the framework of replica symmetry breaking (RSB) of diluted disordered systems [13].
According to our analysis, this simple model provides an optimal compression scheme
for an arbitrary fidelity-criterion degree, though the encoding procedure remains an NPcomplete problem without any practical encoders.
The paper is organized as follows. In Section 2, we briefly review the concept of the
rate distortion theory as well as the main results related to our purpose. In Section 3, we
introduce a toy model. In Section 4, we obtain consistent results with information theory.
Conclusions are given in the last section. Detailed derivations will be reported elsewhere.
2
Review: Rate Distortion Theory
We briefly recall the definitions of the concepts of the rate distortion theory and state the
simplest version of the main result at the end of this section. Let J be a discrete random variable with alphabet J . Assume that we have a source that produces a sequence
J1 , J2 , ? ? ? , JM , where each symbol is randomly drawn from a distribution. We will assume that the alphabet is finit. Throughout this paper we use vector notation to represent
sequences for convenience of explanation: J = (J1 , J2 , ? ? ? , JM )T ? J M . Here, the
encoder describes the source sequence J ? J M by a codeword ? = f (J ) ? X N . The
? = g(?) ? J?M , as illustrated in Figure 1. Note that
decoder represents J by an estimate J
M represents the length of a source sequence, while N represents the length of a codeword.
Here, the rate is defined by R = N/M . Note that the relation N < M always holds when
a compression is considered; therefore, R < 1 also holds.
Definition 2.1 A distortion function is a mapping
d : J ? J? ? R+
(1)
from the set of source alphabet-reproduction alphabet pairs into the set of non-negative
real numbers.
? is a measure of the cost of representing the symbol J by
Intuitively, the distortion d(J, J)
?
the symbol J. This definition is quite general. In most cases, however, the reproduction
alphabet J? is the same as the source alphabet J . Hereafter, we set J? = J and the
following distortion measure is adopted as the fidelity criterion:
Definition 2.2 The Hamming distortion is given by
(
?
? = 0 if J = J ,
d(J, J)
1 if J 6= J?
(2)
,
? = P[J 6= J]
?
which results in a probable error distortion, since the relation E[d(J, J)]
holds, where E[?] represents the expectation and P[?] the probability of its argument. The
distortion measure is so far defined on a symbol-by-symbol basis. We extend the definition
to sequences:
Definition 2.3 The distortion between sequences J , J? ? J M is defined by
M
X
?) = 1
d(Jj , J?j ) .
d(J , J
M j=1
(3)
Therefore, the distortion for a sequence is the average distortion per symbol of the elements
of the sequence.
Definition 2.4 The distortion associated with the code is defined as
? )] ,
D = E[d(J , J
(4)
where the expectation is with respect to the probability distribution on J .
A rate distortion pair (R, D) should be achiebable if a sequence of rate distortion codes
? )] ? D in the limit M ? ?. Moreover, the closure of the set
(f, g) exist with E[d(J , J
of achievable rate distortion pairs is called the rate distortion region for a source. Finally,
we can define a function to describe the boundary:
Definition 2.5 The rate distortion function R(D) is the infimum of rates R, so that (R, D)
is in the rate distortion region of the source for a given distortion D.
As in [7], we restrict ourselves to a binary source J with a Hamming distortion measure
for simplicity. We assume that binary alphabets are drawn randomly, i.e., the source is
not biased to rule out the possiblity of compression due to redundancy. We now find the
description rate R(D) required to describe the source with an expected proportion of errors
less than or equal to D. In this simplified case, according to Shannon, the boundary can be
written as follows.
Theorem 2.1 The rate distortion function for a binary source with Hamming distortion is
given by
1 ? H(D) 0 ? D ? 21
R(D) =
,
(5)
1
0
2 <D
where H(?) represents the binary entropy function.
encoder
decoder
?
J ?? f ?? ? ?? g ?? J
Figure 1: Rate distortion encoder and decoder
3
General Scenario
In this section, we introduce a toy model for lossy compression. We use the inverse problem
of Sourlas-type decoding to realize the optimal encoding scheme [4]. As in the previous
section, we assume that binary alphabets are drawn randomly from a non-biased source
and that the Hamming distortion measure is selected for the fidelity criterion.
We take the Boolean representation of the binary alphabet J , i.e., we set J = {0, 1}.
We also set X = {0, 1} to represent the codewords throughout the rest of this paper.
? an M -bit reproduction
Let J be an M -bit source sequence, ? an N -bit codeword, and J
sequence. Here, the encoding problem can be written as follows. Given a distortion D and
a randomly-constructed Boolean matrix A of dimensionality M ? N , we find the N -bit
codeword sequence ?, which satisfies
? = A?
J
(mod 2) ,
(6)
where the fidelity criterion
? )]
D = E[d(J , J
(7)
holds, according to every M -bit source sequence J . Note that we applied modulo 2 arithmetics for the additive operations in (6). In our framework, decoding will just be a linear
? = A?, while encoding remains a NP-complete problem.
mapping J
Kabashima and Saad recently expanded on the work of Sourlas, which focused on the zerorate limit, to an arbitrary-rate case [5]. We follow their construction of the matrix A, so we
can treat practical cases. Let the Boolean matrix A be characterized by K ones per row
and C per column. The finite, and usually small, numbers K and C define a particular
code. The rate of our codes can be set to an arbitrary value by selecting the combination
of K and C. We also use K and C as control parameters to define the rate R = K/C. If
the value of K is small, i.e., the relation K N holds, the Boolean matrix A results in
a very sparse matrix. By contrast, when we consider densely constructed cases, K must
be extensively big and have a value of O(N ). We can also assume that K is not O(1) but
K N holds. The codes within any parameter region, including the sparsely-constructed
cases, will result in optimal codes as we will conclude in the following section. This is one
new finding of our analysis using statistical physics.
4
Physics of the model: One-step RSB Scheme
The similarity between codes of this type and Ising spin systems was first pointed out by
Sourlas, who formulated the mapping of a code onto an Ising spin system Hamiltonian
in the context of error correction [4]. To facilitate the current investigation, we first map
the problem to that of an Ising model with finite connectivity following Sourlasfmethod.
We use the Ising representation {1, ?1} of the alphabet J and X rather than the Boolean
one {0, 1}; the elements of the source J and the codeword sequences ? are rewritten in
Ising values by mapping only, and the reproduction sequence J? is generated by taking
products of the relevant binary codeword sequence elements in the Ising representation
J?hi1 ,i2 ,??? ,iK i = ?i1 ?i2 ? ? ? ?iK , where the indices i1 , i2 , ? ? ? , iK correspond to the ones per
row A, producing a Ising version of J?. Note that the additive operation in the Boolean
representation is translated into the multiplication in the Ising one. Hereafter, we set
Jj , J?j , ?i = ?1 while we do not change the notations for simplicity. As we use statisticalmechanics techniques, we consider the source and codeword-sequence dimensionality (M
and N , respectively) to be infinite, keeping the rate R = N/M finite. To explore the
system?s capabilities, we examine the Hamiltonian:
X
H(S) = ?
Ahi1 ,??? ,iK i Jhi1 ,??? ,iK i Si1 ? ? ? SiK ,
(8)
hi1 ,??? ,iK i
where we have introduced the dynamical variable Si to find the most suitable Ising codeword sequence ? to provide the reproduction sequence J? in the decoding stage. Elements
of the sparse connectivity tensor Ahi1 ,??? ,iK i take the value one if the corresponding indices
of codeword bits are chosen (i.e., if all corresponding indices of the matrix A are one) and
zero otherwise; C ones per i index represent the system?s degree of connectivity.
For calculating the partition function Z(A, J ) = Tr{S} exp[??H(S)], we apply the
replica method following the calculation of Kabashima and Saad [5]. To calculate replicafree energy, we have to calculate the annealed average of the n-th power of the partition
function by preparing n replicas. Here we introduce the inverse temperature ?, which can
be interpreted as a measure of the system?s sensitivity to distortions. As we see in the following calculation, the optimal value of ? is naturally determined when the consistency of
the replica symmetry breaking scheme is considered [13, 3]. We use integral representations of the Dirac ? function to enforce the restriction, C bonds per index, on A [14]:
?
?
I 2?
X
dZ ?(C+1) Phi ,i ,??? ,i i Ahi,i2 ,??? ,iK i
K
??
Ahi,i2 ,??? ,iK i ? C ? =
Z
Z 2 3
, (9)
2?
0
hi2 ,i3 ,??? ,iK i
giving rise to a set of order parameters
q?,?,??? ,?
N
1 X
Zi Si? Si? ? ? ? Si? ,
=
N i=1
(10)
where ?, ?, ? ? ? , ? represent replica indices, and the average over J is taken with respect
to the probability distribution:
P[Jhi1 ,i2 ,??? ,iK i ] =
1
1
?(Jhi1 ,i2 ,??? ,iK i ? 1) + ?(Jhi1 ,i2 ,??? ,iK i + 1)
2
2
(11)
as we consider the non-biased source sequences for simplicity. Assuming the replica symmetry, we use a different representation for the order parameters and the related conjugate
variables [14]:
Z
q?,?,??? ,? = q dx ?(x) tanhl (?x) ,
(12)
Z
q??,?,??? ,? = q? dx ?
? (?
x) tanhl (? x
?) ,
(13)
where q = [(K ? 1)!N C]1/K and q? = [(K ? 1)!]?1/K [N C](K?1)/K are normalization
constants, and ?(x) and ?
? (?
x) represent probability distributions related to the integration
variables. Here l denotes the number of related replica indices. Throughout this paper,
integrals with unspecified limits denote integrals over the range of (??, +?). We then
obtain an expression for the free energy per source bit expressed in terms of the probability
distributions ?(x) and ?
? (?
x):
1
hhln Z(A, J )ii
M
= ln cosh ?
*
!+
Z Y
K
K
Y
+
dxl ?(xl ) ln 1 + tanh ?J
tanh ?xl
??f =
l=1
?K
Z
l=1
dx ?(x)
J
(14)
Z
d?
x?
? (?
x) ln(1 + tanh ?x tanh ? x
?)
"
#
Z Y
C
C
Y
C
?l ) ,
d?
xl ?
? (?
xl ) ln Tr (1 + S tanh ? x
+
S
K
l=1
l=1
where hh? ? ? ii denotes the average over quenched randomness of A and J . The saddle
point equations with respect to probability distributions provide a set of relations between
?(x) and ?
? (?
x):
#
!
Z "C?1
C?1
Y
X
?(x) =
d?
xl ?
? (?
xl ) ? x ?
x
?l ,
l=1
?
? (?
x) =
Z "C?1
Y
l=1
l=1
dxl ?(xl )
#* "
1
? x
? ? tanh?1
?
tanh ?J
K?1
Y
tanh ?xl
l=1
!#+
(15)
.
J
By using the result obtained for the free energy, we can easily perform further straightforward calculations to find all the other observable thermodynamical quantities, including
internal energy:
EE
1 DD
1 ?
TrS H(S)e??H(S)
hhln Z(A, J )ii ,
(16)
e=
=?
M
M ??
which records reproduction errors. Therefore, in terms of the considered replica symmetric
ansatz, a complete solution of the problem seems to be easily obtainable; unfortunately, it
is not.
This set of equations (15) may be solved numerically for general ?, K, and C. However, there exists an analytical solution of this equations. We first consider this case. Two
dominant solutions emerge that correspond to the paramagnetic and the spin glass phases.
The paramagnetic solution, which is also valid for general ?, K, and C, is in the form of
?(x) = ?(x) and ?
? = ?(?
x); it has the lowest possible free energy per bit fPARA = ?1,
although its entropy sPARA = (R?1) ln 2 is positive only for R ? 1. It means that the true
solution must be somewhere beyond the replica symmetric ansatz. As a first step, which is
called the one-step replica symmetry breaking (RSB), n replicas are usually divided into
n/m groups, each containing m replicas. Pathological aspects due to the replica symmetry
may be avoided making use of the newly-defined freedom m. Actually, this one-step RSB
scheme is considered to provide the exact solutions when the random energy model limit
is considered [15], while our analysis is not restricted to this case.
The spin glass solution can be calculated for both the replica symmetric and the one-step
RSB ansatz. The former reduces to the paramagnetic solution (fRS = fPARA ), which is
unphysical for R < 1, while the latter yields ?1RSB (x) = ?(x), ?
?1RSB (?
x) = ?(?
x) with
m = ?g (R)/? and ?g obtained from the root of the equation enforcing the non-negative
replica symmetric entropy
sRS = ln cosh ?g ? ?g tanh ?g + R ln 2 = 0 ,
(17)
with a free energy
f1RSB = ?
1
R
ln cosh ?g ?
ln 2 .
?g
?g
(18)
Since the target bit of the estimation in this model is Jhi1 ,??? ,iK i and its estimator the product
Si1 ? ? ? SiK , a performance measure for the information corruption could be the per-bond
energy e. According to the one-step RSB framework, the lowest free energy can be calculated from the probability distributions ?1RSB (x) and ?
?1RSB (?
x) satisfying the saddle point
equation (15) at the characteristic inverse temperature ?g , when the replica symmetric entropy sRS disappears. Therefore, f1RSB equals e1RSB . Let the Hamming distortion be our
fidelity criterion. The distortion D associated with this code is given by the fraction of the
free energies that arise in the spin glass phase:
D=
f1RSB ? fRS
1 ? tanh ?g
=
.
2|fRS |
2
(19)
Here, we substitute the spin glass solutions into the expression, making use of the fact that
the replica symmetric entropy sRS disappears at a consistent ?g , which is determined by
(17). Using (17) and (19), simple algebra gives the relation between the rate R = N/M
and the distortion D in the form
R = 1 ? H(D) ,
which coincides with the rate distortion function retrieving Theorem 2.1. Surprisingly, we
do not observe any first-order jumps between analytical solutions. Recently, we have seen
that many approaches to the family of codes, characterized by the linear encoding operations, result in a quite different picture: the optimal boundary is constructed in the random
energy model limit and is well captured by the concept of a first-order jump. Our analysis
of this model, viewed as a kind of inverse problem, provides an exception. Many optimal
conditions in textbook information theory may be well described without the concept of a
first-order phase transitions from a view point of statistical physics.
We will now investigate the possiblity of the other solutions satisfying (15) in the case
of finite K and C. Since the saddle point equations (15) appear difficult for analytical
arguments, we resort to numerical evaluations representing the probability distributions
?1RSB (x) and ?
?1RSB (?
x) by up to 105 bin models and carrying out the integrations by
using Monte Carlo methods. Note that the characteristic inverse temperature ? g is also
evaluated numerically by using (17). We set K = 2 and selected various values of C to
demonstrate the performance of stable solutions. The numerical results obtained by the
one-step RSB senario show suboptimal properties [Figure 2]. This strongly implies that
the analytical solution is not the only stable solution. This conjecture might be verified
elsewhere, carrying out large scale simulations.
5
Conclusions
Two points should be noted. Firstly, we found that the consistency between the rate distortion theory and the Parisi one-step RSB scheme. Secondly, we conjectured that the
analytical solution, which is consistent with the Shannon?s result, is not the only stable
solution for some situations. We are currently working on the verification.
Acknowledgments
We thank Yoshiyuki Kabashima and Shun-ichi Amari for their comments on the
manuscript. We also thank Hiroshi Nagaoka and Te Sun Han for giving us valuable references. This research is supported by the Special Postdoctoral Researchers Program at
RIKEN.
References
[1] H. Nishimori. Statistical Physics of Spin Glasses and Information Processing. Oxford
University Press, 2001.
[2] T. M. Cover and J. A. Thomas. Elements of Information Theory. Wiley, 1991.
[3] V. Dotsenko. Introduction to the Replica Theory of Disordered Statistical Systems.
Cambridge University Press, 2001.
[4] N. Sourlas. Spin-glass models as error-correcting codes. Nature, 339:693?695, 1989.
[5] Y. Kabashima and D. Saad. Statistical mechanics of error-correcting codes. Europhys.
Lett., 45:97?103, 1999.
[6] Y. Kabashima, T. Murayama, and D. Saad. Typical performance of Gallager-type
error-correcting codes. Phys. Rev. Lett., 84:1355?1358, 2000.
1
K=2
R(D)
2
0.8
^ )
?^ ( x
1
R
0.6
? ( x )
0.4
0
-2
-1
0
1
2
0.2
0
0
0.1
0.2
0.3
0.4
0.5
D
Figure 2: Numerically-constructed stable solutions: Stable solutions of (15) for the finite
values of K and L are calculated by using Monte Carlo methods. We use 105 bin models
to approximate the probability distributions ?1RSB (x) and ?
?1RSB (?
x), starting from various
initial conditions. The distributions converge to the continuous ones, giving suboptimal
performance. (?) K = 2 and L = 3, 4, ? ? ? , 12 ; Solid line indicates the rate distortion
function R(D). Inset: Snapshots of the distributions, where L = 3 and ?g = 2.35.
[7] T. Murayama. Statistical mechanics of the data compression theorem. J. Phys. A,
35:L95?L100, 2002.
[8] A. Montanari and N. Sourlas. The statistical mechanics of turbo codes. Eur. Phys. J.
B, 18:107?119, 2000.
[9] C. E. Shannon. Coding theorems for a discrete source with a fidelity criterion. IRE
National Convention Record, Part 4, pages 142?163, 1959.
[10] T. Berger. Rate Distortion Theory: A Mathematical Basis for Data Compression.
Prentice-Hall, 1971.
[11] T. Hosaka, Y. Kabashima, and H. Nishimori. Statistical mechanics of lossy data
compression using a non-monotonic perceptron. cond-mat/0207356.
[12] Y. Matsunaga and H. Yamamoto. A coding theorem for lossy data compression by
LDPC codes. In Proceedings 2002 IEEE International Symposium on Information
Theory, page 461, 2002.
[13] M. Mezard, G. Parisi, and M. Virasoro. Spin-Glass Theory and Beyound. World
Scientific, 1987.
[14] K. Y. M. Wong and D. Sherrington. Graph bipartitioning and spin glasses on a random
network of fixed finite valence. J. Phys. A, 20:L793?L799, 1987.
[15] B. Derrida. The random energy model, an exactly solvable model of disordered systems. Phys. Rev. B, 24:2613?2626, 1981.
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1,283 | 2,169 | Feature Selection by Maximum Marginal
Diversity
Nuno Vasconcelos
Department of Electrical and Computer Engineering
University of California, San Diego
[email protected]
Abstract
We address the question of feature selection in the context of visual
recognition. It is shown that, besides efficient from a computational
standpoint, the infomax principle is nearly optimal in the minimum
Bayes error sense. The concept of marginal diversity is introduced, leading to a generic principle for feature selection (the principle of maximum
marginal diversity) of extreme computational simplicity. The relationships between infomax and the maximization of marginal diversity are
identified, uncovering the existence of a family of classification procedures for which near optimal (in the Bayes error sense) feature selection
does not require combinatorial search. Examination of this family in light
of recent studies on the statistics of natural images suggests that visual
recognition problems are a subset of it.
1 Introduction
It has long been recognized that feature extraction and feature selection are important problems in statistical learning. Given a classification or regression task in some observation
space (typically high-dimensional), the goal is to find the best transform into a feature
space (typically lower dimensional) where learning is easier (e.g. can be performed with
less training data). While in the case of feature extraction there are few constraints on ,
for feature selection the transformation is constrained to be a projection, i.e. the components of a feature vector in are a subset of the components of the associated vector in .
Both feature extraction and selection can be formulated as optimization problems where
the goal is to find the transform that best satisfies a given criteria for ?feature goodness?.
In this paper we concentrate on visual recognition, a subset of the classification problem for
which various optimality criteria have been proposed throughout the years. In this context,
the best feature spaces are those that maximize discrimination, i.e. the separation between
the different image classes to recognize. However, classical discriminant criteria such as
linear discriminant analysis make very specific assumptions regarding class densities, e.g.
Gaussianity, that are unrealistic for most problems involving real data. Recently, various
authors have advocated the use of information theoretic measures for feature extraction or
selection [15, 3, 9, 11, 1]. These can be seen as instantiations of the the infomax principle
of neural organization1 proposed by Linsker [7], which also encompasses information theoretic approaches for independent component analysis and blind-source separation [2]. In
the classification context, infomax recommends the selection of the feature transform that
maximizes the mutual information (MI) between features and class labels.
While searching for the features that preserve the maximum amount of information about
the class is, at an intuitive level, an appealing discriminant criteria, the infomax principle
does not establish a direct connection to the ultimate measure of classification performance
- the probability of error (PE). By noting that to maximize MI between features and class
labels is the same as minimizing the entropy of labels given features, it is possible to establish a connection through Fano?s inequality: that class-posterior entropy (CPE) is a lower
bound on the PE [11, 4]. This connection is, however, weak in the sense that there is little
insight on how tight the bound is, or if minimizing it has any relationship to minimizing
PE. In fact, among all lower bounds on PE, it is not clear that CPE is the most relevant.
An obvious alternative is the Bayes error (BE) which 1) is the tightest possible classifierindependent lower-bound, 2) is an intrinsic measure of the complexity of the discrimination
problem and, 3) like CPE, depends on the feature transformation and class labels alone.
Minimizing BE has been recently proposed for feature extraction in speech problems [10].
The main contribution of this paper is to show that the two strategies (infomax and minimum BE) are very closely related. In particular, it is shown that 1) CPE is a lower bound
on BE and 2) this bound is tight, in the sense that the former is a good approximation to the
latter. It follows that infomax solutions are near-optimal in the minimum BE sense. While
for feature extraction both infomax and BE appear to be difficult to optimize directly, we
show that infomax has clear computational advantages for feature selection, particularly in
the context of the sequential procedures that are prevalent in the feature selection literature [6]. The analysis of some simple classification problems reveals that a quantity which
plays an important role in infomax solutions is the marginal diversity: the average distance
between each of the marginal class-conditional densities and their mean. This serves as
inspiration to a generic principle for feature selection, the principle of maximum marginal
diversity (MMD), that only requires marginal density estimates and can therefore be implemented with extreme computational simplicity. While heuristics that are close to the MMD
principle have been proposed in the past, very little is known regarding their optimality.
In this paper we summarize the main results of a theoretical characterization of the problems for which the principle is guaranteed to be optimal in the infomax sense (see [13] for
further details). This characterization is interesting in two ways. First, it shows that there is
a family of classification problems for which a near-optimal solution, in the BE sense, can
be achieved with a computational procedure that does not involve combinatorial search.
This is a major improvement, from a computational standpoint, to previous solutions for
which some guarantee of optimality (branch and bound search) or near optimality (forward
or backward search) is available [6]. Second, when combined with recent studies on the
statistics of biologically plausible image transformations [8, 5], it suggests that in the context of visual recognition, MMD feature selection will lead to solutions that are optimal in
the infomax sense. Given the computational simplicity of the MMD principle, this is quite
significant. We present experimental evidence in support of these two properties of MMD.
2 Infomax vs minimum Bayes error
In this section we show that, for classification problems, the infomax principle is closely
related to the minimization of Bayes error. We start by defining these quantities.
1
Under the infomax principle, the optimal organization for a complex multi-layered perceptual
system is one where the information that reaches each layer is processed so that the maximum amount
of information is preserved for subsequent layers.
Theorem 1 Given a classification problem with
classes in a feature space , the decision function which minimizes the probability of classification error is the Bayes classifier
!#" , where $ is a random variable that assigns to one of
classes, and &%('*)+-,.,-,#+
/ . Furthermore, the PE is lower bounded by the Bayes error
0
1)32543687 9 !#" ;:;+
(1)
where 436 means expectation with respect to .
Proof: All proofs are omitted due to space considerations. They can be obtained by contacting the author.
Principle 1 (infomax) Consider an
-class classification problem with observations
@?
drawn from random variable < %
, and the set of feature transformations >=
. The best feature space is the one that maximizes the mutual information A $CBD
< , and A $EBFD G
where $ is the class indicator
variable defined above, D
H IKJ ML !N+FOPRQ*1S.TV6U W9Y X 6 L ZYZY#[
the mutual information between D and $ .
S.T9X S.W9X
]^ $ _2`]^ $ " D , where ]^ D \
It is straightforward to show that A D + $ \
2 IKJ ! PRQ* J [ is the entropy of D . Since the class entropy ]a $ does not depend on , infomax is equivalent to the minimization of the CPE ]^ $ " D . We next derive
a bound that plays a central role on the relationship between this quantity and BE.
Lemma 1H Consider a probability mass function b
)+jif and #J
1) . Then,
c' J d +-,.,.,.+ Jfe / such that g5h J h
)
PZQVnm
2o)p
j)325 J ;lk
]^ b N2
)
(2)
ZP Q
PRQ*
q
H
.J PZQ J . Furthermore, the bound is tight in the sense that equality
where ]a b r
s2
holds when
)
+Oixwz
u
y {|,
J t
m
(3)
2u) and J v
m
2o)
The following theorem follows from this bound.
Theorem 2 The BE of an -class classification problem with feature space
indicator variable $ , is lower bounded by
0 }
lk
PRQ*
)
]^ $ " D 2
PZQfnm
PZQ
2o)p
q
)+
%
where
is the random vector
from which
d features are drawn. When
?D ~
0
) this bound reduces to } lk ? ?F? e ]^ $ " D .
(
and class
(4)
is large
It is interesting
to note the relationship between (4) and Fano?s lower bound on the PE
?uk ? ?? d e ]^ $ " D r2 ? ?? d e . The two bounds are equal up to an additive constant
d
?e
( ? ?? e PRQ*?
? eE?xd ) that quickly decreases to zero with the number of classes . It follows
that, at least when the number of classes is large, Fano?s is really a lower bound on BE, not
only on PE. Besides making this clear, Theorem 2 is a relevant contribution in two ways.
First, since constants do not change the location of the bound?s extrema, it shows that infomax minimizes a lower bound on BE. Second, unlike Fano?s bound, it sheds considerable
insight on the relationship between the extrema of the bound and those of the BE.
In fact, it is clear from the derivation of the theorem that, the only reason why the righthand (RHS) and left-hand (LHS) sides of (4) differ is the application of (2). Figure 1
0.6
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
1?max(p)
H(p) ? log(3) + 1
0.2
p2
0
p2
0.4
0.5
0.5
?0.2
?0.4
0.4
0.4
0.3
0.3
0.2
0.2
?0.6
0.1
?0.8
0
0.1
0.2
0.3
0.4
0.5
p
0.6
0.7
0.8
0.9
0
1
0.1
0
0.1
0.2
0.3
1
0.4
0.5
p1
0.6
0.7
0.8
0.9
0
1
0
0.1
0.2
0.3
0.4
0.5
p1
0.6
0.7
0.8
0.9
1
Figure 1: Visualization of (2). Left: LHS and RHS versus for
. Middle:
contours of the LHS versus
! !
. Right: same, for RHS.
for
3
0.5
1
0.4
0.8
0.3
0.6
2
0.1
?y
L
*
H(Y|X)
1
0.2
?1
0.2
0
5
0
0.4
?2
0
5
5
5
?3
0
0
0
?y
?5
?5
?x
0
?y
?5
?5
?3
?x
?2
?1
0
?
1
2
3
x
Figure
"$#% & 2: The LHS of (4)"$as
# % & an approximation to (1) for a two-class Gaussian problem where
('*) ,+.-/
10324 and
('5) 6 +-7
89:2; . All plots are functions of 8 . Left: surface plot
of (1). Middle: surface plot of the LHS of (4). Right: contour plots of the two functions.
%`'m+< / , illustrating three
shows plots of the RHS and LHS of this equation when
interesting properties. First, bound (2) is tight in the sense defined in the lemma. Second,
the maximum of the LHS is co-located with that of the RHS. Finally, (like the RHS) the
LHS is a concave function of b and increasing (decreasing) when the RHS is. Due to these
properties, the LHS is a good approximation to the RHS and, consequently, the LHS of (4)
a good approximation to its RHS. It follows that infomax solutions will, in general, be very
similar to those that minimize the BE . This is illustrated by a simple example in Figure 2.
3 Feature selection
For feature extraction, both infomax and minimum BE are complicated problems that can
only be solved up to approximations [9, 11, 10]. It is therefore not clear which of the
two strategies will be more useful in practice. We now show that the opposite holds for
feature selection, where the minimization of CPE is significantly simpler than that of BE.
We start by recalling that, because the possible number of feature subsets in a feature
selection problem is combinatorial, feature selection techniques rely on sequential search
methods [6]. These methods proceed in a sequence of steps, each adding a set of features
to the current best subset, with the goal of optimizing a given cost function 2. We denote the
current subset by D7= , the added features by D7> and the new subset by D/?
D/> + D/= .
Theorem 3 Consider an -class classification problem with? observations drawn from a
random variable < %
, and a feature transformation =
. is a infomax feature
2
These methods are called forward search techniques. There is also an alternative set of backward
search techniques, where features are successively removed from an initial set containing all features.
We ignore the latter for simplicity, even though all that is said can be applied to them as well.
space if and only if i
0
y
K!&" j."Z"
k 0
K!&" O-"R"
(5)
H
< ,D
< , j
!j !j
where D
6 Y expectation with respect
0 7 J "Z" p:
I J ! PZQ denotes
S X 6 Y [ is the Kullback-Leibler
to the prior class probabilities and
X
divergence between J and . Furthermore, if D7?
D7> + D/= , the infomax cost function
decouples into two terms according to
0 9
? " O-"R"
! ?
0 9 L " +Fj."Z" 9
> =
0 9 . ! " j."Z" ,
=
=
q
! > " =
(6)
Equation (5) exposes the discriminant nature of the infomax criteria. Noting that
!
K!&" j , it clearly favors feature spaces where each class-conditional
density is as distant as possible (in the KL sense) from the average among all classes.
This is a sensible way to quantify the intuition that optimal discriminant transforms are the
ones that best separate the different classes. Equation (6), in turn, leads to an optimal rule
with the current optimal solution D = : the set which
for finding the
0 features
9 D L > to > merge
" = +Fj."Z" 9 ! > " = . The equation also leads to a
minimizes
straightforward procedure for updating the optimal cost once this set is determined. On the
other hand, when the cost function is BE, the equivalent expression is
L p! > " + =
l !#" = O:"! , (7)
! > " =
Note that the non-linearity introduced by the operator, makes it impossible to ex !#" = ;: . For this reason,
press 4 7 l
8!#" ? ;: as a function of 4 7 *9
4
7
!#" ? O:8
4 4r
7
infomax is a better principle for feature selection problems than direct minimization of BE.
4 Maximum marginal diversity
To gain some intuition for infomax solutions, we next consider the Gaussian problem of
Figure 3. Assuming that the two classes have equal prior probabilities j)
m
)$#m , the marginals &%('- r*)9" )p and +%
'# r*)" m* are equal and feature , d does not contain any useful information for classification. On the other hand, because the classes are
clearly separated along the ) ? axis, feature , ? contains all the information available for
discriminating between them. The different discriminating powers of the two variables
are reflected
the infomax costs: while % ' *)V
+%
'# r*)9" )p
-%
'. r.)9" m leads
0 7 +%
by
'
K
*)" j."Z" % ' *)fO:/
%&0 .)f5
y -% 0 *)" ) (
c
y +% 0 r*)" m* it
to
0 7 +% 0 *)" j-"R" %&0 *)Vg ;,: from
g , and (5) recommends the selection of
follows that
, ? . This is unlike energy-based criteria, such as principal component analysis, that would
select , d . The key advantage of infomax is that it emphasizes marginal diversity.
Definition 1 Consider a classification problem on a feature space , and a random vector
feature vectors are drawn. Then, 132 , v o
0 D 7 +%(
4 ., )9" d j.+."Z,-" ,.,#% + ,4 *?)V ;:+from
iswhich
the marginal diversity of feature , v .
The intuition conveyed by the example above can be easily transformed into a generic
principle for feature selection.
Principle 2 (Maximum marginal diversity) The best solution for a feature selection
problem is to select the subset of features that leads to a set of maximally diverse marginal
densities.
2.5
2
0.025
PX |Y(x|1)
1
(x|2)
P
PX
|Y
PX
|Y
(x|1)
2
(x|2)
2
X |Y
1
1.5
2
0.02
1
0.5
1.5
x2
0.015
0
?0.5
0.01
1
0.005
0.5
?1
?1.5
?2
?5
?4
?3
?2
?1
0
x1
1
2
3
4
0
?50
5
?40
?30
?20
Figure 3: Gaussian problem with two classes
Left: contours of
?10
0
x
10
20
30
40
0
?2
50
?1.5
?1
?0.5
0
x
0.5
1
1.5
2
6 , in. Right:
the two-dimensions,
marginals for
.
probability. Middle: marginals for
.
This principle has two attractive properties. First it is inherently discriminant, recommending the elimination of the dimensions along which the projections of the class densities are
most similar. Second, it is straightforward to implement with the following algorithm.
Algorithm 1 (MMD feature selection) For a classification problem with features D
, d +.,.,-,.+ ,? ,
classes $ % ' )+.,-,.,#+
/ and class priors jE
J the following
procedure returns the top MMD features.
- foreach feature w % ' )+-,.,.,.+ / :
'*)+-,.,-,#+
v L of -% 4 *)" j ,
* foreach class &%(
d H v L
/ , compute an histogram estimate
v
e
* compute
,
H .J v L PRQ*| v L , # v
* compute the marginal diversity 1 2 , v
, where both the
and division , # are performed element-wise,
- order the features by decreasing diversity, i.e. find 'w d +-,.,-,.+w ? / such that
1 2 , v Mk 1 2 , v ' , and return ' , v ' +.,-,.,-+ , v / .
In general, there are no guarantees that MMD will lead to the infomax solution. In [13]
we seek a precise characterization of the problems where MMD is indeed equivalent to
infomax. Due to space limitations we present here only the main result of this analysis,
see [13] for a detailed derivation.
Theorem 4 Consider a classification problem with class labels drawn from a random
, d +.,.,-,.+ , ? and let
variable $ and features drawn from a random vector D
r
+
.
,
,
#
,
+
D
, d
, be the optimal feature subset of size in the infomax sense. If
? d
A , v BD d L v ?8d " $ +ji8wE%(' )+-,.,., + /
A , v B D d L v 8
? d / , the set D is also the optimal subset of size
where D d L v ?8d
\' , d +.,.,-,#+ , v x
MMD sense. Furthermore,
0
9
r!&" O-"R"
!
! 1
v " d
2 ,
v ,
(8)
in the
(9)
The theorem states that the MMD and infomax solutions will be identical when the mutual
information between features is not affected by knowledge of the class label. This is an
interesting condition in light of various recent studies that have reported the observationof
consistent patterns of dependence between the features of various biologically plausible
image transformations [8, 5]. Even though the details of feature dependence will vary from
one image class to the next, these studies suggest that the coarse structure of the patterns of
dependence between such features follow universal statistical laws that hold for all types of
images. The potential implications of this conjecture are quite significant. First it implies
1
1
2.2
0.95
1.8
2
0.95
0.8
Cumulative marginal diversity
Classification rate
Jain/Zongker score
0.9
0.85
0.9
0.85
1.6
1.4
1.2
1
0.8
0.75
DCT
PCA
Wavelet
0.8
0.7
DCT
PCA
Wavelet
0.6
0.4
0.65
1
10
2
3
10
10
4
10
Sample size
a)
0.75
0
5
10
15
20
Number of features
25
0.2
30
0
5
10
b)
15
20
Number of features
25
30
35
c)
Figure 4: a) JZ score as a function of sample size for the two-class Gaussian problem discussed
in the text, b) classification accuracy on Brodatz as a function of feature space dimension, and c)
corresponding curves of cumulative marginal density (9). A linear trend was subtracted to all curves
in c) to make the differences more visible.
that, in the context of visual processing, (8) will be approximately true and the MMD
principle will consequently lead to solutions that are very close to optimal, in the minimum
BE sense. Given the simplicity of MMD feature selection, this is quite remarkable. Second,
it implies that when combined with such transformations, the marginal diversity is a close
predictor for the CPE (and consequently the BE) achievable in a given feature space. This
enables quantifying the goodness of the transformation without even having to build the
classifier. See [13] for a more extensive discussion of these issues.
5 Experimental results
In this section we present results showing that 1) MMD feature selection outperforms combinatorial search when (8) holds, and 2) in the context of visual recognition problems,
marginal diversity is a good predictor of PE. We start by reporting results on a synthetic
problem, introduced by Trunk to illustrate the curse of dimensionality [12], and used by
Jain and Zongker (JZ) to evaluate various feature selection procedures
d : It consists of
d d ,.,-, [6].
and is an intwo Gaussian classes of identity covariance and means 7R) ?
?
teresting benchmark for feature selection because it has a clear optimal solution for the
best subset of [ features (the first [ ) for any [ . JZ exploited this property to propose
an automated procedure for testing the performance of feature selection algorithms across
variations in dimensionality of the feature space and sample size. We repeated their experiments, simply replacing the cost function they used (Mahalanobis distance - MDist between the means) by the marginal diversity.
Figure 4 a) presents the JZ score obtained with MMD as a function of the sample size. A
comparison with Figure 5 of [6] shows that these results are superior to all those obtained
by JZ, including the ones relying on branch and bound. This is remarkable, since branch
and bound is guaranteed to find the optimal solution and the Mdist is inversely proportional
to the PE for Gaussian classes. We believe that the superiority of MMD is due to the
fact that it only requires estimates of the marginals, while the MDist requires estimates
of joint densities and is therefore much more susceptible to the curse of dimensionality.
Unfortunately, because in [6] all results are averaged over dimension, we have not been
able to prove this conjecture yet. In any case, this problem is a good example of situations
where, because (8) holds, MMD will find the optimal solution at a computational cost that
is various orders of magnitude smaller than standard procedures based on combinatorial
search (e.g. branch and bound).
Figures 4 b) and c) show that, for problems involving commonly used image transformations, marginal diversity is indeed a good predictor of classification accuracy. The figures
compare, for each space dimension, the recognition accuracy of a complete texture recognition system with the predictions provided by marginal diversity. Recognition accuracy
was measured on the Brodatz texture database ( )) m texture classes) and a dimensional
feature space consisting of the coefficients of a multiresolution decomposition over regions
of pixels. Three transformations were considered: the discrete cosine transform, principal component analysis, and a three-level wavelet decomposition (see [14] for detailed
description of the experimental set up). The classifier was based on Gauss mixtures and
marginal diversity was computed with Algorithm 1. Note that the curves of cumulative
marginal diversity are qualitatively very similar to those of recognition accuracy.
References
[1] S. Basu, C. Micchelli, and P. Olsen. Maximum Entropy and Maximum Likelihood Criteria for
Feature Selection from Multivariate Data. In Proc. IEEE International Symposium on Circuits
and Systems, Geneva, Switzerland,2000.
[2] A. Bell and T. Sejnowski. An Information Maximisation Approach to Blind Separation and
Blind Deconvolution. Neural Computation, 7(6):1129?1159, 1995.
[3] B. Bonnlander and A. Weigand. Selecting Input Variables using Mutual Information and Nonparametric Density Estimation. In Proc. IEEE International ICSC Symposium on Artificial
Neural Networks, Tainan,Taiwan,1994.
[4] D. Erdogmus and J. Principe. Information Transfer Through Classifiers and its Relation to
Probability of Error. In Proc. of the International Joint Conference on Neural Networks, Washington, 2001.
[5] J. Huang and D. Mumford. Statistics of Natural Images and Models. In IEEE Computer Society
Conference on Computer Vision and Pattern Recognition, Fort Collins, Colorado, 1999.
[6] A. Jain and D. Zongker. Feature Selection: Evaluation, Application, and Small Sample Performance. IEEE Trans. on Pattern Analysis and Machine Intelligence, 19(2):153?158, February
1997.
[7] R. Linsker. Self-Organization in a Perceptual Network. IEEE Computer, 21(3):105?117, March
1988.
[8] J. Portilla and E. Simoncelli. Texture Modeling and Synthesis using Joint Statistics of Complex
Wavelet Coefficients. In IEEE Workshop on Statistical and Computational Theories of Vision,
Fort Collins, Colorado, 1999.
[9] J. Principe, D. Xu, and J. Fisher. Information-Theoretic Learning. In S. Haykin, editor, Unsupervised Adaptive Filtering, Volume 1: Blind-Souurce Separation. Wiley, 2000.
[10] G. Saon and M. Padmanabhan. Minimum Bayes Error Feature Selection for Continuous Speech
Recognition. In Proc. Neural Information Proc. Systems, Denver, USA, 2000.
[11] K. Torkolla and W. Campbell. Mutual Information in Learning Feature Transforms. In Proc.
International Conference on Machine Learning, Stanford, USA, 2000.
[12] G. Trunk. A Problem of Dimensionality: a Simple Example. IEEE Trans. on Pattern. Analysis
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[13] N. Vasconcelos. Feature Selection by Maximum Marginal Diversity: Optimality and Implications for Visual Recognition. In submitted, 2002.
[14] N. Vasconcelos and G. Carneiro. What is the Role of Independence for Visual Regognition? In
Proc. European Conference on Computer Vision, Copenhagen, Denmark, 2002.
[15] H. Yang and J. Moody. Data Visualization and Feature Selection: New Algorithms for Nongaussian Data. In Proc. Neural Information Proc. Systems, Denver, USA, 2000.
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1,284 | 217 | VLSI Implementation of a High-Capacity Neural Network
VLSI Implementation of a High-Capacity
Neural Network Associative Memory
Tzi-Dar Chiueh 1 and Rodney M. Goodman
Department of Electrical Engineering (116-81)
California Institute of Technology
Pasadena, CA 91125, USA
ABSTRACT
In this paper we describe the VLSI design and testing of a high
capacity associative memory which we call the exponential correlation associative memory (ECAM). The prototype 3J.'-CMOS
programmable chip is capable of storing 32 memory patterns of
24 bits each. The high capacity of the ECAM is partly due to the
use of special exponentiation neurons, which are implemented via
sub-threshold MOS transistors in this design. The prototype chip
is capable of performing one associative recall in 3 J.'S.
1
ARCHITECTURE
Previously (Chiueh, 1989), we have proposed a general model for correlation-based
associative memories, which includes a variant of the Hopfield memory and highorder correlation memories as special cases. This new exponential correlation associative memory (ECAM) possesses a very large storage capacity, which scales
exponentially with the length of memory patterns (Chiueh, 1988). Furthermore, it
has been shown that the ECAM is asymptotically stable in both synchronous and
1Tzi-Dar Chiueh is now with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan 10764.
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Chiueh and Goodman
asynchronous updating modes (Chiueh, 1989). The model is based on an architecture consisting of binary connection weights, simple hard-limiter neurons, and
specialized nonlinear circuits as shown in Figure 1. The evolution equation of this
general model is
(1)
where u(1), u(2), ... , u(M) are the M memory patterns. x and x, are the current and
the next state patterns of the system respectively, and sgn is the threshold function,
which takes on the value +1 if its argument is nonnegative, and -1 otherwise.
We addressed, in particular, the case where f(?) is in the form of an exponentiation,
namely, when the evolution equation is given by
(2)
and a is a constant greater than unity.
The ECAM chip we have designed is programmable; that is, one can change the
stored memory patterns at will. To perform an associative recall, one first loads a
set of memory patterns into the chip. The chip is then switched to the associative
recall mode, an input pattern is presented to the ECAM chip, and the ECAM chip
then computes the next state pattern according to Equation (2). The components
of the next state pattern appear at the output in parallel after the internal circuits
have settled. Feedback is easily incorporated by connecting the output port to the
input port, in which case the chip will cycle until a fixed point is reached.
2
DESIGN OF THE ECAM CIRCUITS
From the evolution equation of the ECAM, we notice that there are essentially three
circuits that need to be designed in order to build an ECAM chip. They are:
? < u(1:), x >, the
correlation computation circuit;
M
? I:
a<u(k), x>
u(1:),
the exponentiation, multiplication and summing circuit;
1:=1
? sgn( .), the threshold circuit.
We now describe each circuit, present its design, and finally integrate all these
circuits to get the complete design of the ECAM chip.
VLSI Implementation ora High-Capacity Neural Network
2.1
CORRELATION COMPUTATION
In Figure 2, we illustrate a voltage-divider type circuit consisting of NMOS transistors working as controlled resistors (linear resistors or open circuits). This circuit
computes the correlation between the input pattern x and a memory pattern u(l:).
If the ith components of these two patterns are the same, the corresponding XOR
gate outputs a "0" and there is a connection from the node V~~ to VBB; otherwise, there is a connection from V~~ to GND. Hence the output voltage will be
proportional to the number of positions at which x and u(l:) match. The maximum
output voltage is controlled by an externally supplied bias voltage VBB. Normally,
VBB is set to a voltage lower than the threshold voltage of NMOS transistors (VTH)
for a reason that will be explained later. Note that the conductance of an NMOS
transistor in the ON mode is not fixed, but rather depends on its gate-to-source
voltage and its drain-to-source voltage. Thus, some nonlinearity is bound to occur
in the correlation computation circuit, however, simulation shows that this effect is
small.
2.2
EXPONENTIATION, MULTIPLICATION, AND SUMMATION
Figure 4 shows a circuit that computes the exponentiation of V~~, the product of
the u~l:) and the exponential, and the sum of all M products.
The exponentiation function is implemented by an NMOS transistor whose gate
voltage is V~~. Since VBB, the maximum value that V~~ can assume, is set to be
lower than the threshold voltage (VTH); the NMOS transistor is in the subthreshold
region, where its drain current depends exponentially on its gate-to-source voltage
(Mead, 1989). If we temporarily ignore the transistors controlled by u~l:) or the
complement of u~l:), the current flowing through the exponentiation transistor associated with V~~ will scale exponentially with V~~. Therefore, the exponentiation
function is properly computed.
Since the multiplier u~l:) assumes either +1 or -1, the multiplication can be easily
done by forming two branches, each made up of a transmission gate in series with an
exponentiation transistor whose gate voltage is V~~. One of the two transmission
gates is controlled by u~l:), and the other by the complement of u~l:). Consequently,
when u~l:)
1, the positive branch will carry a current that scales exponentially
with the correlation of the input x and the ph memory pattern u(l:) , while the
negative branch is essentially an open circuit, and vice versa.
=
Summation of the M terms in the evolution equation is done by current summing.
The final results are two currents It and Ii, which need to be compared by a
threshold circuit to determine the sign of the ith bit of the next state pattern x~.
In the ECAM a simple differential amplifier (Figure 3) performs the comparison.
795
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Cbiueb and Goodman
2.3
THE BASIC ECAM CELL
The above computational circuits are then combined with a simple static RAM cell,
to make up a basic ECAM cell as illustrated in Figure 5. The final design of an
ECAM that stores M N-bit memory patterns can be obtained by replicating the
basic ECAM cell M times in the horizontal direction and N times in the vertical
direction, together with read/write circuits, sense amplifiers, address decoders, and
I/O multiplexers. The prototype ECAM chip is made up of 32 x 24 ECAM cells,
and stores 32 memory patterns each 24 bits wide.
3
ECAM CHIP TEST RESULTS
The test procedure for the ECAM is to first generate 32 memory patterns at random
and then program the ECAM chip with these 32 patterns. We then pick a memory
pattern at random, flip a specified number of bits randomly, and feed the resulting
pattern to the ECAM as an input pattern (x). The output pattern (x') can then be
fed back to the inputs of the ECAM chip. This iteration continues until the pattern
at the input is the same as that at the the output, at which time the ECAM chip
is said to have reached a stable state. We select 10 sets of 32 memory patterns and
for each set we run the ECAM chip on 100 trial input patterns with a fixed number
of errors. Altogether, the test consists of 1000 trials.
In Figure 6, we illustrate the ECAM chip test results. The number of successes is
plotted against the number of errors in the input patterns for the following four
cases: 1) The ECAM chip with VBB
5V; 2) VBB
2V; 3) VBB
IV; and 4) a
simulated ECAM in which the exponentiation constant a, equals 2. It is apparent
from Figure 6 that as the number of errors increases, the number of successes
decreases, which is expected. Also, one notices that the simulated ECAM is by far
the best one, which is again not unforeseen because the ECAM chip is, after all,
only an approximation of the ideal ECAM model.
=
=
=
What is really unexpected is that the best performance occurs for VBB = 2V rather
than VBB = IV (VTH in this CMOS process). This phenomenon arises because
of two contradictory effects brought about by increasing VBB. On the one hand,
increasing VBB increases the dynamic range of the exponentiation transistors in the
ECAM chip. Suppose that the correlations of two memory patterns u(l) and u(k)
with the input pattern x are tJ and tk, respectively, where tJ > tk; then
V(I) _ (tJ
ux -
+ N) VBB
2N
(k) _
-
,V ux
(tk
+ N)
2N
VBB
.
Therefore, as VBB increases, so does the difference between V~I~ and V~~, and u(l)
becomes more dominant than u(k) in the weighted sum of the evolution equation.
VLSI Implementation or a High?Capacity Neural Network
Hence, as VBB increases, the error correcting ability of the ECAM chip should
improve. On the other hand, as VBB increases beyond the threshold voltage, the
exponentiation transistors leave the subthreshold region and may enter saturation,
where the drain current is approximately proportional to the square of the gateto-source voltage . Since a second-order correlation associative memory in general
possesses a smaller storage capacity than an ECAM, one would expect that with
a fixed number of loaded memory patterns, the ECAM should do better than the
second-order correlation associative memory. Thus one effect tends to enhance the
performance of the ECAM chip, while the other tends to degrade it. A compromise
between these two effects is reached, and the best performance is achieved when
VBB = 2V.
=
For the case when VBB 2V, the drain current versus gate-to-source voltage characteristic of the exponentiation transistors is actually a hybrid of a square function
and an exponentiation function. At the bottom it is of an exponential form, and
it gradually flattens out to a square function, once the gate-to-source voltage becomes larger than the threshold voltage . Therefore, the ECAM chip with VBB
2V is a mixture of the second-order correlation associative memory and the pure
ECAM . According to the convergence theorem for correlation associative memories
(Chiueh, 1989) and the fact that f(?) in the ECAM chip with VBB
2V is still
monotonically nondecreasing, the ECAM chip is still asymptotically stable when
VBB
2V.
=
=
=
We have tested the speed of the ECAM chip using binary image vector quantization
as an example problem. The speed at which the ECAM chip can vector-quantize
binary images is of interest. We find experimentally that the ECAM chip is capable
of doing one associative recall operation, in less than 3 j.ts, ' n 4 x 4 blocks. This
projects to approximately 49 ms for a 512 x 512 binary image, or more than 20
images per second .
4
CONCLUSIONS
In this paper, we have presented a VLSI circuit design for implementing a high
capacity correlation associative memory. The performance of the ECAM chip is
shown to be almost as good as a computer-simulated ECAM . Furthermore, we
believe that the ECAM chip is more robust than an associative memory using a
winner-take-all function, because it obtains its result via iteration, as opposed to
one shot. In conclusion, we believe that the ECAM chip provides a fast and efficient
way for solving many associative recall problems, such as vector quantization and
optical character recognition.
Acknowledgement
This work was supported in part by NSF grant No . MIP - 8711568.
797
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Chiueh and Goodman
References
T. D. Chiueh and R. M. Goodman. (1988) "High Capacity Exponential Associative
Memory," in Proc. of IEEE IeNN, Vol. I, pp. 153-160.
T. D. Chiueh. (1989) "Pattern Classification and Associative Recall by Neural
Networks," Ph. D. dissertation, California Institute of Technology.
C. A. Mead. (1989) Analog VLSI and Neural Systems. Reading, MA : AddisonWesley.
Figure 1: Architecture of the General Correlation-Based Associative Memory
(I<)
u
N-l
X
N-l
Figure 2: The Correlation Computation Circuit
VLSI Implementation or a High-Capacity Neural Network
Voo
X'.
I
I
+
I
Figure 3: The Threshold Circuit
-
Ii
V (1)
ux
(1)
U.
I
V
(2)
ux
(2)
U.
I
?
??
V(M)
ux
(M)
U.
I
Figure 4: The Exponentiation, Multiplication, and Summation Circuit
799
800
Chiueh and Goodman
I 1.
I 1.
r.,
(1<)
U.
1
V(k)
ux
(1<)
RAM
u.1
cell
(1<)
u.1
Figure 5: Circuit Diagram of the Basic ECAM Cell
.....res
1000
G
?
fIl
....
900
8.....
.....
800
.~
0
....0
g
~fIl
~
~
.....0
'"'
Q)
]
Z
700
600
.. Simulation (a=2)
500
... Vbb=5V
400
300
0-
Vbb =2V
-I-
Vbb = lV
200
100
0
0
1
2
3
4
5
6
7
Number of errors in input patterns
Figure 6: Error Correcting Ability of the ECAM Chip with Different
with a Simulated ECAM with a 2
=
VBB
compared
| 217 |@word effect:4 trial:2 implemented:2 divider:1 multiplier:1 build:1 evolution:5 hence:2 direction:2 read:1 open:2 flattens:1 occurs:1 simulation:2 illustrated:1 sgn:2 said:1 pick:1 implementing:1 shot:1 simulated:4 carry:1 capacity:11 m:1 decoder:1 series:1 really:1 degrade:1 complete:1 summation:3 reason:1 performs:1 taiwan:2 current:8 fil:2 length:1 image:4 mo:1 specialized:1 multiplication:4 negative:1 winner:1 designed:2 exponentially:4 analog:1 proc:1 implementation:5 design:7 perform:1 limiter:1 vertical:1 neuron:2 versa:1 enter:1 ith:2 vice:1 dissertation:1 t:1 weighted:1 provides:1 brought:1 node:1 nonlinearity:1 replicating:1 incorporated:1 rather:2 stable:3 voltage:17 differential:1 dominant:1 complement:2 consists:1 namely:1 specified:1 properly:1 connection:3 store:2 california:2 expected:1 binary:4 os:1 sense:1 success:2 address:1 beyond:1 tested:1 greater:1 pattern:31 pasadena:1 vlsi:8 determine:1 reading:1 increasing:2 becomes:2 project:1 monotonically:1 branch:3 ii:2 circuit:23 classification:1 program:1 saturation:1 what:1 memory:29 hybrid:1 match:1 special:2 equal:1 once:1 improve:1 technology:2 controlled:4 variant:1 basic:4 essentially:2 iteration:2 tzi:2 normally:1 grant:1 achieved:1 appear:1 randomly:1 cell:7 acknowledgement:1 positive:1 national:1 engineering:2 addressed:1 drain:4 tends:2 diagram:1 source:6 consisting:2 expect:1 goodman:6 proportional:2 mead:2 posse:2 amplifier:2 conductance:1 versus:1 approximately:2 interest:1 lv:1 switched:1 integrate:1 chiueh:11 call:1 mixture:1 port:2 storing:1 ideal:1 range:1 tj:3 supported:1 testing:1 architecture:3 capable:3 block:1 asynchronous:1 bias:1 prototype:3 institute:2 procedure:1 addisonwesley:1 iv:2 wide:1 synchronous:1 re:1 plotted:1 mip:1 altogether:1 feedback:1 computes:3 made:2 get:1 far:1 storage:2 dar:2 programmable:2 select:1 obtains:1 ignore:1 ph:2 gnd:1 stored:1 summing:2 generate:1 supplied:1 correcting:2 pure:1 nsf:1 combined:1 notice:2 sign:1 per:1 robust:1 write:1 ca:1 enhance:1 connecting:1 together:1 unforeseen:1 vol:1 suppose:1 four:1 again:1 threshold:9 settled:1 quantize:1 opposed:1 recognition:1 updating:1 continues:1 ram:2 asymptotically:2 multiplexer:1 sum:2 bottom:1 run:1 exponentiation:15 electrical:2 includes:1 region:2 almost:1 cycle:1 depends:2 sub:1 decrease:1 later:1 position:1 resistor:2 exponential:5 doing:1 bit:5 reached:3 bound:1 parallel:1 dynamic:1 rodney:1 externally:1 highorder:1 nonnegative:1 theorem:1 solving:1 square:3 occur:1 compromise:1 xor:1 loaded:1 characteristic:1 load:1 subthreshold:2 easily:2 hopfield:1 chip:32 speed:2 argument:1 nmos:5 quantization:2 performing:1 optical:1 fast:1 describe:2 department:2 voo:1 according:2 smaller:1 whose:2 apparent:1 larger:1 against:1 character:1 unity:1 pp:1 otherwise:2 forming:1 vth:3 ability:2 unexpected:1 associated:1 explained:1 gradually:1 static:1 nondecreasing:1 temporarily:1 ux:6 final:2 associative:19 equation:6 recall:6 transistor:12 previously:1 ma:1 consequently:1 product:2 flip:1 actually:1 back:1 fed:1 feed:1 hard:1 change:1 experimentally:1 operation:1 flowing:1 contradictory:1 done:2 furthermore:2 partly:1 correlation:17 convergence:1 until:2 transmission:2 working:1 horizontal:1 hand:2 gate:9 cmos:2 leave:1 nonlinear:1 tk:3 assumes:1 illustrate:2 internal:1 mode:3 arises:1 believe:2 taipei:1 phenomenon:1 usa:1 |
1,285 | 2,170 | Retinal Processing Emulation in a
Programmable 2-Layer Analog Array
Processor CMOS Chip
R. Carmona, F. Jim?
enez-Garrido, R. Dom??nguez-Castro,
S. Espejo, A. Rodr??guez-V?
azquez
Instituto de Microelectr?
onica de Sevilla-CNM-CSIC
Avda. Reina Mercedes s/n 41012 Sevilla (SPAIN)
[email protected]
Abstract
A bio-inspired model for an analog programmable array processor
(APAP), based on studies on the vertebrate retina, has permitted
the realization of complex programmable spatio-temporal dynamics in VLSI. This model mimics the way in which images are processed in the visual pathway, rendering a feasible alternative for
the implementation of early vision applications in standard technologies. A prototype chip has been designed and fabricated in a
0.5?m standard CMOS process. Computing power per area and
power consumption is amongst the highest reported for a single
chip. Design challenges, trade-offs and some experimental results
are presented in this paper.
1
Introduction
The conventional role of analog circuits in mixed-signal VLSI is providing the I/O
interface to the digital core of the chip ?which realizes all the signal processing.
However, this approach may not be optimum for the processing of multi-dimensional
sensory signals, such as those found in vision applications. When massive information flows have to be treated in parallel, it may be advantageous to realize some
preprocessing in the analog domain, at the plane where signals are captured.
During the last years, different authors have focused on the realization of parallel
preprocessing of multi-dimensional signals, using either purely digital techniques [1]
or mixed-signal techniques, like in [2]. The data in Table 1 can help us to compare
these two approaches. Here, the peak computing power (expressed as operations
per second: XPS) per unit area and power is shown. This estimation is realized
by considering the number of arithmetic analog operations that take place per unit
time, in the analog case, or digital instructions per unit time, in the digital case. It
can be seen that the computing power per area featured by chips based in Analog
Programmable Array Processors (APAPs) is much higher than that exhibited by
digital array processors. It can be argued that digital processors feature a larger
accuracy, but accuracy requirements for vision applications are not rarely below 6
Table 1: Parallel processors comparison
Reference
Li?
nan et. al. [2]
Gealow et. al. [1]
This chip
CMOS
process
No.
of cells
Cells/
mm2
XPS/
mm2
XPS/
mW
0.5?m
0.6?m
0.5?m
4096
4096
1024
81.0
66.7
29.2
7.93G
4.00M
6.01G
0.33G
1.00M
1.56G
bits. Also, taking full advantage of the full digital resolution requires highly accurate
A/D converters, what creates additional area and power overhead.
The third row in Table 1 corresponds to the chip presented here. This chip outperforms the one in [2] in terms of functionality as it implements a reduced model
of the biological retina [3]. It is capable of generating complex spatio-temporal dynamic processes, in a fully programmable way and with the possibility of storing
intermediate processing results.
2
2.1
APAP chip architecture
Bio-inspired APAP model
The vertebrate retina has a layered structure [3], consisting, roughly, in a layer of
photodetectors at the top, bipolar cells carrying signals across the retina, affected by
the operation of horizontal and amacrine cells, and ganglion cells in the other end.
There are, in this description, some interesting aspects that markedly resemble
the characteristics of the Cellular Neural Networks (CNNs) [4]: 2D aggregations
of continuous signals, local connectivity between elementary nonlinear processors,
analog weighted interactions between them. Motivated by these coincidences, a
model consisting of 2 layers of processors coupled by some inter-layer weights, and an
additional layer incorporating analog arithmetics, has been developed [5]. Complex
dynamics can be programmed via the intra- and inter-layer coupling strengths and
the relation between the time constants of the layers. The evolution of each cell,
C(i, j), is described by two coupled differential equations, one for each CNN node:
?n
r1
X
dxn,ij
= ?g[xn,ij (t)] +
dt
r1
X
ann,kl ? yn,(i+k)(j+l) +
k=?r1 l=?r1
+bnn,00 ? unn,ij + zn,ij + ano ? yno,ij
(1)
where n and o stand for the node in question and the other node respectively. The
nonlinear losses term and the output function in each layer are those described for
the full-signal range (FSR) model of the CNN [7], in which the state voltage is also
limited and can be identified with the output voltage:
g(xn,ij ) = lim
m??
and:
(
m(xn,ij ? 1) + 1
xn,ij
?m(xn,ij + 1) ? 1
if
xn,ij > 1
if |xn,ij | ? 1
if xn,ij < ?1
(2)
yn,ij = f (xn,ij ) =
1
(|xn,ij + 1| ? |xn,ij ? 1|)
2
(3)
The proposed chip consists in an APAP of 32 ? 32 identical 2nd-order CNN cells
(Fig. 3), surrounded by the circuits implementing the boundary conditions.
Figure 1: (a) Conceptual diagram of the basic cell and (b) internal structure of each
CNN layer node
2.2
Basic processing cell architecture
Each elementary processor includes two coupled continuous-time CNN cores
(Fig. 1(a)). The synaptic connections between processing elements of the same
or different layer are represented by arrows in the diagram. The basic processor
contains also a programmable local logic unit (LLU) and local analog and logic
memories (LAMs and LLMs) to store intermediate results. The blocks in the cell
communicate via an intra-cell data bus, multiplexed to the array interface. Control
bits and switch configuration are passed to the cell from a global programming unit.
The internal structure of each CNN core is depicted in the diagram of Fig. 1(b).
Each core receives contributions from the rest of the processing nodes in the neighbourhood which are summed and integrated in the state capacitor. The two layers
differ in that the first layer has a scalable time constant, controlled by the appropriate binary code, while the second layer has a fixed time constant. The evolution
of the state variable is also driven by self-feedback and by the feedforward action of
the stored input and bias patterns. There is a voltage limiter for implementing the
FSR CNN model. Forcing the state voltage to remain between these limits allows
for using it as the output voltage. Then the state variable, which is now the output, is transmitted in voltage form to the synaptic blocks, in the periphery of the
cell, where weighted contributions to the neighbours? are generated. There is also a
current memory that will be employed for cancellation of the offset of the synaptic
blocks. Initialization of the state, input and/or bias voltages is done through a
mesh of multiplexing analog switches that connect to the cell?s internal data bus.
3
3.1
Analog building blocks for the basic cell
Single-transistor synapse
The synapse is a four-quadrant analog multiplier. Their inputs will be the cell
state, or input, and the weight voltages, while the output will be the cell?s current
contribution to a neighbouring cell. It can be realized by a single transistor biased
in the ohmic region [6]. For a PMOS with gate voltage VX = Vx0 + Vx , and the
p-diffusion terminals at VW = Vw0 + Vw and Vw , the drain-to-source current is:
Vw
Io ? ??p Vw Vx ? ?p Vw Vx0 + |V?Tp | ? Vw0 ?
2
(4)
which is a four-quadrant multiplier with an offset term that is time-invariant ?at
least during the evolution of the network? and not depending on the state. This
offset is eliminated in a calibration step, with a current memory.
For the synapse to operate properly, the input node of the CNN core,
L in Fig. 2,
must be kept at a constant voltage. This is achieved by a current conveyor. Any
difference between the voltage at node
L and the reference V w0 is amplified and
the negative feedback corrects the deviation. Notice that a voltage offset in the
amplifier results in an error of the same order. An offset cancellation mechanism is
provided (Fig. 2).
3.2
S3 I current memory
As it has been referred, the offset term of the synapse current must be removed
for its output current to represent the result of a four-quadrant multiplication. For
this purpose all the synapses are reset to VX = Vxo . Then the resulting current,
which is the sum of the offset currents of all the synapses concurrently connected
to the same node, is memorized. This value will be substracted on-line from the
input current when the CNN loop is closed, resulting in a one-step cancellation of
the errors of all the synapses. The validity of this method relies in the accuracy of
the current memory. For instance, in this chip, the sum of all the contributions will
range, for the applications for which it has been designed, from 18?A to 46?A. On
the other side, the maximum signal to be handled is 1?A. If a signal resolution of
8b is pretended, then 0.5LSB = 2nA. Thus, our current memory must be able to
distinguish 2nA out of 46?A. This represents an equivalent resolution of 14.5b. In
order to achieve such accuracy level, a S3 I current memory is used. It is composed by
three stages (Fig. 2), each one consisting in a switch, a capacitor and a transistor. I B
is the current to be memorized. After memorization the only error left corresponds
to the last stage.
3.3
Time-constant scaling
The differential equation that governs the evolution of the network (1) can be
written as a sum of current contributions injected to the state capacitor. Scaling up/down this sum of currents is equivalent to scaling the capacitor and, thus,
speeding up/down the network dynamics. Therefore, scaling the input current with
the help of a current mirror, for instance, will have the effect of scaling the timeconstant. A circuit for continuously adjusting the current gain of a mirror can be
designed based on a regulated-Cascode current mirror in the ohmic region. But the
strong dependence of the ohmic-region biased transistors on the power rail voltage
causes mismatches in ? between cells in the same layer. An alternative to this is
a digitally programmable current mirror. It trades resolution in ? for robustness,
hence, the mismatch between the time constants of the different cells is now fairly
attenuated.
Figure 2: Input block with current scaling, S3 I memory and offset-corrected OTA
schematic
A new problem arises, though, because of current scaling. If the input current
can be reshaped to a 16-times smaller waveform, then the current memory has to
operate over a larger dynamic range. But, if designed to operate on large currents,
the current memory will not work for the tiny currents of the scaled version of the
input. If it is designed to run on small input currents, long transistors will be needed,
and the operation will be unreliable for the larger currents. One way of avoiding
this situation is to make the S3 I memory to work on the original unscaled version
of the input current. Therefore, the adjustable-time-constant CNN core will be a
current conveyor, followed by the S3 I current memory and then the binary weighted
current mirror. The problem now is that the offsets introduced by the scaling block
add up to the signal and the required accuracy levels can be lost. Our proposal is
depicted in Fig. 2. It consists in placing the scaling block (programmable mirror)
between the current conveyor and the current memory. In this way, any offset error
will be cancelled in the auto-zeroing phase. In the picture, the voltage reference
generated with the current conveyor, the regulated-Cascode current mirrors and
the S3 I memory can be easily identified. The inverter, Ai , driving the gates of the
transistors of the current memory is required for stability.
4
Chip data and experimental results
A prototype chip has been designed and fabricated in a 0.5?m single-poly triplemetal CMOS technology. Its dimensions are 9.27 ? 8.45mm2 (microphotograph in
Fig. 3). The cell density achieved is 29.24cells/mm2, once the overhead circuitry is
detracted from the total chip area ?given that it does not scale linearly with the
number of cells. The power consumption of the whole chip is around 300mW. Data
I/O rates are nominally 10MS/s. Equivalent resolution for the analog images handled by the chip is 7.5 bit (measured). The time constant of the fastest layer (fixed
time constant) is intended to be under 100ns. The peak computing power of this
chip is, therefore, 470GXPS, what means 6.01GXPS/mm2 , and 1.56GXPS/mW.
Figure 3: Prototype chip photograph
The programmable dynamics of the chip permit the observation of different phenomena of the type of propagation of active waves, pattern generation, etc. By
tuning the coefficients that control the interactions between the cells in the array?
i. e. the weights of the synaptic blocks, which are common to every elementary
processor? different dynamics are manifested. Fig. 4 displays the evolution of the
state variables of the two coupled layers when it is programmed to show different
propagative behaviors. In picture (a), the chip is programmed to resemble the socalled wide-field erasure effect observed in the retina. Markers in the fastest layer
(bottom row) trigger wavefronts in this layer and induce slower waves in the other
layer (upper row). These induced spots are fedback, inhibiting the waves propagating in the fast layer, and generating a trailing edge for each wavefront. In picture
(b), a solitary traveling wave is triggered from each corner of the fast layer. This
kind of behavior is proper of waves in active media. Finally, in picture (c), edge
detection is computed by extraction the low frequency components of the image,
obtained by a diffusion in the slower layer, from theoriginal one. The remaining
information is that of the higher frequency components of the image. These phenomena have been widely observed in measurements of the vertebrate retina [3].
They constitute the patterns of activity generated by the presence of visual stimuli.
Controlling the network dynamics and combining the results with the help of the
built-in local logic and arithmetic operators, rather involved image processing tasks
can be programmed like active-contour detection, object-tracking, etc.
5
Conclusions
From the figures obtained, we can state that the proposed approach supposes a
promising alternative to conventional digital image processing for applications re-
lated with early-vision and low-level focal-plane image processing. Based on a simple
but precise model of part of the real biological system, a feasible efficient implementation of an artificial vision device has been designed. The peak operation speed
of the chip outperforms its digital counterparts due to the fully parallel nature of
the processing. This especially so when comparing the computing power per silicon
area unit and per watt.
Acknowledgments
This work has been partially supported by ONR/NICOP Project N00014-00-1-0429,
ESPRIT V Project IST-1999-19007, and by the Spanish CICYT Project TIC-19990826.
References
[1] Gealow, J.C. & Sodini, C.G. (1999) A Pixel Parallel Image Processor Using
Logic Pitch -Matched to Dynamic Memory. IEEE Journal of Solid-State Circuits, Vol. 34, No. 6, pp. 831-839.
[2] Li?
nan, G., Espejo, S., Dom??nguez-Castro, R., Roca, E. and Rodr??guezV?
azquez, A. (1998) A 64 x 64 CNN with Analog and Digital I/O. Proceedings of the IEEE Int. Conf. on Electronics, Circuits and Systems, pp. 203-206,
Lisbon, Portugal.
[3] Werblin, F. (1991) Synaptic Connections, Receptive Fields and Patterns of Activity in the Tiger Salamander Retina. Investigative Ophthalmology and Visual
Science, Vol. 32, No. 3, pp. 459-483.
[4] Werblin, F., Roska, T. and Chua, L.O. (1995) The Analogic Cellular Neural
Network as a Bionic Eye. International Journal of Circuit Theory and Applications, Vol. 23, No. 6, pp. 541-69.
[5] Rekeczky, Cs., Serrano-Gotarredona, T., Roska, T. and Rodr??guez-V?
azquez, A.
(2000) A Stored Program 2nd Order/3- Layer Complex Cell CNN-UM. Proc.
of the Sixth IEEE International Workshop on Cellular Neural Networks and
their Applications, pp. 219-224, Catania, Italy.
[6] Dom??nguez-Castro, R., Rodr??guez-V?
azquez, A., Espejo, S. and Carmona, R.
(1998) Four-Quadrant One-Transistor Synapse for High Density CNN Implementations. Proc. of the Fifth IEEE International Workshop on Cellular Neural
Networks and their Applications, pp. 243-248, London, UK.
[7] Espejo, S., Carmona, R. Carmona, Dom??nguez-Castro, R. and Rodr??guezV?
azquez, A. (1996) A VLSI Oriented Continuous- Time CNN Model. International Journal of Circuits Theory and Applications, Vol. 24, No. 3, pp. 341-356,
John Wiley and Sons Ed.
Figure 4: Examples of the different dynamics that can be programmed on the chip:
(a) wide-field erasure effect, (b) traveling wave accross the layers, and (c) edge
detection.
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1,286 | 2,171 | Reinforcement Learning to Play an Optimal
Nash Equilibrium in Team Markov Games
Xiaofeng Wang
ECE Department
Carnegie Mellon University
Pittsburgh, PA 15213
[email protected]
Tuomas Sandholm
CS Department
Carnegie Mellon University
Pittsburgh, PA 15213
[email protected]
Abstract
Multiagent learning is a key problem in AI. In the presence of multiple Nash equilibria, even agents with non-conflicting interests may not
be able to learn an optimal coordination policy. The problem is exaccerbated if the agents do not know the game and independently receive
noisy payoffs. So, multiagent reinforfcement learning involves two interrelated problems: identifying the game and learning to play. In this paper,
we present optimal adaptive learning, the first algorithm that converges
to an optimal Nash equilibrium with probability 1 in any team Markov
game. We provide a convergence proof, and show that the algorithm?s
parameters are easy to set to meet the convergence conditions.
1 Introduction
Multiagent learning is a key problem in AI. For a decade, computer scientists have worked
on extending reinforcement learning (RL) to multiagent settings [11, 15, 5, 17]. Markov
games (aka. stochastic games) [16] have emerged as the prevalent model of multiagent RL.
An approach called Nash-Q [9, 6, 8] has been proposed for learning the game structure
and the agents? strategies (to a fixed point called Nash equilibrium where no agent can
improve its expected payoff by deviating to a different strategy). Nash-Q converges if a
unique Nash equilibrium exists, but generally there are multiple Nash equilibria. Even
team Markov games (where the agents have common interests) can have multiple Nash
equilibria, only some of which are optimal (that is, maximize sum of the agents? discounted
payoffs). Therefore, learning in this setting is highly nontrivial.
A straightforward solution to this problem is to enforce convention (social law). Boutilier
proposed a tie-breaking scheme where agents choose individual actions in lexicographic
order[1]. However, there are many settings where the designer is unable or unwilling
to impose a convention. In these cases, agents need to learn to coordinate. Claus and
Boutilier introduced fictitious play, an equilibrium selection technique in game theory, to
RL. Their algorithm, joint action learner (JAL) [2], guarantees the convergence to a Nash
equilibrium in a team stage game. However, this equilibrium may not be optimal. The
same problem prevails in other equilibrium-selection approaches in game theory such as
adaptive play [18] and the evolutionary model proposed in [7].
In RL, the agents usually do not know the environmental model (game) up front and receive
noisy payoffs. In this case, even the lexicographic approaches may not work because agents
receive noisy payoffs independently and thus may never perceive a tie. Another significant
problem in previous research is how a nonstationary exploration policy (required by RL) affects the convergence of equilibrium selection approaches?which have been studied under
the assumption that agents either always take the best-response actions or make mistakes at
a constant rate. In RL, learning to play an optimal Nash equilibrium in team Markov games
has been posed as one of the important open problems [9]. While there have been heuristic
approaches to this problm, no existing algorithm has been proposed that is guarenteed to
converge to an optimal Nash equilibrium in this setting.
In this paper, we present optimal adaptive learning (OAL), the first algorithm that converge
to an optimal Nash equilibrium with probability 1 in any team Markov game (Section 3).
We prove its convergence, and show that OAL?s parameters are easy to meet the convergence conditions (Section 4).
2 The setting
2.1 MDPs and reinforcement learning (RL)
In a Markov decision problem, there is one agent
in the environment.
A fully observable
where
is
a
finite
state space;
Markov decision problem (MDP) is a tuple
is the space of actions the agent can take;
is a payoff function ( ' ()+*-,
!"#$%&%
is the expected payoff .for
taking
action
in
state
);
and
is
/ 0 213
41
a transition
function
(
is
the
probability
of
ending
in
state
,
given
that
ac
tion is taken in state ). An agent?s deterministic
policy (aka. strategy) is a mapping
6
from states to actions. We denote by 5
the action that :Apolicy
5 prescribes in state .
@ CB
B
:ED 5 , where : is the pay5 that maximizes 79:<8 ;=?>
The objective is to find a' ()policy
E*2
H
>
G
off at time F , and
is a discount factor. There exists a deterministic optimal
5JI [12]. The Q-function for this policy, K
I , is defined by the set of equations
policy
/ NM/ PO
L 0 1 VUXWZY)[
1 1
R S \ KLI
> 7QARCS/T
KLI
. At any state , the optimal
W ]^UXWZY_[
/
KLI
policy chooses
[10].
Reinforcement learning
can
be
viewed
as a sampling method for
estimating KI when the
/
payoff function and/or
transition
function
are
unknown.
KI
can be approximated
/
by a function K :
calculated from the agent?s experience
up
to
time F . The model
based
approach
uses
samples
to
generate
models
of
and
,
and
then
iteratively
computes
/ V `M
/ PO
/ ab 1 VUXWZY [
1 1
:
:
RcS \ K :edf
> 7
.
K :
Q R S/T
Based on K : , a learning policy assigns probabilities to actions at each state. If the learning
policy has the ?Greedy in the Limit with Infinite Exploration? (GLIE) property, then K :
will converge to K.I (with either a model-based or model-free approach) and the agent will
converge in behavior to an optimal policy [14]. Using GLIE, every state-action pair is
visited infinitely often, and in the limit the action selection is greedy with respect to the
Q-function w.p.1. One common GLIE policy is Boltzmann exploration [14].
2.2 Multiagent RL in team Markov games when the game is unknown
A natural extension of an MDP to multiagent environments is a Markov game (aka.
stochastic game) [16]. In this paper we focus on team Markov games, that are Markov
games where each agent receives the same expected payoff (in the presence of noise, different agent may still receive different payoffs at a particular moment.). In other words,
there are no conflicts between the agents, but learning the game structure and learning to
coordinate are nevertheless highly nontrivial.
g is a tuple
Definition
1 A team Markov
game (aka identical-interest stochastic j
game)
Chi
h
M`lk
;mfonpnpn q is a
, where is a setr
ofsn9agents;
S
is
a
finite
state
space;
tuv
joint w
action
space of n agents;
is the common expected payoff function;
Vxy#z{|' (_E*E,
and
is a transition function.
The objective of the
} agents is to find a deterministic joint policy (aka. joint strategy aka.
k
k ??? k
M#~
/{?
strategy profile) 5
5 ;Jfbnpnpn q0 (where 5
and 5
) so as to maximize the
/
expected sum of their discounted payoffs. The Q-function, K
, is the expected sum
of discounted payoffs given that the agents
play
joint
action
in
state
and follow policy
/ V
5 thereafter. The optimal K -function K.I
is the K -function for (each) optimal policy
5mI . So, KLI captures the game structure. The agents generally do not know KI in advance.
Sometimes, they know neither the payoff structure nor the transition probabilities.
f
~
k
if each individual policy1 is a best
response
A joint policy 5 ;mfonpnpn q is a Nashh equilibrium
/-~
k 2
k
G
G
5 , KLI
5
to the
others.
That
is,
for
all
,
and
any
individual
policy
k 2
k 2
k
/-~
1 2
k
d
d
d
5
KLI
5
5
, where 5
is the joint policy of all agents except
agent
.
(Likewise,
throughout
the
paper,
we
use
to denote all agents but , e.g., by
k
k
d
their joint action, by d their joint action set.) A Nash equilibrium is strict if the
inequality above is strict. An optimal Nash equilibrium 5 I is a Nash equilibrium that gives
the agents the maximal expected sum of discounted payoffs. In team games, each optimal
Nash equilibrium is an optimal joint policy (and there are no other optimal joint policies).
/
/ 1
1
G
A joint
action is optimal in state if
for all
. If we treat
K.I
KLI
/
as the payoff of joint action in state , we obtain a team game
in
matrix
form.
K I
We call such a game a state game for . An optimal joint action in is an optimal Nash
equilibrium of that state game. Thus, the task of optimal coordination in a team Markov
game boils down to having all the agents play an optimal Nash equilibrium in state games.
f f
f
f
f
f
However, a coordination problem arises if there are multiple Nash equilibria. The 3-player
[
[
[
f
10
-20
-20
-20
-20
5
-20
5
-20
-20
-20
5
-20
10
-20
5
-20
-20
-20
5
-20
5
-20
-20
-20
-20
10
Table 1: A three-player coordination game
game in Table 1 has three optimal Nash equilibria and six sub-optimal Nash equilibria. In
this game, no existing equilibrium selection algorithm (e.g.,fictitious play [3]) is guarenteed
to learn to play an optimal Nash equilibrium. Furthermore, if the payoffs are only expectations over each agent?s noisy payoffs and unknown to the agents before playing, even
identification of these sub-optimal Nash equilibria during learning is nontrivial.
3 Optimal adaptive learning (OAL) algorithm
We first consider the case where agents know the game before playing.
This enables the
learning agents to construct a virtual game (VG) for each state of the team Markov
game to eliminate all the strict suboptimal Nash equilibria in that state. Let I
be the/payoff
that
the
agents
receive from/the
VG in state for
a joint
action . We let
V yM
*
9M
WZ] ^`UXW4Y)[
1
M
(
R S \ KLI
LI
if
and LI
otherwise. For example,
the VG~6 for the
game in~6Table
1 gives payoff 1 for each optimal Nash equilibrium
~6 0* 2* 4*
) )
,
, and
), and payoff 0 to every other joint action. The
(
VG in this example is weakly acyclic.
Definition 2 (Weakly acyclic game [18]) Let be an n-player
game in matrix form. The
G
best-response
graph
of
takes
each
joint
action
as
a
vertex
and connects two
1
1
M
1
vertices and with a directed
edge
if
and
only
if
1)
; 2) there exists
k
k
k1
1 k M9
d . We say the game
exactly one agent such that is a best response to d and d
is weakly acyclic if in its best-response graph, from any initial vertex , there exists a
directed path to some vertex I from which there is no outgoing edge.
To tackle the equilibrium selection problem for weakly acyclic games, Young [18]
pro :
G
posed a learning algorithm called adaptive play (AP), which works as follows. Let
be a joint action played at time F over an n-player game in matrix form. Fix integers
Throughout the paper, every Nash equilibrium that we discuss is also a subgame perfect Nash equilibrium. (This refinement
of Nash equilibrium was first introduced in [13] for different games).
. When , each agent randomly chooses its
agent looks back at the most recent plays
and, each
randomly
(without replacement) selects samples
from . Let
be the number of times that a reduced joint action
(a
joint action without agent ?s individual action) appears in the samples at
. Let
be agent ?s payoff given that joint action has been played. Agent calculates its expected
! ,
payoff w.r.t its individual action
as
and then randomly
an action from a set of best responses: "
# chooses
.
*
and
such that
actions.
Starting
from
mf
M? :ed
:ed
:
:
: mf
C
F
M
O
F
*
- :ed?f
d
k
F
@
k
WZ] ^`UXW4Y [
R S \
@
1k
k
d
O{*
C Vke yM
7
[
k ~6 Vk
S \
M
k
d
k
[
ke
d
k:
G
~2 ?k
VkM
D
Young showed that AP in a weakly acyclic game converges to a strict Nash equilibrium
w.p.1. Thus, AP on the VG for the game in Table 1 leads to an equilibrium with payoff 1
which is actually an optimal Nash equilibrium for the original game. Unfortunately, this
does not extend to all VGs because not all VGs are weakly acyclic: in a VG without any
strict Nash equilibrium, AP may not converge to the strategy profile with payoff 1.
In order to address more general settings, we now modify the notion of weakly acyclic
game and adaptive play to accommodate weak optimal Nash equilibria.
Definition 3 (Weakly acyclic game w.r.t a biased set (WAGB)): Let be a set containing some of the Nash equilibria of a game (and no other joint policies). Game is a
WAGB if, from any initial vertex , there exists a directed path to either a Nash equilibrium
inside or a strict Nash equilibrium.
We can convert any VG to a WAGB by setting the biased set to include all joint policies that give payoff 1 (and no other joint policies). To solve such a game, we introduce
a new learning algorithm for equilibrium selection. It enables each agent to deterministically select a best-response action once any Nash equilibrium in the biased set is attained
(even if there exist several best responses when the Nash equilibrium is not strict). This is
different from AP where players randomize their action selection when there are multiple
best-response actions. We call our approach biased adaptive play (BAP). BAP works as
follows. Let
be the biased set composed of some Nash equilibria of a game in matrix
: be the set of
form. Let
samples drawn at time F , without replacement,
from among
1
G
joint
actions.
If
(1)
there
exists
a
joint
action
such
that for all
the
most
recent
k
k
1
: , d
d
G
and
,
and
(2)
there
exists
at
least
one
joint
action
such
that
k
k
: R
: and
G
G
G
,
then
agent
chooses
its
best-response
action
such
that
and
UXW4Ys~6
Vk
1 M
. That is,
D
:
G
G
F
is contained in the most recent play of
a Nash equilibrium inside . On the other hand, if the two conditions above are not met,
then agent chooses its best-response action in the same way as AP. As we will show, BAP
(even with GLIE exploration) on a WAGB converges w.p.1 to either a Nash equilibrium in
or a strict Nash equilibrium.
So far we tackled learning of coordination in team Markov games where the game structure
is known. Our real interests are in learning when the game is unknown. In multiagent
:
reinforcement learning, K
I /is asymptotically approximated with K : . Let
be
: so as to assure
the virtual
game
w.r.t
.
Our
question
is
how
to
construct
K :
:
LI w.p.1. Our method of achieving this makes use of the notion of -optimality.
$
$
$
$
&%
( $
$
'%
*) ( $
$
$
+
+
+
+
+
Definition
4 Let beUXaW4Y)positive
constant. A joint
action a is -optimal at state s and time
/ `O
[
1
1
R K :
G
t if K :
for
all
. We denote the set of -optimal joint
2
actions at state s and time t as
.
2
The idea is to use a decreasing -bound : to estimate
at state and time F . All the
joint actions belonging to the set are treated as optimal Nash equilibria in the virtual game
: which give agents payoff 1. If : converges to zero at a rate slower than K : converges
&,
+
+
+
&,
+
"
' (_E*E,
:
:
G
to KLI , then
LI w.p.1. We make : proportional to a function
:
:
which decreases slowly and monotonically to zero with , where
is the smallest number of times that any state-action pair has been sampled so far. Now, we are ready to present
the entire
optimal
adaptive learning (OAL) algorithm. As we will present thereafter, we
: carefully using an understanding of the convergence rate of a model-based RL
craft
algorithm that is used to learn the game
structure.
k
"
Optimal adaptive learning algorithm (for agent )
,
:J;
=
1. Initialization
;
; \
Q
;
. For all Q ST
\ . ; \ .
[
S
\
and
:
do
q
Q
[
;{f
,
(
[
Q
;
Q R
2. Learning of coordination policy If
, randomly select
[ an action,
[ otherwise do
;yf
Q
[
[
(a) Update the virtual
game
at state Q :
if S \
Q and
;
;
f
Q
Set
.
(b) According to GLIE exploration, with an exploitation probability do
!
Q
and
[
Q
;X=
[
;=
.
otherwise.
i. Randomly select (without replacement) records from
recent observations of others? joint actions
played at state Q .
[
[
Q
)
(
ii. Calculate expected payoff
of individual action '& over
the virtual game
at current state Q
[
[
[
as follows:
Q
;
# %$
"!
7
:
;
response set at state Q and time : ,-
Q
*
[ [
Q
;/.021435.6
+
R
. Construct the best-
"!
Q
[
R
.
iii. If conditions 1) and 2) of BAP are met, choose a best-response action with respect to the biased set .
Q .
Otherwise, randomly select a best-response action from ,-
Otherwise, randomly select an action to explore.
( ( (
(
(
(
,,
,
is the number of times a joint action has been played in state by time .
Here,
is a positive constant (any value works).
is the number of times that a joint
action
appears in agent ?s samples (at time ) from the most recent joint
3. Off-policy learning of game structure 798
[
Q:
Q R and payoff ; under the joint action . Do
(a) Observe state
transition
[ 4
<
[
q
q
.
f
Q [ 4<
Q= [
i.
.
[
(
d
Q
Q
Q
ii.
.[
;
>
$ '&
[
4<
[
M
Q
AQ R
>
/
} :
Q R
( f?d
$ '& [
4<
?A; @ Q R do
Q= Q%?
[
ED
CB
Q= Q R
R
7
[
"D
! q
I
Q
&
.
Q Q R .
(
fd
Q
i
?
S
T
Q
iv. For all
and
> )$ %&
[ 4<
[
35.6 "!
Q=
Q
R
(b) 7
7
<
<
:
: .f
35G H
(c)
.F
(d) If J < , F
(see Section 4.2 for the construction of , F
)
, F [ .
i.
[
ii. Update4
using (b). I!
7 < Q [K for all[ Q
[
Q=
Q= R .
L 35.6 R
7
7
iii. \
Q
Q=
iii.
d
k
G
d
:
k
d
k
Q R
[
[
Q
R
eQ ?
.
.
F
F
actions taken in state .
4 Proof of convergence of OAL
In this section, we prove that OAL converges to an optimal Nash equilibrium. Throughout,
we make the common RL assumptions: payoffs are bounded, and the number of states
and actions is finite. The proof is organized as follows. In Section 4.1 we show that OAL
agents learn optimal coordination if the game is known. Specifically, we show that BAP
against a WAGB with known game structure converges to a Nash equilibrium under GLIE
exploration. Then in Section 4.2 we show that OAL agents will learn the game structure.
Specifically, any virtual game can be converted to a WAGB which will be learned surely.
Finally, these two tracks merge in Section 4.3 which shows that OAL agents will learn the
game structure and optimal coordination. Due to limited space, we omit most proofs. They
can be found at: www.cs.cmu.edu/? sandholm/oal.ps.
4.1 Learning to coordinate in a known game
In this section, we first model our biased adaptive play (BAP) algorithm with best-response
action selection as a stationary Markov chain. In the second half of this section we then
model BAP with GLIE exploration as a nonstationary Markov chain.
4.1.1 BAP as a stationary Markov chain
B
M
Consider
BAP
with
randomly selected initial plays. We take the initial
F
C f
+
as the initial state of the Markov
chain. The definition of the other states is
1
inductive: A successor of state is any state obtained by deleting the left-most element
M
of
and3appending
a new right-most element. The only exception is that all the states
C 0 a
with being either a member of the biased set or a strict Nash equilibrium
are grouped into a unique terminal state
. Any state directing to G
is treated as
directly connected to .
1
Let be
the~6state
transition
matrix of the above Markov chain. Let be a successor of ,
M
-
f
q ( } players) be the new element that was appended to the right
and let
1
(
1
(
R
R
of to get . Let
be the transition probability from to . Now,
k
if and only if for each agent , there exists a sample of size in to which is ?s best
response according to the action-selection rule of BAP. Because agent chooses such a
sample with a probability independent of time F , the Markov chain is stationary. Finally,
due to our clustering of
multiple states into a terminal state
, for any state connected
M
R
7
R
to , we have
.
S
In the above model, once the system reaches the terminal state, each agent?s best response
is M to Crepeat
its
most recent action. This is straightforward if in the actual terminal state
0 0
(which is one of the states that were clustered to form the terminal
state),
is a strict Nash equilibrium. If is only a weak Nash equilibrium (in this case, G
), BAP
biases each agent to choose its most recent action because conditions (1) and (2) of BAP are
satisfied. Therefore, the terminal state
is an absorbing state of the finite Markov chain.
On the other hand, the above analysis shows that
essentially is composed of multiple
absorbing states. Therefore, if agents come into , they will be stuck in a particular state
forever instead of cycling around multiple states in .
in
Theorem 1 Let G be a weakly acyclic game w.r.t. a biased set D. Let L(a) be the length
of the shortest directed path in the best-response graph
of G Cfrom
a joint action
a to
either
M
UXW4Y [
O
an absorbing vertex or a vertex in D, and let
. If
, then,
w.p.1, biased adaptive play in G converges to either a strict Nash equilibrium or a Nash
equilibrium in D.
O
Theorem
1 says that the stationary Markov chain for BAP in a WAGB (given
) has a unique stationary distribution in which only the terminal state appears.
4.1.2 BAP with GLIE exploration as a nonstationary Markov chain
Without knowing game structure, the learners need to use exploration to estimate their payoffs. In this section we show that such exploration does not hurt the convergence of BAP.
We show this by first modeling BAP with GLIE exploration as a non-stationary Markov
chain.
(
$
(
(
(
(
(
$
(
(
(
(
With GLIE exploration, at every time step F , each joint action occurs with positive probability. This means that the system transitions from the state it is in to any of the successor
states with positive probability. On the other hand, the agents? action-selection becomes
increasingly greedy over time. In the limit, with probability one, the transition probabilities
converge to those of BAP with no exploration.
Therefore, we can
model the
learning pro~
U
M
:
cess with a sequence of transition matrices : :<8 ;Jf such that
: :
, where is
8
the transition matrix of the stationary Markov chain describing BAP without exploration.
Akin to how we modeled BAP as a stationary Markov chain above, Young modeled adaptive play (AP) as a stationary
Markov chain [18]. There are two differences. First, unlike AP?s, BAP?s action selection is biased. Second, in Young?s model, it
is possible to have several absorbing states while in our model, at most one absorbing state exists (for any team game, our model
has exactly one absorbing state). This is because we cluster all the absorbing states into one. This allows us to prove our main
convergence theorem.
Our objective here is to show that on a WAGB, BAP with GLIE exploration will converge
to the (?clustered?) terminal state. For that, we use the following lemma (which is a combination of Theorems V4.4 and V4.5 from [4]).
matrix of a stationary Markov chain with a unique
Lemma 2 Let be the finite transition
~
: :<;mf be a sequence of finite transition matrices. Let
stationary distribution . Let
8
M
U
M
: :
:
. If
beU a probability
vector and denote
, then
M
(
8
:
for all .
8
Using this lemma and Theorem 1, we can prove the following theorem.
Theorem
3 O (BAP
with GLIE) On a WAGB G, w.p.1, BAP with GLIE exploration (and
/
) converges to either a strict Nash equilibrium or a Nash equilibrium in D.
4.2 Learning the virtual game
So far, we have shown that if the game structure is known in a WAGB, then BAP will
converge to the terminal state. To prove optimal convergence of the OAL algorithm, we
need to further demonstrate that 1) every virtual game is a WAGB, and 2) in OAL, the
: will converge to the ?correct? virtual game
?temporary? virtual game
I w.p.1.
The first of these two issues is handled by the following lemma:
Lemma 4 The virtual game VG of any n-player team state game is a weakly acyclic game
w.r.t a biased set that contains all the optimal Nash equilibria, and no other joint actions.
(By the definition of a virtual game, there are no strict Nash equilibria other than optimal
ones.) The length of the shortest best-response path
} .
Lemma 4 implies that BAP in a known virtual game with GLIE exploration will converge
to an optimal Nash equilibrium. This is because (by Theorem 3) BAP in a WAGB will
converge to either a Nash equilibrium in a biased set or a strict Nash equilibrium, and
(by Lemma 4) any virtual game is a WAGB with all such Nash equilibria being optimal.
$
The following two lemmas are the last link of our proof chain. They show that OAL will
cause agents to obtain the correct virtual game almost surely.
,
Lemma 5 In any team Markov game, (part 3 of) OAL assures that as F
UXW4Y
Q S/T
[
S \
DK
:
?KLI
/ V
D
'
M
1
F
1
F
for some constant
M
(
w.p.1.
Using Lemma 5, the following lemma is easy to prove.
1
M
: R
the
event that for all F F ,
Lemma 6 Consider any team Markov game. Let : be
:
LI in the OAL algorithm in a given state. If 1)
decreases monotonically to zero
(
U
: :
8
"
`M9(
:
), and 2)
U
: :
8
1
,
"
F
1
F
F
M9(
, then
U
: :
8
B?~
+
:
M#*
.
Lemma 6 states that if the criterion for including a joint action among the -optimal joint
actions in OAL is not made strict too quickly (quicker than the iterated logarithm), then
agents will identify all optimal joint actions with probability one. In this case, they set up
the correct virtual
game. It is easy to make OAL satisfy this condition. E.g., any function
d
f
NM
(
:
:
, will do.
4.3 Main convergence theorem
Now we are ready to prove that OAL converges to an optimal Nash equilibrium in any
team Markov game, even when the game structure is unknown. The idea is to show that
the OAL agents learn the game structure (VGs) and the optimal coordination policy (over
these VGs). OAL tackles these two learning problems simultaneously?specifically, it
interleaves BAP (with GLIE exploration) with learning of game structure. However, the
convergence proof does not make use of this fact. Instead, the proof proceeds by showing
that the VGs are learned first, and coordination second (the learning algorithm does not
even itself know when the switch occurs, but it does occur w.p.1).
"
"
Theorem
7 (Optimal
convergence)
In any team Markov game among } agents, if (1)
O
: satisfies Lemma 6, then the OAL algorithm converges to an
, and (2)
}
optimal Nash equilibrium w.p.1.
Proof. According to [1], a team Markov game can be decomposed into a sequence of state
games. The optimal equilibria of these state games form the optimal policy 5I for the
game. By the definition of GLIE exploration, each state in the finite state space will be
visited infinitely often w.p.1. Thus, it is sufficient to only prove that the OAL algorithm
will converge to the optimal policy over individual state games w.p.1.
M
1
: R
F . Let f be any positive
Let : be the event that
LI at that state for all F
L
f
constant. If B?Condition
(2) of the theorem
is satisfied, by Lemma 6 there exists a time
~
*
.
:b
f if F
f .
such that
If : occurs and Condition (1) of the theorem is satisfied, by Theorem 3, OAL will converge
to either a strict Nash equilibrium or a Nash equilibrium in the biased set w.p.1. Furthermore, by Lemma 4, we know that the biased set contains all of the optimal Nash equilibria
(and nothing else), and there are no strict Nash equilibria outside the biased set. Therefore,
if : occurs, then OAL converges to an optimal Nash equilibrium w.p.1. Let
be any
positive constant, and
let
be
the
event
that
the
agents
play
an
optimal
joint
action
at a
1
given state for all
F
.
With
this
notation,
we
can
reword
the
previous
sentence:
there
B?~
*
exists a time
.
F such that if
F , then
D :
+
+
+
+
+
+
+
+
B?~
Put
there exists
a time
+ f +
such that if
+ f +
, then
B?~ together,
B?~
*
- *
*
:
D :
+ f +
+ f +
. Because + f and +
are only used
in the proof (they are not parameters of the OAL algorithm), we can choose them to be
arbitrarily small. Therefore, OAL converges to an optimal Nash equilibrium w.p.1.
5 Conclusions and future research
With multiple Nash equilibria, multiagent RL becomes difficult even when agents do not
have conflicting interests. In this paper, we present OAL, the first algorithm that converges
to an optimal Nash equilibrium with probability 1 in any team Markov game. In the future
work, we consider extending the algorithm to some general-sum Markov games.
Acknowledgments
Wang is supported by NSF grant IIS-0118767, the DARPA OASIS program, and the PASIS
project at CMU. Sandholm is supported by NSF CAREER Award IRI-9703122, and NSF
grants IIS-9800994, ITR IIS-0081246, and ITR IIS-0121678.
References
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[2]
[3]
[4]
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[14]
[15]
[16]
[17]
[18]
! !
Theorem 3 requires
L
Lemma 4, we do have
L
.
. If Condition (1) of our main theorem is satisfied (
! '
L
q
), then by
| 2171 |@word exploitation:1 open:1 hu:1 accommodate:1 moment:1 initial:5 contains:2 existing:2 current:1 john:2 enables:2 update:1 v:1 stationary:11 greedy:3 half:1 selected:1 record:1 prove:8 inside:2 introduce:1 expected:9 behavior:1 nor:1 planning:1 multi:4 chi:1 terminal:9 discounted:4 decreasing:1 decomposed:1 actual:1 becomes:2 project:1 estimating:1 bounded:1 notation:1 maximizes:1 guarantee:1 every:5 tackle:2 tie:2 exactly:2 grant:2 omit:1 before:2 positive:6 scientist:1 treat:1 modify:1 limit:3 mistake:1 zeitschrift:1 meet:2 path:4 ap:8 merge:1 initialization:1 studied:1 limited:1 directed:4 unique:4 acknowledgment:1 subgame:1 word:1 get:1 selection:12 put:1 www:1 deterministic:3 straightforward:2 iri:1 starting:1 independently:2 ke:1 unwilling:1 identifying:1 assigns:1 perceive:1 rule:1 notion:2 coordinate:5 hurt:1 construction:1 play:19 tan:1 programming:2 us:1 pa:2 assure:1 element:3 approximated:2 centrum:1 cooperative:2 levine:1 quicker:1 wang:2 capture:1 calculate:1 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called:3 ece:1 player:7 craft:1 exception:1 select:5 arises:1 outgoing:1 |
1,287 | 2,172 | VIBES: A Variational Inference
Engine for Bayesian Networks
Christopher M. Bishop
Microsoft Research
Cambridge, CB3 0FB, U.K.
research.microsoft.com/?cmbishop
David Spiegelhalter
MRC Biostatistics Unit
Cambridge, U.K.
[email protected]
John Winn
Department of Physics
University of Cambridge, U.K.
www.inference.phy.cam.ac.uk/jmw39
Abstract
In recent years variational methods have become a popular tool
for approximate inference and learning in a wide variety of probabilistic models. For each new application, however, it is currently
necessary first to derive the variational update equations, and then
to implement them in application-specific code. Each of these steps
is both time consuming and error prone. In this paper we describe a
general purpose inference engine called VIBES (?Variational Inference for Bayesian Networks?) which allows a wide variety of probabilistic models to be implemented and solved variationally without
recourse to coding. New models are specified either through a
simple script or via a graphical interface analogous to a drawing
package. VIBES then automatically generates and solves the variational equations. We illustrate the power and flexibility of VIBES
using examples from Bayesian mixture modelling.
1
Introduction
Variational methods [1, 2] have been used successfully for a wide range of models,
and new applications are constantly being explored. In many ways the variational
framework can be seen as a complementary approach to that of Markov chain Monte
Carlo (MCMC), with different strengths and weaknesses.
For many years there has existed a powerful tool for tackling new problems using
MCMC, called BUGS (?Bayesian inference Using Gibbs Sampling?) [3]. In BUGS
a new probabilistic model, expressed as a directed acyclic graph, can be encoded
using a simple scripting notation, and then samples can be drawn from the posterior
distribution (given some data set of observed values) using Gibbs sampling in a way
that is largely automatic. Furthermore, an extension called WinBUGS provides a
graphical front end to BUGS in which the user draws a pictorial representation of
the directed graph, and this automatically generates the required script.
We have been inspired by the success of BUGS to produce an analogous tool for
the solution of problems using variational methods. The challenge is to build a
system that can handle a wide range of graph structures, a broad variety of common conditional probability distributions at the nodes, and a range of variational
approximating distributions. All of this must be achieved whilst also remaining
computationally efficient.
2
A General Framework for Variational Inference
In this section we briefly review the variational framework, and then we characterise a large class of models for which the variational method can be implemented
automatically. We denote the set of all variables in the model by W = (V, X) where
V are the visible (observed) variables and X are the hidden (latent) variables. As
with BUGS, we focus on models that are specified in terms of an acyclic directed
graph (treatment of undirected graphical models is equally possible and is somewhat more straightforward). The joint distribution P (V, X) is then expressed in
terms of conditional distributions P (Wi |pai ) at each node i, where pai denotes the
set of variables corresponding to the parents of node i, and Wi denotes the variable,
or group of variables, associated with node i. The joint distribution
of all variables
Q
is then given by the product of the conditionals P (V, X) = i P (Wi |pai ).
Our goal is to find a variational distribution Q(X|V ) that approximates the true
posterior distribution P (X|V ). To do this we note the following decomposition of
the log marginal probability of the observed data, which holds for any choice of
distribution Q(X|V )
ln P (V ) = L(Q) + KL(QkP )
(1)
where
L(Q)
=
X
Q(X|V ) ln
X
KL(QkP )
= ?
X
X
P (V, X)
Q(X|V )
Q(X|V ) ln
P (X|V )
Q(X|V )
(2)
(3)
and the sums are replaced by integrals in the case of continuous variables. Here
KL(QkP ) is the Kullback-Leibler divergence between the variational approximation
Q(X|V ) and the true posterior P (X|V ). Since this satisfies KL(QkP ) ? 0 it follows
from (1) that the quantity L(Q) forms a lower bound on ln P (V ).
We now choose some family of distributions to represent Q(X|V ) and then seek a
member of that family that maximizes the lower bound L(Q). If we allow Q(X|V )
to have complete flexibility then we see that the maximum of the lower bound
occurs for Q(X|V ) = P (X|V ) so that the variational posterior distribution equals
the true posterior. In this case the Kullback-Leibler divergence vanishes and L(Q) =
ln P (V ). However, working with the true posterior distribution is computationally
intractable (otherwise we wouldn?t be resorting to variational methods). We must
therefore consider a more restricted family of Q distributions which has the property
that the lower bound (2) can be evaluated and optimized efficiently and yet which
is still sufficiently flexible as to give a good approximation to the true posterior
distribution.
2.1
Factorized Distributions
For the purposes of building VIBES we have focussed attention initially on distributions that factorize with respect to disjoint groups Xi of variables
Y
Q(X|V ) =
Qi (Xi ).
(4)
i
This approximation has been successfully used in many applications of variational
methods [4, 5, 6]. Substituting (4) into (2) we can maximize L(Q) variationally
with respect to Qi (Xi ) keeping all Qj for j 6= i fixed. This leads to the solution
ln Q?i (Xi ) = hln P (V, X)i{j6=i} + const.
(5)
where h?ik denotes an expectation with respect to the distribution Qk (Xk ). Taking
exponentials of both sides and normalizing we obtain
Q?i (Xi ) = P
exphln P (V, X)i{j6=i}
.
Xi exphln P (V, X)i{j6=i}
(6)
Note that these are coupled equations since the solution for each Qi (Xi ) depends on
expectations with respect to the other factors Qj6=i . The variational optimization
proceeds by initializing each of the Qi (Xi ) and then cycling through each factor in
turn replacing the current distribution with a revised estimate given by (6). The
current version of VIBES is based on a factorization of the form (4) in which each
factor Qi (Xi ) corresponds to one of the nodes of the graph (each of which can be a
composite node, as discussed shortly).
An important property of the variational update equations, from the point of view of
VIBES, is that the right hand side of (6) does not depend on all of the conditional
distributions P (Wi |pai ) that define the joint distribution but only on those that
have a functional dependence on Xi , namely the conditional P (Xi |pai ), together
with the conditional distributions for any children of node i since these have X i in
their parent set. Thus the expectations that must be performed on the right hand
side of (6) involve only those variables lying in the Markov blanket of node i, in
other words the parents, children and co-parents of i, as illustrated in Figure 1(a).
This is a key concept in VIBES since it allows the variational update equations to
be expressed in terms of local operations, which can therefore be expressed in terms
of generic code which is independent of the global structure of the graph.
2.2
Conjugate Exponential Models
It has already been noted [4, 5] that important simplifications to the variational
update equations occur when the distributions of the latent variables, conditioned
on their parameters, are drawn from the exponential family and are conjugate with
respect to the prior distributions of the parameters. Here we adopt a somewhat different viewpoint in that we make no distinction between latent variables and model
parameters. In a Bayesian setting these both correspond to unobserved stochastic
variables and can be treated on an equal footing. This allows us to consider conjugacy not just between variables and their parameters, but hierarchically between
all parent-child pairs in the graph.
Thus we consider models in which each conditional distribution takes the standard
exponential family form
ln P (Xi |Y ) = ?i (Y )T ui (Xi ) + fi (Xi ) + gi (Y )
(7)
where the vector ?(Y ) is called the natural parameter of the distribution. Now
(i)
consider a node Zj with parent Xi and co-parents cpj , as indicated in Figure 1(a).
Y1
YK
cpj(i)
{
Xi
Z1
}
Zj
(a)
(b)
Figure 1: (a) A central observation is that the variational update equations for node
Xi depend only on expectations over variables appearing in the Markov blanket of
Xi , namely the set of parents, children and co-parents. (b) Hinton diagram of hW i
from one of the components in the Bayesian PCA model, illustrating how all but
three of the PCA eigenvectors have been suppressed.
As far as the pair of nodes Xi and Zj are concerned, we can think of P (Xi |Y )
(i)
as a prior over Xi and the conditional P (Zj |Xi , cpj ) as a (contribution to) the
likelihood function. Conjugacy requires that, as a function of Xi , the product
of these two conditionals must take the same form as (7). Since the conditional
(i)
P (Zj |Xi , cpj ) is also in the exponential family it can be expressed as
(i)
(i)
(i)
ln P (Zj |Xi , cpj ) = ?j (Xi , cpj )T uj (Zj ) + fj (Zj ) + gj (Xi , cpj ).
(8)
Conjugacy then requires that this be expressible in the form
(i)
(i)
(i)
ln P (Zj |Xi , cpj ) = ?ej?i (Zj , cpj ) T ui (Xi ) + ?(Zj , cpj )
(9)
for some choice of functions ?e and ?. Since this must hold for each of the parents of
(i)
Zj it follows that ln P (Zj |Xi , cpj ) must be a multi-linear function of the uk (Xk ) for
each of the parents Xk of node XZj . Also, we observe from (8) that the dependence
(i)
of ln P (Zj |Xi , cpj ) on Zj is again linear in the function uj (Zj ). We can apply a
similar argument to the conjugate relationship between node Xj and each of its
parents, showing that the contribution from the conditional P (Xi |Y ) can again be
expressed in terms of expectations of the natural parameters for the parent node
distributions. Hence the right hand side of the variational update equation (5) for
a particular node Xi will be a multi-linear function of the expectations hui for each
node in the Markov blanket of Xi .
The variational update equation then takes the form
?
?T
M
?
?
X
(i)
ln Q?i (Xi ) = h?i (Y )iY +
h?ej?i (Zj , cpj )iZj ,cp(i)
ui (Xi ) + const.
j ?
?
(10)
j=1
which involves summation of bottom up ?messages? h?ej?i iZj ,cp(i) from the children
j
together with a top-down message h?i (Y )iY from the parents. Since all of these
messages are expressed in terms of the same basis ui (Xi ), we can write compact,
generic code for updating any type of node, instead of having to take account
explicitly of the many possible combinations of node types in each Markov blanket.
As an example, consider the Gaussian N (X|?, ? ?1 ) for a single variable X with
mean ? and precision (inverse variance) ? . The natural coordinates are uX =
[X, X 2 ]T and the natural parameterization is ? = [??, ?? /2]T . Then hui = [?, ?2 +
? ?1 ]T , and the function fi (Xi ) is simply zero in this case. Conjugacy allows us to
choose a distribution for the parent ? that is Gaussian and a prior for ? that is
a Gamma distribution. The corresponding natural parameterizations and update
messages are given by
?
h? ihXi
?
?h(X ? ?)2 i
e
e
,
h
?
i
=
,
u
=
,
h
?
i
=
.
u? =
X??
?
X??
?h? i/2
ln ?
1/2
?2
We can similarly consider multi-dimensional Gaussian distributions, with a Gaussian prior for the mean and a Wishart prior for the inverse covariance matrix.
A generalization of the Gaussian is the rectified Gaussian which is defined as
P (X|?, ? ) ? N (X|?, ? ) for X ? 0 and P (X|?, ? ) = 0 for X < 0, for which moments
can be expressed in terms of the ?erf? function. This rectification corresponds to
the introduction of a step function, whose logarithm corresponds to fi (Xi ) in (7),
which is carried through the variational update equations unchanged. Similarly, we
can consider doubly truncated Gaussians, which are non-zero only over some finite
interval.
Another example is the discrete distribution for categorical variables. These are
most conveniently represented using the 1-of-K scheme in which S = {Sk } with
P
QK
k = 1, . . . , K, Sk ? {0, 1} and k Sk = 1. This has distribution P (S|?) = k=1 ?kSk
and we can place a conjugate Dirichlet distribution over the parameters {?k }.
2.3
Allowable Distributions
We now characterize the class of models that can be solved by VIBES using the
factorized variational distribution given by (4). First of all we note that, since a
Gaussian variable can have a Gaussian parent for its mean, we can extend this hierarchically to any number of levels to give a sub-graph which is a DAG of Gaussian
nodes of arbitrary topology. Each Gaussian can have Gamma (or Wishart) prior
over its precision.
Next, we observe that discrete variables S = {Sk } can be used to construct ?pick?
functions which choose a particular parent node Yb from amongst several conjugate
QK
parents {Yk }, so that Yb = Yk when sk = 1, which can be written Yb = k=1 YkSk .
QK
Under any non-linear function h(?) we have h(Y )P= k=1 h(Yk )Sk . Furthermore the
expectation under S takes the form hh(Y )iS = k hSk ih(Yk ). Variational inference
will therefore be tractable for this model provided it is tractable for each of the
parents Yk individually.
Thus we can handle the following very general architecture: an arbitrary DAG
of multinomial discrete variables (each having Dirichlet priors) together with an
arbitrary DAG of linear Gaussian nodes (each having Wishart priors) and with
arbitrary pick links from the discrete nodes to the Gaussian nodes. This graph
represents a generalization of the Gaussian mixture model, and includes as special
cases models such as hidden Markov models, Kalman filters, factor analysers and
principal component analysers, as well as mixtures and hierarchical mixtures of all
of these.
There are other classes of models that are tractable under this scheme, for example
Poisson variables having Gamma priors, although these may be of limited interest.
We can further extend the class of tractable models by considering nodes whose
natural parameters are formed from deterministic functions of the states of several
parents. This is a key property of the VIBES approach which, as with BUGS, greatly
extends its applicability. Suppose we have some conditional distribution P (X|Y, . . .)
and we want to make Y some deterministic function of the states of some other nodes
?(Z1 , . . . , ZM ). In effect we have a pseudo-parent that is a deterministic function of
other nodes, and indeed is represented explicitly through additional deterministic
nodes in the graphical interface both to WinBUGS and to VIBES. This will be
tractable under VIBES provided the expectation of u? (?) can be expressed in terms
of the expectations of the corresponding functions uj (Zj ) of the parents. The pick
functions discussed earlier are a special case of these deterministic functions.
Thus for a Gaussian node the mean can be formed from products and sums of the
states of other Gaussian nodes provided the function is linear with respect to each
of the nodes. Similarly, the precision of the Gaussian can comprise the products
(but not sums) of any number of Gamma distributed variables.
Finally, we have seen that continuous nodes can have both discrete and continuous
parents but that discrete nodes can only have discrete parents. We can allow discrete
nodes to have continuous parents by stepping outside the conjugate-exponential
framework by exploiting a variational bound on the logistic sigmoid function [1].
We also wish to be able to evaluate the lower bound (2), both to confirm the
correctness of the variational updates (since the value of the bound should never
decrease), as well as to monitor convergence and set termination criteria. This can
be done efficiently, largely using quantities that have already been calculated during
the variational updates.
3
VIBES: A Software Implementation
Creation of a model in VIBES simply involves drawing the graph (using operations
similar to those in a simple drawing package) and then assigning properties to each
node such as the functional form for the distribution, a list of the other variables
it is conditioned on, and the location of the corresponding data file if the node is
observed. The menu of distributions available to the user is dynamically adjusted
at each stage to ensure that only valid conjugate models can be constructed.
As in WinBUGS we have adopted the convention of making logical (deterministic)
nodes explicit in the graphical representation as this greatly simplifies the specification and interpretation of the model. We also use the ?plate? notation of a
box surrounding one or more nodes to denote that those nodes are replicated some
number of times as specified by the parameter appearing in the bottom right hand
corner of the box.
3.1
Example: Bayesian Mixture Models
We illustrate VIBES using a Bayesian model for a mixture of M probabilistic PCA
distributions, each having maximum intrinsic dimensionality of q, with a sparse
prior [6], for which the VIBES implementation is shown in Figure 2. Here there are
N observations of the vector t whose dimensionality is d, as indicated by the plates.
The dimensionality of the other variables is also determined by which plates they
are contained in (e.g. W has dimension d ? q ? M whereas ? is a scalar). Variables
t, x, W and ? are Gaussian, ? and ? have Gamma distributions, S is discrete and
? is Dirichlet.
Once the model is completed (and the file or files containing the observed variables
Figure 2: Screen shot from VIBES showing the graph for a mixture of probabilistic
PCA distributions. The node t is coloured black to denote that this variable is
observed, and the node ?alpha? has been highlighted and its properties (e.g. the
form of the distribution) can be changed using the menus on the left hand side.
The node labelled ?x.W+mu? is a deterministic node, and the double arrows denote
deterministic relationships.
are specified) it is then ?compiled?, which involves allocation of memory for the
variables and initializing the distributions Qi (which is done using simple heuristics
but which can also be over-ridden by the user). If desired, monitoring of the lower
bound (2) can be switched on (at the expense of slightly increased computation) and
this can also be used to set a termination criterion. Alternatively the variational
optimization can be run for a fixed number of iterations.
Once the optimization is complete various diagnostics can be used to probe the
results, such as the Hinton diagram plot shown in Figure 1(b).
Now suppose we wish to modify the model, for instance by having a single set of
hyper-parameters ? whose values are shared by all of the M components in the
mixture, instead of having a separate set for each component. This simply involved
dragging the ? node outside of the M plate using the mouse and then recompiling
(since ? is now a vector of length q instead of a matrix of size M ? q). This literally
takes a few seconds, in contrast to the effort required to formulate the variational
inference equations, and develop bespoke code, for a new model! The result is then
optimized as before. A screen shot of the corresponding VIBES model is shown in
Figure 3.
4
Discussion
Our early experiences with VIBES have shown that it dramatically simplifies the
construction and testing of new variational models, and readily allows a range of
alternative models to be evaluated on a given problem. Currently we are extending
VIBES to cater for a broader range of variational distributions by allowing the user
to specify a Q distribution defined over a subgraph of the true graph [7].
Finally, there are many possible extensions to the basic VIBES we have described
Figure 3: As in Figure 2 but with the vector ? of hyper-parameters moved outside
the M ?plate?. This causes there to be only q terms in ? which are shared over the
mixture components rather than M ? q. Note that, with no nodes highlighted, the
side menus disappear.
here. For example, in order to broaden the range of models that can be tackled we
can combine variational with other methods such as Gibbs sampling or optimization
(empirical Bayes) to allow for non-conjugate hyper-priors for instance. Similarly,
there is scope for exploiting exact methods where there exist tractable sub-graphs.
References
[1] M. I. Jordan, Z. Ghahramani, T. S. Jaakkola, and L. K. Saul. An introduction
to variational methods for graphical models. In M. I. Jordan, editor, Learning
in Graphical Models, pages 105?162. Kluwer, 1998.
[2] R. M. Neal and G. E. Hinton. A new view of the EM algorithm that justifies
incremental and other variants. In M. I. Jordan, editor, Learning in Graphical
Models, pages 355?368. Kluwer, 1998.
[3] D J Lunn, A Thomas, N G Best, and D J Spiegelhalter. WinBUGS ? a Bayesian
modelling framework: concepts, structure and extensibility. Statistics and Computing, 10:321?333, 2000. http://www.mrc-bsu.cam.ac.uk/bugs/.
[4] Z. Ghahramani and M. J. Beal. Propagation algorithms for variational Bayesian
learning. In T. K. Leen, T. Dietterich, and V. Tresp, editors, Advances in Neural
Information Processing Systems, volume 13, Cambridge MA, 2001. MIT Press.
[5] H. Attias. A variational Bayesian framework for graphical models. In S. Solla,
T. K. Leen, and K-L Muller, editors, Advances in Neural Information Processing
Systems, volume 12, pages 209?215, Cambridge MA, 2000. MIT Press.
[6] C. M. Bishop. Variational principal components. In Proceedings Ninth International Conference on Artificial Neural Networks, ICANN?99, volume 1, pages
509?514. IEE, 1999.
[7] Christopher M. Bishop and John Winn. Structured variational distributions in
VIBES. In Proceedings Artificial Intelligence and Statistics, Key West, Florida,
2003. Accepted for publication.
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1,288 | 2,173 | Incremental Gaussian Processes
?
Joaquin Quinonero-Candela
Informatics and Mathematical Modelling
Technical University of Denmark
DK-2800 Lyngby, Denmark
[email protected]
Ole Winther
Informatics and Mathematical Modelling
Technical University of Denmark
DK-2800 Lyngby, Denmark
[email protected]
Abstract
In this paper, we consider Tipping?s relevance vector machine (RVM)
[1] and formalize an incremental training strategy as a variant of the
expectation-maximization (EM) algorithm that we call Subspace EM
(SSEM). Working with a subset of active basis functions, the sparsity
of the RVM solution will ensure that the number of basis functions and
thereby the computational complexity is kept low. We also introduce
a mean field approach to the intractable classification model that is expected to give a very good approximation to exact Bayesian inference
and contains the Laplace approximation as a special case. We test the
algorithms on two large data sets with O(103 ? 104 ) examples. The results indicate that Bayesian learning of large data sets, e.g. the MNIST
database is realistic.
1 Introduction
Tipping?s relevance vector machine (RVM) both achieves a sparse solution like the support
vector machine (SVM) [2, 3] and the probabilistic predictions of Bayesian kernel machines
based upon a Gaussian process (GP) priors over functions [4, 5, 6, 7, 8]. Sparsity is interesting both with respect to fast training and predictions and ease of interpretation of the
solution. Probabilistic predictions are desirable because inference is most naturally formulated in terms of probability theory, i.e. we can manipulate probabilities through Bayes
theorem, reject uncertain predictions, etc.
It seems that Tipping?s relevance vector machine takes the best of both worlds. It is a GP
with a covariance matrix spanned by a small number of basis functions making the computational expensive matrix inversion operation go from O(N 3 ), where N is the number of
training examples to O(M 2 N ) (M being the number of basis functions). Simulation studies have shown very sparse solutions M N and good test performance [1]. However,
starting the RVM learning with as many basis functions as examples, i.e. one basis function
in each training input point, leads to the same complexity as for Gaussian processes (GP)
since in the initial step no basis functions are removed. That lead Tipping to suggest in
an appendix in Ref. [1] an incremental learning strategy that starts with only a single basis
function and adds basis functions along the iterations, and to formalize it very recently [9].
The total number of basis functions is kept low because basis functions are also removed.
In this paper we formalize this strategy using straightforward expectation-maximization
(EM) [10] arguments to prove that the scheme is the guaranteed convergence to a local
maximum of the likelihood of the model parameters.
Reducing the computational burden of Bayesian kernel learning is a subject of current
interest. This can be achieved by numerical approximations to matrix inversion [11] and
suboptimal projections onto finite subspaces of basis functions without having an explicit
parametric form of such basis functions [12, 13]. Using mixtures of GPs [14, 15] to make
the kernel function input dependent is also a promising technique. None of the Bayesian
methods can currently compete in terms of speed with the efficient SVM optimization
schemes that have been developed, see e.g. [3].
The rest of the paper is organized as follows: In section 2 we present the extended linear
models in a Bayesian perspective, the regression model and the standard EM approach.
In section 3, a variation of the EM algorithm, that we call the Subspace EM (SSEM) is
introduced that works well with sparse solution models. In section 4, we present the second
main contribution of the paper: a mean field approach to RVM classification. Section
5 gives results for the Mackey-Glass time-series and preliminary results on the MNIST
hand-written digits database. We conclude in section 6.
2 Regression
An extended linear model is built by transforming the input space by an arbitrary set of basis functions ?j : RD ? R that performs a non-linear transformation of the D-dimensional
input space. A linear model is applied to the transformed space whose dimension is equal
to the number of basis functions M :
y(xi ) =
M
X
j=1
?j ?j (xi ) = ?(xi ) ? ?
(1)
where ?(xi ) ? [?1 (xi ), . . . , ?M (xi )] denotes the ith row of the design matrix ? and ? =
(?1 , . . . , ?N )T is the weights vector. The output of the model is thus a linear superposition
of completely general basis functions. While it is possible to optimize the parameters of
the basis functions for the problem at hand [1, 16], we will in this paper assume that they
are given.
The simplest possible regression learning scenario can be described as follows: a set of
N input-target training pairs {xi , ti }N
i=1 are assumed to be independent and contaminated
with Gaussian noise of variance ? 2 . The likelihood of the parameters ? is given by
?N/2
1
? , ? 2 ) = 2?? 2
p(t|?
exp ? 2 kt ? ? ? k2
(2)
2?
where t = (t1 , . . . , tN )T is the target vector. Regularization is introduced in Bayesian
learning by means of a prior distribution over the weights. In general, the implied prior
over functions is a very complicated distribution. However, choosing a Gaussian prior on
the weights the prior over functions also becomes Gaussian, i.e. a Gaussian process. For
the specific choice of a factorized distribution with variance ??1
j :
r
1
?j
p(?j |?j ) =
exp ? ?j ?j2
(3)
2?
2
? ) is N (0, ?A?1 ?T ), i.e. a Gaussian process with covariance
the prior over functions p(y|?
function given by
M
X
1
?k (xi )?k (xj )
(4)
Cov(xi , xj ) =
?k
k=1
where ? = (?0 , . . . , ?N )T and A = diag(?0 , . . . , ?N ). We can now see how
sparseness in terms of the basis vectors may arise: if ??1
= 0 the kth basis vector
k
?k ? [?k (x1 ), . . . , ?k (xN )]T , i.e. the kth column in the design matrix, will not contribute
to the model. Associating a basis function with each input point may thus lead to a model
with a sparse representations in the inputs, i.e. the solution is only spanned by a subset of
all input points. This is exactly the idea behind the relevance vector machine, introduced
by Tipping [17]. We will see in the following how this also leads to a lower computational
complexity than using a regular Gaussian process kernel.
The posterior distribution over the weights?obtained through Bayes rule?is a Gaussian distribution
? , ? 2 )p(?
? |?
?)
p(t|?
? |t, ? , ? 2 ) =
? |?
?, ?)
p(?
= N (?
(5)
?, ? 2 )
p(t|?
?, ?) is a Gaussian distribution with mean ? and covariance ? evaluated at t.
where N (t|?
The mean and covariance are given by
? = ? ?2 ??T t
? = (? ?2 ?T ? + A)?1
(6)
(7)
The uncertainty about the optimal value of the weights captured by the posterior distribution (5) can be used to build probabilistic predictions. Given a new input x ? , the model
gives a Gaussian predictive distribution of the corresponding target t ?
Z
? |t, ? , ? 2 ) d?
? = N (t? |y? , ??2 )
p(t? |x? , ? , ? 2 ) = p(t? |x? , ? , ? 2 ) p(?
(8)
where
= ?(x? ) ? ?
y?
??2
2
= ? + ?(x? ) ? ? ? ?(x? )
(9)
T
(10)
For regression it is natural to use y? and ?? as the prediction and the error bar on the
prediction respectively. The computational complexity of making predictions is thus
O(M 2 P + M 3 + M 2 N ), where M is the number of selected basis functions (RVs) and P
is the number of predictions. The two last terms come from the computation of ? in eq.
(7).
The likelihood distribution over the training targets (2) can be ?marginalized? with respect
to the weights to obtain the marginal likelihood, which is also a Gaussian distribution
Z
? , ? 2 ) = p(t|?
? , ? 2 ) p(?
? |?
? ) d?
? = N (t|0, ? 2 I + ?A?1 ?T ) .
p(t|?
(11)
Estimating the hyperparameters {?j } and the noise ? 2 can be achieved by maximizing
(11). This is naturally carried out in the framework of the expectation-maximization (EM)
algorithm since the sufficient statistics of the weights (that act as hidden variables) are
available for this type of model. In other cases e.g. for adapting the length scale of the
kernel [4], gradient methods have to be used. For regression, the E-step is exact (the lower
bound on the marginal likelihood is made equal to the marginal likelihood) and consists in
estimating the mean and variance (6) and (7) of the posterior distribution of the weights
(5). For classification, the E-step will be approximate. In this paper we present a mean
field approach for obtaining the sufficient statistics.
The M-step corresponds to maximizing the expectation of the log marginal likelihood with
respect to the posterior, with respect to ? 2 and ? , which gives the following update
Prules:
1
1
1
2 new
2
2 old
=
,
and
(?
)
=
(||t
?
?
?||
+
(?
)
?new
=
2
2
j
j ?j ),
N
h? i
? +?jj
j
? |t,?
? ,? 2 )
p(?
j
where the quantity ?j ? 1 ? ?j ?jj is a measure of how ?well-determined? each weight ?j
is by the data [18, 1]. One can obtain a different update rule that gives faster convergence.
Although it is suboptimal in the EM sense, we have never observed it decrease the lower
bound on the marginal log-likelihood. The rule, derived in [1], is obtained by differentiation
of (11) and by an arbitrary choice of independent terms as is done by [18]. It makes use of
the terms {?j }:
?new
=
j
?j
?2j
(? 2 )new =
||t ? ? ?||2
P
.
N ? j ?j
(12)
In the optimization process many ?j grow to infinity, which effectively deletes the corresponding weight and basis function. Note that the EM update and the Mackay update
for ?j only implicitly depend upon the likelihood. This means that it is also valid for the
classification model we shall consider below.
A serious limitation of the EM algorithm and variants for problems of this type is that the
complexity of computing the covariance of the weights (7) in the E-step is O(M 3 +M 2 N ).
At least in the first iteration where no basis functions have been deleted M = N and we
are facing the same kind of complexity explosion that limits the applicability of Gaussian
processes to large training set. This has lead Tipping [1] to consider a constructive or
incremental training paradigm where one basis function is added before each E-step and
since basis functions are removed in the M-step, it turns out in practice that the total number
of basis functions and the complexity remain low [9]. In the following section we introduce
a new algorithm that formalizes this procedure that can be proven to increase the marginal
likelihood in each step.
3 Subspace EM
We introduce an incremental approach to the EM algorithm, the Subspace EM (SSEM), that
can be directly applied to training models like the RVM that rely on a linear superposition
of completely general basis functions, both for classification and for regression. Instead of
starting with a full model, i.e. where all the basis functions are present with finite ? values,
we start with a fully pruned model with all ?j set to infinity. Effectively, we start with no
model. The model is grown by iteratively including some ?j previously set to infinity to
the active set of ??s. The active set at iteration n, Rn , contains the indices of the basis
vectors with ? less than infinity:
R1 = 1
Rn = {i | i ? Rn?1 ? ?i ? L} ? {n}
(13)
where L is a finite very large number arbitrarily defined. Observe that R n contains at most
one more element (index) than Rn?1 . If some of the ??s indexed by Rn?1 happen to reach
L at the n-th step, Rn can contain less elements than Rn?1 . In figure 1 we give a schematic
description of the SSEM algorithm.
At iteration n the E-step is taken only in the subspace spanned by the weights whose
indexes are in Rn . This helps reducing the computational complexity of the M-step to
O(M 3 ), where M is the number of relevance vectors.
Since the initial value of ?j is infinity for all j, for regression the E-step yields always
an equality between the log marginal likelihood and its lower bound. At any step n, the
posterior can be exactly projected on to the space spanned by the weights ? j such that
j ? Rn , because the ?k = ? for all k not in Rn . Hence in the regression case, the SSEM
never decreases the log marginal likelihood. Figure 2 illustrates the convergence process
of the SSEM algorithm compared to that of the EM algorithm for regression.
Set ?j = L for all j. (L is a very large number) Set n = 1
Update the set of active indexes Rn
Perform an E-step in subspace ?j such that j ? Rn
Perform the M-step for all ?j such that j ? Rn
If visited all basis functions, end, else go to 2.
1.
2.
3.
4.
5.
Figure 1: Schematics of the SSEM algorithm.
Number of RVs vs. CPU time
Likelihood vs. CPU time
450
standard EM
SSEM
1200
400
1000
350
300
Number of RVs
Log marginal likelihood
800
600
400
200
250
200
150
0
100
?200
50
SSEM
standard EM
?400
0
20
40
60
80
CPU time (seconds)
100
120
0
0
20
40
60
80
CPU time (seconds)
100
120
Figure 2: Training on 400 samples of the Mackey-Glass time series, testing on 2000 cases.
Log marginal likelihood as a function of the elapsed CPU time (left) and corresponding
number of relevance vectors (right) for both SSEM and EM.
We perform one EM step for each time a new basis function is added to the active set. Once
all the examples have been visited, we switch to the batch EM algorithm on the active set
until some convergence criteria has been satisfied, for example until the relative increase in
the likelihood is smaller than a certain threshold. In practice some 50 of these batch EM
iterations are enough.
4 Classification
Unlike the model discussed above, analytical inference is not possible for classification
models. Here, we will discuss the adaptive TAP mean field approach?initially proposed for
Gaussian processes [8]?that are readily translated to RVMs. The mean field approach has
the appealing features that it retains the computational efficiency of RVMs, is exact for the
regression and reduces to the Laplace approximation in the limit where all the variability
comes from the prior distribution.
We consider binary t = ?1 classification using the probit likelihood with ?input? noise ? 2
y(x)
,
(14)
p(t|y(x)) = erf t
?
?
Rx
2
where Dz ? e?z /2 dz/ 2? and erf(x) ? ?? Dz is an error function (or cumulative
Gaussian distribution). The advantage of using this sigmoid rather than the commonly
used 0/1-logistic is that we under the mean field approximation
can derive an analytical
R
expression for the predictive distribution p(t? |x? , t) = p(t? |y)p(y|x? , t)dy needed for
making Bayesian predictions. Both a variational and the advanced mean field approach?
used here?make a Gaussian approximation for p(y|x? , t) [8] with mean and variance given
by regression results y? and ??2 ? ?
? 2 , and y? and ??2 given by eqs. (9) and (10). This leads
to the following approximation for the predictive distribution
Z
y
y?
p(y|x? , t) dy = erf t?
.
p(t? |x? , t) =
erf t?
?
??
(15)
However, the mean and covariance of the weights are no longer found by analytical expressions, but has to be obtained from a set of non-linear mean field equations that also follow
from equivalent assumptions of Gaussianity for the training set outputs y(x i ) in averages
over reduced (or cavity) posterior averages.
In the following, we will only state the results which follows from combining the RVM
Gaussian process kernel (4) with the results of [8]. The sufficient statistics of the weights
are written in terms of a set of O(N ) mean field parameters
? = A?1 ?T ?
T
? =
where ?i ?
?
?yic
Z(yic , Vic
A + ? ??
ln Z(yic , Vic + ? 2 ) and
2
+? )
?
Z
p(ti |yic
(16)
?1
q
+ z Vic + ? 2 ) Dz = erf
(17)
yc
ti p c i
Vi + ? 2
!
. (18)
The last equality holds for the likelihood eq. (14) and yic and Vic are the mean and variance
of the so called cavity field. The mean value is yic = ?(xi ) ? ? ? Vic ?i . The distinction
c
c
between the different approximation
schemes
is solely in the variance V i : Vi = 0 is the
c
?1 T
Laplace approximation, Vi = ?A ? ii is the so called naive mean field theory and
an improved estimate is available from the adaptive TAP mean field theory [8]. Lastly, the
diagonal matrix ? is the equivalent of the noise variance in the regression model (compare
??i
c ??i
eqs. (17) and (7) and is given by ?i = ? ?y
c /(1+Vi ?y c ) . This set of non-linear equations
i
i
are readily solved (i.e. fast and stable) by making Newton-Raphson updates in ? treating
the remaining quantities as help variables:
? = (I + A?1 ?T ??)?1 (A?1 ?T ? ? ? ) = ?(?T ? ? A?
?)
??
(19)
The computational complexity of the E-step for classification is augmented with respect to
the regression case by the fact that it is necessary to construct and invert a M ? M matrix
usually many times (typically 20), once for each step of the iterative Newton method.
5 Simulations
We illustrate the performance of the SSEM for regression on the Mackey-Glass chaotic
time series, which is well-known for its strong non-linearity. In [16] we showed that the
RVM has an order of magnitude superior performance than carefully tuned neural networks
for time series prediction on the Mackey-Glass series. The inputs are formed by L = 16
samples spaced 6 periods from each other xk = [z(k ? 6), z(k ? 12), . . . , z(k ? 6L)] and
the targets are chosen to be tk = z(k) to perform six steps ahead prediction (see [19] for
details). We use Gaussian basis functions of fixed variance ? 2 = 10. The test set comprises
5804 examples.
We perform prediction experiments for different sizes of the training set. We perform in
each case 10 repetitions with different partitions of the data sets into training and test. We
compare the test error, the number of RVs selected and the computer time needed for the
batch and the SSEM method. We present the results obtained with the growth method
relative to the results obtained with the batch method in figure 3. As expected, the relative
Mackey?Glass data
3
growth
Classification on MNIST digits
batch
Ete
/Ete
Tcpugrowth/Tcpubatch
NRVgrowth/NRVbatch
2.5
1.5
1
2
1.5
0.5
1
0
Training error prob.
Test error prob.
Scaled loglik
0.5
0
0
500
1000
1500
2000
Number of training examples
?0.5
0
200
400
Iteration
600
800
Figure 3: Left: Regression, mean values over 10 repetitions of relative test error, number
of RVs and computer time for the Mackey-Glass data, up to 2400 training examples and
5804 test examples. Right: Classification, Log marginal likelihood, test and training errors
while training on one class against all the others, 60000 training and 10000 test examples.
computer time of the growth method compared with the batch method decreases with size
of the training set. For a few thousand examples the SSEM method is an order of magnitude
faster than the batch method. The batch method proved only to be faster for 100 training
examples, and could not be used with data sets of thousands of examples on the machine on
which we run the experiments because of its high memory requirements. This is the reason
why we only ran the comparison for up to 2400 training example for the Mackey-Glass
data set.
Our experiments for classification are at the time of sending this paper to press very premature: we choose a very large data set, the MNIST database of handwritten digits [20],
with 60000 training and 10000 test images. The images are of size 28 ? 28 pixels. We
use PCA to project them down to 16 dimensional vectors. We only perform a preliminary
experiment consisting of a one against all binary classification problem to illustrate that
Bayesian approaches to classification can be used on very large data sets with the SSEM
algorithm. We train on 13484 examples (the 6742 one?s and another 6742 random non-one
digits selected at random from the rest) and we use 800 basis functions for both the batch
and Subspace EM. In figure 3 we show the convergence of the SSEM in terms of the log
marginal likelihood and the training and test probabilities of error. The test probability of
error we obtain is 0.74 percent with the SSEM algorithm and 0.66 percent with the batch
EM. Under the same conditions the SSEM needed 55 minutes to do the job, while the batch
EM needed 186 minutes. The SSEM gives a machine with 28 basis functions and the batch
EM one with 31 basis functions.
6 Conclusion
We have presented a new approach to Bayesian training of linear models, based on a subspace extension of the EM algorithm that we call Subspace EM (SSEM). The new method
iteratively builds models from a potentially big library of basis functions. It is especially
well-suited for models that are constructed such that they yield a sparse solution, i.e. the solution is spanned by small number M of basis functions, which is much smaller than N , the
number of examples. A prime example of this is Tipping?s relevance vector machine that
typically produces solutions that are sparser than those of support vector machines. With
the SSEM algorithm the computational complexity and memory requirement decrease from
O(N 3 ) and O(N 2 ) to O(M 2 N ) (somewhat higher for classification) and O(N M ). For
classification, we have presented a mean field approach that is expected to be a very good
approximation to the exact inference and contains the widely used Laplace approximation
as an extreme case. We have applied the SSEM algorithm to both a large regression and a
large classification data sets. Although preliminary for the latter, we believe that the results
demonstrate that Bayesian learning is possible for very large data sets. Similar methods
should also be applicable beyond supervised learning.
Acknowledgments JQC is funded by the EU Multi-Agent Control Research Training Network - EC
TMR grant HPRNCT-1999-00107. We thank Lars Kai Hansen for very useful discussions.
References
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1,289 | 2,174 | Application of Variational Bayesian Approach to
Speech Recognition
Shinji Watanabe, Yasuhiro Minami, Atsushi Nakamura and Naonori Ueda
NTT Communication Science Laboratories, NTT Corporation
2-4, Hikaridai, Seika-cho, Soraku-gun, Kyoto, Japan
{watanabe,minami,ats,ueda}@cslab.kecl.ntt.co.jp
Abstract
In this paper, we propose a Bayesian framework, which constructs
shared-state triphone HMMs based on a variational Bayesian approach,
and recognizes speech based on the Bayesian prediction classification;
variational Bayesian estimation and clustering for speech recognition
(VBEC). An appropriate model structure with high recognition performance can be found within a VBEC framework. Unlike conventional
methods, including BIC or MDL criterion based on the maximum likelihood approach, the proposed model selection is valid in principle, even
when there are insufficient amounts of data, because it does not use
an asymptotic assumption. In isolated word recognition experiments,
we show the advantage of VBEC over conventional methods, especially
when dealing with small amounts of data.
1 Introduction
A statistical modeling of spectral features of speech (acoustic modeling) is one of the most
crucial parts in the speech recognition. In acoustic modeling, a triphone-based hidden
Markov model (triphone HMM) has been widely employed. The triphone is a context
dependent phoneme unit that considers both the preceding and following phonemes. Although the triphone enables the precise modeling of spectral features, the total number of
triphones is too large to prepare sufficient amounts of training data for each triphone. In
order to deal with the problem of data insufficiency, an HMM state is usually shared among
multiple triphone HMMs, which means the amount of training data per state inflates. Such
shared-state triphone HMMs (SST-HMMs) can be constructed by successively clustering
states based on the phonetic decision tree method [4] [7]. The important practical problem
that must be solved when constructing SST-HMMs is how to optimize the total number
of shared states adaptively to the amounts of available training data. Namely, maintaining
the balance between model complexity and training data size is quite important for high
generalization performance.
The maximum likelihood (ML) is inappropriate as a model selection criterion since ML
increases monotonically as the number of states increases. Some heuristic thresholding
is therefore necessary to terminate the partitioning. To solve this problem, the Bayesian
information criterion (BIC) and minimum description length (MDL) criterion have been
employed to determine the tree structure of SST-HMMs [2] [5] 1 . However, since the
BIC/MDL is based on an asymptotic assumption, it is invalid in principle when the number
of training data is small because of the failure of the assumption.
In this paper, we present a practical method within the Bayesian framework for estimating posterior distributions over parameters and selecting an appropriate model structure of
SST-HMMs (clustering triphone HMM states) based on a variational Bayesian (VB) approach, and recognizing speech based on the Bayesian prediction classification: variational
Bayesian estimation and clustering for speech recognition (VBEC). Unlike the BIC/MDL,
VB does not assume asymptotic normality, and it is therefore applicable in principle, even
when there are insufficient data. The VB approach has been successfully applied to model
selection problems, but mainly for relatively simple mixture models [1] [3] [6] [8]. Here,
we try to apply VB to SST-HMMs with more a complex model structure than the mixture
model and evaluate the effectiveness through a large-scale real speech recognition experiment.
2 Variational Bayesian framework
First, we briefly review the VB framework. Let O be a given data set. In the Bayesian
approach we are interested in posterior distributions over model parameters, p(?|O, m),
and the model structure, p(m|O). Here, ? is a set of model parameters and m is an index
of the model structure. Let us consider a general probabilistic model with latent variables.
Let Z be a set of latent variables. Then the model with a fixed model structure m can be
defined by the joint distribution p(O, Z|?, m).
In VB, variational posteriors q(?|O, m), q(Z|O, m), and q(m|O) are introduced to approximate the true corresponding posteriors. The optimal variational posteriors over ? and
Z, and the appropriate model structure that maximizes the optimal q(m|O) can be obtained
by maximizing the following objective function:
p(O, Z|?, m)p(?|m)
,
(1)
Fm [q] =
log
q(Z|O, m)q(?|O, m) q(Z|O,m),q(? |O,m)
w.r.t. q(?|O, m), q(Z|O, m), and m. Here hf (x)ip(x) denotes the expectation of f (x)
w.r.t. p(x). p(?|m) is a prior distribution. This optimization can be effectively performed
by an EM-like iterative algorithm (see [1] for the details).
3 Applying a VB approach to acoustic models
3.1
Output distributions and prior distributions
We attempt to apply a VB approach to a left-to-right HMM, which has been widely used
to represent a phoneme unit in acoustic models for speech recognition, as shown in Figure
1. Let O = {O t ? RD : t = 1, ..., T } be a sequential data set for a phoneme unit. The
output distribution in an HMM is given by
YT
p(O, S, V |?, m) =
ast?1 st cst vt bst vt (O t ),
(2)
t= 1
where S is a set of sequences of hidden states, V is a set of sequences of Gaussian mixture
components, and st and v t denote the state and mixture components at time t. S and V are
sets of discrete latent variables that correspond to Z mentioned above. aij denotes the state
1
These criteria have been independently proposed, but they are practically the same. Therefore,
we refer to them hereafter as BIC/MDL.
a11
i =1
a33
a22
a12
i=2
Figure 1: Hidden Markov model for each
phoneme unit. A state is represented by
the Gaussian mixture distribution below
the state. There are three states and three
Gaussian components in this figure.
i=3
a 23
Gaussian mixture for state i
transition probability from state i to state j, and cjk is the k-th weight factor of the Gaussian
mixture for state j. bjk (= N (O t |?jk , ?jk )) denotes the Gaussian distribution with mean
vector ?jk and covariance ?jk . ? = {aij , cjk , ?jk , ??1
jk |i, j = 1, ..., J, k = 1, ..., L} is
a set of model parameters. J denotes the number of states in an HMM and L denotes the
number of Gaussian components in a state. In this paper, we restrict covariance matrices in
the Gaussian distribution to diagonal ones. The conjugate prior distributions are assumed
to be as follows:
Y
0
p(?|m) =
D({aij0 }Jj0 = 1 |?0 )D({cjk0 }L
k0 = 1 |? )
i,j,k
YD
0
0
? N (?jk |? 0jk , (? 0 )?1 ?jk )
G(??1
(3)
jk,d |? , Rjk,d ).
d= 1
0
?0 = {?0 , ?0 , ? 0jk , ? 0 , ? 0 , Rjk
} is a set of hyperparameters. We assume the hyperparameters are constants. In Eq.(3), D denotes a Dirichlet distribution and G denotes a gamma
distribution.
3.2
Optimal variational posterior distribution q?(?|O, m)
From the output distributions and prior distributions in section 3.1, the optimal variational
posterior distribution q?(?|O, m) can be obtained as:
D({aij }Jj= 1 |{??ij }Jj= 1 )
D({cjk }L
?jk }L
)
k= 1 |{?
k= 1
QD
?1
? jk,d ),
?jk , R
N (?jk |?
? jk , ??jk ?jk ) d= 1 G(??1
jk,d |?
q?({aij }Jj= 1 |O, m) =
q?({cjk }L
k= 1 |O, m) =
q?(bjk |O, m)
=
(4)
? ?,
? ??, R
? ? {?,
? jk } is a set of posterior distribution parameters defined as:
? jk , ?,
?
? ?
XT
? jk = ? 0 ? 0jk +
??ij = ?0 + ??ij , ??jk = ?0 + ??jk , ??jk = ? 0 + ??jk , ?
??t O t /??jk ,
t= 1 jk
XT
t
? jk,d = R0 + ? 0 (? 0 ? ??jk,d )2 +
??jk = ? 0 + ??jk , R
??jk
(Odt ? ??jk,d )2 .
(5)
jk,d
jk,d
t= 1
t ?t
? is composed of
, ?jk ? q?(st = j, v t =
? q?(s = i, s
= j|O, m), ??ij ? ?Tt= 1 ??ij
?
t
t
k|O, m) and ??jk ? ?Tt= 1 ??jk
. ??ij
denotes the transition probability from state i to state j at
t
?
time t. ?jk denotes the occupation probability on mixture component k in state j at time t.
t
??ij
3.3
t
t+ 1
Optimal variational posterior distribution q?(S, V |O, m)
From the output distributions and prior distributions in section 3.1, the optimal variational
posterior distribution over latent variables q?(S, V |O, m) can be obtained as:
YT
q?(S, V |O, m) ?
a
?st?1 st c?st vt ?bst vt (Ot ),
(6)
t= 1
where
a
?st?1 st
=
c?st vt
=
?bst vt (Ot ) =
XJ
e x p ? (??st?1 st ) ? ? (
?? t?1 st0 ) ,
st0 =1 s
XL
e x p ? (??st vt ) ? ? (
?? t t0 ) ,
v t0 =1 s v
?
t
t
t
t
?s v /2) ?
e x p D/2 log 2? ? 1/?s v + ? (?
XD
? st vt ,d /2) + (Ot ? ??st vt ,d )2 ??st vt /R
? st vt ,d
log(R
. (7)
?1/2
d
d=1
t
? (y) is a digamma function. From these results, transition and occupation probability ??ij
t
and ??ij can be obtained by using either a deterministic assignment via the Viterbi algorithm
or a probabilistic assignment via the Forward-Backward algorithm. Thus, q?(? |O, m) and
q?(S, V |O, m) can be calculated iteratively that result in maximizing Fm .
4 VB training algorithm for acoustic models
Based on the discussion in section 3, a VB training algorithm for an acoustic model based
on an HMM and Gaussian mixture model with a fixed model structure m is as follows:
????????????????????????????????????
t
t
Step 1) Initialize ??ij
[? = 0], ??ij
[? = 0] and set ?0 .
Step 2) Compute q(S, V |O, m)[? + 1] using ?? t [? ], ??t [? ] and ?0 .
ij
t
t
?
Update ??ij [? +1] and ?ij [? +1] using q(S, V
ij
|O, m)[? +1] via the Viterbi algorithm
or Forward-Backward algorithm.
? + 1] using ?? t [? + 1], ??t [? + 1] and ?0 .
Step 4) Compute ?[?
ij
ij
? + 1] and calculate Fm [? ] based on
Step 5) Compute q(? |O, m)[? + 1] using ?[?
q(? |O, m)[? + 1] and q(S, V |O, m)[? + 1].
Step 6) If |(Fm [? + 1] ? Fm [? ])/Fm [? + 1]| ? ?, then stop; otherwise set ? ? ? + 1 and
go to Step 2.
????????????????????????????????????
? denotes an iteration count. In our experiments, we employed the Viterbi algorithm in
Step 3.
Step 3)
5 Variational Bayesian estimation and clustering for speech
recognition
In the previous section, we described a VB training algorithm for HMMs. Here, we explain
VBEC, which constructs an acoustic model based on SST-HMMs and recognizes speech
based on the Bayesian prediction classification. VBEC consists of three phases: model
structure selection, retraining and recognition. The model structure is determined based on
triphone-state clustering by using the phonetic decision tree method [4] [7]. The phonetic
decision tree is a kind of binary tree that has a phonetic ?Yes/No? question attached at each
node, as shown in Figure 2. Let ? (n) denote a set of states held by a tree node n. We
start with only a root node (n = 0), which holds a set of all the triphone HMM states
? (0) for an identical center phoneme. The set of triphone states is then split into two sets,
? (nY ) and ? (nN ), which are held by two new nodes, nY and nN , respectively, as shown
in Figure 3. The partition is determined by an answer to a phonetic question such as ?is
the preceding phoneme a vowel?? or ?is the following phoneme a nasal?? We choose a
particular question for a node that maximize the gain of F m when the node is split into two
*/a(i)/*
Yes
Yes
No
No
Yes
k/a(i)/i
k/a(i)/o
Yes
Yes
?(n)
No
Yes
No
n
No
Phonetic
question
No
root n od e(n=0)
?
ts/a(i)/m
ch/a(i)/n g
nY
?(nY)
nN
?(nN)
leaf n od e
?
Figure 2: A set of all triphone HMM
states */a(i)/* is clustered based on the
phonetic decision tree method.
Figure 3: Splitting a set of triphone
HMM states ?(n) into two sets ?(nY )
?(nN ) by answering phonetic questions
according to an objective function.
nodes, and if all the questions decrease F m after splitting, we stop splitting. We continue
this splitting successively for every new set of states to obtain a binary tree, each leaf node
of which holds a clustered set of triphone states. The states belonging to the same cluster
are merged into a single state. A set of triphones is thus represented by a set of sharedstate triphone HMMs (SST-HMMs). An HMM, which represents a phonemic unit, usually
consists of a linear sequence of three or four states. A decision tree is produced specifically
for each state in the sequence, and the trees are independent of each other.
Note that in the triphone-states clustering mentioned above, we assume the following conditions to reduce computations:
? The state assignments while splitting are fixed.
? A single Gaussian distribution for one state is used.
? Contributions of the transition probabilities to the objective function are ignored.
By using these conditions, latent variables are removed. As a result, all variational posteriors and Fm can be obtained as closed forms without an iterative procedure.
Once we have obtained the model structure, we retrain the posterior distributions using the VB algorithm given in section 4. In recognition, an unknown datum xt for a
frame t is classified as the optimal phoneme class y using the predictive posterior clast
sification probability p(y|xt , O, m)
? ? p(y)p(xt |y, O, m)/p(x
?
) for the estimated model
structure m.
? Here, p(y) is the class prior obtained by language and lexcon models, and
p(xt |y, O, m)
? is the predictive density. If we approximate the true posterior p(?|y, O, m)
?
by the estimated variational
posteriors q?(?|y, O, m),
? p(xt |y, O, m)
? can be calculated by
R
p(xt |y, O, m)
? ? p(xt |y, ?, m)?
? q (?|y, O, m)d?.
?
Therefore, the optimal class y can be
obtained by
Z
0 t
0
y = arg m ax
p(y |x , O, m)
? ? arg m ax
p(y ) p(xt |y 0 , ?, m)?
? q (?|y, O, m)d?.
?
(8)
0
0
y
y
In the calculation of (8), the integral over Gaussian means and covariances for a frame can
be solved analytically to be Student distributions. Therefore, we can compute a Bayesian
predictive score for a frame, and then can compute a phoneme sequence score by using
the Viterbi algorithm. Thus, we can construct a VBEC framework for speech recognition
by selecting an appropriate model structure and estimating posterior distributions with the
VB approach, and then obtaining a recognition result based on the Bayesian prediction
classification.
Table 1: Acoustic conditions
Sampling rate
16 kHz
Quantization
16 bit
12 - order MFCC
Feature vector
with ? MFCC
Hamming
Window
Frame size/shift 25/10 ms
Table 2: Prepared HMM
# of states
# of phoneme categories
Output distribution
3 (Left to right)
27
Single Gaussian
6 Experiments
We conducted two experiments to evaluate the effectiveness of VBEC. The first experiment compared VBEC with the conventional ML-BIC/MDL method for variable amounts
of training data. In the ML-BIC/MDL, retraining and recognition are based on the ML
approach and model structure selection is based on the BIC/MDL. The second experiment
examined the robustness of the recognition performance with preset hyperparameter values
against changes in the amounts of training data.
6.1
VBEC versus ML-BIC/MDL
The experimental conditions are summarized in Tables 1 and 2. As regards the hyperparameters, the mean and variance values of the Gaussian distribution were set at ? 0 and
R0 in each root node, respectively, and the heuristics were removed for ? 0 and R0 . The
determination of ? 0 and ? 0 was still heuristic. We set ? 0 = ? 0 = 0.01, each of which were
determined experimentally. The training and recognition data used in these experiments
are shown in Table 3.
The total training data consisted of about 3,000 Japanese sentences spoken by 30 males.
These sentences were designed so that the phonemic balance was maintained. The total
recognition data consisted of 2,500 Japanese city names spoken by 25 males. Several
subsets were randomly extracted from the training data set, and each subset was used to
construct a set of SST-HMMs. As a result, 40 sets of SST-HMMs were prepared for several
subsets of training data.
Figures 4 and 5 show the recognition rate and the total number of states in a set of SSTHMMs, according to the varying amounts of training data. As shown in Figure 4, when
the number of training sentences was less than 40, VBEC greatly outperformed the MLBIC/MDL (A). With ML-BIC/MDL (A), an appropriate model structure was obtained by
BIC/M DL
maximizing an objective function lm
w.r.t. m based on BIC/MDL defined as:
# (? ? )
log T?(0) ,
(9)
2
where, l(O, m) denotes the likelihood of training data O for a model structure m, # (? ? )
denotes the number of free parameters for a set of states ?, and T?(0) denotes the total
frame number of training data for a set of states ?(0) in a root node, as shown in Figure 2.
The term # (?2 ? ) log T?(0) in Eq.(9) is regarded as a penalty term added to a likelihood, and
is dependent on the number of free parameters # (? ? ) and total frame number T?(0) of the
training data. ML-BIC/MDL (A) was based on the original definitions of BIC/MDL and
has been widely used in speech recognition [2] [5]. With such small amounts of training
data, there was a great difference between the total number of shared states with VBEC and
BIC/M DL
lm
= l(O, m) ?
Table 3: Training and recognition data
Training
Recognition
Continuous speech sentences (Acoustical Society of Japan)
100 city names (Japan Electronic Industry Development Association)
?????
??
??
??
????
??????????????
??????????????
??
?
?
??
???
????
??????????????
?????
Figure 4: Recognition rates according to the
amounts of training data based on the VBEC
and ML-BIC/MDL (A) and (B). The horizontal axis is scaled logarithmically.
???????????
????????????????????
???
????
???
????
??????????????
??????????????
??
?
??
???
????
??????????????
?????
Figure 5: Number of shared states according to the amounts of training data based on
the VBEC and ML-BIC/MDL (A) and (B).
The horizontal and vertical axes are scaled
logarithmically.
ML-BIC/MDL (A) (Figure 5). This suggests that VBEC, which does not use an asymptotic
assumption, determines the model structure more appropriately than the ML-BIC/MDL
(A), when the training data size is small.
Next, we adjusted the penalty term of ML-BIC/MDL in Eq. (9) so that the total numbers of
states for small amounts of data were as close as possible to those of VBEC (ML-BIC/MDL
(B) in Figure 5). Nevertheless, the recognition rates obtained by VBEC were about 15 %
better than those of ML-BIC/MDL (B) with fewer than 15 training sentences (Figure 4).
With such very small amounts of data, the VBEC and ML-BIC/MDL (B) model structures
were almost same (Figure 5). It is assumed that the effects of the posterior estimation and
the Bayesian prediction classification (Eq. (8)) suppressed the over-fitting of the models to
very small amounts of training data compared with the ML estimation and recognition in
ML-BIC/MDL (B).
With more than 100 training sentences, the recognition rates obtained by VBEC converged
asymptotically to those obtained by ML-BIC/MDL methods as the amounts of training data
became large.
In summary, VBEC performed as well or better for every amount of training data. This
advantage was due to the superior properties of VBEC, e.g., the appropriate determination
of the number of states and the suppression effect on over-fitting.
6.2
Influence of hyperparameter values on the quality of SST-HMMs
Throughout the construction of the model structure, the estimation of the posterior distribution, and recognition, we used a fixed combination of hyperparameter values, ? 0 = ? 0 =
0.01. In the small-scale experiments conducted in previous research [1] [3] [6] [8], the
selection of such values was not a major concern. However, when the scale of the target
application is large, the selection of hyperparameter values might affect the quality of the
models. Namely, the best or better values might differ greatly according to the amounts of
training data. Moreover, estimating appropriate hyperparameters with training SST-HMMs
needs so much time that it is impractical in speech recognition. Therefore, we examined
how robustly the SST-HMMs produced by VBEC performed against changes in the hyperparameter values with varying amounts of training data.
We varied the values of hyperparameters ? 0 and ? 0 from 0.0001 to 1, and examined the
speech recognition rates in two typical cases; one in which the amount of data was very
small (10 sentences) and one in which the amount was fairly large (150 sentences). Tables
Table 4: Recognition rates in each prior
distribution parameter when using training data of 10 sentences.
Table 5: Recognition rates in each prior
distribution parameter when using training data of 150 sentences.
?0
?0
100
10?1
10?2
10?3
10?4
100
1.0
2.2
31.2
60.3
66.5
10?1
66.3
65.9
66.1
66.2
66.6
?0
10?2
65.9
66.2
66.5
66.7
66.3
10?3
66.5
66.7
66.3
66.1
65.5
10?4
66.1
66.1
65.5
65.5
64.6
100
10?1
10?2
10?3
10?4
100
22.0
49.3
83.5
92.5
94.1
10?1
93.5
94.3
94.4
93.8
93.2
?0
10?2
94.0
93.9
93.2
93.3
92.3
10?3
93.1
93.3
92.3
92.5
92.3
10?4
92.3
92.5
92.3
92.4
92.2
4 and 5 show the recognition rates for each combination of hyperparameters. We can
see that the hyperparameter values for acceptable performance are broadly distributed for
both very small and fairly large amounts of training data. Moreover, roughly the ten best
recognition rates are highlighted in the tables. The combinations of hyperparameter values
that achieved the highlighted recognition rates were similar for the two different amounts of
training data. Namely, appropriate combinations of hyperparameter values can consistently
provide good performance levels regardless of the varying amounts of training data.
In summary, the hyperparameter values do not greatly influence the quality of the SSTHMMs. This suggests that it is not necessary to select the hyperparameter values very
carefully.
7 Conclusion
In this paper, we proposed VBEC, which constructs SST-HMMs based on the VB approach,
and recognizes speech based on the Bayesian prediction classification. With VBEC, the
model structure of SST-HMMs is adaptively determined according to the amounts of given
training data, and therefore a robust speech recognition system can be constructed. The first
experimental results, obtained by using real speech recognition tasks, showed the effectiveness of VBEC. In particular, when the training data size was small, VBEC significantly
outperformed conventional methods. The second experimental results suggested that it is
not necessary to select the hyperparameter values very carefully. From these results, we
conclude that VBEC provides a completely Bayesian framework for speech recognition
which effectively hundles the sparse data problem.
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1,290 | 2,175 | The RA Scanner: Prediction of Rheumatoid
Joint Inflammation Based on Laser Imaging
1
Anton Schwaighofer1 2
TU Graz, Institute for Theoretical Computer Science
Inffeldgasse 16b, 8010 Graz, Austria
http://www.igi.tugraz.at/aschwaig
Volker Tresp, Peter Mayer
Siemens Corporate Technology, Department of Neural Computation
Otto-Hahn-Ring 6, 81739 Munich, Germany
http://www.tresp.org,[email protected]
2
Alexander K. Scheel, Gerhard Muller
?
University G?ottingen, Department of Medicine, Nephrology and Rheumatology
Robert-Koch-Stra?e 40, 37075 G?ottingen, Germany
[email protected],[email protected]
Abstract
We describe the RA scanner, a novel system for the examination of patients suffering from rheumatoid arthritis. The RA scanner is based on
a novel laser-based imaging technique which is sensitive to the optical
characteristics of finger joint tissue. Based on the laser images, finger
joints are classified according to whether the inflammatory status has
improved or worsened. To perform the classification task, various linear and kernel-based systems were implemented and their performances
were compared. Special emphasis was put on measures to reliably perform parameter tuning and evaluation, since only a very small data set
was available. Based on the results presented in this paper, it was concluded that the RA scanner permits a reliable classification of pathological finger joints, thus paving the way for a further development from
prototype to product stage.
1 Introduction
Rheumatoid arthritis (RA) is the most common inflammatory arthropathy with 1?2% of the
population being affected. This chronic, mostly progressive disease often leads to early disability and joint deformities. Recent studies have convincingly shown that early treatment
and therefore an early diagnosis is mandatory to prevent or at least delay joint destruction [2]. Unfortunately, long-term medication with disease modifying anti-rheumatic drugs
(DMARDs) often acts very slowly on clinical parameters of inflammation, making it difficult to find the right drug for a patient within adequate time. Conventional radiology,
such as magnetic resonance imaging (MRI) and ultrasound, may provide information on
soft tissue changes, yet these techniques are time-consuming and?in the case of MRI?
costly. New imaging techniques for RA diagnosis should thus be non-invasive, of low cost,
examiner independent and easy to use.
Following recent experiments on absorption and scattering coefficients of laser light in
joint tissue [6], a prototype laser imaging technique was developed [7]. As part of the prototype development, it became necessary to analyze if the rheumatic status of a finger joint
can be reliably classified on the basis of the laser images. Aim of this article is to provide an overview of this analysis. Employing different linear and kernel-based classifiers,
we will investigate the performance of the laser imaging technique to predict the status
of the rheumatic joint inflammation. Provided that the accuracy of the overall system is
sufficiently high, the imaging technique and the automatic inflammation classification can
be combined into a novel device that allows an inexpensive and objective assessment of
inflammatory joint changes.
The paper is organized as follows. In Sec. 2 we describe the RA scanner in more detail, as
well as the process of data acquisition. In Sec. 3 we describe the linear and kernel-based
classifiers used in the experiments. In Sec. 4 we describe how the methods were evaluated
and compared. We present experimental results in Sec. 5. Conclusions and an outlook are
given in Sec. 6.
2 The RA Scanner
The rheumatoid arthritis (RA) scanner provides a new medical imaging technique, developed specifically for the diagnosis of RA in finger joints. The RA scanner [7] allows the
in vivo trans-illumination of finger joints with laser light in the near infrared wavelength
range. The scattered light distribution is detected by a camera and is used to assess the
inflammatory status of the finger joint. Example images, taken from an inflamed joint and
from a healthy control, are shown in Fig. 1.
Starting out from the laser images, image pre-processing is used to obtain a description of
each laser image by nine numerical features. A brief description of the features is given in
Fig. 1. Furthermore for each finger joint examined, the circumference is measured using a
conventional measuring tape. The nine image features plus the joint circumference make up
the data that is used in the classification step of the RA scanner to predict the inflammatory
status of the joint.
2.1 Data Acquisition
One of the clinically important questions is to know as early as possible if a prescribed
medication improves the state of rheumatoid arthritis. Therefore the goal of the classification step in the RA scanner is to decide?based on features extracted from the laser
images?if there was an improvement of arthritis activity or if the joint inflammation remained unchanged or worsened.
The data for the development of the RA scanner stems from a study on 22 patients with
rheumatoid arthritis. Data from 72 finger joints were used for the study. All of these 72
finger joints were examined at baseline and during a follow-up visit after a mean duration of
42 days. Earlier data from an additional 20 patients had to be discarded since experimental
conditions were not controlled properly.
Each joint was examined and the clinical arthritis activity was classified from 0 (inactive,
not swollen, tender or warm) to 3 (very active) by a rheumatologist. The characteristics of
joint tissue was recorded by the above described laser imaging technique. In a preprocess-
(a) Laser image of a healthy finger joint
(b) Laser image of an inflamed finger
joint. The inflammation changes the joint
tissue?s absorption coefficient, giving a
darker image.
Figure 1: Two examples of the light distribution captured by the RA scanner. A laser beam
is sent through the finger joint (the finger tip is to the right, the palm is on the left), the light
distribution below the joint is captured by a CCD element. To calculate the features, first
a horizontal line near the vertical center of the finger joint is selected. The distribution of
light intensity along that line is bell-shaped. The features used in the classification task are
the maximum light intensity, the curvature of the light intensity at the maximum and seven
additional features based on higher moments of the intensity curve.
ing step nine features were derived from the distribution of the scattered laser light (see
Fig. 1). The tenth feature is the circumference of the finger joint.
Since there are high inter-individual variations in optical joint characteristics, it is not possible to tell the inflammatory status of a joint from one single image. Instead, special
emphasis was put on the intra-individual comparison of baseline and follow-up data. For
every joint examined, data from baseline and follow-up visit were compared and changes
in arthritis activity were rated as improvement, unchanged or worsening.
This rating divided the data into two classes: Class 1 contains the joints where an improvement of arthritis activity was observed (a total of 46 joints), and class 1 are the
joints that remained unchanged or worsened (a total of 26 joints). For all joints, the differences in feature values between baseline and follow-up visit were computed.
3 Classification Methods
In this section, we describe the employed linear and kernel-based classification methods,
where we focus on design issues.
3.1 Gaussian Process Classification (GPC)
In Gaussian processes, a function
f x
M
? w j K x x j ?
(1)
j 1
is described as a superposition of M kernel functions K x x j ? , defined for each of the
M training data points x j , with weight w j . The kernel functions are parameterized by the
vector ? ?0
?d . In two-class Gaussian process classification, the logistic transfer
function ? f x
1 e f x 1 is applied to the prediction of a Gaussian process to
produce an output which can be interpreted as ? x , the probability of the input x belonging
to class 1 [10].
In the experiment we chose the Gaussian kernel function
K x x j ?
?0 exp
1
x x j T diag ?21
?2d
2
1 x
x j
(2)
with input length scales ? 1 ?d where d is the dimension of the input space. diag ? 21
?2d
denotes a diagonal matrix with entries ? 21
?2d . For training the Gaussian process classifier
(that is, determining the posterior probabilities of the parameters w 1
wM ?0 ?d ) we
used a full Bayesian approach, implemented with Readford Neal?s freely available FBM
software.1
3.2 Gaussian Process Regression (GPR)
In
GPR we treat the classification problem as a regression problem with target values
1 1 , i.e. we do not apply the logistic transfer function as in the last subsection.
Any GP output 0 is treated as indicating an example from class 0, any output 0 as
an indicator for class 1.The disadvantage is that the GPR prediction cannot be treated as
a posterior class probability; the advantage is that the fast and non-iterative training algorithms for GPR can be applied. GPR for classification problems can be considered as
special cases of Fisher discriminant analysis with kernels [4] and of least squares support
vector machines [9].
The parameters ? ? 0
?d of the covariance function Eq. (2) were chosen by maximizing the posterior probability of ?, P ? t X ? P t X ? P ? , via a scaled conjugate
gradient method. Later on, this method will be referred to as ?GPR Bayesian?. Results are
also given for a simplified covariance function with ? 0 1, ?1 ?2
?d r, where
the common length scale r was chosen by cross-validation (later on referred to as ?GPR
crossval?).
3.3 Support Vector Machine (SVM)
The SVM is a maximum margin linear classifier. As in Sec. 3.2, the SVM classifies a
pattern according to the sign of f x in Eq. (1). The difference is that the weights w
w1
wM T in the SVM minimize the particular cost function [8]
wT Kw
M
? Ci 1
yi f xi
(3)
i 1
where sets all negative arguments to zero. Here, y i
1 1 is the class label for
training point x i . Ci 0 is a constant that determines the weight of errors on the training
data, and K is an M
M matrix containing the amplitudes of the kernel functions at the
training data, i.e. K i j K xi x j ? . The motivation for this cost function stems from statistical learning theory [8]. Many authors have previously obtained excellent classification
results by using the SVM. One particular feature of the SVM is the sparsity of the solution
vector w, that is, many elements w i are zero.
In the experiments, we used both an SVM with linear kernel (?SVM linear?) and an SVM
with a Gaussian kernel (?SVM Gaussian?), equivalent to the Gaussian process kernel
Eq. (2), with ? 0 1, ?1 ?2
?d r. The kernel parameter r was chosen by
cross-validation.
1 As a prior distribution for kernel parameter ?
0
we chose a Gamma distribution. ?1 ?d are samples of a hierarchical Gamma distribution. In FBM syntax, the prior is 0.05:0.5 x0.2:0.5:1.
Sampling from the posterior distribution was done by persistent hybrid Monte Carlo, following the
example of a 3-class problem in Neal [5].
To compensate for the unbalanced distribution of classes, the penalty term C i was chosen
to be 0 8 for the examples from the larger class and 1 for the smaller class. This was found
empirically to give the best balance of sensitivity and specificity (cf. Sec. 4). A formal
treatment of this issue can be found in Lin et al. [3].
3.4 Generalized Linear Model (GLM)
A GLM for binary responses is built up from a linear model for the input data, and the
model output f x w T x is in turn input to the link function. For Bernoulli distributions,
the natural link function [1] is the logistic transfer function ? f x 1 e f x
1 . The
overall output of the GLM ? f x computes ? x , the probability of the input x belonging
to class 1. Training of the linear model was done by iteratively re-weighted least squares
(IRLS).
4 Training and Evaluation
One of the challenges in developing the classification system for the RA scanner is the low
number of training examples available. Data was collected through an extensive medical
study, but only data from 72 fingers were found to be suitable for further use. Further
data can only be acquired in carefully controlled future studies, once the initial prototype
method has proven sufficiently successful.
Training From the currently available 72 training examples, classifiers need to be trained
and evaluated reliably. Part of the standard methodology for small data sets is N-fold crossvalidation, where the data are partitioned into N equally sized sets and the system is trained
on N 1 of those sets and tested on the Nth data set left out. Since we wish to make use of
as much training data as possible, N 36 seemed the appropriate choice 2 , giving test sets
with two examples in each iteration. For some of the methods model parameter needed
to be tuned (for example, choosing SVM kernel width), where again cross-validation is
employed. The nested cross-validation ensures that in no case any of the test examples is
used for training or to tune parameters, leading to the following procedure:
Run 36 fold CV
For Bayesian methods or methods without tunable parameters
(SVM linear, GPC, GPR Bayesian, GLM):
Use full training set to tune and train classifier
For Non-Bayesian methods (SVM Gaussian, GPR crossval):
Run 35 fold CV on the training set
choose parameters to minimise CV error
train classifier with chosen parameters
evaluate the classifier on the 2 example test set
Significance Tests In order to compare the performance of two given classification methods, one usually employs statistical hypothesis testing. We use here a test that is best suited
for small test sets, since it takes into account the outcome on the test examples one by one,
thus matching our above described 36-fold cross validation scheme perfectly. A similar
test has been used by Yang and Liu [11] to compare text categorization methods.
Basis of the test are two counts b (how many examples in the test set were correctly classified by method B, but misclassified by method A) and c (number of examples misclassified
by B, correctly classified by A). We assume that examples misclassified (resp. correctly
classified) by both A and B do not contribute to the performance difference. We take the
2 Thus,
it is equivalent to a leave-one-out scheme, yet with only half the time consumption.
Method
Error rate
GLM
GLM, reduced feature set
GPR Bayesian
GPR crossval
GPC
SVM linear
SVM linear, reduced feature set
SVM Gaussian
20 83%
16 67%
13 89%
22 22%
23 61%
22 22%
16 67%
20 83%
Table 1: Error rates of different classification methods on the rheumatoid arthritis prediction problem. All error rates have been computed by 36-fold cross-validation. ?Reduced
feature set? indicates experiments where a priori feature selection has been done
counts b and c as the sufficient statistics of a binomial random variable with parameter ?,
where ? is the proportion of cases where method A performs better than method B.
The null hypothesis H0 is that the parameter ? 0 5, that is, both methods A and B have
the same performance. Hypothesis H 1 is that ? 0 5. The test statistics under the null
hypothesis is the Binomial distribution Bi i b c ?) with parameter ? 0 5. We reject
the null hypothesis if the probability of observing a count k c under the null hypothesis
P k c ?bi cc Bi i b c ? 0 5 is sufficiently small.
ROC Curves In medical diagnosis, biometrics and other areas, the common means of
assessing a classification method is the receiver operating characteristics (ROC) curve. An
ROC curve plots sensitivity versus 1-specificity 3 for different thresholds of the classifier
output. Based on the ROC curve it can be decided how many false positives resp. false
negatives one is willing to tolerate, thus helping to tune the classifier threshold to best suit
a certain application.
Acquiring the ROC curve typically requires the classifier output on an independent test set.
We instead use the union of all test set outputs in the cross-validation routine. This means
that the ROC curve is based on outputs of slightly different models, yet this still seems to
be the most suitable solution for such few data. For all classifiers we assess the area of the
ROC curve and the cross-validation error rate. Here the above mentioned threshold on the
classifier output is chosen such that sensitivity equals specificity.
5 Results
Tab. 1 lists error rates for all methods listed in Sec. 3. Gaussian process regression (GPR
Bayesian) with an error rate of 14% clearly outperforms all other methods, which all
achieve comparable error rates in the range of 20
24%. We attribute the good performance of GPR to its inherent feature relevance detection, which is done by adapting the
length scales ?i in the covariance function Eq. (2), i.e. a large ? i means that the i-th feature
is essentially ignored.
Surprisingly, Gaussian process classification implemented with Markov chain Monte Carlo
sampling (GPC) showed rather poor performance. We currently have no clear explanation
for this fact. We found no indications of convergence problems, furthermore we achieved
similar results with different sampling schemes.
In an additional experiment we wanted to find out if classification results could be improved
3 sensitivity
true positives
true positives false negatives
specificity
true negatives
true negatives false positives
1
0.9
0.8
Sensitivity
0.7
0.6
0.5
0.4
0.3
0.2
GPR Bayesian
GLM, reduced feature set
SVM linear, reduced feature set
0.1
0
0
0.2
0.4
0.6
1?Specificity
0.8
1
Figure 2: ROC curves of the best classification methods, both on the full data set and on
a reduced data set where a priori feature selection was used to retain only the three most
relevant features. Integrating the area under the ROC curves gives similar results for all
three methods, with an area of 0 86 for SVM linear and GLM, and 0 84 for GPR Bayesian
by using only a subset of input features 4 . We found that only the performance of the two
linear classifiers (GLM and SVM linear) could be improved by the input feature selection.
Both now achieve an error rate of 16 67%, which is slightly worse than GPR on the full
feature set (see Tab. 1).
Significance Tests Using the statistical hypothesis test described in the previous section,
we compared all classification methods pairwise. It turned out the three best methods
(GPR Bayesian, and GLM and SVM linear with reduced feature set) perform better than
all other methods at a confidence level of 90% or more. Amongst the three best methods,
no significant difference could be observed.
ROC Curves For the three best classification methods (GPR Bayesian, and GLM and
SVM linear with reduced feature set), we have plotted the receiver operating characteristics
(ROC) curve in Fig. 2. According to the ROC curve a sensitivity of 80% can be achieved
with a specificity at around 90%. GPR Bayesian seems to give best results, both in terms
of error rate and shape of the ROC curve.
Summary To summarize, when the full set of features was used, best performance was
obtained with GPR Bayesian. We attribute this to the inherent input relevance detection
mechanisms of this approach. Comparable yet slightly worse results could be achieved
by performing feature selection a priori and reducing the number of input features to the
three most significant ones. In particular, the error rates of linear classifiers (GLM and
linear SVM) improved by this feature selection, whereas more complex classifiers did not
benefit. We can draw the important conclusion that, using the best classifiers, a sensitivity
of 80% can be reached at a specificity of approximately 90%.
6 Conclusions
In this paper we have reported results of the analysis of a prototype medical imaging system, the RA scanner. Aim of the RA scanner is to detect soft tissue changes in finger joints,
4 This was done with the input relevance detection algorithm of the neural network tool SENN,
a variant of sequential backward elimination where the feature that least affects the neural network
output is removed. The feature set was reduced to the three most relevant ones.
which occur in early stages of rheumatoid arthritis (RA). Basis of the RA scanner is a novel
laser imaging technique that is sensitive to inflammatory soft tissue changes.
We have analyzed whether the laser images are suitable for an accurate prediction of the
inflammatory status of a finger joint, and which classification methods are best suited for
this task. Out of a set of linear and kernel-based classification methods, Gaussian processes
regression performed best, followed closely by generalized linear models and the linear
support vector machine, the latter two operating on a reduced feature set. In particular, we
have shown how parameter tuning and classifier training can be done on basis of the scarce
available data. For the RA prediction task, we achieved a sensitivity of 80% at a specificity
of approximately 90%. These results show that a further development of the RA scanner is
desirable.
In the present study the inflammatory status is assessed by a rheumatologist, taking into
account the patients subjective degree of pain. Thus we may expect a certain degree of label
noise in the data we have trained the classification system on. Further developments of the
classification system in the RA scanner will thus incorporate information from established
medical imaging systems such as magnetic resonance imaging (MRI). MRI is known to
provide accurate information about soft tissue changes in finger joints, yet is too costly to
be routinely used for RA diagnosis. By incorporating MRI results into the RA scanner?s
classification system, we expect to significantly improve the overall accuracy.
Acknowledgments AS gratefully acknowledges support through an Ernst-von-Siemens
scholarship. Thanks go to Radford Neal for making his FBM software available to the
public, and to Ian Nabney and Chris Bishop for the Netlab toolbox.
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1,291 | 2,176 | Automatic Derivation of Statistical Algorithms:
The EM Family and Beyond
Alexander G. Gray
Carnegie Mellon University
[email protected]
Bernd Fischer and Johann Schumann
RIACS / NASA Ames
fisch,schumann @email.arc.nasa.gov
Wray Buntine
Helsinki Institute for IT
[email protected]
Abstract
Machine learning has reached a point where many probabilistic methods can be understood as variations, extensions and combinations of a
much smaller set of abstract themes, e.g., as different instances of the
EM algorithm. This enables the systematic derivation of algorithms customized for different models. Here, we describe the AUTO BAYES system which takes a high-level statistical model specification, uses powerful symbolic techniques based on schema-based program synthesis and
computer algebra to derive an efficient specialized algorithm for learning
that model, and generates executable code implementing that algorithm.
This capability is far beyond that of code collections such as Matlab toolboxes or even tools for model-independent optimization such as BUGS
for Gibbs sampling: complex new algorithms can be generated without new programming, algorithms can be highly specialized and tightly
crafted for the exact structure of the model and data, and efficient and
commented code can be generated for different languages or systems.
We present automatically-derived algorithms ranging from closed-form
solutions of Bayesian textbook problems to recently-proposed EM algorithms for clustering, regression, and a multinomial form of PCA.
1 Automatic Derivation of Statistical Algorithms
Overview. We describe a symbolic program synthesis system which works as a ?statistical
algorithm compiler:? it compiles a statistical model specification into a custom algorithm
design and from that further down into a working program implementing the algorithm
design. This system, AUTO BAYES, can be loosely thought of as ?part theorem prover, part
Mathematica, part learning textbook, and part Numerical Recipes.? It provides much more
flexibility than a fixed code repository such as a Matlab toolbox, and allows the creation
of efficient algorithms which have never before been implemented, or even written down.
AUTO BAYES is intended to automate the more routine application of complex methods in
novel contexts. For example, recent multinomial extensions to PCA [2, 4] can be derived
in this way.
The algorithm design problem. Given a dataset and a task, creating a learning method
can be characterized by two main questions: 1. What is the model? 2. What algorithm will
optimize the model parameters? The statistical algorithm (i.e., a parameter optimization
algorithm for the statistical model) can then be implemented manually. The system in
this paper answers the algorithm question given that the user has chosen a model for the
data,and continues through to implementation. Performing this task at the state-of-the-art
level requires an intertwined meld of probability theory, computational mathematics, and
software engineering. However, a number of factors unite to allow us to solve the algorithm
design problem computationally: 1. The existence of fundamental building blocks (e.g.,
standardized probability distributions, standard optimization procedures, and generic data
structures). 2. The existence of common representations (i.e., graphical models [3, 13] and
program schemas). 3. The formalization of schema applicability constraints as guards. 1
The challenges of algorithm design. The design problem has an inherently combinatorial
nature, since subparts of a function may be optimized recursively and in different ways.
It also involves the use of new data structures or approximations to gain performance. As
the research in statistical algorithms advances, its creative focus should move beyond the
ultimately mechanical aspects and towards extending the abstract applicability of already
existing schemas (algorithmic principles like EM), improving schemas in ways that generalize across anything they can be applied to, and inventing radically new schemas.
2 Combining Schema-based Synthesis and Bayesian Networks
Statistical Models.
Externally,
A
UTO
B
AYES
has
the
look
and feel of
2 const int n_points as ?nr. of data points?
a compiler. Users specify their model
3
with 0 < n_points;
4 const int n_classes := 3 as ?nr. classes?
of interest in a high-level specification
5
with 0 < n_classes
language (as opposed to a program6
with n_classes << n_points;
ming language). The figure shows the
7 double phi(1..n_classes) as ?weights?
specification of the mixture of Gaus8
with 1 = sum(I := 1..n_classes, phi(I));
9 double mu(1..n_classes);
sians example used throughout this
9 double sigma(1..n_classes);
paper.2 Note the constraint that the
10 int c(1..n_points) as ?class labels?;
sum of the class probabilities must
11 c ? disc(vec(I := 1..n_classes, phi(I)));
equal one (line 8) along with others
12 data double x(1..n_points) as ?data?;
(lines 3 and 5) that make optimization
13 x(I) ? gauss(mu(c(I)), sigma(c(I)));
of the model well-defined. Also note
14 max pr(x| phi,mu,sigma ) wrt phi,mu,sigma ;
the ability to specify assumptions of
the kind in line 6, which may be used by some algorithms. The last line specifies the goal
with respect to the painference task: maximize the conditional probability pr
rameters , , and . Note that moving the parameters across to the left of the conditioning
bar converts this from a maximum likelihood to a maximum a posteriori problem.
1 model mog as ?Mixture of Gaussians?;
Computational logic and theorem proving. Internally, AUTO BAYES uses a class of techniques known as computational logic which has its roots in automated theorem proving.
AUTO BAYES begins with an initial goal and a set of initial assertions, or axioms, and adds
new assertions, or theorems, by repeated application of the axioms, until the goal is proven.
In our context, the goal is given by the input model; the derived algorithms are side effects
of constructive theorems proving the existence of algorithms for the goal.
1
Schema guards vary widely; for example, compare Nead-Melder simplex or simulated annealing (which require only function evaluation), conjugate gradient (which require both Jacobian and
Hessian), EM and its variational extension [6] (which require a latent-variable structure model).
2
Here, keywords have been underlined and line numbers have been added for reference in the text.
The as-keyword allows annotations to variables which end up in the generated code?s comments.
Also, n classes has been set to three (line 4), while n points is left unspecified. The class
variable and single data variable are vectors, which defines them as i.i.d.
Computer algebra. The first core element which makes automatic algorithm derivation
feasible is the fact that we can mechanize the required symbol manipulation, using computer algebra methods. General symbolic differentiation and expression simplification are
capabilities fundamental to our approach. AUTO BAYES contains a computer algebra engine using term rewrite rules which are an efficient mechanism for substitution of equal
quantities or expressions and thus well-suited for this task.3
Schema-based synthesis. The computational cost of full-blown theorem proving grinds
simple tasks to a halt while elementary and intermediate facts are reinvented from scratch.
To achieve the scale of deduction required by algorithm derivation, we thus follow a
schema-based synthesis technique which breaks away from strict theorem proving. Instead,
we formalize high-level domain knowledge, such as the general EM strategy, as schemas.
A schema combines a generic code fragment with explicitly specified preconditions which
describe the applicability of the code fragment. The second core element which makes
automatic algorithm derivation feasible is the fact that we can use Bayesian networks to
efficiently encode the preconditions of complex algorithms such as EM.
First-order logic representation of Bayesian netNclasses
works. A first-order logic representation of Bayesian
?
?
networks was developed by Haddawy [7]. In this
framework, random variables are represented by
functor symbols and indexes (i.e., specific instances
?
x
c
of i.i.d. vectors) are represented as functor arguments.
discrete
gauss
Nclasses
Since unknown index values can be represented by
Npoints
implicitly universally quantified Prolog variables, this
approach allows a compact encoding of networks involving i.i.d. variables or plates [3]; the
figure shows the initial network for our running example. Moreover, such networks correspond to backtrack-free datalog programs, allowing the dependencies to be efficiently
computed. We have extended the framework to work with non-ground probability queries
since we seek to determine probabilities over entire i.i.d. vectors and matrices. Tests for independence on these indexed Bayesian networks are easily developed in Lauritzen?s framework which uses ancestral sets and set separation [9] and is more amenable to a theorem
prover than the double negatives of the more widely known d-separation criteria. Given a
Bayesian network, some probabilities can easily be extracted by enumerating the component probabilities at each node:
Lemma
1. Let
be
sets
of variables over a Bayesian network with
. Then
and parents
hold 4 in the corresponding dependency
descendents
graph iff the following probability statement holds:
parents
!
"$#&%
&('
parents
(') +*
Symbolic probabilistic inference. How can probabilities not satisfying these conditions
be converted to symbolic expressions? While many general schemes for inference on networks exist, our principal hurdle is the need to perform this over symbolic expressions incorporating real and integer variables from disparate real or infinite-discrete distributions.
For instance, we might wish to compute the full maximum a posteriori probability for
the mean and variance vectors of a Gaussian mixture model under a Bayesian framework.
While the sum-product framework of [8] is perhaps closer to our formulation, we have out
of necessity developed another scheme that lets us extract probabilities on a large class of
mixed discrete and real, potentially indexed variables, where no integrals are needed and
3
Popular symbolic packages such as Mathematica contain known errors allowing unsound derivations; they also lack the support for reasoning with vector and matrix quantities.
-,
.
/,
.
4
Note that
descendents
and
parents
.
all marginalization is done by summing out discrete variables. We give the non-indexed
case below; this is readily extended to indexed variables (i.e., vectors).
Lemma
2.
descendents
holds and ancestors
is independent of given
iff
there
exists
a
set
of
variables
that Lemma 1 holds if we replace
by
. Moreover, the unique minimal such
set
satisfying
these
conditions
is
given
by
ancestors
ancestors
such that ancestors
is independent
Lemma
be a subset of
descendents
3. Let
of
&
ancestors
given
.
Then
Lemma
2
holds
if
we
replace
by
and
by . Moreover, there is a unique maximal set
satisfying these
conditions.
Lemma 2 lets us evaluate a probability by a summation:
&
' +
& )
" (# Dom %
while Lemma 3 lets us evaluate a probability by a summation and a ratio:
&
Since the lemmas also show minimality of the sets and
&
&
, they also give the minimal
conditions under which a probability can be evaluated by discrete summation without integration. These inference lemmas are operationalized as network decomposition schemas.
However, we usually attempt to decompose a probability into independent components
before applying this schema.
3 The AUTOBAYES System ? Implementation Outline
Levels of representation. Internally, our system uses three conceptually different levels of
representation. Probabilities (including logarithmic and conditional probabilities) are the
most abstract level. They are processed via methods for Bayesian network decomposition
or match with core algorithms such as EM. Formulae are introduced when probabilities of
the form parents
are detected, either in the initial network, or after the application of network decompositions. Atomic probabilities (i.e., is a single variable) are
directly replaced by formulae based on the given distribution and its parameters. General
probabilities are decomposed into sums and products of the respective atomic probabilities. Formulae are ready for immediate optimization using symbolic or numeric methods
but sometimes they can be decomposed further into independent subproblems. Finally, we
use imperative intermediate code as the lowest level to represent both program fragments
within the schemas as well as the completely constructed programs. All transformations
we apply operate on or between these levels.
Transformations for optimization. A number of different kinds of transformations are
available. Decomposition of a problem into independent subproblems is always done. Decomposition of probabilities is driven by the Bayesian network; we have a separate system
for handling decomposition of formulae. A formula can be decomposed along a loop, e.g.,
for "!
? is transformed into a for-loop over subproblems
the problem
?optimize
for !
$#&%
? is transformed
?optimize for !
.? More commonly,
?optimize
into the two subprograms ?optimize for !
? and ?optimize for %
.? The lemmas
given earlier are applied to change the level of representation and are thus used for simplification of probabilities. Examples of general expression simplification include simplifying
the log of a formula, moving a summation inwards, and so on. When necessary, symbolic
differentiation is performed. In the initial specification or in intermediate representations,
likelihoods (i.e., subexpressions of the form %
) are identified and simplified into linear expression with terms such as mean
and mean
. The statistical
algorithm schemas currently implemented include EM, k-means, and discrete model selection. Adding a Gibbs sampling schema would yield functionality comparable to that of
BUGS [14]. Usually, the schemas require a particular form of the probabilities involved;
they are thus tightly coupled to the decomposition and simplification transformations. For
is
example, EM is a way of dealing with situation where Lemma 2 applies but where
indexed identically to the data.
Code and test generation. From the intermediate code, code in a particular target language may be generated. Currently, AUTO BAYES can generate C++ and C which can be
used in a stand-alone fashion or linked into Octave or Matlab (as a mex file). During this
code-generation phase, most of the vector and matrix expressions are converted into forloops, and various code optimizations are performed which are impossible for a standard
compiler. Our tool does not only generate efficient code, but also highly readable, documented programs: model- and algorithm-specific comments are generated automatically
during the synthesis phase. For most examples, roughly 30% of the produced lines are
comments. These comments provide explanation of the algorithm?s derivation. A generated HTML software design document with navigation capabilities facilitates code understanding and reading. AUTO BAYES also automatically generates a program for sampling
from the specified model, so that closed-loop testing with synthetic data of the assumed
distributions can be done. This can be done using simple forward sampling.
4 Example: Deriving the EM Algorithm for Gaussian Mixtures
1. User specifies model. First, the user specifies the model as shown in Section 2.
2. System parses model to obtain underlying Bayes net. From the model, the underlying
Bayesian network is derived and represented internally as a directed graph. For visualization, AUTO BAYES can also produce a graph drawing as shown in Section 2.
3. System observes hidden-variable structure in Bayesian network. The system attempts to decompose the optimization goal into independent parts, but finds that it cannot.
However, it then finds that the probability in the initial optimization statement matches the
conditions of Lemma 2 and that the network describes a latent variable model.
4. System invokes abstract EM
schema max Pr
wrt
family schema. This triggers the
...
EM-schema, whose overall structure
C = ?[initialize
];)
is shown. The syntactic structure of
while
+ wrt ;
the current subproblem must match
/* M-step */ max Pr
the first argument of the schema;
/* E-step */ calculate Pr
;
if additional applicability constraints
?
(not shown here) hold, this schema is
executed. It constructs a piece of code which is returned in the variable . This code fragment can contain recursive calls to other schemas (denoted by !
" ) which return code for
subproblems which then is inserted into the schema, such as converging, a generic con
vergence criterion here imposed over the variables . Note that the schema actually
implements an ME-algorithm (i.e., starts the loop with the M-step) because the initialization already serves as an E-step.
The system identifies the discrete variable # as the
#
. For representation of the distribution of the hidden
single hidden variable, i.e., $
variable a matrix % is generated, where % '& is the probability that the ( -th point falls into
the ) -th class. AUTO BAYES then constructs the new distribution c(I) ? disc(vec(J
:= 1..n classes, q(I, J)) which replaces the original distribution in the following
recursive calls of AUTO BAYES.
while converging
for
for &
Pr &
"
!
!
max Pr
wrt #
5. E-step: System performs marginalization. The freshly introduced distribution for
#
implies that # can be eliminated from
the objective function by summing over % $ .
This gives us the partial program shown in
the internal pseudocode.
6. M-step: System recursively decomposes optimization problem. AUTO BAYES
is recursively
called with the new
goal
9 ;9 = 9 > 9 ? 9
= 9 #> 9 ? 9
wrt
.
max 6,7 8 Pr : <
Now, the Bayesian network
decomposition
1
#
,
schema applies with
max %32 (+* 4%5)& (+*
wrt
, revealing that is independent of
, thus the optimization problem can be decomposed into two optimization subproblems:
9
@9 : 9 #> 9 ;? 9
A9 9
B9 =
=9
max Pr <
wrt > ? and max Pr :
wrt
.
while converging
for
for &
Pr &
for &
max %')& (+* , - .0/ Pr 1 wrt
7. System
unrolls i.i.d. Avectors.
The first subgoal from the decomposition schema,
19 : 9 > 9 ;? 9
9 ;9
max Pr <
wrt > ? , can be unrolled over the independent and identically distributed vector using an index decomposition schema which moves expressions out of
loops (sums or products) when they are not dependent on the loop index. Since # and
are co-indexed, unrolling proceeds over both (also independent and identically distributed)
#9 ;9
A9 9
vectors in parallel: max DE)C FHG Pr < E : E > ?
wrt > ? .
8. System
identifies and solves Gaussian elimination
problem. The probability Pr
#
#
is atomic because parents
. It can thus be replaced by the
appropriately instantiated Gaussian density function. Because the strictly monotone IJ#K L
function can first be applied
to the objective function of the maximization, it becomes
G
> N P
A> 9 ? 9
?
PRQ < E
67S8UT VW
67S8 N wrt
max % EC FHG %5N0M FHG+O E N
. Another application of
index decomposition allows solution for the two scalars & and & . Gaussian elimination
is then used
to solve this subproblem analytically, yielding the sequence of expressions
& % ) X
% & % X
% & and & % X
% '& Y &
% ) X
% & .
9. System
identifies
and solves Lagrange multiplier problem. The second subgoal
B9 = 9
=9
max Pr :
wrt
can be unrolled over the i.i.d. vector # as before. The specifica
[Z
tion condition % &B X
&
creates a constrained maximization problem in the vector
which is solved by an application of the Lagrange
multiplier schema. This in turn results
in two subproblems for a single instance & and for the multiplier which are both solved
symbolically. Thus, the usual EM algorithm for Gaussian mixtures is derived.
10. System checks and optimizes pseudocode.
During the synthesis process,
AUTO BAYES accumulates a number of constraints which have to hold to ensure proper
operation of the code (e.g., absence of divide-by-zero errors). Unless these constraints can
be resolved against the model (e.g., ]\_^ ), AUTO BAYES automatically inserts run-time
checks into the code. Before finally generating C/C++
code, the pseudocode is optimized
`Z
using information from the specification (e.g., % &B X
&
) and the domain. Thus, optimizations well beyond the capability of a regular compiler can be done.
11. System translates pseudocode to real code in desired language. Finally,
AUTO BAYES converts the intermediate code into code of the desired target system. The
source code contains thorough comments detailing the mathematics implemented. A regular compiler containing generic performance optimizations not repeated by AUTO BAYES
turns the code into an executable program. A program for sampling from a mixture of
Gaussians is also produced for testing purposes.
5 Range of Capabilities
Here, we discuss 18 examples which have been successfully handled by AUTO BAYES,
ranging from simple textbook examples to sophisticated EM models and recent multinomial versions of PCA. For each entry, the table below gives a brief description, the number
of lines of the specification and synthesized C++ code (loc), and the runtime to generate the
code (in secs., measured on a 2.2GHz Linux system). Correctness was checked for these
examples using automatically-generated test data and hand-written implementations.
Bayesian textbook examples. Simple textbook examples, like Gaussian with simple
prior
, Gaussian with inverse gamma prior
have
, or Gaussian with conjugate prior
closed-form solutions. The symbolic system of AUTO BAYES can actually find these solutions and thus generate short and efficient
code. However, a slight relaxation of the prior
on (Gaussian with semi-conjugate prior, ) requires an iterative numerical solver.
Gaussians in action.
is a Gaussian change-detection model. A slight extension of
our running example, integrating several features, yields a Gaussian Bayes classifier model
. has been successfully tested on various standard benchmarks [1], e.g., the Abalone
dataset. Currently, the number of expected classes has to be given in advance.
Mixture models and EM. A wide range of -Gaussian mixture models can be handled by
AUTO BAYES, ranging from the simple 1D (
) and 2D with diagonal covariance ( )
and with (conjugate) priors on mean
to 1D models for multi-dimensional classes
or variance . Using only a slight variation in the specification, the Gaussian distribution can be replaced by other distributions (e.g., exponentials,
, for failure analysis) or
combinations (e.g., . Gaussian and Beta,
, or -Cauchy and Poisson
). In the algorithm generated by , the analytic subsolution for the Gaussian case is combined with the
numerical solver. Finally, is a
-Gaussians and -Gaussians two-level hierarchical
mixture model which is solved by a nested instantiation of EM [15]: i.e., the M-step of the
outer EM algorithm is a second EM algorithm nested inside.
Mixtures for Regression. We represented regression with Gaussian error and Legendre
polynomials with full conjugate priors allowing smoothing [10]. Two versions of this were
then done: robust linear regression
replaces the Gaussian error with a mixture of two
Gaussians (one broad, one peaked) both centered at zero. Trajectory clustering replaces
the single regression curve by a mixture of several curves [5]. In both cases an EM algorithm is correctly integrated with the exact regression solutions.
Principal Component Analysis. We also tested a multinomial version of PCA called latent
Dirichlet allocation [2]. AUTO BAYES currently lacks variational support, yet it manages to
combine a -means style outer loop on the component proportions with an EM-style inner
loop on the hidden counts, producing the original algorithm of Hofmann, Lee and Seung,
and others [4].
#
,+
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N >
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G , P -Gauss hierarch
rob. lin. regression
mixture regression
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loc
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Gauss Bayes Classify
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5
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6 Conclusion
Beyond existing systems. Code libraries are common in statistics and learning, but they
lack the high level of automation achievable only by deep symbolic reasoning. The Bayes
Net Toolbox [12] is a Matlab library which allows users to program in models but does not
derive algorithms or generate code. The B UGS system [14] also allows users to program
in models but is specialized for Gibbs sampling. Stochastic parametrized grammars [11]
allow a concise model specification similar to AUTO BAYES ?s specification language, but
are currently only a notational device similar to XML.
Benefits of automated algorithm and code generation. Industrial-strength code. Code
generated by AUTO BAYES is efficient, validated, and commented. Extreme applications.
Extremely complex or critical applications such as spacecraft challenge the reliability limits of human-developed software. Automatically generated software allows for pervasive
condition checking and correctness-by-construction. Fast prototyping and experimentation. For both the data analyst and machine learning researcher, AUTO BAYES can function
as a powerful experimental workbench. New complex algorithms. Even with only the few
elements implemented so far, we showed that algorithms approaching research-level results
[4, 5, 10, 15] can be automatically derived. As more distributions, optimization methods
and generalized learning algorithms are added to the system, an exponentially growing
number of complex new algorithms become possible, including non-trivial variants which
may challenge any single researcher?s particular algorithm design expertise.
Future agenda. The ultimate goal is to give researchers the ability to experiment with the
entire space of complex models and state-of-the-art statistical algorithms, and to allow new
algorithmic ideas, as they appear, to be implicitly generalized to every model and special
case known to be applicable. We have already begun work on generalizing the EM schema
to continuous hidden variables, as well as adding schemas for variational methods, fast
kd-tree and -body algorithms, MCMC, and temporal models.
Availability. A web interface for AUTO BAYES is currently under development. More
information is available at http://ase.arc.nasa.gov/autobayes.
References
[1] C.L. Blake and C.J. Merz. UCI repository of machine learning databases, 1998.
[2] D. Blei, A.Y. Ng, and M. Jordan. Latent Dirichlet allocation. NIPS*14, 2002.
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[4] W.L. Buntine. Variational extensions to EM and multinomial PCA. ECML 2002, pp. 23?34, 2002.
[5] G.S. Gaffney and P. Smyth. Trajectory clustering using mixtures of regression models. In 5th KDD, pp. 63?72, 1999.
[6] Z. Ghahramani and M.J. Beal. Propagation algorithms for variational Bayesian learning. In NIPS*12, pp. 507?513, 2000.
[7] P. Haddawy. Generating Bayesian Networks from Probability Logic Knowledge Bases. In UAI 10, pp. 262?269, 1994.
[8] F. R. Kschischang, B. Frey, and H.-A. Loeliger. Factor graphs and the sum-product algorithm. IEEE Trans. Inform. Theory,
47(2):498?519, 2001.
[9] S.L. Lauritzen, A.P. Dawid, B.N. Larsen, and H.-G. Leimer. Independence properties of directed Markov fields. Networks,
20:491?505, 1990.
[10] D.J.C. Mackay. Bayesian interpolation. Neural Computation, 4(3):415?447, 1991.
[11] E. Mjolsness and M. Turmon. Stochastic parameterized grammars for Bayesian model composition. In NIPS*2000 Workshop on Software Support for Bayesian Analysis Systems, Breckenridge, December 2000.
[12] K. Murphy. Bayes Net Toolbox for Matlab. Interface of Computing Science and Statistics 33, 2001.
[13] P. Smyth, D. Heckerman, and M. Jordan. Probabilistic independence networks for hidden Markov models. Neural Computation, 9(2):227?269, 1997.
[14] A. Thomas, D.J. Spiegelhalter, and W.R. Gilks. BUGS: A program to perform Bayesian inference using Gibbs sampling.
In Bayesian Statistics 4, pp. 837?842, 1992.
[15] D.A. van Dyk. The nested EM algorithm. Statistica Sinica, 10:203-225, 2000.
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1,292 | 2,177 | Learning Sparse Topographic Representations
with Products of Student-t Distributions
Max Welling and Geoffrey Hinton
Department of Computer Science
University of Toronto
10 King?s College Road
Toronto, M5S 3G5 Canada
welling,hinton @cs.toronto.edu
Simon Osindero
Gatsby Unit
University College London
17 Queen Square
London WC1N 3AR, UK
[email protected]
Abstract
We propose a model for natural images in which the probability of an image is proportional to the product of the probabilities of some filter outputs. We encourage the system to find sparse features by using a Studentt distribution to model each filter output. If the t-distribution is used to
model the combined outputs of sets of neurally adjacent filters, the system learns a topographic map in which the orientation, spatial frequency
and location of the filters change smoothly across the map. Even though
maximum likelihood learning is intractable in our model, the product
form allows a relatively efficient learning procedure that works well even
for highly overcomplete sets of filters. Once the model has been learned
it can be used as a prior to derive the ?iterated Wiener filter? for the purpose of denoising images.
1 Introduction
Historically, two different classes of statistical model have been used for natural images.
?Energy-based? models assign to each image a global energy, , that is the sum of a number of local contributions and they define the probability of an image to be proportional to
. This class of models includes Markov Random Fields where combinations of
nearby pixel values contribute local energies, Boltzmann Machines in which binary pixels
are augmented with binary hidden variables that learn to model higher-order statistical interactions and Maximum Entropy methods which learn the appropriate magnitudes for the
energy contributions of heuristically derived features [5] [9]. It is difficult to perform maximum likelihood fitting on most energy-based models because of the normalization term
(the partition function) that is required to convert
to a probability. The normalization term is a sum over all possible images and its derivative w.r.t. the parameters is
required for maximum likelihood fitting. The usual approach is to approximate this derivative by using Markov Chain Monte Carlo (MCMC) to sample from the model, but the large
number of iterations required to reach equilibrium makes learning very slow.
The other class of model uses a ?causal? directed acyclic graph in which the lowest level
nodes correspond to pixels and the probability distribution at a node (in the absence of
any observations) depends only on its parents. When the graph is singly or very sparsely
connected there are efficient algorithms for maximum likelihood fitting but if nodes have
many parents, it is hard to perform maximum likelihood fitting because this requires the
intractable posterior distribution over non-leaf nodes given the pixel values.
There is much debate about which class of model is the most appropriate for natural images.
Is a particular image best characterized by the states of some hidden variables in a causal
generative model? Or is it best characterized by its peculiarities i.e. by saying which of a
very large set of normally satisfied constraints are violated? In this paper we treat violations
of constraints as contributions to a global energy and we show how to learn a large set of
constraints each of which is normally satisfied fairly accurately but occasionally violated
by a lot. The ability to learn efficiently without ever having to generate equilibrium samples
from the model and without having to confront the intractable partition function removes a
major obstacle to the use of energy-based models.
2 The Product of Student-t Model
Products of Experts (PoE) are a restricted class of energy-based model [1]. The distribution
represented by a PoE is simply the normalized product of all the distributions represented
by the individual ?experts?:
(1)
where are un-normalized experts and denotes the overall normalization constant.
In the product
of Student-t (PoT) model, un-normalized experts have the following form,
"!
(2)
where is called a filter and is the # -th column in the filter-matrix . When properly
normalized, this represents a Student-t distribution over the filtered random variable $
. An important feature of the Student-t distribution is its heavy tails, which makes it a
suitable candidate for modelling constraints of the kind that are found in images.
Defining
%
%
&
%
, the energy of the PoT model becomes
'
(
)+*-,
.
(3)
Viewed this way, the model takes the form of a maximum entropy distribution with weights
on real-valued ?features? of the image. Unlike previous maximum entropy models,
however, we can fit both the weights and the features at the same time.
When the number of input dimensions is equal to the number of experts, the normally intractable partition function becomes a determinant and the PoT model becomes equivalent
to a noiseless ICA model with Student-t prior distributions [2]. In that case the rows of
the inverse filters /0 21 will represent independent directions in input space. So noiseless ICA can be viewed as an energy-based model even though it is usually interpeted as
a causal generative model in which the posterior over the hidden variables collapses to a
point. However, when we consider more experts than input dimensions (i.e. an overcomplete representation), the energy-based view and the causal generative view lead to different
generalizations of ICA. The natural causal generalization retains the independence of the
hidden variables in the prior by assuming independent sources. In contrast, the PoT model
simply multiplies together more experts than input dimensions and re-normalizes to get the
total probability.
3 Training the PoT Model with Contrastive Divergence
When training energy-based models we need to shape the energy function so that observed
images have low energy and empty regions in the space of all possible images have high
energy. The maximum likelihood learning rule is given by,
(4)
It is the second term which causes learning to be slow and noisy because it is usually
necessary to use MCMC to compute the average over the equilibrium distribution. A much
more efficient way to fit the model is to use the data distribution itself to initialize a Markov
Chain which then starts moving towards the model?s equilibrium distribution. After just a
few steps, we observe how the chain is diverging from the data and adjust the parameters
to counteract this divergence. This is done by lowering the energy of the data and raising
the energy of the ?confabulations? produced by a few steps of MCMC.
#"
$%"&'
!
(5)
1
It can be shown that the above update rule approximately minimizes a new objective function called the contrastive divergence [1].
As it stands the learning rule will be inefficient if the Markov Chain mixes slowly because
the two terms in equation 5 will almost cancel each other out. To speed up learning we need
a Markov chain that mixes rapidly so that the confabulations will be some distance away
from the data. Rapid mixing can be achieved by alternately Gibbs sampling a set of hidden
variables given the random variables under consideration and vice versa. Fortunately, the
PoT model can be equipped with a number of hidden random variables equal to the number
of experts as follows,
)(#* ,+ %-/1. 0&2
3 4 5 76 82 :5 9=; <> 8 > 6 5
Integrating over the *
1
1
>@? ACB 4 ED
(6)
variables results in the density of the PoT model, i.e. eqns. (1) and
(2). Moreover, the conditional distributions are easy to identify and sample from, namely
M
LK
(7)
J
I
HG 4
S Q
*
(8)
N <PO K RQ
UTPVFWYX 3 Z D
where denotes a Gamma distribution and N a normal distribution. From (8) we see that
* can be interpreted as precision variables in the transformed space [
G
.
the variables
F*
'
!
.
1
In this respect our model resembles a ?Gaussian scale mixture? (GSM) [8] which also
multiplies a positive scaling variable with a normal variate. But GSM is a causal model
while PoT is energy-based.
The (in)dependency relations between the variables in a PoT model are depicted graphically
in figure (1a,b). The hidden variables are independent given , which allows them to be
Gibbs-sampled in parallel. This resembles the way in which brief Gibbs sampling is used
to fit binary ?Restricted Boltzmann Machines? [1].
To learn the parameters of the PoT model we thus propose to iterate the following steps:
*^\ ]
1) Sample
given the data
distribution (7).
_]
for every data-vector according to the Gamma-
u
u
u
T 2
(Jx)
J
x
x
(a)
T 2
(Jx)
1
2
3
4
5
W
J
x
(b)
(c)
(d)
Figure 1: (a)- Undirected graph for the PoT model. (b)-Expanded graph where the
deterministic
relation (dashed lines) between the random variable and the activities of the filters
is made
explicit. (c)-Graph for the PoT model including weights . (d)-Filters with large (decreasing from
left to right) weights into a particular top level unit . Top level units have learned to connect to
filters similar in frequency, location and orientation.
\]
2) Sample reconstructions of the data given the sampled values of
data-vector according to the Normal distribution (8).
*\ ]
for every
R\ ]
3) Update the parameters according to (5) where the ?k-step samples? are now given
by the reconstructions , the energy is given by (3), and the parameters are given
by
.
)(
4 Overcomplete Representations
The above learning rules are still valid for overcomplete representations. However, step-2
of the learning algorithm is much more efficient when the inverse of the filter matrix
exists. In that case we simply draw standard normal random numbers (with the num1
ber of data-vectors) and multiply each of them with 1
. This is efficient because
1
is diagonal while the costly inverse 1 is data indepenthe data dependent matrix
dent. In contrast, for the overcomplete case we ought to perform a Cholesky factorization
on
for each data-vector separately. We have, however, obtained good results by
proceeding as in the complete case and replacing the inverse of the filter matrix with its
pseudo-inverse.
Q ] 82
Q ] 82
RQP]
From experiments we have also
found that in the overcomplete case we should fix the
norm of the filters,
# , in order to prevent some of them from decaying to zero.
This operation is done after every step of learning. Since controlling the norm removes the
ability of the experts to adapt to scale it is necessary to whiten the data first.
4.1 Experiment: Overcomplete Representations for Natural Images
(
We randomly generated ! ! ! ! patches of ! ! pixels from images of natural scenes 1 .
The patches were centered and sphered using PCA and the DC component (eigen-vector
with largest variance) was removed. The algorithm
for overcomplete representations using
"!
the pseudo-inverse was used to train !
experts, i.e. a representation
that is more than
!-!
times overcomplete.
We
fixed
the
weights
to
have
and
the
the
filters to have
a -norm of . A small weight decay term and a momentum
term were included in the
gradient updates of the filters. The learning rate was set so that initially the change in the
filters was approximately !# ! ! . In figure (2a) we show a small subset
of the inverse-filters
given by the pseudo-inverse of %$'&)(%* , where $+&)(%* is the ! !, .-/- matrix used for
sphering the data.
(
1
Collected from http://www.cis.hut.fi/projects/ica/data/images
5 Topographically Ordered Features
In [6] it was shown that linear filtering of natural images is not enough to remove all higher
order dependencies.
In particular,
it was argued that there are residual dependencies among
the activities $ of the filtered inputs. It is therefore desirable to model those
dependencies within the PoT model. By inspection of figure (1b) we note that these dependencies can be modelled through a non-negative weight matrix ! , which connects
the hidden variables with the activities . The resultant model is depicted in figure
(1c).
Depending on how many nonzero weights emanate from a hidden unit (say
), each expert now occupies input dimensions instead of justone.
The expressions
for
these richer experts can be obtained from (2) by replacing,
. We
# ).
have found that learning is assisted by fixing the -norm of the weights (
Moreover, we have found that the sparsity of the weights can be controlled by the following generalization of the experts,
Z
Z
-
(
"!
(
!
(9)
The larger the value for the sparser the distribution of
values.
Joint and conditional distributions over hidden variables are obtained through similar replacements in eqn. (6) and (7) respectively. Sampling the reconstructions given the states of
the hidden variables proceeds by first sampling from independent generalized
Laplace
which are
distributions
with precision parameters 0
subsequently transformed into 1 . Learning in this model therefore proceeds with
only minor modifications to the algorithm described in the previous section.
*
\
When we learn the weight matrix from image data we find that a particular hidden
variable develops weights to the activities of filters similar in frequency, location and
orientation. The variables therefore integrate information from these filters and as a
result develop certain invariances that resemble the behavior of complex cells. A similar
approach was studied in [4] using a related causal model 2 in which a number of scale
variables generate correlated variances for conditionally Gaussian experts. This results in
topography when the scale-generating variables are non-adaptive and connect to a local
neighborhood of filters only.
Z
*
We will now argue that fixed local weights
also give rise to topography in the PoT
model. The reason is that averaging the squares of randomly chosen filter outputs (eqn.9)
produces an approximately Gaussian distribution which is a poor fit to the heavy-tailed
experts. However, this ?smoothing effect? may be largely avoided by averaging squared
filter outputs that are highly correlated (i.e. ones that are similar in location, frequency and
orientation). Since the averaging is local, this results in a topographic layout of the filters.
5.1 Experiment: Topographic Representations for Natural Images
(
!
!
pixels in the
same
For this experiment we collected ! ! !-! image patches of size
!
way as described in section
(4.1).
The
image
data
were
sphered
and
reduced
to
di!
mensions by removing low variance and high variance (DC) direction. We learned
an
.
.
overcomplete representation with ! ! experts which were organized on a square !, !
&
grid. Each expert connects with a fixed weight of
to itself and all its neighbors, where periodic boundary conditions were imposed for the experts on the boundary.
2
Interestingly, the update equations for the filters presented in [4], which minimize a bound on
the log-likelihood of a directed model, reduce to the same equations as our learning rules when the
representation is complete and the filters orthogonal.
(a)
(b)
Figure 2: (a)-Small subset of the learned filters from a
times overcomplete representation for natural image patches. (b)-Topographically ordered filters. The weights were fixed and
connect to neighbors only, using periodic boundary conditions. Neighboring filters have learned to
be similar in frequency, location and orientation. One can observe a pinwheel structure to the left of
the low frequency cluster.
.
We adapted the filters ( -norm ) and used fixed values for
and
. The resulting inverse-filters are shown in figure (2b). We note that the weights have enforced a
topographic ordering on the experts, where location, scale and frequency of the Gabor-like
filters all change smoothly across the map.
!
In another experiment we used the same data to train
a complete representation of
experts where we learned the weights ( -norm ), and the filters (unconstrained),
but with a fixed value of
. The weights and were kept positive by adapting their
can now connect to any other expert we do not expect
logarithm. Since the weights
topography. To study whether the weights
were modelling the dependencies between
the energies of the filter outputs we ordered the filters for each complex cell
according to the strength of the weights connecting to it. For a representative subset of the
complex cells , we show the ! filters with the strongest connections to that cell in figure
(1d). Since the cells connect to similar filters we may conclude that the weights
are
indeed learning the dependencies between the activities of the filter outputs.
Z
*
6 Denoising Images: The Iterated Wiener Filter
If the PoT model provides an accurate description of the statistics of natural image data it
ought to be a good prior for cleaning up noisy images. In the following we will apply this
idea to denoise images contaminated with Gaussian pixel noise. We follow the standard
Bayesian approach which states that the optimal estimate of the original image is given by
the maximum a posteriori (MAP) estimate of , where denotes the noisy image.
For the PoT model this reduces to,
*
&
W X V <
.
1
(
) * ,
(
.
(10)
(a)
(b)
(c)
(d)
Figure 3: (a)- Original ?rock?-image. (b)-Rock-image with noise added. (c)-Denoised image using
Wiener filtering. (d) Denoised image using IWF.
To minimize this we follow a variational procedure where we upper bound the logarithm
%
%
)+*-,
&
using )+*-, %
. The bound is saturated when
. Applying this to
every logarithm in the summation in eqn. (10) and iteratively minimizing this bound over
and we find the following update equations,
*
&
&
(
.
1
T
1
1
(11)
T UTPVFWYX 3
0
D
(12)
where denotes componentwise multiplication. Since the second equation is just a Wiener
filter with noise covariance and a Gaussian prior with covariance 1 we have
named the above denoising equations the iterated Wiener filter (IWF).
T
When the filters are orthonormal, the noise covariance isotropic and the weight matrix the
identity, the minimization in (10) decouples into minimizations over the transformed
variables $ . Defining we can easily derive that $ is the solution of the
following cubic equation (for which analytic solutions exist),
$
$
.
$
.
!
(13)
We note however that constraining the filters to be orthogonal is a rather severe restriction if
the data are not pre-whitened. On the other hand, if we decide to work with whitened data,
the isotropic noise assumption seems unrealistic. Having said that, Hyvarinen?s shrinkage
method for ICA models [3] is based on precisely these assumptions and seems to give good
results. The proposed method is also related to approaches based on the GSM [7].
6.1 Experiment: Denoising
To test the iterated Wiener filter, we trained a complete set of - - experts on the data described in section (4.1). The norm of the filters was unconstrained, the were free to adapt,
but we did not include any weights . The image
shown in figure (3a) was corrupted
with
.
. .
!
Gaussian noise with standard deviation
, which resulted in a PSNR of # ! dB (figure (3b)). We applied the adaptive Wiener filter from matlab (Wiener2.m) with an optimal
neighborhood size and known noise-variance. The denoised image using adaptive
."!
Wiener filtering
has a PSNR of # - dB and is shown in figure (3c). IWF was run on every
possible ! ! patch in the image, after which the results were averaged. Because the
filters were trained on sphered data without a DC component, the same transformations
have to be applied to the test patches before
IWF
is applied. The denoised image using
.
#
dB,
which is a significant improvement of
IWF
is
shown
in
(3d)
and
has
a
PSNR
of
.
#
dB over Wiener filtering. It is our hope that the use of overcomplete representations and
weights will further improve those results.
7 Discussion
It is well known that a wavelet transform de-correlates natural image data in good approximation. In [6] it was found that in the marginal distribution the wavelet coefficients are
sparsely
distributed but that there are significant residual dependencies among their ener
gies $ . In this paper we have shown that the PoT model can learn highly overcomplete filters with sparsely distributed outputs. With a second hidden layer that is locally connected,
it captures the dependencies between filter outputs by learning topographic representations.
Our approach improves upon earlier attempts (e.g. [4],[8]) in a number of ways. In the
PoT model the hidden variables are conditionally independent so perceptual inference is
very easy and does not require iterative settling even when the model is overcomplete.
There is a fairly simple and efficient procedure for learning all the parameters, including
the weights connecting top-level units to filter outputs. Finally, the model leads to an
elegant denoising algorithm which involves iterating a Wiener-filter.
Acknowledgements
This research was funded by NSERC, the Gatsby Charitable Foundation, and the Wellcome Trust.
We thank Yee-Whye Teh for first suggesting a related model and Peter Dayan for encouraging us to
apply products of experts to topography.
References
[1] G.E. Hinton. Training products of experts by minimizing contrastive divergence. Neural Computation, 14:1771?1800, 2002.
[2] G.E. Hinton, M. Welling, Y.W. Teh, and K. Osindero. A new view of ICA. In Int. Conf. on
Independent Component Analysis and Blind Source Separation, 2001.
[3] A. Hyvarinen. Sparse code shrinkage: Denoising of nongaussian data by maximum likelihood
estimation. Neural Computation, 11(7):1739?1768, 1999.
[4] A. Hyvarinen, P.O. Hoyer, and M. Inki. Topographic independent component analysis. Neural
Computation, 13(7):1525?1558, 2001.
[5] S. Della Pietra, V.J. Della Pietra, and J.D. Lafferty. Inducing features of random fields. IEEE
Transactions on Pattern Analysis and Machine Intelligence, 19(4):380?393, 1997.
[6] E.P. Simoncelli. Modeling the joint statistics of images in the wavelet domain. In Proc SPIE,
44th Annual Meeting, volume 3813, pages 188?195, Denver, 1999.
[7] V. Strela, J. Portilla, and E. Simoncelli. Image denoising using a local Gaussian scale mixture
model in the wavelet domain. In Proc. SPIE, 45th Annual Meeting, San Diego, 2000.
[8] M.J. Wainwright and E.P. Simoncelli. Scale mixtures of Gaussians and the statistics of natural
images. In Advances Neural Information Processing Systems, volume 12, pages 855?861, 2000.
[9] S.C. Zhu, Z.N. Wu, and D. Mumford. Minimax entropy principle and its application to texture
modeling. Neural Computation, 9(8):1627?1660, 1997.
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1,293 | 2,178 | Neural Decoding of Cursor Motion Using a Kalman Filter
W. Wu
M. J. Black
Y. Gao
M. Serruya
A. Shaikhouni
E. Bienenstock
J. P. Donoghue
Division of Applied Mathematics, Dept. of Computer Science,
Dept. of Neuroscience, Division of Biology and Medicine,
Brown University, Providence, RI 02912
[email protected], [email protected], [email protected],
[email protected], Mijail [email protected],
Ammar [email protected], john [email protected]
Abstract
The direct neural control of external devices such as computer displays
or prosthetic limbs requires the accurate decoding of neural activity representing continuous movement. We develop a real-time control system
using the spiking activity of approximately 40 neurons recorded with
an electrode array implanted in the arm area of primary motor cortex.
In contrast to previous work, we develop a control-theoretic approach
that explicitly models the motion of the hand and the probabilistic relationship between this motion and the mean firing rates of the cells in
70 bins. We focus on a realistic cursor control task in which the subject must move a cursor to ?hit? randomly placed targets on a computer
monitor. Encoding and decoding of the neural data is achieved with a
Kalman filter which has a number of advantages over previous linear
filtering techniques. In particular, the Kalman filter reconstructions of
hand trajectories in off-line experiments are more accurate than previously reported results and the model provides insights into the nature of
the neural coding of movement.
1 Introduction
Recent results have demonstrated the feasibility of direct neural control of devices such as
computer cursors using implanted electrodes [5, 9, 11, 14]. These results are enabled by a
variety of mathematical ?decoding? methods that produce an estimate of the system ?state?
(e.g. hand position) from a sequence of measurements (e.g. the firing rates of a collection
of cells). Here we argue that such a decoding method should (1) have a sound probabilistic
foundation; (2) explicitly model noise in the data; (3) indicate the uncertainty in estimates
of hand position; (4) make minimal assumptions about the data; (5) require a minimal
amount of ?training? data; (6) provide on-line estimates of hand position with short delay
(less than 200ms); and (7) provide insight into the neural coding of movement. To that
Monitor
12
Target
10
8
6
Tablet
Trajectory
4
2
Manipulandum
a
2
4
6
8
10
12
14
16
b
Figure 1: Reconstructing 2D hand motion. (a) Training: neural spiking activity is recorded
while the subject moves a jointed manipulandum on a 2D plane to control a cursor so that
it hits randomly placed targets. (b) Decoding: true target trajectory (dashed (red): dark to
light) and reconstruction using the Kalman filter (solid (blue): dark to light).
end, we propose a Kalman filtering method that provides a rigorous and well understood
framework that addresses these issues. This approach provides a control-theoretic model
for the encoding of hand movement in motor cortex and for inferring, or decoding, this
movement from the firing rates of a population of cells.
Simultaneous recordings are acquired from an array consisting of microelectrodes [6]
implanted in the arm area of primary motor cortex (MI) of a Macaque monkey; recordings
from this area have been used previously to control devices [5, 9, 10, 11, 14]. The monkey
views a computer monitor while gripping a two-link manipulandum that controls the 2D
motion of a cursor on the monitor (Figure 1a). We use the experimental paradigm of [9], in
which a target dot appears in a random location on the monitor and the task requires moving
a feedback dot with the manipulandum so that it hits the target. When the target is hit, it
jumps to a new random location. The trajectory of the hand and the neural activity of
cells are recorded simultaneously. We compute the position, velocity, and acceleration of
the hand along with the mean firing rate for each of the cells within non-overlapping
time bins. In contrast to related work [8, 15], the motions of the monkey in this task are
quite rapid and more ?natural? in that the actual trajectory of the motion is unconstrained.
The reconstruction of hand trajectory from the mean firing rates can be viewed probabilistically as a problem of inferring behavior from noisy measurements. In [15] we proposed
a Kalman filter framework [3] for modeling the relationship between firing rates in motor
cortex and the position and velocity of the subject?s hand. This work focused on off-line
reconstruction using constrained motions of the hand [8]. Here we consider new data from
the on-line environmental setup [9] which is more natural, varied, and contains rapid motions. With this data we show that, in contrast to our previous results, a model of hand
acceleration (in addition to position and velocity) is important for accurate reconstruction.
In the Kalman framework, the hand movement (position, velocity and acceleration) is modeled as the system state and the neural firing rate is modeled as the observation (measurement). The approach specifies an explicit generative model that assumes the observation
(firing rate in ) is a linear function of the state (hand kinematics) plus Gaussian noise .
Similarly, the hand state at time
is assumed to be a linear function of the hand state at the
previous time instant plus Gaussian noise. The Kalman filter approach provides a recursive,
on-line, estimate of hand kinematics from the firing rate in non-overlapping time bins. The
This is a crude assumption but the firing rates can be square-root transformed [7] making them
more Gaussian and the mean firing rate can be subtracted to achieve zero-mean data.
results of reconstructing hand trajectories from pre-recorded neural firing rates are compared with those obtained using more traditional fixed linear filtering techniques [9, 12]
using overlapping windows. The results indicate that the Kalman filter decoding is
more accurate than that of the fixed linear filter.
1.1 Related Work
Georgopoulos and colleagues [4] showed that hand movement direction may be encoded
by the neural ensemble in the arm area of motor cortex (MI). This early work has resulted
in a number of successful algorithms for decoding neural activity in MI to perform offline reconstruction or on-line control of cursors or robotic arms. Roughly, the primary
methods for decoding MI activity include the population vector algorithm [4, 5, 7, 11],
linear filtering [9, 12], artificial neural networks [14], and probabilistic methods [2, 10, 15].
This population vector approach is the oldest method and it has been used for the real-time
neural control of 3D cursor movement [11]. This work has focused primarily on ?center
out? motions to a discrete set of radial targets (in 2D or 3D) rather than natural, continuous,
motion that we address here.
Linear filtering [8, 12] is a simple statistical method that is effective for real-time neural
control of a 2D cursor [9]. This approach requires the use of data over a long time win
dow (typically to ). The fixed linear filter, like population vectors and neural
networks [14] lack both a clear probabilistic model and a model of the temporal hand kinematics. Additionally, they provide no estimate of uncertainty and hence may be difficult to
extend to the analysis of more complex temporal movement patterns.
We argue that what is needed is a probabilistically grounded method that uses data in small
time windows (e.g. or less) and integrates that information over time in a recursive fashion. The C ONDENSATION algorithm has been recently introduced as a Bayesian
decoding scheme [2], which provides a probabilistic framework for causal estimation and
is shown superior to the performance of linear filtering when sufficient data is available
(e.g. using firing rates for several hundred cells). Note that the C ONDENSATION method is
more general than the Kalman filter proposed here in that it does not assume linear models
and Gaussian noise. While this may be important for neural decoding as suggested in [2],
current technology makes the method impractical for real-time control.
For real-time neural control we exploit the Kalman filter [3, 13] which has been widely
used for estimation problems ranging from target tracking to vehicle control. Here we
apply this well understood theory to the problem of decoding hand kinematics from neural
activity in motor cortex. This builds on the work that uses recursive Bayesian filters to
estimate the position of a rat from the firing activity of hippocampal place cells [1, 16]. In
contrast to the linear filter or population vector methods, this approach provides a measure
of confidence in the resulting estimates. This can be extremely important when the output
of the decoding method is to be used for later stages of analysis.
2 Methods
Decoding involves estimating the state of the hand at the current instant in time; i.e.
x
representing -position, -position, -velocity, -velocity, acceleration, and -acceleration at time
"!
where !
# in our experiments.
The Kalman filter [3, 13] model assumes the state is linearly related to the observations
z %$'&)( which here represents a *,+
vector containing the firing rates at time
for *
observed neurons within . In our experiments, * cells. We briefly review the
Kalman filter algorithm below; for details the reader is referred to [3, 13].
Encoding: We define a generative model of neural firing as
z
x q
(1)
" , is the number of time steps in the trial, and
$ & (
is a
where
matrix that linearly relates the hand state to the neural firing. We assume the noise in the
observations is zero mean and normally distributed, i.e. q
" % $ &)( ( .
The states are assumed to propagate in time according to the system model
x
x w
(2)
where $ & is the coefficient matrix and the noise term w
"
$
& . This states that the hand kinematics (position, velocity, and acceleration) at time
is linearly related to the state at time . Once again we assume these estimates are
normally distributed.
In practice, might change with time step , however, here we make the
common simplifying assumption they are constant. Thus we can estimate the Kalman filter
model from training data using least squares estimation:
!#"
!
$ '& &
$ &*&
'
&
&
)
x &'& )
argmin
x ( x
z
argmin
A
H
%
%
&*& &'&
)
where is the + norm. Given and it is then simple to estimate the noise covariance
matrices and ; details are given in [15].
Decoding: At each
" time step the algorithm has two steps: 1) prediction of the a priori
state estimate x, ; and 2) updating this estimate with new measurement data to produce an
a posteriori state estimate x, . In particular, these steps are:
I. Discrete Kalman filter time update equations:
At each time
, we obtain the" a priori estimate from the previous time
its error covariance matrix, - :
"
x, "
x,
"
- .- "
"
, then compute
(3)
(4)
II. Measurement update equations:
"
Using the estimate x, and firing rate z , we update the estimate using the measurement
and compute the posterior error covariance matrix:
x, /x,
-
"
"
0 z x,
"
21 0 3 -
(5)
(6)
where - represents the state error covariance after taking into account the neural data and
0 is the Kalman gain matrix given by
0
-
"
"
-
#
"
(7)
This 0 produces a state estimate that minimizes the mean squared error of the reconstruction (see [3] for details). Note that is the measurement error matrix and, depending on
the reliability of the data, the gain term, 0 , automatically adjusts the contribution of the
new measurement to the state estimate.
)
Method
Correlation Coefficient
MSE ( )
Kalman (0 lag)
(0.768, 0.912)
7.09
Kalman (70 lag)
(0.785, 0.932)
7.07
Kalman (140 lag)
(0.815, 0.929)
6.28
Kalman (210 lag)
(0.808, 0.891)
6.87
Kalman (no acceleration)
(0.817, 0.914)
6.60
Linear filter
(0.756, 0.915)
8.30
Table 1: Reconstruction results for the fixed linear and recursive Kalman filter. The table
also shows how the Kalman filter results vary with lag times (see text).
3 Experimental Results
To be practical, we must be able to train the model (i.e. estimate , , , ) using a
small amount of data. Experimentally we found that approximately 3.5 minutes of training
data suffices for accurate reconstruction (this is similar to the result for fixed linear filters
reported in [9]). As described in the introduction, the task involves moving a manipulan
dum freely on a + tablet (with a + workspace) to hit randomly
placed targets on the screen. We gather the mean firing rates and actual hand trajectories
for the training data and then learn the models via least squares (the computation time is
negligible). We then test the accuracy of the method by reconstructing test trajectories offline using recorded neural data not present in the training set. The results reported here use
approximately 1 minute of test data.
Optimal Lag: The physical relationship between neural firing and arm movement means
there exists a time lag between them [7, 8]. The introduction of a time lag results in the
measurements, z , at time
, being taken from some previous (or future) instant in time
" for some integer . In the interest of simplicity, we consider a single optimal time
lag for all the cells though evidence suggests that individual time lags may provide better
results [15].
Using time lags of 0, 70, 140, 210 we train the Kalman filter and perform reconstruction
(see Table 1). We report the accuracy of the reconstructions with a variety of error measures
used in the literature including the correlation coefficient ( ) and the mean squared error
(MSE) between the reconstructed and true trajectories. From Table 1 we see that optimal
lag is around two time steps (or 140 ); this lag will be used in the remainder of the
experiments and is similar to our previous findings [15] which suggested that the optimal
lag was between 50-100 .
Decoding: At the beginning of the test trial we let the predicted initial condition equal the
real initial condition. Then the update equations in Section 2 are applied. Some examples of
the reconstructed trajectory are shown in Figure 2 while Figure 3 shows the reconstruction
of each component of the state variable (position, velocity and acceleration in and ).
From Figure 3 and Table 1 we note that the reconstruction in is more accurate than in
the direction (the same is true for the fixed linear filter described below); this requires
further investigation. Note also that the ground truth velocity and acceleration curves are
computed from the position data with simple differencing. As a result these plots are quite
noisy making an evaluation of the reconstruction difficult.
20
20
20
18
18
18
16
16
16
14
14
14
12
12
12
10
10
10
8
8
8
6
6
6
4
4
4
2
2
2
4
6
8
10
12
14
16
18
20
22
2
2
4
6
8
10
12
14
16
18
20
22
2
4
6
8
10
12
14
16
18
20
22
Figure 2: Reconstructed trajectories (portions of 1min test data ? each plot shows 50 time
instants (3.5 )): true target trajectory (dashed (red)) and reconstruction using the Kalman
filter (solid (blue)).
3.1 Comparison with linear filtering
Fixed linear filters reconstruct hand position as a linear combination of the firing rates over
some fixed time period [4, 9, 12]; that is,
$
$
%
"
where is the -position (or, equivalently, the -position) at time
!
!
, , where
is the number of time steps in a trial, is the constant
offset, " is the firing rate of neuron at time
" , and are the filter coefficients. The
coefficients can be learned from training data using a simple least squares technique. In
our experiments here we take which means that the hand position is determined
from firing data over . This is exactly the method described in [9] which provides a fair
comparison for the Kalman filter; for details see [12, 15]. Note that since the linear filter
uses data over a long time window, it does not benefit from the use of time-lagged data.
Note also that it does not explicitly reconstruct velocity or acceleration.
The linear filter reconstruction of position is shown in Figure 4. Compared with Figure 3,
we see that the results are visually similar. Table 1, however, shows that the Kalman filter
gives a more accurate reconstruction than the linear filter (higher correlation coefficient and
lower mean-squared error). While fixed linear filtering is extremely simple, it lacks many
of the desirable properties of the Kalman filter.
Analysis: In our previous work [15], the experimental paradigm involved carefully designed hand motions that were slow and smooth. In that case we showed that acceleration
was redundant and could be removed from the state equation. The data used here is more
?natural?, varied, and rapid and we find that modeling acceleration improves the prediction
of the system state and the accuracy of the reconstruction; Table 1 shows the decrease in
accuracy with only position and velocity in the system state (with 140ms lag).
4 Conclusions
We have described a discrete linear Kalman filter that is appropriate for the neural control
of 2D cursor motion. The model can be easily learned using a few minutes of training data
and provides real-time estimates of hand position every given the firing rates of 42
x-position
y-position
20
10
15
5
10
0
5
5
10
15
20
5
x-velocity
10
15
20
15
20
15
20
y-velocity
2
2
1
0
0
1
2
2
5
10
15
20
5
x-acceleration
10
y-acceleration
2
1
1
0
0
1
1
2
5
10
15
20
5
time (second)
10
time (second)
Figure 3: Reconstruction of each component of the system state variable: true target motion
(dashed (red)) and reconstruction using the Kalman filter (solid (blue)). 20 from a 1min
test sequence are shown.
y-position
x-position
20
10
15
5
10
0
5
5
10
time (second)
15
20
5
10
15
20
time (second)
Figure 4: Reconstruction of position using the linear filter: true target trajectory (dashed
(red)) and reconstruction using the linear filter (solid (blue)).
cells in primary motor cortex. The estimated trajectories are more accurate than the fixed
linear filtering results being used currently.
The Kalman filter proposed here provides a rigorous probabilistic approach with a well
understood theory. By making its assumptions explicit and by providing an estimate of
uncertainty, the Kalman filter offers significant advantages over previous methods. The
method also estimates hand velocity and acceleration in addition to 2D position. In contrast
to previous experiments, we show, for the natural 2D motions in this task, that incorporating acceleration into the system and measurement models improves the accuracy of the
decoding. We also show that, consistent with previous studies, a time lag of
improves the accuracy.
Our future work will evaluate the performance of the Kalman filter for on-line neural control of cursor motion in the task described here. Additionally, we are exploring alternative
measurement noise models, non-linear system models, and non-linear particle filter decod-
ing methods. Finally, to get a complete picture of current methods, we are pursuing further
comparisons with population vector methods [7] and particle filtering techniques [2].
Acknowledgments. This work was supported in part by: the DARPA Brain Machine
Interface Program, NINDS Neural Prosthetics Program and Grant #NS25074, and the National Science Foundation (ITR Program award #0113679). We thank J. Dushanova, C.
Vargas, L. Lennox, and M. Fellows for their assistance.
References
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[14] Wessberg, J., Stambaugh, C., Kralik, J., Beck, P., Laubach, M., Chapin, J., Kim, J., Biggs, S.,
Srinivasan, M., and Nicolelis, M. (2000). Real-time prediction of hand trajectory by ensembles
of cortical neurons in primates. Nature, 408:361?365.
[15] Wu, W., Black, M. J., Gao, Y., Bienenstock, E., Serruya, M., and Donoghue, J. P., Inferring hand motion from multi-cell recordings in motor cortex using a Kalman filter, SAB?02Workshop on Motor Control in Humans and Robots: On the Interplay of Real Brains and
Artificial Devices, Aug. 10, 2002, Edinburgh, Scotland, pp. 66?73.
[16] Zhang, K., Ginzburg, I., McNaughton, B., Sejnowski, T., Interpreting neuronal population
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Neurophysiol. 79:1017?1044, 1998.
| 2178 |@word neurophysiology:3 trial:3 briefly:1 norm:1 propagate:1 carolina:1 simplifying:1 covariance:4 tr:1 solid:4 initial:2 contains:1 nordhausen:1 current:3 must:2 john:1 realistic:1 motor:13 plot:2 designed:1 update:4 generative:2 device:5 manipulandum:4 wessberg:1 plane:1 oldest:1 beginning:1 scotland:1 short:1 provides:9 location:2 zhang:1 mathematical:1 along:1 direct:3 acquired:1 rapid:3 roughly:1 behavior:1 multi:1 brain:4 automatically:1 actual:2 window:3 electroencephalography:1 estimating:1 chapin:1 what:1 argmin:2 minimizes:1 monkey:3 unified:1 finding:1 impractical:1 temporal:3 fellow:3 every:1 exactly:1 hit:5 schwartz:4 control:21 normally:2 grant:1 negligible:1 understood:3 encoding:3 firing:25 approximately:3 black:4 plus:2 might:1 suggests:1 elie:1 practical:1 acknowledgment:1 shaikhouni:2 recursive:4 practice:1 area:4 shoham:1 pre:1 radial:1 confidence:1 get:1 dam:1 demonstrated:1 center:1 focused:2 welch:1 simplicity:1 chapel:2 insight:2 adjusts:1 array:3 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1,294 | 2,179 | Mismatch String Kernels for SVM Protein
Classification
Christina Leslie
Department of Computer Science
Columbia University
[email protected]
Eleazar Eskin
Department of Computer Science
Columbia University
[email protected]
Jason Weston
Max-Planck Institute
Tuebingen, Germany
[email protected]
William Stafford Noble
Department of Genome Sciences
University of Washington
[email protected]
Abstract
We introduce a class of string kernels, called mismatch kernels, for use
with support vector machines (SVMs) in a discriminative approach to
the protein classification problem. These kernels measure sequence similarity based on shared occurrences of -length subsequences, counted
with up to mismatches, and do not rely on any generative model for
the positive training sequences. We compute the kernels efficiently using
a mismatch tree data structure and report experiments on a benchmark
SCOP dataset, where we show that the mismatch kernel used with an
SVM classifier performs as well as the Fisher kernel, the most successful method for remote homology detection, while achieving considerable
computational savings.
1 Introduction
A fundamental problem in computational biology is the classification of proteins into functional and structural classes based on homology (evolutionary similarity) of protein sequence data. Known methods for protein classification and homology detection include
pairwise sequence alignment [1, 2, 3], profiles for protein families [4], consensus patterns
using motifs [5, 6] and profile hidden Markov models [7, 8, 9]. We are most interested
in discriminative methods, where protein sequences are seen as a set of labeled examples
? positive if they are in the protein family or superfamily and negative otherwise ? and
we train a classifier to distinguish between the two classes. We focus on the more difficult
problem of remote homology detection, where we want our classifier to detect (as positives)
test sequences that are only remotely related to the positive training sequences.
One of the most successful discriminative techniques for protein classification ? and the
best performing method for remote homology detection ? is the Fisher-SVM [10, 11] approach of Jaakkola et al. In this method, one first builds a profile hidden Markov model
Formerly William Noble Grundy: see http://www.cs.columbia.edu/?noble/name-change.html
(HMM) for the positive training sequences, defining a log likelihood function
for any protein sequence . If
is the maximum likelihood estimate for the model parameters, then the gradient vector
assigns to each (positive or negative)
training sequence an explicit vector of features called Fisher scores. This feature mapping defines a kernel function, called the Fisher kernel, that can then be used to train a
support vector machine (SVM) [12, 13] classifier. One of the strengths of the Fisher-SVM
approach is that it combines the rich biological information encoded in a hidden Markov
model with the discriminative power of the SVM algorithm. However, one generally needs
a lot of data or sophisticated priors to train the hidden Markov model, and because calculating the Fisher scores requires computing forward and backward probabilities from the
Baum-Welch algorithm (quadratic in sequence length for profile HMMs), in practice it is
very expensive to compute the kernel matrix.
In this paper, we present a new string kernel,"called
the mismatch kernel, for use with an
!
SVM for remote homology detection. The -mismatch kernel is based on a feature
map to a vector space indexed by all possible subsequences of amino acids of a fixed
length ; each instance of a fixed -length subsequence in an input sequence contributes
to all feature coordinates differing from it by at most mismatches. Thus, the mismatch
kernel adds the biologically important idea of mismatching to the computationally simpler
spectrum kernel presented in [14]. In the current work, we also describe how"to
compute
!
the new kernel efficiently using a mismatch tree data structure; for values of useful
in this application, the kernel is fast enough to use on real datasets and is considerably
less expensive than the Fisher kernel. We report results from a benchmark dataset on the
SCOP database [15] assembled by Jaakkola et al. [10] and show that the mismatch kernel
used with an SVM classifier achieves performance equal to the Fisher-SVM method while
outperforming all other methods tested. Finally, we note that the mismatch kernel does
not depend on any generative model and could potentially be used in other sequence-based
classification problems.
2 Spectrum and Mismatch String Kernels
The basis for our approach to protein classification is to represent protein sequences as
vectors in a high-dimensional feature space via a string-based feature map. We then train
a support vector machine (SVM), a large-margin linear classifier, on the feature vectors
representing our training sequences. Since SVMs are a kernel-based learning algorithm,
we do not calculate the feature vectors explicitly but instead compute their pairwise inner
products using a mismatch string kernel, which we define in this section.
2.1 Feature Maps for Strings
"!
The # -mismatch kernel is based on a feature map from the space of all finite sequences
from an alphabet $ of size $% &(' to the ' ) -dimensional vector space indexed by the set
of -length subsequences (? -mers?) from $ . (For protein sequences, $ is the alphabet of
amino
acids, '*&,+- .) For a fixed -mer ./&,0 1204365758590 , with each 0;: a character in $ ,
"!
)
the -neighborhood generated by . is the set of all -length sequences < from $ that
E. .
differ from . by at most mismatches. We denote this set by =?>
)A@ BDC
We define our feature map FG>
)A@ BDC
FH>
as follows: if . is a -mer, then
)A@ BDC
(1)
E. &IKJMLNE. 9 L;OPQ
where JRL E. S&UT if < belongs to =V>
-mer
E. , and JL W.X&Y- otherwise. Thus, a
)A@ BDC
contributes weight to all the coordinates in its mismatch neighborhood.
For a sequence
Z
of any length, we extend the map additively by summing the feature
vectors for all the -mers in Z :
FH>
KZ"
) @ B*C
&
FH>
)
-mers in
W.
)A@ BDC
Note that the < -coordinate of FG>
-mer <
KZR is just a count of all instances of the
"!
)A@ BDC
occurring with up to mismatches in Z . The # -mismatch kernel >
is the inner
) @ B*C
product in feature space of feature vectors:
>
!
KZ
)A@ BDC
KFH>
&
) @ B*C
!
KZ"
FH>
)A@ BDC
25
For
&/
, we retrieve the -spectrum kernel defined in [14].
2.2 Fisher Scores and the Spectrum Kernel
While we define the spectrum and mismatch feature maps without any reference to a generative model for the positive class of sequences, there is some similarity
between the
-spectrum feature map and the Fisher scores associated to an order
T Markov chain
model. More precisely, suppose the generative model for the positive training sequences is
given by
KZ
&
for a string ZV&
1
58575
1 3658575
1
E
&
7
!
!
!
57575
1
!
585759
1
")
&
1
"58575 E
1
)
)
, with parameters
)
!
57585
1
)
1
")
!
575859
1
1
!"" Q #
&
for characters 1 58575
1 in alphabet $ . Denote by
the maximum likelihood es)
timate for
on the positive training set. To calculate the Fisher scores for this model,
$&% ' ( ( ( ' Q#
we follow [10] and define independent variables
@
!"" Q!# & )
$ * % '
' Q!# satisfying
@ "" !Q #
&/
"" !Q # , ,
*
*
Q #
@ ""
.
WZ
@ "" Q#
&
.
&
&
T
. Then the Fisher scores are given by
/
T
!"" Q!#
( ( (
$+*
"" Q!#
"" Q!#
.
"" Q!#
Q!#
@ ""
0
@
!
"" Q#
1
1
.
32
"" !Q #
where .
is the number of instances of the -mer 1 57585
1 in Z , and .
)
4
3 !"" Q#
"" Q!#
is the number of instances of the
T -mer M157585
1 . Thus the Fisher score captures the
)
degree to which the -mer 158575!
16 is over- or under-represented relative to the positive
)5
model. For the -spectrum kernel, the corresponding feature coordinate looks similar but
KZ" &7.
5
simply uses the unweighted count: J
Q!#
Q!#
!""
""
3 Efficient Computation of the Mismatch Kernel
Unlike the Fisher vectors used in [10], our feature vectors are sparse vectors in a very high
dimensional feature space. Thus, instead of calculating and storing the feature vectors, we
directly and efficiently compute the kernel matrix for use with an SVM classifier.
3.1 Mismatch Tree Data Structure
We use a mismatch tree data structure (similar to a trie or suffix tree [16, 17]) to represent
the feature space (the set of all -mers) and perform a lexical traversal of all -mers occurring in the sample dataset match with up to of mismatches; the entire kernel matrix
KZA:
!
Z
,
!
&
for the sample of
58575
T
sequences is computed in one traversal of
the tree.
!
A # -mismatch tree is a rooted tree of depth where each internal node has $ "& '
branches and each branch is labeled with a symbol from $ . A leaf node represents a fixed
-mer in our feature space ? obtained by concatenating the branch symbols along the path
from root to leaf ? and an internal node represents the prefix for those -mer features which
are its descendants in the tree. We use a depth-first search of this tree to store, at each node
that we visit, a set of pointers to all instances of the current prefix pattern that occur with
mismatches in the sample data. Thus at each node of depth , we maintain pointers to
all substrings from the sample data set whose -length prefixes are within mismatches
from the -length prefix represented by the path down from the root. Note that the set of
valid substrings at a node is a subset of the set of valid substrings of its parent. When we
encounter a node with an empty list of pointers (no valid occurrences of the current prefix),
we do not need to search below it in the tree. When we reach a leaf node, we sum the
contributions of all instances occurring in each source sequence to obtain feature! values
corresponding to the current -mer, and we update the kernel matrix entry WZ Z for
each pair of source sequences Z
and Z having non-zero feature values.
0
A
V
L
A
L
K
A
V
A
0
A
V
L
A
L
K
A
V
A
0
A
V
L
A
L
K
A
V
(a)
0
V
L
A
L
K
A
V
L
0
L
A
L
K
A
V
L
L
(b)
0
V
L
A
L
K
A
V
1
L
A
L
K
A
V
L
1
A
L
K
A
V
L
L
0
V
L
A
L
K
A
V
L
0
V
L
A
L
K
A
V
0
L
A
L
K
A
V
L
L
1
L
A
L
K
A
V
L
1
A
L
K
A
V
L
L
0
V
L
A
L
K
A
V
L
0
L
A
L
K
A
V
L
L
L
1
L
A
L
K
A
V
(c)
1
A
L
K
A
V
L
Figure 1: An
-mismatch tree for a sequence AVLALKAVLL, showing valid instances at each
node down a path: (a) at the root node; (b) after expanding the path ; and (c) after expanding the
path . The number of mismatches for each instance is also indicated.
3.2 Efficiency of the Kernel Computation
Since we compute the kernel in one depth-first traversal, we do not actually need to store
the entire mismatch tree but instead compute the kernel using a recursive function, which
makes more efficient use of memory and allows kernel computations for large datasets.
,
B
:
E'
"!
The number
of -mers within
TA
:
mismatches of any given fixed -mer is
&
!
N
'K&
B
'WB
. Thus the effective number of -mer instances that we
B 'WB , where =
need to traverse grows as E=
is the total length of the sample data. At
a leaf node, if exactly input sequences contain valid instances of the current -mer, one
3
performs updates to the kernel matrix. For sequences each of length . (total length
=
&.!
), the worst case for the kernel computation occurs when the feature vectors
are all equal and have the
maximal
number of non-zero entries, giving worst case overall
"!
!
3
3
running time "
.#
'W &
"
. B ' B . For the application we discuss here,
small values of are most useful, and the kernel calculations are quite inexpensive.
When mismatch kernels are used in combination with SVMs, the learned classifier $ WZ"6&
,
!
(where Z : are the training sequences that map to
FH>
WZ"3
)A@ BDC
support vectors, : are labels, and . : are weights) can be implemented by pre-computing
and storing per -mer scores. Then the prediction $ WZ" can be calculated in linear time by
look-up of -mer scores. In practice, one usually wants to use a normalized feature map,
so one would also need to compute the norm of the vector F>
WZ" , with complexity
)A@ BDC
.
B 'WB for a sequence of length . . Simple
9TA normalization schemes, like dividing
by sequence length, can also be used.
:
:W.:
1
FH>
KZ8:K
)A@ BDC
4 Experiments: Remote Protein Homology Detection
We test the mismatch kernel with an SVM classifier on the SCOP [15] (version 1.37)
datasets designed by Jaakkola et al. [10] for the remote homology detection problem. In
these experiments, remote homology is simulated by holding out all members of a target
SCOP family from a given superfamily. Positive training examples are chosen from the
remaining families in the same superfamily, and negative test and training examples are
chosen from disjoint sets of folds outside the target family?s fold. The held-out family
members serve as positive test examples. In order to train HMMs, Jaakkola et al. used the
SAM-T98 algorithm to pull in domain homologs from the non-redundant protein database
and added these sequences as positive examples in the experiments. Details of the datasets
are available at www.soe.ucsc.edu/research/compbio/discriminative.
Because the test sets are
designed ! for remote! homology detection, we use small val"!
ues of . We tested # & ! TA and TA , where we normalized the kernel via
>
Norm
) @ B*C
KZ
!
&
>
#>
)A@ BDC
)A@ D
B C
!
Z
W
Z"
WZ
>
) @ B*C
M!
"!
5
We found that
&
!
TA
gave
slightly
better performance,
though results were similar for the two choices. (Data
"!
!
for & T not shown.) We use a publicly available SVM implementation
(www.cs.columbia.edu/compbio/svm) of the soft margin optimization algorithm described
in [10]. For comparison, we include results from three other methods. These include the
original experimental results from Jaakkola et al. for two methods: the SAM-T98 iterative
HMM, and the Fisher-SVM method. We also test PSI-BLAST [3], an alignment-based
method widely used in the biological community, on the same data using the methodology
described in [14].
Figure 2 illustrates the mismatch-SVM method?s performance relative to three existing
homology detection methods as measured by ROC scores. The figure includes results for
SCOP families, and each series corresponds to one homology detection method.
all
Qualitatively, the curves for Fisher-SVM and mismatch-SVM are quite similar. When
we compare the overall performance of two methods using a two-tailed signed rank test
[18, 19] based on ROC scores over the 33 families with a -value threshold of -M5 - and
including a Bonferroni adjustment to account for multiple comparisons, we find only the
following significant differences: Fisher-SVM and mismatch-SVM perform better than
SAM-T98 (with p-values 1.3e-02 and 2.7e-02, respectively); and these three methods all
perform significantly better than PSI-BLAST in this experiment.
!
Figure 3 shows a family-by-family comparison of performance of the TA -mismatchSVM and Fisher-SVM using ROC scores in plot (A) and ROC-50 scores in plot (B). 1 In
both plots, the points fall approximately evenly above and below the diagonal, indicating
little difference in performance between the two methods. Figure 4 shows the improvement
provided by including mismatches in the SVM kernel. The figures plot ROC scores (plot
1
The ROC-50 score is the area under the graph of the number of true positives as a function of
false positives, up to the first 50 false positives, scaled so that both axes range from 0 to 1. This
score is sometimes preferred in the computational biology community, motivated by the idea that a
biologist might be willing to sift through about 50 false positives.
35
Number of families
30
25
20
15
10
(5,1)-Mismatch-SVM ROC
Fisher-SVM ROC
SAM-T98
PSI-BLAST
5
0
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
ROC
Figure 2: Comparison of four homology detection methods. The graph plots the total number of
families for which a given method exceeds an ROC score threshold.
, & T
(A)) and ROC-50 scores (plot (B)) for two string kernel SVM methods: using &
mismatch kernel, and using &
(no mismatch) spectrum kernel, the best-performing
choice with
&
- . Almost all of the families perform better with mismatching than
without, showing that mismatching gives significantly better generalization performance.
5 Discussion
We have presented a class of string kernels that measure sequence similarity without requiring alignment or depending upon a generative model, and we have given an efficient
method for computing these kernels. For the remote homology detection problem, our discriminative approach ? combining support vector machines with the mismatch kernel ?
performs as well in the SCOP experiments as the most successful known method.
A practical protein classification system would involve fast multi-class prediction ? potentially involving thousands of binary classifiers ? on massive test sets. In such applications,
computational efficiency of the kernel function becomes an important issue. Chris Watkins
[20] and David Haussler [21] have recently defined a set of kernel functions over strings,
and one of these string kernels has been implemented for a text classification problem [22].
However, the cost of computing each kernel entry is . 3 in the length of the input sequences. Similarly, the Fisher kernel of Jaakkola
et al. requires quadratic-time computation
!
for each Fisher vector calculated. The # -mismatch kernel is relatively inexpensive to
compute for values of
that are practical in applications, allows computation of multiple kernel values in one pass, and significantly improves performance over the previously
presented (mismatch-free) spectrum kernel.
Many family-based remote homogy detection algorithms incorporate a method for selecting probable domain homologs from unannotated protein sequence databases for additional
training data. In these experiments, we used the domain homologs that were identified by
SAM-T98 (an iterative HMM-based algorithm) as part of the Fisher-SVM method and included in the datasets; these homologs may be more useful to the Fisher kernel than to the
mismatch kernel. We plan to extend our method by investigating semi-supervised techniques for selecting unannotated sequences for use with the mismatch-SVM.
1
1
0.95
0.8
Fisher-SVM ROC50
Fisher-SVM ROC
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.6
0.4
0.2
0.55
0.5
0.5
0
0.55
0.6
0.65 0.7 0.75 0.8 0.85
(5,1)-Mismatch-SVM ROC
0.9
0.95
1
0
0.2
(A)
0.4
0.6
(5,1)-Mismatch-SVM ROC50
0.8
1
(B)
Figure 3: Family-by-family comparison of -mismatch-SVM with Fisher-SVM. The coordinates of each point in the plot are the ROC scores (plot (A)) or ROC-50 scores (plot (B)) for one
SCOP family, obtained using the mismatch-SVM with
,
(x-axis) and Fisher-SVM
.
(y-axis). The dotted line is
1
1
0.95
k=3 Spectrum-SVM ROC50
k=3 Spectrum-SVM ROC
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.8
0.6
0.4
0.2
0.55
0.5
0.5
0
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
0
0.2
(5,1)-Mismatch-SVM ROC
0.4
0.6
0.8
1
(5,1)-Mismatch-SVM ROC50
(A)
(B)
Figure 4: Family-by-family comparison of -mismatch-SVM with spectrum-SVM. The coordinates of each point in the plot are the ROC scores (plot (A)) or ROC-50 scores (plot (B)) for one
SCOP family, obtained using the mismatch-SVM with
,
(x-axis) and spectrum-SVM
with
(y-axis). The dotted line is
.
Many interesting variations on the mismatch kernel can be explored using the framework
presented here. For example, explicit -mer feature selection can be implemented during calculation of the kernel matrix, based on a criterion enforced at each leaf or internal
node. Potentially, a good feature selection criterion could improve performance in certain
applications while decreasing kernel computation time. In biological applications, it is
also natural to consider weighting each -mer instance contribution to a feature coordinate
by evolutionary substitution probabilities. Finally, one could use linear combinations of
kernels #>
to capture similarity of different length -mers. We believe that further
) @ B EC
experimentation with mismatch string kernels could be fruitful for remote protein homology detection and other biological sequence classification problems.
Acknowledgments
CL is partially supported by NIH grant LM07276-02. WSN is supported by NSF grants
DBI-0078523 and ISI-0093302. We thank Nir Friedman for pointing out the connection
with Fisher scores for Markov chain models.
References
[1] M. S. Waterman, J. Joyce, and M. Eggert. Computer alignment of sequences, chapter Phylogenetic Analysis of DNA Sequences. Oxford, 1991.
[2] S. F. Altschul, W. Gish, W. Miller, E. W. Myers, and D. J. Lipman. A basic local alignment
search tool. Journal of Molecular Biology, 215:403?410, 1990.
[3] S. F. Altschul, T. L. Madden, A. A. Schaffer, J. Zhang, Z. Zhang, W. Miller, and D. J. Lipman. Gapped BLAST and PSI-BLAST: A new generation of protein database search programs.
Nucleic Acids Research, 25:3389?3402, 1997.
[4] Michael Gribskov, Andrew D. McLachlan, and David Eisenberg. Profile analysis: Detection of
distantly related proteins. PNAS, pages 4355?4358, 1987.
[5] A. Bairoch. The PROSITE database, its status in 1995. Nucleic Acids Research, 24:189?196,
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[6] T. K. Attwood, M. E. Beck, D. R. Flower, P. Scordis, and J. N Selley. The PRINTS protein
fingerprint database in its fifth year. Nucleic Acids Research, 26(1):304?308, 1998.
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1531, 1994.
[8] S. R. Eddy. Multiple alignment using hidden markov models. In Proceedings of the Third
International Conference on Intelligent Systems for Molecular Biology, pages 114?120. AAAI
Press, 1995.
[9] P. Baldi, Y. Chauvin, T. Hunkapiller, and M. A. McClure. Hidden markov models of biological
primary sequence information. PNAS, 91(3):1059?1063, 1994.
[10] T. Jaakkola, M. Diekhans, and D. Haussler. A discriminative framework for detecting remote
protein homologies. Journal of Computational Biology, 2000.
[11] T. Jaakkola, M. Diekhans, and D. Haussler. Using the fisher kernel method to detect remote
protein homologies. In Proceedings of the Seventh International Conference on Intelligent
Systems for Molecular Biology, pages 149?158. AAAI Press, 1999.
[12] V. N. Vapnik. Statistical Learning Theory. Springer, 1998.
[13] N. Cristianini and J. Shawe-Taylor. An Introduction to Support Vector Machines. Cambridge,
2000.
[14] C. Leslie, E. Eskin, and W. S. Noble. The spectrum kernel: A string kernel for SVM protein
classification. Proceedings of the Pacific Biocomputing Symposium, 2002.
[15] A. G. Murzin, S. E. Brenner, T. Hubbard, and C. Chothia. SCOP: A structural classification
of proteins database for the investigation of sequences and structures. Journal of Molecular
Biology, 247:536?540, 1995.
[16] M. Sagot. Spelling approximate or repeated motifs using a suffix tree. Lecture Notes in Computer Science, 1380:111?127, 1998.
[17] G. Pavesi, G. Mauri, and G. Pesole. An algorithm for finding signals of unknown length in DNA
sequences. Bioinformatics, 17:S207?S214, July 2001. Proceedings of the Ninth International
Conference on Intelligent Systems for Molecular Biology.
[18] S. Henikoff and J. G. Henikoff. Embedding strategies for effective use of information from
multiple sequence alignments. Protein Science, 6(3):698?705, 1997.
[19] S. L. Salzberg. On comparing classifiers: Pitfalls to avoid and a recommended approach. Data
Mining and Knowledge Discovery, 1:371?328, 1997.
[20] C. Watkins. Dynamic alignment kernels. Technical report, UL Royal Holloway, 1999.
[21] D. Haussler. Convolution kernels on discrete structure. Technical report, UC Santa Cruz, 1999.
[22] Huma Lodhi, John Shawe-Taylor, Nello Cristianini, and Chris Watkins. Text classification using
string kernels. Preprint.
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1,295 | 218 | 750
Koch, Bair, Harris, Horiuchi, Hsu and Luo
Real- Time Computer Vision and Robotics
Using Analog VLSI Circuits
Christof Koch
Wyeth Bair
John G. Harris
Timothy Horiuchi
Andrew Hsu
Jin Luo
Computation and Neural Systems Program
Caltech 216-76
Pasadena, CA 91125
ABSTRACT
The long-term goal of our laboratory is the development of analog
resistive network-based VLSI implementations of early and intermediate vision algorithms. We demonstrate an experimental circuit for smoothing and segmenting noisy and sparse depth data
using the resistive fuse and a 1-D edge-detection circuit for computing zero-crossings using two resistive grids with different spaceconstants. To demonstrate the robustness of our algorithms and
of the fabricated analog CMOS VLSI chips, we are mounting these
circuits onto small mobile vehicles operating in a real-time, laboratory environment.
1
INTRODUCTION
A large number of computer vision algorithms for finding intensity edges, computing motion, depth, and color, and recovering the 3-D shapes of objects have been
developed within the framework of minimizing an associated "energy" functional.
Such a variational formalism is attractive because it allows a priori constraints
to be explicitly stated. The single most important constraint is that the physical
processes underlying image formation, such as depth, orientation and surface reflectance, change slowly in space. For instance, the depths of neighboring points on
a surface are usually very similar. Standard regularization algorithms embody this
smoothness constraint and lead to quadratic variational functionals with a unique
global minimum (Poggio, Torre, and Koch, 1985). These quadratic functionals
Real-Time Computer Vision and Robotics Using Analog VLSI Circuits
G
G
G
G
Rl
G
G
Rl
(a)
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Node
Voltage
(b)
3.0V
I? I?
?
1
1
Edge
Output
?
2
345
Photoreceptor
6
o
7
Figure 1: (a) shows the schematic of the zero-crossing chip. The phototransistors
logarithmically map light intensity to voltages that are applied via a conductance G
onto the nodes of two linear resistive networks. The network resistances Rl and R2
can be arbitrarily adjusted to achieve different space-constants. Transconductance
amplifiers compute the difference of the smoothed network node voltages and report
a current proportional to that difference. The sign of current then drives exclusive-or
circuitry (not shown) between each pair of neighboring pixels. The final output is
a binary signal indicating the positions of the zero-crossings. The linear network
resistances have been implemented using Mead's saturating resistor circuit (Mead,
1989), and the vertical resistors are implemented with transconductance followers.
(b) shows the measured response of a seven-pixel version of the chip to a bright
background with a shadow cast across the middle three photoreceptors. The circles
and triangles show the node voltages on the resistive networks with the smaller and
larger space-constants, respectively. Edges are indicated by the binary output (bar
chart at bottom) corresponding to the locations of zero-crossings.
751
752
Koch, Bair, Harris, Horiuchi, Hsu and Luo
can be mapped onto linear resistive networks, such that the stationary voltage distribution, corresponding to the state of least power dissipation, is equivalent to
the solution of the variational functional (Horn, 1974; Poggio and Koch , 1985).
Smoothness breaks down, however, at discontinuities caused by occlusions or differences in the physical processes underlying image formation (e.g., different surface
reflectance properties). Detecting these discontinuities becomes crucial, not only
because otherwise smoothness is incorrectly applied but also because the locations
of discontinuities are often required for further image analysis and understanding.
We describe two different approaches for finding discontinuities in early vision: (1)
a 1-D edge-detection circuit for computing zero-crossings using two resistive grids
with different space-constants, and (2) a 20 by 20 pixel circuit for smoothing and
segmenting noisy and sparse depth data using the resistive fuse.
Finally, while successfully demonstrating a highly integrated circuit on a stationary
laboratory bench under controlled conditions is already a tremendous success, this
is not the environment in which we ultimately intend them to be used. The jump
from a sterile, well-controlled, and predictable environment such as that of the
laboratory bench to a noisy and physically demanding environment of a mobile
robot can often spell out the true limits of a circuit's robustness. In order to
demonstrate the robustness and real-time performance of these circuits, we have
mounted two such chips onto small toy vehicles.
2
AN EDGE DETECTION CIRCUIT
The zero-crossings of the Laplacian of the Gaussian, V 2 G, are often used for detecting edges. Marr and Hildreth (1980) discovered that the Mexican-hat shape
of the V2G operator can be approximated by the difference of two Gaussians
(DOG). In this spirit, we have built a chip that takes the difference of two resistivenetwork smoothings of photoreceptor input and finds the resulting zero-crossings.
The Green's function of the resistive network, a decaying exponential, differs from
the Gaussian, but simulations with digitized camera images have shown that the
difference of exponentials (DOE) gives results nearly as good as the DOG. Furthermore, resistive nets have a natural implementation in silicon, while implementing
the Gaussian is cumbersome.
The circuit, Figure la, uses two independent resistive networks to smooth the voltages supplied by logarithmic photoreceptors. The voltages on the two networks are
subtracted and exclusive-or circuitry (not shown) is used to detect zero-crossings. In
order to facilitate thresholding of edges, an additional current is computed at each
node indicating the strength of the zero-crossing. This is particularly important
for robust real-world performance where there will be many small (in magnitude
of slope) zero-crossings due to noise. Figure 1b shows the measured response of a
seven-pixel version of the chip to a bright background with a shadow cast across the
middle three photoreceptors. Subtracting the two network voltage traces shown at
the top, we find two zero-crossings, which the chip correctly identifies in the binary
output shown at the bottom.
Real-Time Computer Vision and Robotics Using Analog VLSI Circuits
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Figure 2: (a) Schematic diagram for the 20 by 20 pixel surface interpolation and
smoothing chip. A rectangular mesh of resistive fuse elements (shown as rectangles)
provide the smoothing and segmentation ability of the network. The data are given
as battery values dij with the conductance G connecting the battery to the grid set
to G = 1/2u 2 , where u 2 is the variance of the additive Gaussian noise assumed to
corrupt the data. (b) Measured current-voltage relationship for different settings
of the resistive fuse. For a voltage of less than VT across this two-terminal device,
the circuit acts as a resistor with conductance A. Above VT, the current is either
abruptly set to zero (binary fuse) or smoothly goes to zero (analog fuse). We can
continuously vary the I-V curve from the hyperbolic tangent of Mead's saturating
resistor (HRES) to that of an analog fuse (Fig. 2b), effectively implementing a
continuation method for minimizing the non-convex functional. The I-V curve of a
binary fuse is also illustrated.
753
754
Koch, Bair, Harris, Horiuchi, Hsu and Luo
3
A CIRCUIT FOR SMOOTHING AND SEGMENTING
Many researchers have extended regularization theory to include discontinuities.
Let us consider the problem of interpolating noisy and sparse 1-D data (the 2-D
generalization is straightforward), where the depth data di is given on a discrete
grid. Associated with each lattice point is the value of the recovered surface Ii
and a binary line discontinuity Ii. When the surface is expected to be smooth
(with a first-order, membrane-type stabilizer) except at isolated discontinuities, the
functional to be minimized is given by:
J(f, I) = A~(fi+l - 1i)2(1 -Ii)
+ 2!2 ~(di -
I
I
1i)2 + a
~ Ii
(1)
I
where (]'2 is the variance of the additive Gaussian noise process assumed to corrupt
the data di, and A and a are free parameters. The first term implements the
piecewise smooth constraint: if all variables, with the exception of Ii, Ii+l, and Ii,
are held fixed and A(fi+l - h)2 < a, it is "cheaper" to pay the price A(fi+l - h)2
and set Ii = 0 than to pay the larger price a; if the gradient becomes too steep,
Ii = 1, and the surface is segmented at that location. The second term, with the
sum only including those locations i where data exist, forces the surface I to be
close to the measured data d. How close depends on the estimated magnitude of
the noise, in this case on (]'2. The final surface I is the one that best satisfies the
conflicting demands of piecewise smoothness and fidelity on the measured data.
To minimize the 2-D generalization of eq. (1), we map the functional J onto the
circuit shown in Fig. 2a such that the stationary voltage at every gridpoint then
corresponds to hi. The cost functional J is interpreted as electrical co-content,
the generalization of power for nonlinear networks. We designed a two-terminal
nonlinear device, which we call a resistive fuse, to implement piecewise smoothness
(Fig. 2b). If the magnitude of the voltage drop across the device is less than
VT = (a/A)1/2, the fuse acts as a linear resistor with conductance A. If VT is
exceeded, however, the fuse breaks and the current goes to zero. The operation of
the fuse is fully reversible. We built a 20 by 20 pixel fuse network chip and show
its segmentation and smoothing performance in Figure 3.
4
AUTONOMOUS VEHICLES
Our goal-beyond the design and fabrication of analog resistive-network chips-is
to build mobile testbeds for the evaluation of chips as well as to provide a systems
perspective on the usefulness of certain vision algorithms. Due to the small size
and power requirements of these chips, it is possible to utilize the vast resource of
commercially available toy vehicles. The advantages of toy cars over robotic vehicles
built for research are their low cost, ease of modification, high power-to-weight ratio,
availability, and inherent robustness to the real-world. Accordingly, we integrated
two analog resistive-network chips designed and built in Mead's laboratory onto
small toy cars controlled by a digital microprocessor (see Figure 4).
Real-Time Computer Vision and Robotics Using Analog VLSI Circuits
(c)
(b)
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( f)
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Figure 3: Experimental data from the fuse network chip. We use as input data a
tower (corresponding to dij = 3.0 V) rising from a plane (corresponding to 2.0 V)
with superimposed Gaussian noise. (a) shows the input with the variance of the
noise set to 0.2 V, (b) the voltage output using the fuse configured as a saturating
resistance, and (c) the output when the fuse elements are activated. (d), (e),
and (f) illustrate the same behavior along a horizontal slice across the chip for
(12
0.4 V. We used a hardware deterministic algorithm of varying the fuse I-V
curve of the saturating resistor to that of the analog fuse (following the arroW in
Fig. 2b) as well as increasing the conductance A. This algorithm is closely related
to other deterministic approximations based on continuation methods or a Mean
Field Theory approach (Koch, Marroquin, and Yuille, 1986; Blake and Zisserman,
1987; Geiger and Girosi, 1989). Notice that the amplitude of the noise in the last
case (40% of the amplitude of the voltage step) is so large that a single filtering
step on the input (d) will fail to detect the tower. Cooperativity and hysteresis
are required for optimal performance. Notice the "bad" pixel in the middle of the
tower (in c). Its effect is localized, however, to a single element.
=
755
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1,296 | 2,180 | Automatic Alignment of Local Representations
Yee Whye Teh and Sam Roweis
Department of Computer Science, University of Toronto
ywteh,roweis @cs.toronto.edu
Abstract
We present an automatic alignment procedure which maps the disparate
internal representations learned by several local dimensionality reduction
experts into a single, coherent global coordinate system for the original
data space. Our algorithm can be applied to any set of experts, each
of which produces a low-dimensional local representation of a highdimensional input. Unlike recent efforts to coordinate such models by
modifying their objective functions [1, 2], our algorithm is invoked after
training and applies an efficient eigensolver to post-process the trained
models. The post-processing has no local optima and the size of the system it must solve scales with the number of local models rather than the
number of original data points, making it more efficient than model-free
algorithms such as Isomap [3] or LLE [4].
1 Introduction: Local vs. Global Dimensionality Reduction
Beyond density modelling, an important goal of unsupervised learning is to discover compact, informative representations of high-dimensional data. If the data lie on a smooth low
dimensional manifold, then an excellent encoding is the coordinates internal to that manifold. The process of determining such coordinates is dimensionality reduction. Linear
dimensionality reduction methods such as principal component analysis and factor analysis are easy to train but cannot capture the structure of curved manifolds. Mixtures of these
simple unsupervised models [5, 6, 7, 8] have been used to perform local dimensionality
reduction, and can provide good density models for curved manifolds, but unfortunately
such mixtures cannot do dimensionality reduction. They do not describe a single, coherent low-dimensional coordinate system for the data since there is no pressure for the local
coordinates of each component to agree.
Roweis et al [1] recently proposed a model which performs global coordination of local
coordinate systems in a mixture of factor analyzers (MFA). Their model is trained by maximizing the likelihood of the data, with an additional variational penalty term to encourage
the internal coordinates of the factor analyzers to agree. While their model can trade off
modelling the data and having consistent local coordinate systems, it requires a user given
trade-off parameter, training is quite inefficient (although [2] describes an improved training algorithm for a more constrained model), and it has quite serious local minima problems
(methods like LLE [4] or Isomap [3] have to be used for initialization).
In this paper we describe a novel, automatic way to align the hidden representations used by
each component of a mixture of dimensionality reducers into a single global representation
of the data throughout space. Given an already trained mixture, the alignment is achieved
by applying an eigensolver to a matrix constructed from the internal representations of the
mixture components. Our method is efficient, simple to implement, and has no local optima
in its optimization nor any learning rates or annealing schedules.
2 The Locally Linear Coordination Algorithm
!
#
%$ * '
$,+.'
/ $ * '
$ 10
Suppose we have a set of data points given by the rows of
from
a -dimensional space, which we assume are sampled from a
dimensional manifold. We approximate the manifold coordinates using images
in a
dimensional embedding space. Suppose also that we have already trained, or have been
given, a mixture of local dimensionality reducers. The th reducer produces a dimensional internal representation
for data point
as well as a ?responsibility?
describing how reliable the th reducer?s representation of
is. These satisfy
and can be obtained, for example, using a gating network in a mixture of experts, or the
posterior probabilities in a probabilistic network. Notice that the manifold coordinates and
internal representations need not have the same number of dimensions.
"
# &('
$
)'
Given the data, internal representations, and responsibilities, our algorithm automatically
aligns the various hidden representations into a single global coordinate system. Two key
ideas motivate the method. First, to use a convex cost function whose unique minimum is
attained at the desired global coordinates. Second, to restrict the global coordinates
to
depend on the data
only through the local representations
and responsibilities
,
thereby leveraging the structure of the mixture model to regularize and reduce the effective
size of the optimization problem. In effect, rather than working with individual data points,
we work with large groups of points belonging to particular submodels.
'
3 '$
* ' '2$
& '
$
' & '
$ * '
$ & '
$
?A@ B B
3
* '2$%9 $ & '
$;: 4<5$>= 87 $ 7BDC * '
$>E '
$ 4 $ B F7>G H ' G 4 G B
J J 9LK # = HM5 ' G N*O'
$P& '
$ 4 G N4 $
3 $ E '2B $ K
& '
$ E '
5 $ Q0
J 9LK # = J 9LK R# =
J B J 9K R# =
0
RS%A3 " :,/ G $ $ B
G
J
3
I
HT' F*O'
$ E '
$ I
4 .4 $ 3
in terms of
and
. Given an input
We first parameterize the global coordinates
, each local model infers its internal coordinates
and then applies a linear projection
and offset
to these to obtain its guess at the global coordinates. The final global
is obtained by averaging the guesses using the responsibilities as weights:
coordinates
' 465$
' 87 $
F8I 3
B
4$ K
9LK # = 9LK R# =
(1)
(2)
where
is the th column of
,
is the th entry of
, and
is a bias.
This process is described in figure 1. To simplify our calculations, we have vectorized the
indices
into a single new index
, where
is an invertible mapping from
to
. For compactness, we will write
.
the domain of
Now define the matrices and as
and the th row of as
. Then
(1) becomes a system of linear equations (2) with fixed and unknown parameters .
responsibilities
r nk
unj
high?dimensional
data
xn
alignment
parameters
lj
yn
global
coordinates
z nk
local dimensionality
reduction models
Responsibility?weighted
local representations
local coordinates
Figure 1: Obtaining global coordinates from data via responsibility-weighted local coordinates.
''
B
The key assumption, which we have emphasized by re-expressing
above, is that the
mapping between the local representations and the global coordinates
is linear in each
of
,
and the unknown parameters . Crucially, however, the mapping between the
original data
and the images
is highly non-linear since it depends on the multiplication of responsibilities and internal coordinates which are in turn non-linearly related to
the data
through the inference procedure of the mixture model.
& '
$ * '
$ '
'
9 =
3
'
4$
3 9 9 3 =
9 = =
3
9 = -
I I
3
3 ? ?
3 I 3
I ?
We now consider determining according to some given cost function
. For this we
advocate using a convex
. Notice that since is linear in ,
is convex in
as well, and there is a unique optimum that can be computed efficiently using a variety of
methods. This is still true if we also have feasible convex constraints
on . The
case where the cost and constraints are both quadratic is particularly appealing since we
can use an eigensolver to find the optimal . In particular suppose and are matrices
defining the cost and constraints, and let
and
. This gives:
9 = 3 3
3 I 3 I
3
8 FI I
9 = -
(3)
where
is the trace operator. The matrices
and
are typically obtained from the
original data and summarize the essential geometries among them. The solution to the
constrained minimization above is given by the smallest generalized eigenvectors with
. In particular the columns of are given by these generalized eigenvectors.
3
Below, we investigate a cost function based on the Locally Linear Embedding (LLE) algorithm of Roweis and Saul [4]. We call the resulting algorithm Locally Linear Coordination (LLC). The idea of LLE is to preserve the same locally linear relationships between
the original data points
and their counterparts . We identify for each point
its
nearest-neighbours
and then minimize
'
'
' '
9 , = F7 ' '
/ !#"#$ ' % ( 9 &
' = 9 (
' =
/ ! " $ ' 0
$ '
)
'
9 = * ( 9 &
' = 9 (
' =
+ 0 7 ' ' +-0 0, + 0 7 ' ' ' + 0 *
?
0,
0 3
9 . 3 = % ( I 9 ( 3
= 9 (3
' 3 = I 3 3 M* /O 3 3
0 , MI I I 0 % ?
8FI 9 (
' = 9
= I + 0 I I
(4)
subject to the constraints
. The weights are unique1
with respect to
and can be solved for efficiently using constrained least squares (since solving for
is
decoupled across ). The weights summarize the local geometries relating the data points
to their neighbours, hence to preserve these relationships among the coordinates
we
arrange to minimize the same cost
(5)
but with respect to instead. is invariant to translations and rotations of , and scales as
we scale . In order to break these degeneracies we enforce the following constraints:
(6)
where is a vector of ?s. For this choice, the cost function and constraints above become:
(7)
(8)
with cost and constraint matrices
1
(9)
1
In the unusual case where the number of neighbours is larger than the dimensionality of the data
, simple regularization of the norm of the weights once again makes them unique.
0 , I 3
5 I /,0
5
8 0(
As shown previously, the solution to this problem is given by the smallest generalized
. To satisfy
, we need to find eigenvectors
eigenvectors with
that are orthogonal to the vector
. Fortunately,
is the smallest generalized
eigenvector, correspondingto
an eigenvalue of 0. Hence the solution to the problem is
given by the
to
smallest generalized eigenvectors instead.
S ' ? 9 : 0 =
LLC Alignment Algorithm:
Using data
, compute local linear reconstruction weights
$ '
using (4).
& '
$ '
Train or receive a pre-trained mixture of local dimensionality reducers.
Apply this mixture to , obtaining a local representation
and
responsibility
for each submodel and each data point .
* '
$
Form the matrix
I
with
GH ' * '
$>E '
B #$
: 0
BS : 0
4 F$ I 3
and calculate
0(
Find the eigenvectors corresponding to the smallest
of the generalized eigenvalue system
.
3
J
3
nd
Let be a matrix with columns formed by the
Return the th row of as alignment weight .
Return the global manifold coordinates as
" : / $$
to
and
from (9).
eigenvalues
st
eigenvectors.
.
Note that the edge size of the matrices and whose generalized eigenvectors we seek
which scales with the number of components and dimensions of the local
is
representations but not with the number of data points . As a result, solving for the
alignment weights is much more efficient than the original LLE computation (or those
of Isomap) which requires solving an eigenvalue system of edge size . In effect, we
have leveraged the mixture of local models to collapse large groups of points together and
worked only with those groups rather than the original data points. Notice however that
the computation of the weights
still requires determining the neighbours of the original
in the worse case.
data points, which scales as
+
9 + =
#
+
*P4 5$$
3$
Coordination with LLC also yields a mixture of noiseless factor analyzers over the global
coordinate space , with the th factor analyzer having mean
and factor loading
.
Given any global coordinates , we can infer the responsibilities and the posterior means
over the latent space of each factor analyzer. If our original local dimensionality reducers also supports computing from and , we can now infer the high dimensional mean
data point which corresponds to the global coordinates . This allows us to perform operations like visualization and interpolation using the global coordinate system. This is the
method we used to infer the images in figures 4 and 5 in the next section.
&$
*$ &$
3 Experimental Results using Mixtures of Factor Analyzers
The alignment computation we have described is applicable to any mixture of local dimensionality reducers. In our experiments, we have used the most basic such model: a mixture
of factor analyzers (MFA) [8]. The th factor analyzer in the mixture describes a probabilistic linear mapping from a latent variable
to the data with additive Gaussian noise.
The model assumes that the data manifold is locally linear and it is this local structure that
is captured by each factor analyzer. The non-linearity in the data manifold is handled by
patching multiple factor analyzers together, each handling a locally linear region.
#
&$
MFAs are trained in an unsupervised way by maximizing the marginal log likelihood of
the observed data, and parameter estimation is typically done using the EM algorithm 2.
2
In our experiments, we initialized the parameters by drawing the means from the global covariance of the data and setting the factors to small random values. We also simplified the factor analyzers
to share the same spherical noise covariance
although this is not essential to the process.
A
B
C
D
Figure 2: LLC on the S curve (A). There are 14 factor analyzers in the mixture (B), each with 2 latent
dimensions. Each disk represents one of them with the two black lines being the factor loadings. After
alignment by LLC (C), the curve is successfully unrolled; it is also possible to retroactively align the
original data space models (D).
A
Figure 3: Unknotting the trefoil
B
curve. We generated 6000 noisy
points from the curve. Then we fit
an MFA with 30 components with
1 latent dimension each (A), but
aligned them in a 2D space (B).
We used 10 neighbours to reconstruct each data point.
&$
Since there is no constraint relating the various hidden variables , a MFA trained only
to maximize likelihood cannot learn a global coordinate system for the manifold that is
consistent across every factor analyzer. Hence this is a perfect model on which to apply
automatic alignment. Naturally, we use the mean of conditioned on the data (assuming
the th factor analyzer generated ) as the th local representation of , while we use the
posterior probability that the th factor analyzer generated as the responsibility.
#
#
#
&$
We illustrate LLC on two synthetic toy problems to give some intuition about how it works.
The first problem is the S curve given in figure 2(A). An MFA trained on 1200 points
sampled uniformly from the manifold with added noise (B) is able to model the linear
structure of the curve locally, however the internal coordinates of the factor analyzers are
not aligned properly. We applied LLC to the local representations and aligned them in a 2D
space (C). When solving for local weights, we used 12 neighbours to reconstruct each data
point. We see that LLC has successfully unrolled the S curve onto the 2D space. Further,
given the coordinate transforms produced by LLC, we can retroactively align the latent
spaces of the MFAs (D). This is done by determining directions in the various latent spaces
which get transformed to the same direction in the global space.
To emphasize the topological advantages of aligning representations into a space of higher
dimensionality than the local coordinates used by each submodel, we also trained a MFA
on data sampled from a trefoil curve, as shown in figure 3(A). The trefoil is a circle with a
knot in 3D. As figure 3(B) shows, LLC connects these models into a ring of local topology
faithful to the original data.
We applied LLC to MFAs trained on sets of real images believed to come from a complex
manifold with few degrees of freedom. We studied face images of a single person under
varying pose and expression changes and handwritten digits from the MNIST database.
After training the MFAs, we applied LLC to align the models. The face models were
aligned into a 2D space as shown in figure 4. The first dimension appears to describe
Figure 4: A map of reconstructions of the faces when the global coordinates are specified. Contours
describe the likelihood of the coordinates. Note that some reconstructions around the edge of the map
are not good because the model is extrapolating from the training images to regions of low likelihood.
A MFA with 20 components and 8 latent dimensions each is used. It is trained on 1965 images. The
weights
are calculated using 36 neighbours.
changes in pose, while the second describes changes in expression. The digit models were
aligned into a 3D space. Figure 5 (top) shows maps of reconstructions of the digits. The
first dimension appears to describe the slant of each digit, the second the fatness of each
digit, and the third the relative sizes of the upper to lower loops. Figure 5 (bottom) shows
how LLC can smoothly interpolate between any two digits. In particular, the first row
interpolates between left and right slanting digits, the second between fat and thin digits,
the third between thick and thin line strokes, and the fourth between having a larger bottom
loop and larger top loop.
4 Discussion and Conclusions
Previous work on nonlinear dimensionality reduction has usually emphasized either a parametric approach, which explicitly constructs a (sometimes probabilistic) mapping between
the high-dimensional and low-dimensional spaces, or a nonparametric approach which
merely finds low-dimensional images corresponding to high-dimensional data points but
without probabilistic models or hidden variables. Compared to the global coordination
model [1], the closest parametric approach to ours, our algorithm can be understood as post
coordination, in which the latent spaces are coordinated after they have been fit to data. By
decoupling the data fitting and coordination problems we gain efficiency and avoid local
optima in the coordination phase. Further, since we are just maximizing likelihood when
fitting the original mixture to data, we can use a whole range of known techniques to escape
local minima, and improve efficiency in the first phase as well.
On the nonparametric side, our approach can be compared to two recent algorithms, LLE
Figure 5: Top: maps of reconstructions of digits when two global coordinates are specified, and the
third integrated out. Left: st and nd coordinates specified; right: nd and rd . Bottom: Interpolating
between two digits using LLC. In each row, we interpolate between the upper leftmost and rightmost
digits. The LLC interpolants are spread out evenly along a line connecting the global coordinates of
the two digits. For comparison, we show the 20 training images whose coordinates are closest to the
line segment connecting those of the two digits at each side. A MFA with 50 components, each with
6 latent dimensions is used. It is trained on 6000 randomly chosen digits from the combined training
and test sets of 8?s in MNIST. The weights
were calculated using 36 neighbours.
[4] and Isomap [3]. The cost functions of LLE and Isomap are convex, so they do not
suffer from the local minima problems of earlier methods [9, 10], but these methods must
solve eigenvalue systems of size equal to the number of data points. (Although in LLE the
systems are highly sparse.) Another problem is neither LLE nor Isomap yield a probabilistic model or even a mapping between the data and embedding spaces. Compared to these
models (which are run on individual data points) LLC uses as its primitives descriptions
of the data provided by the individual local models. This makes the eigenvalue system to
be solved much smaller and as a result the computational cost of the coordination phase of
LLC is much less than that for LLE or Isomap. (Note that the construction of the eigenvalue
system still requires finding nearest neighbours for each point, which is costly.) Furthermore, if each local model describes a complete (probabilistic) mapping from data space
to its latent space, the final coordinated model will also describe a (probabilistic) mapping
from the whole data space to the coordinated embedding space.
Our alignment algorithm improves upon local embedding or density models by elevating
their status to full global dimensionality reduction algorithms without requiring any modifications to their training procedures or cost functions. For example, using mixtures of factor
analyzers (MFAs) as a test case, we show how our alignment method can allow a model
previously suited only for density estimation to do complex operations on high dimensional
data such as visualization and interpolation.
Brand [11] has recently proposed an approach, similar to ours, that coordinates local parametric models to obtain a globally valid nonlinear embedding function. Like LLC, his
?charting? method defines a quadratic cost function and finds the optimal coordination directly and efficiently. However, charting is based on a cost function much closer in spirit to
the original global coordination model and it instantiates one local model centred on each
training point, so its scaling is the same as that of LLE and Isomap. In principle, Brand?s
method can be extended to work with fewer local models and our alignment procedure can
be applied using the charting cost rather than the LLE cost we have pursued here.
Automatic alignment procedures emphasizes a powerful but often overlooked interpretation of local mixture models. Rather than considering the output of such systems to be a
single quantity, such as a density estimate or a expert-weighted regression, it is possible
to view them as networks which convert high-dimensional inputs into a vector of internal
coordinates from each submodel, accompanied by responsibilities. As we have shown, this
view can lead to efficient and powerful algorithms which allow separate local models to
learn consistent global representations.
Acknowledgments
We thank Geoffrey Hinton for inspiration and interesting discussions, Brendan Frey and
Yann LeCun for sharing their data sets, and the reviewers for helpful comments.
References
[1] S. Roweis, L. Saul, and G. E. Hinton. Global coordination of local linear models. In Advances
in Neural Information Processing Systems, volume 14, 2002.
[2] J. J. Verbeek, N. Vlassis, and B. Kr?ose. Coordinating principal component analysers. In Proceedings of the International Conference on Artificial Neural Networks, 2002.
[3] J. B. Tenenbaum, V. de Silva, and J. C. Langford. A global geometric framework for nonlinear
dimensionality reduction. Science, 290(5500):2319?2323, December 2000.
[4] S. Roweis and L. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500):2323?2326, December 2000.
[5] K. Fukunaga and D. R. Olsen. An algorithm for finding intrinsic dimensionality of data. IEEE
Transactions on Computers, 20(2):176?193, 1971.
[6] N. Kambhatla and T. K. Leen. Dimension reduction by local principal component analysis.
Neural Computation, 9:1493?1516, 1997.
[7] M. E. Tipping and C. M. Bishop. Mixtures of probabilistic principal component analysers.
Neural Computation, 11(2):443?482, 1999.
[8] Z. Ghahramani and G. E. Hinton. The EM algorithm for mixtures of factor analyzers. Technical
Report CRG-TR-96-1, University of Toronto, Department of Computer Science, 1996.
[9] T. Kohonen. Self-organization and Associative Memory. Springer-Verlag, Berlin, 1988.
[10] C. Bishop, M. Svensen, and C. Williams. GTM: The generative topographic mapping. Neural
Computation, 10:215?234, 1998.
[11] M. Brand. Charting a manifold. This volume, 2003.
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1,297 | 2,181 | Approximate Inference and
Protein-Folding
Chen Yanover and Yair Weiss
School of Computer Science and Engineering
The Hebrew University of J erusalem
91904 Jerusalem, Israel
{cheny,yweiss} @cs.huji.ac.it
Abstract
Side-chain prediction is an important subtask in the protein-folding
problem. We show that finding a minimal energy side-chain configuration is equivalent to performing inference in an undirected
graphical model. The graphical model is relatively sparse yet has
many cycles. We used this equivalence to assess the performance of
approximate inference algorithms in a real-world setting. Specifically we compared belief propagation (BP), generalized BP (GBP)
and naive mean field (MF).
In cases where exact inference was possible, max-product BP always found the global minimum of the energy (except in few cases
where it failed to converge), while other approximation algorithms
of similar complexity did not. In the full protein data set, maxproduct BP always found a lower energy configuration than the
other algorithms, including a widely used protein-folding software
(SCWRL).
1
Introduction
Inference in graphical models scales exponentially with the number of variables.
Since many real-world applications involve hundreds of variables, it has been impossible to utilize the powerful mechanism of probabilistic inference in these applications. Despite the significant progress achieved in approximate inference, some
practical questions still remain open - it is not yet known which algorithm to use
for a given problem nor is it understood what are the advantages and disadvantages of each technique. We address these questions in the context of real-world
protein-folding application - the side-chain prediction problem.
Predicting side-chain conformation given the backbone structure is a central problem in protein-folding and molecular design. It arises both in ab-initio proteinfolding (which can be divided into two sequential tasks - the generation of nativelike backbone folds and the positioning of the side-chains upon these backbones [6])
and in homology modeling schemes (where the backbone and some side-chains are
assumed to be conserved among the homologs but the configuration of the rest of
the side-chains needs to be found).
Figure 1: Cow actin binding protein (PDB code 1pne, top) and closer view of its 6
carboxyl-terminal residues (bottom-left). Given the protein backbone (black) and
amino acid sequence, native side-chain conformation (gray) is searched for. Problem
representation as a graphical model for those carboxyl-terminal residues shown in
the bottom-right figure (nodes located at COl atom positions, edges drawn in black).
In this paper, we show the equivalence between side-chain prediction and inference
in an undirected graphical model. We compare the performance of BP, generalized
BP and naive mean field on this problem as well as comparing to a widely used
protein-folding program called SCWRL.
2
The side-chain prediction problem
Proteins are chains of simpler molecules called amino acids. All amino acids have
a common structure - a central carbon atom (COl) to which a hydrogen atom,
an amino group (N H 2 ) and a carboxyl group (COOH) are bonded. In addition,
each amino acid has a chemical group called the side-chain, bound to COl. This
group distinguishes one amino acid from another and gives its distinctive properties.
Amino acids are joined end to end during protein synthesis by the formation of
peptide bonds. An amino acid unit in a protein is called a residue. The formation
of a succession of peptide bonds generate the backbone (consisting of COl and its
adjacent atoms, N and CO, of each reside), upon which the side-chains are hanged
(Figure 1).
We seek to predict the configuration of all the side-chains relative to the backbone.
The standard approach to this problem is to define an energy function and use the
configuration that achieves the global minimum of the energy as the prediction.
2.1
The energy function
We adopted the van der Waals energy function, used by SCWRL [3], which approximates the repulsive portion of Lennard-Jones 12-6 potential. For a pair of atoms,
a and b, the energy of interaction is given by:
E(a, b) = { -k2
:'0 + k~
d> Ro
Ro ~ d ~ k1Ro
k1Ro > d
Emax
(1)
where Emax = 10, kl = 0.8254 and k2 = ~~k;' d denotes the distance between
Ro is the sum of their radii. Constant radii were used for protein's
atoms (Carbon - 1.6A, Nitrogen and Oxygen - 1.3A, Sulfur - 1.7A). For two sets
of atoms, the interaction energy is a sum of the pairwise atom interactions. The
energy surface of a typical protein in the data set has dozens to thousands local
minima.
a and band
2.2
Rotamers
The configuration of a single side-chain is represented by at most 4 dihedral angles
(denoted Xl,X2,X3 and X4)' Any assignment of X angles for all the residues defines
a protein configuration. Thus the energy minimization problem is a highly nonlinear
continuous optimization problem.
It turns out, however, that side-chains have a small repertoire of energetically preferred conformations, called rotamers. Statistical analysis of those conformations in
well-determined protein structures produce a rotamer library. We used a backbone
dependent rotamer library (by Dunbrack and Kurplus, July 2001 version). Given
the coordinates of the backbone atoms, its dihedral angles ? (defined, for the ith
residue, by Ci - 1 - Ni - Ci - Ci ) and 'IjJ (defined by Ni - Ci - Ci - NHd were
calculated. The library then gives the typical rotamers for each side-chain and their
prior probabilities.
By using the library we convert the continuous optimization problem into a discrete
one. The number of discrete variables is equal to the number of residues and the
possible values each variable can take lies between 2 and 81.
2.3
Graphical model
Since we have a discrete optimization problem and the energy function is a sum of
pairwise interactions, we can transform the problem into a graphical model with
pairwise potentials. Each node corresponds to a residue, and the state of each node
represents the configuration of the side chain of that residue. Denoting by {rd an
assignment of rotamers for all the residues then:
P({ri}) =
! e - +E({r;})
Z
!e -+ L;j E(r;)+E(r;,rj)
Z
1
Z II 'lti(ri) II 'ltijh,rj)
i
(2)
i ,j
where Z is an explicit normalization factor and T is the system "temperature"
(used as free parameter). The local potential 'ltih) takes into account the prior
probability of the rotamer Pi(ri) (taken from the rotamer library) and the energy
of the interactions between that rotamer and the backbone:
\(Ii(ri) = Pi (ri)e-,j,E(ri ,backbone)
(3)
Equation 2 requires multiplying \(I ij for all pairs of residues i, j but note that equation 1 gives zero energy for atoms that are sufficiently far away. Thus we only need
to calculate the pairwise interactions for nearby residues. To define the topology of
the undirected graph, we examine all pairs of residues i, j and check whether there
exists an assignment ri, rj for which the energy is nonzero. If it exists, we connect
nodes i and j in the graph and set the potential to be:
(4)
Figure 1 shows a subgraph of the undirected graph. The graph is relatively sparse
(each node is connected to nodes that are close in 3D space) but contains many
small loops. A typical protein in the data set gives rise to a model with hundreds
of loops of size 3.
3
Experiments
When the protein was small enough we used the max-junction tree algorithm [1] to
find the most likely configuration of the variables (and hence the global minimum
of the energy function). Murphy's implementation of the JT algorithm in his BN
toolbox for Matlab was used [10].
The approximate inference algorithms we tested were loopy belief propagation (BP),
generalized BP (GBP) and naive mean field (MF).
BP is an exact and efficient local message passing algorithm for inference in singly
connected graphs [15]. Its essential idea is replacing the exponential enumeration
(either summation or maximizing) over the unobserved nodes with series of local enumerations (a process called "elimination" or "peeling"). Loopy BP, that is
applying BP to multiply connected graphical models , may not converge due to circulation of messages through the loops [12]. However, many groups have recently
reported excellent results using loopy BP as an approximate inference algorithm
[4, 11, 5]. We used an asynchronous update schedule and ran for 50 iterations or
until numerical convergence.
GBP is a class of approximate inference algorithms that trade complexity for accuracy [15]. A subset of GBP algorithms is equivalent to forming a graph from
clusters of nodes and edges in the original graph and then running ordinary BP on
the cluster graph. We used two large clusters. Both clusters contained all nodes
in the graph but each cluster contained only a subset of the edges. The first cluster contained all edges resulting from residues, for which the difference between
its indices is less than a constant k (typically, 6). All other edges were included
in the second cluster. It can be shown that the cluster graph BP messages can
be computed efficiently using the JT algorithm. Thus this approximation tries to
capture dependencies between a large number of nodes in the original graph while
maintaining computational feasibility.
The naive MF approximation tries to approximate the joint distribution in equation 2 as a product of independent marginals qi(ri) . The marginals qi(ri) can be
found by iterating:
qi(ri)
f-
a\(li(ri) exp
(L L
qj(rj) log \(Iij(ri, rj ))
JENi
rj
(5)
where a denotes a normalization constant and Ni means all nodes neighboring i.
We initialized qi(ri) to \[Ii(ri) and chose a random update ordering for the nodes.
For each protein we repeated this minimization 10 times (each time with a different
update order) and chose the local minimum that gave the lowest energy.
In addition to the approximate inference algorithms described above, we also compared the results to two approaches in use in side-chain prediction: the SCWRL and
DEE algorithms. The Side-Chain placement With a Rotamer Library (SCWRL)
algorithm is considered one of the leading algorithms for predicting side-chain conformations [3]. It uses the energy function described above (equation 1) and a
heuristic search strategy to find a minimal energy conformation in a discrete conformational space (defined using rotamer library).
Dead end elimination (DEE) is a search algorithm that tries to reduce the search
space until it becomes suitable for an exhaustive search. It is based on a simple
condition that identifies rotamers that cannot be members of the global minimum
energy conformation [2]. If enough rotamers can be eliminated, the global minimum energy conformation can be found by an exhaustive search of the remaining
rotamers.
The various inference algorithms were tested on set of 325 X-ray crystal structures
with resolution better than or equal to 2A, R factor below 20% and length up to 300
residues. One representative structure was selected from each cluster of homologous
structures (50% homology or more) . Protein structures were acquired from Protein
Data Bank site (http://www.rcsb.org/pdb).
Many proteins contain Cysteine residues which tend to form strong disulfide bonds
with each other. A standard technique in side-chain prediction (used e.g. in
SCWRL) is to first search for possible disulfide bonds and if they exist to freeze
these residues in that configuration. This essentially reduces the search space. We
repeated our experiments with and without freezing the Cysteine residues.
Side-chain to backbone interaction seems to be much severe than side-chain to sidechain interaction - the backbone is more rigid than side-chains and its structure
assumed to be known. Therefore, the parameter R was introduced into the pairwise
potential equation, as follows:
\[Io(ro
ro) - (e -,f-E(ri ,r;))*
"J ", J -
(6)
Using R > 1 assigns an increased weight for side-chain to backbone interactions
over side-chain to side-chain interactions. We repeated our experiments both with
R = 1 and R > 1. It worth mentioning that SCWRL implicitly adopts a weighting
assumption that assigns an increased weight to side-chain to backbone interactions.
4
Results
In our first set of experiments we wanted to compare approximate inference to
exact inference. In order to make exact inference possible we restricted the possible
rotamers of each residue. Out of the 81 possible states we chose a subset whose
local probability accounted for 90% of the local probability. We constrained the size
of the subset to be at least 2. The resulting graphical model retains only a small
fraction of the loops occurring in the full graphical model (about 7% of the loops
of size 3). However, it still contains many small loops, and in particular, dozens of
loops of size 3.
On these graphs we found that ordinary max-product BP always found the global
minimum of the energy function (except in few cases where it failed to converge).
80
80
70
II!
70
..
80
.!! 50
eo
.!! 50
a.
a.
~ <1l
~
"'~ 30
<1l
"'~ 30
20
I
20
10
0
{;>
I ?
"
"
10
0
,,, 01> ~ {> .?> ..," ." ."
<9 4> <P $'
{;>
"
E(Sum-product BP) - E(Max-product BP)
I.
",,, 01> ~ {>.?>..,"."."
<9 4> <p.<p
E(Mean field) - E(Max-product BP)
80
.
70
eo
100
.!! 50
a.
gOJ
~ <1l
-
98
"'~ 30
20
10
0
{;>
.
.
.
-. - "
"
," 01> ~ {> .?> ..," ." ."
<9 4> <p.*
E(SCWRL) - E(Max-product BP)
t
96
o
94
>
c
(,)
,---
OJ
";J!. 92
90
SCWRL
Sum , R=1 Sum , R>1
nn
Max. R= 1 Max. R>1
Figure 2: Sum-product BP (top-left), naive MF (top-right) and SCWRL (bottomleft) algorithms energies are always higher than or equal to max-product BP energy.
Convergence rates for the various algorithms shown in bottom-right chart.
Sum-product BP failed to find sum-JT conformation in 1% of the graphs only. In
contrast the naive MF algorithm found the global minimum conformation for 38%
of the proteins and on 17% of the runs only. The GBP algorithm gave the same
result as the ordinary BP but it converged more often (e.g. 99.6% and 98.9% for
sum-product GBP and BP, respectively).
In the second set of experiments we used the full graphical models. Since exact
inference is impossible we can only compare the relative energies found by the
different approximate inference algorithms. Results are shown in Figure 2. Note
that, when it converged, max-product BP always found a lower energy configuration
compared to the other algorithms. This finding agrees with the observation that the
max-product solution is a "neighborhood optimum" and therefore guaranteed to be
better than all other assignments in a large region around it [1 3].
We also tried decreasing T , the system "temperature", for sum-product (in the
limit of zero temperature it should approach max-product) . In 96% of the time,
using lower temperature (T = 0.3 instead of T = 1) indeed gave a lower energy
configuration. Even at this reduced temperature, however, max-product always
found a lower energy configuration.
All algorithms converged in more than 90% of the cases. However, sum-product
converged more often than max-product (Figure 2, bottom-right) . Decreasing temperature resulted in lower convergence rate for sum-product BP algorithm (e.g.
95.7% compared to 98.15% in full size graphs using disulfide bonds). It should be
mentioned that SCWRL failed to converge on a single protein in the data set.
Applying the DEE algorithm to the side-chain prediction graphical models dramatically decreased the size of the conformational search space, though, in most cases,
the resulted space was still infeasible. Moreover, max-product BP was indifferent
;::; 3
;::; 3
.
.~
~
e::.
~
e::.
~ 2
u
2
U
:::J
:::J
rn
rn
<1' 1
<1' 1
0
Xl
x2
x3
x4
Xl
X2
Xl
X4
SCWRL buried residues success rates
Xl
X2
X3
X4
85.9% 62.2% 40.3%
25.5%
Figure 3: Inference results - success rate. SCWRL buried residues success rate
subtracted from sum-product BP (light gray), max-product BP (dark gray) and
MF (black) rates when equally weighting side-chain to backbone and side-chain to
side-chain clashes (left) and assigning increased weight for side-chain to backbone
clashes (right).
to that space reduction - it failed to converge for the same models and, when
converged, found the same conformation.
4.1
Success rate
In comparing the performance of the algorithms, we have focused on the energy of
the found configuration since this is the quantity the algorithms seek to optimize.
A more realistic performance measure is: how well do the algorithms predict the
native structure of the protein?
The dihedral angle Xi is deemed correct when it is within 40? of the native (crystal)
structure and Xl to Xi-l are correct. Success rate is defined as the portion of
correctly predicted dihedral angles.
The success rates of the conformations, inferred by both max- and sum-product
outperformed SCWRL's (Figure 3). For buried residues (residues with relative
accessibility lower than 30% [9]) both algorithms added 1% to SCWRL's Xl success
rate. Increasing the weight of side-chain to backbone interactions over side-chain
to side-chain interactions resulted in better success rates (Figure 3, right). Freezing
Cysteine residues to allow the formation of disulfide bonds slightly increased the
success rate.
5
Discussion
Recent years have shown much progress in approximate inference. We believe that
the comparison of different approximate inference algorithms is best done in the
context of a real-world problem. In this paper we have shown that for a realworld problem with many loops, the performance of belief propagation is excellent.
In problems where exact inference was possible max-product BP always found the
global minimum of the energy function and in the full protein data set, max-product
BP always found a lower energy configuration compared to the other algorithms
tested.
SCWRL is considered one of the leading algorithms for modeling side-chain conformations. However, in the last couple of years several groups reported better results
due to more accurate energy function [7], better searching algorithm [8] , or extended
rotamer library [14].
As shown, by using inference algorithms we achieved low energy conformations,
compared to existing algorithms. However, this leads only to a modest increase in
prediction accuracy. Using an energy function, which gives a better approximation
to the "true" physical energy (and particularly, assigns lowest energy to the native
structure) should significantly improve the success rate. A promising direction for
future research is to try and learn the energy function from examples. Inference
algorithms such as BP may play an important role in the learning procedure.
References
[1] R. Cowell. Introduction to inference in Bayesian networks. In Michael I. Jordan,
editor, Learning in Graphical Models. Morgan Kauffmann , 1998.
[2] Johan Desmet , Marc De Maeyer, Bart Hazes, and Ignace Lasters . The dead-end
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[3] Roland L. Dunbrack, Jr. and Martin Kurplus. Back-bone dependent rotamer library
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See also http://www.fccc.edu/research/labs/dunbrack/scwrlj.
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1,298 | 2,182 | Recovering Articulated Model Topology from Observed
Rigid Motion
Leonid Taycher, John W. Fisher III, and Trevor Darrell
Artificial Intelligence Laboratory
Massachusetts Institute of Technology
Cambridge, MA, 02139
{lodrion, fisher, trevor}@ai.mit.edu
Abstract
Accurate representation of articulated motion is a challenging problem
for machine perception. Several successful tracking algorithms have
been developed that model human body as an articulated tree. We propose a learning-based method for creating such articulated models from
observations of multiple rigid motions. This paper is concerned with
recovering topology of the articulated model, when the rigid motion of
constituent segments is known. Our approach is based on finding the
Maximum Likelihood tree shaped factorization of the joint probability
density function (PDF) of rigid segment motions. The topology of graphical model formed from this factorization corresponds to topology of the
underlying articulated body. We demonstrate the performance of our algorithm on both synthetic and real motion capture data.
1 Introduction
Tracking human motion is an integral part of many proposed human-computer interfaces,
surveillance and identification systems, as well as animation and virtual reality systems. A
common approach to this task is to model the body as a kinematic tree, and reformulate
the problem as articulated body tracking[6]. Most of the state-of-the-art systems rely on
predefined kinematic models [16]. Some methods require manual initialization, while other
use heuristics [12], or predefined protocols [10] to adapt the model to observations.
We are interested in a principled way to recover articulated models from observations. The
recovered models may then be used for further tracking and/or recognition. We would
like to approach model estimation as a multistage problem. In the first stage the rigidly
moving segments are tracked independently; at the second stage, the topology of the body
(the connectivity between the segments) is recovered. After the topology is determined, the
joint parameters may be determined.
In this paper we concentrate on the second stage of this task, estimating the underlying
topology of the observed articulated body, when the motion of the constituent rigid bodies
is known. We approach this as a learning problem, in the spirit of [17]. If we assume that
the body may be modeled as a kinematic tree, and motion of a particular rigid segment is
known, then the motions of the rigid segments that are connected through that segment are
independent of each other. That is, we can model a probability distribution of the full body-
pose as a tree-structured graphical model, where each node corresponds to pose of a rigid
segment. This observation allows us to formulate the problem of recovering topology of
an articulated body as finding the tree-shaped graphical model that best (in the Maximum
Likelihood sense) describes the observations.
2 Prior Work
While state-of-the-art tracking algorithms [16] do not address either model creation or
model initialization, the necessity of automating these two steps has been long recognized.
The approach in [10] required a subject to follow a set of predefined movements, and
recovered the descriptions of body parts and body topology from deformations of apparent
contours. Various heuristics were used in [12] to adapt an articulated model of known
topology to 3D observations. Analysis of magnetic motion capture data was used by [14]
to recover limb lengths and joint locations for known topology, it also suggested similar
analysis for topology extraction. A learning based approach for decomposing a set of
observed marker positions and velocities into sets corresponding to various body parts was
described in [17]. Our work builds on the latter two approaches in estimating the topology
of the articulated tree model underlying the observed motion.
Several methods have been used to recover multiple rigid motions from video, such as
factorization [3, 18], RANSAC [7], and learning based methods [9]. In this work we assume that the 3-D rigid motions has been recovered and are represented using a 2-D Scaled
Prismatic Model (SPM).
3 Representing Pose and Motion
A 2-D Scaled Prismatic Model (SPM) was proposed by [15] and is useful for representing
image motion of projections of elongated 3-D objects. It is obtained by orthographically
?projecting? the major axis of the object to the image plane. The SPM has four degrees of
freedom: in-plane translation, rotation, and uniform scale. 3-D rigid motion of an object,
may be simulated by SPM transformations, using in-plane translation for rigid translation,
and rotation and uniform scaling for plane-parallel and out-of-plane rotations respectively.
SPM motion (or pose) may be expressed as a linear transformation in projective space as
!
a ?b e
M= b a f
(1)
0 0 1
Following [13] we have chosen to use exponential coordinates, derived from constant velocity equations, to parameterize motion.
An SPM transformation may be represented as an exponential map
?
M = e?
c
?? = ? ?
0
??
c
0
vx
vy
0
!
? ?
vx
? vy ?
? = ?? ?
?
c
(2)
In this representation vx is a horizontal velocity, vy ? vertical velocity, ? ? angular velocity,
and c is a rate of scale change. ? is analogous to time parameter. Note that there is an
inherent scale ambiguity, since ? and (vx , vy , ?, c)T may be chosen arbitrarily, as long as
?
e? = M.
It can be shown ([13]) that if the SPM transformation is a combination of scaling and
rotation, it may be expressed by the sum of two twists, with coincident centers (u x , uy )T
of rotation and expansion.
?
?
? ?
uy
?ux
?c
??ux ?
??uy ? ???
? = ??
+c?
=
1 ?
0 ? ?
0
1
?
?c
?
?
ux
? ? uy ?
?? ? ?
1
c
??
1
(3)
While ?pure? translation, rotation or scale have intuitive representation with twists, the
combination or rotation and scale does not. We propose a scaled twist representation, that
preserves the intuitiveness of representation for all possible SPM motions. We want to
separate the ?direction? of motion (the direction of translation or the relative amounts of
rotation and scale) from the amount of motion.
If the transformation involves rotation and/or scale, then we choose ? so that ||(?, c)|| 2 = 1,
and then use eq. 3 to compute the center of rotation/expansion. The computation may be
expressed as a linear transformation:
??
?
?
?ux ? ?
? ? ?
? = ? uy ? = ?
??? ?
?
?
c
?
?
?
?
? 2 + c?2
c
?
? ?? 2 +?
c2
?
?
?
? 2 +?
c2
?
? ?? 2?+?
c2
c
?
? ?? 2 +?
c2
?
1
?
? 2 +?
c2
?
1
?
? 2 +?
c2
where ? = (?
vx , v?y , ?
? , c?)T .
? ?
? 1
? ?v?x ?
?? ?
? ?v?y ?
?? ?
? ?
?
?
c?
(4)
The the pure translational motion (? = c = 0) may be regarded as an infinitely small
rotation about a point at infinity, e.g. the translation by l in the direction (u x , uy ) may be
?u
represented as ? = lim??0 (l|?|, ? y , u?x , ?, 0)T , but we choose a direct representation
?p 2
?
v?x + v?y2
?
?
?ux ? ?
? ? ?
? = ?uy ? = ?
?0? ?
?
0
?
In both cases ? = A(1, ??T )T , and
?3
?
det(A) =
??1
? 21 2
v
?x +?
vy
?
?
1
2 +?
2
v
?x
vy
1
1
? ?
? 1
? ?v?x ?
?? ?
? ?v?y ?
???
?
? ?
c?
?=
6 0 ? c 6= 0 (rotation/scaling)
? = 0 ? c = 0 (pure translation)
(5)
(6)
Note that ?I = (0, ux , uy , ?, c)T represents identity transformation for any ux , uy , ?, and
c. It is always reported as ?I = 0.
4 Learning Articulated Topology
We wish to infer the underlying topology of an articulated body from noisy observations
of a set of rigid body motions. Towards that end we will adopt a statistical framework for
fitting a joint probability density. As a practical matter, one must make choices regarding
density models; we discuss one such choice although other choices are also suitable.
We denote the set of observed motions of N rigid bodies at time t, 1 ? t ? F as a set
{Mts |1 ? s ? N }. Graphical models provide a useful methodology for expressing the dependency structure of a set of random variables (cf. [8]). Variables M i with observations
{Mti |1 ? t ? F } are assigned to the vertices of a graph, while edges between nodes indicate dependency. We shall denote presence or absence of an edge between two variables,
Mi and Mj by an index variable Eij , equal to one if an edge is present and zero otherwise.
Furthermore, if the corresponding graphical model is a spanning tree, it can be expressed
as a product of conditional densities (e.g. see [11])
Y
PM (M1 , . . . , MN ) =
PMs |pa(Ms ) (Ms |pa (Ms ))
(7)
Ms
where pa(Ms ) is the parent of Ms . While multiple nodes may have the same parent, each
individual node has only one parent node. Furthermore, in any decomposition one node
(the root node) has no parent. Any node (variable) in the model can serve as the root node
[8]. Consequently, a tree model constrains E. Of the possible tree models (choices of
E), we wish to choose the maximum likelihood tree which is equivalent to the minimum
entropy tree [4]. The entropy of a tree model can be written
X
X
H(M ) =
H(Ms ) ?
I(Mi ; Mj )
(8)
s
Eij =1
where H(Ms ) is the marginal entropy of each variable and I(Mi ; Mj ) is the mutual information between nodes Mi and Mj and quantifies their statistical dependence. Consequently, the minimum entropy tree corresponds to the choice of E which minimizes the
sum of the pairwise mutual informations [1]. The tree denoted by E can be found via the
maximum spanning tree algorithm [2] using I(Mi ; Mj ) for all i, j as the edge weights.
Our conjecture is that if our data are sampled from a variety of motions the topology of
the estimated density model is likely to be the same as the topology of the articulated body
model. It follows from the intuition that when considering only pairwise relationships, the
relative motions of physically connected bodies will be most strongly related.
4.1 Estimation of Mutual Information
Computing the minimum entropy spanning tree requires estimating the pairwise mutual
informations between rigid motions Mi and Mj for all i, j pairs. In order to do so we
must make a choice regarding the parameterization of motion and a probability density
over that parameterization; to estimate articulated topology it is sufficient to use the the
Scaled Prismatic Model with twist parameterization described in Section 3).
4.2 Estimating Motion Entropy
We parameterize rigid motion, Mti , by the vector of quantities ?it (cf. Eq. 2). In general,
H(Mi ) 6= H(?i ),
(9)
but since there is a one-to-one correspondence between the M i ?s and ?i ?s [4], we can
estimate the I(Mi ; Mj ) by first computing ?it , ?jt from Mti , Mtj
I(Mi ; Mj ) = I(?i ; ?j ) = H(?j ) ? H(?j |?i )
Furthermore, if the relative motion Mj|i between segments si and
assumed to be independent of Mi , it can be shown that
sj (Mjt
(10)
=
H(?j |?i ) = H(log Mi Mj|i | log Mi ) = H(log Mj|i ) = H(?j|i ).
t
Mit Mj|i
)
is
(11)
We wish to use scaled twists (Section 3) to compute the entropies involved. Since the involved quantities are in the linear relationship ? = A(1, ??T )T (Eqs. 4 and 5), the entropies
are related,
H(?) = H(? ) ? E[log det(A)],
(12)
where E[log det(A)] may be estimated using Equation 6.
4.3 Estimating the Motion Kernel
In order to estimate the entropy of motion, we need to estimate the probability density
based on the available samples. Since the functional form of the underlying density is not
known we have chosen to use kernel-based density estimator,
X
p?(? ) = ?
K(? ; ?i ).
(13)
i
Since our task is to determine the articulated topology, we wish to concentrate on ?spatial?
features of the transformation, center of rotation for rotational motion, and the direction of
translation for translational, that correspond to two common kinds of joints, spherical and
prismatic. Thus we need to define a kernel function K(?1 ; ?2 ) that captures the following
notion of ?distance? between the motions:
1. If ?1 and ?2 do not represent pure translational motions, then they should be considered to be close if their centers of rotation are close.
2. If ?1 and ?2 are pure translations, then they should be considered close if their
directions are close.
3. If ?1 and ?2 represent different types of motion (i.e. rotation/scale vs. translation),
then they are arbitrarily far apart.
4. The identity transformation (? = 0) is equidistant from all possible transformations (since any (ux , uy , ?, c)T combined with ? = 0 produces identity)
One kernel that satisfies these requirements is the following:
?
KR ((ux1 , uy1 ); (ux2 , uy2 ))
?
?
?
?
?
?
?
?
?
KT ((ux1 , uy1 ); (ux2 , uy2 ))
?
?
?
?
?
?
?
?
0
K(?1 ; ?2 ) =
?
?
?
?
?
?0
?
?
?
?
?
?
?
?
?(0)
?
?
?
Condition 1
(?1 6= 0 ? c1 6= 0) ? (?2 6= 0 ? c2 6= 0)
Condition 2
?1 = 0 ? c 1 = 0 ? ? 2 = 0 ? c 2 = 0
Condition 3
(?1 6= 0 ? c1 6= 0) ? (?2 = 0 ? c2 = 0)
Condition 3
(?1 = 0 ? c1 = 0) ? (?2 6= 0 ? c2 6= 0)
Condition 4.
?1 = 0 ? ? 2 = 0
(14)
where KR and KT are Gaussian kernels with covariances estimated using methods from
[5].
5 Implementation
The input to our algorithm is a set of SPM poses (Section 3) {Pts |1 ? s ? S, 1 ? t ? T },
where S is the number of tracked rigid segments and F is the number of frames. In order
to compute the mutual information between the motion of segments s 1 and s2 , we first
compute motions of segment s1 in frames 1 < t ? F relative to its position in frame
t1 = 1,
Mts11t = Pts1 (Pts11 )?1 ,
(15)
?1
and the transformation of s2 relative to s1 (with the relative pose Ps2 |s1 = (Ps1 ) Ps2 ),
Mts12t|s1 = ((Pts1 )?1 Pts2 )((Pts11 )?1 .Pts12 )?1
(16)
t
The parameter vectors ?st21 t and ?st21|s
are then extracted from the transformation matrices
1
Ms2 and Ms2 |s1 (cf. Section 3), and the mutual information is estimated as described in
Section 4.2.
6 Results
We have tested our algorithm both on synthetic and motion capture data. Two synthetic sequences were generated with the following steps. First, the rigid segments were positioned
by randomly perturbing parameters of the corresponding kinematic tree structure. A set of
feature points was then selected for each segment. At each time step point positions were
computed based on the corresponding segment pose, and perturbed with Gaussian noise
with zero mean and standard deviation of 1 pixel. The inputs to the algorithm were the segment poses re-estimated from the feature point coordinates. In the motion capture-based
experiment, the segment poses were estimated from the marker positions.
The results of the experiments are shown in the Figures 6.1, 6.2 and 6.3. The first experiment involved a simple kinematic chain with 3 segments in order to demonstrate the
operation of the algorithm. The system has a rotational joint between S 1 and S2 and prismatic joint between S2 and S3 .
The sample configurations of the articulated body are shown in the first row of the Figures
6.1. The graph computed using method from Section 4.2 and the corresponding maximum
spanning tree are in Figures 6.1(d, e).
The second experiment involved a humanoid torso-like synthetic model containing 5 rigid
segments. It was processed in a way similar to the first experiment. The results are shown
in Figure 6.2.
For the human motion experiment, we have used motion capture data of a dance sequence
(Figure 6.3(a-c)). The rigid segment motion was extracted from the positions of the markers
tracked across 220 frames (the marker correspondence to the body locations was known).
The algorithm was able to correctly recover the articulated body topology (Compare Figures 6.3(e) and 6.3(a)), when provided only with the extracted segment poses. The dance is
a highly structured activity, so not all degrees of freedom were explored in this sequence,
and mutual information between some unconnected segments (e.g. thighs S 3 and S7 ) was
determined to be relatively large, although this did not impact the final result.
7 Conclusions
We have presented a novel general technique for recovering the underlying articulated
structure from information about rigid segment motion. Our method relies on only a very
weak assumption, that this structure may be represented by a tree with unknown topology.
While the results presented in this paper were obtained using the Scaled Prismatic Model
and non-parametric density estimator, our methodology does not rely on either modeling
assumption.
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1,299 | 2,183 | Half-Lives of EigenFlows for Spectral Clustering
Chakra Chennubhotla & Allan D. Jepson
Department of Computer Science, University of Toronto, Canada M5S 3H5
chakra,jepson @cs.toronto.edu
Abstract
Using a Markov chain perspective of spectral clustering we present an
algorithm to automatically find the number of stable clusters in a dataset.
The Markov chain?s behaviour is characterized by the spectral properties
of the matrix of transition probabilities, from which we derive eigenflows
along with their halflives. An eigenflow describes the flow of probability mass due to the Markov chain, and it is characterized by its eigenvalue, or equivalently, by the halflife of its decay as the Markov chain
is iterated. A ideal stable cluster is one with zero eigenflow and infinite half-life. The key insight in this paper is that bottlenecks between
weakly coupled clusters can be identified by computing the sensitivity
of the eigenflow?s halflife to variations in the edge weights. We propose
a novel E IGEN C UTS algorithm to perform clustering that removes these
identified bottlenecks in an iterative fashion.
1 Introduction
We consider partitioning a weighted undirected graph? corresponding to a given dataset?
into a set of discrete clusters. Ideally, the vertices (i.e. datapoints) in each cluster should
be connected with high-affinity edges, while different clusters are either not connected or
are connnected only by a few edges with low affinity. The practical problem is to identify
these tightly coupled clusters, and cut the inter-cluster edges.
Many techniques have been proposed for this problem, with some recent success being obtained through the use of spectral methods (see, for example, [2, 4, 5, 11, 12]). Here we use
the random walk formulation of [4], where the edge weights are used to construct a Markov
defines a random walk on the graph to
transition probability matrix, . This matrix
be partitioned. The eigenvalues and eigenvectors of
provide the basis for deciding on a
particular segmentation. In particular, it has been shown that for weakly coupled clusters, the leading eigenvectors of
will be roughly piecewise constant [4, 13, 5]. This
result motivates many of the current spectral clustering algorithms. For example in [5], the
number of clusters must be known a priori, and the -means algorithm is used on the
leading eigenvectors of
in an attempt to identify the appropriate piecewise constant
regions.
In this paper we investigate the form of the leading eigenvectors of the Markov matrix .
Using some simple image segmentation examples we confirm that the leading eigenvectors
of
are roughly piecewise constant for problems with well separated clusters. However,
we observe that for several segmentation problems that we might wish to solve, the coupling between the clusters is significantly stronger and, as a result, the piecewise constant
approximation breaks down.
Unlike the piecewise constant approximation, a perfectly general view is that the eigenvectors of
determine particular flows of probability along the edges in the graph. We refer
to these as eigenflows since they are characterized by their associated eigenvalue , which
specifies the flow?s overall rate of decay. Instead of measuring the decay rate in terms of
the eigenvalue , we find it more convenient to use the flow?s halflife , which is simply
defined by
. Here is the number of Markov chain steps needed to reduce
the particular eigenflow to half its initial value. Note that as approaches the half-life
approaches infinity.
From the perspective of eigenflows, a graph representing a set of weakly coupled clusters produces eigenflows between the various clusters which decay with long halflives. In
contrast, the eigenflows within each cluster decay much more rapidly. In order to identify clusters we therefore consider the eigenflows with long halflives. Given such a slowly
decaying eigenflow, we identify particular bottleneck regions in the graph which critically
restrict the flow (cf. [12]). To identify these bottlenecks we propose computing the sensitivity of the flow?s halflife with respect to perturbations in the edge weights.
We implement a simple spectral graph partitioning algorithm which is based on these ideas.
We first compute the eigenvectors for the Markov transition matrix, and select those with
long halflives. For each such eigenvector, we identify bottlenecks by computing the sensitivity of the flow?s halflife with respect to perturbations in the edge weights. In the current
algorithm, we simply select one of these eigenvectors in which a bottleneck has been identified, and cut edges within the bottleneck. The algorithm recomputes the eigenvectors and
eigenvalues for the modified graph, and continues this iterative process until no further
edges are cut.
2 From Affinities to Markov Chains
Following
the formulation in [4], we consider an undirected graph
with vertices , for
, and edges with non-negative weights . Here the weight represents the affinity of vertices and . The edge affinities are assumed to be symmetric,
that is, . A Markov chain is defined using these affinities by setting the transition
probability !" from vertex # to vertex to be proportional to the edge affinity, $ .
That is, !" %'&) (+* where & ,.-./10 gives the normalizing factor which ensures
- /10 !2 23 . In matrix notation, the *affinities are represented by a symmetric 46574
* 8 , with elements $ , and the transition probability matrix 9;: !<
= is given by
matrix
>.8@? (A* B?C
Notice that the 4F524 matrix
diag : & D&
*
is not in general symmetric.
/
=E
(1)
defines the random walk of a particle on the graph
This transition probability matrix
. Suppose the initial probability of the particle being at vertex is GI H , for JK;L4 .
Then, the probability of the particle being initially at vertex and taking edge
is
!"
N MGI H . In matrix notation, the probability of the particle ending
N up anyN of the vertices
N
<O: * P QQQRM / = after one step is given by the distribution G *
G%H , where GTSU
:VGWS EXGYS = .
*
/
For analysis it is convenient to consider the matrix Z[.? (+*\ P ]? *\ P , which is similar to
(where ? is as given
N in Eq. (1)). The matrix Z therefore has the N same spectrum as
and any eigenvector ^ of Z must correspond to an eigenvector ? *\ P ^ of
with the same
eigenvalue. Note that Z_C? (+*\ P ]? *M\ P `? (A*M\ P 8@? (A* ? *\ P `? (A*M\ P 8@? (A*M\ P , and
therefore Z is a symmetric 4a524 matrix since 8 is symmetric while ? is diagonal.
The advantage of considering the matrix Z over
is that the symmetric eigenvalue problem is more stable to small perturbations, and is computationally much more tractable.
Since the matrix Z is symmetric, it has an orthogonal decomposition of the form:
Z[.bTcdbTef
(2)
(a)
(b)
(c)
(d)
(e)
Figure 1: (a-c) Three random images each having an occluder in front of a textured background. (d-e) A pair of eye images.
N N
N
bC ^ * ^ P QQQ+ ^ are the eigenvectors and c is a diagonal matrix of eigenvalQQQ+D / sorted/ in decreasing order. While the eigenvectors have unit length,
* DIP ,Lthe
^ ;
eigenvalues are real and have an absolute value bounded by 1, .
S
S
where
ues
N
The eigenvector representation provides a simple way to capture the Markovian relaxation
process [12]. For example, consider propagating the Markov chain for iterations. The
transition matrix after iterations, namely ] , can be represented as:
'? M* \ P b c b e ? (+*\ P
(3)
Therefore the probability distribution for the particle being at vertex
steps
N after
N
N of
,
G
H
@
G
G
H
the randomN walk, given
that
the
initial
probability
distribution
was
,
is
N
N
? *\ P bTcf H N , where H b e ? (+*\ P G@H provides the expansion coefficients of the initial
distribution G%H in terms of the
of Z . As
, the Markov chain approaches
N eigenvectors
N
the stationary distribution ,
e . Assuming the graph
is connected with edges
having non-zero weights, it is convenient to interpret the Markovian relaxation process as
N
N
N
perturbations to the stationary distribution, G
- / 0 P
, where * is
N
N
P N
associated with the stationary distribution and ? *M\ ^ .
3 EigenFlows
N
Let G@H be an initial probability distribution for a random particle to be at the vertices of
the graph
. By the definition of the Markov chain, recall that the probability of making
the transition from vertex to is the probability of starting in vertex , times the
conditional probability of taking edge given that the particle is at vertex , namely
! GI H . Similarly, the probability of making the transition in the reverse direction is ! D G)H .
The net flow of probability mass along edge from # to is therefore the difference
!" MGI H !U D VG)H .N It then follows that
N the net flow of probability mass from vertex to
is given by
: G,H#= , where : G%H#= is the : J= -element of the 4F5 4 matrix
K: G
N
diag : G
N
N
H = diag : G H = ]ef
N
Notice that
diag : G%H
= , and therefore
e for
). This expresses the fact that the flow from to
e
H =f
(i.e.
isisjustantisymmetric
of the flow
the opposite sign
in the reverse
direction. Furthermore, it can be shown that K: N =T for any
with _ ,
stationary distribution . Therefore, the flow is caused by the eigenvectors
N
and hence we analyze the rate of decay of these eigenflows K
: =.
(4)
For illustration purposes we begin by considering an ensemble of random test images
formed from two independent samples of 2D Gaussian filtered white noise (see Fig. 1a-c).
One sample is used to form the 5a background image, and a cropped 5
fragment
of second sample is used for the foreground region. A small constant bias is added to the
foreground region.
(a)
(b)
(c)
Figure 2: (a) Eigenmode (b) corresponding eigenflow (c) gray value at each pixel corresponds to the maximum of the absolute sensitivities of all the weights on edges connected
to a pixel (not including itself). Dark pixels indicate high absolute sensitivities.
A graph clustering problem is formed where each pixel in a test image is associated with a
vertex of the graph
. The edges in
are defined by the standard 8-neighbourhood of each
pixel (with pixels at the edges and corners of the image only having 5 and 3 neighbours,
respectively). The edge weight
between
neighbouring vertices
and is given by the
N
N
N
affinity N
: Y: = Y: =M= P $: P = , where Y: S = is the test image brightness
, where is the median
at pixel
S and is a grey-level standard deviation. We use
absolute difference of gray levels between all neighbouring pixels and ;L .
This generative process provides an ensemble of clustering problems which we feel are
representative of the structure of typical image segmentation problems. In particular, due
to the smooth variation in gray-levels, there is some variability in the affinities within both
foreground and background regions. Moreover, due to the use of independent samples for
the two regions, there is often a significant step in gray-level across the boundary between
the two regions. Finally, due to the small bias used, there is also a significant chance for
pixels on opposite sides of the boundary to have similar gray-levels, and thus high affinities.
This latter property ensures that there are some edges with significant weights between the
two clusters in the graph associated with
N the foreground and background pixels.
N
along with its eigenflow, K: S = .
In Figure 2 we plot one eigenvector, , of the matrix
S
Notice that the displayed eigenmode is not in general piecewise constant. Rather, the
eigenvector is more like vibrational mode of a non-uniform membrane (in fact, they can
be modeled in precisely that way). Also, for all but the stationary distribution, there is a
significant net flow between neighbours, especially in regions where the magnitude of the
spatial gradient of the eigenmode is larger.
4 Perturbation Analysis of EigenFlows
As discussed in the introduction, we seek to identify bottlenecks in the eigenflows associated with long halflives. This notion of identifying bottlenecks is similar to the well-known
max-flow, min-cut theorem. In particular, for a graph whose edge weights represent maximum flow capacities between pairs of vertices, instead of the current conditional transition
probabilities, the bottleneck edges can be identified as precisely those edges across which
the maximum flow is equal to their maximum capacity. However, in the Markov framework, the flow of probability across an edge is only maximal in the extreme cases for
which the initial probability of being at one of the edge?s endpoints is equal to one, and
zero at the other endpoint. Thus the max-flow criterion is not directly applicable here.
Instead, we show that the desired bottleneck edges can be conveniently identified by considering the sensitivity of the flow?s halflife to perturbations of the edge weights (see Fig.
2c). Intuitively, this sensitivity arises because the flow across a bottleneck will have fewer
alternative routes to take and therefore will be particularly sensitive to changes in the edge
weights within the bottleneck. In comparison, the flow between two vertices in a strongly
coupled cluster will have many alternative routes and therefore will not be particularly
sensitive on the precise weight of any single edge.
In order to pick out larger halflives, we will use one parameter, , which is a rough estimate
H
of the smallest halflife that one wishes to consider. Since we are interested in perturbations
which significantly change the current halflife of a mode, we choose to use a logarithmic
scale
in halflife. A simple choice for a function which combines these two effects is : =
: = , where the halflife of the current eigenmode.
H
N
^ of Z , with eigenvalue . This eigenvector decays with
Suppose we have aneigenvector
:X L=
:D = . Consider the effect on d: = of perturbing the affinity
a halflife of F
for the : J = -edge, to . In particular, we show in the Appendix that the
derivative of : f: =M= with respect to , evaluated at
, satisfies
&
:
H=
&
: ^ M ^ #=
^
&
:X L=
: I= :
=
N
: J= elements of eigenvector ^
^ P
&
:
I=
^ P
&
^ P
&
(5)
are the
and :X&& #= are degrees of nodes
(Eq.1). In Figure 2, for a given eigenvector and its flow, we plot the maximum of
absolute sensitivities of all the weights on edges connected to a pixel (not including itself).
Note that the sensitivities are large in the bottlenecks at the border of the foreground and
background.
Here
: J =
5
E IGEN C UTS : A Basic Clustering Algorithm
We select a simple clustering algorithm to test our proposal of using the derivative of the
eigenmode?s halflife for identifying bottleneck edges. Given a value of , which is roughly
H
the minimum halflife to consider for any eigenmode, we iterate the following:
1. Form the symmetric affinity matrix , and initialize .
2. Set !#" $ &%()+' *-, ." ) , and set a scale factor / to be the median of
,?>+@
,?>+@
Form the symmetric matrix ; =<
.
=<
3. Compute eigenvectors ACD B , 4 D B @ 49E9E9E4FD B
of ;H , with eigenvalues I J ,
'FG
DN
B
for 1$ 325476869694: .
0#"
IFKLI J
@
IM69696KLI J
'
NQP3R
I
.
4. For
each eigenvector
of ; with halflife O
OTS , compute the halflife sensitivities,
U N
) WVX Y:Z\[^]`_\ab]
Mc for each edge in the graph. Here we use R h2\i\j .
#"
V+dFe.f g
5. Do non-maximal
suppression within each of the computed sensitivities.
That is, suppress
U N
U N
U N
the sensitivity ." ) if there is a strictly more negative value k " ) or ." for some vertex
l k
in the neighbourhood of l ) , or some l in the neighbourhood of l . '
N
6. Compute the sum m
of
t
t
i`/ . We use
hnvu6w2 .
n
U
N
#"
)8o )
#"
'
over all non-suppressed edges
for which
p.1:4Cq r
7. Select the eigenmode D B N8x for which m N8x is maximal.
U N8x
8. Cut all edges p.1:4Cq r in y (i.e. set their affinities to 0) for which #" )0s
this sensitivity was not suppressed during non-maximal suppression.
t
i\/
U
N
."
)s
and for which
9. If any new edges have been cut, go to 2. Otherwise stop.
Here steps (z are as described previously, other than computing the scaling constant { ,
which is used in step to provide a scale invariant threshold on the computed sensitivities.
In step 4 we only consider eigenmodes with halflives larger than |D , with | _ } because
H
this typically eliminates the need to compute the sensitivities for many modes with tiny
values of and, because of the term in : = , it is very rare for eigenvectors with
S
H
halflives smaller than | to produce any sensitivity less than ~ .
H
In step 5 we perform a non-maximal suppression on the sensitivities for the ?C? eigenvector.
We have observed that at strong borders the computed sensitivities can be less than ~ in a
band along the border few pixels thick. This non-maximal suppression allows us to thin this
region. Otherwise, many small isolated fragments can be produced in the neighbourhood
of such strong borders.
In step 6 we wish to select one particular eigenmode to base the edge cutting on at this
iteration. The reason for not considering all the modes simultaneously is that we have
found the locations of the cuts can vary by a few pixels for different modes. If nearby
edges are cut as a result of different eigenmodes, then small isolated fragments can result
in the final clustering. Therefore we wish to select just one eigenmode to base cuts on each
iteration. The particular eigenmode selected can, of course, vary from one iteration to the
next.
The selection strategy in step 6 above
picks out the mode which produces the largest
linearized
: S
H = . That is, we compute
increment in d: S =
S
_
, where
is the change of affinities for any edge left to
-
e#f g
otherwise. Other techniques for selecting a particular mode were
be cut, and
also tried, and they all produced similar results.
This iterative cutting process must eventually terminate since, except for the last iteration,
edges are cut each iteration and any cut edges are never uncut. When the process does
terminate, the selected succession of cuts provides a modified affinity matrix 8 which
has well separated clusters. For the final clustering result, we can use either a connected
components algorithm or the -means algorithm of [5] with set to the number of modes
having large halflives.
6 Experiments
We compare the quality of E IGEN C UTS with two other methods: a -means based spectral
clustering algorithm of [5] and an efficient segmentation algorithm proposed in [1] based on
a pairwise region comparison function. Our strategy was to select thresholds that are likely
to generate a small number of stable partitions. We then varied these thresholds to test the
quality of partitions. To allow for comparison with -means, we needed to determine the
number of clusters a priori. We therefore set to be the same as the number of clusters
that E IGEN C UTS generated. The cluster centers were initialized to be as orthogonal as
possible [5].
The first two rows in Fig. 3 show results using E IGEN C UTS. A crucial observation with
E IGEN C UTS is that, although the number of clusters changed slightly with a change in
H , the regions they defined were qualitatively preserved across the thresholds and corresponded to a naive observer?s intuitive segmentation of the image. Notice in the random
images the occluder is found as a cluster clearly separated from the background. The performance on the eye images is also interesting in that the largely uniform regions around
the center of the eye remain as part of one cluster.
In comparison, both the -means algorithm and the image segmentation algorithm of [1]
(rows 3-6 in Fig. 3) show a tendency to divide uniform regions and give partitions that are
neither stable nor intuitive, despite multiple restarts.
7 Discussion
We have demonstrated that the common piecewise constant approximation to eigenvectors arising in spectral clustering problems limits the applicability of previous methods to
situations in which the clusters are only relatively weakly coupled. We have proposed a
new edge cutting criterion which avoids this piecewise constant approximation. Bottleneck
edges between distinct clusters are identified through the observed sensitivity of an eigenflow?s halflife on changes in the edges? affinity weights. The basic algorithm we propose
is computationally demanding in that the eigenvectors of the Markov matrix must be recomputed after each iteration of edge cutting. However, the point of this algorithm is to
simply demonstrate the partitioning that can be achieved through the computation of the
sensitivity of eigenflow halflives to changes in edge weights. More efficient updates of the
eigenvalue computation, taking advantage of low-rank changes in the matrix Z
from one
iteration to the next, or a multi-scale technique, are important areas for further study.
(a)
(b)
(c)
(d)
(e)
Figure 3: Each column refers to a different image in the dataset shown in Fig. 1. Pairs
of rows correspond to results from applying: E IGEN C UTS with L +|
$+~
and H
(Rows 1&2), -Means spectral clustering where , the number
of clusters, is determined by the results of E IGEN C UTS (Rows 3&4) and Falsenszwalb &
(Rows 5&6).
Huttenlocher ~
Acknowledgements
We have benefited from discussions with Sven Dickinson, Sam Roweis, Sageev Oore and Francisco
Estrada.
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Appendix
We compute the derivative of the log of half-life O of an eigenvalue J with respect to an element o )
of the affinity matrix . Half-life is defined as the power to which J must be raised to reduce the
eigenvalue to half, i.e., J ] 28i . What we are interested is in seeing significant changes in those
half-lives O which are relatively large compared to some minimum half-life O S . So eigenvectors with
half-lives smaller than O S are effectively ignored. It is easy to show that,
p
r
;
p O
ObS9r
J
J
o )
o ) 4 and o ) DB
o ) D B 6
(6)
J
pCJbr
pCJ ]
i 5r
Let D B be the corresponding
eigenvector
such that ; D B &J D B , where ; is the modified affinity matrix
,?>M@
,?>M@
(Sec 2). As ; ( <
<
, we can write for all 1 q :
,?>M@
,?>M@
,?>+@
,?>M@
;
)
)
o ) ( <
n <
n
<
4
(7)
<
)
is a matrix of all zeros except for a value of 2 at location p b4 br ; p 4 r are degrees of
where
V!e #"@ %$ &
the nodes 1 and q (stacked as elements on the diagonal matrix see Sec 2); and
) )
V g #"@ %$ &
having non-zero entries only on the diagonal. Simplifying the expression further, we get
,?>+@
,?>M@
,?>+@
,?>M@
,?>+@
;
D
D
D
)
)
o ) DB DB
<
B
B n&B
<
<
<
DB
(8)
,?>M@
,?>+@
,?>+@
D
n&B
<
<
7
DB 6
,?>+@
,?>M@
,?>M@
D
D
D
y <
B
; B
J B , and
Using the fact that <
diagonal, the above equation reduces to:
,?>+@
,?>M@
;
D
D
D
)
)
o ) DB DB
<
B
B n( J B
' <
,?>+@
,?>M@
D
D
D
)
)
<
B
<
B n
J B
The scalar form of this expression is used in Eq.5.
,?>+@
,?>M@
<
,
D
as both
and
are
B
)<
,
) )
D
B 6
(9)
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